Power System Dynamics and Stability
PETER W. SAUER M. A. PAl Department of Electrical and Computer Engineering Univers...
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Power System Dynamics and Stability
PETER W. SAUER M. A. PAl Department of Electrical and Computer Engineering University of Illinois at UrbanaChampaign
Prentice Hall Upper Saddle River, New Jersey 07458
Library of Congress Catalogi ngi nPu blication Daw Sauer, Peler W. Power system dynamics and stabi lity / PeTer W. Sauer and M. A. Pai. p. em . Incl udes bibliographical references and index.
ISBN 01J, + 2;)
(3.78)
(3.73)
(w,t +0, _
U sing the transformation (3.13),
(P
)
(3.79)
0,)
(3 .80)
hVs VBABC) . VBDQ sm 2"0shaft  wst  0,
Vd
=
(
Vq
=
(h~~:BC )
cos (
~ 0shaft 
w,t 
(3.81)
Vo = 0
(PO
. Id = ( hIsIBABC) I Sill 2 shaft  wet BDQ (
(P
'" )
'1',
hI,IBABC) cos Oshaft  w,t  ¢, IBDQ 2
)
(382) (3.83) (3.84)
By the definitions of VB ABC, VBDQ, IBABC, and I BDQ ,
,l2v,vBABC _ v: VBDQ "
hI,IBABC IBDQ
= Is
(3.85)
Using the definition of 0 from (3.39),
Vd= Vs sin (00,)
(3 .86 )
3.4. THE LINEAR MAGNETIC CIRCUIT
35
Vq = V, cos (6  0,)
(3 .87)
I d = I, sin (6  ¢,)
(3.88)
Iq = I, cos (6¢,)
(3.89 )
These algebraic equations can be written as complex equations
+ jVq)ej(5~/2) (Id + jlq)ej(5~ / 2)
(Vd
= V,e j6 ,
(3.90)
= I, e;¢'
(3.91)
These are recognized as the p erunit RMS phasors of (3. 73 ) and (3.76). It is also imp ortant, at t his point , to note that the m odel of (3.64) (3.72) was derived using essentially four general assumptions. These assumptions are summarized as follows. l. Stator has three coils in a balanced symmetrical conflguration centered
120 electrical degrees apart. 2. Rotor has four coils in a balanced symmetrical configuration located in pairs 90 electrical degrees apart. 3. The relationship between the flux linkages and currents m ust reflect a conservative coupling field . 4. The relationships between the flux linkages and curr ents must b e independent of 0shaft when expressed in the dqo coordinat e system. The following section s give t he flux linkage/current relationships t hat satisfy these fOU I assumpt ions and thus complete t he dynamic model.
3.4
The Linear Magnetic Circuit
This section p resents the special case in which the machine flux linkages are assumed to be linear functions of currents:
+ L,,(OshaftJirotor L,,( Oshaft)iabe + LTT(Oshaft)irotor
.Aabe = L.. (Oshaft )i abc .Arotor =
(3.92 ) (3.93)
where
(3 .94)
CHAPTER 3. SYNCHRONOUS MACHINE MODELING
36
If space harmonics are neglected, the entries of these inductance matrices can be written in a form that satisfies assumptions (3) and (4) of the last section. Reference [20] discusses t his formulation and gives the following standard first approximation of the inductances for a Ppole machine. L'.
L,,(Bshaft) = Ll,
+ LA 
LB cos PB shaft
!LA  LB cos(PBshaft _ ';) [
+ 2;) LB cos(PBshaft + 2;)
 tLA  LB cos(PBshaft _ ';) Ll,
+ LA 
j
~LA  LB cos PBshaft Ll, + LA  LB cos(PBshaft _ 2;)
L .. (Bshaft) =
+ ';)
~LA  LB cos PBshaft
~LA  LB cos(PBshaft
~LA 
LB cos(PBshaft
(395)
L~,(Oshaft) ~ L,'d sin tBshaft
L'l dsin(~Bshaft  23")
Lddsin(~Bshaft
+ ';)
L,1q cos tOshaft
°
L dq cos ( ~ shaft  ';) L dq cos( tOshaft + ;.".)
L'.
Lrr(Oshaft) =
(3.96)
Lfdfd
L fdld
0
0
Lfdld 0
L'd'd 0
0
0
L 'qlq
L'q'q
0
0
L 'q2q
L'q'q
(3.97)
The rotor selfinductance matrix Lrr( 0shaft) is independent of Bshaft. Using the transformation of (3.10) , Ad
=
(Ll
X = 2" 
v ... \ mq
(3119 )
3.4. THE LINEAR MAGNETIC CIRCUIT
39
Similarily, we also define (3.120) (3 .121) The resulting scaled
'I/J 
I relationship is
+ X=d1fd + X=dl,d X=d( 1d) + Xfd1fd + CdX=d!,d Xmd( 1d) + cdX=d1fd + X,d!,d
(3.122)
+ Xmq1'q + X=qh q Xmq( 1q) + X'q!,q + cqXmqhq Xmq( 1q) + CqXmq!,q + X 2qh q
(3.125)
'l/Jd = Xd( ld) 'l/Jfd = 'I/J'd =
(3.123) (3.124)
and
'l/Jq = Xq( 1q)
'I/J'q
=
'l/J2q =
(3.126) (3.127)
and (3.128) While several examples [30] have shown that the terms Cd and cq are important in some simulations, it is customary to make the following simplification [20]: (3.129) This assumption makes all of the offdiagonal entries of the decoupled inductance matrices equal. An alternative way to obtain the same structure
without the above simplification would require a different choice of scaling and different definitions ofleakage reactances [30]: Using the previously defined parameters and the simplification (3.129), it is common to define the following parameters [20] (3.130)
(3.131 )
CHAPTER 3. SYNCHRONOUS MACHINE MODELING
40
x; X"q
e:. e:.
, e:.
T'qu e:.
" Tqo
1
+ I Xm.d
Tdo =
" Tdo
Xl,
e:.
Xl,
+
l 1d
1
x ,mq
(3.133)
+ xu:; ' + xa; '
Xfd
(3.134)
w,Rfd X ,q
(3 .135)
W~Rlq
1 W,R'd
e:.
(3.132)
+~ ' + X'
1 WsR2q
( Xlld
+ _, X.rnd
( Xl 2q
1
_1 )
+X
ljd
+ , + 1 I ) xtnq
(3.136 )
(3.137)
x llq
and the following variables E'q
Efd
e:.
Xmd ..p   fd Xfd
(3.138)
e:.
X mdv   fd Rfd
(3. 139)
, e:. Ed = _ Xmq ..p,q X 'q
(3.140)
Adkins [23] and several earlier refer ences define X: as in (3.131) and T~~ as in (3. 135 ). This practice is based on the convention that single primes refer to the socalled "transient" p eriod, while the double primes refer to the supposedly faster "subtransient" period . Thus , when a single damper winding is modeled on the rotor , this is interpreted as a "subtransient" effect and denoted as such with a double prime. Some published models use the notation of Yo ung [35], which uses an E~ definition that is the same as (3. 140) but with a positive sign . In several publications, the terminology Xfd is used to define leakage reactance r a.t her than selfreactance. The symbols Xad and Xaq are common alternatives for the magnetizing reactances Xmd and X mq. The dynamic model can contain at most only seven of the fourteen flux linkages and currents as independent state variables. The natural form of
3.4. THE LINEAR MAGNETIC CIRCUIT
41
the state equations invites the elimination of currents by solving (3.122)(3.128) . Since the terminal constraints have not yet been specified, it is unwise to eliminate la, I q , or 10 at this time. Since the terminal constraints do not affect I fd , lId, Ilq, Iz q, these currents can be eliminated from the dynamic model now . This is done by rearranging (3.122)(3.128) using the newly defined variables and parameters to obtain 7/1d =
X"I 
d d
(X;  Xl,) E' (X~  X;) 01. + (X~ _ Xl,) q + (X~ _ Xl,) 'l'ld
1, Ifd = [Eq X=d X~ 
hd =
+ (Xd 
X;
(X~ _ Xl,»
, Xd)(Id  hd)]
(3.142)
I
[7/1ld
+ (Xd 
(3.141)
I
XI,)Id  Eq]
(3.143)
and
(3.144) 1,  X [Ed =q
+ (Xq 
Xq, Xq" (X' _ X )' [7/1'q q
l,
, Xq)(Iq Izq)] t
I
+ (Xq 
(3.145)
Xl,)Iq
+ Ed]
(3.146)
and
(3.147) Substitution into (3.64)(3.72) gives the dynamic model for a linear magnetic circuit with the terminal constraints (relationship between Vd, IdJ Vcp 1q) VOl 10 ) not yet specified. Since these terminal constraints are not specified, it is necessary to keep the three flux linkage/current algebraic equations involving Id, I q, and 10 , In addition, the variables Efd, TM, and TFW are also as yet unspecified. It would be reasonable, if desired at this point, to lllii.ke Eid alld TM constant inputs, and TFW equal to zero. We will continue to carry them along as variables . With these clarifications, the linear magnetic circuit model is shown in the following boxed set.
CHAPTER 3. SYNCHRONOUS MACHINE MODELING
42
(3. 148) (3.149) 1 d,po d W,
I
t
= R,Io + Vo
dE~
(3.150)
I
d'l/J2q
[
q
+(X~  X t .)lq II
X; )2(,p2q
X~ Xq) 1q  (X' _ X
+ (Xq 
I
Tqod t =  Ed
t,
+ E~)l
(3. 153 )
I
I
Tq0dt = ,p2q  Ed  (Xq  Xt ,)Iq
do =w dt 2H dw
(3.154)
w '
;;;dt = TM ,
(3.155)
(,pd1q
,pqId )


TFW
" (X;  Xt,) , ,pd =  Xd 1d + (X~ _ Xl,) Eq
(3.156) (X~  X;) Xis) ,pld (3.157)
+ (X~ _
(X'.  X") (X'''q X) " l. , q ,pq = Xq 1q  (X' _ X ) Ed + (X ' _ X ) ,p2. (3.158) q
,po
= Xl,Io
b
q
b
(3.159)
Although there are time constants that appear on all of the flux linkage derivatives, the righthand sides contain flux linkages multiplied by constants. Furthermore, the addition of the terminal constraints could add more terms when 1d, I q, and 10 are eliminated. Thus, the time constants shown are not true time constants in the traditional sense, where the respec
3.5. THE NONLINEAR MAGNETIC CIRCUIT
43
tive states appear on the righthand side multiplied only by 1. It is also possible to define a mechanical time constant T, as
T, _J2Hw,
(3. 160)
w, =tl T,(w  w,)
(3.161)
tl
and a scaled transient speed as
to produce the following angle/speed state pair (3 .162) (3.163) While this will prove useful lat er, in the analysis of the timescale prop erties of synchronous machines, the model normally will be used in the form of (3.148)(3.159). This concludes the basic dynamic modeling of synchronous machines if saturation of t he magnetic circuit is no t considered. The next section presents a fairly general m ethod for including such nonlinearities in the flux linkage/current relationships.
3.5
The Nonlinear Magnetic Circuit
In t his section, we propose a fairly generalized treatment of nonlineari ties in the magnetic circuit . The generalization is motivated by the multitude of various representations of saturation that have appeared in the literature. Virtually all methods proposed to date involve the addition of one or more nonlinear terms to the model of (3.148)(3.159). The following treatment retUInS to the original abc variables so that any assumptions or added terms can be traced through the transformation and scaling processes of the last section. It is clear that , as in the last section, the flux linkage/current relationships must satisfy assumptions (3) and (4) at the end of Section 3.3 if the result s here are to be valid for the general model of(3.64)(3.72 ). Toward this end, we propose a flux linkage/ current relationship of the following form:
Aabc = L,,(Oshaft)iabc
+ L,,(Oshaft)irotor
 Sabc( iabc, Arotor. 0shaft)
(3.164)
CHAPTER 3. SYNCHRONOUS MACHINE MODELING
44
Arotor = L!r (Oshaft )iabe + Lrr (Oshaft )irotor Srotor( iabe, Arotor, 0shaft)
(3.165)
where all quantities are as previously defined, and Sabe and Srotor satisfy assumptions (3) and (4) at the end of Section 3.3. The choice of stator currents and rotor flux linkages for t he nonlinearity dependence was made to allow comparsion with traditional choices of function s. With these two assumptions, Sabe and Srotor must be such that when (3.164) and (3.165) are transformed using (3.10), the following nonlinear flux linkage/current relationship is obtained Ad
= (Ll, + L=d)id + L,fdifd + L,ldild 
Sd(idqa, Arotor) (3 .166)
3 Afd = 2 L,fdid+ Ljdjdijd + Ljdldild  SjJ(idqa,Arotor)
(3 .167) (3.168)
(3.170) (3.171) and (3.172) This system includes the possibility of coupling between all of the d, g, and 0 subsystems. Saturation functions that satisfy these two assumptions normally have a balanced symmetrical threephase dependence on shaft position. In terms of the scaled variables of the last two sections , and using (3.129 )
(cd = cq = l ), ..pd = Xd(  Id) + Xmd1jd ,pfd
+ X=dIld 
S~l )(ytl
(3.173)
= X=d(  Id) + Xfd1fd + Xmdhd  S}~(ytl
(3.174)
+ X=d1fd + X1dhd  Sl~)(Yl)
(3.175)
,pld = Xmd(  Id )
3.5. THE NONLINEAR MAGNETIC CIRCUIT
45
and
'1fiq = Xq (  l q)
+ X~qhq + x=q!,q 
S~')(Y')
(3.176)
'1fi' q = X~q (  Iq)
+ X,q 1'q + X~ql2q  S;~)(Yl)
(3.177)
'1fi2q = X=q ( 1q)
+ X~q hq + X 2q12q  S~~lt Yl )
(3 .178 )
and
(3.179) where
(3.180)
S~I)
t. Sd / ABDQ,
S}~ ~ Sfd / ABFD , 5;~)
t. 5 1d/ A BlD
t. 5 q/ A BDQ , 5(1) t. 5 1q / A B1Q , 5(1) t. 5 2q / A B2Q 5 q(1) = 1q = 2q =
(3 .181) wit h each 5 evaluated using Adqo, Arotor written as a function of Y , . Using new variables E~ and E~J and rearranging so th at rot or currents can be
eliminated, gives
(3.183) (3.184) and
CHAPTER 3. SYNCHRONOUS MACHINE MODELING
46
~ _
'l/Jq ~
" _ (Xq" : Xl, ) , Xqlq (X' _ X ) Ed q
l~
( Xq '  " Xq) _ (2) X ) 'l/J2q Sq (Y2 ) (3.18 5)
+ (X'q _
is
(3. 186)
(3.187) and
(3188) where (3.189) and
5(2)
~
d

S(I) _ 5(2 ) _ d
fd
5 (2) b. 5(1) _ 5(2) _
q
q
5( 2) b. 5(1) o
0
lq
(X~  X~) 5(2)
(X~
_ Xl,)
ld'
(X~  X~' ) 5 (2) (X'q _ X I, ) 2q'
(3.190)
with each 5(1) evaluated using Y1 , written as a function of Y2 • Elimination of rot or currents from (3.64) (3.72) gives the final dynamic model with general nonlinearities.
3.5. THE NONLINEAR MAGNE T IC CIR CUIT
47
(3. 191)
1 d.,pq W W dt = R , I q  .,pd + v:q w, ,
(3.192) (3. 193)
X'd  X,"d dE q' / ,.' Tdo =  Eq  (XdXd )[Id ( ' _) (.,pl d+( X r dt X d Ji.1,2 I
I
 E; + S~~)(Y2 ))]  S}~)(1T2) + E Jd " d.,pld Tdod t =
 .,pld
' + Eq'r(Xd 
XI, )Id 
I
I
I
dEd
'
(2)
SId
(3.194)
(Y2 )
(3.195 )
/I
Xq  }(q
,. 1
XI, )Id
I
Tq0dt =  Ed + (Xq  X q) [I q  (X~ _ Xl,F (.,p2,+ (Xq  XI, )I q
+E~ + S~!) (1T2 ))] + S~!\Y2)
(3 .196) (3 .197) (3 198) (3.199)
with the three a.lgebra.ic equations , 01.
'I'd = 
X "I (X ;XI')E' ( X~X; ) ol. d d + ( X~ _ Xl, ) q + ( X~ _ Xl, ) 'l'ld
 S~2)(Y2)
(3.200)
" ( X~'  Xl, ) , ( X~  X~') 1{Iq =  Xq Iq  ( 'K' _ X ) Ed + (X ' _ X )1{I2 q J
q
t.
 S~2) ( Y2 ) .,po
= XI, I o  si2 )( y2 )
1'2 = [Id E~ 1{I,d Iq Ed .,p2q Jo]t
q
llf
(3.201)
(3.2 02 ) (3.203 )
48
CIlAPTER 3. SYNCHRONOUS MACHINrJ MODELING
As in the last section , new speeds could be defined so tha t each dynami c sLate includ es a time constant . It is important to note , however, that, as in t he last section , terms on the righthand side of the dynamic state model imply that these t ime constants do noL necessarily completely identify the speed of res ponse of each variable. This is even more evident with the addition of nonlinearities. One purpose for beginning this sect ion by returning to the abc variables was to trace t he nonlineari!ies through t he transformation and scaling proce~s. This € lLS Ur es that the result ing model with nonlinearit ies is, in some sense, consistent. This was partly motivated by the proliferati on of different met hods to account for saturation in the literature . For example, the lite r ature talks a b out " X~/' sat urating, o r being a function of t.he dynamic states. This could imply that many constants we have defined would change when saturation is considered. With the presentation given above, it is clear that all constants can be left unchanged, while the nonlinearities are included in a set of funct ions to be specified based on some design calculation or test
X=d
procedure.
It is interesting to compare these general nonlinearity functions with
other meth ods that have ap peared in the literature [20,22,23,26,27,35,36]  [50] . Reference [37] discu sses a typical representation that uses: 5(2) d

()
and keeps S}~ and
0 1 5(2 )  0 5(2)  0 5 (2) = 0 q I 2q I 0
5 (2) 
)
ld

(3204 )
sl!) expressed as (3.205 )
(3.206) where (3.207 ) and .,." '/' d
"
(X;  Xl,) E•' + (X~  X;) '/'l d X'  X X' _ X
(3208)
(X:Xt') , (X~X~') ..p2. X' _ X Ed + X ' _ X
(3209)
.1."" '1'.  
.1.
d
q
is
d
ill
is
q
l.f
3.5. THE NONLINEAR MAGNETIC CIRCUIT
49
The saturation function SG should be correct under opencircuit conditions. For steady state with
= Iq = 10 = I 'q = Izq = 0 ..p. =  E~ = ..p~ =  Vd = ..p,. = ..p2. = 0 Id
""d 'I'  E'•  ""~II 'I'd
V;q  ""'l'ld  Efd  S(2) fd

(3.210)
the opencircllit terminal voltage is (3.211) and the field current is E'
I jd
=
+ S(2)
• fd Xmd
(3.212)
From th e saturation representation of (3 .205),
S}~ = SG(V,.J XmdIfd = V, ••
(3.213)
+ SG(Vi.J
(3.214)
The function SG can then be obtained from an opencircuit characteristic , as shown in Figure 3.3. While this illustrates the validity of the satur ation function under opencircuit conditions, it does not totally support its use under load . In addition, it has been shown that this representation does not satisfy the assumption of a conservative coupling field [51J. slope = Xmd
v VI OC
I
~
Sc,(VJ Xmd
lfd
Figure 3.3: Synchronous machine opencircuit characteristic
50
3.6
CHAPTER 3. SYNCHRONOUS MACHINE MODELING
Single Machine Steady State
To introduce steady sta.te, we a.ssume constant states and look at the algebraic equations r esulting from t he dynamic model. We will analyze t he system under the condition of a linear magnetic circuit. Thus, beginning with (3.148)(3 .159), we observe that, for constant stat es, we must h ave constant speed w and const ant angle 5, thus r equiring w = w, and, therefore,
Vd =  R,ld  ,po
(3215)
Vo = R,lq + ,pd
(3.216)
Assuming a balanced threephase operation, all of the "zero" variables a nd damper winding currents are zero. The fact that damperwinding currents are zero can be seen by recalling that the righthand sides of (3.152)(3. 154) are actually scaled damp erwjnding currents. Usjng these t o sjmplify (3. 151 ), (3.153), (3.157), and (3.158), the other algebraic equations to be solved are
o =  E~  (Xd  X~)ld + Efd o =  ,p'd + E~  (X~  Xt.)ld o = E~ + (Xq  X~)Iq o =  ,p2q  E~  (X;  X/,)lq o = TM  (,pdIq  ,pq Id )  Trw
(3.217) (3. 218)
(3.2 19) (3.220 ) (3. 221)
,pd = E;  X~Id
(3.222)
= E~ 
(3.223)
,po
X~Io
Except for (3.221), these are all linear equations that can easHy b e solved for various steadystate repr esentations . Substituting for ,pd and ,pq jn (3.215) and (3.216) gjves
+ E~ + X~Io
(3224 )
Vq =  R , Iq + E;  X~ld
(3.225)
Vd =  R,Id
These two real algebraic equations can be written as one complex equation of the form
(3. 226)
3.6. SINGLEMACHINE STEADY STATE
51
where E

[( E~  (Xq  X~)Iq)

j[(Xq  Xd)Id
+j (E~ + ( Xq 
X~)Id)]ei(Ii "/2)
+ E~] ej(Ii~/ 2)
(3.227)
Clearly, many alternative complex equations can be written from (3.224) and (3 .225), dep ending on what is included in the "internal" voltage E. For b alanced symmetrical sinusoidal steadystate abc voltages and currents, the quantities (Vd + jVq)e i (Ii  ,,/2) and (Id + jIq)ei (Ii" /2) are the perunit RMS phasors for a phase voltage and current (see (3.73 )(3.91 )). This gives considerable physical siguificance to the circuit form of (3.226 ) shown in Figure 3.4. The internal voltage E can be further simplified, using (3.217), as j E = j[(Xq  Xd)Id + Efd]e (Ii"/2l =
[(Xq  Xd )Id
+ Efd]e''Ii
(3.228)
+
E +
F igure 3.4: Synchronous machin e circuit representation in steady state An important observation is
(3 .229)
0 = angle on E Also from (3 .142) and (3 .143),
(3.230)
lfd = Efd/Xmd
Several other p oints are worth noting. First , tlte opencircuit (or zero stator current ) terminal voltage is (Vd + jVq)e i (Ii ,, /2 ) 11d =I.=0= Efde,li
( 3.231)
Therefore, for Efd = 1, t he opencircuit t erminal voltage is 1, and field current is 1/ X md. Also, Vd IIp1.=0
= E~ 11 =I. =0=  1/;q 11 =1. =0= 0
Vq 11d =I.=0 =
d
d
E~ 11.=1,=0= ..pd 11d =1,=0= Efd
(3.232) ( 3.233)
52
CHAPTER 3. SYNCHRONOUS MACHINE MODELING
The electrical torque is (3.234) Tills torque is precisely the "real power" delivered by the controlled source of Figure 3.4. Tha.t is, for 1 = (Id + j1q )ei (6"/ 2), TELEe
= TM = Real[Er]
(3.235)
We can then conclude that the electrical torque from the shaft is equal to the power delivered by the controlled source. In steady state, the electrical torque from the shaft equals TM when Tpw = O. From the circuit with V = (Vd + jVq )e i (6" / 2), (3.236) For zero stator resista.nce and round roteIl
(3.237) or (3.238) where the angle OT is called the torque angle, (3.239) with (3.240) Under these conditions, and this definition of the torque angle, OT motor and liT > 0 for a generator.
< 0 for
Example 3.1 Consider a synchronous machine (without saturation) serving a load with V = ILlO° pu
1=
0.5 L  20 0 pu
a
53
3.6. SINGLEMACHINE STEADY STATE
It has Xd = 1.2, Xq = 1.0, X~d = 1.1, Xd = 0.232, R, = 0 (all in pu). Find 0, OT, Id, I q , Vd , V., ..pd, ..p., E~, Efd, Ifd (all in pu except angles in degrees) .
Solution: (1.2  0.232)Id + E fd
E'
•
(0 .768Id + E~)LO
1L90 x 0 .5 L  20° + 1LlO° _
1.323 L29.1°
so 0=29 .1° OT = 29.1°  10° = 19.1° Id
+ jI. = 0.5 L Id Vd
+ jV. =
20°  29.1° + 90°
= 0.378
= 0.5L40.9°
= 0.327 29.1° + 90° =
1LlOo 
I.
1nO.9°
Vd = 0.327 V. = 0.945 ..pd
= Vq + 01. = 0.945
..p. =  Vd  Old =  0.327
To find E~ and E Id , return to I E
I
0.768 x 0.378 E~
+ E~ =
= 1.033
1.033 = (1.2  0.232) Eld
1.323
X
0.378 + Eld
= 1.399
To find Ifd, it is easy to show that 1 fd  Eld _ 1.399  127 . X=d 1.1 These solutions can be checked by noting that, in scaled per unit , equal to Pou,,· TELEC

..pdI.  ..pqld = 0.4326
POUT

Real (V 7*) = Real (0.5L
+ 30°) =
0_433
TELEC
is
CH APTER J. SYNCHRONOUS MACHINE MODELING
54 Also, QOUT
Imag(V J') = Imag(0 .5 L300) = 0.25
+ jVq)e  j (6 1C/2)(Id Imag((Vd + jVq)(Id  jIq)) ..pdId + ..pql q = 0.25 Imag((Vd
jIq)e;(61C(2))
o The steadystate analysis of a given problem involves certain constraints. For example, depending on what is specified, the sol ution of the ,teadystate equations may be very difficult to solve. T he solution of steadystate in multimachine power systems is usually called load flow, and is discussed in later chapters . The extension of this steadystate analysis to include saturatjon js left as an exercise.
3.7
Operational Impedances and Tes t Data
The synchronous machine mo del derived in this chapter was based on the initial assumption of three stator windings, one field winding, and three damper win dings (ld, lq, 2q) . In addition , the machine reactances and time constants were defined in terms of this machine structure . This is consistent wit h [20J and many other references. It was noted earlier, however, that many of t he machine reactances and time constants have been defined through physical tests or design parameters rather than a presupposed physical structure and model. Regardless of the definition of constants, a given model contains quantities that must be replaced by numbers in a specific simulation . Since designers use considerably more detailed modeling, and physical tests are model independent , there could be at least three different ways to a rrive at a value for a constant denoted by the symbols used in the model of this chapter. For example, a physical test can be used to comput e a value of T~o if T~~ is defined through the outcome of a test. A designer can compute a value of T~o from p hysical pa.rameters such that the value would approximate the test value. The definition of T~o in this chapter was not based on any test and could, therefore, be different from that furnished by a manufacturer. For this reas on, it is important to always verify the definitions of all constants to ensure that the numerical value is a good approximation of t he constant used in the model.
3.7. OPERATIONAL IMPEDANCES AND TEST DATA
55
The concept of operational impedance was introduced as a means for relating test data to model constants. The concept is based on the response of a machine to known test voltages. These test voltages may be either dc or sinusoidal ac of variable frequency. Tbe stator equations in the transformed and scaled variables can be written in the Laplace domain from (3 .64) (3.72) with constant speed (w = w u ) as 
W,u
.5
..pq + ..pd
Vd
= R.Id 
Vq
= R.Iq + ..pd + ..pq Ws W!J
VA
= R.Io + w. ..po

w&

W!J!I 

s
(3.241)
Ws
S 
(3.242) (3.243)
where s is the Laplace domain operator, which, in sinusoidal steady state with frequency W o in radians / sec, is S
= JW o
(3.244)
If we make the assumption that the magnetic circuit has a linear flux linkage/current relationship that satisfies assumptions (3) and (4) of Section 3.3, we can propose that we have the Laplace domain relationship for any number of rotorwindings (or equivalent windings that represent solid iron rotor effects). When scaled, these relationships could be solved for ..pd, ..po' and ..po as functions ofl d , I q, la, all rotor winding voltages and the operator s. For balanced , symmetric windings and alirotor winding voltages zero except for Vfd , the result would be
,pd =  Xdop(s)Id ,pq
=
Xqop(s)Iq
..po = Xooplo
+ Gop(S)Vfd
(3.245) (3.246) (3.247)
To see how this could be done for a specific model, consider the threedamp erwinding model of the previous sections. The scaled Kirchhoff equations are given as (3.67)(3.70), and the scaled linear magnetic circuit equa· tions are given as (3 .122)(3.128). From these equations in t he Laplace domain with operator sand V,d = V 'q = V2q = 0,
CHAPTER 3. SYNCHRONOUS MACHINE MODELING
56
(3.249) S

 [X~q(  Iq)
w, s

 [X~q( Iq)
W,
+ X'qIq + CqXmq1,q ) = R'qI'q
+ cqX=qI,q + X 2qI 2q ] =
R 2q I 2q
(3.250) (3.251) (3.252)
The two d equations can be solved for Ifd and I'd as functions of s times Id and V fd. The two q axis equations can be solved for I'q and 1,q as functions of s times I q . When substituted into (3.1 22) and (3.125), this would produce the following operational functions for tills given model:
X dopeS)
= Xd
;;;X!d(Rld + ;;;X'd  2;;;Cd X =d + Rid + ;;;X/d)] (3.253) [ (Rfd + ':, Xfd)(R'd + ':,X'd )  C,CdX=d )2 _
G (s) = op
[
X~d(R'd + ..L X 'd  ..LCdX=d) ] "" "', (Rfd+ ':,Xfd)(R1d+ ':,X,d)  (':,Cd X =d)2
(3.254)
Xqop(S) = Xq ;;;X!q(R 2q + ~X2q  2 ~cqXmq + R 1q + ttXlq) (R2q + ':,X2q ){R ,q + ':,X,q )  (':.CqX=q)2
X ""p{s ) = Xl, Note that for
S
(3.256 )
= 0 in this model Xdop{O ) = Xd Gop{O) = Xmd Rfd Xqop {O) = Xq
and for
S
=
00
(3.255)
(3.257) (3.258) (3.259)
in this model with Cd = cq = 1
= X;
(3 .260)
Xqop {oo) = X;
(3.261)
Xdop{ oo)
For this model, it is also possible to rewrite (3.253) (3.256) as a ratio of polynomials in s that can b e factored to give a time constant representation.
3.7. OPERATIONAL IMPEDANCES AND TEST DATA
57
The purpose for introducing this concept of operational functions is to show one possible way in which a set of parameters may be obtained from a machine test. At standstill, the Laplace domain equations (3.241)(3.243) and (3.245)(3.247) are
Vd =  (R.
+ ':"'Xdop(S)) Id + ':"'Cop(S)VJd Ws
(3.262)
w~
(R.+:.Xqop(s))Iq
(3.263)
Vo =  (R'+ :,Xoop(S))10
(3.264)
Vq
To see how these can be used with a test, consider the schematic of Figure 3.1 introduced earlier. With all the abc dot ends connected together to form a neutral point, the three abc x ends form the stator terminals. If a scaled voltage Vtest is applied across be, with the a terminal open, the scaled series Ic( h) current establishes an axis that is 90 0 ahead of the original aaxis, as shown in Figure 3.5.
(neutral)
""'P'rLj+
(neutral)
aaxIS
(neutral)
Figure 3.5: Standstill test schematic With this symmetry and 0shaft = "; mech. rad (found by observing the
58
CHAPTER 3. SYNCHRONOUS MACHINE MODELING
field voltage as the rotor is turned), the scaled voltage Va will be zero even for nonzero Ie =  Ib and lfd, since its axis is perpendicular to both the band caxis and the field winding daxis. Also, the unsealed test voltage is Vtest
and, by symmetry with Os haft =
= Vb

(3.265)
llc
't mechanical radians, (3.266)
The unsealed transformed voltages are
v'3
v'3
v'3
Vd = 2Vb  2ve = 2Vtest 'V q
= Va
=0
(3.267) (3.268)
and the unscaled transformed currents are
.
v'3
v'3
'd = 2'b  2'e = iq
=
io
v'3'b = v'3'test
=0
(3.269) (3.270)
For a test set of voltage and current,
v'3v'2v,0 cos(wot + 00) = v'2lto cos(wot + :j
cos °shaft . Po 2" 8m "2 shaft + "3
1
I
V'2
.j2
Show that PjqO = P;;'~ (Pdqo is orthogonal). 3.2 Given the following model v
= 10i +
d>'
dt' >.
= 0.05;
scale v, i, and>' as follows:
to get
V=RI
+ ~d1/J WB dt
'
"'=XI 'I'
Find R and X if VB = 10,000 volts, SB = 5 rad / see, and 1B = SB/VB, AB = VB/WB'
X
106 V A, WB
= 271"60
3.3 Using the Pdqo of Problem 3.1 with
hv cos(w,t + 0) h 2V eos(w,t + 0 h 2Ve08(w,t+O+ and Vd ,
2" 3) 2" 3 )
~Oshaft  w,t ~ , express the phasor V = Ve jB in terms of vq, and 6 that you get from using Pdqo to transform Va, Vb, V c into
6 D.
Vd, V q1 Va'
3.4 Neglect saturation and derive an expression for E' in the following alternative steadystate equivalent circuit:
62
CHAPTER 3. SYNCHRONOUS MACIIINE MODELING
1X'd
R, +
Writ e E' as a function of E~, id, I" , O. 3.5 Given the rnagnetizatioll curve shown in pu, compute X 771d and plot SG( 1/1) for
I I I I
I
u 
I I I I I
~,
I
1.11_
2 .0
1.0
3.0
I
3.6 R epeat t he derivation of the singlemachine steady st a te equivalent circuit using the following saturation function s: 5 d(2) 
S(2)  5(2)  5,(2)  5,(2) ld

q

1q

5}~ = 5G(B~)
2q

5(2)  0 0

3.8. PROBLEMS
63
where SG is obtained from the opencircuit characteristic as given in the text.
3.7 From Ref. [51], show that the saturation model of Problem 3.6 does satisfy all conditions for a conservative coupling field. 3.8 Given (3.122)(3.124) and (3.125)(3.127), together with (3 .64)(3.70), find a circuit representation for these mathematical models . (Hint: Split Xd, Xfd , X'd , etc . into leakage plus magnetizing.) 3.9 Given the following nonlinear magnetic circuit model for a synchronous machine: 1/Id 1/Ifd 1/Iq
1/1,.
Xd(  Id)
+ X=dIfd 
Sd( 1/Id, 1/1 fd)
+ X fdIfd  Sfd( 1/Id, 1/1fd) X.(  Iq) + X=q!,q  Sq( 1/Iq, 1/1,.) X=.( I.) + X'.!,q  S'q(1/Iq , 1/Ilq) X=d(  Id)
(a) Find the constraints on the saturation functions Sd, Sfd, Sq, S'q such that the overall model does not violate the assumption of a conservative coupling field. (b) If the other steadystate equations are Vd =  R.Id  1/Iq Vq =  R.I.
+ 1/Id
Vfd
= RtdItd
0= R .!,q '
find an expression for E, where E is the voltage "behind" Rs+jXq in a circuit that has a terminal voltage
Chapter 4
SYNCHRONOUS MACHINE CONTROL MODELS 4.1
Voltage and Speed Control Overview
The primary objective of an electrical power system is to maintain balanced sinusoidal voltages with virtually constant magnitude and frequency. In the synchronous machine models of the last chapter) the terminal constraints
(relationships between Vd, fa, V q , f q , Vo, and fo) were not specified. These will be discussed in the next chapter. In addition, the two quantities Efa and TM were left as inputs to be specified. Efd is the scaled field voltage, which, if set equal to 1.0 pu, gives 1.0 pu opencircuit terminal voltage . TM is the scaled mechanical torque to the shaft. If it is specified as a constant, the machine terminal constraints will determine the steadystate speed. Specifying Efd and TM to be constants in the model means that the machine does not have voltage or speed (and, hence, frequency) control. IT a synchronous
machine is to be useful for a wide range of operating conditions, it should be capable of participating in the attempt to maintain constant voltage and frequency. This means that E fd and TM should be systematically adjusted to accommodate any change in terminal constraints. The physical device that provides the value of E fd is called the exciter. The physical device that provides the value of TM is called the prime mover. This chapter is devoted to basic mathematical models of these components and their associated control systems.
65
66
4.2
CHAPTER 4. SYNCHRONOUS MACHINE CONTROL MODELS
Exciter Models
One primary reason for using threephase generators is t he constant electrical torque developed in steady state by the interaction of the magnetic fields produced by the armature ac currents wi th the field de curren t. Furthermore , for balanced threephase machines, a de current can be produced in the field winding by a de voltage source. In steady state, adjustment of the field voltage changes the field current and, therefore, the terminal voltage. Perhaps the simplest scheme for voltage control would be a battery with a rheostat adjusted voltage di vider connected to the field winding through slip rings . Manual adjustment of the rheostat could be used to continuously react t o changes in operating conditions to maintain a voltage magnitude at some point. Since large amounts of power are normally required for the field excitation, the control device is usually not a battery, and is referred to as the main exciter. This main exciter may be either a de generator driven off the main shaft (with brushes and slip rings), an inverted ac generator driven off the main shaft (brushless with rotating diodes) , or a static device such as an actodc con verter fed from the synchronous machine terminals or auxiliary power (with slip rings). The main exciter may have a pilot exciter that provides the means for changing the output of the main exciter. In any case, Efd normally is not manipulated directly, but is changed through the actuation of the exciter or pilot exciter .
Consider first the model for rotating de exciter s. One circuit for a separately excited dc generator is shown in Figure 4 .1 [21J. Lal
rfI +
+
Saturation
+

K il l (r)\ 4>01
e on l l = Vfd
Figure 4.1: Separately excited dc machine circuit Its output is the unscaled synchronous machine field voltage small ral and L al , this circuit bas the dynamic model
ein1 Vfd
.
KalWI
For
dAjl
= 'in1rfl + & =
Vfd.
<Pal
(4.1) (4.2)
4.2. EXCITER MODELS
67
with the exciter field flux linkage related to field flux ¢fl by ( 43) Assuming a constant percent leakage (coefficient of di spersion (71)' the armature flux is . 1 ( 4.4) ¢al = ¢j1
[(Xq  Xd)Iq
(5. 134)
The equations for Vd and Vq r emain the same as in the last sections, so that the circuit of Figure 5.3 can be constructed to reflect the algebraic con straints for this oneaxis model.
+
+
V efov,
s
Figure 5.3 : Synchronous machine oneaxis dynamic circuit
It is easy to verify that, as in the last two sections, the "real power from" the internal source is exactly equal to the electrical torque across the air gap (TELEC) for this model.
5.5. THE ONEAXIS (FLUXDECAY) MODEL
105
The final form of this oneaxis model, which has eliminated the stator/ network and all three fast damperwinding dynamics, is obtained by substituting (5.131) into (5 .115) (5.130) to eliminate Ed: I
dE~
I
I
T dodt = Eq  (Xd  Xd)ld
+ Efd
(5.135)
do
 = ww dt '
(5 .136)
dw I 2H w, dt = TM  E q 1g 
(
X q  Xd' ) la1q  TFW
(5.137)
TE d~t = (KE + SE(Efd))Efd + VR dRf
(5.138)
KF
(5 .139) TF = Rf+  Efd dt TF dVR KAKF TA""a;: = VR + KARt TF Efd + KA(Vref  Vi) (5 .140) dTM TCWit = TM
+ Psv
Tsv dPsv = Psv
dt with the limit constraints
(5.141)
+ Pc _ ..2.... (.':".. RD
vRrnin < _
o ::;
w,
1)
(5 .142)
VR < _ VRIDax
(5.143)
Psv ::; P!f'v x
(5.144)
and the required algebraic equations, which come from the circuit of Figure 5.3 , or the solution of the following equations for la, I q:
o = (R. + Re)la  (Xq + Xep)l. + V, sin(o  ilv ,) o = (R, + Re)l. + (Xd + X,p)la  E~ + V. cos(o 
(5.145) 8v ,)
(5.146)
and then substitution into the following equations for Vd, V.:
Vd = R.Id  Xepl.
+ V. sin(o  8.,)
(5.147)
Vq = R.I. + Xepld
+ V, cos(o  Ov,)
(5148)
and finally
Vi =
JVl + Vq2
The quantities Vref and Pc remain as inputs.
(5.149)
106
5 .6
CHAPTER 5. SINGLEMACHINE DYNAMIC MODELS
The Classical Model
The classical model is the simplest of all the synchronous machine models, but it is the hardest to justify. In an effort to derive its basis, we first state what it is. In reference to aU of the dynamic circuits of this chapter , the classical model is also called the constant voltage behind the transient reactance (XdJ model. Return to the twoaxis dynamic circuit (Figure 5.2) and the dynamic model of (5.115)( 5. 130). Rather than assuming T~o = 0, as in the last section, assume that an integral manifold exists for Ed, E~, Eid, Rf, and VR, which, as a first approximation, gives E~ equal to a constant and (Ed (X~  Xd)Iq) equal to a constant. For this constant based on the initial values Edo, I; , and E~o, we define the constant voltage
+
(5.150) and the constant angle
5'0
to ta

1 (
n
E'o
) _ 1f
2
(5 .151)
R,
R,
jXd
go ef(o+l)'O)
q
E'o + (X'q _ X')Io d d q
+
Figure 5.4: Synchronous machine classical model dynamic circuit The classical model dynamic circuit is shown in Figure 5.4. Because the classical model is usually used with the assumption of constant shaft torque (initial TXt) and zero resistance, we assume TCH
=
00,
R.
+ R. = 0
(5 .152)
to
5 + 5'0
(5. 153)
and define
5classical
5.7. DAMPING TORQUES
107
The classical model is then a secondorder system: d°classical dt
(5.154)
2 H rU,., E'°V: d = TXt  (X' ; ) sin(odassical  Bu ,)  TFw (5 .155) w"
d+
t
e.p
This classical model can also be obtained formally from the twoaxis model by setting X; = X~ and TCB = T~o = T~o = 00, or from the oneaxis model by setting Xq = X~, T~o = 0, and TCB = T~o = 00. In the latter case, 0'° is equal to zero, so that 0classical is equal to 0 and E'o is equal to E;o.
5.7
Damping Torques
The dynamic models proposed so far have all included a friction windage torque term TFw. For opposition to rotation, such torque terms should have the form
(5.156) The literature often includes damping torque terms of the form To
= D(w 
w,)
(5 .157)
(w  w,)
(5.158)
or
D'
To = 
w.
These damping torque terms can be positive or negative, depending on the machine speed. Values of D' = 1 or 2 pu have been cited as a reasonable method to account for the turbine windage damping [62] . Such terms look like indu ction motor slip torque terms, and have often been included to model the short circuited damper windings. Thus, if damper windings are modeled through their differential equations, then their effects need not be added in TD . In this case, friction can be modeled through TFW, if desired. If damper windings have b een eliminated, as in the oneaxis or classical models, all of the third damping effects have been lost. To account for their damping without including their differential equations, To can be added to the torque equation (added to TFW), with D appropriately specified to approximate the damperwinding action. We now justify this damping torque
CHAPTER 5. SINGLEMACHINE DYNAMIC MODELS
108
term by returning to the twoaxis model of (5.115)(5 .130), which included one damperwinding differential equation. We would like to show that if Ed is more accurately eliminated, a damping torque term proportional to slip speed should automatically appear in the swing equation. To show this , we propose that the integral manifold for Ed should be found more accurately. To begin, we define the small parameter I" as r:, T'qo (5.159) p. = and propose an integral manifold for
Ed of the form
Ed = "')
Figure 5.6: Synchronous machine subtransient dynamic circuit including saturation
I dEd I Tq0dt = Ed
+
(
')[ X~  X~' (0'. Xq  Xq Iq  (X~ _ X ,)2 'l'2q
I
+ (Xq' 
+Ed + S~!)(Y2))l + Sl!)(Y2)
)
Xl, Iq
(5.198)
Til d.,p2q qo dt
(5.199)
do
(5.200)
dt 2Hdw w, dt
_(X~'  X~)Idlq
+ S~2>tY2)Iq
 S~2)(Y2)Id  TFW dEfd
TS;jt =  (Ks dRJ dt
(5.201)
+ SS(EJd))EJd + VR
TF = RJ
+ KF EJd TF
dVR TA d t =  VR
+ KAR j
dTM T P l ' eHT! =  M+ sv

(5.202) (5.203)
KAKF TF EJd
+ KA(Vref 
V,)
(5.204) (5.205)
5.B. SYNCHRONOUS MACHINE SATURATION
115
dPsv = Psv+Pc1 (W Tsv1) RD
dt
(5.206)
w,
with the limit constraints
vRrnin oi
=
[ v.; ] ~Tdqo. ~:
(6 .4)
i= 1" . . , m
/l BABC' [ 'D; ] e,. Tdqo.Ti.'m. [ •• ] _ T. V>Qi V>o' dq08 1\ V>Oi
(6.3)
;
V>bi
V>ci
i = 1,. "lm
(6 .5)
where Tdqoi is the machine i transformation of Section 3.3, and all base scaling quantities are also as previously defined . For 5; = ~ 0shaft;  w.t, it
6.1. THE SYNCHRONOUSLY ROTATING REFERENCE FRAME
125
is easy to show that sin 0, 1
T.dqo.J Tdqoi 
=
and
TdqoiT;;'~, =
[
:]
cos 0;,
ot. sin Of. 0 0
 cos
["..
co~ 0;
::
 cos 0; sin 0;
0
i=l, ... ,m
(6 .6)
i=l, ... ,m
(6.7)
= l, ... , m
(6.9)
This transformation gives
and i
We now assume that all of the m machine data sets and variables have been scaled by selecting a common systemwide power base and voltage bases t hat are related in accordance with the interconnecting nominal transformer ratings. Applying this transformation to the general model of (3.148)(3.159) with the same scaling of (5.37)(5.40) with E = l/w" the multimachine model (without controls) in the synchronously rotating reference frame is
d..pDi
E~
d..pQi
E~
= R.iID; +..pQi+ Vn;
i=l, . . . ,m
(6.10)
= R,;lQi 
i
= 1, ... , m
(6.1l)
d..pOi
E;U: = I
Td
R.Jiloi
~E~i  . E' 
·
'" dt


qi 
..pOi
+ VOl
+VQ i
(Xdl·  X') di
+ (Xdi 
(6.12)
i = 1, .. . m I
[I .
dl 
Xt,i)Idi  E~i)l
(X~i  XX3J.)2 (,I. . 't'ldt
(X'
di. 
l .u
+ Efdi
i = 1, ... , m (6.13)
i= 1, . . . ,m
(6.14)
CHAPTER 6. MULTIMACillNE D YNAMIC MOD ELS
126
T' . dE~i qo'
~
dt
E'
di
+ (X qi 
(X~i (X ' . _ XX~;) _)2(.1, Y' 2qi
X ' ) [I
qi 
qi
qt
i s).
i = I,. "Jm
Til . d'ifi2qi q01 dt T .d6 i " dt
i = I, . .. , m i
= Wti
=
_ XUI . d, d.
(6.18)
+ (X:J.  Xl,.) E' . + ( Xdi  X:JJ .1, . (X ' _ X .) (X' _ X .) 'l'ld, di
l.n
q.
di
In
i=l,,,.,m II  X.J 
q' q'
(6. 16)
(6 17)
1, . .. , m i = 1, .. . ,m

(6.15)
(6.19 )
( Xq/~'
 Xl,i) I (Xq',  Xq/~) E .+· . nh (X'ql.  X i S l.) d, (X'qt _ X In.) 'l'2qi ~
i= 1, ... ,m
(6.20) (6.21)
i = 1, . .. ,m
where
(VD.
+ jVQi) = (Vdi + jVqi) ej(Oi  ¥)
(ID.
+ jIQi) = (Idi + j Iqi)ej(oi i)
('ifiDi
+ j'ifiQi)
= (,pd.
+ Nqi)ej(Oi i)
= 1, ... , m i = 1, ... , m
(623)
i = 1, .. . , m
(6.24)
i
(6.22)
The terminal constraints for each machine are still unspecified . The next section gives a muJtimacrune set of terminal constraints, which can be analyzed in a manner similar to the last chapter.
6.2
Network .and RL Load Constraints
Rather than intr oduce an infinite bus as a terminal constraint , we propose that all m synchronous machine terminals be interconnected by balanced symmetrical threephase RL elements . These RL elements are e.ither transmission lines wher e capacitive effects have been neglected , or transformers. For now, we assume that all loads are balanced symmetrical threephase RL
6.2. NETWORK AND RL LOAD CONSTRAINTS
127
elements, so that the multimacrune dynamic m odel can be written in a multitimescale form , as in Chapter 5. We assume that all line, transformer , and load variables have been scaled by selecting the same common power base as the machines , and voltage buses that are related in accordance with the interconnecting nominal transformer ratings. The scaled voltage across the line, transformers, and loads is assumed to be related to the scaled current through them by
Vai
= RJai + W,1 d"
[IDl" . 1m]'
(6.52)
Vnbranch
L>
[VDl'" Vm]'
(6.53)
and similarily for Q and O. For a system with a basic branch loop incidence matrix describing the interconnection of these b branches as C b , we define loop flux linkages as ' C' = b1Pnbranch
(6.54)
'
Ct'
Wi. = Wi 
WI
i
= 1, ." , m
(6.240)
so that (6 241 )
w;
and replaces Wi as a dynamic state. In this case, there is no need to include either the angle or the speed equation for machine 1, since all other dynamics would depend only on 6; (i = 2, .. . ,m) and (i = 2, ... , m). This situation also arises when the only speed terms on the righthand side appear in the swing equations with uniform damping (H;/ Di = Hk/ Dk i, k = 1, .. . , m). A common transformation used in transient stability analysis is the centerofinertia (COl) reference. Rather than r eference each angle to a specific machine (i.e. , 6d, the COl reference transformation defines the COl angle and speeds as
w;
(6242)
"
WeDI =
(6.243)
where (6.244)
6.10. ANGLE REFERENCES AND AN INFINITE BUS and
. fJ. 2H, M ,
w,
159
(6.245)
The CO l referenced angles and speeds are
• n
Oi = 0,  OeD J Wi
6.
wiweOl
i
= 1, . _. , m
(6.246)
i= l, ... ,m
(6 .247)
With the introduction of DeDI and WeDI, it is possible to use these as dynamic states together with any other m  1 pairs of COlreferenced mechanical pairs. Choosing 2, ... , m, the new mechanical statespace would consist of OeDI , WeDI , 5i, Wi (i = 2, ... , m). This would require the elimination of 51 and W1 in terms of OeDI, WeDI and 5" Wi (i = 2, .. . ,m). The resulting system would have MT multiplying the time derivative of weD I. Since MT represents the total system inertia, it is usually quite large relative to any single M i . For this reason, it is often taken to be infinity, in which case the COl mechanical pair is eliminated, reducing the dynamic order by 2. It is important to emphasize that simply using COlreferenced variables does' Ilot , in itself, reduce the dynamic order. The reduction requires the use of COlreferenced variables together witb the approximation that MT is infinity. The resulting swing equations are complicated by the COl reference, since an inertiaweighted form of "system" acceleration is subtracted from each machine)s true acceleration.
Chapter 7
MULTIMACHINE SIMULATION In t his chapter, we consider simulation techniques for a multimachine power system using a twoaxis machine model with no saturation and neglecting both the stator and the network transients . The resulting differentialalgebraic model is systematically derived . Both t he partitionedexplicit (PE) and the simultaneousimplicit (SI) methods for integration are discussed . The 51 method is preferred in both research gr ade programs and industry programs, since it can handle "stiff" equations very well. After explaining the 51 method consistent with our analytical development so far, we then explain the equivalent but notationally different method, the wellknown EPRIETMSP (Extended Transient Midterm Stability Program) [70] . A numerical example to illustrate the systematic computation of initial conditions is presented.
7.1
DifferentialAlgebraic Model
We first rewrite the two axis model of Section 6".4 in a form suita ble for simulation after neglecting the subtransient reactances and saturation. We also neglect the turbine governor dynamics resulting in TMi being a constant . The limit constraints on VRi are also deleted, since we wish to concentrate on modeling and simulation . vVe assume a linear damping term TF Wi = D i(Wi  w,). The resulting differentialalgebraic equations follow from (6.196)( 6.209) for the m machine , n bus system with the IEEEType I exciter as
161
CHAPTER 7. MULTIMACHINE SIMULATION
162
1. Differential Equations i=l, ... ,m f dEdi I Tqo, T t =  Ed,
+
(
dOi dt = Wi  w~
Xqi  XqiI ) I q,
Ei
T
(7.4)
i = l , ... ,m
dEfd, dt 
i = 1" "Im
dRfi =  Rfi
F,Tt
+ KTF,Fi Efd,
(7.2) (7.3)
i = 1, ... , Tn
 D,(w,  w,) T
i= 1, ... ,m
(7.1)
i= 1, . . . ,m
(7.5) (7.6)
dVR ' KAKF' TA 1.'dt' =  VR ' + KA 1.'Rf1.' ' • Efdl' + KA t·(VIef 1  v.) TFi t &
i
= 1, . .. ,m
(7.7)
Equation ( 7.4) has dimensions of torque in perunit. When the stator transients were neglected, the electrical torque became equal to the perunit power associated with the internal voltage source. 2. Algebraic Equations The algebra.ic equations consist of the stator algebraic equations and the network equations . The stator algebra.ic equations directly follow from the dynamic equivalent circuit of Figure 6.5 , which is reproduced in Figure 7.1. Application of Kirchhoff's Voltage Law (KVL) to Figure 7.1 yields the stator algebraic equations: (a) Stator algebraic equations 0= V;e ie ,  [Edi
+ (R'i + jXd,)(Idi + jIq,)e i (6,t)
+ (X~i 
i =l, ... ,m
Xd,)Iqi + jE~iJei(6' f) (7.8)
7.1. DIFFERENTIALALGEBRAIC MODEL
163
(b ) Network equations The dynamic circuit, together with the static network and the loads , is shown in Figure 7.2 . The network equations written at the n buses are in complex form. From (6 .208 ) and (6 .209), these network equations are jX~
R" +
[E" + (x~X;)Jqi
(Vd' + jV,, ) ,)lk1rl2)
+ jE', ] ,Kb,l [ t.
1= 6.V
gi ,
and
[ 6V 6TMi 1= 6u., (8.25) ren
8.2. BASIC LINEARJZATION TECHNIQUE
227
can be written as
For the mmachine system, (8.26) can be expressed in matrix form as (8.27) where A B B 2 , and E, are block diagonal matrices. " linearize " We now the stator algebraic equations (8 .1 3) and (8.14):
(8.28) !':.E~i
 cos( 6io  Oio )!':. Vi + Vio sin( 6io  Oio)!':.6,  Vio sin( 6io  O'o)!':.O, i = 1, ... , m ( 8.29)  R,,!':.lq'  Xd,!':.h = a
Writing (8.28) and (8.29) in matrix form, we have
!':.Oi 6Wi
o a 100 o 100 0 + [R~i X~i] Xdi
R"
i
!':.Edi !':.Efdi !':. VRi !':.RF'
[ Md; ] !':.lq.
+ [ V;ocos(6i~ _Oi~)  V,osm(6.0
~]
!':.E~,
0. 0 )
 sin (6io  Ow) ] [ !':.Oi. ] =  cos(6.o0. o) !':.V,
= l"",m
a ( 8.30)
Rewriting (8.30), we obtain (8.31) In matrix notation, (8.31) can be written as (8.32)
CHAPTER 8. SMALLSIGNAL STABILITY
228
where C D and D, are block diagonal. Linearizing the network equations " (8.16), " (8.15) and which pertain to generators, we obtain Via sin(Oio  Bio)fl1di
+ Jdio sin(5io  Bio).6..Vi + IdioV'io C08(Oio  (}io).6..0i
IdioV'iocOS(Oio  Bio)!:l(}i
+ VioCOS(Oio
 Bio).6..Iqi
+Iqiocos(6io  Bio)6Vi IqioViosin(oio  ( 10 ).6..0i
+Iqio Vio sin( 0'0  e'o)t,.ei 
[f
VkoYik cost eio  eko  aik)] t,. Vi
k=l
n
k=l
+
[V. ~ V"y;,
"o('"
,~ 0,,1] 
M,
 Vio L [VkoYik sm(e,o  eko  aik)] t,.ek + n .
ah(Vi) all: t,. Vi = 0
k~l
(8.33)
t
ie'
Via cos( Oio  Bio).6..Idi
+ ldio COs( Oio  8io).6.. Vi  ldio Via sin( 0';0

8io ).6..5,;
+Jdio Via sin{ 6io 
Bio).6..Bi  Via sin( 6io  Bio).6..Iqi Iqiosin(oio  (}io).6.Vi  IqioViocos(6io  8io)1:1 0i
+IqioViocos(oio  eio)t,.ei 
[f
VkoYiksin(eio  eko  aik)] t,.Vi
k=1
n
k=l
 [v.. ~ V~y" m,('.  '"  =1]
l'.O,
+ Vio L [VkoYik cos(eio  eko  "ik)] Mk n
+
aQLi(Vi) aVi t,. Vi = 0
10== 1
ie' i=1,2, ... ,m
(8.34)
B.2. BASIC LINEARIZATION TECHNIQUE
229
Rewriting (8 .33) and (8.34) in matrix form, we obtain
DSl ,=+l
+[
:
(8.35)
D Sm ,Tn+ l
where the various submatrices of (8.35) can be easily identified . In matrix notation, (8.35) is (8.36) where
for the nongenerator buses i = m + 1, . . . ,n. N ote that C 2 , D3 are block diagonal, whereas D 4 , Ds are full matrices. Linearizing network equations (8.17) and (8.18) for the toad bnses (PO buses) , We ob tain
n
ViD L [Y;k COS(Oio  Oko  Uik)) t>Vk k= l n
 Vio L [VkoY;k sin(Oio  Oko  Uik)] 6. Ok 1.=1
;
(8 .37)
CHAPTER 8. SMALL· SIGNAL STABILITY
230
n
 ViD
I: [Yik sin( BiD 
BkD  Ciik)] L':. Vk
k=l n
+ViD I: WkDYik sin( BiD 
BkD  Ciik)] L':.Bk
10=1
# i=m+l, ... ,n
(8.38)
Rewriting (8.37) and (8.38) in matrix form gives
(8.39)
where, again, the various submatrices in (8.39) can be identified from (8.37) and (8.38). Rewriting (8.39) in a compact notation ,
(8AO) where D 6 , D7 are full matrices. Rewriting (8.27), (8.32), (8.36), and (8AO) together ,
+ B1L':.Ig + B 2 L':. Vg + E,L':.u o = C1L':.x + D,L':.Ig + D 2 L':. Vg 0= C 2 L':.x + D3L':.Ig + D4L':.Vg + DsL':.Vl o = D6L':. Vg + D 7 L':. VI
L':.± = A,L':.x
where X
t t = r.xl· , xm
It
(8.41) (8.42)
(8.43 ) (8.44)
8.2. BASIC LINEARIZATION TECHNIQUE Xi
=
[ X2 in the mode ),1 == 5 as . Pu

P2l =
=
,
 ( (3)?) )
,
=
W ll'Un W1'UI W2 1 1112
C4~(1)) 
W 2 'U2
=
3

7
4 7
Letting i 2 and k 1,2, we obtain the participation of the state variables Xl , x, in the mode ),2 == 2 as
P22
4
,

,n

W12'U21
Pl2
7
W 2 'U2
=
W22 11
W 2V2
3 7
The participation matrix is t herefore p == [
Pll P21
P12 P22
3
4
] [ ] 'i 4
'i
'i 3
'i
8.3. PARTICIPATION FACTORS
239
Normalizing tbe largest participation factor as equal to 1 in eacb column results in
Pnorm = [ 0'175
o The ith column entries in tbe P or Pnorm matrix are the sensitivities of the ith eigenvalue with respect to the states. Example 8.2 Compute the participation fact ors corresponding to t he complex eigenvalue of
01.4
A=
0
0
[  1.4 9.8
The eigenvalues are )., = ·0.6565 and ).2,3 = 0.1183 ± jO.3678. The right and left eigenvectors corresponding to the complex eigenvalue ).2 = 0.1183 + jO.3678 are
V2
0.0138  jO.0075 ] =  0.0075  jO.04 , [  0.9918  jO.1203
W2
[0.838  jO.0577 ] = 0.4469 jO.307 0.0061 + jO.0205
+
Using the formula ill the previous section , we obtain
P21 = 0.2332; P22 = 0.3896 , Pn = 0.3772 Note P21 + Pn + P23 = 1. We can normalize with respect to Pn by making it unity, in wh.ich case P21(norm) 0.598, Pn(norm) 1 and P23(nOrm) 0.968.
=
=
o
=
240
CHAPTER 8. SMALL SIGNAL STABILITY
Example 8.3 The numerical Example 7.1 is used to illustrate the eigenvalue computation. Compute the eigenvalues, as well as the participation factors, for the eigenvalues for the nominal loading of Example 7.1. The damping Di '" a (i 1,2,3) . The machine and exciter data are given in Table 7.3 . Loads are assumed as constant power type.
=
Solution Following the linearization procedure results in a 21 X 21 sized Asys matrix_ Because of zerodamping, two zero eigenvalues are obtained. The eigenvalues are shown in Table 8.1. The participation factors associated with the eigenvalues are given in Table 8.2. Only the participation factors greater than 0.2 are listed . Also shown are the state variables and the machines associated with these state variables. From a practical point of view, this information is very useful. Table 8.1: Eigenvalues of t he 3Machine System 0.7209 0.1908 5.4875 5.3236 5.2218 5.1761 3. 3995 0.4445 0.4394 0.4260 0.0 000 0.0000 3.2258
±J·12 .7486 ±j8.3672 ±j7.9487 ± j7.9220 ±j7.8161
± j1.2104 ±jO.7392 ±jO.496 0
8.4. STUDIES ON PARAMETRIC EFFECTS
241
Table 8.2 : Eigenvalues and Their Participation Factors Eigenvalue 0.7209 ± jI2.7486 0.1908
±
;8.3672
5.4875
±
j7.79487
5.3236
±
j7.9220
5.2218
±
j7.8161
5.1761 3 .3995 3.2258 0.4445 ± j1.2104
0.4394
±
jO .7392
0.4260
±
jO.4960
0.0000 0.0000
Machine Number 3 2 2 1 2 2 3 3 1 1 2 3 3 2 1 2 3 1 2 3 3 2 1 2 1 2
Machine Variable
PF
Ii,w 6,w 6,w 6,w
1.0, 1.0 0.22, 0.22 1.0 , 1.0 0.42, 0.42 1.0, 0.98 0.29 1.0, 0.98 0.29 1.0, 0 .97 0.31 1.0 0.92 1.0 0.89 1.0, 0.74 0.67, 0.48 0.38, 0.28 1.0, 0.78 0.78, 0.60 0.22 1.0, 0.83 0.43 , 0.33 1.0, 1.0 0.26, 0.26 1.0, 1.0 0.26, 0.26
VR,Ejd RJ VR,Ejd RJ VR , Ejd RJ E'd
E 'd E'd E'd E~)RJ
E~ l RJ E~ , Rf E;,Rj E;,Rj
,
E'
E~lRJ E~,Rf
6,w .b,w 6,w 6,w
o
8.4
Studies on Parametric Effects
In this section, the effect of various parameters on the smallsignal stability of the system is studied .
CHAPTER 8. SMALL·SIGNAL STABILITY
242
8.4.1
Effect of loading
The WSCC 3machine, 9bus system of Chapter 7 is considered. The real or reac tive loads at a particular bus / b uses are increased continuously. At each step, the initial conditions of the state variables are computed, after running the load flow, and linearization of the equations is done. Ideally, the increase in load is picked up by the generators through the economic load dispatch scheme. To simplify matters, the load increase is allocated among the generators (real power) in prop ortion to their inertias. In the case of increase of reactive power, it is picked up by the PV buses . The Asys matrix is formed, and its eigenvalues are checked for stability. Also dethF and detJ~E are computed. The step·by.step algorithm is as follows: 1. Increase the load at bus/buses for a particular generating unit model. 2. If the real load is increased , then dis tribute the load among the various generators in proportion to their inertias . 3. Run the load flow . 4. Stop, if the load flow fails to converge. 5. Compute the initial conditions of the state variables, as dIscussed in Chapter 7. 6. From the linearized DAE model, compu te the various matrices. 7. Compute dethF, detJ~E' and the eigenvalues of Asys. 8. If Asys is stable, then go to step (1). 9. If unstable, identify the st ates associated with the unstable eigen· valuer s) of Asys using t he participation factor method, and go to step (1). The above algorithm is implemented for mo dels A and B. Nonuniform damping is assumed by choosing Dl = 0 .0254 , D2 = 0.0066, and D3 = 0.0026. The resnlts are summarized in Tables 8.3 and 8.4. It is observed that for constant power load, with model A and the IEEE· Type I slow exciter, it is the voltage control mode that goes unstable at PL5 = 4.5 pu. Examination of the participation factor indicates t hat t he pair of state variables E~, Rj of machine 1 in t he e.."'{citation system is responsible for this m o del. In the case of model B, the mode that goes unst able is due to the electromechanical
243
8.4. STUDIES ON PARAMETlUC EFFECTS
variables 6, w of machine 2 at a load of 4.6 pu. A value of K A = 45 is assumed . The point at which the eigenvalues cross over to the righthalf plane is called the Hopf bifurcation point , and the point at which the detJAE changes sign is the singularityinduced bifurcation. These are discussed in the literature in detail [82, 85}.
Table 8.3: Eigenvalues with Model A, KA = 20 "gn(detJLp )
.•gn(detJ/4.E}5
4.3 4.4 4.5 (A)
+ +
4.6 4.7 4.8
+ + +
+ + +
Load at Bus
4 .9
5.0 (8) 5.1 5.2 5.3 5.35
+ +
+ + +
+ 
· 0. 16]8
0.4 446 ± 1.0825 ± 2 .605 1 ± 17.568,
+ +

5.45
± j1.9769
0.0522 ± j2 .11 02 0.1268 ± ,2.2798
+ +

Criti cal Eigenvalue{s)
,gn(deU~R)
+ + + + + + +
+
5.0 (8) 5.1 5.2 5.3 5.4 5.5
+
+ + + 
+

Eft E,l E', I
0 .3505
"gn(dellLF)
4 .9
,2'
Ed2 1Eql
E~ ll 61 ,02
0 .0496 0.1454 LF d oes not con verge
4.3 4.4 4.5
+ + + +
E~1' E~2' E:a E:n, E;l ,E E~3
1.0 526 0 .6553
L06d a.t Bus 5
4.6 (A) 4.7 4.8
E;l ' RJ1 Eq1.lE~2
;2 .4911 ;2 .7064 ,2.439 2 1. 7849
Table 8.4: EigenvaLues with Model B, KA
+
Associated SLa.tes E ,Rf1 11 E'll,RJl
""
= 45
Critical Eigenvalue(s) 0.1119 ± ,8.8738
± ,8.8401 ± j8 .81 83 0.0901 ± j8 .8421 0 .1587 ± ;8.9371
.0.0729 0.0035
0.1292 ± j9.0538 0 .7565 ± j20.1162 0.0471 ± j9.09 02 14 .7308 7.1144 4.2567 2.3120
0 .0 597 ± jS.7819 LF does not con verge
Associated States 52 , CoI 2 62 ,W2 02,''2
5 2 ,''2 (h,w2
6'2 , toT2 E~l l E~l 62,W2
E~l' E~l Ej l E" E~l 6'2 . w :I
244
8.4.2
CIIAPTER 8. SMALLSIGNAL STABILITY
Effect of KA
It was found that, for m odel A, the increase in KA alone did not lead to any instability. The stabilizing feedback in the IEEEType I exciter was removed, and then an increase in K A led to instability for this model, as well. For model B, a sufficient increase in KA led t o instability even for a nominal load .
8.4.3
Effect of type of load
Appropriate voltagedependent load modeling can be incorporated into the dynamic model by specifying the load functions. The load at any bus i is given by
= 1, .. 'In
(8 .85)
i = 1, .. _, n
(8.86)
i
where PLio and QLio are the nominal real and reactive powers, r espectively, at bus i, with the corresponding voltage magnitude Vio, and npi' no' are the load indices . Three types of load are considered. 1. Constant power type (np
= no = 0)
2. Constant current type (n p
= no =
3. Constant impedance ty pe (np
1)
= no = 2)
The stepbystep procedure of analysis for a given generatingunit model is as follows: 1. Select the type of the load at various buses (i.e., choose values of and nq at each bus) .
np
2. Compute the system ma.trix. 3. Compute the eigenvalues of Asys for sta.bility analysis . For the three types of loads mentioned earlier, the eigenvalues of model A for increased values of load PLo at bus 5 are listed in Tables 8.5 to 8.7. First of all, the relative stability of constant power , constant current , and constant impedancetype load has been shown for a nominal operating point
8.4. STUDIES ON PARAMETRIC EFFECTS
245
= 1.5 pu and QLo =
0.5 pu (Table 8.5) . We observe that the system is dynamically stable for all types of loads. For an increased value of load at bus 5 (PLo 4.5 pu, QLo 0.5 pu), the eigenvalues are listed in Table 8.6. From Ta.ble 8.6 We observe that, for this increased load at bus 5, the system becomes dynamically unstable if the load is treated as a constant power type, whereas for the other two types of loads the system remains stable. 0.5 pu), in which we Finally, we take another case (PLo = 4.6 pu , QLo show that the constant impedance type load is more stable than the constant current type. To demonstrate this condition, we take model B with a high gain of the exciter (KA 175). The eigenvalues for various kind of loads are listed in Table 8.7. Both the constant power and constant current cases are unstable, whereas the constant impedance type is stable . These results corroborate the observation in the literature that constant power gives poor results as far as network loadability is concerned [84]. PLo
=
=
=
=
Table 8.5: Eigenvalues for Different Types of Load at Bus 5 for model A (PLo = 1.5 pu ; QLo = 0.5 pu): (a) constant power; (b) constant current; (c) constant impedance Constant P ower (a) 0.7927 ± j12.7660 0.2849 ± j8.3675 5.51 87 ± j7 .9508 5.3 325 + j7.9240 5.2238 ± j7 .8156 5. 2019 3.4040 0 .4427 ± j1.2241 0 .4404 ± jO.7413 0.0000 0.1975 0.4276 ± jO.4980 3.2258
Constant Current (b) 0. 7904 ± j12 .7686 0.2768 ± j8 .3447 5.5214 ± j7.9516 5 .3335 ± j7.9247 5. 2273 ± j7 .8259 5.2030 3.4462 0 .4537 ± j1.1822 0 .4412 ± jO.7416 0.0000 0.1974 0.4276 ± jO.4980 3 .2258
Constant Impedance (c) 0 .7887 ± jI2.7706 0.2703 ± j8.3271 5 .5236 ± j7 .9523 5.3344 ± j7. 9253 5.2301 ± j7 .8337 5 .2039 3.4801 0.4617 ± j 1.1439 0.4419 ± jO.7418 0.0000 0.1973 0.4277 ± jO.4980 3.2258
CHAPTER 8. SMALLSIGNAL STABILITY
246
Table 8.6: Eigenvalues for Different T ypes of Load at Bus 5 for Model A (PLo = 4.5 pu; QLo = 0.5 pu): (a) constant power; (b ) const ant current ; (c) constant impedance Constant Power (a) 0 .7751 ± j12 .7373 0.2845 ± j8.0723 6.7291 ± j7 .8883 5.6034 ± j7.9238 5.2935 ± j7.6433 5.2541 0.1268 ± j2 .2798 2.5529 0.4858 ± jO .7475 0.0000 0.5341 ± jO .5306 0.1976 3.2258
Constant Current (b) 0.7335 ± j12 .7842 0.2497 ± j8.0650 6. 7669 ± j7.9730 5 .6287 ± j7.9557 5.2812 ± j7 .8419 5.2715 3 .5296 0 .5020 ± j 1. 253 1 0.0000 0.4910 ± jO.7561 0 .5360 ± jO.7561 0.1972 3.2258
Constant Impedance ( c) 0.7285 ± j 12.7936 0.2444 ± j8 .0659 6.7760 ± j7.9895 56338 ± j7 .9639 5 .2938 ± j7.871 2 5 .2790 3.8 105 0 .5303 ± j1.0434 0 .4950 ± jO .7653 0 .5371 ± jO .5336 0.0000 0.1970 3.2258
Table 8.7: Eigenvalues for Different T ypes of Load at Bus 5 for Model B with KA = 175 (PLo = 4.6 pu, QLo = 0.5 pu): (a) constant power; (b) constant currentj (c) constant imp edance Constant Power (a) 1.96 10 ± j19.3137 0.1237 ± j 15.5812 0.4495 ± j9 .1844 3 _1711 ± j8 .2119 2 .7621 ± j7.1753 11 0.0000 ·0 .1987
Const ant Current (b) 0.2039 ± j15.5128 2.2586 ± j12 .4273 0.1441 ± j9 .0636 3.1359 ± j8.1099 2 .7599 ± j7 .1594 0.0000 0.1989
Constant Impedance
(e) 0 .2054 2 .1834 0 .0607 3 .0930 2 .7575 0 .0000 0.1990
± j 15.5285 ± j11.1128
± j9 .1218 ± j8 .0364 ± j7 .1462
8.4. STUDIES ON PARAMETRIC EFFECTS
8.4.4
247
Hopf bifurcation
For model A, when the load is increased at bus 5, it is observed that the critical modes for the unstable eigenvalues are the electrical ones associated with the exciter , and are complex (Table 8.3). At a load of 4.5 pu, the eigenvalues cross the jw axis. This is known as Hopf bifurcation (point A). When the load is increased from 4.8 pu to 4.9 pu, the complex pair of unstable eigenvalues split s into real ones that move in opposite directions along the real axis. The one moving along the positive real axis eventually comes back to tbe lefthalf plane via +00 when the load at bus 5 is increased from 4.9 pu to 5.0 pu (point B). This is the point at which detJAE changes sign. This is also known as singularityinduced bifurcation in the literature [85J . The other unstable real eigenvalue moves to the left, and is sensitive to the varia.ble E~, of the exciter. This eigenvalue returns to the lefthalf plane at a loading of approximately 5.4 pu, and the system is again dynamically stable (poin t C). For the load at bus 5 = 5.5 pu, the load flow does not converge . It is possible through other algebraic techniques to reach the nose of the PV curve or the saddle node bifurcation. This phenomenon is pi ctorially indicated in the PV curve for model A and also is the locus of critical eigenvalue(s) in the .plane (Figures 8.5 and 8.6). 1.2
~
;.
A B
c
0.8
0 .6
0 .4
0.2
0 0
2
3
4
5
Real Power at Bus 5
Figure 8.5: PV curve for bus 5 with model A
6
248
CHAPTER 8. SMALLSIGNAL STABILITY
I I
IA
,1 ...  ....  .. , I
I
CI
'\
\ B
o    U ~ l A' I
o Real Part
Figure 8.6: The qualitative behavior of the critical modes of Asys as a function of the load at bus 5 (model A) At point A, Hopf bifurcation occurs , and it has been shown to be subcritical, i.e. , the limit cycle corresponding to the E~  R f pair is unstable [86J. However , loadflow solution still exists. In this region, E~ and Rf state variables are clearly dominant initially. As the eigenvalues become real and positive, other state variables st art participating subst antially in the unstable eigenvalues, as indicated in Table 8.3. For model B , which has the fast static exciter with a single time constant, the modes that go unstable are the electromechanical ones (Table 8.4). When the Hopf bifurcation phenomenon in power systems was first discussed in the literature for a singlemachine case, the electromechanical mode was considered as tbe critical one [78, 87J. It was called lowfrequency oscillatory instability. In studies relating to voltage collapse, it was shown that the exciter mode may go unstable first [82J . From Tables 8.3 and 8. 4, it is seen that both the excit er modes and tbe electromechanical modes are critical in steadystate stability and voltage collapse , and that t hey both participate in the dynamic instability depending on t he machine and exciter models . Hence, decoupling the QV dynamics from t he PC dynamics as suggested in the li terature may not always hold. It may be t rue for special system configuration / operating conditions . Load dynamics, if included, can be considered as fast dynamics, and the phenomenon of detJ~E changing sign will still exist. In conventional bifurcationt heory terms, one can think of solving g( x , y) = 0 for y = h( x) in (8.2) and substituting this in the differential equation (8.1) to get x = lex, h(x)). The change in sign of detJ~E (which generally agrees with the sign of JAE in (8 .5)) is the instant at which
B.5. ELECTROMECHANICAL OSCILLATORY MODES
249
the solution of y is no longer possible. This is also tied in with the concept of the implicit fun ction theorem in singular perturbation theory.
8.5
Electromechanical Oscillatory Modes
These are the modes associated with the rotor angles of the machines. These can be identified through a participation factor analysis in the detailed model. In the classical model with internal node description , we have only the rotor angle modes. We now discuss the computation of these modes as a special case . The equations have been derived in Chapter 7 and are reproduced below. db;
dt 2H.
i=l, ... ,m

WiW"

TMiPei
(8.87)
dWi
w, dt
(8.88)
'i= l, ... ,m
Linearization around a n operating point
0
gives
~6.0
(8.89)
dt ' d dt 6.w;
(8.90)
Because TMi = constant , and 6.P,. =
2::7=1
a:';i 6. OJ , we get (8.91)
6. W i
( 8.92) where J
In matrix form we ha.ve
=,
(8.93)
250
CHAPTER 8. SMALLSIGNAL STABILITY
(8.94 )
The elements of Aw as
aTe given by 2~: aC:;:.i. The matrix Aw can be written
w,
2H,
of,,) a5,
w . apel
2Hl 80m.
Aw =
(8.95) w.~ 2H~
85 ,
~!2Lm. 2H~
a5~
0
The elements of Aw can also be expressed in a polar nota.tion by noting that
and
Hence, in (8.93)
(8.96) Therefore
ap,; 
a5
J
J
=,
(8 .97)
8.5. ELECTROMECHANICAL OSCILLATORY MODES
251
For the 3machine case
211 (B)B21'12 si n{ O:12 6f2 ) +8 1 B 3 1'13 .in(a:13 6
Aid
13 )]
~Elg2Y12
sin{<X12  6f2)
~r81B3Y13.in(1X13  6fall
= "'. 2d31 83Bl Y 13 .in(oc31 +E2 82 Y23
 oj!)
'in ( ""32 6~2»)
(8.98) although (Xij = (Xji_ However, the sum of the three columns = 0, hence, the rank of Aw = 2. This means that one eigenvalue of Aw = O. Aw can be expressed as Aw = Ml K, where M'  [diag(frt)] and K = [K;j ]: Aw is not symmet ric , SInce O;,j
=
 Oij1
=
I:
Kij
EiEj Yij sin( ex;,j  6?) '1
J = ,
(8.99)
j :::::; ly!.i
EiEjYtj sin (CXij _6°) '1
jf.i
(8.100)
Ki/S are called the synchronizing coefficients, and are equivalent to those in
(8.93). Eigenvalues of A and Aw Case (a): Transfer conductances are neglected, i.e.,
Gij =
O.
This implies (Xij = ~ and Kij = Kji by inspection from (8.99) and (8.100). Hence, the matrix]( is symmetric. It can be shown mathematically that (1) the eigenvalues of
~~]
A = [M':=O:"1.E~
]
+
[ V"" cos 6° ] V"" sin 60
f'>.6 (8.123)
Now
R. [ (X, +X~)
 (Xc +Xq) ]  ' Rc
=~[ f'>.
R.  (Xc + X~)
(Xc
+ Xq)
]
R, (8 .124)
where the determinan t f'>. is given by
f'>. = R~
+ (Xc + Xq) (Xc + X d)
Solve for f'>.Jd. f'>.Iq in (8 .123). to get (after simplifi.cation)
R.V"" cos 5° + V"" sin 5°(Xq + Xc) ] R.V"" sin 50 + Voo cos 5°(X d + Xc) (8 .125)
CHAPTER 8. SMA LLSIGNAL STABILITY
258 Step 4
Linearize the differential equations (8.108) (8.110). We introduce the normalized frequency v = .!!!.. so that the linearized differ ential equations become w.
+
[T~o ~ 1
(8. 126)
2H
Step 5 Substitut e for 6.Id , 6.1. from (8. 125) into (8. 126) to obtain
,K.
., 1 6.Eq =  K T' 6.E q 3

do
 , Tdo
6.6
1
+ T'do 6.Efd
(8. 128)
6.5 = w.6.v
.
K2
(8.127)
,K,
6.v =  2H 6.E.  2H 6.6 
Dw.
2H 6.v
1
+ 2H 6.TM
(8 129)
where 1
K3 =
1+
(Xd 
X~)(Xq
+ X ,)
(8. 130)
6.
K.
= Voo(Xd6. X~) [( X.. + X. ) SIn.
K2
= ~ [1;6. 
5°  R, cos 5°1
(8.131)
1;(Xd  Xq )(Xq + X,)  R,(Xd  X.)ld + R,E~oJ (8. 132)
8.6. POWER SYSTEM STABILIZERS tVoo{( X~
259
 Xq)I~  E~O}{(X~ t X,) cos 50 t R, sin aD}]
(8.133 ) Since
Vi
JV} t Vq'
=
v:'t 
V'd t V'q
2V,t:. V, = 2VJ t:. Vd
+ 2V; t:. V,
VO
VO
V,
V,
t:. V, = ~ t:. Vd t ...'L t:. Vq
(8.134)
Substituting (8. 125) in (8.121) r esults in
[~~:] = ~ [:~ ~,] X, t X, [ R,
t:.t:.~~ ] + [ t:.:~ 1
[
 6. _
 R, Voo cos 00 t Voo(X, + X,) sin 5°) ] R,Voosin 5° t Voo cos 5°(X~ + X,)
1 [
X,R, X~(X, t X,)
Xq ( R eVoo sin 00 + Voo cos aO(X~ + X, )  X~(  ReVoo cos 00 + Voo(Xq t Xe)sin oo )
[ ~~~ 1t [ 6.:~ 1
1 (8.135)
Substituting (8.135) in (8.134) gives t:. Vi = Ks6.o t K6t:.E~
(8.1 36)
where
(8.1 37) (8.1 38)
CHAPTER 8. SMALLSIGNAL STABILITY
260
The constants th at we have derived are called the K1K6, developed by Heffron Phillips [88], and later by DeMelloConcordia [78], for the study of local lowfrequency oscillations. Example 8.5
In Figure 8.7 , assume that R e = 0, X e = 0.5 pu, V,LB = 1 L15° pu, and Voo LO° = 1.05 LO° pu. The machine data are H = 3.2 sec, T/u, = 9.6 sec, KA = 400, TA = 0.2 sec , R, = 0.0 pu, Xq = 2.1 pu, Xd = 2.5 pu , X~ = 0.39 pu, n 0, and w, 377. Using the flux decay model, fin d (1) the initial values of state and algebraic variables , as well as Vref , TM , and (2) KlK6 constants.
=
=
1. Computation of Initial Conditions
The technique discussed in Section 7.6 is followed. The superscript on the algebraic and state variables is omitted. IGeh = (Id
0
+ jIq )ej (0  "/2) = lLl5° ~ 1.05 LO° = 0.5443Ll8° JO.5
6(0)
= angle of E
= V,e j8 + (Rs + jXq)IG eh 1Ll5° + (j2.1)(0.5443 Ll 8°)
where E E =
.
= 1.4788L65.52°
Therefore 6(0) = 65 .520. Id + jIq = Ia ei'r e j(6.. / 2) = 0 .5443 L42.48°, Id = 0.4014, and Iq = 0.3676 . Vd
+ jV
q
= Ve j8 ,j(6,,/ 2) = 1L39.48°
Hence
Vd
= 0.77185,
Vq = 0.6358 1
From (8. 116): E~ = Vq +X~Id
= 0.63581 +
(0.39 )(04014) = 0.7924
8.6. POWER SYSTEM STABILIZERS
261
From (8.62)(8.65), setting derivatives = 0,
Efd = E~
+ (Xd 
= 0.7924
Efd
Vref = V,
+ KA
w, = 377,
I'M
XWd
+ (2.5 
0.39)0.4014 = 1.6394
1.6394 = 1 + 400 = 1.0041 =
E~lq
+ (Xq 
X~)ldlq
= (0.7924)(0.3676)
+ (2.1 
0.39)(0.4014)(0.3676)
= 0.5436 This completes the calculation of the initial values .
2. Computation of K1K6 Constants The formulas given in (8.130)(8.134) and (8.137)(8.138) are used. /::,. = R~
+ (Xe + Xq)(Xe + X~)
= 2.314 ~ = 1 + (Xd  X~)(Xq K3 /::,.
+ X e)
= 3.3707
K3 = 0.296667 K 4 = Voo(Xd6. X~) [(X q
cO + X)· e SIn u 
Re
COS
CO]
U
= 2.26555 1
K, = /::,. [I;/::"  I~(X~  Xq)(Xq
+ Xe) 
Re(X~  Xq)I'd
= 1.0739
Similiarly, K
"
K s , and K6 are calculated as K, = 0.9224
Ks = 0.005 K6 = 0.3572
o
+ ReE~oJ
262
8.6.3
CHAPTER 8. SMALL·SIGNAL STABILITY
Synchronizing and damping torques
To the single. machine infinite· bus system of Figure 8.7 , we add a fast exciter wbose state· space equation is (8 .13 9) The linearized form of (8.139) is (8 .140 ) Then the machine differential equations (8.127)(8.129), the exciter equation (8.140), and the algebraic equation (8.136) can be put in t he block diagram form shown in Figure 8.8. Both the normalized frequency 1/ and w in rad/ sec are shown.
(fad/sec) ~1'

A
~CO
r,
!.
1:;+_ 115 (fad)
o !:.E'q
~
I + SfA
Figure 8.8: Block diagram of the incremental flux.decay model with fast ex· citer (dotted portion represents the exciter) System loading, as well as the external network p arameter X., affect the parameters Kl K 6. Generally these are > 0, but under heavy loading, K s might become negative, contributing to negative damping and instability, as we explain b elow .
8.6. POWER SYSTEM STABILIZERS
263
Damping of Electromechanical Modes There are two ways to explain the damping phenomena: 1. Statespace analysis [89J
2. Frequencydomain analysis [78J 1. Statespace analysis (assume that ClTM == 0)
Equations (8 .127)(8.129) are rewritten in matrix form as
I [
[
ClE~ Cl6 = Clv
K,~,;. 0
If·
0
0
w,
/if'
,Iii
_ Dw . 2H
T""
I[~: I tI +[
U" (8.141)
Note that, instead of Clw, we have used the normalized frequency deviation Clv = Clw / w, . Hence, the last row in (8. 141) is Clv = _ K 2 ClE _ Kl Cl6 _ D w, Clv ' 2H q 2H 2H
(8.142)
ClEfd is the perturbation in the field voltage. Without the exciter, t he
machine is said to be on "manual control." The matrix generally has a pair of complex eigenvalues and a negative real eigenvalue. The former corresponds to the electromechanical mode (1 to 3Hz range), and the latter the fiuxdecay mode. Without the exciter (i.e., K A = 0), there are three loops in the block diagram (Figure 8.8), the top two loops corresponding to the complex pair of eigenvalues, and the bottom loop due to ClE~ through K., resulting in the real eigenvalue. Note tha.t the bottom loop contributes to positive feedback . Hence , the torqueangle eigenva.lues tend to move to the lefthalf plane , and the negative real eigenvalue to the right . Thus , with constant Eld, there is "natural" damping. With enough gain , the real pole may go to the righthalf plane . Tills is referred to as monotonic instability . In the power literature, this twofold effect is described in more graphic physical terms. The effect of the lag associated with the time constant T~o is to increase the dam ping torque but to decrease the synchronizing tor que. Now, if we add the exciter through the simplified representation, the statespace equation
CHAPTER 8. SMALLSIGNAL STABILITY
264
will now be modified by making 6.E f d a state variable . The equation for 6.Efd is given by (8. 140) as . 1 KA 6.Efd =~ 6.Efd+ (6.V f ~ 6.v,)
TA
TA
re
=~ ~6.Efd ~ KA K 5 6
K .4 K 66.E' TA 6. TA q
TA
6.V + KA TA ref
(8. 143)
Ignoring the dynamics of the exciter for the moment, if Ks < 0 and KA i5 large enough, then the gain through T,i" is approximately  (K4 + KA K 5)K 3 . Thi5 gain may become positive, resulting in negative feedback for the torqueangle loop and pushing the complex pair to the righthalf plane. Hence, this complicated action should be studied carefully. The overall statespace model for Figure 8.8 becomes  1 K3 T~Q
6. E'q 6.6 6.i;
~
6. Efd
.p,
1
0
T'do
6.E'q
do
0
0
w,
0
6.6
;11
if!'
_Dw~
0
6.11
 K ~K ,
K~K.
0
1
6.Efd
1A
1A
2H
TA
0
+
0 0
6. Vref
~ (8 .144)
The exciter introduces an additional n egative real eigenvalue. Example 8.6 For the following two test syst ems whose K,  K6 constants and other paramet ers are given, find the eigenvalues for K A = 50. Plot the root locus for varying K A. Note t hat in system 1 K 5 > 0, and in system 2 K5 < o. Test System 1
Kl = 3.7585 K2 = 3.6816 K3 = 0.2162 K4 = 2.6582 K s = 0.0544 K6 = 0.3616 T~o = 5 sec H = 6 sec TA = 0.2 sec
8.6. POWER SYSTEM STABILIZERS
265
Test System 2
Kl = 0.9831 K3 = 0.3864 Ks =  0.1103 T!w = 5 sec
K2
= 1.0923
K. = 1.4746 K6 = 0.4477 H = 6 sec TA = 0.2 sec
The eigenvalues for KA = 50 are shown below using (8 .144). Test System 1 0.353 ± j10.946 2.61 ± j3.22
Test System 2 0.015 ± j5.38 2 .77 ± j2 .88
Notice that test system 2 is unstable for this value of gain. The root loci for the two systems can be drawn using MATLAB. An alternative way to draw the root locus is to remove the exciter and compute the transfer function AA~l(J) = H(s) in Figure 8.8. Note t hat H(s ) includes all the dynamics except that of the exciter. With the G(s) = feedback transfer function , as in Figure 8.9. /). Vret +
1:0""
we can view H(s ) as a
KA
I + sTA
'
l1Efd
l!.V,
If(s)
Figure 8.9: Smallsignal model viewed as a feedback system
H (s ) can be computed for each of t he two systems as System 1
+
H s _ ( ) 
S3
0.0723s 2 7.2811 + 0.9251s 2 + 118.0795. + 47.74
CHAPTER 8. SMALLSIGNAL STABILITY
266 System 2
2
H(s) _ 
S3
+
+
0.0895s 3.5225 0.5176s 2 30.886s 2
+
+ 5.866
The closedloop characteristic equation is given by 1 + G(s)H(s) = 0, where G( s) = The root locus for each of the two test systems is shown in Figure 8.10. System 1 is stable for all values of gain, whereas system 2 becomes unstable for K A = 22 .108.
1.:'0·2..
10
5
 10
 IS LJc,"""'':':~':':',~+,  $ 4 .5 4 3 .S _3  l.S  2 1 .5 ~1 O.S 0 Real Axis
(. )
, 6
, 2
~• 11
0
, .A
.;;
...
5
J
_2
.
o
(»)
Figure 8.10: Root loci for (aJ test system 1 and (b) test system:2
o
8.6. POWER SYSTEM STABILIZERS
267
2. Frequencydoma.in analysis through block diagram
For simplicity, assume that the exciter is simply a high constant ga.in [(A, i.e., assume TA = in Figure 8.8. Now, we can compute the transfer function I1E~(s) / 115(s) as
°
I1E~(s )
(8 .145)
M (s)
This assumes that 11 v"ef = 0. The effect of the feed.back around TJo is to reduce the time constant. If Ks > 0, the overall situation does not differ qualitatively from the case without the exciter, i.e., the system has three open loop poles , with one of them being complex and positive feedback . Thus, the real pole tends to move into the righthalf plane . If [(s < and, consequently, [(, + KAKS < 0, the feedback from 115 to I1T. changes from positive to negative, and, with a large enough gain K A , the electromechanical modes may move to tbe righthalf plane and the real eigenvalue to the left on the real axis . The situation is changed in detail, but not in its general features , if a more detailed exciter model is considered. Thus , a fastacting exciter is bad for damping , but it has beneficial effects also . It minimizes voltage fluctuations , increases t he synchronizing torque , and improves transient stability. With the time constant T A present ,
°
_ [(K,(1 + sTA ) + KA[(S)]K3 115(5)  KAK6K3 + (1 + K3Tdos)(1 + sTA )
I1E~ ( 5 )
(8.146)
The contribu tion of this expression to the torqueangle loop is given by
I1T, (5) = K I1E~(5 ) ~ H( ) M(s)
2
M(s)
s
(8.147)
TorqueAn gle Loop
=
Letting I1TM 0, the torqueangle loop is given by Figure 8.11. The undamped frequency of the torqueangle loop (D '" 0) is given by the roots of the characteristic equation 2H 2 5 w,

+K, Sl,2

(8.148)
°
±./K,W, J
2H
Tad/sec
(8.149)
CHAPTER 8. SMALLSIGNAL STABILITY
268
,
K,
IV
;/ Te ,,;.
,
/
ro,
1
,
2Hs
s
/
D
dO)
I
, /
1
 dT, )
211 ,
.~ +
W.
D5 + K 1
Figure 8.11: Torqueangle loop
With ahlgher synchronizing torque coefficient K , and lower H , 51 ,2 is hlgher. K, is a comp li cated expression involving loading conditions and external reactances . The value of D is generally small and, hence, neglected. We wish to compute the damping due to E~ . The overall block diagram neglecting damping is shown in Figure 8.12. From this diagram and the closedloop t ransfer function, it can be verified th at the characteristic equation is given by
To rqueangle loop
I \..
/
dT,
1
,
0
I
2II s~ +K I
"'.,
I' H(s)
Figure 8.12: Torqueangle loop with other dynamics added
269
8.6. POWER SYSTEM STABILIZERS
(8.150)
H ( s )6.ij therefore con tributes to both the synchronizing torque and the damping torque. The contributions are now computed approximately. At oscillation frequencies of 1 to 3 Hz , it can be shown that K4 has negligible effect . Neglecting the effecl of K4 in Figure 8.8, we get from (8.147)
(8.151) Let s = jw. Then
H (jw) =
 K2KA K S
c::":~~:,.,::
(i, + KAKe 
+ jW(it, + T~o)
w2T~oKA)
 K2KAKs(x  jy)
(8 .152)
+ y2
",2
where X
=
1
2
I
Ka + KAKe  w TdoTA
Y = w
(~ + T~o)
(8. 153) (8 .154)
From (8 .150), it is clear that, at the oscillation frequency, if Im[H (jw) ) > 0, p ositive damping is implied, i.e., the roots move to the lefthalf plane. If I m [H (jw)] < 0, it tends to make the system unstable, i.e., negative damping results . Thus
' ) ] =  K 2KA K SX" ' . = contn'b' utlOll to t he synch rOlllzmg R e [H (JW ",2 + y2 torque component du e to H (s)
(8155 )
K sY" ' )] = + K 2KA ' . I m [H (JW 2 2 = contn'butlOn to d amplllg x
+y
torque component due to H(s)
(8 .156)
CHAPTER 8. SMALLSIGNAL STABILITY
270 Synchronizing Torque
For low frequencies, we set w "" O. Thus, from (8.155): K2 K S
Ks
for high KA
(8.157)
Thus , the to tal synchronizing component is Kl  Kk~' > O. Kl is usually high, so that even with Ks > 0 (low to medium external impedance and lowtomedium loadings), K, > o. With Ks < 0 (moderate to high external impedance and heavy loa.dings), the synchronizing torque is enhanced posit ively.
£:K.
Damping torque
(8. 158) This expression contributes to positive damping for Ks > 0 but negative damping for Ks < 0, which is a cause for concern . Further, with Ks < 0, a higher KA spells trouble (see Figure 8.10). This may offset the inherent machine damping torque D. To introduce damping, a power system stabilizer (PSS) is therefore introduced. The stabilizing signal may be Ll.v, tJ.Paoo , or a combination of both. We discuss this briefly next. For an extensive discussion of PSS design, the reader is referred to the literature [77, 78].
8.6.4
Power system stabilizer design
Speed Input PSS Stabilizing signals derived from machine speed, terminal frequency, or power are processed through a device called the power system stabilizer (PSS) with a transfer function G(s) and its output connect ed to the input of the exciter. Figure 8.13 shows the PSS with speed input and the signal path from tJ.v to the torqueangle loop.
8.6. POWER SYSTEM STABILIZERS
271
tl.TpSS
PSS
K2 +
K3
KA
7d;
I +.r TA
1+ s K3
+ ~Vrcf
Figure 8.13 : Speed input PSS FrequencyDomain Approach [78] From Figure 8.13, the contribution of the PSS to the torqueangle loop is (assuming ~Vref 0 and ~5 0)
=
=
~Tpss
(f, + KA K 6) + s (ft + Tdo) + s2TdoTA = C(s)CEP(s)
(8 .159)
For the usual range of constants [78], the above expression can be approximated as (8. 160) For large values of KA (high giUn exciter) , this is further approximated by ~Tpss
~v
K2 C(s) = K s[I + s(T.io I KAK6) ][ I + sTA ]
(8.161)
If this were to provide pure damping throughout the frequency range, then C (.) should be a pure lead function with zeros, i.e., C(.) = Kpss[1 + s(Tdol KAK6)](1 + .TA) where Kpss = gain of the PSS. Such a function is not physically realizable. Hence, we have a compromise resulting in what
CHAPTER 8. SMALLSIGNAL STABILITY
272
is called a leadlag type transfer function such that it provides enough phase lead over the expected range of frequencies. For design purposes, G(s) is of the fo rm
G(s) = K
(1
+ .IT,) (1 + sT3 )
pss (1 + sT2 ) (1
sTw
+ sT.) (1 + sTw)
= K
pss
G (s) 1
(8.162)
The t ime constants T" T 2 , T 3 , T. should be set t o provide damping over the range of frequencies at which oscillations are likely to occur . Over this range they sho uld compensate for the phase lag introduced by the machine and the regulator. A typical technique [77J is to compensate for t he phase lag in t he absence of PSS such that the net phase lag is: 1. Between 0 to 4S 0 from 0.3 to 1 Hz
2. Less t han 90 0 up to 3 Hz Typical values of the parameters are:
K pss is in t he range of 0.1 to SO T, is T 2 is T.l is T. is
the the the the
lead time constan t, 0.2 lag time constant, 0.0 2 lead time constant, 0.2 lag time constant, 0.02
to to to to
1.S sec 0.15 sec 1.S sec O.IS sec
The desired sta.bilizer gain is obtained by fir st finding the gain at which the system becomes unstable. This m ay be obtained by act ual test or by root locus study. Tw , called the washout time constant, i s set at 10 sec. The purpose of this constant is to ensure that there is no st eady st ate error of voltage r eference due to sp eed deviation . Kpss is set a t K pss, where Kpss is the gain at which the system becomes unstable [77J . It is important t o avoid interaction between the PSS an d the tor sional modes of vibration . Analysis h as revealed that such interaction can occur on nearly all modern excitation systems, as they have rela ti vely high gain at high frequencies. A stabilizertorsional instability with a highresp onse excit ation system m ay res ult in shaft damage, p articularly at light generator loads where the inherent mechanical damping is small . Even if shaft damage does not occur, such an instabilit y can cause saturation of the stabilizer output, causing it to be ineffective , and possibly causing saturation of the voltage regulator, resulting in loss of synchronism and tripping the unit . It is imperative that stabilizers do not induce torsion al instabilities. Hence,
3
8.6. POWER SYSTEM STABILIZERS
273
the PSS is put in series with another transfer function FILT(s) [77] . A typical value of FILT(s) '" 570;;~'+ " . The overall transfer function of PSS is G( s )FILT( s). Design Procedure Using the FrequencyDomain Method The following procedure is adapted from [90]. In Figure 8.13, let
6.Tpss = GEP(s)G(s) 6.v
(8.163)
where GEP(s) from (8.159) is given by
GEP(s) =
K2KA K 3 KAK3KS + (1 + sTdoK3)(1
+ sTA )
(8.164)
Step 1 Neglecting the damping due to all other sources, find the undamped natural frequency Wn in rad/sec of the torqueangle loop from
2H _s2 W,
+ Kl = 0
(8.165)
) l.e.
Step 2 Find the phase lag of GEP(s) at s = jWn in (8. 164). Step 3 Adjust the phase lead of G(. ) in (8.163) such that
LG(s) I.=iwn +LGEP(s)
I.=iw= n
0
(8.166)
Let
G(s) =
J(
pss
(1 ++ ST1)k 2 1
81
(8.167)
ignoring the washout filter whose net phase contribution is approximately zero . k = 1 or 2 with Tl > T 2 . Thus, if k = 1: Ll
+ jwnTl
= Ll + jwnT2  LGEP(jwn )
(8.1 68)
274
CHAPTER 8. SMALLSIGNAL STABILITY
Knowing Wn and LGEP(jwn ) , we can select T,. T, can be chosen as some value between 0.02 to 0.15 sec. Step 4 To compute Kpss, we can compute Kpss , Le., the gain at which the system becomes unstable using the root locus, and th en have Kpss = ~ Kpss . An alternative procedure that avoids having to do the root locus is to design for a damping ratio ~ due to PSS alone . In a secondorder sys tem whose characteristic equation is 2H _5' + Ds + K, w,
= 0
(8.1 69 )
The dampi ng ratio is ~ = ~ D / VM K, where M = 2H /w,. This is shown in Figure 8.14. ~  I ,
I
'
,
, / / /
/
/ / / / /
I /
X
Figure 8. 14: Damping ratio The charac teristic roots of (8.169) are
 it ± J(it)' 1fI 81,2
2
2~d ± J~  C~I)' if ~wn ±jw V e

j
n
1
4K, M
B.6. POWER SYSTEM STABILIZERS
We note that
Wn
=
J1fi..
~=
275
Therefore
D
2M Wn
= _D_
I._M_
2M 'V K,
=
D
C7 ;;2..; rK 'i7,=;M
(8.170)
Verify that
To revert to step 4, since the phase lead of G( s) cancels phase lag due to GEP(s) at the oscillation frequency, the contribution of the PSS through GEP(s) is a pure damping torque with a damping coefficient Dpss . Thus, again ignoring the phase contribution of the washout filter , (8.171) Therefore, the characteristic equation is 2
5
i.e .,
52
+ 2(Wn 5 + W~ =
+ Dpss + K, M 5 M
= 0
(8.172)
O. As a result, (8 .173)
We can thus find Kpss , knowing for ( is between 0.1 and 0.3.
Wn
and the desired ( . A reasonable choice
Step 5 Design of the washout time constant is now discussed . The PSS should be activated only when lowfrequency oscillations develop and should be automatically terminated when the system oscillation ceases . It should not interfere with the regular function of the excitation syst em during steadystate operation of the system frequency. The washout stage has the transfer function
Gw(s) =
sTw
1 + sTw
(8 .174)
CHAPTER 8. SMALLSIGNAL STABILITY
276
Since the washout filter should not have any effect on phase shift or gain at the osciilating frequency, it can be a chieved by choosing a large value of Tw so that sTw is much larger than unity. (8. 175) Hen ce , its phase contribution is close to zero . The PSS wiil not have any effect on the steady state of the system since, in st eady state,
6.v = 0
(8.176)
Example 8.7 The purpose of this example is to show that the int roduction of the P SS will improve the damping of the electromechanical mode. Without the PSS, the A m atrix, for example, 8.5 , is calculated as  0.3511
o  0.1678  714 .4
0.236 0 0.144 10
0 0.104 377 0 0 0
o
5
Tb.e eigenvalues are >'1,2 = 0 .0875 ± j7 .11 , >'3,4 = 2.588 ± j8 .495 . The electromechanical mode >' 1,2 is poorly damped. Instead of a twostage lag lead compensator , we will have a singlestage laglead PS S. Assume th at the damping D in the torqueangle loop is zero. The input to t he stabilizer is 6.v . An extra state equation will be added. The washout stage is omitted , since its objective is to offset only the de steadystate error. Hence, it does not play any role in the design . The block diagram in Figure 8.15 shows a single laglead stage of the PSS . The added state equation due to the PSS is 1
6.iJ = T 6.y 2
+ Kpss 6.v + Kpss TJ 6.v. T2
T2
 1  6.y + Kpss 6.v T2
T2
(K + Kpss TJ   2 6.E , T2 2H q
Kl 6.0 ) (8.177) 2H
The new A matrix is given as  1 K3T~
1.ffd.'
0
0
0
377
0
0
.J!'
;ftl
0
0
0
K1o;;'
 KpK,
0
J
TA
{f,:
 K,T, (~)
~
0
K, T, 12
(4Jr) 2
1..4.
1~
2
1
Tdo
0
1
T,
8 . 7. CONCLUSION
277 rd
( 1+57, J
uv
KpSS

( IH T2)
Il)"
f' , +
K"
+''/ 
1+ t'J : around the equilibrium point "0" and the derivative V( x) < 0, then the equilibrium is asymptotically stable. V(",) is obtained as ~i=1 ~~:i:i = ~i=1 ~~f;(x) \7VT . f(x) wher e n is the order of the system in (9.6). Thus, f(x) enters directly in the computation ofV(x). The condition V(x) < 0 can be relaxed
°
=
CHAPTER 9. ENERGY F UNCTION METHODS
288
to Vex) < 0, provided that Vex) does not vanish along any other solution with the exception of x = o. Vex) is act ually a generalization of the concept of t he energy of a system . Since 1948, when the results of LyapunoV' appeared in the English language together with potential applica.tions, there has been extensive liter ature surrouncling this topic . Application of the energy function metho d to power system stability began with the early work of Magnusson [102J and Aylet t [103], followed by a formal application of the m ore general Lyapunov's method by EIAbiad and N agappan [99J. Reference [99J provided an algorithmic procedure to compute the criti cal clearing time . It us ed t he lowest energy u.e.p metho d to compute V= . Although many different Lyapunov fun ctions have been t ried since t hen , the first integral of motion, which is the sum of kinetic and potential energies, seemed to have provided the bes t result. In the power literature, Lyapunov's method has become synonymous with t he transient energy funct ion (TEF) method and has been applied successfully [93 , 98J. Today, this technique has proved to be a pract ical tool in dy namic security assessment . To m ake it a practical tool , it is necessary to compute the region of stability of the equilibrium p oint of (9 .5). In physical sys tems , it is finite and not the whole statespace . An estimate of the region of stability or attraction is characterized by an inequality of the type V(x) < Vcr. The computation ofV= remained a form id a ble barrier for a long time. In the case of a multimachine classical model with loads b eing treated as constant impedances, there are wellproved algorithms. Extensions to multimachine systems with detailed m odels have b een m ade [104, 106J .
9.4
Modeling Issues
In applying the TEF technique, we must consider the model in two time frames, as follows: 1. Faulted system
(9.7) 2. Postfault system :i;
= f(x(t)) , t > tel
(9.8)
In reality, the model is a set of differenti alalgebraic equations (DAE), i.e.,
9.4. MODELING ISSUES
289
x
fF(X(t),y(t))
o
gF(X(t), y(t)) , 0 < t ::: tel
(9.9) (9.10)
and
x
f(x(t), y(t))
(9.11)
o
g(x(t), y(t)), t > tel
(9.12)
The function 9 represents the nonlinear algebraic equations of the stator and the network, while the differential equations represent the dynamics of the generating unit and its controls. In Chapter 7, the modeling of equations in the form of (9.9) and (9.10) or (9 .11) and (9.12) has been covered extensively. Reducedorder models , such as a fluxdecay model and a . classical model, have also been discussed. In the classical model representation , we can either preserve t he network structure (structurepreserving model) or eliminate the load buses (assunling constant impedance load) to obtain the internalnode modeL These have also been discussed in Chapter 7. Structurepreserving models involve nonlinear algebraic equations in addition to dynanlic equations, and can incorporate nonlinear load models leading t o the concept of structurepreserving energy function (SPEF) V(x, y), while models consisting of differential equations lead only to closedform types of energy functions V(x). The work on SPEF by Bergen and Hill [104] has been extended to more detailed models in [105][108]. It is not clear at this stage whether a more detailed generator or load model will lead to more accurate estimates of t er . What appears to be true , however) from extensive simulation studies by researchers is that , for the socalled firstswing stability (i.e., instability occurring in 1 to 2 sec interval ), the classical model with the loads represented as constant impedance will suffice. This results in only differential equations, as opposed to DAE equations. Both the PEBS and BCU methods give satisfactory resnlts for this modeL We first discuss this in the mult imachine context. The swing equations have been derived in Section 7.9.3 (using P=, = TMi) as
(9.13)
290
CHAPTER 9. ENERGY FUNCTION METHODS
where
m
Poi = EtGii
+ 2) Gij sin Oij + Dij cos Oij)
(9.14)
;==1 f.i
Denoting 2H; 11,.1,
II Jv[.1
d2/j. Jv[;
dt 2'
and p'.1.
do .
II
Pml.  E~G " we get \ Ul
m
+ D; dt' = Pi  2::=
(C,j sin oii
+ Dij cos Oij)
(9.15)
;=1#
which can be written as
(9.16) be the rotor angle with respect to a fixed reference. Then 0, = D:i Wat. 6; =
D:,
(9.17)
Equations (9.17) and (9.18) are applicable both to the faulted state and the posHault state, with the difference that Poi is different in each case, because the internal node admittance matrix is different for the faulted and postfault system. The model corresponding to (9 .17)(9.18) is known as the internalnode model since the physical buses have been eliminated by network reduction.
9. 5. ENERGY FUNCTION FORMULATION
9.5
291
Energy Function Formulation
Prior to 1979 , there was considerable research in constructing a Lyapunov function for the system (9.15) using the statespace model given by (9.1 7) and (9.18) [109]  [11 2]. However, analytical Lyapunov fun ctions can be constructed only if the transfer conductances are zero, i.e., D ,j ;: O. Since these terms have to be accounted for properly, the first integrals of motion of the system are constructed , and these are called energy functions . We have two op tions, to use either t he relative rotor angle formulation or the center of inertia formulation . We use the latter, since there are some advantages. Since the angles are referred to a center of inertia, the resulting energy functi on is called the transient energy function (TEF). In this formulati on , the angle of the center of inertia (COl ) is used as the reference angle, since it represents the Hmean moti on" of the system. Although the resulting energy function is iden tical to V( 5, w) (using relative rotor angles), it has the advantage of being more sy=etric and easier t o handle in terms of the pathdependent t erms . Synchronous st ability of all machines is j udged by examining the angles referenced ouly to COl instead of relative rotor angles. Modern literature invariably uses the cor formulation. The energy function in the COl notation, including D ,j terms (transfer conductances) , was first proposed by Athay et a1. [97]. We derive the transient energy fun ction for the conservative system (assuming Di = 0). The cent er of inertia (COl) for t he whole system is defined as 1
60
1
m
=  I:: Mi 6i and MT .=1
the center of speed as Wo
m
=   I:: MiWi
(9.1 9)
MT i=1
where MT = 2::Z;1 Mi . We then transform the variables 6i , Wi to the variables as (Ji = 6i  00 l Wi = Wi  Wo' It is easy to verify
cor
ei = 6.  50
The swing equations (9. 15) with Di
Pi 10.
t
. J::::l:;>!i
= fire)
i
= 0 become (omitting the algebra):
(Cij sin eij
+ Dij cos eij) 
= 1, ... , m
MM; Pea J T
(9. 20)
292
CHAPTER 9. ENERGY FUNCTION METHODS
where m
m
Pi=Pmi  E!G,i ; Peo/=L Pi  2 L i=l
m
L
DijCOSOij
i = l j=i+ l
If one of the machines is an infinite bus, say, m whose inertia cons tant Mm is very large , then tJ~ Peol '" 0 (i '" m) and also 00 ' " Om and Wo '" "'m . The COl variables become Oi = OiOm and Wi = Wi  Wm' In the literature where the BeU method is discussed [98], Om is simply taken as zero. Equation (9 .20) is modified accordingly, and there will be only (m  1) equations after omitting the equation for machine m. We consider the general case in which all MIs are finite . Corresp onding to the faulted and t he postfault states, we have two sets of differential equations ,
MdWi = fF(O) I dt 1
dO. = dt
Wi ,
O < t ~ tel
i = 1,2 , . .. , m
(9.21)
and
dw ' M ,' dt' = li (O) dOi dt
= Wi
J
t > tel i = 1,2 , . "lm
(9 22)
Let the p ostfault system given by (9 .22) have the stable equilibrium point at 0 = 0' , w = O. 0' is obtained by solving the nonlinear algebraic equations
Ii(O) = 0 , i = 1, . . . , m
(9 .23)
Since E:', M;O; = 0, Om can be expressed in terms of the other Ois and substituted in (9.23), which is then equivalent to
h(O" . . . , Omi) = 0 , i = 1, . .. , m  1
(9.24)
The basic procedure for computing the critical clearing time con sists of the following steps : 1. Construct an energy or Lyapunov function V( O,W) for the system (9.22) , i.e., t he postfault system.
9.5. ENERGY FUNCTION FORMULATION
293
2. Find the critical value of V(O,w) for a given fault denoted by Vcr . 3. Integrate (9.21) , i.e., the faulted equations, until V(O,w) = Vcr . This instan t of time is called the critical clearing time tor .
While this procedure is common to all the methods , they differ from one another in steps 2 and 3, i.e., finding Vcr and integrating the swing equations . There is general agreement that the first integral of motion of (9.22) constitutes a proper energy function and is derived as follows [94J. From (9.22) we have, for i =1 , ... ,m
dt
= Midwi = dOl = M 2dW 2 = d0 2 = .. . Mmdwm = dO m h(0)
WI
fl(O)
W2
fm (B)
Wm
(9.25)
Integrating the pairs of equations for each machine between (0:, 0), the postfault s.e.p to (0., Wi) results in
I1;(B,..;:,) =
~Miwl  ].8 2
6~,
i
MB)dB;, i
= 1, .. . , m
(9.26)
This is known in the literature as the individual mach.ine energy functon [113]. Adding these functions for all the machines, we obtain the first integral of motion for the system as (omitting the algebra):
(9.27) 1
m
 "2 L M;w; 8.+8 . ].
L
p.( fJ;  Oil 
'
J
8/+9;
= VKE (W)
m
L L
[C;j(cos B;j  cos 0lj )
t= l j=i+ 1
i=l
i= l
_
ml
Tn
]
D '].. cos 0·,·d(fJ , 1· + fJ ]·)
+ VPE(fJ)
SInce
M . ].8i L ' PCO[dO; = 0 m
i==l
MT 81
(9.28)
(9.29)
CHAP TER 9. ENERGY FUNCTION METHODS
294
Note that (9.28) contains pathdependent integral terms. In view of trus, we cannot assert that Vi and V are positivedefinite. If D ,j _ 0, it can be shown that V(B,w) constitutes a proper Lyapunov fu nction [93, 109,110).
9.6
Pot e ntial Energy Boundary Surface (PEBS)
We first discuss the PEBS method because of its simplicity and its natural relationship t o the equalarea criterion. Ever since it was first proposed by Kakimoto et al. [100] and Athay et al. [97), the m ethod has received wide attention by researchers b ecause it avoids computing the con trolling (relevant) u .e.p and requires only a quick fault on system integration to compute Vcr. We can even avoid computing the postfault s.e.p., as discussed in Section 9.6 .5. In trus section, we will first motivate the method through application to a singlemachine infinitebus system , establish the equivalence between t he energy function and the equalarea criterion, and , finally, explain the multimacrune PEES method.
9 .6.1
Singlemachine infinit ebus system
Consider a singlemachine infinitebus system (Figure 9.3). Two parallel lines each having a reactance of XI connect a generator having t ransient reactance of Xd through a transformer wjth a reactan ce of X, to an infinite bus whose voltage is EzLO". A threephase fault occurs at t he ruiddle of one of the lines at t = 0, and is subsequently cleared at t = tet by opening the circuit breakers at both ends of the faulted line. The prefault, fault ed, and post fault configurations and their reduction to a twomachine equivalent are shown in Figures 9.4, 9.5, and 9.6. The electric p ower P e during and postfault states are E,X ~" sin 6' X E, E, ,in, and E, E, .in' faulted P refaultl ' P X I r espectively. The computation of Xl for t he prefault system and X for the postfault system is straightforward, as shown in Figures 9.4 and 9.6. 1
M, j Xd
•
HDi ....:J_' x','_n~___4_
~ 0, then the system is unstable if E > V(6 U). The points OU and Ju constitute the zerodimensional PEBS for the singlemachine system. Some researchers restate the above idea by saying that if the VPE is initialized to zero at oct, VtfE represent s the excess kinetic energy injected into the system. Then stability of the system is determined by the ability of the postfault system to absorb this excess kinetic energy (i.e. , the system is stable if VPE(OU)  VPE (Ocl) > VtfE) ' Most of the stability concepts can be interpreted as if the moment of inertia M is assumed as a particle that slides without friction within a " hill" with the shape VPE(O ). Motions within a p otential "well" are bounded and, hence, stable. It is interesting to relate the p otential "well" concept to the stability of equilibrium points for small disturbances. Using (9.31), (9.30) can be written as
a=
oct
=
d2 0 M= dt 2
(9.37)
We can expand the righthand side of (9.37) in a Taylor series about an equilibrium point 0', i.e. , 0 = 0* + /';0 , and retain only the linear term. Then
(9.38)
s· i.e. ,
9.6. POTENTIAL ENERGY BOUNDARY SURFACE (PEES)
301
/':,0 = 0
(9.39)
If 8':,~B < 0, the equilibrium is unstable. If 8':,~E > 0, then it is an oscillatory system, and the oscillations around 0' are b ounded. Since there is always some positive damping, we may call it stable. In the case of (9 .30), it can be verified that 0' is a stable equilibrium point and that both OU and 6u are unstable equilibrium points using this criterion . The energy function, Lyapunov function, and the PEBS are thus all equivalent in the case of a singlemachine infinitebus system. It is in the case of multimachioe systems and oonconservative systems that each method gives only approximations to the true stability boundary! In the case of multimachine systems, the second derivative of Vp E is the Hessian matrix.
15'
15'
Example 9.2 Consider an SMIB system whose postfault equation is given by d25 . do 0.2= 1  2 Sill 0  0.02dt 2 dt The equilibrium points are given by 5' 7r 
~
= 56'11", gu = 
71'" 
~
=
= sin 1 (n. Hence,
0'
=
~, 5u =
~7r . Linearizing around an equilibrium point
"0" results in
d2 /':,0 0 d/':,o 0.2  2 =  2 cos 0 /':,0  0.02  dt dt
This can be put in the statespace form by defining /':,0 , /':,w state variables.
o  lOcos5°
/':,5
as the
1  0.1
For 5° = 0' = ~, eigenvalues of this matrix are A1,2 =  0.05 ± j2.942. It is a stable equilibrium point called the focus. For 5° = 5u or 6u , the eigenvalues
CHAPTER 9. ENERGY FUNCTION METHODS
302
are )" = 2.993 and )" = 2 .8 93. Both are saddle points. Since there is only one eigenvalue in the righthalf plane , it is called a T ype 1 u.e.p.
o Example 9.3 Construct the energy function for Example 9.2. Verify the stability of the equilibrium points by using (9.39) . T he energy function is constructed for the undamped system, i.e., the ~oefficient of ~~ is set equal to zero . M = 0.2, P= = 1, p:;nax = 2, 0' = ~. The energy function is 7r
1
V(o,w) = 2 (0 .2)w'  1(07r/6)  2(cosOcos"6) = O.lw'  (0 
~)  2(coso  (0.866) )
i) 2(coso ° 0' °< o.
VPE (O,O') =  (0 2
8 VPE(0 , 0')
80
= 2 cos
At 0 = 0' = 7r / 6, 2 cos 0 > 0; hence = 0" = S6~ or g" = ~" , 2 cos equilibrium points.
0.866 )
°
is a stable equilibrium point . At Hence , both 0" and gu are unstable
o 9.6.3
Equalarea criterion and the energy function
The prefault , faulted, and postfault power angle curves P e for the singlemachine infinitebus system are shown in Figure 9.9. The system is initially at = 0° . We shall now show that the area A, represent s the kinetic energy injected into the sy stem during the fault, which is the same as V!fE in Figure 9.8. A, represents the ability of the postfault system to absorb th.is energy. In terms of Figure 9.8, A2 represents VPE(O") VpE (6cl). By the equalarea criterion, the system is stable if A, < A 2 . Let the faulted and postfault equations, respectively, be
°
2
M ddt 0 = Pm. 2

pF·
c e Sllu
(9.40 )
9.6. POTENTIAL ENERGY BOUNDARY SURFACE (PEBS) /
303
Prefault
P,max
Pos lfault
Faulted
8U
Figure 9.9: Equalarea criterion for the SMIB case and
(9. 41 ) where
and
pmax = E,E2 e
X
The area A, is given by
(9.42)
CHAPTER 9. ENERCY FUNCTION METHODS
301
Hence, AJ is the kineti c energy injected into the system due to the fault. Area A2 is gi ven by
 P~(b"
_ VPE(a U )

ael )

VPE(a el )
from (9 .34). If we add area A3 to both sides of the criterion AJ < A 2 , the result is
(9.43 ) Now
A3 =
1"'·'
=  Pm Changing bel, wel to any
A,
+ A3
( P;'"' sin a
P~)do
(ocl _ b')  r;'ax (cosa el a, w iUld ad ding A,
cosb')
(9 .44 )
to A 3 , gives
1
= 2Mw2  P~(o  6·')  p;nax( coso  coso' )
(9.45 )
This is the same as V(b , w) as in (9.35) . Now, frolll Figure 9.9:
A2 I A3 = r  "(P;naXsino  Pm)db
1"
=:
2P;laXcos6s  Pm(7r 
2o~)
(946)
The righthand side of (9.46 ) is also verified to be the sum of the areas A2 and A 3 , for which analytical express ions have been derived. It may be verified from (9.35) that
V(8,w)
I,.," w:..:O
(9.4 7)
9.6. POTENTIAL ENERGY BOUNDARY SURFACE (PEBS)
305
+ A2
< A2 + A 3 ,
Thus, the equalarea criterion Al < A2 is equivalent to Al which in turn is equivalent to V(o,w) 0 [93]. In the absence of transfer conductances, f(8) = aV~:(8). When 8 is away from 0', within the potential multidinlensional "well," aV',;~(8) (which is the
0'»
gradient of the potential energy function) and iJ (i.e., (8 are both > O. Hence, fT(O) . iJ < 0 inside the "well." Outside the "well," 0  0' is > 0 and av (e) a~ < 0, resulting in f T (0)·0• > O. On the PEBS, the product f T (0)·0• is equal to zero. The steps to compute t~ using the PEBS method are as follows: 1. Compute the postfault s.e.p 0' by solving (9.23) .
2. Compute the faulton trajectory given by (9.21). 3. Monitor fT(O) . ii and VPE(O) at each time step. The parameters in f( 0) and VPE( 0) pertain to the postfault configuration. 4. Inside the potential "well" fT(O). {) < O. Continue steps 2 and 3 until fT(O). (} = O. This is the PEBS crossing (t·,O·,w*). At this point, find VPE(O*). This is a good estimate of V~ for that fault. 5. Find when V(e,w) = Vcr from the faulton trajectory. This gives a good estimate of ter. One can replace steps 3 and 4 by monitoring when VfJ'iX(O) is reached, and taking it as Ver. There will be some error in either of the algorithms.
9.6 .5
Initialization of VPECB) and its use in PEBS method
In this section , we outline a further simplification of the PEBS method that works well in m any cases, particularly when 0' is "close" to 80 • While integrating the faulted trajectory given by (9.21), the initial conditions on the states are 0;(0) = Of and Wi(O) = O. In the energy function (9.29), the reference angle and velocity variables are 0: and W;(O) = O. Thus, at t = 0, we evaluate VPE(O) in (9.29) as
CHAPTER 9. ENERGY FUNCTION METHODS
310
The pathdependent integral term in (9.54) is evaluated using the trapezoidal rule:
~Dij
[,j(O) =
[cos( Oi'  OJ)
+ cos(O; 
Oil] [(Oi'
+ OJ) 
(Oi
+ OJ )]
(9.55)
If the postfanlt network is the same as the prefault network , then K = O. Ot herwise, this value of K should be included in the energy function. If one uses the potential energy boundary surface (PEBS) method, then when the postfault network is not equal to the prefault network, this term can be subtracted from the energy func tion , i.e. ,
V (O,w)
=
VKE (W)
+VPE(O) 
VPE(OO)
(9.56)
Hence, the potential energy can be defined, with 0° as the datum, as
VPE(O)
6
VPE (O)  VPE (OO)
=
[t, t
= 
L 18'' f,(O)dO, m
i=l
J;(O)dO, 
t, l;r
f,(O )dO,] (9.57)
8° i
If the pathdependent integral term in (9.57) is evaluated, using trapezoidal integration as in (9.51), [,j(O) = O. At the PEBS crossing 0* , VPE(O*) gives a good approximation to Vcr' The PEBS crossing has been shown as approximately the p oint at which the p otential energy VPE reaches a maximum value. Hence, one can directly monitor VPE and thus avoid having
9.6.
POTENTIAL ENER.GY BOUNDARY SURFACE (PEBS)
311
to m onitor the dot product fT (0) . (0  0') as in step 4 of the previous sectio n. This leads t o the importan t advantage of not having to compute Ir. In fas t screening of contingencies , this coulcl result in a significant saving of compu tation . On largescale system s, this has not been investigated in the literature so fa r. Example 9.5 Compute the Yint for Example 7.1, usin g the classical mo del for the fault at bus 7 follo wed clearing lines of 7 5. Using the PEES method, comput e ler. Use fT(O) . iJ as the criterion for PEBS crossing.
Y int wit h fault at bus 7 (faulted system )
=
In the y,:'ue; m a tri x of Example 7.6, si nce Vs 0, we delete the row and column corresponding t o b us 5 . Then eliminate aU buses except the internal no des 10 , 11, and 12. The resul t is
yfnt
0.6588  j3.8 175 0.0000  jO.OOOO = 0.0000  j O.OOOO 0.0000  j5.4855 [ 0.0 714 j jO.6296 0.0000  jO.OOOO
0.0714 + jO.6296 0.0000  jO.OOOO 0.1750  j2 .7966
J
Y int with lines 7 5 cleared (postrault system) Ybll s = Y N is firs t computed with lincs 7 5 removed, and the rest of the steps a re as in Example 7.6 . Buses 1 to 9 are eliminated, resulting in
+
1.1411   j2.2980
0.1323 jO.7035 Yint = 0.1323 + jO.7035 0.3810  j2 .0202 [ 0. 1854 + j 1.0611 0.1965 + j 1. 2031
=
0.1854 I· j1.0611 0. 1965 + j1.2031 0.2723  j2.3544
=
J
The initial rotor angles are 51 (0) 0.0396 rad , 52 (0) 0.344 rad , and 63 (0) = 0.23 rad. The COA is calculated as 50 = T 2:l=l M;6;(0) = 0.116 rad, where MT = Ml + M2 + M 3 · Hence, we have 01 (0) = 61 (0)  60 = 0.0764 rad, O2 (0) 52 (0)  50 0.229 Tad, and B3 ( 0) = 53(0)  50 0.114 rad, W, (0) = W2(0) = W3(0) = O. The postfault s.e.p is calculated as OJ = 0.1649, O2 = 0.4987, B~ = 0.2344 . The steps in computing ter are given below.
=
=
J
=
CHAPTER 9. ENERGY FUNCTION METHODS
312
1. From the entries in Yint for faulted and postfault systems, the a.ppropriate C.; and D,;'s are calculated to pu t the equations in the form of
(9 .21) and 19.22).
2. V(O,w) is given by VKE+ VPE(O), where VKE = tM,w; and VPE(O) is given by (9.49) . The pathintegral term is evaluated as in (9.51), with 1,;(0) = 0, and the term (9.55) is added to VPE(O) . 3. The faulted system corresponding to (9 .21) is integrated and at each time step V(O, w) as well as VPE(O) are computed . Also the dot product p '(U) ·8 is monitored. The plots of V(O,W) an d VPE(O) are shown in Figure 9.11. Figure 9.12 shows the plot of jT(O) . O.
4. Vf!kax = 1.3269 is reached at approximate by 0.36 sec. Note from Figure 9.12 that the zero crossing of JT(O) . 0 occurs at approximately the same time. 5. From the graph for V(O,w),
tCT
= 0.199 sec when V(O,w) = 1.3269.
I
: . . '" ...._.i S ......_..... ...! ...... ......•.•.....  ."''" '" .  " ' ,i ..  ,.
. 4 _ .. __.,/_ ...,
3
.... ... ...... : ...
H
••• _
' ! j
i .'····
,.(
i ,.._. t···__·.. . . L .............. ~........ ·······"i'··:~;/:.: . . 1···· .
iI
r
.1'.'
H
•
I'
· ··r·_··+· . ·,······· r)::.··/+· · ··1. . I
/
.
······t_····· ·······T ...... ....; ...... _ .......... :.•",~'+. . . .··. _·t_· .. 1"'" . _, :
2
;
i,
1
I
I
' ,
:.i~l,~t~~ t~. Time(scc)
Total and potential energies: (a) V(6, w) (dashed line); (b) VPE(O) (solid line) Figure 9.11:
9.7. THE BOUNDARY CONTROLLING U.E.P (BCU) METHOD
313
3
...... . + .. .. . . . . . . . .... . . j
2.5 2
1.5
.... !
T' ............_. ". ..
'? . . _.
•••• _H"
... _, ._.
.....
.§
j ~
til
"
j "
0.5
~
.
..
0
 ._
0.5
.... .;. ....
",.

•......... _.. .....•.~ .
. . ... .
;. ..... .
~
..... t; ... ,' ...........: .... ,..
1
M
• • • • • ••
 1. 5 ':::.;:.,..,...::'::::'::~:_;__:_:_:__::_:__;:
o
0.05
11.1
0.1 5
0.2
0.25
0.3
0.35
(),4
Tillle(sec)
Figure 9.12: The monitoring of the PEBS crossing by fT(O) . iJ
o
9.7
The Boundary Controlling u.e.p (BCU) Method
This method [98] provided another breakthrough in ap plying energy function m ethods to stability analysis aft er the work of Athay et al. [97], which originally proposed the con trolling u.e.p method. The equations of the postfault system (9.22) can be put in t he statespace form as
() = Wi
Mi ~i = fi (O)
aVPE(O) aOi
i = 1) .. " m
Now
.
av·
av .
V(O,w) = ao 8 + aw w
(9.58 )
314
CHAPTER 9. ENERGY FUNCTION METHODS
= 
m
m
i= ]
i=l
2: fi(IJ)Oi + 2: M,w/;)i
m
(9.59)
=0
Hence, V( 0, w), is a valid energy function. The equilibrium points of (9.58) lie on the subspace IJ,w such that e.Rm, w o. In the previous section, we have qualitatively characterized the PEBS as the hypersurfaces connecting the u .e.p's. We make it somewhat more precise now. Consider the gradient system
=
. IJ
=
iWPE(O)
(9.60 )
8IJ
Note that the gradient system has dimension m, which is half the order of the system (9.58 ). It has been shown by Chiang et al. [98J that the region of attraction of (9.58) is the union of the stable manifolds of u .e.p's lying on the stability boundary. If this region of attraction is projected onto the angle space, it can be characterized by
(9.61) where IJF is an u.e.p on the stability boundary in the angle space. The stable manifold W'(IJr" ) of Or" is defined as the set of trajectories that converge to IJr" as t > +00. Since the gradient of VPE(O) is a vector orthogonal to t he level surfaces VPE(O) = constant in the direction of increased values of VPE (O), t h e PEES in the direction of decreasing values of VPE(O) can be described by the differential equations iJ aV~~(8) f(IJ) . Hence , when the fault on trajectory reaches the PEBS at e = 0* corresponding to t = t*, we can integrate the set of equations for t > t* as
=
iJ
=
frO), B(t*) = IJ*
=
(9 .62)
where f(IJ) pertains to the postfault system . This will take IJ(t) along the PEBS to the saddle points (u.e .p 's U1 or Uz in Figure 9.10 depending on
9.7. THE BOUNDARY CONTROLLING U.E.P (BCU) METHOD
315
0') . The integration of (9.62 ) requires very small time steps since it is "stiff." Hence , we stop the integration until II f(O) " is minimum. At this point , let 0 =0 0ltpp ' If we need the exact 0" , we can solve for f (O) =0 0 in (9.23) using Oltpp as an initial guess . The BCU method is now explained in an algorithmic manner. Algorithm 1. For a given contingency that in volves ei ther line switching or load/ generation change, compute the postdisturbance s.e.p . 0' as follows. The s.e.p and u.e.p 's are solutions of tbe real power equations
MO) Since Om
=0
=0
M. p.  p•• (O)   Peor (O)
M~
MT
=0
0 i
=0
1, .. . , m
(9.63)
z=:::;;:1 M.O. , it is sufficient t o solve for 1;(0) = 0 i=ol, ... , ml
(9.64)
witb Om being substituted in (9 .64) in terms of 01 , ... ,Om 1. Generally, the s.e.p. 0' is close to 0° the prefault e.p . Hence, using 0° as the starting point , (9.64) can be solved using the NewtonRaphson method. 2. Next , compute the controlling u.e.p . 0" as follows:
(a) IntegTatethe faulted system (9 .21 ) and compute V(O,W) = VKE(W) + VPE (O) given in (9.52) at each time step. As in the PEBS algori thm of the previous section, determine when the PEBS is crossed at 0 =0 0' corresponding t o t =0 t'. Tills is best done by ftnding when fT(O) . (0  0') =0 O. (b) After the PEBS is crossed, the faulted swing equations aTe no longer integrated . Instead, the gTadient system (9 .62) of the postfault system is used. This is a reducedorder system in that ouly the 0 dynamics are considered as explained earlier, i.e., for t > t'
iJ = f(O) ,
ott')
=0
o·
(9 .65)
CHAPTER 9. ENERGY FUNCTION METHODS
316
Equation (9.65) is integrated while looking for a minimum of m
E l /i(B) 1
(9.66)
i=l
=
At the first minimum of the norm given by (9.66), B Bapp and VPE(B app ) Vcr is a good approximation to the critical energy of the system. The value of /1':i p p is almost the relevant or the controlling u.e.p.
=
=
(c) The exact u.e.p can be obtained by solving I(B) 0 and using Oapp as a starting point to arrive at B" . Note that since I( B) is nonlinear, some type of minimization routine must be used to arrive at B". Generally, Bapp is so close to B" that it makes very little difference in the value of Vcr whether B" or Bapp is used .
3. Vcr is approximated as Vcr = V(O",O) = VPE(B") . Because of the path.dependent integral term in VPE, this compu· tation also involves approximation. Unlike computing VPE(B) from the faulted trajectory where B was known, here we do not know the trajectory from the full system . Hence, an approximation has to be used. The most convenient one is the straight·line path of integration. VPE(B") is evaluated as [97]:
m
rn
ml
E
Pi(Or 
On  E
E
[eij cos (Bij  cos Bfj)
i=l j=i+l
( B1'  0» + (B"  B') ( )] (Br  Bn _ (OJ _ OJ) Dij sm Bij  sm °ij 1.
1.
' ]
'U'6
(9 .67)
We derive the third term of (9.67) as follows. Assume a ray from Of to and then any point on the ray is Bi = Bi + p(B; is discussed in [120].
=
o
CHAPTER 9. ENERGY FTJNCTION METHODS
318
J.:! ].1 5
11
.
.~ . _ .. .......
..L.
... , ...
lJJ.~
..................
~
0
E 0
"/.
.1..., ,
o.<JS
.1
I). !J
!
.. _,
OX:;
.j....
n , /{
0..:;:
1t.7
0.6
tl.l