Handbook of Networks in Power Systems II

Energy Systems Series Editor: Panos M. Pardalos, University of Florida, USA

For further volumes: http://www.springer.com/series/8368

.

Alexey Sorokin Steffen Rebennack Panos M. Pardalos Niko A. Iliadis Mario V.F. Pereira l

l

l

l

Editors

Handbook of Networks in Power Systems II

Editors Alexey Sorokin University of Florida Industrial and Systems Engineering Weil Hall 303 32611 Gainesville Florida USA [email protected]

Steffen Rebennack Colorado School of Mines Division of Economics and Business Engineering Hall 15th Street 816 80401 Golden Colorado USA [email protected]

Panos M. Pardalos University of Florida Dept. Industrial & Systems Engineering Weil Hall 303 32611-6595 Gainesville Florida USA [email protected]

Niko A. Iliadis EnerCoRD - Energy Consulting Research & Development Plastira Street 4 171 21 Athens Nea Smyrni Greece [email protected]

Mario V.F. Pereira Centro Empresarial Rio Praia de Botafogo -A-Botafogo 2281701 22250-040 Rio de Janeiro Rio de Janeiro Brazil [email protected]

ISSN 1867-8998 e-ISSN 1867-9005 ISBN 978-3-642-23405-7 e-ISBN 978-3-642-23406-4 DOI 10.1007/978-3-642-23406-4 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2012930379 # Springer-Verlag Berlin Heidelberg 2012 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Handbook of Networks in Power Systems: Optimization, Modeling, Simulation and Economic Aspects

This handbook is a continuation of our efforts to gather state-of-the-art research on power systems topics in Operations Research. Specifically, this handbook focuses on aspects of power system networks optimization and is, as such, a specialization of the broader “Handbook of Power Systems I & II,” published by Springer in 2010. For decades, power systems have been playing an important role in humanity. Industrialization has made energy consumption an inevitable part of daily life. Due to our dependence on fuel sources and our large demand for energy, power systems have become interdependent networks rather than remaining independent energy producers. Such dependence has revealed many potential economic and operational challenges with energy usage and the need for scientific research in this area. In addition to fundamental difficulties arising in power systems operation, the industry has experienced significant economic changes; specifically, the power industry has transformed from being controlled by government monopolies to becoming deregulated in many countries. Such substantial changes have brought new challenges in that many market participants maximize their own profit. The challenges mentioned above are categorized in this book according to network type: Electricity Network, Gas Network, and Network Interactions. Electricity Networks constitute the largest and most varied section of the handbook. Electricity has become an inevitable component of human life. An overwhelming human dependence on electricity presents the challenge of determining a reliable and secure energy supply. The deregulation of the electricity sector in many countries introduces financial aspects such as forecasting electricity prices, determining future investments and increasing the efficiency of the current power grid through network expansion and transmission switching. The Gas Networks section of the book addresses the problem of modeling gas flow, based on the type of gas, through a pipeline network. The section describes the

v

vi

Handbook of Networks in Power Systems

problem of long-term network expansion as well as the optimal location of network supplies. Deregulation of the gas sector is becoming common in many countries. The deregulation presents new decisions to the gas industry including determining optimal market dispatch and nodal prices. Network Interactions are common in power systems. This section of the book addresses the interaction between gas and electricity networks. The development of natural gas fired power plants has significantly increased interdependence between these two types of networks. This handbook is divided into two volumes. The first volume focuses solely on electricity networks, while the second volume covers gas networks, and network interactions. We thank all contributors and anonymous referees for their expertise in providing constructive comments, which helped to improve the quality of this volume. Furthermore, we thank the publisher for helping to produce this handbook. Alexey Sorokin Steffen Rebennack Panos M. Pardalos Niko A. Iliadis Mario V.F. Pereira

Contents

Part I

Gas Network

Implementation of a Scheduling and Pricing Model for Natural Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 W. Pepper, B.J. Ring, E.G. Read, and S.R. Starkey Long-Term Pressure-Stage Comprehensive Planning of Natural Gas Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Michael Hu¨bner and Hans-Ju¨rgen Haubrich Optimal Location of Gas Supply Units in Natural Gas System Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Teresa Nogueira and Zita Vale An LP Based Market Design for Natural Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 E.G. Read, B.J. Ring, S.R. Starkey, and W. Pepper Part II

Network Interactions

Energy Carrier Networks: Interactions and Integrated Operational Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Ricardo Rubio-Barros, Diego Ojeda-Esteybar, and Alberto Vargas

vii

viii

Contents

Costs and Constraints of Transporting and Storing Primary Energy for Electricity Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Sarah M. Ryan and Yan Wang Integrated Optimization of Grid-Bound Energy Supply Systems . . . . . . . . 187 Simon Prousch, Hans-Ju¨rgen Haubrich, and Albert Moser Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

.

Part I

Gas Network

Implementation of a Scheduling and Pricing Model for Natural Gas W. Pepper, B.J. Ring, E.G. Read, and S.R. Starkey

Abstract Since 1999, the Australian state of Victoria has operated a natural gas spot market to both determine daily prices for natural gas and develop an optimal schedule for the market based on an LP (Linear Programming) approximation to the underlying inter-temporal nonlinear aspects of the gas flow optimization problem. This market employs a dispatch optimization model and a related market clearing model. Here we present the model employed for both the operational scheduling and price determination. The basic dispatch optimization formulation covers the key physical relationships between pressure, flow, storage, with flow controlled by valves, and assisted by compressors, where flow and storage are measured with respect to energy rather than in terms of mass. But we also discuss a range of sophisticated mathematical techniques which have had to be employed to create a practical dispatch tool, including iterating between piecewise and successive linearization; iterating between barrier and simplex algorithms to manage numerical accuracy and solution speed issues, and special methods developed to deal with scheduling flexibility. The market clearing model is a variation on the dispatch optimization model which replaces the gas network with an infinite storage tank with unlimited transport capacity. We address the performance of the model including accuracy and run time.

The authors wish to thank the Australian Energy Market Operator (AEMO) for providing information used in this chapter and for review of the content. W. Pepper ICF International, Fairfax, Virginia, USA e-mail: [email protected] B.J. Ring Market Reform, Melbourne, Australia e-mail: [email protected] E.G. Read (*) • S.R. Starkey University of Canterbury, Christchurch, New Zealand e-mail: [email protected]; [email protected]; [email protected] A. Sorokin et al. (eds.), Handbook of Networks in Power Systems II, Energy Systems, DOI 10.1007/978-3-642-23406-4_1, # Springer-Verlag Berlin Heidelberg 2012

3

4

W. Pepper et al.

Keywords Linear Programming (LP) • Linearization • Market • Natural gas • Optimization • Pipelines • Prices

1 Introduction Since 1999 a gas market has been operating in the Australian state of Victoria, based on a “common carriage”1 model, similar to that employed in many electricity markets. The motivation for introduction of that market concept has been discussed by Read et al. [1], who demonstrate that, conceptually, such a market can be based on an LP formulation analogous to that employed in electricity markets. That paper develops and interprets an LP formulation based on a textbook representation of the gas flow equations, and does not deal fully with some real-life complexities, such as the potential non-convexity of some functional relationships. Thus that formulation is not the one actually employed for dispatch purposes in the Victorian gas market. Further, while the gas market concept developed by Read et al., like the electricity market concepts on which it was based, would produce different prices for each system injection/extraction point in each trading interval, market participants in Victoria have preferred to trade on a much simpler basis, which captures some of the inter-temporal variation, but treats all gas, anywhere in the system, as interchangeable, within each day. While Read et al. [1] describes a conceptual model this paper describes the model implemented in the working market and operational dispatch models as previously presented by Pepper and Lo [20]. The Victorian market produces both the physical dispatch schedules and the market price are determined in accordance with the Market Clearing Logic described by Ruff [2] which called for optimization of the physical gas dispatch in the context of market driven supply and demand to form a realistic constrained schedule of hourly injections and withdrawals of natural gas into/from the pipeline system, alongside calculation of another hypothetical unconstrained schedule to determine the daily market prices and schedules used in the settlement of the market. These features of the Market Clearing Logic were implemented within the Market Clearing Engine (MCE) developed by ICF International [3]. The MCE can be run to produce a physical gas dispatch, the so called ‘Operational Model’, or to determine market prices and trading schedules, the so called ‘Market Model’. Both of these models are run by the gas system operator, originally VenCorp, now absorbed into the Australian Energy Market Operator (AEMO). The Operational Model includes a detailed representation of the physical gas system and can optimize supply, based on supplier injection bids, purchaser withdrawal bids,

1 The use of common carriage is due to the similarities with electricity markets. Victoria actually use the term “market carriage” to represent its pool based market, contrasting with “contract carriage” which relates to more traditional scheduling under bilateral contracts for access to each individual transmission pipelines.

Implementation of a Scheduling and Pricing Model for Natural Gas

5

and non-price sensitive uncontrollable withdrawal, by hour and location (node) over the gas day. The Operational Model represents network storage (linepack) constraints, constraints on minimum and maximum allowable pressures along the pipeline system, network devices such as compressors, regulators, and check valves, as well as physical and operational constraints on supply and demand side network entry and exit points (reflecting the fact that there can be multiple participants trading through these points so that individual participants cannot manage these limits). The Operational Model produces an operational schedule which is the primary basis for scheduling gas. All gas flow and storage quantities are represented in terms of energy rather than mass, since trading mass is inadequate in a market context if different gas sources have different energy content. The basic underlying equations, as derived in the Appendix, are based on kg and kg/s, but all calculated quantities are converted to GJ and GJ/h before use in the optimization. The Operational Model also produces indicative nodal prices for natural gas, and these could be used as a basis for market trading, as discussed by Read et al. On the other hand, the Operational Model has also proved to be a highly effective dispatch tool in its own right. Thus this kind of LP formulation represents a viable approach to gas system dispatch optimization, anywhere, and irrespective of any market developments. The Market Model is functionally very similar to the Operational Model except that the gas network is represented by an ‘infinite tank’. In effect, no storage limits are represented and negative storage inventories are even allowed during the middle of the gas day. Gas is assumed to be able to move from any point in the network, at a given point in time, to any other point in the network at any other time within the gas day. This has the effect that gas prices are the same throughout the network in all time periods. This price is used to settle the market, with constrained-on payments funded through an uplift charge on participants, being made when constraints force higher cost gas to be supplied or lower value demand to occur in the operational schedule. There are no constrained-off payments made. From 1999 to 2007 the Operational Model was run at the start of the gas day, and as required during the gas day to provide updated schedules for the remainder of that day. The Market Model was run to produce indicative prices whenever the Operational Model was run, but settlement energy prices were based on an ex post run of the Market Model after the gas day, with demand based on the actual on-theday conditions that had occurred. Thus there was one price for the entire gas day and this was based on actual gas flows. This concept of ex post pricing is used in numerous markets, and is described in more detail in the context of the New Zealand electricity market by. Since 2007, the gas day has been divided into five intervals with a different price applying in each interval. But prices are not determined for all five intervals when the model is first run for each day. The first interval commences at 6 a.m. and lasts 24 h. The Operational Model and the Market Model are both run for that interval, with the Market Model’s ex ante price being used to settle the market, based on the gas scheduled for the entire 24 h (not gas as eventually flowed). The second interval

6

W. Pepper et al.

runs from 10 a.m. for 20 h. Both models are run again, with a new schedule and a new ex ante price determined. This price is used to settle differences between actual gas flows and scheduled gas flows from 6 a.m. to 10 a.m., and settles all changes in scheduled gas flows for the 20 h from 10 a.m.. In effect, the 6 a.m. schedule provides a form of forward market for the 10 a.m. schedule. Similar processes are repeated at 2 p.m., 6 p.m. and 10 p.m., the last schedule covering the remaining hours from 10 p.m. to 6 a.m. the next day. The Operational Model can also be run at other times if required. The MCE seems unique in many ways. Aspects of its function are captured in other models, however. Zhu et al. [4] note that ‘the literature on gas pipeline control is rather sparse’, yet they do make note of two commercial software packages for dynamic gas pipeline simulation, the Hyprotech PYPESYS and the Gregg Engineering WinTran model. Johnson et al. [5] discuss the use of WinTran model for the Tennessee Gas Pipeline which consists of over 15,000 miles of pipeline and 70 compressor stations. These models determine the (operational) gas flows, and WinTran optimizes compressor fuel use, yet they have no explicit concept of costs or economics. ICF International has two other models of gas pipeline markets that have a much different focus than the MCE. The ICF Gas Market Model [6] represents the North American Gas Market focusing on finding a market equilibrium and uses costs and economics but the daily pipeline capacities are fixed. Similarly, the ICF RIAMS model provides detailed regional modeling of pipeline capacity and determines a market equilibrium using costs and economics but again uses fixed estimates of daily pipeline capacity. The Scheduling and Pricing Engine (SPE) employed in AEMO’s Short Term Trading Model (STTM) [7] for gas trading at hubs in Adelaide and Sydney (Australia) can be thought of as a form of Market Model which determines a single price and schedule for the day, but only represents a simple capacity limit on the ability of transmission pipelines to supply a hub. Reviews of recent literature may be found in Read et al. [1] and Zheng et al. [8]. The latter note optimization models being applied to gas production, and gas pipeline network development and operations, including optimization of gas compressor operation to minimize fuel use, which may be seen as a sub-problem of the application discussed here. Both Peretti and Toth [19] and Rios-Mercado [21] use Dynamic Programming to optimize fuel use. In particular, Wu et al. [9] discuss the issues in considerable detail, although they take a different convexification approach than we do. Zheng et al also discuss formulation of the “least cost gas purchasing problem”, which deals with the same physical flow requirements as our problem. De Wolf and Smeers [10] suggest a solution strategy broadly similar to our own, based on successive application and refinement of a piece-wise linearization approach. But their approach to piece-wise linearization [11] is different from ours. Several more recent papers describe practical models developed for the Norwegian Gas Industry. Somewhat like Read et al. [1], Tomasgard et al. [12] and Midthun et al. [13] both discuss linearization strategies to deal with non-linear pipe flow losses using Taylor’s expansions, and note that this strategy “makes analyses computationally feasible even for large networks”.

Implementation of a Scheduling and Pricing Model for Natural Gas

7

But the linearization approach we have adopted here uses the “lambda”, or “convex combination” method, as discussed by Martin et al. [14]. For a recent PhD thesis on gas flow linearization techniques see Van der Hoeven [15]. The current paper differs from all of the above in that it describes a Market Clearing Engine that has been specifically implemented for the purpose of determining operational schedules, and clearing short term gas market trading on a routine basis. After describing the physical gas system involved, we present an outline of the Operational Model formulation. This model was developed on a different and more pragmatic basis than the formulation of Read et al. [1], partly because it was unclear how easy it would be to tune the more theoretical formulation of Read et al to produce accurate results in an acceptable computation time. Thus the priority for the MCE was that it had to be developed commercially in a limited timeframe to meet specific performance requirements. But the MCE, as implemented in its “Operational Model” form, is capable of producing hourly nodal prices, as discussed by Read et al. On the other hand, we later describe the “Market Model” form of the MCE, which determines the much more aggregated prices actually used for trading by solving a highly simplified special case of the Operational Model.

2 The Physical Gas Transmission System The Victorian Gas Market is a spot market developed to facilitate the trade of gas between privately owned wholesale suppliers of gas and privately owned retailers and industrial users of gas across the Principal Transmission System (PTS), the core gas transmission network.2 As shown in Fig. 1, this is a meshed network rather than a single long pipeline. The major market participants are both suppliers of gas and purchasers of gas so are only exposed to spot prices to the extent that their supply and usage are out of balance. At its commencement, the market was characterized by a large demand around Melbourne supplied by a single gas production facility, Longford (represented by ESSO at the Pre-Longford node), approximately 160 km from Melbourne. The capacity of the Longford to Melbourne pipeline is small enough that the pipe’s throughput could be constrained within a day. If this occurred, typically during peak demand winter days, then the only alternative supply source was from an LNG facility, with limited storage, at the Dandenong City Gate (DCG Inlet) within Melbourne. Since market commencement, the PTS has become interconnected with South Australia (SEAGas Pipeline connected at Pre-Iona), New South Wales (EAPL Pipeline connected at Pre-Culcairn) and, via the VicHub (at the PreLongford node), with pipelines (not shown) linking to Tasmania (Tasmanian Gas Pipeline) and New South Wales (Eastern Gas Pipeline). An underground storage

2 As of the Victorian gas market recently coming under Australia’ National Gas Rules the PTS is now called the DTS, or Determined Transmission System.

8

W. Pepper et al.

EAPL

Brooklyn Compressor Station and City Gates

Supply Source

Northern Zone

Bendigo

Brooklyn BL

Legend

Node yn kl oo Br C ok

Bro

Check Valve

Wodonga

Brooklyn – Melb

) 2B (M

eg R

eg

R lyn

Regulator

Clonbinane

Bendigo Junc

S

BBP Reg

P BL

Brooklyn CS (G2B)

BLP Reg Inlet

Springhurst Outlet Inlet

Compressor

Kilmore Wollert

Brooklyn GL

Outlet

Inlet

Melbourne Zone Ballarat Junc

Pre-Iona

LNG

BASSGas

Gooding

Brooklyn Compressor Station and City Gates

Pakenham

Ballarat

DCG Inlet

Ballarat Zone

Traralgon West Dup Outlet

P BL Lara Reg

Inlet

Lara Iona

Lurgi

Geelong Zone

Pre-Longford

ESSO VicHub

Geelong

DTS Inlet

WUGS SEAGas Otway

Lara Reg Inlet

Longford

Gippsland Zone

Fig. 1 Victorian gas network schematic (Source: AEMO)

field (WUGS connected at pre-Iona) which takes in gas at times of low prices and releases it at times of high price has also been established. Gas fields off the south cost of Victoria have also connected (Ottway at Pre-Iona and BassGas at Pakenham) to the PTS. The major supply node point is still Pre-Longford, though. Major compressor stations, which effectively pump gas between regions of the PTS, are installed at Brooklyn, Wollert, Springhurst, and Gooding. These compressors are powered by a small quantity of the gas withdrawn from the network. According to Wu et al. [9] “compressor stations typically consume about 3–5% of the transported gas”. In this case, though, total compressor gas usage is relatively small. For a 1,200 TJ peak demand scenario with all compressor stations running no more than 3 TJ, or 0.25% of daily demand, is consumed by compressors. Since the commencement of the market, the demand for gas has increased, particularly in the form of exports to surrounding jurisdictions and increased use of gas fired power generation. Gas powered generators (GPGs) are located at Traralgon (Jeerlang and Valley Power GPGs), Brooklyn-Melbourne (Newport GPG), Brooklyn-Geelong (Laverton GPG) and Wollerton Outlet (AGL Somerton GPG). With all generators running, the GPG demand is about 5–10% of daily demand, and 20% of demand during the critical evening period, on a winter peak demand day.3 AEMO is both a system and market operator. It runs the market and

3

Based on information supplied by AEMO for winters 2007 to 2009 inclusive

Implementation of a Scheduling and Pricing Model for Natural Gas

9

has the duty to manage the system so as to maintain secure operation of the gas system. Principally this entails managing gas stored within the PTS so as to ensure that supply and demand match. The core issues which AEMO variously deal with are forecasting demand, dealing with the characteristics of various supply subsystems, managing the use of the transmission system, and determining an appropriate end of day storage pattern. Participants in the Victorian Gas Market are typically retailers who have contracts with physical suppliers of gas. When scheduled in the Victorian Gas Market, these participants must nominate gas inflow levels from physical suppliers in accordance with their contracts. Operators of gas processing plants and interconnected pipelines typically like to keep gas flows relatively constant over the day, but may allow gas flow rates to be changed several times during the day. This means that AEMO must match relatively inflexible supply rates with diurnally varying gas demand in Melbourne. The gas day begins at 6 a.m. with supply exceeding demand, causing the pressures in the PTS to rise through the morning. During the late afternoon, demand typically outstrips supply, and remains higher than supply rates until late in the evening, causing pressures to drop. Hence AEMO must manage the system to ensure that enough gas is supplied early enough to meet demand, and to keep pressures above minimum pressure levels especially during the evening peak period. Demand in the Victorian Gas System is quite variable. During the summer months the peak demand can be as low as 400 TJ per day, unless gas fired power generators are operating. At the peak of winter, demand can be in the region of 1,200 TJ per day. A sudden and significant weather change can produce large changes in demand. If enough gas has not been scheduled to flow to Melbourne from the various supply sources in advance of the change, then the time may not be available to schedule new supplies from producers or interconnected pipelines before minimum pipeline pressures are reached. Such ‘surprise’ events can force AEMO to call on LNG to meet demand. While typical gas prices are in the region of $3/GJ – $4/GJ, LNG may cost $10/GJ or as much as $800/GJ, which is the maximum market price. Hence the use of LNG imposes significant costs on the market. If LNG becomes exhausted, or cannot be vaporized fast enough,4 then demand curtailment of industrial users must be employed. If LNG runs out then the (constrained) market may be in shortage situations on peak days, and curtailment is priced at $800, hence the highest prices for LNG tend to approach $800. AEMO forecasts the demand for gas (aided by knowledge of participant forecasts, for which participants are financially accountable) at the start of the day, and updates that forecast throughout the day as the situation changes.

4 Storage of LNG is limited, and it is possible to use a significant proportion of the storage quite quickly if LNG is used too freely. Peak LNG vaporization for 1 h can take about 1 day to replace at the (emergency) maximum rate of LNG production, though more typically takes several days to replace.

10

W. Pepper et al.

Sections of the PTS can become constrained depending upon the diurnal schedule and location of injections and withdrawals. These constraints are typically pressure related and can reduce both the ability to supply consumers and the ability to take injections from suppliers. The minimum pressure of a pipeline reflects the pressure associated with minimum linepack storage. These limits cannot readily be breached – in 1998 an explosion at the Longford gas processing plant required the system to be shut down for several weeks to protect the system’s minimum pressures. The maximum pressure of a pipeline is related to the physical capabilities of the pipeline, though lower limits may be set for broader system operation reasons. Gas flows are primarily driven by the pressure differences across a pipeline, where a pipeline may be tens or hundreds of kilometers long. If gas is to flow in the desired direction, at the desired rate, then a specific pressure difference must be achieved. If demand is too great, relative to the scheduled supply, and no operator actions are taken, then consumption can literally suck gas from the pipeline at a rate which lowers the outlet pressure to the point where the outlet pressure hits its minimum level. If demand is too low, relative to the specific pressure difference, then the pressure at the demand end will rise and the pressure difference will drop. This will reduce the ability for gas to flow away from the injection point and may limit the ability to inject gas into the PTS. This can make it difficult to move enough gas through the system in time to meet increased consumption later. AEMO use gas powered compressor stations to pump gas around the network. The PTS is divided into zones (see Fig. 1), and compressors allow gas to be transferred from low to high pressure zones, effectively making gas flow against the natural direction of flow. They can also be used to ‘pump up’ a zone during offpeak periods so that gas will flow from it to Melbourne during peak periods. Pipelines are often configured with other useful fittings, such as regulator valves which are designed to either restrict flow, or adjust pressure differentials. Check valves may also be used in a pipe to restrict the gas flow to a single direction. At the commencement of the market, transmission constraints were rare, impacting only a few days per year and creating relatively small costs. The incidence and severity of constraints grew as demand grew, particularly from gas fired generation. More recently, network augmentation has increased the storage capacity of the PTS, thus increasing the ability of the system to meet demand at peak times, and significantly reduced the incidence and severity of constraints. A major consideration for AEMO in scheduling the system is the end of day storage. There is a relatively small range of end storage levels within which AEMO seeks to finish the day. A minimum storage must be maintained for system security reasons. Too high a storage level equates to high pressures around Melbourne and this may limit the ability of gas to be supplied from locations like Longford on the next day. In summary, then, AEMO must forecast demand and manage the scheduling of gas to manage both the daily peak demand and the end of day conditions so as to minimize the risk of surprise events requiring the use of LNG and to manage the timing of gas delivery and the positioning of stored gas to minimize transmission congestion.

Implementation of a Scheduling and Pricing Model for Natural Gas

11

3 Operational Model Formulation Read et al. [1] develop a general formulation based on a theoretical representation of gas mass moving through adjacent pipe-line segments, described as nodes. They also discuss the treatment of practical complexities including bi-directional flows, fitting losses, compressor behavior, and flow continuity through junctions. Here we describe the actual model, which addresses these issues in a generally more pragmatic fashion. Any operational system must balance a range of computational and practical limitations. Read et al. explicitly define the physics within pipe segments by layers of interconnected equations. This is useful in terms of providing a variety of insights, especially for engineers and pipeline operators, when looking at specific issues, such as flow losses in a given pipe segment, or compressor losses in a given zone. The challenge here, though, is not to micro-manage specific system issues, but to economically allocate the gas across the entire PTS, whilst ensuring compliance with key physical and operational pipeline and market constraints. Thus the major innovation of this formulation is to use rather longer pipe segments than those envisaged by Read et al., and to approximate the key pressure flow relationships over each pipeline segment, or system element, by fitting a convex linearization to what is essentially empirical performance data, as discussed later in Sect. 4. A comprehensive formulation for the Operational Model is beyond the space available in this paper. However, many Operational Model constraints are unrelated to the core issues, instead implementing constraints on the operation of particular injection or withdrawal points, or market rule requirements. Here we focus on the core gas flow equations, with some nonlinear relationships depicted generically, and assuming generic forms of bid constraints, and pipeline constraints. Notationally, this formulation is rather different from that of Read et al, but the basic relationships are essentially the same, and we will focus mainly on points of real difference.5 This formulation represents each pipeline segment as an arc, not a node. The nodes in this formulation occur at either end of a length of pipe. This change in the construction of the formulation allows us to more readily deal with a multiple interconnected network of meshed pipes and loops. It also allows key pressure/flow relationships to be defined over the length of each pipeline arc. Flows into a node include injections (from supply bids) and flows into that node from pipe segments. The flow into the node must balance flows out of the node, into pipelines, and withdrawals from the node, as in Fig. 2. Thus the mass conservation equation (1) is expressed in a different way from that of Read et al. Specifically, gas moves through a pipe segment from an origin node

5 In this formulation, variables are generally upper case, and constants lower case, whereas Read et al. [1] use lower case for variables, and upper case for constants. Upper and lower limits are still represented by over and under bars, respectively. Unless otherwise stated, all variables in this formulation are positive.

12

W. Pepper et al.

Fig. 2 Mass balance representation

of the pipe segment l, identified by o(l), to the corresponding destination node d(l). We denote the flow into and out of pipe segment l, in period t, as FIlt and FOtl respectively. Stm is gas scheduled to be supplied according to injection offer m, in period t while Dtc is gas scheduled to be withdrawn according to withdrawal bid c. Hence the operational MCE defines the pipe network mass balance as6: 0¼

X m:nðmÞ¼n

Stm 

X c:nðcÞ¼n

Dtc  UDtn þ

X l:dðlÞ¼n

FOtl 

X

FIlt þ slack : 8n; t

(1)

l:oðlÞ¼n

Here the expressions n(m) and n(c) respectively indicate the node associated with injection offer m and withdrawal bid c. Given each pipe segment has an origin node and a destination node, the MCE can model pipe segments as allowing flow to only move from the origin to the destination; the MCE can also allow for reversible flows by allowing the origin and destination pressures to be reversed. Although not evident in (1), the formulation also differs from that in Read et al. in that it is expressed in terms of energies not gas masses. Extra variables, UD, are added to represent a base load of “uncontrollable” demand. This accounts for a large proportion of total market demand. Here, and elsewhere in the formulation, the “slack” term indicates high priced violation variables, which ensure that the optimization can still solve, if all constraints on the problem cannot be satisfied.7 This allows the model to produce useful solutions

6

slack is a variable, yet it is also a vector, and as such it is stated in lower case. Although this terminology is common in such models these are not really “slack” variables in the traditional sense. They do indicate how far the final solution point is from the constraint, but it lies outside the feasible region, not inside. Thus they have the form of generalized slack variables, which are positive when the constraint is slack, but negative when it is violated. 7

Implementation of a Scheduling and Pricing Model for Natural Gas

13

in that situation, giving the dispatch schedule that most nearly meets all requirements, and indicating the location and extent of potential violation. Read et al. introduce specific gas flow relationships for segments within a pipeline, describing pressure/flow relationships, and also the way in which those relationships will change over time. It can be seen that, through a series of substitutions, flows along any length of pipeline will be related to the change in pressure across the length of the pipe. For the operational formulation; the flow into pipe segment l, in period t, is represented as a nonlinear function, Qo, of the origin and destination pressures, PBtoðlÞ and PBtdðlÞ respectively, for that pipe segment. Thus we implicitly define the pipe inflow as: FIlt ¼ Qo ðPBtoðlÞ ; PBtdðlÞ Þ

: 8l; t

(2)

The flow out of pipe segment l is defined in a similar fashion, being dependent again on the origin and destination pressures. The function Qd differs from Qo in that it also accounts for compressor fuel usage for pipe segments that have compressors, and the net change in useable linepack in the pipeline from period t-1 to t. If we let KNlt:t¼0 be the linepack in pipe segment l measured (or extrapolated) at the start of the optimization, we define flow out of the pipe segment via: FOtl ¼ Qd ðPBtoðlÞ ; PBtdðlÞ Þ  KNlt þ KNlt1

: 8l; t

(3)

FOtl will tend to differ from FIlt because of changes in linepack stored within the pipe segment or, in the case of compressors, because of fuel used in the segment. FIlt and FOtl will be non-negative for directional pipe segments, but may be negative for reversible pipe segments. Useable linepack in pipe segment l can also be expressed as a nonlinear function of the origin and destination pressures. If a pipe segment has a pressure regulator, then the origin pressure is modified by a throttle variable, TRtl . Regulators are modeled such that they are situated between an origin node and the corresponding pipe segment l, and are subject to a minimum regulator outlet pressure limit, opl . If the pressure at the node is PBtoðlÞ then this is the inlet pressure to a regulator, but the outlet pressure of the regulator, and hence the inlet pressure of the attached pipeline is PBtoðlÞ  TRtl . With TRtl set to zero for pipe segments without regulators the general equation for linepack on a pipe segment is KNlt ¼ MðPBtoðlÞ  TRtl ; PBtdðlÞ Þ : 8l; t

(4)

In addition, if “reg” denotes the set of pipe segments with regulators then for those pipe segments that have regulators the following limit applies. opl  PBtoðlÞ  TRtl þ slack

: l 2 reg

(5)

To manage linepack, we allow pipe segments to be assigned to zones, which may overlap, thus allowing us to state upper/lower linepack limits for each zone,

14

W. Pepper et al.

including an aggregate system linepack limit for the whole system, Zone 0. We can add “slack” variables to allow targeted end of day minimum linepack constraints for a given zone, mz , to be violated at a high penalty cost, and also a variable MLz that discourages, but does not prevent, end-of-day linepack exceeding targeted levels – it is non-negative and has a small penalty cost, ps, applied to it. So we have: mz ¼

X

KNlT þ

l2z

X

ðFIlt  FOtl Þ  MLz þ slack

: 8z

(6)

t;l2z

Read et al. [1] discuss a number of further constraints that may be imposed on pressures in pipeline segments, or at the end of segments, or on pressure differences (and hence flow velocity) across segments. Here we simply represent these constraints generically, in terms of the underlying pressure variables, as: PCðPBtoðlÞ ; PBtdðlÞ ; TRtl ; slackÞ ¼ 0

: 8l; t

(7)

A market bidding model is overlaid on the physical gas flow representation, using bids and offers essentially as described by Read et al. We let SPms be the amount of gas scheduled from step m8 of injection bid s, at price cms, and Dcd be the amount of gas scheduled from step c of withdrawal bid d, at price bcd. While uncontrollable demand, UDtn is taken to be non-price responsive, it can be curtailed by the Operational Model at the ‘value of lost load’, n, which is currently set to $800/GJ. This defines an effective bid price for uncontrollable withdrawal. Finally PenaltyðSlackÞ is a generic function defining the penalty for using “slack” (i.e. violation) variables on each constraint. Penalties are discussed further in Sect. 9. Thus the objective function may be stated as follows: Minimise X m;s

cms Sms 

X c;d

bcd Dcd 

X n;t

vUDtn þ

X

psMLz þ PenaltyðSlackÞ

(8)

z

Note that while Read et al. discuss the possibility of trading end-of-day linepack, it is not explicitly represented in this objective because it was decided that such trading was not appropriate. There is, in fact, only a small range of end-of-day linepack levels that will leave the system in a state from which the next day’s supply requirements can reliably be met. Thus linepack requirements are specified by the System Operator, via constraint (6). Further, the offer/bid terms here have been expressed in a very generic way, not necessarily tied to particular hours as in Read et al. In fact, when run to determine market prices, using the simplified infinite tank assumption as in Sect. 8, the same

8 Market participants submit supply offers to the market, these ‘bids’ are composed of up to 10 price and quantity tranches, where we call each individual tranche a ‘step’.

Implementation of a Scheduling and Pricing Model for Natural Gas

15

offers and bids are assumed to persist over the entire remaining gas day, and corresponding (pseudo-)dispatch schedules will be produced. Conversely, in reality, injections and withdrawals at particular points in the PTS may be constrained by a number of constraints not discussed by Read et al. These include physical constraints, such as maximum hourly quantities, ramp limits, and restrictions on the frequency and timing of changes to gas flow rates, and constraints implied by market rules. For example, where possible, the rules require pro-rating the schedules of tied offers or tied bids.9 Thus we describe supplier offers and consumer bids by their aggregated injection or withdrawal price and quantity step combinations. But these must also be related to the hourly injection/withdrawal variables employed in the gas flow modeling representation above. Rather than detail those relationships here, we will express them in a very general way, using a set of generic “Market Constraints” (MC), defining relationships between uncontrollable demand in each period, UDtn , the supply and demand scheduled to be dispatched in each period from offers and bids, Stm and Dtc , and the aggregate quantities cleared from those bids and offers, Sms and Dcd 10: MCðUDtn 8n; t; Stm 8m; t; Dtc 8c; t; Dcd 8c; d; Sms 8m; s; slackÞ ¼ 0

(9)

4 Linearized Flow and Linepack Representation The Operational Model is built around a physical gas flow model, as was the conceptual formulation developed by Read et al. [1]. But Read et al. basically only stated that the nonlinear gas flow/linepack relationships should be linearized, without specifying how that should be done in practice. Read and Whaley [16] did explore a linearization approach based on analytical differentiation of the theoretical gas flow equations, but the implemented MCE described here adopted a very different, and more pragmatic, approach. In this section we discuss linearization of the basic gas flow model without considering specialized equipment like compressors. We begin by developing the relationship between flow rate and the pressure change across a pipe segment, a relationship which is then employed within both piece-wise and successive linear representations of the gas flow problem. The pipeline system is divided into a number of pipe segments where the physical characteristics of the pipe segments and constraints on the pipe segment are relatively uniform within the pipe segment. This includes the pipe

9 The “slack” variables applied to such constraints, have very small penalties, so as to encourage, but not force, tie-breaking. 10 Parameters associated with a number of these constraints limits/bounds may also be scaled in a pre-processing step so as to resolve conflicts between quantities of gas previously scheduled and quntities actually observed, for example.

16

W. Pepper et al.

segment diameter and minimum and maximum constraints on pressure in the pipe segment. The flow of gas along a pipe segment is a direct function of the average pressure of the gas in the pipe segment and the pressure decline from one end of the pipe segment to the other, and hence of the inlet and outlet pressures, as indicated by Eq. 2 and 3 for FI and FO above. But that relationship is nonlinear, and approximately related to the square root of the average pressure times the pressure decline. The allowable pressure drop along a pipe is actually nonlinear, but our aim is to develop an LP based formulation. For a short enough pipe segments, and time intervals, an approximately linear loss function could be considered accurate enough, as suggested by Read et al. [1], and this strategy is employed by iterative pipeline simulation models such as Dorin and Toma-Leonida [17]. But our model of the Victorian Gas System use a piece-wise linearization of performance over longer pipe segment lengths, and has been calibrated to within acceptable tolerances against a more general physical gas flow model. Similar experience was recently reported by Midthun et al. [13], whose linearised approximation, a Taylor series expansion about multiple-points, was deemed acceptable for work undertaken for North Sea gas system users and its operator, Gassco. The gas flows within the pipe segments are derived assuming isothermal conditions, uniform pipeline diameter and surface conditions, and a constant altitude (sea level). While not described in this paper, the MCE includes preprocessing steps to translate measured pressures and pipeline pressure limits at pipeline altitude to sea-level equivalents. It scales flow and linepack equations to reflect the higher density of gas at sea level, and then converts pressures derived from the optimization back to be applicable at the relevant altitude. The Appendix contains the detailed equations used to estimate flow rates and linepack, given specified values of the (sea-level) pressure at both ends of a pipe segment. The MCE went through a rigorous calibration exercise during its development, and the model results for pressures closely match measured pressures from actual gas days given the initial conditions on the pipeline system, hourly injections and withdrawals, and final linepack on a given gas day. Pipelines, valves, and compressors control the flow of gas between injection/ withdrawal nodes. Additionally, a pipe segment stores gas, so a key parameter is the linepack stored at the end of each hour and the change in linepack over the hour, which impacts the flow from the pipe segment. The gas storage in a pipe segment is primarily a function of the pressures in the pipe segment and its length and diameter (See Appendix). For pipe segments in which the flow direction is specified, this pressure delta must be non-negative and less than a maximum pressure delta. But, for most pipe segments, the pressure decline required to produce maximum allowable flow rates is significantly less than the difference between the maximum allowed inlet pressure and the minimum allowed outlet pressure. So the MCE allows for a user specified maximum pressure decline on each pipe segment set, so that the estimated flow rate corresponding to that decline will be no less than the maximum flow rate expected at any time on the pipe segment. Limiting the allowable pressure decline

Implementation of a Scheduling and Pricing Model for Natural Gas

17

in this way allows the MCE to provide a more accurate linear representation of the nonlinear pressure flow relationship, by linearizing over a narrower pressure range. For bi-directional pipe segments the same logic applies, but the roles of the origin and destination nodes are expanded to allow for the possibility that the pressure conditions will be reversed.

4.1

Piece-Wise Linearization Using Convex Combinations

An optimization based on this continuous feasible region would need to directly handle the nonlinearity of gas flow and linepack equations. To avoid this complexity, a piece-wise linearization is used. A classic piece-wise linear scheme would use separate pressure variables for each block and sum these to form the actual pressure, however the approach we used here was a convex weighting approach, similar to that subsequently described by Martin et al. [14]. To implement this method, feasible combinations of inlet pressure and pressure deltas are first chosen to form a discrete “grid”, as shown for a uni-directional flow pipeline in Fig. 3. In principle this method will work for any randomly chosen set of “grid” points, with the feasible region being implicitly determined by the convex hull of those points. In practice, though, a (non-rectangular) grid is formed using “families” of points, arranged in rows. Four ‘families’ of points are shown. The family forming the top row (circular points) corresponds to maintaining the maximum inlet pressure across different pressure deltas, while the family at the bottom (square points) corresponds to maintaining minimum inlet pressures. The intermediate families (triangular points) are evenly distributed between them. Typically, the range of pressure deltas is very large and the rate at which the flow rate changes declines significantly as the pressure decline increases. Thus the MCE

Fig. 3 Discrete representation of pressures and pressure deltas

18

W. Pepper et al.

uses 5 pressure deltas (d ¼ 0 to 4 ¼ PS11) for each family specified, or 20 points in total, so that the differences in pressure deltas increase as the pressure delta increases. Obviously, it is desirable to allocate those points to the most strongly curved regions, thus giving greater accuracy in that region. This is achieved by placing point d of each row at a level 1-((PS-d)/PS)1.75 of the maximum pressure delta. For 5 pressure deltas, this implies pressure deltas at 0%, 9%, 30% and 60% and 100% of the maximum pressure delta. For each pipeline segment, j, we have an origin pressure PO(j) and a pressure delta DPj. Figure 3 presents a discrete grid of points, each defined by a specific value PO(j) and for DPj. But we can represent any other point in the convex hull of these grid points as a weighted sum of grid points.12 by weighting point (PO(j), DPj)i, with a variable, aji, and require that these weights be positive, and collectively sum to 1.13 Mathematically, any general point, (PO(j), DPj), in the implicit feasible region for pipe segment j can then be expressed as: X aji POðjÞi (10) POðjÞ ¼ i

k

(11)

P where aji  0 8j; i and aji ¼ 1 i But note that other characteristics, such as the destination pressure, the steady state gas flow (FO and FI), and the linepack stored in the pipe (KN) can be determined as functions of PO and DP, as described by the Appendix. Thus, while the location of each point shown in Fig. 3 is defined with respect to only two variables (Pj(o), DPj), it may have associated with it a vector giving the values for any number of characteristics that can be calculated as a function of those two variables. We denote these characteristics by xkj for k ¼ 1,. . .K. We can then create an effective piece-wise linear representation of any one of these characteristics by applying the same weights used to represent (PO(j), DPj) in terms of the underlying grid point values. X xkj ¼ aji xkji 8j; k (12) i

Thus the generalized functions stated in Eqs. 2, 3 and 4 for flow in and out of a pipe segment, and linepack in that segment, are actually defined by essentially empirical value determined for a set of grid points. And the fundamental LP variables in this representation are not actually flows, pressures etc, but the

11

PS indicates the number of “pressure states”, although actually there are PS + 1, including state 0. 12 For a 2 dimensional representation like this, every point can actually be defined as a weighted sum of three particular grid points. But, in order to define a convex feasible region, we need to leave the model to determine which grid points it prefers to use. 13 Together, these two constraints actually mean all weights must lie in the range [0,1].

Implementation of a Scheduling and Pricing Model for Natural Gas

19

weighting variables, aji. This allowed the model to be implemented quickly, and readily tuned to match observed empirical relationships. Subsequently the same approach was used to improve modeling of losses on the HVDC inter-island link in the New Zealand electricity market clearing engine, and to model losses on all transmission lines in the Singapore market clearing engine. While it is not immediately obvious whether this type of formulation will increase or decrease computational times, it has proven at least competitive with more traditional piece-wise linearization in the latter context. And here it has the advantage of increasing the length of pipeline segments, and hence reducing their number, relative to using explicitly defined gas flow equations of the type proposed by Read et al.14

4.2

Successive Linearization to Deal with Non-convexities

The piece-wise linearization approach works by creating a convex LP feasible region, and implicitly assumes that the underlying physical equation set also forms a convex feasible region. Unfortunately the real problem here is non-convex, even if the LP defines a convex feasible region. This is because the physical equations require that the solution lie on the boundary of the LP feasible region, and that boundary is itself a non-convex set. In many cases this does not matter because the objective function makes it desirable for the optimal solution to lie on the physically feasible boundary of the feasible region anyway. Thus piece-wise linearization is often applied without problems to optimization problems where the physical feasible region is actually a non-convex boundary set. That is not always the case here, though. Figure 4 highlights the kind of issue that can be associated with the pressure delta, although there can be a similar problem with respect to the relationship between flow and inlet pressure. In the diagram, the flow should be on the top part of the curve (at the point marked Realistic) but the LP formulation allows for any flow in the shaded ‘Initial Solution Area’. If the marginal value of natural gas is less at the pipe segment outlet than at the inlet, the model will initially solve in the shaded area or on the bottom line (at the point marked Not Realistic), because it then becomes desirable to retard flow as much as possible. Cases where the MCE will want to solve in the shaded area or on the bottom line are not uncommon. They can occur when the pipeline system will be constrained during an hour later in the day and it would be beneficial to send gas to some part of

14 Another potential advantage of the convex combination approach to piece-wise linearization is that, if and when convexity issues arise, they could be dealt with by employing an integer type formulation, employing Special Ordered Sets (SOS2) as in Martin et al. [14], to force the model to apply weights only to adjacent grid points. A similar strategy is mentioned by DeWolf and Smeers (2000a), but dismissed on computational efficiency grounds. Still, that approach has been applied successfully in modeling HVDC link losses in the New Zealand electricity market model.

20

W. Pepper et al.

Fig. 4 Non-feasible flow solution

the system earlier in the day so that it is available as linepack during the constrained period. Or they can occur when the system is reaching minimum pressures at one point on the system but the pressure/flow equations imply that upstream gas will be diverted to another direction due to pressure differentials in that part of the system. Cases where the MCE will want to solve in the shaded area or on the bottom line are not uncommon. They can occur when the pipeline system will be constrained during an hour later in the day and it would be beneficial to send gas to some part of the system earlier in the day so that it is available as linepack during the constrained period. Or they can occur when the system is reaching minimum pressures at one point on the system but the pressure/flow equations imply that upstream gas will be diverted to another direction due to pressure differentials in that part of the system. There can also be an issue when negative prices arise, but this is not common. They occur when either the bid or pipeline constraints force too much gas into the system, or where the system is constrained during the day, perhaps during the evening peak. If the bid and pipeline constraints are treated as being inflexible, they can end up forcing more gas into the PTS during the night, thus producing end-ofday linepack greater than AEMO would want, and has specified. Compressor fuel use can seem desirable to the optimization during this period, simply to use up negatively priced gas, but the impact is usually limited by constraints on pressures and flows. Whatever the cause may be, the MCE implements a successive iteration process to correct the problem whenever the LP optimization recommends solutions that are not physically feasible. At the first iteration, the problem is solved using piece-wise linearization, with no special features added. This might produce the iteration one solution shown in Fig. 4, where the modeled flow is ‘Not Realistic’, and less than

Implementation of a Scheduling and Pricing Model for Natural Gas

21

Fig. 5 Revised solution at second iteration

the flow that would actually occur, given that pressure delta. In successive iteration mode, the MCE will then do three things: • Restrict the inlet pressures used in the model from PS4 to PS3 based upon the three inlet pressures that are closest to the solution • Restrict the pressure deltas included in the model from PS5 to PS3 based upon the three pressure deltas closest to the solution • Create a slope line that restricts the flow as a function of the pressure All of these are illustrated in Fig. 5 below, where the choice of three inlet pressures and three pressure deltas allows flexibility of the model to adjust if one of the constraints (let’s say pressure delta) on one pipeline segment causes the inlet pressure required on another pressure delta to adjust up or down, possibly making the constraints on that pipeline segment infeasible. Small deviations from the constraint line are allowed without penalty. But larger deviations are subject to a small cost penalty. Additional system wide constraints limit the total deviation allowed across all pipe segments. These constraints can be relaxed at subsequent iterations if found to be overly restrictive. The stopping condition is that the deviation is small enough, or that a maximum of 14 iterations are completed.15 Also, in each successive iteration, the three inlet pressures and three pressure deltas are reviewed to see if they are still appropriate, and alternatives chosen if required.

15

Early experience with the model showed that most cases solved within 14 iterations. Also, cases which did not solve showed negligible improvement in convergence after 14 successive approximations.

22

W. Pepper et al.

5 Modeling Compressors Pipelines move gas from node to node, and through the process of transporting the gas there are pressure losses to the gas. Unlike gas pipeline segments, compressors move gas from a lower pressure state to a higher pressure state, while consuming gas in the process. Depending on the inlet gas pressure, a compressor can be run over a range of speeds (rpm) to achieve various combinations of gas flow and pressurization (pressure delta), and hence of outlet pressures and gas consumption. We can think of the feasible region as being defined by three primary (LP) variables: Flow, Inlet Pressure, and Outlet Pressure, or equivalently, Pressure Delta. Figure 6 simplifies this representation by projecting the feasible region down on to only two dimensions, Flow and Pressure Delta, with Inlet Pressure and/or Outlet pressure being only implicit. This representation allows us to define the maximal feasible region over which the compressor could operate, in terms of the extreme low/delta combinations that can be achieved at various rpm, if the pressure (and hence the density) of gas flowing through the compressor were also allowed to vary all the way from its maximum (typically limited by the maximum output pressure  pout ) down to its minimum (typically limited by the minimum input pressure pin ). Our approximation methodology requires identifying performance at a number of key points, both within the (3-dimensional) range of variations, and at its corner points. Figure 6 shows some of the key characteristics of each compressor, in terms of their projection onto this 2-dimensional representation:

Fig. 6 Compressor operational envelope16

16 In practice, it is usual for the axes to be reversed to match compressor manufactures data relating to efficiency curves, however we present them in this orientation for consistency

Implementation of a Scheduling and Pricing Model for Natural Gas

23

• Minimum and Maximum Compressor Power curves (cp and cp) • A Null State (NS) corresponding to no pressure delta, and no flow17 • A “Surge Limit” line, modeled as a ray passing through the maximum pressure delta that can be achieved at cp, and pin • A ‘Stonewall Limit’ line, modeled as a ray passing through the minimum pressure delta that can be achieved at cp and  pout . • A maximum efficiency line, assumed to be a ray passing through the most efficient operating point on the Maximum Compressor Power curve. • Maximum (ol) and Minimum (ol) operating lines and their associated, actual, nonlinear feasible operating curves are also shown. Since the work required from the compressor to achieve a fixed flow is a function of the compression ratio (Outlet pressure divided by inlet pressure), the maximum, minimum, and most-efficient pressure deltas are scaled to estimate the performance when solving for inlet pressures greater than the minimum inlet pressure. We model the compressor in the LP using the same convex combination approach as described in relation to Eqs. 10, 11 and 12 above. In this case, we use six operational states to represent combinations of the maximum, minimum, and most efficient pressure deltas with the minimum inlet pressure and maximum outlet pressure, all assuming the maximum compressor operating level. As above, we can require that all solution weights lie between 0 and 1, and that they must sum to 1. This would limit the solution to lie within the feasible region defined by the (3-dimensional) convex hull of this set of points at which performance parameters such as gas consumption have been assessed. Figure 6 shows the limits of feasible compressor operation as being linearised, and this was deemed to be a good enough approximation in this case. But the upper limit could be replaced with a piecewise frontier at points selected along the nonlinear maximum power curve. Further points within the running region could also be modeled as in Martin et al. [14]. The lower limit looks more problematic, because it is non-convex, but this linearization is less restrictive than it may appear. There are also two null states, one for minimum inlet pressure and maximum outlet pressure, both representing zero flow, zero pressure delta, and zero operating level. So, if we ignore the minimum running curve and allow non-zero weights to be placed on the null operational states, the model can choose to run the compressors at zero, if that seems optimal. But it also allows the model to choose low operating states that are not physically feasible.18 But, as the solution is refined, and as real time approaches, we can impose conventional minimum or maximum constraints

17 There are actually two null operational states one for the minimum inlet pressure, and the other for the maximum outlet pressure, although they appear as one point in Fig. 6 18 In theory, operating states between zero and the minimum operating level could actually be achieved, on average, by operating efficiently for only part of the time. If so, the convex approximation to performance in that region could actually be valid. Hydro generators, for example, can be validly operated, and represented, in that way. In this case, though, the savings in operational efficiency would probably be outweighed by increased startup/shutdown costs.

24

W. Pepper et al.

on the LP variables, representing Maximum (ol) and Minimum (ol) compressor operating levels for any given hour. These can be used to force the compressor to be in the null operational state in some periods, or above its minimum running level in others. Or they can simply “trim” the feasible region to have linearized boundaries, as shown in Fig. 6. These constraints could have been imposed on the pressure/flow variables formed by the convex combination process, but are actually implemented by restricting the sum of the weights applied in the convex combination approximation to a narrower range than [0,1], as follows. Consider a case where the inlet pressure is at the minimum, and the null state for the minimum inlet pressure is assigned a solution weight, NS p in , of say 0.25. This means that the weight being placed on the points involving maximum compressor power must be 0.75, and that implies that the flow rate and pressure delta values will be both 0.75 of the values that would be produced at the maximum compressor power. Now, at a constant inlet pressure, the work required by the compressor is approximately proportional to the flow rate multiplied by the pressure delta. So the compressor power required to achieve this flow/delta combination must be approximately 0.752 ¼ 0.5625 of the maximum compressor power. This approximation also works well when weight is put on states at the maximum outlet pressure, or on any linear combination of minimum inlet pressure and maximum outlet pressure. So we can generalize this approach to apply across the full set of inlet/outlet pressures and pressure deltas, and to apply to upper and lower operating limits, both defined in terms of the power output required by the compressor.19 Thus we have: ol ol  ð1  NS p in  NSp out Þ2  cp cp

(13)

Or, to make this constraint linear in the LP variables:  0:5  0:5 ol ol  ð1  NS p in  NSpout Þ  cp cp

(14)

In the LP, the compressor operating level (i.e. its power requirement and hence its gas consumption), is determined by linear interpolation between the values determined by the set of points for which it has been pre-computed. Once the LP solution is known, though, the operating level can be better approximated using the quadratic formula: ð1  NS p in  NSpout Þ2  cp

(15)

19 This lower limit here defines the minimum running boundary of the feasible region if we force the compressor to be “on”, thus eliminating the null states. As shown, this boundary is actually non-convex, but it can reasonably be approximated by a straight line, at the cost of eliminating a relatively small set of operating states that are seldom utilized, in practice.

Implementation of a Scheduling and Pricing Model for Natural Gas

25

6 Modelling Pressure Regulator Valves and Check Valves The gas pipeline system in Victoria uses pressure regulation valves at a number of key points in the system, and check valves to regulate flow and optimize the daily operation of the system. A pressure regulator controls the decline in pressure across it, while maintaining the rate of flow consistent with that for the corresponding supplying and receiving pipe segments.20 A pressure regulator often has a maximum pressure at its outlet, which can be pre-set at the valve, and in most cases this value is also the desired pressure to be achieved. The MCE models the pressure regulator and the pipe segment at the outlet of the regulator as a single entity with both a maximum and minimum pressure at the outlet of the regulator. The pipe segment inlet pressure is allowed to be less than the inlet node pressure, but must be between the maximum and minimum outlet pressure of the regulator. Check valves generally constrain flow across them to a single direction, their operation depends on their being a positive pressure difference between corresponding inlet and outlet pipeline segment pressures. Check valves are often installed with by-pass lines, or can be manually and temporarily by-passed, the model includes such by-passes to allow reverse flow when the pressure difference is in the opposite direction. Although such an abnormal-operation attracts a high penalty cost. The modeling of a check valve is more complex. We model this by determining the flow and pressure delta combinations that occur at each combination of maximum and minimum inlet and outlet pressures and corresponding flow, or lack of, across the check valve. This results in there being five potential limiting operational points, as shown in Table 1. The four points labeled F allow the MCE to cover the range of possible forward flows and pressure combinations, while the three points labeled C allow the MCE to represent zero flow at any combination of pressures where the inlet pressure is less than the outlet pressure (that is, where the flow is “checked”). The convex combination approach to piece-wise linearization is then applied, as above. The problem in modeling check valves is that some linear combinations of the weights applied to pressure flow states can produce physically infeasible results,

Table 1 Check valve point modeling Outlet pressure Zero flow Inlet pressure

Minimum Maximum

Minimum F,C

Maximum flow Maximum C F,C

Minimum F

Maximum F

20 In other words, the pressure can decline significantly without implying the increase in flow rate one would otherwise expect for such a decline.

26

W. Pepper et al.

as for the pipe segment flow equations discussed earlier, but much more severe. We address this problem, if necessary, using the successive iteration logic discussed earlier.

7 Modeling Injection and Off-take Restrictions Participants in the Victorian Gas Market can make injection bids to supply gas, or withdrawal bids to buy gas, or specify uncontrollable withdrawal, being gas which will be purchased at any price. Injection bids are associated with supply points, and withdrawal bids and uncontrollable withdrawal are associated with withdrawal points, represented as “nodes” in the network. AEMO can over-ride the aggregate uncontrollable withdrawal if the cumulative participant forecast differs significantly from its own. AEMO also controls how the aggregate uncontrollable withdrawal is profiled across the network and across the day. These profiles can be varied during a day as weather conditions or observed demand patterns change. In combination, the following features allow the Operational Model to more closely represent how gas is physically injected into the system over the gas day. Collectively they define the constraint set referred to as MC in Eq. 9. Bids include ten price-quantity steps as well as a response time, an expiration time, and ramp limits. While participants can freely submit revised price-quantity steps between daily scheduling intervals, these other parameters must be approved by AEMO and are only changed infrequently. There may be multiple participants trading at a supply point. For instance, there are multiple participants bidding to supply gas to the market at the Longford supply point. This supply point represents a physical gas production facility. A set of constraints can be imposed on aggregate supply or demand at a point, and these may over-ride constraints on individual bids at those points (with logic included to resolve conflicts). Supply and demand point constraints can impose minimum hourly flow rates, maximum hourly flow rates, minimum daily flow rates, maximum daily flow rates, response times, expiration times, and ramp limits. Ramp limits restrict the rate of increase and decrease in schedules between hours. A response time can be set for each hour during the gas day and indicates how long before that time a schedule must be issued for the participant to match that schedule. Thus if response times are 2 h for all hours after 10 a.m., and a schedule is issued at 6 a.m., but then revised at 10 a.m., the 10 a.m. schedule will still match the schedule issued at 6 a.m. for the first 2 h after 10 a.m., with the schedule only changing to match the 10 a.m. market-clearing dispatch schedule from noon on. The expiration time is a time during the day after which the participant will no longer respond to re-schedules. A participant with an expiration time of 8 a.m. could be scheduled for the day at 6 a.m., but would then retain that same schedule for the remainder of the day, irrespective of how prices, and other parties’ dispatch schedules, may change when the market is subsequently re-cleared during the day.

Implementation of a Scheduling and Pricing Model for Natural Gas

27

The Victorian Gas Market also has a form of capacity right called Authorized Maximum Daily Quantity (AMDQ). This reflects an amount of gas that a participant can supply at one point, or receive at one point, without being deemed to have “caused” any constraints.21 Withdrawal bids or injection bids within a participant’s AMDQ limits are given priority in scheduling if tied with withdrawal bids or injection bids with the same price that are not covered by AMDQ. The market model also allows for directional flow constraints. These can be imposed across a supply point and withdrawal point so as to constrain the net flow. This can be useful in modeling gas storage fields which can only either take in gas or release gas each day. The directional flow constraint allows limits to be imposed on the net supply from a group of points. Minimum hourly net flow rates, maximum hourly net flow rates, minimum daily net flow rates, and maximum daily net flow rates can be implemented. A problem that existed in the early history of the market was that the Operational Model was not fully reflecting how gas facility operators managed the flow of gas. They typically manage gas flow at a constant hourly rate, changing rates several times during the day. The Operational Model imposes small costs on ramping to encourage flat scheduling, but this feature failed to account for the specific times at which schedules could be changed. These could be over-ridden when strong economic incentives existed to encourage more varied scheduling. While it might be thought desirable to have more variable flow rates, the operators of gas production and supply facilities are not generally participants in the market, and ignoring their physical operating characteristics in scheduling has the potential to create costs for participants which are beyond their control (e.g. for not following their schedule). Thus the MCE could produce schedules which were not implementable in the real-world, and in extreme cases could create system security risks. This problem has been largely resolved in recent years by allowing bids and supply and demand point constraints to have defined flexibility limits. If flow at a bid or pipeline point is specified to be completely inflexible, then the optimization will determine an hourly flow rate which is constant over the time window for which inflexibility applies. If an aggregate supply point flow is inflexible, but the injection bids there are flexible, then the total supply through that pipeline point must maintain a constant rate, but the participant schedules at that pipeline point can vary, while maintaining the same total flow. Inflexibility restrictions are relaxed for the times at which major constraints, such as minimum and maximum hourly flow limits or directional flow constraints, change, and for a sufficient period after that, so as to allow prior schedules to be followed within the response time, and then for the dispatch schedule to be ramped to minimum or maximum hourly schedule quantities, given the applicable ramp limits. In practice, most ramp rates are quite fast, so these intervals of flexibility can be quite short.

21 This being important because it determines who faces penalties and/or receives constrained-on/ off payments when the actual dispatch schedule differs from the market trading schedule.

28

W. Pepper et al.

8 The Market Model Formulation Each time it is run, the Market Model determines the single market price to be applied to trades (including deviations from trades cleared earlier) over the remainder of the gas day. It also determines the set of injection and off-take schedules that would occur if there were no limits on the ability of the gas transmission system to move gas to, or from, any node at any time within the optimization horizon. The differences between these unconstrained schedules and the schedules produced by the Operational Model runs, over the same optimization horizon, are used to determine compensation for the costs created by the constraints. The Market Model LP is structured similarly to that for the operating schedule with a few rather significant exceptions: • All injections and withdrawals are treated as if they were at a single node so that there are no constraints modeled on the ability to flow gas between locations. Consequently there is no representation of pipelines, compressors, check valves or regulators, although pipeline point constraints and directional flow constraints – which are applied at injection and off-take points, not pipe segments – still apply. • With no pipeline pressure-flow relationships modeled, successive iterations is not required. • As pipe segements are not modeled A system wide total initial linepack is used rather than pipe segment specific linepack • The market schedule imposes no upper or lower limits on linepack levels during the day, other than an end-of-day minimum linepack constraint. • The end-of-day minimum linepack constraint differs slightly from that used in the operational schedule formulation. Compressors are modelled in the Operational Model, and consume gas, but are not modeled in the Market Model. To ensure that each model has the same change in linepack over the day, and hence the same pricing in an unconstrained case, the compressor fuel usage determined in the Operational Model is subtracted from the minimum linepack limits used in the Operational Model but not from the minimum linpeack limits in the Market Model. As both models include cost penalties to discourage linepack above minimum, these minimum levels are effectively target end-of-day linepack levels. When the Operational Model schedules gas supplies run compressors then it must supply correspondingly less gas to achieve the end-of-day linepack target, ensuring that total supply over the day matches that in the Market Model.

9 Solution Methods The MCE is currently executed as a set of LP problems solved with CPLEX version 9.1. Early versions of the MCE developed in the late 1990s solved the problem using the Simplex method, as this is well suited to reliably and accurately determining prices in markets. The MCE solved the problem up to 14 times in executing success

Implementation of a Scheduling and Pricing Model for Natural Gas

29

iterations for the Operational Model. Even using an advanced basis for each iteration, the Simplex method was found to perform too slowly on some problems, taking several hours using the technology of the day, so a switch was made to using the barrier method. The problem then encountered was that the barrier method had difficulty maintaining adequate numerical accuracy to cope with the range of numerical values used in a typical MCE problem. This is because the MCE employs a hierarchy of penalties; which have been adjusted over the years to minimise inappropriate interactions between them. Within this hierarchy there are many very small penalties in implementing tie-breaking, penalizing infeasible flow rates during successive iterations, and encouraging some other desirable outcomes. Since the highest market price is $800/GJ, all infeasibility penalties are set greater than this. Penalties on operator configurable constraints, which can be modified if infeasible, typically have penalties in the region of $3,000/GJ and $4,000/GJ.There are also some very large penalties to address physical infeasibilities. For example, the mass balance constraint, Eq. 8, has a violation penalty value of $9,999.9/GJ. As at 2010, the smallest penalty used is $2  106/GJ while the largest used is $10,099.9/GJ, a variation of nine orders of magnitude. To resolve the precision problem, while using the barrier method, an additional three iteration phases were introduced to both the Operational Model and the Unconstrained Market Model. Broadly, the first phase involves solving the problem with only the large penalties activated, and all small adders removed. This determines physically feasible aggregate quantities of gas to schedule over the day. The second phase fixes the aggregate schedules over the day, thus not requiring large penalties, but uses small penalties to allocate the gas flows optimally across time. Both phases 1 and 2 are performed for each successive iteration. Finally, Phase 3 is only performed at the completion of the problem – it involves effectively fixing the problem to a tight region around the optimal solution, with some minor penalty costs set to zero, and re-solving the problem using the simplex method to determine nodal prices (for the Operational Model) and the single market clearing price (for the Unconstrained Market Model). The market timelines require that the software be able to reliably produce a schedule in the time between the window for bids and offers closing and the deadline for publishing schedules. While this time frame is about 1 h, both the Operational Model and the Unconstrained Market Model must be solved, and it may be necessary to solve multiple schedules in that time to allow the operators of the system to modify compressor commitments and other constraints under their control to correct issues seen in prior solutions. At the commencement of the market the Operational Model could be solved, including data input and output processing, in approximately 15 min using the simplex approach and earlier versions of CPLEX. The Unconstrained Market Model, which was then solved after the Operational Model only took seconds to solve. Since 2007 the Unconstrained Market Model has been solved before running the Operational Model, providing the operators with insights about the schedule before needing to determine compressor commitments to be used in the Operational

30

W. Pepper et al.

Model. Today, using CPLEX 9.1 on faster workstations, and despite a significantly more complicated PTS than existed in the late 1990s, the Operational Model can be solved within 3 min while the Unconstrained Market Model solves well within 1 min.

10

Conclusions

The Market Clearing Engine described here has been used in the Victorian Gas Market since 1999, and demonstrates the practical use of optimization techniques to schedule dispatch, and determine prices, for a complex gas market. In principle it could be used to support market trading based on determine hourly nodal prices, as originally proposed by Read and Whaley [16], and explained by Read et al. [1]. In practice this has not eventuated and the market has evolved along somewhat different lines. This has occurred for various reasons, but it should be clear that it is not because it proved impossible to determine hourly nodal prices, since the MCE model does, in fact, determine such prices. Nor is it because the prices determined by the MCE model are always the same, everywhere in the network, as may be seen from the example discussed by Read et al. Thus while, in Victoria, the full nodal version of the MCE has proved most successful as a dispatch optimization tool, a model of this type could also be used to clear markets and support trading, and this paradigm may well prove more beneficial elsewhere, where congestion is more prevalent, and there is greater economic value at stake.

Appendix: Detailed Pressure Flow Equations for Flow Rates and Linepack In this Appendix we present the key equations for deriving natural gas flows and pressures in the Victorian pipe network. The derivation is based on six initial equations described in Eqs. 16, 17, 18, 19, 20, 21) below. Pl ¼ rl RTz

(16)

The ideal gas law equation 1 describes pressure, Pl at a point l along a pipeline of length 0 < l < L. R is the ideal gas constant in units of kPa*m3/( K  kg), T is the temperature of the pipeline in  K, and z is the supercompressibility of gas, while rl is the density in units of kg/m3. q l ¼ rl v l A

(17)

Implementation of a Scheduling and Pricing Model for Natural Gas

31

The gas flow rate ql at a point l, in kg/s, is described in (17). Where A is the pipe cross-sectional area in units of m2, nl is the velocity of gas in m/s in the pipe at point l, at the point l measured at sea level. dP f  1000 ¼  r v2 dl 2D l l

(18)

The Fanning equation 18 describes the rate of change of pressure at position l along a pipeline. D is the pipe diameter in units of m while f is the Fanning friction factor. Re ¼

rl vl D m

(19)

The Reynolds Number, Re, is defined by (19). In this equation m is the viscosity of gas measured in kg/ms.   !0:25 0:316 1 fturb 4 f ¼ þ  Re 0:316

(20)

The Fanning friction factor formulation in (20) utilizes the Blasius formulation [18], where Z is the pipeline efficiency (a fraction). It describes friction as gas flows along a pipe. This makes use of a turbulent friction factor, fturb. fturb ¼ 0:0053 þ 0:1662



e 0:35 1000D

(21)

Smooth pipelines have no turbulence while rough pipelines have more turbulence. Turbulence becomes more important as the Reynolds Number increases. Given e, a measure of the roughness of the pipeline in mm, then the relative roughness of a pipeline can be defined with respect to its diameter as (e/D). Using empirical data the form of the turbulent friction factor shown in (21) was developed by gas system engineers working on the development of the MCE. Now, using Eq. 17 to substitute for nl in (18) we get:   dP f q2 ¼  2 l  1000 dx D A rl  2

(22)

Equation 22 can be further refined by using Eq. 16: dP 1 ¼ 0:5Gf q2l dx Pl

(23)

32

W. Pepper et al.

Where: Gf ¼

fRTz 1,000DA2

(24)

By integrating (23) with respect to l and observing that for l ¼ 0, Pl ¼ Po, the origin pressure of the pipeline gives:  0:5 Pl ¼ P20  Gf q2l  l

(25)

Using (25) to define the value of Pl where l ¼ L, i.e. the destination pressure, and assuming a constant flow (ql ¼ q), friction factor, and supercompressibility, then the steady state flow rate can be derived as: q¼

 2 0:5 P0  P2L Gf L

(26)

Here Eq. 26 is closely related to the Weymouth panhandle equation referred to by Zheng et al. [8] and Midthun et al. [13]. It is used later to define friction factors as a function of flow rate. However, we can derive another flow rate equation by assuming no friction arises from turbulence (fturb ¼ 0), and substituting the Reynolds Number from (20) into (19) and the resultant equation for f into (22) to give: Pl 

dPl ¼ 0:5G  q1:75 l dl

(27)

G=

0:316m0:25 RTz 1,000ZDia1:25 A1:75

(28)

Where:

Integrating (27) with the assumption that the flow is constant along the pipe segment and the requirement that for l ¼ 0, Pl ¼ Po (the origin pressure) we get:  0:5 Pl ¼ P20  Gq1:75  l

(29)

Using (29) to define the value of Pl where l ¼ L, i.e. destination pressure, then for a non-constant friction factor and constant supercompressibility the steady state flow rate can be derived as:  2 ð1=1:75Þ P0  P2L ql ¼ GL

(30)

Implementation of a Scheduling and Pricing Model for Natural Gas

33

Equations 6 and 30 describe flow rates under different assumptions about friction factors. These are used later to define more general flow rate equations. However, before exploring that, it is necessary to consider the linepack equations. The linepack, I, in a pipe segment can be calculated by integrating the volume of gas in each slice of pipeline along the length of the pipeline: ðL

ðL

I ¼ rl Adl ¼ 0

Pl A dl RTz

(31)

0

Assuming a constant friction factor this can be rewritten as: ðL A I¼  Pl dl RTz

(32)

0

We also have22: dFðlÞ ¼ Pl dl

(33)

Substituting this into (32), we derive: I¼

A  fFðLÞ  Fð0Þg RTz

(34)

Substituting the expression for Pl from (25), which was based on a constant fraction factor, into (33), then integrating over l, we get, for a non-zero flow rate:  0:5 dFðlÞ ¼ Pl dl ¼ P20  Gf q21  l dl

(35)

Given the rule (f(x))n differentiated by x gives n  df/dx  f(x)n-1 then we can integrate dF(l) by the reverse transformation to give:  FðlÞ ¼

P20  Gf q2  l 1:5Gf q2

1:5 (36)

Evaluation of (36) for F(0) and for F(L), and substituting these into (34) gives:  A P30  P3L I¼ RTz 1:5Gf q2

(37)

Energy ¼ Work ¼ Force  Displacement and Pressure ¼ ForceArea; so combining these two results in Pressure  Length ¼ Force  DisplacementArea which then equals the flow of energy past a point, which is WorkArea ¼ EnergyArea 22

34

W. Pepper et al.

Further, substituting for q from (26) gives linepack for non-zero flow of:  3 P0  P3L AL  I¼ 1.5RTz P20  P2L

(38)

Where the flow rate is zero, then Pl ¼ P0 for all l, and (16) implies I ¼ P0

AL AL P0 ¼ RTz RT z

(39)

Equations 38 and 39 can be represented in terms of an average pressure on the pipeline Pa: I¼

AL Pa RT z

(40)

Hence  2 P30  P3L Pa ¼  2 3 P0  P2L

: if

P0 > P L

(41)

Or: Pa ¼ P0

: if

P0 ¼ P L

(42)

Given user defined values of typical low and high pressures in the system, Plow and Phigh, and corresponding supercompressibility values zlow and zhigh it is possible to define an average supercompressability za as: za ðPa Þ ¼ zlow þ

ðPa  Plow Þ  ðzhigh  zlow Þ ðPhigh  Plow Þ

(43)

To take advantage of the linear relationship between linepack and pressure in (40), the MCE formulation uses this equation to compute linepack for all cases, including the case when the pipeline inlet and outlet pressure values are different. The term (P0/z) is replaced by the average value of the supercompressibilityadjusted-pressures at the inlet and outlet nodes. Further adjustments are made to these equations to allow for altitude. The model also combines the results of (26) and (30) to determine flows which address the impact of a varying friction factor, non-zero fturb, and varying supercompressibility along the pipeline, for given pressures at the origin and destination of the pipe. All of these values are refined using an iterative approach that converges quickly and tests have demonstrated that a further iteration past the current stopping point would typically impact final flows by less than 0.2%.

Implementation of a Scheduling and Pricing Model for Natural Gas

35

References 1. Read EG, Ring BJ, Starkey SR, Pepper W (2012) An LP formulation for a natural gas market. A. Sorokin et al. (eds.), Handbook of Networks in Power Systems II, Energy Systems, Springer-Verlag Berlin Heidelberg 2. Ruff LE (1997) Victorian Gas Market Clearing Logic Version MCL 1.2. Released by Energy Projects Division, Department of Treasury and Finance, 30 October. 3. Pepper W (1999) AEMO market clearing engine: mathematical formulation. Version 1.20.4 Draft Report by ICF Consulting to VENCorp (now AEMO) 4. Zhu G, Henson MA, Megan L (2001) Dynamic modeling and linear model predictive control of gas pipeline networks. J Process Control 11(2):129–148 5. Johnson AT, Marquart BC, Istre ML, Walloopillai RK (2000) Integrating an expert system and pipeline simulation to enhance gas pipeline operation, profitability and safety. www.psig.org/ papers/2000/0007.pdf. Accessed 23 Nov 2010 6. Hogan WW (1992) Contract networks for electric power transmission. J Reg Econ 4 (3):211–242. ICFI (2010) ICFI gas market model. www.icfi.com/markets/energy/doc_files/ nangasweb.pdf. Accessed 28 Oct 2010 7. AEMO (2010) National gas rules, Part 20 at http://www.aemc.gov.au/Gas/National-GasRules/Current-Rules.html. Accessed 2 Sep 2010 8. Zheng QP, Rebennack S, Iliadis N, Pardalos PM(2010) Optimization models in the natural gas industry. In: Rebennack IS, Pardalos PM, Pereira MVF, Iliadis NA (eds) Handbook of power systems I. Springer-Verlag Berlin Heidelberg, pp 121–148 9. Wu S, Rios-Mercado RZ, Boyd EA, Scott LR (2000) Model relaxations for the fuel cost minimization of steady-state gas pipeline networks. Math Comput Model 31:197–220 10. De Wolf D, Smeers Y (2000a) The gas transmission problem solved by an extension of the simplex algorithm. Manag Sci 46:1454–1465 11. De Wolf D, Smeers Y (2000b) The simplex algorithm extended to piecewise-linearly constrained problems. http://citeseerx.ist.psu.edu/viewdoc/summary?. doi¼10.1.1.14.8090. Accessed 23 Nov 2010 12. Tomasgard A, Rømo F, Fodstad M, Midthun KT (2007) Optimization models for the natural gas value chain, In: Hasle G, Lie KA, Quak E (eds), Geometric modelling, numerical simulation, and optimization: applied mathematics at SINTEF, Springer-Verlag Berlin Heidelberg, pp 521–558 13. Midthun KT, Bjørndal M, Tomasgard A (2009) Modeling optimal economic dispatch and system effects in natural gas networks. Energy J 30(4):155–180 14. Martin A, Moller M, Moritz S (2006) Mixed integer models for the stationary case of gas network optimization. Math Program Ser B 105:563–582 15. Van der Hoeven, T. (2004), Math in gas and the art of linearization, PhD thesis International Business School and Research Center for Natural Gas, Groningen, The Netherlands 16. Read EG, Whaley R (1997) A gas market model for Victoria: dispatch/pricing formulation. Report by Putnam, Hayes & Bartlett – Asia Pacific Ltd. to VENCorp (now AEMO) 17. Dorin B, Toma-Leonida D (2008) On modelling and simulating natural gas transmission systems (part I). J Contr Eng Appl Infor 10(3):27 18. Hager W (2003) Blasius: a life in research and education. Exp Fluids 34(5):566–571 19. Peretti A, Toth P (1982) Optimization of a pipeline for the natural gas transport. Eur J Oper Res 11:247–254 20. Pepper W, Lo G (1999). Application of linear program to gas market and pipeline model in the state of Victoria, Australia. In: INFORMS conference, Cincinnati, May 1999 21. Rios-Mercado RZ (2002) Natural gas pipeline optimization. In: Pardalos PM, Resende MGC (eds) Handbook of applied optimization. Oxford University Press, New York, pp 813–826

Long-Term Pressure-Stage Comprehensive Planning of Natural Gas Networks Michael H€ ubner and Hans-J€ urgen Haubrich

Abstract Due to the forthcoming regulation schemes throughout Europe, new challenges for natural gas network operators arise. The pressure for realizing and operating cost-efficient network structures increases as the regulation is based on a comparison of different network operators with the network operator setting the minimal allowable costs. Optimization methods, which will also be applied by regulatory authorities as part of the analytical cost models for calculating the efficiency of natural gas networks, provide the opportunity to identify long-term cost-efficient network structures, so called reference networks. Boundary conditions of natural gas networks, which concern the system’s technical safety and thus need to be regarded during network planning, are given by the rules set by each country’s technical assembly for natural gas supply. Degrees of freedom exist in alternative network structures, the number and degree of pressure stages and for the dimensioning of equipment. Therefore, optimization methods are required for solving the extensive optimization problem. Especially heuristic optimization algorithms have proved to deliver an optimal performance for the determination of cost-efficient network structures. Their essential advantages over exact methods are a reduced computational effort, leading to computing times of typically few hours for real natural gas systems while simultaneous delivering several similar cost-efficient network structures. These advantages allow sensitivity analysis by a variation of boundary conditions and supply tasks on network structure and network costs and lead to a greater flexibility for the future network development. Therefore, an optimization method for natural gas distribution networks based on Genetic Algorithms is proposed. The method is capable of calculating cost-efficient

M. H€ubner • H.-J. Haubrich (*) Institute of Power Systems and Power Economics (IAEW), RWTH Aachen University, Aachen, Germany e-mail: [email protected]; [email protected] A. Sorokin et al. (eds.), Handbook of Networks in Power Systems II, Energy Systems, DOI 10.1007/978-3-642-23406-4_2, # Springer-Verlag Berlin Heidelberg 2012

37

38

M. H€ ubner and H.-J. Haubrich

network structures with regard to all technical and economic boundary conditions and is also used by the German Federal Network Agency for calculating reference networks with minimum costs for given supply tasks. Exemplary applications demonstrate the method’s capability and the advantages through applying this method for long-term planning of natural gas networks. Keywords Combinatorial optimization • Distribution networks • Genetic algorithms • Minimal network costs • Natural gas • Network planning • Parallel optimization

1 Introduction and Motivation The cost pressure on distribution companies has increased because of the upcoming regulatory framework in the European power and gas markets. Therefore, in particular gas network operators, who focused mainly on the compliance of the technical safety proposals by national technical and scientific associations for gas and water networks, need to reveal potentials for a cut down of costs in the field of network planning. The reduction of directly controllable costs in the short-term, e.g. a decline in maintenance work or an abandonment of investments, provides a limited lowering of costs, if the high reliability and safety standards of existing networks should be maintained. Another option for lowering network costs is the operational optimization. The aim of this method is to supply all customers via the existing network at minimal costs with regard to contractual obligations [1, 2]. Nevertheless, due to the narrow time planning horizon, advancements in efficiency are constricted to shortterm manipulable expense factors. A sensible approach seems to be the refinement of the network planning process in order to determine efficient networks, which provide for an equivalent level of reliability and technical safety compared to the existing network and at the same time lower costs. Therefore, the application of a computer-based network-planning instrument for the objective determination of long-term cost-efficient networks is necessary. At present, there exists no instrument for the integrated planning of natural gas distribution networks with different pressure stages. Hence, the primary objective of this paper is to introduce the status quo of current developments of a computer-based technique for the integrated planning of natural gas distribution networks with variable pressure stages. In case of the long-term planning of electrical networks, computer-based procedures are deployed that optimize the network structure as well as the dimensioning of necessary operating facilities with regard to the given supply task. All relevant technical and company internal planning criteria need to be considered. Numerous studies have proven the overall functionality of the aforementioned procedures for realistic planning tasks [3]. On the other hand, in the past, the planning of natural gas networks has been vastly driven by the company’s success

Long-Term Pressure-Stage Comprehensive Planning of Natural Gas Networks

39

of acquiring new customers. Therefore, apparently the long-term strategy has depended on subjective expert knowledge of the planning engineers in charge [4]. The objectification of the network planning process by means of long-term costoptimal networks (target networks) gains in importance against the background of the liberalization and regulation of the European power markets. Thus, the application of computer-aided optimization procedures is indispensable for natural gas networks.

2 Network Planning Process Due to the complexity of the network planning process, it seems both necessary and beneficial to subdivide it into two separate planning stages [5], see also Fig. 1.

2.1

Long-Term Planning

Regarding a planning horizon of several decades, the existing network structure and operating facilities may be ignored. The long-term planning is based on the aforementioned assumption that is often referred to as Greenfield development. Therefore, the long-term planning will identify cost-efficient network structures neglecting the existing facilities.

2.2

Expansion Planning

The expansion planning determines cost-efficient systematic development strategies starting with the existing network structure and aiming at the identified

Existing Network

Uncertainties

Demand/Supply Scenarios

Expansion Plan

Target Networks t

Possible Planning Projects Expansion Planning

Fig. 1 Expansion and long-term planning

Long-Term Planning

40

M. H€ ubner and H.-J. Haubrich

target networks based on the outcomes of the long-term planning [6]. Uncertainties concerning, e.g. load growth, changes in interest rates, etc., may be considered in this planning stage. As the long-term planning resembles in many respects the determination of reference networks and is therefore attached with special consideration in the regulatory context, this article will focus exclusively on the long-term planning of natural gas networks.

2.3

Boundary Conditions and Degrees of Freedom

A complete description of the underlying degrees of freedom and boundary conditions of the network planning is fundamental to the development of efficient network structures. In the following, the main aspects of both will be presented. For a summary, see also Fig. 2. The supply task is defined by the geographical location of network customers as well as their demand and feed-in respectively. Topographic characteristics of the supply area restrict the degrees of freedom considering the choice of possible pipeline trenches and location of stations. Technical restrictions are given by each operating facility’s operating characteristics and safety requirements regarding both the system’s safety and individual customer specifications. They primarily consist of maximum and minimum admissible connection pressures for every nodal point, maximum gas flow speeds in pipelines for a limitation of noise emissions and maximum volumetric flow rates in pressure regulators. Company-internal planning guidelines that exceed technical minimum requirements correspond to a demanded retention capacity that is necessary for a safe operation especially in cases of critical network loadings. It may also be used,

Degrees of Freedom Pressure Stages

Technical Boundaries/ Supply Task

Degrees of Freedom Equipment Pressure Regulating Stations

Pressure Stages Optimization Procedure Network Structure

Piping Trenches Operating Equipment

Natural Gas Network at Optimal Costs

Fig. 2 Boundary conditions and degrees of freedom for network planning

Long-Term Pressure-Stage Comprehensive Planning of Natural Gas Networks

Station X Node Pressure Stage: A

41

Station Y Trench Admissible Types

Regulator A/B Admissible Types Node Pressure Stage: B

Pipeline,Pressure Stage B Admissible Types

Node Pressure Stage B Regulator B/C Admissible Types

Customer (B) Demand, Min./max.Pressure

Node Pressure Stage: C Customer (C) Demand, Min./max.Pressure

Fig. 3 Network model

especially in networks of high operating pressures, for an optimization of the gas acquisition strategy. Information on the calculation as well as dimensioning of retention capacities is published by the national technical and scientific organizations [7]. Degrees of freedom exist in the choice of different network structures, the number and degree of pressure stages and for the dimensioning of equipment. Figure 3 depicts the chosen network model that forms the basis for the proposed optimization approach. It is composed of stations and interconnecting trenches on the primary level. Stations are specified by their associated coordinates and likewise serve as geographical positioning reference of pressure regulators and network customers. Trenches may contain pipelines of different pressure stages in order to connect two stations.

2.4

System Model

The model allows for a definition of multiple pressure stages connected via pressure regulators within a station. Each node represents a pressure stage in each station. The closed analysis of possible dislocations of network customers between pressure stages as well as the fundamental determination of efficient vertical network configurations, realization and positioning of pressure regulators, is made possible by the presented network model.

42

M. H€ ubner and H.-J. Haubrich

As the pressure level of each pressure stage is depending on the location and discharge pressure of the used pressure regulators that is determined within the optimization procedure, the resulting operating pressure is an outcome of the optimization. Therefore, it is possible to determine the optimal combination of pressure stages objectively. Nodes are used to offer both an assignment of operating equipment to dedicated pressure levels and furthermore a junction of pipelines as well as pressure regulators. Trenches are composed of a combination of pipelines with potentially differing technical properties and pressure stage assignments.

3 Methodology Methods for a solution of combinatorial optimization problems can be divided into exact and heuristic proceedings [8]. Approaches with a mathematically verified optimality for a given objective target are ranked among the group of exact methods. The group of heuristic methods incorporates proceedings that feature a systematic search adapted to the problem statement.

3.1

Exact Methods

All relevant exact optimization proceedings are based on the iterative resolution and modification of a relaxation – broken down simplified problem – with a solution set containing all solutions of the initial problem. The branch-and-bound search and the cutting plane method apply a relaxation into a linear optimization problem and disregard all integrity requirements in the first instance. Since in principle, a greater number of solutions are accepted in the relaxation than in the initial problem, the optimal value of the relaxed problem is at least as good as the optimal value for the initial problem, hence providing an upper or lower bound. The value of each admissible integral solution defines just as well a boundary for the optimal solution of the initial problem as per definition. The comparison of both upper and lower boundary yields the maximum distance between a found solution and the optimum one, even though it is unknown.

3.2

Heuristic Methods

Heuristic methods are recognized as approaches to identify reasonable and admissible solutions to a problem statement and help map real decision problems mathematically. They proceed according to specific rules for solution finding or advancements that shall exploit the given model structure effectively while simultaneously lowering the time necessary for computing. Heuristic methods come into operation in situations where exact methods are inapplicable to real life planning problems because of their immoderate need for computing power.

Long-Term Pressure-Stage Comprehensive Planning of Natural Gas Networks

43

Local search algorithms generally start with an admissible solution of the problem, which has been identified either stochastically or by applying approximation algorithms. Transitions between adjacent solutions in the solution space are initiated in each step of the process. The neighborhood consists of all solutions that can be derived from the actual solution by a once-only execution of a transition directive. Pure local search algorithms end as soon as no further superior adjacent solution exists. The best-achieved solution represents, for the chosen neighborhood, the local optimum. Its objective function value may be considerably lower than the objective function value of the global optimum. In the past, a multitude of meta strategies have been developed to overcome the aforementioned problem of local optimality. The combination of local and global search techniques summarized in abstract meta strategies are used to enhance advancement heuristics by navigating to promising solution spaces in order to avoid local optima. For solving combinatorial optimization problems, applied problem solving models of specific adaption mechanisms in natural systems are being applied cumulatively. Common ground of the developed adaption analogies is, besides their origin from the study of natural phenomena, the principle as to converge to a problem’s solution by manipulating potential solution candidates, assess each manipulation and in conclusion extract new solution candidates based on the previous assessment.

3.3

Selection of Optimization Procedure

A guarantee for optimality, mathematically proven, is the essential advantage of exact methods. Although the branch and bound algorithm as well as all thereof derived variants exhibit a considerably improved run-time characteristic in contrast to alternative exact methods for combinatorial optimization, only problems with restricted extent can be solved [9–12]. The handling of extensive problems accrued from real life applications is rarely possible with justifiable computing power because the number of integer variables is exponentially increasing the complexity. Considering the aforementioned network model and the resulting degrees of freedom, computer-based optimization methods are required for solving the extensive optimization problem. Heuristic optimization algorithms have proved to deliver an optimal performance for the determination of cost-efficient network structures [13–16]. Their essential advantages are a reduced computational effort leading to computing times of typically few hours for realistic natural gas systems while simultaneously delivering several similar cost-efficient network structures. These advantages allow sensitivity analysis by a variation of boundary conditions and supply tasks on network structure and network costs and lead to a greater flexibility for the future network development. In contrast to exact algorithms, heuristic algorithms yield no guarantee for optimality. Nevertheless, numerous studies indicate that the difference between these nearly optimal results and the absolute optimum is typically

44

M. H€ ubner and H.-J. Haubrich

below 1% [3] and therefore negligible for realistic problem formulations. Genetic algorithms are, especially for combinatorial optimization problems, widely adopted heuristic optimization approach that will be applied for the introduced optimization problem. They refer to the natural evolution of animals over multiple generations. They additionally offer the possibility to exploit their population based design with a parallel execution to achieve further computing time reductions [17].

3.4

Optimization Procedure

In the following, the newly developed optimization procedure will be described in detail. For that purpose, the heuristic optimization approach will be explained in general before the necessary specific adaption to the problem of natural gas network planning will be introduced.

3.4.1

Genetic Algorithms

The nomenclature of genetic algorithms is based on the evolution theory and population genetics. The main elements that will be used frequently in the following algorithms are specified in Table 1. At the bottom of genetic algorithms is the idea, likewise based on the evolution theory, that a population made up of many individuals is generated stochastically and of these individuals who conform best to a dedicated quality criterion, i.e. have a high fitness value, are selected. The properties of the selected individuals that are encoded in each genestring are afterwards recombined and stochastically altered with the help of the genetic operators selection, crossover and mutation, in order to generate a population for a new generation. This procedure will be repeated many times for each newly generated population until an optimal conformance with the problem structure is reached. A schematic illustration of the process sequence of genetic algorithms is shown in Fig. 4. At the beginning of the procedure, an adequate and problem-adapted coding scheme has to be chosen, defining the composition of genestrings. A single gene with a generally arbitrary number range will map each decision variable of the underlying problem. These genes act as distinct identification objects of the Table 1 Terms of genetic algorithms Term Individual Population Fitness Generation Gene Genestring

Explanation Potential problem solution Set of potential problem solutions Objective function subject Iteration Variable of decision Encoding of problem statement

Long-Term Pressure-Stage Comprehensive Planning of Natural Gas Networks

45

Initial Population New Population

Fitness Appraisal

Mutation

Crossover

Selection Break?

no

yes Result

Fig. 4 Process sequence of genetic algorithms

decision variables and therefore make a decoding of the problem for a validation and assessment of the individual possible. Subsequently, an upfront-predefined number of individuals will be chosen from the search space stochastically or in combination with already known solutions. These individuals compose the initial population of the procedure. Afterwards, the quality of all individuals within the present population will be assessed with the help of a fitness function. The fitness function includes all relevant parameters as investment cost input and technical specifications so that a concrete roundup of each individual’s quality is possible. The termination criterion will be checked at the end of every generation. For example, a termination criterion may be a maximum number of iterations or a stagnant convergence over several iterations. If no termination criterion is met, new individuals are created by means of genetic operators that constitute the new population. Selection, crossover, and mutation are counted among the genetic operators. Figure 5 shows an exemplary application to illustrate the general approach of genetic operators. The genetic operator selection is chosen to arrange a gene pool from individuals of a population. All individuals are considered with a probability according to their fitness values. Therefore, well-adapted individuals receive a higher share in the gene pool than individuals with a poor fitness value do. Thus, the main functions of selection operators within genetic algorithms are: • Selection of individuals participating in the crossover • Selection of a predefined number of individuals that form a successive population from a set of individuals • Selection of a predefined number of individuals from the set of individuals resulting from a crossover that shall be further regarded

46

M. H€ ubner and H.-J. Haubrich Initial Population

Selection

Crossover

Mutation

Fig. 5 Exemplary application of genetic operators

Following the selection, at least two individuals are extracted from the gene pool and recombined with a definable crossover operator to form new individuals. The main goal of this recombination is ideally to transfer the positive attributes of the different parent individuals so that the succession unifies these attributes. The concept of a genetic operator mutation has been introduced to genetic algorithms in order to replicate the naturally occurring stochastical modification of genes. For this purpose, individuals from the actual population will be selected and their genestrings altered at sparse positions. In comparison to the crossover operator, the mutation operator should be applied the more infrequent the more advanced the genetic algorithms get as otherwise individuals in the near of a potential optimum will be altered too excessively. Depending on the chosen mutation function, one ore more genes of a genestring will be mutated. Finally, the population generated by the genetic operators and tested for the termination criterion will replace the actual population. The genetic algorithm does not necessarily need to replace the old population entirely but can also take over the best individuals from the old population. If no termination criterion is fulfilled, all genetic operators will be applied to the new population. Otherwise, a predefined number of the best individuals will be exported.

3.4.2

Application to the Planning of Natural Gas Networks

A fundamental advantage of genetic algorithms is they are in large part problemindependent frameworks that allow for an application different formulated problems. The approach simultaneously offers a great flexibility to consider problem specific requirements. Essential adaptions are in particular: • Used encoding • Definition of a fitness function

Long-Term Pressure-Stage Comprehensive Planning of Natural Gas Networks Table 2 Representation of decision variables Network level Pressure regulator Range 1–ℓ 0–m Representation Realization and choice Realization and choice of of structural operating resource parameter category

47

Trench 0–n Realization and choice of operating resource category

The encoding is determined by the individual approach’s implementation and defines the meaning of each decision variable. All degrees of freedom, such as the choice of operating equipment for pressure regulators and pipelines, are ranked among the decision variables. For the planning of natural gas systems, only integer values are acceptable for decision variables. Table 2 depicts the chosen gene encoding for the degrees of freedom for pressure stages, pressure regulators, and trenches. The encoding for trenches contains both the surface works and all possible pipelines that may be installed on the trench with specification of type and pressure stages. Basis for the appraisal of an individual (the fitness function) is the economic evaluation of each decision variable so that predominantly cost-efficient network concepts with a consequentially high fitness value will be brought forward. Additionally, violations of technical minimum requirements that are uncorrectable with repair functions may be considered via a penalty function. The penalty function makes sure that network concepts with malfunctions receive a lower fitness value.

Optimization Procedure An overview of the applied optimization procedure is illustrated in Fig. 6. First, possible optimization variables consisting of usable trenches, pressure regulating stations and types of operating facilities are assembled. They enter the optimization procedure in combination with the given boundary conditions. Of the closed optimization core, a multitude of different possible network concepts is drafted stochastically within each iteration. First, each network concept will be analyzed regarding its technical feasibility. If any boundaries are violated, a multitude of repair functions are applied in order to transfer the draft into the admissible solution space. Local search algorithms improve the network structure with limited changes and thus support a faster convergence towards optimal network structures. In the next step, network costs for each concept will be determined and attributed to the corresponding concept as a quality rating. All concepts will be arranged according to their quality in order to compose a new genetic pool for the stochastic generation of new network concepts. The best network concepts will be transferred unchanged into the new population. Thereby, the already achieved level of quality will likewise be transferred combined with completely new concepts. Genetic operators, e.g., selection, crossover, and mutation, will be applied on the whole population. If no significant advancement in quality is perceived the iterative process will be aborted and the best network structure will be exported.

48

M. H€ ubner and H.-J. Haubrich

Degrees of Freedom and Boundary Conditions Parameterization and Initialization

Closed Optimisation of Pressure Stages and Network Structure

Pressure Stage x Technical Verification, Repair Functions Local Search for Convergence Enhancement Application of Genetic Operators (Selection, Cross-Over, Mutation)

Target Networks at Optimal Cost under Consideration of Technical Specifications

Fig. 6 Optimization procedure scheme

Technical Assessment The compliance with all required technical minimum requirements is checked in a two-stage process. If a violation is detected, the repair functions are used to recover a state without these violations. The first step comprises structural inspections that in essence target the security of supply. Afterwards, all operational constraints are checked by calculating nodal pressures and all thereof derived values like gas velocities. Based on this flow calculation, technically admissible and sensible planning alternatives in the neighborhood of the network structure under consideration are analyzed.

Operational Assessment Substantial operational requirements for the planning of natural gas networks are limits for flow velocities in pipelines, flow rates in pipelines and through pressure regulators and as obligatory limit maximum and minimum pressure levels for all nodes. The compliance of these limits is likewise checked with the results of a flow calculation. Required maximum flow rates are depending on the type of each used pressure regulator. If the actual flow rate exceeds the maximum admissible value, a potentially more expensive pressure regulator with broader limits or a pressure regulator with a lower outlet pressure may be inserted. Should the aforementioned possibility be unfeasible, pressure regulators at different positions within the pressure stage may be amplified to relieve the pressure regulator under consideration.

Long-Term Pressure-Stage Comprehensive Planning of Natural Gas Networks

49

In principle, it is possible to give a minimum and maximum pressure limit for every network node. In the majority of cases, such stipulations suffice for nodes connecting network customers or storages. In the case of a violation of pressure limits in the flow calculation, adjacent pipelines in the direction of pressure regulators will be amplified in order to increase the node’s pressure. Similarly, violations of maximum flow velocities can be intercepted with an increase of the pipeline diameter of the section under consideration. As high flow velocities coincide with high-pressure gradients due to undersized pipeline diameters, the examination of minimum pressure limits will have a positive impact on the compliance of maximum flow velocities.

Local Search for Convergence Improvement The stochastic components of genetic algorithms and the starting population that in general has been generated without consideration of any constraints cause the individuals in the beginning generations to be far from the searched optimum. The violations of boundary conditions due to the aforementioned stochastics furthermore demand the use of repair functions that in either case lead to a worsening of the fitness value. Therefore, it seems advisable to conduct costreducing structural refinements in the form of a local search that does not change the fundamental structural character of a network significantly. Because of marginal efforts for technical and economic evaluations of network concepts, a great part of the surrounding solution space of each solution can be examined easily. If a solution possibility with a higher fitness than the initial solution can be found, a transfer to this solution will be carried out. Possible solution changeovers include modifications of operating equipment or the dismantling of several – from technical point of view not necessary – equipment. For that purpose, a network concept from a restricted neighborhood of the actual concept is chosen with the help of two heuristics. An obvious approach is based upon inspection of exceptionally low loaded operating equipment. For instance, diameters of pipelines with low flow velocities are reduced and pressure regulator types are adjusted to the real flow rate. A second heuristic, based on sensitivity matrices, considers the impact of single structural changes on the residual network. Having preferred equipment with low sensitivity coefficients, the network will be adapted adequately with a minimum number of structural changes and with a minimum operational influence. The immediate verification of the technical characteristics for all solution transitions provides information on its suitability and possibly its economic advancement. Each new solution, summing up all carried out transitions, will be included in the population and technically and economically evaluated. This approach allows for a systematic search for structural changes to improve the convergence characteristics of the genetic algorithm.

50

M. H€ ubner and H.-J. Haubrich

4 Exemplary Results In this chapter, the approach’s performance and functionality will be demonstrated with the help of an exemplary application for a real supply area. For that purpose, a medium-scale urban supply task will be optimized with regard to pressure stage composition, network structure design, dimensioning of operating equipment, and compliance with state-of-the-art technology requirements. Initially, the system under investigation will be described and the already existing natural gas network assessed from a technical as well as economic point of view. The findings will provide a standard of comparison for the optimized cost-efficient network structures. A survey of all necessary degrees of freedom and boundary conditions will be subsequently defined. In the following section, the results of a pressure stage comprehensive network optimization for the considered area will be presented as well as technical and economical assessments. The integrated optimization of pressure stages, network structures, and operating equipment is a fundamental innovation in the field of network planning. Furthermore, the convergence characteristic of the genetic algorithm will be explored with the use of a sensitivity analysis. Finally, prospects for a significant reduction of computing time by the use of a parallel execution of the genetic algorithm on multiple processors shall be analyzed.

4.1

Data Models

The considered supply area (see Fig. 7) exhibits a predominant urban character. The total area is approximately 100 km2 in size. Two connection points to the overlaying 16 bar regional distribution grid provide the opportunity to feed natural gas into the supply area. With a number of 30,000 inhabitants, the total power output adds up to 150.8 MW. A share of 68% of the power output falls upon the low-pressure household customers and the remaining output falls upon business customers in the medium pressure stage. In a prognosis for a long-range planning horizon, stagnation on today’s level of the demand situation has been determined (cp. Table 4). Because of the general decline of demand for natural gas due to a replacement of natural gas for alternative energy sources and a reducing population, a scenario of a halving of the total demand will be analyzed additionally. Three potential pressure stages will be considered for the exemplary results: one high-pressure stage operated at four bar (HP4), one medium-pressure stage (MP), and one low-pressure stage (LP). 4.1.1

Boundary Conditions and Degrees of Freedom

The degrees of freedom are composed as follows: • The existing network features three pressure stages with the low and medium pressure stage predominantly used for the supply of network customers. Due to

Long-Term Pressure-Stage Comprehensive Planning of Natural Gas Networks

Connection HP16

Regulator HP4/MP

MP-Customer

Possible Trench HP4/MP/LP

Regulator HP4/LP

LP-Customer

Regulator HP16/HP4

Regulator MP/LP

Branching Point

51

Fig. 7 Network for exemplary investigation

• •

• •

the small geographical dimension of the supply area, the number of pressure stages will be limited to these three pressure stages. Each combination of pipelines of all possible pressure stages may be placed in every trench connecting two stations. The positioning of pressure regulators will be restricted to adequate areas according to the place requirements (see Fig. 7). Overall, the optimization procedure may choose from 64 pressure regulators connecting the three pressure stages. Additionally, there are two possible transfer stations connecting to the overlaying 16 bar transmission network that bypasses the urban area (see Fig. 7). In accordance to the planning practice, the number of selectable equipment categories will be limited. For instance, pipeline types may be chosen from six categories representing diameters between DN 80 and DN 500 (cp. Table 3). Furthermore, the following technical requirements are considered:

• Customers are located in the low and medium-pressure stage. The assignment of connection points is evident in Fig. 7. • The requirements concerning the pressure level in each pressure stage result from maximum infeed pressure and minimum pressure at customer as well as pressure regulator connections (cp. Table 4).

52

M. H€ ubner and H.-J. Haubrich

Table 3 Possible equipment categories Type 1 Type 2 Regulator pd ¼ 4 bar pd ¼ 0.8 bar Pipeline DN 80 DN 100 Trench One pipeline Two pipelines

Type 3 pd ¼ 0.05 bar DN 150 Three pipelines

Type 4

Type 5

Type 6

DN 300

DN 400

DN 500

Table 4 Supply area properties HP4 0 4 1.6 2

Discharge power [MW] Feed-in pressure [bar] Minimum pressure [bar] Number of possible regulators

HP4

Trench

MP

Feed-in PressureRegulator Take-off PressureRegulator

MP 48.4 0.8 0.2 22

LP 102.4 0.05 0.03 42

LP

MP-Customer LP-Customer

Fig. 8 Structure of existing network

• In order to restrict noise disturbances for residents in the urban area, the flow velocity will be limited to 20 m/s. • The distributed natural gas is a H-gas with caloric value of 11.3 kWh/mN3 and a medium temperature of 5
C for the relevant peak load case. 4.1.2

Existing Network

Figure 8 illustrates the pressure stage design as well as the network structure for each pressure stage of the existing network, featuring two connection points with the overlaying 16 bar high-pressure network along with distributing the natural gas over three pressure stages to the end customers. While the high pressure stage (HP4) exhibits a wide-meshed network structure, the medium and low pressure stages (MP and LP) exhibit a fine-meshed network structure. Besides the absolute minimum pressures, Table 4 equally provides bandwidths of the admissible pressure within each pressure stage as the zone between the feed-in pressure and the minimum outlet pressure for customers or pressure

Long-Term Pressure-Stage Comprehensive Planning of Natural Gas Networks HP4

Chosen Trench

MP

Feed-in Pressure Regulator Take-off Pressure Regulator

53

LP

MP-Customer LP-Customer

Fig. 9 Target network for a stagnating demand

regulators feeding lower pressure stages. In the existing network, the resulting total bandwidths are utilized only to a low extent indicating that taking into account the technical requirements, a substantial reduction of overall pipeline length and equipment dimensioning could be achieved. In Fig. 11, the annual network costs of the existing networks (differentiated between investment and operating costs) for pressure regulators, pipelines, and trenches are depicted for each pressure stage. The total annual network costs aggregate to 3.2 Mio. € for an interest rate of 9.5% and are mainly influenced by the necessity to connect household and business customers to the medium and low pressure stage respectively leading to big pipeline lengths in these stages.

4.2

Determination of Long-Term Cost-Efficient Networks

In the following, long-term cost-efficient networks (also referred to as target networks) for the given supply task will be determined with the use of the proposed optimization approach. The results will exhibit optimal functional segmentations in pressure stages and optimal network structures – with an integrated dimensioning of operating equipments – for each used pressure stage. Figure 9 (scenario stagnating demand) and Fig. 10 (scenario halved demand) give an overview of both obtained target networks for the projected demand scenarios. Both target networks have a significantly different network structure, especially in the low pressure stage, compared to the existing network but also among one another. The pipeline length in both target networks has been considerably reduced (stagnating demand: 17 km, halved demand: 24 km with a total of 79 km in the existing network). Furthermore, the pipelines in the target networks are on average of smaller dimension. The low-pressure stage is, adjusted to the connection points of end customers, wider meshed than the existing network leading to the formation of island networks. Island networks can be found in the medium pressure stage as well.

54

M. H€ ubner and H.-J. Haubrich HP4

Chosen Trench

MP

Feed-in Pressure Regulator Take-off Pressure Regulator

LP

MP-Customer LP-Customer

Fig. 10 Target network for a halved demand

The target network for the scenario halved demand exhibits less isolated networks in comparison to the target network for scenario of stagnating demand. This is due to a decreased need for pressure regulators. Furthermore, all target networks have significantly lower annual pipeline costs compared to the existing network. On the other hand, costs for pressure regulators are only slightly reduced. At the same time, the results demonstrate that the existing network is over dimensioned for the fulfillment of the supply task. The following results are achieved: • Positions of pressure regulators for a connection of two pressure stages are, due to the integrated optimization approach, optimally adjusted. This yields drastically reduced costs for pipelines with simultaneously maintaining a nearly equal number of pressure regulators. • The northern pressure regulator feeding in from the 16 bar grid is not necessary for an economic connection of the considered network area. • With exception of the medium-pressure stage in the scenario halved demand, the nodal pressures in all pressure stages max out the admissible bandwidth of pressures. Consequently, both target networks do not possess spare transfer capacities beyond the peak demand case. • The medium-pressure stage in the scenario halved demand is used almost exclusively for the supply of network customers. Therefore, an investigation on altered pressure connections of medium and low-pressure customers seems sensible and has been included in the economic evaluation.

4.2.1

Evaluation of the Pressure Stage Comprehensive Optimization

Figure 11 illustrates the comparison of all executed evaluations and the existing network in terms of annual investment and operating costs. The majority of annual network costs of the existing as well as of the target networks fall upon the distribution level of household customers. In contrast to the

Long-Term Pressure-Stage Comprehensive Planning of Natural Gas Networks LP

Annual Network Costs [M /a]

3,5 3,0

HD4

MP

HP4

55

Household Pressure Regulators

–40.2 %

–46.1 %

–30.7 % –58.7 %

–46.3 %

2,5 MD

2,0 1,5 1,0

ND 0,5 0,0 Existing Network

Stagnating Demand

Halved Demand Abandonment LP

Abandonment MP

Fig. 11 Comparison of annual network costs

low-pressure stage that is mainly influenced by the given points of supply, advanced degrees of freedom exist in overlaying pressure stages leading to significant differences in their costs. The medium and high-pressure stage may take over different functionalities such as a wide-ranging transportation or localized supply of natural gas. The theoretical cost cutting potential of all target networks are in-between 30,7% with an abandonment of the low-pressure stage and the assumption that each then required household pressure regulator will incur expenses of a further 500€ and 58,7% if the expenses for household pressure regulators are neglected. Because of the low number of medium-pressure customers, additional expenses for pressure regulators are neglectable concerning the target network with an abandonment of the medium-pressure stage. The optimal annual network costs in the scenario halved demand are reduced just slightly by approximately 6% compared with the scenario stagnating demand. If the target networks for a reduced demand are analyzed it is obvious that the expenses for three pressure stages is almost equal to the expenses with regard to an abandonment of the medium-pressure stage. Nevertheless, three pressure stages offer major flexibilities for a deviation from the projected demand. An increase of the connection pressure for household customers to medium-pressure leads to considerably increased expenses due to the necessary household pressure regulators. As the results have shown, the proposed optimization procedure is capable of identifying inefficiencies in the existing network and recommends measures for correction.

4.3

Appraisal of the Optimization Procedure

Genetic algorithms offer a multitude of possibilities to parameterize, with each having an influence on the convergence characteristic to a greater or lesser extent

56

M. H€ ubner and H.-J. Haubrich 3

Fitness [M /a]

2,8 2,6 2,4 2,2 2 1,8

0

20

40

60

80

100

Iteration

Fig. 12 Convergence characteristic

and consequently on the computing time. The iteration number and population size have a direct influence on the computing time as well as the quality of the achievable results. The iteration number specifies how many populations are generated and evaluated. With choosing a higher iteration number, in general, a greater part of the solution space will be explored. In order to analyze the influence of the iteration number in detail, an exemplary optimization for the identification of the cost-efficient network structure for the already described planning task is executed. Figure 12 depicts the bandwidth of individuals’ fitness values evolving with increasing iterations with a population of 1,000 individuals. As shown in the diagram, it is apparent that due to the pure stochastic nature of the generation scheme at the beginning of the optimization and therefore high distance to the optimum, very fast advancements in the solution quality are achieved. After approximately 50 iterations, only minor improvements are necessary for the present case. The required number of iterations mainly depends on the number of degrees of freedom of each planning task. Application of the optimization procedure on different real supply tasks demonstrated that typically an iteration number in the same magnitude as the degrees of freedom provide good solutions even for huge problems. By using additional supporting local search algorithms, the iteration number can be decreased relative to comparable genetic algorithms without local search. The population size specifies how many different individuals have to be created and evaluated during a single iteration of the genetic algorithm. An increasing population size leads to a greater diversity amongst the individuals and helps in preventing a premature convergence to local minima. For a detailed evaluation of this correlation, population sizes between 50 and 2,000 for 100 iterations are chosen. Figure 13 illustrates the resulting means and standard deviations of the annual network costs of the most cost-efficient network structures after 100 iterations for the specified population sizes. With an increasing population size,

Long-Term Pressure-Stage Comprehensive Planning of Natural Gas Networks Expected Annual Costs

Standard Deviation

3

0,2

2,5

0,16

2 0,12 1,5 0,08 1 0,04

0,5 0

Standard Deviation [M /a]

Annual Network Costs [M /a]

57

0 50

100

250

500

750

1000

2000

Population Size

Fig. 13 Influence of population size on the solution quality

an initial drastic reduction of the annual network costs is achieved, but eventually converging to a nearly global optimum.

4.3.1

Parallel Optimization

A major issue of the demonstrated optimization procedure for practical planning tasks is the increasing computing time in consequence of size and complexity of the optimization problem. Hence, possibilities for accelerating the genetic algorithm with the use of parallel programming are analyzed. Genetic algorithms are particularly suitable for parallelization due to their design [18]. Creation, assessment, and selection of individuals during the genetic algorithm require the majority of all computing time. An adequate parallelization approach is the segmentation of each population’s fitness assessment process. High computing time benefits are expected because of the time-consuming evaluation of the merit function, including a large number of flow calculations, for the planning of natural gas networks [17]. For the chosen parallelization approach, a primary process administrates the population and assigns further processes with the assessment of one individual each. Hence, the process flow remains unaltered in comparison to a sequential genetic algorithm, but high computational benefits can still be achieved (see also Fig. 14). The parallel computation on multiple processors yields considerable computing time benefits in comparison to a sequential computation. However, the speedup – meaning the relative computing time benefit relating to the serial computation – does not ascent linearly, but converges with a rising number of processors. This results from higher coordination efforts towards the communication of all processes and their differing run-times. As the coordination effort is

58

M. H€ ubner and H.-J. Haubrich Real

Ideal

Speedup(N)

100%

14 12 10

60%

8

40%

6

Speedup

Computing Time

80%

4 20%

2

0%

0 1

10

30 50 70 Number of Processors

90

Fig. 14 Computing time benefits by parallelization

directly dependent on the population size, the number of utilized processors has to be adapted accordingly.

5 Conclusions Due to the forthcoming regulation schemes throughout Europe, new challenges for natural gas network operators arise. In the past natural gas networks have been planned without the specification of long-term objectives and therefore often exhibit oversized dimensioning of the underlying network structure as well as the utilized operating facilities. This in consequence leads to corresponding higher network costs. Computer-aided optimization procedures provide important assistance for the retrieval of potentials for future cost reductions. While the development and application of such optimization procedures for the planning of electricity networks is widely adopted and state-of-the-art, the planning of natural gas networks has previously been accomplished by predominantly experience-based example investigations and thus without the necessary objective future orientation. The presented approach enables the long-term planning of natural gas networks under consideration of all relevant technical and furthermore company internal guidelines. It yields multiple alternatives as well as cost-efficient network structures with considerable low computational time. With the aid of sensitivity analyses, possible effects of varying boundary conditions on structure and costs of cost-efficient networks may be quantified. These sensitivity analyses allow for a fundamental determination of optimal planning principles and are therefore able to explain minimal network costs for a given set of boundary conditions.

Long-Term Pressure-Stage Comprehensive Planning of Natural Gas Networks

59

References 1. Ehrhardt K, Steinbach MC (2005) In: Bock HG, Kostina E, Pu HX, Rannacher R (eds) Nonlinear optimization in gas networks. Modeling, simulation and optimization of complex processes. Springer, Berlin/Heidelberg/New York 2. H€ubner M, Haubrich HJ (2008) Long-term planning of natural gas networks. In: IEEE 5th international conference on the European electricity market, Lissabon, 2008 3. Paulun T, Haubrich H-J, Maurer Ch (2008) Calculating the efficiency of electricity and natural gas networks in regulated energy markets. In: IEEE 5th international conference on the European electricity market, Lissabon, Portugal, 2008 4. Hofbauer M, Anders S, Sigrist R, Weßing W (2003) Modell einer kostenoptimierten Gasverteilung. In: GWF-Gas/Erdgas 144, No. 5, 2003 5. H€ubner M (2009) Druckebenen€ ubergreifende Grundsatzplanung von Gasverteilungsnetzen, vol 127. Aachener Beitr€age zur Energieversorgung, Klinkenberg/Aachen 6. Paulun T (2006) Strategic expansion planning for electrical networks considering uncertainties. Eur Trans Electrical Power 16(6):661–671 7. German technical and scientific association for gas and water e.V. (DVGW). Code of practice G 2000, “Gasbeschaffenheit”, Bonn, 2006 8. Domschke W, Drexel A (2007) Einf€ uhrung in operations research, 7th edn. Springer-Verlag, Berlin 9. Rothfarb B, Frank H, Rosenbaum DM, Steiglitz K, Kleitman DJ (1970) Optimal design of offshore natural-gas pipeline systems. Oper Res 18:992–1020 10. Osiadacz AJ, Go´recki M (1995) Optimization of pipe sizes for distribution gas network design. In: Proceedings of the 27th PSIG Annual Meeting, Albuquerque, Pipeline Simulation Interest Group, 1995 11. Hoeven Tvd (2004) Math in gas and the art of linearization. Dissertation, Groningen: Energy Delta Institute, Rijksuniversiteit Groningen, 2004 12. Handschin E, Waniek D, Martin A, Mahlke E, Zelmer E (2007) Gekoppelte optimale Auslegung von Strom-, Gas- und W€armenetzen. In: VDI-Berichte Nr. 2018, Leverkusen, 2007 13. Boyd ID, Surry PD, Radcliffe NJ (1994) Constrained gas network pipe sizing with genetic algorithms. In: EPCC-TR94-11, 1994 14. Cunha MC, Ribeiro L (2004) Tabu search algorithms for water network optimization. Eur J Oper Res 3(175):746–758 15. Duarte H, Goldbarg EFG, Goldbarg MC (2006) A tabu search algorithm for optimization of gas distribution networks. Evol Comput Comb Optimization 3906:37–48 16. Castillo L, Gonza´les A (1998) Distribution network optimization: finding the most economic solution by using genetic algorithms. EurJ Oper Res 108(3):527–537 17. Rajan SD, Nguyen DT (2004) Design optimization of discrete structural systems using MPIenabled genetic algorithm. Struct Multidisciplinary Optimization 28(5):340–348 18. Gottlieb J (2000) Evolutionary algorithms for constrained optimization problems, 1st edn. Shaker Verlag, Aachen

Optimal Location of Gas Supply Units in Natural Gas System Network Teresa Nogueira and Zita Vale

Abstract Natural gas industry has been confronted with big challenges: great growth in demand, investments on new GSUs – gas supply units, and efficient technical system management. The right number of GSUs, their best location on networks and the optimal allocation to loads is a decision problem that can be formulated as a combinatorial programming problem, with the objective of minimizing system expenses. Our emphasis is on the formulation, interpretation and development of a solution algorithm that will analyze the trade-off between infrastructure investment expenditure and operating system costs. The location model was applied to a 12 node natural gas network, and its effectiveness was tested in five different operating scenarios. Keywords Constrained capacities • Fixed costs • Gas supply units • GSU • Natural gas • Optimal location • Optimization • Transportation costs

1 Introduction The constant increases in oil prices, the governments’ commitments to the Kyoto Protocol, and the social and political concerns with the adoption of sustainable development strategies, have led to the increasing search of new technologies and alternative energies. Natural gas fits in those specifications, because it’s a clean energy and provides an important contribution to the security of supply. The European energy policies encourage the production and consumption of natural gas. The natural gas market growth implies a significant reinforcement on

T. Nogueira (*) • Z. Vale Institute of Engineering, Polytechnic Institute of Porto Rua, Dr. Anto´nio Benardino de Almeida, 431, Porto, Portugal e-mail: [email protected]; [email protected] A. Sorokin et al. (eds.), Handbook of Networks in Power Systems II, Energy Systems, DOI 10.1007/978-3-642-23406-4_3, # Springer-Verlag Berlin Heidelberg 2012

61

62

T. Nogueira and Z. Vale

gas supply units – GSUs – and on transport networks, in order to properly meet the gas load demands. On one hand, the investment in GSUs is significant, and on the other hand, pipelines also require a large investment whereas they have to face serious physical and technical constraints. Thus, the investment decision in GSUs is of major importance: how many of these facilities should be placed and where, in order to meet gas demand, at the lowest possible costs? This question is addressed in this article by using an optimization technique – specifically, using a location problem and model. A simple location problem can be formulated as a linear programming problem in which some of the variables are constrained to assume only integer values. Such problems are called integer linear programming problems. The added complexity in these problems results from the fact that we usually are interested not only on location, but also on the optimal allocation of GSUs to gas loads. This led us to choose an optimal combinatorial problems’ approach [1]. In the case of small dimension natural gas networks, location problems can be supported by optimization solver tools, which are commercially available. The analytical location methodology is heavily dependent on the system network size [2]. In literature [3] we can find different location methodology approaches. For large networks, Beasly [4] describes very effective heuristics dealing with location problems. To deal with the realistic dimension of a gas system, the mathematical models need to be extended, observing regulatory and technical restrictions. To better understand the general concepts of the location problem methodology, we will apply it to small networks, as a basis for further comprehension.

2 Natural Gas Industry 2.1

Development and Trends

Natural gas is a gift from nature, and finds its origin in high-pressure conversion of biomass into coal and hydrocarbons. Part of this million-of-years process is due to intensive underground dynamics, which brought the natural gas through porous layers deeper and deeper, until it became captured in an impermeable layer. Widely used all over the word, natural gas is an attractive fuel, because it is easy to transport (in its liquefied form), cheaper and clean. It has the highest hydrogen carbon ratio of all fossil fuels, and when burned or used as feedstock the exhaust gases hold less CO2 compared to the conversion of oil and coil. Natural gas incorporates highly important national interests, in terms of economy and social welfare. In gas producing countries, natural gas reserves are considered a national asset and have been exploited for the national benefit. For gas consuming countries, access to gas reserves is also of crucial importance. The development of natural gas markets in Europe has so far been guided by national policies that take into account each country’s access position to gas reserves.

Optimal Location of Gas Supply Units in Natural Gas System Network

63

We can distinguish two different models for the gas market development: a public property focused model and a public utility focused model [5]. The first model is usually found in countries with voluminous gas fields, and the second model, in countries without gas fields. Only a few European countries have gas reserves voluminous enough to allow for both domestic consumption and export. In these countries gas reserves are considered as national assets and have been exploited for the national benefit, with the ambition to capitalize economic value. This became a guiding principle in the exploitation of gas resources. The organization of markets based on a public property model focuses in the upstream activities – production and transmission, controlling access to a limited number of players, and the ownership structure is public dominant. This model assures maximization of state revenues and pursues the national welfare and prosperity. Net importing countries – those without gas reserves – have concentrated on gas consumption and gas demand. Policy for the public utility model draws on the idea that natural gas is a vital national energy resource that should be available nationwide. In this regard, the state takes an active role in developing the utility-oriented gas market for economic and political reasons. State control and monopoly regulation was meant to reduce investment risk and to ascertain long-term domestic consumption by developing a domestic consumer market. The public utility model is therefore consumer oriented and heavily relies on specific regulation. Unlike the public property model, it focuses in the downstream activities, also controlling the access to a limited number of actors. As a consequence, the model pursues reasonable consumer tariffs and selective services. In addition to these two models, there is what one could name a commodity focused model, referring to a gas market where natural gas is perceived as a freely tradable commodity – reflecting the ideal of a competition-based gas market. This commodity-oriented model is now challenging all European countries due to the EU gas Directive (see Sect. 2.3). Because of its import dependency and expected strong growth in demand, continental Europe gas industry faces important challenges and opportunities [6]. The growth of the European natural gas market will depend, to a large extent, of gas price competitiveness as compared to other energy sources and technologies.

2.2

Technological Infrastructures of Natural Gas

A complex infrastructure, including upstream and downstream activities, is necessary to comply with the natural gas demand. The production, supply, transmission and distribution are the main activities in the gas chain, which are complementary, technically as well as economically (see Fig. 1). Strict coordination of such activities is therefore of paramount importance to safeguard high quality supply of natural gas from a technical point of view. Production activity is found only in countries which have gas fields. The exploitation of gas fields is far from easy, but the similarity to the oil extraction

64

T. Nogueira and Z. Vale

PRODUCTION Exploitation

SUPPLY Import

TRANSMISSION Storage

DISTRIBUTION Usage

Fig. 1 Natural gas chain

process has allowed the development of efficient technologies and techniques to locate and explore gas reserves. While only a few countries actually have production activity, most countries rely on security of supply provided by importation. After extraction and refinery processes, usually far away from consumers, natural gas is transported in ships in liquid form, that is, liquified natural gas – LNG, and delivered to national marine terminals and stored in appropriate infrastructures. Storage facilities, such as liquefied natural gas storage facilities, salt cavities and depleted fields or aquifers, require specific physical and geological characteristics which are difficult to meet. According to load demands, natural gas is later regasified and injected in pipelines to be transported and delivered to distribution centers or directly to end consumers. For non producing countries, these storage facilities constitute the gas sources of the natural gas network. It is of major interest to carefully plan the most adequate location for these facilities, minimizing expenses and maximizing throughput and security of supply, observing regulatory, technical and safety restrictions. LNG is natural gas that has been cooled to about
260 F for shipment and/or storage as a liquid. LNG is more compact than the gaseous equivalent with a volumetric difference of approximately 610–1. Liquefaction provides the opportunity to store large quantities of gas for use according to the demand periods. Current LNG facilities reflect different applications of LNG-related technology: marine terminals receive ship delivered imports of LNG and have on-site storage (non producer countries), underground storage facilities and other operations to serve demand [7]. Underground reservoirs are geological structures that have unique features. There is a porous medium having some degree of permeability. The porosity allows natural gas to be contained within the medium. The permeability allows the gas to move from point to point within the medium [8]. For non producing countries, LNG facilities play the role of gas supply units, the first link of the natural gas chain, and provide operational flexibility during times of high demand. Usually, the transportation pipeline companies also own and operate LNG facilities in much the same way as they own and operate underground storage facilities as part of their integrated systems. Growth in the use of LNG technology by natural gas industry depends on expansion of current facilities and new construction. The need for additional GSUs to meet projected demand and their optimal location on the gas network is a major issue of the natural gas industry. To comply with the demand growth, the natural gas markets need to organize an efficient upstream infrastructure with the appropriate investment on new GSUs facilities.

Optimal Location of Gas Supply Units in Natural Gas System Network

2.3

65

Liberalization of Natural Gas Market

Some differences on the national responses to the emerging EU gas market liberalization can be found. A point of departure is the obvious difference among European countries with respect to access to gas resources. Liberalization has affected the image and perception of natural gas, which can no longer be considered only as a public property or a public utility, as the ownership structure is open to private dominance. So far, the commodity-based model can be found, where some competition requirements are met. To be able to perform, a commodity based model requires both international and national matured upstream and downstream gas systems. In an attempt to harmonize the differences, European markets are oriented by Directive 2003/55/EC [9] of the European Parliament and of the European Council, concerning common rules for the internal market of natural gas. The new current law facilitates competition by introducing third party access of transmission and distribution pipelines and financially unbundling transport and trade activities. In order to ensure efficient and non-discriminatory network access it is appropriate that the transmission and distribution systems are operated through legally separate entities where vertically integrated undertakings exist. It is also appropriate that the transmission and distribution system operators have effective decision making rights with respect to assets necessary to maintain, operate and develop networks when the assets in question are owned and operated by vertically integrated undertakings. National regulatory authorities should be able to fix or approve tariffs, or the methodologies underlying the calculation of the tariffs, on the basis of a proposal by the transmission system operator or distribution system operator(s) or LNG system operator, or on the basis of a proposal agreed between these operator(s) and the users of the network. In carrying out these tasks, national regulatory authorities should ensure that transmission and distribution tariffs are non-discriminatory and cost-reflective, and should take account of the long-term, marginal, avoided network costs from demand-side management measures. In the liberalized natural gas market, gas customers should be able to freely choose their supplier. Nonetheless, a phased approach should be taken to completing the internal market for gas, coupled with a specific deadline, to enable industry to adjust and ensure that adequate measures and systems are in place to protect the interests of customers and ensure they have a real and effective right to choose their supplier. The progressive opening of markets towards full competition is almost complete in European countries with the implementation of appropriate measures to achieve the Directive’s objectives. Such measures include adequate economic incentives, using, where appropriate, all existing national and Community tools, for the maintenance and construction of necessary network infrastructure, including interconnection capacity and investment on new gas supply units – GSUs [10].

66

T. Nogueira and Z. Vale

These arguments reinforce the need of GSUs investment, to ensure the security of supply, and the existence of new natural gas players within a liberalized market. Having the GSUs economic feasibility in mind, we’ll study the GSUs’ location problem, which plays a key role in supporting the sustainable development of the natural gas industry.

3 Gas Supply Units Location Problem 3.1

Concepts and Notation

A major determinant of the level of effective natural gas supply is the ease to feed customers, with the correct balancing between the infrastructure investment and the ongoing transportation costs through the gas network. The number of GSUs and their optimal location in a gas network is a decision problem that can be formulated in order to minimize a cost function [11]. In practice, because of economic constraints, if we want to improve a gas system supply, we must try and find the best location sites for a fixed number of GSUs. We often optimize an objective function subject to the constraint that we should locate a given number P of facilities, reaching a solution that is usually different from the one we find in the case where such a constraint does not exist. Both situations will be considered in this study: no P value constraint and P value constraint. In the first case, P is an output of the problem; in the second case, P is an input. If the GSUs are marine terminals or other kind of aerial storage facilities, we can consider that they have an unlimited capacity. This means that we can consider all the supply capacity we need to feed properly gas demand. In case of underground storage facilities, because of physical issues, maximum specified capacities of infrastructures are important, so GSUs capacitated fixed charges are given as constraints of the location problem. For geological reasons, some GSUs infrastructures constructions are not suitable or feasible in certain locations. Similar GSUs structures can have different fixed implantation costs in different locations. To include this particularity in this study, we will consider an explicit cost, fi, of locating a GSU at each candidate location i. In this study we consider that these i candidates location are nodes on network. According Hakimi [12], the optimal places to locate facilities on network can be found at nodes. This author proved that by relaxing the problem to allow facility locations on network arcs, total costs would not be reduced. Location problems include not only the study of the best place implementation in the network, but also the optimal assignment from sources to loads. The transportation costs – Cij, are the costs of serving demands at each node j from the sources i, they are treated as demand-weighted distances. We can understand the importance of accounting for these costs, by observing Fig. 2. The long dashed curve shows the routing cost as a function of the number of facilities that are to be located. The long dashed dot-dot curve reflects the

Optimal Location of Gas Supply Units in Natural Gas System Network

67

Fig. 2 Fixed, transporting and total costs

accumulated fixed cost of locating GSUs. The fixed facility cost is added to the routing cost to obtain the total cost, as shown by the solid curve [13]. The total cost initially declines as the reduction in routing cost that results from the addition of more gas sources more than offsets the additional facility location costs. At some point, the cost of additional GSUs exceeds the savings in routing costs and the total cost increases as we add more facilities. Our goal is to determine the turning point, which correspond to the optimal number of the natural gas sources to be placed in the gas system in order to minimize overall costs. The model developed in this study is versatile and accounts for the referred natural gas technical and economic constraints. In Sects. 3.3, 3.4 and 3.5, we will analyze five operating scenarios: 1. 2. 3. 4. 5.

No P imposed, capacitated GSUs No P imposed, relax capacitated GSUs No P imposed, unlimited GSUs No P imposed, increase unitary cost Imposed P, capacitated GSUs

3.2

Analytical Model

The location problem methodology consists of locating P GSUs to minimize the sum of facility location costs and transport costs from GSUs to gas loads, subject to constraints that stipulate that all demands must be served, GSUs capacities must not be exceeded and customers can only be served from open facilities. The problem can be formalized as a linear problem, with m origins and n destinies. Considering

68

T. Nogueira and Z. Vale

that GSUs will be implanted only on the gas network’s nodes, our model considers a network with m GSUs potential sites and n demand points. The m value can be equal to n, if we assume that all network nodes are potential sites lo locate GSUs. In this case, we have m location decision variables and m x n transport decision variables, totalizing m + m x n unknown variables or decision variables, being Yi binary location variables and Xij transportation variables. The problem can be formulated as minimizing the sum of the facility locating cost, fi and the routing or transportation costs, Cij. These transportation costs are the total demand-weighted distance (dij. bj) multiplied by a, the cost per unit distance per unit demand, Cij ¼ a.dij.bj. Being, ai the maximum capacity of a GSU facility at candidate node i and bj the demand at demand node j, the problem can be formalized as a combinatorial problem, stated as follows: Minimize m X

f i :Yi þ

i¼1

m X n X

a:dij :bj :Xij

(1)

i¼1 j¼1

Subject to n X j¼1

Xij
Yi

n X

b j b0

j¼1

n X

! Xij bai :Yi

(2)

j¼1

m X

Xij ¼bj

(3)

Xij = p

(4)

i¼j n X i¼1

Xij r0

(5)

Yi 2 f0; 1g

(6)

The objective function (1) minimizes the total cost which is the sum of the fixed GSUs costs and the routing costs. Offer constraint (2) means that if Yi ¼ 0, the GSU is not located in i, if Yi ¼ 1, the gas quantity from i source is limited by total demand. This is applied to unconstrained capacity. The term in parenthesis, applied for fixed charge problems, requires that the demand for the source i will not go beyond its maximum capacity and no client will be allocated to an unavailable natural gas source. Demand constraint (3) stipulates that each client should be completely served and constraint (4) states that P facilities will be located (for P imposed cases). If we remove this constraint from the problem, P number will be calculated as an output. Constraint (5) states that the decision variables are positive

Optimal Location of Gas Supply Units in Natural Gas System Network

69

quantities and constraint (6) is the binary integer condition: the variable Yi is 1 if a GSU is located in i or 0, if not. The optimal solution is the one that minimizes the objective function (1). As binary variables, Yi will assume only the value 1 or 0, which means a GSU is located at i node or not, respectively. The transportation variables Xij will give us the information about the gas quantity that should be supplied from i source (if it was installed a GSU) to j load consumer.

3.3

Application to a Gas Network

In this section the location model is applied to a 12-node gas network (Fig. 3). Gas network nodes are marked from 1 to 12. The triangular boxes are the gas demands – bj, in million cubic meters unit (Mm3).

Fig. 3 Gas network

70

T. Nogueira and Z. Vale

The distances between nodes – dij, stated on the branches in kilometers, are the pipelines in which the natural gas is transported. Assuming that the network is known, we can build the distance matrix (Table 1), an important input of the location problem. Each node can be a potential GSU location site, it has its explicit cost location fi and it could be imposed its maximum supply capacity ai (as seen in Table 2). Another important assumption to the problem is the a value, which is the cost per unit distance per unit demand. As an input, its value will have a crucial impact on the problem result. In our operating scenarios, we consider two different costs, for a ¼ 0.009 and 0.027. The problem we want to solve is to find the number and the optimal GSUs location-allocation sites on the 12-node network so that the total cost is minimized (Eq. 1). For this size gas network, the problem can be solved by optimization software, such as the Risk Solver Platform. In this minimization problem we have 156 total decision variables, 12 unknown location variables, Y1 to Y12, and 12  12 ¼ 144 unknown transportation variables – Xij. The results depend on the constraint assumptions, as seen in the next sections. Table 1 Network distance matrix, dij j i 1 2 3 4 1 0 150 370 550 2 150 0 220 400 3 370 220 0 180 4 550 400 180 0 5 240 380 160 340 6 600 520 300 120 7 180 330 410 590 8 330 480 280 460 9 480 420 200 240 10 400 550 580 620 11 580 610 390 430 12 670 610 390 340

Table 2 Location site data Node Demand (Mm3) – bj 1 15 2 10 3 12 4 18 5 5 6 24 7 11 8 16 9 13 10 22 11 19 12 20

5 240 380 160 340 0 360 250 120 240 470 370 430

6 600 520 300 120 360 0 570 420 120 500 310 220

7 180 330 410 590 250 570 0 150 450 220 400 610

8 330 480 280 460 120 430 150 0 300 370 250 460

Fixed cost (M€) – fi 1,000 2,000 1,300 1,500 2,250 1,750 1,900 2,100 1,650 2,300 1,250 2,150

9 480 420 200 240 240 120 450 300 0 380 190 190

10 400 550 580 620 470 500 220 370 380 0 190 400

11 580 610 390 430 370 310 400 250 190 190 0 210

12 670 610 390 340 430 220 610 460 190 400 210 0

Max. capacity (Mm3) – ai 25 55 30 40 70 35 50 40 55 55 35 30

Optimal Location of Gas Supply Units in Natural Gas System Network

3.4 3.4.1

71

Capacitated GSUs Versus Uncapacitated GSUs Scenario 1

We begin by considering that there is a maximum capacity stated for each potential GSU installed (Eq. 2 in parenthesis), as well as its local implantation cost (Table 2). In this scenario, P is an output (Eq. 4 not included), the GSUs have fixed charge capacities and a ¼ 0.009. Results are shown in Table 3. In Table 3, we have the decision variables’ results for the location variables Yi and transportation variables Xij. The program optimized the result installing 5 GSUs (P ¼ 5), located at nodes 1, 4, 7, 9 and 11. With this choice, fixed costs are: 1,000 + 1,500 + 1,900 + 1,650 + 1,250 ¼ 7,300 M€ (see Table 4, third column). When location variables Yi have 0 value, it means that node i won’t have a GSU installed and therefore the respective transportation variables Xij are null as well. For example, let’s consider node 2, Y2 ¼ 0; if there isn’t any gas source located on node 2, no gas quantity is transported from node 2 to any other consumer on the network. As a result, the transportation variables X21 to X2.12 are all zero. This node Table 3 Decision variables solution Y2 Y3 Y4 Y5 Y1 1 0 0 1 0 X11 X12 X13 X14 X15 15 10 0 0 0 X21 X22 X23 X24 X25 0 0 0 0 0 X31 X32 X33 X34 X35 0 0 0 0 0 X41 X42 X43 X44 X45 0 0 12 18 0 X51 X52 X53 X54 X55 0 0 0 0 0 X61 X62 X63 X64 X65 0 0 0 0 0 X71 X72 X73 X74 X75 0 0 0 0 0 X81 X82 X83 X84 X85 0 0 0 0 0 X91 X92 X93 X94 X95 0 0 0 0 5 X10.1 X10.2 X10.3 X10.4 X10.5 0 0 0 0 0 X11.1 X11.2 X11.3 X11.4 X11.5 0 0 0 0 0 X12.1 X12.2 X12.3 X12.4 X12.5 0 0 0 0 0

Y6 0 X16 0 X26 0 X36 0 X46 10 X56 0 X66 0 X76 0 X86 0 X96 14 X10.6 0 X11.6 0 X12.6 0

Y7 1 X17 0 X27 0 X37 0 X47 0 X57 0 X67 0 X77 11 X87 0 X97 0 X10.7 0 X11.7 0 X12.7 0

Y8 0 X18 0 X28 0 X38 0 X48 0 X58 0 X68 0 X78 16 X88 0 X98 0 X10.8 0 X11.8 0 X12.8 0

Y9 1 X19 0 X29 0 X39 0 X49 0 X59 0 X69 0 X79 0 X89 0 X99 13 X10.9 0 X11.9 0 X12.9 0

Y10 0 X1.10 0 X2.10 0 X3.10 0 X4.10 0 X5.10 0 X6.10 0 X7.10 6 X8.10 0 X9.10 0 X10.10 0 X11.10 16 X12.10 0

Y11 1 X1.11 0 X2.11 0 X3.11 0 X4.11 0 X5.11 0 X6.11 0 X7.11 0 X8.11 0 X9.11 0 X10.11 0 X11.11 19 X12.12 0

Y12 0 X1.12 0 X2.12 0 X3.12 0 X4.12 0 X5.12 0 X6.12 0 X7.12 0 X8.12 0 X9.12 20 X10.12 0 X11.12 0 X12.12 0

72

T. Nogueira and Z. Vale

Table 4 Scenario 1 results Optimal location Allocated loads 1 1, 2 4 3, 4, 6 7 7, 8, 10 9 5, 6, 9, 12 11 10, 11 (P ¼ 5)

Fixed costs (M€) Transport costs (M€) Total costs (M€) 1,000 135.00 1,135.00 1,500 492.48 1,992.48 1,900 606.96 2,506.96 1,650 1,100.88 2,750.88 1,250 601.92 1,851.92 7,300 2,937.24 10,237.24

2 is not a source point, but it is a demand node and it is supplied by the GSU located in node 1, as seen in Table 2: X12 ¼ 10 Mm3, which is exactly its demand. The GSU located in node 1 will supply its own demand plus the node 2 demand, which is exactly the maximum quantity it can give, 15 + 10 ¼ 25 Mm3, as seen in the first line, fourth column, in Table 2. However, to respect the maximum capacities, some gas loads need to be supplied by two different sources, which is the case of nodes 6 and 10. The demand node 6 is supplied by GSU 4 with 10 Mm3 (X46) and by GSU 9 with 14 Mm3 (X96), totalizing 24 Mm3, which is exactly the node 6 load demand. The GSU 4 also supplies nodes 3 and 4, whose demands are 12 and 18 Mm3, respectively. These demands plus the node 6 total demand will put GSU 4 out of its maximum capacity. So, the optimal solution suggests that GSU 4 will supply 12 + 18 + 10 ¼ 40 Mm3, which is its fixed charge capacity. Transportation costs are the costs of the optimal transport solution from those GSUs installed to the assigned gas loads. As an example, GSU 1 supplies its own demand (X11 ¼ 15) and the demand in node 2 (X12 ¼ 10). There is no transportation cost to supply one node’s own demand, of course, but for GSU 1 to supply load 2 it costs: 0.009  150  10  10 ¼ 135 M€, as presented in Table 4. Let’s take now GSU 4: this unit supplies its own demand, as well as nodes 3 and 6 demand. Transportation costs from GSU 4 to node 3 are: 0.009  180  12  12 ¼ 233.28; transportation costs from GSU 4 to node 6 are: 0.009  120  24  10 ¼ 259.2. Total transportation costs from GSU 4 to loads are 492.48 M€, as shown in Table 4. The location-allocation results can be summarized in Table 4. The objective function is 10237.24 M€, which are the optimized system total costs. As we can notice in Table 4, all gas demands are completely satisfied by open natural gas sources and the maximum GSUs’ capacities were not exceeded. 3.4.2

Scenario 2

If we increase maximum GSUs capacities, the location program has more flexibility and shows different optimal results. Let’s now consider that maximum GSUs capacities were relaxed to 60 Mm3, and the same for each potential node. This means that all values of the fourth column in Table 2 have the same ai value. The location-allocation results can be shown in Table 5.

Optimal Location of Gas Supply Units in Natural Gas System Network

73

Table 5 Scenario 2 results Optimal location Allocated loads 1 1, 2, 7 3 3, 5, 8 6 4, 6, 9, 12 11 8, 10, 11, 12 (P ¼ 4)

Fixed costs (M€) Transport costs (M€) 1,000 331.02 1,300 519.84 1,750 730.44 1,250 1,538.64 5,300 3,119.94

Total costs (M€) 1,331.02 1,819.84 2,480.44 2,788.64 8,419.94

Table 6 Scenario 3 results Optimal location Allocated loads 1 1, 2, 5, 7 4 3, 4, 6 11 8, 9, 10, 11, 12 (P ¼ 3)

Fixed costs (M€) Transport costs (M€) 1,000 385.02 1,500 855.36 1,250 2,448.63 3,750 3,689.01

Total costs (M€) 1,385.02 2,355.36 3,698.63 7,439.01

In this scenario, the optimal solution is to locate 4 GSUs on nodes 1, 3, 6 and 11, and total costs decrease. This is due to lower fixed location costs, but instead transportation costs increase. We have less GSUs, so gas need to “travel” more kilometers, which increases the transportation distances, and consequently the transportation costs. 3.4.3

Scenario 3

If we continue to relax the maximum GSUs’ capacities, now assuming an unconstrained capacities (Eq. 2 out of parenthesis), the location program finds a better solution, with total costs even smaller, as shown in Table 6. In previous scenarios it was assumed that a value was 0.009. Let’s see the influence of a on the total cost, by increasing it to 0.027. Results are presented ahead, in scenario 4. 3.4.4

Scenario 4

This scenario is similar to scenario 3 (unconstrained capacities), but now a ¼ 0.027. The location-allocation results are shown in Table 7. Because of the increase on a value, transportation costs are increased, so the program minimizes the solution by adding more GSUs. This is an attempt to minimize transportation cost, but fixed cost has a big influence, so, the result is a higher total cost. In natural gas networks, it is very important to keep a small in order to have a more efficient gas system operation. If P increases too much, we can see that some GSUs are installed with the only objective to supply its own node demand – this is the case of GSUs 10 and 12. The gas from these sources will not supply any load demand, so transportation costs are zero (notice that dii ¼ 0).

74

T. Nogueira and Z. Vale

Table 7 Scenario 4 results Optimal location Allocated loads 1 1, 2, 5, 7 4 3, 4 6 6, 9 10 10 11 8, 11 12 12 (P ¼ 6)

Fixed costs (M€) 1,000 1,500 1,750 2,300 1,250 2,150 9,950

Table 8 Scenario 5 results Optimal location (P ¼ 5)

Allocated loads

1 4 9 10 11

1, 2 3, 4, 6 5, 6, 9, 12 7, 10 8, 11

3.5

Transport costs (M€) Total costs (M€) 1,155.06 2,155.06 699.84 2,199.84 547.56 2,297.56 0.00 2,300.00 1,728.00 2,978.00 0.00 2,150.00 4,130.46 14,080.46

Fixed costs (M€) 1,000 1,500 1,650 2,300 1,250 7,700

Transport costs (M€) 135.00 492.48 1,100.88 239.58 576.00 2,543.94

Total costs (M€) 1,135.00 1,992.48 2,750.88 2,539.58 1,826.00 10,243.94

Fixed Number of GSUs

In the previous section it was assumed that the number P of GSUs installed on the gas network is endogenously calculated, therefore an output of the problem. Now we’ll consider that P is taken as a proxy of the number of GSUs, thus constituting an input imposed by user. In this case, the Eq. 4 is included as a problem constraint. We assume that P is known, but the location problem will solve its optimal locations on the network. 3.5.1

Scenario 5

This scenario is similar to scenario 1 (constrained GSUs capacities, a ¼ 0.009), but now imposing P ¼ 5. The location-allocation result is given in Table 8. Comparing these results to the scenario 1 results (Table 4), we can see a little difference on the five location nodes. By imposing P ¼ 5, the program makes a different rearrangement, choosing other optimal locations and a little different objective function solution. In scenario 1, total costs were 10,237.24 M€, and in scenario 5 they are 10,243.94 M€. This is explained by the different initial constraints of the location problem.

4 Conclusions The natural gas industry in Europe is currently facing great challenges: European harmonization, integrated operations of pipeline system and studying optimal economic locations of gas supply units.

Optimal Location of Gas Supply Units in Natural Gas System Network

75

Because of expected growth in demand, large infrastructure investments are necessary, with special focus on GSUs – gas supply units. Our work supports GSUs location decision making and their optimal allocation to gas loads. In these location-allocation problems there is an inherent trade-off between the fixed costs of GSUs and the transportation costs. As additional GSUs are added, the fixed costs increase, but the transport costs decrease and vice-versa. The approach was illustrated using a small example problem involving 12 candidate locations, for different operating scenarios. The location algorithm provided optimal solutions, subject to user parameters and initial constraints, with accuracy and effectiveness. Natural gas networks need to deal with a condition characterizing pressure drop on the calculated network sections. Due to the location problem study carried, that constraint becomes irrelevant and it will be lost in the course of gas flow. Under this assumption, the developed approach can be valid. In future research, an extension of the location field should be made and its application to increasingly realistic problems, preferably using the Lagrangean heuristic approach [14], which is beyond the scope of this text.

References 1. Dresner Z (1995) Facility location: a survey of applications and methods, Series in operations research. Springer, New York 2. Nogueira T, Vale Z et al (2006) Advanced techniques for facility location problem in natural gas network. ICKEDS’06, Lisbon, pp 347–351 3. Lorena L, Narciso M (1996) Relaxation heuristics for a generalized assignment problem. Eur J Oper Res 91:600–610 4. Beasly E (1993) Lagrangean heuristics for location problems. Eur J Oper Res 65:383–399 5. Arentsen M, Kunneke R (2003) National reforms in European gas. Elsevier, Oxford 6. Nogueira T, Vale Z et al (2005) Natural gas market in Europe: developments and trends. In: CAIP2005 – congresso interamericano de computation aplicada a la industria de processos, Vila Real, Portugal, pp 115–118 7. Jensen J (2003) The LNG revolution. Energy Journal 24(2): 1–37 8. Flanigan O (1995) Underground gas storage facilities. Gulf Publishing Company, Houston, pp 40–53 9. Directive 2003/55/EC of the European Parliament and of the Council of 26 June 2003 concerning common rules for the internal market in natural gas and repealing Directive 98/30/EC 10. Nogueira T, Mendes R, Vale Z, Cardoso J (2006) Heuristic model for Iberian natural gas source location. J WSEAS Trans, 1:1343–1349 11. Nogueira T, Mendes R, Vale Z, Cardoso J (2007) A heuristic approach for optimal location of gas supply units in transportation system. In: 22nd International scientific meeting of gas experts, vol 1, Opatija, pp 303–311 12. Hakimi S (1964) Optimum locations of switching centers and the absolute centers and medians of a graph. Oper Res 12:450–459 13. Daskin M (1995) Network and discrete location: models algorithms and applications. Wiley, New York 14. Nogueira T, Vale Z (2008) Natural gas system operation: Lagrangean optimization techniques. IGRC – International gas union research conference, ID 147, Category: Transmission, IGRC Foundation

An LP Based Market Design for Natural Gas E.G. Read, B. J. Ring, S.R. Starkey, and W. Pepper

Abstract Many electricity markets are now cleared using Linear Programming (LP) formulations that simultaneously determine an optimal dispatch and corresponding nodal prices, for each market dispatch interval. Although natural gas markets have traditionally operated in a very different fashion, the same basic concept can be applied. Since 1999, the Australian state of Victoria has operated a gas market based on an LP approximation to the underlying gas flow optimization problem. Here we discuss market design issues, using a formulation derived from the key gas flow equations. Dual variables on key constraints imply prices which vary by location, as for electricity markets, but also by time. But gas is both delayed and stored within the transportation system itself. This raises a number of operational, pricing, and hedging issues which could be ignored in the case of electricity, but become important when operating this kind of market for gas, or other commodities, such as water, in a supply network where there are delays and storage. Keywords Linear Programming (LP) • Linearization • Market • Natural gas • Optimization • Pipelines • Prices

The authors wish to thank AEMO for information used in this chapter, Ray Whaley for assistance with the initial formulation, and Peter Jackson for his input. E.G. Read (*) • S.R. Starkey University of Canterbury, Christchurch, New Zealand e-mail: [email protected]; [email protected] B.J. Ring Market Reform, Australia e-mail: [email protected] W. Pepper ICF International, Virginia, USA e-mail: [email protected] A. Sorokin et al. (eds.), Handbook of Networks in Power Systems II, Energy Systems, DOI 10.1007/978-3-642-23406-4_4, # Springer-Verlag Berlin Heidelberg 2012

77

78

E.G. Read et al.

1 Introduction Many electricity markets are now cleared using an LP-based “Market Clearing Engine” (MCE) to optimize the value of trade, as determined by participant bids and offers. An MCE formulation typically optimizes power system operations at quite a detailed level, using a detailed representation of the transmission system, and simultaneously optimizing dispatch of generation, transmission power flows, and often ancillary services. This level of detail also enables the MCE to simultaneously determine corresponding prices for energy at each “node” in the transmission system, and often for each ancillary service, in each market dispatch interval. In 1996, the first market of this type was developed in New Zealand [2]. The Australian “National Electricity Market” (NEM) followed soon after, in 1997, using a re-developed version of the same software, NEM1. That development was also partly built upon the Victorian electricity market, which had been successfully operating in the State of Victoria since 1994, with interconnected trade between Victoria and New South Wales commencing mid 1997 [39]. Broadly similar electricity markets were being discussed or developed in many parts of the world [14], and are now widespread [36]. Thus an extensive literature was developing on topics such as the design of such markets, how market participants might behave in that environment, and how they might hedge their risks over time and space. Hogan [15] had also proposed an auction based model for allocation of gas transport capacity, while McCabe et al. [43] had performed simulations of simple market structures. These developments lead the Victorian State Government to consider the desirability of developing a similar kind of market for trading natural gas, in that State [42]. The gas system operates in a broadly analogous way to electricity, with a variety of suppliers and consumers, simultaneously injecting gas into, or withdrawing gas from, various points in an interconnected “transmission network”. As reported by DPI [10], the Victorian gas system currently has six suppliers, three major retail buyers, and multiple wholesale traders. As shown in Fig. 2, the gas transmission system is a meshed network with multiple inter-regional connectors and active underground gas storages. In 1996, though, there was only one supplier and buyer, operating under a single long term contract, with a simpler network, and no interconnections. The market design discussed here was developed to allow that monopolistic arrangement to be decomposed, and to support evolution towards a more dynamic trading environment. Convergence towards a consistent market framework also seemed desirable in view of the way in which natural gas interacts with electricity, both as a fuel for generation and a competing supplier of end demand. Thus, one goal of the initial design was to try and create a gas market framework which drew on experience with electricity markets, aligned with the Australian electricity market, and could develop towards greater integration over time. The key question was whether the basic concepts developed for electricity markets could also be applied to gas, in the sense that an analogous “nodal” market-clearing formulation could be developed, and form a practical basis for trading. Further, if that was possible, under what circumstances might it be worthwhile to do so?

An LP Based Market Design for Natural Gas

79

This paper focuses on the conceptual market design issues, using an LP formulation analogous to those employed in the electricity sector, based on a representation of the gas system dispatch problem in nonlinear form. In reality, from its inception in March 1999 [1], the Victorian gas market has been dispatched using an MCE based on an LP model developed by ICF International. As described in Pepper et al. [26], the MCE incorporates a number of advanced features to deal with various physical and computational issues, and takes a different approach to linearizing the gas flow equations. That model has proved accurate and reliable, but its fundamental variables are a set of convex weights, which does not make for any intuitive discussion of the forces driving pricing effects in a way likely to prove meaningful to potential developers of gas markets. (In the same way, meaningful discussion of pricing effects in electricity markets must be based on a representation of the fundamental electrical relationships, in a full nodal model, not on the kind of implicit representation some markets use in practice). The reader is referred to Pepper et al. for a description of the implemented model. Since our goal in this paper is really to discuss the implications of adopting this kind of approach to gas market trading, we abstract away from some of the detail and base our exposition on the original formulation originally developed by Read and Whaley [32], as part of the Putnam Hayes and Bartlett (PHB) team responsible for the market design. Even though this simplified formulation was not implemented in practice, it provides a more accessible introduction to the concepts, starting from a standard “textbook” representation of the gas flow equations, and produces a dual that is more readily interpreted. It will be seen that the resultant formulation is really no more complex than some of the formulations that have been discussed or applied to form electricity markets, particularly if AC power flow equations are modeled [16], and/or ancillary services co-optimized [31], and/or inter-temporal unit commitment constraints represented [17]. Thus this kind of market development does not seem infeasible on grounds of complexity. The major complication is that the gas flow equations imply that gas is both delayed and stored within the transportation system itself. This raises a number of operational, pricing, and hedging issues which could be ignored in the case of electricity, but become important when operating this kind of market in a gas supply network. One major motivation for discussing these issues, at this time, is that similar issues are likely to be important in markets for other commodities, such as water [28], where delays and storage also occur within the “transportation system” over which the market operates. On the other hand, while Pepper et al. [26] describes a market dispatch process that has proceeded down the path of increasing sophistication and precision, commercial gas market trading is actually based on a highly simplified version of the formulation. In fact, the initial market clearing logic of Ruff [35] only involved a daily clearing of a market in daily gas delivery, while also accounting for overnight storage in the system. For reasons discussed later, the market has still only moved forward to the point of re-clearing to determine prices for the remainder of the day at four-hourly intervals. The studies reported by Frontier Economics [11] did

80

E.G. Read et al.

show the potential for significant spatio-temporal differentiation in the marginal value of gas, suggesting the potential value of this kind of market design. The approach described here was developed in 1997, when there seemed to be little or no literature on the application of optimization models to support, or analyze, trading in gas markets. Since that time, gas market deregulation has proceeded in many places, and the literature has developed accordingly, although to a significantly lesser degree than for electricity markets. Zheng et al. [40] survey gas sector optimization models being applied to optimization of various aspects of gas production, and of gas pipeline network development and operations, but most of those papers do not deal with gas markets, per se. O’Neil et al. [24] and Gabriel et al. [12] model economic equilibrium in gas markets, broadly defined, while Cremer et al. [6] seek to characterize pricing patterns in pipeline networks, with and without a cost recovery requirement. But all these models deal with the issues at a high level, on a much broader scale and longer time frame than envisaged here, and were not intended to form a basis for actual spot gas trading. De Wolf and Smeers [7], Breton and Zaccour [4], and Gabriel et al. [13] all deal with strategic gaming issues in deregulated gas markets, although only the last models the physical gas transportation system, again on a relatively broad scale. Although the potential for gaming is certainly an important issue for the Victorian system, with its relatively small group of participants, it lies beyond our present scope. Our goal was simply to produce a market framework in which prices are closely aligned with physical system realities, and economic costs. In this respect, the closest approach to ours is probably that of Midthun et al. [21], who discuss a piece-wise linearization approach to modeling the nonlinear pipeline transportation dynamics, and Midthun et al. [21] who apply that approach to consider a gaming problem in which the pipe system operator plays an active role, rather than being a passive “system operator” as in our market paradigm.

2 Market Concepts Most gas pipelines in the world operate under a contract carriage model. Historical overviews of the emergence of competition in the US may be found in Vany and Walls [37] and Doanne and Spulber [8]. Under this model, the pipeline operator funds its pipeline by selling access to shippers of gas with varying levels of priority. Those with the greatest priority have firm access to the pipeline and tend to pay the most for their access. Those with less priority pay less, but only get access to the pipeline to the extent that it has otherwise unused capacity. Open access regimes may be imposed by regulators who require some transparency to these arrangements, but the basic access arrangement is still via bilateral contracts. Markets for gas around the world generally operate at hubs between pipelines. Gas can be delivered to these hubs in accordance with pipeline usage contracts, traded at the hub, and hauled away on other pipelines or consumed at the hub. In the US an unregulated natural gas market trades over the New York Mercantile

An LP Based Market Design for Natural Gas

81

Exchange (NYME). Futures contracts are traded relative to the principal Henry Hub, in Louisiana. Two equivalent virtual trading point markets are the National Balancing Point (NBP) system in the UK, and the Title Transfer Facility (TTF) in the Netherlands. A Short Term Trading Market (STTM) for gas along these lines operates in Australia at hubs in Adelaide and Sydney, and soon in Brisbane. The Victorian Gas Market is different in that it operates instead on the concept of “market carriage”. The Primary Transmission System (PTS) is funded through Transmission Use of System (TUOS) charges, rather than under bilateral contracting arrangements. This makes it very similar to how electricity transmission systems are funded. With the network funded in this manner, the Victorian Gas Market can operate a “commodity only” market for the trade of gas, and has done so since 1999. Further, with AEMO operating the network, the trade of gas can be used to determine day-to-day transmission access, and (in principle) point-to-point transport charges. Given the obvious analogies between the gas and electricity systems it seems natural to ask whether the concepts that have been applied to design electricity markets might also be applied to design gas markets. Ignoring a simplified marginal loss adjustment, the Australian electricity market differentiates spot prices by region, so there is only one spot electricity price for all of Victoria. Another point of reference was the New Zealand electricity market, which determines spot prices for each node at which physical injection or off-take occurs [2]. Both operate on the basis of prices for half-hourly trading intervals. Buying and selling (wholesale) electricity is done through a “pool”, where electricity generators offer electricity to the marketplace for dispatch through the electricity transmission network. A central market coordinator receives generation offers (and potentially load bids), determines which of those should be accepted (i.e., “clears the market”), implements the optimal dispatch, and announces the corresponding spot prices, all in real time. Thus the goal here was to develop an analogous market design for gas, in which a “market-clearing solution” is determined by an LP optimization model that simultaneously determines: • A “dispatch” schedule for all gas “injections” and “off-takes” that is optimal, in the sense that it maximize the “value of trade” defined as the benefits delivered to loads (as determined by their bids), minus the costs incurred by suppliers (as determined by their offers). • A matching set of “nodal” spot prices, varying over time and across network locations, defined by the marginal cost of meeting a (possibly hypothetical) load at each time and location, and applied symmetrically to buy gas from suppliers, and sell gas to consumers.

3 Gas Flow Modeling In principle, we can divide pipelines up into arbitrarily small cells, and it becomes somewhat arbitrary as to whether primary variables are defined at cell midpoints, or cell boundaries. Our basic nonlinear formulation is based on the initial market

82

E.G. Read et al.

design formulation developed by Read and Whaley [32]. They developed their formulation in terms of average midpoint cell values, assuming velocity and pressure changes to be implicitly defined at cell boundaries. That approach was designed to be applied to a fairly discretized representation of a uniform pipeline, though. Pepper et al. [26] model a more general network, approximating over quite long pipeline segments, and modeling nodes where multiple pipes may connect, and various pipe fittings that may induce step changes in velocity and pressure. We briefly touch on such issues in an Appendix, but here take an intermediate approach, developing a formulation for a single pipeline, with variables primarily defined at cell boundaries, which can be thought of as nodes. The major complication is that natural gas is compressible, unlike some other piped fluids such as water, so its density (and hence pressure) varies, with gas flows being driven by pressure differences. We start with general equations, which allow pressures etc. to differ on each side of each junction, but later assume a single pressure variable at each junction, so as to develop a simplified conceptual formulation. Pepper et al. provide a more detailed development of the actual formulation employed. The pipeline is split into segments as shown above in Fig. 1. A variable at the centre point of the subscripted nth element represents the average value. We discuss flow reversals later but, for simplicity, assume flow is from cell n
1, to n, to n + 1 etc. For simplicity we also assume time periods to be of unit length. The key variables are gas pressure and flow, at the beginning of each pipe segment (or cell), n ¼ 1,. . .,N and at, or from, the beginning of time period, t ¼ 1,. . .,T. Other variables are derived from these, as necessary. Notationally, constants are represented using normal fonts (e.g. R or H below), whereas variables and indices are represented using italic fonts (e.g. qi, mn etc.).

Fig. 1 Key variables for gas pipeline modeling

An LP Based Market Design for Natural Gas

83

If there were no compression, the problem could be formulated in terms of fixed volumes of gas travelling from one cell, n, to the next cell, n + 1, with some delay. In reality gas density/pressure can vary, and flows are driven by pressure differences but, eventually, balancing forces act to equalize those pressure differences. Gas stored in the transport system is known as “linepack”, and this plays an important role in system operation. Daily demands can generally not be met unless linepack storage is built up substantially by pressurizing the gas pipeline overnight. An increase in cell pressure means that more mass is stored in that pipe cell. Thus the mass in the nth cell, at time t, is proportional to the average pressure in the cell, and given by: mtn ¼ Ln Gn p~tn

(1)

p~tn ¼ ðptin þ pto n Þ=2

(2)

Gn ¼An /RH

(3)

Here the nth pipeline cell has diameter Dn, cross-sectional area An, and length Ln. H is the gas temperature , assumed to be constant, and R is known as the “specific gas constant” for the particular gas composition in the pipeline. The rate of gas (mass) flow at any point (e.g. qtn ) is determined by the pressure of the gas, and its velocity. We will apply this relationship to the midpoint flow/pressure/velocity values, as in (4). This allows us to state the mass conservation equation for each cell, as in (6). Note that, in this equation, injection (ytn ) is treated interchangeably with mass (mtn ), and mass is a midpoint value, reflecting average pressure across the cell. Thus injection is implicitly treated as if it were occurring at the midpoint of a cell, increasing pressures at both ends. This is not likely to happen, in practice, but the cell in which injection is assumed to occur can be made arbitrarily short, or represented by a “node” as in Pepper et al. q~tn ¼ Gn p~tn v~tn

(4)

q~tn ¼ ðqtin þ qto n Þ=2

(5)

t mtþ1 ¼ mtn þ qtin
qto n n þ yn

(6)

Modisette and Modisette [22] discuss fluid forces in pipes, initially defining the forces for a single element. By applying the conservation of momentum, they then sum forces and flows across all time periods and pipe elements. We use their results to present what is known as the Bernoulli equation, in its more general form for unsteady flows, which we state for the midpoint velocity/pressure pair:1

1 In this formulation, the pipe is assumed to be horizontal, thus eliminating the gravitational term ‘g  siny’, for an elevation angle of y. The effect of this force is negligible because natural gas is nearly twice as light as air, at standard conditions. Superscripts i and o are dropped because this equation applies at any point.

84

E.G. Read et al.


Dvtn þ v~tn rvtn þ Bn ð~ vtn Þ2 ¼
RHrptn p~tn

(7)

Bn = f = 2Dn

(8)

Here f is the “Moody friction factor”, which we assume to be a fixed parameter. This Bernoulli equation describes energies within the pipe system, at a given point in time and space. On the LHS, the equation describes the rate of change of gas velocity in time and then in space, and the final LHS term represents viscous losses. These dynamic velocity terms equate to the RHS proportional pressure change term, i.e. the absolute pressure gradient divided by the actual pressure value. In practice we can make the time periods as short as required to allow the model to solve with sufficient accuracy, and replace derivatives with differences. To allow Eq. 7 to be expressed for the midpoint of a cell, we define: v~tn ¼ ðvtin þ vto n Þ=2

(9)

t
1 Dvtn ¼ vtþ1 n
vn

(10)

ti rvtn ¼ ðvto n
vn Þ=Ln

(11)

ti rptn ¼ ðpto n
pn Þ=Ln

(12)

Each pipe section will have a working pressure range, and may also have maximum velocity limits. Since flow in a network, especially on loops, can be bidirectional the physical lower bound is likely to have a negative value. Later, we set this lower bound to zero, so as to keep solutions within the range where solutions are convex.2 In reality, both bounds are more likely to bind at the inlet end of a cell, but for now we impose them on midpoint flows, because this simplifies the dual of our formulation, and discuss variations later: t

Ptn  p~tin  Pn

(13)

t Vn

(14)

V tn  v~tin 

Equations 1, 2, 4, 5, 6, 7 and 9, 10, 11, 12, 13, 14 describe the key physical gas flow relationships, and consequently feature prominently in the optimization model, with the associated dual variables generating the information required to create consistent market trading prices. Apart from this, an initial pressure (mass) and/ flow profile must be assumed, and a target (range) specified for the final period. But the above model is incomplete, because we have not specified how the input variables

2

Direction of flow is relative to a conventional direction, which for simplicity we define as being from i (inlet) to o (outlet). As with electricity networks this can be generalized by defining a conventional direction for all arcs, then allowing the flow to take a +/
value in that direction.

An LP Based Market Design for Natural Gas

85

for each pipe cell link to the outlet variables for the upstream cell. It should be clear that the flow out from cell n
1 to cell n equals the flow in to cell n from cell n
1, in mass terms. Both pressure and velocity may change, though, if there is some kind of fitting, or compressor, or just a change in diameter at the junction of two pipelines. A reasonably simple formulation can be produced by assuming that that there will be a proportional change in pressure at such a junction, at least locally around some likely solution level with velocity adjusting to match. But we will simplify further, by assuming that there are no special fittings, or pipeline diameter changes, at junctions. Thus not only mass flow, but pressure and velocity have the same value immediately upstream and downstream from each junction. This means that we can drop the distinction between inlet and outlet variables, with ptin etc. just becoming ptn etc. and t pto n etc. just becoming pnþ1 etc. Thus, for example: ptn ¼ ptin ¼ pto n
1

(15)

4 Basic Market Clearing Formulation We can now state a formulation for a single pipeline, over a gas trading day. We assume that flow always occurs in a uniform direction, in the direction from cell n
1 to n to n + 1, and use the endpoint formulation, assuming pipelines of uniform cross-section, with no abrupt pressure changes due to fittings, as discussed above. Thus Gn becomes simply G from here on. An appendix discusses modeling of complications such as compressors, fittings, and junctions. Initial linepack in the pipeline is inherited from the previous day’s end condition, as specified in (20).3 The end state of the system, at time T + 1, must be set to ensure sufficient linepack carryover; in the required sections of pipeline, to meet next trading day’s requirements. In the limit we could try to force final linepack in each cell to a specific value, representing a desired pressure/flow profile for the start of the next trading day, between defined limits. But this is too restrictive, and could lead to extreme price impacts, if not infeasibility, as the system will struggle to meet any exact profile. Still, we may group pipe segments into “zones”, z 2 Z, for which the aggregate end of day linepack must be between defined limits, as in (21).4

3

We could also specify initial flows to get a more accurate representation of the nonlinear equations involved. But that increased accuracy would come at the cost of increasing the likelihood that the LP could not actually find any feasible initial flow/pressure pattern to exactly match the specified parameters. 4 A combination of upper and lower bounds may suffice to ensure that pressure differentials are also large enough to create sufficient flow in the next trading day. But pressure differential constraints may also be added to ensure this directly, and independently of aggregate linepack levels.

86

E.G. Read et al.

The formulation seeks to maximize the value from allocating gas across time t, and space n, as expressed by Eq. 16. To the extent possible, the optimization balances trade between time periods from t ¼ 1,. . .,T, and over all nodes from n ¼ 1,. . .,N. Participants submit bids and offers as price (Bidtn and Offertn ) and quantity (xtdi and xtsi ) combinations. Individual participant involvement depends on the physical configuration of the system, at each specific location. Many pipe cells will only have extractions, while many others only injection, but most will have neither. For each individual consumer or supplier spot market bidding is represented by Eqs. 16 and 17, where the index subscript i indicates a single bid or offer tranche from a demand (d) or supply (s) side participant.5 The aggregate net injection of gas into the system into cell n, at time t, is given by Eq. 18, and upper and lower limits are imposed on this via Eq. 19. Combining all these market bid/ offer curves with the gas flow and pressure equations, and ignoring linepack bidding, generates the following nonlinear dispatch formulation. Maximise XX x; y; m; p; p~; q; q~; rp; v; v~; rv t n

XX d2Dn

Bidtdi xtdi


i

XX s2Sn

! Offertsi xtsi

(16)

i

Subject to:6 Mass, pressure, flow and velocity relationships, within a cell: mtn ¼ Ln G p~tn p~tn ¼ ðptn þ ptnþ1 Þ=2 :

(1) ctn

(2a)

q~tn ¼ G p~tn v~tn q~tn ¼ ðqtn þ qtnþ1 Þ=2 :

(4) tn

(5a)

v~tn ¼ ðvtn þ vtnþ1 Þ=2

(9a)

Mass conservation equation: ¼ mtn þ qtn
qtnþ1 þ ytn : mtþ1 n

mtn

(6a)

Bernoulli energy conservation equation with substitution from (10), over cell:

The notation i,d ∈ n means i or d is located at node n. i ∈ D(n) means i is a bid from a demand side participant at n. i ∈ d means i is a bid/offer from d, etc. 6 All constraints are 8 n ¼ 1,. . .,N and t ¼ 1,. . .,T unless otherwise stated. Greek symbols associated with equation numbers indicate the key dual variables which will be significant in later discussion of pricing relationships. 5

An LP Based Market Design for Natural Gas

87


~t
1 ~tn rvtn þ Bn ð~ vtn Þ2 þ RHrptn p~tn ¼ 0 : v~tþ1 n
v n þv

btn

(7a)

Velocity and pressure gradients: rvtn ¼ ðvtnþ1
vtn Þ=Ln

(11a)

rptn ¼ ðptnþ1
ptn Þ=Ln

(12a)

Pressure bounds: t : Ptn  p~tn  P n

t ftn ; f n

(13a)

wtn ; wtn

(14a)

Velocity bounds: t

Vtn  v~tn  V n : Bounds on offer/bid tranches:

gti ; gti

0  xti  Xti :

(17)

Net injection into a cell: ytn ¼

X

xts


s2n

X

xtd :

ltn

(18)

d2n

Net injection bounds: dtn ;dtn

t

Ytn  ytn  Yn :

(19)

Initial linepack status: m0n ¼ Mn0 :

8n 2 N

(20)

Final linepack bounds: MTþ1  z

X

Tþ1

mTþ1  Mz n

:

8z2Z

(21)

n2z

Alternatively, or additionally, terms could be included in the objective function representing a set of net demand curves for linepack in various zones. These would consist of a set of bid tranches, xTþ1 each bidding to buy linepack in zone z, at a zi price of PackBidTþ1 . Thus (ignoring the possibility that participants might already zi own linepack rights that they wish to sell) the following could be added to (16) above:

88

E.G. Read et al.

þ

XX z2Z

Tþ1 PackBidTþ1 zi xzi

(22)

i

The total system linepack at the end of the period T, across all nodes n in z, would then have to match the linepack purchased for the final period: ¼ mTþ1 z

X

mTþ1 ¼ n

n2z

X

xTþ1 zi

(23)

i2z

5 Simplification and Linearization If we are to clear the market using LP software, we must first linearize the nonlinear constraints in the formulation of Sect. 4. Read and Whaley [32] originally proposed to substitute the mass/pressure and flow/velocity relationship in Eqs. 1 and 4 into Eq. 6a, then linearize the resultant nonlinear flow/ pressure equation directly. They also showed how to re-arrange the Bernoulli equation (7a) into two expressions, one involving only velocities, and the other only pressures, and apply Taylor’s expansion separately to each. The relevant derivatives can certainly be formed, but they will not be discussed here, partly because that approach was not actually adopted in practice, and partly because it creates a dual formulation from which it is not particularly easy to deduce pricing relationships. But there were also concerns with respect to the accuracy and convexity of the implied approximation. One proposal was to employ an iterative successive linearization scheme, with each iteration solving an LP linearized around the solution from the previous iteration. A coarse discretization can yield a poor result in this kind of modeling, because small errors can propagate and compound though the equation set. The goal of the representation introduced in Fig. 1 was to re-express the underlying nonlinear differential equations by a set of linear difference equations, with both time and space discretized on a sufficiently fine grid to make the linear assumption reasonable, and to refine the grid further, around a proposed solution, if the results were deemed to be too inaccurate. But this approach is similar to employing Euler’s method to solve the underlying differential equations. In practice, that first order method is known to have stability issues, and a fine discretization may be required. Thus higher order methods are generally applied, as described by Dorin and Toma-Leonida [9], for example. Thus piece-wise linearization within the LP seemed preferable to successive linearization. It was proposed that a piece-wise linear model could be produced using Taylor’s expansions, as above, to create “supporting hyper-planes” around a set of points spanning the feasible region. A critical issue, though, was whether the piece-wise linearization so produced would actually form a convex LP feasible region. It can be shown that the flow equations are not actually convex if flows are allowed to reverse, but Read and Whaley argued that an acceptable convex

An LP Based Market Design for Natural Gas

89

approximation could be found in the vicinity of any likely optimum. There are actually two possible issues here, and a later section discusses why piece-wise linearization may break down in situations where gas, or gas flow, turn out to have negative value. But, concerns about the potential non-convexity of the feasible region itself focused on the first two terms in the Bernoulli equation, relating to velocity changes and kinetic energy, and to the possibility of flow reversal. Read and Whaley wished to retain these terms in the Bernoulli equation because, at the time, it was unclear whether they would have any significant pricing implications. But almost all other authors, including Pepper et al. [26], have considered those terms small enough to be ignored, on the grounds that a gas, being very light, has little kinetic energy or momentum. Thus it is reasonable to assume that changes to gas injection or withdrawal rates will primarily be reflected in changes to pressure/flow relationships. Friction losses will slow the process, but velocities will quickly respond without significant expenditure of energy to reflect this new “steady state”, which then evolves over a longer time frame in accordance with the flow and mass balance equations. Thus most authors use a steady state version of the Bernoulli equation, in which the first (time derivative) term is dropped, and most authors also drop the kinetic energy term. Since the pressure in (7a) in the average pressure over that whole cell, which is proportional to the mass in the cell, Eq. (7a) can be simplified to:7 q~tn ¼

hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi An =Bn Ln 
rptn mtn :

btn

(7b)

This form of the equation clearly encloses a convex feasible region.8 In fact it forms a convex cone, being linear in mtn along any ray where
mtn =rptn is constant. Although we can now drop (11a), defining rvtn , because it no longer appears in (7b), v~tn itself is still defined by (4) and appears in the velocity bounds (14a). But v~tn can be eliminated from the formulation, along with (4) and (9a), by substituting (4) into (14a) and re-arranging to express those bounds as constraints on flow, as a function of pressure.9 GVtn p~tn  q~tn  GVtn p~tn :




wtn ; wtn

(14b)

7 A substitution for velocity in terms of flow and pressure is made from (4) and the equation is rearranged in terms of q with coefficients grouped. This steady sate equation assumes that the mass flow rate is uniform across cell n, and period t. While this approximation is commonly employed, it is not quite consistent with (4), which allows the mass in the cell to change, implying different flow rates at each end. 8 t mn is always positive and (12a) ensures that
rptn is positive, since we are excluding solutions where flows reverse. Thus the RHS is just a constant times their geometric mean which is known to enclose a convex set [3]. 9 Note that, if the lower velocity limit is only used to prevent flow reversal, Vtn will be set to zero, so the lower bound on mass flow is also set to zero, irrespective of pressure.

90

E.G. Read et al.

If we use (1) to substitute for m in terms of endpoint pressures, ptn and ptnþ1 , while also substituting for rptn from (12a) (which can then be dropped from the formulation) we get: q~tn

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r h  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii ¼ An = 2Bn RHLn  ðptn Þ2
ðptnþ1 Þ2 :

btn

(7c)

This is just the Weymouth equation employed by many other authors, in various forms. If we let s be the ratio of the downstream and upstream pressures, then substituting sptnþ1 for ptn makes it clear that this expression is linear in upstream pressure (or downstream pressure) along any ray from the origin, in the upstream/ downstream pressure plane, over the range of interest. (That is for 0 < s < 1, since otherwise downstream pressure would be higher than upstream pressure, causing flow to reverse.) Over that range, this expression also forms part of a convex cone, as discussed by Tomasgard et al. [38] and Midthun et al. [21]. Zhou and Adewumi [41] take a different route. They form a steady state version of (7) by dropping the first term with its time derivative, but show how to obtain analytic expressions for flow which account for the kinetic energy term. Rearranging their equation (12), for a horizontal pipe, and re-expressing it in our notation gives a modified form of Eq. (7c).10 2

3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  t 2   p 2 t Þ2
ðpt 5 ðp Þ : q~tn ¼ 4An = 2Bn RHLn
RH ln nþ1 n nþ1 ptn

btn

(7d)

This approximation can not hold for outlet pressures (and hence outlet/inlet pressure ratios) near zero, because then the logarithm in the divisor tends to infinity, and the predicted flow falls rapidly to zero. For more realistic outlet/inlet pressure ratios, closer to unity, the logarithmic term is close to zero, and this equation creates only a modest adjustment to the Weymouth formula, implying a slightly greater resistance to flow. Substituting sptnþ1 for ptn shows that this, too, is linear in pressure along any ray with constant outlet/inlet pressure ratio, and letting that ratio range from a small value up to 1 also forms part of a convex cone. Although (7d) provides a convex formulation which accounts for the kinetic energy term, we will use the much more common Weymouth type equation in (7c). Martin et al. [19] discuss a piece-wise linearization that could be applied to either equation, using “convex combinations”, in the context of a mixed integer formulation. In Victoria a hybrid approach was adopted, using convex combinations to form a piece-wise linear formulation most of the time, but reverting to successive linearization when required to deal with “convexity issues”. The detail may be

10 Assuming a compressibility of Z ¼ 1, using the Specific Gas Constant rather than the Universal Gas Constant, and assessing the pressure drop over a cells length Ln.

An LP Based Market Design for Natural Gas

91

found in Pepper et al. [26]. But Tomasgard et al. [38] and Midthun et al. [21] discuss piece-wise linearization using hyper-planes created around a set of points, as suggested by Read and Whaley [32]. Using Maple# (see www.maplesoft.com), and re-arranging provides (7e) as the linearization of (7c) around a point denoted by superscript *. Here constant terms are enclosed in square brackets, and the simplification of Ft n reflects the fact that the square root term on the top line is just a t t constant times the expression for q~t n , the flow corresponding to (pn ,pnþ1 ), while the ( ) in the divisor is just the square of the same term. q~tn ¼ Ft n 



t  t t  pt n pn
pnþ1 pnþ1 :

btn

(7e)

Where ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r  3 t Þ2
ðpt Þ2

A  ðp n n nþ1 6 7 A2n 6 7¼   ¼ Ft n 4pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 5 t 2Bn RHLn q~t n 2Bn RHLn  ðpt n Þ
ðpnþ1 Þ 2

We will adopt this linearization approach here because it provides a dual formulation from which pricing relationships can be readily be deduced. In the context of a successive linearization scheme, (7e) can be left in equality form, representing the current linearization about a specific point. To create supporting hyper-planes for piece-wise linearization, though, we need to form several copies of (7e), each linearized around a different point, and treats these as inequalities as follows: h i  t t t t q~tn  ptn  Ft nk pnk
pnþ1  Fnk pnþ1;k : k ¼ 1; :::; K :

btnk

(7f)

Thus the final simplified LP formulation consists of equations (2), (5) (6a), (7f), (13a), (14b), and (16, 17, 18, 19, 20, 21).

6 Pricing Implications Although Cremer et al. [6] present a high level analysis of some pricing relationships, we have not seen any systematic analysis of the kind of price patterns that could arise as a result of modeling gas transport dynamics on the time and distance scales discussed here. As with any LP, a complete dual formulation could be stated, and solution of the primal problem will automatically determine the solution of that dual. But market participants, and market designers, will want to understand how those prices are driven by offers and bids, and the kind of pricing patterns that will be produced. To generate that insight, we focus on the key dual relationships determining the way in which spatio-temporal price information

92

E.G. Read et al.

generated by solution of the LP reflects the opportunity costs of having one more unit of natural gas available to supply at any time and place. As always, there will be one dual pricing constraint for each primal variable, and we can generate that constraint simply by collecting terms and summing them. Thus if variable xi appears with coefficient aik in constraint k, then the shadow price on constraint k will appear in the pricing constraint for commodity xi, with the same coefficient. Also note that if a primal constraint relating to cell n and period t contains primal variables relating to, say, cell n + 1 and/or period t + 1, then the corresponding primal variables for period n and t must appear in primal constraints relating to cell n
1 and/or period t
1. So the pricing equation for that primal variable will involve shadow prices computed for those constraints. Ultimately we are really only concerned to price commodities traded in the market, in this case gas injected/extracted, and possibly end-of-day linepack.11 In other words, ultimately, we are mainly interested in the shadow prices on constraints (18). However, these prices depend on the prices of other (non-traded) commodities, and all will ultimately be determined by what is effectively the solution of a set of simultaneous equations, in the LP solution process. Thus we need to consider some other pricing relationships as well. First note that for a maximization objective, standard duality theory implies that the shadow prices on < constraints will be positive, while those on > constraints will be negative. We have expressed all upper bounds in our formulation as < constraints, and all lower bounds as > constraints. So the shadow prices on all upper bounds will be positive, while those on all lower bounds will be negative. Thus adding the shadow prices on upper and lower bounds effectively creates a composite shadow price, which will be positive if the upper limit binds, and negative if the lower limit binds. (Similarly. the shadow price on an equality constraint will be positive if it binds as an upper limit and negative if it binds as a lower limit.). With that convention in mind, the pricing equations corresponding to the variables for traded quantities, xtdi and xtsi , and for net injection ytn , are:12   ltn ¼ Bidtdi
gtdi þ gtdi : (24a) xtdi   ltn ¼ Offertsi þ gtsi þ gtsi :

 dtn : ltn ¼ mtn
dtn þ 

xtsi ytn

(24b) (25)

11 By way of analogy, an electricity market formulation such as that in Alvey et al. [2] may be used to determine prices for line capacity, and even phase angles, but we really only focus on prices for electricity injected/extracted. 12 Primal variables are associated with each dual equation, just as dual variables were associated with each primal equation.

An LP Based Market Design for Natural Gas

93

These conditions are easily interpreted. First, (24a) implies that the local price, l, will equal the price for some step of the local bid/offer stack if, and only if, that bid/ offer is “marginal” at the optimum. That is, if and only if the market is free to take one more, or one less, unit from that step at the offer/bid price because it is not up against either the upper or lower limit of that step. The whole price system is driven by these (marginal) bid/offer prices. Otherwise, one of the g prices will be non-zero. For offers, the upper (or lower) limit will bind when the market price, l, is above (or below) the offer price and (24b) merely shows how gti (or gti ) adjust to reflect that difference. For bids to buy gas, the situation is reversed, because a positive buy variable corresponds to a decrease in net supply, and has the opposite coefficient in the objective function. Second, we can take mtn , the shadow price on the mass conservation constraint (6a) to be the system price for gas injected into the main transmission system at that time and place i.e. ytn .13 From (25), the local price, l, will equal m if, but only if, the market is free to take one more, or one less, unit from that location, at the local price, because it is not up against either the upper or lower injection/extraction limit at that location. Otherwise, if multiple participants want to inject (or extract) more gas at n than the bottleneck constraints (19) will accommodate, d (or d) will be nonzero, and m will be higher (lower) than l, in order to throttle injection (extraction) back to the bottleneck capacity limit. Local participants could trade between themselves at the l price, in order to ration limited injection/extraction capacity, but that local price will not impact on prices anywhere else in the system. The issue is, though, to determine the system price, m, for non-marginal locations, where injection/extraction is constrained, or limited by upper/lower offer/bid limits. Fundamentally, the price of a unit of gas at any point in the system, and in time, is determined by the marginal value that gas may have in meeting future requirements (or reducing the need for future supply) at some time and place. In this deterministic market-clearing formulation the gas price will also be the marginal cost of supplying gas to that point in time and space, from whatever sources are marginal. Looking at the issue either way, the value of gas at each time and place must be consistent with prices at adjacent times and places, which must be consistent to prices at times and places adjacent to them, and so on, until

13 This is a slight simplification, with respect to the original formulation, because injected gas will also have a velocity of its own, and if this was modeled it would have some small impact on the solution of the gas transport equations. Thus, in principle, we could have differing prices for “fast gas” and “slow gas” injected at the same time and place. But no such distinction arises here, because the simplified formulation ignores gas velocity. This situation is conceptually similar to that with respect to electricity injected or consumed with differing “power factors” in electricity markets. Hogan et al. [16] discuss a regime that would explicitly price the active and reactive components determining power factor. But real electricity markets typically only price and trade active power, using a DC approximation to the power flow equations, while relying on other agreements to control power factor within acceptable limits. We assume the same to be true here, with respect to the setting of injection pressure differentials, and hence velocities.

94

E.G. Read et al.

ultimately the entire price system is driven by a small set of marginal offer/bids. In our simplified model these marginal prices will be for gas bought or sold in some node in some period of the day. (If linepack trading were allowed for in the formulation, the marginal prices be prices for end-of-day linepack, as determined by some Packbid) Just as for electricity markets, the equations linking all these system prices together are the duals of the equations linking the physical quantities together in the gas transmission system. These are the mass balance equations, the shadow prices on which are the system gas prices, m, and the Bernoulli equations which determine how gas flows, with shadow prices b. And the key variables, in our simplified LP formulation, are the pressures, ptn , which (with volumes fixed) effectively measure the mass of gas available at each point in the system, and qtn , measuring the flow rates between adjacent locations. Thus we must consider the dual equations associated with these variables. We first discuss the impact which simple limits on flows through space and time would be expected to have, as in the case in a market for stored water, for example. We then discuss how these results will be affected by terms arising from the Bernoulli equation and velocity limits, so as to produce pricing effects which may not be immediately intuitive. First, gas prices will vary over time. If gas was being stored in a static fashion, like water in a reservoir, then we would have a simple equation linking the price of gas in successive periods to the shadow prices on the upper/lower storage bounds. That is, the price of gas stored in the cell would be the same in each successive period, unless a pressure (i.e. storage) limit was binding. Conversely, an upper (lower) mass/pressure limit would be binding if the price for gas in the next period was higher (lower) than in the current period, giving the system incentives to maximize (minimize) gas carried forward. Note that Eq. 1 is really only a convenience, allowing simplification of some of the equations. Simplistically, each unit of pressure in pipe cell n implies GLn units of mass there. Thus, dividing by GLn  t ), to be compatible converts prices associated with pressure in cell n, (ftn and f n with prices associated with mass variables for that cell (mtn from the mass balance equation). So, in this simplified model, the price (mtn ) for injected gas (ytn ) would be inferred from the following equation, describing the way in which those prices evolve over time:   t t mtn ¼ mt
1 n þ fn þ fn =Ln G :

p~tn

(26)

Second, gas prices will also vary over space. Simplistically, we might expect the price of gas moving through the pipe to be the same in each successive cell, unless a flow limit is binding. And we might expect an upper (lower) flow limit to be binding if (and only if) the price for gas in the next cell is higher (lower) than in the current cell, giving the system incentives to maximize (minimize) gas flowed forward. This would produce inter-nodal pricing impacts analogous to those arising in electricity markets. But we do not have (mass) flow limits, per se, in this formulation, only velocity limits in (14b), and a “friction” term in the simplified Bernoulli (Weymouth)

An LP Based Market Design for Natural Gas

95

equation (7f) which slows flow, but does not ultimately limit it. And the gas “stored” at one time and place also influences the rate at which gas flows to other places, over time. Thus Eq. 26 is too simplistic. To develop a more accurate representation, of the way in which these pressure relationships affect prices, we first eliminate mass, m, from the formulation, by substituting (1) into Eq. 4,14 to get: Ln G p~tþ1 ¼ Ln G p~tn þ qtn
qtnþ1 þ ytn : n

mtn

(6b)

This leaves us with two sets of pressure/quantity variables in the formulation, one for cell endpoints and one for cell midpoints. In the dual, there will be separate “pricing equations” for each set, but the prices will be linked by the shadow prices on the equations which, in the primal, define the relationships between midpoint and endpoint variables, i.e. (2a) and (5a). Since both midpoint and endpoint variables appear in those equations, their shadow prices appear in pricing equations for both types of variable. First, inter-locational price interactions are primarily determined by the dual equations for the flow variables. Because the endpoint flow variable, qtn , actually appears in the mass balance constraints for cells n and n
1, and also in the flow averaging equation for both cells (5a), the corresponding dual equation (27) relates the prices on all four of those constraints. But we also have pricing equation (28), for the midpoint flow, (~ qtn ), which appears in the Bernoulli equation (7f), and the cell velocity bounds (14b). And (28) can be substituted into (27) to give Eq. (29). mtn ¼ mtn
1 þ ðtn
1 þ tn Þ=2 : tn ¼

X

btnk þ






!  t

wtn þ  wn

qtn :

q~tn

(27)

(28)

k

mtn ¼ mtn
1 þ

!  X btn;k þ btn
1;k þ
wtn
1 þ wtn
1 þ
wtn þ wtn =2 : k

(29)

qtn This equation tells us that the price of gas in cell n reflects the price of gas in cell n
1, upstream, plus the implied cost of moving gas from cell n
1 into cell n. That cost is determined by the shadow prices on constraints limiting flows between adjacent cells. Since the mtn prices are for cell midpoints, the price difference is half determined by conditions in each cell. The friction term in the Bernoulli equation (priced at btn ) has a pervasive effect in terms of limiting and slowing inter-cell

14

And also in Eq. (21), for t ¼ T + 1, but that is not relevant here.

96

E.G. Read et al.

flows.15 The (velocity) flow limits can also play an important role, if binding. If the upper flow limit binds,  wtn will be positive, and the value of gas in the downstream cell will be higher than in the upstream cell. But if the lower limit binds (typically at zero) downstream gas will be worth less than upstream gas, and this change can be quite abrupt. This may be a proper reflection of the situation, if reverse flow is physically blocked by a check valve. Otherwise the model solution may not be physically feasible, because there is no way to physically stop flow from reversing, and alternative solutions should be explored, with flow restricted to be in the opposite direction. Second, inter-temporal price interactions are primarily determined by the dual equations for the pressure variables.16 Because the endpoint pressure variable, ptn , actually appears in the Bernoulli constraints for both cells n and n
1, and also in the pressure averaging equation for both cells, the corresponding dual equation (30) relates the prices on all four of those constraints. But we also have a pricing equation for the midpoint pressure, (~ ptn ), which appears in the pressure averaging equation (2a), and in the pressure and velocity bounds (13a) and (14b) for the midpoint of cell n. So the prices for those constraints appear in the corresponding dual equation (31), which can be re-arranged and re-scaled, to give equation (32), describing the way in which the price of gas (mass) in cell n evolves over time. ðctn þ ctn
1 Þ=2 ¼

i X h  t t t  t t Ft : n
1;k pnk bn
1;k
Fnk pnk bnk

ptn

(30)

k

   t t  t t  t t :  þ V w þ V f þ f ctn ¼ G ½Ln mtn
½Ln mt
1
w n n n n n n n mtn ¼ mt
1 n þ

n

  o  t t  t
G V t wt þ Vt  ftn þ f n n n n wn þ cn =GLn :

p~tn

(31)

mtn (32)

The first pair of shadow prices in { } reflects the impact of pressure (storage) bounds in period t, as for the hypothetical simplified model discussed above. The middle pair of terms in { } reflects the fact that having more gas in a cell increases the pressure and hence, for a constant velocity, the rate at which gas (mass) can flow through the cell. If either velocity limit (Vtn ,Vtn ) is binding then, it will have a nonzero shadow price (wtn , wtn ). If the lower velocity limit is only used to prevent flow t reversal, Vn will be set to zero, so that term disappears from this equation, so that constraint plays no role in determining inter-temporal price differentials. Its shadow price, wtn , may still contribute to inter-locational price differentials, though, via 15

As noted earlier, the Zhou and Adewumi [41] equation (7d) effectively implies a small increase to this term, and hence a small increase to inter-spatial differentials, but the pricing effect will be small enough to ignore if the physical impact is small enough to ignore. 16 In principle, the time derivative terms in the full Bernoulli equation, (7), would create a further inter-temporal link between prices. But we consider that influence to be small enough to ignore if the terms themselves are small enough to ignore.

An LP Based Market Design for Natural Gas

97

Eq. (29) above. If the upper velocity limit is binding, the wtn term will reflect a benefit from increasing pressure so as to increase mass flow. But that strategy may  t ) term be constrained by an upper pressure limit in this cell, in which case the (f n t associated with the upper pressure limit in Eq. 31 will rise to offset the wn term here. If the binding pressure limit is in a different cell, its impact will be reflected by the price of gas delivered to cell n rising high enough to make any further pressure increase there unattractive. The last term in { }, ctn , summarizes the inter-temporal pricing impacts of the Bernoulli equation, as determined by Eq. 30. The RHS of (30) reflects the way in which higher gas pressure at the input end of cell n, ptn , speeds the flow of gas though cell n to cell n + 1, while inhibiting the flow of gas through to cell n from cell n
1, in accordance with the Bernoulli equations for cells n and n
1. Summation over k captures the possibility that more than one supporting hyper-plane from the piecewise linearization may be binding. The LHS of (30) reflects the fact that higher gas pressure at the input end of cell n, increases midpoint pressures in both adjacent cells. Note that (30) involves prices for two adjacent cells, and we can not simply substitute (30) into (31) to get a complete and explicit expression for the way in which prices in cell n evolve over time without any reference to effects in other cells. This reflects the chain-like way in which mid- and end-point variables, and hence prices, are linked along the pipeline. But we can substitute (31) into (30) though, and re-arrange to get: ( ptn

¼

pt
1 n

þ

2

X h

i  t t t  t t Ft p
Fnk pnk bnk b n
1;k n
1;k nk

k

   t
G Vtn ðwn þ  wtn Þ þ Vtn
1 ðwtn
1 þ wtn
1 Þ  o  t þ ft þ f t þ ft þ f = G½Ln þLn
1  n
1

n
1

n

(30a)

n

ptn ¼ ðmtn Ln þ mtn
1 Ln
1 Þ=ðLn þ Ln
1 Þ

(33)

We can think of ptn as the average price for gas in a “pseudo-cell” centered on the boundary between cells n
1 and n, and running from the midpoint of one cell to the midpoint of the next. This price is arguably the correct price for gas injected at a cell boundary, and we can create a cell boundary at any point where gas is to be priced. Then Eq. 30a gives an explicit expression for the way in which the price of gas injected at that point evolves over time, in terms of the impact gas injected there has in both adjacent cells. Other variants can be produced by manipulating the dual equations and/or varying the primal assumptions. For example, we may impose pressure or velocity limits at cell boundaries, rather than at midpoints. But, since the cells modeled can be arbitrarily short, that kind of change does not fundamentally alter the nature of the physical outcomes, or the pricing impact of these equations. The corresponding price terms (ftn and wtn ) just appear in basically the same form, but in Eqs. 27 and 30 rather than in Eqs. 28 and 31.

98

E.G. Read et al.

Whatever variant of these equations is preferred, they define the key pricing relationships, linking prices over time and space, to create a pattern of gas prices, all driven by marginal offers and bids as discussed previously. The price of gas at any time and place will not only reflect the value that gas will have when it is finally delivered to the location at which it will be consumed, but also the indirect value it may have (positive or negative) in terms of assisting or resisting the flow of gas to other places, at other times, where it may prove to be more, or less valuable. The Bernoulli terms mean that the price may vary from period to period, and from place to place, even if there are no “local” or “immediate” pressure bounds limiting the amount of gas that can be “stored” from one period to the next, and no (absolute) flow bounds limiting the amount of gas that can be “moved” from place to place. All other shadow prices in the dual (including shadow prices on initial and final storage constraints, (20) and (21)), merely adjust to match that pattern. This situation is analogous to that arising in electricity markets, where a flow constraint on a single link will generate a distinctive “spring washer” price pattern, first described by Ring and Read [33], implying price differentials across all links involved in any loops in which that constraint is involved. These effects arise because power flows according to the laws of physics, splitting across all possible parallel paths in inverse proportion to their impedance. Prices must reflect the fact that some part of any incremental flow will travel over the over-loaded circuit, because it is not possible to “direct” flows to take alternative parallel paths avoiding it. The gas system is similar in that, while valves and compressors give some degree of control over how gas will flow, gas will flow though much of the system entirely according to the laws of physics, not economics. Thus a single binding constraint, at some time and place, will cause difficulty in delivering gas to various downstream locations at, or over, various subsequent periods. Thus it will generate price differentials across space, as in electricity markets, but also across time. Price differentials could become quite extreme if, as sometimes happens, extreme measures must be taken to keep the system operating. In Victoria, an LNG stockpile is maintained near Melbourne, with stocks being gradually built up over an extended period, so as to be available for release when required in order to maintain pressures when demand is too high to be met by continuous supply through the main pipeline system. The operation of that stockpile is optimized outside the market, as is the operation of other storage facilities, such as the Western Underground Storage Facility (WUGS). This gas is all purchased at market prices, when they are relatively low, then re-sold at times when the marginal value of gas, and hence the optimal gas price, must be very much higher, at the LNG facility. Prices may be even higher at the critical time and place which actually creates the need for such release, if that is not the LNG facility. In electricity networks, constraint pricing effects are not the only possible drivers of price differentials, though. If transmission system losses are modeled, as in Alvey et al. [2], they will cause pervasive price differentials between all locations, even when no constraints are binding. These transmission system losses are not really analogous to the “friction loss” terms in the Bernoulli equation, though. This is not a loss of gas, but a loss of energy, and its effect is to slow and

An LP Based Market Design for Natural Gas

99

delay gas delivery, rather than ultimately to limit it. This does represent a potential barrier to the efficient and timely transfer of gas from producers to consumers, but that will only imply inter-locational, or inter-temporal, price differentials in situations where the delay, in combination with insufficient linepack in the right part of the system, forces some other constraints, such as pressure or velocity limits, to bind. When price differentials do occur, these terms, of themselves, typically imply gradual change, as the cumulative effect of the Bernoulli equation on each segment of the pipeline means that flow rates gradually decrease as distance increases. Compressor operation does imply gas losses, though, and these are accounted for in Eq. 6c, in the Appendix. Thus the price of gas downstream from a compressor must rise in proportion to the marginal gas consumption of the compressor. If achieving the desired pressurization requires a compressor to consume 1% of the gas passing through it, the compressor effectively converts 100 units of upstream gas, at the upstream pressure, to 99 downstream units, at the higher downstream pressure. So, if there were no other costs or constraints involved, they would have the same total value, with the marginal value therefore needing to be (approximately) 1% higher on the downstream side. Abrupt price change can occur at compressors where flow is constrained by a minimum or maximum flow limit, though.

7 An Example While the equations in the previous section allow us to infer how prices relate in adjacent cells, and periods, we have not made, or seen, any systematic attempt to determine, the variety of system wide price patterns that might emerge. But Annex 3 of Frontier Economics [11] presents some empirical analysis, based on the results produced by the MCE of Pepper et al. [26] for a number of scenarios. That model was developed as a pragmatic replacement for the original conceptual formulation developed by Read and Whaley, and reported here. It ignores the time derivative and kinetic energy terms included in the original formulation but, if these are small, they will also have little impact on prices. While it is linearized in a different way, the pressure/flow relationship employed in this model is essentially the same as that in Eq. 7c above. This approximation has proved sufficient to produce a very good approximation to physical gas flows in the system. Thus we believe the price patterns produced by that the implemented MCE model to be indicative of the kind of price patterns likely to emerge from any implementation of the fundamental market design concept developed here. That is a nodal market based on an LP representation of the underlying network realities, on a short time scale. Here we discuss the price patterns produced for just one of the scenarios considered by Frontier, as reported by Pepper [25]. Figure 2 gives a general locational overview of the Victorian Gas System in terms of main pipelines and nodes. Figure 3, taken from Pepper shows the kind of

100

E.G. Read et al.

Fig. 2 Victorian gas system: network overview

Fig. 3 Victorian gas system example price pattern

price pattern that could occur in this system for a day in which a constraint binds. Pepper notes that prices almost always decline to a flat off-peak value at around 10 PM, with some lag in the outer portions of the system. (Hence, prices after midnight are omitted from Fig. 3). This behavior after the evening peak reflects the fact that

An LP Based Market Design for Natural Gas

101

the intra-day linepack constraints are no long binding and the MCE only needs to achieve a minimum system-wide linepack constraint by the end of the gas day. If the optimization were changed to a 48 h optimization and the next gas day was also constrained, this flattening of prices may not always occur. The relatively high prices during this particular gas day stem from two underlying causes. First, the system did not start the day from a position of unconstrained equilibrium, but inherited a gas pressure/flow pattern which made it difficult to meet the day’s requirements. This may be seen by the fact that the model assigns differing values to gas in different locations, even at the start of the day. Second, demand for gas during the peak period of the day exceeded the ability of the system to deliver gas from the low cost supply at Longford, which was constrained by production and gas processing plant capacity. Thus, in this solution, pressures were expected to reach minimum allowed levels early in the evening peak period, at Bendigo Junction and at other key points, such as the Dandenong City Gate (DCG), towards the end of the evening peak period. Some higher cost supplies such as LNG or stored gas are thus required to keep pressure at, or above, the minimum pressures. Thus by 1 PM, the system is clearly struggling to get enough gas through to Melbourne to cover requirements over the rest of the day. So prices rise over the day, then collapse after the critical hour. Prices also rise at Longford, the main injection point, but prices there fall earlier because gas can not reach the critical areas in time to make any difference. Or, more exactly, the value of injecting more gas at that place, in terms of maintaining a pressure differential to increase flow through to the critical area by the end of the critical period falls gradually over several hours as the end of that period approaches. Iona is, in sense, at the opposite end of the system, at the end of the South-West pipeline. But prices at Iona follow a similar pattern to those at Longford, presumably because Iona also acts as a source from which gas can flow to Melbourne within the critical period. Prices on this pipeline start falling earlier in the day, though, presumably because extra gas at Iona will only have a positive impact on Melbourne delivery if injected early in the day. Bendigo also lies at the end of a pipeline, off the opposite side of the outer pipeline loop from Longford. But Bendigo is not a source, and prices there follow a similar pattern to those at Melbourne. Pepper [25] reported that there is a capacity constraint between Melbourne (DCG) and Bendigo, restricting Bendigo gas availability during the peak period. The Bendigo price drops sharply after 3 PM because changes in supply or demand of gas at Bendigo no longer have much impact on the ability to meet demand at Melbourne through to the end of the evening peak. This fall below the end-of day value is much more marked for Springhurst, near the neighboring state of New South Wales. There the price drops so early, and so low, as to apparently exhibit almost the opposite of the Melbourne pattern, only rising slowly to match the end-of-day price at the end. Pepper reports that gas is actually flowing north, away from Melbourne, even when prices are higher in Melbourne, at that time. As noted earlier, gas will flow in accordance with the laws of physics and, where there is no valve to control flows, there is no reason why that flow should necessarily enhance economic value. Exactly the same situation

102

E.G. Read et al.

arises with respect to loop flows in electricity networks, where counter-price flows are common. In any case, it is not the price difference at any particular time that determines optimality. The critical issue here is maximizing flow to the major load, at Melbourne, over the critical period. Extra gas in the Northern pipeline, early in the day, is actually assigned a similar value to Melbourne gas because the increased pressure can inhibit northward flows during the critical period, thus allowing the system to meet Melbourne requirements. After 3 PM, though, the value of extra gas there actually falls below the end-of day value. This is because the impact of one unit of additional supply, or reduced demand at Springhurst, requires a reduction in the flow which the model achieves by reducing pressure at the supply end of that pipeline, thus impacting the ability to meet demand on the Northern section of the pipeline.17

8 Issues for Market Design and Implementation 8.1

Non-physical Flows and Flow Reversal

The discussion of convexity in Sect. 5 glosses over one significant point which can prove troublesome in implementing this kind of model, whether or not it is used for market clearing. We focused on the convexity of the LP feasible region. But (7c) is an equality, not an inequality, and the actual physical flows are confined to lie exactly on the boundary defined by that equation, rather than within the region bounded by it. As with any nonlinear equation, the set of points it defines can never be convex, even if the equation defines the boundary of a convex set. So the physical feasible region for this problem is definitely not convex. Still, piece-wise linearization is often used to model this kind of situation in LP models, including those of Thomasgard, Midthun and others. This does not create a problem so long as the objective function implies that points on the boundary are preferred to physically infeasible interior points. It will break down, though, if that is not the case. A similar situation arises when a piece-wise linear representation is used to model quadratic losses for electricity markets, as in Alvey et al. [2]. This creates a convex LP feasible region, and solutions will lie on the appropriate boundary provided losses are economically undesirable. This is almost always the case, but Ring and Read [33] note that a switch must be made from piece-wise linearization

17 In this case, though, the effect is at least partly due to the fact that this section of pipeline is represented with one pipeline segment. If the representation of the pipeline to Springhurst was divided into multiple pipeline sections, the MCE would be better able to account for the dynamics of the flow relationships and these prices would not fall so far.

An LP Based Market Design for Natural Gas

103

to successive linearization if the model determines that the optimal electricity price at some point in the network would be negative. This can actually occur, in situations where increased load (or losses) would relieve pressure on constrained lines in a loop. And that will make it seem desirable for the optimization to propose “solutions” which are not physically feasible, because they imply losses greater than would actually occur, for the specified flow level. Analogous situations could occur here, if the value of having more gas pressure, or a greater pressure differential, becomes negative at some time and place, most likely because it forces gas to flow away from where it is needed, and there are no check valves available to stop that occurring.18 In practice, this situation is handled by switching to successive linearization, as discussed by Pepper et al. [26]. But a closely related situation occurs when the model determines that flow reversal would be desirable. We have restricted velocity, and hence pressure differentials to be positive in a defined direction, partly because the Weymouth equation is not convex if extended into the range where the inlet/outlet pressure difference, and hence the flow direction, reverses. This gives us a convex optimization problem, with a unique optimum. There are cases, though, in which a quite different alternative optimum could be considered, with the flow on some pipe segments reversed. Gas could be compressed into a dead-end pipeline segment, for example, and then allowed to flow back out to meet peak demand. And the existence of a ring structure in the DTS suggests that some locations could be supplied sometimes from one direction, and sometimes from the other. In many cases this may not matter, in the sense that the alternative strategies do not greatly affect economic value. But the gas system operator may face some real “integer” choices between significantly different operating strategies. Ideally, an integer optimization, such as that in Martin et al. [19], could be employed to ensure that the true optimum is found. In reality, the plausible range of operating strategies is quite restricted, at least for this relatively simple system. If the model is observed to force some flows to their lower limits, the operator may make integer decisions with respect to valve and compressor settings, or just with respect to desired flow direction on certain pipeline segments. Given those decisions we can set the limits in (14b) so as to maintain minimum flows in the desired direction, and re-solve using the convex linearization valid for that flow direction but that does mean that the prices determined by the model potentially depend on some high level strategic choices made by the operator, and that may of significant concern, from a participant perspective.19

18

One could imagine this happening in a more extreme case of the Springhurst example above. It has been suggested, though, that at least some of the observed price effects could have resulted from sub-optimal compressor settings and from inter-temporal constraints on the bids such as overly constrained hourly ramp rates or minimums on hourly injection quantities. 19

104

8.2

E.G. Read et al.

Rents and Cost Recovery

The existence of price differences, over both space and time, means that there will be a potentially significant “settlement surplus” remaining after all accepted bids and offers have been cleared at the prices produced by the LP optimization. This settlement will be the sum of “rents” collected on all binding constraints. Even when constraints do occur, though, differentials will typically still be small if those constraints can readily be worked around, for example by adjusting compressor settings. Running compressors creates what is effectively only a small loss of gas from the system, and implies equally small differentials, as discussed earlier. If the gas loss factor were constant, this price differential would be just enough to pay for gas losses, thus making no contribution to recovering the cost of the compressor itself. Since the price difference across a compressor reflects marginal losses, and compressor loss functions are convex (see [26]), rents will be generated equal to the price of gas at that point in the network, times the difference between marginal and average losses.20 But this rent is also small. Larger price differentials, and hence larger rents, will arise when compressors reach their throughput limits, and/or flows are limited by the other constraints discussed in Sect. 6. As discussed there, a single constraint, binding in a single period, may generate price differences between various locations at various times, and between various times at various locations. Indeed price differences can arise even when no pipeline segment is constrained at all, in terms of absolute flow capacity. So, rent will be collected across a great many links, and periods, where no constraint is binding. This is analogous to the situation in electricity markets, where a single line constraint in a loop will generate price differences, and hence rents on all lines involved in that loop. In both cases, though, the total rent generated by each binding constraint must equal its RHS value times its shadow price. In the electricity market literature, there has been much debate about the extent to which nodal price differentials, and rents, could or should signal, incentivize, and perhaps fund, transmission network expansion. The desire to signal and incentivize gas network expansion was a significant consideration in developing this gas market framework, too. But we should caution against assuming that rents derived from inter-nodal, or inter-temporal price differences will prove sufficient, of themselves, to fund all optimal network enhancement. If we were to solve a joint operation/expansion optimization, assuming compressor and pipeline, capacity to be continuously expandable with convex costs, we would find an optimum at which the marginal cost of expansion equaled the marginal rent assigned to compressor capacity, by the market clearing prices, on average in NPV terms.

20

Analogously, a quadratic loss function for electricity transmission implies that marginal losses are always twice average losses, thus generating rents equal to half the loss-induced price differential, even for un-constrained transmission lines.

An LP Based Market Design for Natural Gas

105

But neither pipeline nor compressor capacity can be expanded continuously. And, rather than being convex, capacity costs are likely to exhibit significant scale economies, just as for transmission lines. In that case, Read [29, 30] showed that optimal capacity expansion policy implies that the line capacity should be sized so that over its lifetime, it recovered just enough rents to cover the marginal cost of making the line larger, given that a decision had been made to build the line. Put another way, if we approximate transmission capacity costs as having a fixed cost component, plus a variable cost per capacity unit equal to the marginal cost of building more capacity (at the time of construction), then the rents implied by optimal market prices should only recover the “variable” portion, not the “fixed” portion. For transmission lines, Read calculated that this marginal cost component, and hence direct cost recovery from nodal price differentials, was unlikely to be more than 30% of the total cost in an optimally expanded system, while empirical evidence from New Zealand suggested it could be as low as 10% in practice. Rudnick et al. [34] reached similar conclusions. Depending on the strength of scale economies for gas pipeline networks, and compressor equipment similar conclusions are likely to apply. This is not to say that a theoretically optimal transmission expansion/pricing regime could not be driven by these spatio-temporal price differentials. But that regime must rely on forward contracting, prior to expansion, rather than simply on collecting rents from the expanded network, as outlined by Read [30]. In practice, though, such a regime has proved difficult to establish, and supplementary funding, e.g. from industry levies or access charges, is still likely to be required.

8.3

Hedging

One major factor inhibiting further development of the gas market towards a nodal pricing paradigm is that participants fear that they could be exposed to significant price differentials, and not be able to purchase any form of insurance to cover the implied trading risks. Following the electricity market analogy, the development of hedging instruments similar to the “Financial Transmission Rights” (FTRs) developed by Hogan [15] gas been proposed. But a key requirement for an FTR regime to work is that FTRs not be issued beyond what the system is (expected to be) physically capable of delivering. Otherwise, Hogan shows that a revenue adequacy problem arises, because the rents generated on the binding constraints will not be sufficient to support the payments demanded by FTR holders.21 Conversely, if the flow pattern corresponding to the set of all FTRs held by participants lies within the convex feasible region of the market clearing formulation, the implied “FTR flows”

21

We do not expect “revenue adequacy” to cover the cost of gas actually consumed by compressors, any more than the cost of actual losses is covered for electricity markets. In both cases these costs must be face by traders as a residual differential, or covered in some other way.

106

E.G. Read et al.

on the lines which turn out to have binding limits in the spot market clearing solution must be no more than their capacity. The rent required to support FTR payments matches the FTR flows times the shadow prices on those constraints, which must be no more than the rent collected as settlement surplus, that as determined by the full RHS capacity of the binding limits, times their respective shadow prices.22 The situation is essentially the same for the gas market formulation described here, except that the “transport network” allows gas to be transferred over both time and space, and constraints may thus be on flows from one cell to another, or on one time period to another.23 Ignoring the possibility of line-pack, what could clearly be supported would be FTRs with a defined delay time, hedging the difference between the gas price in one cell and “start” period and that in another cell in that start period, plus a specific delivery delay. If no constraints bind in such a way as to (directly or indirectly) limit that flow we expect the two prices to be identical. But otherwise, just as for Hogan’s electricity model, the rent collected on the flow limiting constraints should suffice to provide hedging for the volume that can be physically transported, with that delay. Alternatively, we could decompose each “delayed flow” FTR into two components: An instantaneous inter-locational FTR, as in electricity markets, and a locationally specific inter-temporal FTR, hedging between the prices at two different times, for the same location. Ignoring line-pack, neither of these FTR components needs to be physically feasible, on its own. The situation is not really very different from that arising in an electricity market for which FTRs are all expressed with respect to some reference hub. In such a market an FTR from A to B can be decomposed into an “A-to-Hub” component and a “Hub-to-B” component. But the transmission system does not need to be able to support the requested volume of flows from A to the hub, or from the hub to B, only the net flow pattern after all requested flows have been accounted for. In the gas market case we can think of “cell j at time t” as being analogous to a hub. Thus we can define and issue instantaneous inter-locational FTRs, from “cell i at time t” to “cell j at time t”,

22 This holds even though a single binding constraint may generate price differentials, and hence rents, across all lines involved in any loop in which it is involved. One way to see this is to solve the simultaneous equation system defining power flows in terms of net nodal injections so as to express the line flow directly in terms of net nodal injections. Since a binding constraint holds with equality, the total rent collected on the RHS side of the constraint will be broken down into a set of “nodal rents” on the LHS of the constraint. These nodal rents correspond to the rent collected on that part of a notional flow from the node to a reference node which passes over the constrained line. This representation of the constraint rents making up the settlement surplus can be used to construct constraint based “flow gate rights”, as in Chao et al. [5], or classic FTRs, as in Hogan [15]. 23 Convexity issues will arise with respect to “integer” decisions, such as valve or compressor settings, and possibly flow directions. But that is also true with respect to the “integer” decisions, such as breaker or transformer settings, determining the configuration of electricity networks. In both cases any issuer of FTRs must take care to assess the feasibility of supporting those FTRs across the range of network configurations that might apply on the day.

An LP Based Market Design for Natural Gas

107

provided we also define and issue locational inter-temporal FTRs from “cell j at time t” to “cell j at time t + delay”. Once issued, such instruments could not be traded independently, but they could be traded using a market clearing optimization that guarantees simultaneous feasibility. The gas system can support a much wider range of FTRs than this, though, because gas stored as line-pack can typically be released over a wide range of intervals. Thus there is no fixed delay between the time at which gas is injected at i, and the time it is extracted at j. So, for convenience, we could define and issue instantaneous inter-locational FTRs, from “cell i at time t” to “cell j at time t”, and we could also define and issue a wide variety of locational inter-temporal FTRs from “cell j at time t” to “cell j at time r”. Here r may be greater than t, but it could be less than t. In other words it may be possible to extract an incremental unit of gas earlier in the day, provided we know that it will eventually be replaced by a unit injected at time t, and arriving some time later, and that no constraints will actually be violated in the meantime. In particular, instantaneous “trade” will often be possible, even though instantaneous “transport” is not. The feasible range of such transactions will be limited by binding constraints on pressures or flows, in which case the system will need to incur the costs of redispatching other injection, extraction, or compression, in order to make this transaction possible. But the marginal cost of such re-dispatch is exactly what the inter-nodal and inter-temporal price differences measure. And the shadow prices on the binding constraints that determine the inter-nodal and inter-temporal price differences will also generate the rents required to support any simultaneously feasible pattern of inter-nodal and inter-temporal FTRs. More generally, the rents should support any simultaneously feasible pattern of mixed spatio-temporal FTRs, whether or not they are decomposed as in this discussion. To date, difficulties in conceptualizing hedging arrangements have proven to be a significant deterrent to introducing greater spatio-temporal price differentiation into this relatively small market. But the mathematics of hedging in this kind of gas market seem closely analogous to that in nodal electricity markets. It may seem complex to determine simultaneous feasibility, over both space and time, but all that is required is to notionally clear the proposed FTR trades through a version of the spot market clearing optimization, just as in electricity markets.

9 Experience and Conclusions In principle, the spatio-temporal prices determined by the formulation discussed in Sect. 6 could be used to coordinate the market at all times, and particularly when congestion creates significant spatio-temporal price differentials. It should be said, though, that the nodal market paradigm described here has not revolutionized markets to anything like the same extent as the analogous electricity market design. This is partly due to inherent differences between the sectors. Valves and compressors make gas flows relatively more controllable than electricity flows,

108

E.G. Read et al.

and limit the potential for troublesome “loop flow” effects of the type that at least partially motivated electricity market reform in the US, for example. Thus traditional market paradigms may be relatively more effective in the gas sector than they were in the electricity sector. Nor is there so much need for absolute real-time coordination. In this particular case, the net volume traded between participants is also not very large, since much of the gas is effectively transported on behalf of vertically integrated participants, who inject their own (contracted) gas at one location, and extract it at another. Still, Pepper et al. [26] describe a detailed LP optimization model that does calculate spatio-temporal prices as above, and the dispatch schedules associated with those prices are used. But they also describe how actual trading prices are determined using a simplified version of the model, in which the gas system is modeled like a simple “tank”. That is, gas injected at any location, at any time during the day, is assumed to be able to supply demand at any other point, and time of day. Intra-day price differentials arise because the tank model is re-run several times during the day, but a single gas trading price is calculated each time the model is run for (the remainder of) each trading day.24 If no transmission system constraints ever bound, the tank model would always suffice to clear the market. Congestion certainly can occur in Victoria, and give rise to significant pricing effects when it does, as in the example above. The tank model under-estimates the cost of operating the real market at such times, and determines a price which is not consistent with the costs of all suppliers or consumers. Commercially, this is dealt with by “uplift” payments to compensate participants mis-dispatched relative to the daily gas price. But Frontier [11] found that this did not happen often enough to justify moving towards full inter-temporal pricing framework developed here. And we understand that subsequent network developments may have reduced congestion, and averted the need for further market development along these lines. But the reason the industry has not proceeded further with a more granular market design is definitely not because the experimental evidence suggests that optimal market-clearing prices would always be the same at all times and all locations. On days when constraints bind, price differentials would appear to be of a similar order of magnitude to those found in electricity markets, over both space and time. But that raise a different barrier to further development, because participants are reluctant to expose themselves to the risk implied by potentially significant spatio-temporal price variations that may not well understood, and can not be hedged without development of FTR instruments for which there is no internationally established theoretical framework, or precedent. This concern is

24 Originally, the market design included end-of-day linepack trading, thus including a version of (22) in the objective function. However, the concept was dropped, due to concerns over price manipulation, and because the feasible end-of-day target linepack range was considered too small and sensitive to be managed by participants.

An LP Based Market Design for Natural Gas

109

particularly strong when price variations may be significantly influenced by decisions made by the gas system operator, with respect to compressor settings, etc. We would argue that the situation is not really very different from that in the electricity sector, where radically different pricing patterns can arise depending on the system operator’s decisions about which circuits will operate, and how they will be connected, in each dispatch interval. But there is now significant experience with electricity markets, and protocols have been developed which tend to restrict operator freedom, but deliver benefits to the sector as a whole. A similar process may be expected to evolve in gas markets. Even without full market implementation, the pricing information generated by the model gives a clear measure of the economic costs being imposed by constraints, and the value that might be released by investment in equipment and/or operating practices that could relieve those constraints. But the process of developing appropriate protocols could be controversial and expensive, and possibly not worthwhile in a small gas market such as this. At this stage, then, the Victorian gas market does not actually employ the full potential of the formulation described here, and the success of the Victorian gas market development, per se, provides only limited evidence with respect to the potential value of an LP-based market-clearing approach. That market has not fully exploited the paradigm’s potential, partly due to its small size, degree of vertical integration, and relative lack of congestion. Many markets trade a much greater volume and value of gas than Victoria, though, and congestion seems not uncommon. And, at least in Victoria the lack of any international experience with, or literature on, this type of market structure in the natural gas industry has been a major factor inhibiting further development of a market based on the nodal pricing paradigm. Thus the major intended contribution of this paper is to report that the concepts have actually been developed, tested, and to some extent applied, in the context of a market which has now operated successfully for over a decade. This demonstrates that it is not too difficult to develop a spatio-temporal MCE formulation for a gas market. And experience with that model also reveals the potential for price differences large enough to imply significant potential for economic gains from trading. Thus we consider the paradigm developed here could well prove more fully applicable in larger and more diversified markets, elsewhere. Just as importantly, there is an increasing interest in the application of so-called “smart” markets [18] to a wide variety of situations. Many of these situations involve storage of some “commodity”, such as water e.g. [23, 28], or some form of pollution [27], within a “transportation system”, where it may, or may not, be fully or partially controlled by participants and/or in some centralized fashion. This gas market example seems highly relevant to all such developments, because all such markets are likely to exhibit broadly similar spatio-temporal price patterns to those found here, and may need to overcome many of the same conceptual and practical challenges before successful implementation can be expected. In particular, the way in which stock in transit and/or storage needs to be priced represents a significant step beyond established electricity market practice. And the need to account for the possibility that, as perceptions change, stock will need to be

110

E.G. Read et al.

re-priced, perhaps radically, raises significant questions about the validity of deterministic formulations of the type discussed here. Unlike electricity markets we can not rely on participants to manage this in-transit stock, and the relationship between current and future price is determined by the “hard” mathematics of physics and duality, not by trading in futures market reliant on “softer” participant judgments. Thus it may be that market-clearing concepts will eventually have to be developed further, to incorporate stochastic formulations.

Appendix: Modelling Junctions, Fittings, and Compressors Pepper et al. [26] shows discusses how to deal with several complications ignored in our simplified formulation via simple extensions of the approaches discussed in Sect. 5. Compressors play an important role in many gas systems. By compressing gas at one location they not only allow increased linepack storage, but increase pressure differentials, thus increasing gas flows from one location to another. Obviously, gas compression requires energy input, and in the Victorian system the compressors are themselves powered by gas drawn from the gas transport system. Although compressor fuel use is relatively low, it may be modeled as follows. A gas powered compressor is driven by a proportion of the gas that flows through it; increasing throughput in a pipe requires an increase in the compressor pressure to offset the dynamic losses down the pipe. Increasing Dp necessitates speeding up the compressor and hence increases fuel consumption. For centrifugal gas compressors a quadratic equation relates the change in head (pressure), Dp, volumetric flow at the compressor inlet qic , and impeller speed, RPM, as follows25: Dp ¼ C1  ðqic Þ2 þ C2  qic  RPM þ C3 þ C4  RPM2

(34)

We can not use this equation directly in the formulation, though, because RPM is not an LP variable and the equation in this form is nonlinear. Still, we can reasonably assume that compressor operation rules will have been externally optimized and that optimal operation will imply equations giving the minimum gas consumption required to achieve any desired pressure/flow trade-off. Read and Whaley [32] present a number of detailed equations and steps to determine this loss in gas mass, during compressor operation. Basically, if we know the desired flow rate and pressure increase across the compressor, we can calculate the required running speed of the compressor, in RPM, from which we can determine the rate of fuel usage, and hence the actual mass “lost” in the compressor. This loss, which is a function of volumetric flow and pressure change, is the cost associated with

25

Constants C1
C4 are normally stated by compressor manufactures as standard data.

An LP Based Market Design for Natural Gas

111

pressurization of the gas in the downstream section of pipeline. So mass conservation must be revised to account for the fuel usage, Loss, represented as an effective reduction in mass flow on the discharge side of the compressor:26 t t ti t mtþ1 ¼ mtn þ qtin
qto n n þ yn
Lossn ðqn ; Dpn Þ

(6c)

Ultimately, ignoring commitment of compressor units, this loss is the only specific aspect of compressor operation that needs to be included in the market clearing formulation. The compressor loss term in Eq. 6c) can be reasonably approximated by a convex differentiable function over a convex feasible operating region. Thus it can be linearized as a function of the LP variables, i.e.qtn , ptn , ptin and pto n . Apart from this, and the fact that pressure rises, rather than falling in the direction of flow, compressors can be treated like other “fittings”. An implementable solution for a real pipe network must also generalize Eq. 6 to represent mass flow balance in situations where multiple inflow pipes of varying diameter and length connect to a similar variety of outflow pipes. Since the mass flow rate is equivalent on each side of “fittings”, such as valves, tees and bends, flow through them can be determined by the pressure difference between the two adjacent cells. But constraints and variables may be required to represent pressure drops of specific forms implied by particular fitting types. Some valves basically increase the friction factor in the Bernoulli equation, for a short pipe segment, with a closed valve implying infinite resistance. Pressure reducing valves are designed to reduce pressure to a specific level. This can be enforced by an upper pressure limit, but a slack variable is also required to represent the drop from the upstream pressure level to the specified level. Or a ratio constraint can be used to represent proportional pressure change as may occur when pipelines of different sizes are joined. For proportional changes, injecting at a junction increases pressure in both adjacent cells, and produces the same kind of pricing patterns. A more detailed formulation, modeling both input and output pressures and assuming constant pressure ratios at boundaries, produces essentially the same pricing equations. In (30), though, the weights on the price terms (b and c), for cell n
1 now involve the cell boundary pressure ratio. Pressure reducing valves create a pressure discontinuity, though, and a pricing discontinuity can be expected. But, while a more complex formulation may make the pricing relationships more difficult for participants to understand and verify, LP optimization will always ensure that price relationships correctly reflect physical realities, to the extent that they are represented in the LP.

26 In this simplified representation we are assuming that compressors can be dispatched continuously right down to (near) zero, with no “commitment” costs, penalties, or restrictions. This allows us to form an LP representation with a convex feasible region. In reality there is an integer “unit commitment” problem here, as discussed by Pepper et al. [26].

112

E.G. Read et al.

References 1. AEMO (2010) A technical guide to the Victorian gas wholesale market. http://www.aemo. com.au/corporate/0000-0264.pdf. Accessed 7 Aug 2010 2. Alvey T, Goodwin D, Ma X, Streiffert D, Sun D (1998) A security-constrained bid-clearing system for the New Zealand wholesale electricity market. IEEE Trans Power Syst 13 (2):340–346 3. Boyd SP, Vandenberghe L (2004) Convex optimization. Cambridge University Press, New York 4. Breton N, Zaccour Z (2001) Equilibria in an asymmetric duopoly facing a security constraint. Energ Econ 25:457–475 5. Chao H, Peck S, Oren S, Wilson R (2000) Flow-based transmission rights and congestion management. Electric J 13(8):38–58 6. Cremer H, Gasmi F, Laffont JJ (2003) Access to pipelines in competitive gas markets. J Regul Econ 24(1):5–33 7. De Wolf D, Smeers Y (1997) A stochastic version of a Stackelberg Nash-Cournot equilibrium model. Manag Sci 43(2):190–197 8. Doane MJ, Spulber DF (1994) Open access and the evolution of the US spot market for natural gas. JLE 37(2):477–517 9. Dorin B, Toma-Leonida D (2008) On modelling and simulating natural gas transmission systems (Part I). J Control Eng Appl Inform 10(3):27 10. DPI: Victorian Government Department of Primary Industries (2009) The Victorian gas market. http://new.dpi.vic.gov.au/earth-resources/industries/oil-gas/petroleum-explorers-guide-to-victoria/ the-victorian-gas-market. Accessed 2 Sep 2010 11. Frontier Economics (2003) Analysis of high level design directions for the Victorian gas market. Report to VENCorp 12. Gabriel SA, Manik J, Vikas S (2003) Computational experience with a large-scale, multi period, spatial equilibrium model of the North America natural gas system. Netw Spat Econ 3:97–122 13. Gabriel SA, Kiet S, Zhuang J (2005) A mixed complementarity-based equilibrium model of natural gas markets. Oper Res 53(5):799–818 14. Gilbert RJ, Kahn EP (1996) International comparisons of electricity regulation. Cambridge University Press, Cambridge, UK 15. Hogan WW (1992) An Efficient Concurrent Auction Model for Firm Natural Gas Transportation Capacity. Information Systems and Operational Research, Vol 30, No. 3 16. Hogan WW, Read EG, Ring BJ (1996) Using mathematical programming for electricity spot pricing. Int Trans Oper Res 3(4):209–221 17. Johnson RB, Oren SS, Svoboda AJ (1997) Equity and efficiency of unit commitment in competitive electricity markets. Utilities Policy 6(1):9–19 18. McCabe KA, Rassenti SJ, Smith VL (1989) Designing ‘smart’ computer-assisted markets: an experimental auction for gas networks. Eur J Polit Econ 5(2–3):259–283 19. Martin A, Moller M, Moritz S (2006) Mixed integer models for the stationary case of gas network optimization. Math Program Ser B 105:563–582 20. Midthun KT, Bjørndal M, Tomasgard A, Smeers Y (2007) Paper IV Capacity booking in a Transportation Network with stochastic demand and a secondary market for Transportation Capacity. www.iot.ntnu.no/winterschool11/web/material/tomasgard_paper.pdf. Accessed 27 Nov 2011 21. Midthun KT, Bjørndal M, Tomasgard A (2009) Modeling optimal economic dispatch and system effects in natural gas networks. Energy J 30(4):155–180 22. Modisette J, Modisette J (2003) Physics of pipeline flow: energy solutions. www.energysolutions.com/pdf/tech_paper_Modisette_Physics_of_Pipeline_Flow.pdf. Accessed 11 July 2010 23. Murphy JJ, Dinar A, Howitt RE, Rassenti SJ, Smith VL (2000) The design of “smart” water market institutions using laboratory experiments. Environ Resour Econ 17(4):375–394

An LP Based Market Design for Natural Gas

113

24. O’Neil RP, Williard M, Wilkins B, Pike R (1979) A mathematical programming model for allocation of natural gas. Oper Res 27(5):857–873 25. Pepper W (2002) Stage 2-evaluation of market design packages: detailed report. Report by ICF Consulting and Pacific Economics Group to VENCorp 26. Pepper W, Ring BJ, Read EG Starkey SR (2012) Implementation of a scheduling and pricing model for natural gas. A. Sorokin et al. (eds.), Handbook of Networks in Power Systems II, Energy Systems, Springer-Verlag Berlin Heidelberg 27. Prabodanie RA, Raffensperger JF, Milke MW (2009) Simulation-optimization approach for trading point and non-point source nutrient permits. Paper presented at the 18th World IMACS congress and MODSIM09 international congress on modelling and simulation, Cairns, Australia, 13–17 July 2009 28. Raffensperger JF, Milke MW, Read EG (2009) A deterministic smart market model for groundwater. Oper Res 57(6):1333–1346 29. Read EG (1989) Pricing and operation of transmission services: long run aspects. In: Turner A (ed) Principles for pricing electricity transmission. Trans Power New Zealand, Wellington, NZ 30. Read EG (1997) Transmission pricing in New Zealand. Utilities Policy 6(3):227–236 31. Read EG (2010) Co-optimization of energy and ancillary service markets. In: Rebennack IS, Pardalos PM, Pereira MVF, Iliadis NA (eds) Handbook of power systems. Springer-Verlag Berlin Heidelberg, pp 307–327 32. Read EG, Whaley R (1997) A gas market model for Victoria: dispatch/pricing formulation. Report by Putnam, Hayes & Bartlett–Asia Pacific Ltd. to VENCorp 33. Ring BJ, Read EG (1996) A dispatch based pricing model for the New Zealand electricity market. In: Einhorn MA, Siddiqi R (eds) Electricity transmission pricing and technology. Kluwer Academic, Boston, pp 183–206 34. Rudnick H, Palma R, Fernandez JE (1995) Marginal pricing and supplement cost allocation in transmission open access. IEEE Tran Power Syst 10(2):1125–1132 35. Ruff LE (1997) Victorian gas market clearing logic. Report by Putnam, Hayes & Bartlett Asia Pacific Ltd to VENCorp 36. Sioshansi F, Pfaffenberger W (2006) Electricity market reform: an international perspective. Elsevier, Amsterdam 37. Vany AS, De Walls WD (1994) Open access and the emergence of a competitive natural gas market. Contemp Econ Policy 12(2):77–96 38. Tomasgard A, Rømo F, Fodstad M, Midthun KT (2007) Optimization models for the natural gas value chain, In: Hasle G, Lie KA, Quak E (eds) Geometric modelling, numerical simulation, and optimization: applied mathematics at SINTEF, Springer-Verlag, Berlin Heidelberg, pp 521–558 39. Wolak FA (2000) An empirical analysis of the impact of hedge contracts on bidding behavior in a competitive electricity market. Int Econ J 14(2):1–39 40. Zheng QP, Rebennack S, Iliadis N, Pardalos PM (2010) Optimization models in the natural gas industry. In: Rebennack IS, Pardalos PM, Pereira MVF, Iliadis NA (eds) Handbook of power systems. Springer-Verlag Berlin Heidelberg, pp 121–148 41. Zhou J, Adewumi MA (1990) The development and testing of a new flow equation: the Pennsylvania State University. www.psig.org/papers/1990/9504.pdf. Accessed 1 Feb 2011 42. Energy Projects Division (EPD), Dept. of Treasury & Finance, (1998) Victoria’s Gas Industry: Implementing a competitive Structure. Information Paper No. 3, 2nd Edition, April 43. McCabe KA, Rassenti SJ, Smith VL (1990) Auction Design for Composite Goods: The Natural Gas Industry. Journal of Economic Behavior and Organization 14:127–149. Elsevier Science, North-Holland

Part II

Network Interactions

Energy Carrier Networks: Interactions and Integrated Operational Planning Ricardo Rubio-Barros, Diego Ojeda-Esteybar, and Alberto Vargas

Abstract The integration of natural gas (NG) and electricity sectors has rapidly increased as a consequence of the growing installation of natural gas fired power plants (NGFPP). This has driven the need to model the interactions among the energy carriers and to optimize energy resources management from a centralized planning perspective. Currently, electricity and NG systems are considered in a decoupled manner. NG prices and availabilities for the electric power generation are used as fixed parameters for the needed coordination between both energy sectors. This chapter presents a comprehensive literature survey of previous research on integrated electricity and NG operational planning. The relevant characteristics of NG and electricity systems are compared considering the physical laws that govern the flows of these energy carriers through dedicated networks. The interactions among the energy carriers and their networks are modeled with different levels of detail according to the evaluated time horizon. The integrated operational planning problem of multiple energy carriers systems is comprehensively described and formulated, covering from the long/medium-term energy resource scheduling to the single period economic dispatch. Finally, a contribution is made about the economic interactions between different energy carriers (electricity, NG, and hydro energy) through opportunity costs such as water and NG values. Keywords Economic dispatch • Electricity-gas integration • Energy carrier networks • Energy systems modeling • Integrated operational planning • Natural gas system optimization

R. Rubio-Barros • D. Ojeda-Esteybar • A. Vargas (*) Instituto de Energı´a Ele´ctrica, Facultad de Ingenierı´a, Universidad Nacional de San Juan, San Juan, Argentina e-mail: [email protected]; [email protected]; [email protected] A. Sorokin et al. (eds.), Handbook of Networks in Power Systems II, Energy Systems, DOI 10.1007/978-3-642-23406-4_5, # Springer-Verlag Berlin Heidelberg 2012

117

118

R. Rubio-Barros et al.

1 Introduction Energy plays an all-encompassing and critically key role in economics and social development. Therefore, energy planning is an essential and strategic part of national/regional economic planning. This energy planning is driven by a set of energy polices, which are developed to meet many interrelated and often conflicting objectives, such as least cost of supply, security of supply and preserving the environment [1]. The energy systems infrastructure needed to achieve these goals is determined within the planning procedures. This complex infrastructure, which is composed by all the technical energy systems, allows providing the consumers with the “useful energies” or “energy services”. Technical energy systems include production, conversion, treatment, transport and storage facilities which comprise the supply chains from the primary energy sources (e.g., oil, coal, natural gas, nuclear, solar, wind) to final energy carriers required by the consumers (e.g., electricity, natural gas, water district heating). Particularly, electric power systems perform two important functions in the energy supply: allowing the use of primary energy sources, such as nuclear, hydropower, and wind that otherwise they were mostly unusable; and providing for flexibility since most of the energy sources can be converted to electricity. While the required energy systems infrastructure is a product of the energy planning, it does not imply some rigid action plan as in a centrally planned economy. Conversely, because market mechanisms furnish the needed coordination in the economy, and then in the energy sectors, energy planning implementation relies mainly on market incentives, pricing methods and decentralized competitive forces.

1.1

Energy Systems Planning Background

Before the energy crisis of the 1970s, energy was relatively cheap, and the emphasis was more on the engineering and technological aspects. The energy system planning was confined to various energy sectors such as electricity, oil, and coal, with almost no coordination among them. The increasing demands were invariably dealt with by augmenting the supply without taking into account possible substitutions between the energy carriers [1]. Nowadays, there is consensus among policy makers that energy sector investment planning, pricing, operation and management should be carried out in an integrated and coordinated manner in order to achieve an economic, reliable and environmentally sustainable energy supply. A hierarchical and sequential procedure is typically used to tackle this huge and complex decision-making problem. The so-called energy models are the first stage in this hierarchical energy planning procedure. In such models, all (or most) energy carriers are considered in an integrated approach. Several of these energy models have been developed to

Energy Carrier Networks: Interactions and Integrated Operational Planning

119

analyze a range of energy policies and their impacts on the energy system infrastructures and on the environment. Others are focused on the energy services demands forecast. An overview and a classification of some of the most relevant energy models, like TIMES (integrated MARKAL-EFOM system) [2], MESSAGE [3], ENPEP-BALANCE [4] and LEAP [5], is included in [6]. These models are focused on a long term planning horizon (more than 10 years) and can be tailored to cover local, national, regional or world energy systems. The interactions between the energy sectors and the other sectors of the economy (e.g., transport, industry, commerce, agriculture) can be taken into account through model extensions or represented by means of constraints. Basically, energy models can be classified in two types according to the analytical approach used: aggregated general equilibrium (top-down) and technology explicit (bottom-up) models. In top-down models, each energy sector is represented by a single production function including macroeconomic variables (capital, labor) and elasticities of substitution. Bottom-up models have a detailed description (capacities, costs, efficiencies) of the available technologies for each energy sector, thus the production functions are implicitly constructed [2]. Because the dimensions of the problem, energy models are not developed to represent the characteristics of different transport modes, neither the complex physical laws that governs electric power and natural gas (NG) systems. Only nodal energy balances are considered. Another limitation of these models is related to energy storage, which is oversimplified or disregarded. More recently, new approaches to energy system planning have been presented. They are focused on a higher technical description of some energy sectors and their transport modes. The model presented in [7] includes the topology of several energy systems, and the technical and economic properties of different investment alternatives. Among other energy modes of transport, simplified electricity and NG networks are considered. In [8] and [9] specific methodologies and tools are proposed to address the integrated NG and electric power systems planning in particular. The results of energy models provide the framework for the following stages, in which each energy carrier system is planned and operated in a decoupled manner. Thus, specific procedures and strategies are implemented according to specific value system, e.g., economic, technical, political, and environmental context. Usually, single energy carrier system expansion and operation planning are carried out considering the other energy carriers availabilities and prices as coordinating parameters. Electric power systems are a good example of this approach: they are planned and operated without taking into account the integrated dynamics of the fuel infrastructures and markets, i.e., costs and capacities of fuel production, as well as storage and transportation. The main assumption, in which this decoupled planning and operation approach, is based on the fact that there have not been significant energy exchanges between the energy carriers if they are compared with the total amount of energy supplied by each energy carrier.

120

1.2

R. Rubio-Barros et al.

Increasing Interactions Between Electric Power and Natural Gas Systems

Natural gas-fired power plants (NGFPPs) are the linkage between electric power and NG systems, being producers for the former and consumers for the latter. The growing installation and utilization of NGFPPs over the last two decades has lead to increasing interactions between electricity and NG industries. From 1990 to 2005, the worldwide share of NGFPPs in the power generation mix has almost doubled, from around 10% to nearly 19%; reaching, for instance, the 54% in Argentina, the 42% in Italy and the 32% in UK [10, 11]. The installation of NGFPP has been driven by technical, economic, and environmental reasons. The high thermal efficiency of combined-cycle gas turbine (CCGT) power plants and combined heat and power (CHP) units, their relatively low investment costs, short construction lead time and the prevailing low natural gas prices until 2004 have made NGFPPs more attractive than traditional coal, oil and nuclear power plants, particularly in liberalized electricity markets. Additionally, burning NG has a smaller environmental footprint and a lower carbon emission than any other fossil fuel. The increasing use of NGFPPs has had a great impact on NG market. Power generation accounted for around half of growth in gas use from 1990 to 2004; over the most recent 5 years, this proportion rose to nearly 80% [10]. This fact is especially notable in those countries where large capacities of NGFPPs have been installed. Therefore, NG demand for power generation as a share of the total NG consumption has constantly augmented during these years, reaching globally the 39%, in 2006 [12]. The other index, which also indicates the level of interrelations between electric power and NG systems, is the share of electrical energy produced by NGFPPs. This share depends not only on the NGFPPs installed capacity, but also on the fuel prices and the availability of other energy resources (hydroelectricity, wind power). For instance, in 2006, this share was globally the 20%, the 50% in Argentina and in Italy, the 35% in UK, and the 20% in USA [11]. From an operational point of view, the NGFPPs dispatch affects the total amount of NG consumption and its flows through the pipelines. On the other hand, the maximum capacity of gas injection for all the system, the limited transmission capacity of pipeline network and the priorities of use in case of NG shortages (usually not assigned to NGFPPs), impose constraints on NGFPPs generation. The described interactions are closely related to the regulatory frameworks of the markets implemented in electric power and NG systems. NGFPP generation companies participate simultaneously in both markets, therefore, they are best suited for price arbitrage between both commodities. This means that according to electricity and NG market prices, and its marginal production heat rate, NGFPPs can decide to use the gas, previously contracted, and sell electricity in the power market or resell the gas in the NG market instead of generating electricity. Since 2005 to the first half of 2008, the raising NG prices have eroded the competiveness of NGFPPs, decreasing the pace of growth in NG use for electricity generation and reducing the incentives for future investments in these technologies.

Energy Carrier Networks: Interactions and Integrated Operational Planning

121

Nevertheless, as NG prices have converged to lower levels during 2009, the NGFPPs have recovered their investment attractiveness. For the coming decade, NGFPP capacity is estimated to continue to account for the bulk of capacity additions, thus, the interactions between electricity and NG sectors will keep rising [13].

1.3

Integrated Operational Planning of Energy Carrier Systems: State of the Art

In this new context, in which the energy exchanges between electric power and NG systems are significant, is essential to consider an integrated modeling of both energy carrier systems for the planning and operation tasks. In more general terms, the “boundary conditions” and energy exchanges between all energy carrier systems must be carefully analyzed to determine the most suitable modeling of these interactions. Particularly, it is important to verify if coordinating parameters can appropriately represent these interrelations, maintaining the systems in decoupled way; or a more detailed description is needed, leading to an integrated modeling of the energy carrier systems. Among all the issues, which arise from this new integrated perspective of energy carrier systems, this chapter is focused on the modeling and operational planning aspects. Several approaches that address the integrated modeling and analysis of energy systems in a more comprehensive and generalized way have been presented. These approaches consider multiple energy carriers; particularly electricity and NG systems interactions and combined operation have been investigated. An assessment of the impact of NG prices and NG infrastructure contingencies on the operation of electric power systems is presented in [14] from a decoupled modeling approach. From the market perspective, different methodologies have been proposed to support the decision making related to: optimize NG supply contract portfolios [15], pricing NG supply contracts [16], and price hedging strategies between the gas and electricity markets [17]. The effects of applying different transmission cost allocations to electricity and NG networks are analyzed in [18, 19]. A review of the main approaches and models, which deal with the integrated operational planning of multiple energy carrier systems, is presented in following subsections. This review is based on the survey shown in [20]. The different approaches are conveniently grouped according to the time horizon considered.

1.3.1

Long and Medium Term

Reference [21] proposes a generalized network flow model of an integrated energy system that incorporates the production; storage (where applicable); and transportation of coal, natural gas, and electricity in a single mathematical framework, for a medium-term (several month to 2–3 years) operational optimization.

122

R. Rubio-Barros et al.

The integrated energy system is readily recognized as a network defined by a collection of nodes and arcs. Fuel production facilities, electric power plants and storage facilities are also modeled as arcs. A piecewise linear functions are applied to represent all cost and efficiencies. Since the problem is entirely modeled as a network and linear costs, a more efficient generalized network simplex algorithm is applied, than ordinary linear programming. The total costs considered are defined as the sum of the fossil fuel production costs, fuel transportation costs, fuel storage costs, electricity generation costs (operation and maintenance costs), and the electric power transmission costs. The objective of the generalized minimum cost flow problem is to satisfy electric energy demands with the available fossil fuel supplies at the minimum total cost, subject to nodal balances, maximum and minimum flow in each arc and emission (sulfur dioxide) constraint. Additionally, the hydroelectric systems (hydropower plants and reservoirs) are also taken into account in [22], but the emission constraints are not considered in this model. Reference [23] presents also a generalized network flow model including only hydroelectric, NG and sugar cane bagasse as energy resources. Bezerra et al. [24] presents a methodology for representing the NG supply, demand and transmission network within a stochastic hydrothermal scheduling model. The NG demand at each node is given by the sum of forecasted non-power gas and NGFPPs consumptions. The gas network modeling comprise: a gas balance at each node; maximum and minimum gas production, pipelines flow limits; and loss factors applied to gas flows (to represent the gas consumed by compressor stations). NG storage facilities are not been taking into account in this approach. The stochastic dual dynamic programming (SDDP) algorithm is used to determine the optimal hydrothermal system operation strategy, which minimize the expected value of total operating cost along the time horizon (2–3 years typically). While the total cost includes the fuel and shortage costs relating to electricity supply, the shortage costs associated to non-for-power NG load shedding are not considered. The NG prices are fixed from the outset and they are not results of the optimization process.

1.3.2

Short Term

Reference [25] presents a new formulation in order to include a NG system model in the short-term hydrothermal scheduling and unit commitment. NG wells, pipelines and storage facilities are considered, while nodal balances and pipelines loss factors are taking into account for a simplified gas network modeling. Gas storages are modeled similarly to water reservoirs. A constant conversion factor is used as input–output conversion characteristic for NGFPPs. A DC power flow modeling without losses is applied to determine electric power flows. The problem is formulated as a multi-stage optimization problem, whose objective function is to minimize the total cost to meet the gas and electricity demand forecast. This total cost is the sum of the non-gas fired generators fuel costs, the startup costs of thermal units and the NG costs calculated at each gas well. The optimization procedure is

Energy Carrier Networks: Interactions and Integrated Operational Planning

123

subject to the following constraints: (a) Electric power balance at each node; (b) Hydraulic balance at each water reservoir; (c) NG balance at each node and gas storage; (d) Initial and final water and gas volumes at reservoirs; (e) Electric power generation limits; (f) Maximum electric power flow through lines; (g) NG wells injection limits; (h) Pipelines maximum transport capacity; (i) Bounds on storage and turbined water volumes; (j) Bounds on storage and outflow gas volumes; (k) Minimum up and down time of thermal units; (l) Minimum spinning reserve requirement. To solve the integrated electricity-gas optimal short-term planning problem an approach based on Dual Decomposition, Lagrangian Relaxation, and Dynamic Programming is employed. More recently, in [26] and [27], the NG network model is included in the electric power security-constrained unit commitment problem. While in [26] the NG flows are calculated through a nodal gas balance model, the steady-state physical laws (pressure differences) that govern NG flows are modeled in [27]. In both approaches, local NG storages at each NGFPP are considered. Particular and detailed modeling of fuel switching capabilities is described in [26]. A multi-period combined electricity and NG optimization problem is presented in [28]. The modeling in this approach takes into account not only NG storages facilities, but also the NG contained in the NG network, so-called line pack.

1.3.3

Single Period - Snapshot

Reference [29] presents a combined NG and electricity optimal power flow. The authors deal with the fundamental modeling of NG network, i.e., the steady-state nonlinear flow equations and detailed gas consumption functions in compressor stations. A complete formulation of the NG load flow problem and its similarities with power flows are shown in detail. AC power flow modeling is applied to determine power flows in the electricity network. The objective function is formulated in terms of social welfare. While generation costs due to non-gas electrical plants and gas supply costs account for the total costs, benefit to electrical consumers and benefit to gas consumer, except NGFPPs, account for the total benefits. Reference [30] also deals with the integrated NG and electricity optimal power flow. Nonlinear steady-state pipelines flows and compression station are modeled. However, the gas consumption in compressor stations is not considered. The objective function in this approach is to minimize the sum of generation cost due to non-gas electrical plants and cost of gas supply. References [31] and [32] present a model to compute the maximum amount of electric power that can be supplied by NGFPPs, subject to NG systems constraints. Nonlinear steady-state pipelines flows and the effect of compressor stations to enlarge the transmission capacity are included in the gas network modeling. Like in [30], the amount of gas consumed in the compressor stations is neglected. A comprehensive and generalized optimal power flow of multiple energy carriers is presented in [33]. This paper presents an approach for combined optimization of coupled power flows of different energy infrastructures such as electricity,

124

R. Rubio-Barros et al.

gas, and district heating systems. A steady-state power flow model is presented that includes conversion and transmission of an arbitrary number of energy carriers. The couplings between the different infrastructures are explicitly taken into account based on the new concept of energy hubs. With this model, combined economic dispatch and optimal power flow problems are stated covering transmission and conversion of energy. Additionally, the optimality conditions for multiple energy carriers’ dispatch are derived, and the approach is compared against the standard method used for electrical power systems. Reference [34] is an extension of [33], and it also considers hydrogen as another energy carrier. The impacts of NG transport cost in the electric power economic dispatch problem are presented in [35]. The NG flows are modeled through the steadystate nonlinear equations and transport cost is defined as the sum of NG consumption in compression stations. In [36], the decoupled and the combined approach for the optimal dispatch of electric power and NG systems are compared. It is shown that a higher economic efficiency is achieved if both energy systems are considered in an integrated manner. Reference [37] presents an extensive analysis of the coordinating parameters, which are the reasons for the inefficiencies in the decoupled approach.

1.4

Market Issues

During the last two decades, market mechanisms have been implemented in some energy sectors to provide the essential coordination required for the efficient planning and operation of these sectors. Under this scheme, the optimal expansion and operational planning is mainly achieved by means of the decentralized decision that are taken by the companies and organizations involved in these markets. In particular, electricity and gas sectors have been liberalized to a certain extent in many countries, introducing competition at varying degrees and at various levels of the value chain. Essentially, this restructuring has been accomplished by unbundling the different segments of the industries. In the electricity sector, the production segment (generation) was separated from the service segments (transmission and distribution). In the same way, the natural gas sector was split up into a production segment (upstream) and pipeline network services (midstream and downstream). One of the major features to ensure competition in the production segment is that transmission and distribution companies provide open access on the transmission network to other market participants for energy delivery that has permitted producers to sell their product (electricity, NG) directly to end users and marketers. Different types of markets have been established, thereby allowing the interaction between the production sectors (supply) and the demand at different time frames, ranging from long-term contracts to hourly spot markets. Since a significant share of total NG consumption is used to produce electricity, both energy carrier market prices are linked. Then, the NGFPPs play a key role in the electricity and gas price dynamic because they are the market participants that

Energy Carrier Networks: Interactions and Integrated Operational Planning

125

allow the arbitrage between the two commodities. Liberalized markets for both commodities promote the arbitrage and, therefore, contribute to price convergence. The increasing links between NG and electricity also offer both a threat and an opportunity regarding energy supply security. Flexible facilities, such as energy storage (e.g., NG storage, water reservoirs) and fuels switching (in NGFPPs or steam power plants) are important resources to assure the gas and electricity supply security and to reduce prices volatility. Additionally, efficient gas and electricity markets tend to reduce gas demand as prices increase, thus saving gas at times of high demand or low supply. In liberalized electricity markets, experiences show that one of the most powerful instruments for consumers to oppose the market power of suppliers is the presence of a well-functioning, transparent and liquid wholesale market. It is likely that a liquid and competitive wholesale market for NG is also a powerful tool to counterbalance potential upstream market power in gas. There are numerous policy challenges in establishing well-functioning gas and electricity markets to ensure affordable and reliable energy supply. The growth of world oil prices has produced also an increase or readjustment of NG price. This correlation is mainly because both fuels are substitutes of each other especially in the electricity sector. Because of that, hardly we could talk about a disengagement of these two fuel prices. In contrast, we could consider gas prices as regional prices, due to the lack of a unified world NG market, but a partitioned market in which prices are subject to regional supplies and demands. However, the trend is that NG will be a global commodity because of the development of LNG technology and its associated markets. The economic theory shows that in a competitive market, like a mature NG market (U.S.A., England), the price maker is defined by short-term prices (spot price on Henry Hub or National Balancing Point) or by standard quotations in a Stock Exchange (NYMEX, IPE). This price reflects instantaneous match of supply and demand. However, in NG markets still a significant part of the volumes is traded through long-term supply or demand contracts with indexed prices over the time and penalties in any case of lack (called deliver-or-pay or take-or-pay contracts). In NG monopolies, NG prices are calculated deducting the transmission and distribution costs from the equivalent NG price in the final convergent energy market (usually the electricity market). For NGFPPs, active and liquid electricity and NG markets represent a great opportunity to increase their revenues taking advantages of all arbitrage possibilities. NGFPPs can now purchase NG with great flexibility, through bilateral contracts or through the spot market. On the other hand, the wholesale electricity market price is an important part in the decision-making process for NGFPPs. When the market implied marginal heat rate (which is the equivalent heat-rate calculated using the clearing price for electricity divided by the prevailing natural gas price) is lower than the marginal production heat rate of the NGFPP, the NGFPP might want to purchase power instead of generating it itself and sell the natural gas in the spot market [15]. Another way of looking at the same problem is through the so-called spark spread, which is defined as the difference, at a particular location and time

126

R. Rubio-Barros et al.

between the price of one MWh of electricity and the fuel cost of generating that MWh. As a result, a positive spark spread indicates the power generator should generate electricity rather than resell the previously contracted gas in the spot market. Other service that can be provided by NGFPPs is called tolling, where a power generator receives NG from a beneficiary and delivers electric power to the same beneficiary in return for a service fee.

1.5

Outline of the Chapter

The introduction section presents the context in which the interactions among energy carrier networks are analyzed. An overview of how energy systems are currently planned and operated is presented. The existing interactions between electric power and NG systems and their prospect for the coming decade are explained. A complete review of the state of the art in the integrated operational planning of energy carrier systems discusses about the proposed approaches to deal with these increasing interactions. Some issues related to the introduction of market mechanisms in electric power and NG sectors are described, along with the role of markets in the provision of coordination for the efficient operation and planning. Section 2 describes those energy carrier systems which are inherently associated with dedicated networks, i.e., electric power systems, NG systems and hydrological networks. General aspects, similarities and differences between electric power and NG systems are discussed in detail. The modeling of electric power and NG flows in their respective networks are extensively described and analyzed. The characteristics of the hydrological networks are also presented. In Sect. 3, the integrated operational planning problem of multiple energy carriers systems is comprehensively described and formulated. This huge problem is tackled by means of a hierarchical sequential optimization scheme, ranging from the long/medium-term to the short-term time horizons. The single period economic dispatch of electric power and NG systems is thoroughly discussed. Two case studies are presented to demonstrate the advantages of the integrated operational planning over the decoupled approach. It is important to point out the integrated operational planning presented throughout this chapter takes into account a centralized perspective of the issue. Consequently, the different type of markets (spot, forward, futures) and different forms of trading (bilateral, multilateral, pool) developed within electricity and NG markets are not modeled.

2 Characterization of Energy Carrier Networks Electricity and natural gas are energy carriers, i.e., a substance or phenomenon that can be used to produce mechanical work or heat or to operate chemical or physical processes [38]. While natural gas is a primary energy because exists in a naturally

Energy Carrier Networks: Interactions and Integrated Operational Planning

127

occurring form and has not undergone any technical transformation, electrical energy is a secondary energy, which is the result of the conversion of primary energy sources. Some energy carriers are inherently associated with dedicated networks. Then, an energy carrier network is a set of elements that allows the transport of certain energy between the production/conversion points to the consumption points. Energy flows in these networks complies with physical laws, such as the relation between voltage and current in electricity, pressures and flows in gaseous substances. These relationships are particular for each energy system. Water reservoirs and rivers constitute hydrological networks which are also other type of energy carrier network. This chapter is focused in transmission networks, which cover extensive areas, therefore, electricity, NG and hydrological energy carrier networks are studied. Some general aspects and the mathematical formulation of energy flows in these networks are presented in the following subsections.

2.1

General Aspects

Even though natural gas and electricity are different energy carriers, they have a similar organization structure (Table 1). Like in the electricity production segment, a great diversity of technical characteristics can be found among gas suppliers. Gas wells (commonly located at sites far from load centers) and LNG regasification terminals (harbor locations) have capacity and operating constraints. Natural gas transmission and distribution networks provide the same services as their electricity counterparts. Transmission pipelines undertake the responsibility of transporting natural gas from producers to local distribution companies or directly to large consumers. Distribution networks generally provide the final link in the natural gas delivery chain, taking natural gas from city gate stations and other gas supply sources to large and small customers. Generally, there is a priority scheme for the supply of natural gas in which residential and commercial customers take precedence over large consumers (NGFPPs included).

Table 1 Natural gas and electricity sector organization [9] Segments Natural gas sector Production (suppliers) Gas wells LNG regasification terminals Transmission High pressure network Distribution Medium/low pressure network Consumption Small consumers (commercial and residential customers) Large consumers (NGFPPs, industries, liquefaction trains)

Electricity sector Electrical power plants (coal, nuclear, gas, hydro) High voltage network Medium/low voltage network Small consumers Large consumers

128

R. Rubio-Barros et al.

NG Supply C

Thermal Generation (coal, oil, nuclear)

Pressure Regulator

Hydro Generation

NG-fired Unit Distribution Pipelines

NG Demand

C

Compressor

C

Transmission lines

Transmission Pipelines

Electricity demand

C

NG Storage

NG Demand LNG Regas Terminal

NG-fired Unit

Wind Generation Electricity demand

Fig. 1 Natural gas and electricity systems

Delivering the natural gas from gas wellhead to end customers entails pipelines, storage facilities, compressors and valves. Figure 1 illustrates NG and electricity systems schematically. Gas flow paths from a gas wellhead to gas-fired units and other gas demands are shown. The electric power system includes a 3-bus network representing the electric transmission system, hydroelectric power plants, non-gas electrical power plants and electricity demands. NG is transported from gas producers to customers at various locations. Three basic types of entities are considered for the modeling of NG transmission network: pipelines (branches), compression stations (branches) and interconnections points (nodes). Compressors are placed at specific locations in order to increase the pressures in these points and thus, increase the transportation capacity of the gas network. Typically, compressors are installed not only at gas wellheads (or regasification terminals), but also at many other transmission system locations. However, the operating pressures are constrained by the maximum pressure allowed in pipelines and the minimum pressure required at gate stations. Therefore, the transmission capacity of a gas pipeline is limited. Valves are protective and control devices whose functions are similar to breakers, fuses, and switches in electric power systems. Isolating valves are used to interrupt the flow and shut-off a network section. Pressure relief valves can prevent equipment damage caused by excessive pressure. Pressure regulators can vary the gas flow through a pipeline and maintain a preset outlet pressure. Unlike electricity, which large scale storage is not yet technically or economically feasible, natural gas can be stored for later consumption. There are three major types of NG storage facilities which are different in terms of capacity (working volume) and maximum withdrawal rates:

Energy Carrier Networks: Interactions and Integrated Operational Planning

129

(a) Underground storages: 1. Depleted gas/oil fields and aquifers: long/medium-term storage facilities that usually have high working volumes but limited withdrawal rates. 2. Salt caverns: short-term storage facilities that offer a high withdrawal rates but much lower volumes of gas than depleted fields. (b) LNG tanks: buffers located at regasification terminals that provide large storage volumes combined with high withdrawal rates. Other small LNG storages, called peak shaving units, are located at demand centers and also provide high withdrawal rates. (c) Pipelines themselves: the amount of gas contained in the pipes is called “linepack” and can be controlled raising and lowering the pressure.

2.2

Electric Power Flows

In electrical networks, the steady-state electric power flows are governed by Ohm’s and Kirchhoff’s laws. These laws can be expressed by means of nodal power balances and line power flows. The complex power balance at node j in an electrical AC network can be stated as X Sj
Sjn ¼ 0 (1) n2N j

where Sj is the net complex power injected at node j (sum of the production of local generators less the local demand), N j is the set of nodes connected to node j, and Sjn is the outgoing complex power from node j to any other node n. The outgoing complex power Sjk from node j to k is a function of the complex nodal voltages Uj and Uk and the parameters of the line [39]  2 U j  Uj U  Sjk ¼
k (2)  Znjk Zjk where   Yjk
1 1 þ (3) Znjk ¼ Zjk 2 where Zjk is the series impedances and Yjk is the shunt admittance of the Pequivalent of the line m (with Yjk/2 at each line end). The admittance Yjk models the capacitive effect in electric power lines and the small losses due to corona effect [39]. Equations (1) and (2) accounts for the so-called AC power flow model which is the most accurate representation of electric power flows. Figure 2 shows a scheme of this model. Active and reactive power flows can be calculated as the real and imaginary parts of the complex power (2), respectively. In multiple energy carrier flow

130 Fig. 2 Transmission line and AC power flow model

R. Rubio-Barros et al. Sjn

j

k

Sjk

Sj

m

Skn

Skj

Sk

analysis, the modeling is focus on the active power flows because these are the real amounts of the power transferred from one node to another. The active power flow Pjk can be expressed as:       o 1 n  2 Pjk ¼  2 Uj  Rjk
Uj  jUk j Rjk cos yj
yk
Xjk sin yj
yk (4) Zjk  where Rjk and Xjk are the resistance and reactance the line m (Zjk ¼ Rjk + j Xjk), and yj and yk are the phase angles of voltages Uj and Uk. Other factor that should be analyzed is the efficiency of transporting energy in the form of electricity. Therefore, losses are usually taken into account in energy systems studies, particularly when extensive systems are considered. The series losses in the line m, which neglect shunt losses associated to Yjk, can be stated as:

   i 1 h  2 Ljk ¼ Pjk þ Pkj ¼  2 Uj  þ jUk j2 Rjk
2Uj  jUk jRjk cos yj
yk (5) Zjk  Even though the active power flow Pjk through a line is a nonlinear function of the complex voltages at the line ends, a simplified model can be reasonably used to represent electric power flows. This is the so-called DC or MW-only power flow, which can be obtained from (4) via a sequence of approximations [40–43]; thus: • Losses are neglected, Ljk ¼ 0, Pjk ¼ Pkj • Voltages at all nodes are assumed equal to 1 per unit, Ui ¼ 1 p.u. 8i • Phase angle differences corresponding to adjacent buses are small, cos(yj – yk)  1 and sin(yj – yk)  yj – yk • The reactance Xjk is much greater than the resistance Rjk for all the lines m, reactance Xjk > > Rjk 8m, thus Xjk//(R2jk + X2jk)  1/Xjk Under these approximations, the active power flow (4) from node j to k in p.u. can be rewritten as:  1  (6) yj
yk Fm ¼ Xjk This is the classical dc model which accuracy and applications for different purposes are extensively discussed in [42]. Equation (6) can be extended to all system’s branches as follows: F ¼
BD AT u

(7)

Energy Carrier Networks: Interactions and Integrated Operational Planning

131

where F is the vector of lines active power flows, BD is the diagonal matrix whose elements are 1/Xjk, A is the node-branch incidence matrix whose Ajm element is +1 if line m enters node j, -1 if line m leaves node j, or 0 if line m is not connected to node j and u is the vector of phase angles. Another implicit approximation assumed in the extension of (6)–(7) is that the relation Rjk/Xjk is a constant for all the lines in the system. On the other hand, (1) can also be expressed in its matrix form as P ¼
A F

(8)

in which P is the vector of net active power injections. The combination of (7) and (8) leads to 0

P ¼ A BD AT u ¼ Bbus u

(9)

where matrix B’bus has the same structure (sparse and symmetric) as the systems or nodal admittance matrix, but its values being computed solely in terms of branch reactances. Therefore, in order to calculate the active power flows through the lines, the procedure consist in solving (9) to obtain u since P is known, and then use u in (7) to obtain F. Alternatively, (7) and (9) can be combined, leading to a linear relationship between active power injections and active power flows h 0 i
1  
1 F ¼
BD AT A BD AT P ¼
BD AT Bbus P ¼ HP

(10)

where H is the so-called sensitivity matrix. In the implementation of the model described by (7, 8, 9, 10), it is important bear in mind that in electric power systems, not all the power injections are independent variables. One of the injections is a dependent variable since its value equals the sum of all injections to zero. The node where this injection is located is called slack node. In other words, the injection in the slack node provides the balance between the total production of generators and the total demand (and losses if they are modeled). The phase angle at slack node is used as reference; therefore, its value is set as a parameter. The elements Hmj of matrix H are called sensitivity factors or power transfer distribution factors (PTDF) [40] and [42]. Each factor relates the change in the power flow in the line m to an increase in the injection at node j, assuming the slack injection compensates for the injection in any node. The calculation of matrix H requires to make B’bus not nonsingular by replacing the row and column corresponding to the selected slack node with null vectors. Although the classical DC model is lossless, actual power losses can be approximated and introduced to obtain a more general DC modeling. Applying the same approximations used to derive the DC model except the neglect of losses, obviously, (5) in per unit (p.u.) can be rewritten as: Lm ¼ Rjk F2m

(11)

132

R. Rubio-Barros et al.

Fig. 3 Transmission line and DC power flow model with losses

Pjn

j

k

Pkn

Fm m Pj

½Lm

½Lm

Pk

The simplest manner to allocate theses losses is to assign all of them to the slack node, however, this leads to cumulative error in the power flows in the vicinity of the slack node. In order to preserve the system generation-load-loss MW balance, the losses have to be allocated among all the nodes of the network modeling them as equivalent injections. One way to do this is to assign half of the series losses in the line to each line ends. Other more accurate ways to distribute losses are discussed in [42]. Thus, the vector of nodal active power injection includes: P ¼ PG
PD
P L

(12)

where PG, PD and PL are the vectors of generating units’ productions, demands and losses, respectively. Each element PLj of PL is calculated as: PLj ¼

X 1 Lm 2 m2M

(13)

j

where Mj is the set of lines connected to node j. The shunt losses can also be considered and added in (11). Figure 3 shows a representation of DC power flow with losses. Finally, if it is required to model different sending and receiving MW flows in each line, it can be considered that Fij ¼ Fm þ Lm/2 and –Fji ¼ Fm
Lm/2 according to the flow references in Figs. 2 and 3.

2.3

Natural Gas Flows in Pipeline Networks

NG flows in pipeline networks can also be described by means of nodal flow balances and pipeline flows. In analogous way as in (1), the flow balance at node j can be formulated as: Wj


X

Qgjn ¼ 0

(14)

n2N j

where Wj is the net volume flow injected at node j, N j is the set of nodes connected to node j, and Qgjn is the outgoing NG flow from node j to any other node n.

Energy Carrier Networks: Interactions and Integrated Operational Planning

133

The steady-state isothermal NG flow, Qgm, through a horizontal pipeline m is a function of the upstream pressure sj and downstream pressure sk; the NG properties and the pipeline characteristic represented by the constant Km [44]. Qgjk ¼ Qgm ¼Km dm

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   dm sj 2
sk 2

(15)

Where ( dm ¼

þ1;

si

s j rs k


1;

si

sj <sk

rffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2 Rair Tst D5m Km ¼ 64 sst fm T G Z L m

(16)

(17)

where Rair is the ideal gas constant of dry air, Tst and sst are the standard or normal conditions of temperature and pressure, Dm the is the inside diameter of the pipe m, fm is the dimensionless friction factor of the pipe m, T is the average gas flowing temperature, G the is gas gravity relative to air, Z is the dimensionless gas compressibility factor and Lm is the length of the pipe m. An additional term can be introduced to (15) if a significant elevation difference exists between the upstream and downstream locations  of the pipe [44] and [45]. The parameter dm is the sign of the difference sj
sk which defines the actual direction of the NG flow. This auxiliary parameter must be included to avoid an imaginary number as result of (15) when the conventional defined as downstream pressure sk is higher than the upstream pressure sj. Several flow equations are in use in the NG industry such as Weymouth and Panhandle equations, and all of them are modifications of (15), which is called the general flow equation for steady-state gas flow. In fact, (15) denotes that the pipeline transmission capacity is constrained to the allowed maximum and minimum operating pressures in the different sections of the NG network. During transportation of NG in pipelines, the gas flow loses a part of its initial energy due to frictional resistance which results in a loss of pressure. To compensate these pressure losses and maximize the pipeline transport capacity, compressor stations are installed in different network locations. Compressor stations consist of compressors, which are arranged in series and parallel configurations depending on the requirements. Several types of compressor can be used, being centrifugal compressors the most commonly used. Many types of prime movers (e.g. electric motors, gas turbines, gas engines, steam turbines) have found application as compressor drivers. However, an amount of NG is withdrawn from the network, and eventually transformed on site into electricity or heat (steam), to provide the energy needed for the prime movers. Despite electric motors typically drive compressors and main electric power supply is available in the compression station

134

R. Rubio-Barros et al.

Fig. 4 Model of NG transmission link m with a compressor station Cm and a pipeline Pm

u su

Qgsj

j Qgjk Cm

Lgm

sj

Qgm k Pm

sk

area, the electric power is not extracted from electricity network because reliability reasons. Thus, the NG consumed in a compression station Lgm can be considered as an additional NG flow into the pipeline section, as shown in Fig. 4. This amount of NG basically depends on the pressure added to the fluid and the volume flow rate through the compressor. In extensive NG networks, the total NG consumed in compressors stations can reach up to 6% of the total NG transported through the network. The value of Lgm can be approximated as follows [33]:   Lgm ¼ KCm  Qgm  sj
su

(18)

where KCm is a constant characterizing the compression station; su and sj are the suction and discharge pressures, respectively; Qgm ¼ Qgjk ¼ Qgsj is the flow through the compressor. It is important to point out that (18) represents an equivalent and approximate model of an entire compression station. More advanced compression station modeling considering detailed compressor and prime movers characteristics are presented in [29, 44] and [45]. As shown in Fig. 4, compressor stations, like pipelines, can be modeled as branches in NG networks. In order to make NG flows dimensionally suitable for comparison with any other energy carrier, they should be converted to power using the NG higher heating value. In the same way, a given volume of NG can be converted to an amount of energy.

2.4

Energy Flows in Hydrological Networks

Hydrological networks consist in a complex system of watersheds with different types of rivers (rain, snowmelt or combined rain-snowmelt) concatenated with an arrangement of multiple reservoirs. The regulating capacities of the reservoirs are diverse, and they may range from several tens of minutes to years. As it was introduced before, hydro system can be modeled as an energy carrier similar to that presented for the NG and electrical systems as a water network, taking into account certain characteristics that stand out from other networks, among which we highlight two of them: the watersheds are formed by radial and unidirectional water flows, and comply with laws of continuity considering the delay of water use, which can reach several minutes to few days. It may be mentioned that the law accomplished by the water network is the law of gravity, since the water stored is equal to stored potential energy, which will be converted into electrical energy by the hydro plant production coefficient, according the requirement and resource availability. The water flowing downstream the river represents the kinetic energy,

Energy Carrier Networks: Interactions and Integrated Operational Planning

135

which is lost in transit as it loses height until reaching the next reservoir below. Because of that, we can relate water flows and stored water volumes as power flows and stored energy, respectively using the potential energy equation applied to the volume of water stored. In general, hydrological networks have certain characteristics that must be considered, depending on the study to be carried out. In the medium and long term planning, the main considerations to take into account are: (a) Temporal forecasting of lateral streamflow, (b) Regulation capacity of reservoirs (annual, seasonal, monthly, weekly, and hourly), (c) Priority use of water resources other than energy use (ecological minimum outflows, agricultural irrigation requirements, etc.), and (d) Evaporation and filtration losses. In the short term dispatch, in addition, there are another features become relevant, such as: (a) Time delay of water between coupled reservoirs, (b) Reservoirs with small regulation capacity (weekly, daily, hourly, and several minutes), and (c) Pumped storage facilities. Hydrological networks are modeled by means of an series-parallel arrangement of water reservoirs. Thus, water balances should be introduced for each reservoir at each stage. These sets of equations (19) are called continuity equations, including the water flow contributions of upstream hydro power plants. Vetþ1 ¼ Vet þ Ate T


X i2 I he

Qwti T
Ste T þ

X i2 I hu

Qwti T þ

X u2 E e

Stu T

(19)

Vetþ1 and Vet are state variables that represent the stored volume of the water reservoir e at the end of stage t (beginning of stage t + 1) and beginning of stage t, Ate is the lateral streamflow arriving at the reservoir e, Ste is a slack variable that represents the spilled outflow from reservoir e, I he is the subset of hydroelectric power plants i associated with the water reservoir e, T is the total duration of stage t, I hu is the subset of the upstream hydropower plants and E e is the subset of reservoirs u that spill water into reservoir e.

2.5

Similarities and Differences

Natural gas and electric power systems have a remarkable common feature which is that extensive networks are used to transport the energy carrier from suppliers to customers. Natural gas can also be carried in the form of Liquefied Natural Gas (LNG), being required liquefaction trains, LNG ships and regasification terminals to accomplish with transport and inject the gas into the network. Pipelines transportation is more cost-effective over short and onshore distances, while LNG is typically used in transcontinental carriage [10]. Although some analogies can be drawn between the electricity and NG networks, there are several qualitative differences between both systems. Table 2 shows the key differences between electricity and NG systems.

136

R. Rubio-Barros et al.

Table 2 Key differences between natural gas and electricity systems Characteristic Power electric system Natural gas system Energy type Secondary Primary State variables Voltages Pressures Transmission Joule effect up to 3% Gas consumed in compressors up to 6% losses (extensive networks) Transmission Meshing scheme (improves Radial scheme (low failure rates, high system levels of reliability and construction costs) security of supply) Flow modelling Steady-state can be assumed for Transient-state is required for operational operational simulations simulation (time steps shorter than several hours) Supply hierarchy Not required in normal Frequently required in normal operation operation state state. Usually NGFPPs and industries have lower priority Individual flow Currently neither economic nor By means of compressors and pressure controllability practical (FACTS) regulators Storage facilities Not yet technically or Widely used in Europe and USA, not commercially feasible common in Latin America

While steady-state operation of electric power systems requires a constant balance between supply and demand; gas storage facilities are typically used to load balancing at any time, hourly, daily, weekly, or seasonally, keeping a constant supply. Additionally, large underground storages perform, principally, a supply security function (strategic stock). Other important difference between natural gas and electricity systems is that electricity moves at speed of light, while natural gas travels through the transmission network at up to maximum speed of 30 km/h [14, 44]. This fact implies that steady-state flows assumption cannot be assumed strictly for multi-period simulations with time steps shorter than several hours. Rigorously, gas flows simulation requires pipeline distributed-parameters and transient models, [46]. However, many simplified models have been developed to gas flows simulation [46] and transmission system optimization [47, 48].

3 Integrated Operational Planning of Multiple Energy Carriers Systems This section presents the mathematical formulation of the economic dispatch (ED) of electricity an NG systems, from a centralized point of view. The formulation considers a complete horizon, i.e. from long/medium to short term planning, including also the single time period ED or simplified optimal power flow (OPF). At first time these problems are considered, as state-of-the-art report, where each energy system is optimized independently, using at most coordinating parameters like NG

Energy Carrier Networks: Interactions and Integrated Operational Planning

137

prices or quantities for electricity generation. These coordinating parameters allows to divide the problem into two decoupled and independent subproblems, thus reducing the great complexity (dimensionality, nonlinearity, non-convexity) but losing economic synergy. At second time this work contributes with combined ED of electric power and NG systems where it is possible to see the benefits of the planning integration and its contribution to link electricity and NG prices at different nodes of both energies carriers. The methodology is based on the typical hierarchical sequential planning scheme where the results of one stage are the inputs to the following one and so on; always going from longer to shorter time horizons. The coordinating parameters between stages ensure the consistency of the overall optimization process. These parameters are the water values and also the NG values and/or the quantities of stored water and NG to be used in a given time frame. This scheme of optimizations is essential when one or both energy carriers have water and NG reservoirs with different regulating capacities considering also the line-pack in the NG network. Other important issues, that must be considered, are the uncertainties involved in the problem: lateral water inflows (rivers), electricity and NG demands, and the operationfail behavior of generation units. The hierarchical scheme allows dividing the complex problem into a natural and logical way to organize energy systems planning and operation; and thus, the optimization methodology follows the same scheme. This structure, abstractly defined here, is more specifically described in following sections.

3.1

Economic Dispatch of Electric Power Systems

The ED problem consists basically in determining the power output of each generating unit to supply the forecasted demand at minimum cost. This can be formulated as an optimization problem with an objective function stated as: ( ) X X X X   Ci ðpgi Þ þ CGi ðpgni Þ þ CHi ðpghi Þ þ CSE psj min Co ¼ (20) i

i

i

j

where Ci(pgi) is the production cost of a thermal generating unit i which do not use NG as fuel (i.e., coal, nuclear, fuel oil); CGi(pgni) is the production cost of the NGFPP i; CHi(pghi) is the opportunity cost of production of a reservoir-associated hydropower plant i and CSE(psj) is the shortage cost related to a fictitious unit which output power psj equals the load shedding at node j. The opportunity costs of hydropower plants which are associated to a reservoir are given values in the ED, and they can be calculated from the water values obtained in the medium-term operational planning. The output powers of run-of river power plants are also considered as given in the ED, and their operating costs usually neglected. The cost minimization stated in (20) is subject to the following constraints: P i2Gj

pgi þ

P i2Gj

pgni þ

P i2Gj

pghi þ psj þ

P m

Ajm Fm ¼ Dej þ PLj 8j

(21)

138

R. Rubio-Barros et al.


Fmax m b

X

2

0

4Hmj @

j

X

13 ðpgi þ pgni þ pghi Þ þ psj
Dej
PLj A5

i2Gj

(22)

b Fmax 8m m max Pmin i bpgi ; pgni ; pghi bPi

8i

(23)

where (21) is the extended expression of (8) denoting the power balance at each node j. The subset Gj includes all the generating units connected to node j, thus, the first four terms of (21) accounts for the positive injection at this node; Dej is the electricity demand at node j. The maximum transmission capacity limits for each line m are set in (22) in which the power flows are calculated according to (10) and (12). The generation limits are established in (23). Usually, in the ED, the minimum output power of generating units is set to zero if it is not known which units are committed in the analyzed time period. It is important to note that although Hmj depend on the choice of slack node, the result of the optimization problem is indifferent to this choice. Some thermal generating units are capable of use a mix of fuels. If it assumed that any fuel proportion can be burned, the fuel mixing feature can be modeled as: X

pgi;c þ pgni bPmax i

(24)

c

where pgi,c is the partial output power produced by unit i with the fuel c different from NG. Each generating unit that is able to fuel mixing is modeled by as many output power variables as fuels the unit can burn. But the sum of all these variables must be less than the unit’s maximum output power. Equation (24) presents the simplest way of modeling fuel mixing in a generating unit. More complicated formulations, like fuel switching, would require integer variables.

3.2

Economic Dispatch of Natural Gas Systems

The ED of NG systems, which is a simplified steady-state optimal NG flow problem, can be formulated in analogous way to the ED of electric power systems. The objective function consists in the minimization of NG supply costs Cg and NG shortage costs CSG: ( min Co ¼

X

  X   Cg wg þ CSG wsj

g

j

) (25)

Energy Carrier Networks: Interactions and Integrated Operational Planning

139

where wg is the NG injection in network of the supplier g and wsj is the fictitious NG injection which value equals the NG load shedding at node j. This costs minimization is constrained by the following equations P

wg þ wsj þ

g2W j

P m2I j

Qgm


P

Qgm


m2Oj

Wgmin bwg bWgmax max Smin j bs j bS j

P

Lgm ¼ Dgj

m2Cj

8g 8j

su
sj b0 8s 2 ðN Cm \ N u Þ; 8j 2 ðN Cm \ N d Þ

8j

(26) (27) (28) (29)

In addition to these constraints, Eqs. (15), (16) and (18) must be taken into account for all pipelines and for all compression stations, respectively. The NG balance at each node j is stated in (26); where Dgj is the gas demand, W j is the subset of gas supplier injecting in node j; I j and Oj are the subsets of pipelines which flows are incoming and outgoing node j, respectively; and Cj is the subset of compression stations whose suction side is connected to node j. The injection limits for each gas supplier are set in (27). The maximum and minimum pressures allowed at each node are established in (28). These pressure limits can vary throughout the NG network according to pipelines characteristics and security operating rules. The constraint (28) also applies to all the nodes in the NG network, including those nodes which are connected to the suction side of a compression station. The gas flow irreversibility at each compression station, which is a typical operating constraint, is stated in (29); where N u and N d are the subsets of nodes associated to the suction and discharge of any compressor station, respectively; and N cm is the subset of nodes associated to the compression station Cm. In this problem, the decision variables are the suppliers’ injections and the pressure at each node, which determine the NG flows.

3.3 3.3.1

Integrated Electricity and Natural Gas Economic Dispatch Energy Hubs

For the simultaneous modeling and optimization of two or more energy carrier, it is useful to combine different energy carrier nodes in a single entity. This unit, called energy hub, provides the basic features in- and output, conversion, and storage of the converging energy carriers [33]. Thus, energy hubs represent a generalization of a network node, leading to an integrated view of energy carrier systems. In an integrated electricity and NG ED, NGFPPs are the links between both systems. On the other hand, since the ED problem considers a single time period, the hydropower plants are modeled as thermal generating units, whose equivalent

140

R. Rubio-Barros et al. Wind Fossil fuels Nuclear

Hydro

Energy hub

HR2

HRi

NGFPPi

HR1

NGFPP2

Sgj

NGFPP1

Sej

Electric power network

Dej

Natural gas network

Dgj NG wells

Fig. 5 Energy hub as a link between electric power and NG systems

operating costs are the opportunity costs of production, as it is described in (20). Thus, the hydrological networks associated to the hydropower plants are not represented explicitly in the ED. Therefore, in the integrated electricity and NG ED, the energy hubs only include the NGFPPs as conversion units. Figure 5 shows a schematic representation of an energy hub and their links with the electric power and NG systems.

3.3.2

Problem Formulation

From a centralized perspective, the aim of the integrated ED of electric power and NG systems is the minimization of the total operating and shortage costs: X X X 8   9 Co ¼ Ci ðpgi Þ þ CHi ðpghi Þ þ CSE psj þ > > > > = < i i j X   X min   > Cg wg þ CSG wsj > > > ; : g

(30)

j

The integrated total operating cost of both systems is stated as the sum of each energy carrier system cost (20) and (25), less the production costs of NGFPPs. The formulation of the objective function (30) assumes that only the fuel costs account for the production costs of thermal power plants, neglecting other variable production costs (e.g., operation and maintenance costs). If all CGi(pgni) were also considered, this would imply double counting these costs. The cost minimization expressed in (30) is subject to all the constraints described for the ED of electric power and NG systems, i.e., (15), (16), (18), (21), (22), (25), (26), (27), (28) and (29). The interactions between the both systems can be represented by means of two additional constraints Sgj


X i2Gj

HRi ðpgni Þ ¼ Dgj

(31)

Energy Carrier Networks: Interactions and Integrated Operational Planning

Sej þ

X

pgni ¼ Dej

141

(32)

i2Gj

where HRi is the heat rate function or input–output curve of NGFPP i according to [40]. Equations (31) and (32) state the NG and electric power balances at each energy hub j, respectively. In (31), HRi(pgni), which is the amount of NG burned in the NGFPP i, is added as a NG demand, while in (32), the pgni produced by the same NGFPP is an injection in the electric power system. It is important to point out that in the integrated ED of electric power and NG systems, Dgj includes only NG demanded for uses different to electric power generation.

3.4

Coordinating Parameters

Nowadays, as it was described before, the operational planning of electric power and NG systems is carried out in decoupled manner. This does not mean that both systems are totally independent. In fact, the existing interactions are modeled by means of fixed coordinating parameters. Typically, three types of parameters can be identified. One of them, is the NG prices considered in the production cost functions, CGi(pgni), of each NGFPP. Another parameter is the total NG availability for electricity generation (NGA), which is particularly relevant in NG-intensive systems. This parameter can be introduced into the electric power dispatch through the following constraint: X

HRi ðpgni ÞbNGA

(33)

i

The third type of parameter, which is needed to calculate the NG dispatch, is the NG consumption at each NGFPP. Thus, the decoupled approach consists of two stages. First, an ED of the electric power system is performed, being NG consumption at each NGFPP a by-product of this procedure. Then, the ED of the NG systems can be carried out. However, fixed coordinating parameters cannot be suitable to represent the existing interactions, especially if the shares of NGFPPs in the electricity supply and in the total NG consumption are significant. The following conditions can occur: 1. The total NG supply is not sufficient to meet the total NG demand, including the NGFPPs’ demands. The NG supply to NGFPPs can be curtailed before than other demands, since NGFPPs usually have lower priority of supply. 2. The limited transmission capacity in the NG network can imply that same situation described in (1) occurs at a specific node. 3. The fixed NG prices, which determine the NGFPPs production costs, cannot match with the NG marginal costs at nodes where NGFPPs are placed. These

142

R. Rubio-Barros et al.

marginal costs depend on NG consumption in the compressor stations (NG network losses) and binding pipelines (transmission) capacity constraints. The conditions described in (1) and (2) require a re-dispatch of the electric power considering an adjustment in the NGA indicated in (33). Additionally, it is possible to consider NG availabilities at each node, which means that NG flows should be estimated. Therefore, both dispatch models must be run iteratively. The convergence of this procedure is slow and may be hard to reach when NG consumption in NGFPPs is a significant share of the total NG required. The fixed NG prices above-mentioned in (3) can correspond to the NG tariff or the prices established in supply contracts signed between NGFPPs and NG marketers. If nodal spot prices of NG were set at the NG short-run marginal costs at each node, then there will be differences between these prices and the fixed values in (3). These differences constitute benefits (or losses) redistribution among the involved market agents (NGFPP, NG marketer, NG Disco). However, usually, the fixed NG prices do not match the NG marginal cost. Therefore, the dispatch of NGFPPs is based on wrong economic signals, which leads to economic inefficiencies (deadweight loss), i.e., higher electricity and NG operating costs, if price inelastic demands are considered. This is based on welfare economics theory [49], which states that a Pareto-efficient outcome situation is achieved in a competitive equilibrium where prices are set by the marginal costs of supply. Reference [29] shows an assessment of the social welfare losses when different fixed NG prices are considered. The authors demonstrate in a case study that these losses are only avoided when fixed NG prices match the NG marginal costs. On the other hand, in an integrated electricity and NG dispatch, the described coordinating parameters are endogenous results of the optimization problem, which ensure that economic efficiency is achieved. However, currently dispatch procedures in electricity and NG sectors handle the interactions between both sectors by means of coordinating parameters, i.e., using the decoupled approach. This has been based on the fact that in the past the interdependencies between both energy carriers were weak. Nevertheless, as it was described before, this situation has changed over the last years and increasing interactions are expected for the future.

3.5

Case Study: Integrated Versus Decoupled Approach

In this section, a simple case study is presented to analyze the effect of coordinating parameters and compare the performance of integrated and decoupled approaches to electric power and NG economic dispatch. 3.5.1

Two-Hub System

Figure 6 depicts a two-hub electric power and NG systems. Both hubs are connected through an electric power line and a pipeline. The hub’s limits have

Energy Carrier Networks: Interactions and Integrated Operational Planning

pg1

143

pg2

F12

ps1

ps2 pgn1

pgn2 De2

De1

HUB 2

HUB 1

w2

ws1

ws2 Qg21

Dg1

sd Dg2

Fig. 6 Two-hub electric power and NG systems

Table 3 Production/conversion infrastructure Component Production/supply cost 2

pg1 pg2 w2 ps1, ps2 ws1, ws2

pgn1 pgn2

a ($/MW h) 0.0037 0.0058 0.0005 – – Heat rate a (MW
1) 0.0013 0.0014

b ($/MWh) 8.5 8.8 4 500 250 b (no-dim.) 1.7 1.85

Production/supply limits Max (MW) Min (MW) 100 0 80 0 500 250 inf. 0 inf. 0 Production/supply limits Max (MW) Min (MW) 60 0 60 0

been extended for a more comprehensive representation that includes all the production/conversion facilities located in the node. However, the modeling is kept as described in Sect. 3.3.2. The basic information about the production/conversion infrastructure is shown in Table 3. The single-period ED of electric power and NG systems considered in this example last 1 hour. NG flows (production rates, demands) are expressed as powers (MW) instead of volumetric rates (m3/s). For this conversion, the NG higher heating value (HHV) has been used. For instance, the Argentinean HHV is 9,300 kcal/m3 (38,937 MJ/m3), which means that 1 MWh of NG is equivalent to 92.46 m3 (1 MWh ¼ 859,845.23 kcal). The pressure limits in the hub 1 are 1.4 p.u. and 0.6 p.u. The maximum and minimum values of the discharge pressure in the compression station (sd) are 1.4 p.u. and 1 p.u., respectively. The reference pressure at hub 2 is 0.85 p.u. The pipeline and compression station constants (K and KC) are 220 MW and 0.12 (dimensionless). Under these conditions, the maximum NG flow, Qg21, is 278.28 MW, while the maximum electric power flow through F12 is 150 MW.

144

R. Rubio-Barros et al.

Table 4 Electric power and NG demands

Demands De1 (MW) De2 (MW) Dg1 (MW) Dg2 (MW)

Summer scenario 120 80 150 100

Winter scenario 120 80 240 160

Summer and winter scenarios describe two different conditions for the electric power and NG dispatch. The demands for those scenarios are shown in Table 4. 3.5.2

Test Cases and Results

To analyze the impacts of the coordinating parameters in ED of electric power and NG systems, two different scenarios must be considered. One of them is called summer scenario, which corresponds with a high NG availability for NGFPPs. In this scenario, NGFPPs dispatch is not constrained by a maximum NGA, and it can be the typical situation during the summer period. In contrast, the other scenario is called winter scenario, and it is characterized by low NG availability for NGFPPs. This means that the supply of NG is sufficient to meet the high priority NG demands (not for electricity generation), and only some remaining availability is possible to be use by NGFPPs. The summer scenario is suitable to assess the impact of fixed NG prices as coordinating parameters. For this purpose, four cases are proposed: • The integrated electricity and NG dispatch (Integrated) • Three cases of the decoupled electricity and NG dispatch considering different NG prices for NGFPPs at hub 1: • NG price at hub 1 5% lower than the NG marginal cost at hub 1 obtained in the integrated dispatch (Decoupled –5%) • NG price at hub 1 5% higher than the NG marginal cost at hub 1 obtained in the integrated dispatch (Decoupled +5%) • NG price at hub 1 10% higher than the NG marginal cost at hub 1 obtained in the integrated dispatch (Decoupled +10%) For each decoupled dispatch, the NG price for electricity generation at node 2 is maintained at the NG marginal cost calculated in the integrated dispatch. The main results are shown in Table 5. In order to ensure consistency in the analysis, the total operation costs are calculated according to (30) for the integrated and decoupled approaches. Otherwise, the costs of the NG used by the NGFPPs would be double-counted. As it was stated before, the origins of the economic inefficiencies in the decoupled dispatches are due to the mismatch between the NG prices considered in ED of the electric power system and actual NG marginal costs obtained in the ED of the NG system. As it can be noted, lower NG prices for electricity generation at hub 1 (5.05 $/MWh to

Energy Carrier Networks: Interactions and Integrated Operational Planning

145

Table 5 Summer scenario main results

Total operating cost ($) NG price hub 1 ($/MWh) NG marginal cost hub 1 ($/ MWh) pgn1 (MW)

Integrated 2,758.57 5.05 5.05

Decoupled
5% 2,767.87 4.80 5.29

Decoupled +5% 2,769.18 5.30 4.47

Decoupled +10% 2,791.69 5.56 4.45

38.11

60

14.59

0.86

4.80 $/MWh) lead to more output power in pgn1 than in the integrated case (38.11 MW to 60 MW) and to higher NG marginal at hub 1 (5.05 $/MWh to 5.29 $/MWh). The converse situation occurs for the +5% and +10% cases. It should be noticed is that the pgn1 ranges from maximum to minimum output power with a
5% to +10% change in the NG price at hub 1, showing major changes in the operating schedule. Only a combined ED of electric power and NG systems can ensure that NG prices for electricity generation equal NG marginal costs at each node, and therefore, the minimum operating cost is achieved. Theoretically, the same results could be obtained using the decoupled approach, i.e., ED of electric power and NG systems run iteratively; however, the convergence of this procedure to a final solution is not guaranteed. The NG marginal cost obtained in each iteration can oscillate in a divergent manner. Thus, the convergence of the decoupled approach is highly dependent on the starting point, i.e., the NG prices and the NGA considered in the ED of the electric power system. The presented case study is an example that the procedure can diverge. If the NG price at hub 1 considered in the first iteration was 4.8 $/MWh (decoupled
5% case), the NG marginal cost at hub 1 obtained from the NG ED would be 5.29 $/MWh. If in the next iteration, 5.29 $/MWh was used as the NG price at hub 1 for the electric power ED, then NG marginal cost at hub 1 obtained from the NG ED would be around 4.47 $/MWh (decoupled +5% case). Therefore, it can be noticed that the procedure do not converge to 5.05 $/ MWh, i.e., the NG marginal cost at hub 1, obtained in the integrated approach. The deadweight loss is the difference between the total costs obtained under the decoupled ED and the integrated ED of electric power and NG systems. These total operating costs differences (over-cost) are shown in the Fig. 7, in which decoupled
2.5% and +2.5% cases are also added. For the decoupled +10% case, the deadweight loss reaches the 1.2%. However, if the production costs of thermal units pg1 and pg2 were 20% higher than the presented in Table 3, the deadweight loss would reach the 4.66% when the NG price at hub 1 was 27% higher than the NG marginal cost at hub 1 calculated in the integrated case. On the other hand, the winter scenario is suitable to evaluate the impact of total NG availability for electricity generation NGA as coordinating parameter. Other four cases are also presented to explain the influence of NGA: • The integrated electricity and NG dispatch (Integrated) • Three cases of the decoupled electricity and NG dispatch considering different NGA:

146

R. Rubio-Barros et al. 1.30% 1.10%

Over-cost (%)

0.90% 0.70% 0.50% 0.30% 0.10% -0.10%

-5.0%

-2.5%

0.0%

2.5%

5.0%

7.5%

10.0%

NG price difference at hub 1

Fig. 7 Total operating cost differences for summer scenario

Table 6 Winter scenario main results Integrated Total operating cost ($) 3,501.57 pgn1+ pgn2 (MW) 46.35 500 w2 (MW) NGA (MWh) 87.09 NG shortage (MW) –

Decoup. +10% 5,429.39 50.87 500 95.79 7.88 (ws1)

Decoup.
20% 3,513.90 37.07 483 69.67 –

Decoup.
40% 3,517.97 27.76 463.86 52.25 –

• NGA 10% higher than the optimal NGA obtained in the integrated dispatch (Decoupled +10%) • NGA 20% lower than the optimal NGA obtained in the integrated dispatch (Decoupled
20%) • NGA 40% lower than the optimal NGA obtained in the integrated dispatch (Decoupled
40%) For each decoupled dispatch, the NG price for electricity generation at node 1 and 2 are maintained at the corresponding NG marginal cost calculated in the integrated dispatch. These NG prices provide the signals for the optimal allocation of the NGA between pgn1 and pgn2. The most important results are shown in Table 6. As it can be expected, a 10% higher NGA than the obtained in the integrated approach leads to an infeasible NG dispatch, which is shown by 7.88 MW NG shortage at node 1. In the cases with a 20% and 40% lower NGA than the obtained in the integrated approach, the total operating cost are 0.35% and 0.47% higher than integrated total operating cost, respectively. However, the deadweight loss increases if the

Energy Carrier Networks: Interactions and Integrated Operational Planning

147

production costs of thermal units that substitute NGFPPs production were higher. For instance, if the production costs of thermal units pg1 and pg2 were 20% higher than the presented in Table 3, the deadweight loss would reach the 1.45% in the cases decoupled
40%. The lower NGA impacts in a lower NGFPPs’ output power (pgn1 + pgn2) which also means an underutilization of NG supply w2. The difference between the NGA used in the electric power dispatch and the actual NGA is the reason of the over-costs in the decoupled dispatches. In the integrated modeling and dispatch of electric power and NG systems, the NGA is a by-product (not a parameter) of the optimization; thus, the maximum economic efficiency is achieved. In the decoupled approach, the actual NGA could be obtained within an iterative runs of electric power and NG dispatch, however again the convergence is not guaranteed. Moreover, in the presented cases, the estimation of the actual NGA only requires to know the NG consumption in the compressor station (12.91 MW in the integrated case) because no NG transmission capacities are reached, otherwise nodal NGA will be needed instead of a total NGA. Finally, it is important to bear in mind that real cases include fixed NG prices and NGA as coordinating parameters. Therefore the over-costs (deadweight losses) in the decoupled approach can be higher than the presented in this case study since the deviations have analyzed in only one coordinating parameter at each time. Also, NG price deviations can lead to transport constraints in NG or electric power networks, and thus, higher cost differences between the integrated and the decoupled dispatch. However, in order to present a clear exposition, neither combined deviations nor active transport constraints have been included in this chapter.

3.6

Integrated Medium-Term Operational Planning

As it was described before, the integrated operational planning of multiple energy systems is divided into stages according to the considered time horizons. In this section, the integrated medium-term planning of energy systems (electric and NG) is presented, including the problem statement, the basic mathematical formulation, the applied solution method and a case study example. 3.6.1

Purposes of the Medium-Term Optimization

The main objectives underlying the integrated medium-term planning of energy systems are the optimal management of energy resources and the electricity and NG prices forecast. These tasks include the fuel procurement and the optimal scheduling of the energy storages within the time horizon considered. The solution of the optimal operational planning is affected by availabilities and prices of primary energy resources (NG, oil fuels, coal, nuclear), the estimated availability of hydroelectric resources and, to a lesser extent, the foreseen alternative energy resources (wind, solar, etc.). Naturally, the different methodologies applied to solve this problem must deal with the uncertainties associated with many of the input parameters, such as, lateral streamflows, energy demands and fuel prices.

148

R. Rubio-Barros et al.

Note that in energy systems with large controllable hydropower contribution, the scheduling of water movements between reservoirs, the availabilities and prices of fuels, as well as the strategic network and generation maintenance plan, have critical influences on the integrated operational planning. Similarly, NG storages, in particular those with large working volumes, play a relevant role in seeking and achieving the medium-term optimal decisions. The integrated optimization of electric power and NG systems is consistent with the optimal use of available energy resources. Thus, the combined social benefit (overall consumer and producer surplus) obtained through this combined optimization is greater than the sum of the benefits obtained when electric power and NG systems are decoupled optimized.

3.6.2

Temporal Couplings

The integrated energy systems planning with no storage units is solved by dispatching NG suppliers/electric power generation with lower production costs to supply the energy demands and the network losses. While there are additional factors that make the problem complex (energy losses, transport limitations, nonlinear costs, etc.), it has two basic characteristics: first, the dispatch is uncoupled in time, i.e. an operational decision in the present does not affect the operating cost in the near future, thereby making the problem separable; second, the units have a direct cost of production that depends mainly on their own level of generation, rather than the generation level of the other units. Conversely, the obvious feature of an integrated energy system with storage units is to use the energy stored in the reservoirs of hydroelectric and/or NG storage units to supply the demand, thereby avoiding the use of expensive oil fuels in thermal power plants. However, the availability of limited amounts of energy makes the optimal operation of integrated systems with energy storage a complex mathematical problem, as the energy reservoirs create a coupling between operational decisions taken in the present and the future consequences of these decisions. Temporal couplings make the problem harder to solve, since all the stages within the time horizon must be included in a single optimization problem. This condition leads to the use of a large number of variables and constraints, increasing the computing time. This complex and large-scale problem can be easier solved by means of decomposition techniques (e.g., Benders method or Dantzig-Wolfe method), which make possible to split the whole problem into several and smaller problems.

3.6.3

Typical Modeling

The typical assumptions chosen to achieve an appropriate level of compromise between precision and calculating time are listed below: • The objective function and the non linear constraints are linearized in order to solve the problem using linear programming models.

Energy Carrier Networks: Interactions and Integrated Operational Planning

149

• The study time horizon (1–3 years) is divided into monthly (12 months) or weekly (52 weeks) stages. Each stage is divided into subperiods (3–5 blocks) of different time duration. • The electricity and NG demands are represented by load duration curves approximated with a step function. Although, there is a relationship between both demands, usually it is assumed that they are independent and have similar profiles. • The demands and fuel prices (fuel oil, gas oil, carbon, etc., except NG) have a random nature, and therefore theses exogenous parameters are estimated by means of appropriate forecast models. • NG production costs are also uncertain, but they typically have a lower level of dispersion. • The watersheds have a random uncertainty on the lateral streamflows, making necessary to apply appropriate forecasting models for each watershed. • The production of NGFPPs is simulated by means of a linear model, using a constant net heat rate for each NGFPP. For all other thermal units, operating costs are also linearly modeled, taking into account the fuel prices and their net heat rates and the corresponding fuel prices. • The electricity transmission system is modeled through a DC flow model without losses, which takes into account the Kirchhoff laws and the limitations imposed by the network. • The NG transmission system is formulated through a nodal energy balance model, which takes into account the limits imposed by NG pipeline network, but ignores the nodal pressures of the system. • The preventive maintenance scheduling of generating units is considered. • The probability of random outages of thermal generating units is taken into account derating their maximum capacities using the forced outage rates (FOR) • Spinning reserve requirements are also modeled derating the maximum capacities of the generating units according to a prefixed reserve rate.

3.6.4

Mathematical Formulation

The integrated operational planning of electric power and NG systems relies on a centralized coordination and decision-making process of the both energy systems. From this perspective, the objective function (34) corresponds to the minimization of the expected value of NG production costs and operating costs of the electric power systems over the study horizon. Mathematically, it can be expressed as: 2

0

0X

6 X BX B B 6 B OF : min VE6 at B bk B @ 4 t @ k



X

1 Ci pgk;t CSE psk;t þ þ i j C i j C

C X X k;t A k;t C w CSG ws þ g

g

13 C7 C7 C 7 (34) A5

j

g

j

k

t

150

R. Rubio-Barros et al.

where k is the block index, bk is the number of hours at each block, t is the stage index and at is a discount factor of stage t affected by a predetermined discount rate. The stated objective function is subject to a set of electric systems constraints for each block k of the stage t:
Fmax m

b

P

" Hmj

j

P k;t k;t k;t pgi þ pgnk;t þ pgh þ psk;t i i j
Dej

!#

i2Gj

bFmax m

8m (35)

X X k;t X k;t k;t pgi þ pgnk;t þ pgh psk;t Dej þ i i j ¼ i

j

(36)

j

k;t pghk;t i ¼ ri ðQw; H ÞQwi

8i

k;t k;t max 0bpgk;t i ; pgni ; pghi bPi

8i

(37) (38)

Constraints (35) and (36) model the electricity transmission system through a DC flow model detailed above. Constraint (37) is the power equation of the hydroelectric plant i where Qwk;t i represents the flow rate through the turbine of the hydroelectric plant i in block k of stage t and ri the production ratio of the hydroelectric power plant i in terms of the flow rate Qw and the hydraulic head H. To simplify the problem, ri is modeled as a constant average production ratio independent of the variables H and Qw. Finally, constraint (38) models the maximum capacity of power plants. Also there are a set of constraints for each block k of stage t of the NG system: X X 0 X k;t 1 wg þ wsk;t Qgm
Qgm þ j þ B g2W j C m2I j m2Oj B C ¼ Dgk;t 8j X X X (39) j @ A t t QOp
QIp
HRi ðpgni Þ p2P j

p2P j

i2Gj

max Wgmin bwk;t g bW g

8g

k;t max
Qgmax m bQgm bQgm

0bQIpt bQIpmax 0bQOtp bQOmax p

8p 8p

8m

(40) (41) (42) (43)

Equation (39) represents the nodal NG flow balance including storage inflows and outflows, where QOtp and QIpt are the outflows (withdrawal rate) and inflows (injection rate) of the NG storage p located on node j. P j is the subset of NG

Energy Carrier Networks: Interactions and Integrated Operational Planning

151

reservoirs located on the node j. In (39), HRi is modeled in the simplest way as a constant and average net heat rate independent of the NGFPP production pgn. Constraint (40) shows the limits of injection capacity of NG suppliers where Wgmax and Wgmin are the maximum and minimum flow rate of the NG supplier g. Constraint (41) shows the maximum carrying capacity Qgmax m of the pipeline m. QIpmax and QOmax are the peak operating inflows (injection) and outflows (withp drawal) of the NG storage p. These parameters change as the level of NG varies within the facility, but they are modeled as constant values to simplify the problem (constraints (42) and (43)). In addition to the operational conditions outlined above, the complete set of constraints also includes the time coupling constraints. These equations are expressed through the balance equations of NG storages (44) and the continuity equations of water reservoirs (45). Vgtþ1 ¼ Vgtp
QOtp T þ QIpt T p Vetþ1 ¼ Vet þ Ate T


P

t Qwk;t i bk
Se T þ

P

k

k

i 2 I he

8p Qwk;t i bk þ

(44) P u2E e

Stu T

8e

i 2 I hu

(45) where Vgtp is a state variable that represents the stored volume of the NG reservoir p at the beginning of stage t, Vet is a state variable that represent the stored volume of the water reservoir e at the beginning of stage t, Ate is the lateral streamflow arriving at the plant e, Ste is a slack variable that represents the spilled outflow of the water reservoir e, \he is the subset of hydroelectric power plants i associated with the water reservoir e, T is the total duration of stage t, \hu is the subset of the upstream hydropower plants and ℰe is the subset of reservoirs u that spill into the reservoir e. The line-pack of a pipeline, which is its intrinsic storage capacity, can be modeled by a reservoir of small capacity (weekly, monthly capacity) in the medium-term planning. This can be applied in very long pipelines in which the line pack is sufficient to store a volume of NG from one stage to the following one. Finally, constraints (46) and (47) model the capacity of NG and water reservoirs: t max Vgmin p bVgp bVgp

Vemin bVet bVemax

8p 8e

(46) (47)

where Vgmin p is the base gas (or cushion gas) of the NG storage, that is the volume of gas intended as permanent inventory in a storage reservoir to maintain adequate pressure and deliverability rate throughout the withdrawal season. Vgmax is the total p gas capacity of the storage. The difference between the total gas capacity and the

152

R. Rubio-Barros et al.

base gas is the working gas capacity of the storage, represented in constraint (46). Vemin and Vemax are the minimum and maximum volumes of the water reservoir e. The presented mathematical formulation address the so-called deterministic problem, since all input parameters (demands, lateral streamflows, fuel prices, etc.) are known values. However, as it was described before, these parameters are uncertain. This means that the medium-term optimal operation planning is, in fact, a stochastic optimization problem. Among other methodologies, Monte Carlo simulation method can be implemented to deal with these uncertainties. Therefore, a finite number of independent trials or simulations of the deterministic problem are solved in order to achieve a satisfactory level of confidence in the resulting probability distribution functions (PDF). This number depends on the tolerable error specified for the planner which, in turn, greatly depends on the characteristics of the problem. A detailed explanation of Monte Carlo simulation method can be found in [55].

3.6.5

Water and Natural Gas Economic Values

When a hydroelectric power plant produces power, not only incurs in a direct monetary expenditure, but also incurs in an indirect cost associated with the economic value of used amount of water that was previously stored. If this volume of water was not used, it would be available for future utilization, an then, it would save thermal generating costs or possible energy shortages. Thus, the optimal management of the stored water must be consistent with the minimum variable cost of thermal generation over the study period, which it is closely related to the water opportunity cost, since any deviation from the optimal use of water at each period produce additional thermal generation costs within the period. This opportunity cost is known as expected water value or simply water value [50, 51] and represents the economic value of the stored water. The water value defines the economic suitability of accumulating water in the present for future use. One advantage of this parameter is that it can be used, after its conversion to monetary cost per unit of hydro-generated energy, to dispatch hydroelectric plants together with thermal units in the short-term operation planning. NG storage facilities introduce to the modeling of the problem similar complexities as water reservoirs. NG storages allows the accumulation of NG in time periods of reduced demands and/or low prices (typically during summer), for it use in time periods of higher demands and/or prices (typically during winter). Therefore, the optimal use of stored NG is aligned with the condition that minimizes the sum of the present cost function (PCF) and the future cost function (FCF). The PCF represents the production costs of NG in a given stage t and it increases as the stored volume rise. The FCF is associated with the expected cost of NG production, exports and shortages, starting from the end of stage t (beginning of stage t þ 1) to the end of the study horizon. This cost, contrary to PCF, decrease as the stored volume increase because a larger amount of stored

Energy Carrier Networks: Interactions and Integrated Operational Planning

153

Production costs

PCF+FCF

FCF

PCF Marginal cost of production

Optimal decision

Vmax

Vmin

CProdmin

Expected value of NG

NG stored volume

Fig. 8 PCF, FCF and expected value of natural gas for a stage

NG implies more available NG for future consumption. The FCF is achieved through probabilistic simulations of the economic dispatch using different NG demand scenarios. Figure 8 shows the PCF, FCF and PCF þ FCF curves based on the stored volume in a NG reservoir. The minimum cost point is where the absolute derivative of PCF and FCF are equal. The derivative of PCF is called the marginal cost of NG production at stage t, while the derivative of the FCF represents the economic value of storage and is called expected value of natural gas, which relates the cost of using the NG stored in the present with the cost of NG production and shortage in the future. This economic value can be used for decision making between injecting stored NG to the system or saving it, depending on the NG price at each stage. Thus, it reflects the competition for provision between the stored NG in and the different NG suppliers. A major difference from hydrothermal power systems, in which the economic value of water that cannot be stored (spilled water) is considered null, the NG does have a cost in the present; therefore, the FCF will have a nonzero value even when the reservoir is at its maximum volume.

3.6.6

Case Study

The example illustrated in Fig. 9 consists of two energy hubs (N1, N2), an electrical system with two nodes (N1E, N2E), a transmission line (F12) connecting them, a natural gas system with two nodes (N1G, N2G), a pipeline (Qg12) connecting them and a hydroelectric system with two reservoirs in cascade. This example is used to simulate the annual operational planning divided into 12 monthly stages, where

154

R. Rubio-Barros et al. A1 V1

A2

N2E

pgh1

V2

N1E

pg2

pgh2

ps

De

F12

pgn2

N2

pg1

pgn2

N1

Qg12 Dg

w1

w2

FIOR

ws

N2G

N1G VR

Fig. 9 Study example

demands at each stage are divided into 4 blocks of 8, 100, 450 and 172 hours duration. All systems are interconnected and are interdependent in their operation. The link between electric and NG system occurs through the pgn1 and pgn2. The link between the hydroelectric system and the electrical system is through pgh1, and pgh2. The electric power system consists of thermal generating units with liquid fuels (pg1 and pg2) associated with each node, whose costs are associated with the cost of the liquid fuel used and the rate of specific fuel consumption of the unit. Additionally, it uses an artificial unit of shortage (ps) to model the deficit of electric power generation. The electricity load De is concentrated at the node N2E. The system consists of NG production fields (w1, w2) associated with each node, whose production costs are related to the cost of NG extraction. Field ws is used to simulate NG shortages. Also, the model includes a NG reservoir VR that can deliver NG to N2G or withdraw NG from this node steadily during each monthly period. The hydroelectric system consists of a hydro plant V1 and run-of-water plant V2 topologically located in cascade, with lateral streamflows A1 and A2. Each hydroelectric generating unit (pgh1, pgh2) has an associated constant average production ratio. Table 7 presents the data used in the problem. Solved Cases The example with all integrated energy systems is solved deterministically (one scenario) using the objective function (34) and constraints (35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47). In order to show the optimal use of storable resources, we present the following study cases: • Case 1: Integrated system without NG storage and without hydro reservoirs (runof-river power plant).

Max capacity (MW) 129.6 158.4

1 2 3 4 5

Stage

pg1 77.81 83.56 85.69 77.29 78.58

pg2 117.81 123.97 126.52 118.53 120.24

TG Costs ($/MWh)

Transmission system F12

Pipeline Qg12

pg1 70.00 75.17 77.09 69.53 70.69

Fuel Costs ($/t) pg2 75.00 78.92 80.54 75.45 76.55

Max volume (dam3) 20,000

A1 84.00 42.00 36.00 24.00 48.00

A2 11.00 6.50 5.00 6.00 9.00

Lateral streamflows (m3/s)

Min volume (dam3) 0

End volume (hm3) 600 150

w1 108.45 112.37 113.73 107.74 108.53

w2 228.45 232.86 234.70 229.22 230.50 (continued)

NG Costs ($/dam3)

End volume (dam3) 5,000

Natural gas shortage cost ($/dam3) 8,000

Initial volume (hm3) 600 150

Electric shortage cost ($/MWh) 1,500

Initial volume (dam3) 5,000

Max reservoir volume (hm3) 1,200 100

Electricity shortage Cps

Natural gas shortage Cws

Min reservoir volume (hm3) 200 200

Max power flow (MW) 500

Max NG flow (dam3/day) 1,000

Average production ratio (MW.s/m3) 0.648 0.792

Max injection capacity Max withdrawal capacity (dam3/day) (dam3/day) 300 400

Max production (dam3/day) 1,200 400

Costs and inflows

NG storage VR

NG fields w1 w2

Natural gas system

Hydro generation pgh1 pgh2

Specific fuel consumption 1.1116 t/MWh 1.5708 t/MWh 0.1935 dam3/MWh 0.2150 dam3/MWh

Max turbined outflow (m3/s) 200 200

Max Capacity (MW) 80 150 50 80

Hydroelectric system

Thermal generation pg1 pg2 pgn1 pgn2

Table 7 Example data Electrical system

Energy Carrier Networks: Interactions and Integrated Operational Planning 155

560.00 549.43 590.88 735.68 899.25 1,044.93 1,124.93 1,072.19 896.85 769.41 681.44 612.90

2 440.00 431.70 464.26 578.03 706.55 821.02 883.87 842.44 704.67 604.54 535.42 481.57

1

Block 1 2 3 4 5 6 7 8 9 10 11 12

pg2 128.17 126.22 131.21 132.98 133.99 137.38 136.59

NG Demand Dg (dam3/day)

pg1 86.08 83.70 88.25 89.59 90.15 93.09 91.85

TG Costs ($/MWh)

Stage

Energy loads

6 7 8 9 10 11 12

Stage

Table 7 (continued) Costs and inflows

3 388.00 380.68 409.40 509.72 623.05 723.99 779.41 742.88 621.39 533.09 472.14 424.65

pg1 77.44 75.29 79.39 80.59 81.10 83.75 82.63

Fuel Costs ($/t) pg2 81.59 80.35 83.53 84.66 85.30 87.46 86.95

4 296.00 290.41 312.32 388.86 475.32 552.32 594.60 566.73 474.05 406.69 360.19 323.96

A1 66.00 78.00 102.00 132.00 144.00 150.00 114.00

1 369.60 349.12 354.55 363.77 369.31 390.04 403.36 393.85 384.24 374.53 384.11 401.57

2 311.85 294.57 299.15 306.93 311.61 329.10 340.33 332.31 324.20 316.01 324.09 338.83

Electricity demand De (MW)

A2 13.00 14.00 15.60 21.00 18.00 15.00 13.20

Lateral streamflows (m3/s)

3 281.82 266.20 270.34 277.38 281.60 297.41 307.56 300.31 292.98 285.58 292.88 306.20

w1 113.56 111.80 114.76 115.52 115.75 117.57 116.59

4 196.35 185.47 188.35 193.25 196.20 207.21 214.28 209.23 204.13 198.97 204.06 213.34

w2 236.03 234.76 238.23 239.49 240.22 242.55 242.08

NG Costs ($/dam3)

156 R. Rubio-Barros et al.

Energy Carrier Networks: Interactions and Integrated Operational Planning

• • • •

157

Case 2: Integrated system with NG storage, but with no hydro reservoirs. Case 3: Integrated system without NG storage, but with hydro reservoir V1. Case 4: Integrated system with NG storage and hydro reservoir V1. Case 5: Integrated system with NG storage and hydro reservoirs (V1 and V2).

Note that when the hydroelectric plant has no storage capacity, the lateral streamflows A1 and A2 do not produce water spillage, so the amount of available water (available energy) to be stored is the same in all the cases. This condition is essential because if any of the proposed cases some water spillage was verified, the total available water volumes would not be equal, and therefore the cases could not be compared. Table 8 shows that the optimal use of energy reservoirs results in lower electricity generation and NG production costs. This is due to these storages are used to accumulate energy reserves during the periods of low production/generation costs (low energy demands) for their later use in periods when production costs are higher. This effect can also be noticed in the nodal average prices of both energy carriers (Fig. 10).

Table 8 Total production costs Case 1 Case 2 Case 3 Electric power system total cost 22,701,173 22,667,088 8,624,420 ($) NG system total cost ($) 47,966,322 45,922,170 52,070,531 Total cost ($) 70,667,495 68,589,258 60,694,951

Case 4 8,592,179

50,028,470 50,539,970 58,620,649 58,276,213

Fig. 10 Weighted average prices of electricity and NG in each stage1

1

Case 5 7,736,243

Average price calculated as the price weighted by the duration of the block.

158

R. Rubio-Barros et al.

• The average prices of N1G and N2G in Case 2 show a relatively constant value over the study horizon (Fig. 10a) due to the use of the NG reservoir; whereas in Case 1, having no capacity to store NG for high demands periods, NG supply with higher cost (w2) is needed, and thus total generation/production cost are higher than in Case 2 (Table 8). Figure 11 clearly shows that the use of the NG storage results in a decrease of injection of w2, reducing total NG production costs. • The average prices of N1E and N2E in Case 3 are relatively constant during the year due to the use of water stored in V1 (Fig. 10b), decoupling the time of occurrence of inflows regarding the time when water is required for power generation; conversely, in Case 1, having no possibility to dam water for times of high demands and/or less inflows, the water inflows should be used as run-of-river power plant, generating all the natural river water flow at each given time (Fig. 12), increasing the total operating cost. It should be noted that the amount of hydro

Fig. 11 NG demand and supply with and without NG storage

Fig.12 Electric demand and pgh1 supply with and without reservoir V1

Energy Carrier Networks: Interactions and Integrated Operational Planning

159

energy are the same for the two cases because there was no spilling of water in the study period, thus the both cases can be compared. • Case 4 and Case 5 combine the effects discussed above. Table 8 shows that the integrated operational planning of water and NG reservoirs results in further benefits for the system as a whole since the lowest operating costs (of the simulated cases) are obtained. Moreover, in Case 4 electricity and NG prices are relatively more constant over time than the prices obtained for first three cases, even though these prices are higher in the final stages (Fig. 10). • Case 5 shows a small decrease in total costs compared with Case 4 (Table 8), thereby indicating that in Case 4 more hydro storage capacity is still needed. This analysis, despite its simplicity, shows the importance of the operational planning of energy carriers and the advantages coming from their combined optimization and the regional integration of these resources. The optimal scheduling of energy storages (water reservoirs, NG reservoirs) allows to achieve an overall benefit for all the involved energy systems considering the limited availability of the energy resources and the technical operating constraints of the energy systems.

3.7

Integrated Short-Term Operational Planning

The integrated short-term operational planning of multiple energy carrier systems is essentially a multi-period optimal energy flows subject to time coupling constraints. Short-term decision-making is referred to time horizons ranging from a day up to a month, being the weekly scale the most common time period analyzed. It consists of determining the chronological operating schedule, on either hourly or half hourly bases, of all generating units and NG injections for each day of the week. This plan must take into account the medium-term framework, described in the previous section, which provides the higher-level decisions related to the optimal scheduling of energy storages (water and NG reservoirs), the preventive maintenance plan and the fuel quotas. From the centralized perspective analyzed along this chapter, the objective of this short-term operational planning is the minimization of the total operating and shortage costs. At this level, a high detail modeling of energy system’s components is extremely important. The relevant technical characteristics and systems’ operating constraints, typically included in short-term energy systems optimization, are described in the following subsections. 3.7.1

Unit Commitment and Time Coupling Constraints

The integrated short-term operational planning can be presented as an extension of the integrated ED, presented in Sect. 3.3, for all time periods. In this case, Ci(pgi),

160

R. Rubio-Barros et al.

Cg (wg), and HRi(pgni) are usually modeled as quadratic functions. All generating units have a minimum output power, in particular for thermal units, it can reach up to 50% of the maximum output. Thermal generating units also have some technical features that must be considered: (a) significant startup cost; (b) minimum up- and down-time constraints; (c) maximum ramp-up and ramp-down limits. Under these conditions, the start-ups and shutdowns are explicitly modeled by means of binary variables, thus the problem is so-called unit commitment. The startup costs are added to the objective function (30). Technical characteristics (b) and (c) constitute time coupling constraints in this short-term optimization problem. There are two alternative approaches to employ the results of the medium-term operational planning as coordinating parameters for the optimal use of the energy storages in the short-term scheduling [52]. One of them, called primal approach, consists in using the optimal amount of energy (stored water and stored NG), determined in the medium-term model, for the time horizon to be optimized in the short-term model. This means, that (37), (45) and (47) must be included to consider the hydro-system. Equation (45) is re-defined on an hourly basis, and the Vet for the last simulated hour is fixed to same value obtained in the medium-term model. In this primal approach, the CHi(pghi) are equal to zero. To consider NG storages, QOtp and QIpt are added to (26) if NG storage facilities are located in the corresponding nodes. Equations (42), (43), and (46) must be also introduced. The NG storage balance constraint (44) is re-defined on hourly time steps, and the Vgpt for the last simulated hour is fixed to same value obtained in the medium-term model. The other approach for coordinating medium-term and short-term optimizations is called the dual approach [53]. It consists in using the water and NG economic values described in Sect. 3.6.5 Water values can be utilized to calculate the opportunity cost functions CHi(pghi) according to the layout of hydrological network. In a similar way, NG storages can be modelled as equivalent NG suppliers considering supplying cost functions Cg based on the NG values. Another complex unit commitment problem that exists is with compression stations. Since they include several compressor units, there is an optimal operating point that minimizes the amount of NG consumed. However, this specific issue is out of the scope of the presented integrated short-term operational planning.

3.7.2

Modeling of Energy Carrier Network Flows

Electric power flows can be represented with the highest possible detail, i.e., the AC power flow model described in Sect. 2.2. This implies that reactive power resources (e.g., generating units; shunt, synchronized and static VAR compensators) should be introduced in the modeling, as well as reactive power loads. However, if electric power system has sufficient and well-distributed reactive power resources and adequate reactive power flow controlling devices to keep the voltage within the operating limits, AC power flows model can be disregarded. This assumption is in particular suitable to deal with the integrated energy systems operational planning, which is focused on the real quantities of energy transferred within the networks. In

Energy Carrier Networks: Interactions and Integrated Operational Planning

161

this context, the DC model considering losses, also described in Sect. 2.2, is appropriate to model the electric power flows. As it was introduced in Sect. 2.5, hourly NG flows with changing demands cannot be accurately modeled by means of steady-state gas flow equations. A transient NG flow model must be considered to take into account the typical imbalanced operation of the NG systems, i.e., for a given hour the amount of NG injected in the NG transmission systems does not match the total NG withdrawals. This transient model allows to represent the line-pack, i.e., the amount of NG stored in pipeline network, and to optimize its use for the minimization of the total operating and shortage costs. Essentially, simplified isothermal transient NG flows can be described by means of the followings partial differential equations [44]. @s 4 sst T Z @Qg ¼
@t p Tst D2 @x

(48)

@s2 64 s2 f G T Z ¼
2 st2 5 Qg2 p Tst D Rair @x

(49)

@ 2 s2 fGQg @s2 ¼ 2 pD3 Rair T Z @t @x

(50)

where (48) and (49) are the continuity and the momentum equations. These equations can be written as (50) if it assumed that the NG flow at standard conditions Qg is averaged over length in every time interval Dt. To solve these partial differential equations several finite differences schemes can be used [44]. In [48], Eqs. (48) and (49) are applied to the optimization of a single pipeline operation. The classical implicit method to solve (50) can be expressed as [54, 44] Stþ1
Stj j Dt

¼a

tþ1 Stþ1 þ Stþ1 jþ1
2 Sj j
1

ðDxÞ2

(51)

where S ¼ s2, a is the constant indicated in (50). The index j-1, j and j + 1 are the contiguous nodes resulting of the segmentation of each pipeline (principle of finite difference methods). Equation (51) must be introduced as constraints to model transient NG flows within the integrated short-term energy systems optimization. Regarding water flows, if the water continuity constraint (45) is considered in the short-term multi-energy carrier optimization (primal approach), then it is important to take into account the time delays existing since the water is released in the upstream water reservoir up to this water arrives to downstream water reservoir. Finally, the short-term operational planning must also include systems’ security constraints. While minimum spinning (operating) reserve is considered for electric power systems, minimum total line-pack is taken into account for NG systems.

162

R. Rubio-Barros et al.

4 Conclusions The integration of NG and electricity energy carriers is a reality, and their strongest interdependence must be mandatory considered in the centralized planning modeling of energy resources management. The NGFPPs are the main protagonists in this integration. They link both energy carriers because they can arbitrate their participation in both markets to the best suited price from social welfare viewpoint; otherwise, the link is through fixed parameters, losing the economic synergy. In this context, the complexity of the centralized planning optimization problem increases. Therefore, this challenge requires new approaches to model the energy carrier interactions and new solution procedures to divide the entire problems into coordinated subproblems. This chapter presents a comprehensive literature survey of state of the art on integrated NG and electricity operational planning. The relevant characteristics of NG and electricity systems are compared considering the physical laws that govern the flows of these energy carriers through dedicated networks. Also, the interactions among the energy carriers and their networks are modeled with different levels of detail according to the evaluated time horizon. Finally, some contributions about economic interactions between different energy carriers (electricity, NG, and hydro energy carriers) are presented with emphasis on the concept of water values that can be extended to NG values when storage facilities are available. An example of a medium-term planning is presented to show the temporal interdependencies introduced by hydro and NG reservoirs. The formulation of integrated operational planning presented in this chapter addresses the problem from a centralized perspective, and thus it is based on the economic theory of perfect competitive markets. Although the economic concept of perfect markets is never satisfied in reality, the proposed approach provides a benchmark against which actual electricity and NG markets can be compared, since the results obtained from a centralized optimization reflect the efficient use of scarce energy resources.

Appendix: Symbols Used Acronyms AC CCGT DC ED FACTS FCF LHV LNG NGFPP

Alternating current Combined-cycle gas turbine Direct current Economic dispatch Flexible alternating current transmission system Future cost function Lower heating value Liquefied natural gas Natural gas fired power plant (continued)

Energy Carrier Networks: Interactions and Integrated Operational Planning

NG OPF PCF PDF PTDF SDDP UK USA Variables F F L Lg P P pg pgh pgn PG ps PD PL Qg QI QO Qw s S U V Vg w ws y u Functions C(pg) CG(pgn) CH(pgh) CSE(ps) C(w) CSG(ws) HR(pgn) r(Qw,H)

163

Natural gas Optimal power flow Present cost function Probability distribution functions Power transfer distribution factors Stochastic dual dynamic programming United Kingdom United States of America Active power flow of electric power lines Vector of lines active power flows Shunt losses of electric power lines NG consumed in a compression station Active power flow injections Vector of active power injections Production of a thermal generating unit which do not use NG as fuel (i.e., coal, nuclear, fuel oil) Hydropower plant production NGFPP production Vector of generating unit’s production Load shedding Vector of demands Vector of losses NG flow at a pipeline Inflow (injection rate) of NG storage Outflow (withdrawal rate) of NG storage Flow rate through the turbine of the hydroelectric plant Nodal pressure Spilled outflow of the water reservoir/complex power injection Complex voltage Stored volume of the water reservoir Stored volume of the NG reservoir NG injection of the supplier NG load shedding at node Phase angle of complex voltage Vector of voltage phase angles Production cost of a thermal generating unit which do not use NG as fuel (i.e., coal, nuclear, fuel oil) Production cost of the NGFPP Opportunity cost of production of a reservoir-associated hydropower plant Electric power shortage cost NG supply costs NG shortage costs Heat rate function or input-output curve of NGFPP Production ratio of the hydroelectric power plant (continued)

164

Parameters A A b BD D De Dg f FOR G H K KC L NGA Qgmax QImax QOmax R Rair T Tst, sst Vmax, Vmin Vgmin Vgmax Wmax, Wmin X Y Z a

R. Rubio-Barros et al.

Node-branch incidence matrix Lateral streamflow arriving at the water reservoir Number of hours at each block Diagonal matrix whose elements are 1/X Inside diameter of the pipe Electricity demand NG demand Dimensionless friction factor of the pipe Force outage rate of generating unit Gas gravity relative to air Sensitivity matrix or power transfer distribution matrix Factor representing pipeline characteristics and NG properties Constant of the compression station Length of the pipe NG availability for electricity generation Maximum transport capacity of the pipeline Peak operating inflow (injection) of the NG storage Peak operating outflows (withdrawal) of the NG storage Resistance of the line Ideal gas constant of dry air Total duration of stage Standard or normal conditions of temperature and pressure Maximum and minimum volumes of the water reservoir Base gas (or cushion gas) of the NG storage Total gas capacity of the NG storage Maximum and minimum flow rate of the NG supplier Reactance of the line Shunt admittance of the P-equivalent line Series impedance of the P-equivalent line Discount factor of the stage

Subscripts and superscripts c Type of fuel (coal, fuel oil, gas oil) g NG supplier i Generation unit (hydro, NGFPP, thermal) j,k,n Electrical/natural gas node k Block (superscript) m Branch: electric power line, pipeline, compressor station (Cm) t Stage u Natural gas node associated to a suction side of a compressor station Sets Cj Ee Gj Ij I hu I he

Compression stations whose suction side is connected to node j Reservoirs that spill into the reservoir e Generating units connected to node j Pipelines which flows are incoming node j Upstream hydropower plants Hydroelectric power plants associated with the water reservoir e (continued)

Energy Carrier Networks: Interactions and Integrated Operational Planning Mj N Cm Nd Nj Ns Oj Pj Wj

165

Lines connected to node j Nodes associated to the compression station Cm Nodes associated to the discharge of compressor stations Nodes connected to node j Nodes associated to the suction of compressor stations Pipelines which flows are outgoing node j NG reservoirs located on the node j Gas supplier injecting in node j

References 1. Munasinghe M, Meier P (1993) Energy policy analysis and modeling. Cambridge University Press, Cambridge, MA 2. Loulou R, Remne U, Kanudia A, Lehtila A, Goldstein G (2005) Documentation for the TIMES Model, Part I. Energy Technology Systems Analysis Programme [Online]. Available: http:// www.etsap.org. [Accessed: 28 Agu 2009] 3. Messner S, Schrattenholzer L (2005) MESSAGE-MACRO: linking an energy supply model with a macroeconomic model and solving it inter-actively. Energy 25:267–282 4. CEEESA (2008) Overview of the Energy and Power Evaluation Program (ENPEP-BALANCE). Center for Energy, Environmental, and Economic Systems Analysis (CEEESA), Argonne National Laboratory, [Online]. http://www.dis.anl.gov/projects/Enpepwin.html [Accessed: 15 Nov 2008] 5. SEI (2006) LEAP: User Guide. Stockholm Environmental Institute, [Online]. Available: www. energycommunity.org/documents/Leap2006UserGuideEnglish.pdf [Accessed: 08 Mar 2008] 6. van Beeck N (1999) Classification of energy models. Tilburg University and Eindhoven University of Technology, The Netherlands 7. Bakken H, Skjelbred HI, Wolfgang O (2007) eTransport: investment planning in energy supply systems with multiple energy carriers. Energy 32:1676–1689 8. Hecq S, Bouffioulx Y, Doulliez P, Saintes P (2001) The integrated planning of the natural gas and electricity systems under market conditions. In: Proceedings of the IEEE power engineering society PowerTech, Porto, 2001 9. Unsihuay C, Marangon-Lima JW, Zambroni de Souza AC (2007) Integrated power generation and natural gas expansion planning. In Proceedings of the IEEE power engineering society PowerTech, Lausanne, 2007 10. International Energy Agency (2007) Natural gas market review 2007. IEA/OECD Publications, Paris 11. International Energy Agency (2008) Electricity information 2008. IEA/OECD Publications, Paris 12. International Energy Agency (2008) World energy outlook 2008. IEA/OECD Publications, Paris 13. International Energy Agency (2009) Natural gas market review 2009. IEA/OECD Publications, Paris 14. Shahidehpour M, Fu Y, Wiedman T (2005) Impact of natural gas infrastructure on electric power systems. Proc IEEE 93(5):1042–1056 15. Chen H, Baldick R (2007) Optimizing short-term natural gas supply portfolio for electric utility companies. IEEE Trans Power Syst 22:232–239 16. Street A, Barroso LA, Chabar R, Mendes ATS, Pereira MV (2008) Pricing flexible natural gas supply contracts under uncertainty in hydrothermal market. IEEE Trans Power Syst 23:1009–1017

166

R. Rubio-Barros et al.

17. Takriti S, Supatgiat C, Wu LS-Y (2001) Coordination fuel inventory and electric power generation under uncertainty. IEEE Trans Power Syst 16:603–608 18. Morais MS, Marangon Lima JW (2003) Natural gas network pricing and its influence on electricity and gas markets. In: Proceedings of the IEEE power engineering society PowerTech, Bologna, 2003 19. Morais MS, Marangon Lima JW (2007) Combined natural gas and electricity network pricing. Elec Power Syst Res 77:712–719 20. Rubio R, Ojeda-Esteybar D, An˜o´ O, Vargas A (2008) Integrated natural gas and electricity market: a survey of the state of the art in operation planning and market issues. In: Proceedings of 2008 IEEE/PES transmission and distribution conference and exposition: Latin America, Bogota´, 2008, pp 1–8 21. Quelhas A, Gil E, McCalley JD, Ryan SM (2007) A multiperiod generalized network flow model of U.S. Integrated energy system: part I – model description. IEEE Trans Power Syst 22:829–836 22. Gil EM, Quelhas AM, McCalley JD, Voorhis TV (2003) Modeling integrated energy transportation networks for analysis of economic efficiency and network interdependencies. In: Proceedings of North American power symposium (NAPS), Rolla, 2003 23. Correia P, Lyra C (1992) Optimal scheduling of a multi-branched interconnected energy system. IEEE Trans Power Syst 7:1225–1231 24. Bezerra B, Kelman R, Barroso LA, Flash B, Latore ML, Campodo´nico N, Pereira MVF (2006) Integrated electricity-gas operations planning in hydrothermal systems. In: Proceedings of X SEPOPE, Floriano´polis 2006 25. Unsihuay C, Marangon-Lima JW, Zambroni de Souza AC (2007) Short-term operation planning of integrated hydrothermal and natural gas systems. In: Proceedings of the IEEE power engineering society PowerTech, Lausanne, 2007 26. Li T, Erima M, Shahidehpour M (2008) Interdependency of natural gas network and power system security. IEEE Trans Power Syst 23:1817–1824 27. Liu C, Shahidehpour M, Fu Y, Li Z (2009) Security-constrained unit commitment with natural gas transmission constraints. IEEE Trans Power Syst 24:1523–1536 28. Chaudry M, Jenkins N, Strbac G (2008) Multi-time period combined gas and electricity network optimisation. Elec Power Syst Res 78:1265–1279 29. An S, Li Q, Gedra TW (2003) Natural gas and electricity optimal power flow. In: Proceedings of the IEEE power engineering society transmission and distribution conference, Dallas, 2003 30. Unsihuay C, Marangon Lima JW, Zambroni de Souza AC (2007) Modeling the integrated natural gas and electricity optimal power flow. In: Proceedings of the IEEE power engineering society general meeting, Tampa, 2007 31. Mello OD, Ohishi T (2006) An integrated dispatch model of gas supply and thermoelectric generation with constraints on the gas supply. In: Proceedings of X SEPOPE, Floriano´polis, 2006 32. Munoz J, Jimenez-Redondo N, Perez-Ruiz J, Barquin J (2003) Natural gas network modeling for power systems reliability studies. In: Proceedings of the IEEE power engineering society PowerTech, Bologna, 2003 33. Geidl M, Andersson G (2007) Optimal power flow of multiple energy carriers. IEEE Trans Power Syst 22:145–155 34. Hajimiragha A, Canizares C, Fowler M, Geidl M, Andersson G (2007) Optimal energy flow of integrated energy systems with hydrogen economy considerations. In: Proceedings of bulk power system dynamics and control – VII, Charlestone, 2007 35. Rajabi H, Mohtashasmi S (2009) Economic dispatch problem considering natural gas transportation cost. Proc World Acad Sci Eng Technol 38:1482–1487 36. Ojeda-Esteybar D, Rubio-Barros R, An˜o´ O, Vargas A (2009) Despacho o´ptimo integrado de sistemas de gas natural y electricidad: comparacio´n con un despacho desacoplado y aplicacio´n al sistema argentino. In: Proceedings of the XIII ERIAC, Puerto Iguazu´, 2009

Energy Carrier Networks: Interactions and Integrated Operational Planning

167

37. Rubio-Barros R, Ojeda-Esteybar D, An˜o´ O, Vargas A (2009) Identificacio´n de los para´metros para la coordinacio´n de los despachos de los sistemas ele´ctricos y de gas natural. In: Proceedings of the XIII CLAGTEE, Ubatuba, 2009 38. ISO 13600:1997 (1997) Technical energy systems – basic concepts, International Organization for Standardization 39. Bergen AR, Vittal V (2000) Power systems analysis, 2nd edn. Prentice-Hall, Englewood Cliffs 40. Wood AJ, Wollenberg BF (1996) Power generation, operation and control, 2nd edn. Wiley, New York 41. Go´mez-Expo´sito A, Conejo AJ, Can˜izares CA (2009) Electric energy systems analysis and operation. CRC, Boca Raton 42. Stott B, Jardim J, Alsac¸ O (2009) DC power flow revisited. IEEE Trans Power Syst 24:1290–1300 43. Scheweppe FC, Caramanis MC, Tabors RD, Bohn RE (1988) Spot pricing of electricity. Kluwer, Norwell 44. Osiadacz AJ (1987) Simulation and analysis of gas networks. E. & F. N. Spon, London 45. Menon ES (2004) Gas pipelines hydraulics. Marcel Dekker, New York 46. Osiadacz AJ (1996) Different transient models- limitations, advantages and disadvantages. In: Proceedings of the PSIG, 28th annual meeting, San Francisco, 1996 47. Osiadacz AJ (1994) Dynamic optimization of high pressure gas networks using hierarchical system theory. In: Proceedings of the PSIG, 26th annual meeting, San Diego, 1994 48. Wong PJ, Larson RE (1968) Optimization of natural-gas pipeline systems via dynamic programming. IEEE Trans Autom Control AC-13(5):475–481 49. Varian HR (2006) Intermediate microeconomics, 7th edn. W. W. Norton, New York 50. Ferrero R, Vargas A, An˜o´ O, Rivera JF (1995) Valor del agua: Marco conceptual. In: Proceedings of VI ERLAC (CIGRE´), Foz do Iguacu, 1995 51. Pereira M, Campodo´nico N, Kelman R (1998) Long-term hydro scheduling based on Stochastic models. In: EPSOM’98, Zurich, 23–25 Sept 1998 52. Gardner J, Hobbs W, Lee FN, Leslie E, Streiffert D, Todd D (1995) Summary of the panel session “Coordination between short-term operation scheduling and annual resource allocations”. IEEE Trans Power Syst 10:1879–1889 53. Reneses J, Centeno E, Barquı´n J (2006) Coordination between medium-term generation planning and short-term operation in electricity markets. IEEE Trans Power Syst 21:43–52 54. Osiadacz AJ (1983) Optimal numerical method for simulating dynamic flow of gas in pipelines. Int J Numer Meth Fluids 3:125–135 55. Fishmann GS (1999) Monte Carlo, concepts, algorithms and applications. Springer, New York/Berlin/Heidelberg, pp 19–26

Costs and Constraints of Transporting and Storing Primary Energy for Electricity Generation Implications for Optimization Models Sarah M. Ryan and Yan Wang

Abstract This article describes the fuel transportation and storage components of the supply chain for electricity. We focus on dispatchable generation based on transportable fuels. Coal has very flexible transportation and storage requirements. Natural gas requires pressurized pipelines and storage facilities; or it can be liquefied, then stored and transported at very low temperatures, and then revaporized. Biomass presents logistical challenges related to its relatively low energy intensity and seasonality of supply. We review ways to model the physical constraints and cost characteristics that govern the transportation and storage of these fuels and examine their implications for decision models in restructured electricity markets. Keywords Dispatchable generation • Electricity supply chain • Equilibrium • Fuel transportation • Optimization

1 Introduction Delivery of electricity to end-users is the final step in a series of processes that begins with extraction and refinement of fuels, includes fuel transportation and storage, continues with electricity generation and transmission over long distances at high voltage, and ends with distribution in local networks. While local generation can be advantageous for industrial users, particularly where power can be generated simultaneously with large amounts of needed heat, economies of scale dictate that most electric power is generated in large centralized facilities. In 2007, 42% of the worldwide electricity was generated from coal, 21% from natural gas, 18% from renewables including hydropower, and 14% in nuclear reactors. Excluding wind

S.M. Ryan (*) • Y. Wang Iowa State University, Ames, IA, USA e-mail: [email protected]; [email protected] A. Sorokin et al. (eds.), Handbook of Networks in Power Systems II, Energy Systems, DOI 10.1007/978-3-642-23406-4_6, # Springer-Verlag Berlin Heidelberg 2012

169

170

S.M. Ryan and Y. Wang

and hydropower, the largest source of renewable electricity generation worldwide was biomass at 1.3% [1]. In the U.S. in 2008, nearly half of the electricity generated came from coal and just over 20% was generated from natural gas. Nuclear power accounted for nearly 20%, hydroelectric plants generated 6%, and other renewables totaled 3% of generation, including 0.4% from biomass. Fuel costs accounted for 80–90% of the operating costs of generation in fossil steam, gas turbine and smallscale plants [2]. Current concerns with climate change and recent developments in deregulated energy markets have prompted reexaminations of how primary energy sources are used to generate electricity. Resolutions and regulations to reduce carbon emissions have contributed to increased development of renewable generation capacity as well as motivation to switch from coal-fired to gas-fired generation [3]. Deregulation of energy markets has increased the volatility of fuel prices, introduced profit motives for generators, and increased competition for both electricity demand and fuel supplies. At the same time, underinvestment in infrastructure during decades of electricity demand growth has resulted in tighter constraints in transmission capacity, while accidents and disasters have drawn attention to constraints in fuel transportation. For example, a natural gas pipeline disruption in El Paso in 2000 was judged to be a contributing factor to the California energy crisis of 2000–2001 [4]. Electricity prices in New England rose in 2005 as a result of disruption in the supply of natural gas by hurricanes Katrina and Rita in 2005 [5] and the U.S. Federal Energy Regulatory Commission (FERC) observed that high natural gas prices in the Northeast signaled the need for expanded delivery infrastructure [6]. The vulnerability extends to coal transportation as well; in June, 2006, due partly to a train derailment in Wyoming’s Powder River basin the previous year, FERC held a public meeting to discuss potential bottlenecks in the rail delivery of coal to the electric utilities [7]. This article describes optimization-relevant characteristics of the supply chain for electricity that consists of fuel transportation, electricity generation and power transmission. We focus on dispatchable generation based on transportable fuels. Dispatchable generators are those that can be turned on and off as needed (possibly over a period of several hours) to respond to changes in demand, satisfy reliability requirements, and exploit market prices for fuels and electricity. Nuclear reactors are not considered to be dispatchable because they cannot be started up or shut down easily or even adjust their output significantly when running. Wind and solar power rely on intermittent sources of primary energy and typically are used as much as possible when those sources are available. They also must be located where the wind or solar energy exists. Hydropower is dispatchable but must be located where the water resources exist – its “fuel” is not transportable. The remaining renewable resource, biomass, joins coal, natural gas, and oil as fuels that can be transported economically to dispatchable generators. Because liquid oil is not used in significant amounts in the U.S. and its share of total generation internationally is predicted to decline in the future [1], we do not discuss it here. Coal, natural gas, and biomass together account for over two-thirds of net generation in the U.S. and nearly that proportion worldwide. Optimization models

Costs and Constraints of Transporting and Storing Primary Energy

171

could help determine how to best exploit these resources to meet residual demand after nuclear, hydro and intermittent generation is taken into account. Considering fuel transportation along with electricity transmission allows comprehensive comparisons of the economic and environmental impacts of different methods for moving energy from the location of fuel extraction to the point of its end-use as electric energy. For example, to satisfy incremental demand for electricity using coal, the options include “coal by rail”, in which coal is transported to the load center where it is used in a coal generation plant; “coal by wire”, in which electricity is delivered by high-voltage transmission line from a mine-mouth plant; “coal to gas by pipeline”, in which coal is gasified and the synthetic gas is delivered by pipeline to a combined cycle plant near the load; and “coal to gas by wire”, in which electricity is transmitted to the load from a gasifier and combined cycle plant at mine-mouth [8]. We examine the physical, economic and environmental characteristics of transporting each fuel and generating electricity from it. Where applicable, we also survey optimization models that have included these features. Economies of scale influence the tradeoffs between investment and operating costs as well as the prevalence of fuel purchases in long term contracts rather than spot markets. Fuel transportation costs must be weighed against electricity transmission losses when determining generating plant size and location. Uncertainties concerning future voluntary efforts and regulations to limit carbon and other emissions affect capacity investment decisions. Deregulated fuel and electricity markets increase the volatility of prices both upstream and downstream of each generator. Understanding these characteristics and interactions helps the modeler decide where and to what extent to include discrete, nonlinear and stochastic components in electricity supply chain optimization models.

2 Existing Supply Chain Models Models have been constructed in government and academia to simulate the energy system in the U.S. and used to study the interactions between fuel and electricity subsystems. Beginning in the mid-1970s, the U.S. Department of Energy’s Energy Information Administration (EIA) and its predecessor, the Federal Energy Administration (FEA), developed a sequence of computer-based, medium-term energy modeling systems to analyze domestic energy markets and the relationships among electric energy and fuels. The first of these, the Project Independence Evaluation System (PIES) [9] was employed by the FEA prior to 1982. It was designed to provide a framework for the development of a national energy policy through quantitative analysis and projections of the energy system. PIES incorporated effects of fuel price sensitivity, fuel competition, technology restriction or improvement, resource limitations, economic factors, and regional variations on the energy system. In 1982, PIES was updated to the Intermediate Future Forecasting System (IFFS) [10].

172

S.M. Ryan and Y. Wang

In contrast to the PIES, IFFS was partitioned by fuel to avoid complex integration issues and balance the workload among the staff in charge of submodels. In 1993, the IFFS was replaced by the National Energy Modeling System (NEMS) [11]. Coupling advanced modeling and optimization techniques with the latest computing technology, NEMS combines and processes more energy information than its predecessors and therefore is more capable with projections. The system is used to test different assumptions about energy markets and to evaluate the potential impacts of new and advanced energy production, conversion and consumption technologies. Some recent efforts have been made to formulate and analyze optimization or equilibrium models of the supply chain composed of both fuel and electricity. Quelhas et al. [12] formulated a generalized minimum cost network flow model of the bulk energy transportation system in the U.S. When applied in a case study of a recent year, the results suggested that the cost to supply a fixed demand for electricity could be lowered by substituting more coal for natural gas and increasing interregional electricity trade [13]. However, the authors acknowledged that gas turbines may have been underutilized as peaking generators in the case study due to load aggregation. The deterministic nature of the model also may have misrepresented decision-making in the context of fuel price volatility. Wang and Ryan [14] implemented a two-stage stochastic programming version of the model in a rolling horizon simulation with gas price forecast updates and found that the simulated results matched the actual fuel mix much more closely. Liu and Nagurney [15] formulated a network equilibrium model of fuel and electricity markets and applied it in case studies of the New England region of the U.S. They assumed fuel prices would be constant over the medium-term study period while electricity prices could fluctuate, and that the electric power market was perfectly competitive. The results showed the effects of fuel price variations on electricity price variations; revealed interactions among the markets for power, natural gas and fuel oil; and illustrated the impacts of electricity demand changes on the markets for both electric power and natural gas. Ryan et al. [16] combined a simple fuel transportation model with an equilibrium model for the electricity market in which generators could strategically set their output levels based on both their marginal fuel costs and the geographic variation of electricity locational marginal prices. The equilibrium model was validated by comparison with a detailed computational agent simulation [17] and the effect of demand price sensitivity on generator market power was investigated [18]. While NEMS is very comprehensive, it is not available for researchers outside of the U.S. government to use or modify for their own purposes. Many sophisticated models of electricity markets have been developed by academic researchers, but few incorporate the supply, transportation, and storage of fuels in any detail. In the next section, for several different fuels we highlight characteristics of the supply chain and review examples of optimization or equilibrium models for both the fuel itself and its use in generating electricity.

Costs and Constraints of Transporting and Storing Primary Energy

173

3 Characteristics of Transportable Fuels for Dispatchable Generation Fuel supply and transportation arrangements vary according to the physical characteristics of the fuels, the handling and preparation needed to use them in electricity generation, and the economic impacts of regulation. Discrete, nonlinear, and lead time effects have been incorporated to varying extents in analyses and optimization models.

3.1

Coal

In the U.S. in 2009, 94% of coal consumption was for electric power. Over half of the production was of Western, low-sulphur coal, while 32% came from the Appalachia region in the Eastern U.S. and 14% from the country’s interior [19]. Coal is easy to transport and store. It is transported mostly by rail but also barge, truck, and other means including slurry pipeline. Multimodal transportation is used for the longest distances of over 1,000 miles, while rail is used for shipments averaging 800 miles, barge for distances of approximately 300 miles, and truck or other means for the shortest distances [20]. Most rail shipments are in dedicated “unit trains” of over 100 cars that transport only coal to a single location [21]. At the end of 2009, coal stocks in the electric power industry equaled 19% of the amount consumed for the year, but this was a record high level attributable to low demand during the year caused by economic downturn, weather patterns that reduced demand, and lower prices for natural gas [19]. Most coal consumed by electric utilities is purchased in long-term contracts, which by definition span at least 1 year but can last up to 50 years. In 1997 the average length of a utility coal contract was nearly 16 years [20]. Joskow conducted a series of studies on such contracts negotiated in the 1960s and 1970s to determine the reasons for their prevalence and the strength of their provisions for price adjustments when spot market prices dropped. He found that contract duration increased with the level of relationship-specific investment, including (1) site specificity as in mine-mouth generation plants “typically developed simultaneously with the mines themselves”; (2) physical asset specificity such as generators designed to use a specific type of coal; and (3) dedicated assets such as capacity built by the supplier to meet demand from a specific customer [22]. The most common type of contract was “base price plus escalation (BPE),” which specified a price based on market conditions at contract signing and an adjustment formula that incorporated exogenous input-price indexes, such as for labor, materials and supplies, explosives, electricity, and general inflation [23]. The contracts also had “take-or-pay” provisions; i.e., minimum and maximum quantities that the buyers were obligated to take [24]. MacDonald studied how long term coal contracts were modified in the 1980’s in response to falling prices for both coal itself and rail transportation. Economies of

174

S.M. Ryan and Y. Wang

scale are significant in rail transportation as rail costs per ton moved in a dedicated unit train decrease with the train size [25]. Dennis studied the relationship between coal delivered prices and transportation rates before and after railroad deregulation occurred in 1980. Based on a spatial equilibrium model, under which the variation in prices across regions depends only on the variation in transportation rate, he showed that the standard deviation (across regions) of delivered coal prices had dropped mostly steadily from the mid-1970s to 1996, and concluded that rail deregulation had resulted in lower transportation rates for coal [26]. The prevalence of long-term coal contracts results from economies of scale in both transportation and generation and the resulting need for both supplier and consumer to make significant investments in their capability to handle large volumes of material. It follows that generation capacity investment models should include fuel purchase and transportation contracts along with physical plant decisions. An example of such a model is that of Bienstock and Shapiro [27], who included the acquisition of coal contracts in their stochastic programming study of electric utility capacity expansion. Possible investments in the first stage of a 10-year study horizon included coal plants with and without flue gas desulphurization, a gas plant, an oil plant, a peaking plant, two different longterm coal contracts, and a slurry pipeline. Coal purchases on the spot market were included in the second-stage, operational variables. Uncertainties included demand, air pollution (SO2) control limits, the date the slurry pipeline would be completed, and fuel prices. Integer decision variables were used to model fixed and nonlinear costs for adding plants and evaluating fuel purchase contracts.

3.2

Natural Gas

Natural gas is a major source of electricity generation through the use of gas turbines and steam turbines. Natural gas burns more cleanly than other fossil fuels, such as oil and coal, and produces no sulfur dioxide and less carbon dioxide than coal. In addition, gas-fired generating units have distinct advantages of high efficiency, fast response and shorter installation time. It is projected that, by 2035, 46% of the new generation capacity in the U.S. will use natural gas as the preferred fuel [28]. It is transported primarily by pipeline and has many uses other than electricity generation – only 29% is currently used for electricity generation in the U.S. Restructuring in the past decades in Europe and by the Federal Energy Regulatory Commission (FERC) in the U.S. has reduced the vertical integration in the gas industry.

3.2.1

Pipeline Transmission and Distribution System

The natural gas transportation system is composed of the transmission network, which transmits gas at high pressure from gas wells to regional demand points, and

Costs and Constraints of Transporting and Storing Primary Energy

175

the distribution network, which distributes gas to consumers at medium or low pressure from the regional demand points. Researchers often investigate these two systems separately given their distinct functions and features. Optimization of the allocation of natural gas flows is non-trivial because the relevant models usually involve nonlinear constraints that represent physical requirements of the pressure drop at each pipe segment, compressor and valve within the system. O’Neill et al. [29] built a network model of the U.S. intrastate pipeline system of gas suppliers, transporters and end-users, in order to simulate gas reallocation in response to state legislation that partitioned all users of natural gas into nine priority classes and required that the curtailment of gas supplies to higher priority groups, usually residential users, is implemented only after full curtailment of supplies to lower priority groups (industrial and commercial demands). The model minimizes total gas transferred in the system while satisfying mass flow balance constraints and the nonlinear constraints posed by pressure conditions. Two different techniques are used to linearize the nonlinear constraints for pipe segments and compressors, respectively. In the iterative allocation procedure, gas is allocated according to priorities from high to low, then linearized constraints are checked (if they are not satisfied, a new set of linearized constraints are generated using the current solution), until the conversion criteria is met. Results obtained from this model advised the Louisiana Department of Conservation how gas should be optimally allocated to ensure supplies to the nine categories of consumers. DeWolf and Smeers [30] considered a similar problem but at a smaller scale. In this paper, a gas company that operates a transmission network and wants to minimize its total cost of supplying gas must decide the quantities of gas it buys from each supplier and satisfy the demands of the distribution nodes at certain minimum pressures that are pre-determined. Instead of linearizing the nonlinear constraints, they later proposed a new method, the first step of which searches for a good starting point by ignoring the nonlinear constraints. In the second step, the constraints are linearized piecewise according to the initiation points obtained and a new optimal point is identified. The program iterates between updating linearization of the nonlinear constraints and solving for another optimal point until the critical pressure is within a certain tolerance [31]. Within the transmission network, pressure is lost due to friction between the gas and the pipes. Compressors, which consume about 2% of the gas flowing through, are used to compensate for the pressure loss and ensure that gas is supplied to distribution systems at the required volume and pressure. The gas consumption at each compressor depends on the inlet (suction) pressure, the outlet (discharge) pressure and the mass flow rate through the compressor. The physics of gas flowing through a pipe is described by three sets of partial differential equations, of which the momentum equation is most important and, in the stationary case of long-term gas network planning, reduces to a nonlinear constraint in terms of squares of the pressures. A great amount of effort has been devoted to solving for the optimal flow rates and pressures that minimize operational costs (total fuel consumption at the compressors) while satisfying demands of sub systems. With the nonlinear constraints, this problem is proved to have a non-convex feasible region. In

176

S.M. Ryan and Y. Wang

addition, the presence of combinatorial decision variables for compressor status makes it even more difficult to find the optimal solution. Rı´os-Mercado et al. [32] explored the pipeline network properties and developed a technique based on graph theory to reduce the dimension of the problem, which greatly facilitated the search for a solution. Martin et al. [33] constructed piecewise linear approximations of the nonlinear functions that are linked by sub-polyhedra. The authors came up with an exact separation algorithm for the non-linear constraints and implemented an appropriate branching strategy that is able to discover the global optimal solution. Chebouba et al. [34] proposed an ant colony optimization algorithm and compared the results with those obtained from traditional approaches such as dynamic programming. There are also studies looking beyond operational planning to optimize the investment cost of gas pipeline networks. DeWolf and Smeers [31] designed a two-stage problem to decide the optimal diameter for each pipe that minimize the sum of investment and operation costs in a gas transmission network of which the topology is known. It is proved that the operational cost is nondifferentiable and nonconvex in the investment decision. A nonsmooth optimization technique, the bundle method, is applied to solve for the optimal diameters. The model is applied to a real world problem of building a new trunkline in Belgium in order to deliver Norwegian gas to Belgium and northern France. Andre et al. [35] presented techniques for deciding the location and size of new parallel pipelines on the existing network in order to satisfy future demand increase and minimize total investment cost. A two phase approach first solves a continuous relaxation of the problem to locate the pipelines and then applies a branch and bound scheme to choose best diameters within the given set of available sizes. The design of the distribution network is less well studied than that of the transmission network. Djebedjian et al. [36] focused on developing a computer program to simulate and optimize gas distribution networks with medium- or lowpressure gas flow inside. The objective is to find the minimum diameter sizes that fulfill the requirements of maximum link velocity and minimum node pressure. The gradient algorithm used for the gas network simulation significantly reduces the computational time when compared to other numerical schemes. A genetic algorithm was employed for optimization. Steinbach [37] looked into the extremely complex and mostly unexplored problem of the short-term (24–48 h) natural gas operational planning problem. The large-scale model includes both partial differential equation constraints for the transient status of gas flows and combinatorial decisions indicating the start-up and shut-down statues of compressors. A custom solution algorithm based on structural properties of Karush-Kuhn-Tucker (KKT) systems was proposed. Midthun et al. [38] were the first to include physical features of the gas network in analysis of natural gas markets. With multiple objectives such as maximizing social surplus and consumer surplus and constraints accounting for gas pressure requirements, this paper illustrated the importance of combining economics with system effects implied by the physical structure of the network.

Costs and Constraints of Transporting and Storing Primary Energy

3.2.2

177

Contracts for Gas Purchase, Transportation and Storage

Gas distribution utilities formerly purchased most of their gas from pipeline transmission companies through long-term contracts which typically were composed of commodity charge, demand charge and winter requirement charge. The latter two charges, together with a “take-or-pay” clause, induce distribution utilities to use storage facilities and add interruptible industrial customers to increase the ratio of the average daily purchase to the peak-day purchase [39]. The restructuring of the natural gas industry that began with FERC Order 436 (1985) and was substantially completed with Order 636 (1992) has changed gas transportation patterns and rates [40]. The role of pipeline companies was transformed from merchants to nondiscriminatory carriers so that local distribution companies (LDCs) can now purchase gas directly from the producers. Duann [41] discussed the economic effect of drastic increases in the amount of gas directly purchased by LDCs, which induced more responsive adjustments of gas production and transportation capacity. New market flexibility has contributed to greater efficiency in the use of the gas industry infrastructure. As the unbundling of pipeline sales, transportation, and storage increased competition among gas suppliers, LDCs face more complicated options in gas procurement, transportation and storage. However, they see opportunities to achieve cost savings by building a least-cost gas supply portfolio [42]. Structural changes in the natural gas industry have increased the importance of systematic analysis of supply combinations by LDCs. Optimization models designed for this problem are usually multi-period cost-minimizing linear programs with supply, demand and storage constraints. Choices are made between firm and interruptible pipeline reservations, firm and interruptible sales, spot purchase and long term contract, injections and withdrawals from storage [43]. The natural gas demand market is divided into submarkets of residential, commercial, and industrial end users. Gas demand mostly depends on weather conditions and is therefore fundamentally highly variable. Guldmann and Wang [42] used a two-stage stochastic mixed-integer linear program to incorporate uncertain demand in the gas procurement and delivery decision process, where contracts are determined in the first stage and the operational variables are selected as recourse to the demand scenarios. The authors proposed a simulation method which approximates the total cost as a response function of demand by sampling and regression. This approach provides a convenient way to conduct sensitivity analyses over parameters other than demands. Unlike most prevailing models where deliverability, the maximum amount of gas that can be withdrawn from or injected into storage each day, is given as exogenous, another linear stochastic program proposed by Bopp et al. [44] explicitly considers it as a decision variable that closely ties and impacts other variables. A case study of the LDC at Huntsville, Alabama, showed that in order to achieve minimum cost, deliverability must be carefully chosen according to the minimum level of firm transportation requirements and the availability of spot purchases. The total costs decrease if

178

S.M. Ryan and Y. Wang

more storage facilities (thus more rates of deliverability) are available. The model suggests that LDCs negotiate for a rate of deliverability close to the optimal rate. In a 1 year horizon model that aids a utility in selecting gas contracts, Butler and Dyer [45] adopted a daily-weekly-monthly method to aggregate future periods and yet still obtain insights regarding current daily operational decisions. While most studies in the literature focus on the U.S. natural gas industry, Contesse et al. [46] developed a mixed-integer model for Metrogas, a Chilean LDC, in support of its daily operations. This model incorporates special features that do not appear in the U.S. gas system, including make-up generation and recovery, the aging of generated make-up, the adoption of linepack to handle demand fluctuation and the inclusion of fines for different types of deficits and overruns. Evidently, the use of linear programs in selecting an optimal gas supply portfolio is easily adapted to various industrial specifications. Restrictions on the desired level of detail provided are usually due to constraints on computational power available.

3.2.3

Restructured Markets for Natural Gas

Restructuring of the gas markets in the U.S. and Europe has reduced the amount of vertical integration and provided opportunities for different entities in the supply chain to pursue their own objectives. Because decision-making is distributed among multiple agents whose objectives may not be perfectly aligned, equilibrium models are more appropriate than system optimization from a single perspective. Moreover, while a major goal of restructuring was to increase competition, the large market shares held by a few entities have allowed those players to exert oligopolistic power. Gabriel et al. [47] reviewed models of gas markets in the U.S. and formulated a nonlinear complementarity model in which gas marketers compete as Nash-Cournot players and other participants including pipeline, production, storage reservoir and peak gas operators as well as consumers are price-takers. Gabriel and Smeers [48] noted that the mixture of regulated and non-regulated behavior that is present in both U.S. and European markets is naturally captured in complementarity models. In the EU, market power is held by gas producers rather than marketers, as captured in equilibrium models reviewed and extended by Lise and Hobbs [49].

3.2.4

Interdependency of Natural Gas Network and Power System

The market price of natural gas directly affects the commitment, dispatch and cost of power supplies. Interruptions or outages in gas network could greatly reduce the available supply of electricity and jeopardize the power system security. The power system, on the other hand, also has a significant impact on the natural gas market because it uses 29% of total gas consumption. When the demands for electricity and natural gas peak at the same time because of, for example, severe weather

Costs and Constraints of Transporting and Storing Primary Energy

179

condition, the price could rise sharply in the gas market, which results in higher generation costs. Shahidehpour et al. [50] summarized the structure of the U.S. natural gas transportation system and analyzed its interdependency with the electricity system. The authors introduced constraints of gas supply capacities and emission limits into the security-constrained unit commitment (SCUC) model. They studied the fast response of combined-cycle units compared to thermal units, the impact of gas price on gas-fired generation and the impact of gas infrastructure outages on the locational marginal price (LMP) of electricity. Li et al. [51] further investigated the inter-relationship of the two systems by incorporating linear natural gas network constraints into the optimization of SCUC for a vertically integrated utility’s daily operation planning. Liu et al. [52] improved the model in [51] by using nonlinear equations to represent constraints of the natural gas transmission system. The complex nonlinear mixed-integer problem was solved by decomposition techniques. Takriti et al. [53] presented a dynamic program for unit commitment which also accounts for fuel constraints and prices that vary with uncertain future power demand and price. In [54], the authors focused on hedging between the natural gas market and electric power market. A common strategy for a gas utility to take this advantage is to purchase power generating assets so that it can turn some of its supply into electricity to profit from soaring electricity prices in the summer when gas prices are low. Scenarios of gas price and demand and electricity price are produced as input for a stochastic program. The optimal solution advises a utility on when to buy or sell natural gas and when to convert gas into electric power. Instead of a cost-minimization-based framework, a risk-cost trade-off was considered by Chen and Baldick [55]. An integrated simulation-optimization algorithm was proposed to solve for optimal gas supply portfolio that maximized the utility function for a power generation entity. Geidl and Andersson [56] introduced a new concept of “energy hub”, which includes basic elements of connections, converters and storage. An energy hub can represent real facilities such as power plants and large buildings. It enables the integration and optimal dispatch of multiple energy carriers and products.

3.2.5

Liquefied Natural Gas

Liquefaction has emerged recently as an alternative means of transporting natural gas over long distances. In general, gas is difficult to store or transport (compared to oil, for example) because it requires high pressure and/or low temperature to increase the bulk density [57]. Natural gas liquefies at a temperature around 162 C and has a volume around 1/600 that of gas at room temperature. The production chain for liquefied natural gas (LNG) consists of a short pipeline from the gas well to a liquefaction terminal that consists of a liquefaction plant and storage tank, vessels to carry the LNG, and a vaporization (regasification) terminal, which has a LNG tank and vaporization plant [58], from which it can enter the

180

S.M. Ryan and Y. Wang

natural gas distribution system. The chain is very capital intensive. Huge cryogenic tanks, around 70 m in diameter and 45 m high, holding over 100,000 m3, are required to store LNG, and specialized tanker ships are required to transport it [57]. These ships are powered by steam turbines that use boil-off gas in a gas boiler [59], supplemented if necessary by fuel oil. Tanker capacity was 130,000 m3 in 2004 but projected to increase to 200,000 m3 by 2020 [60]. In 2007, there were five LNG terminals in the U.S. and 45 proposed ones, 18 of which had been approved by FERC. LNG is attractive to suppliers as a way to monetize natural gas for export. For both thermodynamic efficiency and low cost, LNG facilities must be large scale. Long contracts, on the order of decades, are required to justify the investments. Transport and liquefaction account for approximately 85% of the cost to deliver LNG to the vaporization terminal. The cost is less dependent on distance than is the cost of pipeline gas [57]. Although natural gas-based generation is promoted for environmental reasons, Jaramillo et al. [59] found that the life-cycle air emissions of gas are not much lower than those of coal when LNG is included, due to emissions caused by the liquefaction and transportation stages. With advanced generation technology and carbon capture and storage, coal would have similar emissions to gas in various forms. Cayrade [60] studied pipeline and LNG gas infrastructure necessary to meet Europe’s growing demand, motivated by GHG reduction. He found that LNG is competitive with pipelines at long distances; however, clean coal could also be competitive. Because of the large scales involved, infrastructure investment must overcome uncertainty about future gas prices as well as other factors. However, LNG at competitive prices was available in Europe in 2009 [61]. Zheng and Pardalos [62] formulated a two-stage stochastic mixed integer program for simultaneously expanding a transmission pipeline network and locating LNG terminals. Discrete expansion sizes and choices among candidate facility locations were modeled with binary variables while uncertain demands and supply limits were formulated as discrete probabilistic scenarios. For either a risk-neutral or a riskconstrained objective, the presence of integer variables in the second stage required the development of a specialized solution algorithm. Despite the significance of economies of scale in its associated facilities, LNG can be a viable alternative at smaller scales as well. Kuwuhara et al. [58] developed a model to optimize the supply of LNG to remote Amazonian regions of Brazil as a possibility to replace diesel generators by gas turbines. Transport would be by ship on local rivers to towns located between 1,000 and 1,500 km from the liquefaction terminal, with travel times of 72 h or more. The model was a linear program with a single nonlinear constraint that also involved integer variables representing the number of trips per month and the number of ships required. It found the optimal capacities of the liquefaction and vaporization plants, including their storage tanks, and the optimal capacities of the ships. The cost was most sensitive to variation in cost of the liquefaction plant and least sensitive to changes in cost of the storage tanks. The conclusion was that LNG could be economically competitive to diesel for supplying electricity to small, stand-alone networks.

Costs and Constraints of Transporting and Storing Primary Energy

3.3

181

Biomass

Biomass is increasing in importance as a fuel for electricity production because of its environmental benefits. Sources include agricultural residues, forestry residues, and dedicated plantings such as poplar forests or switchgrass. It generates lower emissions of SO2, NOx and CO2 than coal-based plants and, when gasified, also has lower particulate emissions [63]. There are significant scale effects in both generation and fuel transportation and considerable uncertainty about when advanced technologies might become economically viable. To answer basic questions about viability in relation to scale, Dornburg and Faaij [64] compared biomass energy generation systems at different scales based on fossil primary energy savings and total costs per unit of primary energy saved. Energy savings increase with scale and total costs per unit of primary energy saved mostly decrease with scale. There are substantial scale effects in investment costs, efficiencies, and transportation of the biomass (forestry residues). The most efficient units, available at scales from about 50–300 MW, are biomass integrated gasification combined cycle (BIGCC) units, but these were still in the demonstration stage as of 2001. The results were sensitive to biomass costs, which are highly uncertain. Despite the uncertainty, biomass is one of the bigger sources of renewable energy – in 2000, it provided 48% of the energy coming from all renewable sources in the US [63]. Using NEMS, which can represent dedicated biomass or biomass gasification, co-firing biomass with coal, open-loop combustion of biomass, and use in industrial cogeneration, Haq studied how biomass and other renewable generation could meet different renewable portfolio standards (RPS). He found that a low RPS requirement, mandating that 10% of electricity would be generated from renewable energy sources other than hydro, would first be met by wind, but a 20% RPS would involve more biomass gasification. Uncertainties associated with supply of biomass include the value of competing uses, the impact of removal of agricultural residues on the soil, and the impact of forest fire prevention policies on forestry residue availability. BIGCC has a high capital cost compared to coal or natural gas – particularly, higher than coal IGCC because of the need for additional feed preparation. At the time of the study, the main impediment to biomass utilization had been the cost of obtaining the feedstock. The logistics of fuel supply are important considerations in the viability of biomass generation [65]. Problems include the limited availability due to seasonality and the scattered geographical distribution. Both combustion for steam units, considered a mature technology, and gasification for combined cycle units, still an emerging technology, show strong economies of scale in total capacity investment. The combustion approach generates higher logistic costs than gasification, and these logistic costs play a major role in making biomass more expensive than coal or gas. However, the constraints are less restrictive for larger plants. Whether biomass can compete with coal depends on the extent to which carbon emissions are regulated or taxed. As part of the Third Assessment Report of the

182

S.M. Ryan and Y. Wang

Intergovernmental Panel on Climate Change, Sims et al. [66] examined the carbon emission and mitigation costs of fossil fuel, nuclear power, and renewable energy resources. They projected that renewable energy would continue to grow but, in the absence of significant government intervention, still would provide less than 2% of the electricity market share in the year 2020. Gasification improves thermal efficiency and, because biomass fuels contain oxygen and little sulphur, they are easier and cheaper to gasify than coal. Biomass fuels could provide rural areas in particular with renewable energy. The report concluded that in the US, biomass could contribute to CO2 mitigation, especially where forest or agricultural residues are available at very low or negative costs. Gan and Smith [67] used a computable general equilibrium model to simulate coal prices under CO2 emission reduction and tax schemes and compared the costs of electricity generation by woody biomass with coal – specifically, between BIGCC, conventional pulverized coal, and integrated coal gasification combined cycle units. For sources of biomass they considered logging residues and poplar plantations. The generic BIGCC unit studied was smaller than either of the coal plants because of the biomass transportation costs. Without any mandated reductions in CO2 emissions, the biomass production costs would have to be reduced substantially to compete with coal, but imposing a CO2 emission tax would lower that requirement. Woody biomass energy is nearly CO2-neutral but improving the productivity of production, harvesting and transport systems is necessary for adoption on a large or wide scale. However, in some locations, electricity generation by biomass alone or cofiring biomass is already competitive on a small scale. A recent study by the EIA examined a combined 25% RPS with a 25% renewable fuel standard for transportation by 2025 [68]. In addition to requiring that at least 25% of electricity sales be produced from renewable sources by 2025, the RPS would establish a market for renewable energy credits. The NEMS-based results showed a dramatic shift from coal and natural gas to biomass and wind power. Advanced biomass generation (i.e., BIGCC) would be required, but when full-scale commercial plants will be available was still uncertain. The major impediment is the front end handling and processing of the feedstocks, which limits utilization to below 60% because of frequent jams. Also, the supply and cost of biomass energy crops will be critical. Because renewable capacity would offset new capacity of more efficient coal and natural gas generation, the RPS “actually increases the amounts of coal and natural gas consumed per kilowatt hour of electricity generated” [68, p.26]. The impact of the RPS was found to be very sensitive to assumptions about prices for fossil fuels and availability of advanced biomass technology. Despite the challenges posed by economies of scale, logistics of fuel transportation, and uncertainties about fuel price and technology development, optimization models for expanding biomass generation or integrating it with other forms of electricity generation seem to be absent in the literature. We expect that, as concern for reduction of carbon emission increases, such studies will be very valuable for determining how to most efficiently increase the utilization of this renewable and relatively clean energy source.

Costs and Constraints of Transporting and Storing Primary Energy

183

4 Conclusions and Implications for Research Despite the importance of fuel availability and transportation in determining the cost of producing electricity, relatively few attempts have been made to model the fuel supply chain together with electricity generation and transmission. Instead, models of the electricity system have treated generation costs as exogenous and typically having simple functional forms, while models of fuel transportation and allocation have represented demand from electricity producers in similarly approximate ways. Restructuring of both electricity and fuel markets has sparked recent development of distributed decision-making models of each sector separately. As these have gained in sophistication to include strategic behavior and nonlinear physical constraints, the ground has been prepared for building models in which fuel sector decisions interact with electricity supply and demand. For example, systematic optimization models can help electricity providers to design their fuel portfolios, composed of long-term contracts and spot market purchases. Equilibrium models can be used to include different participants and to simulate their behaviors in the deregulated energy systems. The emergence of LNG, biomass and other alternative energy sources promotes the necessity of using optimization models to examine the economic efficiency of generating electricity from newer fuels. Great uncertainties associated with technology advances and regulatory policies must be considered in making capacity investment decisions. As greater computational capacity becomes available and more widely exploited, large-scale optimization models can be implemented for both shortterm operational planning with desired details and long-term investment decisions under uncertainties such as climate change, fuel availability and policy enactment. Judicious use of nonlinear functions and discrete variables can more accurately represent physical transportation constraints, economies of scale, contractual arrangements, and market conditions. More accurate and comprehensive models for the dispatchable supply of electricity from transportable fuels will also help to determine how to integrate increasing supplies of generation that is nondispatchable or based on geographically-bound sources of primary energy.

References 1. Energy Information Administration (2010) International energy outlook 2010. U.S. Department of Energy, Washington, DC 2. Energy Information Administration (2010) Electric power annual 2008. Department of Energy, Washington, DC 3. Casten S (2010) Fuel swap: natural gas as a near-term CO2 mitigation strategy. Public Utilities Fortn 148(4):40–44 4. Federal Energy Regulatory Commission (2005) The Western energy crisis, the Enron bankruptcy, and FERC’s response. http://www.ferc.gov/industries/electric/indus-act/wec/chron/ chronology.pdf. Accessed Sept 2009

184

S.M. Ryan and Y. Wang

5. Commonwealth of Massachusetts (2006) Electricity price, reliability, and markets report 2005. 6. Federal Energy Regulatory Commission (2008) 2007 state of the markets report 7. Clayton M (2006) Enough coal on hand to keep US cool? Christian Science Monitor, 25 May 2006 8. Bergerson JA, Lave LB (2005) Should we transport coal, gas or electricity: cost, efficiency and environmental implications. Environ Sci Technol 39(16):5905–5910 9. Hogan WW (1975) Energy policy models for project independence. Comput Oper Res 2:251–271 10. Murphy FH, Conti JJ, Shaw SH, Sanders R (1988) Modeling and forecasting energy markets with the intermediate future forecasting system. Oper Res 36(3):406–420 11. Energy Information Administration (2003) The national energy modeling system: an overview. U.S. Department of Energy, Washington, DC 12. Quelhas A, Gil E, McCalley JD, Ryan SM (2007) A multiperiod generalized network flow model of the U.S. Integrated energy system: part I – model description. IEEE Trans Power Syst 22(2):829–836 13. Quelhas A, Gil E, McCalley JD (2007) A multiperiod generalized network flow model of the U.S. Integrated energy system: part II – simulation results. IEEE Trans Power Syst 22(2): 837–844 14. Wang Y, Ryan SM (2010) Effects of uncertain fuel costs on optimal energy flows in the U.S. Energy Syst 1:209–243 15. Liu Z, Nagurney A (2009) An integrated electric power supply chain and fuel market network framework: theoretical modeling with empirical analysis for New England. Nav Res Log 56(7):600–624 16. Ryan SM, Downward A, Philpott AB, Zakeri G (2010) Welfare effects of expansions in equilibrium models of an electricity market with fuel network. IEEE Trans Power Syst 25 (3):1337–1349 17. Ryan SM (2009) Market outcomes in a congested electricity system with fuel supply network. In: IEEE Power Engineering Society General Meeting, Calgary 18. Ryan SM (2009) Demand price sensitivity and market power on a congested fuel and electricity network. In: IEEE Power & Energy Society General Meeting, Minneapolis, 25–29 July 2010 19. Freme F (2010) U.S. coal supply and demand: 2009 review (trans: Administration EI). U.S. Department of Energy, Washington, DC 20. U. S. Energy Information Administration Coal transportation information. http://www.eia.doe. gov/cneaf/coal/ctrdb/ctrdb.html. Accessed Jul 2010 21. U.S. Energy Information Administration (2006) Coal production in the United States – an historical overview. U.S. Department of Energy, Washington, DC 22. Joskow PL (1987) Contract duration and relationship-specific investments: empirical evidence from coal markets. Am Econ Rev 77(1):168–185 23. Joskow PL (1988) Price adjustments in long-term contracts: the case of coal. J Law Econ 31:47–83 24. Joskow PL (1990) The performance of long-term contracts: further evidence from coal markets. Rand J Econ 21(2):251–274 25. MacDonald JM (1994) Transactions costs and the governance of coal supply and transportation agreements. J Transp Res Forum 34(1):63–74 26. Dennis SM (1999) Using spatial equilibrium models to analyze transportation rates: an application to steam coal in the United States. Transp Res Part E 35:145–154 27. Bienstock D, Shapiro JF (1988) Optimizing resource acquisition decisions by stochastic programming. Manage Sci 34(2):215–229 28. Energy Information Administration (2010) Annual energy outlook 2010. Energy Information Administration, Washington, DC 29. O’Neill RP, Williard M, Wilkins B, Pike R (1979) A mathematical programming model for allocation of natural gas. Oper Res 27(5):857–873

Costs and Constraints of Transporting and Storing Primary Energy

185

30. De Wolf D, Smeers Y (2000) The gas transmission problem solved by an extension of the simplex algorithm. Manage Sci 46(11):1454–1465 31. De Wolf D, Smeers Y (1996) Optimal dimensioning of pipe networks with application to gas transmission networks. Oper Res 44(4):596–608 32. Rı´os-Mercado RZ, Wu S, Scott LR, Body EA (2002) A reduction technique for natural gas transmission network optimization problems. Annals Oper Res 117:217–234 33. Martin A, M€oller M, Moritz S (2006) Mixed integer models for the stationary case of gas network optimization. Math Program 105:563–582 34. Chebouba A, Yalaoui F, Smati A, Amodeo L, Younsi K, Tairi A (2009) Optimization of natural gas pipeline transportation using ant colony optimization. Comput Oper Res 36:1916–1923 35. Andre J, Bonnans F, Cornibert L (2009) Optimization of capacity expansion planning for gas transportation networks. Eur J Oper Res 197:1019–1027 36. Djebedjian B, Shahin I, El-Naggar M (2008) Gas distribution network optimization by genetic algorithm. In: Ninth International Congress of Fluid Dynamics & Propulsion, Alexandria, 2008 37. Steinbach MC (2007) On PDE solution in transient optimization of gas networks. J Comput Appl Math 203:345–361 38. Midthun KT, Bjorndal M, Tomasgard A (2009) Modeling optimal economic dispatch and system effects in natural gas networks. Energy J 30(4):155–180 39. Guldmann J-M (1983) Supply, storage, and service reliability decisions by gas distribution utilities: a chance-constrained approach. Manage Sci 29(8):884–906 40. Energy Information Administration (2009) Major legislative and regulatory actions (1935–2008). http://www.eia.doe.gov/oil_gas/natural_gas/analysis_publications/ngmajorleg/ ngmajorleg.html. Accessed Sep 2010 41. Duann DJ (1991) Direct gas purchases by local distribution companies: supply reliability and cost implications. J Energy Dev 15(1):61–91 42. Guldmann J-M, Wang F (1999) Optimizing the natural gas supply mix of local distribution utilities. Eur J Oper Res 112:598–612 43. Avery W, Brown GG, Rosenkranz JA, Wood RK (1992) Optimization of purchase, storage and transmission contracts for natural gas utilities. Oper Res 40(3):446–462 44. Bopp AE, Kannan VR, Palocsay SW, Stevens SP (1996) An optimization model for planning natural gas purchases, transportation, storage and deliverability. Omega 24(5):511–522 45. Butler JC, Dyer JS (1999) Optimizing natural gas flows with linear programming and scenarios. Decis Sci 30(2):563–580 46. Contesse L, Ferrer JC, Maturana S (2005) A mixed-integer programming model for gas purchase and transportation. Annals Oper Res 139:39–63 47. Gabriel SA, Kiet S, Zhuang J (2005) A mixed complementarity-based equilibrium model of natural gas markets. Oper Res 53(5):799–818 48. Gabriel S, Smeers Y (2006) Complementarity problems in restructured natural gas markets. In: Seeger A (ed) Recent advances in optimization, Lecture notes in economics and mathematical systems. Springer, Berlin, pp 343–373 49. Lise W, Hobbs BF (2009) A dynamic simulation of market power in the liberalised European natural gas market. Energy J 46(Special Issue):119–135 50. Shahidehpour M, Fu Y, Wiedman T (2005) Impact of natural gas infrastructure on electric power systems. Proc IEEE 93(5):1042–1056 51. Li T, Eremia M, Shahidehpour M (2008) Interdependency of natural gas network and power system security. IEEE Trans Power Syst 23(4):1817–1824 52. Liu C, Shahidehpour M, Fu Y, Li Z (2009) Security-constrained unit commitment with natural gas transmission constraints. IEEE Trans Power Syst 24(3):1523–1536 53. Takriti S, Krasenbrink B, Wu LS-Y (2000) Incorporating fuel constraints and electricity spot prices into the stochastic unit commitment problem. Oper Res 48(2):268–280 54. Takriti S, Supatgiat C, Wu LS-Y (2002) Coordinating fuel inventory and electric power generation under uncertainty. IEEE Trans Power Syst 17(1):13–18

186

S.M. Ryan and Y. Wang

55. Chen H, Baldick R (2007) Optimizing short-term natural gas supply portfolio for electric utility companies. IEEE Trans Power Syst 22(1):232–239 56. Geidl M, Andersson G (2007) Optimal power flow of multiple energy carriers. IEEE Trans Power Syst 22(1):145–155 57. Thomas S, Dawe RA (2003) Review of ways to transport natural gas energy from countries which do not need the gas for domestic use. Energy Convers Manage 28:1461–1477 58. Kuwahara N, Bajay SV, Castro LN (2000) Liquefied natural gas supply optimisation. Energy Convers Manage 41:153–161 59. Jaramillo P, Griffin WM, Matthews HS (2007) Comparative life-cycle air emissions of coal, domestic natural gas, LNG, and SNG for electricity generation. Environ Sci Technol 41(17): 6290–6296 60. Cayrade P (2004) Investments in gas pipelines and liquefied natural gas infrastructure. What is the impact on the security of supply. The Fondazione Eni Enrico Mattei Note di Lavoro Series. Fondazione Eni Enrico Mattei 61. BP (2010) BP statistical review of world energy 62. Zheng QP, Pardalos PM (2010) Stochastic and risk management models and solution algorithm for natural gas transmission network expansion and LNG terminal location planning. J Optim Theory Appl 147:337–357 63. Haq Z (2002) Biomass for electricity generation. Energy Information Administration, Washington, DC 64. Dornburg V, Faaij APC (2001) Efficiency and economy of wood-fired biomass energy systems in relation to scale regarding heat and power generation using combustion and gasification technologies. Biomass and Bioenergy 21:91–108 65. Caputo AC, Palumbo M, Pelagagge PM, Scacchia F (2005) Economics of biomass energy utilization in combustion and gasification plants: effects of logistic variables. Biomass and Bioenergy 28:35–51 66. Sims REH, Rogner H-H, Gregory K (2003) Carbon emission and mitigation cost comparisons between fossil fuel, nuclear and renewable energy cost resources for electricity generation. Energy Policy 31:1315–1326 67. Gan J, Smith CT (2006) A comparative analysis of woody biomass and coal for electricity generation under various CO2 emission reductions and taxes. Biomass and Bioenergy 30:296–303 68. Energy Information Administration (2007) Energy and economic impacts of implementing both a 25-percent renewable portfolio standard and a 25-percent renewable fuel standard by 2025. Energy Information Administration, Washington, DC

Integrated Optimization of Grid-Bound Energy Supply Systems Simon Prousch, Hans-J€ urgen Haubrich, and Albert Moser

Abstract Energy supply is facing the challenge to be secure, competitive and sustainable, in spite of a shortage of fossil energy sources, while worldwide demand is growing, prices for oil and gas are increasing and important regions in the word are politically instable. The European Board as well as the Federal Republic of Germany have passed laws and set regulations to intensify competition, to reduce dependency of imports of recourses, to increase energy-efficiency and to support new technological solutions. Therefore, an optimization of existing energy supply systems is necessary, taking into account alternative energy supply concepts. The concepts, which are investigated in this article, consist of grids, distributed generation units and heating systems. Combinations of these components define different energy supply concepts. All energy supply concepts have an electricity supply via the electrical grid in common. Three energy carriers, and their associated networks, are commonly used to supply heat demand: electricity, natural gas and district heating. Since the complexity of the planning process increases with the number of new technologies and possible energy supply concepts, handling complex planning tasks is challenging. Due to the aforementioned high pressure to reduce costs and to increase energy-efficiency, a solution should be found that is as cost and energy efficient as possible. Computer-based optimization methods provide the opportunity to identify long-term cost- and energy-efficient energy supply systems. Especially heuristic optimization algorithms have shown a good performance in related network optimization problems. Their essential advantages are a reduced computational effort leading to computing times allowing investigation of largescale supply tasks while simultaneously delivering several, similarly cost-efficient energy supply systems.

S. Prousch (*) • H.-J. Haubrich • A. Moser Institute of Power Systems and Power Economics (IAEW), RWTH Aachen University, 52056 Aachen, Germany e-mail: [email protected]; [email protected]; [email protected] A. Sorokin et al. (eds.), Handbook of Networks in Power Systems II, Energy Systems, DOI 10.1007/978-3-642-23406-4_7, # Springer-Verlag Berlin Heidelberg 2012

187

188

S. Prousch et al.

Therefore, an optimization method for grid-bound energy supply systems based on Genetic Algorithms is proposed. For a given supply task the method is capable of calculating cost-efficient energy supply systems with regard to all technical and environmental constrains in an integrated planning process. Exemplary applications demonstrate the method’s capability and the advantages of applying this method for long-term planning of energy supply systems. Comparing optimized energy supply systems with existing systems allows direct conclusions for necessary adjustments and possible gains of efficiency. Keywords District heat • Energy supply • Genetic algorithms • Local search • Natural gas

1 Introduction and Motivation Energy supply is facing the challenge to guarantee a secure, competitive and sustainable energy supply, in spite of a shortage of fossil energy sources, while worldwide demand is growing, prices for oil and gas are increasing and important regions in the world are politically instable. Against the background of these challenges, the European Board as well as the Federal Republic of Germany have passed laws and set regulations to intensify competition, to reduce dependency of imports of recourses, to increase energy efficiency and to support new technological solutions.

1.1

Planning of Grid-Bound Energy Supply Systems

At present, large-scale power plants generate electricity, which is delivered to the customers via transmission and distribution grids. The thermal energy demand of customers is supplied locally by mostly small-scale units like boilers run on oil or natural gas. Planning of municipal, grid-bound energy supply systems is predominantly based on experiences of planning engineers and generally accepted rules of technology. Due to their very different structures, electricity and heat supply were designed separately, as cost saving as possible, under consideration of all technical constrains. Result of this separate planning process is a cost-efficient energy supply for each sector, which does not necessarily represent the global optimum of an integrated approach. Due to growing penetration of combined heat and power (CHP) units and electric heating, increasing interdependencies between electricity and heat supply are occurring. In addition, designing energy supply systems based on subjective experiences of planning engineers turns out to be disadvantageous since this approach neglects crucial, but experientially uncovered areas of the solution space. Furthermore, complexity of the planning process increases exponentially with the number of

Integrated Optimization of Grid-Bound Energy Supply Systems

189

available technologies and possible energy supply concepts. Therefore, there is a high risk that manually obtained solutions deviate severely from the optimum. Different optimization algorithms may be employed to improve the planning process. A common approach to reduce complexity is to divide the planning process into different stages, the long-term planning and the expansion planning, each with different objectives. The long-term planning determines energy supply systems as a target for future grid development. The planning horizon in the far future allows neglecting existing equipment. The expansion planning determines optimal development paths for existing energy supply systems, to ones with a higher cost- and energy-efficiency determined by long-term planning in the previous stage. Long-term planning turned out to be appropriate To evaluate energy supply systems and to analyze fundamental technical, economical and environmental influencing factors [1]. By neglecting existing equipment in long-term planning, reconstruction costs are not considered. Nevertheless, an evaluation of different reconstruction and expansion measures is possible, since measures that develop an existing energy supply system in direction of a long-term cost-efficient systems, is always sensible.

1.2

Previous and Related Work

As described above, electricity and heat supply could be considered separately in the past, which led to a separate planning of these sections. To support this planning process, numerous, decoupled methods, to optimize electricity [2], natural gas [3] and district heating grids [4] have been developed. Thereby exact methods of mixed-integer linear optimization as well as heuristic methods and combinations of these approaches are used. Exact methods guarantee the optimal solution of a problem, but since calculating grows exponentially with the number of variables, they only allow solving smallscale problems. Furthermore, the consideration of technical constrains within an exact optimization method is challenging Consequently exact approaches make vast assumptions necessary, which often lead to an insufficient representation of the reality [5]. By contrast, heuristic methods already support the network planning process successfully, since they allow solution of large-scale problems with an adequate precision [3]. Aforementioned developments in municipal energy supply suggest an integrated planning of electricity and heat supply, since from distributed conversion of primary energy into electricity as well as of electricity into heat, results in a coupling of the different sectors. This coupling leads to technical and economical interdependencies between the concerned infrastructures. Investigation of such phenomena demands integrated models and methods, which has recently come into the focus of research [5–13].

190

S. Prousch et al.

Approaches to optimize grid-bound energy supply systems introduced so far are showing deficits, which conflict with a practical application. In [6] an integrated consideration of electricity, gas and heat supply is done, but with focus on operational optimization of distributed generation units and energy storages. An optimization of grid structures is not realized. The method introduced in [7] basically allows an integrated planning of electricity, gas and district heating networks. However, the use of an exact optimization method demands vast calculation times and the violation of technical restriction can only be checked insufficiently. To identify potentials for cost reduction as well as possibilities to increase energy efficiency and to assure an objective determination of evaluation criteria, the use of optimization methods is necessary. The method introduced in this article is the first that allows an integrated optimization of electricity and heat supply for large-scale problems, which makes an objective determination of economical and environmental evaluation criteria possible. Existing methods only allow optimizing single components of an energy supply system, but not an integrated investigation of complete energy supply systems for large-scale supply tasks.

2 Analysis of Grid-Bound Energy Supply Systems 2.1

Definition of System Boundaries

Energy demand in municipal areas represents the highest share of the total energy demand. Therefore, municipal areas have a key role in fulfilling political aims. Additionally, the same company usually operates the different energy supply grids in municipal areas. Thus, an integrated optimization of these grids offers a high potential for cost reductions. To optimize and evaluate municipal energy supply concepts, the scope of observation must be chosen in a way that all technical, economical and environmental consequences of planning decisions can be detected. Therefore, the technical system boundaries include all equipment needed to build a municipal energy supply system. This article focuses on low voltage electricity grids, mediumpressure natural gas grids and district heating grids. For the separate planning of energy supply grids, electrical and thermal demand of customers are exogenous data. When planning whole energy supply systems, the energy demand of customers can be crucially influenced by the choice of their heating systems. Therefore, in this article, different types of heating systems are also within the technical system boundaries. Economical effects resulting from planning decisions include investment costs, operational costs as well as energy costs discussed in subsequent section of this paper.

Integrated Optimization of Grid-Bound Energy Supply Systems

191

Environmental effects resulting from planning decisions are evaluated by the demand of resources and caused emissions when converting primary energy into electrical and thermal energy.

2.2

Supply Task

Fulfilling the supply task is the basis and aim of energy supply system planning. Geographical positions of customers, their electrical and thermal energy demand as well as the topology of the supply area characterize the supply tasks. Customers of municipal areas can be subdivided into distributed generation and loads, which deliver respectively demand electrical and/or thermal energy. While the optimal positioning of local network stations and pressure regulating stations as well as distributed generation units is a task of network planning, geographical positions of loads are fixed. Differences between various customers are given by their electrical and thermal characteristics. Especially, the individual load profiles are of relevance. Due to political incentives, penetration of distributed generation is increasing. It generally concerns systems with co-generation or the use of renewable energy sources like global radiation, wind or hydro energy. For municipal energy supply, especially CHP-units are well suited, since they are usually run by fossil energy carriers and therefore not depended on fluctuating availability of renewable energy sources. Furthermore, it is possible to erect these units within municipal areas in contrast to wind energy generation and hydro power plants. Due to their different characteristics, loads in municipal areas can be divided into the categories business and household. Load profiles of business customers are usually measured und therefore well known for network planning. However, annual energy demand and rated power are the only known information for household customers. Furthermore, household customers have a highly stochastic load behavior, which differs from customer to customer. Therefore, in [2] stochastic models have been developed to determine the relevant load for dimensioning grids.

2.3

Grid-Bound Energy Supply Concepts

In general, energy supply systems consist of energy supply grids, distributed generation units and heating systems. Combinations of these components define different energy supply concepts. All energy supply concepts have an electricity supply via the electrical grid in common. Heat supply can be realized either gridbound or autarkic. Natural gas supply, district heating supply and all electric supply are the fundamental categories of grid-bound energy supply concepts. Utilization of oil or biomass for heating in rural areas is more economical compared to, grid-bound energy supply in low load density areas.

192

S. Prousch et al.

In the category natural gas supply, conventionally common boilers cover the heat demand. Alternative concepts with gas-run heat pumps or solar units allow using renewable energies. Solar heating systems can only contribute a fraction of the total heat demand. Conventional boilers usually cover the significantly higher part. Gas-run heat pumps are commonly operated in combination with geothermal units and are capable of covering the total energy demand without the necessity of additional boilers. Using renewable energy sources for heat supply instead of natural gas leads to a reduced utilization of existing natural gas grids and thereby to a reduced economical efficiency there off. Covering heat demand and part of electricity demand via small-scale CHP-units describes another alternative concept in the category natural gas supply. Generated electricity is either fed into the grid or used locally. The district heating supply concept consists of co-generators providing thermal energy, which is distributed via district heating grids. Generated electricity is fed into the medium voltage grid. Besides an electrical grid, all electric energy supply concepts require no further grids. In these concepts electric heating systems e.g. electrical heat pumps supply the heat demand. Table 1 gives an overview of the considered concepts in this article.

2.3.1

Energy Supply Grids

Energy supply grids serve to transport and distribute different energy carriers such as electricity, natural gas and district heat. On one hand, the supply task is influencing the design of energy supply grids, on the other hand, degrees of freedom in network planning as network structures or choice of equipments as well as technical constrains have major influence. There are several analogies between electricity, natural gas and district heating grids. Underlying physical laws are generally different, though transferring models Table 1 Energy supply concepts Energy supply Construction concepts Natural gas supply Conventional Solar

District heating supply All-electric supply

Heat pump (natural gas) Small-scale CHP District heat

Electricity/natural gas grid condensing boiler Electricity/natural gas grid condensing boiler, solar heating Electricity/natural gas grid heat pump

Electricity/natural gas grid Small-scale CHP unit Co-generator, electricity/district heating grid heat exchanger Heat pump (electrical) Electricity grid, heat pump Electric heating Electricity grid, electric heating

Integrated Optimization of Grid-Bound Energy Supply Systems

193

and methods is often possible. This section discusses analogies and differences of electricity, natural gas and district heating networks. The fundamental problem structure of network calculation is comparable for the different grids. The load flow occurs from source to drain via conductors. The analogy to the voltage u of an electricity network is the pressure p in natural gas and district heating networks. There is also an analogy between current I and flow rate w. A significant difference exists in the physical laws. In contrast to electricity, in natural gas and heat grids active and reactive power is inexistent. Additionally, gas is compressible. Therefore, above-mentioned analogies between voltage, current, pressure and flow rate are only valid under the assumption of incompressible gas (constant density). An additional state variable in district heating grids is the temperature T. For a given load situation and network structure (e.g. as depicted in Fig. 1), a network calculation determines the unknown state variables (Table 3). Voltages and pressures must fulfill the load flow equations introduced in Table 2. With the knowledge of all mass flows in the investigated grid, calculating temperatures of the network nodes is possible. Technical grid constrains can be verified with the results of the network calculation.

2.3.2

Technical Constrains in Low Voltage Grids

The permissible voltage range in distribution grids – standardized in EN 50160 – is set to
10% of nominal voltage. Furthermore, there are special guidelines for Electricity S1

Natural Gas

.

u1

V1

m Δp

.

u2

V2

.

Φ2

p2

Table 2 Load flow equations Energy carrier Load flow equation Electricity S12 ¼ P12 þ jQ12 P12 ¼ ðU1 Q12 ¼ Natural Gas

District heat

V_12

p1, T1

.

m Δ u, Δ S

S2

.

Φ1

p1

.

i

Fig. 1 State variable in energy supply grids

District Heat

jU1 jjU2 j cosd12 ÞR12 þjU1 jjU2 j sind12 X12 R12 2 þX12 2 ðU1 2 jU1 jjU2 j cosd12 ÞX12 þjU1 jjU2 j sind12 R12 R12 2 þX12 2 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2 d 5 Tn ¼ ðp21  p22 Þ  16lTlp m r Km

.

Δ p, Δ Φ

p2, T2

(1) (2) (3) (4)

n

with Km ¼ 1  450pmbar and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 5 p d F_ 12 ¼ ðp1  p2 Þ  8llr n

pm ¼ 23 

p31 p32 p21 p22

(5), (6) (7)

194

S. Prousch et al.

Table 3 Variables and indexes

Variable/index S P Q U d X R V_

Description Apparent power Active power Reactive power Voltage Phase angle Reactance Resistance Flow rate

p d K m F_

Pressure Pipeline diameter Compressionability factor Molar Heat flow

l l T n

Friction coefficient Length of line Temperature Nominal

operating distributed generation units in low voltage grids. According to these guidelines the maximum permissible voltage increase caused by feed-in of distributed generation units is limited to 2% compared to the voltage without this feed-in. In low voltage grids maximum rated currents of equipments and limitations regarding short-circuit currents have to be observed. Complying with maximum rated currents of equipments guarantees that no damages or reductions of useful lifetime caused by overloading occur. Failures in power systems can lead to shortcircuits. Resulting short-circuit currents exceed the maximum rated current of equipments at a multitude and usually come along with an electric arc which can destroy equipments and harm operating personal. Thus, for all relevant technical and operational states of the grid, minimum and maximum values of short-circuit currents have to be observed.

2.3.3

Technical Constrains in Natural Gas Grids

Pressure controlling elements in natural gas distribution grids guarantee an adequate pressure at all nodes. Assuming that overlying pressure levels deliver contractually guaranteed pressures and natural gas volumes, pressure maintenance in natural gas distribution grids is vastly decoupled from overlaying pressure levels. Within a pressure level, specific minimum and maximum pressure limits of network customers have to be maintained. Furthermore, obtaining maximal gas speeds in pipelines for a limitation of noise emissions and maximal volumetric flow rates in pressure regulators is mandatory.

Integrated Optimization of Grid-Bound Energy Supply Systems

195

Company-internal planning guidelines that exceed technical minimum requirements correspond to a demanded retention capacity that is necessary for a safe operation in cases of critical network loadings. 2.3.4

Technical Constrains in District Heating Grids

The pressure within a district heating grid has to be above a minimum pressure guaranteed to customers and has to be above vapor pressure of the heat carrier with a safety margin. This is also valid for the static pressure, which has to be regulated by the pressure maintenance. Furthermore, to prevent damaging of equipments, the upper pressure limit must not be exceeded. Depending on the type of pipeline, considering different inlet temperatures is mandatory. Besides limitations due to temperature resistance of equipments, in view of economical advantages (heat dissipation, heat production costs) an inlet temperature of 90 C should not be exceeded. Besides inlet temperature, return temperature is an important factor for the efficiency of district heating grids. Transmission capacity, energy demand for pumping, heat dissipation and efficiency of co-generators depend on this parameter. Furthermore, a minimum flow rate of about 0.8 m/s is necessary to guarantee a minimum recirculation preventing a cooling of the heat carrier. Observing a maximum flow rate of about 2 m/s prevents objectionable noise emissions.

2.4

Evaluation of Energy Supply Systems

Fulfilling all technical constrains introduced in the prior section is a fundamental condition for an economical and environmental evaluation of energy supply systems. Economical and environmental criteria to evaluate energy, supply systems are described in the subsequent chapter. An economical evaluation includes capital and operational expenditures. Capital expenditures of an energy supply system include investments (project planning, land acquisition, system acquisition and construction as well as commissioning), operational expenditures include costs for maintenance of equipments, taxes, insurance and network losses as well as energy expenditures (acquisition of primary and secondary energy). Accordingly, the energy supply system with lowest total costs is preferred. Supplying electricity and heat requires resources and causes emissions. The demand of resources consists of the annual demand of different primary energy carriers. Carbon dioxide emissions, which are mainly responsible for the greenhouse effect, make up the highest share of resulting emissions when transferring primary energy carriers in electricity and heat. To consider other greenhouse gases as nitrogen, methane or carbon monoxide, so-called carbon dioxide equivalents are used. They correspond to the amount of carbon dioxide, which causes the same greenhouse effect like the aggregation of all caused emissions.

196

2.5

S. Prousch et al.

Degrees of Freedom and Requirements on Optimization Method

Degrees of freedom of the optimization method are number and position of local network stations, pressure regulation stations and co-generators as well as the choice of routes and equipments. In addition, the heating system of customers is a degree of freedom. Practical purposes (e.g. stock keeping, acquisition) as well as specifications for equipments are limiting the number of available types of equipments in network planning. Furthermore, heating systems and distributed generation units are offered in standardized sizes, thus investigations in this article are limited to typical systems for different types of customers (e.g. household in one family house or apartment house, business customers). The aim of the optimization method introduced in this article is finding an economically optimal, technically valid and energy efficient development strategy for municipal energy supply systems. This requires considering parameters of the supply task and their future development as well as all degrees of freedom and boundary conditions when planning energy supply systems. Existing methods to optimize energy supply systems either only allow planning electricity, natural gas or district heating grids separately or are only able to solve small-scale problems due to high computing times. A fundamental advantage of the developed method is an appropriate modeling of the supply task as well as of technical units while allowing an integrated planning of realistic energy supply systems. Figure 2 illustrates the requirements on the optimization method including all relevant constrains. Computation time of the optimization method should not increase exponentially with the number of degrees of freedom. Assumptions and simplifications to reduce computation time are only valid as long as the quality of the solution is not affected. Degrees of Freedom

Constrains

Location of stations Choice of routes Heating systems

Supply task Optimization method

Technical constrains Ecological constrains Planning guidelines

Equipments

Cost and energy-efficient, technically valid grid-bounded energy supply systems

Fig. 2 Degrees of freedom and constrains

Integrated Optimization of Grid-Bound Energy Supply Systems Table 4 Variables and indexes

Variable/index C_ CAPEX OPEX a_ b k P el th LNS PRS Co-Gen. PL heat PL NG _ Tm HS

197

Description Annual expenditures Capital expenditures Operational expenditures Annuity factor Specific costs for maintenance and insurance Specific costs Power Electric Thermal Local network station Pressure regulating station Co-generator Pipeline district heating Pipeline natural gas Annual utilization period Heating system

The target function of the optimization is minimizing the capital and operational expenditures (Table 4). TF ¼ min½C_ CAPEX þ C_ OPEX 

(8)

with C_ CAPEX ¼

NLNS X 

 a_ LNS; i þ bLNS; i  cLNS; i  SLNS; i

i¼1

þ

NPRS X 

 a_ LNS; i þ bLNS;i  cPRS;i  V_ PRS; i

i¼1

þ

NCoGen: X



 a_ CoGen:; i þ bCoGen:; i  cCoGen:; i  Pth; CoGen:; i

i¼1

þ

N cabel X

  a_ cabel; i þ bcabel; i  li  ccabel; i

i¼1

þ

NX PL heat



 a_ PL heat; i þ bPL heat; i  li  cPL heat; i

i¼1

þ

NX PL NG



 a_ PL NG; i þ bPL NG; i  li  cPL NG; i

i¼1

þ

NHS X  i¼1

 a_ HS; i þ bHS; i  cHS; i  Pth;HS; i

(9)

198

S. Prousch et al.

and C_ OPEX ¼

NCustomer X

 T_ m; el:; Customer; i  Pel:; Customer; i  kel:; Customer; i

i¼1

þT_ m; th:; Customer; i  Pth:; Customer; i  k_th:; Customer; i



NCoGen:;i X



T_ m; el:; CoGen:; i; i  Pel:; CoGen:; i; i  kel:; feedin; i



(10)



i¼1

3 Modeling of Energy Supply Systems The newly developed system model bases on the system model to design electric grids introduced in [1]. This model has been successfully applied in long term planning of electric grids and shows that it can easily be adapted for designing other energy supply grids [3]. This allows a consistent modeling of all relevant grids in energy supply systems and makes an integrated planning of grid-bound energy supply systems possible that uses synergies and considers cost-shifting between the different sectors. A hierarchical model with different levels of abstraction is developed to represent a real energy supply system. The highest level in the developed model consists of streets of houses and junction points. A street of houses consists of customers that are connected via routes. For all customers within a street of houses a uniform demand of electricity and heat is assumed. Junction points contain network nodes of a single or several energy supply grids. They represent the connection between streets of houses and connection points for transformers, pressure regulators and cogenerators. As mentioned above all energy supply systems have an electricity supply via the electrical distribution grid. Therefore, in this model every customer is connected to an electrical network node. To supply the heat demand of customers several different heating systems can be allocated. Each of them requires a connection to a corresponding network node. The model assumes that all customers within the same street of houses are equipped with a uniform heating system. Routes are differentiated in cable and pipeline routes as well as overhead line routes. Cable and pipeline routes consist of a ditch with lines of single or several energy supply grids. Overhead line routes contain electric lines exclusively. Figure 3 illustrates the developed system model. Input data of the optimization is a complete system model containing all degrees of freedom and constrains. However, the result of the optimization describes a specific condition of the model representing an energy supply system. The algorithm delivers information describing if specific components (lines, transformers, pressure regulators, heating systems etc.) are realized and which type should be

Integrated Optimization of Grid-Bound Energy Supply Systems street of houses

customer

PR

199

LNS heating system

customer heat

route

electricity

heating system

customer

heating system

JP

heating system

customer

JP

electricity line JP

JP natural gas line

heating system LNS

customer

LNS: local network station PR: pressure regulator

co-gen.

district heat line

co-gen.: co-generator JP: junction point

Fig. 3 System model

used. Thereby, every component can have a variety of different conditions of which one is optimal in terms of the planning task. Attached to every condition are the technical data as well as the capital and operational expenditures of the corresponding system. The optimization method defines conditions for every component in a way that the resulting energy supply system complies with all constrains and has minimal total cost simultaneously. The presented task corresponds with a combinatorial optimization since all optimization variables can only have integer values and the solution space contains a limited number of possible combinations.

4 Optimization Method The system model described in the prior section allows transferring the planning of municipal energy supply systems into a combinatory optimization problem. Solution of such problems require high computing times, which is why even solving small-scale problems within a feasible computing time is already challenging. Generally, methods to solve combinatory optimization problems are divided in exact and heuristic methods. The category of exact methods includes methods, which determine solutions that can be proven to be mathematically optimal. The category of heuristics contains methods, which have a methodological search adapted to the problem structure. The fundamental advantage of an exact optimization method is the guaranteed optimality when a solution is found. Although the branch and bound method as well as various derived variants of it offer run time advantages compared with

200 Table 5 Nomenclature of genetic algorithms

S. Prousch et al.

Term Individual Population Fitness Generation Gene Gene string

Annotation Potential problem solution Set of potential problem solutions Objective function subject Iteration Variable of decision Encoding of problem statement

alternative exact methods, only small-scale problems can be solved. The system model results in such a complex combinatory optimization problem, that the use of exact methods is infeasible. Therefore, a heuristic optimization method is chosen to allow solving large-scale problems. Since in long-term planning of energy supply grids it turned out to be advantageous to have the possibility to choose out of a range of similar cost efficient network structures, population based approaches proved to be particularly suitable. Furthermore, they offer the possibility to realize parallel processing for a additional reduction of computing time. For integrated planning of energy supply grids, genetic algorithms seem to be advantageous since their fundamental structure is mainly problem independent, which allows their application in various different problems. This allows an integrated planning of different energy supply. The nomenclature of genetic algorithms is very similar to that of evolution theory and genetics. Table 5 provides an overview of the nomenclature. Genetic algorithms solve an optimization problem by iteratively evaluating all individuals of a population and selecting the ones with the highest fitness as parents for the next generation. By combining attributes of parent individuals with a high fitness, the fitness of individuals is advancing from generation to generation. Therefore, the gen string of a new individual is a composition of the gen strings of parent individuals. [14] gives a detailed description of genetic algorithms. The fundamental concept of genetic algorithms does not allow an implicit consideration of constrains in the problem structure. However, as aforementioned, when designing energy supply systems a variety of constrains and planning guidelines have to be considered. Consequently, methodologies are needed to deal with invalid solutions. An obvious approach is excluding individuals that violate boundary conditions from the solution space with the result that all considered individuals comply with the given boundary conditions. This is not feasible for problems with a large number of constrains like the problem at hand. A more suitable approach to consider boundary conditions in genetic algorithms is the application of penalty functions. Invalid solutions are panelized in the fitness determination, e.g. with a penalty function proportional to extend and type of violation. This reduces the fitness of invalid individuals and consequently their probability of selection. For optimization problems with a large number of invalid solutions with regard to the solution space, application of repair functions to convert invalid to valid

Integrated Optimization of Grid-Bound Energy Supply Systems

201

solutions is advantageous. To prevent converging to a local minimum, the fundamental properties of an individual must still be maintained.

4.1

Genetic Algorithms in Optimization of Grid-Bound Energy Supply Systems

As aforementioned, a main advantage of genetic algorithms is their high flexibility. An adaption of their parameterization allows considering problem specific requirements. For an adaption, the definition of the problem representation as well as the definition of an evaluation function is necessary. The coding of genes of an individual defines the problem representation, which is determined by the decision variables of the planning problem. Every degree of freedom is a variable, e.g. the selection of lines and heating systems, and every realization is encoded and represented by integer values. Tables 6–8 show the chosen coding of genes for the degrees of freedom including selection of locations for local network stations, pressure regulation stations and co-generators, selection of routes as well as heating systems. The coding of the location for local network stations, pressure regulation stations and co-generators includes the realization of the corresponding systems at the different locations. The coding of routes contains the possibility to choose the type of ditch as well as the type of cables and pipelines. The coding of heating systems includes different technologies. Basis of the evaluation function is the economical evaluation of every planning variant in a way that particular cost efficient designs benefit from their high fitness. Additionally penalty functions reduce the fitness of designs that violate technical and economical constrains. Table 6 Representation of decision variables of stations Local network station Pressure regulating station Range 0–m 0–n Representation Realization & type of Realization & type of transformer pressure regulator

Co-generator 0–o Realization & type cogenerator

Table 7 Representation of decision variables of routes Electric line Natural gas line Range 0–x 0–y Representation Realization & type of cable// Realization & type of over heat line pipeline

District heating line 0–z Realization & type of pipeline

Table 8 Representation of decision variables of heating systems

Range Representation

Heating system 1–k Realization & type of heating system

202

4.1.1

S. Prousch et al.

Overview of the Optimization Method

The schematic procedure of the developed method is depicted in Fig. 4. Genetic algorithms are applied to find the optimal number and location of stations, to find the optimal pathway of routes as well as to determine the optimal dimensioning of lines. In the initializing phase of the method, firstly number and location of stations as well as the heating systems of customers are initialized, representing the origin of the following optimization of the pathways of routes and dimensioning of conductors. To optimize the pathways of routes and dimensioning of conductors the second entity of genetic algorithms under consideration of results of the first entity is initialized. After the initializing phase the iterative optimization starts. The iterative optimization consists of several steps. At first, each individual of a population is analyzed with respect to its technical feasibility. If any constrains are violated, a multitude of repair functions is applied in order to transfer the draft into the admissible solution space. The following local search determines marginal structural modifications that increase the fitness of the solution and thus support a faster convergence towards the optimal solution. Parameterization and initializing Technical evaluation Local search

Repair Solution valid? no yes Genetic operators

Local search done? no yes Economical evaluation

Abort? yes

no

Economical-ecological evaluation

Abort? no yes Solution output

Fig. 4 Optimization procedure scheme

Integrated Optimization of Grid-Bound Energy Supply Systems

203

In the next step, costs for each energy supply system are determined and attributed to the corresponding systems as quality rating. All systems are arranged according to their quality in order to compose a new genetic pool for the stochastic generation of new system drafts. After completion of a generation the best individuals are transferred unchanged into the new population, new individuals replace the others. Therefore, genetic operators – e.g. selection, crossover, mutation – are applied. In case no improvement of the solution is achieved for several generations, the most cost efficient energy supply systems, which comply with all technical constrains are handed over to the overlaying genetic algorithm. In the overlaying genetic algorithm, the economical and environmental evaluation determines the overall fitness of every energy supply system. After completion of a generation, an abortion criterion is checked. In case it is not fulfilled, again genetic operators are applied otherwise the most efficient energy supply systems are exported.

4.1.2

Initial Solution

Generally, genetic algorithms are initialized with stochastically generated initial solutions. Accordingly, the random choice of equipments and network structures guarantees that no limitations of the solution space occur. However, a fundamental condition for such an initialization is an appropriate number of initial solutions, which is predominantly determined by the extend of the optimization problem. A suitable number of initial solutions according to the appropriate parameterization of genetic algorithms would lead to extensive computing times especially for supply tasks with a large number of degrees of freedom. Therefore, a selection of initial solutions is necessary. As aforementioned, genetic algorithms are applied to determine the heating systems of customers as well as to design energy supply grids at optimal costs. This includes the determination of the optimal number and position of stations. To achieve manageable computing times, functions are necessary to determine initial solutions for both optimization loops that are already close to a known solution of high quality. Since the position of stations has a major influence on the structure of energy supply grids, a preselection of potentially optimal positions can improve the convergence behavior considerably. In the developed system model junction points represent optional positions for stations. In accordance to generally accepted planning conventions, the minimal required number of stations to fulfill the supply task is chosen and located in the load centers. Thereby, the number of optional positions are reduced to potentially optimal positions. Therefore, the electrical and thermal load of streets of houses directly bordering a junction point as well as the load of an optional number of additional surrounding streets of houses is aggregated and attached to the corresponding junction point. A comparison of the resulting load values for each junction point allows identifying load centers within the supply

204

S. Prousch et al.

area. In the identified load centers, the minimum necessary number of stations is allocated. The selection of optimal energy supply concepts mainly depends on the supply task, especially on the load density. Thus, to determine initial solutions for the first optimization loop these parameters are determined. Based on empirical values or estimations regarding the economy of different energy supply concepts depending on the mentioned parameters, potentially optimal concepts for each street of houses can be determined. The optimal network structure for low voltage and district heating grids is a radial network structure. A minimum spanning tree connecting all customers with the shortest path represents the optimal design of radial networks. Therefore, it is obvious to design low voltage and district heating grids based on the theoretically achievable length of a minimum spanning tree. However, a minimum spanning tree can conflict with technical constrains. Thus, an adequate number of initial solutions, with structures similar to the minimum spanning tree, is necessary. To determine a minimum spanning tree different algorithms are available since it is a common problem of graph theory. In the developed method, the prim algorithm is applied. To determine an adequate large number of initial solutions spanning trees with restriction of single routes are generated. Thus, according to a given probability in every step of the prim algorithm the shortest possible path is restricted. Thereby, numerous degenerated minimum spanning trees can be generated. To guarantee that the selection of initial solutions does not limit the solution space improperly, further initial solutions are stochastically generated.

4.1.3

Technical Evaluation

The compliance with all technical minimum requirements is checked in a two-stage process. If a violation gets detected repair functions are applied to recover a state without any violations. The first step comprises structural inspections that in essence target the network structure. Afterwards, aforementioned operational constraints are checked by load flow calculations. If the stress of equipments exceed maximum admissible values, potentially more expensive equipments with broader limits are realized. Should the aforementioned possibility be unfeasible, additional equipments are inserted to release the stress. In the case of a violation of voltage or pressure limits, adjacent pipelines in the direction of local network stations or pressure regulators will be amplified to increase the node’s voltage respectively pressure. Similarly, violations of maximum flow velocities can be intercepted with an increase of the pipeline diameter of the section under consideration. As high flow velocities coincide with high pressure gradients due to undersized pipeline diameters, the examination of minimum pressure limits will have a positive impact on the compliance of maximum flow velocities.

Integrated Optimization of Grid-Bound Energy Supply Systems

4.1.4

205

Local Search

In general, the starting population of genetic algorithms is completely stochastic and consequently the individuals of the beginning generations are far away from the searched optimum. The violations of boundary conditions because of the aforementioned stochastic furthermore demands the use of repair functions that in either case lead to a worsening of the fitness value. Therefore, it seems advisable to conduct cost-reducing structural refinements in the form of a local search that does not change the fundamental structure of an energy supply system significantly. Because of marginal efforts for technical and economic evaluations, a great part of the surrounding solution space of each solution can be examined easily. If a solution possibility with a higher fitness than the initial solution can be found, a transfer to this solution will be carried out. Possible solution changeovers include modifications of equipment or dismantling of several – from technical point of view not necessary – equipment. 4.1.5

Parallel Optimization

Due to size and complexity of the optimization problem, achieving acceptable computing times is a major problem when developing methods for practical planning tasks. Hence, measures to reduce computing times are necessary. Since genetic algorithms are particularly suitable for parallelization due to their design, parallel computing is realized for the introduced method. Creation, assessment, and selection of individuals during the genetic algorithm require the predominant part of computing time. An adequate parallelization approach is the segmentation of each population’s fitness assessment process. High computing time benefits can be achieved due to the time-consuming evaluation of the merit function – including a large number of flow calculations – for the planning of energy supply grids. For the chosen parallelization approach, a primary process administrates the population and assigns further processes with the assessment of one individual each. Hence, the process flow remains unaltered in comparison to a sequential genetic algorithm but still high computational benefits can be achieved.

4.2

Exemplary Results

In this section the functionality and performance of the newly developed method is demonstrated by an exemplary application to a close to reality planning problem. Therefore, for a modeled supply task, reflecting all relevant features of a real supply task, the optimal network structures, dimensions of equipments and heating systems are determined. After introducing the modeled supply task, the conventional energy supply system is designed at optimal costs. It serves as a reference to evaluate the potential

206

S. Prousch et al.

for gains in cost and energy efficiency of an integrated optimization of energy supply systems including grids and heating systems.

4.2.1

Supply Task

The considered supply area (see Fig. 5) exhibits a predominant urban character. The total area is approximately 0.6 km2 in size. It includes 5,118 inhabitants, whose electrical and thermal energy demand is distributed on 224 nodes. A very heterogeneous load density characterizes the supply task. While a very dense building structure predominate in the center of the supply area, single-family houses are dominating in the peripheral area resulting in a comparable low load density. Assuming a conventional energy supply the total electrical load amounts 4.4 MW and the total thermal load 15.2 MW. The electrical energy demand amounts 4.4 GWh/a and the thermal 12.4 GWh/a. The different customer types are described in Table 9.

4.2.2

Constrains and Degrees of Freedom

Due to the geographically limited expansion of the supply area, only a single network level for each grid is considered. Only low voltage part of the electricity grid is considered. The design of natural gas supply is limited to medium pressure level.

Customer type 1 Customer type 2 Customer type 3 Junction point Optional routes 100 m

Fig. 5 Supply task Table 9 Types of customers Customer type ~

● ■

Pel [kW] 6 20 54

W_ el [MWh/a] 6 21.6 48

Pth [kW] 20 70 180

W_ th [MWh/a] 17.8 57.8 143.2

Integrated Optimization of Grid-Bound Energy Supply Systems

207

Usable routes are limited to the pathway of roads. On every route, realizing lines of every energy supply grid is allowed. Optional positions of stations for the connection to overlaid network levels or feed-in of district heating are the 40 junction points depicted in Fig. 5. In accordance with the praxis of network planning, the number of possible types of equipments is limited. Table 10 shows technical data of the optional equipments. Furthermore, technical constrains listed in Table 11 are considered. For natural gas, h-gas with a calorific value of 11.3 kWh/mN3 and a medium temperature of 5 C in the relevant peak load situation is assumed. The district heating grid is operated with a temperature of 90 C and a pressure of eight bars.

Table 10 Technical and economical data of equipment Equipment Types Local network station Pressure regulating station Co-generator Cable Pipeline (natural gas) Pipeline (district heat) Heating systems

250–630 kVA

Capital expenditures [tsd. €], [tsd. €/km] 24. . .28

0.8 bar; 100–400 m/h

16. . .19

0.3–1.5 MWel; 0.5–2.5 MWth

KCAPEX 30 50 43 77 110 60 110 160 KCAPEX KCAPEX KCAPEX

VPE AL 4  95 VPE AL 4  150 DN50 DN100 DN150 DN32 DN40 DN50 Condensing boiler heat exchanger (district heat) electric heat pump

¼ 4361 

pffi 2 ½3PKWK;el

¼ 0; 184  Pth þ 4; 77 ¼ 0; 057  Pth þ 3; 654 ¼ 0; 798  Pth þ 2; 083

Table 11 Technical constrains Electricity

Natural gas District heat

Parameter Voltage range Thermal current Pressure range Flow rate Pressure range Flow rate Inlet temperature Return temperature

Value Un
10% 215A (VPE AL 4  95) 275A (VPE AL 4  150) 0.8. . .0.2 bar