Computer Aided Chemical Engineering, Volume 21, Part B: 16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering

16TH EUROPEAN SYMPOSIUM ON COMPUTER AIDED PROCESS ENGINEERING AND 9TH INTERNATIONAL SYMPOSIUM ON PROCESS SYSTEMS ENGINE...

Author: Marquardt | W.; Pantelides | C. (Editors)

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16TH EUROPEAN SYMPOSIUM ON COMPUTER AIDED PROCESS ENGINEERING AND 9TH INTERNATIONAL SYMPOSIUM ON PROCESS SYSTEMS ENGINEERING

COMPUTER-AIDED CHEMICAL ENGINEERING Advisory Editor: R. Gani Volume Volume Volume Volume

1: 2: 3: 4:

Volume 5:

Volume 6: Volume 7: Volume 8: Volume 9: Volume 10 Volume 11 Volume 12 Volume 13 Volume 14 Volume 15 Volume 16 Volume 17 Volume 18 Volume 19 Volume 20 Volume 21

Distillation Design in Practice (L.M. Rose) The Art of Chemical Process Design (G.L. Wells and L.M. Rose) Computer Programming Examples for Chemical Engineers (G. Ross) Analysis and Synthesis of Chemical Process Systems (K. Hartmann and K. Kaplick) Studies in Computer-Aided Modelling. Design and Operation Part A: Unite Operations (I. Pallai and Z. Fonyo, Editors) Part B: Systems (I. Pallai and G.E. Veress, Editors) Neural Networks for Chemical Engineers (A.B. Bulsari, Editor) Material and Energy Balancing in the Process Industries - From Microscopic Balances to Large Plants (V.V. Veverka and F. Madron) European Symposium on Computer Aided Process Engineering-10 (S. Pierucci, Editor) European Symposium on Computer Aided Process Engineering-11 (R. Gani and S.B. Jorgensen, Editors) European Symposium on Computer Aided Process Engineering-12 (J. Grievink and J. van Schijndel, Editors) Software Architectures and Tools for Computer Aided Process Engineering (B. Braunschweig and R. Gani, Editors) Computer Aided Molecular Design: Theory and Practice (L.E.K. Achenie, R. Gani and V. Venkatasubramanian, Editors) Integrated Design and Simulation of Chemical Processes (A.C. Dimian) European Symposium on Computer Aided Process Engineering-13 (A. Kraslawski and I. Turunen, Editors) Process Systems Engineering 2003 (Bingzhen Chen and A.W. Westerberg, Editors) Dynamic Model Development: Methods, Theory and Applications (S.P. Asprey and S. Macchietto, Editors) The Integration of Process Design and Control (P. Seferlis and M.C. Georgiadis, Editors) European Symposium on Computer-Aided Process Engineering-14 (A. Barbosa-Povoa and H. Mates, Editors) Computer Aided Property Estimation for Process and Product Design (M. Kontogeorgis and R. Gani, Editors) European Symposium on Computer-Aided Process Engineering-15 (L. Puigjaner and A. Espuha, Editors) 16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering (W. Marquardt and C. Pantelides)

COMPUTER-AIDED CHEMICAL ENGINEERING, 21B

16TH EUROPEAN SYMPOSIUM ON COMPUTER AIDED PROCESS ENGINEERING AND 9TH INTERNATIONAL SYMPOSIUM ON PROCESS SYSTEMS ENGINEERING Edited by

W. Marquardt RWTH Aachen University, Leiirstulil fur Prozesstechnil| l 0 t S 1 4omi^ 1 1 «iRrm ( 8 ^ H

1 < ^ 11 '' ''>^NpXNbj^p,\

••• X '

,Nbi,Nsi ' * " ' ^''J'^'

* * *' ^NpXNbNpA

\

(4)

^P'^SNp,NbNp,NsNp -'

' "^'^Np,Ns^p,NbNp,NsNp

-*'

(5)

where c(u) is the Cudras and Auges copula density [Nelson, 1999] which is a function of cumulative distributions of x,y,^, Uy^k, and correlation parameters among them, and Oij^k and bij^k are the parameters of the prior Beta distribution for the failure probabilities. Spearman's correlation matrices [Meel and Seider, 2005c], which give the correlation

Real-Time Failure Prediction for Chemical Processes: Plantwide Framework

nil

parameters for the copula, are defined for the failure probabilities of: (i) individual safety systems under various conditions, (ii) safety systems within a plant unit, (iii) safety systems of several units. The posterior joint failure probability distribution is: Np Nsi Nbij

fix I Data) oc n n i=l j=\

Il(^,Mf"'~'i^~î.Mt'"~'fix) k=\

Np Nsj Nbjj

(6)

II Np NSi Nbjj ''i,j,k-

/=i j=i

/=1 y=i k=i

k=i

Nsj Nhi^q

11 Np NspNbp^q

n(i-^--,M-.-,M..)|| n n(i-^.-,M-...)

""ij^k

nnn(i-^.-,M-... >

(7)

where Kij^k and Ltj^k are the cumulative number of failures and successes, respectively, at the end of each time interval and Oij^k-p,q,r is the correlation parameter between x^^k and Xp^q^r- The marginal posterior distribution of each component of x, x^^k, is obtained from fix\Data) through Markov-Chain Monte-Carlo integration [Robert, 2001]. Again, an introduction to the use of prior and posterior distributions in the Bayesian analysis of accident precursor data is provided by Meel and Seider [Meel and Seider, 2005b]. In Figure 3a, failure probabilities for operator diagnosis for Rl at two branch points, with jci 3,2 > xi,3,i, are shown that account for their correlation with other safety systems associated with the same and other plant units. Similarly, Figure 3b shows the failure probabilities of the high-temperature alarm for Rl at five branch points with xi,5,5 > X\^s,3 > ^1,5,4 >-^1,5,2 > ^1,5,1- The probabilities of consequences, CO, SD, REL, and EXP for Rl are shown in Figure 3c. Observe that the ASP data gives higher probabilities of shutdown than continued operation. It is also important that a finite probability for EXP is estimated, although no occurrences of this end-event appear in the ASP data. Note that the variations in failure probabilities as a fiinction of time interval exceed those for the Bayesian analysis of individual plant units, yielding failure probabilities that account for plantwide interactions. Furthermore, the membership values in green, blue, orange, and red zones, associated with increasing safety threats, formulated by Meel and Seider [Meel and Seider, 2005b] for Rl are graphed in Figure 3d, with higher memberships in the blue and orange zones. Similar results were obtained for the other plant units. '->^

'

^

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>r

'

-^^^.3.^

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A / \

P sx 5 0.6 o a.

^^0

\ /

V

\

\ /

JK ^•^y^-^ ?\

1 T k \

•o

r\

°-4

SiO^ i +2H^0

In this the model derived by Bogush et al.(1988) is implemented to predict the diameters of Si02 particles given the concentrations of TEOS solution and NH3 solution as following: I d = A [Hfif Qxp(-B [HÔY) A = [TE0S]2{S2-I5l [NH,] + 1200[ NH,f -366

[NH,f)

5 = 1.05 + 0.523 [NH,]-0.12S [NH,f

where d is silica particle diameter (nm), [H2O] is water condensation (M), [TEOS] is TEOS condensation (M), [NH3J is ammonia condensation (M). Hence the particle diameter is affected by ammonia condensation, water condensation, TEOS condensation, reactive temperature, reactive time, and water/TEOS condensation proportion. The above model is linearized and the linear model is, in turn, implemented to control the particle sizes of the SiOi particles.

Theoretical Analysis and Experimental Studies of Mixed Product Run-to-Run Control 1189 3.2. The Experimental Setup TEOS*Ethanol

V

NH3*H20*Ethan«l

/

A,B measuring

Two open flask

flask

J ^ Q ^pgn

fl^sk

A,B measuring f

OO (b)

Fig.l Experimental steps Set [TEOS]=QAlM,

[H20J=5M

Temperature=35 °C . V= 100ml. The experimental steps are shown in Fig.l. After paricles preparing experiment, SEM is used to observe 3 drop liquid and 2 pictures are taken each drop. The diameter of 100 particles as shown in Fig.2 can be analyzed using software OPTIMAS. The average and the standard deviation of the particle diameters can be observed.

Fig.2 The particle analysis graph 3.3. Experimental results In this work, two different sizes particles (d=200nm and d=250nm) are prepared at one experimental setup as depicted the previous section. Fig.3 and Fig.4 show control results of the particle diameters of two products using the "tool based" and the "product-based" approach respectively. As shown in these two figures, it is found that "product-based" approach is superior to "tool based" approach. Fig.5 shows the error of the output using the two approaches. It is clear that the product output is convergent using "product-based" approach, while divergent using the "tool based" approach, and this is consistent to our theoretical derivation in the previous section.

Y. Zheng et al.

1190

290-

290-

280-

280-

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270260•g" 240-

270260-

230-

5 230-

1

220-

6

1

1' t 11t ' I

180170160150-

T

220-

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160150-

Fig.3 Two products controlled by "tool based" approach

Fig.4 Two products controlled by "product based" approach m

J * 1

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Tool (T=250nm) Too! (T=200nm) Product {T=250nm) Product (T=200nm)

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Fig.5 Errors of the products using two different approaches

4. Conclusion In this work, a rigorous analysis of two different runto-run control strategies in a high mixed production is performed. We found that a tool-based approach is some times unstable while a product-based control is stable. The experimental study of the preparation of

nano scale Si02 particles verifies the theoretical analysis. References Bogush G..H., Tracy M.A., Zukoski IV C.F. (1988). Preparation of monodisperse silica particles: control of size and mass fraction, Journal ofNon-Crystalline Solids , 104 , 95-106 Del Castillo, E (2002). Statistical Process Adjustment for Quality Control. John-Wiley and Sons, New York. Moyne, J., E.D.Castillo, and A.M.Hurwitz (2001). Run-to-Run Control in Semiconductor Manufacturing. CRC Press, Florida. Stober W., Fink A., Bohn E. (1968). Controlled Growth of Monodisperse Silica Spheres in the Micro Size Range, Journal of Colloid and Interface Science, 26, 62-69 Zheng, Y., Lin, A.H., Wang,D.S.H., Jang, S.S. and Hui,K.(2005). Stability and Performance Analysis of Mixed Product Run-To-Run Control. J. ofProcess Control, In Press.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Methods of State Estimation for Particulate Processes M. Mangold,^ C. Steyer,^ B. Niemann,^ A. Voigt,^ K. Sundmacher ^'^ ^Max-Planck-Institutfur Dynamik komplexer technischer Systeme, Sandtorstr. 1, 39106 Magdeburg, Germany Ôtto-von-Guericke- Universitdt, Lehrstuhlfur Systemverfahrenstechnik, Universitdtsplatz 2, 39106 Magdeburg, Germany Abstract Determining property distributions of particles online by measurement is difficult in many cases, especially if the particles are in the nanometre range. An alternative may be state estimation techniques, which use information from process simulations in addition to the measurement signals. Two examples of state estimators for particulate processes are presented in this contribution. The first one is an extended Kalman filter based on a population balance model. The second one is a bootstrap filter based on a Monte Carlo simulation. Keywords: particulate processes; state estimation; population balance; extended Kalman filter; bootstrap filter 1. Introduction A large amount of chemical products are generated in the form of particles. Very often, property distributions have a strong influence on the product quality. Therefore, online information on a property distribution like the particle size is desirable. However, it may be difficult to obtain such information directly from measurements, especially if the particles are very small, i.e. in the nanometre range. State estimator techniques may be a solution to this problem. A state estimator consists of a simulator part containing a process model and of a corrector part, which uses the measurement information to let the estimated state converge towards the actual state. As an example, a state estimator using a population balance model of a bulk precipitation process is presented in Section 2. After discussing the question of observability for this system, an extended Kalman filter is designed based on the population balance model. As an alternative to population balance models, Monte Carlo simulations are gaining increasing attention in the field of particulate processes. In order to incorporate a Monte Carlo simulation model into a state estimation framework, suitable correction techniques are required. Such techniques have been developed over the last years (Doucet et al., 2001). An example is the state estimator for a micro emulsion process discussed in Section 3. 2. Population Balance Model of a Bulk Precipitation Process The precipitation of nanoparticles in a batch process is considered in the following. Only the key features of the model are presented here. For details on the derivation of a population balance model, see e.g. (Oncul et al., 2006) and references therein. In the liquid phase, two components A and B react to a component C that forms solid particles. In the model, the liquid phase is assumed to be perfectly mixed. The

1191

M Mangold et al

1192

nucleation and growth of the particles is described by a one-dimensional population balance model, where the particle size is used as the property coordinate. Further model assumptions are isothermal conditions and negligible agglomeration and breakage. 2.1. Model Equations The mass balances of the liquid phase read dc

-

k^C^Cg

(1)

dt dCr dt

= ^VCACB - CG

\(ÎG{CC,dp

)fddp - C^^^c^,

(2)

dp=dp.

where C/ denotes the molar concentration of component /, r{cA,CB) is the rate of the liquid phase reaction; / is the number density function of the particles; dpo and dp^rnax are the minimum and the maximum particle size, respectively; G is the particle growth rate, and CG , Cnuc J K are constant factors. A population balance for the particles of component C leads to the following equation for the number density function/

f =- ^ ^ ^ ^ ^ ^ ;

G{c,,dpMd..,t)

= B{c^

(3)

The following expressions are used for the growth rate G and the nucleation rate B: ( G{cc,dp) = k^ ^c-^êxpl -11 (4) ydp J

B{cc) = k„

f r ^^ ^C

^C,c

exp

\^dpQ J J

(5)

(^G, l^nuc, b, C2, and c^^^ are constant parameters). It is assumed that the total number of particles is measurable, i.e. the following relation holds for the measurement variable y: **r,max

y =

\fddp

(6)

dp=dp

2.2. Investigation of Observability A system is observable, if it is possible to reconstruct its complete initial state from measurements taken over a finite time interval; see e.g. (Levine, 1996). Observability is a necessary condition for successful state estimation. A strict proof of observability is very difficult for a nonlinear distributed system. In the following, some preliminary results on the observability of the bulk precipitation model Eq. (l)-(6) will be given. If the system is observable, it must be possible to solve the equations for y, dy/dt, (fy/d^,... uniquely for the system states (Zeitz, 1977; RothfuB and Zeitz, 1995). From Eq. (6) and(3) one obtains for f\dp^^^ J)=0:

f=.fc)

(7)

Methods of State Estimation for Particulate

1193

Processes

Eq. (7) can be solved uniquely for cc, i.e. CQ is observable. Assuming for the moment that the integral in Eq. (2), which is denoted as I(t) in the following, is known, algebraic equations for CA and CB can be derived from Eq. (1) and (2):

do c, = Cgk^

dt

- + I + C^

(8)

f^^C n

dcc + I + C„^c, - c , dt

( A^

d'cr

dl

dr

dt

+ C„

dt (9)

Jdc, V dt

+ J + C„^^c^ = 0

Eq. (8) and (9) can be solved at least locally for Q and CB . This means that CA and CB are observable if it is possible to reconstruct / and / from measurements. In order to prove observability of the number density function / one has to show that / can be determined uniquely from the equations for j^, cfy/dt^, (fy/dt^,..., which have not been used yet in the observability analysis. By deriving Eq. (6) with respect to time, inserting the balance equations (l)-(3), and dissmmng f\dp ^^^j)=

0 , one obtains expressions

of the form \Pifddp =F.{y,dy/dt,...,dy/dt\c^,Cs,Cc\

with PQ =l;P2 = d^Gip.

i = 0,2,3,...

(10)

= 3p._i IddpG (/ > 3).

In order to be able to reconstruct/from the projections Eq. (10), the functions/>/ should form a complete set of basis functions. By developing G into a Taylor polynomial in dp and evaluating the expressions for pi, it is possible to see that the left-hand side of (10) contains arbitrary powers of dp, if G depends at least quadratically on dp, as it is the case for the model used here. This is a necessary condition that the model of the bulk precipitation process is observable. A further hint that the system is observable is the (numerically evaluated) full rank of the observability matrix of the linearised version of the discretised system. 2.3. Extended Kalman Filter For the numerical solution of the bulk precipitation model, the population balance (3) is discretised along the property coordinate dp, using a finite volume method. Based on the discretised model, a continuous-discrete extended Kalman filter with a continuous simulator part and time discrete filter update part is developed. The design equations for such a filter are well-known, see e.g. (Gelb, 1974), and are not listed here. The spectral density matrix of the process noise Q and the covariance matrix of the measurement noise R are chosen as diagonal matrices with entries on the main diagonal only. The filter is tested in simulations. Virtual measurement data are used, which are generated by adding normal distributed artificial noise to the simulation results of the bulk precipitation model. A time interval of 1 s between two measurements is assumed. A simulation result is shown in Fig. 1. The initial concentrations of ^ and B are chosen intentionally 5 % too high in the filter model. If the filter update is deactivated

1194

M Mangold et al.

(diagrams in the left column), then this small error in the initial conditions causes the estimated number density function to deviate considerably from the number density ftmction of the real system. If the filter update is active (diagrams in the right column), then the estimated particle size distribution converges rapidly towards the particle size distribution of the real system.

without filter correction •

with Kalman filter

estimate L measurement |.

^ 0.8

0)

E 3 c o ^

E 0.6 ^0.4

CO

_cg o

20

40 timet/s

60

^0

20

40 timet/s

60

with Kalman filter

without filter correction

— —

o 4

real value estimate |.

^-—__18s

a 2 F. 1- E

10"" 10" particle size d / m

10

So E10

_-36s 54s K Nv/\rs70s 18s^ 10"° 10"' particle size d / m

10"'

Figure 1. Test of the extended Kalman filter for the bulk precipitation process in simulations; lefthand side column: comparison of reference states and estimated states, if no filter update is used (pure simulation); right-hand side column: estimates of the extended Kalman filter and comparison with reference states; top row: estimated and reference measurement value; bottom row: estimated and reference particle size distribution.

3. Monte Carlo Simulation of a Micro Emulsion Process Models of particulate processes become computationally expensive, if several property coordinates have to be taken into account. In such a case, the use of stochastic Monte Carlo simulation approaches may be advantageous. Therefore, it seems worthwhile studying the application of Monte Carlo simulations to state estimation problems. As an example, the BaS04 precipitation in micro emulsion is considered in the following, where two reactants BaCb and K2SO4 dissolved in aqueous droplets react to BaS04. A Monte Carlo model of the process was presented in (Adityawarman et al. 2005). Its main features are summarized in Section 3.1. In order to use such a model in a state estimation framework, suitable filter correction techniques are required. Various

Methods of State Estimation for Particulate Processes

1195

recursive Bayesian state estimation techniques have been developed over the last decade (Doucet et al., 2001), but -with few exceptions (Goffaux and Vande Wouver, 2005)have hardly been used in chemical engineering applications. The concept of these methods is illustrated by a bootstrap filter for the micro emulsion process in Section 3.2. 3.L Model Description The Monte Carlo simulation is initialised with a given number of droplets. The droplets are filled with one of the two reactants BaCl2 (component A) or K2SO4 (component E). The droplet coalescence and redispersion are the basic steps of the Monte Carlo simulation. For a coalescence event, a pair of droplets is randomly chosen and mixed. A and B in the droplets are assumed to react instantaneously to BaS04. The number of BaS04 molecules present in the droplet is compared to critical value Xûci related to supersaturation. If the number of BaS04 molecules exceeds Xnuch a particle nucleation can occur, depending on a rate coefficient for such an event. In the redispersion step, the remaining liquid reactants are distributed randomly back into two identical droplets. The particle is put randomly into just one of the droplets. If a particle is already present in one of the droplets then all dissolved BaS04 will be used to let that particle grow. As in the previous example, it is assumed that only the total number of particles is measurable. 3.2. Bootstrap Filter The basic idea of recursive Bayesian estimation is as follows (Doucet et al. 2001, Gordon et al., 1993). One starts from the model description X, = f ( x , _ l , ^ , _ l )

(11)

y = h(x,,vj,

(12)

where x is the state vector, y is the measurement vector, co is the process noise, and v is the measurement noise (in the example of the micro emulsion process, f is given implicitly and evaluated by Monte Carlo simulation). The initial probability density distribution (PDF) of the state vector /?(xo) is assumed to be known. The PDF at a later time t can then be predicted from the previous estimate of the state vector Xt.i and past measurements according to:

p{^, I y,-i) = \p{^, I \-x )p{^,-i I y 1.-1 )^r-i

(13)

When a new measurement j ^ becomes available, the predicted PDF can be updated using the Bayes rule:

/

p\^t

.

N

p{yt\\)p{^t\yv,-x)

I y 1:^) - T ^ — \ — ^ 7 — ]

r ^

(14)

lp{yt\^t)p{^t\yi.t-i)d^t The evaluation of the high-dimensional integrals in Eq. (8) and (9) is very difficult in practice. However, Gordon et al. (1993) proposed a simple sequential Monte Carlo algorithm for an approximate solution, the so-called bootstrap filter. To test the bootstrap filter for the micro emulsion example, it is assumed that the initial ratio between the amount of ^ and the amount of B is non-stoichiometric and unknown. The filter is tested with virtual measurement data obtained by adding noise to the result of a previous Monte Carlo simulation. The initial conditions of the N samples used by the filter differ in the A/B ratio that goes from 0 to 0.5. Figure 2 shows the state of the filter after 10 measurement intervals and update step. From Figure 2 (a) it can be seen that after a few time steps most of the samples group rapidly around the reference A-to-B

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M Mangold et al

ratio of 0.4. Figure 2 (b) shows the estimated particle size distribution at time step 20 which agrees well with the reference size distribution. S 0.2 40 estimate reference

.-^0.15

30 Q. CD

(D •D

S20 E

n

0

E

!^B

10

I 0.05 "co E

^0

0.1 0.2 0.3 0.4 initial A-to-B ratio in droplets

0.1

0.5

A

L

1 2 particle diameter / nm

Figure 2. Bootstrap filter for a microemulsion process; (a) distribution of the samples at different time steps; (b) comparison between reference size distribution and estimate at time step 20.

4. Conclusions State estimation techniques or model based measurements may be a useful tool for the process operation of particulate processes. They can serve as software sensors in cases where a direct measurement of property distributions is difficult or not possible, e.g. due to a small particle size. If population balance models of moderate complexity are available, classical observer design methods can be used to set up a state estimator, as was illustrated in Section 2. However, the solution of population balance models in real time may be difficult, especially for models with several property coordinates. In such a case, Monte Carlo simulations may be used as a viable alternative. The use of Monte Carlo simulations for state estimation requires appropriate filter correction techniques. The results of Section 3 indicate that recursive Bayesian filter techniques are a promising approach to this problem.

References D. Adityawarman, A. Voigt, P. Veit, K. Sundmacher, 2005, Precipitation of BaS04 nanoparticles in a non-ionic microemulsion: Identification of suitable control parameters, Chem. Engng. Sci., 60, 3371-3383. A. Doucet et al. (eds.), 2001, Sequential Monte Carlo Methods in Practice, Springer. A. Gelb (ed.), 1974, Applied Optimal Estimation, MIT Press, Cambridge. G. Goffaux, A. Vande Wouver, 2005, Bioprocess State Estimation: Some Classical and Less Classical Approaches, In: T. Meurer, K. Graichen, E.D. Gilles (eds.). Control and Observer Design for Finite and Infinite Dimensional Systems, Springer, Berlin, 111-130. N.J. Gordon, D. J. Salmon, A.F.M. Smith, 1993, Novel approach to nonlinear / non-Gaussian Bayesian state estimation, lEE Proceedings-F, 140:107-113. W.S. Levine (ed.), 1996, The Control Handbook, CRC Press, Boca Raton. A. Onciil, K. Sundmacher, A. Seidel-Morgenstem, D. Thevenin, 2006, Numerical and analytical investigation of barium sulphate crystallization, Chem. Eng. Sci., 61, 652-664. R. RothfuB and M. Zeitz, 1995, Einfuhnmg in die Analyse nichtlinearer Systeme, In: S. Engell (ed.), Entwurf nichtlinearer Regelungen, Oldenbourg, 3-22. M. Zeitz, 1977, Nichtlineare Beobachter fur chemische Reaktoren, VDI-Verlag, Dtisseldorf


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Simultaneous Scheduling and Optimization of a Copper Plant liro Harjunkoski, Hans Werner Borchers and Marco Fahl ABB Corporate Research, Wallstadter Str. 59, 68526 Ladenburg, Germany Abstract This work presents a novel batch scheduling solution that has been developed for a copper plant. The major characteristics of the copper production process with some specific interdependencies are briefly summarized, the main modeling challenges and some key elements of the resulting mixed-integer optimization problem are presented, followed by a concrete example problem highlighting the benefits of the solution. Keywords: Copper production, planning, scheduling, recipe optimization 1. Introduction The copper production process is difficult to plan and schedule due to the lack of sufficient and exact measurement data, complex logistics, high raw-material variability, frequently occurring disturbances and maintenance operations that heavily affect the ideal process cycles. Normally, the scheduling is done manually and a "global" overview of the overall production process is usually missing. Thus, typically each unit ends up running at "full speed", i.e. trying to produce as much as possible as this is the intuitive local optimum from an individual equipment perspective. This results in productivity losses since the overall process efficiency may be far from optimal as many batches end up unnecessarily waiting for equipment in the next production stage. Only very few contributions on copper plant scheduling problems can be found in the literature, see e.g. Pradenas et al. (2003), where a heuristic algorithm is used to maximize the copper production at the Chuquicamata Copper Smelter in Chile. The method first generates combinations of three possible production cycles and then tests the resulting schedule against the best existing solution. The objective is to maximize the troughput. The main constraints are operational, metallurgical, mass balance related, environmental or related to the timing of loading and unloading operations. The scheduling and optimization approach that is presented here considers simultaneously, and in a rigorous way, the most important aspects affecting the production process. It enables a more efficient production, better overall coordination and visibility of the process, faster recovery from disturbances and supports optimal maintenance planning. The resulting batch scheduling model is solved as an MILP and covers the main processing steps, making it possible to optimally relate input material properties to corresponding processing times. By taking into account main chemical reactions, recipe optimization is performed simultaneously ensuring that the schedule is always optimal with regard to varying raw-material quality. In the following, we briefly summarize the major characteristics of the copper production process.

/. Harjunkoski et ah

1198

2. The copper production process The copper production process discussed is a batch process that covers two parallel production lines, with the typical requirement to synchronize the lines such that they serve the caster in an alternating way with anode copper, see Fig. 1. The processing is done under strictly controlled temperatures where several chemical reactions take place to enable the separation of copper from other material. The processing in each stage consists of multiple phases, e.g. the four major converter operation phases are: Charging, slag blowing, copper blowing and discharging. Typically, it takes 14-18 hours for one batch to go through the whole process.

Converter Primary Furnace

pranes

Converter

Anode Furnace

O:

Caster

Anode Furnace

Figure 1. Parallel processing equipment A special aspect on the production process is that there is only one unique end product copper anodes with the final copper content of 99.6%. The major production challenge originates from highly varying raw-material properties, which affect the optimal production recipe. An example of variations is the copper content of the matte (melt copper from the primary furnace), which may vary between 60% and 70%, depending on the chemical composition of the raw-material (copper concentrate) used. A high copper amount reduces significantly the required processing time at the converter and furthermore also results in less slag that needs to be collected along the processing. Low copper content, on the contrary leads to longer processing times, large slag amounts and a need for more so-called cooling material, i.e. recycling material with high Cu-content. The raw-material quality variations have a high impact on the scheduling decisions. For this reason, the scheduling model must also comprise main mass-balances, most important dependencies between the material amounts and purities, and the processing times. Consequently, an appropriate scheduling model also needs to encapsulate a simultaneous recipe optimization for each batch. As in most copper plants, the material is transported in ladles using cranes. Thus, the process must also be synchronized from the production logistics perspective such that parallel activities do neither overload nor block the cranes. This poses additional constraints to the scheduling problem and highlights the requirement of having a complete overview on the plant when doing the detailed production scheduling. 3. Scheduling problem According to the process description above, the scheduling problem at hand can be classified as a single-product multi-stage batch-scheduling problem with parallel processing equipment and the requirement of simultaneous recipe optimization. The main goal of the targeted solution has been to generate a detailed and optimized schedule for the production process that (from the model perspective) explicitly considers:

Simultaneous Scheduling and Optimization of a Copper Plant

1199

• two parallel converters, • two parallel anode furnaces, and, • one continuous caster that is directly fed by the anode furnaces. The primary objective is to maximize the throughput. From a practical perspective and in the light of re-scheduling requirements, the computing time for generating a valid optimized schedule must not exceed a few seconds. Therefore the scheduling model is not allowed to be too complex but only to capture the essentials, i.e. major issues that affect the production timings. Extreme operating conditions create regular maintenance needs and cause unexpected equipment breakdowns. Therefore, optimal maintenance scheduling is needed in order to minimize the impact of maintenance actions and corresponding process interruptions with respect to throughput reduction. Summarizing the above, the main decisions to be taken are: • Exact timings for the processing steps (parallel lines, maintenance) • Material amounts (melt copper, scrap material, slag) • Production cycles (converter and anode furnace) • Utilization of limited resources (timing critical) • Reserving sufficient time for necessary logistics (e.g. cranes) Since we are dealing with a single-product problem, the sequence of batches is not relevant and can therefore be given a priori. However, this refers purely to the batch numbers. The integrated recipe optimization will define individual characteristics (size, composition) for each batch. The batch routes may vary as indicated in Fig. 1. and the continuous-casting stage is treated as a batch operation. The solution that has been developed builds on a valid and robust process model that captures the main chemical reactions and is able to link the variable material amounts with predicted processing times. 4. Solution approach A mathematical programming solution approach has been selected to ensure that an optimal solution can be obtained. This poses two main challenges: • The resulting process model is in its pure form intractable, mainly due to the chemical reaction kinetics and • Logic decisions that need to be made in scheduling (commonly precedence and equipment assignment) must be represented by discrete or binary variables. In order to make the problem solvable, a linearized process model has been derived. This enables the use of standard Mixed Integer Linear Programming (MILP) techniques, for which several robust commercial solvers are available. For ensuring the validity of the linearization approach, the process model has been verified with a significant amount of plant data. A standard continuous-time job-shop scheduling formulation (Pinto and Grossmann, 1998) is used to provide the basic aspects of the production decisions, such as sequencing and assignment of jobs. The key of the mathematical solution is to build an optimal schedule by capturing the durations of each processing step and to relate it to the material amounts. Therefore, a top-down approach will be presented here to illustrate some main principles of the model only.

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First, some constraints related to the converter operations are shown. In the following, we consider the four major converter operation phases. The index s is used to denote a certain phase and p refers to a batch. The corresponding upper case letters refer to the respective sets, e.g. *S^ refers to the phases that are valid for the converting. Two different timings must be included. The time when a batch, p, actually is in the converter is defined by the start and end times, f^p and f^p. These are relevant for keeping track on the equipment availability. This aggregated timing is then subdivided into more detailed timings, which define the start and end time of each converter phase, t^^^p,s and f^^p,s' These, again are necessary for the synchronization of parallel tasks. Here, we consider primarily the converting phases.

C.=C

V;,eP,V.e5^|.,. = !

.CE

.CSE

^p =^p,s

W^ ^

(4)

D „

I cC I

/c\

ypeP,s=\S

I

(5)

Constraint (1) specifies the relation between the start and end times of a phase, where Tp^s is the processing time needed. Constraint (2) specifies that the two first phases must be followed immediately by the next one. Some flexibility is allowed for the rest of the phases, as defined by the delta-variable in Eq. (3). The relationship between first and last phases and the converter beginning and ending times are defined in Eqs. (4-5). They specify exactly the time when a converter is assigned to each batch. Similar relations are given for each phase of all equipment operations. Below, an example is shown where a phase combines two equipment. This is e.g. the case when a converter is emptied to an anode furnace (requires that both equipment are available simultaneously) or during the continuous casting, which actually is done by discharging an anode furnace. The former case is illustrated by Eqs. (6-7). fCSB p,s

_

\/peP,s=\S^ \,s' mm{S'}

(6)

T

=T , \/peP,s=^S^ \,s' mm{S'} p,s

(7)

p,s

.ASB p,s'

The start time of the last converter phase (emptying) is equal to the start time of the first anode furnace phase (filling), as expressed by constraint (6), where s' denotes the first phase of the anode fumace. The durations are equal, as shown in Eq. (7). A minor time shift, due to crane movements and initial filling at the converter, could be included but since this is typically in the order of a few minutes, it has been omitted here. Another important issue is to avoid the overlapping of crane operations. Avoiding simultaneous filling/emptying operations can be modeled by the constraints that are shown below. We assume that batch/? is immediately followed by batchp+7. Here, the major converter phases are enumerated by 1-4.

Simultaneous Scheduling and Optimization of a Copper Plant

tlfu^^?,i^K

1201

yp^p\ptfl,+^^^-^,,

(8)

^p^p\p 0; Rk = G ^ MV

(2)

^—^

where J is a weighted sum of the feed rates to the plant. The manipulated variables (MVs) at the coordinator level are typically the external feed rates and crossovers in the plant. G represents the dynamic influence from each MV to Rk. The coordinator MPC should operate such that it is possible to keep each unit specification. However, unmeasured disturbances and slow responses may require some back-off in the unit when the disturbances occur. The constraint back-off should be set according to the controller performance and the acceptable constraints violations. The magnitude of the back-off depends on the expected size of the disturbances and how strict the product specifications are. If the product is mixed on tanks before sale, violating the product specifications for a shorter period may be acceptable. The use of back-off reduces the value of J, but makes the coordinator more robust. Control target/range changes in the local MPC, like MV and CV limit changes, have a direct infiuence on the remaining capacity measure and must also be handled by feedback with the given coordinator design. Nonlinear effects in the process causes modeling error in the coordinator and must also be handled by feedback. All these effects argue for a fast feedback sampling in the coordinator MPC. The process dynamics seen by the coordinator MPC includes the local MPCs. Local MV saturation should be avoided so the local MPCs are more robust to handle disturbances and to linearize the process seen by the coordinator. To avoid local MV saturation, some back-off on each MV is included in the calculation of remaining capacity.

5. K A R S T 0 GAS PROCESSING CASE STUDY The coordinator MPC approach has been tested with good results using the Karst0 Whole Plant simulator. This is a dynamic simulator built in the software D - S P I C E ® L 5.1. The case To demonstrate the applicability of the coordinator MPC, we use a detailed simulation model of parts of the Karst0 plant. The two fractionation trains, T-lOO and T-300, both have a deethanizer, depropanizer, debutanizer and a butane splitter. In addition T-300 has two stabilizers in parallel. The simulated parts of the plant are shown in Figure 1. There are two separate train feeds, a liquid stream from a dew point control unit (DPCU) that is divided between the two trains, and a crossover. The five streams are MVs in the coordinator MPC and indicated by valves in Figure 1. The local MPCs and the coordinator are implemented in SEPTIC^ MPC software [7]. For description of the local MPCs, see [8]. 5.2. Coordinator M P C The coordinator task is to maximize the plant throughput within feasible operation. Maximizing flow rate can be realized with our standard quadratic objective function by a total plant feed as a CV with a high (not reachable) set point with lower priority than the capacity constraints. The inputs and outputs of the coordinator MPC are as follows: • CVs: Remaining capacity in each column, 10 in total (ETIOO, PTIOO, BTIOO, BSlOO, STABl, STAB2, ET300, PT300, BT300, BS300), T-lOO deethanizer sump level controller output (LC OUTLET) and total plant feed (PLANT FEED) ^Statoil E s t i m a t i o n a n d Prediction Tool for Identification a n d Control

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21FC5288VWA

Figure 1. The simulated parts of the Karst0 plant

• MVs: Feed flow from DPCU to T-lOO and T-300 (21FC5334VWA, 21FC5288VWA), train feed flow T-lOO (21FR1005VWA), train feed flow T-300 (FEEDT300VWA) and crossover flow from T-lOO to T-300 (24FC5074VWA) The total plant feed is defined as the sum of the train feeds and DPCU feeds. The level controller output as a CV follows to avoid emptying or filling up the sump level in the deethanizer T-lOO when manipulating the crossover. The remaining column capacity is calculated in each local MFC as an LP problem considering CV and MV constraints around the column. The column feed is a free variable in the LP formulation and the fiooding point is represented by a high limit on the column differential pressure. The execution is set to a slower rate than the local MFCs to ensure robustness in the feedback loop and is here chosen to be 3 minutes. The column capacity depends both on the column feed fiow and the feed composition. At the Karst0 plant, only the feed flow is manipulative. The composition is measured with gas chromatograph (GC) at the plant feed inlets and in top of the distillation columns. However, the dead time in the GC sampling makes the measurements unsuitable for control. The feed composition changes are therefore characterized as unmeasured disturbances. The coordinator models are experimental step-response models, and are found in the same way as in the local MFCs. The models were obtained at 80-95% of the maximum throughput which is a common operation area for the real plant. The coordinator MFC tuning is a trade-off between MV variation and CV constraint violation. Some constraint violation cannot be avoided due to the process response times, unmeasured disturbances and model errors. The tuning should not be so aggressive that model errors are amplified, which means that some constraint back-off will be necessary. 5.3. Results from the simulator case study The coordinator performance is illustrated with tree different cases, and the CVs in the coordinator MFC are shown in Figure 2 whereas the MVs are displayed in Figure 3. Move the plant to maximum throughput. The coordinator is turned on at t = 0 minutes to move the plant operation from a non-optimal to an optimal operation point. Figure 2 shows that the deeethanizer in T-lOO and the stabilizers are bottlenecks at the optimal operation point. The butane splitter in T300 reaches its capacity limit, however, there is available capacity in the depropanizer and downstream columns of T-lOO and the coordinator uses the crossover to reroute, removing the T-300 butane splitter bottleneck. Change in feed composition. A momentary feed composition change is introduced in the T-lOO feed at t = 360 minutes. The feed composition change increases the remaining capacity of the T-lOO deethanizer and makes it possible for the coordinator to increase the train feed. The disturbance reduces the remaining capacity in the butane splitter so the coordinator uses the crossover to keep the column within its capacity. However, the butane splitter in T-lOO is not a plant bottleneck yet, since there is still capacity for rerouting to T-300. Change in a local MPC CV limit. With the butane splitter in T-lOO operating at its capacity limit, the

Coordinator MPC with Focus on Maximizing Throughput 50

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50

^.L^^J

^

1000

1. 11.. \*

§ 520

110

g 500 lU

100

^ 480 1000

I 30 S 25

in LL 20

*^ 15

1000 minutes

Figure 3. MVs in the coordinator MPC, flows in t / h

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operator reduces the bottom quality high limit in the local MPC at t = 600 minutes. The coordinator increases the crossover since there is available capacity in the T-300 string, but must also reduce the T-lOO feed some. Both the butane splitters are now bottlenecks in the plant, together with the stabilizers. 6. D I S C U S S I O N The coordinator MPC is demonstrated on a dynamic simulator and performs well for the simulated challenges. Back-off reduces the optimal function value, but is necessary due to unmeasured disturbances and the long process response times. An improvement of the coordinator MPC is to include some feed forward especially from feed composition changes. Split fraction in the column can be used, then both feed composition changes and upstream processing units operation changes will be detected. Including feed forward information, the back-off in the coordinator could then be reduced leading to a larger plant throughput. The coordinator MPC uses linear models while the process is nonlinear. In cases where the nonlinearities mostly are reflected in model gains, gain scheduling of the model improve the performance. Gain scheduling is possible to include in the current model form. Significantly model changes including dynamics, other model types in the coordinator MPC should be evaluated. Another weakness with the coordinator MPC is the simplified maximum throughput objective function. This is a special case, and if the feed turns to be limited for a period, the economic optimum will be different since energy costs and product prices should be included in the objective function. In such a case the coordinator will not lead the plant to optimal operation. Further, the buffer capacity in the distillation columns are not exploited. Faster responses can be obtained by active use of the buffer volumes, leading to a smaller loss in the economic objective function. In this simulated case the buffer volumes are limited, however, in other plants with larger buffer volumes this should be considered. Other linking variables between the units can also be considered, like increasing impurity in a column to decrease the load to the downstream unit. At last, the simplified coordinator MPC will be inadequate for longer-term planning purposes, where a more traditional RTO model will give valuable information. 7. C O N C L U S I O N In this paper we suggest to use a coordinator MPC with experimental step response models to optimize a plant with disturbances of dynamic character. The plant economic object function is simplified to maximize throughput in this case with a gas processing plant as case study. The coordinator MPC is designed and set up in a simulator together with unit MPC applications and performs well for the simulated challenges. REFERENCES [1] Y. Zhang and J.F. Forbes. Extended design cost: a performance criterion for real-time optimization systems. Comp. Chem. Eng., 24:1829-1841, 2000. [2] T. Tosukhowong, J.M. Lee, J.H Lee, and J. Lu. An introduction to a dynamic plant-wide optimization strategy for an integrated plant. Comp. Chem. Eng., 29:199-208, 2004. [3] M. Tvrzska de Gouvea and D. Odloak. One-layer real time optimization of L P G production in the F C C unit; procedure, advantages and disadvantages. Comp. Chem. Eng., 22:S191-S198, 1998. [4] J.V. Kadam, M. Schlegel, and W. Marquardt. A two-level strategy of integrated dynamic optimization and control of industrial processes - a case study. ESCAPE-12, The Hague, The Netherlands, pages 511-516, 2002. [5] J.Z. Lu. Challenging control problems and emerging technologies in enterprise optimization. Control Engineering Practice, 11:847-858, 2003. [6] S. Skogestad. Control structure design for complete chemical plants. Comp. Chem. Eng., 2S:219-234, 2004. [7] S. Strand and J.R. Sagfi. M P C in Statoil - Advantages with in-house technology. ADCHEM 2003,Hong Kong, pages 97-103, 2003. [8] E.M.B Aske, S. Strand, and S Skogestad. Implementation of M P C on a deethanizer at Karst0 gas plant. 16th IFAC World Congress, Prague, Czech Republic, 2005.


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Fault diagnosis based on support vector machines and systematic comparison to existing approaches Ignacio Yelamos,^^ Gerard Escudero,^^ Moises Graells,^^ Luis Puigjaner^^ ""Chemical Engineering Department-CEPIMA, ^Software Department, ÊTSEIB Diagonal 647, 08028-Barcelona (Universitat Politecnica de Catalunya) ÊUETIB Comte d'Urgell 187, 08036-Barcelona (Universitat Politecnica de Catalunya) Abstract An innovative data based fault diagnosis system (FDS) using Support Vector Machines (SVM) is applied on a standard chemical process. Besides its simpler design and implementation, this technique allows dealing better with complex and large data sets. For that reason, it was expected to improve usual pattern classifiers performance reported in chemical engineering literature, such as artificial neural networks or PCA modeling techniques. In order to compare results with previously reported works, a standard case study such as the Tennessee Eastman (TE) process benchmark was considered and SVM achieved consistent and promising results. Besides, the difficulties encountered when comparing the results reported are discussed and a FDS comparison methodology is proposed based on reliability and accuracy of each FDS. In that sense, this study establishes a reference framework for future comparisons. Keywords: Fault diagnosis, support vector machines. 1. Introduction Parallel to the growing computational power, diverse statistical and machine learning techniques have been developed for fault detection and diagnosis (FDD). Approaches based on pattern recognition techniques (neural networks, fuzzy systems, etc) or qualitative reasoning (signed directed graph) have been used as diagnosis tools. Yet, this issue is still an open field requiring further work. Automatic data interpretation and diagnosis required by process industries must primarily prove enough confidence when classifying different plant states. SVM is a kernel based method from the Statistical Learning Theory (V. N. Vapnik, 1998) that has succeeded in many pattern recognition problems. Recently, it has been applied to fault diagnosis in chemical plants and the TE benchmark proposed by Downs and Vogel (1992) was used for studying the data sets overlapping problem (L. H. Chiang et al., 2004). However, in that work only three of the 20 TE faults were considered, so that the global diagnosis problem was not analyzed. 2. On the comparison of Support Vector Machines performance According to the review by Venkatasubramanian et al. (2003), three main methodologies have been followed in fault diagnosis of chemical processes: I) Quantitative model-based methods, II) Qualitative model-based methods and III) Process history based methods. The third approach uses historic data bases for generating models regardless of any previous expertise or first-principles knowledge

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requirements. Furthermore, these models allow dealing with the intrinsic non-linearities of most chemical processes with no additional cost. Several approaches based on the so called pattern recognition techniques (e.g. neural networks or fuzzy systems) have been implemented to overcome the limitations of the two first FDS approaches. Although data-based models are also limited by difficulties on class generalization, multiple fault diagnosis and data uncertainties, they are popular techniques in industry because of their simplicity and non-linearities processing. SVM falls in this wide group, having proved more successful than previously reported pattern classifiers. The TE case is used for comparative purposes because of its complexity and because it has become a benchmark in this topic as shown by the number of works where it is reported. The most complete fault diagnosis results on the TE process were given by Raich and înar, who used angular relations between data clusters (Raich and Qinar, 1997) and techniques for PCA model overlapping quantification (Raich and înar, 1996). The success of their fault diagnosis systems were reported fault by fault (Table 1). However, results on the TE diagnosis do not globally consider the problem and partial results reported do not allow a rigorous comparison between different methodologies. For instance, M. R. Maurya et al. (2004), propose a hierarchical SDG (signed directed graph) based on qualitative knowledge requiring further quantitative information to reduce the inaccurate fault candidate sets diagnosed; Chiang et al. (2000), show that Fisher Discriminant Analysis (FDA) gives a good lower dimensional representation in terms of maximizing the separation between different data clusters, but the results on misclassification are not easily comparable. Table 1 also includes the results by Chen and Liao (2002), who addressed the detection partial problem, but without extracting the abnormal situations cause. 2.1. Accuracy and Reliability of the FDS Accuracy and reliability are useful and complementary measures of diagnosis success and misclassification that allow rigorous FDS comparison. Accuracy: It is a measure of FDS correctness when diagnosing each state. It is estimated independently for each faulty state considered. Most studies on process fault diagnosis report results just using this index. Reliability: It is a confidence measure on the FDS. It is estimated globally for all faulty states but reliability for each fault may be also given. This crucial aspect has been hardly considered in quantitative terms. Next, the required equations to calculate both measures for each fault are shown. First, the number of FDS correct responses and the misclassification errors must be defined, Send

Cr = I,R, i = Sini

JL.

(1)

^

^f^XXEj, / =!

(2)

' = 'ini

where SM and Send correspond to the initial and final samples considered for each fault, N is the total faults considered, Rt is the FDS response diagnosing fault/on sample / {Ri = 1 if there is a correct diagnosis in sample i and Ri = 0 otherwise), Cf is the total correct responses for all the studied samples when fault/is happening, Ejt is the FDS response

Fault Diagnosis

1211

when occurring faulty in sample / (E,/= 1 if there is a /diagnosis and Ey/= 0 otherwise) and so that Vfis the total number of/diagnosis when the rest of considered states have occurred. Then the accuracy and reliability corresponding to each fault can be also defined by,

where 7/^ is the total number of samples (Sgnd) when fault/is occurring and Afand 7?/are the FDS accuracy and reliability diagnosing fault/ C/+]^ represents the FDS "votes" to the/class when all the considered faults are tested. 3. Support Vector Machines Introduced in 1992 (B. Boser et. al., 1992), SVM have been gaining popularity among the learning community since 1996. Based on the Structural Risk Minimization principle by the Statistical Learning Theory, they learn a linear hyperplane that separates a set of positive examples from a set of negative examples with maximum margin (the margin is defined by the distance of the hyperplane to the nearest of the positive and negative examples). This learning bias has proved to have good properties in terms of generalization bounds for the induced classifiers. Learning the maximal margin hyperplane is a convex quadratic optimization problem with a unique solution. When examples are not linearly separable or, simply, a perfect hyperplane is not needed, it is preferable to allow some errors in the training set so as to maintain a "better" solution hyperplane. This is achieved by a variant of the optimization problem, referred to as soft margin, in which the contribution to the objective function of the margin maximization and the training errors can be balanced through the use of a parameter called C. Despite the linearity of the basic algorithm, SVM can be converted into a dual form, allowing the use of kernel functions to produce non-linear classifiers. Kernel functions make SVM work more efficiently in high dimensional spaces, where new features can be expressed as combinations of the basic ones (N. Christianini and Shawe-Taylor, 2000). In this work, the SVM^^^* software (T. Joachims, 1999) was used. We have worked only with linear kernels, performing a tuning of the C parameter. It has to be noted that no significant improvements were achieved by using non-linear kernels. 4. Results To compare the SVM performance against other diagnosis techniques, works offering most complete quantitative results on the TE case were considered, showing the lack of comparable data. A first work [1] (Chen and Liao, 2002) only presents detection results for the 1 to 15 TE faults. A second one [2] (Raich and (înar, 1996), includes accuracy and reliability as defined previously, but the values can not be compared as they are referred to previously detected faults instead of actual faults happening. Not taking into account any of the non-detected samples, overall reliability and accuracy are increased by removing possible wrong diagnosis. The last work analyzed [3] (Raich and înar, 1997), also refers results to previously detected faults and, despite the better accuracy value obtained, offers less information.

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A similar comparative analysis is used in this work, but including complete information. Results are obtained after SVM learning with treated data sets fi'om TE process simulations. Data are collected at second intervals during 24-hour runs. Aimed at enhancing performance, the exponential weighted moving average (EWMA) is used to remove useless noise and principal component analysis (PCA) is applied as a fault detection technique, taking advantage of its calculated statistics (latent scores and contributions to SPE) in fault diagnosis. Reduced training and testing sets from these treated data are randomly and independently selected to ensure rigorous results. In that way, 300 representative treated samples define each faulty data set used for training, whereas each faulty testing data set consists of 100 different observations. For comparative purposes between SVM and referred works, the total number of samples (S), the number of detected samples (D), the times the system votes to each class (V, C/+ FJr described in section 2.1) and the accuracy (A) and reliability (R) for each fault are shown in Table 1 columns. In rows, mean values for accuracy and reliability are introduced as comparative calculated measures. These measures are: the average (M, ftinction of A or R), the average weighted by the number of detected samples (Ml, that is, the rate of the total FDS correct responses and the total detected samples) and the average weighted by the number of actual samples for each case (M2, that is, the rate of the total FDS correct responses and the total samples tested). For comparative purpose also, two different computational experiments were carried out, SVM-1 and SVM-2. SVM-1 addresses the 21 faults implementing a fault detection previous SVM classification (Figure 1) in order to be able of comparing results with [2] and [3]. Besides, an extra calculation regarding all input data was done to show the actual diagnosis accuracy (M2). SVM-2 does not include information on detected faults, since this work proposes to assess performance just regarding FDS input and output (Figure 1). It takes advantage of statistics from PCA in the fault classification and only uses three columns for the complete analysis of the diagnosis performance. Moreover, the 21st TE fault is not analyzed in SVM-2, since it is a simultaneous fault that should not be treated as another single one. This work limits to the single fault problem, although the study of the multilabel problem given by simultaneous faults is underway. Results analysis shows that SVM-1 not only produces better accuracy (M), but the detection weighted average (Ml), which is much more significant, widely improves the accuracy and reliability of other works. Besides, it offers a general measure for global accuracy (M2) that is not known in [2] and [3]. SVM-2 gives consistent and promising results, and it is the only one that is able to give a global diagnosis reliability of the FDS, as it is considering all the input data in the classifier module. \

y '

Input : Data 1

DETECTION

FDS Detected

' "N

CLASSIFIER

1 Figure 1. FDS approaches regarding Input data or only detected data

! Output

1 1

Fault Diagnosis

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5. Conclusion and future work Support Vector Machines (SVM), a novel computational technique for data based modeling has been applied for fault diagnosis in chemical processes. The Tennessee Eastman benchmark has been used as the case study for obtaining comparative results. The results are very promising, but comparing these results has proved tough despite the number of works reporting data for this benchmark, which are incomplete or referred to previously detected faults. This work also proposes measuring and comparing the FDS performance by referring the output to the very input, which is the raw data supplied to the FDS. Discarding the intermediate step given by fault detection, a weighted average has been proposed and calculated regarding each single fault happening. Simultaneous faults and the multilabel related problem are being investigated. Further work also includes meta-leaming, combining different kernels related to similar variable sets and the use and comparison of other machine learning algorithms. These techniques are expected to improve the current SVM results. Acknowledgements Financial support from "Spanish Ministerio de Educacion y Ciencia" through the FPI program and the European community through contracts: No MRTN-CT-2004-512233, RFC-CR-04006, INCA-CT-2005-013359, is thankfully appreciated. References B. Boser, I. Guy on, V. Vapnik, 1992, A Training Algorithm for Optimal Margin. Classifiers. In Proceedings of the Workshop on Computational Learning Theory, COLT. J. Chen, Chien-Mao Liao, 2002, Dynamic process fault monitoring based on neural network and PCA. Journal of Process Control, 12, 277-289 L. H. Chiang, E. L. Russell, R. D. Braatz, 2000, Fault diagnosis in chemical processes using Fisher discriminant analysis, discriminant partial least squares, and principal component analysis,, Chemometrics and Intelligent Laboratory Systems, 50, 243-252 L. H. Chiang, M. E. Kotancheck, A. K. Kordon, 2004, Fault diagnosis based on Fisher discriminant analysis and support vector machines. Comput. Chem. Engng. 28, 1389-1401. N. Christianini, J. Shawe-Taylor, 2000, An Introduction to Support Vector Machines. Cambridge University Press. J. Downs, Vogel, E, 1992, A plant-wide industrial process control problem. Comput. Chem. Engng. 17 (3), 245-255 T. Joachims, 1999, Making large-Scale SVM Leaming Practical. Advances in Kemel Methods Support Vector Leaming, B. Scholkopf, C. Burges and A. Smola (ed.), MIT-Press. M. R. Maurya, R. Rengaswamy, V. Venkatasubramaian. 2004, Application of signed digraphs based analysis for fault diagnosis of chemical flowsheets. Engng. Aplications of Artificial Intelligence. Vol 17, 501-518. A. Raich and A. înar, 1996, Statistical process monitoring and disturbance diagnosis in multivariable continuous processes. Aiche J., 42, 995-1009. A. Raich, A Cinar, 1997, Diagnosis of process disturbances by statistical distance and angle measures. Computers chem. Engng, Vol. 21, No. 6, pp. 661 - 673. V. N. Vapnik, 1998, Statistical Leaming Theory, John Wiley. V. Venkatasubramanian, R. Rengaswamy, K.Y, Surya N. Kavury, 2003, A review of process fault detection and diagnosis. Comput. Chem. Engng. 27, 293-311, 313-326, 327-346.

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2 Tablel. V: Votes to the class; D: Detected samples; S: Total number of samples; A: Accuracy; R: Rehbility; M:Average; M1: Wuglhed average by D; M2: Wdgthed average by S.*:Considerated state. **: Variable

!3-

2

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Explicit Parametric Controller for a Batch Polymerization System Mariano Asteasuain'^, Konstantinos Kouramas^, Vassilis Sakizlis"^ and Efstratios N. Pistikopoulos^ ""PLAPIQUI (UNS-CONICET), C. Carrindanga km 7, (8000) Bahia Blanca, Argentina. ^CPSE, Imperial College London, London SW7 2AZ, UK ^Bechtel, Hammersmith Road, London W6 8DP, UK

Abstract This work is focused on the design and evaluation of an explicit parametric controller for a batch polymerization process. A parametric controller is a novel control method that has been recently applied to a number of continuous processes but never to a batch system. In this work we aim at exploiting the properties of explicit parametric controllers and demonstrate the potential benefits of this control method in batch polymerization processes. Keywords: styrene polymerization, batch reactor, explicit parametric control 1. Introduction Control of batch polymerization reactors is a challenging task due to the non-linear and complex dynamics (heat transfer and fluid dynamics) and the varying operating conditions of these processes (Rho et al., 1998). Usually, the control objective in this type of systems is to ensure the tight tracking of a desired reactor temperature profile despite the possible perturbations of the nominal operating conditions. The problem has already been treated with the use of classical PID control or via dynamic optimization methods (see Bonvin et al, 2005 and references within); however, the research on process systems is steadily focusing on the exploitation of advanced control methods such as Model Predictive Control (MPC) that can guarantee optimal performance, constraint satisfaction and robustness (Morari and Lee, 1999). There are few works of MPC implementation for batch (polymerization) processes. Recently, Christofides and co-workers developed nonlinear MPCs for batch crystallization systems (Shi et al, 2005, 2006), based on appropriate reduced order models to achieve relatively fast calculation of the control action. However, the repeated computation of the MPC optimization problem is still a critical issue. Explicit parametric or ParOS controllers is a novel, wellestablished control method which has the ability to perform all the MPC computations off-line, thus reducing control action computations to simple function evaluations. At the same time, all other benefits of MPC are preserved. ParOS controllers have been recently applied to a number of control problems involving continuous processes (Dua et al., 2004; Grancharova et al., 2003), but never to a batch system. The aim of this work is to design an explicit parametric controller for a batch polymerization process and demonstrate the potential benefits of this control method in such process control problem. This is of particular interest since there are currently no reported case studies or implementations of the explicit parametric controller for batch

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M Asteasuain et ah

polymerization processes. First the non-linear mathematical process model is obtained and the optimal temperature profile is designed for which final product qualities are achieved. A PI controller is also designed whose behavior is compared with the one of the explicit parametric controller. A linear input output ARX approximation of the nonlinear model is then obtained and the explicit parametric controller is designed. Finally, the performances of both the PI and parametric controller are presented with an evaluation of the behavior of the two controllers in nominal and perturbed operating conditions. 2. Process description and mathematical model The system under consideration is a styrene batch polymerization reactor. The system consists of a jacketed reactor, in which the monomer (styrene) and a solvent (toluene) are charged with a proportion 70% to 30% in volume, respectively. The mixture is preheated to an initial reactor temperature, obtained for proper initiator performance. The initiator dibenzoyl peroxide (BPO) is added next to start the reaction. The kinetic mechanism considered in the polymerization model involves the following reactions: initiation by peroxide decomposition, monomer thermal initiation, propagation and termination by combination. Constant reaction volume and negligible gel effect are assumed. The kinetic parameters for the monomer thermal initiation constant were taken from Asteasuain et al. (2004), and all other kinetic parameters, physical properties and correlation parameter values were taken from Erdogan et al. (2002). The differential algebraic model obtained from the mass and energy balances is shown in Table 1. Table 1. Differential algebraic model of the polymerization system clI/dt = -KJ

(1)

Mn = Mw„„„y,/)'„

dM/dt = -K^MA, - 2fKJ - 3K^M'

(2)

Mw =

dA,/dt = 2fKJ - KX + 2K^M'

(3)

Pd = Mw/Mn

(4)

dT^ _(-AH)K^J{,M dt ~ pCp

dX,ldt = 2fKJ + K^M\

-KX^

dX,ldt = 2fKJ + K^M\ + 2K^M\ (5)

(9) (10)

Mw^rJri

dT, F.{T^-T)

(11) UA(T,-T,) pVCp

UA{T^-T.)

(12)

(13)

djjdt = Q.5KX

(6)

x = iM,-M)/M,

(14)

n = M,x

(7)

U = U,-ax

(15)

drJdt = KXA,Z,+A,')

(8)

In Table 1, / is the initiator, Mthe monomer, Aa and y^, a = 0, 1, 2, are the ath order moments of the radical and polymer molecular weight distributions. K^, K^, K^, and K^ are the kinetic constants, a n d / i s the initiator efficiency. Mn and Mw are the polymer number and weight average molecular weights, and Pd is the polydispersity index. Mwnom is the monomer molecular weight and x is the monomer conversion. Fj and T^^ are the coolant flow rate and inlet temperature. Lô and a are constant parameters.

Explicit Parametric Controller for a Batch Polymerization System

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2.1. Optimal temperature profile It is a common practice in most polymerization problems to control a batch reactor by pre-programmed recipe implementation and temperature control based on heat balances. Therefore, this work proceeds with the design of a temperature profile that minimizes the batch process time and ensures satisfaction of a number of constraints on the reactor and jacket temperature, process yield and final product (polymer) qualities. The polymer quaHties are expressed as its desired number average molecular weight Mn and polydispersity Pd. The following open-loop optimization problem is solved mmF = tf Differential algebraic equations

7 ; ( 0 < i i o ' c , 25^c;,_i +1.7633>;,_3 -0.624;;,_3 +0.165557,_4 -0.0085766W,+0.0086875w,_i

.^0)

Aw, =w,-w,_i, 0<w, <Sf{X)

(3)

Uk-i = GcCk-i+fk-i

(4)

ek = ysp-Jk x{t) = ^{u{t),x{t-l),d)

(5) (6)

{ypred)k

=

GpXk-l

(7)

{ypred)k+p

=

GpXk+p-l k+p Y. {{ypred)i-ysp? i=k+l

(8)

ISEY

Var{X)

=

= variance{xi),i = k-{-l....{k-^p-l)

^{X) = 100* ( 1 - - ^ ) , p-l X

=

{Xk,Xk-\-l,....:Xk+p-l)

Fk

=

{fk,fk+\i'-"ifk+m-\)

(9)

(10) (11)

where, p is the prediction horizon, m is the control horizon, x denotes the predicted stem movement using the nonlinear transformation 9{^ (simple stiction model given by Equation 1) and Ut is the total number of switches in the signal between Xt and Xt^p. The stem aggressiveness denoted as (|)(X), is computed as a signal friendliness factor (see [7] for definition). The friendliness factor for the stem denotes how fast the stem changes, a value of (|) ?^ 0 indicates that the stem changes its position at every time instant and a value (j) ?^ 1 indicates that the stem changes it position only at few time instances over the time duration considered. At every time instant, a set of 'm' compensator moves are computed and similar to the MPC concept, only the first compensator move is added to the PID controller output. Since, the stiction model fAîs discontinuous and the stem friendliness factor ((|)(X)) does not have a closed form, an approximate numerical approach is used to solve the above nonlinear optimization problem. 3. Simulation results 3.1. Simulation example Consider a closed loop process with ^ , , 0.009U-i-h0.0082z-2 ^ ^ ^ 16-27.53^-1 + 11.81^-2 ^^(^^^l-1.724z-i+0.748z-i ^^^^^ ^ T^F^

^^'^

Integrating Stiction Diagnosis and Stiction

Compensation

^/WVVVV---=Tr^ Compensation started at time = 120 seconds

jlj

1237

""•

Compensation started at tinfe t = 120 S(

set-npint Runi Run 2

Umm^ 9 (sacs)'w

toJ ^"rimo (secsy^

^^^

I ^"îmefsecsy^

Figure 4. Results for optimization approach for compensation, (a) Process output and (b) ControUer output. The compensation started at time 120 seconds

Figure 5. Results for optimization approach for compensation, (a) Compensator signal, (b) Valve input obtained after addition of compensating signal to controller output and (c) Stem position.

sampled at every second. The loop oscillates due to the presence of stiction. The stiction is simulated using the simple stiction model given by Equation 1, with a stiction band d = 0.5. 3.2. Results Two simulations were performed to highlight the usefulness of the optimization approach. The optimization approach used the 'fmincon' algorithm of the MATLAB optimization toolbox used. The parameters for both the simulations are given in Table 1. Figures 4 and 5 show the comparison of the results obtained for the two simulations. It is seen that, based on the penalty imposed for each term in the objective function, the duration for the process output to reach its set-point varies. Also, the amount of variability and stem friendliness factor varies based on the weights of X.

Table 1 Parameters used in the simulation for the Optimization approach. Parameters Prediction horizon (p) Control horizon (m) ^1

yî

h

Runl 40 2 100 0 0

Run 2 40 2 0 100 100

3.3. Discussion It was seen that the optimization approach has parameters that need to be tuned to attain efficient stiction compensation. Two main drawbacks are seen with the optimization approach:

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R. Srinivasan and R.

Rengaswamy

1. As the objective function (Equation 3) is non-smooth, the optimizer was not able to attain the global minimum; instead a local solution was obtained. This is evident from Figure 4, where for both runs, the process output failed to reach the set-point. This is because the stem position did not move to the correct steady state value, instead moved the stem close to it with an offset. Also the objective function values obtained for the various compensating signals (simulated as a grid of values for the next two moves) showed that the objective function is generally non-smooth but convex. 2. When the optimization approach was tried on an experimental level system at Clarkson University, the Simulink interface could not solve the optimization formulation between each iteration, due to real-time issues. Alternate non-gradient based optimization techniques that use function evaluation such as DIRECT (Divide RECTangle method). Implicit filtering can be studied to overcome the real-time issues. 4. Summary In this paper, a model based optimization approach for stiction compensation was presented. The optimization approach was observed to provide more trade-off than the 'knocker' approach. Preliminary studies suggest that the model based compensation method can be an useful strategy for stiction compensation. Further analysis of the effect of model plant mismatch, incorrect stiction measure, real time issues on the proposed stiction compensation approach needs to be done before these methods can be implemented online. REFERENCES 1. L. D. Desborough and R. M. Miller. Increasing customer value of industrial control performance monitoring - honey well's experience. Arizona, USA, 2001. CPC-VI. 2. B. Armstrong-Helouvry, R Dupont, and C. C. De Wit. A survey of models, analysis tools and compensation methods for the control of machines with friction. Automatica., 30(7): 1083-1138, 1994. 3. A. Kayihan and F. J. Doyle. Friction compensation for a process control valve. Control Engg. Practice, 8:799-812, 2000. 4. T. Hagglund. A friction compensator for pneumatic control valves. Journal of Process Control, 12:897-904, 2002. 5. R. Srinivasan and R. Rengaswamy. Stiction compensation in process control loops: A framework for integrating stiction measure and compensation. Accepted for publication in Industrial and Engg. Chemistry Research, 2005d. 6. R. Srinivasan, R. Rengaswamy, S. Narasimhan, and R. M. Miller. Control loop performance assessment 2: Hammerstein model approach for stiction diagnosis. Industrial and Engg. Chemistry Research, 44:6719-28, 2005b. 7. S. Narasimhan, R. Srinivasan, and R. Rengaswamy. Multi-objective signal design for plant friendly identification. 13th IFAC Symposium on System Identification, 2003.


1239

Differential Recurrent Neural Network based Predictive Control Rihab Al Seyab, Yi Cao Cranfield University, Bedford, MK43 OAL, UK

Abstract An efficient algorithm to train general differential recurrent neural networks is proposed. The trained network can be directly used as the internal model of a predictive controller. The efficiency and effectiveness of the approach are demonstrated through a two-CSTR case study, where a multi-layer perceptron differential recurrent network is adopted. Keywords: Predictive control, Recurrent neural networks. Nonlinear identification. Nonlinear control

system

1. Introduction Model predictive control (MPC) strategies have been well received by industry because they are intuitive and can explicitly handle MIMO systems with input and output constraints. Until recently, industrial applications of MPC have relied on linear dynamic models even though most processes are nonlinear. MPC based on linear models is acceptable when the process operates at a single setpoint and the primary use of the controller is the rejection of small disturbances. Operating points of modem chemical processes vary over large regions and cannot be modelled adequately using linear models. These conditions are observed in many situations such as, change over in continuous processes, tracking problems in start-up and batch processes and the control of nonlinear reactors. To properly control these processes a nonlinear dynamic process model should be used. Predictive control with a nonlinear model is referred as nonlinear model predictive control (NMPC). One of main difficulties to use NMPC is to develop a reliable nonlinear dynamic model. Most existing techniques for nonlinear dynamic model identification are only for discrete-time models although most process systems are continuous in time. A discretetime model can only work for a particular sampling frequency. If the sampling frequency is to change, the model has to be re-built. Hence, it is not convenient to use discrete-time model for multi-rate control. In contrast, once a continuous-time model has been created, it can be used for any sampling frequency, even for continuous-time NMPC. Although a continuous-time model has clear advantages, it has rare been used in NMPC except those based on first principles. The main reason is due to the difficulty to solve the differential parameter optimization problem associated with the continuoustime nonlinear model identification problem. In recent work, we have developed an efficient NMPC formulation using automatic differentiation (AD)\ This formulation has been successfully extended to train

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R. Al Seyab and Y. Cao

differential recurrent neural networks (DRNN). The developed DRNN can be directly used for NMPC within the above formulation. The training and control algorithms of DRNN are suitable for most process systems. A two-CSTR case study is presented to demonstrate the usage and benefit of the algorithms proposed. With the DRNN model developed, we can freely change the sampHng frequency to improve control performance. The paper is organized as follows. In section 2, a training algorithm is proposed using AD techniques. Section 3 presents formulations for predictive control to use a DRNN as internal model with AD techniques. These algorithms are applied to the case study in section 4 and in section 5 some conclusions of the work are provided. 2. DRNN training Assume a model-unknown continuous-time nonlinear dynamic system has riu inputs and Hy outputs. Npoints of input, u{k), ^ 0 , ,..,N-\ and output, y^{k), ^ 0 , ...,N data are collected at sampling rate h. These data are used to train a recurrent neural network (RNN) so that the RNN trained can predict the dynamic behaviour of the system in a reasonable accuracy. According to the universal approximation theory of artificial neural networks, there are many types of neural networksfi*ommulti-layer perceptrons (MLP) to radial basis functions (RBF), which can be constructed as recurrent networks to approximate the nonlinear system. The training algorithm to be discussed is suitable for any kind of networks. Hence, the DRNN to be considered is represented in the following general form. dx — = /(x,w,^), y = Cx dt

(1)

where XG R"' is the hidden state of the DRNN, we R"" the input, J^G R"' the output, ^G 7?"^ the parameters of DRNN to be trained, and C = [/ 0], i.e. the outputs are equal to the first Uy states. The collected input data are directly applied to the DRNN by assuming constant input between two sampling instants. The initial state of the DRNN is x(0) = [yp(0) 0]. Then, for a given set of parameters, 0, output can be predicted fi-om (1). The training algorithm to be proposed aims to minimize the total prediction error: (2) ^

k=l

^

To solve this optimization problem efficiently, the gradient, (p^=îs

to be calculated

using the AD techniques described as follows. Let the state of (1) be a Taylor series up to d terms, i.e. ^(0 = X/=o^['i^' ' where ^[i]~'n^' techniques^

Then, the Taylor expansion o f / c a n be directly obtained using AD as

/ = X/=0'^']^'^^^^^ Taylor coefficients,

coefficients, Xy^, with j 0}>a

(1) x(to) = Xo

where / is the objective function, E und D are its expected value and variance, respectively. The vectors g and h represents the equality (model equations) and inequality constraints, x, u and ^ are the vectors of state, control and uncertain parameters. Pr {h(x,u, ^)>0} represents the reliability of complying with the inequality constraints h, while a is the user-predefmed confidence level (0 0 Even if the total current, /

is fixed by the demand, the total electric power

consumption, W in eq. (4) can be reduced by adjusting the operating temperature, T.. W = E'I

= {EO.-^R.I.)I

(4)

The values of the parameters Eo-, a- and b- in eqs. (2) and (3) change as a result of the deterioration of the ion exchange membranes and electrodes.

3. Formulation of an Optimization Problem By introducing a variable ^ = J _ , the optimisation problem can be expressed as follows. n I +

W=

0
/ ; _ W - / ^ , + Z K -Eo)x.x, -I^,^±Xj 7=1

(9)

7=1

The boundaries are hyper-planes, which are composed of hyperbolic curves and lines between the two variables X. and x •. If the starting voltages of the electrolysis, Eo. and EOj , are different from each other, the relationship between X- and x • is a hyperbolic line. All of the curves and lines go through the origin of the coordinate of X-. The hyperbolic line shows a monotonic increase in the first quadrant of the space of x.. The feasible region, which satisfies all constraints in eqs. (8) and (9), is conical and its upper vertex is the origin of the coordinate of X.. The performance index shows a monotonic decrease with respect to each value of X-. I+ dW dx,

t,{Eoj-Eo,)xj 7=1

(10) V7=l

—/ Uoadx + Tioadx + Toperationx, the latest being the intermediate operation period for the current feeding conditions and it is formed by sum of different stages duration, included Tcookx- These periods can be computed as before from interpolation in a table rj{B,P) (i=every stage) that has also been obtained off-line. In order to complete the vacuum pan model, other constraints must be added reflecting its logic

of

operation,

such

as

tioadl

>

hoadX +

TioadX +

ToperattonX +

tunloadX +

TunloadX t h a t

indicates that the next batch 2 must start after the previous batch 1 has been unloaded. Obviously, these two constraints are necessary for each batch predicted and for each vacuum pan. Also additional tables are needed, see Fig. 3 c), such as the ones relating brix and purity in the feeding tank with: the total cooked mass obtained Mcm(B,P), the brix Bnii(B,P) and purity Pmi(B,P) of mother liquor, and percentage Acin(B,P) of crystals of cooked mass in the unload stage of a tacha.

Hybrid Model Predictive Control of a Sugar End Section

1273

In relation with the subjacent time model and the scheduling policy, the classical approach considers the time axis divided in sampling periods, where each sampling time j has an associated integer variable indicating if unit i starts or not it operation in period j . The scheduler solves a MIP problem to determine the optimal starting and ending times of each batch units. In this paper we have applied an alternative approach that is coherent with the use of the temporal patterns shown in Fig. 2 c) and d). It assumes as unknowns the time of occurrence of the events, tioadx and tunhadu which are real variables, instead of using integer variables in every sampling period [4]. In this way, all the decision variables of the internal model are continuous. Notice that this approach means that the scheduling problem is not computed separately but it is integrated into the overall predictive control and the need for solving mix integer optimization problems is avoided, being substituted by an NLP one that includes among its decision variables the time instants in which every tacha must load and unload along the prediction horizon. 3.1. NMPC controller Before the non-linear model predictive control problem can be solved, it is necessary to adapt some concepts used in standard continuous MPC to the context of mix continuous-batch processes. The first one is the prediction horizon (N2) that will be translated into Np minimum number of full cycles performed for all batch units (a cycle is the evolution of each batch unit from the state measured, the instant of called to controller, to the next future same state). The concept of control horizon (Nu) is split into batch control horizon (Nbi) and continuous control horizon (Nc). The first refers to the number of cycles of each batch unit i in which the decision variables tioad and tunhad will be computed. From Nbi until the end of the prediction horizon (Np), these values will be equal to the ones of the last cycle. Notice that this implies the assumption that a stable cyclic pattern will be reached at the end of the prediction horizon, in a similar way to how the future control signal is treated in continuous MPC. Each Nbi will fix the number of unknown time instants tioad and tunhad, two per cycle and per unit. Finally the Nc horizon has the classical meaning for the classical continuous manipulated variables. The control decisions are computed solving an NLP optimization problem where the aim is to minimize a quadratic cost function J:

J=f"{â,(y,(t)-yfr

+^^jAuj(tf}dt

(1)

with the usual constraints yi"'^^ < yi(t) < yi"'"^ and Uj"'^ < Uj(t) < Uj"'^^ where the yi's extend to brixes and purities in the feeding tanks as well as the levels in these tanks and the two malaxadors, and tstop is the total time of prediction. Respect to the future manipulated variables, Uj are times of load and unload vacuum pans plus total flow and proportion of higher and lower purity syrup in the centrifugal separators of section A and B. The optimization is subjected to the internal model of the process and additional constraints imposed by the range and operation of the vacuum pans and other units. 4. Simulation results and conclusions The control strategy described in the previous sections was tested in simulation using the state-of-the-art EcosimPro environment. The process was represented by a detailed simulation built using validated models of the Sugar Processes Library [5] including sequential and local controls of all units. The MPC controller was programmed in C++ and contained the SQP algorithm, which, in turn, was able to call another EcosimPro simulation with the internal model for computing the cost function J (integration of

1274

D. Sarabia et al

dynamical internal model) each time it was needed. The sampling period was chosen as 15 min. We present an experiment of 69.4 hours (250000 sec), with an inflow of feed syrup of 6 Kg/sec. with 94.4 of purity and 72 of brix. All control horizons (Ncbi) were fixed in 2 and prediction horizon (Np) was fixed in 3 (25 hours of prediction). Fig. 4 a) and b) shows levels of the melter and the malaxador A, and its minimum and maximum values allowed, Fig. 4 c) shows purity and brix in the melter. Fig. 4 d), e) and f) shows the same variables but for the section B. The sequence of stages of vacuum pans Al, A2, A3 and B can be seen in Fig. 4 g). Time of stage 1 is the manipulated variable for load syrup and time of stage 9 is the manipulated variable to unload cooked mass, cooking stage correspond with stage 8 and load and unload stages with 3 and 11. g) Sequence of stages of tachas Al, A2, A3 and B a) Level of melter (in %)

120000

d) Level of tank B (in %)

160000

200000

240000 0

b) Level of malaxador A (in %)

Wfii^^ 0

40000

80000

80000

120000

160000

200000

240000 0

40000

80000

120000

160000

^Purity [-•'"^

80000

200000

240000 lo

200000

240000

5

3

Brix

Purity

40000

160000

f) Purity and Brix in Tank B

/ 0

120000

NNNNNNN\ N^

c) Purity and Brix in melter

]

40000

e) Level of malaxador B (in %)

120000

160000

200000

240000 0

Brix

/

]_ 40000

80000

120000

160000

200000

240000

"

40000

80000

120000

160000

200000

240000

Fig. 4: a), b) and c) Controlled variables for section A. d), e) and f) The same variables but for section B. g) Sequence of stages of tachas Al, A2, A3 and B. In this paper a plant-wide control strategy for the crystallization section of a beet sugar factory has been presented. It is based in a hierarchical view of the problem and, in the use of MPC with a simplified model that combines material balances of the continuous units and an abstract model of the batch ones. This is described in terms of tables computed off-line and prescribed patterns of the batch units variables and time of occurrence of the events, instead of using integer variables, which allows to use NLP algorithms instead of MIP ones. The strategy has proved to perform well in a simulated environment and opens the door to practical implementations at industrial scale. References A. Bemporad, M. Morari, 1999, Control of systems integrating logic, dynamics, and constraints, Automatica, 35:,407-427 K. T. Erickson, J. L. Hedrick, 1999, Plant-Wide Process Control. Wiley Publishers C. A. Floudas, 1995, Non-linear and Mix-Integer Optimization. Oxford Univ. Press C. de Prada, A. Merino, S. Pelayo, F. Acebes, R. Alves, 2003, A simulator of sugar factories for operators training, AFoT 2003, ISBN: 84-95999-46-3, 97-100 C. de Prada, S. Cristea, D. Sarabia, W. Colmenares, 2004, Hybrid control of a mixed continuousbatch process, ESCAPE14, ISBN: 0-444-51694-8, 739-744

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Pubhshed by Elsevier B.V.

1275

Systematic Methodology for Reproducible Optimizing Batch Operation S. Bay J0rgensen^* and D. Bonne^ ^CAPEC, Department of Chemical Engineering, Technical University of Denmark, S0ltofts Plads, 2800 Kgs. Lyngby, Denmark This contribution presents a systematic methodology for rapid acquirement of discretetime state space model representations of batch processes based on their historical operation data. These state space models are parsimoniously parameterized as a set of local, interdependent models. The present contribution furthermore presents how the asymptotic convergence of Iterative Learning Control is combined with the closed-loop performance of Model Predictive Control to form a robust and asymptotically stable optimal controller for ensuring reliable and reproducible operation of batch processes. This controller may also be used for Optimizing control. The modeling and control performance is demonstrated on a fed-batch protein cultivation example. The presented methodologies lend themselves directly for application as Process Analytical Technologies (PAT). 1. I n t r o d u c t i o n The batch processing types covered in this paper includes Batch, Fed-batch and periodic operation which all have the common traits of a repeated operation which start from nearly the same initial conditions. Thus the time within the bacth and the batch number are the two characteristic independent variables. Batch processing is subject to variations in raw material properties, in start-up initialization and other disturbances during execution. These different disturbances introduce variations in the final product quality. Compensating for these disturbances have been difficult in the past due to the nonlinear and time-varying behavior of batch processing and to the fact that reliable on- or in-line sensors for monitoring final product quality rarely are available. Consequently development of a systematic methodology which can ensure reliable reproducible operation may provide significant bennefits for batch processing. Each batch operation may be defined as a series of operational tasks, i.e. mixing, reaction and separation. Within each task a set of subtasks, e.g. heating/cooling, (dis)charging is handled. There may be more than one feasible set of operational tasks that can produce the specified product (-s). Consequently an optimal sequence of tasks and subtasks with respect to. a defined objective needs to be identified. This set of operational tasks is labeled the optimal batch operations model Thus the Batch Operations Model combines the batch processing tasks normally specified in a generic recipe with the batch equipment under availability and other resource constraints. *[email protected]

1276

S.B. Jorgensen and D. Bonne

Several research groups have used general empirical model knowledge to develop methods for control of batch processes including an experimental adaptation (optimization) of the Batch Operations Model, labelled "the solution model" through tracking the Necessary Conditions of Optimality (NCO) Srinivasan and Bonvin (2004). Another approach Akesson et al. (2001) exploits the knowledge that optimal batch operation consists of a sequence of operations to determine the presently (most) constrained variable. A data driven approach develops prediction of end of run properties (Flores-Cerrillo and MacGregor, 2003), while asimmilar inspiration lead to reconstruction of approximate time series models from available process data (Gregersen and J0rgensen, 2001; Bonne and J0rgensen, 2003). This paper presents methodologies based upon a model for batch or periodic operation which can ensure reliable reproducible operation and which may enable optimizing operation. The contribution comprises data driven time series modeling of batch processes and a learning model predictive control methodology. The modelling methodology produces both a Linear Time-Invariant state space model representation for inter-batch prediction and a Linear Time-Varying state space model representation for intra-batch prediction. The modelling approximates the non-stationary and nonlinear behavior of batch processes with a set of local but interdependent linear regression models parameterized as AutoRegressive Moving Average models with eXogenous inputs (ARMAX). Tikhonov Regularization is apphed to estimate the parameters of this model set. Learning Model Predictive Control is presented for control of repeated operation of stochastic Linear Time-Varying systems with finite time horizons together with tuning requirements for ensuring guaranteed convergence and hence closed-loop stability. The methodologies have been implemented as a Matlab toolbox Grid of Linear Models (GoLM). 2. M e t h o d s Batch processes are modeled with the toolbox GoLM as a sets of N LTI models. Such a set of LTI models could also be referred to as one LTV batch model. These LTI models can be parameterized in a number of ways, but in the present contribution an ARM AX parameterization was chosen. This choice of parameterization offers a simple multivariable system description with a moderate number of model parameters. The objective of the model set is to quantify the causal correlations between the process outputs y^î G M^^, inputs Ukî-i G W^"", and distrubances Vi G W^y, for i = 1 , . . . , t, at times t = 1 , . . . , A/" in batch k. To simphfy notation, define the input Uk, output y^, shifted output y°, and disturbance Vk profiles in batch k as Uk= [ < 0 yk=

w^S'^w' where it; is a discriminant direction. If 5^ is nonsingular, the optimal discriminant direction w maximizing J{w) is identical to the eigenvector which corresponds to the maximal eigenvalue of the conventional eigenvalue problem {S^'^Slw = Xw. 2.2. XmR Charts The XmR charts are used when samples are individual measurements, e.g., the sample size=l. The upper control limit(UCL) and lower control limit(LCL) for monitoring are calculated from the moving range of two successive observations. Given n observations xi, X2,..., x^,..., a::^, then the moving range is defined as MRi = \xi-x^_i\.

(4)

Let X be the average of all the observations and MR be the average of all the moving ranges, then UCL and LCL are defined as MR MR LCL L128' ^ "L128' If an observation x satisfies LCL < x < UCL, the observation is judged as normal. UCL = x + 3

(5)

Discriminant Analysis and Control Chart for Fault Detection and Identification

1283

^ Fault 1 -* ^ A^

pFault 2

^ ^2

t

Fault N N ^

Fault 2 •< A

V

A • Fault N

>

Normal observations

^ ^ Fault 1-1

A

Fault i+1

A

Fault i-1
y,{k

+ \)

y,{k)>y,{k

+ \)

where Xset is the set point value of x and M is large positive number. The first constraint implies that iiyx= 1, x(k) < Xset and xfy^ = 0, x(k) = xêu an alternative formulation which does not involve square terms and relies on introducing slack variables can also be employed (Floudas, 1995). Since we are interested in minimizing the time period the elements oiy^ which take a value of 1 will indicate the time intervals during which x has not reached the set point value. To ensure that the time periods are arranged in a sequential order third and fourth constraints are introduced. The second constraint is similar to the first one except that it indicates that at the end of the time period for control variables the control variables do not change; an alternative formulation involving set point value of control variables, similar to first constraint for state variables, can also be used. The stability of the isothermal CSTR example presented earlier was analyzed by using the proposed formulation and the same results as reported in the open literature were obtained - for the case when the system was unstable integer infeasible solution were obtained and feasible solutions were obtained for stable systems; all the cases were solved by using GAMS/DICOPT (Brooke et al., 1998). The key advantage of the proposed formulation is that one does not have to try different horizon lengths and instead solving a mixed-integer program provides the answer regarding the stability of the nonlinear MPC. The proposed approach provides a tool for stability analysis but does not give optimal horizon lengths and this is being currently investigated.

3. Concluding Remarks A mixed-integer programming approach, where 0-1 binary variables represent discrete time intervals, for stability analysis of nonlinear model predictive controllers was presented. The issues pertaining to global optimization and guaranteeing stability for all the values of state variables given within certain lower and upper bounds will be the subject of future work. The proposed approach does not require testing different values of horizon lengths, is quite general and can be extended to other systems.

References A. Brooke, D. Kendrick, A. Meeraus and R. Raman, 1998, GAMS: a user's guide, GAMS development corporation, Washington. A. Bemporad, M. Morari, V. Dua and E.N. Pistikopoulos, 2002, The expHcit Unear quadratic regulator for constrained systems, Automatica, 38, 3. C.A. Floudas, 1995, Nonlinear and mixed-integer optimization, Oxford University Press, New York. C.E. Garcia and M. Morari, 1982, Internal model control . L a unifying review and some new results. Industrial & Engineering Chemistry Process Design and Development, 21, 308.

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S.S. Keerthi and E.G. Gilbert, 1988, Optimal infinite horizon feedback laws for a general class of constrained discrete-time systems: stability and moving horizon approximations. Journal of Optimization Theory and Applications, 57, 265. D.Q. Mayne and H. Michalska, 1990, Receding horizon control of nonlinear systems. IEEE Transaction on Automatic Control, 35, 814. E.S. Meadows and J.B. Rawlings, 1997, Model predictive control, in: Nonlinear process control, M.A. Henson and D.E. Seborg (Editors), Prentice Hall, New Jersey. E.N. Pistikopoulos, V. Dua, N.A. Bozinis, A. Bemporad and M. Morari, 2002, On-line optimization via off-line parametric optimization tools, Computers and Chemical Engineering, 26, 175. J.B. Rawlings and K. Muske, 1993, The stability of constrained receding horizon control, IEEE Transactions on Automatic Control, 38, 1512. P.B. Sistu and W. Bequette, 1995, Model predictive control of processes with input multiplicities. Chemical Engineering Science, 50, 921. J. Viswanathan and I.E. Grossmann, 1990, A combined penalty ftinction and outer-approximation method for MINLP optimization. Computers and Chemical Engineering, 14, 769.


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AN OPTIMIZATION FRAMEWORK TO COMPUTER-AIDED DESIGN OF RELIABLE MEASUREMENT SYSTEMS Raffaele Angelini, Carlos A. Mendez, Estanislao Musulin, Luis Puigjaner* Chemical Engineering Department-CEPIMA, Universitat Politecnica de Catalunya ETSEIB, Av.Diagonal 647, E-08028, Barcelona, Spain Abstract The optimal design and upgrade of sensor networks have been receiving an increase in attention for the last few years. This work presents a new methodology based on a MINLP formulation to the optimal design of reliable measurement systems with minimum cost. The applicability of the MINLP model is illustrated through the resolution of a benchmark case study taken from literature and its computational performance is compared with a genetic algorithm-based approach. Keywords: Sensor networks, design and retrofit, MINLP model. 1. Introduction A wide-ranging variety of approaches relying on exhaustive enumeration, algorithmic procedures and meta-heuristic techniques have been developed to address the complexity of the sensor network placement problem, usually assuming a steady-state system at the nominal operating conditions. Within this context, the sensor placement can be regarded as a highly combinatorial optimization problem where the main goal is to find the optimal balance between the performance indicators and the cost of the data acquisition system. The performance of the whole measurement system as well as of individual process variables is typically determined through multiple criterions such as precision, flexibility, reliability, robustness and so on, to which minimum requirements are usually to be satisfied. The concept of observability is also a necessary condition of any sensor network since that the state of the entire process must be known, which means that a numerical value needs to be determined for every process variable by using direct measurement or deduction from other process variables. On the other hand, the total cost of the measurement system can comprise the investment cost of new sensors, the installation and maintenance charges and, in the case of retrofitting, the sensor reallocation cost. The design of a sensor network that allows for the observation of all process variables was first addressed by Vaclaveck and Loucka (1976). Later, this problem was solved by Madron and Veverka (1992) regarding the minimum total cost. However, sensor failure can lead to a reduction of measurements, thereby seriously affecting the whole process performance. Thus, it is necessary to assure process observability even after sensor failures. The observation of a process variable can be expressed mathematically by making non-null the probability of estimating this variable at a given time t (reliability). The evaluation of this probability is closely related to the different ways of estimating a process variable given a sensor failure probability and a

Author to whom correspondence should be addressed: [email protected]

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specific sensor network. On the basis of these concepts, a method for optimal sensor location in a pure flow process was developed using graph-theory by Ali and Narasimhan (1993). This preliminary approach did not directly consider either the network cost or the reliability of the individual process variables. However, these key topics were later addressed by Bagajewicz and Sanchez (2000). Additionally, these authors transformed the problem presented by Ali and Narasimhan into a mathematical programming problem. Later on, Benqlilou et al. (2004) proposed an approach for evaluating the reliability of process variable estimation taking into account all the redundancies offered by the system in terms of functional as well as hardware. In this evaluation, both quantitative process knowledge and fault tree analysis are considered and combined, which leads to a more suitable and practical evaluation of reliability. The reliability of estimating each one of the key process variables is then used to determine the sensor network reliability, which, in turn, is used as a set of sensor placement constraints in the design and retrofitting procedure. Thus, a sensor placement method was initially proposed to consider the number (hardware redundancy or multiplicity) of sensors of a given type (reliability) that are to be assigned to a given process variable while satisfying the reliability requirements at the minimum total cost. This proposal was applied for network design as well as for retrofitting. Although the formulated sensor placement optimization problem was successfully solved using genetic algorithms, it showed some typical limitations of stochastic optimization methods, such as the impossibility to assure a global optimum and the high difficulty in dealing with hard-constrained problems. The optimization problem as posed involves a very large number of discrete decisions to determine the number and type of sensors to measure each process variable. The main challenge is that the problem size scales exponentially in the number of process variables and sensors. An additional complexity is the lack of direct algebraic equations to compute the reliability of process variable estimation which was typically calculated by algorithmic procedures. This work is focused on the optimal design and upgrade of reliable sensor networks. A novel MESFLP-based approach that takes into account sensor redundancy as well as multiple sensor types is presented. Different objectives functions besides cost can be easily implemented. The successful applicability of the proposed method is illustrated through a challenging case study taken from literature. 2. Reliability of process variable estimation The probability, rj{t), of estimating a process variable j at time t must simultaneously consider sensor, hardware and functional reliabilities (see Figure 1). For the sake of simplicity, the sensor reliability is assumed to be constant over time and the time reference is omitted from here on. As an example, consider a simple process unit with one feed stream 1 and two product streams 2 and 3, as shown in Figure 1. For this process unit, the three flow variables (Fi, F2 and F3) are related to each other through the pure mass balance expressed by the equality Fi = F2 + F^. Let us assume that all sensor failures occur randomly and independently and that the mass flow of all streams can be measured using similar flow-meters k = SI, SI, S3, SA, S5 and S6, all of them with a sensor reliability equal to 0.80. Therefore, using the property that if PI and Fl are the probabilities of two independent events, the logical condition P\ ox Fl is equal to P\+P2-P\.F2. The reliability of estimating variable Fi is summarized in Table 1. For these three illustrating cases (sensor reliability, hardware reliability and functional reliability), vpi > 0 and variable Fi is said to be observable. Moreover, in the case of hardware reliability, the reliability of estimating variable Fi is greater than the reliability

Optimization Framework to Computer-Aided Design of Reliable Measurement Systems 1295 of each one of the sensors measuring it, showing the existence of more than one way to estimate Fi. Thus, r^j is directly related to the degree of redundancy and can still be estimated even if one sensor fails. However, in cases of sensor reliability and functional reliability, no sensor failure is permitted. These results illustrate the direct effect of the sensor placement on the reliability of measuring or estimating the process variables. Sensor reliabilily

Hardware relrability

1

^

Funcfonai reliability

^ %

Figure 1. Reliability of process variable estimation Table 1. Calculation of reliabilities depending on sensor placement Reliability equation r'si ^S3 o r r^S4 = ^S3^ /S5^rid/s6=

^S4 - f^S3 • ^S4

f^S5-r^S6

Reliability rFj{t) Reliability type 0.80

Sensor

0.96

Hardware

0.64

Functional

The lowest reliability associated to the estimation of a process variable is used to determine the entire sensor network reliability, which in turn, is used as a criterion for the design and retrofit of the instrumentation system. Thus, a reliable sensor network design and retrofit problem can be formulated and solved through a rigorous mathematical programming model. Data, variables and constraints involved in the mathematical representation of this problem are introduced in next section. 3. The proposed MINLP-based approach 5^/5: J (process variables); Jk (process variables that can be measured through sensor k); Je (process variables involved in the estimation e); K (sensor types); Kj (sensor types that can be used to measure process variable y); N (sensor multiplicities); Njk (allowable sensor multiplicities for measuring process variable j through sensor k); E (alternative ways of process variable estimation); Ej (alternative ways for estimating process variable y) Parameters: c/""^ (acquisition cost of sensor k)\ Ck"^ (installation cost of sensor K)\ c^^'"^^ (deinstallation cost of sensor A:); n^"" (minimum number of sensors for measuring variable y); nf""^ (maximum number of sensors for measuring variable 7); njî""^ (number of sensors k already installed in variable 7); r^ (reliability of sensor K)\ rj'"'" (minimum reliability requirement for variable7); c"""^ (maximum total cost of the sensor network) Continuous variables: F^'^jXri (hardware failure probability for measuring process variable 7 through n sensors keKj); F^^'^^j (hardware failure probability for measuring variable 7); F^^'^'j (functional failure probability for estimating variable 7); C"'j^k (sensor installation cost); C^^'"'j^k (sensor de-installation cost); Cost (total sensor network cost); i?""'" (minimum reliability of the sensor network)

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Binary variable: Yj^n (binary variable denoting that n sensors A:GX, are used to measure process variable y) 3.1. Model constraints • Sensors already installed in the process (Only needed in case of retrofitting).

Z«MÊ j

Z«^y,M

Vfc

(1)

jeJ^ riENj^^

• Minimum and maximum allowable sensor redundancy.

«f"c^^cq

\/k

(7)

• Total sensor installation cost.

C j,kms

v^ ins ~ ^k

ll"YjXr,-4l

yj,keKj

(8)

V"^^;.*

Total sensor de-installation cost.

C j,kdeins

-^

deins ^^k

\/J,keKj neN

j,k

(9)

Optimization Framework to Computer-Aided Design of Reliable Measurement Systems 1297 • Sensor network cost including acquisition, installation and de-installation costs. Cost = ^

Cf^ + 2 ^ ^ +

^jf'

(10)

• Reliability of the whole sensor network as a problem variable to be maximized. X-pfi^'^ ^Jl^riin

Vy

(11)

4. The ammonia plant case study The dynamic case of the ammonia plant initially introduced in Benqlilou et al. (2004) is addressed in this section to illustrate the performance of the proposed MINLP model. This problem comprises eight flows and five level process variables which can be measured through a catalogue of 5 different flow-meters and 3 level meters, respectively. A schematic representation of this plant is depicted in Figure 2. The available flow-meters have reliabilities of 0.70, 0.75, 0.80, 0.85 and 0.90 and their associated costs are 1500, 1700, 2000, 2300 and 2800 euros, respectively. For the level meters, reliabilities are 0.60, 0.70 and 0.80 having a cost of 2000, 3500 and 5000 euros. A maximum hardware redundancy of 2 sensors was considered. A large spectrum of reliability requirements ranging from 0.6 to 0.99 was evaluated in order to test the performance of the proposed deterministic model under different scenarios. The design and retrofit problems of the sensor network of the ammonia plant were solved through the MINLP formulation using the optimization codes BARON (2004) and DICOPT (2004). For each scenario, maximum computational times of 1 minute, 10 minutes and 1 hour were enforced to BARON. In these cases, the reliability was gradually increased from 0.6 to 0.99 by 0.01, 0.03 and 0.13 steps, respectively. The initial point for each minimum reliability requirement was taken from the best solution of the previous step. No time restriction was needed to DICOP because of the low CPU time required to prove local optimality. Figure 3 shows the sensor network cost as a ftinction of the minimum reliability requirements. Solutions obtained throilglTttie different solvers with different time restrictions are compared in this Figure. For the sake of comparison, an enhanced version of the proposed GA-based approach introduced in Benqlilou et al. (2004) was also utilized. Figure 4 presents the results obtained by means of the determinist and the stochastic approaches. In order to perform a fair comparison, a time limit of 10 minutes was enforced both for BARON and GA. The results put clearly on evidence the better performance of the MILP approach, which was able satisfy the minimum reliability requirements with lower cost in most of the cases.

Figure 2. Schematic representation of the ammonia plant.

R. Angelini et al.

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Figure 3. Comparison of solutions obtained with different time restrictions, steps and solvers GA-MINLP COMPARISON AMMONIA PLANT (Design and Retrofit) _ —.*•— 10 min - 0.03 step - BARON • —•

10 min - 0.03 step - 5000 ind - GA O.03 Step - DICOFT

design

Reliability

Figure 4. Comparison of solutions obtained through MINLP and GA-based approaches 5. Conclusions An efficient MINLP-based approach for the optimal design and retrofit of reliable sensor networks with minimum cost has been presented. The formulation takes into account most of the problem features such as sensor redundancy, different sensor types and sensor network cost. The reliability of estimating individual process variables is explicitly computed through a set of constraints that considers sensor, hardware and functional reliabilities. The performance of the deterministic model was successfully compared with a GA-based approach. Results showed a more stable and robust behavior in the MIL? method which was able to find better solutions than the stochastic approach in most of the cases. Global optimality could also be proved through a global optimizer such as BARON with reasonable computational effort.

Acknowledgments Financial support received from the European Community (MRTN-CT-2004-512233; RFC-CR04006; INCO-CT-2005-013359) isfrillyappreciated. Literature AH, Y. & Narasimahan, S. (1993). AIChEJ. 39, 820 - 826. Bagajewicz, M. J.& Sanchez, M. C. (2000). Comp. & Chem. Eng., 23, 1757 - 1762. Baron 7.2. (2004). Solver Manual. Benquilou, C; Graells, M.; Musulin, E. & Puigjaner, L. (2004). Industrial & Engineering Chemistry Research, 43, 8026 - 8036. Dicopt2x-C. (2004). Solver Manual. Madron, F. & Veverka, V. (1992). AIChEJ. 38, 227 - 236. Vaclaveck, V. & Loucka, M. (1976). Chemical Engineering Science. 1199 - 1205.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantehdes (Editors) © 2006 Published by Elsevier B.V.

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An approach to linear control of nonlinear processes T. Schweickhardt^, F. Allgower^ Înstitute for Systems Theory and Automatic Control, University of Stuttgart, 70569 Stuttgart, Germany Nonlinear controller design is a demanding task. Therefore, linear methods are favored over nonlinear ones in practice although it is not always clear a priori whether the former are appropriate in face of a nonlinear process. Nonlinearity measures are a means to quantify the nonlinearity inherent to a process. However, as yet there were no rule on how to design a controller based on the results of nonlinearity assessment. In this paper, an approach is presented that integrates nonlinearity assessment and the design of linear controllers for nonlinear systems by using methods from linear robust control. 1. I N T R O D U C T I O N Linear techniques for systems analysis and controller design are well developed. For many control-related engineering problems, methods are available that are theoretically sound as well as practically implement able. Also for mildly nonlinear systems, one can attempt to use a linear model and linear controller design methods, hoping that the nonlinear effects are too small to destabilize the closed-loop system or to deteriorate closed-loop performance. The most common approach is to use the local linearization as a linear model, to design a linear controller on the basis of this model, and to analyse (e.g. by simulation) the stability and performance of the closed loop with the nonhnear plant. However, no stability and performance guarantees are given this way and the nonlinearity of the plant is not taken into account. In order to decide whether linear controller design is adequate, nonlinearity measures can give important insight [1-3]. Some results exist on the suitabihty of linear controllers for nonlinear systems [4-7], but no practical methods are available how to use the results of nonlinearity assessment in the design of linear controllers for nonlinear processes. In this paper, an approach is presented that integrates nonhnearity assessment and the design of linear controllers for nonhnear systems in order to (i) decide for a given control problem whether linear controller design is adequate, (ii) derive a suitable linear model (not necessarily the local linearization) and (iii) describe a linear controller design procedure that guarantees stability. This paper is structured as follows. We first give a brief introduction to nonlinearity assessment in Section 2. In Section 3 an approach is presented that integrates nonlinearity assessment and the design of linear controllers for nonlinear systems based on linear robust control techniques and based on the nonlinearity measures introduced earlier. An example is given in Section 4. The paper concludes with Section 5.

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2. LINEAR MODELS OF NONLINEAR SYSTEMS The fundamental idea of the error-gain nonlinearity measure as defined in [1,2,8] is depicted in Fig. 1.

A= u

N-G

G L

+r

1 2/ = i V p

+ 1

.JN

Figure 1. A general nonhnear system N can be decomposed into a linear model G and a nonlinear uncertainty A. The best linear model is defined as the linear model for which the associated uncertainty has the least system gain.

A general nonhnear (i.e. not necessarily linear) stable dynamical system N, described by the transfer operator N : u \-^ y = Nu is modeled by a linear model G described by the linear transfer operator G \ uv-^ y = Gu. The error system A = N — G represents the plant/model mismatch and its gain is a measure of the model quality of G. We therefore define the error-gain nonlinearity measure of N onU as u A . .

7 ^ = mf sup

\\Nu-Gu\\

-—n—n—

(1) .1/2

where ||x|| = (j^ \^{t)\ dtj is the L2-signal norm, the symbol |-| denotes the Euclidean vector norm and the plant N is required to be (finite-gain) L2-stable [9]. This nonlinearity measure gives the gain of the error system A, when the worst case input signal u Û is considered. The best linear approximation G is chosen among the set of all causal stable linear transfer operators Q such that the resulting worst case gain is minimized. The set U usually describes the region of operation in which the nonlinearity of the system N is to be assessed. In this case U contains only signals not exceeding a certain maximal amplitude. Note that the nonlinearity measure can not decrease when a larger operating regime is considered. Furthermore, the error gain nonhnearity measure is bounded by 0 < TJV < ||A^||f = sup^^^^^ô} | l ^ where the right-hand side is the system gain of N on hi. A value of 7^ close to zero indicates a negligible nonlinearity, whereas a large value suggests that also the best linear model results in a significant model/plant mismatch. In order to quickly get a lower bound, one can for instance consider the steady-state behaviour of the process [8]. Different computational schemes are available to get approximations of the dynamic nonlinearity measure [2,3,10]. We will not treat the computational details here, but we mention that the computation of exact values of the nonlinearity measures can be quite demanding.

An Approach to Linear Control of Nonlinear Processes

1301

3. A N A P P R O A C H T O L I N E A R C O N T R O L O F N O N L I N E A R S Y S T E M S Once we have decided that Hnear controller design is adequate for the nonlinear control problem at hand, we turn the attention to the design of a suitable controller. The usual approach to the design of linear controllers for nonlinear systems is to use the linearization around the operating point as a linear model, design a hnear controller and analyze (e.g. by simulation) the stability and performance of the closed loop with the nonlinear plant. By this method, no stability and performance guarantees can be made and the degree of nonlinearity of the plant is not taken into account in the controller design step. In this Section, a novel approach is presented that integrates nonlinearity analysis and linear controller design for nonlinear systems in order to 1. decide for a given control problem whether linear controller design is adequate, 2. derive a suitable linear model (not necessarily equivalent to the local hnearization) 3. describe a linear controller design procedure that guarantees stability of the closed loop containing the nonlinear process. The suggested method in this Section will deal with stable nonlinear systems exhibiting a finite gain onU: \\N\\^ < oc. The idea is to use the small gain theorem in order to maintain stability despite a nonlinear uncertainty (for the small gain theorem, see e.g. [9]). With the help of the small gain theorem, a controller C that satisfies \\N\\^ • ||C||- < 1, where N is the plant, will achieve closed loop stability, see Fig. 2 (a). While stability can be achieved this way, it remains unclear how performance requirements can be met and how the controller should look like apart from the requirement ||C||. < ||A^||~^ In particular, this approach excludes controllers with integral action as the controller has to be stable itself. To circumvent these problems, a different approach is taken. We therefore spht the nonlinear system N into a (stable) linear part G and a (stable) nonlinear part A = N — G and use linear techniques to design a (not necessarily stable) linear controller C for the linear part G. With the help of the loop transformation theorem [9, Ch. 6], we can reorder the A^ — C—structure of the closed loop into the M — A—structure as depicted in Fig. 2 (b). Now, let the linear closed loop transfer operator be M = CS = C{I + GC)~^. The small gain theorem then states that the closed loop system is stable if the gains of M and A satisfy ||A||. • ||M||. < 1. The linear controller can then be designed such that the performance requirements are met at least for the linear model G and integral action is possible. In order to get the best results, the linear model must be chosen such that the gain of the model uncertainty|| A||. takes the least possible value. This is achieved by choosing G such that ||A||. = ||A^ — G||- is minimized. But this procedure exactly corresponds to the definition of the error gain nonlinearity measure 7 ^ - MG \\N -G\\^, where a model G* with ||iV - G*||f = 7 ^ is a best linear model. The nonlinearity assessment then consists of fi) computing 'j^ and G* and (ii) check (e.g. with linear i/oo-techniques) whether ||M||. < 1/7^ is achievable. If so, G* gives a suitable linear model and hnear controller design is applicable (as long as the constraint -/^ • ||M||f < 1 is respected). Up to now, we have neglected the fact that the nonlinearity measure is computed for a restricted region of operation (given by a range of admissible amplitudes), whereas the

1302

T. Schweickhardt and F. Allgower N

u,

A = A^ - Gh

+ êi

?

yi

^î^nfi-

1—•

2/2

62

CH

U2

(a) Linear closed loop with nonlinear uncertainty

^l

(

2/1 - G e i

A = A^-G

J

G

C

'_ -H.y^

\t" ^2 — Gwi rO4T

(b) Reformulation for use with small gain theorem

Figure 2. Formulation of the nonlinear control problem as linear control problem with norm bounded nonhnear uncertainty.

framework of the small gain theorem allows for signals that are unbounded (in amplitude). In virtually all applications, control engineers have to deal with processes whose manipulated variables can take values in a certain range only. But for such processes, an anti-windup control strategy has to be applied anyway, and we can combine it with nonlinearity assessment in the following way. T h e o r e m [11] Consider the system M = CS = C{I + GC)~^ where the linear model G has a saturating input. Let the saturation bounds correspond to the bounds for which the error-gain nonlinearity measure 7 ^ was derived. Under these conditions, stability of the closed control loop in Fig. 2 a) is achieved if ||M||f < 1/75^. An upper bound on the gain of the linear saturated feedback subsystem M can for instance be derived using the result from [12], 4.

EXAMPLE

We will illustrate the presented approach with a small example. We therefore study the ultra-filtration membrane reactor given in [13]. A chemical reaction is conducted in loop with a filtration unit in order to keep the biocatalyst in the reactor. We make the simphfying assumptions that no enzyme deactivation occurs and we neglect any transport delays between reactor and filtration unit. We thus obtain the first-order model y

VR

+ KM

VM {CSLO -

+ {csLQ -y)

y)

+ KMpy

where the manipulated variable u is the inlet flow rate and the controlled variable y the product concentration. We consider here a steady state input of uss = 1.5 cm^ min~^ and an input range oi u = uss i 1-0 cm*^ min~^. The corresponding steady state output is y — 22.21 mmol dm~^. The nonlinearity measure is computed to be "^ — 3.82. We

An Approach to Linear Control of Nonlinear

1303

Processes

80

Figure 3. Responses of the closed loops to steps in the reference signal of different height. The solid line corresponds to the controller based on the best linear model, the dotted line to the controller based on the local linearization.

obtain for the best linear model G*{s) — ^f^q 04'5^ as opposed to the linearization at the operating point which yields Giin{s) = -^^^^ Two important differences become apparent when comparing the two models. Firstly, the linearization has no transmission zeros, while the best model has a non-minimumphase zero. And secondly, the steady-state gains are different. The steady-state gain of the best model leads to a better steady-state approximation. But the real plant does not show inverse response behaviour, and we would expect that the best model G* should be minimum-phase. But the best model is not unique, and nothing in our setup requires G* to be preferably minimum-phase. Obviously the algorithm leaves space for improvement so that the most suited model out of the pool of best models is chosen. For our simple example, however, this fact does not play an important role as will become clear. For the model G* we design a P/—controller with anti-windup and an additional first-order filter. The controller parameters are chosen such that the gain of the resulting M—system (with saturating plant) has an upper bound of 0.184. The required condition ||M||^ - 7 ^ = 0.18-3.82 = 0.70 < 1 is satisfied and closed-loop stability is guaranteed. For purposes of comparison, a similar P/—controller with anti-windup was designed for the plant model Gun so that the same bandwidth was achieved. Step responses of the closed loop with the "real" plant are shown in Fig. 3, the solid line corresponding to the first controller (based on the best model) and the dashed line corresponding to the second controller. For this simple example, the performance of the two controllers is similar, but note that considering the design process of the second controller, closed-loop stability was not guaranteed. Due to integral action, both controllers achieve vanishing steady state errors, while the anti-windup schemes prevent the control systems from going unstable.

13 04 5.

T. Schweickhardt

and F. A llgower

CONCLUSIONS

Linear controller design can be implemented much more easily than nonlinear controller design. One therefore requires tools that allow to determine prior to controller design whether linear controller design is adequate or whether nonlinear techniques have to be used. In this work, we presented a novel approach that combines nonlinearity assessment and controller design by using linear robust control methods and nonlinearity measures. The key features of this method are that (i) closed-loop stability is guaranteed and (ii) conservatism is reduced by considering the nonlinearity of the plant only for the actual operating regime. This is an important advantage, as global control of nonlinear systems is usually neither desirable nor feasible. For the future, further developments are desirable to obtain more robust and more easily realizable methods that also guarantee a certain level of performance. REFERENCES 1. 2. 3. 4. 5. 6.

7. 8.

9. 10.

11. 12.

13.

C. A. Desoer and Y.-T. Wang. Foundations of feedback theory for nonlinear dynamical systems. IEEE Trans, on Circuits and System, CAS-27(2): 104-123, February 1980. F. Allgower. Definition and computation of a nonlinearity megisure. In 3rd IFAC Nonlinear Control Systems Design Symposium, pages 279-284, Lake Tahoe, CA, 1995. D. Sourlas and V. Manousiouthakis. On the computation of the nonlinearity measure. In Proc. 37th IEEE Conf. Decision Contr., pages 1434-39, 1998. A. J. Stack and F. J. Doyle III. The optimal control structure: an approach to measuring control-law nonlinearity. Comp. and Chem. Eng., 21 (9): 1009-1019, 1997. S. A. Eker and M. Nikolaou. Linear control of nonlinear systems: Interplay between nonlinearity and feedback. AIChE J., 48(9): 1957-80, 2002. T. Schweickhardt, F. Allgower, and F . J. Doyle III. Nonlinearity quantification for the optimal state feedback controller. In European Control Conference (ECC), Cambridge, U.K., September 2003. Paper #056. M. Quay, R. Dier, J. Hahn, and P. J. McLellan. Effect of process nonlinearity on linear quadratic regulator performance. J. Proc. Contr., 15(1): 113-124, 2005. T. Schweickhardt and F . Allgower. Linear modeling error and steady-state behaviour of nonlinear dynamical systems. In Proc. 44th IEEE Conf. Decision Contr., pages 8150-8155, Seville, Spain, December 2005. M. Vidyasagar. Nonlinear Systems Analysis (2nd ed.). SIAM, Philadelphia, PA, 2002. D. Kihas and H. J. Marquez. Computing the distance between a nonlinear model and its linear approximation: an I2 approach. Comp. and Chem. Eng., 28(12):2659-2666, 2004. T. Schweickhardt and F. Allgower. A robustness approach to linear control of nonlinear processes. 2006. submitted. B. G. Romanchuk and M. C. Smith. Incremental gain analysis of piecewise linear systems and application to the antiwindup problem. Automatica, 35(7): 1275-1283, 1999. A. Bodalo, J. L. Gomez, M. Fuensanta Maximo, and M. C. Montiel. Development and experimental checking of an unsteady-state model for ultrafiltration continuous stirred tank reactors. Chem. Eng. ScL, 60(15):4225-4232, 2005.


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Control of the Synthesis Section of a Urea Plant by means of an MFC Controller Oscar Mauricio Agudelo Manozca ^ Jairo Jose Espinosa ^, Joos Vandewalle ^ ^ Katholieke Universiteit Leuven, Electrical Engineering Department (ESAT), Research Group SCD, Kasteelpark 10, B-3001 Leuven, Belgium. {mauricio. agudelo, joos. vandwalle} @esat. kuleuven. be ^ IPCOSN. VISMC office, Technologielaan 11/0101 B-3001 Leuven, Belgium. jairo. espinosa@ipcos. com. Abstract The control of the synthesis section of a urea plant is a challenge due to delays, recirculation flows and large residence times. This paper presents the results of applying MPC control techniches for controlling the synthesis section of a urea process. The control system stabilizes the syntehsis section while maximizing the throughput of the synthesis section by maximizing the feeding flows. This control scheme keeps the main variables (NH3/CO2 ratio, reactor level, reactor pressure and reactor temperature) on their desired values or within a defined operating range. The results obtained with this control scheme showed improved stability and increased production. Keywords: Predictive control, Process Control, Chemical Industry, Multivariable systems, PID controllers. 1. Introduction In the last decades, model predictive control (MPC) has been widely accepted by the industry, mainly because of its ability to handle constraints explicitly and the natural way in which it can be applied to multivariable processes [6]. Initially, MPC was applied to meet the specialized control needs of power plants and petroleum refineries, nowadays this control technology can be found in a wide variety of application areas including chemicals, food processing, automotive, aerospace, metallurgy, and pulp and paper. This paper presents the results of using an MPC controller to control the synthesis section of a urea plant. The control integrates the reactor, condenser and stripper units providing maximum integration of the plant via the control system. The controller acts dynamically improving the operating point according to the dynamic limits of the process and exploits the embedded optimization of the MPC to maximize the production. In order to test the control scheme, a complete and very accurate urea model has been used. Such model was developed by Protomation and it has been implemented in a software package called Process Studio, which was created by the same company. For the Implementation of the MPC strategy, an industrial MPC controller developed by IPCOS has been used. This MPC controller is called INCA Suite[l], and it was interconnected with Process Studio via OPC (OLE for Process Control) to control the urea model.

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2. Description of the Process The urea simulator developed by Protomation, models a "Stamicarbon Urea CO2 Stripping Process" [2], where we can distinguish 4 sections: synthesis, recirculation, evaporation and finishing technique and waste water treatment. In a CO2 stripping process, ammonia (NH3) and carbon dioxide (CO2) are fed directly into the synthesis section, where at high pressure (13-30 MPa) and high temperature (170-200°C) the urea formation takes place, according to the following reactions: 2NH3 + CO2 ^ NH2COONH4 + (heat) NH2COONH4 log2 {AS).

Multiscale SPC in the Presence of Multiresolution Data

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Figure 4. Plots for Q statistic in MS SPC with unif discretization (a) and classical PCA-SPC (b). 3.2. ARL characterization study As MR-MS SPC involves a certain delay in the calculation of coefficients, since it is based on an orthogonal wavelet transformation, we studied whether the benefits illustrated in the previous example come at the cost of higher detection times, through an ARL study where two situations regarding the resolution of variable X^ were contemplated, J^=2 and ^4 = 3, along with several shift magnitudes. The following methods were considered in the study: MR-MSSPC, Dyadic-MSSPC (similar to MRMSSPC, with a dyadic discretization strategy, but using data at a single resolution - the finest one), Unif-MSSPC (i.e., MSSPC with uniform discretization) and PCA-SPC. Figure 5 presents the ARL results obtained, where we can see that any time delay associated with MR-MSSPC only becomes an issue for shifts of magnitude greater than 2, after which stabilizes around 0.5, an acceptable value for most applications.

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-H^ ^ ^

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2

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4. Conclusions In this paper we present an approach for integrating multiresolution data in MSSPC. The approach was tested using a simulated latent variable system, leading to improvements in the definition of the region where faults occur, without compromising detection speed.

References 1. R.L. Motard, B. Joseph (eds.), 1994, Wavelet Applications in Chemical Engineering, Kluwer, Boston. 2. B.R. Bakshi, 1998, Multiscale PCA with Application to Multivariate Statistical Process Control, AIChE J. 44, 7, 1596-1610. 3. J.E. Jackson, 1991, A User's Guide to Principal Components, Wiley, New York. 4. S. Mallat, 1998, A Wavelet Tour of Signal Processing. Academic Press, San Diego. 5. M. Vetterli, J. Kovacevic, 1995, Wavelets andSubband Coding. Prentice Hall, New Jersey. 6. B.K. Alsberg, A.M. Woodward, D.B. Kell, 1997, An Introduction to Wavelet Transforms for Chemometricians: A Time-Frequency Approach, Chemom. Intell. Lab. Syst. 37, 215. 7. H.B. Aradhye, B.R. Bakshi, R. Strauss, J.F. Davis, 2003, Multiscale SPC Using Wavelets: Theoretical Analysis and Properties, AIChE J., 49,4, 939-958. 8. N. Lu, Y. Yang, F. Gao, F. Wang, 2004, Multirate Dynamic Inferential Modeling for Multivariable Processes, Chem. Eng. Sci., 59, 855-864. 9. P.R.C. Nelson, P.A. Taylor, J.F. MacGregor, 1996, Missing Data Methods in PCA and PLS: Score Calculations with Incomplete Observations, Chemom. Intell. Lab. Syst., 35, 45. 10. A.S. Willsky, 2002, Multiresolution Markov Models for Signal and Image Processing, Proceedings of the IEEE, 90, 8, 1396-1458. 11. J.F. MacGregor, 1995, T. Kourti. Statistical Process Control of Multivariate Processes, Control Eng. Pract. 3, 3, 403-414.

Acknowledgments Financial support from FCT (POCTI/EQU/4763 8/2002) is gratefully acknowledged.


Nonlinear Model Predictive Control of the Wastewater Treatment Plant Mircea V. Cristea and Serban P. Agachi "Babe§-Bolyai" University ofCluj-Napoca, Faculty of Chemistry and Chemical Engineering, 11 Arany Janos, 400028 Cluj-Napoca, Romania, e-mail: mcristea@chem. ubbcluj. ro Abstract Sustainable development of the modem society implies special care of the water resources and their efficient management. The biological Wastewater Treatment (WWT) plants play a major role for preserving the health of the environment. This work presents alternatives for the design of the control system for the aerobic suspended growth wastewater treatment plant. Model Predictive Control (MPC) of the WWT plant soluble substrate and dissolved oxygen are investigated in the presence of typical disturbances. The proposed combination of feedback-feedforward MPC control design improves the control performance, compared to traditional control or simple feedback MPC, for both disturbance rejection and setpoint tracking. The control performance presents short settling time and reduced overshoot for the multivariable approach of MPC with output and manipulated variables constraints, showing benefits for industrial implementation. Keywords: Biological wastewater treatment, model predictive control. 1. Introduction The main task of the biological treatment is to convert soluble organic contaminants into insoluble organic and inorganic constituents or to CO2 and H2O. Aerobic or anaerobic treatment is based on the work of enzymes that transform hydrocarbons into food for the bacteria, breaking down the undesired hydrocarbons. The activated sludge process is widely used and consists in an effective wastewater treatment for the removal of dissolved and colloidal biodegradable organics. This is a treatment technique well suited for cases when organically contaminated wastewater exists. A wide range of municipalities and industries that treat wastewater containing organic chemicals, petroleum refining wastes, textile wastes, and municipal sewage use the activated sludge process. Wastewater treatment processes usually include: flocculation, neutralization, and biological processes. The active sludge process converts dissolved and colloidal organic contaminants into biological sludge, further removed by settling. The microorganisms usually found in the activated sludge consist of bacteria, fungi, protozoa, and rotifers. With proper detention, activated sludge systems have good performance for Biochemical Oxygen Demand (BOD) removal [1]. Success of the continuous biological treatment process depends on the way that suitable conditions are achieved for bacterial growth, death and endogenous respiration phase. These requirements may be only obtained, especially for large wastewater treatment units, by appropriate control techniques intended to keep the operation of the unit at the most efficient working regime. Model Predictive Control is a good candidate for these demanding control tasks. The way this control algorithm may be used for the control of a suspended growth aerobic system is presented in the present work.

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2. Model description and model predictive control approach The aerobic activated sludge (suspended growth) wastewater treatment system has been considered in this study. Organic wastes are introduced in a (completely mixed assumed) reactor where the bacterial culture is held in suspension. Aerobic environment is achieved by diffused or mechanical aeration. The biological solids form a floe, settle out, and separate from the treated water. A portion of the settled cells is recycled to maintain the desired concentration of organisms in the reactor. A satisfactory floe is necessary for effective separation of the biological solids. Important for settling are proper design of a settling unit and proper operation to prevent the presence of filamentous organisms and fungi. Bacteria in activated sludge are capable of performing hydrolysis and oxidation reactions. Both the oxidation of complex hydrocarbons and the hydrolysis of polysaccharides occur outside the cell. They are catalysed by exoenzymes secreted from the cell wall into the surrounding aqueous environment. Oxidation is conducted by aerobic organisms which use dissolved oxygen present in the biological system. Hydrolytic reactions are caused by aerobic organisms using water present in the biological system [1]. For this completely mixed continuous-flow aeration unit, presented in Fig. 1, the influent is fed uniformly along the entire length of the basin. AIR SUPPLY

WASTEWATER

SLUGE WASTE

Figure 1: Schematic representation of the continuous-flow aeration unit. The aeration is essentially homogenous, resulting in uniform oxygen demand throughout the basin [2]. This results in a homogeneous concentration of solids and substrates in the basin. This system is stable and is less affected by disturbances. Disturbances will be uniformly distributed in the basin and subsequently diluted, because of the relatively uniform population of organisms. The first principle model presented in the following equations has been used both for the dynamic behaviour description of the unit and for building the simulator on which the model based control strategies have been investigated [3]. The mass balance for the concentration of biomass X(t), substrate S(t) and dissolved oxygen DO(t) are described in the Eq. (1) to (3): • = ju (t)X(t) - D(t)(l + r)X(t) + rD(t)X^ (t) dt

(1)

Nonlinear Model Predictive Control of the Wastewater Treatment Plant

— = -^X(t)-D(t)(l+r)S(t) dt Y dDO = dt

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dt

= ^[D(t)(l V

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... ^^^

The dynamic model succeeds to capture the main dynamic features of the biological wastewater treatment unit, needed for the control investigations. Model Predictive Control, also referred as moving or receding horizon control, has become an attractive control strategy especially for linear but also for nonlinear systems subject to input, state or output constraints [4, 5]. MPC determines the control action based on the prediction of future dynamics of the system, allowing early control action to be taken in order to accomplish the control performance based on the expected future behaviour. The incentives of MPC algorithm, compared to traditional control algorithms, are associated to the need the system usually has for satisfying input, state or output constraints. These constraints put limitations on the achievable control performance, but this task is systematically managed by the MPC optimisation objective (with associated constraints), compared with the ad-hoc solutions frequently used in conventional control. The MPC algorithm used for the control of the wastewater treatment unit uses the combination of feedback and feedforward control design in order to reduce the effects of the large time constants presented by the process. 3. Results of the MPC approach The main control objective of the wastewater treatment unit has been considered the maintenance of the effluent soluble substrate (pollutant) concentration S(t) at the desired level of 43.5 mg/l. Three most important disturbances have been taken into account: the inlet soluble substrate concentration Sin(t), the inlet dissolved oxygen concentration Sin(t) and the inlet flowrate of the feed, the last disturbance being taken into account by the dilution rate D(t) variable of the previously presented model. The control performance has been first investigated for the cases when two of the aforementioned disturbances act stepwise. Namely, at the time moment of t=10 h acts the inlet soluble substrate concentration step, with an amplitude change of +12.5 %

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(from 200 mgA to 225 mg/l), and for the second case acts the dilution rate step disturbance with an amphtude change of -\-14.2 % (from 0.07 h'^ to 0.08 h'^). The first investigated MPC control approach was to control the dissolved oxygen DO value in the aeration basin at a desired setpoint value of 3.21 mg/l, in the presence of the considered disturbances. Results of the dissolved oxygen DO(t) and soluble substrate S(t) concentration change in the WWT unit are presented in Fig. 2 and Fig. 3, when the inlet soluble substrate concentration disturbance is acting stepwise.

Figure 2: Dissolved oxygen variation for MPC Figure 3: Soluble substrate concentration control of DO in the presence of inlet soluble variation for MPC control of DO in the presence of inlet soluble substrate substrate concentration Sjn disturbance. concentration Sin disturbance. It may be noticed that both soluble substrate S and dissolved oxygen DO concentration are kept, by the MPC control of the DO concentration, at their desired values after the disturbance is acting, although DO is the only controlled variable. This result is usefiil for the control system design of WWT units where the inlet soluble substrate concentration Sjn is the most important disturbance and the dilution rate may be straightforwardly maintained at constant value. Unfortunately, for this first control approach and when the dilution rate D disturbance is acting, the DO variable is maintained at the desired setpoint but the soluble substrate S is irreversibly disturbed from its desired value. Due to this undesirable behaviour, the MPC control of both dissolved oxygen DO and soluble substrate S has been fiirther investigated. Disturbance rejection and setpoint following performance have been studied. Additionally, in order to improve the MPC performance, the combined feedback-feedforward control has been proposed and tested. For this second MPC approach the simulation scenario consists in putting into operation three disturbances: the dilution rate D step disturbance acting at time moment t=20 h (changing from the initial value of 0.07 h'^ to the final value of 0.075 h'^X followed by the inlet soluble substrate Si„ concentration step disturbance applied at time moment t=100 h (changing from the initial value of 200 mg/l to the final value of 275 mg/l) and ending by the dissolved oxygen Z)0,„ inlet concentration step disturbance applied at time moment t=250 h (changing from the initial value of 0.5 mg/l to the final value of 0.1 mg/l). The comparative results showing feedback MPC and combined feedback-feedforward MPC, both acting for the disturbances rejection, are presented in Fig. 4 and Fig. 5.

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Furthermore, the results of the combined feedback-feedforward MPC, for the case of two a priori known (programmed) setpoint changes (one for the dissolved oxygen concentration setpoint change of +9.3%, changing from 3.21 mg/l to 3.51 mg/l, at the time moment t=100 h, and one for the soluble substrate concentration change of +11.5%, changing from 43.47 mg/l to 48.47 mg/l, at the time moment t=225 h) are presented in Fig. 6 and Fig. 7.

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V. Kariwala and S. Skogestad

1374

CV and f^ MV, respectively. Then the variable selection problem requires solving 7/ 7/

>

0

(6)

where a^, Pj = {0, 1}. The i^^ CV is selected if a^ = 1 and vice versa. For selecting m variables out of (ny+Uu) CVs and MVs, we need the additional constraint ^^ • a^ + ^ f3j = m. An upper bound is obtained by relaxing the decision variables as 0 < 0;^, ^^ < 1 and a lower bound is obtained by rounding-off a, (3 that optimally solves the convex program. Note that the assumption of distinct unstable poles can be satisfied by numerically perturbing the poles. 3.2. Application There are many software packages available for solving integer semi-definite programs. In this paper, we use the B&B algorithm available with Yalmip [8], where the package Sedumi 1.05 [9] is used for solving the LMI problems resulting on relaxation of binary variables.

(a) Total number of CVs and MVs

fixed

(b) Number of CVs and MVs fixed individually

Figure 1. Selection of variables for stabilization of Tennessee Eastman process. The algorithm is applied to Tennessee Eastman process with six unstable poles. The model with 22 CVs and 12 MVs is scaled as outlined in [4]. The variation of maximum HSV with the total number of CVs and MVs fixed is shown in Figure 1(a). Cao and Saha [4] suggest using m = ^, for which maximum HSV is 926.65. The bounding strategy used in [4] is similar to the strategy used for bounding MSV Out of 1.72 x 10^° alternatives, the proposed method evaluates only 1441 alternatives cumulatively for all m, which is an order of magnitude less than reported in [4]. It is also possible to fix the number of CVs and MVs individually through the constraints Yliî — ^y» S j A? — ^ u ^^^ the results are shown in Figure 1(b). For vrty — rriu = 3, the optimal solution is same as for m = 6, as expected. 4. Other problems Similar to the problems discussed earlier, the combinatorial nature of many other problems arising in CSD can be handled using B&B methods. Usually, the challenging task is to find a tight upper bound on the objective function. Next, we discuss some problems requiring novel manipulations of the partial solution to get monotonic relationships useful as upper bounds. The first problem consists of selecting pairings such that iJ.{E) is minimized (or —iJi{E) is maximized), where /x is the structured singular value [1] and E = {G — G^diag)^diig ^^^^ ^diag

Branch and Bound Methods for Control Structure Des ign

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consisting of diagonal elements of G. Selection of pairings with minimal IJ^{E) (equivalent to finding optimal permutation of G) puts least restrictions on decentralized controller synthesis using independent designs [1]. When there exists a pairing with /x(^) < 1, iterative-RGA provides the optimal solution explicitly [1], otherwise a B&B method is required and the bounding strategy is discussed next. For any given partial solution (node), a lower bound on —fi{E) can be obtained using the sequential approach discussed in § 2. Now, assume that G is permuted such that the (1,1)-block of permuted G correspond to gain of partially selected pairings. By partitioning E similar to G, we then have —/i(E'ii) > —iJi{E) [1], which serves as the upper bound on all the alternatives that can be generated by expanding on this node. This happens as any expansions on this node do not affect En. Another problem consists of selecting pairings such that the achievable decentralized output performance (Jdecen) is minimal. Though exact characterization of Jdecen is an open problem, sub-optimal methods providing upper bounds on Jdecen are available. Then, at any node, the lower bound on —Jdecen is obtained by completing the partial solution using any reasonable heuristic and the sub-optimal methods. For a given pairing, an upper bound on —Jdecen was recently presented in [10]. This method can be adapted for partially completed solutions by using a block decentralized controller, where the remaining (unpaired) CVs and MVs are controlled using a full multivariable controller. As only bounds on Jdecen can be computed, B&B method may end with a number of alternatives, for all of which the upper bound on —Jdecen is larger than the best available lower bound. Nevertheless, B&B method is useful for eliminating inferior pairing alternatives. 5. Conclusions B&B methods can efficiently handle the combinatorial problems arising in CSD. In comparison with standard mixed integer non-linear programs, CSD problems require novel bounding and search strategies. We demontrated the use of monotonic relationships and integer semi-definite programs for obtaining tight upper bounds on partial solutions. The algorithms scale well with problem dimensions and are shown to provide optimal solutions for large-scale benchmark problems. The future work will focus on extending these ideas to multi-objective optimization problems. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.

S. Skogestad and I. Postlethwaite. John Wiley & Sons Ltd., Chichester, UK, 2005. Y. Cao, D. Rossiter and D. H. Owens, Proceedings of DYCOPS 5, Korfu, Greece, 1998. I. K. Kookos and J. D. Perkins. Ind. Eng. Chem. Res., 40, 2001. Y. Cao and P Saha, Chem. Engg. Sci., 60, 2005. R. Horn and C. Johnson. Cambridge University Press, New York, NY, USA, 1985. V. Kariwala, S. Skogestad, J. F. Forbes and E. S. Meadows. Intl. J. Control, 78, 2005. S. Boyd., L. Ghaoui, E. Feron and V. Balakrishnan. SIAM, Philadelphia, PA, USA, 1994. J. Lofberg. Proceedings of CACSD, Taipei, Taiwan, 2004. (Available from h t t p : / / c o n t r o l . ee . e t h z . c h / ~ j o l o e f / y a l m i p . p h p ) . 9. J. F. Sturm, Optimization Methods and Software, 11-12, 1999. (Available from h t t p : //sedumi .mcmaster. ca/). 10. V. Kariwala, J. F. Forbes and E. S. Meadows. Proceedings of ACC, Boston, MA, 2005.

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6. Appendix: Average performance We further evaluate the performance of B&B methods for maximization of MSV and HSV using randomly generated examples. For maximization of MSV, the method is applied to 1000 randomly generated matrices of different sizes and the results are shown in Table 1. Table 1 Performance of B&B method for maximization of MSV for randomly generated matrices n m Alternatives % of alternatives evaluated Mean CPU Time Mean Median Max Min (sec) 10 3 120 44.1 40.8 132.5 11.7 0.0042 20 3 1140 11.1 8.8 2.1 66.5 0.0067 15504 75.2 20 13.7 10.9 0.6 5 0.1188 2.8 30 5 142510 3.8 37.1 0.1 0.3409 For maximization of HSV, the efficiency of this algorithm is tested using 100 random square transfer matrices of different dimensions generated using Matlab command rss and the results are shown in Table 2. For the last two cases, the maximum number of alternatives evaluated are not reported, as they were restricted to 5000 and 20000, respectively, and there exists examples that exceed these bounds. Table 2 Performance of B&B method for maximization of HSV for randomly generated systems rip n m Alternatives % of alternatives evaluated Mean CPU Time Mean Median Max Min (sec) 3 5 3 120 43.3 32.5 155.8 0.8 4.0657 11.7 120 84.5 87.5 142.5 10.858 5 5 3 5 1140 63.3 70.5 0.6 10 3 134.6 75.631 15504 4.17 1.5 0.0 56.057 5 10 5 0.2 0.0 298.67 5 142510 1.7 15 5 Some of the salient observations are: 1. The efficacy of the method, measured in terms of the number of the alternatives evaluated, increases with problem dimensions. This is reasonable, as many more branches can be pruned, when less variables are selected. 2. There exist cases that are worse than complete enumeration. As most combinatorial problems are AfV-hard, the worst-case performance is likely to increase nonpolynomially with problem dimensions, when any ingenious B&B strategies are used. 3. The average performance is much worse than the large-scale benchmark problems considered previously. This happens as real problems often show more structure than randomly generated problems. For example, for the HDA process, the scaled gain for 23 out of 129 candidate CVs is nearly zero, which are eliminated in the first step of the B&B method resulting in substantially reduced search space. 4. As we need to solve an LMI problem iteratively for finding an upper bound for the HSV maximization problem, the mean CPU time is much larger than the MSV maximization problem. An advantage of this formulation is that there exist cases, for which the optimal integer solution is obtained through evaluations of only one alternative.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantehdes (Editors) © 2006 PubHshed by Elsevier B.V.

A mathematical programming approach including flexible recipes to batch operation rescheduling Sergio Ferrer-Nadal, Carlos A. Mendez, Moises Graells, Luis Puigjaner* Chemical Engineering Department-CEPIMA, Universitat Politecnica de Catalunya ETSEIB, Av.Diagonal 647, E-08028, Barcelona, Spain Abstract The inherent dynamic nature of industrial environments often needs not only the execution of the required rescheduling actions but also the proper adjustment of the production recipe to the current process conditions. Therefore, the concept of flexible recipe becomes an important part of the rescheduling fi-amework that allows ftiU exploitation of the batch plant intrinsic flexibility. This work introduces a rigorous mathematical approach that incorporates the concept of recipe flexibility to batch operation rescheduling. Keywords: rescheduling, flexible recipe, batch operations, MILP model. 1. Introduction Batch processes have received great attention over the last years because of their higher flexibility compared to continuous processes and the increasing demand for specialty, high added-value chemical and pharmaceutical products. Within this context, the shortterm scheduling deals with the optimal allocation of a set of scarce plant resources over time to manufacture one or more products following a batch recipe. Most of the scheduling approaches assume that batch processes are operated at nominal conditions following predefined fixed production recipes (Mendez and Cerda, 2003 a). However, in many cases a flexible recipe operation may result a more suitable way of incorporating systematic recipe adaptations depending on the actual process conditions. The flexible recipe concept was originally introduced by Rijnsdorp (1991) as a set of adaptable elements that controls the process output. Afterwards, Verwater-Lukszo (1994) presented a flexible recipe approach for the adjustment of control recipes during production which has been applied to several case studies (Sel et al., 1999; Rutten and Bertrand, 1999). One of the first attempts to extent the flexible recipe approach to a plant-wide scheduling problem was carried out by Romero et al. (2001). These authors proposed to integrate a linear flexible recipe model into a multipurpose batch process scheduling model based on an S-graph algorithm. In addition to changes in nominal process conditions, frequent unexpected events can also take place during the normal batch plant operation (equipment failures, late order arrivals, order cancellations and so on). These unforeseen changes may lead the inprogress schedule to become suboptimal or even infeasible. Although rescheduling techniques have a central role in process operations, only a few developments have focused their attention on this challenging problem (Cott and Macchietto, 1988; Mendez and Cerda, 2003b). These rescheduling approaches allow performing certain corrective actions such as partial resource re-allocation, re-sequencing and re-timing assuming a Author to whom correspondence should be addressed: [email protected].

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fixed batch production recipe. This work introduces a MILP-based framework to batch operation rescheduHng including the concept of recipe flexibihty. 2. The flexible rescheduling framework The proposed rescheduhng approach is based on the MILP model for single-stage multiproduct reactive scheduling introduced in Mendez and Cerda (2003b). The original model was adapted and extended to address the rescheduling problem of multistage multipurpose batch plants involving different storage policies, non-zero transfer times and flexible recipes. This model relies on the notion of general precedence which reduces the number of binary variables and so the computational effort, as reported in Mendez and Cerda (2003a). Flexible recipe constraints are incorporated in this model to account for the possibility of changing the processing time of some tasks tweaking the rest of the parameters of the product recipe. The cost for modifying these process variables from their optimal economic conditions is taken into account to represent how plant productivity is increased regardless of the cost of altering the nominal plant conditions. Different incidences during the processing horizon can be considered such as insertion of new orders, equipment failures, due dates changes, delay in arrivals, variations in the cost of the raw materials or products, etc. Therefore, when an unexpected event arises, the following groups of tasks can be defined: • Executed tasks (T^^^^) already processed at the rescheduling point which are not included in the rescheduling formulation since they are past events with no influence in the remaining schedule. • Non-directly affected tasks (T"^^) by the unexpected event that are being processed or still have to be processed at the rescheduling point. • Directly affected tasks (T^^) by the unexpected event. They include rejected tasks , running at the rescheduling point that have to be transferred to an alternative unit in order to be reprocessed. Successive stages of these rejected tasks in the processing sequence are also included in this group. • ^New task (T"®^) from late order arrivals to be scheduled. All these different tasks are depicted in Figure 1 which represents a production scenario where two products (dark grey and light grey) are manufactured. rnda

I

Unit U l Unit U 2 Unit U 3 Rescheduling point

Time 'Cxec rpnda rpda ,

Figure 1. Basic representation of task types: r^^^ T'^, T"^ and T"'^. Rather than re-optimizing the sequence of the remaining tasks, only local changes are allowed in order to reduce the impact over the schedule in progress. Partial rescheduling actions linked to each group of tasks are summarized in Table 1.

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3. Case study The proposed rescheduling strategy under the flexible recipe framework will be illustrated by solving a modified version of the case study proposed in Romero et al. (2003). Five products are manufactured in four stages being available alternative process equipment units The flexible recipe for the production of benzyl alcohol (PI) is introduced within this production scenario. Transfer times as a 5 % of the processing time of each processing task have been also considered. Production orders comprise a single batch of one or several products with specific due dates. Batch processing times, available processing units and order due dates are reported in Table 2. Table 1. Allowable rescheduling actions (X: No ; V: Yes) ^ , Task type

.^ .... (Re)Alloc.

.^ s^ (Re)Seq.

.^ xr^. . (Re)Timing

1. Executed tasks, T""^

X

X

X

X

2. Non directly affected tasks, T"^

X

V

V

V

V V

V V

V V

V V

3.Directly affected tasks, T''* 4. New tasks, T™

Recipe adjustment ^ i / ^ ur . i N (only for flexible tasks)

The crossed-Cannizaro reaction for the batchwise production of benzylalcohol (second stage of product PI at unit U2) from the reduction of benzaldehyde has been studied by Keesman (1993). This author proposed a quadratic model to predict the yield of the reaction for a priori known disturbances in the process inputs. Equation 1 shows the linearized model required to be incorporated into the MILP formulation, assuming a small flexibility region around the current operating conditions (see Table 3). y = 4A X i + 4 JC2+95 X3+95 X4

(1)

Additional flexibility has been considered in the first stage of product PI (unit Ul). This is a preheating stage where the temperature is directly proportional to the processing duration and the temperature for the reaction in the next stage. The selected objective function to be minimized is the total cost associated to the order earliness (1 m.u./h), tardiness (5 m.u./h) as well as the corresponding cost for manipulating the process conditions (See Table 3). The proposed approach is implemented within the modelling language GAMS (Brooke et al., 1998) using CPLEX version 7.5. 4. Results This rescheduling approach is applied to the schedule in progress shown in Figure 2 which has to be updated at time 3h in order to face the breakdown of unit 7 with a repairing time of 17 hours. In addition, new batches corresponding to the arrival of late orders must be also inserted in the on-going schedule (see Table 2). Figure 3 shows the proposed reschedule plan without considering the recipe flexibility. Finally, Figure 4 depicts a flexible reschedule that, despite the recipe modification cost, results in a better solution in terms of the proposed objective function. This improvement comes not only from the recipe changes but also from the several modifications of sequencing decisions, which can be easily observed in Figures 3 and 4. It is worth nothing that the

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computational effort for updating the current schedule remains very low, even in the case that additional rescheduling actions are considered. Short reaction times are highly important in real industrial environments. Table 4 summarizes the main features of the schedules generated without and with recipe flexibility. In order to improve the customer satisfaction in the flexible production environment, the second stage of the first, third, fifth and sixth batch of product PI (flexible reaction task) reduces their processing time at the expense of increasing the amount of formaldehyde Table 2. Process data for the case study Product PI Stage

Product P5

Unit

PT,h

Unit

PT,h

Unit

PT, h

Unit

Ul

0.5

Ul

1

U7

2

U2

1.5

U8

U2

0.75

U9

2.5

U8

2

U8

2

U3

2

U3

2

U3

1

U9

2

U4

1

U9

2.5

U6

1

U7

2

U4

2

U5

1.5

1.75

(Flexible stage)

4

Product P4

PT,h

U2

3

Product P3

Unit

(Flexible stage)

2

Product P2

U3

2

U4

1.5

U9

1.75

U7

2

U4

0.5

U6

1

U7

0.75

U5

1

PT, h

U5

1

U2

1.25

Ul

1

Due dates (h) Order 1

10

10

9

10

15

Order 2

10

20*

9

22*

15

Order 3

10

15

Order 4

15

20*

Order 5

19*

Order 6

19*

(*) New ordei

Table 3. Recipe parameters, flexibility region and cost for deviation from nominal conditions Flexible process variable

Flexibility Region Lower bound

Deviation cost

Upper bound No deviation is allowed in quality

5y

reaction yield

5xi

reaction temperature

-0.7'^C

0.5 °C

3 m.u.AC

8x2

reaction duration

-0.3 h

0.1 h

2m.u./C

5x3

amount of KOH

-2.7 g

8.5 g

5 m.u./g

6x4

amount of Formaldehyde

-30 g

7.5 g

4 m.u./g

A Mathematical Programming Approach

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and KOH. However, in none of the batches of PI, flexibility is exploited for the first stage. This situation arises because flexibility has always a cost and this stage does not suppose a bottleneck in the process and, consequently, no improvement can be achieved. Therefore, this example clearly reflects the high importance of recipe flexibility in the rescheduling process of critical and hard-constrained batch operations. I

Product PI ] Product P2 ] Product P3 Product P4 Product P5 111^^81 Waiting time WMMi Rejected task Figure 2. Schedule in progress at the rescheduling point. U1 U2 tJ3 LJ^

US U6 UT-

US U9

Figure 3. Optimal rescheduling considering fixed production recipe at nominal conditions.

Figure 4. Optimal rescheduling consideringflexibleproduction recipes. 5. Conclusions An efficient MILP-based reschedulingfi*ameworkthat incorporates the recipe flexibility as an additional rescheduling opportunity has been presented. The approach is based on

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a continuous-time domain representation and the generalized notion of precedence. The rescheduling strategy considers the on-going production schedule and the current process conditions in order to simultaneously adapt the production recipe to the new scenario and re-optimize the schedule of the batches still to be processed. Different objective functions can be employed to regain feasibility or optimality at minimum cost. Efficiency and applicability of the proposed strategy is demonstrated by successfully solving a complex rescheduling problem in a multipurpose batch plant with reasonable computational effort. The results reported put clearly on evidence the significant benefits of exploiting the inherent flexibility of batch plants. Table 4. Comparison between fixed and flexible rescheduling Fixed reschedule

Flexible reschedule

Tardiness cost, m.u.

131.81

116.44

Earliness cost, m.u.

7.93

11.54

Process conditions change cost, m.u.

0.00

1.95

Objective function, m.u.

139.74

Binary vars., cont. vars., constraints

5, 555, 974

CPU time, s (AMD Athlon 2600 MHz, CPLEX 7.5 )

68

129.93

283,676,1131 230

Acknowledgements Financial support received from the European Community projects (MRTN-CT-2004512233; RFC-CR-04006; INCO-CT-2005-013359) and the Generalitat de Catalunya with the European Social Fund (FI grant) is fully appreciated. References Brooke, A., Kendrick, D., Meeraus, A. & Raman, R. (1998). GAMS - A user's guide. The Scientific Press. San Francisco. Cott B.J. & Macchietto S. (1989). Minimizing the effects of batch process variability using online schedule modification. Computers & Chemical Engineering. 13, 105 - 113. Mendez C.A. & Cerda, J. (2003a). A MINLP continuous-time framework for short-term scheduling of multipurpose batch processes under different operation strategies. Optimization & Engineering. 4, 7 - 22. Mendez C.A. & Cerda, J. (2003b). Dynamic scheduling in multiproduct batch plants, Computers & Chemical Engineering. 27, 1247 - 1259. Rijnsdorp, J.E. (1991). Integrated Process Control and Automation, Elsevier, Amsterdam. Romero J., Espuna A., Friedler F. & Puigjaner L. (2003). A new framework for batch process optimization using the flexible recipe. Industrial & engineering chemistry research. 42, 370 379. Rutten, W. G. M. M. & Bertrand, J. W. M. (1998). Balancing stocks, flexible recipe costs and high service level requirements in a batch process industry: A study of a small scale model. European Journal of Operational Research. 110, 626 - 642. Sel, D., Hvala, N., Strmcnik, S., Milanic, S. & Suk-Lubej, B. (1999). Experimental testing of flexible recipe control based on a hybrid model. Control Engineering Practice. 7, 1191-1208. Verwater-Lukszo, Z. (1998). A practical approach to recipe improvement and optimization in the batch processing industry. Computers in Industry. 36, 279 - 300.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 PubUshed by Elsevier B.V.

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Using multi sensor data fusion for level estimation in a separator Nils-Olav Skeie^, Saba Mylvaganam^, Bernt Lie^ ^Telemark Technological R&D Centre, Kj0lnes Ring 30, N-3918 Porsgrunn, Norway ^Telemark University College, R O . Box 203, N-3901 Porsgrunn, Norway Abstract A data driven model is developed to be used as a soft sensor to predict the hquid and interface levels in an oil / water separator. The methodology uses a set of absolute pressure sensors together with multi sensor data fusion for estimation of the levels. Experimental results are provided for model validation. Key^vords: Multi sensor data fusion, soft sensor, oil / water separator, interface level. 1. Introduction The product from oil drilling consists of a mixture of oil, gas, water, and sand. Oil / water separators are used to separate these components: gas is vented off at the top, sand is removed from the bottom, while settling is used to separate oil and water. Settling requires a certain residence time to be completed, and it is necessary with a certain oil volume in the separator. The water feed can be controlled to achieve the required oil volume and settling time, but it is necessary to have a measure of the oil / water interface level. Further complications for the measurement are foaming, and sand in the water. A number of principles for level measurement in separators exist, but today mostly sensors based on nuclear principles are used. Since operators work in the neighbourhood of separators, it is desirable to avoid using nuclear based sensors. In this work, we consider the possibility to combine information from several sensors in a multi sensor data fusion (MSDF) algorithm to develop a soft sensor for oil / water interface level. The paper is organized as follows: in section 2, sensor types are discussed, and a specific sensor type is chosen in this initial study. In section 3, a linear regression model is proposed for describing the relationship between sensor data and interface level. Then in section 4, the experimental separator rig is described. In section 5, experiments are described, and a soft sensor is developed. Finally, the results are discussed. 2. Sensor selection In this study only standard sensors will be considered, and if possible, used. Using MSDF, many sensors can be used, both of the same type and of different types. The following types of sensors have been considered [1]: • floats: in an oil/water separator two floats must be used, but due to mechanical solutions these sensors are not desirable in the separator.

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• hydrostatic devices: differential or absolute pressure devices; - differential pressure devices are not suitable due to the possibility of mud, condensation rates, and plugging in the dry-leg, — absolute pressure devices can be used, using a flush diaphragm to avoid plugging. The challenges regarding absolute pressure devices are dependency of density, and that each device must be able to measure the whole pressure range of the separator and still be accurate for density measurement, • load cells: difficult to make a reliable mechanical solution for weighing the separator, • ultrasonic transmitters: will not work due to sand in water, and foam on top of oil, • radar transmitters: will not work due to foam on top of the oil, and a low dielectrical constant of the oil, • guided wave radar: working, but not stable in real life. Very dependent on the thickness of oil layer, the dielectrical constant of the oil, and the thickness of the emulsion layer between oil and water. The separation is working due to different densities of oil and water. Normally the density of oil is 65% — 90% of the density of water. Using more than one pressure sensor, it is possible to measure the density difference between the sensors. The relationship between pressure and density is:

r pgdh

(1)

J/ho he,

where p is the value measured by the sensor, po is the gauge pressure, p is the density, g is acceleration due to gravity, and h is the height of the liquid. Using more then one pressure sensor, po will be the pressure from the sensor above, and h is the height between these sensors. In this first study, five absolute pressure sensors are used, together with a top pressure sensor measuring the gauge pressure in the separator. An overview of the sensor locations are shown in Figure 1. Due to a small distance between the pressure sensors, sensor fusion is selected instead of single sensor values and a logic system for estimation of the liquid and interface levels. Particularly the interface level is difficult to estimate. 3. Linear regression model A software application was developed using .NET technology and C # as a programming language, for communication with the sensors, logging sensor data, configuration, and user interaction. Using absolute pressure sensors, an atmosphere pressure sensor is needed to be able to display the gauge pressure. The relative pressures of the sensors are calculated

Using Multi Sensor Data Fusion for Level Estimation in a Separator

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• the gauge pressure in the separator: Vtop

Ptopsens

Patmsens

(2)

• the pressures for the sensors that can be covered by Hquid: Preli ^^ Psensori

Ptop

Patm_sens

Psensori

Ptop sens

(3)

Patm sens IS the valuc from the atmosphere pressure sensor, ptopsens the value from the top sensor, and Psenson the value from pressure sensor i that can be covered by liquid. Using the Prei values for the pressure sensors that can be covered by liquid, these values will be independent of the gauge pressure in the separator. The experiment can then be done for any value oiptopThe relative pressures will be used as inputs to a data driven model of the separator. A linear relationship between the relative pressure values and the liquid level and oil / water interface is assumed: y = bx -^ e

(4)

where y contains the estimated levels, b is the regression coefficients, x contains the relative pressures, e is the error / noise. Let matrix X be composed of x vectors from N experiments, and Y be composed of y vectors from the same experiments. Then equation 4 can be written as:

Y = XB + E

(5)

and the least squares solution of B is implicitly given by the normal equation [2]:

X^XB

- X^Y

(6)

To calculate the B matrix in Equation 6 requires that the X^X matrix is nonsingular. If the X^X matrix is singular, Principal Component Regression (PCR) will be used. In PCR, Singular Value Decomposition (SVD) may be used to factor the X matrix into the matrixes X = UEV^ [3]. Matrix E gives the singular values Ci > ... > cr^ > 0. The singular values tell us if any of the rows in X is linear dependent on other rows, in this study meaning that one or more sensors may be redundant. The soft sensor model for estimating the liquid level and the oil / water interface of the separator will then be:

Y=XB

(7)

4. Separator rig A separator rig is built as a 1:3 scale model of a first stage vertical separator rig in a real separator train. The separator rig used for the experiment is equipped with pressure sensors, valves, pumps, and buffer tanks as shown in Figure 1.

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N.-O. Skeieet al ^ - j Over pressure valve

Top pressure sensor Well input

Figure 1. Process overview of the separator rig.

5. Soft sensor from experimental data Water and olive oil are used in the experiment, and the input pumps are used to set up different filling fractions of water and oil in the separator. The sensor data x are logged using the C # based software application, while the output data y are measured manually by visual reading against a centimeter tape which is glued to the separator. MATLAB was used for further processing of the logged data. The relative pressure data were mean-centered [4], and a MATLAB script was used to generate the meancentered X matrix, and SVD of the X matrix is performed. The E matrix from the SVD gives the singular values cr^ G {594.5, 83.4, 42.4, 28.1, 11.1}. The X matrix is thus of full rank, and the least squares method can be used; no pressure sensors are redundant, as used with different heights shown in figure 1. Equations 6 and 7 are then used, and figure 2 displays the estimated levels and the manually read levels from the training set used for the model development. The levels are the liquid level (water 4- oil) and the interface (water) level. This is the first model developed in this study, and a clear relation is found between the estimated and manually read levels. For validation of the model, a validation set was acquired. The estimated and manually read values of the validation set using equation 7 are shown in Figure 3, still showing a reasonably good relationship between the values. Manually read levels and the estimated soft sensor levels relationship and residual of the validation set are shown in Figure 4.

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Manually read levels and soft sensor levels using the training set 800 r

. 650h

L

\

i

i

:; ;

;: i

.: i

;i t

^

•;

^

z

"[5

.;

l

(1)

? ' ' * 9

lu(p)_ u(2*p) u(l)

' u{k*p + l) ~ u(k*p + 2)

u(2)

u{{k + r)*p)_ uik)

is an integer. Therefore, the lifted input sequence U has the same period

as the slowly sampled output. A process model with a slower sampling period,

pT^,

can be identified straightforwardly. In order to obtain the inter-sample value of the slow-sampled process measurements, {j^CA:)}, a fast model is commonly extracted according to the relationship between the slow and the fast system. Given a discrete linear system with a sampling period of 7^: x^+i=Ax,+Bu, (2) A new discrete linear system is obtained using lifted data:

Product Quality Estimation Using Multi-Rate Sampled Data

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y^=cxt where the parameter relations can be derived:

B = [A^

B

A^'B

AB

(4)

B]

C=C It is straightforward to extract matrices B and C from the lifted system. The system matrix A can be obtained using at least one of the following two approaches:(Wang et al, 2004) The first approach starts from the equation A = A^ where A is calculated j_

directly as A = A ^ . It is necessary to obtain a real-valued matrix. Therefore, A is j_

j_

approximated with the real part of A ^ . That is, A = Re( A ^ ) . The second approach derives A

B = [A^

B

AB

through a linear regression. Given

BJ,

B, = A ' - ^ B / = 1,2,...,;7

(5)

which is the i^^ block of B . The following relationship holds: B.î=AB. z = l,2,-..,;7-l

(6)

AB = AB^

(7)

and rewrite in a matrix form:

[AB B^

B^_,

-

BJ=A[B^

r

B^_,

B^_,

-

BJ

(8)

Y

Then,

A = (rY^)(YY^)"'

(9)

A third approach is proposed here in which the extraction of fast models is reformulated within an optimization framework. Assume the system matrix of the fast system A is expressed as: A = Re(A) + zIm(A) (1) the objective function is expressed as the deviation between A^ and A , plus the norm of the imaginary part of A :

argmin|A^-AII +||lm(A)|U where

L can be taken as 1, 2 or the oo norm.

(1)

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The objective function can also be defined as the deviation between the impulse response of the lifted and fast models:

?ir%mm^{h'{i)-h{pi)y

(12)

An unconstrained optimization problem is then formulated, and solved, e.g. with the "fininsearch'' function of MATLAB or any other unconstrained minimizer. fminsearch finds the minimum of an unconstrained nonlinear scalar function with simplex search. Since the quality measurements are sampled at a slower rate than other process measurements, the same value is inserted in the log system till a new lab analysis is obtained. Therefore, a weighting vector {w(A:)}: w(^) = 1^

,,

0

.

(13)

otherwise

is used to downweight the inter-sample value of the quality measurement as follows: M(l)

y{2) yip

"(2) uip - 1) u{p) u{p + \)

-1)

yip) yiP + 1)

f

(14)

uik)

yik) VL ' J

L

•

J y

A regression relationship is developed using the weighted matrices of regressors and the dependent variable, which is then applied to all available samples of regressors in order to obtain the inter-sample estimation. A soft sensor derived using the weighted partial least squares (WPLS) approach is compared with those developed by extracting the fast model from a lifted system. 3. Case studies The proposed approaches to design data driven soft sensors from combining fast process measurements with slowly sampled quality data are compared for a cement kiln simulation system. The product quality of a cement kiln is indicated by the amount of CaO (free lime) in clinker. In practice, the quality measurement is generally only available every 1 or 2 hours with a time delay of about 40 minutes. It is desirable to develop a soft sensor that can accurately predict the content of CaO in real time for effective quality control. Data collected fi:om a Cemulator® simulation system which was based on a first principles model (Delfter, 1982), are used in this study. Pseudo Random Binary Sequence (PRBS) signals are applied to the kiln speed, feed, tertiary air damper and IDFan power to excite the process. The free lime content of the clinker is available every hour, while other process measurements are logged every 10 minutes, which include kiln speed, kiln feed, position of the tertiary air damper, induced draft fan

Product Quality Estimation Using Multi-Rate Sampled Data

60

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Sample Figure 1. Free lime estimation obtained with quality measurements and lifted data (IDFan) power, fuels to calciner and kiln, plus several temperature measurements within the kiln system. A data block of 2000 samples is selected: 1800 samples to derive the model (300 quality measurements) and 150 samples for validation. A state space model is identified from lifted data using the identification toolbox of MATLAB. As shown in Figure 1 {start - quality measurement; solid line state space model estimation), the free lime soft sensor obtained with lifted data is able to predict the trend of the quality measurement reasonably well. In order to estimate a model for the inter-sample behaviour the three approaches described above are investigated. The fast model obtained from the first approach is unstable, due to the omission of imaginary part of A^^. The second approach yields a solution, however, the validation shows significant sensitivity as seen in Figure 2. The PRESS (one-step-ahead prediction residual sum of squared errors) between the extracted model and quality measurement is 31.29. Since the simplex search method is a local optimization approach, the final result depends on the initial guess. 10 runs are performed for approach 3 using both objective functions (eqs. 11 and eql2). The best solution of 10 runs performs similarly to the 2"^ approach. However, large variations are observed in the results of 10 tests, which indicate the high probability of arriving at a locally optimal solution. Therefore, a more reliable algorithm is desirable to obtain a consistent solution.

60

80

Sample Figure 2. Free lime estimation with fast model extracted by approach 2 A CaO soft sensor is developed using WPLS approach. As mentioned before, weighting vectors are multiplied onto inter-sample inputs and outputs of modelling data. During the model validation phase, this regression model is applied to all input samples. Therefore, the inter-sample behaviour of the soft sensor can be calculated.

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2

60

80

Sample Figure 3. Free lime estimation with WPLS approach during vahdation period As shown in Figure 3, the soft sensor follows the trend well, however, deviations are observed in a couple of periods. Comparing with the lifting technique based approaches (see Figure 2), WPLS approach shows improved performance for providing inter-sample estimation. The PRESS between the extracted model and quality measurement is reduced to 7.25. Therefore lifting with regularization should be developed to investigate which regularization is to be preferred. 4. Conclusions This paper investigates the approaches of developing soft sensors fi-om multi-rate sampled data sets. Since product quality is normally infrequently sampled and only available with a time delay, inter-sample information is most valuable. A two-step approach based on data-lifting techniques is investigated: the identification of a system with slow sampling rate followed by the extraction of a fast model. An optimization reformulation is provided to overcome the shortcomings associated with direct extraction methods. The proposed approach is applied to design a free lime soft sensor for product quality. Data collected from a simulator based on first principle models are used to compare the approaches based on data-lifting techniques and WPLS method. Preliminary results reveal that the soft sensor from the WPLS approach outperforms those of the investigated data-lifting techniques. One reason might be the large ratio between the frequency of standard measurements and that of the product quality. More importantly, the soft sensor developed with the WPLS approach is able to provide reasonable prediction for the free lime, demonstrating potential of using regularization combined with lifting to provide estimates for effective quality control and optimizing process operation. References Delfter, O. (1982). Simulation for Start-up and Operation of a Rotary Dryprocess Kiln,(in Danish), Thesis, Technical University of Denmark. Khargonekar, P.P., K. Poolla and A. Tannenbaum (1985). "Robust Control of Linear TimeInvariant Plants Using Periodic Compensation." IEEE Transactions on Automatic Control 30(11): 1088-1096. Lin, B., B. Recke, J. Knudsen and S.B. Jorgensen (2005). A Systematic Approach for Soft Sensor Development. European Symposium on Computer Aided Process Engineering -15, Barcelona. Lu, W. and D.G. Fisher (1989). "Least-Squares Output Estimation with Multirate Sampling." IEEE Trans. Automat. Control 34(6): 669-672. Srinivasaro, M., R.D. Gudi and S.C. Patwardhan (2005). Identification of Fast-Rate Nonlinear Output Error ModelsfromMulti-Rate Data. IF AC World Congress, Prague. Wang, J., T. Chen and B. Huang (2004). "Multirate Sampled-Data Systems: Computing Fast-Rate Models." Journal ofProcess Control 14(1): 79-88.


A Novel Solution Approach for Quality-based Retrimming Optimization liro Harjunkoski and Marco Fahl ABB Corporate Research, Wallstadter Str. 59, 68526 Ladenburg, Germany Abstract In this work, we present a novel mathematical programming formulation for solving the re-trimming problem. The proposed approach leads to optimal or close-to-optimal solutions with the effect of a significant reduction in quality loss, i.e. the economical loss based on degraded quality. The model introduced is able to consider quality profiles along the jumbo-reel as well as the requirements attached to each roll providing a geometric representation on the trim-loss problem. Keywords: Trim-loss problem, cutting stock, jumbo-reel quality, re-trimming. 1. Introduction The primary objective for cutting problems in paper production is trim loss minimization. A typical example is the cutting of a wide paper reel (jumbo reel) into smaller paper rolls, which are either end-customer rolls or intermediate products waiting for further processing, see Fig. 1. The general problem formulation tries to find a cutting strategy for producing the required roll widths using as little material as possible, i.e. by minimizing the trim loss. A secondary objective may also be to minimize the number of different patterns and to sequence them such that knife-setup actions are avoided.

Figure 1. Jumbo-reel cutting & the trim-loss problem The trim-loss problem as such is a discrete problem. Thus, the problem size explodes fast with the number of rolls considered due to the large number of alternative cutting strategies. Consequently, there exist lots of heuristic-mathematical approaches to solve the problem efficiently, without guaranteeing global optimality of a solution. These include rounding heuristics, column generation, solving problem only partially and other knapsack-type of algorithms. In modem paper mills, the trim-loss problem for the paper machine/winder is commonly solved as an integrated part of production planning already long before the jumbo-reels are actually being produced. Thus, uniform quality distribution of the paper to be produced is assumed. During the paper-making process, product quality data is collected from various on-line measurement devices. Comparing the actual quality distribution with the original trim patterns of a jumbo reel, the predetermined cutting plan may be far from optimal. For instance, the most valuable customer roll might have been assigned to the worst position with respect to product quality of the reel and must

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thus be rejected. Many providers of trim optimization packages/solutions therefore allow the user to perform a quality check on the trim solution and to manually adjust the cutting patterns to better meet the customer requirements and to react to quality deviations or defects. Today's modeling approaches of the trim optimization problem do not support qualitybased trim optimization, since the standard trim-loss (or cutting stock) problem does not take into account the exact position of each roll in a pattern, but merely focuses on total amounts. Only very few contributions into this direction are reported in the literature (e.g. Bergman et al., 2002). 2. A Novel mathematical programming approach Since adding the quality aspect to existing standard models would increase the complexity such that the problem becomes intractable (additional non-convexities, high amount of new discrete variables), an alternative modeling approach for quality-based trim optimization is needed. Here, we present a novel mathematical programming approach for automatic computation of an optimized solution of the re-trimming problem. The model covers quality profiles of a jumbo-reel through a geometric representation of the trim-loss problem. Driven by tight performance requirements, the mathematical model presented here considers only one cutting pattern at a time. A repositioning of the rolls on the jumbo reel is suggested taking into account the exact geometrical position as well as the quality information along the reel-width. The approach results in optimal or close-to-optimal solutions significantly reducing the quality loss, i.e. the economical loss based on degraded quality. In the following, we propose a two-step approach. First, a discretization-based method is used to generate a good approximation for the optimum solution, In a second step, a continuous exact approach is applied in order to ensure the feasibility of the final solution. This allows us to overcome non-convexities of the problem and to ensure that a close-to-optimal solution is obtained fast. From a conceptual point of view, a close relationship to general scheduling methodology and corresponding model formulations can be observed. Both approaches are first presented separately, before the combination of them is discussed. 2.1. The discrete model (discretization-based model) The main idea of the discretization-based approach is to divide the jumbo-reel width into fixed "slices". For each slice a respective cost coefficient for every roll is defined. The cost coefficient is based on quality classes, e.g. A-, B-, or C-quality (see Fig. 2). The final quality class for each roll is calculated by combining the quality mapping with key product-roll requirements.

AjA; A'A|A|A;AiiÂ|BiBiBiB|Ai A'AiA|/J

Figure 2. Discrete approach illustration The approach leads to solving an MILP and is similar to the one proposed by Ronnqvist (1995) for cutting wooden boards. However, to meet the requirements of the particular problem at hand, some aspects of the Ronnqvist formulation (e.g. considering different cut options) can be removed. It is worth noting that the simplified Ronnqvist problem is almost equivalent to the State-Task Network problem formulation for short-term

A Novel Solution Approach for Quality-Based Retrimming Optimization

1397

scheduling by Shah et al. (1993). There, the scheduling time-horizon is discretized by a fixed number of gridpoints, which are assigned through binary variables. In the retrim optimization context, the width takes the corresponding role of the time. Here, the binary variable, xdrp equals one at the jumbo-reel position, 7, in which the roll, r, starts. For the full problem formulation we also need a roll value coefficient for each potential roll position, Crp and the width of a roll, Wr, as well as a discretized roll width, Wdr. The problem formulation follows.

max^c^^. W^ 'Xd^j

r

(1)

j\{jp-Wd,+\

Y,xd,j=l

Vr

(3)

J

xd^je {0,1} The objective function (1) maximizes the quality yield, i.e. ensures that the total value of all rolls is as high as possible. Equation (2) ensures that no rolls should (virtually) overlap and Eq. (3) states that each roll must occur exactly once. The cost coefficients in (1) are determined by matching existing quality data with customer/product requirements, also taking into consideration the amount of material. The problem results in an optimal cutting plan with respect to a chosen grid density. For typical jumbo-reel widths of up to 8000 mm an exact grid (e.g. deltaw = 1 mm) would make the problem size intractable. Therefore, a coarser grid (delta_w >10-20 mm) must be selected and all widths need to be rounded to match the chosen discretization. In this case the roll widths have to be rounded down in order to maintain the feasibility of the problem (e.g. 578 mm becomes 570 mm when using a 10 mm grid). Due to the rounding error, a post-processing step is needed in order to ensure that the final solution is realistic and can be implemented on the winder. 2.2. The continuous model (slot-based model) In the continuous/slot-based approach, the jumbo-reel width is divided into quality sectors, s, beginning and ending at pre-defined points, ^S/ and 5'/. Each sector has a respective quality class coefficient for every roll (again: A, B, C), see Fig. 3. The quality value is calculated by combining the quality mapping with some key product roll parameters.

i

1540 mm I

2300 mm

Figure 3. Continuous approach illustration The model is based on the continuous time-slot scheduling approach by Pinto and Grossmann (1995). The slots are ordered from left to right and the borders between the

1398

/. Harjunkoski and M. Fahl slots, Wn^ and fF/, are continuous, i.e. able to adapt to the respective customer roll widths, Wr. Each roll is assigned to a slot and quality sector through binary variables, Xm and Xrs. The sector is selected with respect to the mid-point of the roll, r^'"'^, in order to simplify the quality mapping.

"^^^Ya^rsK'^rs r,s

(4)

Z^™=1 ^'^

(5)

n

r

Wf=K"+I,Xrr.-K

V«

(7)

r

Wf=W„l,

V«|«Wf+^-W^-M-{l-xJ

\fr,n

X^„=l

Vr

(9) (10) (11)

^m/J

<Sf + M-{\-xJ

\fr,s

(12)

^m/J

>5f-M-(l-x„)

Vr,^

(13)

W„\W„\rr''e^'

^.,^„€{0,1}

The objective function (4) maximizes the quality yield, as in (1) but here with respect to sectors and not with respect to single grid points. Constraints (5) and (6) ensure that each roll is assigned to exactly one designated slot. The distance between the start and end point of a slot depends on the assigned roll, as stated in Eq. (7). Constraint (8) hinders (virtual) overlaps of rolls and defines that the next slot must start exactly at the end point of the previous one (i.e. slots are adjacent). The mid-roll position is defined in Eqs. (9) and (10) and depends on the slot assignment, as well as, the slot starting point. In Eq. (11) each roll is assigned to one quality sector. Finally, constraints (12) and (13) make the decision to which quality sector the mid-point of a roll belongs to. This approach results in a global optimal cutting strategy taking into account the quality distribution along the jumbo-reel. However, the computation times may be very high -

A Novel Solution Approach for Quality-Based Retrimming Optimization

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with some test problems a first solution was found relatively fast (< 1 CPU-min) and finding the optimal solution did also not take more than a few minutes, however, proving the optimality took typically around 1-2 CPU-hours. As the complexity of this approach also grows with an increasing number of quality sectors, a combination of the discretization-based and the slot-based approaches is proposed. 2.3. Combined discrete-continuous approach The proposed two-step strategy makes use of both of the abovementioned approaches. The resulting steps are: 1. Solve the discretization-based problem with a grid of e.g. delta_w=10 or deltaw = 20 [mm] (inexact, but fast). 2. Fix the sequence of the rolls (variables, Xm defining the slot assignments) and solve the exact slot-based problem. Using this strategy, the slot-based problem can be solved within afi-actionof a second and the resulting solution is exact, feasible and close-to-optimal. The mathematical models presented above remain unchanged.

3. Example results In the following some examples are given in order to illustrate the method. The example roll widths are shown in Table 1. Table 1. Example problem data (roll widths) Roll

1

2

3

4

5

6

7

8

9

10

Width (mm)

1790

1100

825

485

770

750

650

580

580

385

The considered jumbo-reel trim width is 8000 mm. The sum of the roll widths to be trimmed is 7915 mm, resulting in a trim loss of 85 mm. Each roll is assumed to have exactly the same quality requirements. Therefore, the example can be simplified by directly dividing the jumbo-reel into various quality zones. If a roll spans over several quality zones, it is valued according to the worst quality. For illustration purposes, three different quality distributions are shown in Fig. 4 (case 1 on the top, followed by case 2 and case 3), where the quality is expressed in terms of A-, B-, and C-quality. For the optimization problems reported below, we calculate the value of each roll based on the following assumptions: Paper weight = 80 g W , price = 500 €/ton, A-quality (full quality) = 100% of price, B-quality (minor defects) = 70% of price and C-quality (rejected) = 0% of price, i.e. no value. 3760

1

1210

1700

A

B

900

1210

C ' B

2020

2220

B

A

A

1985

2345

1900

440

A

B

760

c

B

A

1185 38£

A

c

1160

820

A ' B

Figure 4. Example jumbo-reel quality mapping The example cases are solved using GAMS/CPLEX 9.1 and the results are shown in Table 2. The size of the discretization-based models is constant (1611 constraints, 14417 0-1 variables) and is therefore not shown in the table to save space. The

/. Harjunkoski and M. Fahl

1400

optimization time for all optimization runs was limited to 1000 CPU-s (optimality criteria 0,1 %) and the results obtained until then are reported. Table 2. Example results Discretization-based (deltaw = 5 mm) Case

CPU-s

Combined (deltaw = 20 mm)

Slot-based (continuous)

Obj

con

var(c/d)

CPU-s

Obj

1

130,2

1672,10

310

31/130

1000

1723,05

0,7

1702,88

2

1000

1426,95

348

31/149

1000

1429,43

259,9

1426,95

105,1

1364,45

406

31/178

1000

1364,45 22,8

CPU-s

Obj

1208,44

The specially selected examples show various aspects of the problem: The highly varying CPU-times and the fact that the combined approach is not always giving the best solution can be seen in case 3. In this case the 20 mm discretization causes a rounding error such that a quality sector change is ignored. This behaviour is avoided when using a 10 mm discretization. However, the combined strategy gives a good answer within a reasonable time. An example solution for case 2 is illustrated in Fig. 5.

Figure 5. Resulting trim set (quality: A=white, B=yellow, C=red) 4. Conclusion The two-step quality-based re-trimming solution is able to efficiently generate close-tooptimal solutions. The algorithm can be further tuned for performance/solution quality by e.g. changing the width discretization, CPU-limits and optimality criteria. The same methodology can also be applied to quality-driven sequencing of trim sets in machine direction. Furthermore, the combination of the optimization approaches for width and machine direction can result in an automatic quality-based re-trimming of an entire jumbo-reel.

References Bergman J., Flisberg P. and Ronnqvist M. (2002). Roll cutting at paper mills. Control Systems 2002, pp 159-163. Pinto, J.M. & Grossmann, I.E. (1995). A continuous time mixed integer linear programming model for short-term scheduling of multistage batch plants. Industrial and Engineering Chemistry Research, 34, 3037 - 3051. Ronnqvist M. (1995). A method for the cutting stock problem with different qualities. European Journal of Operational Research, 83, Issue 1, pp. 57-68. Shah, N.E., Pantelides, C.C. and Sargent, R. (1993). A General Algorithm for Short-Term Scheduling of Batch Operations - II. Computational Issues. Computers & Chemical Engineering, 17, pp. 229-244.


Chapter 1

An Integrated Framework Based on Data Driven Techniques for Process Supervision B. Bhushan,^ Jose A. Romagnoli^ ^ Department of Chemical Engineering, University of Sydney, Sydney, 2006, Australia ^Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA 70803, USA. Abstract An integrated framework for process monitoring and supervision is proposed. Firstly, the data is freed from outliers using mean minimum distance clustering technique. A novel technique for unsupervised pattern classification is proposed. It is applied for simultaneous fault detection and diagnosis. A continuous pilot plant is used to check the efficiency of the proposed strategy. The result shows that the proposed framework can be used for process supervision of real time, non-linear systems. Keywords: Fault detection, Fault diagnosis. Unsupervised pattern classification, Mean minimum distance (MMD) clustering 1. Introduction The advent of faster and more reliable computer systems has revolutionized the manner in which industrial processes are monitored and controlled. These advances have resulted in the generation of a large amount of process data. These process data may sometime contain outliers due to faulty sensors, equipment failure or other unmeasured disturbances. For reliable assessment of the process status, it is necessary to remove these outliers from the measurement data. Fault detection and diagnosis are two most important tasks of process supervision. Principal component analysis (PCA) is extensively used for feature extraction and fault detection (Jackson, 1991). On the other hand, all fault diagnosis methods can be broadly classified into two groups; model based methods (observers, SDG, parity space etc.) and process history based methods (PCA/PLS, neural networks, expert systems etc.). Model based methods is been used by many researchers for fault diagnosis (Vaidhyanathan et al., 1995; Vedam et al., 1997). The main disadvantage of these methods is that the complexity of the model increases with increase in size and complexity of the plant. Although research and development in each of these areas is active, little has been done to establish an effective framework for process supervision. Research in interfacing multivariate statistical tools with fault diagnosis has taken off in the last few years (Vedam et al, 1999; Norvilas et al. 2000), yet these integration attempts are qualitative in nature. Though, the main contribution of this work is the integrated framework; self organizing self clustering network (SOSCN) for unsupervised classification is also a novel contribution which is applied for simultaneous fault detection and diagnosis.

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B. Bhushan and J. A. Romagnoli

1402

The remaining part of the paper is organized as follows. In section 2, framework for process supervision is presented. Outlier detection based minimum distance clustering is explained in section 3. Section 4 contains SOSCN in detail followed by case study and conclusions in section respectively.

proposed on mean proposed 5 and 6

2. Integrated Framework In this integrated approach for process supervision, following steps are applied in succession: (1) Outlier detection using mean median distance (MMD) clustering, (2) Feature extraction using PCA and (3) Simultaneous fault detection and diagnosis using SOSCN. The overall framework is shown in fig. 1. Process Measurement

Data Preprocessing

^

M

Feature Extraction

) \

Fault Detection and Diagnosis

Fig. 1. Integratedframeworkfor process supervision In data preprocessing step, the outliers are detected and removed from the normal data. These refined data is projected from measurement space to the reduced feature space using PCA. Furthermore, the proposed SOSCN is used for simultaneous fault detection and diagnosis. These steps are discussed in detail in next sections. 3. Outlier Detection Outliers are defined as those measurements in which error does not follow the statistical distribution of the bulk of the data. Normally, outliers are a small fraction of the whole data and have little or no relation with the main structure. In other words, the outlier lies outside the pattern or cluster formed by the normal data. In this work, the non-iterative clustering based on mean minimum distance is used (Yin et al., 1994). Since the variation of individual variables may be different, each variable should be weighted by its own variance (Chen et al., 1998). Therefore, for a given window size M, '\fx^^\ x^^\ ..., x^^ are M measurements in N-dimensional space, the mean minimum distance VM is defined as 1

.i^f-^f?^

^

'M /=1

]*l

(1)

\k=\

where Vk is the k^^ diagonal element of the covariance matrix V. A moving window is used and if minimum distance of any measurement is more that ITM it is considered as an outlier.

4. Self Organizing Self Clustering Network (SOSCN) Let S = {x %..., xJQ) ^ 7 is a sample of Q, n-dimensional feature vectors from a population P such that x "^ = /x;'(I)\ ...,x„'(1)7. The process begins by randomly organizing the order of

An Integrated Framework Based on Data Driven Techniques for Process Supervision 1403 the feature vectors. A node /^^ is generated with the center c^^^ = {x/^\...,xf^} i.e. by assigning the first sample feature vector as the center of the first node and spread a^^^ = { (JP\ ..., aj^^} where oP^ , i = 1, ..-, n are predefined constants. The belongingness or the membership of a vector x^^ to a node /^^ is defined as

F,(x^>) = exp(-7/2)5;

(2)

(f'y

In this approach, the winner is decided based on the highest inclusion of the vector to the node and both center and the standard deviation of the node are learnt. The winner node k* for a sample vector x^^ is defined as ^*=arg max Fj^(x^^)

(3)

i r then/7,7 = 1 else p,y = 0, where jUk(/") is the membership value of m^^ node with respect to k node and r is the threshold defined by the user and determines the spread of the clusters. Once the matrix is created, column searching method is applied for finding out the classes. It is a recursive process that comes back to the remaining column after it runs out of rows for the current column. When no more columns remain for searching, all vectors are assigned to the classes. If a new data point is presented to the network, its membership value with each node is calculated. If the membership value is greater than a pre-specified value the data belong to the class of the node, else the data belongs to a new class. 5. Case Study The proposed strategy is applied to a section of an existing twin-CSTR pilot plant. This contains two CSTRs, a mixer, a feed tank and a number of heat exchangers. Each CSTR consists of a reaction vessel, a steam jacket, a cooling coil and a stirrer. Material from the feed tank is heated before being fed to the first reactor and the mixer. The effluent from the first reactor is then mixed with the material in the mixer before being fed to the second reactor. The effluent from the second reactor is, fed back to the feed tank and the cycle continues. Nine variables Fin (feed flow rate in). Tin (temperature of feed in), Tc,in (temperature of cooling water in), Ts,in(temperature of steam in), Lvl (level of the reactor), Fout (feed flow rate out). Tout (temperature of feed out), Tc,out (temperature of cooling

An Integrated Framework Based on Data Driven Techniques for Process Supervision 1405 water out), Ts,out (temperature of condensate) related to first CSTR is considered for this study.

il^^^w^ ^*'SvJU^vV;Â^W

50

100

t50

200

rTn Fig. 2: Outlier detection of all the nine variables using MMD algorithm (* - outliers). Plant was first run under normal conditions followed by six different types of abnormalities. 200 data points at 5 seconds interval were collected for all these condition. Outliers of magnitude 8-10 times the standard deviation of the data was randomly added to the actual measurements.

25 J 20 J 15

+

*

+ o

10

Trx-biaslO T-bias60

• * •

Cool-foul Cat-deact Normal Fin-dist10 T-bias-60

5J

oJ -5 J

Fig. 3. Classification of all the data points in 7 different classes using SOSCN.

1406

B. Bhushan and J. A. Romagnoli

Mean minimum distance clustering was applied to detect the outliers and the result is shown in figure 2. It can be noted that the algorithm could correctly detect all the outliers. Furthermore, PC A was applied to the preprocessed data for feature extraction. Three principal components could explain 62% of the variance in the data. Proposed SOSCN was applied for classification of the data without specifying the classes. The result is shown in figure 3. It can be seen that all the data were correctly classified in 8 classes. 25 data points of all these conditions were used for validating the capability of the strategy for fault detection and diagnosis. The membership value of these points with the nodes was calculated. If the membership value was greater than 0.75 the point was assigned to that class. All the points were assigned correctly to the appropriate class. 6. Conclusion An integrated framework for process monitoring and supervision is proposed in this work. Firstly, the outliers are removed using mean minimum distance clustering algorithm followed by PCA for dimensionality reduction. A novel technique is proposed in this work for unsupervised classification. The advantage of the proposed method over existing techniques is its ability to automatically determine the number of clusters. Since the methodology is based on learning, it is computationally less expensive and the result is not affected by the initial guesses. This technique is applied for simultaneously fault detection and diagnosis. The proposed fi-amework is applied for process supervision of a continuous pilot plant and the results show promising direction for real time process monitoring and supervision. References A. Norvilas, A. Negiz, J. DeCicco and A. Cinar (2000). Intelligent process monitoring by interfacing knowledge-based systems and multivariate statistical monitoring. Journal of Process Control, 10, 341-350. C. G. Looney (1997). Pattern recognition using neural networks: theory and algorithm for engineers and scientists. Oxford University Press, New York. H. Vedam, and V. Venkatasubramanian (1997). Signed diagraph based multiple fault diagnosis. Computer and Chemical Engineering, 21, S655-S660. H. Vedam and V. venkatasubramanian (1999). PCA-SDG based process monitoring and fault diagnosis. Control Engineering Practice, 7, 903-917. J. Chen, and J. A. Romagnoli (1998). A strategy for simultaneous dynamic data reconciliation and outlier detection. Computer and Chemical Engineering, 22, 559-562. J. E. Jackson (1991). A user's guide to principal components. Wiley, New York. P. Y. Yin and L. H. Chen (1994). A new non-iterative approach for clustering. Pattern Recognition Letters, 15, 125-133. R. Vaidhyanathan. and V. Venkatasubramanian (1995). Diagraph-based models for automated HAZOP analysis. Relaibility Engineering and Systems safety 50(1), 33-49. T. Nomura and T. Miyoshi (1995). An adaptive rule extraction with the fuzzy self organising map and a comparison with other methods. Proceedings of ISUMA-NAFIPS '95, 311-316.


Reliable multi-objective optimal control of batch processes based on stacked neural network models Ankur Mukherjee and Jie Zhang School of Chemical Engineering and Advanced Materials, University ofNewcastle, Newcastle upon TyneNEl 7RU, UK, E-mail:[email protected] Abstract This paper presents a stacked neural network based multi-objective optimal control method for batch processes. Stacked neural networks not only give better generalisation performance than single neural networks but also provide model prediction confidence bounds. In addition to the process operation objectives, the reliability of model prediction is incorporated in multi-objective optimisation in order to improve the reliability of the obtained optimal control policy. The standard error of the individual neural network predictions is taken as the indication of model prediction reliability. The proposed method is demonstrated on a simulated fed-batch process. Keywords: Batch processes, multi-objective optimisation, neural networks 1. Introduction Batch or semi-batch processes are suitable for the responsive manufacturing of high value added products [1]. In the operation of batch processes, it is desirable to meet a number of objectives, which are usually conflicting to each other. The relative importance of the individual objectives usually changes with market conditions. To maximise the profit from batch process manufacturing, multi-objective optimal control should be applied to batch processes. The performance of multi-objective optimal control depends on the accuracy of the process model. Developing detailed mechanistic models is usually very time consuming and may not be feasible for agile responsive manufacturing. Data based empirical models have to be utilised. Stacked neural networks have been shown to possess better generalisation capability than single neural networks [2, 3] and are used in this paper to model batch processes. An additional feature of stacked neural networks is that they can also provide prediction confidence bounds indicating the reliability of the corresponding model predictions. Due to model-plant mismatches, the "optimal" control policy calculated from a neural network model may not be optimal when applied to the actual process [4]. Thus it is important that the calculated optimal control policy should be reliable. This paper presents a reliable multi-objective optimal control method for batch processes. In addition to the process operation objectives, the reliability of model prediction is incorporated in multi-objective optimisation in order to improve the reliability of the obtained optimal control policy. The standard error of the individual neural network predictions is taken as the indication of model prediction reliability. The proposed method is demonstrated on a simulated fed-batch process. It is shown that by incorporating model prediction reliability in the optimisation criteria, reliable control policy is obtained.

1407

A. Mukherjee and J. Zhang

1408

2. A fed-batch process The fed-batch reactor is taken from [5]. The following reaction system

5 + 5-

-^D

is conducted in an isothermal semi-batch reactor. The objective in operating this reactor is, through addition of reactant B, to convert as much as possible of reactant A to the desired product, C, in a specified time tf= 120 min. It would not be optimal to add all B initially as the second order side-reaction yielding the undesired species D will be favoured at high concentration of B. To keep this undesired species low, the reactor is operated in semi-batch mode where B is added in a feed stream with concentration bfeed = 0.2. At the start of reaction, the reactor contains [^](0) = 0.2 moles/litter of ^, no B ([B](0) = 0) and is fed to 50% (F(0)=0.5). 3. Modelling of the fed-batch process using stacked neural networks Fig. 1 shows a stacked neural network where several networks are developed to model the same relationship and are combined together. Earlier studies show that an advantage of stacked neural networks is that they can not only give better generalisation performance than single neural networks, but also provide model prediction confidence measures [2]. Disturbances Goal _ J Multi-objective optimiser |

_L Batch Process

^^^--l y^-'

Fig. 1. A stacked neural network

Fig. 2. The control system structure

In this study, a fixed batch time of 120 minutes is considered as in [6]. Since it is usually difficult to measure the product quality variables frequently during a batch, it is a general practice to measure the product quality variables only at the end of a batch. The batch duration is divided into 10 equal intervals and within each interval the reactant feed rate is kept constant. The objective in operating this process is to maximise the amount of the final product Ccitf)V(tf) and simultaneously minimise the amount of undesired species CD{tf)V{tf). Neural network model for the prediction of concentration variables Cc(tf}V(tf) and CD{tf)V(tJ) at the final batch time are of the form: (1) y2=f2(U)

(2)

where yi = Cdtf} F(//), y2 = Coitj) V(tf}, (7 = [wi W2 • • • wio]^ is a vector of the reactant feed rates,/i and^ are nonlinear functions represented by neural networks. In this study, simulated process operational data from 50 batch runs were generated with the reactant feed rate randomly distributed in the range [0, 0.01]. Of the 50 batches of

Reliable Multi-Objective Optimal Control of Batch Processes

1409

data, 40 batches were used to develop neural network models and the remaining 10 batches were used as unseen testing data. Gaussian noise with zero mean and a variance of 2.5x10''^ was added to the reactant feed rate to simulate the effect of measurement noise. Two stacked neural networks each containing 20 neural networks were developed for predicting Cc{tj)V{tf) and CoitJ)V{tf). Each individual neural network has a single hidden layer with 10 hidden neurons. Hidden neurons use the sigmoid activation function whereas the output layer neuron uses the linear activation function. The LevenbergMarquardt training algorithm with "early stopping" was used in the study to train the networks. For training each network, bootstrap re-sampling with replacement [7] was used to generate a replication of the 40 batches of process data. Half of the replication was used as training data while the other half was used as the validation data. 4. Reliable multi-objective control of the fed-batch process The objective in operating the process is to maximise the amount of the final desired product Cc(tf)V(tf) and simultaneously minimise the amount of the final undesired species Co(tf)V(tJ). In order to obtain reliable control policyfi*omthe stacked neural network model, minimisations of the standard errors of individual network predictions are introduced as additional objectives in the optimisation problem. This may be formulated in terms of a multi-objective optimisation problem which is solved using the goal attainment method [8].

-c.it^Witf) F(U) =

(3)

e.cAtf) min u. subject to

(4)

F,{U)-W,

C are taking place. However, we have added a stochastic behavior to the kinetics parameters ki (internal parameter), so that there are time dependent parameters in the formulation and the resulting problem is a stochastic optimal control problem. The temperature of the reactor, T, is the state variable, the cooling water temperature, w, is the control variable and the objective is to maximize the concentration of B, Q , after an operation time tf. Eq. 13 provides the Ito process expression used for ki. The state variables are defined by

Stochastic Optimal Control in Batch Reactive

1423

Systems

Eq. 15. The problem includes state constraints (E[T(t)JMûmw)' The rest of the model equations are:

iêi •

i^^di +

^/ -

max^ J ^ E "

dC.=

,Uo/exp

(13) RT (14)

C^{ty)

- 2 ,^„, e x p l - ^ IC! dt-2

Aiexp|-^|ci-

dCr,

dT =

and control variable

(15)

,dw,

A2exp|

RT

(-^^) , , ^ ^ e x p f ^ V >C„^ C„

^tfJ ,ci*;.

\C C

\dt +

^Cl^dw] - ^^BtCaC^wf

A.^-p(^\C.C.. RT

UA _ ^ VC„

^^ VC„

dt

i^ ,C,,Q,^w;

4.1. Results Table 1 provides the data for all of the parameters included in the formulation. The initial conditions are T(0)=2{) C, CA(0)= Imol/lt, CB(0)=Q mol/lt and Cc^t/; =0.005mol/l; tf is 1 hr. Concentration of C is calculated through a mass balance equation. This is a problem with internal uncertainties, in which the optimal control profile cannot be explicitly obtained from the optimality conditions. So, the numerical implementation of stochastic dynamic programming was used. Figure 2 shows the results. In case (b) the state variable is constrained to be less or equal to 35 degrees, and in case (a) does not exist such a constraint. By applying Eq. 11, the control profile due to the state path constraints can be obtained.

INTEGRATED SOLUTION

APPROACH

MODEL FORMULATION STAGE

SOLUTION STAGE

Uncertainty

Optimal Control Problem

Non Singular

Static (time independent) Uncertainties

Time Dependent Uncertainties

Stochastic Optimal Control Problem

Ito Process

^ ^

Stochastic Maximum Principle Dynamic Programming Singular

Figure 1. An integrated solution approach to stochastic optimal control problems

V. Rico-Ramirez et al.

1424 Table l.Data Parameter

ai

0C2

Value

0.95

0.9

Parameter Value

0.05

Hi

^mm

^max

10 C

100 C

35 C

Cp

0.10

H2

500

19

1/molh

1/molh

V

U

A

EAI

EA2

12500

5000

-30

-40

900

4.2

1000

10^

5

J/mol

J/mol

KJ/mol

KJ/mol

g/lt

J/gK

It

J/hm^K

m'

u("C)

u ("C) to

Tij^^.vi'v'HWVW^^ Tlempo (hrs)

U

\jÂVMHWVW^/^

Tlempo (hrs)

Figure 2. Optimal control profiles, (a) No path constraints (b) With path constraints (T 0 via the transformation: Xo -^ (PpM = [^Ni,pNz,PN3,liN4,PHR,TR,V/P]. As a result, we can estimate the plant state by the G-invariant observer, and see more details in [2]. After making a conventional design of two PI controllers - one from the error TRset ~TR{t) to theflowrate of the cooling water /Vw(0 ^^^ the other from the error Vset — V(^) to the outlet flow rate Fout{t) - the controller states are respectively denoted as Xpn G 9t and Xpi2 G 5t. Next, we obtain the formulation of a constrained nonlinear control system (1). The state vector is defined as X = [Ni,N2,N^,N4,HR,AR,Xpi\,Xpi2] G St^, the set-point reference as w = [Vset^ TRset] ^ ^^» the output as zi = [V,TR] G 9t^, and the constrained variable as zo — \y'>TR^Fcw',T'out\ ^ ^'^• In this simulation, we set an initial state JCQ = [2.5 x 10^,2.5 x 10^, 1.2 x 10^, 1.2 x 10^, 1.0 x 10^,2.284 X 10~^5.0 x 10,2.0 x 10"^]. These controllers can stabilize the equilibrium state jc^ - [6.342X 102,6.342x102,5.266xl0^5.266xl0^,5.234xlO^3.255 xlO-^5.821,9.993 X 10] corresponding to the set-point reference We = \yset->TRset\' However, when a set-point with rapid changes is input to a constrained system (1) without any reference management by a reference governor, the constraints will be violated in the time-responses from the initial state JCQ to the goal state Xe. A simulation result of such a time-response is shown in Figure 3. Figure 3 respectively shows for the given set-points We = [Vset, TRset] the time-responses of liquid volume V and the reaction temperature TR under the prescribed constraints. From these figures, we can see that the constraints are violated, where in this simulation the constraints are considered soft constraints. In the case of the saturation of time-responses of V and TR by the upper bound, we numerically confirm that the CSTR control system is destabilized due to saturation. On the other hand, the set-point reference Vset and TRset are managed by the reference governor into the actual input signals Vmset and TRmset, which are both thin solid lines in Figure 4 (a)(b). As a result, from Figure 4 we can see that both constraints about V and TR are fulfilled, where the no time-responses of Few and Fout are introduced, but where we confirmed the constraint fulfillment. From the simulation results, we can see that the governor only manages the given set-point and that constraint violations do not occur. Therefore, it is obvious that our proposed control technique with a reference governor is more effective for control systems in which constraint

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K. Kogiso et ah

h^ aj 398 Q.

Vset

I

'

E *1 3961

V upper bound on V 1000 time

upper bound on TJj

1500 [s]

1000 time

15.00 [s]

2000

(a) Set-point Vêt of liquid volume V(t) and (b) Set-point TRêt of reaction temperature tank capacity. TR (t) and temperature limitation. Figure 3. Simulation results of CSTR time-responses without a reference governor. 4021

^

i

2^ 400[

I r S.398 T/jmsef(managed Taset)

/

(1)

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G. Lee et ah

Where r/ is the residual of output variable /, and yt and y. are the measured and estimated values of variable /, respectively. A qualitative state, which corresponds to ranges of possible values for the residual, becomes an attribute of the residual. We will consider methods that use three ranges: low, to which the qualitative state (-) is assigned; normal, assigned (0); and high, assigned (+). If a fault occurs, the qualitative state for the residual may be (+) or (-). The abnormal qualitative state for the residual becomes a symptom, which is expressed as the pair of the target variable and the qualitative state of the residual. Those faults inducing the abnormality of each residual are classified along with their symptoms, and the classified faults are stored in a set (called a fault set). Also, faults can be classified into two types: one is the faults added to the target variable and the other is the sensor faults that occur in the sensor corresponding to the source variables in the DPLS model. The first step of on-line fault diagnosis is the monitoring of the residuals, in order to detect their qualitative change of state. The detected residual of a variable becomes an element in the set. The fault of sensor degradation can make the signs of the symptoms fluctuate between (+) and (-), which greatly decreases the diagnosis accuracy. In order to make a stable diagnosis, CUSUM monitors the squared residuals as well as the residuals of each variable. The next step of fault diagnosis is to obtain the minimum set of faults that can explain all of the detected symptoms. 3.3. Incorporation of Time Delay into the DPLS Model Time delay can be defined as the time interval between the start of an event in one point and its resulting action at another point. It is also referred transportation lag, time delay, dead time or distance-velocity lag. In the target process, large time delay can be found in the equipments such as the digester, the storage tank, and the D2 tower. As the model assumes that the change of the source variable effects instantly the target variable, these large time delay may make the estimation model inaccurate. In order to increase the accuracy of the estimation, the information on the time delayfi*omthe input variables to the output variables should be incorporated into the dynamic PLS model. The input matrix, X of the DPLS model for the variable / is modified as follows. x..(/t

+ T

- T^^^ •

0,y }

(2)

x=

îN:

Where, T

V "*" ^MD,Ni

MD,i

^^MAX(t

^MD,i

-lAt^\

^Ti

is the measurement delay of /, r ^ is the time delayfi-om/ to /, and t ^ . •'

TD,ij

•'

J

7

" MD,i

, r Twr. )• The measured value of variable / to calculate the residual / is

\ MD,j=\,Ni'

MD,i I

yk'^'^MDi~'^1!î)' ^^^ study used the data obtained with the set-point change in order to determine the dead time.

Fault Detection and Diagnosis of Pulp Mill Process

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4. Result and Discussion Consider the example of Dl Stuck (sticking of Dl CIO2 flow control valve). While the +10% setpoint change of wood chips flow at 600 minutes, the Dl CIO2 flow control valve stuck also at the same time. Using the DPLS model, the detection sequence of symptoms is KN22(+) and KN22^ from 915 minutes, T21(+) from 920 minutes, T21^ from 935 minutes, BR26^ from 1185 minutes, and BR26(-) from 1190 minutes (Figure 1 (a)). The bounds of Figure 1 are the minimal jump size of CUSUM. There is no false detection, and DlCm5Bias, DlCStuck, WLlBias, WLStuck, and WoodDens obtained as the solution from 920 minutes. Although the resolution is 5, operator may easily judge that WLlBias, WLStuck, and WoodDens are not fault candidates because there are no detected variables in the digester and oxygen reactor section. Also, the resolution can be reduced by using the dynamics of the residuals. Figure 1 (b) shows the residual obtained by the previous hybrid model without time delay. KN22(+) is detected from 995 minutes, KN22^ and T21(+) from 1005 minutes, T21^ from 1035 minutes, BR26^ from 1055 minutes, and BR26(-) from 1105 minutes. The detection is 80 minutes later than the proposed method with time delay. Figure 1 shows evidently that the models with time delay generate clearer and more accurate residuals denoting fault occurrence than the ones without time delay. 0.05 0.04 H o

g 0.02

0.03

•2 0.01

1 0.02

'So

f2

'% 0.01 0 -0.01

, 500

700

0 -0.01

900 1100 Time (min)

500

1300 1500

700

900

1100

1300

1500

1300

1500

Time (min)

(a)

700

900

1100

1300

900

1500

1100

Time (min)

Time (min)

(b)

Fig. 1 (a) residuals obtained by the DPLS models with time delay, (b) residuals obtained by the DPLS models without time delay for DlCStuck.

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G. Lee et al Table 1. Diagnosis result

fault BR0025Drift BRlODeg BR20Bias BRm5Bias CFmSBias DlCStuck DlCmSBias EbackTemp ECausComp ECausTemp KN0025Drift KNlODeg KN20Bias KNmSBias

detection delays 215/40/560 20/20 20/65/65 315/395 20/20 15/15 65/65 15/15 115/105 55/45 35/25 35/35

accuracy

fault

lA 1/1 1/1 1/0 1/1 1/1 0.99/0.94 0.22/0.1 1/1 1/1 1/1 1/1 1/0.97 1/1

OH0025Drift OH20Bias 0Hm5Bias PR0025Drift PR20Bias PRm5Bias WClBias WCStuck WCm5Bias WLlBias WLStuck WLm5Bias WoodDens WoodTemp

detection delays 65/85 25/25 25/25 145/175 35/35 45/45 35/35 35/35 15/35 25/25 10/10 10/20 110/60 115/70

accuracy

iTl 0.99/1 1/1 1/1 1/1 1/1 1/1 1/1 0.02/0.0 1/0.94 1/1 0.85/0.28 0.26/0.17 1/1

To compare the diagnostic performance, accuracy and detection delay are used. The accuracy is 1 if the diagnosis is accuate; that is, the true fault is included in the final fault candidates set. Otherwise, the accuracy is 0. The detection delay refers to the time from fault occurrence to fault diagnosis. In Table 1, the former performance parameter value is the diagnosis result obtained by the proposed method, and the latter is the one by the previous hybrid method not considering time delay. The 1% faults of BRlDeg, CFlBias, DlClBias, KNlDeg, and PRlDeg are too small to be diagnosed by two methods, and are not shown in Table 1. In addition, the method failed to diagnose CFStuck and PRlODeg. Though T15 is independent with WCm5Bias and WoodDens, the proposed method detected wrongly T15 for these faults. Because WLStuck can explain all detected symptoms, the accuracy was very low. The diagnostic performance by the new method for all cases except WoodDens and WoodTemp are much better than the previous one. The faster detections of two cases are due to fast and wrong detection of T12. In WoodTemp case, wood temperature increases and the symptom of T12(+) should be detected. However, the previous method detected T12(-), and the detection was earlier than the new method.

Acknowledgement This work was supported by grant No. (R05-2002-000-00057-0) from the Basic Research Program of the Korea Science & Engineering Foundation.

References [1] V. Venkatasubramanian, R. Rengaswamy, K. Yin and S.N. Kavuri, Comp. Chem. Engng., 27, (2003) 293 [2] JJ. Downs and E.F. Vogel, 1993, Comp. and Chem. Engng., 17, (1993) 245 [3] J.J. Castro and F.J. Doyle III, Journal of Process Control, 14, (2004) 17 [4] G. Lee, S.-O. Song, and E.S. Yoon, 2003, Ind. Engng. Chem. Res., 42, (2003) 6145 [5] http://www.chemengr.ucsb.edu/~ceweb/faculty/doyle/docs/benchmarks/mill


Improving Observability of Large-Scale Systems by Iterative Weighting Adjustment Richard Faber^, Harvey Arellano-Garcia^, Pu Li^, Gunter Wozny ^ ^Berlin University of Technology, Department of Process Dynamics and Operation Sekr. KWT-9, Strasse des 17. Juni 135, 10623 Berlin, Germany ^Technische Universitdt Ilmenau, Simulation and Optimal Processes Department, P. O. Box 10 05 65, 98684 Ilmenau, Germany Abstract An optimization-based approach is proposed to improve the observabihty of large-scale systems with iterative adjustment of the weighting matrix. The approach is based on a rigorous process model making it applicable to nonlinear systems. The result of the state estimation is improved by introducing sensitivity information into the weighting of the objective function. The needed sensitivity information is iteratively computed and adjusted during the optimization run. The approach has been applied to a large-scale nonlinear industrial process to estimate the unknown feed composition from scarce measurement data. Keywords: model-based optimization, state estimation, weighting adjustment 1. Introduction Many advanced approaches have been developed for on-line optimization and control of industrial processes. The realization of these approaches requires the information about the current state of the processes. It is usually assumed that the state can be gained through measuring essential process variables. In many cases, however, it is not possible to measure all required variables, especially in on-line applications where a frequent update of measurements is necessary. Therefore, it is necessary to use techniques which are able to estimate as much unmeasured variables as possible from the set of available measurements. One possibility to determine the unmeasured variables is to use the concept of observability classification to identify the observable variables from a given set of measurements using balance equations. A broad variety of methods have been proposed to address this problem. Basically two categories can be distinguished in equation oriented observability analysis: structural and non-structural techniques. The nonstructural techniques are based on calculations made using model coefficients. In general, the nonstructural equation-oriented techniques can be applied to linear [1] and bilinear relationships [2]. On the other hand, structural techniques use the process occurrence matrix to classify the variables into observable and unobservable [3,4]. The occurrence matrix is rearranged and observable subsets are generated from the whole variable space. Therefore, these methods are very useful if complex industrial plants with a large number of units are analyzed. A broad variety of state observers such as the extended Kalman filter (EKF) have been developed to estimate unmeasured process states continuously from frequent measurements [e.g. 5]. A weakness of the standard EKF is that it performs poorly when applied to highly nonlinear processes. Another approach uses the techniques of parameter estimation where nonlinear optimization techniques are used to minimize the difference between the experimentally

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measured output and the predictive output of the rigorous mathematical model. In this case the result of the optimization strongly depends on the definition of the objective fimction and the nonlinearity of the process model. In this work, we propose an optimization-based approach where the weighting matrix in the objective function is iteratively adjusted to improve the result for state estimation of nonlinear large-scale systems. The basic idea is to utilize the available measured variables to infer unmeasured variables based on a detailed nonlinear process model. The residual of the measured data to the computed values according to the model is to be minimized. Due to the nonlinearity of the model the sensitivity of the measured variables to the unmeasured variables strongly influences the estimation quality and a poor scaling may even lead to a wrong result. Therefore, the weighting matrix in the objective ftmction plays a key role and has to be carefully chosen. We include the information of the sensitivity of the measured variables to the unmeasured variables in the estimation procedure. The approach has been applied to small-scale example problems as well as to an industrial gas purification process described by a rigorous non-equilibrium model. 2. Optimization-based state estimation The problem of estimating the unmeasured variables firom a set of available measurements can be stated as an optimization problem where the unknown variables are the optimization variables. An objective function consisting of the difference between the experimentally measured output and the predictive output of the rigorous mathematical model is to be minimized. For the case of unknown input variables the optimization problem takes following form: min / = Ay'W;^ Ay + Aw'W;Aw u

^

s.t. g(x,u,0) = O h(x,u,0)>O

where u e U c S t ' " are the (independent) input variables of the process model, X c X G 9t' are the (dependent) output variables, y c xe 9t'~'" is the measured subset of the output variables and w c u e 9t'"^ is the measured subset of the input variables. Ay is the residual of the measured data to the computed values according to the model for the output variables and Aw is the residual of the measured data to the estimated values of the input variables. W^ and W^ are weighting matrices, g and h are the equality and inequality constraints, representing the model equations and restrictions and 0 are model parameters which in this case are assumed to be known. If the optimization problem is analyzed it is obvious that if the number of measured variables is smaller than the degree offi*eedomof the optimization problem (number of inputs of the process model) the solution is not unique and it strongly depends on the initial values given. On the other hand, for highly nonlinear processes even if there are theoretically enough measurements available the result of the optimization might be still very poor because the sensitivity of the measured variables to the estimated variables is

Improving Observability of Large-Scale Systems by Iterative Weighting Adjustment

1469

very small. In this case the result strongly depends on the nonlinearity of the process model and the scaling of the objective function introduced by the weighting matrices. The standard procedure for weighting the individual terms of the objective function is to use the variance-covariance matrix of the measurement errors to compensate for different scaling of the variables and different accuracies of the measurements. If the measurements are assumed to be independent from each other, the diagonal elements of the weighting matrix consist of the standard deviation of the individual measurements: ^/ag(W) = ^/ag(WJ = G

(2)

with a being the vector of standard deviations of the measurements. In this formulation the sensitivity information that links the measured and the unmeasured variables is only available in the process model g. The objective function only depends on the residuals of the measured variables and the (mostly constant) weighting of the individual variables. It is, however, also possible to include information about the measurement variances, the variances of the measured variables and the sensitivity of these variables with respect to the unknown variables. 2.1. Using the variance-covariance matrix for model predictions In several research works for model discrimination and experimental design an alternative weighting matrix formulation is used [e.g. 6]: W = (W,+W,J

(3)

where W^^ is the variance-covariance matrix for model predictions: W,, =S-V„-'-S^

(4)

V^ is the variance-covariance matrix for the optimization variables and S being the sensitivity matrix of the measured variables with respect to the unknown variables:

s = | ^ z-[y.wr

(5)

an By using the information of W^^, the sensitivity information of the measured variables to the unknown variables is additionally introduced into the objective function. In previous applications V^ has to be approximated using historical experimental data or has to be calculated at the solution of a previous optimization [7]. In the presented approach the variance-covariance matrix V„ has been iteratively calculated using first order approximations during runtime following eq. 6:

dn da' W. dz dz

(6) f)ll

The needed sensitivities — are calculated at each iteration of the optimization using first order approximations of the Hessian matrix of the objective fimction [8]. This approximation is strictly speaking only valid at convergence u = u*, but with the

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R. Faber et al

iterative recalculation during runtime the approximation of V„ converges to the actual solution. The model-based optimization with iterative weighting adjustment has been used to identify unmeasured quantities from only few measurements for the separation process presented in the next section. 3. The process model The estimation procedure described in the previous section has been applied to the ammonia hydrogen sulfide circulation scrubbing process which is a common absorption/desorption process for coke-oven-gas purification. This process selectively removes NH3, HCN and H2S in the effluent gases from the coking oven while simultaneously suppressing the absorption of C02. A rigorous rate-based tray-by-tray model for the mass and heat transfer is used to describe the process. The model is composed of a large-scale highly nonlinear equation system with up to 636 variables per ab-/desorption unit. The enhanced mass transfer due to the chemical reactions in the liquid phase is accounted for by the use of enhancement factors. The mass transfer is described by a mass transfer coefficient and the interfacial area. The correlations of the mass transfer coefficients and the interfacial area are taken from Billet and Schulte's correlations [9]. A detailed description of the process model can be found in [10]. 3.1. Estimation setup To test the performance of the proposed estimation approach the absorption part of the described coke-oven-gas purification process has been chosen. The setup was chosen according to the specifications of the deacidifier unit in the real scrubbing plant where the enriched water from the H2S washing unit is freed from sour components. The absorption plant with the in- and outgoing streams is shown in Fig. 1.

V^.x^.D^J,

Fig. 1: Absorption column setup The enriched water enters at the top of the column containing 6 components (H2O, CO2, NH3, H2S, HCN and traces of coke oven gas-COG). The same components can be found in the vapor stream entering at the bottom of the column. The aim of this investigation was to determine the unknown variables for the mass streams (Vj, V2) and component concentrations (Xj,X2) of the ingoing streams. It was assumed that the temperatures of all streams can be measured (T1-T4) and that 5 additional temperature measurements inside the column are available (T5-T9). In addition mass stream of 3 streams ( X - V3) and the densities of the liquid streams (DijDs) are assumed to be known. This means

Improving Observability of Large-Scale Systems by Iterative Weighting Adjustment

1471

that 14 measurement can be used to estimate 14 variables (Note that although Vj, V2 can be measured they are also included in the set of variables to be estimated. This is reasonable to account for measurement errors in these variables). Although enough measurements are available to reduce the degree of freedom of the problem to 0 the result is still very poor because the temperature measurements hold very little information about the composition of the feed streams especially in multicomponent absorption case. 3.2. Decision about additional measurements To decide which measurements are to be additionally included, the Fischer-InformationMatrix is used:

F = s'-w;^-s

(7)

The Fischer-Information-Matrix is often used in experimental design to gain more information about the estimated parameters. In this case measurements have been chosen which maximally increase the determinant of the information matrix. This is equivalent to making the elements of the variance-covariance matrix V^ small. In this investigation additional heuristical information about possible measurement in the real plant have been used to decide on which variables should be additionally measured. Combining both information, the NH3 and CO2 concentration of the outgoing vapor stream (4) and the H2O saturation of the ingoing vapor stream (2) have been chosen. The measurements have been used to determine the unknown quantities using the described optimization procedure. 4. Results To investigate the effect of introducing sensitivity information into the objective ftinction by iteratively adjusting the weighting matrix using Wj,^, the process model setup described in section 3 is used and the input variables given in Table 1 and Table 2 are to be estimated. The values in Table 1 and Table 2 are given in mole fractions and kmol/h. Table 1. Vapor feed conditions H20

NH3

CO2

H2S

HCN

COG

V

2.43E-2

7.93E-3

1.72E-2

3.28E-3

1.14E-3

0.946

3.66

vap

Table 2. Liquid feed conditions H2O

NH3

CO2

H2S

HCN

COG

Zp — {1 — A)po ajx < 6^ -h (1 - X)pi, ^ == 1,2,..., m X > 0, A > 0, A < 1 3. Initial problem The considered process can be found in an oil refinery, where many units produce different refined products in a continuous way. These products should be stored on intermediary tanks until they are ready to be sent to final customers. An overview of a storage and distribution area is illustrated in Fig. 1. In this scenario, a transfer for receiving a continuous production occurs among units and tanks or among tanks, for releasing space or sending products to a customer. Initially,

Scheduling of Storage and Transfer Tasks in Oil Refineries

1485

a mathematical programming model in discrete time is established, considering tanks as resources and a product demand as an input parameter [10]. The output generated yields a scheduling of operations. Such scheduling considers the refinery mass balance and it defines the starting pumping associated with a target tank (see Fig. 2). In this decision level, the necessary pipes for a transfer are not considered. This assumption is made to simplify the model, since the association of a pipe for each pumping operation may increase the number of binary variables. Starting from this scheduling, we propose an approach to deal with pipe allocation through a job-shop formulation [11] with additional constraints. Hence, tasks can be eventually reordered to adequate available pipes. Our approach also considers time delays in tasks, yielding a model with uncertainties in parameters. This results in a very different scheduling from the initial one. Time Horizon Pi storage

1

m-9

• : • : • : • , • : : • : • : • ; . : : : : :

1

J

^

1

J

,,,

'^fZiii^yyyyyy^Mii^ WizÂ^yyAiyyyyyyyyyyAyAyAyyyyyyyy

1

p Local Market

Production Line ~ Pipeline Receivership

ta

yyyyyy)

TQ5

Pipeline Disoatch TQ8

Pices

[—]

Taskl

Task 3

I Task 2

Task 4

P2 Storage I

Figure 1. A storage and distribution area.

Figure 2. Initial scheduhng.

Fig. 2 shows an initial scheduling obtained by another optimization program [10] that only considers mass balance among tanks and demand requirements. In this case, no pipe allocation is considered. For example. Task 2, which is for sending product P I to a customer, is composed by five operations ( O P l to 0 P 5 ) . Each of these operations is related to each transfer of P I from tanks T Q l , TQ7, TQ6, TQ2, and T Q 7 in order to provide the whole demand. This is the initial scheduling to be executed. Note that it yields five simultaneous operations, which obviously need five pipes. 4. O p t i m i z a t i o n m o d e l As established in sections 2 and 3, t h e general model of hnear fuzzy programming, is reduced to a model of mixed integer linear programming (MILP) in continuous time. The decisions are modeled by binary and continuous variables, linked by linear restrictions. The objective is to minimize the total execution time or makespan with operations starting as soon as possible. The restrictions basically define the order among operations and the allocation of resources, tanks, and pipes in the considered scenario [12]. The mathematical model is henceforth explained:

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L. C. Felizari and R. Luders

The cost function (expression 4) is defined by the minimization of makespan and the sum of starting times (TSoj) for all operations. fuzzy min

{Mak + X l X l ^ ' ^ ^ ' ^ ^ OEO

(^)

jeJ

The optimization model is also subjected to constraints, stated in Eq. 5 to Eq. 16, as follows: The finishing time (TFoj) of any operation must be lower than the makespan: Mak - TFoj >0

yoeO.jeJ

(5)

The order among operations, for the same task, must be assured: TSoj > TFo-i,j

V o > 1 G O, j G J

(6)

The duration time of the operations {TPoj) is subjected to uncertainties. The fuzzy restriction allows us to consider possible delays in the processing time of the operations: TFoj - TSoj = TPoj

yoeO.jeJ

(7)

The order among operations of different tasks, for the same tank, must be respected: TFp,, - TSoj > - ( 1 - Do,j,p,k)' M

\/o,p e O, (j ^k)eJ,

Qoj = Qp,k

(8)

TFoj - TSp^k > -Doj,p,k 'M

ô.peO,

{j ^k)eJ,

Qoj = Qp,k

(9)

TFp^k - TSo,j < -Do,j,p,k 'M

"io.peO,

{j ^k)eJ,

Qoj = Qp,k (10)

TFoj - TSp,k < (1 - Doj,p,k) • M

"ô.peO,

{j ^k)eJ,

Doj,p,k + Dp^k,oj = 1

yo,peO,

{j ^k)e

Qo,j - Qp,k (11) J, Qoj = Qp,k (12)

The allocation order of the pipes of the same type among the operations, for different tasks, must be considered: TFoj - TSoj

< (1 - Doj,p,k) X M + TD,,d,p,fc +

(1 - K,d,p,fc) X M + (1 - K,d,o,j) X M -

Wo.p e O, (j ^k)eJ,ne

TDn,d,oJ

NDd, de

D

TFp^k - TSp,k < Doj,p,k X M + TDr,,d,oj + (1 - VnAoj) X M + (1 - Vn,d,p,k) X M - TDn,d,p,k

\fo,p e O, (j ^k)eJ,ne

(13)

NDa, de

(14)

D

For the occurrence of a transference operation, a tank and a pipe must be allocated at the same instant of time: TDn4,oj = TSoj

\/oeO,jeJ,ne

NDd, deD

(15)

At any given time, it must be assured that only one pipe is allocated for each operation:

Y,

VnAoj = i

yoeO,jeJ,deD

(16)

neNDd

As follows, the main results obtained for the case study with four pipes and flexibihzation in the operating time of operation O P l of the task T2 are presented.

1487

Scheduling of Storage and Transfer Tasks in Oil Refineries 5. R e s u l t s

Returning to the initial scheduling presented in Fig. 2, we now consider a delay in the dispatch process of product P I to a customer. It will be considered that operation O P l of task T2 will be able to be delayed up to 3 time units {TAi^2 — 3), representing uncertain processing time during a product pumping operation. The time delay can represent the difficulty of the involved maneuvers, established by the operator's skills. Using the proposal of [9], and the Eq. 3, we get the following model of linear programming: maximize subject to:

Mak + ^

^

A

(17)

TSoj < 75 + (1 - A) x (84 - 75)

o£0 jeJ

TFi,2 - T5'i,2 > 5 - (1 - A) X (5 - 2) A > 0,

A< 1

The complete model is composed by the Eq. 17, in addition to Eq. 5 to Eq. 16, with exception of the Eq. 7 that is replaced by the time restriction already in the Eq. 17. Note that the initial solution of Fig. 2 requires five pipes to be implemented. Assuming that only four pipes are available. Fig. 3 shows the obtained scheduling. The variation in the processing time was considered to the fuzzy optimization proposal, resulting in A = 0, 5833. In this case, the start task T2 was postponed and the solution considered an intermediate duration for O P l of T2. Moreover, if the duration of O P l changes from 2 up to 4 time units, the programming will remain feasible and it will have little impact on the total scheduling time. The allocation of pipes during the scheduling horizon is shown in Fig. 4. Time Horizon

Time Horizon —^

1

^J

^—r 1 "'l' '

V r' •

Y^/yy//;/xxyy//)yx/>

1 1 1 1 1 1 1 ^-^

'\

y/y//)''////Aoyyyyy /////////////////////////>

'^^2^^^m22m^^^m^^^^^ E zz^

^^1

iillii Task 1

Tasks

l:;::x::;:::;l Task2

Task 4

E Eia

D4 1 1

I Task 5

T ' T" 'f'''x"'l ' Y''^Y^Y"*\ "i

liilM^^^ Task 1

Tasks

l;^;^:^;^;^:;^;l Task2

Task 4

Figure 3. Gantt graph for the final scheduling.

Figure 4. Pipes allocation.

The solver Lingo/PC release 8.0, installed in a computer Pentium III lGHz/256MB RAM was used for the solution of the optimization. In a general way, the models used about 37 continuous variables, 328 binary variables and 1,700 restrictions, consuming an average processing time of some minutes.

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L.C. Felizari and R. Luders

6. Conclusions This paper considers the use of fuzzy optimization in scheduling of storage and transfer tasks characterized by delays in their processing times. Another considered aspect is the allocation of shared resources. In particular, the allocation of pipes that allow parallelism of tasks. Starting from an initial scheduling, obtained by not considering pipe allocation, the introduction of operational delays yields a significant increase on the makespan. However, as the maximum delay may not occur, an intermediate solution would be more suitable. Therefore, the solution considering processing times as fuzzy quantities make it possible to obtain a scheduling that admits some flexibility, since an intermediate case is found. REFERENCES 1. J. M. Pinto, M. Joly and L. F. L. Moro (2000). Planning and scheduling models for refinery operations. Computer and Chemical Engineering (Oct), Vol. 24, pp. 22592276. 2. J. Balasubramanian and I. E. Grossmann (2002). A Novel Branch and Bound Algorithm for ScheduHng Flowshop Plants with Uncertain Processing Times. Computers & Chemical Engineering (Jan), Vol. 26, No. 1, pp. 41-57. 3. J. Balasubramanian and I. E. Grossmann (2003). Scheduling Optimization Under Uncertainty - An Alternative Approach. Computers & Chemical Engineering (Apr), Vol. 27, No. 4, pp. 469-490. 4. X. Lin, S. L. Janak; C. A. Floudas (2004). A new robust optimization approach for scheduling under uncertainty: I. Bounded uncertainty. Computers & Chemical Engineering (Jun), Vol. 28, No. 6-7, pp. 1069-1085. 5. L. A. Zadeh (1965). Fuzzy sets. Information and Control (Jun), Vol. 8, No. 3, pp. 338-353. 6. W. Pedrycz and F. Gomide (1998). An Introduction to Fuzzy Sets: Analysis and Design. MIT Press, Cambridge. 7. U. Kaymak and J. M. Sousa (2003). Weighted constraint in fuzzy optimization. Constraints (Jan), Vol. 8, No. 1, pp. 61-78. 8. R. E. Bellman and L. A. Zadeh (1970). Decision-making in a fuzzy environment. Management Science (Dec), Vol. 17, No. 4, pp. 141-164. 9. H. J. Zimmermann (1993). Fuzzy Set Theory and its Application. Kluwer Academic Press, Boston. 10. S. L. Stebel (2001). Modelagem da Estocagem e Distribuigao de GLP de uma Refinaria de Petroleo. Masters thesis. CEFET-PR/CPGEI, Curitiba, Parana, Brazil, (in Portuguese). 11. E. Teixeira and A. Mendes (1998). An Extension of the Model for the Problem of Workpieces Scheduhng in a Flexible Manufacturing Cell. Production Planning & Control, Vol. 9, No. 2, pp. 176-180. 12. L. C. Felizari and R. Liiders (2004). Estudo do escalonamento de operagoes de transferencia de produtos em refinarias usando otimizagao fuzzy. CBA 2004 - XV Brazilian Automation Conference (Sep), Gramado, RS, Brazil, (in Portuguese).

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 PubUshed by Elsevier B.V.

1489

Degrees of freedom analysis for process control M.RodriguezJ.A.Gayoso Universidad Politecnica de Madrid Jose Gutierrez Abascal, 2 Madrid 28006, Spain Abstract In defining the control structure of a system it is very important to knovs^ how many variables we can regulate. The degrees of freedom (DOF) analysis of the system allows establishing the maximum variables that need to be fixed to have a completely determined system. Of all the DOF some of them will be disturbances (i.e. they are fixed externally) and the rest indicates the maximum number of variables to control. The procedure developed in this paper is based on, and extends and generalizes, the one presented by Ponton (Degrees of freedom analysis in process control.). The procedure presented derives, from the analysis of a general unit (system), a formula to compute the DOF. This formula is then generalised to be applied to any process. Keywords: Process control, degrees of freedom. 1. Introduction The degrees of freedom analysis is used in the development of control (and plantwide control) strategies. Writing all the equations for a process and counting the variables is a tedious and error prone process, so it is important to have a simple but generic method to compute the degrees of freedom for control of a whole process. The first approach in this direction was developed by Kwauk [1] in the context of process design. The procedure developed in this paper follows somehow the one described by Ponton [2] but instead of applying the Kwauk method as in Ponton's it starts from a generic system and applies on it first principles equations.. Degrees of freedom (DOF) can be defined as: DOF = Number of variables of the system - Number of equations of the system The degrees of freedom are the variables that have to be set to have a completely determined system DOF can be set by the environment (disturbances) or by the control system. The following method will compute the overall DOF of the system and will differentiate between disturbances and control variables. An expression is deduced to compute in a simple way all the available DOF for control. 2. Degrees of freedom analysis The degrees of freedom analysis will be performed on a generic system as the one described in figure 1. First the amount of system variables will be calculated and then the equations that can be applied to the system. 2.1. System variables (streams and unit) • Input streams There are Si input streams, each stream has C variables corresponding to the components and one pressure and temperature, so the total amount of variables in input streams are: Si*(C+2)

M Rodriguez andJ.A. Gayoso

1490 C components

Su Hi!

PiT]

vapor

-•So.l —Êi

liquid 1

->So,2

liquid2

•^So.3

Si.; ni2 P2,T2

Sio ni3 P3,T3 Si,p

-•s,

•o,J

InipPiTf %"*'""

-^Saf —

•

*

•

-»-Ek

\ y Energy Fig 1. Generic system. Where Si are the input streams, So are the output streams (one per phase), E are additional output streams from a phase, nij are the moles of component i in phase j . Pi and Ti are the pressure and temperature of phase i. The system has C different species and F different phases. One or more energy streams can interact with the system. • Unit (system) There are C variables corresponding to component accumulation and a pressure and a temperature in each phase. Total variables in the unit are: F*(C+2) • Output streams We have So plus E output streams from the system, but the only new (not accounted for) variables that they add are the flows, because composition, pressure and temperatures have been taken into account in the unit (and they are the same as the composition, pressure and temperature of the outlet streams) Total variables in output streams: So+E • Energy stream If there is an energy stream to (or out of) the system then an additional variable has to be taken into consideration. This energy flow can be thermal or mechanic... It only adds one variable (even if more than one energy stream exists) from a control point of view and it is the amount of energy that will be added in the propoer balance (this will affect to system temperature or pressure). If the energy is transferred inside the system boundaries as it happens with a process to process heat exchanger then this variable will not add any DOF.. To consider the energy a variable called H is added, which will take the value 1 if there is energy flow to (out of) the system and 0 otherwise. Variables in the system: Si*(C+2) + F*(C+2) + Sout + H 2.2. System equations • Mass balance There is one mass balance per component, so we have C equations.

Degrees of Freedom Analysis for Process Control

1491

• Energy balance There is one energy balance to be applied to the system • Momentum balance Although the momentum balance is vectorial and three individual balances can be established, in the process industry only one of them is generally significant (in the flow line, Bemouilli's equation) so we have an additional equation (if more balances are applied more variables have to be considered so it doesn't affect to the degrees of freedom computation) • Equilibrium or transport equations We can have the system in equilibrium or not, in any case the same amount of equations arise. If we have equilibrium we can establish composition, pressure and temperature equalities in the interfaces. If we don't have equilibrium we can establish in each interface C mass transport equations (Pick's law) for the compositions, one heat transport equation (Fourier's law) for temperature and one momentum (only one out of the three possible is generally meaningfiil) transport equation for pressure (Newton's law). Total equilibrium or transport equations (there are F-1 interfaces) : (F-l)*(C+2) Equations in the system: C+l+l+(F-l)*(C+2) (In the case of vapor phase we have an additional equation, the gas law, but we have another variable, V. so it doesn't change the degrees of freedom) 2.3. Degrees of freedom The computation of the degrees of freedom using the above decomposition results in: DOF= Si*(C+2) + F*(C+2) + So„t + H-[ C+l+l+(F-l)*(C+2)]= Si*(C+2) + Sout + H From these degrees of freedom in the input streams we can only act upon the flow so the remaining degrees of freedom of these streams are disturbances (from a control point of view). This results in: DOF= Si + Sout + H Up to this point all the inventories have been considered, but many times we are not interested in or we cannot control them (like when a pipe is splitted into two or when the output of a tank is through a weir). We introduce now an additional variable, A. This variable is the amount of inventories (liquid or gas) that are not considered. This variable (A) removes one DOF if there is one process variable available to control the inventory that is not used (or cannot be used). For example if we have a tank with one input and one output and both are flow controlled then A does not remove any degree of freedom as the inventory is taken into account indirectly. The final DOF equation is: DOF= Sj + Sout + H-A We have not considered the case of the existence of a reaction as it does not vary the DOF analysis (it adds one variable the extent of the reaction, and one equation, the kinetic expression which is composed of variables that are already considered). One final consideration must be noticed. In the system only a temperature, pressure,., is considered in each phase. This is true only in completely mixed systems, in the case where a gradient of variables exists it doesn't modify the DOF analysis. We can decompose each phase in N compartments, this will add N-1 new variables for temperature, pressure and each composition. But we can set an equation for each of the new N-1 interfaces for temperature, pressure and each composition through the transport laws, so the DOF is not altered as stated before.

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M Rodriguez

and J. A.

Gayoso

2.4. Examples The expression is applied to some common process units. Table 1. Process units degrees of freedom Unit

Degrees of freedom Si + Sout + H-A

Heater

D0F-1 + 1 + 1 - 1 = 2

Process heat exchanger

D0F=2 + 2 + 0 - 2 = 2 (in this case the inventories in shell and tubes are not controlled) Energy is cero because the flow is between process streams.

Pump Compressor

DOF=l + 1 + 1-1=2^^^ DOF=l + 1 + 1 - 1 = 2 ^ ^ ^

Vaporizer DOF= 1+1 + 1 - 0

CSTR DOF=2+1+0-0 =3

Distillation Column: 1+3+0-2 = 2 Condenser: 1+2+1-0 = 4 Reboiler: 1+1+1-1 = 2 (pressure is not controlled) The distillation has been decomposed as a process so the general expression (next section) is applied: DOF= l+[3-2]+[2+l]+[l+l-l]= 6 Furnace D0F=3+ 2 + 0 - 1 = 4 The inventory in the tubes is not considered, and the energy term is cero because it both are considered process streams. (1) Although it is possible to control (specify) two different variables in pumps, the flow and the head, it is not desirable from a process point of view. The speed of the pump and the impulsion valve can be manipulated to fix the flow and the head, but it doesn't make sense to work with other than the minimum necessary head so only one degree of freedom is used in pumps. (2) The same happens to compressors. Usually the pressure is the only variable to control, but both pressure and flow can be controlled.

3. Degrees of freedom analysis of a process 3.1. DOF expression The expression deduced can be extended to compute the degrees of freedom of a complete process. The total DOF of the process will be the sum of the DOF of the units, but removing all the DOF related to input streams but the inputs to the process (any input of a unit is the output of an upstream unit so it is already accounted for in the DOF expression).

Degrees of Freedom Analysis for Process Control

1493

The DOF expression for a complete process is: DOF=Sip+X"(Sout + H-A) Where Sip are the inputs to the process, and S is the sum of all the units in the process. This is the maximum SISO control loops that can be established, although other process constraints can make this number lower. This expression is easily applied to any process, and it finally derives in counting all the process streams, adding all the energy flows (one per unit) and removing all the inventories not to be considered. Following the expression is applied to several processes. 3.2. Example 1. Vinyl Acetate process The flowsheet of fig. 2 represents the industrial process for the vapor phase manufacture of vinyl acetate monomer. The process is based on the description in [3] and [4]. The process has three feeds: oxygen, ethylene and acetic acid. The main reaction takes place in a plug flow reactor and the heat is removed generating steam. This process is described by Luyben in [5] when exposing his methodology for plantwide control. He presents 26 degrees of freedom.

Figure 2. Vinyl acetate monomer process Degrees of fi-eedom: Number of process streams: 39 Number of inventories not accounted: 20 (9 mixers or splitters, 4 heaters, 2 heat exchanger, reactor, CO2 removal, 1 reboiler, 2 distillation column) Number of energy streams: 8 (4 heaters, 1 vaporizer, 1 reactor, 1 reboiler, 1 condenser) DOF=39-20+8=27 The difference with Luyben is the compressor. Usually (as in this case) only the pressure is controlled, so one degree offi"eedomis removed (as explained in the previous section when commenting the DOF of pumps and compressors). One possible set of manipulated variables is shown in the figure. 3.3. Example 2. Vinyl Chloride Monomer(VCM) process Vinyl chloride (VCM), which is made from ethylene and chlorine, is used as feedstock in the production of the standard plastic material PVC. Figure 3 shows the balanced process with no net consumption or generation of HCl as described in [6] and [7]. It combines direct chlorination, ethylene dichloride (EDC) pyrolysis and oxychlorination (which consumes all the HCl generated in EDC pyrolysis. Chlorination and oxychlorination reactions takes place in tubular reactors and the pyrolysis takes place in a furnace type reactor.

1494

M Rodriguez and J.A. Gayoso

Figure 3. Vinyl chloride monomer process Degrees of freedom: Number of process streams: 57 Number of inventories not accounted: 24 (4 in mixers or splitters, 2 in heaters, 3 in reactors, 5 in reboilers, 10 (2*5) in distillation columns) Number of energy streams: 14 (2 in reactors, 2 in heaters, 5 in reboilers, 5 in condensers) DOF=57-24+14=47 One possible set of manipulated variables is shown in figure 3.

4. Conclusions In this paper a simple expression to compute the (maximum) degrees of freedom available for control of any process. There is no need to write any equations as the expression has been obtained through a rigorous application of the available physicochemical equations (conservation laws, transport laws, equilibrium constraints) to a generic system. This method takes into consideration the inventories which are very important in establishing any control strategy for a process. The expression has been tested in numerous processes as the two presented in this paper. The expression is easily implemented in a software application and can be used as an initial step in any Plantwide control method. It can help to sort out the available controlled variables of the process. This expression is currently implemented and is being used in a plantwide control methodology that is being developed by the authors and that will be presented elsewhere.

References [l]Kwauk,M. AIChE Journal 2 (1952), 40 [2]Ponton, J.W., Degrees of Freedom Analysis in Process Control, Chemical Engineering Science, Vol. 49 (1994), No. 13, pp 1089- 1095. [3]Vinyl acetate, Stanford Research Insititute (SRI) Report 15B, 1994 [4]Ullmann' s Encyclopedia of Industrial Chemistry, 2003. [5]Luyben,L., Tyreus, B. and Luyben, M.. Plantwide Process Control. Mc Graw-Hill, 1999, pp. 321 [6] Kirk-Othmer Encyclopedia of Chemical Technology, 4^^ edition, 2001, Vol 24 [7] http://www.uhde.biz/informationen/broschueren.en.html. Technical Report on Vinylchoride and Polyvinylchloride.


1495

Autothermal Reactors for Hydrogen Production: Dynamics and Model Reduction Michael Baldea^, Prodromos Daoutidis^'^ ^Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455, USA. Email: daoutidi&,cems. umn. edu ^currently at Aristotle University of Thessaloniki, Thessaloniki, Greece Abstract In the present work, we propose a generic dynamic analysis and model reduction framework for autothermal reactors. We document the existence of a two time scale dynamic behavior, and identify the variables that are associated with each time scale. We derive reduced-order, nonlinear models for the dynamics in each time scale. We show that the derived slow model corresponds to generally accepted empirically derived simplified models for the class of reactors considered. Subsequently, we present illustrative simulation results on a hydrogen production reactor. Keywords: Nonlinear Model Reduction, Autothermal Reactors, DAE systems. 1. Introduction The interest in economically efficient hydrogen production has been steadily increasing, and even more so recently, given the progress made in the development and implementation of fuel cell technologies. Autothermal reactors, combining exothermic and endothermic reactions are one of the most promising hydrogen production technologies, featuring in-situ heat generation, which allows for increased fuel efficiency and a compact size. From a design and operation point of view, autothermal reactors rely either on a constant, unidirectional flow, whereby the raw material for hydrogen production and the necessary fuel flow in different, parallel channels of the reactor (either in co-current or counter-current), or on flow reversal, in which case the catalyst bed within the reactor acts as a heat trap (Frauhammer et al, 2000). The majority of the research studies concerning the design and operation of autotermal reactors of either category investigate the steady-state behavior of the system. However, in the context of integrating such reactors in larger systems that include fuel cells for power production, the transient operation of the autothermal reactor, enabling variable levels of hydrogen supply in response to varying power requirements to a fuel cell downstream becomes much more interesting (Gorgun et al, 2005). Thus, the availability of accurate, reliable and, at the same time, computationally efficient models is of great importance for dynamic analysis and control (Vanden Bussche et al., 1993). Due to the inherent multiple-time scale behavior of autothermal reactors (Kaisare et al., 2005), their dynamic models are ill conditioned and challenging to simulate, and recent efforts have been aimed at providing an approach for deriving models of reduced dimensions that capture the salient dynamic features of the original system (Gorbach et a/., 2005). In the present work, we propose a generic dynamic analysis and model reduction framework for autothermal unidirectional flow reactors. Using mathematically rigorous

1496

M. BaldeaandP. Daoutidis

arguments, we document the existence of two distinct time scales in their dynamic behavior, and identify the variables that are associated with each time scale. We derive reduced-order, nonlinear models for the dynamics in each time scale. In particular, the derived slow model corresponds to generally accepted empirically derived simplified models for the class of reactors considered. Subsequently, we present illustrative simulation results on a hydrogen production reactor. 2. Modeling and Model Reduction of Autothermal Reactors We consider 2 channels of an autothermal reactor with catalytic walls (Fig. 1).

a

b

Figure lAutothemal reactor (a) countercurrent flow; (b) co-current flow A gaseous reacting mixture with components j

of molecullar mass M.,

and a

composition given by the weight fractions W-, density p ^, and temperature T^ ^ enters channel A:, /r = 1,2 at an inlet velocity K^. We assume that reactions r., with stoechiometric coefficients l>.. only occur on the catalytic wall, and that the reaction rates account for the diffusion of components to and from the catalyst surface (diffusion is not modeled explicitly). Under these assumptions, the mass and energy balance equations describing the reactor behavior take the form:

dw,.

^w,.

-

d

H^;

U)

Ot

OZ

k=l,2

k=l,2 1=1

which, by defining the dimensionless timeT = F j / L / , the dimensionless spatial coordinate ^ = zlL

(with L

temperature 0 = T IT .

kj^ = M^ IM^, ^wj k ~^jk^^jk^

being the spatial coordinate), the dimensionless

and the quantities A:^. = Kj / K2 , A:^ = Jêff,! ^ êff,i»

with M ^ being the average molecular mass in channel k,

and

^^^^ ^j k being the mole fraction of component I in channel k ,

and using the standard definitions of the Peclet number Pe, the Stanton number

St,

Autothermal Reactors for Hydrogen Production: Dynamics and Model Reduction

1497

the Fourier number for the gas, Fo^^, the Fourier number for the soUd, Fo^, the Damkohler number Da. and the adiabatic temperature rise B^ for the I th reaction, takes the following dimensionless form:

£

OT

OT

it=l,2

iâ

k=\,2 i=l

with £ = PgiCpg^ ^ Ps^ps- Taking into account that ky.kj^.k^.k^,

kj^ are typically

0 ( 1 ) quantities, the more compact form arises:

^=-MA|^+^'''|^+^^'('?.-^.)

€

oT

oT

k=\,i

^'-^

k=i,2 1=1

where the prime superscript denotes appropriately modified dimensionelss numbers. Generally, due to the large differences in heat capacity and density between the gas phase and the solid catalyst, € -/'>(|G,4y«('>)|-l)

(5)

where j ^ > 0 is the convergence rate and Y^^p \JC0 j is computed by \GW (M" )| = \GLP (M")e-^-^"''"

I=If

y{t)e-^'^"'dl\l\

f

uXt)e-''^"'dt\

(6)

Notice that we have A^ = 0 if ^^ = 1, and, in general, A^ increases as A^ and P increase. Thus, the value of A

is suggested as (^^ — l ) P / 6 . In addition, j'^^ is

chosen as:

/ o = (A-rly r.. ^.nu-MQ. \A ! « 0 < fankllevel < 600) && viopen == false) ; P ' 0 p « r 1 y h •i.-Û-iued

\A ! « 0 < tankaeve! < 60)) && v2open == false) ipifiQ-^riK' ri

'•.:iti'-Jlf"i.

i A O !«0 < tankaevel < 600) && v^pen == false)

Figure 8. Verification results of example 4.2 5. C o n c l u s i o n s This study represents a novel method to verify the safety of chemical processes using graphical modeling and simulations. This method is applied to two batch processes and the safety and operability is verified. The graphical description of the process is model in system editor. The simulator provides visual sequence of symbolic states of the processes, calculates the dynamic variables, and records the possible behaviors of the process models at the same time. The model checker using CTL specifications verifies the safety and properties of the processes automatically. This study represents an effective and automatic technique for the safety verification using a graphical modeling and simulation. References I. Moon, 1994, Modeling PLCs for logic verification, IEEE Control Systems, Vol.14, No.2, pp. 53-59 J. Kim, M. Kim and I. Moon, 1999, Improved Search Algorithm for the Efficient Verification of Chemical Processes, Computers and Chemical Engineering, Vol. 23, SuppL, pp. S601-604 R. Alur and D. Dill, 1994, A Theory for Timed Automata, Theoretical Computer Science, Vol. 125, pp. 193-235 A. Furfaro and L. Nigro, 2003, Temporal verification of communicating real-time state machines using uppaal Industrial Technology, 2003 IEEE International Conference on, Vol.1, pp.399404 J. Kim and I. Moon, 2000, Synthesis of Safe Operation Procedure for Multi-purpose Bath Processes using SMV, Computers and Chemical Engineering, Vol. 24, pp.385-392 B. Justin, G. L. Kim, P. Paul, and Y. Wang, 2000, UPPAAL: A tool suite for automatic verification of real-time systems. Lecture notes in computer science. No. 1066, pp. 232-236 A. Bums, 1998, How to Verify a Safe Real-Time System: The Application of Model Checking and Timed Automata to the Production Cell Case Study, Real-time systems. Vol. 24, pp. 135151 S. Cha, H. Son, J. Yoo, E. Jee, and P. Seong, 2003, Systematic evaluation of fault trees using realtime model checker UPPAAL, Reliability engineering & system safety. Vol. 82, pp. 11-20 T. A. Henzinger and W. K. Peter, 1999, Discrete-time control for rectangular hybrid automata. Theoretical Computer Science, Vol. 221, Issues 1-2, pp. 369-392 G. L. Kim, P. Paul, and Y. Wang, 1995, Model-Checking for Real-Time Systems, Lecture Notes in Computer Science, Vol. 965, pp. 62-88 S. Yovine. Kronos, 1997, A Verification Tool for real-Time Systems, Journal of Software Tools for Technology Transfer, Vol.1, pp. 123-133 T. A. Henzinger, X. NicoUin, J. Sifakis, and S. Yovine, 1994, Symbolic model checking for realtime systems, Information and Computation, Vol. 111, pp. 193-244.


1515

Application of A Hybrid Control Approach to Highly Nonlinear Chemical Processes Yoshiyasu Sakakura^, Masaru Noda^, Hirokazu Nishitani ^, Yoshiyuki Yamashita^, Masatoshi Yoshida^, Sigeru Matsumoto ^ ^Graduate School of Information science, Nara Institute of Science and Technology, Takayama, Ikoma, Nara, 630-0192, Japan ^Department of Chemical Engineering, Tohoku University, Aramaki, Aoba, Sendai, Miyagi, 980-8579, Japan This paper proposes a new control approach for highly nonlinear chemical processes with operational mode changes. We combine a successive linearization and an MLD (mixed logical dynamical) formulation to transform an MPC (model predictive control) problem for a multimodal nonlinear dynamical system into an MIQP problem. We apply the proposed approach to the temperature control problem of CSTR (continuous stirred tank reactor), which is modeled as a bimodal nonlinear hybrid system. The simulation result shows a good control capability in the control of highly nonlinear hybrid systems. Key words: Process Control, Hybrid System, Model Predictive Control, MLD Formulation, CSTR, 1. INTRODUCTION In plant operations, operational modes change frequently due to product changes, emergency evacuations, and so on. The plant dynamics with operational mode changes are usually formulated as a nonlinear hybrid dynamical system. There are two problems to be overcome in nonlinear hybrid dynamical system control: nonlinearity of the process dynamics and logical rules in the hybrid dynamical system. Many approaches for nonlinear control have been proposed so far. One of the most effective is an MPC(model predictive control) using successive model linearization [2]. Shikanai et al. proposed a predictive control framework using successive linearization of a highly nonlinear process [4]. Many control approaches also have been proposed for hybrid dynamical systems. One has been proposed by Mhaskar et al. [3], where a Lyapunov-based predictive controller is designed for a each mode that allows for an explicit characterization of its stability region. Mode changes are then appropriately incorporated to the controller implementation. Another, and the most remarkable, approach was proposed by Bemporad et al [1], where a linear hybrid dynamical system is described as the MLD (mixed logical dynamical) model. This model is used in an MPC scheme. Since the method can be applied only to a linear discrete-time model, hnearization of the nonlinear process plays an important

1516

Y. Sakakura et al

role when the MLD model is used. In this study, we propose a new control method for nonUnear hybrid dynamical systems based on MLD formulation. We employ the modified linearization method proposed by Shikanai et al. [4] in a control scheme, and apply it to temperature control of a CSTR with operational mode changes. 2. METHOD 2.1. Problem description We consider a bimodal nonlinear hybrid system as given by Eqs. (1) and (2). X = fi{x,u) ^ = /2(^, u) y = Zx

if 5a: < 0 (mode 1) ii Sx >0 (mode 2) (2)

along with u G [t^min^'^max], l^t /i and /2 be nonlinear functions, where S and Z are coefficient matrices, and x and u denote state and input vector variables. Conditional inequalities in Eq. (1) indicate that the operational mode changes according to the state variables of the process. We can formulate a control problem of a bimodal nonlinear hybrid system as an MPC problem through the following two steps: 1. Discretization and linearization of nonlinear continuous ODEs in Eq. (1). 2. Formulate logical constraints in Eq. (1) as linear inequality constrains with binary variables. Through the above two steps, the control problem of a nonlinear hybrid system is transformed into an MILP problem, which is solved at every control period. 2.2. Discretization and linearization methods The nonlinear differential equations in Eq. (1) are discretized and linearized at every control period to transform a nonlinear hybrid system into an MLD model. In general, the conventional hnearized and discrete-time models of Eqs. (1) and (2) are given by Eqs. (3) and (4), jxk+j+i = Ai^Xk+j + BiÛk+j + Ci ]^Xk+j+i = A2uXk+j + B2uUk+j + C2 yk+j = Zxk^j

if Sxk+j < 0 (mode 1) if Sxk+j > 0 (mode 2)

(3) (4)

where k denotes the present time step, j{= {0,1, 2 • • • }) represents the number of steps from the present step, x^+j is the state vector variable, Uk+j is the input vector variable, Ann, Bnu ^rc coustaut matriccs, and Cn denote constant vectors in a control period. The subscript u indicates the conventional linearization method and n represents the mode of the system (1 or 2).

Application of a Hybrid Control Approach to Highly Nonlinear Chemical Processes

1517

In this study, we use the modified linearization method [4] in our control scheme. According to this method, Eq. (1) can be formulated as follows: ^fc+j+i = AiiXk+j + Ai2Xk+j-i + BiAuk+j

if Sxk+j < 0 (mode 1)

ock^j+i = A2iXk+j + A22Xk+j-i + B2Auk+j

if Sxk+j > 0 (mode 2)

where Auk+j = Uk+j — Uk+j-i- Because Eq. (5) is a finite difference model, the equation is calculated recursively to estimate state vector variables x at any time step j using both Xk and Xk-i as known values. We finally obtain a simple linear discrete prediction model in every control period as follows: \xk+j+i = ^ 1 + BiAuk+j

if Sxk+j < 0 (mode 1)

yxk+j+i = ^2 + B2Auk+j

if Sxk+j > 0 (mode 2)

where Ai = AnXk+j + Ai2Xk+j-i A2 = A2lXk+j

+

A22Xk+j-l

2.3. M L D formulation Bemporad et al. proposed a systematic approach for the control problems of hybrid dynamical systems [1]. According to the MLD formulation, Eq. (6) is formulated as follows: Xk+j+i = Al + Sk+j{A2 - Al) + {Bi + 6k+j{B2 -m

Sk+j + m

320

• '

0.99

300

>

300 280

280

0.98 C

10

20 30 Time[min]

40

5C

Figure 2. Control response of the proposed method.

C

10

20 30 Time[min]

40

50

Figure 3. Control response of the comparative approach.

All parameters in this model are summarized in notation. The MFC controller is constructed based on this model. In controller design, we use the following variables: y = {T}, u = { T U N } , X = {if, C A , T , Ti,T2}. It is assumed that all state variables are observed at every sampling period. 3.2. Results We applied the MIQP formulation represented by Eq. (10) to the problem. Control performance was simulated by using Mathematica 5.1. We changed the flow rate of the reactant (FIN) from 0.0366 m^/min to 0.0426 m^/min to cause mode shifts. In an MFC, we used the following reference trajectory Tref given by Eq. (13). Tref k+j = (1 - a')

T{k)

+ a^ Zet k+j

(13)

The set point Tget is 325 K and the MFC parameters are given as follows: Hy^ — 1, Ht = Hu = 5, Q{j) = 10, R(j) = 0.001, a = 0.8, Control interval = 1 min (see subsection 2.3). Figure 2 shows the control responses. The mode changes occur at time t = IS min (mode 1 -^ mode 2) and t = 41 min (mode 2 —> mode 1). In Figure 3, the control responses of MFC based on the conventional linearized model (Eq. (3)) are shown for comparison. Although the response shows overshoots, this system does provide good control capabilities, meaning an adequate sampling interval can give good control performance in nonlinear hybrid dynamical systems. 4. CONCLUSION We proposed a new control method for nonlinear hybrid systems. Modified successive linearization was used to transform each nonlinear continuous-time system into a linear discrete-time model. Hybrid hnear discrete-time models were then formulated as an MLD model and this model was employed in the MFC scheme. We applied this method to a highly nonlinear CSTR with operation mode changes. Result revealed that the MFC controller with modified linearization performed well in controlling nonlinear hybrid systems.

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REFERENCES 1. Bemporad, A. & Morari, M. (1999). Control of systems integrating logic, dynamics, and constraints. Automatical 35^ 407-427 2. Lee, J.H. & Ricker, N.L. (1994). Extended Kalman filter based nonlinear model predictive control, Ind.Eng.Chem.Res., 33, 1530-1541 Mhaskar, P.,E1-Farra, N. H. , & Christofides, P. D. (2005) Predictive Control of Switched Nonlinear Systems with Scheduled Mode Transitions, IEEE Trans. Autom. Contr. , 50, 1670-1680. Shikanai, Y., Yamashita, Y., & Suzuki, M. (2002). Design of a robust model predictive controller. Proceedings of the 4^th conference of the automatic control association, 571-572 (in Japanese) NOTATION Parameters of a case study plant. description reactor temperature liquid level of the reactor concentration of reactant A CA temperature Ti,T2 feed temperature Tl IN feed temperature T2 IN feed ÎN cross-section of the reactor A heat transfer area Ahi heat transfer area Ah2 overall heat transfer coefficient UuU2 c constant specific heat of liquid mixture Op specific heat of heat transfer liquid Cpi,Cp2 density of reactor solution P density of thermal medium Pl^P2 flow late Fi flow late F2 concentration of reactant A (feed) CA IN feed temperature of reactor TIN frequency factor ko activation energy E AH heat of reaction gas constant R temperature of hot water Thot temperature of cold water ^cold setpoint of reactor temperature ^set Subscript 1 Temperature control system Subscript 2 O N / O F F cooling system

Variable ~T H

Value controlled state variable state variable state variable manipulated 278 3.66-4.26 3.14 6.28 3.14 5.11 X 10^ 0.04 3.14 X 10^ 4.18 X 10^ 800.9 997.9 0.5 5 8.24 294.4 1.18 X 10^ 69.8 69.8 8.3145 X 10^ 368 278 325

Unit K m kmol/m^ K K K lQ-2

j^^/^niYi

m^ m^ m^ J / m i n K m^ J/kgK J/kgK kg/m^ kg/m^ m^/min m^/min kmol/m*^ K 1/min MJ/kmol MJ/kmol J / K kmol K K K


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Dynamic optimization of dead-end membrane filtration Bastiaan Blankert, Ben H.L. Betlem and Brian Roffel Faculty of science and technology, University ofTwente, P.O. Box 217, 7500 AE Enschede, The Netherlands 1. Abstract An operating strategy aimed at minimizing the energy consumption during the filtration phase of dead-end membrane filtration has been formulated. A method allowing fast calculation of trajectories is used to allow incorporation in a hierarchical optimization scheme. The optimal trajectory can be approximated closely by constant power filtration, which allows robust implementation of the optimal strategy. Under typical conditions, the relative saving in energy, is small compared to constant flux (0.1%) or constant pressure filtration (4.1%). Keywords: Dynamic optimization, dead-end filtration 2. Introduction Dead-end ultra filtration is applied in the purification of surface water to produce either process water or drinking water. Due to its high selectivity, economic scalability and low chemicals consumption, it is a promising technology in this field. However, the performance of membrane systems is often limited by fouling phenomena. Accumulation of retained particulates at the membrane surface increases the hydraulic resistance of the system. This increases the operating costs due to extra energy consumption and the necessity of periodic cleaning. Hence, dealing with membrane fouling is one of the main challenges in the application of this technology. Since the process settings are currently based on rules of thumb and pilot plant studies, it is believed that optimization will result in a reduction of the operational costs. Dead-end filtration is a cyclic process which consists of three phases. During the filtration phase clean water is produced and the membrane is subject to fouling. This is followed by the backflush phase, in which the flow is reversed in order to remove the fouling. After a number of alternating filtrations and backflushes, chemical cleaning is performed to remove "irreversible" fouling. The evolution of the fouling state during the sequence of filtrations and backwashes is illustrated in the left of fig. 1. This study is concerned with the sequence of alternating filtrations and backflushes. Since filtration and backflushing are fundamentally different, they need to be described by separate models. Both have the flux as control variable and the amount of fouling as state. Since the filtration and backflush phases alternate, the initial state and the cost of the final state are difficult to determine.

B. Blankert et al

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Therefore, a hierarchical structure with two layers is used. The top level coordinates the initial and final states of subsequent phases. It searches for a trajectory of initial and final conditions for which the total costs of the series of subsequent phases are minimal. The bottom level is concerned with reaching the final state at minimal costs. This is illustrated in the right of figure 1. A bottom up approach is followed to construct the hierarchical structure, starting with optimization of the filtration phase. This is a dynamic optimization problem which aims to minimize the energy consumption. As it is part of the hierarchical structure, a requirement on the final state and time must be satisfied. Furthermore, each iteration towards the total optimum at the top level involves a dynamic optimization at the bottom level. Hence, the optimal trajectory should be calculated fast. Cycle

Cycle w0,i "^T.!

^F,l

TF,i\

I

^0,1+1

Cs,,

.11

TB

Filtration J^t)

Time (t)

A

4

Backflush JB(t)

'r

T

Figure 1. Left: Semi continuous operation of dead-end filtration consists of consecutive filtration and backflush phases. Right: Hierarchical control structure which corresponds to the cyclic operation.

3. Theory 3.1. Model In dead-end filtration the fouling state (w) is proportional to the filtrated volume. The flux (J) is the control variable, which is also the production rate. The model parameters are given in table 1.

dw ~dt

= J

(1)

The driving force of the filtration process is the trans-membrane pressure, which is related to the flux and the hydraulic resistance of the system by Darcy's law:

AP = rjJRM7F

(2)

in which yp is the relative increase in the resistance due to fouHng, which can be given by (Blankert):

rF = i + — mvO(l + mvO rjfij) RM with O a correction factor for the geometry of the fiber, given by (Belfort):

(3)

Dynamic Optimization ofDead-End Membrane Filtration

1523

0 = - ^ l n l - ^ 2wx

\

(4)

r .

The relative increase in energy consumption due to the pump efficiency can be approximated by (Karasik):

v

f rp =

' /i,. /

(5)

4

Table 1. Model parameters and their values Specific cake resistance Compressibility Cake volume fraction Viscosity Membrane resistance Fiber radius Maximum pump pressure Maximum pump efficiency Flux at maximum efficiency

a

m' Pa^ Pas m-^ m Pa m/s

p X

ri RM

r p •*• m a x

VP,max

J.

1.00x10'' 5.00 X 10-^ 1.00x10-^ 1.01x10-^ 7.00 xlO^' 4.00 X 10-^ 1.33x10^ 0.50 4.16x10-^

3.2. Optimization The energy consumption per unit area is equal to the integral over the power per unit area (JAPyf), which can be given by:

Cp = \{vvp,m^RMrFrp'^^)^t

(6)

For this process the Hamiltonian can be given by:

H{w,

j,X)Âj+mp,m.ArFrp'^'

(7)

In the Hamiltonian the adjoined state (X) is introduced. The first necessary condition for optimality states that the optimal flux minimizes the Hamiltonian, thus: (

J dYp

— = ^^+vnp,m.,RMrFrpJ\2 + YP V

J dfp ^ • + •

dJ

YF

=0

(8)

^J

This equation allows us to calculate the optimal flux as function of the state and the adjoined state. However, here it is used to eliminate X from eqn. 7. The result is the minimum value of the Hamiltonian as function of the flux and the state. ^

1^

Yp dJ

YF 3 /

(9)

This equation leads to two approaches which are discussed in the following paragraphs.

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3.2.1. Simplified system One approach is simplification of the system. The effect of compressibility and pump efficiency are neglected (y5 = 0, j ^ = 1). In that case the right-hand term of eqn. 9 vanishes and the Hamiltonian is proportional to the power. Since the Hamiltonian is constant along an optimal trajectory, constant power filtration is optimal. From this consideration, constant gross power filtration is introduced as alternative for the dynamically optimal trajectory. The main advantage of this approach is that it can be implemented in a robust way. The power, which can be easily measured, can be kept constant by a feedback controller, which is part of a cascade control structure. The master controller uses the setpoint of the power to ensure the final condition (produced volume) is met. 3.2.2. Predefined trajectories The second approach also makes use of the consideration that for optimal trajectories the Hamiltonian is constant. With eqn. 9 the Hamiltonian is calculated for a large number of states and fluxes on a grid. The calculated points (J, w, H) are sorted in a table such that each row corresponds to a value of the state and each column corresponds with a value of the Hamiltonian. Each column contains a trajectory, which is optimal for some final condition. Since the state and the flux are known, the time and costs of each point in a column can be calculated and added to the table. This is done at the moment the model parameters are estimated. Then at each filtration phase, for a given final time and fmal state, the optimization problem is reduced to finding the correct row indices and column index. The row indices follow directly from the initial and final condition for the state. The column index follows fi*om the final time (duration) condition. It is equal to the difference between the initial and final column. The costs can be found in a similar way. 4. Results The optimal trajectories were calculated for model parameters shown in table 1. Fig. 2 shows these trajectories for WQ = Om, WT = 0.0375m and T = 1800s. The common operating strategies, constant flux (flow) and constant trans-membrane pressure (driving force), are shown in the figure as well. It can be seen that the constant gross power trajectory is close to the optimum. The constant flux trajectory is also close. The relative difference in costs between the optimal and suboptimal strategies are shown in table 2.

Table 2. Potential savings of reference strategies Complete model Final time (s) Final state (m) Constant flux Constant pressure Constant gross power

1800 3.75X IQ-^ 0.1 % 4.1 % < 0.1 %

3600 7.50X 10"^ 1.1 % 16.0 % 120

140

160

time [h]

Figure 1. CVs for nominal scenario, and Figure 2. MVs for nominal scenario; dashed desired setpoints (dashed). are the control bounds. during the transition from one problem to the next, a shift in the optimization variables is performed to account for the movement in time due to the fixed past control trajectory u{t). MHE setup: In the practical simulations, the samphng time for the measurements is 5 minutes. The MHE horizon is 20 minutes long and thus comprises five measurement samples, which proved sufficient to recover the states and, additionally, estimate the two reaction kinetic parameters p. The MHE weighting matrix is V = diag(y5s)~^. In the EKF W = diag(xss)~^ is used as weighting matrix for the state noise. The subscript ss denotes the steady state at grade C. 5. N U M E R I C A L S I M U L A T I O N S We present simulation results for a control scenario comprising several production grades: start in grade C (t = 0), setpoint change to grade B {t = 600 min), measured feed temperature disturbance from Tf = 353.0 K to Tf = 357.0 K (t = 1200 min), setpoint change to grade A {t = 3000 min), setpoint change to grade C {t = 6000 min), and back to grade A {t = 8000 min). The process in closed loop has been tested under different conditions: (i) nominal process model and no noise, and (ii) process model with two perturbed initial parameters and noisy measurements. The performance of the closed loop under conditions (i) is shown in Figures 1 and 2. In test (ii), both reaction parameters had been initialized largely wrong, one order of magnitude lower than their true values. They had to be determined by the estimator online, see Figure 5. In addition, measurement errors were introduced, in the form of zero-mean white noise with standard deviation 1% of the steady state value of grade C. The resulting closed loop simulation is shown in Figures 3 and 4 The overall control performance and duration of transients does not deteriorate much compared to the nominal scenario, even though the manipulated variables oscillate considerably. Optimizing NMPC: In a further test, we aim to maximize the production rate (CVl) while maintaining the grade specifications (CV2, CV3). Instead of the pure least squares objective (la) we use the cost rto+T

mm

-M-ycvi + Wvcvit) - ycvWq + Jto

Mt)\\l dt

(3)

Combined Nonlinear Model Predictive Control and Moving Horizon Estimation CV1: production rate 0.4 mw»^*imi»%' •9) ^

0.2 0.1

>

1

1531

MV1: feed rate monomer B

h JSN^_

M

Figure 3. CVs for noisy scenario: measure- Figure 4. MVs for noisy scenario; the ments (light) and estimates (solid). dashed line denotes the control bounds. Controlled Variables ^ 10= Reaction parameter I

Manipulated Variables

— Least Squares 1 - Economic |

r:

1 0-5 — 0.4 1^ 0.3

ta. 0.2

\l D

1000

2000

3000

4000

50

D

1000

2000

3000

4000

50

^

I 340

Figure 5. Estimates (light grey) vs. true Figure 6. Comparison of tracking and ecovalues (solid) for non measured quantities. nomic NMPC within a transient.

in problem (1) with factor /x := 10~^kg/min, and a very low weight within matrix Q on setpoint deviations in ycvi- Such a formulation leads to increased production rates that adapt to the current state and parameter estimate. A comparison of least-squares and "economic" objective is shown for one grade transition in detail in Figure 6. The production rate at both grades is increased by 7-12%, without deterioration of product specifications (CV2, CVS) or transition times. It is interesting to note that for MV2 the lower bound is touched, as the optimal operating point lies at the boundary of the feasible region. The lower bound on MV2 only becomes inactive during the transitions, where the additional degree of freedom is used to achieve a fast transient. Note that the resulting controller at the boundary of the feasible region has one sided gains and is highly nonlinear. Acknowledgements: The first four authors gratefully acknowledge financial support by the DFG under grant BO864/10-1.

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REFERENCES 1.

2.

3.

4. 5.

6.

7.

8.

9.

10. 11.

12. 13. 14.

15. 16.

F. Allgower, T. A. Badgwell, J. S. Qin, J. B. Rawlings, and S. J. Wright. Nonlinear Predictive Control and Moving Horizon Estimation - An Introductory Overview. In Paul M. Frank, editor, Advances in Control^ pages 391-449, Springer, 1999. R. D. Bartusiak. NMPC: A platform for optimal control of feed- or product-flexible manufacturing Proc. Assessment and Future Directions of NMPC, Freudenstadt-Lauterbad, Germany, pages 3-14, 2005. L.T. Biegler. Efficient solution of dynamic optimization and NMPC problems. In F. Allgdwer and A. Zheng, editors. Nonlinear Predictive Control, volume 26 of Progress in Systems Theory, pages 219-244, Basel, 2000. Birkhauser. Rahul Bindlish. Modelling and Control of Polymerization Processes. PhD-thesis, University of Wisconsin-Madison, 1999. H.G. Bock. Numerical treatment of inverse problems in chemical reaction kinetics. In K.H. Ebert, P. Deuflhard, and W. Jager, editors. Modelling of Chemical Reaction Systems, volume 18 of Springer Series in Chemical Physics, pages 102-125. Springer, Heidelberg, 1981. H. G. Bock and K.-J. Plitt. A multiple shooting algorithm for direct solution of optimal control problems. In Proceedings of the 9th IF AC World Congress, Budapest. Pergamon Press, 1984. M. Diehl, H.G. Bock, J.P. Schloder, R. Findeisen, Z. Nagy, and F. Allgower. Real-time optimization and nonlinear model predictive control of processes governed by differentialalgebraic equations. J. Proc. Contr., 12(4):577-585, 2002. M. Diehl, H. G. Bock, and J. P. Schloder. A real-time iteration scheme for nonlinear optimization in optimal feedback control. SI AM J. of Control and Optimization, 43(5), pages 1714-1736, 2005. M. Diehl, R. Findeisen, F. Allgower, H. G. Bock, and J. P. Schloder. Nominal stability of real-time iteration scheme for nonlinear model predictive control lEE Proceedings - Control Theory and Applications, 152(3), pages 296-308, 2005. J. Kallrath, H.G. Bock, and J.P. Schloder. Least squares parameter estimation in chaotic differential equations. Celestial Mechanics and Dynamical Astronomy, 56:353-371, 1993. D. B. Leineweber, I. Bauer, H. G. Bock, and J. P. Schloder. An efficient multiple shooting based reduced SQP strategy for large-scale dynamic process optimization. Part 1: theoretical aspects. Comp. & Chem. Eng., 27, pages 157-166, 2003. L. Magni and R. Scattolini. Stabilizing model predictive control of nonlinear continuous time systems. Annual Reviews in Control, 28, pages 1-11, 2004. S. J. Qin and T. A. Badgewell. A survey of industrial model predictive control technology Contr. Eng. Pract, 11, pages 733-764, 2003. C. V. Rao, J. B. Rawlings, and D. Q. Mayne. Constrained state estimation for nonlinear discrete-time systems: Stability and moving horizon approximations. IEEE Trans. Auto. Cont, 48(2):246-258, 2003. L.O. Santos. Multivariable Predictive Control of Nonlinear Chemical Processes. PhD thesis, Universidade de Coimbra, 2000. A. Zheng. Does nonlinear dynamic matrix control provide integral control? Comp. & Chem. Eng., 23, pages 1753-1756, 2000.


Automatic adjustment of data compression in process information management systems Frank Alsmeyer AixCAPE e.V., Intzestr. 1, 52072 Aachen, Germany Abstract Process information management systems (PIMS) use data compression methods where appropriate parameters must be set for each process variable. In this work, we describe algorithms to adjust these parameters automatically by analyzing raw process data. Our algorithms are based on wavelet analysis of the raw data and perform a conservative estimation of the noise variance. Based on these noise estimates, parameters for the compression algorithms are derived. The possible incorporation of prior information about the signal characteristics, like finite resolution of the AID converter, is discussed. Finally, we describe the validation of the proposed algorithms with real process data from the industrial members of the AixCAPE consortium. Keywords: data historian, compression algorithms, variance estimation, denoising, wavelets. 1. Introduction The compression methods in process information management systems (PIMS) are needed to store the masses of data measured in large-scale production processes. In practice, the quality of the compressed data is often unsatisfactory due to inadequate compression settings. This can lead to significant delays and high cost when production problems occur, because the compression settings must be corrected, and more data must be collected before the problem can be analysed and solved. The adverse effects of overcompression on data analysis have been examined in detail by Thomhill et al. (2004). The reason for inadequate compression settings is the high cost associated with the PIMS configuration, as it is mostly done manually. The piecewise linear compression methods in use today, like the swinging door algorithm (SDA; Bristol, 1990) or the modified Boxcar-Backs lope algorithm (MBBA; Aspentech, 2002) must be parameterized for each process variable separately. As a consequence, the parameters are often left at their default values, typically 1% of the maximum value. This is often larger than the actual noise variance in the data, resulting in loss of information. It has been shown that compression methods based on wavelets are superior to piecewise linear methods (Watson et al., 1998). These methods are adaptive to the signal itself and parameterization is, therefore, less critical than in the piecewise linear case. Unfortunately, these methods are not yet available in PIMS. As a pragmatic solution, we combine in this work the strengths of wavelet methods with the technical possibilities of current PIMS. We describe an algorithm to calculate adequate compression parameters for piecewise linear methods automatically. It is based on the analysis of uncompressed raw data in the wavelet domain and builds upon previous work on trend detection in process data (Flehmig, 2005). In the next section, we discuss possible configuration strategies by looking at the full signal path from sensor to data archive. Section 3 describes the algorithms used and

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discusses proper choice of wavelet bases and thresholding strategies. Finally, we validate the approach using real plant data from two commercial PIMS that implement the piecewise linear compression algorithms SDA and MBBA.

2. Strategy for automatic compression tuning Generally, we suggest using a conservative compression strategy as missing data can never be reconstructed again, and storage costs are constantly decreasing. To derive such a strategy, it is usefiil to consider the signal path from sensor to archive, as shown in Fig. 1. The electrical signal (1) at the sensor, usually linearized, is digitized in an analog/digital converter (2), resulting in a signal of finite resolution. It is not uncommon that this resolution, often not more than 12 or 16 bit, can be clearly seen as steps ("quantization") of the recorded signal (cf Figure 2a). The calibration model (3) that converts from the electrical to the underlying physical property is often linear, as in the case of a PT-100 temperature sensor, or only slightly nonlinear. The next two data transformations (4) and (5) depend on the PIMS at hand and on its configuration. Step (4) is used either as a preprocessing step to reduce the computational load of (5) by letting pass only those new values yi that differ significantly from the previous value yi.i (|yi - Yi-il < devpre)', or it is used alternatively to the piecewise linear compression step (5), which is, in this paper, either SDA or MBBA. These linear compression algorithms have in common that they keep only a subset of the original values. These values are chosen such that all discarded values lie within a specified band around the linear segments connecting the kept values. The acceptable deviation from the recorded linear segments dev is the essential compression parameter and the one that we derive below from the analysis of raw data alone. By raw data we mean the data entering step (4), because they are centrally accessible via the control system or the PIMS itself The idea is to analyze the data in an external configuration tool called Alanda, which then suggests compression parameters for steps (4) and (5). PIMS 1. linearized signal at sensor

2.A/D converter

4. Step compression

Derive compression parameters (wavelets analysis)

5. Piecewise linear compression

Archive

ALANDA

Figure 1: Signal path and interaction between PIMS and the configuration tool Alanda For the subsequent discussion, let A be the smallest step change entering (4), i.e. the visible data resolution, and a the standard deviation of the signal noise. We can always determine A, but a may be unaccessible if the resolution is too coarse. Figure 2 shows the effect of data resolution on the raw data. Both examples are fi'om real plants; details are not given for reasons of confidentiality. In Figure 2a, resolution dominates over process noise (o « A), creating a block-like structure when the measured data points are connected. The reason is that there are steady state periods where the measurement assumes one out of two or three adjacent values determined by the data resolution. Figure 2b shows a situation where process noise and resolution are

Automatic Adjustment of Data Compression

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comparable (a ~ A). Both situations may cause difficulties when a is to be determined from raw data. This is why we distinguish two types of data: Resolution and noise dominated data. The distinction is made using a heuristic: If more than half of the samples has the same value as its predecessor, a data set is considered resolution dominated. Resolution dominated data show a behavior as in Figure 2a or 2b. Wavelet denoising methods for noise estimation may not work as expected, but the resolution is a good starting point to suggest a reasonable compression parameter dev. Typically, one will avoid recording adjacent points within block-like steady states, i.e. dev should be larger than A, e.g. dev = 1.5A. We often observed block-like steady states with a greater width and, therefore, suggest dev = 2.5A. Note that integer multiples of A are not a good idea because unwanted samples may still be kept due to finite precision arithmetics. For noise dominated data, we can obtain a good estimate of a. How should dev be chosen in this case? Assuming a noisy steady state, where ideally we would want to record only the first and last sample, we can use standard statistical reasoning. We suggest dev = 2c which means that 95.5% of the samples will be discarded if we have white noise. The reasoning above applies to the piecewise linear compression step (5). For the precompression step (4), a reasonable parameterization is devpre = A/2. This discards only unchanged values and leaves the handling of the rest to linear compression (5) that is more effective.

£> 71.45

Figure 2: Influence of data resolution: a) resolution is dominating, block-like structure b) limiting case where noise level is similar to resolution

3. Algorithms and implementation The following algorithms have initially been implemented in Matlab and later made available in a standalone configuration tool called ALANDA. 3.1. Estimation of data resolution A The following algorithm is used to find the resolution A for a data vector yO of length m=l..M: 1. Sort the entries in yO (increasing values), resulting in y 1. 2. Remove the minimum and maximum entries in yl to avoid problems with data on bounds, resulting in y2 with length M2 < M. 3. Calculate the vector dy of absolute differences of y2, of length M2-1. 4. Remove all zero entries in dy, resulting in a vector dyl. 5. Use the minimum entry of dyl as A.

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Note that this algorithm will not work if there are not enough changes in the data. This can happen for small data sets or in the case of setpoint values. Our implementation checks if the data are adequate. 3.2. Estimation of standard deviation a using wavelets Wavelets are families of mathematical basis functions that are non-zero only on a finite interval. They have the ability to capture the essence of a signal y(t) in a small number of coefficients (Strang and Nguyen, 1996), by summing over translated and dilated basis functions. A signal y(t) is decomposed into

k

j=l

k

where Cj,k are approximation coefficients, dj,k detail coefficients, Oj k are the scaling and ^j,k the wavelet functions. J is the number of dyadic scales and j the scale index, and k is the translation index. For signal denoising or data compression, coefficients below a threshold t are set to zero. In practice, only very few non-zero coefficients are needed to achieve a good approximation of the original signal, which makes wavelets useful for data compression. A reasonable threshold can be inferred from the signal itself (Donoho and Johnstone, 1995) - called the Waveshrink method. There is one parameter - the decomposition level J - that must be adapted to the signal characteristics. A useful value is half of the maximum possible depth, yielding good results in practice (Nounou and Bakshi, 1999). Tona et al. (2005) have proposed an alternative method called Levashrink where J is also adapted to the signal itself In this work, we are using the biorthogonal spline wavelet of order 3.3 that has proven useful in previous applications on a wide range of industrial data sets (Alsmeyer, 2005). 4. Validation and Results 4.1. Implementation of original compression algorithms The original compression algorithms as implemented in the PIMS are needed to test the effectiveness of the tuning algorithms developed in this work. The algorithms were implemented in Matlab using the available documentation from the PIMS vendors. Then we used pairs of uncompressed snapshot data from the PIMS and the corresponding compressed data from the PIMS archive to validate our implementation. 4.1.1. Modified Boxcar-Backslope algorithm The Modified Boxcar-Backslope algorithm (MBBA) is described in the data base manual of InfoPlus.21 (AspenTech, 2002). The description is somewhat ambiguous, but we could reproduce the IP compression results after we introduced rounding of the time elapsed to the nearest fiill second. 4.1.2. Swinging Door algorithm The Swinging Door algorithm (SDA) is described in OSI's PI server reference manual (OSISoft, 2003). Note that the SDA here differs from the one in the original publication (Bristol, 1990). In the original SDA, a newly arriving point is not required to lie in the center of the parallelogram spanning the region of values that can be discarded. The algorithm in the PI system is, therefore, less efficient than the original one. In the PI system, there is an additional boxcar compression that is applied before the actual SDA (step 4 in Figure 1). We have implemented both compression steps.

Automatic Adjustment of Data Compression

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4.2. Validation with plant data We have used various pairs of compressed and uncompressed data sets from real plants to validate our approach for compression parameter adjustment. The data sets included all relevant types of measurement like temperature, pressure, flow, or level, as well as different categories like actual values, setpoints, and controlled variables. Below are two examples how existing compression settings can be corrected using our approach. Figure 3 shows a noise-dominated temperature measurement that is overcompressed when the current settings are used, with a mean absolute deviation MAD=1,24K The transient phenomenon after the step change on the left of the Figure is not well reproduced. With the corrected compression settings, the oscillations appear more naturally (MAD=0,28X), and it seems possible to use the reconstructed signal for tasks like controller performance assessment. The compression factor CF, which is the number of samples in the uncompressed signal divided by the number of retained samples, is 18 vs. 56 in the current scheme. 170 c 165

CO

160

CO

> 155

uncompressed automatic compression settings current compression 06:00 time

Figure 3: Overcompression in a temperature measurement and automatic correction Figure 4 shows the reverse case of undercompression in a resolution-dominated pressure measurement. The current compression captures oscillations in blocks and in fact does not compresses at all (CF=1, MAD=OmZ?ar). After correction, the compression factor raises to CF=7.5, using less space for adequate reproduction with a small MAD=6,09wZ?ar. Generally, our approach yields compression settings that generate a reasonable signal when inspected carefiilly. In the data sets we examined, overcompression seemed to be somewhat more common than undercompression. In some cases, after correction, too many of the samples are retained, but this is by design: We definitely want to avoid overcompression as it is detrimental to data quality.

5. Conclusions We have developed an algorithm for automatic adjustment of compression parameters in process data archives (PIMS), based on the analysis of data resolution and noise in raw process measurements. We have validated this approach with extensive compressed and uncompressed data sets from real plants. The application of the automatic algorithm improves data quality from PIMS. In some cases the method has a tendency to

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undercompress, because it has been designed to yield conservative settings. This is not seen as a problem as storage costs will continue to decrease. Nevertheless, we are hoping to further improve the algorithms by incorporation of prior information about the process variables (e.g. setpoints). This is under current investigation.

3600 „

3550

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uncompressed automatic compression settings current compression time

Figure 4: Undercompression in a pressure measurement and automatic correction

References Alsmeyer, F., 2005, Trend-based Treatment of Process Data: Application to Practical Problems, 7* World congress of chemical engineering, Glasgow, Scotland AspenTech, 2002, InfoPlus.2FM 5.0 Database User's Manual Bristol, E.H. 1990, Swinging door trending: Adaptive trend recording, in: ISA National Conference Proceedings, pp. 749-753 Cao, S. and R. R. PUiinehart, 1995, An efficient method for on-line identification of steady state, J. Proc. Cont., 5(6), pp. 363-374 Donoho, D. L. and I. M. Johnstone, 1995, J. Am. Stat. Assoc, 90, pp. 1200 Flehmig, F., 2005, Automatische Erkennung von Trends in Prozessgroi3en (automatic detection of trends in process quantities), PhD Thesis, RWTH Aachen Nounou, M. N. and B. R. Bakshi, 1999, AIChE J., 45(5), pp. 1041 OSISoft, Inc., 2003, PI Server Reference Guide, Version 3.4 Strang, G. and T. Nguyen, 1996, Wavelets and filter banks, Wellesley-Cambridge Press, Wellesney, MA, USA Thomhill, N. F., M. A. A. S. Choudhury and S. L. Shah, 2004, The impact of compression on data-driven process analyses, J. Proc. Cont., 14, pp. 389-398 Tona, R. V., A. Espu~na and L. Puigianer, 2005, Improving of wavelets filtering approaches. In: Proc. European Symposium on Computer Aided Process Engineering - 15 Watson, M. J., A. Liakopoulos, D. Brzakovic and C. Georgakis, 1998, A practical assessment of process data compression techniques, Ind. Eng. Chem. Res. 37, pp. 267-274

Acknowledgements Funding of this work by the AixCAPE member companies BASF, Bayer Technology Services and Degussa is gratefully appreciated.


Virtual Plant, New Paradigm for Future Production Management Hossam A.Gabbar, Kimitoshi Nishiyama, Ikeda Shingo, Teruo Ooto, Kazuhiko Suzuki Graduate School of Natural Science & Technology, Division of Industrial Innovation Sciences, Okayama University, 3-1-1 Tsushima-Naka, 700-853OOkayama, Japan Abstract Operator is a key player in plant operation. However, still operator working environment is limited to traditional interfaces and monitoring systems, which include actual plant, sensors, alarms, and other process and operation condition monitoring systems. Providing operator with virtual environment that integrates plant and process conditions in actual and virtual modes will support operator decisions in normal and abnormal situations. This research work discusses current limitations and proposes future generation virtual plant environment that enables operator to comprehend current plant and process condition and predict future states for safe and optimum operation. To achieve such target, process modeling is proposed to analyze operator activities in view of process design and operation practices. Keywords: virtual plant, plant operation environment, operator activity modeling. 1. Introduction Plant operation is a complex process where there are critical decisions and activities that need to be taken in timely and highly accurate manner. The system approach is widely used to improve plant operation starting from the concept, design, and operation engineering till the control and management stages. However, still there are major limitations in operator environment that requires further investigations and considerations. Chemical plant operation requires complete understanding and management of activities related to process chemistry, environmental impacts, product and production systems, etc. In addition, plant operation requires carrying out management activities related to maintenance, production scheduling, operation planning, human resources management, and financial matters. In such complex domain, operator requires extensive information in different forms to be able to perform the different tasks accurately and easily. Till to date, there are different solutions that have been proposed to address these needs, such as intelligent and integrated software systems, i.e. ERP, or by enabling new technologies such as multimedia and virtual realities. In current practices, operator is carrying out most of these activities efficiently using some automated tools via intelligent user interfaces, which provide operator with the required information. However, still accidents are happening due to operator errors, especially within the control room domain. In addition, there is always delay in production schedules, which causes costs and affects business reputation. In addition, there are other critical factors, such as those related to environment, recycling, energy management, and occupational safety, which are not supported by current operation management and interface systems.

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Process system modeling and engineering approaches are widely used to provide practical solutions that link process engineering practices with software / hardware, and technologies. Such system engineering approaches are based on process modeling where elements of plant operation are modeled in an integrated view so that operator environment can be reengineered and realized in practical manner, while establishing the links among different elements of operational environment. This research paper discusses current limitations of operational environment and their impact on plant safety. From such discussions, a proposed virtual plant framework, design, and its realization are illustrated, and system architecture of the proposed virtual plant environment is illustrated. In the following section, issues around current operational environment will be discussed. In section three, system approach is proposed to design target virtual environment, which is used to describe proposed virtual plant, as illustrated in section 4. 2. Current Operational Environment Currently, operator is carrying out different activities such as process monitoring, operation execution, maintenance tasks, and safety assessment activities. In addition, he is carrying other activities such as operation scheduling, planning, human resources management, scheduling, procurement, and financial. In such multidimensional environment, he is required to do job accurately, timely, optimally, and safely. Moreover, there is another constraint of increasing complexity of chemical plants. Most of these activities are carried out concurrently by operator with direct communication with operation engineers and management. In most of chemical and oil & gas production plants, operator works in control room and directly with equipment interaction in the plant process area, i.e. shop floor. In the control room, there are set of computers, DCS, monitors and other monitoring devices that reflect plant process condition. Usually, operator spend most of the time with DCS and HIS or human interface systems to monitor plant condition in terms of sensors and alarm. In most of production plants, plant and process condition monitoring systems are used to reflect real time condition for better operation decision and management. And for operation execution and monitoring, SFC, or sequential fiinction chart, is used to show the current step of plant operation. Operator needs to make mental work to link DCS values with SFC. In addition, operator carry out operation planning and scheduling activities in separate PC with a separate monitor based on received instructions of production planning and quality. From the incoming operation schedules, operation execution and control setup points will be defined and executed. Such switch among different computation environment and monitors might lead to errors, delay, and fiiistration. In addition, it might lead to exerting more efforts which has negative impact on physical and mental operator limits. In addition to these requirements, operator needs to carry out safety and environmental assessment regularly where some paper work is done on P&ID and engineering diagrams to write down list of possible hazards and causes, consequences, and safeguards for safe operation. Moreover, factors from waste and water treatment and energy management are considered as part of plant operation, which will ensure clean and cheap energy and less impact on environment. This caused additional work to be carried out by operator to

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reduce environmental impacts from energy utilization and water and waste treatment. To manage these activities, i.e. water, waste, and energy management, the different automated systems are usually executed in different workstations and interface systems. These limitations can be eliminated by using sophisticated virtual plant environment that augment the different monitoring, control, and interface systems. The proposed approach to achieve such virtual plant is via system approach, which is explained in the following section. 3. Process System Modeling Approach to Design Virtual Plant In view of current limitations of operator interface and support environment, this paper proposes a robust process system modeling approach that links the different automated systems, their information models, and operator activities and tasks, as shown in figure 1. The proposed process system approach is based on defining flowchart for the different activities that operator performs. These activities are classified in terms of process operation modes such as startup, shutdown, recovery, etc. In each mode, more detailed activities are developed such as in case of detected alarms or process deviations. In figure 1, operator activities are defined for a case when fault or abnormal situation is detected. In such case, operator validates sensor data, simulation data, operation history, and use incident / accident data to confirm possible and root causes and consequences. FDS or fault diagnosis system is proposed by Gabbar (2006) to do such task and provide list of possible fault propagation scenarios. For each activity, list of tasks are identified and associated information elements are listed along with the reference systems that owns / maintains such information elements, i.e. alarm data from DCS. Providing such list will enable the reengineering of operator interface systems and engineering of virtual plant environment with suitable visualization and interaction mechanism, such as 2D / 3D diagrams, voice, animation, textual, video, images, etc. In addition, human factors are analyzed for each task to provide suitable visualization and interaction mechanism. Human factors are analyzed using predefined list of main and detailed factors that are related to operation task, including physical and mental models (Jamieson, 2005). These factors are considered in each task / activity in process model diagrams, which provides input specification for virtual plant and operator interface system design. 4. Proposed Virtual Plant Virtual plant has set of key features that are proposed in view of system requirement and the reported limitations of current operator interface systems. One key feature is the synchronization between actual and simulation models where operator is required to perform real time operation by evaluating different operation scenarios using dynamic simulation. To overcome this requirement, integrated simulation environment is proposed where real time integration module is developed using VB. In this experiment dynamic simulator software is used to capture real time DCS data, which is stored in CSV or excel format and transferred every 20 seconds from DCS to simulator workstation. Such sensor data are compared with simulation data and used to tune simulation models and propose changes to operation control and execution. Detailed about this mechanism is explained in separate research report. Another feature of the target virtual plant environment is the use of latest technologies such as GIS or geographical information system, to capture and visualize plant and process condition in the corresponding geographical locations. Operator will be able to view high temperature locations in 2D / 3D in physical plant location for faster response and

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counteraction. In addition, CFD or computational fluid dynamics software is proposed to be integrated within virtual plant to visualize the inside of process equipments and facility locations. Task

Activity

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References M. Aoki, 1990, State Space Modeling of Time Series, 2^^ edition. Springer-Verlag, Berlin Heidelberg New York H. Akaike, 1974, Markovian representation of stochastic processes and its application to the analysis of autoregressive moving average processes. Ann. Inst. Statist. Math., 26, 363-387 S. Wold, 1994, Exponentially weighted moving principal components analysis and projections to latent structures, Chemometrics and Intelligent Laboratory Systems, 23, 149-161 X. Liu, T. Chen, S.M. Thornton, 2003, Eigenspace updating for non-stationary process and its appUcation to face recognition. Pattern Recognition, 36, 1945-1959 C.-T. Pan, R.J. Plemmons, 1989, Least squares modifications with inverse factorizations Parallel implications. Journal of Computational and Applied Mathematics, 27, 109-127 S. W. Choi, E. B. Martin, A. J. Morris, I.-B. Lee, 2005, Adaptive Multivariate Statistical Process Control for Monitoring Time-varying Processes, Journal of Process Control, accepted. P. Hall, D. Marshall, R. Martin, 2000, Merging and splitting Eigenspace models, IEEE Transactions on Pattern Analysis and Machine Intelligence, 22, 1042-1049. S. W. Choi, C. K. Yoo, I.-B. Lee, 2003, Overall statistical monitoring of static and dynamic patterns. Industrial & Engineering Chemistry Research , 42, 108-117


Scheduling of make and pack plants: a case study Cord-Ulrich Fundeling^, Norbert Trautmann^ Înstitute for Economic Theory and Operations Research, University ofKarlsruhe Kaiserstrafie 12, 76128 Karlsruhe, Germany ^Department of Business Administration, University of Bern Engehaldenstrafie 4, 3012 Bern, Switzerland Abstract We consider a case study published by Honkomp et al. (2000) that refers to the scheduling of a make and pack plant. Due to the problem size, the approaches known from literature for this kind of scheduling problem seem not to be capable of computing a feasible solution. We therefore present a new priority rule method where operations are scheduled successively taking into account the technological constraints. The order of the operations is determined by priority values assigned to groups of operations. Using this approach we are able to compute a feasible production schedule for the case study within less than one second of CPU time. To the best of our knowledge, no solution to this case study has been reported before. Keywords: case study, make and pack plant, scheduling, priority-rule based solution method 1. Introduction In the case study published by Honkomp et al. (2000) a make and pack plant is considered. A make and pack plant is a two-stage production plant where the two stages are linked by intermediate storage tanks with limited capacities. On the first stage (make stage), the production process takes place whereas on the second stage (pack stage), the products are packed into consumer packages for shipping. To produce and to package products, different operations have to be performed for which a set of parallel processing units (mixing tanks and packing lines) is available on each stage. Several technological constraints have to be respected: After the production process products have to be stored at least for a given quality release time. Processing units as well as the intermediate storage tanks have to be cleaned between the processing of different operations where the cleaning time depends on the products involved. In the case study, demand data of 10 weeks is given for the different combinations of products and consumer packages leading to a total of more than 650 operations per week to be scheduled. The problem consists in determining a production schedule for each week such that the demand is met, the technological constraints are respected, and the makespan is minimized. In literature, numerous models and solution procedures have been presented for a large variety of scheduling problems; for a review, we refer to Brucker (2004). There are only a few publications dealing explicitly with the scheduling of make and pack plants. Belarbi and Hindi (1992) develop a heuristic procedure for scheduling make and pack plants where intermediate storage capacity is limited. They illustrate their approach by solving two instances containing about 30 operations to be scheduled each. Ramudhin and Ratliff (1995) present a priority rule as well as a langrangian relaxation-based

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heuristic in order to solve a case study from food industries. Due to hygienic reasons no intermediate storage tanks have to be taken into account. Instances with up to 300 operations to be scheduled are solved to feasibility within at most 140 seconds. Mendez and Cerda (2002) develop a mixed-integer linear optimization model for the scheduling of make and pack plants. Intermediate storage tanks are assumed to have unlimited capacity. The largest instance that can be solved to feasibility within a few seconds of computing time contains 53 operations to be scheduled. Eventually, Gupta and Karimi (2003) state an alternative mixed-integer linear program. The capacity of intermediate storage tanks is again assumed to be unlimited. Due to the large number of binary variables 40 minutes of computing time are needed in order to solve an instance containing 90 operations to be scheduled. Since none of these solution approaches seems to be capable of solving problem instances of the size given in the case study we do not expect that the case study can be solved to optimality within reasonable time. We therefore present a new priority rule method where operations are scheduled one after the other taking into account the technological constraints. Priority values assigned to groups of operations determine the order in which operations are scheduled. Priority rule methods have been very successftil in finding near-optimal feasible solutions for various kinds of scheduling problems (cf. Franck et al., 2001).

2. Case study The data of the case study considered stems from the Procter & Gamble company (cf Honkomp et al., 2000). The make stage consists of 3 pre-mix as well as 6 final-mix tanks. Two final-mix tanks have been assigned to each pre-mix tank. 80 intermediate storage tanks are located between the make and the pack stage, the latter consisting of 7 packing lines. In total, 59 different final products can be produced and packed in various package types. In the case study, demand data is given for 203 combinations of final product and package type and varies unsystematically between 0 and 228,11 product units. A planning horizon of 10 weeks is considered. The production and packaging of an arbitrary final product runs off as follows (cf Figure 1): • Production (make stage). Raw material is converted into some intermediate product by executing an operation of type 01 on an appropriate pre-mix tank. The intermediate product is transferred (TF) to a final-mix tank immediately after the completion of this operation. By executing a second operation of type 02 on the final-mix tank, 10 units of some intermediate product together with some raw material are transformed into some final product. For some final products, no operation of type 01 needs to be executed. All operations are discontinuous, i.e., no material is added or removed during their execution, whereas the transfer of the intermediate product is a continuous process. Therefore, both the pre-mix as well as the final-mix tank are occupied during the transfer. Intermediate and final products are assigned to several product groups. Between the processing of products belonging to different product groups the pre-mix or final-mix tank, respectively, has to be cleaned. The processing times of the operations depend solely on the product in question but not on the pre-mix or final-mix tank. • Intermediate storage. After the termination of an operation of type 02 the final product is pumped out to one or two intermediate storage tanks (PO). The capacity of 6 of these tanks is 10 units whereas the remaining 74 tanks may contain only 5 units of some final product. No restriction is imposed on the choice of the tank. Different

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products, however, may not be stored in one tank at the same time. Final products can only be filled into an empty tank. Before being filled, each tank has to be cleaned. Filling, cleaning and unloading processes may not overlap. Each final product has to stay in some intermediate storage tank at least during a given quality release time t^^ depending on the product. 01

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Figure 1: Production and packaging process at the make and pack plant • Packaging (pack stage). By executing operations of type 03 and 04, respectively, 5 units of some final product are packed on one of the packing lines. Not all package types can be handled on each packing line. The processing times for packaging depend on the packing lines. Final products are taken from the intermediate storage tanks and packed continuously. Between the packaging of final products belonging to different product groups or packed in different package types packing lines have to be cleaned or set up, respectively. An unlimited amount of each raw material is available. Capacities for the storage of packed final products are not considered. The total number of operations to be processed can be computed easily starting from the given demand quantities and varies between 679 and 917 per week. The problem tackled in this paper consists in assigning a processing unit as well as a starting time to each operation to be processed such that all technological constraints are met and the overall cycle time is minimized. Because the time horizon for processing the operations corresponding to each week is limited the problem of determining a feasible solution is already NP-hard. 3. Solution method The basic principle of the solution method is to schedule the operations to be processed successively taking into account the technological constraints. At the beginning of the

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scheduling process, priority values are assigned to each operation on the pack stage. An iteration of the solution method runs as follows: • In step 1, some operation of type 03 or 04 with the smallest priority value is scheduled during its latest execution time interval (packaging of 5 units of some final product). We thereby ensure that there is an intermediate storage tank Si whose allocation allows for storing 5 units of the final product to be packed for a sufficient time period. The fact that the final product has to be produced and stored in advance, however, is neglected. This constraint will be enforced in step 3 of the solution method. • In step 2, a further operation of type 03 or 04 with smallest priority value is scheduled during its latest execution time interval (packaging of another 5 units of the same final product). We again ensure that there is some intermediate storage tank S2 whose allocation allows for storing 5 units of the final product to be packed for a sufficient time period. Intermediate storage tanks Si and S2 may be different (two tanks with a capacity of 5 final product units) or identical (one tank with a capacity of 10 final product units). It has to be taken into account that the 10 final product units to be packed must be filled into tanks Si and S2 at the same time since they are produced as a whole. The material availability constraint is again ignored in step 2. • In step 3, two operations of types 01 and 02, respectively, are scheduled during its latest execution time interval (packaging of 10 units of the same final product). In steps 1 to 3, all technological constraints with the exception of the material availability constraint are taken into account. If in some step, no feasible execution time interval is found for some operation, the operations scheduled in the preceding steps are shifted to an earlier execution time interval. Steps 1 to 3 are repeated until the demand given for each final product is fiilfilled by processing the scheduled operations. The solution's overall cycle time depends critically on the order of scheduling und thus on the used priority rule. In unfavorable cases the overall cycle time may exceed the given planning horizon, i.e., no feasible solution is found. We make use of a multi-stage priority rule that applies different sorting criteria lexikographically to the operations to be scheduled where random numbers are used as tie-breakers. The following sorting criteria have been examined: • SWF (Smallest Wash-out Family): Operations are sorted according to different product groups. • SCF (Smallest Change-over Family): Operations are sorted according to different package types. • LMS (Least Mixing tanks Suitable): Operations are sorted according to the number of mixing tanks capable to produce the final product in demand. • LPS (Least Packing lines Suitable): Operations are sorted according to the number of packing lines capable to pack the desired package type.

4. Experimental performance analysis The priority rule method has been implemented in ANSI-C. In an experimental performance analysis we have examined which combinations of different sorting criteria lead to good feasible solutions for the case study described in section 2. The computational experiments have been performed on a PC with 2,8 GHz clock pulse and 256 MB RAM. A feasible solution has been found after 0,1 seconds on the average. For each out of the 10 weeks for which demand data is available, and for each out of the 64 possible combinations of sorting criteria, 10 runs of the solution method have been

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performed. As an example, Figure 2 shows the cycle times found by the 5 best as well as the 5 worst prioity rules for the demand data given for week 1. Significant differences exist between the solutions found by the different priority rules. The best schedules have been determined using sorting criteria LMS and SCF on stage 1 and 2 of the priority rule, respectively. For certain priority rules, however, the resulting schedules do not fulfill the given demand though there is enough production capacity. The results for weeks 2 to 10 are similar to those obtained for week 1.

__Planning "" horizon

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Figure 2: Overall cycle times found by the 5 best as well as the 5 worst priority rules (week 1) 5. Conclusions and Outlook In the present paper we have presented a new priority rule method for the scheduling of make and pack plants. Priorities are assigned to operations on the pack stage. Operations on the make stage are scheduled on demand by the pack stage. Hence, the main new characteristic of the solution approach is that priority values are assigned to groups of operations implicitly. Computational experiments have shown the efficiency of the method for a real-world case study with respect to computational times as well as solution quality. To the best of our knowledge no feasible solution to the case study has been reported before. One possible direction for future research consists in applying the solution method to generalized problem settings incorporating fiirther objective functions or constraints such as limited shelf life times. Furthermore, the performance of the solution approach should be analyzed for more complex process networks incorporating, e.g., more than two stages or convergent material flows. Moreover, it would be interesting to apply other more general priority rule approaches as well as commercial production scheduling software to the case study and to compare the solutions, if found, to the solutions determined by the presented priority rule method.

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References K. Belarbi, K.S. Hindi, 1992, Detailed scheduling for a class of two-stage intermittent manufacturing systems, Production Planning & Control, 3, 3 6 ^ 7 P. Brucker, 2004, Scheduling Algorithms (4* ed.), Berlin: Springer B. Franck, K. Neumann, C. Schwindt, 2001, Truncated branch-and-bound, schedule-construction, and schedule-improvement procedures for resource-constrained project scheduling, OR Spektrum, 23, 297-324 S. Gupta, LA. Karimi, 2003, Scheduling a two-stage multiproduct process with limited product shelf life in intermediate storage. Industrial Engineering & Chemistry Research, 42, 490-508 S.J. Honkomp, S. Lombardo, O. Rosen, J.F. Pekny, 2000, The curse of reality — why process scheduling optimization problems are difficult in practice. Computers and Chemical Engineering, 24, 323-328 C.A. Mendez, J. Cerda, 2002, An MILP-based approach to the short-term scheduling of makeand-pack continuous production plants, OR Spectrum, 24, 403^29 A. Ramudhin, H.D. Ratliff, 1995, Generating daily production schedules in process industries, HE Transactions, 27, 646-656


Detection of abnormal alumina feed rate in aluminium electrolysis cells using state and parameter estimation. Kristin Hestetun and Morten Hovd'' "Norwegian University of Science and Technology, Department of Engineering Cybernetics, N-7491 Trondheim, Norway

Abstract The concentration of dissolved alumina in the electrolyte is one factor influencing the current efficiency in aluminium electrolysis cells. Too low concentration might lead to serious operational problems, like anode effect. In this paper measurements of the current through anodes in different parts of the cell are used to estimate states and parameters using an extended Kalman Filter. Faulty alumina feed rate causing abnormal alumina distribution might then be detected by examining how the expected model output differs from the state estimate from the Kalman Filter in combination with drift in the estimated parameter values. Keywords: Fault detection and diagnosis, state and parameter estimation, Extended Kalman filter 1. Introduction Process operation that was 'good enough' some years ago might no longer meet the required standards for economic and environmental performance. Fortunately, computer power and software have developed rapidly in the last decades introducing new possibilities for better monitoring and control of processes. Fault detection and diagnosis are the central components of abnormal event management (AEM) and have become a standard part of most process control systems. AEM deals with the timely detection, diagnosis and correction of abnormal process conditions or faults in a process (Venkatasubramanian, 2003). More precisely, a fault can be defined as an unpermitted deviation of at least one characteristic property or variable of the system (Isermann and Ball, 1996). Early detection, diagnosis and correction of process faults might prevent the process from entering an uncontrollable operating region thereby reducing production losses and increase performance. In aluminium electrolysis one of the largest expenses is the costs of electric energy and maximizing energy efficiency is an important control objective. One factor influencing this efficiency is the concentration of dissolved alumina in the electrolyte. Both too high and too low concentration of alumina will lead to serious operational problems. Anode effect is one such problem that occurs at low alumina concentration and is a major process disturbance. Besides giving low energy efficiency, anode effect severely disturbs the cell energy balance and produces significant amounts of environmentally harmful CFC-gases. Despite the importance of controlling the alumina concentration, tight control are difficult because of the lack of reliable online information. Key control variables like temperature and electrolyte concentration are usually not available, forcing most control strategies to rely heavily on measurements of cell pseudo

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resistance calculated from total line amperage and change in cell voltage. No cost efficient sensor for measuring alumina concentration in the electrolyte is presently available for industrial use. Beame (1999) states that the correlation between change in pseudo resistance and alumina concentration probably will continue to be used to control alumina concentration and alumina feed rate. On these premises, any additional measurements that can give information about what goes on in the electrolyte has potential for improving control of alumina concentration and reduce spatial variations within the cell. Due to improved sensing equipment and computer control, process information that was once considered of no practical use can now give additional information that can be used to optimize process performance. If the current through each anode, or the anode current distribution, is available, this might give valuable information about alumina distribution and consumption in the cell and thereby help prevent anode effects. Rye et al. (1998) showed a correlation between areas with low alumina concentration and anodes with less current when the alumina concentration was low. This paper continues the work started in a previous paper (Hestetun and Hovd, 2005) where this correlation is used to detect abnormal alumina concentration due to faulty alumina feed rate. Measurements of current through anodes in different sections of the cell are used in a state estimator (extended Kalman Filter) to estimate alumina concentration at different locations. The goal is to detect abnormal alumina feed rate early enough to prevent an oncoming anode effect. Hestetun and Hovd (2005) focused on using the state estimate to detect differences between expected and observed rates. This paper focus on what additional information can be gained from also utilising residuals generated based on variation in the estimated parameters values.

2. Short on aluminium electrolysis Liquid aluminium is produced by electrochemical reduction of alumina (AI2O3) as current passes through a high-temperature electrolyte in which alumina is dissolved. Liquid aluminium is formed at the metal/bath interface acting as the cathode while carbon-oxide gases are produced at the anodes, according to the main cell reaction Al Ô .{diss

) + —C{s)^

lAl

{I) + —CO 2 ( g )

^^^

A sketch of a modem prebake electrolysis cell is given in Fig. 1. 20-30 individual prebaked anodes are positioned in two rows and connected in parallel to the horizontal bus bars. The high amperage DC current enters the cell through the bus bars and distributes among the individual anodes. According to Ohm's law, more current will go where there is less resistance and how the current distributes between the anodes depends upon the resistance in the anode and in the inter-electrode gap. Since resistance in the electrolyte strongly depends on alumina concentration, measurements of the individual anode currents do give information about how alumina is distributed throughout the cell. Alumina is fed to the electrolyte through two feeders, one in each end of the cell. The electrolyte is covered by a layer of frozen electrolyte and alumina powder, so before each feeding operations a hole must be made in the top crust with a bar, allowing a controlled amount of alumina powder to be dropped into and dissolved in the electrolyte.

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Cells

3. Modeling, state and parameter estimation A simple model for estimation of alumina concentration at different locations in the cell has been developed. The resulting model is a non-linear state-space model with a number of unknown parameters: X = fix,u,0) (2) y = g{x ,w,6>) Both/and g are nonlinear functions, g calculates the total voltage drop for the cell. Several terms contribute to cell voltage drop, some of which depend on the alumina concentration. Equations for calculation oig are found from literature data (Solheim, 1998; Haupin, 1998)./is derived from the mass balance of dissolved and undissolved alumina and change in anode-cathode distance. For each control volume, the states are x = [\|/ c 6], where \|/ and c are the concentration of undissolved and dissolved alumina respectively, and 6 is the anode-cathode distance. The parameter vector 6 = [ki_^2 ki-^s kmtf Me] where ki^2 and k2-^3 are dispersion parameters describing transport of alumina between adjacent control volumes, k^tf is the mass transfer coefficient for dissolution of alumina and Mg is the mass of electrolyte in each control volume. The rate of dissolution of alumina is assumed to be proportional to both the concentration of undissolved alumina and the degree of under-saturation of alumina in the electrolyte. The rate of metal production (and alumina consumption) is given by Faraday's law and the cell current efficiency factor. The model inputs u = [Fl F2 ACD II .... I J are the feed rate of alumina for the two feeders, the change in anode-cathode distance and the anode currents through the n sections in the cell. A9

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Fig. 1: Layout of prebake electrolysis cell used in the experiments. Placement of anodes (A), control volumes (V), Fig. 2: Sketch of prebake electrolysis cell, as seen feeders (F) and measurement points are from the shorter end. indicated. The objective of this work has been to detect abnormal situations rather than to accurately estimate concentration profile throughout the electrolyte. Earlier work (Jakobsen et al., 2001) dealt with the problem of estimating alumina concentration underneath each anode. However, to achieve good results with this model, reliable information about electrolyte flow pattern is required. This is in general not available during normal operation and to avoid the problem of the unknown flow pattern, the

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number of control volumes has been reduced to three. This allows all mass transport of alumina between adjacent control volumes to be approximated as dispersion while the differences between different parts of the cell can still be detected. The state and parameter estimation is build around an extended Kalman Filter (EKF) representation implemented in Matlab. In general,

yk = gi^kûj^,ej^)

+ w,

(3)

where Kk is the Kalman filter gain matrix. The parameter vector 6 and the initial states are unknown and must be estimated from experimental data. The program SENIT ModelFit was used to simultaneously determine initial values for states parameters by fitting the model response to measurement data collected from industrial electrolysis cells at Elkem Aluminium's smelter in Mosjoen, Norway. This program utilizes an algorithm based on a modified extended Kalman filter (Schei, 97) to estimate states and parameters in nonlinear state space models. An outer SQP-type optimization loop optimizes the initial values for the parameters and states by minimizing the difference between the measured and modeled output, here alumina concentration and total cell voltage drop for each control volume. 0"'' = min VAî,,) (4) 1 ^ ^7V ( ^ 7 , A : ) ~

î,k

= yTk^'

~\j~ 2^

-yi,k

\ ^ v,k^

v^

v,k

"•" ^ Al20^,k'^

Al jOj^

Al20i,k

)

'• = { v , ^ / 2 < ^ 3 }

Fig. 3 illustrates how the resulting model with optimal parameters and initial states (no Kalman filter update) fit the experimental data of measured alumina concentration. 4. Fault detection and diagnosis, results and comments. 4.1. Experimental data Faulty equipment, like a leak in one of the feeders or a malfunctioning bar, could mean that the amount of alumina that is actually dissolved in the electrolyte is considerably different from what is assumed by the process control system. If this goes on unnoticed it may lead to severe process disturbances. The experimental data collected from aluminium electrolysis cells at Elkem Aluminium ANS' smelter in Mosjoen, Norway contains data from periods with normal operations as well as data containing abnormal feed rate that eventually leads to anode effect. I. e., after a period with normal feed rate, the feed rate to the cell was reduced, either by closing one or both of the feeders in the cell (exp. 1 and 2 respectively) or by keeping the feed rate at a reduced rate too long (exp. 3). During the experiments alumina concentration was measured at six positions in the cell while the current through each individual anode were logged in addition to the standard measurements recorded in the process database. An overview of the layout of the cell can be seen in Fig. 2. 4.2. State and parameter estimation. Fig. 4 illustrates how measurements of anode current distribution in addition to standard measurements can be used in the EKF filter to indicate abnormal alumina distribution.

Detection of Abnormal Alumina Feed Rate in Aluminium Electrolysis Here (exp.l), the actual feed rate of the cell is only half of what is assumed (and recorded in the process database) since feeder Fl is actually closed down. Note that the measurements of alumina concentration are not used to update the estimates in the Kalman filter. These measurements are usually not available. Only measurements of cell voltage and anode current distribution in addition to assumed feed rate and change in anode cathode distance from the process database are used to produce the results in Fig. 4. The residual generated from the difference between the expected and estimated concentration (Fig. 4, third plot from the top) clearly indicates that something is wrong.

•.*..,-—-->.:.+

2.5

1561

Cells

•

•;+

-

2 Model output section 1

1.5 1

+

Measurement section 1

•*

'

'

+

'

^""~^'^"^^-~t Model output section 3 *

Measurement section 3

Time [sample index]

In previous work (Hestetun and Hovd, 2005) the model parameters were assumed to be Fig. 3: Measured alumina concentration constant. In reality the parameters represent compared to model output using optimal physical properties that vary with changing parameter values and initial states for exo. 1. process conditions. Even when the measured input rate is correct, the parameters might vary somewhat. For example will a changes in temperature most likely affect all parameters. Here the parameters are modeled as integrated white noise and estimated together with the states using the extended Kalman filter. When the model inputs are "faulty", parameters estimated "online" might also try to compensate for the faulty conditions. By monitoring the parameter behavior and detect out-of-normal variation, we can use the parameters in a fault detection and identification scheme in combination with the results indicated in Fig. 4. Fig. 5 and Fig. 6 show residuals generated from the Time [sample index] difference between the nominal and estimated parameters in exp. 1. The deviation in parameters can serve as a basis Fig. 4: Detection of abnormal feed rate using for detection and isolation of faults. The state and parameter estimation. From the top: change in parameter values in the last 200 Model response (no KF update) with faulty feed rate, state estimate from KF with faulty samples does clearly indicate an abnormal feed rate, difference in estimated and situation. The increase in mass of electrolyte modeled (No KF update) alumina feed rate, (Fig. 5) might indicate that there has been a and last assumed, faulty feed rate. temperature increase, which is consistent with both temperature measurements during the experiment and expected behavior prior

1562

K. Hestetun and M. Hovd

to an anode effect. In Fig. 6 the estimated diffusion parameters diverge in different directions. As the model believes that both Fl and F2 have the same, normal feed rate, the estimated diffusion parameters try to reflect the observed difference between the two sections of the cell. When an abnormal situation is detected, this additional information from the parameters might help to identify a likely cause of the fault. 5. Concluding remarks Even though much work remains in order to make this a robust fault detection and identification scheme, the observed results are interesting as they do give indication of what goes on inside the electrolyte. By considering both the parameters and the states when trying to identify an abnormal situation more information of the process can be obtained. It should be noted that the experimental data presented her all involve rather large manipulations compared to normal feed rate. Though we were able to detect the faulty feed rate with these data, data from normal operation containing more realistic abnormal situations should be analyzed. Other issues that should be adressed involve robust tuning of the Kalman filter matrices, how to ensure proper initial estimates and how to handle changes in anode-cathode distance.

References V. Venkatasubramanian, R. Rengaswamy, K. Yin, and S.N. Kavuri, 2003, A review of process fault detection and diagnosis Part I: Quantitative model-based methods. Computers & Chemical Engineering, 27, pp. 293-311. R. Isermann, and P. Ball, 1996, Trends in the application of model based fault detection and diagnosis of technical processes, In Proc. Of the 13^*^ IF AC World Congress, volume N, pages 1-12, Piscataway, NJ. G.P. Beame, 1999, The development of aluminium reduction cell process control. Journal of the Minerals, Metals and Materials Society, 51(5), pp-16-22. K.A. Rye, M. Konigsson, and I. Solberg, 1998, Current redistribution among individual anode carbons in a Hall-Heroult prebake cell at low alumina concentration, In: TMS Light Metals 1998, pp. 241-246. K. Hestetun, and M. Hovd, 2005, Detecting abnormal feed rate in aluminium electrolysis using extended Kalman filer, IFAC World Congress 2005, Prague. S.R. Jakobsen, K. Hestetun, M.Hovd and I.Solberg, 2001, Estimating alumina concentration distribution in aluminum electrolysis cells. In Proceeding of the 10* IFAC Symposium on Automation in Mining, Mineral and Metal Processing, pp. 253-258. W. Haupin, 1998, Interpreting the components of cell voltage. In: TMS Light Metals 1998, pp. 531-537. A. Solheim,1998, Reviderte aktivitetsdata for NaF, A1F3 and A1203, Technical Report SINTEF. T.S. Schei, 1997, Afinite-differencemethod for linearization in non-linear estimation algorithms, Automatica, 33(11), pp. 2053-2058.

/-J T.'>*

--^ât/

3.^!!!s-«<Sft;ij-if

,'/.v\ •'vAf

Time (sample intdexl

Fig. 5: Change in total mass of electrolyte during exp. 1.

Tim© (s^mpte iind#x|

Fig. 6: Change in dispersion parameter values, during exp. 1.


Thermodynamic diagram based estimation structure design for ternary distillation column Anna Pulis^ Carlos Fernandez^, Roberto Baratti^ Jesus Alvarez^ ^ Dipartimento di Ingegneria Chimica e del Materiali, Universitd di Cagliari, Piazza D'Armi, 09123 Cagliari, Italy ^ Departamento de Ingenieria de Procesos e Hidrdulica, Universidad Autonoma Metropolitana-Iztapalapa Apdo. 55534, 09340 Mexico, D.F., Mexico

Abstract In this work the ternary distillation column estimation problem is addressed, using an ethanol-tertbutanol-water pilot experimental column. The estimation problem is more difficult and less studied that the one of the binary case. The combination of a thermodynamic framework with results drawn in a previous structure-oriented geometric estimation study yields a procedure with criteria with physical meaning to jointly design the estimation structure and algorithm. The approach is illustrated and tested with an adjustable-structure geometric estimator implementation. Keywords: ternary distillation, estimation problem, geometric observer.

1. Introduction The need of designing or redesigning the separation process for better compromises between productivity, quality, safety and costs in conjunction with the availability of computing capability motivate the need of developing more systematic and reliable distillation column monitoring and control design methodologies, and this in turn justifies studies on the development of on-line estimation techniques to infer compositions on the basis of temperature measurements. The observability problems of binary and multicomponent distillation columns have been extensively studied and tested with a diversity of techniques. The related state of the art can be seen elsewhere [1-8], and here we circumscribe ourselves to mention that only a few studies have considered the multi-component case, that is considerably more difficult than the binary case, for the following reasons: (i) the relationship between temperature and composition is not uniquely defined, (ii) some colurmi trays may lie in stage zones with low temperature changes with compositions, and/or small composition changes, (iii) these features may change along a transient behavior, and (iv) there column model exhibits strong sensitivity with respect to the choice of the vapor-liquid equilibrium description. In a recent study [8], the multicomponent column estimator design problem was successfully addressed within an adjustable-structure geometric estimation framework [9]. However, the results are highly systems theory oriented and devoted from interpretation in the light of standard thermodynamic arguments, and this motivates the scope of the present work: the improvement of the ternary column estimator design by incorporating thermodynamic tools and arguments that are commonly used in the field of distillation column engineering. In this work, the ternary distillation column estimation problem is addressed, using an ethanol-tertbutanol-water pilot experimental ternary column. The combination of a thermodynamic fi'amework with results drawn in the above mentioned nonlinear geometric estimation study yields a combined structure algorithm estimation design in terms of concepts and criteria with

1563

1564

A.Pulisetal

clear physical meaning. The approach is illustrated and tested with an implementation on the basis of experimental data. 2. Estimation problem Consider a continuous ternary distillation column with, feed rate F at composition Cp, distillate {D) and bottoms {B) at composition CD or CB, heat load Q (proportional to vapor flow rate V), and reflux flow rate R. Under the standard assumptions (liquid-vapor equilibrium at each stage, linear pressure drop along the column, quasi steady-state hydraulics and equimolar flow) the column behavior is described by the following equations [10, 11]: Stripping section (7 . \ ^ ^- :

NG • - • LIQ . . . SUB — LIQcold

180 170 160 150

a2oo

140

195

130

190

120 185

)

50

100 Position (c) NG2

1^)0

( (d) NG3

Figure 2. Temperature profiles

5. Control s t r u c t u r e d e s i g n In the section above we where able to identify the optimum for the process, but how should this optimum be implemented in practice? First we need to control the active constraints: • • • •

Degree of super-heating (4 locations): For this we may use the corresponding choke valve opening Pi is for each of the 3 cycles: For this we may use "active charge" (see discussion above) Maximum cooling in 4 SW coolers: SW flow at maximum LNG outlet temperature at -155°C: May use first compressor stage in SUB

The four remaining degrees of freedom should be used to control variables which have good self optimizing properties: "Self optimizing control is when we can achieve acceptable loss with constant setpoint values for the controlled variables (without the need to re-optimize when disturbances occur)" [5].

Optimal Operation of a Mixed Fluid Cascade LNG Plant

1573

Table 1 Optimal operation of a MFC process PREl Pi ^m

Ph

c, C2 C3 N2

Flow

w.

[Pa] [Pa] [Pa]

[%] [%] [%] [%] [mol/s] [MW]

PRE2

LIQ

SUB

2.00 6.45 15.03 15.03 0.00 0.00 37.70 37.70 62.30 62.30 0.00 0.00 464 685 1.2565 + 2.644

2.00

2.00 28.38 56.99 52.99 42.45 0.00 ' 4.55 627 3.780+1.086

6.45

-

20.58 4.02 82.96 13.02 0.00 390 2.128

SW(

Figure 3. Suggested control structure for the MFC process. SH are degree of super-heating controllers, P C and T C are pressure and temperature controllers respectively. Not shown: Three pressure controllers on the low pressure side using the active charge in each cycle

To evaluate the loss one needs to consider the effect of disturbances and implementation errors. A steady-state analysis is usually sufficient because the economics are primarily determined by the steadystate. Based on physical insight the following four variables may been suggested » Tout • ^NGIA ^ rpout

• Pm

1574

J.B. Jensen and S. Skogestad

A possible control structure with these four variables and the active constraints controlled is shown in Figure 3. However, note that the "pairings" of controlled and manipulated inputs are included primarily to illustrate that we have available degrees of freedom, as this does not matter for evaluating self-optimizing control at steady-state. It will be the subject of future work to compare this choice of controlled variables with one that follows from a systematic procedure. 6. Conclusion We have shown that the degrees of freedom in vapour compression cycles are equal to the number of compressors and valves plus one. The extra degree of freedom is related to the "active charge" in the system, and a tank with variable holdup should be included to gain this degree of freedom. A detailed degree of freedom analysis for the MFC process reveals that there are four unconstrained degrees of freedom in operation (not considering manipulating refrigerant compositions). To fully achieve the potentially high thermodynamic efficiency of the MFC process it is important that these four unconstrained degrees of freedom are utilized optimally. REFERENCES 1. W. A. Bach. Developments in the mixed fluid cascade process (MFCP) for LNG baseload plants. Reports on science and technology Linde, 63, 2002. 2. W. Forg, W. Bach, R. Stockmann, R. S. Heiersted, P. Paurola, and A. 0. Fredheim. A new LNG baseload process and manufacturing of the main heat exchanger. Reports on science and technology Linde, 61, 1999. 3. Statoil. Sn0hvit homepage, www.statoil.com/snohvit. 4. J. B. Jensen and S. Skogestad. Control and optimal operation of simple heat pump cycles. In European Symposium on Computer Aided Process Engineering (ESCAPE) 15, Barcelona^ 2005. 5. S. Skogestad. Plantwide control: the search for the self-optimizing control structure. J. Process Contr., 10(5):487-507, 2000. 6. h t t p : / /wiTV. psenterpr i s e . com/product s_gproms. html.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. PanteHdes (Editors) © 2006 PubUshed by Elsevier B.V.

Multiplicity of steady states in an UOP FCC unit with high efficiency regenerator Joana L. Femandes,^ Carla I.C. Pinheiro,^ Nuno Oliveira,^ F. Ramoa Ribeiro^ ^CEBQ, Department of Chemical Engineering, Institute Superior Tecnico Av. Rovisco Pais, 1, 1049 - 001 Lisboa, Portugal ^GEPSI-PSE Group, Department of Chemical Engineering, University ofCoimbra Pinhal de Marrocos, 3030-290 Coimbra, Portugal Abstract Static bifurcation in a FCC unit is a problem that arises whenever studying the control of a FCC unit. The origin of this behavior is usually due to the exothermicity of the catalyst regeneration reactions and to important phenomena of backmixing in the regenerator. For this reason the geometrical and operational design of the regenerator unit plays an important role in the overall performance and dynamic stability of FCCs. Prior work has focused on model and control problems of different operating FCC units. However, none of these studies have considered a FCC unit with high efficiency regenerator. This paper presents an analysis of the static biftircation behavior of an UOP FCC unit with high efficiency regenerator. The results show that the high efficiency regenerator presents static bifurcation exhibiting multiple steady states, depending on the operating conditions. Keywords: Fluidized Catalytic Cracking; high efficiency regenerator; steady state multiplicity; nonlinear dynamics. 1. Introduction Fluidized Catalytic Cracking is an important refinery process, not just because of its economical relevance but also due to its complex nature and from a strong interaction between the reactor and regenerator units. The complexity of the FCC process originates a highly nonlinear system, quite interesting for control studies. A comprehensive analysis of its steady state behavior and detailed knowledge of the relationships between state and manipulated variables in the operating range is therefore essential for the design of the control systems for these processes. According to Arbel et al. (1995b) all FCCs should exhibit multiple steady states since they are autothermic reactors that require heating to start up the system. Another author (Elnashaie et al., 2004) refers the feedback effects resulting from complex nonlinear interactions between the regenerator and the reactor as the cause for static bifurcation. This author has investigated several industrial units and states that the behavior of the different industrial FCC configurations quantitatively differs, but qualitatively they are very similar. Even though configurations with riser type regenerator, such as the UOP high efficiency regenerator, as well as riser type reactor are expected to have bifurcation behavior albeit both parts of the unit as stand-alone, will not exhibit such behavior because they are in plug flow. In this paper the existence of static bifurcation behavior in a FCC unit with a riser type regenerator will be investigated.

1575

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J.L. Fernandes et al

To minimize coke-on-regenerated catalyst, regenerators were designed with a long residence time. Thus by means of a long residence time and low regenerator temperature, coke-on-regenerated catalyst could be reduced to relatively low values. To reduce the time required for coke combustion and backmixing effects UOP has designed a high efficiency regenerator (see Fig. 1) that operates in a fast-fluidized flow followed by a plug flow as opposed to conventional back-mixed regenerators. The burning takes place in a fast-fluidized combustor. By conducting regeneration in this manner at high temperature, regeneration can be accomplished in a fraction of the time required in the back-mixed regenerators (Grace Davison, 1996). Combustoi" Regenerator Hue Gas

Dilute Hiase Dense Pliase

Fig. 1. Design of a high efficiency regenerator from UOP. (Couch et al., 2003) 2. High Efficiency Regenerator Model The FCC model used in this study was developed by Fernandes et al (2005a and 2005b) in what refers to the riser, stripper and standpipes models. The high efficiency regenerator was modeled in three parts: • combustor - lower part of the regenerator, where combustion air enters the unit and mixes with catalyst coming from the reactor (spent catalyst) and the regenerator vessel (recirculated regenerated catalyst); • lift or regenerator riser - takes the catalyst from the combustor to the upper regenerator vessel. • regenerator vessel - upper part of the regenerator where combustion gases disengage from catalyst and a dense phase of catalyst is formed. Most of the combustion reactions occur in the combustor and lift at normal operating conditions. The mixing of catalyst and gases occurs in the lower part of the combustor in the absence of reactions. The combustion kinetics considered in this model was previously studied at IFF by Vale (2002). The coke is considered to be composed mainly of carbon and hydrogen. Sulfur and nitrogen are also present but in small quantities. Therefore, it is only considered the combustion of carbon and hydrogen. Carbon combustion and hydrogen combustion can then be given by the following reactions:

" ' ^ p -

'

a+1

'co,t'

'

a+1

CO

(1)

Multiplicity of Steady States in an UOP FCC Unit with High Efficiency Regenerator

lb'/'/

Where a is the intrinsic molar ratio between CO2/CO. /Z + I Q

3,gas-solid ÎHÔ

(2)

The combustor and lift were modeled as a plug-flow reactor where combustion reactions occur and are considered to be in a pseudo steady state, since the residence time in the combustor and lift is much smaller than in the regenerator vessel. The regenerator vessel was modeled as a CSTR in dynamic state, with combustion reactions occurring in the dense phase. The equations for the regenerator vessel are the same as presented in Femandes et al. (2005a). The mass and energy balances for the combustor and lift are presented below: 2,1. Gaseous Species Molar Balance

dz

^,l(r/t;|.)+^.p,l(r>;,)

(3)

2.2. Carbon and Hydrogen Mass Balance dN

^=^^cPcj:(r;v;,) J

dz

(4)

2.3. Energy Balance dT 3^

Q.

^,S(r/A//f)+^,/.,Z(r;A//j)

TîCp,

(5)

3. Simulation Results and Discussion The results shown in this paper where obtained through simulation by using a static version of the model presented. The operating conditions are the same as used by Femandes et al. (2005b). Combustion gases and coke heats of formation are the same as used by Han et al. (2001) and vaporization enthalpy is calculated through correlations found in Grace Davison (1996). The results presented in Figure 2 and Figure 3 where obtained by fixing all the inputs and changing respectively the catalyst-to-oil (COR ratio) and air-to-oil ratio (Air/Oil). As it can be seen the regenerator temperature goes through a maximum when going from low COR to high COR, which agrees with results presented by other authors (Han et al, 2001 an Arbel et al., 1995a). The air to-oil ratio has also the same effect on regenerator temperature; however the decrease in regenerator temperature after passing the maximum is very slightly. These effects are related to the quantities of coke formed and to the regime of combustion: partial or full combustion that is represented by the ratio CO/CO2. Partial combustion leads to lower temperatures and higher ratios of CO/CO2, since the oxidation of CO to CO2 occurs with low extension and is highly exothermic. At low COR the conversion is low since only small quantities of coke are formed due to low quantity of catalyst, in that way the quantity of coke to be burned is small and the temperature decreases due to a lower heat generation. At low air-to-oil ratio, since there

1578

J.L. Fernandes et al

is a deficient oxygen concentration, the combustion is more incomplete leading to lower temperatures. 0.7

800

0.6

750

- Air/Oil=0.7 -AirO=0.8

0.5

2 700 -I

O 0.4

t

S 0.3

650

0.2 600 0.1 550

0 4

6

8

COR (kg catatystl^g oil)

10

4

6

8

10

CORCkgcutab/stkgoil)

Fig. 2. Effect of COR ratio in regenerator temperature and partial or full combustion.

0.6

0.7

0.8

0.9

Air/Oil (kg air/kg oil)

0.7

0.8

0.9

Air;t»il (kg airikg oil)

Fig. 3. Effect of air/oil ratio in regenerator temperature and partial or full combustion. To investigate the existence of multiple steady states all the inputs were fixed and the heat balance was centered on the regenerator. Two simulations with different coke compositions with typical industrial hydrogen-to-carbon molar ratio (H/C) were made to investigate the existence of multiple steady states. By analyzing Figure 4 and Figure 5 it can be seen that there is always a cold steady state that is unconditionally stable and that resultsfi*omthe impossibility of preheating the feed sufficiently to achieve self-ignition. This cold steady state is physically impossible since it would result in the absence of feed vaporization, and only exists from a mathematical point of view. In Figure 4 three more steady states were obtained. The upper and lower ones are stable and the middle one unstable. All this three states appear at high regenerator temperatures, even though the lower one would take to smaller conversions not interesting from an economical point of view. Figure 5 shows that for higher hydrogen content on coke there is only one steady state at high temperatures besides the cold steady state. This results fiôm higher heat generation through the combustion of hydrogen. Different contents of hydrogen in coke are caused by feed composition and stripping efficiency, which depends on the stripper configuration, stripping steam flow and temperature.

Multiplicity of Steady States in an UOP FCC Unit with High Efficiency Regenerator 1579 The breakdown by sources (Figure 5) of the heat generated Hne shows that the extension of coke combustion in each section (combustor, lift and regenerator vessel) of the high efficiency regenerator varies significantly with temperature and in a nonlinear way which means that all the three components of the high efficiency regenerator contribute to the existence of multiple steady states. 2500

700

800

1000

Teinperatiiie |K)

Fig. 4. Heat generation and heat removal lines for coke combustion with H/C = 0.8. 2500 Hregenerator vessel

- H generated

J

• - - -Hcombustor

-Hremo\ed

^^"^ \

2000 - — - -Hlitl

^'

^'^

Hlost

m i f 1500 -

Hremoved

^'•''

•^

1

J J

^

1000 -

j.,,''^

'

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J^'^ "'^ jT r

a» X 500 -

'

^^''^"^^^^ f^ j^s.

/ /

''

/

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0 - r.^TT^-..^. 500

600

700

800

Temperature |K)

900

100C

*

y'

05

o where j)(r) represents the vector of estimated functions (reconciled measurements, model parameters and non-measured variables), yj are the discrete measured values, F i s a diagonal matrix of measurement variances,/is a vector of dynamic constraints, h and g are vectors of equality and inequality algebraic constraints, respectively. Several strategies have been proposed to solve similar constrained nonlinear dynamic optimization problems (Biegler and Grossman, 2004). In this work, a sequential strategy is applied to a time moving window. For every sampling time, the differential equation system and the resulting nonlinear optimization problem are solved sequentially using the measured data over the window, until convergence is reached. The solver ESTIMA (Noronha et al., 1993), which is based on a Gauss-Newton method, is employed here to solve the problem. The studied process consists of the bulk polymerization of propylene in a recycled single CSTR (LIPP-SHAC Shell Technology), using high-activity fourth generation Ziegler-Natta catalyst (TiCU/MgCh+PEEB+TEA) to produce polypropylene in liquid propylene (liquid pool or bulk reactor). A simplified model of this process was developed to be used as the dynamic constraints and is given in the appendix, with the corresponding nomenclature. The studied problem involves 15 input variables (m^, rricat, ^TEA> ^PEEB, i^bieed, rrirec, rrieA, rrieB, m^, Teu Terea TeA, TeB, Tû PH2 ), 7 output variablcs {Ca, nipoh XS, MI, L, T, Tô\ 8 initial conditions for each window {Peo, Pao, Cato, TEAQ, NO, XSo, Lo, To) and 2 process parameters {Kp, Q . The values of the parameters and operational conditions can be found in (Prata, 2005) and (Prata et al., 2005).

3. Results Six sets of real plant data were obtained for different production campaigns during distinct periods of time. Some of the obtained results are presented below.

On-Line Data Reconciliation and Parameter Estimation

1583

Figures 1 to 6 present results obtained from a data set recorded over a time interval of 7h (29h to 36h), with a sample time of 5 min, using a time window of2h. 3.0E+04

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if

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6. Conclusions and future work An MILP continuous-time formulation for scheduUng of multipurpose, make-to-order industries with recirculation and assembly is presented, and a Reactive Scheduling Algorithm proposed to address the problem of inserting new orders in a previously determined schedule. The usefulness of the approach as a decision support tool for scheduling is illustrated by creating different scenarios for new order insertion in a medium size example. In the future, the model should be extended to the case where multiple units are available per operation. The improvements of Liao and You (1992) to the model of Manne (lower and upper bound calculations) should be considered to increase model performance when applied to larger problems. Other situations should be modelled like temporary unavailability of processing units and changes in order specifications (due date, demand) and order canceling, which are relevant in the make-to-order sector.

References 1. 2. 3. 4. 5. 6.

Harjunkoski, I. and Grossmann, I.E. (2002), Comp. & Chem. Eng., 26, 1533-1552. Liao, C.-J. and You, C.-T. (1992), J. Operational Research Soc, 43, 1047-1054. Manne, A. S. (I960), Operations Research, 8, 219-223. Mendez, C.A. and Cerda, J. (2002), Comp. & Chem. Eng., 26, 687-695. Mendez, C.A., Henning, G.P. and Cerda, J. (2001), Comp.&Chem.Eng, 25, 701-711. Zhu, Z. and Heady, R.B. (2000), Comp. & Industrial Eng, 38, 297-305.


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Optimal steady-state transitions under constrained predictive control David K Lanf and Christopher L.E. Swartz^ "^Department of Chemical Engineering, McMaster University, 1280 Main Street West, Hamilton L8S 4L7, Canada There has been an increasing focus in recent years on the design and operation of chemical process plants that need to respond rapidly to frequently changing market demands. We consider in this paper the transition from one steady-state operating point to another, with constrained predictive control in use as the regulatory control system. An automation structure is proposed in which the optimal transitions are implemented through specification of appropriate set-point trajectories determined at an upper level. A strategy for computing optimal reference trajectories is presented that takes into account the dynamics of the underlying closed-loop control system. Its effectiveness for determining optimal transitions is illustrated through example problems. 1. Introduction There are increasing economic incentives for demand driven operation with product diversification in the chemical industry, which requires flexible operation in responsive plants [1]. In particular, the ability to optimize transitions between product grades in response to changes in demand while satisfying operational, safety and product quality constraints, is a key component to maximizing economic performance in a competitive market. Backx et al. [1] advocate the use of model-based regulatory control in conjunction with dynamic real-time optimization for such intentionally dynamic, market-driven operations. McAuley and MacGregor [2] consider optimal grade transitions in a gas phase polyethylene reactor. They compute optimal input trajectories as the solution of a dynamic optimization problem that minimizes an economic based objective function. Feedback control on quality variables is not included, which the authors demonstrate could lead to suboptimal policies and offset in the presence of plant/model mismatch or disturbances. Chatzidoukas et al. [3] pose an optimization problem for optimal grade transitions in which input trajectories, a control structure and controller tuning parameters are simultaneously computed. Integer variables are used for the control structure selection, resulting in a mixed-integer dynamic optimization (MIDO) problem. Multi-loop PI control is used to control four output variables, with the density and melt index (product quality variables) controlled via open-loop optimal control.

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In this paper, we consider optimal steady-state transitions where the product quality variables are controlled via constrained model predictive control (MPC). Optimal setpoint trajectories are computed, taking into account the dynamics of the underlying closed-loop control system. This is illustrated in Figure 1 as an additional layer in the standard process automation hierarchy [4,5], in which optimal set-point trajectories are determined to move the plant to a target steady-state determined at the real-time optimization (RTO) level. This results in a multi-level optimization problem which we solve using an interior point solution approach. In the sequel, we present the mathematical formulation and illustrate its application through two example problems.

C V measurements and disturbance estimates

Real time optimizer Determines optimal economic setpoint targets subject to slow disturbances, input and output constraints, and a steaj^-state process model CV targets

Dynamic optimizer CV setpoints

Determines the optimal and feasible setpoint trajectory subject to input and output constraints, constrained MPC and a dynamic process model

Model predictive controller Minimization of control objective function subject to fast disturbances, input and output constraints and dynamic process model Optimal MVs

Process Under mvestigation for steady-state transitions mmmM

Figure 1. Proposed optimization and control hierarchy for steady-state transitions. 2. Mathematical formulation The steady-state transition problem that computes optimal set-point trajectories subject to constraints on the closed-loop response of a system controlled using constrained MPC may be posed as follows:

min 0(y(*),u(*))

(1)

S.t.

ymin < y(k) < Ym:

(2)

Umin < U ( ^ ) < U„

(3)

rmin 0,

(14)

which transforms the multi-level optimization problem into a single mathematical programming problem with complementarity constraints (MPCC). We then solve the resulting system using an interior point approach in which the complementarity constraints are relaxed as w^X/ < Sju with ju driven to zero as the algorithm iterates toward the solution. The implementation in IPOPT-C [8] was found to be highly effective for problems of this type [9], and was used in this work.

D.K. Lam and C.L.E. Swartz

1596 3. Application

3.1. Example 1 - SISO System A single-input single-output system comprising three continuously stirred tank reactors in series, based on a problem in [10], was considered. It is desired to change the target mole fraction of component ^ in the product stream, XAS, from 3% to 4%, with an output constraint on product quality ofxAs < 4%. The output, XA3, is controlled by manipulating the valve position of an inlet stream, with the plant transfer function given by

As)=

0.039 v(.)(5^ + 1)^

(15)

The initial steady-state valve position was taken to be v = 20% open. The system was controlled using constrained model predictive control with the manipulated variables constrained to [0,80] % open, executed every 2 minutes with a prediction horizon of 30 and an input horizon of 10. The initial tuning with an output to input move suppression weighting ratio of 100:1 resulted in a 7% overshoot relative to the set-point change. Detuning the controller results in a feasible response, but with an increased settling time. However, a feasible transition can be achieved while maintaining the more aggressive controller tuning by using the reference trajectory optimization formulation given in the previous section. Figure 2 shows the set-point trajectory and resulting closed-loop response using a transition objective function comprised of the sum of squared deviations between the output and set-point target at each time step over the simulation horizon. 4.2

'

'

^

'

100

'

4.0 ^ £

3.8 • i

1 3-6

£ 2 3.4 3.2 3.0

j

/

/

/

"'i iy

C

'

_

80

-

^

60

-

1. 40

-

/

-i i .i i

'

-

/

/ /

-

/ 10

20

30 Time (min)

40

,

50

60

>

-

20 0

[]

10

20

30 Time (min)

40

50

6

Figure 2. Optimal reference trajectory and corresponding closed-loop response.

3.2. Example 2 - MIMO System The multi-input multi-output application considered involves a styrene polymerization reaction process based on [11] and [12], and shown in Figure 3. The nonlinear system was approximated by a linear transfer function model, which is used in this study. The number average molecular weight (NAMW) and reactor temperature (7) are controlled by manipulating the initiator flow rate (Qt) and the coolant flow rate (Qc), consistent with the control structure in [11]. The objective in our study is to determine an optimal set-point trajectory for a change in NAMW, with set-

Optimal Steady-State Transitions Under Constrained Predictive Control

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point tracking implemented through constrained MPC. The following constraints are imposed, a G [0,150], e,G [0,500],

NAMW^

[50,80],

T E [323,324]

(16)

with only the input constraints applied at the MPC level.

> Effluent Q (Ml [I] T

Figure 3. Styrene polymerization in continuously stirred tank reactor. The response using set-point trajectory optimization is shown in Figure 4, where a weighted sum of squared deviations of the outputs from their targets is minimized, subject to path constraints on the closed-loop response. Direct application of the single target set-point to the MPC controller (with the same tuning) without reference trajectory optimization results in a temperature constraint violation. 4. Conclusions The cost of transitions between steady-state operating points becomes a significant factor in the economics of process operations that are required to respond to rapidly changing market demands. In this paper, a mathematical formulation for reference trajectory optimization with consideration of the closed-loop dynamics of constrained model predictive control was presented. The supervisory controller aims to achieve feasible and optimal operation, while enabling the regulatory controller to remain unaltered both in structure and tuning, thus retaining capability for disturbance rejection. The application of the methodology was illustrated on two example problems with linear dynamics. Extensions currently under development include application to nonlinear systems, explicit incorporation of economics in the reference trajectory optimization problem, and the use of feedback at the supervisory level to compensate for plant/model mismatch.

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20

30 Time (h)

Swartz

40

Figure 4. Optimal reference trajectory and corresponding closed-loop response.

References 1. T. Backx, O. Bosgra and W. Marquardt, IF AC Symposium Advanced Control of Chemical Processes, Vol. 1, 2000, pp. 249-260. 2. K.B. McAuley and J.F. MacGregor, AIChE J. 38 (1992) 1564. 3. C. Chatzidoukas, J.D. Perkins, E.N. Pistikopoulos and C. Kiparissides, Chem. Eng. Sci. 58 (2003)3643. 4. T.E. Marlin and A.N. Hrymak, In: J.C. Kantor, C.E. Garcia and B. Camahan (Eds.), Fifth International Conference on Chemical Process Control, AIChE Symposium Series Vol. 97, 1997, pp. 156-164. 5. S.J. Qin and T.A. Badgwell, Control Engineering Practice 11 (2003) 733. 6. C.E. Garcia and M. Morshedi, Chem. Eng. Commun. 46 (1986) 73. 7. E. Zafiriou and A.L. Marchal, AIChE J. 37 (1991) 1550. 8. A.U. Raghunathan and L.T. Biegler, Comp. Chem. Eng. 27 (2003) 1381. 9. R. Baker and C.L.E. Swartz, AIChE Annual Meeting, Cincinnati, 2005. 10. T.E. Marlin, Process control: Designing processes and control systems for dynamic performance 2"^ edition, McGraw-Hill Inc., New York, 2000. 11. B.R. Maner, F.J. Doyle, B.A. Ogunnaike and R.K. Pearson, Automatica 32 (1996) 1285. 12. P.M. Hidalgo and C.B. Brosilow, Comp. Chem. Eng. 14 (1990) 481.


Optimal Configuration of Artificial Neural Networks Vivek Dua Centre for Process Systems Engineering, Department of Chemical Engineering, University College London, Torrington Place, London JVCIE 7JE, United Kingdom Abstract In this work a mixed-integer programming approach for training the network is presented. This approach reHes on modelhng the existence or non-existence of nodes by introducing 0-1 binary variables. The interconnection between the nodes and the layers is also similarly modelled. This results in a mixed integer program where the objective is not only the minimization of the error but also the number of nodes. The key advantage of this approach is that a reduced number of nodes and a much simplified network are obtained, for a similar performance in terms of minimization of the error. Keywords: Training of Artificial Neural Networks, Optimization, Mixed Integer Programming

1. Introduction Artificial Neural Networks (ANN) are mathematical models that mimic simple biological nervous systems and have been extensively used in many applications including system identification and model reduction (Prasad and Bequette, 2003), control of chemical processes (Hussain and Kershenbaum, 2000; Hussain et al., 2003) and fault detection and diagnosis (Rengaswamy and Venkatasubramanian, 2000). A network consists of a number of layers and each layer consists of a number of neurodes where neurodes are considered to be similar to neurons. Each neurode receives a number of inputs each of which is weighted, summed and then modified by an internal transfer function to give an output. This output then becomes either an input to another neurode or the result itself. A typical architecture of ANN which consists of interconnected neurodes is shown in Figure 1 where neurodes are denoted by circles. There is an input layer which receives the data and an output layer which gives the response of the network to the data. The transformation of the input data to the output response is facilitated via an extra neurode, known as bias, and one or more hidden layers. The output of an ANN is determined by the architecture of the network, internal transfer functions of the neurodes and the weights on the nodes. For a fixed architecture and internal transfer functions, the error between the output response of the network and the correct output is minimised by iteratively computing the weights and biases. This is known as training the network. Shang and Wah (1996) presented a global optimization technique for training the network. Teng and Wah (1996) presented cascade-correlation learning and population-based learning mechanisms for reducing the number of hidden units for the case when there are binary outputs and Kim and Park (1995) proposed an expand-and-tmncate learning algorithm that guarantees convergence for any binary-tobinary mapping and automatically determines the number of nodes required in the hidden layer. In this paper a mixed-integer programming approach for training the network is presented, where the existence or non-existence of the nodes is modeled by

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introducing 0-1 binary variables and the objectives are to minimize the number of nodes and minimize the error between the ANN prediction and desired output. This can result in reduction in the computational time required by the ANN to compute the output for the given inputs. The rest of the paper is organized as follows: the next section presents the proposed approach, an illustrative example is given in section 3 and concluding remarks are presented in section 4.

•

OUTPUT

BIAS

Input Layer

Hidden Layer

Output Layer

Figure 1. Artificial Neural Network

2. Mixed-Integer Programming Approach for ANN Let Xi denote the input values to the network where / = l,...,A^jc combinations of these inputs gives the activation variables:

and Nn linear

i=\

where A^„ is the number of nodes in the hidden layer, the superscript 1 denotes the index of the hidden layer, Wji are the weights and bj the biases. These activation variables are then transformed non-linearly to provide: h)=i2ir)h{a)) where /?| is the output of the first hidden layer; note that nonlinear transformations other than tanh are also used in the literature, /zj becomes the input to next hidden layer such that:

optimal Configuration of Artificial Neural Networks

1601

where a^ denote the activation variables of the second hidden layer which are also transformed nonlinearly and then become the input to third hidden layer. Similarly the activation variables, a^^, for Nh, the last hidden layer, are given by:

af^='zwJ/hf^-'+bf^

j = l,...,N„

i=\

which are transformed nonlinearly to obtain:

hf' = tmh(af') which are then combined to provide the outputs:

u,='zWk,hf^+B,

k = l,...,N,

where No is the number of outputs and Wki and B^ are the weights and biases respectively. Let w^ denote the desired output, the training of the network can then be formulated as the following optimization problem:

mm

E=

a,b,w,n,W ,B,u

Tiuk-uj,) j^î

subject to equations described earlier, where E is the error function. In this formulation it is assumed that number of hidden layers and the number of nodes in each of the layers is given. The configuration of the network can be optimized so that the sum of the number of the layers and nodes is minimized and yet E is within a pre-specified tolerance. This can be achieved by introducing 0-1 binary variables as follows: -M'y] 0 and 0 otherwise. The values of Uk are similarly decided. Consider a function of three inputs such that if inputs are {000, 010, Oil, 111} then the output is 1; if the inputs are {001, 100, 110} then the output is 0; if the input is {101} then we are not concerned what the output is. Kim et al. (1998) reported a solution for this problem which is given in Table 1 and Figure 2 where the numbers on the interconnections are the weights and the numbers in the circles are the biases; the interconnections with zero weight are not shown; the problem was formulated such that the weights and the biases are integer variables. Table 1. Input-Output Mapping for 3-bit Binary Function (Kim et al., 1998) Xi

000,010,011 001,100, n o 111

u 1 0 1

ai

Cl2

1 0 0

1 1 0

u 1 0 1

In this work the problem was formulated so as to minimize the number of interconnections for one hidden layer such that the output of the network is equal to the desired output and the weights and biases are integer variables; this was modeled by

Optimal Configuration of Artificial Neural

Xi

Networks

X2

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X3

Figure 2. The configuration of a three-layer network for 3-bit example (Kim et al., 1998)

Xi

X2

Figure 3. Optimal Network Configuration for 3-bit Example

X3

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V. Dua

expressing weights and biases in terms of binary variables (Floudas, 1995, p i l l ) . The lower bounds of -6 and 0 and upper bounds of 9 and 7 were considered on weights and biases respectively. This resulted in a mixed-integer linear program which was solved by using GAMS/CPLEX (Brooke et al., 1998). The results are shown in Figure 3, a reduced configuration with one interconnection less than that reported by Kim et al. (1998) is obtained.

4. Concluding Remarks A mixed-integer programming approach for optimizing the configuration of neural networks has been presented. The basic idea is to model the existence of nodes and interconnections by introducing 0-1 binary variables and minimizing these binary variables. This results in a reduced configuration that still satisfies the error criteria specified by the user. The reduced configuration would take less computational effort for computing the outputs for the given inputs. Although not discussed in this paper, it is recognized that more than one network configuration may be optimal and they can be obtained by recursively introducing integer cuts. Another important issue is that logical conditions relating certain inputs and outputs can also be incorporated in the proposed formulations.

References A. Brooke, D. Kendrick, A. Meeraus and R. Raman, 1998, GAMS: a user's guide, GAMS development corporation, Washington. C.A. Floudas, 1995, Nonlinear and mixed-integer optimization, Oxford University Press, New York. M.A. Hussain and L.S. Kershenbaum, 2000, Implementation of inverse-model-based control strategy using neural networks on a partially simulated exothermic reactor, Trasanctions of the Institution of Chemical Engineers (Part A), 78, 299. M.A. Hussain, C. Ng, N. Aziz and I.M. Mujtaba, 2003, Neural network techniques and applications in chemical process control systems. Intelligent systems techniques and applications, 5, 326-362, CRC Press. J.H. Kim and S.-W. Park, 1995, The geometrical learning of binary neural networks, IEEE Trasanctions on Neural Networks, 6(1), 237-247. J.W. Kim, S.-W. Park, H. Oh and Y. Han, 1998, Synthesis of three-layer threshold networks, in Algorithms and Architectures, C.T. Leondes (Editor), Academic Press, San Diego. V. Prasad and B.W. Bequette, 2003, Nonlinear system identification and model reduction using artificial neural networks. Computers and Chemical Engineering, 27, 1741-1754. R. Rengaswamy and V. Venkatasubramanian, 2000, A fast training neural network and its updation for incipient fault detection and diagnosis. Computers and Chemical Engineering, 24(2-7), 431-437. Y. Shang and B.W. Wah, 1996, Global optimization for neural netwrok training, IEEE Computer, 29(3), 45-54. C.-C. Teng and B.W. Wah, 1996, Automated learning for reducing the configuration of a feedforward neural network, IEEE Transactions on Neural Networks, 7(5), 1072-1085. J. Viswanathan and I.E. Grossmann, 1990, A combined penalty function and outer-approximation method for MINLP optimization. Computers and Chemical Engineering, 14, 769.


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Diagnosis of oscillations in process control loops Yoshiyuki Yamashita ^ ^Department of Chemical Engineering, Tohoku University, 6-6-07 Aramaki Aoba, Sendai 980-8579, Japan Abstract Valve stiction is the most common cause of the problem on control loops in process industry. To improve the productivity and quality, industrial engineers demand a tool to facilitate loop monitoring. This paper present a new algorithm to detect valve stiction for diagnosis of oscillation in control loops. The method is for the level control loops and based on a statistical analysis in phase plane by using controller output and level signals, those are available for all the level control loops. The usefulness of the method are successfully demonstrated on a simulation data set and several industrial data sets. Keywords: fault diagnosis, process monitoring, control performance 1. INTRODUCTION Among many control loops in a process plant, quite a few loops have oscillatory behavior. Notwithstanding, it is usually too time consuming and sometimes too difficult to maintain all the control loops in proper working order. Therefore, to improve the productivity and quality, industrial engineers demand a tool to facilitate loop monitoring. These oscillations can be caused by improper controller tuning, external disturbances, and so-called stiction in a control valve. For example, more than 20% of all control loops in paper mills reportedly oscillate because of valve stiction. Detection of oscillations in process control loops have been investigated by many researchers. The next step of the loop monitoring is to indicate likely causes of the oscillations. Several methods have been reported to identify the causes. When the valve position or the corresponding flow rate is available, plot of controller output v.s. valve position or corresponding flow rate represents characteristic pattern for oscillatory loops caused by valve stiction and therefore this pattern can be used for the identification of the cause [1]. Unfortunately the valve position or the corresponding flow rate is not often measured in real plant. Therefore, a method is highly required to detect valve stiction without using this information. Several methods have been proposed for the detection of valve stiction without requiring the valve position. Horch and Isaksson developed a detection method based on the probabiUty density function of the second derivative of the process output [2]. Singhal and Salsbury proposed a simple method to check if the shape of the process output signal is similar to

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Controller Output

(b) Figure 1. IVpical plots for a valve stiction loop

\ -\9

^ (a)

(b)

Figure 2. Typical plots for a non-stiction loop

sinusoidal or not [4] . Rossi and Scali used square sum of the differences between typical oscillatory patterns and the observed data [5]. In this study, a method is proposed to detect stiction in a control loop by using only information of controller input and output. As the result of investigation on the behavior of the controller input and output in polar coordinate, the distribution of the sampling points was found to be lopsided for the oscillation caused by valve stiction, although it is almost symmetrical for the loop with bad tuning. This characteristics can be well displayed in an angle histogram of the controller input and output. Moreover an index to detect stiction is developed based on the skewness of the histogram. The method is illustrated with several level control loops of simulation and industrial plant. 2. Method 2.1. Observation Atfirst,oscillating data in level control loops are visually inspected. If the valve position or correspondingflowrateis available, it is relatively easy to find valve stiction because the plot of controller output and the valve position shows typical parallelogram shape as shown in Fig. 1(a). Automated method for detection of stiction based on these variables is also proposed [1]. Figure 2 shows plots of another oscillatory control loop, which does not include stiction. In this loop, corresponding input-output plots does not show parallelogram shape (Fig 2(b)). However, valve position or correspondingflowrateis not always measured in the industrial plant. Even if the valve position is not available, controller output and controlled measurement variable are always available. Comparing the Figures 1(b) and 2(b), it may be difficult to find typical characteristics for stiction in a plot between controller output and level signal. To find characteristic features for stiction, other plots are investigated. Let the gravity

Diagnosis of Oscillations in Process Control Loops

^

1607

'•^•^.

Si

- 4 - 3 - 2 - 1 0 1 2 3 4

(a) stiction

(b) non-stiction

Figure 3. Plots converted in polar coordinate

center of the controller-output and the level plots be the new origin, the original plots can be converted into polar coordinate. Figure 3 shows the polar plots of the two loops. By comparing these twofigures,distributions of the sampling points is found to be lopsided for the oscillation caused by valve stiction, although it was almost symmetrical for the loop with bad tuning. This characteristics can be well displayed in an angle histogram of the controller input and output. Fast motion in the slip jump probably causes this uneven distribution of stiction pattern. 2.2. Algorithm Based on the above mentioned observation, a method to identify stiction is developed. The following steps shows the details of the process. 1. Normalization: Let A?i and A^ be each time differences of controller output {u) and tank level {y) during the sampling period. Both of the values are normalized between zero and one by using their mean and standard deviation. 2. Polar conversion: Plots in /S.u v.s. Ay plane are converted in polar coordinate r v.s. 6 as r = vÂu2~+~Ay2, i9 = tan"^(A2//A'?x)

(1) (2)

3. Histogram: To clarify the distribution of the sampling points in polar coordinate, number of samplings are counted for each interval of 6, which is divided into 100 intervals. If the histogram is symmetric, the loop oscillate because of poor tuning. If it is asymmetric, the loop will have stiction. The same information can also be visually shown in rose plot, which is a polar plot showing the distribution of values grouped according to their numeric range. Rose plot is useful to find asymmetrical distribution by human observation. Since the trajectory in polar coordinate is periodic in the 6 direction it is possible to fold into half plane and get 0: if(9>0, e if(9% 60 development of more • effective processes and , ^ ^ L 40concomitant product quality Time (min) control. Using on-line Fig 6 Crystal length evolution, plotted against imaging and image analysis, supersaturation, temperature and turbidity. Each point a preliminary study was represents the average of previous 60 seconds containing conducted on the estimation 300 images of the growth rates of needle-shaped crystals in two dimensions for p-form L-glutamic acid in cooling crystallization under a cooling rate of O.lO^C/min (Calderon De Anda et al. 2005b). The length and width of each needle-shaped crystal were measured every 60 seconds, ranging from 100 to nearly 200 |Lim in length and from 30 to 45 |Lim in width, and the values were used to estimate growth rates on both directions (Fig. 6). The growth rate in length was found to be 4 to 6 times greater than for the width. The (101) plane was found to be the fastest growing surface of the morphology studied and an attempt was also made to estimate its growthkinetics parameters from measurements of length, whilst it was harder to estimate kinetics from measurements of width for other crystal faces. In the temperature range between 68.34^C to 67.5 l^C, the length growth rate is estimated as between 2.440x10'^ ~ 2.995x10"^ m/s, while the growth rate for the width is between 0.558x10"^ ^ 0.502x10" ^ m/s. The capability to measure crystal growth rates in different directions could be used to estimate the parameters associated with growth kinetics in multi-dimensional directions. If a semi-empirical kinetic model is used, R = kG"", k ^1.761x10'' m / s , an d /; «2.61. It was assumed for [3 L-glutamic acid, the growth rate in length is very close to the growth rate of the faces {101}.

5. Model Predictive Control of Crystals Morphology 5.1. Model Predicted Crystal Morphology Control The recently developed imaging and image analysis technique for real-time measurement of crystal morphology makes inroads into automatic control of the morphology of crystals grown from solution. Since once an unfavorable polymorph or morphology is formed, to reverse the process e.g. through heating to dissolve and regrowing the crystals may not be a preferred option in industrial operation, advanced control based on shape predictive models is therefore attractive. Since morphology prediction is formulated for a single crystal, in order to consider the whole population of crystals in a reactor to give a statistical measure of the shape distribution, morphology

Advances and Future Directions in Morphology Monitoring

1615

modeling should be integrated with population balance (PB) modeling (as indicated in the next sub-section, this gap has yet to be filled), and very importantly, PB has to be conducted based on multi-dimensions, instead of being based on the On-line 3D Imaging assumption of an equivalent spherical Image Analysis shape with a single dimension of the I diameter to represent the size Shape Recognition distribution. Multi-dimensional PB Multi-dimensional modeling depends on the number of Faceted Growth Rates Population and Faceted Population facets of a crystal, each with its own Balance Modelling Balance growth rates. Some researchers (Briesen 2006; Puel et al. 2003) have experimented the idea of two dimensional PB modeling for needle shaped crystals. In early part of this Morphology paper we showed results of measuring Control Based on the growth rates in two dimensions for Predictive Models needle-shape crystals. Not only being Fig 7 The necessary components for developing restricted to two dimensions, work so model predictive control strategy for crystal far has also assumed that there is only morphology one polymorph in a reactor. If more than one polymorphs exist at the same time in the reactor, the problem clearly becomes more complicated because the transition between two polymorphs also needs to be taken into account. Fig. 7 shows a conceptual framework highlighting the necessary components for developing a model predictive control strategy of crystal morphology. Key future research needs are highlighted below. 5.2. Shape Measurement Although proved to be effective, so far the measurement of shape using on-line imaging has been restricted to giving 2-D images. Li et al (Li et al. 2005) proposed to use a camera model to construct the 3-D shape from 2-D images, there are still a few obstacles to be overcome before it can become practical. We are experimenting the use of two video cameras to simultaneously image the same object so that the 3-D shape can be fully constructed. 5.3. Morphology modeling Traditionally, morphology prediction of crystals had almost assumed that the crystals were growing in vacuum without (a) (b) adequately considering process Fig. 7 (a) traditional modelling and control: based on a spherical assumption, e.g. operational conditions such as cooling growth rate in m/s of diameter; (b) new rates, supersaturation, solvents and modelling and control: based on impurity or additives etc though these multidimensional model, e.g. growth rates, factors are known to affect the polymorph m/s for each surface. and morphology. Progress are being made in improving the morphology models to take into account these factors and the work has been reviewed (Clydesdale et al. 1996; Gadewar and Doherty 2004; Liu and Bennema 1996; Liu et al. 1995; Winn and Doherty 2000). Success has been achieved on predicting the effect of solvent on the shape of several organic crystal systems. Despite

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X.Z. Wangetal

the progress, there is still some way to go before robust models available that can fully take into account process conditions to give confident prediction of crystals shape. This does not mean however, this represents the weakest link that will jeopardize the whole framework for model predictive morphology control research. In fact, on-line shape measurement techniques could contribute to the development of morphology prediction models since they can be used for model validation. 53.1. Integration with CFD The effect of mixing on crystal growth has long been recognized and computational fluid dynamics (CFD) has been integrated into PB modeling (though only for one-dimension based on the spherical assumption) (Kulikov et al. 2005) so that PB modeling can be conducted in different zones with varied mixing characteristics. A major challenge is computational speed. Research on either model reduction or data compression is needed in order to be able to effectively use the CFD data into crystal morphology model predictive control.

6. Final Remarks The recent developments in on-line shape measurement as well as in crystal morphology prediction and multi-dimensional population balance modeling opens the way for developing model-predictive control of the morphology as well as size of crystals grown form solution. This will need integration of on-line real-time 3-D shape measurement, multi-scale modeling of morphology, multi-dimensional PB modeling and CFD. Future research needs towards this goal are highlighted.

Acknowledgements Financial supports from EPSRC for the Shape project (EP/C009541) and for the Chemicals Behaving Badly project (GR/R43860), and from Malvern Instruments Ltd for the Vision and IntelliSense projects are greatly appreciated. The first author thanks Malvern for sponsoring his Readership in Intelligent Measurement and Control. References Barrett, P., Glennon, B., 2002, Chem. Eng. Res. Des., 80, 799-805. Braatz, R. D., Fujiwara, M., Ma, D. L., Togkalidou, T., Tafti, D. K., 2002, Int. J. Modem Physics B, 16, 346-353. Briesen, H., 2006, Chem. Eng. Sci., 61, 104-112. Calderon De Anda, J., Wang, X. Z., Roberts, K. J., 2005a, Chem. Eng. Sci., 60, 1053-1065. Calderon De Anda, J., Wang, X. Z., Roberts, K. J., 2005b, J. Phar. Sci. Calderon De Anda, J., Wang, X. Z., Lai, X., Roberts, K. J., 2005c, J. Proce. Cont., 15, 785-797. Calderon De Anda, J., Wang, X. Z., Lai, X., Roberts, K. J., Jennings, K. H., Wilkinson, M. J., Watson, D., Roberts, D., 2005d, AIChE J., 51, 1406-1414. Clydesdale, G., Roberts, K. J., Docherty, R, 1996, J. Cryst. Growth, 166, 78-83. Gadewar, S. B., Doherty, M. F., 2004, J. Cryst. Growth, 267, 239-250. Kulikov, v., Briesen, H., Marquardt, W., 2005, Chem. Eng. Res. Des., 83, 706-717. Li, R. F., Thomson, G. B., White, G., Calderon De Anda, J., Wang, X. Z., Roberts, K. J., AIChE J, in press, 2006. Liu, X. Y., Bennema, P., 1996, J. Cryst. Growth, 166, 117-123. Liu, X. Y., Boek, E. S., Briels, W. J., Bennema, P., 1995, Nature, 374, 342-345. Ma, Z. H., Merkus, H. G., Scarlett, B., 2001, Powder Techno!., 118, 180-187. Patience, D. B., Rawlings, J. B., 2001, AIChE J., 47, 2125-2130. Puel, F., Fevotte, G., Klein, J. P., 2003, Chem. Eng. Sci., 58, 3715-3727. Wilkinson, M. J., Jennings, K. H., Hardy, M., 2000, Microscopy Microanal., 6, 996-997. Winn, D., Doherty, M. F., 2000, AIChE J., 46, 1348-1367. Yamamoto, H., Matsuyama, T., Wada, M., 2002, Powder Technol., 122, 205-211.


Molecular Weight Control in Acrylonitrile Polymerization with Neural Network Based Controllers Atasoy I, Yuceer M., Berber R. Department of Chemical Engineering, Faculty of Engineering, Ankara University, Tandogan, 06100 Ankara, Turkey

Abstract Acrylic fiber is commercially produced by free radical polymerization, initiated by a redox system. Industrial production of polyacrylonitrile is a variant of aqueous dispersion polymerization, which takes place in homogenous phase under isothermal conditions with perfect mixing. The fact that the kinetics is a lot more complicated than that of ordinary polymerization systems makes the problem of controlling molecular weight a difficult one. On the other hand, abundant data is being gathered in industrial polymerization systems, and this information makes the neural network based controllers a good candidate for a difficult control problem. In this work, neural network based control of continuous acrylonitrile polymerization is studied, based on our previously developed new rigorous dynamic model for the polymerization of acrylonitrile. Two typical neural network controllers are investigated: model predictive control and NARMA-L2 control. These controllers are representative of the variety of common ways in which multilayer networks are used in control systems. As with most neural controllers, they are based on standard linear control architectures. The concentration of bisulfite fed to the reactor as the manipulated variable and weight average molecular weight of the polymer as an output function are used in control studies. The results present a comparison of two common neural network controllers, and indicate that the model predictive controller requires larger computational time. Furthermore, the model predictive controller involves difficulties in determining the values for the weighting factor and the prediction horizons. The NARMA-L2 controller requires minimal online computation. Keywords: Acrylonitrile polymerization, NN predictive control, NARMA-L2 control. 1. Introduction Neural networks have been applied successfully in the identification and control of dynamic systems (Hagan et al., 2002). Rather than attempt to survey the many ways in which multilayer networks have been used in control systems, we concentrated on two typical neural network controllers: model predictive control (Narendra et al., 1997), NARMA-L2 control (Narendra et al., 1990). These controllers are representative of the variety of common ways in which multilayer networks are used in control systems. As with most neural controllers, they are based on standard linear control architectures. There are typically two steps involved when using neural networks for control: system identification and control design. In the system identification stage, we develop a neural network model of the plant that we want to control. In the control design stage, we use the neural network plant model to design (or train) the controller. In each of the two

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control architectures described in this paper, the system identification stage is identical. The control design stage, however, is different for each architecture. 2. NN predictive control The aim of the predictive control strategy using neural predictors is two step: (a) to estimate the future output of the plant: The controller then calculates the control input that will optimize plant performance over a specified future time horizon. (h) to minimize a cost function based on the error between the predicted output of the processes and the reference trajectory. The cost function, which may be different from case to case, is minimized in order to obtain the optimum control input that is applied to the nonlinear plant. In most of the predictive control algorithms a quadratic form is utilized for the cost function. 2.1. System identification The first stage of model predictive control and NARMA-L2 control are to train a neural network to represent the forward dynamics of the plant. The prediction error between the plant output and the neural network output is used as the neural network training signal. 2.2. Predictive control This controller uses a neural network model to predict future plant responses to potential control signals. An optimization algorithm then computes the control signals that optimize future plant performance. The neural network plant model is trained offline, in batch form, using any of the training algorithms. The controller, however, requires a significant amount of on-line computation, since an optimization algorithm is performed at each sample time to compute the optimal control input. The model predictive control method is based on the receding horizon technique. This controller uses a neural network model to predict future plant responses to potential control signals. The neural network plant model is trained offline, in batch form, using any of the training algorithms. The predictions are used by a numerical optimization program to determine the control signal that minimizes the following performance criterion over the specified horizon:

^=i(>'.(^+y)-7j^+7)f+/'i:(«'(^+7-i)-«'(^+;-2F j=N,

0)

7=11

where Ni; N2 and Nu define the horizons over which the tracking error and the control increments are evaluated. The u' variable is the tentative control signal, yr is the desired response and ym is the network model response. The value of p determines the contribution that the sum of the squares of the control increments has on the performance index. Figure 1 shows block diagram that illustrates the model predictive control process. The controller consists of the neural network plant model and the optimization block. The optimization block determines the values of u' that minimize J; and then the optimal u is input to the plant.

Molecular Weight Control in Acrylonitrile Polymerization

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"A

Coiitix>Ilei' I

yr Optirrdzatioti

Nesuml Network Model

Plant

Figure 1. NN predictive control 3. NARMA-L2 control The controller is simply a rearrangement of the neural network plant model, which is trained offline, in batch form. The only online computation is a forward pass through the neural network controller. The main idea of this type of control is to transform nonlinear system dynamics into linear dynamics by canceling the nonlinearities. As with the model predictive control, the first step in using NARMA-L2 control is to identify the system to be controlled. The nonlinear autoregressive moving average model is used to represent the system: Kk + d) = f[y{k\y{k - \\..,,y{k -n + l),...,i/(A: - m +1)] + g{y(k\y(k - l)v..X^ - « + l),w(A: - \\..M{k -m + \)lu(k)

^2)

where d < 2. Using the NARMA-L2 model, the controller is obtained of the form: U(k + 1):

where

yr(k + d)-f[Y,U] g[Y,U]

Y=[y(k),...,y(k-n + l)] U = [u(k),u(k-l),..,u(k-n + l)]

(3)

(4)

This model is in companion form, where the next controller input u(k) is not contained inside the nonlinearity. Figure 2 is a block diagram of the NARMA-L2 controller.

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1 Rediel«n.ce 1 lodel 1

-'

1 ^ -{- ^

Contra

+ J Iter

.1

i

ÂC^

Sir 1 / 1 1s 1 i

—

>U-

^ IL

1 T 1 1 1^ 1L 1

Figure 2. NARMA-L2 controller. 4. Mathematical model The dynamic model comprises 9 state variables, namely the concentrations of monomer, sulfate radical, bisulfite radical, bisiilfite ion, persulfate ion, active radical of one monomer with sulfate end group, active radical of one monomer with sulfite end group, active radical of n monomers with sulfate end group, active radical of n monomers with sulfite end group. The mathematical model developed was based on the following simplifying assumptions: (i) Polymerization takes place in homogenous phase under isothermal conditions with perfect mixing and all species are soluble in the solvent, (ii) Reactivities of sulfate and bisulfite radicals are the same, (iii) All radicals propagate, terminate and transfer with the same velocity constant, (iv) Branching reactions do not take place, (v) Equilibrium is established after 5 to 6 dwell time, (vi) Addition and propagation steps proceed with the same velocity constant, (vii) All reactions are of second order, (viii) All radicals propagate with the same velocity. Addition and growing rate of monomers to the polymer chain were taken to be equal. As our dynamic studies indicated that 6 of those variables were not changing considerably, we assumed that they would hold constant in control studies, and we were thus able to obtain a simplified model comprising of 3 differential equations coupled with the outputfiinctionas follows:

^=-t,M^(Iso,-]+ko;])*ML-M at

V — OC

u

(5)

u

(6)

at Probability of growth;

\-a

6

6

Molecular Weight Control in Acrylonitrile Polymerization ^._

KM

_

1621

K{^] (o)

z

The average (number and weight) molecular weights of the pol5niier are defined by, _ r„ =

2kp[M] p

_

:(l + 2y)

'""

M =W *m

;

kp[M][3 + 2y] z(l + y)'

M =W *r

^p^ (10)

where y and z are ÂHSO;^^

z = (2k,k,[Fe-]M-])>^

^^^^

The manipulated variable was the concentration of bisulfite fed to the reactor. 5. Results and Conclusions A neural network model of the plant was constructed using data from the simulation model. A multilayer perceptron neural network with 3 neurons in the hidden layer. The Acrylonitrile Polymerization model is solved MATLAB environment using variable order ODE solver. The model generated input data considered of 10 000 sample points. It was divided into three subsets, namely; training set (5 000 sample points), validation (2 500 sample points) and testing (2 500 sample points). Then through the optimization routine, the model predictive controller provides a control action to the system depending on the predictive horizon and the control weighting factor. A model predictive controller with a longer prediction horizon and a small control-weighting factor provides good performance in terms of reduced error. The controller employs an online optimization algorithm, which requires more computation than the NARMA-L2 control. The performance of the NARMA-L2 controller depends on the identification of the system with a neural network. An accurate neural network model of the system provides good results in terms of set point tracking, hence reduced error values. However, as there is no factor to adjust the control weighting, use of a limiter on the control action appears to be necessary. The performances are hence reduced but the control signal variation is also reduced. The controller is a rearrangement of the plant model, and requires minimal online computation. Table 1 shows the final NN-Predictive Controller and NARMA-L2 Controller parameters for acrylonitrile polymerization control application. The performances of neural network controllers are highly dependent on the accuracy of the plant identification. For our applications, we typically collect training data while applying random inputs which consist of a series of pulses of random amplitude and duration. The duration and amplitude of the pulses must be chosen carefully to produce accurate identification. If the identification is poor, then the resulting control system may fail. Controller performance tends to fail in either steady-state operation, or transient operation, or both. When steady-state performance is poor, it is useful to increase the duration of the input pulses. Unfortunately, within a training data set, if we

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have too much data in steady-state conditions, the training data may not be representative of typical plant behaviour. This is due to the fact that the input and output signals do not adequately cover the region that is going to be controlled. This will result in poor transient performance. Figure 3 shows the performance of controllers for parameters given in Table 1. Table 1. Parameters for NN-MPC and NARMA-L2 Parameters

NN-MPC

NARMA-L2

Size of hidden layers

3

3

Training function Training epochs

Levenberg-Marquardt 1000

1000

Cost horizon (N2)

7

-

Control horizon (NJ

2

-

Minimization routine

Minimization with backtracking

-

Control weighting factor (p)

0.001

-

Search parameter (a)

0.01

-

1200

1000

800

—\

r^ 1

1

1

r-

600

• NARMA-L2 • Setpoint • NN-MPC 5 time, s

7

8

9

10

xio"^

Figure 3. Response of molecular weight controllers to set point changes References T. M. Hagan., H. B. Demuth and O. D. Jesus, An introduction to the use of neural networks in control systems. Int. J. Robust and Nonlinear Control, 2002; 12, 959-985. K. S. Narendra, S. Mukhopadhyay, Adaptive control using neural networks and approximate models, IEEE Transactions on Neural Networks, 1997, 8, 475-485. K. S. Narendra, K. Parthasarathy, Identification and control of dynamical systems using neural networks, IEEE Transactions on Neural Networks, 1990; 1, 4-27.


A New Approach to Chance Constrained Process Optimization and Control under Time-dependent Uncertainties Harvey Arellano-Garcia*, Tilman Barz, Walter Martini, Gunter Wozny Department of Process Dynamics and Operation, Berlin University of Technology, Sekr. KWT-9, Str. Des 17. Juni 135, Berlin 10623, Germany

Abstract In this work, a novel approach to solving nonlinear chance-constrained dynamic optimization problems under time-dependent uncertainties is proposed. The approach considers a nonlinear relation between the uncertain input and the constrained output variables. In fact, the approach is relevant to all cases when uncertainty can be described by any kind of joint correlated multivariate distribution function. The essential challenge lies in the efficient computation of the probabilities of holding the constraints, as well as their gradients. However, the main novelties of this approach are that nonlinear chance constrained dynamic optimization can now also be realized efficiently even for those cases where no monotonic relation between uncertain input and constrained output exists. This is necessary, particularly, for those process systems where the decision variables are critical to the question of whether there is monotony or not. Furthermore, novel efficient algorithms are proposed to consider dynamic random variables. Thus, the solution of the problem has the feature of prediction, robustness and being closed-loop. The performance of the proposed approach will be demonstrated through application to the optimal operation and control of a high pressure column embedded in a heat integrated column system. In addition, a novel chance constrained nonlinear MFC scheme is introduced to show the efficiency and potential of the chance constrained approach for online optimization and control under uncertainty. Keywords: Time-dependent uncertainty, chance constraints, dynamic optimization, NMPC.

1. Introduction Optimization and control under uncertainty is deemed to be of ftindamental significance in several discipline and application areas. In dynamic processes, in particular, there are parameters which are usually uncertain, but may have a large impact on the targets like the objective value and the constrained outputs. Thus, explicit consideration of the stochastic property of the uncertainties in the optimization approach is necessary for robust process operation and control. Uncertainty and variability are inherent characteristics of any process system. Moreover, measurements often contain random errors that invalidate the process model used for optimization and control. This implies that neither the magnitude nor the sign of the error can be predicted with certainty. However, the uncertainties considered are continuous variables, not results of discrete events. This means that there is infinity of possible "discrete" values for the events associated with continuous time-dependent variables. The only possible way these weaknesses can be characterized is by use of probability distributions. To whom correspondence should be addressed at [email protected]

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tf

to

tf

Figure 1. Constant und time-dependent uncertainty The stochastic distribution of the uncertain variables may have different forms. The values of mean and variance are usually available. Uncertain variables can be constant or time-dependent in the future horizon (Fig. 1). They are, however, undetermined before their realization. Moreover, usually only a subset of variables can be measured. The unmeasured variables are, however, open loop but should be constrained under uncertain disturbances. 2. Chance Constrained Approach In this work, chance constrained programming for process optimization and control under uncertainty is proposed. The stochastic property of the uncertainties is included in the problem formulation such that the output constraints are to be complied with a predefined confidence (probability) level. The resulting problem is then transformed to an equivalent deterministic NLP problem. Here, the basic idea is to map the probabilistic constrained output region back to a bounded region of the uncertain inputs. Hence, the output probabilities and, simultaneously, their gradients can be calculated through multivariate integration of the density function of the uncertain inputs by collocation in finite elements. Recently, a new promising optimization fi-amework for dynamic systems under uncertainty was introduced for the off-line optimization under probabilistic constraints and successfully applied to a large-scale nonlinear dynamic chemical process where the monotony of the constrained output to at least one uncertain input is utilized (Arellano-Garcia et al. 2003, 2004). However, this approach can only be used, if the required monotony exists. In this contribution, we extend the chance constrained approach to allow for such stochastic dynamic optimization problems where no monotone relation between constrained output and any uncertain input variable can be guaranteed. Moreover, the novel approach explicitly considers time-dependent uncertainties. The entire optimization framework also involves efficient algorithms for the computation of the required (mapping) reverse projection and is relevant to all cases when the uncertainties can be described by any kind of joint correlated multivariate distribution function.

superior layor

multivariate integration

.,_^ (dhrnamic solvir

^ ^ ^ e f . er Figure 2. Chance constrained optimization framework.

A New Approach to Chance Constrained Process Optimization and Control

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The proposed approach uses a two-stage computation framework (Fig. 2). The upper stage is a superior optimizer following the sequential strategy. Inside the simulation layer there is a two-layer structure to compute the probabilistic constraints. One is the superior layer, where the probabilities and their gradients are finally calculated by multivariate integration. The inferior sub-layer is the key to the computation of the chance constraints with non-monotonous relation. In case of strict monotony the optimization step in the sub-layer becomes unnecessary. 3. An Approach to Time-dependent Uncertainty In this work, a dynamic process with 7V7 time-dependent output state variables y(t), NU time-dependent control variables u(t), and A/O time-dependent uncertain parameters ^(t) is considered. The probability Pr of complying with a certain restriction yi^ which corresponds to the output state variable y^^ at every time point t during the process operation is to be calculated and formulated by the following expression: Pr{/^(/,^.,f)<X^ VfE [/„/,]}

(1)

In order to transform the infinite number of time points to a finite number of representing values, the entire time horizon is divided into several short time intervals where both the control variables and the uncertain variables are piecewise constant. Due to the monotony between the restricted output y^^ and at least one uncertain parameter, the value of this uncertain parameter ^^^, which corresponds to the bound of the constrained output yi^, can be calculated for every time interval JT according to the following equation: ^'l^MM= f(^w^^JTA^'"^^NMM-xy[]

îth NMM = JTxN^

(2)

However, for NT time intervals, the probability of complying with the constraint for all time intervals can be computed by multivariate integration of a probability density function over all uncertain parameters as follows: psp

pp

Pr= j . . . j j J... J j p{^x.^^^,^NMM)d^NMM'-d^x

(3)

with NMM = NTxN^ Each integration bound of the uncertain parameter (^^^ corresponds to the bound of the constrained output yi^ within the corresponding interval. All the other uncertain parameters will be integrated over their entire space. Since only few discretization points are required for the integration over a relatively large integration space with an acceptable accuracy, the orthogonal collocation method on finite elements has been proved to be very efficient. Thus, in this work we propose a calculation scheme where the first uncertain parameter î is discretized in the first interval. Then, for each resulting collocation point, a value for the second uncertain parameter ^2^^ can be obtained which exactly corresponds to the bound of the constrained output yi^ within this interval and, thus, forming the bound for the second integration layer. Over the new derived integration space, the second uncertain parameter can be discretized. Thus, for instance, in case of two dynamic random variables, NK^ collocation points result for the first interval. This procedure will then be repeated until the next-to-last integration layer.

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first interval. This procedure will then be repeated until the next-to-last integration layer.

Figure 3. Solution strategy for time-dependent uncertainties. Consequently, the last time interval is reached. This approach leads to the computation (tree) structure described in Fig. 3. Since the values below the integration bound are only used for calculations in the following integrals, only one value which corresponds to the bound of the constrained output is required for the last uncertain parameter of the last time interval. Thus, the probability of complying with the restrictions of the last interval under given values of the other uncertain variables, and all control variables will be obtained. This value can also be seen as a part of the probability density function of the next-to-last integration layer. The integration along this layer leads to the probability calculation concerning this layer. This procedure will be carried out up to the most superior integration layer and, thus, we finally obtain the originally wanted probability for fulfilling the constraints of the entire time horizon. Furthermore, to solve the NLP-problem with a standard NLP solver such as SQP, gradients of the objective function and the constraints w.r.t. the control parameters u(t) are additionally required. These are computed simultaneously based on the proposed solution strategy in Fig. 3. 4. Optimal Process Operation under Chance Constraints Model-based optimization has widely been used to develope operation points and operating trajectories for industrial processes. The task of process control systems has been to mantain these predetermined operating points, or follow the given operating trajectory. However, optimization and control have generally been considered individually. The major drawback of performing the two issues separately is the discrepancy in treating process disturbances. Furthermore, the true process optimum lies on the boundary of the feasible region defined by the active constraint or equipment capacities (e.g. maximum allowable pressure, temperature, etc.). Due to the uncertainty in the parameters and the measurement errors, the process optimum and the set-point would be infeasible (e.g. if the pressure of the plant swings). Thus, usually a back-off from the active constraints in the optimization is introduced such that the region around the set-point is moved inside the feasible region of the process to ensure a feasible operation as closely to the true optimum as possible. This can lead to a conservative operation with a much higher product purity than required. Our main concern is that there are variables which are often monitoring for the sake of safety but not close-loop controlled. They should, however, be constrained under uncertain disturbances. This


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leads to a nonlinear optimization problem under chance constraints. To follow these setpoints, the manipulated variables have to be varied corresponding to the realized disturbances. 1 0.998 ^

0.4 0.6 X-Acetonitrile

126

0.8

0.996

127 128 129 Temperature "C

0.59

0.6

0.61 0.62 0.63 0.64

D i s t i l l a t e X Acetonitnle

Figure 4. a) y-x diagram; b) operating set-point; c) constrained output variable The disturbance variables are described with stochastic distributions which can be achieved based on historical data. The novel approach is applied to the optimal operation and control of one column embedded in a coupled two-pressure column system for the separation of an azeotropic mixture (acetonitrile/water). The operating point is defined by the distillate x^^ and bottom product jcf^ specifications, cooling outlet temperature limitations T^^ , as well as the maximum pressure of the considered high-pressure column Ptop (Fig. 4a). The expected disturbances and implementation errors concern the maximal allowable system pressure, the sensitive tray in the stripping column section r/^^' as well as the feed flow rate and its dynamically changing composition xêed' However, the values of the setpoints and controls are adjusted so that the target area will be modified according to the changing disturbances. The objective fiinction is defined by the minimization of the total energy in the considered time horizon. However, the uncertain parameters also have an impact on the objective function. The usual way is to reformulate it to its expected value (Darlington et al., 1999). On the other hand, for practical application, it is more convenient to assure a certain reliability of the realization of the calculated objective value. This can be achieved by minimizing an upper bound p, and the compliance of it, can be guaranteed with a certain reliability by formulating an additional chance constraint. Thus, the entire stochastic optimization problem will be formulated as follows with / as the probability levels: mm s.t. model equations, direct adjusted state variables: rpOUt

rpOUt

,

1

^CW ~ ^CW,ref "*" ^ 7 |_

/ rpOUt

\

yCW

jj '

indirect adjusted constrained output variables: Prjjc^ >x^ ^ ^ \Âc

]>

— -Âc,spec\ —

1 »

originally replaced objective fiinction as chance constraint uncertainties:

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and finally regulatory deviation variance. The formulation of individual pre-defined probability limits of complying with the restrictions incorporates the issue of feasibility and the contemplation of trade-off between profitability and reliability. However, the computation results demonstrate the efficieny of the proposed approach (Fig. 4c).

5. Robust Nonlinear MPC Under Chance Constraints Predictive control is well developed in terms of handling constraints and bounded uncertainty but there is still a need for a efficient framework addressing problems involving stochastic objectives and constraints (Batina et al., 2002). However, since the prediction of future process outputs within an NMPC moving horizon is based on a process model involving the effects of manipulated inputs and disturbances on process outputs, the compliance with constraints on process outputs is more challenging than these on process inputs. Furthermore, as the model involves uncertainty, process output predictions are also uncertain. This leads to output constraints violation by the closeloop system, even though predicted outputs over the moving horizon might have been properly constrained. Thus, a robust predictice control strategy is proposed, namely a nonlinear MPC scheme where the output constraints are to be held with a predefined probability with respect to the entire horizon. Due to the property of the moving horizon approach the control strategy can be extended to on-line optimization under uncertainty. Here, different confidence levels can be assigned to different time periods within the moving horizon by using single chance constraints. Consequently, a decreasing factor, i.e., a lower confidence level for the future periods in the horizon can be introduced. The outcomes of the application to a semibatch reactor with safety restrictions will show the potential of the new approach. 6. Concluding remarks In this work, a new chance constrained approach is proposed. The developed optimization framework demonstrates to be promising to address optimization and control problems under uncertainties. Furthermore, novel efficient algorithms have been integrated to consider time-dependent uncertainties. The solution strategy has been applied to the optimal operation of a high pressure column. The solution provides a robust operation strategy in the future time horizon. Moreover, the relationship between the probability levels and the corresponding values of the objective function can be used for a suitable trade-off decision between profitability and robustness. Tuning the value of / is also an issue of the relation between feasibility and profitability. In addition, a novel robust NMPC scheme will be introduced for the online optimization of a semibatch reactor under hard constraints.

References Arellano-Garcia H., Martini W., Wendt M., Li P., Wozny G., 2003, Chance Constrained Batch Distillation Process Optimization under Uncertainty. In: I. E. Grossmann, C. M. McDonald (Eds.): FOCAPO, pp. 609-612. Arellano-Garcia H., Martini W., Wendt M., Wozny G., 2004, A New Optimization Framework for Dynamic Systems under Uncertainty, In: A. Barbarosa-Povoa, H. Matos (Eds.): Computer Aided Process Engineering -14, Elsevier, 2004, 553-558.


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Batina, I., Stoorvogel, A.A., Weiland, S., 2002, Optimal control of linear, stochastic systems with state and input constraints, Proc. IEEE Conf. Decision & Control, 1564-1569. Darlington J., Pantelides C.C, Rustem B., Tanyi, B.A., 1999, An algorithm for constrained nonlinear optimization under uncertainty. Automatica, 35, 217.

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A Lab-on-a-Chip Simulation Framework A. J. Pfeiffer^, X. He^ T. Mukherjee^ and S. Hauan^* ^Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh PA 15213 ^Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh PA 15213 A Lab-on-a-chip (LoC) miniaturizes and integrates chemical synthesis and analysis capabilities onto a microchip structure. Since LoCs are both physically and chemically complex, a fast, accurate simulation methodology is required for design. Here we present an efficient, configurable LoC simulation framework that can be readily incorporated within design automation and synthesis methods. We compare our simulator to a general computational fluid dynamics (CFD) tool using a LoC design benchmark. Our simulator results in less than 5% error and over 3 orders of magnitude speedup. We demonstrate the efficiency of our approach by redesigning an experimentally generated design from the Uterature [1]. 1. INTRODUCTION A Lab-on-a-Chip (LoC) is essentially a miniaturized microchip implementation of an analytical chemistry laboratory. LoCs are typically fabricated in glass or plastic and are about the size of a credit card. They have been used in the life-science and biomedical industries for applications in genomics, drug discovery, point-of-care analysis and in-vivo diagnostics because they are fast, accurate, readily automatable and inexpensive to fabricate. Microscale unit operations such as mixing, reaction and separation can be constructed entirely on-chip [2]. However, the widespread use of LoC technology has been hindered by the lack of adequate design tools. LoC design combines complex physiochemical phenomena with challenging chip layout and channel interconnectivity issues. The chemistry that takes place during chip operation, as well as the chip layout and manufacturing process must be understood so that the appropriate design trade-offs and constraints are considered. Channel geometry, and the system's channel topology have been shown to contribute a great deal to the overall performance of the final LoC design [3]. Current LoC design practices employ laboratory experiments or iterative computational fluid dynamics (CFD) simulations [4] which are both time consuming and difficult to automate. Simulators using reduced order models [5] have been created, but these tools require CFD pre-solves to extract model parameters. Fast, accurate models have been implemented in a commercial circuit simulator [6]. However, black-box simulators are difficult to integrate within automated design tools. We are developing Computer-Aided Design (CAD) tools to address the complex nature of LoC design by combining electrical circuit simulation [7] and chemical process simulation * Corresponding author email: [email protected]

A J. Pfeiffer et al

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Sample: [Ag*, Ag]

Figure 1. Four phases of LoC operation: (a) Sample and reagent are introduced (shaded area) (b) Steady-state loading phase in which fluids are mixed (Jl) and reacted (Jl - J2). Flow direction shown with dotted arrows, (c) Transient injection phase where a narrow band of material is injected into the separation channel, (d) Fluids are separated electrophoretically into unique species bands.

strategies [8] with fast, accurate, models [9-11] derived from the coupled partial differential equations that describe LoC physiochemistry. We present an efficient, configurable, systemlevel simulation framework for channel-based microfluidic LoCs that can be incorporated into optimization and synthesis tools. 1.1. Basic LoC Operation Figure 1 illustrates the operation of a canonical LoC-based immunoassay [12] where we wish to determine the presence and concentration of a desirable antigen Ag* in a mixture of undesirable antigen Ag. First, the mixture of Ag*, Ag and an appropriate antibody Ab are input into the sample and reagent wells respectively (Fig. 1(a)). In the second phase (Fig. 1(b)), voltages VI and V2 are applied while V4 is grounded. This voltage drop generates an electric field resulting in theflowof sample and reagent to the sample waste well in a continuous fashion. The mixture of Ag* and Ag is contacted with Ab at junction Jl. Between Jl and J2, Ag* and Ab react according to Ag* + Ab v^ Ag*Ab. At J2, the fluid stream is compressed or pinched by applying voltages V3 and V5 to aid in downstream separation. We refer to this collective set of operations as the loading phase. We call the third phase the injection phase (Fig. 1(c)). Here, the voltages are instantaneously switched such that a narrow band of material is injected into the separation channel by applying voltage V3, grounding V5 and setting VI, V2 and V4 to draw back the excess material. Finally, in the time-dependent separation phase (Fig. 1(d)), the species within the injected band undergo electrophoresis and separate into unique species bands as they travel toward the buffer waste well. The antigen-antibody complex Ag*Ab can then be differentiated from the excess reactants. We use this canonical example as our LoC benchmark because it illustrates many of the complexities common in most LoC applications. Fluid mixing, reaction, injection and separation are all integrated into a single, multi-function LoC. The loading phase, a steady-state process, is followed by the injection phase, a transient process, after a discrete voltage switching action.

A Lab-on-a-Chip Simulation Framework

Function: Well Mixer Splitter

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Type:

•

Reactor Separator l\/lixer Injector

H D •r+

(a)

Figure 2. The LoC simulation process: (a) Partial library of LoC units, (b) Channel topology constructed from the library of units, (c) Resistor network representation of the channel topology, (d) Directed Acyclic Graph representation of the channel topology.

We have developed a system-level LoC simulator that accounts for these complexities by combining Kirchoffian network analysis and topological sorting from electrical circuit simulation [7] with the sequential-modular structure of process flowsheet simulation [8]. Our simulator employs fast, accurate physiochemical models [9-11] and allows us to simulate complex designs in only seconds. 2. SIMULATION FRAMEWORK In our approach, we construct LoC channel topologies from a library of microfluidic unit operations. Figure 2(a) illustrates part of our current library of unit operations. A unit is defined in terms of its function or physiochemical process, and in terms of its type or physical geometry. Units communicate using four standard interface objects. The flow = {FC, a'^, Cmax, t} object contains phenomenological information for each chemical species in the system. It is updated by each unit and passed to the appropriate downstream unit, flow contains an array of Fourier coefficients FC that describe species' concentration distributions during the loading phase, and species' band-shapes during the separation phase . flow also contains the variance (j^ or band broadness, the peak concentration Cmax, and the cumulative transit time t for each species exiting the current unit. We also pass a geom object which contains the length L, width uj, and depth d of a. unit, a props object which contains the diffusivity Z), and mobility // of each species, and a buffer object which contains the electrical and thermal conductivity (A and hi respectively) and the concentration Cb of the background electrolyte or buffer solution. The information in flow can be used to calculate a separation performance metric known as resolution R between any two species in the system. Although ^ = 1.5 corresponds to baseline resolution, our simulator also accurately predicts the higher values commonly reported in the literature [12]. We use R as one way to assess overall design quality. In Fig. 2(b) we construct the channel topology of our benchmark. Our simulation approach involves translating a given channel topology into a resistor network (RN) and a directed acycUc graph (DAG). The RN and DAG for the topology shown in Fig. 2(b) are shown in Fig. 2(c) and Fig. 2(d) respectively. The mapping between the DAG and the RN is as follows: nodes with

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Table 1 Comparison between FEMLAB and simulator for the channel topology shown in Fig. 2(b). FEMLAB Simulator % Difference Ag* Ag*Ab Ag* Ag*Ab Ag* Ag*Ab a2 4.9 • 10^ 6.4 . 10^ 5.1-10^ 6.3-10^ 4.1 1.6 (M) 0.83 1.12 0.81 1.08 2.4 3.6 24.1 24.0 0.4 R {sec.) CPW ~ 7200.0 - 1.5 4800 X speedup t 2GHz CPU, 1GB RAM degree one in the DAG correspond to nodes of degree one in the RN, degree two nodes in the DAG correspond to a resistor with nodes at either end in the RN, and nodes with degree greater than two in the DAG correspond to a set of resistors equal to the number of incident edges in the DAG with nodes at each resistor endpoint and a single internal node (eg. node 13 in 2(c)). The RN can be used to calculate the flows and potentials for the loading phase (iioad and vioad) and the injection phase (iinj and Vinj) as shown in Eq. 1. R

Bo\

Bl

0 ;

lîioad\ ^ fyCioad\

f

\vioad)

\Bl

V 0

; '

R

B, 0 l-Hnj ^ yCinj\ 0 ) \VinjJ \ 0 J

... ^^

Here R is the resistance matrix, which in our work is the electrical resistance of each channel section. It is straight forward to include static pressure effects in R. B is the edge-node incidence matrix of the RN. BQ is the matrix formed after removing the columns that correspond to specified nodes from B so that linear independence is maintained [7]. The computational overhead of constructing these matrices is only incurred once for a proposed topology and thus enables an efficient search of applied potentials vcioad and vcinj. In the final step, the DAG is processed in topological order and each unit behaves in a sequential-modular or signal-flow fashion [8]. The flows and potentials calculated from Eq. 1 are passed to the appropriate units as required. This approach works well because LoCs do not contain recycle loops, and a stream-tearing approach [8] is not required. Since the models within each unit are highly nonlinear, our approach avoids the computational expense of simulating the system using a simultaneous approach. Further, our simulator does not require a priori initialization as is often the case with Newton-type solvers. By decomposing the simulation into a simultaneous linear part and a sequential nonlinear part, we can simulate the entire system in the time required for approximately one Newton iteration. 2.1. Example Results To test the accuracy and performance of our simulator, we compare it against an equivalent FEMLAB [13] simulation. In Table. 1 we compare the simulation results generated by FEMLAB and our simulator for our 9 unit benchmark (Fig. 2(b)). Our simulator and FEMLAB are in good agreement for the variance cr^, peak concentration Cmax^ and resolution R between Ag* and Ag*Ab. However, our simulator is over three orders of magnitude faster and thus enables iterative design and optimization. We can simulate designs with 50 units in under 3 seconds, but have been unable to simulate designs of the same complexity in FEMLAB. Fig.3(a) is a LoC from the literature [1] designed to perform a competitive immunoassay where labeled and unlabeled TheophylUne, Th* and Th, compete for antibody as shown in the

A Lab-on-a-Chip

Buffer

CD

(1:

Simulation

111

Tb*

C5]

®

Framework

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I sample waste

^—• detector \ buffer waste Sample VvaMc

1.97cm

(b)

(a)

Figure 3. Competitive immunoassay design comparison: (a) Design from literature with a 7.6cm X 7.6cm footprint [1]. (b) Simulator-based design with a 1.97cm x 2.61cm footprint.

Table 2 Comparison between literature design and simulation-based design.

a^ Sep. time R Area

(/im^)

{sec.) {cw?)

Literature [1,6] Th* Th*Ab 3.0 • 10^ 6.7 • 10^ 29.1 18.6 24.0 57.8

Double T Th* Th*Ab 2.6 • 10^ 6.0 • 10^ 2.7 1.7 27.0 5.1

Cross Th* Th*Ab 2.0 • 10^ 5.4 . 10^ 2.7 1.7 29.4 5.1

following reactions: Th* + Ab # Th*Ab and Th + Ab ^ ThAb. This multi-function chip incorporates mixing, reaction, injection and separation and reportedly fits within a 7.6cm x 7.6cm glass microchip. We can simulate this chip in approximately 2 seconds and our results are in good agreement with a previously published analysis [6]. Fig. 3(b) shows our new design of this chip. It achieves better performance than the original chip, and requires approximately 11 times less area. The design in Fig. 3(b) maintains the same reactor length and voltages as the original. We are also able to quickly evaluate the influence that interchanging particular units in the design has on the overall system performance. Table 2 shows a comparison between the original chip, which uses a double T injector (see Fig. 2(a)), and our new design, where we investigate both a double T and a cross injector. We are able to investigate design alternatives that would be prohibitively time consuming using CFD and experimental approaches. 3. Conclusion We have demonstrated that our simulator is capable of efficiently and accurately evaluating complex LoC systems. The ability to embed a modular simulation tool within our CAD method-

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ologies has been a key to the success of our earher work. We have explored design automation approaches for capillary electrophoresis (CE) LoCs using a distributed agent-based computing approach [14] and a tailored strategy based on Very Large Scale Integration (VLSI) circuit design and standard optimization algorithms [15]. As our library of LoC unit operations grows, we are able to address more complex LoC systems. The simulator presented here allows us to extend our design automation approaches to multi-function LoC systems like the immunoassay LoC benchmark discussed previously. Acknowledgments This research effort is sponsored by the Defense Advanced Research Projects Agency (DARPA) and U. S. Air Force Research Laboratory under agreement number F30602-01-2-0987 and by the National Science Foundation (NSF) under award CCR-0325344. The authors would like to thank members of the S YNBIOS YS group at Carnegie Mellon University.

REFERENCES 1. N.H. Chiem and DJ. Harrison. Microchip systems for immunoassay: an integrated inmiunoreactor with electrophoretic separation for serum theophylline determination. Clin. Chem., 44(3):591-598,1998. 2. A. Auroux, D. lossifidis, D.R. Reyes, and A Manz. Micro Total Analysis Systems. 2. Analytical standard operations and applications. Anal. Chem., 74:2637-2652, June 2002. 3. A.J. Pfeiffer, T. Mukherjee, and S. Hauan. Design and optimization of compact microscale electrophoretic separation systems. Ind. Eng. Chem. Res., 43:3539-3553,2004. 4. O. Geschke, H. Klank, and R Tellemann. Microsystem Engineering of Lab-on-a-Chip Devices. Wiley-VCH, 2004. 5. T. Korsmeyer, J. Zeng, and K. Geiner. Design tools for BioMEMS. In Design Automation Conference (DAC '04), pages 622-627,2004. 6. Y. Wang, R. Magargle, Q. Lin, J. Hoburg, and T. Mukherjee. System-oriented modeling and simulation of biofluidic lab-on-a-chip. In TRANSDUCERS '05, pages 1280-1283,2005. 7. T.L. Pillage, R.A. Rohrer, and C. Visweswariah. Electronic circuit and system simulation methods. McGraw Hill, 1998. 8. L.T. Biegler, I.E. Grossmann, and A.W Westerberg. Systematic Methods of Chemical Process Design. Prentice Hall, Upper Saddle River, NJ 07458,1997. 9. Y. Wang, Q. Lin, and T. Mukherjee. System-oriented dispersion models of general-shaped electrophoresis microchannels. Lab-on-a-chip, 4:453^63,2004. 10. Y. Wang, Q. Lin, and T. Mukherjee. A model for complex electrokinetic passive micromixers. RSC Lab-ona-Chip, 5(8):877-887,2005. 11. R. Magargle, J.F. Hoburg, and T. Mukherjee. Microfluidic injector models based on neural networks. In NanoTech (MSM '05), pages 616-619,2005. 12. WS.B. Yeung, G.A. Luo, Q.G. Wang, and J.R Ou. Capillary electrophoresis-based immunoassay. Journal of Chromatography B, pages 217-228,2003. 13. The Mathworks Inc. FEMLAB - Finite Element Modelling LABoratory version 3.2. http://www.femlab.com. 14. A.J. Pfeiffer, J.D. Siirola, and S. Hauan. Optimal design of microscale separation systems using distributed agents. In Foundations of Computer-Aided Process Design (FOCAPD '04), pages 381-384,2004. 15. A.J. Pfeiffer, T. Mukherjee, and S. Hauan. Simultaneous design and placement of multiplexed chemical processing systems on microchips. In International Conference on Computer-Aided Design (ICCAD '04), pages 229-236. 2004.


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Chapter 1

Two level control of the sequence fed batch continuous hybridoma bioreactor Irina D. Ofiteru, Alexandru Woinaroschy, Vasile Lavric Chemical Engineering Department, University Politehnica of Bucharest, Polizu 1-7, 011061 Bucharest, Romania

In the present study a recirculation system for monoclonal antibodies production, operated consecutively fed batch and continuously, was modelled and subjected to optimal control. The optimization procedure uses a two level approach: one regarding the overall process, and two inner ones, concerning the fed batch and continuous steps. The best switch time between the two operating modes was calculated, together with the best control variable profile for each section. Keywords: hybridoma, two-level optimization, optimal control, genetic algorithms 1. Introduction The monoclonal antibodies (MAbs) are produced in the last years in large quantities and still the potential growth of the market is more than 25% per year. There is a high demand for these products (used especially against cancer) but the current production costs are very high (by factors of 20 to 200 times per gram) compared with the ones from classical chemical synthesis (Sommerfeld and Strube, 2005). The main efforts are now focused on cutting down the operating costs. This can be achieved using the large potential for optimizing the processes; not only its upstream (cells, culture medium, supplements) and downstream (separation and purification) sections, but also the bioreactor itself. For small product concentrations in the fermentation step, the cost distribution between upstream and downstream is approximately 50-50%. Once the product concentration rises, both the overall and the upstream costs decrease (with more than 20%)), although the proportion of downstream costs in the overall costs increases (Sommerfeld and Strube, 2005).

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In the upstream section, one of the pertinent challenges is reducing the formation of large amounts of secondary metabolic products, which have inhibitory effects on cell growth and production rates. It is therefore important to maintain the cells in a physiological state characterized by a minimum production of waste metabolites and a maximum production of valuable products. This goal implies the development of an optimal nutrients supplying strategy which modifies the growth medium in such a way that the cells alter their metabolism to produce as much MAbs as possible, with minimal waste. Optimization studies have been made both for fed batch (Dhir et al., 2000; Sarkar and Modak, 2003; Woinaroschy et al., 2004) and continuous processes (Ofiteru et al., 2005). Our previous studies (Woinaroschy et al., 2004; Ofiteru et al., 2005) in optimal control of the hybridoma bioreactor treated separately the fed batch and the continuous operating modes, pointing out their benefits and drawbacks. The results suggested a combined approach, namely using the sequence fed batch - continuous. In the present study, a recirculation system composed of a perfectly mixed bioreactor operated consecutively fed batch and continuously, respectively, a cell separator, a mixer and a purge was modelled then subjected to optimal control. The optimization procedure uses a two level approach. The outer optimization, searching for the proper time switch between fed batch and continuous operation modes, is based upon the direct search algorithm of Luus and Jaakola. Basically, in the feasible region of parameters takes place a random search in discs with given radius having as centres the proposed values for these parameters. As a result, a possibly new optimum point could be reached, the centres of discs being moved accordingly. After each jump, a contraction of the discs' radii is imposed, till either the minimum value for the objective function is found or the allowed number of iterations is reached. The inner optimization procedure is based upon genetic algorithms, which are applied for the optimal glutamine set point computation for the fed batch operating mode(Woinaroschy et al., 2004) and the determination of the inlet flow profile in time for the continuous mode, both aiming to maximize the MAb production (Ofiteru et al., 2005). 2. Mathematical model The representation of the process is given in Figure 1, together with the main notations. Since the concentration of the cells is rather low, the recirculation fraction, a, was set to 0.15, while the purge fraction, p, to 0.005. For the first operating stage, which is fed batch, there is no recirculation, and that the process is formed only by the reactor, together with the feeding. The Nielsen kinetic model (Ryszczuc and Emborg, 1997) was used, such as in the aforementioned studies for the optimal control of the fed batch, respectively continuous bioreactors. This kinetic is a one-compartment model assuming amino acids as a limiting factor and saturated glucose metabolism. The cells produce monoclonal antibody (P), lactate (L), ammonia (M) and alanine (A)

Two Level Control of the Sequence Fed Batch — Continuous Hybridoma Bioreactor

1639

using a medium which has glucose (S) and glutamine (Q) as substrates. The production rate for MAb was modified (Ofiteru et al., 2003) to be computed the same way the specific growth rate is. For the detailed mathematical model used for each of the stages of the process, see Woinaroschy et al. (2004) and Ofiteru et al. (2005), respectively. aDv,Nv,r,N:D^S,OAI-rM,P

Figure 1. Sketch of the process, together with the main notations used in the mathematical model

2.1. The objective function The objective function should encode the search for the maximum MAb production through an optimum switch time between fed batch and continuous operation, an optimum glutamine set-point profile for the fed batch stage and an optimum flowrate profile for the continuous stage. The overall operating period for the system is fixed. Since we have a two level optimization problem, a specific objective function has to be used for each level. For the fed batch stage, as there is no dead cells removal, optimum glutamine set-point profile should be sought such as to keep their concentration as low as possible, besides the maximum MAb production. After some preliminary studies, two objective functions were employed (subject to minimization): Fobj^j,

=~

P ^ FB

P ^ FR

V ^ FB

V

.^ K

(1)

(2)

' Fl

The objective function described by eq, (1) takes into account the dead cells concentration. But it was found that the overall performance of the process was lowered by the request of keeping this concentration small, so that the objective function was simplified (see eq. (2)). The results obtained will be compared and the best compromise between keeping the dead cells concentration low and increase the overall performance of the process will be selected. For the continuous stage, the objective function should encode the search for the maximum MAb production through an optimum flowrate profile, Dv(t), for a given operating period:

1640

LD. Ofiteruetal

^Cont

(3) For the whole process, we seek for the maximum MAb production, so the natural choice of the objective function for the second level is: ^^^Process

= ^FB ' ^FB + Pcont

f/^\

3. Results and discussions 3.1. Solving procedure The fixed-time two-level optimization problem had to be solved off-line, due to the long computation times, using Luus and Jaakola algorithm to calculate the proper time switch between fed batch and continuous operating modes and genetic algorithms to optimize each individual sequence. The process starts as fed batch, with glucose and glutamine as substrates, until the proposed switch time is reached. After that, the continuous operation begins, starting from the cells' and metabolites' concentrations achieved in the fed batch period. For the fed batch step the command variable is the glutamine set-point (Woinaroschy et al., 2004). The command time profile is encoded into a chromosome, every glutamine set-point corresponding to a time stage being represented as a gene. For the continuous step the command variable is the inlet flow. The operating period was divided into the same number of intervals as the fed batch time and the same encoding procedure was applied (Ofiteru et al., 2005). In order to test the method's convergence, three replicas were done for both cases: with or without dead cells concentration in the objective function. Each run started fi-om the same conditions as in our previous studies. Also, some supplemental runs were done with higher values of substrates concentration in the inlet flow. 3.2. Discussion of results The maximum MAb obtained using the objective function without dead cells was 4385 mg, while the maximum MAb obtained considering them was 3686 mg. Clearly, this difference is the result of the higher concentration (respectively mass) of MAb after the fed batch step: 10.408 mg/1 (1766 mg) for the former case, 8.356 mg/1 (1067 mg) for the later case, respectively. Another significant difference is between time switches: 562.312 h for the former case, respectively 441 h for the later. The request of keeping the dead cells concentration optimally low, while maximizing the living cells concentration, slows down the MAb synthesis process. This is inherent, since the cells' death is the results, partly, of adversely medium conditions, which in turn favour the valu-

Two Level Control of the Sequence Fed Batch - Continuous Hybridoma Bioreactor

1641

able MAb production. As is already know, the glutamine decomposes spontaneously forming ammonia, which is toxic for the cells. This influences the command profile for the fed batch step (glutamine set point), which can vary in the interval 0.3 - 1.0 g/1. When the objective function does not take into account the dead cells concentration, the glutamine set point has values constantly higher then 0.6 g/1, with one exception (see Figure 2a). When the dead cells concentration enters the objective fUnction, the variation is more abrupt and the minimum allowed value (0.3 g/1) is reached several times. Comparing the results, in terms of productivity gain (expressed in mg MAb/h), we observed that the growing rate for the dead cells' concentration, 1.27 (the ratio between the final concentrations when using eq. (2) and (1) as objective function) is lower than productivity rate increase, 1.3, showing that the constraint imposed by eq. (1) are somehow relaxed. A side effect when using eq. (1) as objective function is the length of the fed batch process, lower than for the other case. It should be mentioned that the fed batch step is much longer than in the previous cases, while the concentration of the dead cells is rather higher (Woinaroschy et al. 2004; Ofiteru et al., 2003), irrespective of the objective function used. Definitely, this proves that the system's optimality does not imply the optimal behaviour of its parts. Command continue

Figure 2. The main state variable profiles for the system, a) objective function (1) and (3); b) objective function (2) and (3). Ny/No - viable/dead cells, cP/mP - MAb concentration/ mass

During the continuous period of the system, the dead cells concentration decreases constantly, due to the beneficial effect of the purge (see Figure 2, on the RHS of the dotted line). Around 800 h system time, the MAb concentration reaches a pseudo-stationary value which is maintained till the end of the process and, concomitantly, the living cells concentration starts to grow exponentially. This is supported by the high values of the inlet flow, which reaches constantly the maximum value allowed, thus ensuring a high level of substrates concentrations. One side effect of these high inflows is an increased ammonia concentration (data not shown), the only by-product (and in the same time the most deleterious one for the cells) which rises above the MAb concentration. Analysing the results obtained, we observed the low values for the valuable product concentration and concluded that some run-tests with higher glucose and glutamine concentration in the inlet flow should be done, so we increased

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LD. Ofiteruetal

them five times. Several interesting differences compared with the "normal" situation were noticed (no figure is presented, due to the lack of space). First, the switching time between the fed batch and continuous steps has closer values: 257.8 h for the objective function (1), respectively 277.3 h for the objective function (2). Second, the amount of MAb obtained is significantly higher, 10400 mg for the former case and respectively 9629 mg, for the later. Third, the dead cells concentration does not decrease after switching to the continuous step, but keeps increasing, its growth rate surpassing the viable cells concentration's one. This effect is more pronounced when using the objective function (1), although its influence upon the system's performance is indirect. This can explain why the inlet flow is kept to smaller values in this case. 4. Conclusions The analysis of a recirculation system (Figure 1) was done, first modelling it, and then searching for its optimal control. Two approaches were used, one more conservative, that considers maintaining the dead cells concentration at lowest level, the other more relaxed, taking into consideration only the highest MAb production. Although it could seem surprising, the relaxed objective function gave better results, in terms of MAb production, but at the cost of higher byproducts concentrations. A final selection of the variable command profile would not be proper without considering also the performance of the downstream separation processes. Based only upon the MAb production and byproducts outlet concentrations and without a sound economic analysis, we concluded that the advantage of using higher substrate concentrations in the feed (a bigger amount of MAb produced) is counter-balanced by the increase in dead cells and by-products concentration. But, if the market price for the MAb is high enough, this could be the recommended way of increasing the productivity. References Dhir, S., K.J. Morrow, R.R. Rhinehart, T. Wiesner, 2000. Dynamic optimization of hybridoma in a fed - batch bioreactor. Biotechnol. Bioeng. 67, 197 - 205. Ofiteru, I.D., Lavric, V.and A. Woinaroschy, A., 2003. Sensitivity analysis of the fed-batch animal cell bioreactor. Chemical Engineering Transactions 3 (3), 1845 - 1850. Ofiteru, I.D., Woinaroschy, A., Lavric, V., 2005. Optimal control of a continuous perfectly mixed hybridoma bioreactor. ESCAPE 15*, May 29 - June 1, Barcelona, Spain Ryszczuc, A., Emborg, C , 1997. Evaluation of mammalian fed - batch cultivation by two different models. Bioprocess Eng. 16, 185 - 191. Sarkar, D. and J.M. Modak, 2003. Optimization of fed-batch bioreactors using genetic algorithms. Chem. Eng. Sci. 58, 2284 - 2296. Sommerfeld, S., J. Strube, 2005. Challenges in biotechnology production - generic processes and process optimization for monoclonal antibodies. Chem. Eng. and processing 44, 1123 - 1137. Woinaroschy, A., Ofiteru, I.D., V. Lavric, 2004. Time-free schedule optimization of an animal cell fed-batch bioreactor. ESCAPE 14*, 16-19 May, Lisbon.


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Optimal Delivery of Chemotherapeutic Agents in Cancer Pinky Dua^, Vivek Dua^, Efstratios N. Pistikopoulos^ ^Centre for Process Systems Engineering, Department of Chemical Engineering, Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom ^Centre for Process Systems Engineering, Department of Chemical Engineering, University College London, Torrington Place, London JVCIE 7JE, United Kingdom Abstract In this paper, derivation of the optimal chemotherapy schedule is formulated and solved as a dynamic optimization problem. For this purpose two models representing the tumour growth kinetics are considered. The dynamic optimization problem for the first model, which is cell cycle non-specific, takes into account multiple time characteristics, drug resistance and toxicity. The discontinuity in the model is formulated by introducing integer variables. For the second model, which is cell cycle specific, the tumour growth is modelled via two compartments: proliferating and resting compartment. Keywords: Drug Delivery Systems, Cancer, Chemotherapy, Optimal Control, Mixed Integer Dynamic Optimization 1. Introduction Cancer is a collective term that describes a group of diseases characterized by uncontrolled and unregulated growth of cells leading to invasion of surrounding tissues and spreading to the parts of the body that are distant from the site of origin. There are around 200 types of cancer and cancers of lungs, breast, bowel and prostrate are the most common ones. There are three main stages in the process of carcinogenesis: initiation, promotion and progression. The normal cell changes to an initiated cell and then to cancer differentiated cell and finally invades and spreads to the surrounding cells. The simplest mathematical model describes the entire cell cycle as a uniform entity, where all the cells contained in a tumour are of the same type. The cell cycle nonspecific models consist of one compartment so that the effect of the anticancer agents is same on all the cells. However these models fail to describe the action of cycle specific drugs due to their over-simplified nature. The more detailed multi-compartment models (cell cycle specific models) are considered for this purpose. Here the cell cycle is divided into compartments depending on the types of cells that are affected by the drug. Chemotherapy is one of the most commonly used treatments for cancer that uses anticancer or cytotoxic drugs to destroy or kill cancer cells. The suitability of chemotherapy and the choice of drugs depend on many factors including the type of cancer, the location of the origin of the cancer in the body, how mature the cancer cells are and whether the cancer cells have spread to the other parts of the body. Chemotherapy targets dividing cells which does not only include cancer cells but any

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P.Duaetal

normal cells that are dividing such as the hair producing cells, cells that line mouth and digestive system and those in the bone marrow and the skin. Harrold and Parker (2004) recently proposed a mixed integer linear programming approach for the derivation of optimal chemotherapy schedule. In this paper, optimal chemotherapy schedules are derived with the objective of minimizing the final number of tumour cells at the end of the treatment. The rest of the paper is orgainzed as follows: in Section 2 optimal cancer chemotherapy schedule is derived for a cell cycle non-specific model whereas a cell cycle specific model is considered in Section 3; concluding remarks are presented in Section 4. 2. Optimal Control for Cell Cycle non-Specific Model A pharmacokinetic/pharmacodynamic model given in Martin (1992) is used for the derivation of the schedule. The objective is to obtain the drug dosage over a fixed period of time so as to minimize the number of cancer cells at the end of the period subject to constraints on toxicity and resistance of the drugs. The optimal control problem is formulated as follows: min J(w) = -z{T) s.t. z(t) = -h{t) + k{v{t)-Vth)y

z(0) = l n [ ^ ^ i){t) = u{t) - yv{t) t;(0) = i;o=0 0 ln(400) z(63) > ln(800) vit)>yVth,

forall/e[0,r]

v(t) - Vfh < yM,

for all t e [0, T]

where u = [wi,....,w„]ê9t" is the vector of the rate of delivery of drug, z(t) is the nondimensional tumour size, yt is a growth parameter, k is the proportion of the tumour cells killed per unit time per unit drug concentration, v(t) is the concentration of the anticancer drug at time t, Vth is the therapeutic concentration of the drug, >; is a 0-1 binary variable, 0 is the plateau population or the carrying capacity of the tumour, Aô is the initial tumour cell population, Vmax is the maximum amount of drug in the plasma, Vcum is the cumulative toxicity and Mis a large positive number. Note that in this formulation the binary variable, y, is introduced to model the discontinuity so that y takes the value 0, if 0 < tXO ^ '^th, or the value 1, if v{t) > Vth and the following transformation is introduced to make the model tractable and tumour size dimensionless (Dua, 2005):

Optimal Delivery of Chemotherapeutic Agents in Cancer

1645

where A^ is the number of tumour cells. The data for the parameters in the model is taken from Martin (1992). This problem is a mixed-integer dynamic optimization problem (Bansal et al, 2003) and is solved using gPROMS (gPROMS, 2003) by taking the control interval of one day and a time period of 84 days. The profiles of the optimal chemotherapy and tumour growth are shown in Figures 1 and 2 and are consistent with those reported in the open literature on optimal control strategies for cancer chemotherapy.

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Time (days) Figure 1 Optimal chemotherapy protocol for the cell cycle non specific model

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0

1

t

1

i

10

20

30

40

1 ^WiiiiiifctAÂA^

50

60

A^^^^^^^.>Â.>^^.>

70

80

90

Time (days) Figure 2 Predicted tumour growth using the optimal treatment protocol Initially no drug is delivered until the time approaches 21 days, the first interval of the muhiple characteristic time constraints, z(21) > ln(200), which are introduced to model

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drug resistance. This can be attributed to the fact that an initial high dose may not necessarily result in an overall decrease in the tumour size at the end of the time period as compared to the minimum tumour size obtained by the optimal chemotherapy protocol, while the constraints on toxicity and drug resistance are satisfied. Moreover, the drugs are given in large doses intermittently rather than in small doses continuously. The tumour follows the Gompertz growth until the first injection of the drug that results in a reduction of 70% of the initial tumour size. High intensity chemotherapy, from time 40 days till the end of the therapy, results in an increase in the rate of tumour reduction. The value of the objective function at the end of the treatment is given by 7.5x10"^ cancer cells. As a conclusion, the optimal way to reduce the tumour size was to apply high intensity therapy towards the end of the chemotherapy period. The optimal chemotherapy protocol shown in Figure 1 produces a 99.9% reduction of the initial tumour size.

3. Optimal Control for Cell Cycle Specific Model The model of Panetta and Adam (1995) describes the administration of the anticancer drug in the case of cell cycle specific chemotherapy. In this model, the effect of the drug depends only on the duration of the injection and not on the amount of the drug that is injected. For this reason, this model is modified so as to relate the effect of the drug with the rate of delivery of the drug (Dua, 2005) and minimization of the final tumour population is formulated as the following optimal control problem: mmJ{u) = P{T) + Q{T) u{t)

s.t. P{t) = {a-m-

n)P(t) + bQ(t) - git)P(tl

P(0) = PQ

Q(t) = mP(t)-bQ(tl

Q(0) = Qo

y(t) = ^(01 1 - ^ 1 - g(t)y(tl

y{0) = yo

v(t) = u(t)-lî(t% g(t) = k^Vi(t) y^^::0 + f ^ - | q - r - ^ l H , +qNH3 +rH,S + tHCl (4) ATrj ranges and averages obtained with both stoichiometric formulations, for the twelve samples of set B (NH3 formation, the water gas shift and methane reforming reactions, naphthalene, anthracene and char formation) are indicated in Table 2. Duret et al (2005) reported similar values for the average ATrj of the shift and methane reforming reactions (40 and -224 K respectively). However, results indicate that the spread of the shift reaction is high. Also, most ATrj spreads are larger for Eq. (3) than for Eq. (4). Table 2. ATrj averages and ranges & linear correlation coefficient p-values for certain reactions React. mnAT mxAT avAT H

C N O S CI FC V hm As T ER Eq. (3) stoichiometrv NH3 -657 -482 -565 1. 0. 16. 1. 51. 32. 66. 0. 50. 0. 21. 5. Shift -120 939 159 1. 1. 31. 3. 87. 40. 86. 1. 5. 1. 97. 0. Refo. -367 -230 -281 34. 39. 2. 55. 34. 5. 53. 39. 44. 32. 6. 3. Naph. -494 -383 -428 43. 48. 3. 61. 38. 3. 62. 46. 41. 39. 5. 4. Anth. -440 -299 -378 95. 79. 12. 74. 66. 40. 12. 91. 54. 90. 41. 48. Char -650 -568 -605 69. 79. 2. 84. 27. 1. 68. 70. 70. 65. 1. 13. Eq. (4) stoichiometry (same stoichiometry as above for NH3 and water gas shift reactions) Refo. -299 -213 -248 67. 74. 3. 88. 13. 6. 46. 74. 92. 65. 0. 19. Naph. -386 -331 -359 81. 77. 10. 74. 12. 4. 61. 81. 56. 87. 0. 61. Anth. -371 -249 -312 46. 64. 38. 31. 56. 70. 12. 36. 77. 49. 35. 64. Char -536 -480 -509 56. 49. 12. 56. 12. 3. 74. 58. 38. 61. 0. 92. Notes, mn: minimum; mx: maximum; av: average; FC:fixedcarbon; VM: volatile matter; hm: humidity; ER: equivalence ratio. Units. Temperature differences [K], p-values [%] 2.3. Nature of correlation between temperature differences and independent variables The ATrj are strongly dependent of fiiel composition and operating condition variables. However, the validity of the temperature difference model relies on the assumption that

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certain ATrj parameters are not affected by changes in processing conditions. Hence, sample correlation coefficients between each ATrj and input variables have been computed to determine whether any uncorrelated variables exist. Linear and nonlinear correlations (e.g. logarithms, exponentials, and inverses) of input variables have been tested. The p-values of the linear correlation coefficients are also in Table 2. This Fisher distribution variable is the probability of randomly obtaining a correlation as large as the one observed. T and ER (both independent of fuel properties) are of particular interest. At a 10% significance level (single digit percentages in bold), it appears that, • The ATrj are independent of T for the shift and ammonia (2NH3 4-> 3H2 + N2) reactions, and of ER for most other equations of Eq. (4). (i.e. CO2) stoichiometry. • Fixed carbon is uncorrelated to all reactions; humidity, volatile matter and ash are correlated to the NH3 and shift reactions, as is the ash content to tar formation. • The ATrj are strongly correlated to major elements {C H 0 } for the NH3 and shift reactions, minor elements {N CI} for most other reactions; and weakly to N for NH3 formation, and {CHS} for HC, tars, and char reactions (exponential of inverse test). 2.4. Modelling the temperature difference using artificial neural networks The ATrj represent a relationship between several operational variables, that was not physically modelled, but approximated instead by a nonlinear regression. Multilayer feed forward artificial neural network (NN) models have been used to represent the variation of each ATrj as a fiinction of the operational variables. Having established that strongly correlated variables are not the same for each reaction, a fiilly connected two layer NN is defined for each reaction. Each NN has a number of hidden sigmoid nodes that vary in fiinction of the number of inputs, and a single linear node as the output. As suggested by Sarle, (1994) direct input/output layer connections are added to account for the lower order effects noted in Table 2. The problem formulation is.

mmY,[^Tr.^-^fr.}j s

(5)

with. mh

^%

mf

I/=!

W

=^- +£w^J l + exp b,+Y,Wfl,x^

+£vf.x J

^='

(6)

The problem is solved with standard backpropagation of errors to the hidden layer. Incomplete target vectors are assigned a null error for unmeasured target values. 2.5. Network training and validation NNs can be estimators of arbitrary square-integrable fiinctions (White, 1990), however their major drawback is the high dimensionality of their weight space, which implies the risk of obtaining poor interpolations between training points. Generally speaking, large data samples, i.e. a number a least superior to the number of weights and biases, are needed to obtain good interpolation properties (termed generalisation). With the twelve observations of set B, as indicted in Table 3. (in bold), there would be fewer parameters than observations only in single input networks, or with two variables and a single hidden node per variable. Obviously a larger data sample would be preferable, but it can be costly in practice to generate a sample of several hundred or thousands of entries.

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4 6 Number of iterations

4 6 Number of iterations

Figure 1. Impact of input vars. on shift reaction

Figure 2. Size of hidden layer (Anthracene)

Preliminary validation tests and results from the literature allow reducing the number of input variables and of hidden nodes per input. Fixed carbon (FC) and volatile matter (VM) can be considered as dependent variables because the FC ratio is proportional to both the H/C and O/C ratios (van Krevelen, 1950; Jenkins et al, 1998). We used the preliminary correlation analysis (Table 2.) to further reduce the number of model parameters. For instance, Fig. 1. shows that, for the shift reaction, in concordance with p-value tests, error minimisation is more difficult with apparently uncorrelated inputs (but even more without ER). Fig. 2. indicates that to correlate the (Eq. (4)) set of ATrj of anthracene, there can be less than two hidden nodes per input, but that a single node is insufficient. The results obtained from the training set B have been assessed with another set of data (set A). The quality of generalisation for certain reactions is given in Table 4. as the relative error between modeled and calculated ATrj. Generalisation is particularly poor for the shift reaction while it is better for char and HC reactions. Table 3. Number of parameters (w & b) per input variables and hidden nodes per input. h nodes/vars.

1

2

1

4

5

6

7

8

9

10

11

12

1 2 3

5 8 11

11 19 27

19 34 49

29 53 77

41 76 111

55 103 151

71 134 197

89 169 249

109 208 307

131 251 371

155 298 441

181 349 517

Table 4. Errors on calculatec (Eq. (4) stoichiometry) and interpolated (100000 iterations) ATrj h/vars 3 3 6 6

Reaction

Input variables

NH3 Shift Shift Reform. Char

{C, H, O, hum, ash, ER} {C, H, O, hum, ash, ER} All 12 input variables 10 vars. (all but FC&VM) 10 vars. (all but FC&VM) 3

av B 0.000%

A.1 -17%

A.2 -17%

A.3 -22%

A.4 -19%

0.77% 0.000% 0.020% 0.002%

-150% -71% 29% 14%

-79% -125% 34%

-58% -131% 15%

-50% -123%

8.1%

-0.1%

29% 6.4%

3. Conclusion and recommendations A reaction model has been developed for the rapid computation of product compositions of biomass gasification. An NS equilibrium model based on total tar measurements, is first applied to estimate the distribution of tar species. The product distribution is then formulated as a stoichiometric equilibrium model with reaction equilibrium temperature

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differences. Although certain temperature differences appear to be uncorrected to independent variables such as T and ER, other temperature differences are strongly correlated to these variables. Since there is no clear evidence of any single or characteristic relationship between operational variables and temperature differences, the use of an appropriately designed N N appears as a solution to parameterise the reaction temperature differences, even with a data sample of limited size. Future improvements to generalisation could include adding data samples and/or additional constraints to improve the smoothness of the NN regression, e.g. weight decay (Krogh and Hertz, 1991), and using prediction intervals (De Veaux et al, 1998). Nomenclature ajk or A: number of atomic elements k in molecular specie i, and element molecule formula matrix b: network biases C: rank of formula matrix (usually C = M, the number of atomic elements) F: number of stoichiometric degrees of freedom (of linearly independent equations) G: Gibbs function [kJ/kmol]; I: identity matrix ni or n: quantity of molecular specie i (or of all species) at equilibrium [kmol] N: number of molecular species; N: complete stoichiometric matrix m: number of terms in a sum; M: number of elements P: reaction pressure [kPa]; R: gas constant [kJ/kmol-K]; T: reaction temperature [K] ATrji temperature difference between equilibrium and actual composition for reaction j [K] tar & tars: reconciled measurement of tar concentration and subset of tar species Xf! input variable f to network; w: network weight Z: matrix of dimension C x F when the only compositional constraint are element abundances Greek letters |x: chemical potential [kJ/kmol]; v: stoichiometric coefficient; L,y extent of reaction j Indexes f: input variables to network; i: molecular species; I: isomers h: hidden nodes; j : chemical reactions; k: atomic elements; s: observations

References E.H. Battley and J. R. Stone, 2000. Thermochimica Acta., 349, 153. A.G. Buekens and J.G. Schoeters 1985. in Fundamentals of Thermochemical Biomass Conversion, R.P Overend, T.A Milne and L.K Mudge eds., Elsevier Applied Science (Ch. 35). R.D. De Veaux, J. Schumi, J. Schweinsberg, L.H. Ungar, 1998. Technometrics., 40, 273. A. Duret, C. Friedli and F. Marechal, 2005. Journal of Cleaner Production., 13-15, 1434. R.J. Evans and T.A. Milne, 1987. Energy & Fuels., 1, 123. L. Fagbemi, L. Khezami, R. Capart, 2001. Applied Energy., 69,293. P. Garcia-lbanez, A. Cabanillas and J.M. Sanchez, 2004. Biomass & Bioenergy., 27, 183. B.M. Jenkins, L.L. Baxter, T.R. Miles Jr. and T.R Miles, 1998. Fuel Proc. Technol., 54, 17. CM. Kinoshita, Y. Wang, J.Zhou, 1994. J. Analytical and Applied Pyrolysis., 29, 169. A. Krogh and J.A. Hertz, 1991. Neural Information Processing Systems., 4, 950. D. Merrick, 1983. Fuel., 62, 535. S.A. Patel and L. E. Erickson, 1981. Biotechnology & Bioengineering., 23, 2051. W.S. Sarle, 1994. Proc. 19th annual SAS users group international conference., 1538. F. Shafizadeh, 1982. Journal of Analytical and Applied Pyrolysis., 3, 283. W.R. Smith and R. Missen, 1982. Chemical Reaction Analysis: Theory and Algorithms., USA.: John Wiley and Sons (Ch. 2 & Ch. 6) W.M. Thornton, 1917. Phil Mag., 33, 196. A. van der Drift, J. van Doom and J.W. Vermeulen, 2001. Biomass & Bioenergy., 20, 45. D.W.van Krevelen, 1950. Fuel., 29, 269. H. White, 1990. Neural Networks., 3, 535. Acknowledgements Funding provided by the Ministry of Education, Culture, Sports, Science and Technology of Japan is gratefully acknowledged.


Hybrid metabolic flux analysis/data-driven modelling of bioprocesses A. Teixeira", C.M.L. Alves", P. M. Alves^ M. J. T. Carrondo"'^ R. Oliveira^ "^ FCT/UNL, Laboratorio de Engenharia Bioquimica, P-2825 Monte da Caparica, Portugal ÎBET/ITQB, Apartado 12, P-2781-901 Oeiras, Portugal Abstract This work proposes a hybrid modeUng method that combines data-driven modeHng, metaboUc flux analysis, and bioreactor transport phenomena models. The main objective was to understand the time evolution of metabolism in fedbatch cultivations and to identify favorable conditions for product formation. The overall metabolic network is simplified using the elementary flux modes method. The hybrid modeling scheme was implemented in HYBMOD, an inhouse developed software. Preliminary simulation studies confirmed that HYBMOD could identify the "true" kinetic fianctions of elementary flux modes provided that a given set of state variables are measured. The method was applied to analyze the data of recombinant BHK-21 cultures. Keywords: Metabolic Flux Analysis, Elementary Flux Modes, Hybrid Modelling, BHK21 culture. 1. Introduction The main metabolic pathways of many biological systems with industrial interest are today well known. In principle, classical bioreactor dynamic optimization schemes could profit by the incorporation of this knowledge. Under the balanced growth condition, networks of metabolic reactions are reflected in large stoichiometric matrices and flux kinetics that must obey the constraint that net formation of intracellular components is zero. The overall reactions network may be simplified to a smaller size network using the elementary flux modes method [1]. Elementary flux modes (EFM) are the simplest paths within a metabolic network that connect substrates with endproducts. Each flux mode lumps all the reactions in the particular path, and is governed by the rate-limiting step. The reaction mechanism within the biological system may thus be viewed as a collection of EFMs. Bioreactor dynamical models can be established from material balance equations of EFM input/output compounds in a standard fashion [2]. The main problem of this method is the establishment of reliable and accurate kinetics of EFM, which should obviously reflect the "true" intracellular modulation mechanisms. Provided that a given minimum set of state variables is experimentally

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available, the identification of kinetic functions from measured data becomes possible. In this work, we studied the identification of elementary flux mode kinetics with artificial neural networks (ANN). Thus the approach proposed is a hybrid modeling method that combines data-driven modeling (in our case ANNs), metabolic flux analysis, and bioreactor transport phenomena models. This framework was applied to a recombinant baby hamster kidney BHK-21 cell line expressing the fusion glycoprotein IgGl-IL2 for cancer therapy [3]. The main objective was to understand the time evolution of metabolism in fedbatch cultivations and to identify favourable conditions for product formation. BHK-21 elementary flux modes were obtained with the FluxAnalyser software [4]. The hybrid modelling scheme was implemented in HYBMOD, an in-house developed software. Preliminary simulation studies confirmed that HYBMOD could identify the "true" kinetic functions of elementary flux modes. The application of the same method to experimental data provided the metabolic flux distribution over the time. In future studies, bioreactor dynamic optimization supported by the hybrid model will be performed. 2. Proposed method 2.1. Elementary flux modes The first step is to analyze the metabolic network structure of the biological system under study. The objective is to extract only the relevant knowledge to be used for bioreactor performance optimization. Here we use a metabolic flux analysis (MFA) method called "elementary flux modes". Elementary flux modes are the simplest paths within a metabolic network that connect substrates with end-products [1], thus they define the minimum set ofn species that must be considered for modeling and how they are connected in a simplified reaction mechanism. If a given system has m elementary flux modes, then the result of this analysis is a nxm stoichiometric matrix K. 2.2. General bioreactor hybrid model Once a reaction mechanism has been established using the EFM method, the next problem is to identify the kinetics of EFM from data. Here we adopted the hybrid model structure represented in Figure 1, where the main hypothesis is that reaction kinetics of EFM are partially known. This model structure can be formulated mathematically by the following two equations: - - = r(c,w)-Dc + u at r(c,w) = K((pj(c)xpj(c,w)).^^^ ^^

(la) (lb)

with c a vector of n state variables, r a vector of n volumetric reaction rates, K a nxm coefficients matrix obtained from the elementary flux modes analysis, ^(c) are m known kinetic ftinctions established from mechanistic and/or empirical knowledge, /?Xc,w) are m unknown kinetic functions, D is the dilution rate, u is a vector ofn volumetric input rates (control inputs).

Hybrid Metabolic Flux Analysis/Data-Driven Modelling of Bioprocesses

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Biological system Bioreactor system

Mechanistic/empiric kinetics w

Unknown kinetics: w nonparametric model

^ Material balance equations

Kinetic model Figure 1. General hybrid model for bioprocesses Functions /7y(c,w) are to be identified form data using nonparametric modeling techniques with w a vector of parameters that must be estimated from data. A 3-layered backpropagation neural network with sigmoidal activation function was used:

p(c,w) = Pâx^(w2^(WiC + bi) + b2)

(2)

with Pmax a vector of scaling factors with dim(pn,ax)='w, Wi, bi, W2, b2 are parameter matrices associated with connections between the nodes of the network, w is a vectored form of Wi, bi, W2, b2 and s(.) the sigmoid activation function defined as follows: s(x) = -

—

1 + e ""

(3)

^ ^

2.3. Identification of unknown kinetics The parameters vector w must be estimated from measured data of control inputs and process state over time: {Z)(t), u(t), c(t)}. For the identification of w, a least squares criteria of residuals in concentrations was adopted. The least squares problem was solved using the quasi-Newton algorithm with conjugate gradient with line search (MATLAB^^ optimization toolbox). This optimizer requires analytical residuals gradients, which were computed using the sensitivities method (see [5] for details) 3. Case study: recombinant BHK-21 culture 3.1. Process description A BHK-21 A cell line (a subclone of ATCC accesion CCL-10) expressing the fusion glycoprotein IgGl-IL2 was used. The experiments were carried out in serum free and protein free medium (SMIF6, Life Technology, Glasgow, UK). The batch culture was set up in a 2 1 reactor volume and the fed-batch cultures were set up at 3 different volume scales (2, 8 and 24 1). Sparger aeration was

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employed. Dissolved oxygen concentration was set at 15% of air saturation. Agitation rate used was 60 rpm; pH was set as 7.2 and controlled through the addition of CO2. Analysis techniques are described elsewhere [3]. Experimental data of viable cells cconcentration and six extracellular species (glucose, glutamine, lactate, ammonia, alanine and desired product) was collected. 3.2. Elementary flux modes The metabolic network considered takes into account the most relevant pathways involving the two main nutrients (glucose and glutamine) whithin the central metabolism of BHK cells, which are glycolysis, glutaminolysis, TCA cycle and nucleotides synthesis. The FluxAnalyzer software [4] was used to determine the EFM of BHK metabolic network. The information that must be provided to the algorithm is (i) all metabolites of the system including information whether they are internal or external and (ii) all reactions including information whether reactions are reversible or irreversible. There are seven EFMs describing the BHK metabolic network. Each one is a collection of reaction steps. The hypothesis of balanced growth allows the elimination of the intermediate metabolites resulting in a simplified set of reactions connecting extracellular substrates (glucose and glutamine) with end-products (lactate, ammonia, alanine, carbon dioxide, purine and pyrimidine). Furthermore, some assumptions concerning the fluxes were made based on literature, resulting in five EFMs. The following stoichiometric matrix was obtained |l-kd ri [ 1 0

X2 ra XA rs rigo 0 0 0 0 0

0 - 1 - 1 0 0 - 2

0

Glc

0

0 0

-1 -1 -5

0

Gin

K= 0

2 0

0

0

0

0

Lac

0

0 0

1

2

2

0

Amm

0

1

0

0

0

Ala

0 0

0

0

0

1

IgG

0 0 0

(4^

Note that K also accounts for cell growth and product formation as completely independent fluxes since the stoichiometry of theses reactions is not available. 3.3. Identification of EFM kinetics Off-line measurements of the seven state variables from five experiments were used for model training and validation. The neural network had three inputs: glucose and glutamine, the main limiting nutrients and ammonia, the main toxic by-product. The output vector was formed by the following seven specific kinetics: |i-kd, ri, r2, rs, r4, rs, rigo- The structure of the artificial neural network was selected by trial-and-error. The criterion to stop the parameter estimation algorithm was the minimum modeling error for the validation data set. The best result was obtained with a single hidden layer and five hidden nodes. Figure 2 presents the hybrid modeling results for one of the training and

Hybrid Metabolic Flux Analysis/Data-Driven Modelling of Bioprocesses

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one of the validation sets. A relevant result is the fact that the hybrid model was able to describe simultaneously all five experiments with high accuracy. The flux distribution identified by the hybrid model for one of the fed-batch cultures is shown in figure 3. 2.5

100 200 time(h)

300 0

50

100 150 time(h)

200

Figure 2. Hybrid model results for a training data set (a) and a validation data set (b).

0.07

0.05

^

0.03

0.01

-0.01

^

130

160

190

time (h) -0.03

Figure 3. Elementary flux modes identified by the hybrid model

220 > .

250

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4. Conclusions Analyzing the intracellular flux distribution over the time course of the bioreaction, conclusions can be taken concerning cellular control mechanisms, which in this case are consistent with published works of other mammalian cell lines. The agreement between theory and model predictions confirms that the proposed framework can produce valid conclusions. The proposed hybrid modelling approach, which integrates knowledge from measured data, metabolic flux analysis, and bioreactor transport phenomena, will be used to develop optimal control strategies for fed-batch bioprocesses.

Acknowledgements The authors acknowledge the financial support provided by the Fundaâo para a Ciencia e Tecnologia through project POCTI/BIO/57927/2004 and PhD grant SFRH/BD/13712/2003.

References [1] S. Schuster, D.A. Fell, T. Dandekar. (2000) A general definition of metabolic pathways usefiil for systematic organization and analysis of complex metabolic networks. Nature Biotech., 18: 326-332. [2] A. Provost and G. Bastin. (2004) Dynamic metabolic modeling under balanced growth condition. J, Process Control, 14: 717-728. [3] A. Teixeira, A.E. Cunha, J.J. Clemente, J.L. Moreira, H.J. Cruz, P.M. Alves, M.J.T. Carrondo, R. Oliveira. (2005) Modelling and optimization of a recombinant BHK-21 cultivation process using hybrid grey-box systems. J. Biotechnol. 118: 290-303. [4] S. Klamt, J. Stelling, M. Ginkel, E.D. Gilles. (2003) FluxAnalyser: exploring structure, pathways, and flux distributions in metabolic networks on interactive flux maps. Bioinformatics, 19(2):261-269. [5] R. Oliveira. (2004) Combining first principles modelling and artificial neural networks: a general framework. Comp. Chem. Engn. 28, 55-766.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantehdes (Editors) © 2006 PubUshed by Elsevier B.V.

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Rotavirus-Like Particle Production: Simulation of Protein Production and Particle Assembly Antonio Roldao , Helena L. A. Vieira , Manuel J.T. Carrondo ' , Paula M. a

u

Alves and R. Oliveira

""IBET/ITQB, Apartado 12, P-2781-901 Oeiras, Portugal * FCT/UNL, Laboratorio de Engenharia Bioquimica, P-2825 Monte da Caparica, Portugal Abstract In this work we study the production of rotavirus virus-like particles (VLP) using the baculovirus expression vector system (BEVS). A model-based optimization of the infection strategy was performed in order to maximize the production of correctly assembled VLPs. A structured mathematical model describing the relevant intracellular processes (baculovirus adsorption and trafficking, DNA replication and gene expression) was employed. Some intracellular processes take several hours for completion and may be regarded as pure time delays. A modified 4^V5^^ order RungeKutta solver was employed to integrate the ODEs system with pure time delays. A coinfection program using different combinations of multiplicity of infection (MOI) for each gene {vp2, vp6 and vp7) was investigated. The best results were obtained for MOI combinations of 2 (vp2)H-5 (vp6)+8 {vp7) or 5+2+8. It was also concluded that viral protein 7 (VP7) was the limiting component for VLPs assembly. This study highlights the usefulness of mathematical modeling in the design of improved infection strategies for VLPs production. Keywords: Rotavirus, VLP, modeling, simulation, assembly. 1. Introduction In the last decades. Rotavirus disease (RVD) emerged worldwide to become the primer cause of severe gastrointestinal illness in children, with an incidence estimated at 111 million episodes and a total of 440.000 deaths per year in children with less than 5 years of age [1]. Rotavirus, a non-enveloped virus can be mimicked for vaccine purposes by a triple-layered concentric VLP protein structure: the innermost layer composed by 60 dimers of VP2 (102.7 kDa) [2]; the middle shell formed by 260 trimers of VP6 (44.9 kDa) [3] and the third, outer layer composed by 780 monomers of glycoprotein VP7 (37.2 kDa) [3]. The viral proteins VP2, VP6 and VP7 (and the corresponding VLP assembly) may be effectively produced in Spodoptera frugiperda Sf-9 cells by infecting with recombinant baculovirus containing the three genes of interest, bvp2, bvp6 and bvp7. The process main steps involved in VLPs production are summarized schematically in Fig. 1 for a co-infection strategy using three monocistronic baculovirus vectors, each one expressing the bvp2, bvp6 and bvp7 genes individually. Step 1 encloses four main events: i) the binding of extracellular baculovirus containing the

A. Rolddo et al

1674

genes of interest to the plasmatic membrane and their entry by adsorptive endocytosis [4]; ii) the release of viruses' nucleocapsids (containing the viral genes) to the cytoplasmic space due to the pH decrease inside the endosomme; iii) the nucleocapsids migration to the cell nucleus and iv) the nucleocapsids binding to the nucleus membrane (thereby inserting the genes into the cell nucleus) that triggers the viral DNA (vDNA) replication. The vDNA transcription is the main event in Step 2. In the following step, Step 3, the corresponding mRNA leaves the nucleus and migrates to ribosomes where the coded VPs are synthesized. The synthesized proteins, VP2, VP6 and VP7, assemble into triple layered VLP in Step 4 according to mechanisms that are not yet fully known. Finally, Step 5 consists in the VLPs release to the extracellular medium. The main aim of this study is to optimize the VLPs production using a detailed structured mathematical model describing all five steps presented in Fig. 1. A co-infection program, in which Sf-9 cells were infected which three monocistronic baculovirus vectors (each one expressing the vp2, vp6 and vp7 genes individually) was investigated. 2. Mathematical model The mathematical model used in this work (Table 1) has been previously calibrated/validated by our group with original data [5]. A brief description of the model is provided below. A set of simplifying assumptions were taken: i) MOIs grater than two correspond to synchronized virus infection [6]; ii) infection kinetics are independent of the expression vector; iii) negligible virus budding and release [6-8] and also iv) VLPs assembly is a fast process ruled by the underlying stoichiometry. Plasmatic Membrane Nuclear Membrane mRNA

Baculovirus DNA

Protein VP7 Protein VP2 Protein VP6

VP7-Q

TrimersofVP6 Monomers of VP7 Monomers of VP6 VLP Uncompleted double layered 2/6VLP

Figure 1. VLPs production steps for co-infection strategy using three monocistronic baculovirus vectors each one expressing vp2, vp6 and vp7 genes individually.

Rotavirus-Like Particle Production

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Depletion of extracellular virus due to binding to insect cells surface is expressed by Eq. (1), where Vj represents the extracellular concentration of virus7 (copiesDNAj.ml"^), with subscript index 7=2,6,7 corresponding to vp2, vp6 and vp7 genes respectively, ka the adsorption rate equal to 1.3x10'^ ml.cell^min'^ [6] and Ni the concentration of infected cells (celLml'^). The infected cells equation has two terms (Eq. (2)): the first, accounts for the increase in infected cells concentration due to binding of baculovirus to uninfected cells while the second, represents the cells death rate (Eq. (4)). Table 1. Mathematical Model Equations -^ = -kNy. dt

Eq.(l)

a I J

—L = kNV.-k,N, ^f

a u t

Eq. (2) a I

'-^-Wr^.r'û

Eq.(3)

kd=kdx+k^^{TOl-r,)

Eq. (4)

— = 1^ A V.{t-r^ r)^k^^.,,

.

^

J

fo..At) = [Tt~

.(0

Eq. (5)

DNAJ

'-'^'-^'^

^ = ksRNA jI^NA Y fRNA Jit) - knmA jKNA j dt dRNA dVP ^=kypjRNAj(t-Typj)fypjit)N^ dt

f^.,,

Eq.(6) Eq. (7)

Eq. (8)

The uninfected cell population, Nu (celLml"^), in Eq. (3) has two terms: the first term is the "conversion" of uninfected cells into infected cells due to virus binding and the second term is the intrinsic cell death rate. The intracellular dynamics of vDNA, DNAf""" (DNA.ceir^), is given by Eq. (5). The first term characterizes the transport of genes from extracellular virus into the cell nucleus, the trafficking efficiency and the virus trafficking time. The second term reports the vDNA replication kinetics, which is assumed to be of Michaelis-Menten type. The term^Â^^j (t) accounts for the intrinsic metabolic decay due to infection (Eq. (6)). The transcription of vDNA is defined by first order kinetics in Eq. (7), with RNAj (RNA.cell"^) the concentration of mRNA from gene j \ ksRNAj (h'^) the first order kinetic constant and koRNAj the first order mRNA degradation rate (h'^). The fmAjit) represents the progressive loss of capacity to synthesize mRNA, here similar to Eq. (6). Concerning the VP synthesis, previous models [9,10] have used zero order kinetics to predict/simulate VPs production. The zero-order kinetic constant has been defined empirically as a function of number of viruses infecting the cells. In this study, intracellular mRNA is calculated and therefore

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a more mechanistic relationship between VP synthesis kinetics and mRNA concentration is possible. Therefore, the protein synthesis, VPj (jigj.mr^) was defined by first order kinetics on mRNA (Eq. 8), with kypj the first order kinetic constant (|ig.RNA" \h"^), Tvpj the time delay accounting for glycosilation in protein VP7, equal to 0.5h [11] and/^p/O the linear decay function of VP synthesis capacity, analogous to Eq. (6) for DNA. The mathematical model has pure time delays associated due to protein glycosilation and virus trafficking inside the cell. Therefore, a modified 4^/5 order Runge-Kutta solver, in which all intermediate integration results are stored, was used to integrate the ODE system with pure time delays. All simulations were performed in MATLAB™^ (The MathWorks, Inc., US, 1994-2006).

3. Infection strategy optimization A co-infection program in which the insect cells were infected with three monocistronic baculovirus vectors (each one expressing the vp2, vp6 and vp7 genes individually) was evaluated through simulations. VLPs production was calculated based on total VP2, VP^ and VP7 synthesized and the VLPs stoichiometric composition. The theoretical mass stoichiometric ratio VP2:VP6:VP7 of correctly assembled particle is known to be 1:2.8:2.4 (w/w). The co-infection program investigated is given in Table 2. The individual MOIs varied (either 2, 5 or 8) while the total MOI was kept constant to be 15 (thus guaranteeing that the level of DNA polymerase is similar to that of the experiments used to validate the model). Table 2. Co-infection program MOI = 5 {vp2) MOI = 5 {vp6); MOI = 5 (vpT)

Prog. (1)

MOI = 2 (vp2) MOI = 8 {vp6); MOI = 5 {vp7)

Prog. (2)

MOI = 2 (vp2) MOI = 5 {vp6); MOI = 8 (vp7)

Prog. (3)

MOI = 5 {vp2) MOI = 8 {vp6); MOI = 2{yp7)

Prog. (4)

MOI = 5 {vp2) MOI = 2 {vp6); MOI = 8 (vp7)

Prog. (5)

MOI = 8 {vp2) MOI = 2 {vp6); MOI = 5 (vp7)

Prog. (6)

MOI = 8 (vp2) MOI = 5 {vp6); MOI = 2 (vp7)

Prog. (7)

Figure 2 shows predicted protein profiles for the different infection strategies. As expected, higher MOIs promote higher expression levels of the corresponding viral protein. However, higher intracellular concentration of a particular VP does not necessarily imply higher VLPs yield. Viral proteins must be synthesized at correct stoichiometric ratios. Figure 3 shows the total production of correctly assembled VLPs for the different infection strategies. The higher VLPs yield was obtained for the co-infection strategies where the MOI of VP7 was higher, i.e., the combination MOI = 2, 5 and 8 for vp2, vp6 and vp7 respectively and also MOI = 5, 2 and 8. The results obtained clearly indicate

Rotavirus-Like Particle Production

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that VP7 is the limiting VP in the process of particle assembly. Therefore, infection strategies favoring the formation of VP7 seem to be the key for VLPs production optimization.

(>n -

i

80 • 70 •

/ 1 /

605040 -

solo •

/ / / /

10 • 0• 48

72

96

/ .

. 48

Time (hpi)

.

.

72

96

B 120

144

Time (hpi)

48

72 96 Time (hpi)

Figure 2. Simulations of intracellular VP production in co-infection strategy. Pattem for VP2 (A): the full line (—) represents Prog. (1), the dot line ( ) represent Prog. (2-3), the dash line ( ) indicates Prog. (4-5) and dot and dash line (—-) Prog. (6-7). Pattem for VPg (B): the fiill line (—) represents Prog. (1), the dot line ( ) represent Prog. (5-6), the dash line ( ) indicates Prog. (3-7) and dot and dash line (—-) Prog. (2-4). Pattem for VP7 (C): the full line (—) represents Prog. (1), the dot line ( ) represent Prog. (4-7), the dash line ( ) indicates Prog. (2-6) and dot and dash line (—) Prog. (3-5). 4. Conclusions and future work The production of Rotavirus-like particles (VLPs) adopting a co-infection scheme was assessed using the baculovirus expression vector system (BEVS). A detailed structured model accounting for intracellular processes such as vDNA replication and transcription, usually not taken into consideration in modeling this type of systems [9,10,12,13], was used. The model allowed investigating the effect of MOI on each VLPs production step and on its final VP and VLP yield. The best resuhs were obtained for MOI combinations of 2 (vp2)+5 (vp6)+S (vpT) or 5+2+8. It was also concluded that viral protein 7 (VP7) was the limiting component for VLPs assembly.

A. Rolddo

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et al

Figure 3. VLPs content simulations for each co-infection strategy reported in table 1.

This study highlights the usefulness of this mathematical model in the design of improved infection strategies for VLPs production. In future studies, the model will be improved in order to consider infection statistics, which is known to follow a Poisson distribution. Bibliography [I] [2] [3] [4]

[5]

[6]

[7]

[8] [9]

[10] [II] [12] [13]

U.D. Parashar, E.G. Hummelman, J.S. Bresee, M.A. Miller and R.I. Glass, 2003, Global illness and deaths caused by rotavirus disease in children, Emerg Infect Dis, 9, 5, 565. M. Labbe, A. Charpilienne, S.E. Crawford, M.K. Estes and J. Cohen, 1991, Expression of rotavirus VP2 produces empty corelike particles, J Virol, 65, 6, 2946. B.V. Prasad, G.J. Wang, J.P. Clerx and W. Chiu, 1988, Three-dimensional structure of rotavirus, J Mol Biol, 199,2,269. L.E. Volkman and P.A. Goldsmith, 1985, Mechanism of neutralization of budded Autographa Califomica Nuclear Polyhedrosis Virus by a monoclonal antibody: Inhibition of entry by adsorptive endocytosis. Virology, 143, 185. A. Roldao, H.L.A. Vieira, M.J.T. Carrondo, P.M. Alves and R. Oliveira, 2006, Intracellular dynamics in Rotavirus-like particles production: Evaluation of multigene and monocistronic infection strategies. Submitted, K.U. Dee and M.L. Shuler, 1997, A mathematical model of the trafficking of acid-dependent enveloped viruses: Application to the binding, uptake, and nuclear accumulation of baculovirus, Biotechnol Bioeng, 54, 5, 468. J.F. Power, S. Reid, K.M. Radford, P.P. Greenfield and L.K. Nielsen, 1994, Modeling and optimization of the baculovirus expression vector system in batch suspension culture, Biotechnol Bioeng, 44, 6, 710. M. Rosinski, S. Reid and L.K. Nielsen, 2002, Kinetics of baculovirus replication and release using real-time quantitative polymerase chain reaction, Biotechnol Bioeng, 77, 4,476. Y.C. Hu and W.E. Bentley, 2001, Effect of MOI ratio on the composition and yield of chimeric infectious bursal disease virus-like particles by baculovirus co-infection: deterministic predictions and experimental results, Biotechniques, 75,1, 104. Y.-C. Hu and W.E. Bentley, 2000, A kinetic and statistical-thermodynamic model for baculovirus infection and virus-like particle assembly in suspended insect cells, Chem Eng Sci, 55, 3991. H. Lodish, D. Baltimore and J. Darnell (eds.), Molecular Cell Biology, Scientific American Books, Inc., New York, USA, 1986. G. Enden, Y.H. Zhang and J.C. Merchuk, 2005, A model of the dynamics of insect cell infection at low multiplicity of infection, J Theor Biol, 237, 3, 257. B. Jiang, M.K. Estes, C. Barone, V. Bamiak, CM. O'Neal, A. Ottaiano, H.P. Madore and M.E. Conner, 1999, Heterotypic protection from rotavirus infection in mice vaccinated with virus-like particles. Vaccine, 17, 7-8, 1005.


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Prediction of Secondary Structures of Proteins Using a Two-Stage Method Metin Turkay and Ozlem Yilmaz and Fadime Uney Yuksektepe College of Engineering, Kog University, Rumelifeneri Yolu, Sariyer, 34450 Istanbul, TURKEY Abstract Protein structure determination and prediction has been a focal research subject in life sciences due to the importance of protein structure in understanding the biological and chemical activities in any organism. The experimental methods used to determine the structures of proteins demand sophisticated equipment and time. In order to overcome the shortcomings of the experimental methods, a host of algorithms aimed at predicting the location of secondary structure elements using statistical and computational methods are developed. However, prediction accuracies of these methods rarely exceeded 70%. In this paper a novel two-stage method to predict the location of secondary structure elements in a protein using the primary structure data only is presented. In the first stage of the proposed method, folding type of a protein is determined using a novel classification model for multi-class problems. The second stage of the method utilizes data available in the Protein Data Bank and determines the possible location of secondary structure elements in a probabilistic search algorithm. It is shown that the average accuracy of the predictions increased to 74.1%. Keywords: Protein Structure, Data Classification, Mixed-Integer Linear Programming 1. Introduction Proteins are large molecules indispensable for existence and proper functioning of biological organisms. Proteins are used in structure of cells, which are main constituents of larger formations like tissues and organs. Bones, muscles, skin and hair of organisms are made basically up of proteins. Besides their necessity for structure, they are also required for proper functioning and regulation of organisms such as enzymes, hormones, antibodies. Understanding functions of proteins is crucial for discovery of drugs to treat various diseases and disorders. A protein molecule is the chain(s) of amino acids also called residues. A typical protein contains 200 - 300 amino acids but this may increase up to approximately 30,000 in a single chain. There are 4 basic structural phases in proteins: primary structure, secondary structure, tertiary structure and quaternary structure. The primary structure is the sequence of amino acids that make up the protein. The secondary structure of a segment of polypeptide chain is the local spatial arrangement of its main-chain atoms without regard to the conformation of its side chains or to its relationship with other segments. This is the shape formed by amino acid sequences due to interactions between different parts of molecules. There are mainly three types of secondary structural shapes: a-helices, P-sheets and other structures connecting these such as loops, turns or coils. Alpha-helices are spiral strings formed by hydrogen bonds between CO and NH groups in residues Beta-sheets are plain strands formed by stretched polypeptide backbone. Connecting structures do not have regular shapes; they connect a-helices and P-sheets to each other. Turns enable parts of polypeptide chain to

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fold onto itself reversing the direction of the polypeptide chain to form its threedimensional shape. Proteins are classified according to their secondary structure content, considering ahelices and P-sheets. Levitt and Chothia ^^^ were the first to propose such a classification with four basic types. "All-a" proteins consist almost entirely (at least 90%) of ahelices. "All-p" proteins are composed mostly of P-sheets (at least 90%) in their secondary structures. There are two intermediate classes which have mixed a-helices and P-sheets. "a/P" proteins have approximately alternating, mainly parallel segments of a-helices and P-sheets. The last class, "a+P" has mixture of all-a and all-P regions, mostly in an antiparallel fashion.^^^ Due to bottlenecks in experimental methods to determine protein structures, computational approaches to predict protein structures are developed. All structure prediction methods basically rely on the idea that there is a correlation between residue sequence and structure. Most methods to predict protein structure from residue sequence utilize information on known protein structures. Databases are formed and examined for relationships between amino acid sequence and protein structure. First predictions were made in 1970s with a few dozen structures available. Currently structures of about 33,500 (as of November 2005) proteins are identified that means vast amount of data supporting more reliable predictions with better accuracy is available. Protein structures are stored in and accessible from Protein Data Bank.^^^ Among different computational methods developed to predict protein structures, the most successfiil ones include neural network models, database search tools, multiple sequence alignment, local sequence alignment, threading, hidden Markov model-based prediction, nearest neighbor methods, molecular dynamic simulation, and approaches combining different prediction methods. Neural networks are parallel, distributed information processing structures and the method tries to solve the problem by training the network.^"^"^^ The most successfiil ones are Copenhagen^^^ PSI-BLAST^^^, PHD^^'^^ and SSpro^^^l The multiple sequence alignment method aligns each sequence such that one base in a sequence corresponds to bases in the other sequences to reveal the similarity of genetic code, evolutionary history, and common biological fiinctions of the strings. Consensus is one of the latest approaches utilizing this method with significant performance.^^ ^^ The local sequence alignment approach utilizes local pair-wise alignment of the sequence to be predicted and the most significant method developed with this approach is named PREDATOR.^^^^ Threading maps the unknown structure to the most similar known sequence.^^^^ Hidden Markov Model-Based Prediction of Secondary Structure (HMMSTR) considers similarity of unknown protein to segments of known structures.^^"*^ The nearest neighbor methods operate by matching segments of the sequence with segments within a database of known structures, and making a prediction based on the observed secondary structures of the best matches.^^^"^^^ There are two significant approaches: combination of GOR Algorithm and Multiple Sequence Alignment Method and combination of Nearest-Neighbor Algorithms and Multiple Sequence Alignment Method. The combination of GOR algorithm and multiple sequence alignment method^^^^ starts with selection of a set of proteins (12 proteins) with well-determined structures, none of which belonging to or having identity to any of proteins in databank of GOR program. Next, multiple sequence alignment of these proteins is carried out and the results are the inputs for GOR algorithm. A scoring system considering sequence-similarity matrix, local structural environment scoring scheme and N and C-terminal positions of secondary structure types is utilized with a restricted database of a small subset of

Prediction of Secondary Structures of Proteins Using a Two-Stage Method

1681

proteins that are similar in the combination of nearest-neighbor algorithms and multiple sequence alignment method. Nearest-neighbor algorithmic part is followed by multiple sequence alignments J^^^ The comparison of various methods for predicting secondary structure of globular proteins in general are tested on 195 proteins and three state per residue performances (Q3: helix, sheet or other), and Qs are measuredJ^^^ Prediction accuracies of these methods are given in Table 1. Table 1: Average three-state accuracy indices calculated for six prediction algorithms based on 396proteins^^^l IVfpflinH

n.(o/^\

PHn[6'9]

71 Qsn

NNSSpt^^l

71.400

DsrJ^^^

68.4n

PRKDATORt^^l

68.602

ZPRFD^^^^

59.637

Consensus^^^^

72.707

In this paper, a two-stage algorithm for the secondary structure prediction of proteins is presented. The algorithm has a probabilistic approach utilizing data on all structurally identified proteins having the same folding type with the new unknown protein. The first stage in the method is determination of class of unknown protein. This is accomplished by solving a mixed integer linear program (MILP) problem with 100% accuracy. The objective of the first stage is to reveal some of the uncertainties in the protein structure by determining the folding type accurately. The second stage involves decomposition of the amino acid sequence to overlapping sequential groups of 3 to 7 residues. A local database is formed for each folding type by extracting structural data from PDB files. After matrix of frequency of occurrences of each sequential group of new residue chain is generated, probabilities of being in an ahelix, a P-sheet or a connecting structure are calculated for each residue. The structure with maximum probability is accepted as the structure. 2. The Two-Stage Method The two-stage algorithm decomposes the secondary structure prediction problem into two steps: first the overall folding type of the protein is predicted, and then the secondary structure is predicted using the refined statistical data from the first step. 2.1. Prediction of Folding Type The overall folding type of a protein depends on amino acid composition.^^"^^ Several methods are developed to exploit this theory in the prediction of folding type of proteins.^^^"^^^ These methods use statistical analysis and separate multi-dimensional amino acid composition data into several folding types. The prediction of protein folding type is a typical multi-class data classification problem. Classification of multidimensional data plays an important role in the decision determining main characteristics of a set. Support vector machines is a data mining method to classify data into different groups.^^^^ Although this method can be efficient in classifying data into two groups, it is inaccurate and inefficient when the data needs to be classified into more than two sets. Mixed-integer programming allows the use of hyper boxes for

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M Turkay et al

defining boundaries of the sets that include all or some of the points in that set. Therefore, the efficiency and accuracy of multi-class data classification can be improved significantly compared to traditional methodsJ^^'^^^ Another approach is to define piecewise linear functions to separate the data that belongs to different classes from each otherP^^ The main differences between these three approaches are illustrated in Figure 1.

;

•

i

•

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.-

• • •• 3_ "^*

• . \ * \

*

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••

i

AN

• ^

:

m

A

. A

• • ••

(a) Hyperlanes

(b) Hyper-boxes

(c) Piecewtse-Linear Functions

Figure 1. Three approaches to multi-class data classification problems: (a) support vector machines, (b) MILP using hyper-boxes, (c) piecewise-linear functions. The protein folding type prediction problem is considered in two parts: training and testing. The objective of the training part is to determine the characteristics of the proteins that belong to a certain class and differentiate them from the data points that belong to other classes. After the distinguishing characteristics of the classes are determined, then the effectiveness of the prediction must be tested. The prediction accuracies with different methods for the data set given in are summarized in Table 2. Table 2: Average prediction accuracies with different methods for the folding type problem. Method SVD^^*J NN'^'l SVMP'J

ccf^î Hyper-boxes'^^'^ Piecewise-Linear Functions''"^

al-a 66.7% 68.6% 74.3% 84.3% 87.5% 100%

al-3 90.1% 85.2% 82% 82% 85.7% 100%

a+3 81% 86.4% 87.7% 81.5% 91.3% 100%

a/B 66.7% 56.9% 72.3% 67.7% 50% 100%

Overall 81% 74.7% 79.4% 79.1% 83.3% 100%

2.2. Prediction of the Secondary Structure The basis for the algorithm is searching segments of its residue sequence in pool of known protein structures and predicting structure for each residue on the basis of frequency of occurrence. To determine the number of residues in each segment to be searched, two facts are considered: the segment should be long enough to have a legitimate reason to search considering interactions and bonds formed between amino acids to shape their structures. Every structure is searched in the relevant database whose dimensions were stated in previous section. The algorithm considers 3 to 7residues-long segments of this chain in the database as illustrated on a sample primary sequence in Figure 2.


a

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V

1

a v|

akv

II o y ci 1

raq

hsy aqhs

akvr

akvraq akvraqh

l\

1 III

qkl

aft

O

1

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1

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II aty

fhn

qkim

yaft qhsya

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L \f

1683

naty

srfh

naty.

msrfh fhnaty

qklmsr syaftqk

aty...

Figure 2. Representation of overlapping 3,4,5,6, and 7 residue segments. Then, the probability for a particular residue to be in a secondary element for each residue segment is calculated according to its folding type. (1) where Pyk represents the probability of residue / being of structure type j in A:-residue segments, ryk is the total number of occurrences of residue i in structure type j in kresidue segments, and ttk is the total number of occurrences of residue / in A:-residue segments. Then, the ultimate probability, Qy, for residue / to be in structure type j is calculated as follows. (2)

Qv=l>kP,k

The weights, Wk, are determined for each folding type using the data available in SCOP^^^^ database with least squares regression.^^^^ The three ultimate probabilities calculated using Eq. (2) are compared and the secondary structure type that has the highest ultimate probability is selected as the secondary structure of the residue. The results of the algorithm are given in Table 3. Table 3: Results of secondary structure prediction. Training Set

#ofnroteins # of stnictiires

All-a All-P

o/p a+p Total

2000 2766 3375 2866 11007

41139 67373 100255 62935 271702

Test Set

# of nroteins

419 579 707 601 2306

Qs (%)

# of residues

203302 313844 505693 279355 1302194

80.5 72.4 71.9 75.5 74.1

3. Conclusions A novel two-stage method to predict the secondary structure of proteins is presented in this paper. The objective of the first stage is to reveal some of the uncertainties in the protein structure by determining the folding type accurately. The second stage involves decomposition of the amino acid sequence to overlapping sequential groups of 3 to 7 residues and calculation of probability for a particular residue to be in a secondary

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structure. It is shown that the novel two-stage method performs better compared to the state-of-the-art general methods for globular proteins. References [I] Levitt,M, and Chotia,C. (1976) Structural patterns in globular proteins. Nature, 261, 552-558. [2] Mount, D. W. (2001), Bioinformatics: Sequence & Genome Analysis, Cold Spring Harbor Laboratory Press, Woodbury, New York. [3] Protein Data Bank, htttp://www.pdb.org/. [4] Bohr, H., Bohr, J., Brunak, S., Cotterill, R. M J., Fredholm, H., Lautrup, B., and Petersen, S. B. (1990), A novel approach to prediction of the 3-dimensional structures of protein backbones by neural networks, FEBS Letters 261, 43-46. [5] Cai, Y. D., Liu, X. J., Xu, X. B., and Chou, K. C. (2002), Artificial neural network method for predicting protein secondary structure content. Computers and Chemistry 26, 347-350. [6]Rost, B. (2001) Review: Protein secondary structure prediction continues to rise. Journal of Structural Biology 134, 204-218. [7] Petersen, T. N., Lundegaard, C , Nielsen, M., Bohr, H., Bohr, J., Brunak, S., Gippert, G. P., and Lund, O. (2000), Prediction of protein secondary structure at 80% accuracy. Proteins 41, 17-20. [8] Altschul, S., Madden, T., Shaffer, A., Zhang, J., Zhang, Z., Miller, W., and Lipman, D. (1997) Gapped Blast and PSI-Blast: A new generation of protein database search programs. Nucleic Acids Res. 25, 3389-3402. [9] Rost, B., and Sander, C. (1993), Prediciton of protein secondary structure at better than 70% accuracy, J. Mol. Biol. 232, 584-599. [10] Baldi, P., Brunak, S., Frasconi, P., Soda, G., and PoUastrt, G. (1999) Exploiting the past and the future in protein secondary structure prediction, Bioinformatics 15, 937-946. [II] Cuff, J. A., Barton, G. (1999), Evaluation and improvement of multiple sequence methods for protein secondary structure prediction. Proteins 34, 508-519. [12] Frishman, D., and Argos, P. (1997), Seventy-five percent accuracy in secondary structure prediction, PROTEINS: Structure, Function and Genetics 27, 329-335. [13] Thiele, R., Zimmer, R., and Lengauer, T. (1999), Protein threading by recursive dynamic programming. Journal of Molecular Biology 290, 757-779. [14] Bystroff, C , Thorsson, V., and Baker, D. (2000), HMMSTR: A hidden Markov model for local sequence - structure correlations in proteins, J. Mol. Biol. 301, 173-190. [15] Sen, S. (2003), Statistical analysis of pair-wise compatibility of spatially nearest neighbor and adjacent residues in a-helix and p-strands: Application to a minimal model for secondary structure prediction. Biophysical Chemistry 103, 35-49. [16] Westhead, D. R., and Thornton, J. M. (1998), Protein structure prediction, Current Opinion in Biotechnology 9, 383-389. [17] Yi, T. M., and Lander, E. S. (1993), Protein Secondary Structure Prediction Using Nearestneighbor Methods, Journal of Molecular Biology 232, 1117-1129. [18] Salamov, A. A., and Soloveyev, V. V. (1997) Protein secondary structure prediction using local alignments, J. Mol. Biol. 268, 31-36. [19] Kloczkovski, A., Ting, K.-L., Jemigan, R.L., and Gamier, J. (2002), Protein secondary structure prediction based on the GOR algorithm incorporating multiple sequence alignment information. Polymer 43, 441-449. [20] Salamov, A. A., and Soloveyev, V. V. (1995), Prediction of Protein Secondary Structure by Combining Nearest-neighbor Algorithms and Multiple Sequence Alignments, J. Mol. Biol. 247,11-15. [21] SCOP, http://scop.mrc-lmb.cam.ac.uk/ [22] King, R. D., and Stemberg, M. J. E. (1996), Identification and application of the concepts important for accurate and reliable protein secondary structure prediction. Protein Science 5, 2298-2310. [23] Zvelebil, M. J. J. M., Barton, G. J., Taylor, W. R., and Stemberg, M. J. E. (1987), Prediction of protein secondary stmcture and active sites using the alignment of homologous sequences. Journal of Molecular Biology 195, 957-961. [24] Nakashima, H., Nishikawa, K., and Ooi, T. (1986), J. Biochem. 99, 152-162. [25] Chou, K.C. (1995), Does the folding type of a protein depend on its amino acid composition?, FEBS Letters 363, 127-131. [26] Bahar, L, Atilgan, A.R., Jemigan, R.L., and Erman, B. (1997), Understanding the Recognition of Protein Stmctural Classes by Amino Acid Composition, Proteins: Stmcture, Function, and Genetics 29,172-185.


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[27] Cai, Y.D., Liu, X.J., Xu, X.B., and Zhou, G.P. (2001), Support Vector Machines for predicting protein structural class, BMC Bioinformatics 2, 3. [28] Uney, F., and Turkay, M. (2005)"A Mixed-Integer Programming Approach to Multi-Class Data Classification Problem", European Journal of Operational Research, in print. [29] Turkay, M., Uney, F. and Yilmaz, O. (2005), Prediction of Folding Type of Proteins Using Mixed-Integer Linear Programming, Computer-Aided Chem. Eng., vol 20A: ESCAPE-15, L. Puigjaner and A. Espuna (Eds.), 523-528, Elsevier, Amsterdam. [30] Turkay, M., Bagirov, A. and Uney, F. (2005), Prediction of Folding Type of Proteins Using Piecewise-Linear Functions, manuscript under preparation. [31] Cai, Y.D., Zhou, G.P. (2000), Prediction of protein structural classes by neural network, Biochemie, 82, 783-785. [32] Chou, K.C., Liu, W.M., Maggiora, G.M., Zhang, C.T. (1998), Prediction and classification of domain structural classes, ProteinsL Structure, Function, and Genetics, 31, 97-103. [33] Yilmaz, O, (2003), A Twô-stage mathematical programming algorithm for predicting secondary structures of proteins, MS Thesis, Koc University, Istanbul, Turkey.

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Reconstruction of Transcriptional Regulatory Networks via Integer Linear Programming Joao M. S. Natali, Jose M. Pinto Polytechnic University, 6 Metrotech Center, Brooklyn, NY 11201, USA

Abstract Much effort has been recently dedicated to the identification of genome-wide transcription regulatory networks by means of comprehensive high-throughput experiments that are capable of capturing the systemic behavior of the transcription coordination phenomenon. The present work comprises the development of Linear Programming (LP) and Integer Linear Programming (ILP) approaches to model and analyze the gene regulatory network of Saccharomyces cerevisiae, centered on a logic inference based representation of regulatory events and on a direct evaluation of experimental quantitative results. Our models are based in a simple representation of regulatory logic. Initial results show coherence to published data and improvements on the logical representation of regulatory events are currently under development. Keywords: Integer Linear Programming, Transcriptional Regulatory Networks, Optimization, Logical Modeling. 1. Introduction The concept that proteic molecules are produced via the initial transcription of the genetic code into mRNA strands and the following translation of this mRNA into a sequence of amino acids - which is often referred to as the Central Dogma of Biology has been widely known and accepted by the scientific community for decades. However, the mechanisms underlying the regulation of these processes, which would ultimately explain why proteins are produced in such differing quantities under diverse metabolic conditions, are far from being fully understood. The transcription of a gene relies, among many factors, on the activities of a class of enzymes called RNA-polymerases. The binding of the RNA polymerase to the genetic code may depend on the existence of other chromatin binding proteins and complexes, known as transcription factors (or simply TF's), which can aid or obstruct the enzyme's binding and fiirther genetic transcription. Therefore, the affinities and activities of transcription factors are key elements of the cell's transcription regulation. In recent years significant effort has been put into deciphering transcription regulatory elements and regulatory networks on a genomic scale. This attempt has been founded on the insight that the information that can be extracted from the establishment of a coordinated network of regulatory interactions may reach far broader scopes of understanding than the usual recognition of individual regulatory elements alone. One of the most successful experiments aiming at the identification of a genome-wide regulatory network was carried out by Richard Young's group (Lee et al., 2002) based upon the ideal eukaryotic microorganism Saccharomyces cerevisiae. These authors relied on Chromatin-Immunoprecipitation (ChIP) and Microarrays techniques to identify all gene promoter regions that were physically bounded by a comprehensive set of transcription factors. The outcome from this approach was an array of p-values that provided information on the likelihood of each of the promoter regions from the whole

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J.M.S. Natali andJ.M. Pinto

studied genome to be bound (and, thus, potentially regulated) by each of the transcription factors considered. Additionally, a number of authors undertook the effort of defining mathematical methods and computational procedures that would be capable of providing information on gene regulatory networks in a in silico manner. Gupta et al. (2005) used linear and non-linear dynamical models of mRNA production and gene expression to obtain regulatory patterns from a set of expression profiles. Hasty et al. (2001) provided a comprehensive review on computational studies of gene regulatory networks. Using a procedure based on statistical analysis and relying on the results from Lee et al. (2002) and on the vast amount of expression data currently available, Gao et al. (2003) proposed the confrontation of the information extracted fi-om transcription factor occupancy data and gene expression data to obtain a compendious set of TF-Gene interactions that represent a concise and coherent regulatory network. For that purpose, however, a number of simplifications based on statistical calculations were performed. These simplifications are justifiably expected not to exert great influence on the general outcome of the method; however, they significantly limit the reproducibility of the results while inserting information that is not exclusively provenient of the biological phenomenon studied.

2. Proposed Approaches We propose two approaches for the Regulatory Network Reconstruction problem, one based on Linear Programming (LP) and another on Integer Linear Programming (ILP). The formulated problems involve the modeling of regulatory events and the automated decision making regarding which of the interactions between transcription factors and intergenic regions, previously pointed by genomic location analysis (Lee et al., 2002), are indeed relevant to the global regulation of transcription in yeast and, therefore, belong to its regulatory network. 2.1. Linear Programming/Minimum Cost Network Flow Model The proposed LP formulation is a Minimum Cost Network Flow (MCNF) model in which supply nodes are set to represent transcription factors, demand nodes are regarded as genes, and arcs represent the paths through which regulatory signals flow. The model is based on the representation of regulatory elements and signalling pathways as a network comprised of a bipartite graph and input/output flows. A graphical representation of such network is presented in figure 1. F(TFxRG)

I flows: M flows:

signaling strength required for each transcription factor in an experiment, log-ratios from microarray experiments - measure of the signaling flow required by each regulated gene

Figure 1: Regulatory interaction network in LP model.. The model is defined in the following manner: let

TF = {l,2,...,nj.p] be the set of

transcription factors, and i?G = {7,2,...,«^^} the assumed set of regulated genes. The bipartite graph that represents the interconnections between transcription factors and

Reconstruction of Transcriptional Regulatory Networks

1689

genes is given byF = (F^,£'^), where V^ =TFuRG and Ep=TFxRG. The network model is based on the straightforward concept of flow balance around each defined node, in which microarray experiment results are used as a measure of the signal intensity of positive regulation required by each gene and, thus, determine the overall intensity and units of the global flow through the network. This signal flow is distributed to every gene from each transcription factor through the network F. A cost parameter C. \/(i,j)s [TF,RG) is further associated with each unitary flow through arcs in F, and is defined as the p-value for the existence of an interaction found by Lee at al. (2002). Finally, a demand for flux Mj V/'E RG is assigned to each gene in the network, given by the base-2 logarithm of the ratio of scanned luminescence intensity between the test and control media in each run of the microarrays experiments. The problem is, then, formulated as a MCNF problem (Ahuja et al., 1993). Only the datafi-oma single microarray experiment is considered in the definition of the problem. Therefore, the comparison between different solutions using dissimilar expression data is important for the interpretation of the results. The resulting optimization problem is as follows:

min Z=XZQy-^.y ieTF jeRG

s.t. j F,_j Vi e TF, Vy s RG \ Y, F,j >Mr,Y,F,^>0;0< [

ieTF

f;., < f"

(1)

jeRG

2.2. Integer Linear Programming/Logic Inference Based Representation Let EX = {l,2,...,n^] be the set of experiments used as input data, and TF and RG be defined as previously. Furthermore, we define Xj,^ e [True,False] V/'G RG,\/ke EX as a Boolean variable which is true if and only if gene J is expressed in experiment k, and Y.j^ V/e TF,\/ke EX as a Boolean variable true if and only if a transcription factor / is produced in an experiment k. Moreover, we define two sets of binary variables representing the topological characteristics of the reconstructed transcriptional regulatory network. Let Sp. j \/[i,j)e [TF,RG) be a set of Boolean variables which are true if and only if the transcription factor i activates the transcription of gene y, and, concordantly, Sn.j \/[i,j)e [TF.RG) which are true if and only if the transcription factor /• represses the transcription of geney. Using this formalism, the logical relationship between the variables that represent the regulatory network topology and the inferences that connect TF's and genes can be posed as follows: SP,J

—iSp. J A —tSn. J

No Regulation

y{i,j)e{TF,RG)

(2)

In disjunction (2), L^(Xjj^,Y.A represents a set of logical propositions that describe relationships between the expression of a gene and the binary activity of a transcription factor under activation interactions, and I^ (z^^, 1^ ^) , similarly, for repression.

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J.MX Natali andJ.M. Pinto

The present model is based on the simple logical relationships between the activities of transcription factors and the genes that are regulated by them as below: Sp,j

\/{i,j,k)s{TF,RG,EX)

and

(3)

which can be converted to integer constraints (Raman and Grossmann, 1991). The OF is the maximization of the existence of activation interactions, weighted by the log ratio values of each interactions shown below, where R is the set of log ratio values for the interactions between each pair of transcription factors and genes found by Lee et al. (2002). The optimization model can be defined by: max

Z=i;Z^..y(%v+'5"M)

(4)

ieTF jeRG

sJ. Y.,-X^,>{l-Sp,j)

yieTFyjeRG,keEX

Y,,+Xjj^-l(x,t,P)-e(x,t))

(1)

s.t. Convection-Diffusion equation (2) dt Du _ dp + pVû + pg + S Transport of the bulk fluid (3) Dt ~ dx In problem (l)-(3), 0(x,/) represents the imaging data in space (x^) and as a fiinction of time. (/>(xJ,P) represents the predicted concentration fields of the drug as a function of the unknown transport and reaction properties, P = {r(x),A:(x)} as given in Eq (2). The term R in Eq (2) represents the metabolic uptake of the drug. The diffusivity, r(x) as well as the kinetic rates may be a function of the spatial region of the brain. For example, the diffusion coefficient is known to vary along the orientation of the axons in the white matter, thus qualifying the porous white matter as anisotropic medium. Eq. (3) represents the momentum equations for flow through the porous brain parenchyma and S is the additional pressure drop incurred (Dullien, 1979). The TKIP approach is composed of three steps: First, discretization of the distributed system obtained from the reconstructed computational grid using discretization schemes such as the finite volume approach (Patankar, 1980). For accommodating complex geometry like the brain, the generalized curvilinear coordinate transformation and unstructured computational meshes are used (Fletcher, 1991; Date, 2005). Second, response surfaces are needed in the optimization algorithm for an efficient and robust evaluation of the first derivative as given in Eq. (4). Finally, we

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have developed inexact trust regions methods to solve large scale inversion problem (Zhang and Linninger, 2005; Conn et. al., 2000). With the use of response surface approximation and the trust region radius y, problem (l)-(3) is converted into subproblem (5)-(6). ^(x,^P + AP) = ^(x,/,P) + - ^ A P + 0(AP^) dP

(4)

min ij/{P^ + AP) = (^(x,t,P^)-\-^'AP-0(x,0)^

(5)

Ap

dP

s.t. \\AP\\

0.08 -

1

0.06 -

» 0.04 J

^LJI^ Caudate

Striatum

Gray

Injectkxi Site

1

White

Total

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M.Xenosetal

drop causing tissue stresses and strains. This may cause local injury of the brain tissue. Use of higher flow-rates of the drug may also produce backflow along the catheter shaft, thus distributing the drug uselessly into the peripheral regions as opposed to the areas of interest close to the catheter tip (Morrison et al, 1999). 5.

Conclusions A systematic approach for the design of drug delivery policies has been introduced. It will provide physicians with valuable insight in the selection of proper invasive drug therapies. The novel method consisted of three steps (i) accurate reconstruction of the brain geometry, (ii) extraction of unknown transport and kinetic properties from experimental data and (iii) prediction of treatment volumes based on site specific drug delivery. Injection near the thalamic nucleus was found to the best inftision site for administering the drug effectively into the caudate nucleus. Treatment volume was found to be highly sensitive to regional and structural heterogeneity and anisotropy. Accurate catheter placement based on systematic drug delivery design could provide decision support to neurosurgeons for performing clinical trails and to provide innovative therapies to the patient. Better design of therapies through computational methods can minimize the cost of experimentation and can provide better alternatives to the current approaches. References Bohm G., Galuppo P., Vesnaver A.A., 2000, 3D adaptive tomography using Delaunay triangles and Voronoi polygons. Geophysical Prospecting, 48, 723-744. Conn A.R., Gould N.I.M. and Toint P.L., 2000, Trust Region Methods, SIAM, Philadelphia, PA. Date A.W., 2005, Introduction to Computational Fluid Dyanamics, Cambride University Press: New York. DuUien F.A.L., 1979, Porous Media Fluid Transport and Pore Structure, Academic Press Inc: New York. Fletcher C.A.J., 1988, Computational Fluid Dynamics, vol 1, Springer-Verlag: New York. Hamilton J.F., Morrison P.F., Chen M.Y., Harvey-White J., Pemaute R.S., Phillips H., Oldfield E., Bankiewicz K.S., 2001, Heparin Coinflision during Convection-Enchanced Delivery (CED) Increases the Distribution of the Glial-Derived Neurotrophic Factor (GDNF) Ligand Family in Rat Striatum and Enhances the Pharmocological Activity of Neurturin, Experimental Neurology, 168, 155-161. Linninger A.A., Somayaji R.M.B., Xenos M., Kondapalli S., 2005, Drug delivery into the human brain. Foundations of Systems Biology and Engineering, Corwin Pavillion, Univ. of California Santa Barbara Campus, August 7-10. Materialise, Inc., (Mimics), 2005, http://www.materialise.be/mimics/main ENG.html. Morrison, P.F., et al., 1994, High-flow microinfusion: tissue penetration and pharmacodynamics, American Journal of Physiology, 266, R292-R305. Morrison, P.F., et al, 1999, Focal delivery during direct infusion into the brain: role of flow rate, catheter diameter and tissue mechanics. Am. J. Physiol, 277, R1218- R1229. NIH, NINDS, 2005, www.nih.gov. Patankar S.V., 1980, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corporation: Washington. Reisfeld B., Kalyanasundaram S., Leong K., 1995, A mathematical model of polymeric controlled drug release and transport in the brain. Journal of Controlled Release, 36, 199-207. Warner J.J., 2001, Atlas of Neuroanatomy, Butterworth-Heinmann, Boston. Zhang L. and Linninger A., 2005, Newton, Steepest Descent and Trust Region Methods: Application in Unconstrained optimization and Solution of Nonlinear Equations, UIC-LPPD021805.


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Mathematical modelling of three-dimensional cell cultures in perfusion bioreactors. Part II Francesco Coletti ^, Sandro Macchietto ^, Nicola Elvassore ^ ^ Dept. of Chemical Engineering, Imperial College London, London SW7 2BY, UK ^ DIPIC, Universita diPadova, via Marzolo 9, Padova 1-35131, Italy Abstract Some applications of tissue engineering require growing cells within supporting scaffolds to obtain structures with adequate functionality for in vivo implantation. A novel general dynamic mathematical model of cells growth and oxygen consumption in a three-dimensional perfusion bioreactor was recently proposed, which includes convection and diffusion, the two main transport phenomena affecting cells growth, combined with cells growth kinetics and oxygen supply. That model is used and modified here to analyze various geometric and operational configurations, including medium flow inversion, initial cell seeding procedures, and a simple vascularization model, represented by micro channels in the scaffold matrix. The significant impact on cell growth, density and density distribution within the threedimensional scaffold is evidenced by the simulation results presented. Keywords: Tissue engineering, Perftision bioreactor. Mathematical modelling. Cells culture. Bio-process simulation. 1. Introduction Tissue engineering, the production of engineered grafts for tissue replacement, presents several challenges, of which achieving significant cells growth in supporting scaffolds is one. Problems include substrate delivery to inner cell layers and non homogeneous cell distribution within a 3D scaffold. In order to help the understanding of the complex interacting phenomena in a scaffold-bioreactor system, several mathematical models have been proposed (Galban and Locke 1999; Pisu et al. 2004; Radisic et al. 2005; Radisic et al. 2006). Most published studies take into account only steady-state behaviour or just a small subset of the relevant transport and cellular processes. A comprehensive mathematical model, presented in a recent paper (Coletti et al, 2006) describes the dynamical behavior of a three dimensional cells culture in a perfusion bioreactor. Figure 1 shows the reference geometry considered. The model (Table 1) includes the fluid dynamics of the nutrient medium flowing by both convection and diffusion through a scaffold, modelled as a porous medium, with cells growth models which account for cell proliferation and contact inhibition, both of which in turn affect scaffold permeability. Realistic parameters for a culture of immortalized rat cells C2C12 on a collagen scaffold are reported in Coletti et al. (2006), together with assumptions, boundary conditions and initial conditions for two typical operations. In this work the above model is used to test some new scaffold configurations and operating conditions aimed at achieving faster growth, higher density and more uniform cells density distribution within the scaffold. These include inversion of the nutrients medium flow direction (Section 2), non uniform initial cells seeding (Section 3) and an initial study of the effects of vascularization, represented as micro channels inside the scaffold (Section 4).

1700

F. Colettietal

Figure 1 Reference geometry and system co-ordinates for a total perfusion bioreactor. In the base case the medium flows upwards from entry section ^ i through scaffold Q.^ ^^^ ^ît section Q.^.

Table 1. Model equations and parameters for the three domains Q.^, Q2 ^^^ ^3 • Domain Q.^ and Q3 bioreactor

W

ot

A

c,.,v-

^ = -(V-c,v°) + V-(Z).Vc,) dt

po

=0

D^ constant

p.p

No body force

p

p.j^

No body force

Pcell

Qm^^m

Domain Q.^ • scaffold ^

= _(V.c,v) + V.(A^Vc,) + /?,

Ti ^0,

M-cell ~

— n -Hcell'

^m ^

-v,

r^cell

1^ ^

rr

'^'A^, ^Ri

O2

r'cell

^

C^

,^

P-cell

^d

D^^=eD.lr

€(z, r, 0 = €(z, r, 0) - F,,„p,,, (z, r, 0

max f^cell » ' ^ c ' ^cell ->

e ,T

No cells death: k•^ =0

Pc^K

A ^(2,r,0)

Performance is assessed in terms of the following metrics: 1) Percentage of the scaffold volume with cells density higher than the reference value for physiological human tissues, 2) Percentage of the scaffold volume with oxygen concentration higher than the reference value for cells viability, 3) Maximum cells density, 4) Minimum cells density and 5) Average cells density within the scaffold. Unless stated otherwise, all conditions are the same as in the total perfusion base case of Coletti et al (2006). All calculations were done in Femlab as detailed by those authors.

Mathematical Modelling of Three-Dimensional Cell Cultures

1701

2. Medium flow inversion In perfusion reactors, as evidenced by simulations in the cited work, often cells growth well in the first few millimeters of the scaffold, but oxygen concentration rapidly decreases after a very short distance fi-om the entry face of the scaffold, compromising cells growth and viability. A possible alternative is to periodically invert the direction of the medium flow. The medium, with oxygen concentration c = c^", first enters the bioreactor at z =0 and perfuses through the scaffold (direct flow) until inversion time t.. At this time the flux is inverted by feeding the medium at z= L^ + / / + L^ (inverse flow). At time 2 ^. the direction switches back to direct flow, and so on. Numerically, inversion of the flow is simulated by using the solution at time t^ as initial point for a simulation between t. and 21., while also switching the inlet/outlet sections boundary conditions for the material and momentum balances, given in Table 2. Table 2. Boundary conditions at inlet/outlet sections Coordinate z=0

z = Ij + // + Lj

Balance

Inverse flow

Direct flow

material

^=
'4 + ^'^Z,,£,,y,

+ W,q, " 0 . 6 H ; 2 ^ 2 " 0.41^3^3 = " 4 . 4 9 9 8

0.308Ji -0.482^2 +0.177^3 +0.4^4 -0.37z2iî -0.37z22'2 -^-'^^z^.s^^y^ -0.37Z24^24>'4 ~ 0.245^346*34^4 - 0.385^436*43^3 + 0.4^53^53^3 + 0.6^64^54^4 +

+0.245^3^3+0.3851^4^4-0.4^5^5

Q31w^q^

-0.6w^q^=\Ji6?>

-0.14ji + 0.409^2 - 0.817373 -0.014^4 -0.455^53^53^3 +0.287z34^34j;4 -0.28x^3^3 +0.455H^5^5 =-4.3212

0.041^2 +0.026^3 -0.799^4 -0.405^^4^64^4 +^^26z,,£,,y, -().26ws,+Â{)5w,q, Bounds on Xj

y = 1,2,3

ln(4.9) > J, > ln(6.0) Bounds o n P ,

=-4.5122

/ = 1,....,6

ln(192) > y2> ln(234)

ln(2176) >y^> ln(2660)

ln(P,'^)>P/ >ln(P/'')

where î\l — U 56) and 1^13'^14'^21'^22'^23*^24'^34'^43'^53'^54 j are binary variables, Jy (/ = l v i 4 ) and ^/ (/ = 1,...,6) are continuous variables; L represents lower bound, L'^ represents upper bound.

Y. Zheng et al

1708

original GA .

+ IF mutation

n Fig. 1. XMP and GMP synthesis pathway

Fig.2. Comparisons of the performance four different GA approaches

Fig.2 compares the performance of four different types GA (the original GA, GA+local search, GA+IF mutation, and GA+local search+ IF mutation) for this particular problem. In these four cases, maximum generation number is set to be 2600, local search is performed in every 400 generations, and the population size is set to be 14. The premature detector parameter, D is set to 1. As shown in Fig.2, it is obvious that by including IF entropy and/or local search, the performance of GA can be drastically improved. After 2600 generations 463 feasible solutions are found. Among them we select 84 high score solutions, which satisfy the following conditions: 1) ^5 "^ ^ ; 2) i.e. x^ > 49021 According to the 16 binary ^ 6 > 0 ; 3) J 4 > 1 0 - 8 parameters >V/(' = U 96) and l^n'^14'^21'^22^^23^^24^^34^^43'^53'^64] . We compute Euclidean distance between these solutions and create hierarchical cluster tree. The dendrogram graph is shown in Fig.3. We can find five solutions of different type according to Fig.3. PR

•XI P2+

lâ

3 29 17 10 21 14 28 30 24 25 2 22 6 13 23 15 27 26 9 5 1 4 8 11 12 16 18 20 19 7

Fig.3. The dendrogram graph

A

/\

Fig.4. The solution when ^ =21

P4+

Metabolic Regulatory Network Optimization

1709

Let t represent the leaf node in Fig.3. Fig.4-Fig.8 represent five typical solutions based on different t's. Fig.4 shows the solution when /=21. The values of variables H'/, {^13 5 ^14 9 ^219 ^22 9 ^23 ^^24 9 ^34 9 ^43 9 ^53'^641

J y (/ = U«»»94) a u d ^ / ( / = 1 , . . . , 6 ) a r e 1 1

0 1 0 0, 0 1 0 1 1 1 0 0 0 1, 1.7411 5.4461' 7.6865 10.87, and 1.4722 1.2906 -1.3646 0.42012 0.17219 -0.29841. The maximum concentration of X4 is 52575. Fig.5 shows the solution when /=13. The values of variables are 1 1 1 0 1 0 , 0 1 0 1 1 1 1 0 0 1 , 1.7912 5.3068 7.6894 11.084,1.2309 1.6094-1.6094 0.60501 -0.23515 -0.17344. The maximum concentration of X4 is 65121. Fig.6 shows the solution when t=5. The values of variables are 1 1 0 0 0 1, 0 1 1 1 1 1 0 1 0 1, 1.783 5.4288 7.7101 10.944, and 1.6094 1.6094 1.0534 1.5246 1.6027 0.29228. The maximum concentration of X4 is 56613. Fig.7 shows the solution when /=12. The values of variables are 1 1 1 0 1 1, 0 10 1 1 1 0 10 1, 1.786 5.4192 7.7012 11.151, and 1.534 1.6072 -0.45455 0.77799 0.20221 0.65065. The maximum concentration of X4 is 69633. Fig.8 shows the solution when t=l. The values of variables are 1 1 0 1 0 0, 0 10 0 1 1 0 0 0 1, 1.7408 5.2757 7.688 10.816, and 1.126 1.4727 -0.1548 0.4643 0.036097 -1.2034. The maximum concentration of X4 is 49811. PI+

/\ Fig.5. The solution when ^ =13

7\ Fig.6. The solution when t =5

1710

Y. Zheng etal Fig.7. The solution when t =12

Fig.8. The solution when t =1

According to the above analysis, it is clear that the local optima at t=12 and 13 give the higher yields (x4= 69633 and X4=65121 respectively) compared with other cases. Notably, the original yield by Hatzimanikatis et al.,1996 was X4=55015.6. Further, the network structures suggested by these two solutions (see Fig.5 and Fig.7) are much simpler than the original work without many contradictory modifications of the networks. Note that five different structures of the above solutions based on our clustering analysis represent different modification of the original metabolic network (Figure 1). As indicated by the original work (Hatzimanikatis et al.,1996), it is undesirable to perform many modifications of the original pathway. In our solution set, some of our solutions require more modification with higher yield and the other solutions require less modification but result lower yield. This is very valuable to metabolic engineers. 4. Conclusion Solution of optimal metabolic regulatory network problems can be formulated into a mixed integer nonlinear programming problem (MINLP). In this work, genetic algorithm is implemented to solve this MINLP problem. The information entropy and local search method are implemented to improve these solution approaches for MINLP problem. Furthermore, clustering analysis is implemented to allocate physically meaningful local optima. Different types of solutions are given after cluster analysis. The results show that the approach is valid and efficient. References Bjork, K. M., Nordman, R. (2005). Solving large-scale retrofit heat exchanger network synthesis problems with mathematical optimization methods. Chemical Engineering and Processing, Vol. 44, No. 8, pp.869-876. Cardoso M.F., Salcedo R.L. and Barbosa D. (1997).A Simulated Annealing Approach to the Solution of MINLP Problems.Computers Chem. Eng. ,Vol. 21,No. 12,PP1349-1364. Deb, K. (2000). An efficient constraint handling method for genetic algorithms. Comput. Methods Appl. Mech. Engrg., Vol. 186, pp.311-338. Grossmann, I.E., Sargent, W.H. (1979) Optimum Design of Multipurpose Chemical Plants. Ind. Eng. Chem. Process Des. Dev.,Vol. 18,No.2, pp.343-348 Hatzimanikatis, V., Floudas, C.A. and Bailey, I.E. (1996). Optimization of Regulatory Architectures in Metabolic Reaction Networks. Biotechnol. Bioeng., Vol.52, pp.485-500. Ombuki, B.M. and Ventresca, M.(2004).Local Search Genetic Algorithms for Job Shop Scheduling Problem. Applied Intelligence, Vol. 21, pp. 99-109. Shannon, C. E. (1948). A mathematical theory of communication. Bell Syst. Tech. Journal, Vol 27, pp. 379-423. Voit, E.O.(1992). Optimization in Integrated Biochemical Systems. Biotechnol. Bioeng., Vol 40, pp.572-582. Yeh, C.W. and Jang, S.S. (2005). The Development of Information Guided Genetic Algorithm for Global Optimization", Journal of Global Optimization, In Press.


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Minimal Reaction Sets and Metabolic Pathways for Cultured Hepatocytes Hong Yang, Martin L. Yarmush, Charles M. Roth and Marianthi lerapetritou^ Rutgers, the State University of New Jersey, Piscataway, NJ, 08854, USA. Abstract Extracorporeal bioartificial liver (BAL) devices are the most promising technology for treatment of the liver failure. However, when exposed to plasma from the patient, hepatocytes are prone to accumulate intracellular lipids and exhibit poor liver-specific functions. Our work focuses on understanding the metabolism of cultured hepatocytes used for BAL. In this paper, a logic based programming is used to determine the important reactions in cultured hepatocytes by systemically analyzing the hepatic metabolic network, and investigating whether insulin, amino acid and hormone supplementations upregulate or downregulate certain pathways that control important liver specific functions, such as urea and albumin production. Using elementary mode analysis we were able to obtain 32 independent pathways, which are then used to analyze the results of the logic-based programming approach. Keywords: Hepatocyte metabolism, integer programming, elementary mode analysis. 1. Introduction Liver transplantation remains the best long-term option for the approximately 30,000 patients per year in the United States alone, while roughly 20% die of acute liver failure due to the shortage of organ donors. Extracorporeal bioartificial liver (BAL) devices employ primary hepatocytes or hepatoma cell lines to provide a whole complement of liver specific functions for the treatment of liver failure, but significant technical challenges remain to develop systems with sufficient function capacities. One key limitation is that, during BAL operation, hepatocytes are exposed to plasma from the patient and are prone to accumulate intracellular lipids and to depress liver-specific functions. It has been shown that pre-conditioning hepatocytes in low insulin plasma with supplemented amino acid dramatically improves the hepatic metabolism and increases liver-specific functions (Chan et al., 2003). Further manipulation of the culture environment is likely needed to allow hepatocytes to function at the levels necessary for productive BAL operation. Metabolic pathway analysis has become an important tool of bioinformatics and biotechnology. Without the knowledge of kinetic parameters, it is used to determine the maximal molar yield of biotransformation and as a guideline for reconstruction of metabolic networks based on special cell functions. One of the methods that have been proposed to analyze metabolic networks is elementary mode analysis (Schuster et al., 2000). The elementary modes correspond to the pathways connecting inputs to outputs and comprise a minimal set of enzymes allowing the mass balance for each intermediate metabolite at steady state. Any steady state flux pattern can then be expressed as a nonnegative linear combination of these modes to form a particular pathway. The first aim of this work is to identify the minimal reaction set by logic-based programming for six different cultured conditions after exposure to plasma and to investigate the effects of insulin, amino acid supplementation and hormone in metabolic

1712

H.Yangetal

network. We then use the elementary mode analysis framework to analyze the hepatic metabolic network and to investigate the relations between elementary modes and the results of logic-based programming. 2. Modeling and Computational Protocol 2.1. Logic-Based Programming For a metabolic network comprising M metabolites and N reactions, the material balances result in the following set of equations: dX ^ - Z ^ = ZSijVj,

i = lv..,M

(1)

where Xj is the concentration of metabolite i\ Sjj is the stoichiometric coefficient for metabolite i in reaction], and Vj is the flux of reaction j . The sum of the fluxes entering and exiting of the metabolic network can be assumed to be zero based on pseudo steady-state assumption (Schilling, 2000):

Xs,.v.=0,i = l,...,M

(2)

j=i

The metabolic network considered in this work is based on the model developed for cultured hepatocytes (Chan et al., 2003). It consists of 43 unknown fluxes and 34 measured fluxes (tryptophan uptake is also added), and 45 linearly independent mass balance equations. The unknown fluxes are calculated by using the least-square method of minimizing the square of errors of the measured fluxes. Based on this model we used mathematical programming ideas to develop the following optimization model in order to determine the minimal number of reactions required to maintain the mass balances of cultured hepatocytes. Expressing the presence/absence of reactions by logic 0-1 variables, problem (3) is obtained: min^ Subject to: ^SyV. = 0, i = 1,...,M, vpA^ < v^ < vÂ^, j = 1,...,N

(3)

j=i

where X.^ is the logic variable that correspond to the value of 1 if the reaction is active and 0 otherwise; Vj is considered as variable between a lower and an upper bound v""", Vj"**"" , respectively; which are determined based on the available experimental conditions (Chan et al., 2003). The problem is modeled in GAMS and solved using GAMS/CPLEX as the optimization solver since the problem corresponds to Mixed Integer Linear Program (MILP/ 2.2. Elementary Mode Analysis One approach for finding qualitatively distinct pathways is to calculate the elementary modes (Schuster et al., 1994). The determination of the elementary modes in a given network depends on the classification of metabolites as internal or external and the reactions as irreversible and reversible.

Minimal Reaction Sets and Metabolic Pathways for Cultured Hepatocytes

1713

In this paper, the internal and external metabolites are chosen in a similar fashion as for the logic-based programming approach in the previous section. We thus consider 45 independent internal metabolites, which have to be balanced in any pathway, and 29 external metabolites, which are assumed to be unaffected by the reactions in the network, such as energy cofactors ATP, ADP, AMP and CO2, lactate, glucose, urea etc. We treat all of the reactions reversible except reactions 15 producing urea and reaction 69 producing albumin are irreversible. The dimension of the null space depends on the number of free variables in the original stoichiometric matrix (Schilling et al., 2000): dim(Null(S)) = N - rank(S)

(4)

For a frill rank matrix, the dimension of the null space is equal to the difference between the number of reactions and internal metaboHtes. In the hepatocyte network we consider, the dimension is equal to 32 (d = N - M) . Through the application of convex analysis, the solution of the steady-state eqn. (2) must lie in the nonnegtive orthant of the space spanned by all the reactions. Due to the nonnegtive constraints on the fluxes, the unique set of elementary modes is found. If the number of elementary modes is equal to the dimension of the null space, those modes are systematically independent. Because the elementary mode vectors, e^^^, are uniquely determined, any real flux distribution can be expressed as a linear combination of these vectors with coefficients as follows: where \ = {y^Y^^...) is the vector of fluxes of reactions. If all of the reactions participating in one mode are reversible, this mode is reversible and the corresponding weight parameter X can be of any sign whereas if one or more irreversible reactions are included in one mode, this mode is irreversible and in this case the weight X should be positive. As will be shown in the next section, elementary modes can be used to interpret metabolic frmctioning and identify the importance of pathways and reactions in the production of specific metabolites. 3. Results and Discussion The logic-based programming approach described in the previous section is used to estimate the minimum reactions required for maintaining the hepatic frinctions in different cultured conditions. In particular we investigate the following conditions for which experimental data are available (Chan et al., 2003): (a) high/low insulin preconditioned unsupplemented plasma cultures (HIP/LIP), (b) high/low insulin preconditioned with amino acid supplemented plasma (HIPAA/LIPAA), and (c) high insulin preconditioned with hormone supplemented plasma / low insulin with amino acid and hormone supplemented plasma ( H I P H / L I P A A H ) . The results shown in Table 1 illustrate the dependence of the minimal reaction set on the different cultured conditions. Table 1: Number of Minimal Reaction Set (MRS) under Various Cultured Conditions Number of MRS

HIP 60

LIP 51

HIP_AA 64

LIP_AA 64

HIP_H 56

LIP_AA_H 63

Combining logic-based programming with stoichiometric balancing can clarify the effects of insulin on hepatic metabolism. Figure 1 (a) shows that insulin inhibits gluconeogenesis (Flux no.2 to 6) predominantly by suppressing the expression of

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H. Yangetal

PEPCK (reaction 6) and G-6-Pase (reaction 1) enzymes. Specifically in HIP and HIPAA, the flux of reaction 6 is decreased to 0.372 and 2.05 compared with 0.865 and 2.17 in the LIP and LIPAA, repectively, and also no glucose is released out of the system in high insulin preconditioned plasma. Figure 1(a) also shows that amino acid supplementation significantly increases gluconeogenesis in HIPAA and LIPAA. The difference is that almost all of G-6-P is consumed to produce glycogen and no glucose is released under high insulin preconditionsing, compared to about 28 percent used to produce glucose in low insulin plasma. We also found that hormone addition plays an important role in different culture conditions. In high insulin preconditioned plasma, hormones act similarly to insulin, inhibiting gluconeogenesis and changing the direction of reaction 1 to glycolysis. However, in the low insulin preconditioned plasma supplemented with amino acid, hormones increase the gluconeogenic pathway and almost all of G-6-P is used to produce Glycogen. In summary it is found that low insulin preconditioning with amino acid supplementation is the best culture conditon for glucose production. Furthermore we determined that insulin and hormones have no obvious effect on the TCA cycle (flux no. 9 to 14) of the hepatic network, whereas amino acid supplementation increases significantly the TCA cycle (Figure lb). These results from logic-based programming are in agreement with the experimental data (Chan et al., 2003) in different cultured conditions. As Table 2 shows, fluxes throughout the urea cycle were significantly upregulated by amino acid supplementation, but the albumin synthesis rate decreased to the minimum value. These conditions can be further improved using multiobjective optimization that results in Pareto optimal solutions for urea and albumin optimization (Sharma et al., 2005). (a) (b) TCA Cycle

Gluconeogenic or Glycolytic pathway

•g

""^"

o Q.

A--A

•--•---HIP

A"--^

A

11 E = 1-5 •

& s •il

HIP_M

--*--HIP_H

1

A

E .2 • ^ S 11 -

UP_AA_H

1I 2 « ^

--~*~-UP_M_H

1

/

2

3

4

1

Q.

9-

i

0 -0.5

HIP_H

- - - • - - - LIP

5

2 c

i

9

10

11

12

13

14

7

Reaction number

Reaction number

-1

Fig.l Effect of insulin, amino acid and hormone supplementation on (a) Gluconeogenic/glycolysis pathway, (b) TCA cycle. Table 2: Urea cycle and Albumin Synthesis in the Minimal Reaction Set 15 16 17 69

Arginase Ureal Urea2 Albumin Syn

LIP 0.007 0 0 MAX

LIP AA 1.95 1.822 1.824 MIN

LIP AA H 2.201 2.021 2.033 MIN

Minimal Reaction Sets and Metabolic Pathways for Cultured Hepatocytes

1715

To provide a comprehensive pathway-oriented view of hepatic metabolism, the elementary modes were obtained using FluxAnalyzer (Klamt et al., 2003), which implements the iterative algorithm described by Schuster et al. (2000). Analysis of the hepatic metabolism results in 32 elementary modes. The shortest mode is mode 1, triglyceride storage, the central reaction of which does not include any internal metabolites. Mode 2 consumes glucose to produce glycogen. Modes 4, 14, 15, 16, 17, 18, 24, 25, 26 and 29 include the TCA cycle, serine-pyruvate-cysteine cycle and valinepropionylCOA-methionine cycle. One difference is that these modes include ketone bodies except mode 4. Another difference is that modes 4, 14, 15, 16, 17, 24, 25, 26, 29 include isoleucine uptake, palmitate uptake, cholesterol esterase, glycerol update, lipid, threonine uptake, lysine uptake, tryptophan uptake, and leucine uptake, respectively. Modes 6, 7, 8, 11, 22 and 31 include parts of the TCA cycle and valine-propionylCoAmethionine cycle. Modes 19 and 30 involve the complete electron transport system and gluconeogenesis, respectively. By considering the reactions to produce urea and albumin as irreversible, modes 3, 27, 28 involving the urea production (Figure 2) and mode 32 involving albumin productions become irreversible modes. Since any self-consistent flux distribution can be expressed as a non-negative linear combination of elementary modes, the minimal reaction network obtained by logicbased programming can be reconstructed using the elementary modes. The coefficients of the linear combinations for the different conditions are calculated minimizing the least square error of Eqn. (5). The weights of elementary modes 7, 21 and 22 are negative in all different conditions, which mean that these three pathways will be reversed in these specific conditions investigated in cultured hepatocytes. From the values of coefficients in different culture conditions (data are not shown), we also determined the relative importance of each pathway across the various experimental conditions. For example, modes 3, 27, and 28 involve urea production, so we can determine the importance of urea production by calculating the corresponding coefficients. The sum of these coefficients under the LIP condition is 0.0097; compared to a sum of 0.1853 for L I P A A which means that cultured hepatocytes with amino acid supplementation exhibit significantly greater urea production compared to those cultured in unsupplemented plasma. By comparing the different weight values of modes involving urea production in L I P A A conditions, we found modes 3 and 27 are more important to produce urea than mode 28. In addition, the most important reactions in urea production can be determined to be the inhibition of enzymes carbamoyl-P synthetase I, ornithine transcarbamylase, (reaction 16) argininosuccinate synthetase, and argininosuccinase (reaction 17). Using the results of this analysis, we can investigate the effects of enzyme deletion. For example, blocking fumarase or malate dehydrogenase enzymes will lead to disruption of the TCA cycle and elimination of 19 elementary modes, including all of the modes of 9 amino acid exchange fluxes and albumin production. On the other hand, inhibiting lactate dehydrogenase is likely less important, since it results in deletion of a single elementary mode that may not be essential since its metabolites (e.g. pyruvate) can be produced from alternative pathways. 4. Summary and Future Work In this paper we present an analysis of hepatocyte metabolism based on logic programming and elementary mode analysis. The proposed approach determines the minimal set of reactions required in various conditions, and the importance of individual enzymes. Future work involves the experimental verification of the results obtained as well as the implementation of metabolic control analysis in the hepatocytes network to

1716

Jti.

YangetaL

determine the main branch points and compare with the results of the logic programming model. Arginlne

18 [

(

Ornithine

Argjnine

V

y ,

\

- t » /

n

Qucose-e-P

_54^''^

^^^

^

^

Ornithine

•

•

^

^

^

15

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(b) mode27 ACAC 49 Beta-OHixityrate

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1

Glucose

L 60 O D)

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40 20

50 100 sample point

0

50 sample point

100

5000

4000

3000

1 P' < /

A

f

^ 50 100 sample point

50 100 sample point

50 100 sample point

Figure 2. Results for the validation partition: (a) Gaussian gating network outputs: gi {'' ')^ 92 (~) versus concentrations of 5'o25(o)-(b)-(i) concentrations estimates with a ME with 2 experts (198 parameters): measured values (o), estimated values (-).

experts were MLP networks with 8 inputs (all state variables except XpAo), 5 hidden nodes and 9 outputs corresponding to the specific consumption rates in vector q. The ME equations are the following: qi(c) = W2,i tank (wi,i c + bi,i) + b2,i

(4)

q2(c) ^ W2,2 tank (wi,2 c + bi,2) + b2,2

(5)

q = 9i{c) qi(c) + ^2(c) q2(c)

(6)

with q i the expert 1 outputs, q2 the expert 2 outputs, Wij and hij are MLP parameters, gi and ^2 are the gating system outputs, which are scalar quantities defining the relative contribution of expert 1 and 2 for the evaluation of the specific reaction kinetics q. The training method employed was the EM algorithm and the parameters were identified by cross validation. The modelling results for the validation partition are presented in Figs. 2(b-i). Models and measurements show a good agreement except in what concerns the concentrations of SA and Spo4- An important and most interesting feature of the ME was observed: the ME network was able to detect the switch between the anaerobiose

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and aerobiose as illustrated in Fig. 2(a), and the experts developed expertise in modelling the kinetics of the one or the other metabolic state. The switch between the experts occurs precisely at 0.66 hours in the transition between the anaerobiose and aerobiose. It has succeeded in all batches, either in training or validation batches. This feature was repeatedly observed in several other tests with other processes (not treated in this work). 4.

CONCLUSIONS

The main objective of this work was to explore the possibility of using complex modular network architectures for modelling cells reaction kinetics in a wastewater phosphorus removal process. This idea was motivated by the fact that the metabolism of cells consists of a highly complex modular network of metabolic pathways. As main conclusions it can be stated that hybrid modular mechanistic/MEs trained with the Expectation Maximisation algorithm are able to detect metabolic shifts with the individual experts developing expertise in describing the individual pathways. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

13.

14. 15. 16. 17.

J. Schubert, R. Simutis, M. Dors, I. Havlik and A. Lubbert, Chem. Eng. Technol., 17(1) (1994) 10. R. Simutis, R. Oliveira, M. Manikowski, S. F. de Azevedo and A. Lubbert, J. Biotechnol., 59(1-2) (1997) 73. D. C. Psichogios and L. H. Ungar, AIChE J., 38(10) (1992) 1499. M. L. Thompson and M. A. Kramer, AIChE J., 40(8) (1994) 1328. G. Montague and J. Morris, Trends BiotechnoL, 12(8) (1994) 312. L. Chen, O. Bernard, G. Bastin and R Angelov, Control Eng. Pract., 8(7) (2000) 821. B. Sonnleitner and O. KappeU, BiotechnoL and Bioeng., 28(6) (1986) 927. M. Henze, W. Gujer, T. Mino, T. Matsuo, M. C. Wentzel, G. V. R. Marais, and M. C. M. Van Loosdrecht, Water Sci. Technol., 39(1) (1999) 165. B. Eikens and M. N. Karim, Int. J. Control, 72(7-8) (1999) 576. R. A. Jacobs, M. I. Jordan, S. J. Nowlan and G. E. Hinton., Neural Comput., 3 (1991) 79. S. Haykin., Neural Networks: A comprehensive foundation, Macmillan College Publishing Company, Inc., 1994. R. Oliveira, J. Peres and S. Feyo de Azevedo, M. Pons and J. F. M. van Impe (Eds), Computer Applications in Biotechnology 2004, Elsevier (ISBN: 0-08-044251X), (2005) 195. J. Peres,R. Oliveira and S. Feyo de Azevedo, M. Pons and J. F. M. van Impe (Eds), Computer Applications in Biotechnology 2004, Elsevier (ISBN: 0-08-044251X), (2005) 293. D. M. Titterington, A. F. M. Smith and U. E. Makov, Analysis of Finite Mixture Distributions, New York: Wiley, 1985. C. M. Bishop., Neural Networks for Pattern Recognition, Oxford University Press, 1995. V. Ramamurti and J. Ghosh, IEEE T. Neural Networ., 10(1) (1999) 152. M. I. Jordan and R. A. Jacobs, Neural Comput., 6(2) (1994) 181.


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Modelling Morphological Change in Endothelial Cells induced by Shear Stress R.J. Allen^ D. Bogle^ and A.J. Ridley" ^ C O M P L E X , University College London, Wolfson House, 4 Stephenson Way, London, N W l 2HE, UK ^Department of Chemical Engineering, University College London, Torrington Place, London, W C I E 7JE, UK "Ludwig Institute for Cancer Research, Royal Free and University College School of Medicine, London W I W TBS, UK It is well known that following the onset of fluid flow over endothelial cells (ECs) they polarize and elongate in the direction of the flow. Here the cell is described using a continuum approximation and viewed as a passive object. This is the first step to a broader model incorporating a mathematical description of the signal transduction network of interacting molecules which govern the morphological change involving internal cellular structures such as the cytoskeleton. 1. Introduction One of the motivations for this research is to contribute to our understanding of the development of atherosclerosis. This leads to cardiovascular disease which in turn is the leading cause of death in the western world, [1]. The development of atherosclerosis in an artery is characterised by the deposition of lipids and accumulation of cholesterol rich macrophages, which eventually forms a complex 'plaque' in the artery wall. The early stages of atherosclerosis is characteristically deposition of cholesterol in the artery wall associated with lipoproteins, particularly low density lipoproteins (LDLs). LDLs accumulate and are retained in the vessel wall, triggering the ECs to initiate an inflammatory response and inducing monocytes in the blood to cross the layer of ECs lining the blood vessel and differentiate into macrophages, contributing to the forming lesion. These macrophages rapidly take up a modified form of the LDL, leading to the formation of 'foam cells'. Once the foam cells apoptose they deposit their lipid rich contents within the vessel wall. Smooth muscle cells also migrate into the lesion and proliferate there, adding to its growth and development. Heart disease and stroke can occur if a lesion ruptures into the artery. 1.1. E n d o t h e l i u m The endothehum consists of a monolayer of ECs that line the entire vasculature. The ECs are attached to each other by intercellular junctions that confer mechanical properties

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RJ. Allen et al

upon the endothelium. More importantly (in this context) these junctions are selectively permeable to materials (for example ions) and cells (for example monocytes) crucial to biological function. Certain points of the vasculature are more susceptible to atherosclerosis: areas of branching or curvature where the pattern of flow is complex are more at risk than tubular areas (where the flow is approximately laminar). Furthermore in the branched areas the endothelium has been shown to be more permeable to larger molecules (such as LDL) [1], which means these areas are more likely to develop atheroscelerotic lesions, [2]. It is well documented in in vitro studies that when endothelial cells are exposed to laminar flow they elongate and align with the direction of flow. Conversely, in regions of more complex flow in vivo the cells do not elongate or align with each other and the junctions between cells are more 'leaky', [1,3]. 1.2. Shear Stress Here we discuss an approach to modelling the passive response of the cell to the mechanical force the fluid flow exerts on the cell membrane. The cell also undoubtedly responds to fluid flow through the activation of mechanosensing receptors and subsequent signal transduction. However, the manner in which the signal from the applied shear stress is received by cells is currently unclear, although several different molecules have been implicated as sensors for the shear stress, [4]. For example, shear stress induces potassium channel activation and G protein activation (within the first minute of the application of shear stress), activation of the GTPases, RhoA, Racl and Cdc42, [4]. as well as mitogen activated protein kinase signalling and activation of transcription factor NF/cB (within the first hour, see [4,5]). These shear stress-activated responses contribute to cell shape change and altered gene expression. Internally the cytoskeleton is remodelled to bestow greater mechanical stiffness to the cell, [5]. This is achieved by induction of actin stress fibre growth and alignment in the direction of flow, [6,7]. 2. Modelling t h e Passive R e s p o n s e . In addition to activation of signal transduction pathways, fluid flow will induce cell shape change as a consequence of the mechanical force imposed on the cell. This is the effect of fluid flow that we address here. The general approach taken here is to model the flow of the blood in the artery via the non-dimensionalised Navier-Stokes equations which describe the velocity of the fluid u at a given point in time and space:

where Re is the Reynolds number of the flow and Fr is Froude number of the flow. This expression along with the continuity equation (V.u = 0), define the flow. The NavierStokes equations are valid for a homogeneous, incompressible Newtonian fluid. These assumptions are questionable when applied to blood, however for the region of interest here (i.e. near the arterial walls) they are more suitable since it has been widely shown that there is a thin layer of the flow that can be modelled as homogenous, and for similar reasoning it also approaches a Newtonian fluid at this point [8,9]. It also - in large arteries is well approximated by the incompressible case. The artery is treated as a simple cylinder. In this region Navier-Stokes can be solved (by imposing boundary conditions of no flow at

Modelling Morphological

Change in Endothelial Cells Induced by Shear Stress

1725

Side View

j

Top View

I I

Figure 1. Cartoons of both profiles of an endothelial cell. In the side profile the cell is sitting on a substratum (for example the arterial wall), in the top view the substratum is in a plane parallel (and below) the plane of the page

the arterial wall) to give (via experimental measurements of parameters such as common flow rates and typical artery diameter in common carotid arteries) an estimation - to an order of magnitude - of the Reynolds number as 10~^, and the Navier-Stokes equations (in the absence of a body force, b) can be reduced to the Stokes equation: V P = Vû

(2)

In order to calculate the Reynolds number an estimation as to the characteristic velocity of the flow has to be made. The cell is modelled as a two dimensional ellipse sitting above the arterial substratum as in the bottom representation in figure 1. This section is taken to be 1/im above the substratum, and the characteristic velocity is calculated as the flow velocity at this radius. A useful way of approaching the problem is the boundary integral representation (BIR) of equation (2). In physical terms the BIR expresses the total velocity as a sum of velocities resulting from a concentrated point force, with the sum constructed in such a fashion that the boundary conditions are satisfied. By modelling the interior of the cell as a fluid with the same velocity as the exterior fluid the BIR becomes particularly simple, [10]: ^,(xo) = 2/^(xo) - ^

^ A/,G,,(x,xo)dl(x)

(3)

here Gij^K, XQ) is the free space Green's function, u°° is the flow far away from the interface (here we non-dimensionalise by this characteristic velocity so that we can take u ^ as a unit vector in the a;-direction), Af = {a^ — (j^).n is the discontinuity in the interfacial surface force (cr^ and cr^ are the stress tensors for each of the fluids and n is the normal to the membrane) and the dl is an element of the boundary defined by c. In order to close the problem assumptions are needed on the nature of the interface between the two fluids. Here we choose some of the simplest assumptions, namely, that it is an incompressible, inextensible material with an isotropic surface tension, r, acting

1726

RJ. Allen et al.

parallel to the 1-D boundary. In two dimensions these assumptions lead (see [10] )to the expressions:

->-V

(^)

where t is a vector tangent to the interface, and dl is a differential element of the interface. This expression is from balancing the forces on the differential element, dl. The second expression is:

'•£ = «

,5,

A finite element method is used to describe the two dimensional boundary (i.e. the cell membrane) and equation. The strategy for evolving the boundary is to solve for the velocity on the boundary at every time step. In order to do this expression (4) is substituted into equation (3) and the resulting expression is substituted into (5). Once the problem has been discretised over the boundary elements this leads to a system of linear equations which can be solved for the tension r:

here the square brackets denote averaging (by integrating along the element) over the zth boundary element. Then the tension of each of the elements can be substituted into a discrete version of equation 3 to solve for the velocity of the boundary elements which can be updated at every time step. Here the boundary initially is an ellipse of a non-dimensionalised height relative to the cell's length perpendicular to the fluid flow. The nodes defining the boundary elements are restricted to move perpendicular to the interface, and every node is updated at every time step via: xr^=x|+

TU2

1 S5

—50.89-»i

Fig. 2 Optimal solution for a 2 Process unit - 2 Treatment unit system operating under uncertainty Acknowledgement. The authors gratefiilly acknowledge financial support from the National Science Foundation under Grant CTS-0521769.

REFERENCES 1. I.E. Grossmann, K.P. Halemane and R.E. Swaney, Computers and Chemical Engineering, 7(1983) 439. 2. J. Acevedo and E.N. Pistikopoulos, Computers and Chemical Engineering, 22(1998) 647. 3. M.L. Liu and N.V. Sahinidis, Industrial and Engineering Chemistry Research, 35(1996)4154. 4. N.V. Sahinidis, Computers and Chemical Engineering, 28(2004) 971. 5. N. Sahinidis, Joumal of Global Optimization, 8(1996) 201. 6. R. Karuppiah and I.E. Grossmann, Global Optimization for the Synthesis of Integrated Water Systems in Chemical Processes, to appear in Computers and Chemical Engineering (2006).


Hierarchical Markov Reliability / Availability Models for Energy & Industrial Infrastructure Systems Conceptual Design A.N. Ajah,"'^ P. M. Herder," J. Grievink,^ and M. P.C. Weijnen," "Energy and Industry Group, Faculty of Technology, Policy and Management, Process Systems Engineering Group, Faculty of Applied Science, Delft University of Technology, 2600 GA, Delft, Netherlands. Abstract Modelling infrastructure system's deterioration and repair behaviours through a markov model is not only essential for accurately predicting the system's future reliability condition but also act as key inputs for effective infrastructure systems maintenance. During the conceptual (re)design of these systems, myriads of components are usually involved. A multi-state markov modelling of this component is emphasised in this work. However, the exponential explosion in the size of the markov model as the number of such components increase may pose great limitation in its application at this stage of design. We also present a hierarchical modelling approach that could aid the designer in not only overcoming this limitation but also in the detailed screening and analysis of the reliability and availability of such infrastructure systems' components. The application effectiveness and utility of the proposed approach is tested by means of a case study, the reliability modelling of a proton exchange membrane (PEM) fuel cell power plant. Keywords: Reliability & Availability, Markov model. Conceptual Design, Infrastructure systems. 1. Introduction Infrastructure system's (energy, gas, water) reliability, just like the reliability of any engineered system, is increasingly becoming an important performance indicator. This increasing importance and the associated pressure on the system designers calls for a drastic change in the ways the infrastructure systems are currently conceptually designed. Most of the common practices of estimating infrastructure system's reliability and overall availability at the conceptual design stage are either done on an ad hoc basis or basically rely on some predefined and assumed component availability (usually 80-95%). Two major reliability analysis can be distinguished; measurement and model-based (Sathaye et al., 2000). During systems design, measurement based analysis may be infeasible and expensive, hence model based approaches are often relied upon. However, most of such reliability models utilize the non-state-space model which assumes that the components are independent of one another in their failure and repair behaviours. The markov model could well predict the dependencies and future conditions of these infrastructure components, systems and networks through the characterization of such deterioration and repairs in a probabilistic continuous or discrete-state manner. Capturing such infrastructure system's deterioration and repair behaviours and their effects on the overall system, through markov model is not only essential for accurately predicting the system's future reliability condition but act as key inputs for effective system maintenance (corrective and preventive). However, the explosion in the size of the markov model

1753

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A.N. Ajah et al.

as the number of such systems, components or networks increases is being perceived as a major hmitation in its apphcation during the early phase of conceptual design where time is often a constraining factor. Due to this exponential growth, it may not be easy to markov model complex infrastructure systems reliability as a once-through entity. What is needed is an approach that can decompose the reliability problem into manageable levels of abstraction, address the reliability issues separately at the decomposed levels and then aggregate the results at each level into final system reliability. This work builds on this concept as it dwells on a hierarchical markov modeling approach to circumvent this limitation and also aid the designer in the proper screening and analysis of the reliability and availability of infrastructure systems at the early phase of the conceptual design process. 2. Hierarchical markov modeling approach In the hierarchical markov modeling being proposed, the reliability problem is decomposed into three manageable markov levels (components, units and system), based on the structural and behavioral complexity of the system. This decomposition reduces the number of state space problems to a size that could be easily handled. At the component level, each component is markov-modeled separately. This gives the designer, extra degree of insight into the dynamic performance of the components and hence in the selection of components to feature in the design. At the unit (sub-system) level, for a given flow diagram of an infrastructure system, an aggregation of the equipment based on functional and structural similarities is carried out. This aggregation reduces the number of equipment to be markov-modeled. Lastly, at the higher (system) level, the evaluated subsystems markov reliability and availability are combined into the total system reliability and availability. In this way, the size of the problem as well as the computational efforts is significantly reduced. The attributes of decomposition levels is as shown in table 1. Table 1: attributes of decomposed levels Attributes Analysis type States

Level of decomposition Component

Unit

System

Markov

Markov

RBD + Markov

Multi-states

Reduced

Reduced

Structural model

No connectivity

State space diagram connectivity

Series-parallel

Behavioral model

Independent

Dependence of units

Dependence of units

Reliability and availability modeling at the component level: At the component level, the essence of reliability and availability analysis being proposed is to assess the reliability characteristics and states of each of the components with a view to selecting the most reliable and promising ones. However, the conventional attitude is to assume that a component has binary basic states-operable and failed states normally designated as 1 and 0 respectively. In most infrastructure systems, there are some groups of system failures that may not be immediately observed upon occurrence. Such system disorder do manifest as small errors or defects such as pump fouling, pipeline partial blockage etc that do not cause immediate total or catastrophic failure. Nonetheless, if left undetected and unrepaired, such failures still grow to cause a larger failure that result in unscheduled downtime. Before the catastrophic or total system downtime, occurrence of such menial disorder may force the system into a state of reduced functionality vis-a-vis its incipient operational state. These sorts of failures that do not result into immediate catastrophic failure of the system but can lead to diminishing functionality have been modeled as transient failures and the states at which they occur, as transient states, states r and d in

Hierarchical Markov Reliability/Availability

Models

1755

figure7. Considering and modeling these transient states using the markov approach will give a more realistic steady state and dynamic reliability characteristics of the components and will help in the proper screening and selection of components to be featured in the system being designed. The transition diagram as depicted below covers possible states and transitions that a component can realistically experience. Transition from state u (fully operational) to transient states r or d may be caused by environmental and human induced problems respectively while the transition from state r or d back to state u is caused by the removal of such a failures (repairs). If at states r or d, no action is taken, the component might finally transit to the state / (permanent failure) which may be restored to either states u (full restoration) or to states r and d (partial restoration), depending on the nature of the maintenance action taken. The direct transition from states u to state / depicts mechanical and other unknown problems which lead to instantaneous permanent failures.

,<S^

<S>--

Fully operational state

Reduced Functionality states

Fully unoperatJonaf state

Figure 1: Transition diagram for a multi-state markov modeled component It has been reported that human errors and environmental factors contribute about 40% of total equipment downtimes (w^ww.plantweb.emersonprocess.com); hence, we have differentiated between states r and d to highlight the effects of operators and environmental factors on the availability of components of especially energy infrastructures. Assuming there is enough data, we think that considering them early in the design process; will aid the designers in the critical assessment of these factors and thus the screening, differentiation and better choice of the more reliable components. The probability of components (Pi) to be in any of the M states could be obtained from the solutions of sets of equation 1. X and ji are components failure and repair rates.

dP^(t)_ dt

^

^

^

^

Y,î^j \PXt) + \ Z / ^ ; - h W

V J

i = 0,...,M;M>2

(1)

V J*i

Reliability and availability modeling at the unit level: Having identified the most reliable components to feature in the design based on the detailed component level markov modeling, at the unit (sub-system) level, the combinatorial dependencies and redundancies associated with these components is also markov-modeled. It is envisaged that for complex infrastructure systems, involving units with components in the order of tens and hundreds, state space explosion problem will be imminent. To circumvent this, we propose unit's component aggregation. The basic idea behind this is to reduce the number of equipment in the units of such complex large-scale systems by substituting some sets of equipment whose individual unavailability does not clearly affect the system performance, or whose failure and repair rates are similar, with a single representative component. For a unit with N number of components, the state space size is estimated by 2^, but with the approximate state space aggregation technique (Lefebvre, 2002) the size reduces to:

ri(A^,+i)

(2)

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A.N. Ajah et al.

where Nt is the number of components of the ith family, k is the total number of different aggregate families. Assuming there are a total of 6 components (i.e. N=6 with 2 components aggregated into each family (Ni=2) then k becomes 3); the total number of unreduced state space is 64 while the approximate reduced state space is 27. However, such aggregation is valid for set of equipment which have single (or multiple) input(s) and single output to the remaining components of the system, and for certain types of multiple input-output subsystems (Van Rossen, 1994). Since aggregation policies and rules are often based on dedicated structural and behavioral heuristics, caution, experience and sound engineering judgment are needed of the designer in the application of these aggregation techniques. Reliability and availability modeling at the system level: At the system level, using a reliability block diagram depicting the connections(series, parallel, series-parallel etc) and other reliability characteristics the aggregated individual units of the system is drawn and the overall system structure and reliability, analyzed using the network reduction (Sahner and Trivedi, 1986; Knegtering and Brombacher,2000) approach. From the network reduction technique, given the combined markov models and RBD constructs, if a system has a series-parallel structure, its overall reliability can be obtained using :

(3)

Rs,s(t)=Â^-Kr.uM,aralM{^-K,uM,ar.lM-\^-Kr,uM,araneH^^

Where Rgys (t) is the system overall reliability at time t, Ruint i (t) is the reliability of unit i at time t as obtained from the unit markov reliability modeling. If all the units in the system have parallel connection structure, the last factor of equation 3 can be neglected. 3. Illustrative Case study As an illustration, the proposed decomposed markov reliability model is applied to the analysis of the reliability characteristics of a proton exchange membrane (PEM) ftiel cell power plant as shown in a condensed block diagram of figure 2. Air (oxygen) Hydrogen

DC Power FCP Stack,

REFORMSRt

Natural Gas

Fuel Converter (REFORMER)

Fuel cell i^wer Stack

DC/AC Power Inverter

Transfrnmer

REFOWeR j

(a)

FCPStackj

(b)

Figure 2: a) Condensed block diagram of a PEMFC power plant, b :) Aggregated units of the PEMFC Fuel cell power plant is a heterogeneous system featuring the interactions between chemical, mechanical, thermal and electrical components and subsystems for converting fuels such as gasoline and natural gas to alternating current. Natural gas flows into the reformer where it is converted into hydrogen. The hydrogen produced together with oxygen from air are then routed into the (PEM) cell stack assembly where the O2 and H2 streams are electrochemically converted into electrical power, steam and water. The steam is recycled to the reformer for the reforming process. The electrical power generated, in the form of DC power flows to the power inverter for inversion into an AC power before being routed to grid.

Hierarchical Markov Reliability/Availability Models

1757

Component level: The case study described above has more than ten components such as compressors, pumps, humidifiers, reactors, transformers, membrane, control equipments etc. For the compressor, table 2 shows the state probabilities. Steady state is assumed at 250 hrs. Table 2: compressor steady-state probabilities Time(hrs)

State u

State /

State;

State 1

0

Compressor 1&2

1

0

0

0

250

Compressor 1 Compressor 2

0.9679 0.9736

0.0093 0.0085

0.0152 0.0113

0.0076 0.0066

From the foregoing, it could be deduced that the compressor 2, with higher chances of being operational and lower chances of environmental and human failures (states / 8c j) outperforms compressor 1 and thus would win the selection process. Unit (subsystem) level: Using the concept of components aggregations as discussed in section 2, and assuming redundancy, the components of the PEMFC power plant have been lumped to form the basic unit as shown in figure 2b. In all, five units with 32 state and 80 transition paths as depicted in figure 4a is identified. If a second degree aggregation is carried out, the number of states reduces to about 18, with 32 transition paths (figure 4b). From figure 4a, sets of markov differential equations (equation 5) results, assuming different failure and repair rates for the various transitions to and fro states. The solution of these differential equations, gives the probability of the system to be in each depicted state. Such solution could be obtained using any of the numerical methods of Euler, Runge Kutta and LU decomposition or with software such as MATLAB. Table 3 shows the results of the state probabilities for the second degree aggregated units of 18 state spaces.

r - ^ (/^sto~"^ - - * • - / V-*---. >-—;—-ji " > - " ^ ^ (,pnio))^X'ai'2i ([.i?ioi) ((11^001 ; Ct5U0iiJ (ifllnj

X---^-/ (tofilij)

V" Ctio

(b) Figure 4: a) 1st degree reduced state space [Si denotes State i, jUij and /lij denote repair and failure transition ratesfromstate / to statey]; b) 2"** degree reduced state space of the illustrative case study.

A.N. Ajah et al.

1758

-z\
» 0) th«n P = PI; h - hpHM_Port_al.h; HI = hpHM_Port_al.h*Ml; •Is* P = P2; h = hpHM_Port_bl.h; HI = hpHM_Port_bl.h*Ml; •nd if;

•ad Mass Heat Store; •nd OpenL;

Fig. 4 Ideal gas law.

Fig. 5 Control volume.

Fig. 6 Bidirec. orifice eq.

Numerical difficulties arise when the orifice equation is used at flows speeds close to zero due to the singular derivative of the root fiinction. This can be avoided by using a linear interpolation instead of the root fiinction in a user-definable region around zero flow.

Library for Modeling and Simulating the Thermal Dynamics of Buildings

1781

3. CONSTRUCTIVE AND COMPLEX ELEMENTS 3.1. Building more complex components More complex components like a wall addressing radiation, multi-layer conduction and convection can be easily constructed through aggregation and inheritance. Ventilation, doors, windows, ceiling, floor, junctions and cracks can be defined firom the basic components already defined. As an example a wall is constructed following the UML diagram shown in Fig. 3. In general, each constructive component can be open and its internal components can be further customized. 3.2. Room template A simple room component is shown in Fig. 7, where constructive components have been used to compose it as a general template-like component. Internal component parameters are directly defined through the room parameters. Although the room component has a fixed geometry, it can be easily modified by adding the necessary additional constructive components.

D

g = --D

CK D-

-D

L>i

Fig. 7 Room internal elements.

Fig. 8 Building with boundary conditions.

SIMULATION A simple ten room building is used to illustrate the building library.(see Fig. 8). Boundary conditions accounting for daily temperature and pressure variation are used. The building is left passive without any HVAC system acting and without any agent model. As seen in Fig. 9, the pressures rapidly converge to a common value producing an enthalpy flow (inter-zone convection). After this initial transient response, the slower thermal dynamic of the building reaches the stationary response after a few days, as seen in Fig. 10 for two neighbor rooms. The thermal response has been compared with the ATPlus library (Feigner et al. 2002) and both libraries show similar transient behavior but different stationary response; one reason can be the mathematical solution used to solve the ID-PDEs (e.g. level of discretization of each layer) representing conduction, or the view factors used in the long-wave radiative exchange within the enclosure. 5. CONCLUSIONS While simulation tools gain generality and more computational power is available, more detailed dynamic models, advanced controllers, and energy management strategies can be studied. Using a library of models is a widespread way to structure building simulation and modeling

J.I. Videla andB. Lie

1782

applications, but since a few of these tools expose the underlying models, typically the user has to rely on the quality and completeness of the provided documentation (Tummescheit & Ebom 1998). The primary goal of this paper is to present a base library with common model parts which are user-extensible with a consistent logical structure, that is, neither too abstract and difficult to use, nor obvious and over-crowded of components. The presented library can be easily modified and customized to special needs. Room 1.T (HiTLib)

Room 1 .T (ATPIus) 290 ••
=^Xk), possibly of much higher dimension. This gives a possibility to express the functional mapping J{.) as a parametric model, which in our case is the linear regression for one scalar response >^k y,=0'(/>,ê,

(1)

where ^^ is a vector of regression coefficients, ^ = ^Xk) is a regressor vector, and 8k is a stochastic component of the model. Bayesian prediction. The data set composed of explanatory and response variables is supposed to be a sample from the conditional density/7(y|x,^. The unknown parameters 0 can be eliminated using standard probability calculus rule that enable to evaluate the predictive probability density function of >ô given XQ and past data (x^, j ^ ) as follows:

Piyo\xo,x\y'')=

lpiyo\xo,y\x\0)pîO)da,

PAO) - PoiO) n Piyi^h' ^y • ^k=K(||x - X,I)

(2) (3)

k=l

whQYQPQ(0) = p(0)is

the prior density and /?(J^QXQ,X , y , ^ ) i s the sampling density (local model likelihood) and K[.),K{0) = 1, is a suitable kernel function. As illustrated in (Kulhavy, 2003), in case of the polynomial regression and presumption of normal distribution of the stochastic component £k, the conditional density takes the form p(y,\x„0) = [im'Y^ e x p j - ^ (y, - d'{x,)f^

(4)

which allows an explicit evaluation of the unknown regression coefficients. Locality. The locality of the method is guaranteed by an appropriate choice of the neighborhood, which is defined by bandwidth vector h=(hj,..,hm)' All historical observations x with normalized Eucliedean distance D^ less than 1 are considered as neighbors of query vector XQ.

\x-x^\^=D^={\x-x^\lh)(i,Wi>^

in,i

Wy z out,i)

+ (1 - (o)'-^:^^^ ^ total

/

-•

^-^ ^total

i,j

(1)

Cost Versus Network Length Criteria in Water Network Optimal

1823

Design

In equation (1), A^ stands for the number of unit operations; Ftotal is the maximum supply water the network would employ in the absence of any internal reuse; Lin,i and Lout,i are the lengths of the supplying and discharging pipes, respectively, while Ltj is the length of the pipe from unit i to unit j ; Fi>0, Wi>0 and Xij>0 are the active pipe conditions, such that its length to be considered in the summation process; Ltotai is the overall length of the wastewater network's pipe system; co is the weighting factor. Details regarding the complete development of the flow term of the objective ftinction can be found in Lavric et al (2004; 2005a). Solving the associated optimization problems is not trivial, since the unknowns' number outcomes the equations' number. We employed an improved genetic algorithm which uses each internal flow as a gene, defining a chromosome from all these flows (Lavric et al 2004). The restrictions are coped with during the population generation eliminating these individuals outside the feasible domain. The individuals are interbreeding according to their selection frequency, using one-point crossover method, and then mutation is applied to randomly selected ones. 4. Results and discussions 4.1. Cost versus Network Length Criteria - no thresholdfor inner flows In order to verify if the Network Length Criterion (NLC), as resulting from (1) putting co = 0, gives comparable results with the economic objective function (EOF) used in Lavric et al. (2005b), the same runs were carried out, the results being presented in Table 1, for both criteria. These results are compared against those obtained when only fresh/supply water (F/SW) was employed as objective function, as resulting from (1) putting o) = l. We used two ranking methods, by fresh flow needed when no internal reuse is envisaged or by maximum pollutant load, to comply with the principle of driving force equipartition, as describe in Lavric et al. (2004). Irrespective of ranking procedure, NLC gives better results than F/SW but worse than EOF, when no threshold value for the internal flows is imposed, although using maximum pollutant load as ranking criterion lowers the significantly the active pipes' length. The drawback of using NLC instead of F/SW is a slight increase in the water consumption, with a maximum of 1.786 t/h for the worst case. But, this is compensated largely since the investments are lower (48.07 km against 54.42 km and 39.7 km against 53.88 km) and so does the pumping cost, since the frictions in a smaller network will be lesser. We present in Figure 1 the best NLC topology, against the best EOF one, both corresponding to the maximum pollutant load ranking procedure. The surplus of Table 2 Results of the design of a wastewater network with 15 unit operations and 6 contaminants; all four resources, as presented in Lavric et al. (2005a), are used to supply the network Objective Function

>>

.o o> CO

a:

lo Z3 •o tn 0)

"o CD

o> CD

Supply Water

Network Length & Supplyf Water

Total Cost

Inlet Pipe's flow, t/h length, km

N

54.42

Inlet Cost, Pipe's Inlet Pipe's Inlet Pipe's k$/year flow, t/h length, km flow, t/h length, km flow, t/h length, km 207.4

27.75

48.08 457.194

LI-

CO O "O _l

Y

N

co = 0

CO = 0.5

457.194

48.07 458.88

15.21

37.44

204.6

53.88

173.5

30.18

458.044

39.79

458.62

39.7

39.6

176.0

9.72

457.194

10.28

458.14

9.76

457.194

12.09

15.33

1824

P. lancu et al

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Figure 1 Optimal network topology using as objective function the total costs or the active pipes' length (bold-italic figures) - all flows are taken into consideration; Resources A, B, C and D are the same as in Lavric et al. (2005a) 9.52 km of pipes serves to a more uniform internal flow distribution, at the expense of operating and investments costs. But, in the same time, the NLC topology is quite independent to the market fluctuations. Combining both F/SW and NLC criteria into a single multi-objective function, putting CO = 0.5 in equation (1), made no improvement into the topology of the optimal network, irrespective of the ranking procedure we used. Still the cost based optimized network is at least 30% smaller. The network changed, in terms of active pipes and internal flows, but its total length did not. However, it is worth mentioning that the fresh/supply water consumption decreased either to its minimum value, whenfi*eshflow needed when no internal reuse was used for ranking, or close to this value, when maximum pollutant load was used instead. 4.2. Cost versus Network Length Criteria -threshold of 1 t/hfor inner flows A completely different behavior of the optimization results was observed when we imposed a threshold value of 1 t/h to the internal flows (the lines corresponding to Y in the second column of Table 1), even when only the F/SW criterion was used, although in this case, after reaching the minimum of 457.194 t/h fresh/supply water consumption, the algorithm ceases to search for a better topology. The lengths of 37.44 t/h and

Cost Versus Network Length Criteria in Water Network Optimal Design

1825

39.6 t/h, respectively, were obtained after several runs with the genetic algorithm optimization starting from random internal flows. It must be stated that the final active lengths were all in the vicinity of these values. Quite remarkably, putting a threshold value for the internal flows improved dramatically the topology of the EOF optimized wastewater network (see Figures 1 and 2 for details). In both these Figures, the EOF optimized topology corresponds to the normal written numbers. What is really surprising is the way the internal water reuse simplifies when there is this threshold value - see the conNLCtions between unit operations below 11 in Figure 2 in comparison with Figure 1. Although only two links have actually their flows below 1 t/h (0.964 t/h from 1^12 and 0.205 t/h from 9^12), their disappearance completely changed the topology. The internal flows grow bigger and its distribution between the first half units of the network changed such that the input pollutant concentrations approach to a greater extent the imposed limits, decreasing the need for internal reuse among the terminal units. This has as result a reduction of the active network length to 32.2% of the previously optimum value, at the expense of an increase with 1.44% of the costs. This increase is due to the raise of pipe diameters and fluid velocities, but the simplified network will imply lower maintenance costs, not included in the EOF.

•(8)375,008

Â) 32.436™-*4

• CD| 7232

-(CH2.457--"-*!

- f ^ 0 J4g.

Figure 2 Optimal network topology using as objective fiinction the total costs or the total pipes' length {bold-italic figures) - the flows under 1 t/h are disregarded; Resources A, B, C and D are the same as in Lavric et al. (2005a)

1826

P. lancu et al

When NLC objective function is used (with co = 0) to find the optimum wastewater network topology, disregarding the flows under 1 t/h, the changes are even more dramatic, the result obtained being very close to the one owed to EOF (see Figure 2 for details). Not only had the supplemental fresh water added to the network through unit 13 dropped with 33.71%, but also the internal network architecture simplified almost as much as when EOF was used, under the same circumstances. There are only one supplemental flow introduced (10.071 t/h from 4^'14) and one discharge instead of internal reuse (6.552 t/h from 11-^12 is directly sent to treatment from unit 11). The main differences come from neglecting several internal water reuses and modifying the internal flows, accordingly. But the important benefice is that we obtained the same simplification of the internal network for the last half units. Using a multi-objective optimization fimction does not improve the resulting topology. On the contrary, the beneficial effect of NLC is hindered partially by the use of F/SW, the system increasing slightly the active pipes' length, although the fresh/supply water consumption reaches the aforementioned lowest value. 5. Conclusions In this paper, we presented a multi-objective optimization criterion which can be successfully used to find the best network topology. The parameter of the function permits its use for composite demands, starting from the plain S/FW minimization and ending with optimal NLC. As expected from our previous researches largely presented in Lavric et al (2004; 2005a&b), ranking the network by the maximum pollutant load gives better topologies, no matter the parameter's value. Another important finding is that the resulting complexity of the network is heavily lowered when a given threshold value is imposed upon the internal flows, situation in which both EOF and NLC gave almost the same results. So, a straightforward continuation should be a thorough investigation of this threshold value upon the optimal topology. References M. Bagajewicz, 2000, A review of recent design procedures for water networks in refineries and process plants. Computers and Chemical Engineering, 24, 2093-113 X. Feng and W.D. Seider, 2001, A new structure and design methodology for water networks, Ind. Eng. Chem. Res., 40(26), 6140-6 X. Feng and K.H. Chu, 2004, Cost optimization of industrial wastewater reuse systems, Trans IChemE, Part B, Process Safety and Environmental Protection, 82(B3), 249-55 N. Hallale, 2002, A new graphical targeting method for water minimisation, Advances in Environmental Research, 6, 377-90 V. Lavric, P. lancu and V. Ple§u, 2004, Optimal Water System Topology through Genetic Algorithm under Multiple Contaminated-Water Sources Constraint. In Computer-Aided Chemical Engineering (Barbosa-Povoa A, Matos H, Editors), 18, Elsevier, 433-8 V. Lavric, P. lancu and V. Ple§u, 2005a, Genetic Algorithm Optimization of Water Consumption and Wastewater Network Topology, Journal of Cleaner Production, 13(15), 1405-15 V. Lavric, P. lancu, V. Ple§u, I. Ivanescu and M. Hie, 2005b, Cost-Based Water Network Optimization by Genetic Algorithm, Chem. Engng. Transactions 7, 755-60 M.-H.Suh and T.-Y. Lee, 2002, Robust Optimal Design of Wastewater Reuse Network of Plating Process, Journal of Chemical Engineering of Japan, 9, 863-73 S. Thevendiraraj, J. Klemes, D. Paz, G. Aso and G.J. Cardenas, 2003, Water and wastewater minimization study of a citrus plant. Resources, Conservation and Recycling 37,227-50 Y.H. Yang, H.H. Lou and Y.L. Huang, 2000, Synthesis of an optimal wastewater reuse network. Waste Management 20: 311-9


Synergy Analysis of Collaboration with Biofuel Use for Environmentally Conscious Energy Systems Metin Turkay and Ahu Soylu^ College ofEngineering, Kog University, Rumelifeneri Yolu, Sariyer, 34450 Istanbul, TURKEY

Abstract The energy sector is a high-cost and high-emission sector. With the demanding environmental regulations, the energy producing companies must find new solutions continuously to decrease emissions while satisfying the energy demand. It was shown in previous studies that collaboration by exchanging steam among the energy companies can create synergy both in environmental and economical criteria. In addition to collaboration, transition to new technologies is necessary to satisfy environmental regulations. This paper presents an analysis of the expected gains in a collaborative setting with an environmentally friendly fuel alternative, biodiesel. The problem is modeled as a Mixed-Integer Linear Programming. The results of the solutions are analyzed indicating the benefits of collaboration. Keywords: Collaborative Planning, Energy Planning, Supply Chain Optimization, Biodiesel, Mixed-Integer Linear Programming 1. Introduction Energy is a fiindamental entity of modem life that has strong influence on social, industrial and economic activities in a society. Among the various forms of energy electricity constitutes a major proportion of the energy requirements. Fossil fiiel based energy systems are the dominating electricity production technologies in the world. An important concern with the fossil fiiel based electricity production systems is the release of large quantities of environmentally harmful substances such as, SOx and Green House Gases (GHG). Environmental protection must be treated as an important factor in the energy supply chain due to growing awareness of the environmental problems and enforcement of stricter environmental regulations. The Kyoto Protocol [1] demands for reductions in greenhouse gas emissions by the industrialized countries. The energy sector will be seriously affected with Kyoto Protocol since it requires countries to have an air pollution management strategy. The amount of substances released to environment will be restricted. This restriction can be achieved by reducing the energy consumption, changing energy production technologies to environmentally fi-iendlier ones or taking some remedy actions in energy supply chain management. With the increasing demand of energy by growing population and requirements for higher quality of life, the projections in EIA's report [2] suggest that without transition to new technologies, it is impossible to reduce the emissions to the year 1990 levels or less. Renewable energy technologies are promising in the sense that they release almost no emissions to the atmosphere [3]. They constitute a large group of technologies including, solar photovoltaic cells, wind turbines, and biodiesel usage in energy production. The weakness of renewable energy sources is that their high costs of ^ Current address: Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL,USA

1827

1828

M TurkayandA. Soylu

investment and unreliable output, changing with the shining rate, wind speed, etc. As a consequence, the solutions that integrate the renewable energy technologies together with conventional electricity production techniques are emerging. There are usually a number of energy production companies in an industrial area. Since these companies are in geographical proximity, process integration and collaboration among them is viable. It has already been shown that collaboration by exchanging steam among the energy producing companies may create synergy both in environmental and economical criteria by Tiirkay et. al.[4] and Oru9 [5], Soylu et al. [6] and Soylu [7]. Energy production is the major source of the environmentally harmful emissions and the emission reduction technologies have been studied within the context of energy systems interactions. Barreto and Kypreos [8] model the CO2 emissions trading in an energy systems model in a world-wide basis. They include technology learning in their model that affects the technology choice and emissions of regions. Hanson and Laitner [9] analyze the policies in order to select the most appropriate advanced energyefficient low-carbon technology. One of the environmentally friendly fuels is the biodiesel. Biodiesel is a nontoxic alternative fuel made from renewable fats and vegetable oils with a relatively very small fifference in performance than the petroleum-based diesel. Free of sulfur and aromatics, it can be used in engines and systems with few or no modifications. A biodiesel blend is pure biodiesel blended with petrodiesel. Blends up to 20 % biodiesel are compatible with all known oil tanks and systems. The compatibility of higher biodiesel blends depends on the properties of the materials of the tanks, pumps and fiiel lines. The usage of biodiesel as an urgent action in emission reduction by blending to existing fuels is examined and suggested in this study. 2. Analysis of Energy Production Systems A typical energy production system consists of storage tanks, boilers that convert fuel into steam at high pressures, turbines that expand higher pressure steam to lower pressure steam and convert the mechanical energy released during this expansion in the electricity and mixing equipment for mixing compatible materials originating from different sources in the system. Energy systems utilize fuel, air and other materials to generate electricity and steam. Companies can collaborate by exchanging High Pressure (HP), Medium Pressure (MP) and Low Pressure (LP) steam. There is an investment cost for such inter-company material exchanges, i.e. pipeline construction. The energy production systems that collaborate in order to improve their financial and environmental performance can exchange steam while satisfying the demand for HP, LP, MP steam and electricity. The model consists of MILP models for boilers, turbines, fiiel tanks, mixers, exchange structures and environmental constraints with an objective function of minimizing cost. The generation of HP steam is accomplished in the boilers by burning fiiel, which results in emission of harmful substances such as GHG or SOx. The boilers can be supplied with different fiiels as raw material with minimal adjustments in the operating conditions. This requires the selection of economically and/or environmentally attractive fuel among the available alternatives. The alternatives may be sulfurless oil, heavy oil, etc. which differ in calorie content, harmful emissions and cost. A model on energy production systems should include environmental limits. When environmental constraints appear, companies try to find new alternatives for producing energy with minimum emissions.

Synergy Analysis of Collaboration with Biofuel Use

1829

It is essential to include the possibility of using renewable energy technologies in the energy production systems in order to improve environmental performance. For that purpose, a new renewable energy source, biodiesel is introduced to the model with its own limitations and constraints. While modeling the following properties of biodiesel are taken into account: • The purchasing cost of biodiesel is a little higher than petrodiesel and holding cost is higher because of its material properties [10]. • The biodiesel can be mixed to only one type of the fuel and the other fiiels cannot be mixed to each other. The complete model is not given here for the sake of briefness. The constraints added for biodiesel blending are as follows: (1)

'-^^îjkjuJ

îJhuelL

(2) ArGFuel

z

(S)

îjklco,

iteBioFuel

keVuel

XHF,.^<MxYFU,^^

(4)

E yFu,.,). Eq. (2) models that the amount of HP steam produced in a boiler is equal to the sum of HP produced from different fuels in that boiler. Eq. (3) restricts the amount of biodiesel usage to maximum 20% of the blend used in that period. According to Eq. (4), if a fuel type is used in a boiler in that period YFUijkt becomes 1 where M is large number. Eq. (5) states that only one type of fuel can be used and mixed to biodiesel in a period. 2.1. Collaboration for Financial Performance In order to understand the model behavior, the model is solved for two energy producing companies whose schematic flowsheet is given in Figure 1. As can be seen, both companies have three fixel tanks, two boilers, two turbines and one mixer for each pressure level of steam. The continuous lines represent material flows within units and the discontinuous one represent steam exchanges between companies. For the sake of simplicity, the data used in the solution are not presented here. The problem is solved under non-collaborated and collaborated scenarios. The models are coded in GAMS [11] and solved with CPLEX solver [12]. The objective function consists of costs of fiiel (purchasing, ordering and holding), electricity, SOx penalty, unit operating, unit startup and construction of exchange structure. Figure 2 shows the changes in the portions of the total costs with collaboration.

1830

M Turkay and A. Soylu

Company 1

ifsZ: IP iXxER

LP I^XEI

Company! Eleciricity

I

Y

1

HP MIXER ET3

~' c

rg

E

P

Figure 1: Schematic Flowsheet of Energy Production Systems with Two Companies.

m Non-Collaborated • Collaborated

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1838

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15

Time (weeks)

Figure 5. Ethane storage level and storage target level. Figure 4 shows ethylene production and sales profiles, as well as ethylene demand and ethylene tank storage level. The model determines the production levels required to meet demands, which are satisfied in all but the last time period. Ethane storage level as compared to a target level is shown in Fig. 5. As deviations from the target level are economically penalized in the objective function, the storage level fluctuates around the storage target level. The problem has been solved in GAMS (Brooke et al., 1992) with DICOPT++ (Viswanathan and Grossmann, 1990) in four major iterations, with C 0 N 0 P T 3 and CPLEX, in 2016 sec. in a Pentium IV. 5. Conclusions A multiperiod MINLP model has been formulated for the optimal schedule of the cleanup shutdowns of parallel furnaces with decaying performance in an ethylene plant. Two continuous variables are linearly dependent on operation time: coil internal roughness and fimiace heat load. The model includes the entire plant description through nonlinear correlations and mass balances, as well as inventory management in ethane and products tanks. Two sets of binary variables have been introduced to model the effect of cleaning in the furnace performance. Despite the model has been run at academic level, numerical results show a good agreement with plant historical data. A model extension to allow for two or more fixmace shutdovms in a given time horizon is part of current work. References Brooke, A., D. Kendrick, A. Meeraus, 1992. GAMS: A users guide. Scientific Press, Palo Alto. Houze M.; Juhasz N.; Grossmann 1. E, 2003. Optimization model for production and scheduling of catalyst replacement in process with decaying performance. 4* Intemational Conference of Foundations of Computer - Aided Process Operations, Florida, EE.UU., 311 - 314. Jain v., Grossmann I. E, 1998. Cyclic scheduling of continuous parallel - process units with decaying performance. AIChE Joumal, 44, 1623-1636. Sahinidis, N. V.; Grossmann, I. E., 1991, MINLP Model for Cyclic Multiproduct Scheduling on Continuous Parallel Lines, Comp. Chem. Engng., 15,2,85 - 103 Schulz, E., Diaz, M.S., Bandoni, A., 2005, Supply chain optimisation of large-scale continuous processes, Comp. Chem. Engng, 29, 1305-1316. Viswanathan J.; Grossmann I. E., 1990. A combined penalty function and outer- approximation method for MINLP optimization. Comp. Chem. Engng., 14, 769 - 782.


Operational Optimization of the Thermoelectric System of an Oil Refinery Sandra R. Micheletto^'^ Jose M. Pinto*''" " Petrohras - Petroleo Brasileiro S.A., Brazil ^Department of Chemical Engineering, University of Sao Paulo, Sao Paulo SP, Brazil Ôthmer-Jacobs Department of Chemical and Biological Engineering, Polytechnic University, Six Metrotech Center, Brooklyn NY, 11201, USA. Abstract The objective of this work is to develop a mathematical programming model applied to operational planning of the thermoelectric plant of the RECAP Refinery (Sao Paulo Brazil) as well as its interconnections with the process units. The problem is formulated as a Mixed Integer Linear Programming (MILP) model where the mass and energy balances, the operational status of each unit, and the demand satisfaction are defined in multiple time periods. The model determines the operational configuration of the plant by minimizing utility costs, and identifies steam losses as well as inefficient units by comparing the optimal solution with the current operation. The MILP is able to accurately represent the topology and optimize the operation of the real-world system under different utility demands and abnormal situations, achieving a 5% cost reduction. The MILP is currently integrated with the refinery database and used for the planning of the refinery utility system. Keywords: MILP, thermoelectric plant, utility planning, refinery, optimization. 1. Introduction Demand fluctuations are typical of thermoelectric plants in petroleum refineries. Among the main causes are changes in the oil feed properties, multiple operational campaigns, maintenance of process units, interruption of electric energy supply, and cost variations of fuels and electric energy. On the other hand, utility systems are essential for the operational feasibility of refineries and must continually adapt to satisfy such dynamic demands. There are important contributions that address similar problems within the optimization framework, both in academia and in industry (see Hobbs (1995) for a review). Papoulias and Grossmann (1983) proposed an MILP framework for the design and synthesis of chemical plants, including utility systems. An MINLP approach for the optimal energetic planning of chemical plants was presented by Kalitventzeff (1991). More recent applications include Iyer and Grossmann (1997), Papalexandri et al. (1998), and Strouvalis et al. (2000); an interesting industrial application of MILP planning to a petrochemical plant was developed by Hui and Natori (1996). The objective of this work is to develop a mathematical programming model applied to the thermoelectric plant of the Capuava Refinery (RECAP, Sao Paulo - Brazil) as well as its interconnections with the process units, which solves the structural and parameter optimization and satisfies demand and allocates energy at minimum cost. Several process constraints are imposed, such as mass and energy balances, operating limits of the units, demand constraints and electric energy imports. The expected results are the optimal operational level of each unit at minimal operating cost.

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2. Description of the Thermoelectric System Figure 1 presents the simplified flowchart of the RECAP central thermoelectric system in which each unit connected to header (collector) UT is represented by a rectangle, as well as its identification; this configuration considers only the systems that are subject to optimization. The initials J are used to identify pumps and ejectors; GV or L for furnaces and boilers; Mfor heat exchangers, O ox P for vessels, TF for valves and Ffor blowers and compressors. The utilities are superheated steam at 30 kgf/cm^ (V30S), saturated steam at 30 kgf/cm^ (V30) and its condensate (CV30) , 5 kgf/cm^ steam (V5) and condensate (CV5), 1 kgf/cm^ steam (VI) and condensate (CVl), ; there are also two other collectors for condensate that are CVV and CM, which originatefi*omthe vacuum system and firom vessels, respectively. The water streams are denoted by Ax, where x denotes B=brute, C=clarified, T^cooling to process, Q=cooling from process, D=dearated, G=make-up to boiler, A=feed to boiler. Some of the electric energy is produced by the refinery from the effluent gas of the catalytic cracking unit, in the turbo-expander/electric generator (S-571TE); however, some of the energy is acquiredfi*omthe local company (denoted by BE). Finally, the fuel gas and natural gas network, is denoted by GCGN; it receives natural gas from the desulphurization unit (UDS) and pipeline GASAN and fuel gas from the catalytic cracking unit For instance, the CL_V30S headers receive V30S from the GV-6301, GV-6302 and L402 boilers and from the L-572 the heat recovery boiler. The headers distribute the steam to counter-pressure turbines, condensation of the process units, URFCC, ejectors and flare system.

3. Mixed Integer Optimization Model The steam networks are designed so to allow headers to transfer utility to headers at lower pressure levels, through a pressure reduction valve. In this case, a controller regulates the water injection so to maintain the quality of the steam. Hence, pressure and temperature can be considered constant and equal to the values obtained from the data acquisition system; a MILP representation of the system is sufficiently accurate. It is important to note that design and synthesis of these systems would require an MINLP representation, as in Bruno et al. (1998). Although the model developed in this work is based on the approach of Papoulias and Grossmann (1983), who address the design of thermoelectric units in chemical plants, it relies on the conceptual modeling framework of Pinto et al. (2000) and Neiro and Pinto (2005) that was developed for production planning problems, in which the elements of the utility systems are represented according to figure 2. Moreover, the topology of the thermoelectric system is defined by the interconnection energy generator-collector and collector-consumer. In Figure 2, the utility stream ut that is generated in unit eq is sent to collector cl_ut at time t with flow rate F^^ ^^^ ^z ut,t • This collector receives all the streams of utilities that are produced in the same time period from all units [eq^ ,eq^, -e^ J that are mixed at the temperature and pressure conditions of the network. In the collectors in which pressure and temperature are variable, the resulting temperature is defined by the energy balance. Similarly, collector clut sends utility ut at time period t to consumption units. A decision variable y^^^ is associated with a flow rate variable F^^^^^^ ^^ .

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The model relies on Mixed Integer Linear Programming (MILP) techniques where the mass and energy balances, the operational status of each unit, and the demand satisfaction are defined by mixed integer linear constraints, for each time period t.

\—[&M]

KEY:

•

CL_XX

Figure 1. Schematic representation of the RECAP thermo electric unit.

UNIT HEADER

S.R. Micheletto andJ.M. Pinto

1842 Vol ut,ut,eq_dl,t 'êql,ut,clut,t

^

eqi

Vel ut,ut,eq_d2,t f^ eq2,ut,cl_ut,t

eq2

Vcl ut,ut,eq_d3,t

^ eq_d] W ^

W

eq_d2

^

eqjds

^ W

eq_d

W

f^ eqn,ut,cl_ut,t

eqn

w

Vcl ut,ut,eq_dm,t

Figure 2. Schematic representation of collector of utility clut The following are the sets, indices, parameters and variables: Sets and Indices: CEeqjl

I

CSeqd

EQ: eq I eq_d EUiit,ut'

UT: ut I uf UTCeq UTPeq

T:t Parameters:

enthalpy of utility ut at the T-P conditions of the collector at time t price of utility ut in time period t heat received from external sources at time t (and it is only positive for the boilers; zero for all other units) lower / upper bounds of unit eqd

Put.t Qeqdj

nt

units that feed / receive streams unit eqd units / collectors units eqdihdii consume utility ut and generate utility ut' utility consumed / generated utilities consumed in unit eq utilities produced in unit eq time period

u^. u

Variables: continuous - flow rate of utility ut from eq to eqd in period t

" eq,ut,eq _d,t

binary - 1 if unit eq operates in time period t\ 0, otherwise

yeq,t

The objective is to minimize the overall utility cost at all periods that is given by the production costs, among others, of 30 kgf/cm^ steam, of fiiel consumed in the boilers as well as of electric energy. Moreover, there is an additional cost for the production of demineralized water in the reverse osmosis unit. MinC = Z ut

2-( 2^'êq,ut,eq_d,t \êq_d eq

(1)

The main constraints of the model are defined in general form in the sequence. Firstly, the material and energy balances are generated for each collector and for each unit in the system (2 and 3). Then the operational state of each unit is also established in (4)

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and it relies on the binary variable yeqj.t, as well as lower and upper bounds Q.. Finally, utility demand satisfaction constraints are presented in (5). ^

eq,ut,eq_d,t

~

^

êq_d,ut\eq,t

~^

y(ut,ut')G EU^,^^,.,Mieq_d,cl_ut\\/t 2J *êq,ut,eq_d,t'ût,t'^êq_d,t~ eg^CE^q_d

êq_dyeq_d,t

D,,, < Z eq_d

—

^

I

êq,ut,eq_d,t

F,^,,,,, ,,

2J êq_d,ut\eq,t'ût\t ^^^CS^q_d

^
-" [7]. This empirically chosen value permits the operating thermal power

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units to replace the wind power contribution without overloading problems or electrical system voltage and frequency fluctuations and is directly related with the current back-up power of the operating thermal engines. More specifically, the operator maintains full spinning reserve of the existing thermal power units to avoid loss of load in cases of an unexpected loss of available wind power. On top of this, an additional dynamic penetration limit is necessary, directly related with the rate (kW/sec) that the thermal units in operation can replace any wind power decrease without jeopardizing the local grid stability. In this context, the maximum approved by the local EGS wind energy production "Nw*(t)" can be estimated according to the following equations, i.e.: //

N,{t)T, given the state is s^,, take action x^. according to x^, = arg max U ^, (s^,, x)

(4)

X

Where \j ^,{s^,,x) is the approximated state-action pseudo-utility function for time T'. After compiling the results of the n simulations, compute the expected retumQ^(^^,x^),the riskR^(^^,x^),and the pseudo-utility \J^{s^,x^) with U(^, ,xJ = Q(s,,xJ-

AR(s^, X,)

(5)

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3. Fit an approximate state-action pseudo-utility function U^ (•,) based on the m points \5^{s^,x^),z = \,"',m. The multi-stage back-propagation principle is reflected in step 2 where simulation is conducted from T to T instead of from x to x+l as in the one stage back-propagation. This algorithm builds T-1 approximation state-action pseudo-utility functions, one for each stage except for stage T. Those approximate functions greatly reduce the computational overhead since a deterministic optimization problem represented by Eq. (4) is solved instead of the much more complex stochastic optimization problem represented by Eq. (1) at each state in the simulation. 2.2. Optimal policy approximation algorithm The number of deterministic optimizations in the form of Eq. (4) in the revised backpropagation algorithm is proportional to the number states visited in the simulation, thus the curse-of-dimensionality persists. The following optimal policy approximation algorithm tackles the dimensionality issue with another level of function approximation. In stochastic dynamic programming, a policy is a fiinction returning optimal actions for any state. Let n^(^) denote the approximation optimal policy at the decision time x, then for the decision time x 1. Sample m state-action pairs and obtain {s^, x^), z = 1, • • •, m . 2. For each pair {s^ ,x^), generate realization 4^^, z = 1 ,•••,«; simulate the n realizations from x to T. For each realization, at the decision point x'>x, given the state iss^., take action x^. according to

x,.=il,.K.) After

(6)

compiling the results of the n simulations, compute the expected

return ^ri^^^^^) \J{s^,x^) =

^ nsk ^ ^ ^-^^' ^^ \ and pseudo-utility ^ ^ (^^, x^) ^^ .^^ Q_{s^,x^)-XK{s^,x^)

3. Fit an approximate state-action pseudo-utility function U^ (•,) based on the m points U^(^^,x^),z = l , - - , w . 4. For each states^,z = \,"',m, find its optimal action x^{s^) via x^(^^) = argmaxU^(^^,x)

(7)

Fit an approximation optimal policy function n^(^) with the m points (^^,x^(5^)),z=l,-,m. In this algorithm, T-2 approximate functions for optimal actions are built from stage 2 to state T-1. The number of deterministic optimization performed in the form of Eq. (7) is O(Txm), which is independent of the number of states visited in the simulation. As a result, the curse-of-dimensionality is avoided. In our research, least squares support vector machine (LSSVM) (Wan et al., 2005) is adopted to build all the approximation functions.

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arrives at 0

1

1

P2 arrives;^tt2

r

~ ^

J

V

A competitor arrives

\

t Demand

t+1 1 ^

t

Produce & fill demand

Figure 1: Scheme of the case problem

3. Risk management in pharmaceutical capacity expansion When a pharmaceutical company expands its manufacturing capacity upon new drugs exiting its development pipeline, it may increase the capacity just enough to meet the forecasted demand or it may purchase more capacity for future drugs to reduce the setup cost. However, purchasing capacity for future drugs inevitably incurs risk: the capacity may not match the demands of the future drugs, extra-capacity will reduce the return of investment, while a capacity shortage will necessitate another purchase which incurs an additional undesirable setup cost. The uncertain exit time of future drugs may also make it cost effective to perform additional capacity expansion. Other important factors include competitors: there exists the possibility that competitors will enter the market in the future to take away part of the demand. The right capacity level can only be identified through solving a multi-stage risk management problem. 3.1. Case Study: Capacity expansion in a pharmaceutical company A pharmaceutical company A has a new drug (PI) exiting its development pipeline at the beginning of the horizon, and the initial available capacity is 0. The demand for the drug is stationary, following a normal distribution N (20, 9) in each period. The total horizon considered is 40 periods. Within the horizon and with a probability 0.5, a second new drug (P2) will exit the pipeline. The exit time follows a triangular distribution Tri (10, 20, and 30). The demand for the second drug is also stationary; with normal distribution in each period with mean N (20, 9) (i.e. the mean demand is uncertain) and the coefficient of variation the same as that of the first drug. Assume the second drug is similar to the first drug: they have the same production cost, market price, etc. A single competitor B exists whose product will share the demand of the first drug if it enters the market but does not affect the demand of the second drug. The arrival of B's product follows an exponential distribution with expected arrival time 45. If B enters the market, its product will take away normal distributed market share N (0.4, 0.01)fi-omA's first product. 3.2. Implementation of the optimal policy approximation algorithm As shown in Fig. 1, this case problem is a dynamic optimization problem with capacity decision at 0 and t2, and contingent production decisions at each period. In accordance with the proposed optimal policy approximation algorithm, the problem is approached as follows: sample the state-action space at t2, build the state-action pseudo-utility function and consequently the state-optimal action function (i.e. the policy) after simulating the sampled points; sample the state-action space at 0, simulate and build the corresponding state-action pseudo-utility function; obtain the optimal capacity decision at 0 by optimizing the corresponding state-action pseudo-utility function. The default number of sampled points is 40 for the first stage and 650 for the second stage; the

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630

15

20

30

Figure 2: NPV and semi-norm of the first stage capacity decision

0,004

0.008

0.012

640

650

35

0.016

0.02

risk aversion parameter

Figure 4: Optimal first-stage capacity decisions under different risk aversion parameters

Figure 3: NPV and semi-norm efficient frontier

0.008 0.012 risl< aversion parameter

0.016

Figure 5: The effect of the demand variance on first-stage capacity decisions

default number of sample path simulated for each sampled point is 4000. Those default numbers are chosen such that they provide satisfactory results for this case, and there is no significant improvement with larger values. The implementation of the revised back-propagation algorithm is similar to the above procedure except that the state-optimal action surrogate model is not constructed and a deterministic optimization problem is solved at t2 while simulating a sample path from 0 toT. S.3. Results and discussion For the non-separable risk measure semi-norm, Fig. 2 shows that the optimal decisions for NPV and semi-norm are 26 and 22 respectively when A is equal to 0.001, indicating

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that the optimal decision for pseudo-utility must lie between 26 and 22 to balance their trade-off. For other values of ^ , there will be similar relations between the first stage capacity and the NPVs as well as the semi-norms, from which the corresponding optimal decisions together with the NPVs and semi-norms under these decisions can be calculated. The results are presented as the efficient frontier in Fig. 3. This figure simply states that capacity expansion under dynamic conditions demonstrates the NPV and risk trade-off, well known in stock portfolio management: higher NPVs are necessarily associated with higher risks. Fig. 4 shows that the optimal first-stage capacity level decreases as the risk aversion parameter increases to avoid the risk of lower demand either due to possible arrival of the competitor or non-materization of expected fiiture drugs. Fig. 5 studies the effect of demand variance under different risk aversion parameters on the first-stage capacity level. Clearly, larger demand variances lead to higher capacity levels to avoid the cost of missing demand. 4. Conclusions Two algorithms are proposed for risk management in dynamic optimization based on multi-stage back-propagation scheme and function approximation. The algorithms are the first of their kind to be valid for arbitrary risk measures. Their effectiveness is illustrated by computing the NPV vs. risk efficient frontier for a dynamic capacity expansion case problem. References Bertsekas, D. P., Tsitsiklis, J. N., 1996. Neuro-dynamic programming. Athena Scientific. Cheng, L., Subrahmanian, E., Westerberg, A. W., 2003. Design and planning under uncertainty: issues on problem formulation and solution. Comp. Chem. Engng. 27, 781-801. Cheng, L., Subrahmanian, E., Westerberg, A., 2004a. A comparison of optimal control and stochastic programming from a formulation and computation perspective. Comp. Chem. Engng 29(1). Cheng, L., Subrahmanian, E., Westerberg, A., 2004b. Multi-objective decisions on capacity planning and production-inventory control under uncertainty. Ind. Eng. Chem. Res. 43, 21922208. Li, D., 1990. Multiple objectives and non-separability in stochastic dynamic programming. International Journal of System Science 21 (5), 933-950. Fu, M. C, 2002. Optimization for simulation: theory vs. practice. INFORMS Journal on Computing 14 (3), 192-215 Takriti, S., Ahmed, S., 2004. On robust optimization of two-stage systems. Mathematical Programming 99, 109-126. Wan, X., Pekny, J. F., Reklaitis, G.V., 2005. Simulation-based optimization with surrogate models—application to supply chain management. Comp. Chem. Engng 29 (6), 1317-1328


A decision support tool for process optimization of sulphur free diesel production Z. Lukszo^, M. Salverda,^P. Bosman^ ''Delft University of Technology,P.O. Box 5015, 2600 GA Delft, The Netherlands ^Shell NederlandRaffinaderij BV, Rotterdam, The Netherlands Abstract This work is a contribution to the Sulphur Free Diesel (SFD) production. The research presented in this paper is performed in one of the Dutch refineries, which is going to be one of the front producers of SFD from high sulphur crude. The challenge is to adapt the process operation to changes in the specification requirements without capital investments, i.e. to find optimal process settings which maximize the margin taking into account blend values, deactivation costs of the catalyst, shutdown margin losses, product specifications and quality margins and considering the organizational complexity, too. The decision support tool, called SFD optimizer, determines production settings on a weekly basis to decide on the weekly schedule by Economics and Planning Department and in case of disturbances. The NLP-optimization problem with two types of decision variables (desulphurization depths for three HDS and five quantities of components sent to product pools) and with 22 (non)-linear (in)-equality constraints is solved with generalized reduced gradient method. The optimal solution results in maximal SFD margin, i.e. the total blend value minus the deactivation costs. The SFD decision support system presented in this paper was proven to effectively assist decision makers at the Economics and Planning Department. Keywords: sulphur free diesel, process optimization, decision support tool 1. Introduction Since the seventies the European Union, together with the petroleum industry, aims to reduce the negative environmental impact of the use of hydrocarbon fuels [European Standard, 2004]. A part of this effort focuses on the reduction of sulphur in friels. In 2005 fiscal incentives are imposed to stimulate the production of sulphur free diesel (SFD), a diesel grade which contains at most ten sulphur parts per million. One of the Dutch refineries that are going to produce sulphur free diesel for the Dutch market is Oil & Co Refinery (OCR). To be capable to produce SFD some plants need to be driven to more extreme operational conditions [Torrisi, 2004]. Besides operational aspects the introduction of SFD also has some economic consequences. Revenues drop as many former diesel components cannot be used for the diesel production anymore and should be downgraded. Moreover, the operational costs go up as the costs of desulphurization rise. To cope with the critical operational conditions and minimise the loss of margin the refinery's Economics and Planning department would like a decision support tool that optimizes the diesel production. To be successfiil the process optimization should, besides concentrating on the optimization aspects, also take into account the organisational complexity and consider the chance of successfiil incorporation of the decision support tool. The decision support tool, called SFD optimizer, determines the production settings that maximise profit, including catalyst lifecycle economics in the decision-making process and assuring that at all times all product specifications are met.

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2. Sulphur Free Diesel production At present the European norm for sulphur in diesel corresponds to the ULSD specification, being 50 ppm, but in some European countries the SFD specification of 10 ppm is already operative. In the beginning of 2005 OCR has chosen to import SFD to satisfy these demands rather than producing it themselves. Currently the full diesel production capacity of OCR is used to produce SFD, as presented in Figure 1. Kerosene

HDS/4

JetA1 3000 ppm S

HDS/1

Light gas oil

HDS/2

J

SFD

RBLAGO

10 ppm S

blender HDS^

SBP Kero 2ppmS

HCU gas oil

IGO 2000 ppm S t'-

LO feed

Figure 1. Present diesel production with four Hydro Desulphuisers (HDS) For the SFD production three input flows are processed (see Figure 1), i.e. Kerosene (Kero), Light Gas Oil (LGO) and Hydro Cracking Unit Gas Oil (HCU gas oil) to make the following products: Jet Al: fiiel for airplanes Sulphur Free Diesel (SFD): light gas oil (LGO) is desulphurized by the HDS's. The desulphurization of kerosene has a 12-day cycle: during 10 days the, by HDS/4, desulphurized Kero (Des-Kero) goes to the Jet Al pool, the two following days the Kerosene is desulphurized to 7 ppm. This Des-Kero is stored in tanks. SBP (Special Biling Product) Kero: HDS/3 is, besides for the desulphurization of light gas oil, also utilised to desulphurize kerosene to 2 ppm. HDS/3 is alternately used for the production of diesel and SBP Kero. The number of days a month that HDS/3 is utilised for SBP Kero production varies, but is generally speaking smaller than the number of days this unit is utilised for the desulphurization of light gas oil. Low Olefins (LO) feed: depending on the value of HCU gas oil as feed for the Olefins plant, it can be profitable to run down the HCU gas oil to LO. Also in case the HCU gas oil cannot go to the diesel pool, HCU gas oil can be used as LO feed, rather than downgrading it to industrial gas oil. Industrial gas oil (IGO): the price of industrial gas oil is normally lower than the prices of all other products. That is the reason why downgrading to the IGO pool is something to be prevented.

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3. SFD Optimizer The site-wide diesel optimizer, from now on called the SFD optimizer, aims to support the operation of diesel production. It takes in blend values and properties of the diesel components and then, taking into account the deactivation costs and SFD specifications, it determines the production settings for the Des-Kero production (HDS/4), the desulphurization of light gas oil (HDS/1/2/3) and the blending (RBL AGO blender; RBL stands for Refinery Blending and Logistics). This optimizer is important for two reasons. Firstly, it can optimize the economic performance regarding the diesel production, as both costs of the diesel production, e.g. desulphurization, and the revenues are included in the decision-making. Secondly, the run length can be expanded if the level of desulphurization on the HDS's is adjusted to the situation, rather than a steady state operation. Below important optimizer aspects are mentioned. 3.1. Model goal The main goal of the optimization model is to support the decision-making concerning the diesel production by calculating the solution that maximises the economic performance (SFD optimizer margin) i.e. the total blend value of diesel components minus the costs related to the desulphurization of light gas oil. 3.2. Model requirements The main functional model requirements are functionality and reliability. Functionality is determined by the degree to which the model generates optimal and practical production settings (blending quantities and desired levels of desulphurization). The model is considered reliable if the experts involved in the sulphur free diesel production have confidence in the SFD optimizer and its outcome. The main non-functional requirements are user-friendliness and feasibility. 3.3. Level of aggregation The model has a high level of aggregation, as the overall OCR performance needs to be optimized, satisfying the cost-focus of the production units and the margin-focus Economics and Planning department. 3.4. Model scope The model scope is determined by which factors are included in the model as model variables and which factors are not included. The factors that are included are: Properties of diesel components (i.e. sulphur content, density and flash point) Product specifications (i.e. sulphur content, density and flash point) Desired level of desulphurization on HDS/1/2/3 Quality margins for sulphur content, density and flash point Blending quantities. Run lengths HDS catalysts SFD optimizer margin: total blend value minus deactivation costs Total blend value of diesel components Deactivation costs HDS/1/2/3: replacement costs plus loss of income Blend values of diesel components (kerosene, light gas oil and HCU gas oil) Below the reasoning for excluding some other factors is given: Deactivation: there is already a model in place that predicts the HDS catalyst run length. This model, mastered by the HDS technologist, extrapolates the previous degradation to estimate the remaining run length. HDS settings: the Economics and Planning Department determines the desired properties of the HDS effluent and communicates these to the production unit responsible for desulphurization of light gas oil, Refinery Treating and Conver-

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sion (RTC). To respect this operational freedom the decision support tool should indicate the desired HDS effluent properties, rather than the operational settings. 3.5. Time horizon Although market prices change continuously, the refinery apphes estimates on a weekly basis. Given that, the time horizon of the model is a week. 3.6. Desired output The model should provide the optimal solution for next week's SFD production. Moreover, it should be transparent how the solution is formed and the user should be supported to convert the optimal solution into a feasible one applicable in the real plant, e.g. rounding of the optimal solution. Hence the user can change the decision variables found by the optimizer. 4. Optimization problem This chapter contains the formulation of the optimization problem as described above. 4.1. Optimization criterion [Edgar, 2001] 5

^ {

C •

-\-R LGO MsFD Pi Cj

SFD optimizer margin (thousands of US dollar per week = k$/w) value of product stream i (US dollar per ton = $/ton) deactivation costs, i.e. replacement costs plus loss of income, for a change of the catalyst in HDS/j (k$) - RLj(Xj) run length of the HDS/j catalyst, if the coming week the desulphurization depth on that HDS is going to be x ppm (weeks) RLGO revenues of the light gas oil (LGO) stream (k$/w) 4.2. Decision variables Eight decision variables are identified: qi quantity of Kero to Jet Al pool (kt/w) q2 quantity of Des-Kero to SFD pool (kt/w) qs quantity of HCU gas oil to SFD pool (kt/w) q4 quantity ofHCU gas oil to LO feed (kt/w) qs quantity ofHCU gas oil to IGO pool (kt/w) Xi desulphurization depth on HDS/1 (sulphur parts per million = ppm S) X2 desulphurization depth on HDS/2 (ppm S) X3 desulphurization depth on HDS/3 (ppm S) The decision variables are constrained by upper and / or lower limits and system constraints as: SsFD 5 SsFD spec" ^s (Sulphur content of SFD is smaller than the SFD sulphur specification minus the quality margin for sulphur) psFD spec min + CJp < psFD < PsFD spec max" CJp (Dcusity of SFD is bctwcen the upper and lower SFD density specification taking into account the density quality margin). The sigma a represents the quality margin necessary to guarantee that the product is within specification at the gas station. This quality margin is determined by the inaccuracy of the measurement equipment and the chance of the contamination after production. FPsFD > FPsFD spec + c^FP (Flash poiut of SFD is higher than the SFD flash point specification plus the quality margin for flash point)

Decision Support Tool for Process Optimization

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The type of the optimization problem is determined by the following: The criterion is a nonlinear continuous function with eight decision variables Fourteen linear constraints on decision variables (lower and upper limits) Four nonlinear equality and four nonlinear inequality constraints. The resulting Non-Linear Programming (NLP)-optimization problem with two types of decision variables (desulphurization depths for three HDS and five quantities of components sent to product pools) and with 22 (non)-linear (in)-equality constraints is solved with generalized reduced gradient method [Floudas, 1995]. The optimal solution results in maximal SFD margin, i.e. the total blend value minus the deactivation costs. 5. Decision support by SFD optimizer To be successful the decision support tool should generate additional margin and improve the decision-making process of the Economics and Planning department. To estimate the monetary gains of the SFD optimizer a reference scenario is used without the SFD. In this case the HDS's would desulphurize in a steady state, to a sulphur content of 7 ppm. The production of Des-Kero would be characterised by a 12-day cycle. Ten days the HDS/4 would be utilised for Jet Al production and two days for the desulphurization of kerosene for the diesel pool. Given the maximum throughput of HDS/4 the weekly available quantity of Des-Kero that can be blend into the diesel pool would be X kton. The SFD optimizer does not consider these two factors fixed but rather flexible depending on economics and component properties. Scenario

HDS depth HCU gas oil in Des-Kero in (ppm) SFD (kt/w) SFD (kt/w) Max X 5ppmS Ref 7 HCU gas oil Opt. Max 9,6 X+1 Max Ref 7 X 10 ppm S HCU gas oil Opt. Max 8,3 x+1 17 ppm S X-0,4 Ref. 7 Max-3 HCU gas oil Opt. Max X+1 6,6 Max-9,4 33 ppm S Ref 7 X-3,7 HCU gas oil Opt. 5 Max-5,1 X-1,5 50 ppm S 7 X-4,4 Ref Max-10,8 HCU gas oil Opt. 5 Max-8,2 X-3,1 Table 1. Optimized advice for the HDS depth for five scenarios

Delta margin (k$/w) 0 117 0 105 , 0 233 0 329 0 180

1892

Z Lukszo et al

To obtain a better idea of the potential gain of the SFD optimizer the reference case above, i.e. fixed HDS depths and Des-Kero production, is compared with the advice generated by the SFD optimizer. Five scenarios were run with ascending levels of sulphur content in the HCU GO, see Table 1. On the basis of these scenarios, we can conclude that the SFD optimizer maximises the margin, which is the total blend value of the diesel components minus the deactivation costs of the desulphurization catalysts. The decision support tool is going to be used by the refinery's Economics and Planning department once a week for the formation of the week schedule and in case of disruptions in the diesel production process. Taking the average over the four year run length of the HCU catalyst, the estimation of annual gain is approximately 9 million dollars a year. 6. Final remarks Conclusions regarding the outcome of the SFD optimizer are: Desulphurization in HDS/3 is more expensive than in HDS/2; and desulphurization in HDS/2 is more expensive than in HDS/1. Decreasing the quality margin on sulphur leads to significant gains. Additional margin due to the SFD optimizer varies between 100 and 300 thousand $/w, depending on diesel properties and the desulphurization depth. The SFD decision support system was proven to effectively assist decision makers at the Economics and Planning Department. It was not only proven to be effective in decision support on process settings and the schedule, but also to be effective in improving conmiunication between the relevant company departments. It should be mentioned, that the level of communication and cooperation between the production units and the Economics and Planning Department become more important. The SFD optimizer leads to less steady state operations, smaller margins for compensations for process disruptions, shared responsibility for the diesel quality, and potential conflicts in the HDS utilisation. This imposes a communicational challenge. The clarity of the graphical presentation and the quantitative results of the optimization model and its user friendly interface evidently contributed to breaking down information and communication barriers between the involved production units, planning and logistics departments. Moreover, the refinery realized that not only the actor performance itself but also the way this performance is achieved should be considered. A kind of flexibility is built in the performance assessment system that takes into consideration the sacrifices some production units make to maximise the site-wide performance. At the moment the implementation phase of the decision support system in the refinery has started. It should be stressed, that although the SFD optimizer is developed for a specific refinery, the approach aimed at the formulation of the optimization problem can be applied to a wide variety of production processes for fuels. References European Standard EN 590:2004, Automotive feuels. Diesel-requirements and test methods. Technical Committee CEN/TC 19, Petroleum products, lubricants and related products, 2004 Torrisi, S., M.P. Gunter, Keyftmdamentalsof ultra-low sulphur diesel production: The four C's, Cataly Catalysts & Technologies, 2004 T.F.Edgar, D. M. Himmelblau, L.Lasdon, Optimization of Chemical Processes, McGraw Hill, 2001 Floudas, C.A., Nonlinear and mixed-integer optimization: fundamentals and applications, Oxford University Press, 1995


1893

An Attainable Region Approach for Effective Production Planning Charles Sung, Christos T. Maravelias Department of Chemical and Biological Engineering, University of Wisconsin-Madison, 1415 Engineering Drive, Madison, WI53706, USA Abstract A novel approach for the solution of production planning problems is presented. A detailed scheduling model is analyzed off-line to obtain a convex approximation of the feasible production targets, i.e. the production attainable region (PAR) of each manufacturing facility. The PAR is expressed via simple inequalities that involve only planning variables, lending itself to effective integration with production planning formulations. Moreover, PAR contains all the relevant scheduling information necessary to solve the planning problem with high quality. Keywords: Production Planning, Scheduling, Production Attainable Region. 1. Introduction To remain healthy in today s competitive global environment, chemical firms must have an integrated view of all their operations and use advanced modeling and optimization methods to achieve enterprise-wide optimal solutions. At the tactical and the operational levels, decisions are integrated and should be simultaneously optimized due to interconnections between the various nodes of the supply chain (SC) and the interdependence of the decisions at the various planning levels and geographical locations. In this paper we develop a novel approach for the integration of medium-term (3-6 months) production planning with short-term (1-2 weeks) scheduling. In the production planning problem, we seek to satisfy the (stochastic) demand at the customer-facing nodes of the SC (Fig. 1) at minimum cost. The SC is usually represented as a time-extended network with nodes i,JEN, arcs (i,j)eA, products keK and time periods t ET, and the production planning problem is solved as a multi-period min-cost network flow problem. The decision variables include the inventory levels (likt) and the shipments between nodes (F^,^). Sources, sinks and intermediate nodes are modeled via constraints for supply availability (eq. A), demand satisfaction (eq. E), and material conservation (eq. D), respectively. Since planning problems result in complex multi-period, multi-product (stochastic) mixed-integer programming (MIP) formulations, manufacturing facilities are often simplified via aggregate capacity (eq. B) and material conversion (eq. C) constraints. This simplified representation, however, does not capture the complexities of chemical process networks, leading to infeasible or suboptimal production targets. Several researchers have developed integrated planning-scheduling schemes, where a detailed scheduling model provides information for the planning decisions (Kallrath, 2002; Shah, 2005). The proposed integrated planning-scheduling schemes, however, are hard to solve and used only for problems with short planning horizons, few manufacturing facilities, and simple production recipes. In this paper we develop an approach that allows us to effectively solve large planning problems without compromising the quality of the solution.

C Sung and C. T. Maravelias

1894

Supply availability: Z^*'-•'^* v/es,v/t,v/ (A)

Suppliers

Manufacturing facilities: ZZ^«^*^ Infeasible path

Fig. 3. The conceptual structure of the PSS. [La'zaro et al., 2005;Shen et al., 2005] are performed in matlab and envisioned to users via ArcGIS data visualization for evaluating the supply chains for the decision-making.

5. System Users and Application When designing any planning support system one should consider vs^ho will use the system and the types of functionality the system will provide. In following sections summarize these two important issues. 5.1. User classification The target users of the system include governmental policy designers of energy systems in the national level, individuals related to eco-management (environmental protection, environmental accounting and reporting) at the enterprise and governmental level, and university and government researchers. 5.2. Functionality The proposed PSS is designed to provide the following functions • GIS data visualization methods to identify the geographic distribution of the economically exploited biomass potential; • data analysis methods (e.g. Fuzzy C-means clustering methods and decision trees) for the assessment of biomass potential as theoretical, available and economically exploitable respectively; • the determination of the economical biomass collection points, storage points and bioenergy conversion plants positions; and • simulation methods based on cost modeling, thereby providing optimal decision-making for bioenergy generation at the national and local levels. 6. Case Study: Forestry Residues Exploitation in Japan Planning for forestry residues supply chains for power generation in Japan has different characteristics than in other countries e.g. European countries. These differences originate from the nature of the land, forest residues density, forestry population and weather. For example, Japanese weather is mostly wet and the rain occur at any time, which creates a need for covered material storage. Also, the mountainous topography prevents high capacity equipments from being employed. Limited access distances and

A Planning Support System for Biomass-Based Power Generation

1903

low-density forests results in difficulties in estimating the feasible amount of forestry residues that used as a source of bioenergy. In this case study the Japanese Geographical Information System Data Base (GISDB) for forestry residues is applied to demonstrate that the PSS is efficient in planning for bioenergy production. With the support of PSS, feasible amount, and environmental, economical and social impacts of forestry residues are estimated as follows: 1- The resource quantity in tons, area in square meters and centroid in Cartesian coordinates of each polygon, which represents the city where the resource is located, were determined using the ArcGIS 9 software [ESRJ]. 2- Using Matlab® Software, Version 7.0.1 with service pack 1 [The MathWorks], the (x, y) coordinates in meter of the collection points were generated randomly around the centroid of the polygon, assuming the area of each polygon is a circle. 3- The system asks the user to define; (a) the average access distance inside the forest from the road in the country, (b) total requirements fiôm electricity in the country, (c) efficiencies and expected number of power plants in the country, (d) maximum traveling distance between the storage location and the collection points and (e) the maximum allowable distance between the storage and the power plant. Default values are provided for system users who do not have sufficient information for the required inputs. 4- Once the inputs are defined, the system omits all collection points located at distances greater than the access distance. This resulted in the feasible number of collection points. Employing the resulting feasible collection points locations, the system calculates the suitable number of clusters (storage houses) based on the Validity Index (VI) [Shen et al., 2005]. The VI is defined as the ratio between the intra (average distance between the storage points and the collection points) and the inter (minimum distance between the storage houses). 5- A table of clusters (storage house) numbers versus the VI is automatically generated by the system and displayed for the users thus allowing them to select the desired clusters number or to assist the system in automatically defining an optimum number of clusters. 6- The collection points are then clustered to the pre-defined number of clusters in step 5 using fuzzy C-means clustering method in a fuzzy toolbox provided in Matlab Software.

Fig. 4. Economic locations (cities) of plant installation defined by PSS.

1904 7-

8-

N.Ayoubetal The storage locations are mapped to Japanese map and visualized for the user to define the locations of the desired power plants or help the system to locate them via clustering the storage points. Once the power plants locations are defined, the system start calculating all information about costs, emissions, number of labors required for establishing the business and total investments required for realizing the biomass-based electricity. All calculated information then included in the power plants layer in the ArcGIS9 where the data attributes can be visualized. Fig. 4 shows the power plant layer viewing the city names where power plants of forestry residues conversion should be located in the Japanese case.

7. Conclusions and Future W o r k This paper proposes an explanation of the concepts of the Planning Support System for Power Generation. The PSS helps in solving the real biomass planning problems at the national level. The authors are currently developing system support for local level planners and executers. At that level a more robust model is required which can support detailed information about biomass types, properties and technical data evaluating the most suitable technology for the local resources. This work also is going to be expanded to a Web-based Decision Support System called general Bioenergy Decision System (gBEDS) supporting the maintenance of the system's technological information base. This paper is restricted to only the procedure for the system application. A more complete research description and associated results will be published as a journal paper. Acknowledgments The authors wish to thank the Ministry of Education, Culture, Sports, Science and Technology of Japan for the financial support of this work and the Central Research Institute of Electric Power Industry in Japan for providing the biomass GIS database. References A. K. Akella, M. P. Sharma, R. P. Saini, 2005, Optimum utilization of renewable energy sources in a remote area, Renewable and Sustainable Energy Reviews, In Press, 1-15. ESRI GIS and Mapping Software, www.esri.com. D. O. Hall, J. I.Scrase, 1998, Will biomass be the environmentally friendly fuel of the future?, Biomass and Bioenergy, 15, 6, 451-456. L. C. Kuiper, S. R. Sikkema, 1998, Establishment needs for short rotation forestry in the EU to meet the goals of the commission's white paper on renewable energy, Biomass and Bioenergy, 15,4-5,367-375. J. La'zaro, J. Arias, J. L. Marti'n, C. Cuadrado, A. Astarloa, 2005, Implementation of a modified Fuzzy C-Means clustering algorithm for real-time applications. Microprocessors and Microsystems, 29, 8-9, 375-380. C. P. Mitchell, 2000, Development of decision support system for bioenergy applications, Biomass and Bioenergy, 18, 4, 265-278. Y. Naka, M. Hirao, Y. Shimizu, M. Muraki and Y. kondo, 2000, Technological information infrastructure for product life cycle engineering, Proc. of 7th International Symposium on Process Systems Engineering, Keystone, Colorado, USA, 665-670. L. Pari, 2001, Energy production from biomass: the case of Italy, Renewable Energy, 22,1, 21-30. J. Shen, S. I. Chang, E. S. Lee, Y. Deng, S. J. Brown, 2005, Determination of cluster number in clustering microarray data. Applied Mathematics and Computation, 169, 2, 1172-1185. The MathWorks Inc., www.mathworks.com I. Ushiyma, 1999,Renewable energy in Japan, Renewable Energy, 16, \-A, 1174-1179.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 PubHshed by Elsevier B.V.

Semantic Analysis for Identification of Portfolio of R&D projects. Example of Microencapsulation Andrzej Kraslawski Lappeenranta University of Technology, Department of Chemical Technology, P.O. Box 20 53851 Lappeenranta, Finland [email protected] Abstract The paper introduces a method for building portfoUo of R&D projects. The proposed method consists in comparison of the technological problems addressed in the patents with those referred in the papers. The problems are identified using the semantic analysis for the determination of the structures subject-action-object. As an illustration, there is given an example of the identification of problems related to research and application studies of microencapsulation. Keywords: R&D management, technology forecast, semantic analysis, microencapsulation Introduction Answering the questions about future technology developments supports the decision makers in administration and business communities with rationale to allocate resources in order to better address emerging social, environmental and economic issues. The substitutive resources, directly connected to capital, like equipment or information repositories are accompanied by the non-substitutive ones like highly qualified research and technical staff. The wrong allocation of any resources is always a loss but a misuse of non-substitutive resources is especially painfull as top-class specialists are usually in a very short supply. In consequence, the wrong decisions lead to the lost of time which makes the gap between the needs and the technical possibilities of their satisfying to grow very fast. Therefore the systematic methods of technology forecast are of paramount importance for the decision makers. One of the major goals of the technological forecast is to build a portfolio of R&D projects. It means to identify a group of projects which should be funded in a given period of time. There are used various methods of portfolio creation but usually there are three objectives to be fulfilled by an optimal portfolio of R&D projects, effectiveness: it means to identify the projects contributing to reahsation of the strategic goals of the organisation efficiency: it is to identify a group of the projects able to ensure, with the highest degree of probability, the fulfilment of the performance measures like shareholders value, long-term profitabilify, return-on-investment etc. diversification: it is to identify the group of the projects for which there is a balance ensured between the possible gains and losses. When analysing those objectives, there is clearly visible a lack of method for the identification of the most promising technologies in terms of their innovative potential.

1905

1906

A. Kraslawski

The objective of this paper is to propose a method for the support in building portfolio of R&D projects basing on the analysis of the scientific publications and patents. Information analysis There is a broad spectrum of technology forecast methods. For example a group of authors (Technology Futures, 2004) list 50 methods divided into nine classes. The most common methods used are: analysis of citation of papers, patents, web pages etc., visualisation of the knowledge domain, statistics-based analysis of the texts (words occurrence), scenarios and Delphi method. They have serious drawbacks strongly limiting their usefuhiess. For example, the citation and co-citation is strongly criticised as it is not related to the context of the document, e.g. some very often cited references are given as an example of wrong or erroneous interpretation of data. It is a generally accepted conclusion that there is actually a lack of broadly acknowledged forecasting method. Moreover, the growing complexity of technologies and their interactions with the social and natural environment lequires the development of new methods for forecasting of technological developments. The characteristic features of the contemporary research are high complexity of the studied issues, their interdisciplinary character and the overwhelming amount of the generated information. The high complexity is related to the high cost of research in financial and non-financial terms and hence a must of a very carefiil selection of research subjects. The interdisciplinarity requires the analysis of the reference materials which is based on the information context and not only on the mechanical application of the citation indexes or occurrence of the words as the different fields use their specific vocabularies to represent the identical entities. The amount of information forces the researchers to look for new methods of their analysis. As a result the methods able to handle three above mentioned issues are of great interest to decision makers. The problems of the subjectsD complexity, their interdisciplinary character and amount of the available information make the task of the automatic text analysis an important research issue. The orthographic, semantic, statistical, syntactic and usage analysis are the basic methods used to handle this issue, (Losiewicz et al. 2000). An approach, based on the semantic analysis of the text, leading to the identification of the structure subject-action-object has been applied in this work. Identification of portfolio of R&D projects The proposed method for identification of the development trends of given technology, and finally the proposed portfolio of R&D projects, consists in comparison of the problems addresses in the patents with those referred in the papers dealing with the technology under consideration. The research problems encountered exclusively in the papers are subject of actual great scientific interest but do not have any practical application In this paper they are referred as set A problems. The problems existing only in the patents seem to be less interesting for the research. They are named as set B problems. The problems common for the patents and scientific papers are specified as set C problems. The discovery of the issues common for the patents and the papers presents the areas which are extensively studied scientifically and have the great practical interest. The set B problems are usually the issues of the minor research interestfi-omthe point of view of planning the allocation of R&D resources as they are right now practically solved The start of the research for solving of the set C problems requires a considerable analysis in order to position own skills and resources. The wrong decision of starting such research

Semantic Analysis for Identification of Portfolio of R&D Projects

1907

in the organisation without the sufficient experience and resources would lead to the situation called "me-too strategy". It means the satisfactory level of expertise and interesting results could be obtained too late. The competitors would much earlier successfully address the problems and patent the methods of their solving. The set A problems are generally the issues to be addressed. However, a careful analysis is required to study the cost, required duration of the research and eventual applicability of the obtained results. The presented method is a preliminary step in the determination of the portfolio of R&D projects. It is composed of the following steps: 1. Determination of the set PP of problems in patent literature 2. Determination of the set RP of problems in the scientific literature 3. Identification of the sets A, B and C by the comparison of sets PP and RP 4. Discovery of the frequency of occurrence of the problems in sets A, B, C 5. Identification of the Hst of the interesting R&D subjects and the inventory of the subject which should not be further investigated The determination of the sets PP and RP is realised by use of semantic analysis of the texts aimed at the determination of the structures subject-action-object. The identification of the sets A, B, C is realised by the analysis of PP and RP sets using the strings conârison and filters for the identification of the synonyms. The identification of the frequency of the problems occurrence is done using the standard tools used in the databases (e.g. ACCESS). The most commonly encountered problems are used for the automatic generation of the list of problems. The rarest problems are analysed individually by the experts. The lists of the problems for the detailed consideration are next visualised (Pasteur's quadrant, Fig. 1) in order to facilitate the selection process. Analysis Subject-Action-Object The structure subject-action-object is a universal template of all sentences of any natural language. The use of the specialised knowledge bases allows to organize the concepts into a problem-solution relationships. The semantic analysis of the text leading to the determination of the structure subject-action-object (SAO), and finally to the identification of the pair problem-solution (PS), is based on the method presented in the set of the patents and patent applications (e.g. Tsourikov et al.2000). The coinputer implementation of the method has been realised as the program Knowledgist ^ . Its concept has been presented in several patents (e.g. Batchilo et al. 2003). The outline of the analysis performed by Knowledgist is presented in Fig. 2. The linguistic knowledge base is composed of the dictionaries, classifiers, and database for linguistic models recognition (text-to- word splitting, rule for determination of cause-effect relationship). application inspired I 1 yes 1 curiosity inspired no

pure bas basic research Set A

use-inspired research

SetC

I

purely applied research

SetB

1

Fig.l Pasteur's Quadrant. The pure basic research is called Bohr's approach; purely applied research- Edison approach and use-inspired R&D is called Pasteur's method.

1908

A. Kraslawski

The knowledge base is used to perform the following operations: Preformatting- the document is divided into small parts for the purpose of analysis. It is split into the sentences and words. linguistic analysis - consists in tagging, parsing and extraction of knowledge bits, corresponding to objects, facts and rules of the knowledge domain. sentence weighting - used to quantitatively evaluate importance of information contained in each sentence of the analyzed document. summary generation is used to produce the digest in the form of a list of keywords, topics, or SAOs Linguistic Knowledge Base

/ Preformatting

Linguistic analysis

\ Sentence weighting

Summary generation

Error detection and correction

Part -of -speech tagging

Statistical weighting

Keyword summary

Document to word splitting

Parsing

Cue weighting

Document to sentence splitting

Semantic recognition

Cause-effect weighting

Topic-oriented summary

Document fields recognition

Syntactic recognition

SAO summary

Fig. 2. The structure of Knowledgist TM Example The determination of the portfolio of R&D projects in the field of microencapsulation has been studied in this example. An introduction to mircoencapsulation techniques and applications has been given by Gouin (2004). The research of patents addressing any aspect of microencapsulation has been performed on US Patent and Trademark Office Full Text Database. The search performed with Knowledgist ^^ resulted in 2606 structures subject-action-object identified in 95 patents. The pairs action-object are treated as problem while subject corresponds to the solution. The identified 2606 pairs problem-solution constitute the PP set. The screen of Knowledgist^^ presenting the results of search in the patent database is given in Fig. 3. The analogical search in the Elsevier database ScienceDirect of the scientific papers, years 1995-2005, have resulted in 1656 papers. The 5529 structures problem-solution have been identified. They constitute the RP set. The comparison of the both sets has resulted in the identification of the sets A, B. and C. The problems of pure basic research (classified here as mentioned only in the papers) are given as a set A They are related to the following subjects: production of chitosan microspheres, treatment of heavy metal contaminated sites, application of microcapsules to copper plating, studies on toxicity related to use of microencapsulated materials, improvement of loading capacity of microcapsules, magnetic properties of microcapsules, bioadhesion, stability of encapsulated pigments, pulmonary delivery, immobilized ruthenium catalysts, photoluminescence of microcapsules, thermal stability, application to wound dressing, microcapsules for the cells entrapment studies of swelling stress, behaviour in electric field and use of ultrasonic atomization for the production of micro capsules. The given-above list is not

Semantic Analysis for Identification of Portfolio of R&D Projects

1909

here due to the lack of space. The problems of the purely applied research (classified here as mentioned only in the patents) are given as a set B. They are related to the following subjects: encapsulation of explosives, anesthetics and cements and use of light for hardening of microcapsules. The problems common for the research papers and patents are given as the set C. They are visualised and specified in Fig 4. The problems to be considered for R&D portfolio are given as the elements of the set A and possibly C. The problems of the set B should not be considered for the further research. m^\m\y\

43

T(«ics

Start

SditHons

SAO

mechahicaT strength of '[ j o r e material j '

cz ^ Jfj j*j ^ ip JQ ^ ^ ^ ^ ^ J3 JS jg _*] Sj ^ j3 0 1*3 ^

allow tot production of high potency vitamins present in alter • surface charge or zeta potential of microcapsules - anionic , cat or or amr.hntpr'C ^-omnoNr alter • surface of microcapsules • treatment apply • compositions • standard methods apply - effective amount of dispersion • harmful fungi apply - effective amount of dispersion - method of controling pests apply • liners - many other devices apply • microcapsules • method apply - SAS technique • methods arise from - use of organic solvents - problems assay • amount of protein • Coomassie assemble - said ingredients and cooking or baking - flavoring material assess - permeability • efflux method assess • photoprotection • Comparison of methods assist - fluids - interface assist in - formation of microcapsules - instability attain • controlled/sustained release of permeant • microcapsules attempt - present inventors • combining of preferred platinum catalyst compositions composition attempt - present inventors - emulsifying of preferred platinum catalyst composilions resultant composition attempt • present inventors - evaporating of preferred platinum catalyst compositions water immiscible liquid attract - additional potential customers - increasing its potential

^ ]*] ^ ^ »1

avoid - aggregation of microcapsules - certain precautions become - core - size of solid substance become - core substance - average particle diameter of said solid substance belong to - class of autonomous self-copying chemicals • field of papers belong to - group - at least of chemical functions

^spPlilL.

Concept: mechanical strength of core material - augment strength of wall Ttiese load bearing microcapsules are non-rupturable during storage, tr^niportation, and handling of CB sheets coated thereon vîth the load bearing microcapsules due to the mechanical strength of the core material augmenting the strength of the wall. S u s Patent 5.002.924 Carbonless copy paper coating containing microencapsulated load bearers

HE f»ag»1of3~

O

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