INDIVIDUAL WELFARE MAXIMIZATION IN ELECTRICITY MARKETS INCLUDING CONSUMER AND FULL TRANSMISSION SYSTEM MODELING JAMES DANIEL WEBER

Transcription

1 INDIVIDUAL WELFARE MAXIMIZATION IN ELECTRICITY MARKETS INCLUDING CONSUMER AND FULL TRANSMISSION SYSTEM MODELING BY JAMES DANIEL WEBER BS, University of Wisconsin - Platteville, 1995 MS, University of Illinois at Urbana-Chamaign, 1997 THESIS Submitted in artial fulfillment of the requirements for the degree of Doctor of Philosohy in Electrical Engineering in the Graduate College of the University of Illinois at Urbana-Chamaign, 1999 Urbana, Illinois

2 RED-BORDERED FORM ii

3 ABSTRACT This dissertation resents a new algorithm that allows a market articiant to maximize its individual welfare in the electricity sot market The use of such an algorithm in determining market equilibrium oints, called Nash equilibria, is also demonstrated The start of the algorithm is a sot market model that uses the otimal ower flow (OPF), with a full reresentation of the transmission system The OPF is also extended to model consumer behavior, and a thorough mathematical justification for the inclusion of the consumer model in the OPF is resented The algorithm utilizes rice and disatch sensitivities, available from the Hessian matrix of the OPF, to hel determine an otimal change in an individual s bid The algorithm is shown to be successful in determining local welfare maxima, and the rosects for scaling the algorithm u to realistically sized systems are very good Assuming a market in which all articiants maximize their individual welfare, economic equilibrium oints, called Nash equilibria, are investigated This is done by iteratively solving the individual welfare maximization algorithm for each articiant until a oint is reached where all individuals sto modifying their bids It is shown that these Nash equilibria can be located in this manner However, it is also demonstrated that equilibria do not always exist, and are not always unique when they do exist It is also shown that individual welfare is a highly nonconcave function resulting in many local maxima As a result, a more global otimization technique, using a genetic algorithm (GA), is investigated The genetic algorithm is successfully demonstrated on several systems It is also shown that a GA can be develoed using secial niche methods, which allow a GA to converge to several local otima at once iii

4 Finally, the last chater of this dissertation covers the develoment of a new comuter visualization routine for ower system analysis: contouring The contouring algorithm is demonstrated to be useful in visualizing bus-based and transmission line-based quantities iv

5 ACKNOWLEDGMENTS I would like to thank Professor Thomas J Overbye for his knowledge, guidance, and suort throughout my doctoral studies He and his family have been a great hel to me I would also like to thank all the other rofessors who have heled and continue to hel me in my academic career I also say thanks the University of Illinois Power Affiliates Program and Grainger Foundation for their generous financial suort PowerWorld Cororation also deserves my thanks Working there has given me great insight into turning research into a commercial roduct, and has also insired much of the direction of my research Also, thank you for roviding me the flexibility to comlete my doctoral studies Lastly, I would like to thank my family My brothers Scott and Brian, and sister-in-law Andrea continue to be a source of insiration, suort and friendshi My mother and father, Jan and Gene, deserve great thanks for their unconditional love and suort which have allowed me and my brothers continued success in our ersonal lives and careers Finally, I say thank you to my fiancé Amelia Fuller who has made the last two years of my doctoral work a joy I look forward to our future life together v

6 TABLE OF CONTENTS Page 1 INTRODUCTION1 11 Electricity Market Structures 1 1 History of the Electricity Industry in the United States6 13 History of Electricity Industry Reform throughout the World United Kingdom 8 13 South America Australia The Need for Simulation Tools1 15 Overview The sot market model14 15 Maximizing individual welfare in a sot market Contouring visualization 17 A MODEL OF BIDDING IN A POOL-BASED ELECTRICITY MARKET18 1 The Electric Power Pool Market Structure18 The Otimal Power Flow18 3 A Market Model from the OPF 4 Market Examles 41 Examle market model for a four-sulier lossless system 4 Demonstration of market ower in a 54-sulier market 7 43 Effect of transmission constraints on the market model3 3 INCORPORATION OF CONSUMER MODELS INTO AN OPF [4] Maximizing Social Welfare Using the OPF 37 3 Modeling Consumers as Price-Deendent Loads Modeling Suliers as Price-Deendent Generators4 34 Examle Player Models A rice-deendent real ower load as a model of a consumer41 34 A rice-deendent real ower generation as a model of a sulier Incororation of the benefit of reactive ower into the consumer model44 35 Imlementation of Real and Reactive Price-Deendent Loads into the OPF49 4 INDIVIDUAL PLAYER OPTIMIZATION IN ELECTRICITY MARKETS Individual Welfare Maximization Problem 54 4 Solution of Individual Welfare Maximization by the Direct Alication of Kuhn-Tucker Conditions56 43 Solution of Individual Welfare Maximization through Iterative Means Finding a Nash Equilibrium Using Individual Welfare Maximization Examle market with no constraints two sulier cometition Examle market with constraints two sulier cometition Examle market with no constraints sulier and consumer cometition Examle market with constraints sulier and consumer cometition71 45 Generalizing the Bid Curve74 vi

7 46 Examle Nine-Bus System Illustrating Potential Horizontal Market Power Comutational Requirements8 48 Individual Welfare Maximization Algorithm Convergence83 49 Find Nash Equilibrium Algorithm Convergence 84 5 APPLICATION OF GENETIC ALGORITHMS TO INDIVIDUAL WELFARE MAXIMIZATION86 51 Brief Overview of Simle Genetic Algorithms Selection88 51 Crossover Mutation Genetic algorithm rogram flow What is a GA looking for? 89 5 Alication of a Simle Genetic Algorithm to Individual Welfare Maximization9 51 Fitness and fitness scaling 9 5 Coding91 53 Examle of simle genetic algorithm used for individual welfare maximization9 53 Secies Creation through Fitness Sharing Fitness sharing Mating restrictions Alication of a GA with Fitness Sharing to Individual Welfare Maximization Examle with two local maxima Examle with three local maxima 1 6 USING CONTOURING VISUALIZATION FOR ANALYSIS AND MONITORING OF ELECTRICITY MARKETS [75] Motivation for the Develoment of New Visualization Techniques18 6 Previous Voltage Visualization Work Overview of Contouring Basics Fast Contouring Algorithm Contouring in Animation1 66 Usability Test Contouring of Branch-Based Values18 7 SUMMARY13 APPENDIX ASPARSITY STRUCTURE OF THE DERIVATIVES OF THE HESSIAN AND GRADIENT WITH RESPECT TO K133 REFERENCES135 VITA 14 vii

8 LIST OF TABLES Table Page 1 Profit for each Sulier in the Four-Sulier Market (Assuming Others Bid Marginal)5 Effect of Sulier Bid Increase on the Pool Price6 3 True Cost Curves and True Marginal Bid Curves8 4 Profit for Suliers in the 118-Bus, 54-Sulier Market (Assuming Others Bid Marginal)8 5 Profits for Suliers at Buses 69 and 89 when One Individual Controls Their Bids3 6 Changes in Disatch and Profits Due to Line Constraints3 7 Profits for Bus 4 with Line Limits Enforced34 41 Variation of the Market Solution along the Nash Equilibrium Continuum7 4 Cost and Benefit Equation Coefficient for Illustrative Examle Economic Results When All Bids Corresond to True Marginal Cost77 44 Nash Equilibrium Results When Suliers 7 and 8 Both Raise Prices79 45 Nash Equilibrium Results When Suliers 7 and 8 Try to Overload a Line79 51 Samle Fitness Function for a 3-bit Coding 9 5 Cost and Benefit Equation Coefficient for New Illustrative Examle 1 53 Economic Results When All Bids Corresond to True Marginal Cost1 54 Final Poulation of GA13 55 Local Otimum With No Lines at a Limit14 56 Local Otimum With One Line at a Limit15 57 Local Otimum With Two Lines at a Limit 16 viii

9 LIST OF FIGURES Figure Page 11 Bilateral Market Model 1 Last Acceted Bid Disatch Model5 1 Power Pool Information Flow1 Linear Price-Deendent Suly Bids 3 Six-Bus System with Four Generators 3 4 Bidding k Times Higher Than Marginal Cost4 5 IEEE 118-Bus Case 7 6 Six-Bus System with 1-MVA Limit on Each Line31 7 Six-Bus OPF Solution with Line Constraints Enforced 31 8 Profits versus Percent Over True Marginal Bid33 9 Revenues, Costs, and Profit versus Bid While Enforcing Line Thermal Limits Consumer Benefit and Derivative of Benefit41 3 Consumer Demand for Real Power4 33 From a Quadratic Cost to a Linear Suly Curve44 34 Concave k(x) A Load Model with Reactive Control Hessian for Minimizing Costs53 37 Hessian Matrix for IEEE 118-Bus System with Price-Deendent Real Power Loads Bidding Variation for Suly and Demand 55 4 Binding Inequality Change in One Dimension6 43 Two-Bus System: Two Suliers and a Consumer64 44 Two Suliers Bid Progression with No Line Limit Two Suliers Otimal Resonses with No Line Limits65 46 Two Suliers Bid Progression with 8-MVA Line Limit Two Suliers Otimal Resonses with 8-MVA Line Limit67 48 Sulier Profit-versus-Bid on Either Side of the Discontinuous Point Otimal Bids for Sulier 1 in Resonse to a Mixed-Strategy by Sulier A Sulier s and a Consumer s Bid Progression with No Line Limits7 411 A Sulier s and Consumer s Otimal Resonses with No Line Limits7 41 A Sulier s and a Consumer s Bid Progression with an 8-MVA Line Limit A Sulier s and Consumer s Otimal Resonses with 8-MVA Line Limit Sulier Profit and Consumer Surlus along the Nash Equilibrium Continuum Otimal Curves for Several Fixed Sloes Nine-Bus Electricity Market Contour Plots of Combined Profit of Sulier 7 and Three-Dimensional Plot of Combined Profit of Sulier 7 and Bid Progression Using the Individual Welfare Maximization Algorithm for the Nine-Bus Examle83 4 Bid Progression Suerimosed on a Contour Plot of Combined Profit 84 ix

10 41 Convergence of the Find Nash Equilibrium Algorithm for the Nine-Bus Examle85 51 Examles of Simle Crossover 88 5 The Maing of k Values to a Binary Grey Code91 53 Sulier Profit versus Bid93 54 Fitness Progression of a Simle GA for the Two-Bus System94 55 Illustration of Niche Formation in Parameter Sace97 56 Fitness Progression of a GA with Fitness Sharing for the Two-Bus System Final Poulation Suerimosed on the Welfare Function99 58 Social Welfare Maximum When All Consumer and Suliers Bid Marginal11 59 Fitness Progression of a GA with Fitness Sharing for the Nine-Bus System1 51 Local Otimum with No Lines at a Limit Local Otimum with One Line at a Limit Local Otimum with Two Lines at a Limit Basic Thermometer Grahic 11 6 Ta-Changing Transformers Effects Calculation of the Virtual Value Samle Voltage Contour of Six-Bus System Color-Maing for Red = High, Blue = Low Color-Maing with a Discrete Change at a Limit Voltage Magnitudes with Limits Highlighted Bus Influence Regions Reduction of Comutation by a Factor of ECAR 345 kv-bus Voltages MAIN 115/138/161 kv-bus Voltages1 61 MAIN 115/138/161 kv-bus Angles TVA 161 kv-bus Voltages IEEE 118-Bus System Marginal Costs First Test Case with IEEE 118-Bus System Second Test Case with IEEE 118-Bus System Seven Points (X) Reresenting a Transmission Line Eastern Interconnect PTDFs for a Transfer from Florida to Wisconsin Line MVA Loading Percentages for the Eastern Interconnect 19 A1 Tensor for the Derivative of H with Resect to k134 x

11 NOTATION AND TERMINOLOGY General conventions on notation All vector and matrix variables are in bold (eg, h is a vector and h is a scalar) All vectors are column vectors Subscrit and subscrit q signify a relation to real and reactive ower, resectively Variable Definitions x = state variables and other controls (eg, ta ratios) T T s = [ s ] T sq = the suly vector T T d = [ d ] T dq = the demand vector T T ŝ = [ sˆ ] T sˆ q = augmented suly vector including zeros where no suliers exists T T dˆ = [ dˆ ] T dˆ q = augmented demand vector including zeros where no loads exist C(s)= C(s, s q ) = C ( s, s ) = suliers cost B(d)= B(d, d q ) = k all suliers k all consumers B ( d q, d q ) = consumers benefit hˆ (x) sˆ + dˆ h (x, s, d ) h(x, s, d) = =,, ) = equality h q (x sq dq h(x) constraints ) h(x h (x, s, d ) = h ˆ (x) sˆ + dˆ = real ower flow equations h q(x, sq, dq) = h ˆ q(x) sˆ + q dˆ q = reactive ower flow equations smin s s smax g(x, s, d) = d min d = inequality constraints d d max g(x) L, L = Lagrange functions T T T =[ ] T h g T T h =[ ]T h f = Lagrange multilier vector = Lagrange multilier vector for ower flow equations h hq and other equality constraints T T T T T g = [ ] T gs min gs max gd min gd max g = Lagrange multilier vector for inequality constraints ~ T ~ T ĥs hˆ s hsq ˆ = reduced Lagrange multilier vector, including only entries for ower flow equations that include a suly of real or reactive ower ~ T ~ T ĥd hˆ d hdq ˆ = reduced Lagrange multilier vector, including only entries for ower flow equations that include a demand of real or reactive ower T T T =[ ] T ĥ ˆ ˆ h ~ =[ ] T ~ = [ ] T xi

12 ~ = [ ] T f ~ T f ~ T fq = Lagrange multilier vector for additional constraints T T [ ] T T T T T = s =[ ] T d s sq d dq = rice vector for variable suliers and variable consumers (includes real and reactive rices) B( d) D() = the functional inverse of d C( s) S() = the functional inverse of s Terminology Sulier : Economic entity that offers the sale of roducts to the market (eg, a generator) Suly : The roducts the sulier offers to the market (eg, outut of generator in MWh) Generation : A articular kind of suly Consumer : Economic entity that requests the urchase of roducts from the market (eg, a load) Demand : The roducts the consumer urchases from the market (eg, inut to a load in MWh) Load : A articular kind of demand xii

13 1 INTRODUCTION Electricity markets throughout the world continue to be oened to cometitive forces The underlying objective of introducing cometition to these markets is to make them more efficient It is a general concet of caitalism that if fair and equitable market structures are created, which give all market articiants incentives to maximize their own individual welfare, then the market as a whole will behave in a manner which maximizes welfare for everyone If this objective is to be achieved, the electricity industry needs new algorithms to hel market articiants behave in an efficient manner, heling them maximize their own welfare It is imerative that these algorithms model the economics of the electricity market while maintaining the detail necessary to reresent the underlying engineering requirements This dissertation resents a new algorithm which allows a market articiant to maximize its individual welfare, while taking into account the full model of the electrical transmission system Before continuing further, it is useful to introduce general market structures and to look at the history of the electricity industry to motivate the desire for restructuring Section 11 will introduce electricity market structures, while Sections 1 and 13 will discuss the history of the electricity industry throughout the world leading u to the creation of these market structures 11 Electricity Market Structures In most industries, cometitive market structures give suliers of the commodity direct access to the consumer as shown in Figure 11 1

14 Sulier A Sulier B Sulier C Consumer 1 Consumer Consumer 3 Consumer 4 Figure 11 Bilateral Market Model As an examle consider the commodity of long distance telehone service In this industry, suliers such as MCI, Srint, and AT&T sell their commodity directly to the retail consumer The sulier and consumer then enter into a bilateral contract, where the sulier agrees to rovide long distance telehone service, while the consumer agrees to a fee schedule for this service The advantage of this bilateral market model is that no restrictions are laced on the structure of the contract This gives flexibility, allowing all articiants in the market the oortunity to find the exact kind of service they desire at the rice they desire While this model is aroriate for most markets, the electricity market exhibits two distinguishing characteristics that make the formation of a bilateral market model difficult: (1) the inability to store electrical energy ractically, and () the sharing of a common, essentially uncontrollable, transmission network The inability to ractically store electrical energy can result in extreme rice volatility due to shortages during times of eak demand This market behavior has already been exhibited in the limited wholesale markets of today, with rices during times of high demand reaching more than 1 times normal market rices [1] Markets with this kind of rice volatility will have difficulty reaching equilibrium

15 The sharing of a common transmission network is common to other markets, such as the long distance telehone market What distinguishes the electricity market network, however, is the uncontrollability of the network flows In telehone networks, information flow is controlled through comuter software which erforms call routing Routers determine the exact hysical ath of the hone call, giving the system oerator recise control of the network flow [] There are nearly no network flow controllers available in electricity markets, with a small number of hase-shifting transformers and flexible ac transmission devices (FACTs) as excetions Because the network flows are not controllable, the action of one layer in the market directly affects the ability of other layers to articiate in the market In the long distance telehone market this is not seen If AT&T sells service to your neighbor, this does not revent MCI from selling service to you With electricity, however, there are many ancillary services [3],[4] that must be sulied and limits that must be adhered to along with delivery of the roduct in order for the electric transmission system to maintain integrity One examle of this is ensuring that the transmission line limits are adhered to at all times In systems where transmission congestion is encountered, the actions of layers directly affect who can get ower to the local market This creates a otential for cooerative behavior among several generators and loads that would limit the cometition in a local area of the system Additionally, electric transmission system losses are affected by an individual layer s actions that, as a result, can affect all layers in the market How to account for losses can be very difficult because of their nonlinear nature Because of these characteristics, all the work in setting u new market structures in the electric ower industry throughout the world has involved the develoment of ower ools In this market model, a ool oerator takes bids from suliers and consumers and disatches generation and load in an economic manner based on the bids This economic manner will be 3

16 described shortly The suliers and consumers do not directly interact with one another, but only indirectly through the ool oerator The advantage of this arrangement is that the ool oerator can internalize the roblems of congestion management, loss allocation, and other ancillary services The ool oerator in this model is resonsible for the economic infrastructure of acceting and awarding bids, as well as the engineering system infrastructure of maintaining the transmission system In some markets, such as in California, these two tasks are assigned to searate organizations The economic organization is called the ower ool, or ower exchange, while the engineering organization is called the indeendent system oerator (ISO) The methods which have been used to disatch generation and load in an economic manner have been based on one of two methods: last acceted bid and sot ricing theory In the last acceted bid method, market articiants submit blocks of generation and sometimes load along with associated rices All the suly bids are then aggregated and sorted by rice in ascending order to create the aggregate suly curve If consumer bidding is included, then all the demand bids are aggregated and sorted by rice in descending order to create the aggregate demand curve The curves are then lotted against one another, and the oint were they cross defines the market clearing rice (MCP) All bids to the left of this oint are then acceted and all suliers are aid the MCP for the blocks of generation which were bid, and all consumers must ay the MCP for the blocks of demand which were bid This is deicted in Figure 1 Examles of markets which used the last acceted bid methods are England and Wales [5], New Zealand [6], and the state of California in the Western United States [7] 4

17 Demand Bidding Included Price Fixed Demand Requested Price Aggregate Demand Curve Aggregate Suly Curve Fixed Amount of Demand is Requested Aggregate Suly Curve Market Clearing Price Market Clearing Price Total Suly and Demand Disatched Total Suly and Demand Disatched MW MW Figure 1 Last Acceted Bid Disatch Model The other method used to disatch generation and load in an economic manner is to use sot ricing theory as described in [8] This is the method which is used throughout this dissertation and will be discussed in great detail throughout In this method suliers and sometimes consumers submit bid curves to the ool oerator and an otimization routine is used to determine the disatch results Suliers are then aid a rice according to their bids and consumers must ay a rice according to their bids Examles of markets which using sot ricing methods are Australia [5], the PJM interconnection in the Eastern United States [9], Argentina [5], and the New England region of the United States [1] Electricity market structures have been introduced, but the imetus for creating them is still not clear Sections 1 and 13 discuss the history of the electricity industry, concentrating on what has lead to the creation of market structures throughout the world 5

18 1 History of the Electricity Industry in the United States After the formation of the electric utility industry in the early 19s in the United States, utilities went through a eriod of merging and consolidation u through the 19s During this time the US began forming what would ultimately become the interconnected electric system that exists today From the middle 193s through the early 197s the industry went through an incredible exansion, with electricity demand growing at rates of aroximately 7% each year Increasing economies of scale marked this eriod of time as ever larger lants were built at lower costs er megawatt of caacity Throughout this time the rice of electricity continued to dro even as the demand increased During the 197s this began to change Oil rices increased dramatically during and after the oil embargo This, along with high inflation, resulted in increased energy costs As a result of these events, the United States began striving for energy self-sufficiency Energy conservation rograms increased in number and the develoment of renewable resources such as solar, wind, waste, or geothermal was begun in earnest These events led to new legislation in the United States to hel achieve the United States energy goals This new legislation was encasulated in the Public Utility Regulatory Policy Act (PURPA) of 1978 [11] The goals of PURPA were to encourage (1) conservation of energy sulied by electric utilities; () the otimization of the efficiency of use of facilities and resources by electric utilities; and (3) equitable rates to electric consumers (16 USC Sec 611 of [11]) In order to achieve these goals, PURPA created a new class of electricity suly called the qualifying facility (QF) QFs were intended to be small generation lants that used alternative fuel sources such as wind, solar, waste, or geothermal Under PURPA, the electric utilities were required to urchase electricity from the QFs PURPA stated that 6

19 the rates for such urchase (1) shall be just and reasonable to the electric consumers of the electric utility and in the ublic interest, and () shall not discriminate against qualifying cogenerators or qualifying small ower roducers (16 USC Sec 84-3 of [11]) QFs were the beginning of cometition in the United States Traditionally, electric utilities had been vertically integrated monoolies The utility built the generators that sulied the ower, built the wires that transorted the electricity to each community and to each individual consumer, and directly billed the consumer for this bundled service QFs became a threat to the generation side of this monooly However, the utility still maintained its monooly as it was in control of the method of distributing the roduct to the consumer: the transmission and distribution systems Throughout the 197s and 198s the United States government began to deregulate many industries including telecommunications (1969), trucking (1977), air travel (1978), and natural gas (late 198s) This ush to remove direct government control from markets was undertaken to develo more efficient market-based economies With this trend in government olicy, the move toward a cometitive market in the electricity industry was on the horizon If the industry were to move towards a truly cometitive marketlace, access to the roduct distribution channel needed to be oened The first ste toward this end was another legislative act, the 199 Energy Policy Act [1] This was followed in 1996 by orders from the Federal Energy Regulatory Commission (FERC): FERC Order 888 [13] and FERC Order 889 [14] The stated goal of FERC Order 888 was to romote wholesale cometition through oen access non-discriminatory transmission services by ublic utilities How best to set u markets to achieve this goal is being decided in the federal and state governments throughout the United States [15] Three markets have been created in the United States thus far in California [7], the Pennsylvania-New Jersey-Maryland (PJM) Interconnection [9], and New England [1] 7

20 13 History of Electricity Industry Reform throughout the World While energy conservation and energy self-sufficiency were the motivating goals in the United States, electricity industry reform in other arts of the world have been for vastly different reasons Throughout much of the world, the electricity industry has always been a state-owned monooly Problems due to the inefficiency of state-owned industries have been the motivation behind reform in many countries These roblems have resulted in oor electricity service along with financial shortages of caital investment Several examle exeriences are discussed below 131 United Kingdom The United Kingdom (UK) electricity industry was formerly a state-owned industry In 1947, many imortant industries throughout the UK became government owned and oerated, including the electricity industry In 1957, the government control was extended by the assage of the Central Electricity Generating Board (CEGB) which gave the government control over the generation and transmission systems and all related investment decisions While this may have been an imortant ste in rebuilding the ost-war UK, several decades later industry inefficiencies became a heavy burden on the government One major examle of inefficiency was the government olicy of sustaining the national coal industry through urchases from the national electricity industry from 1957 through the 198s Throughout this time, the electricity industry was ressured and eventually required to urchase British coal at rices which were well above world rices With the electric utilities forced to ay such high rices, electricity rices became excessively high, and the British coal industry in essence became deendent on the electricity industry for survival [5] Eventually due to inefficiencies like these, Britain began to consider the viability of continuing the government-owned industries 8

21 In 1979, under the leadershi of Margaret Thatcher, a conservative government was elected to ower in Britain initiating the rivatization of the government-owned industries Industries were auctioned off and several resent-day comanies were formed such as British Aerosace (1981), Cable and Wireless (1981), British Telecommunications (1984), British Airways (1987), and British Steel (1988) [5] The last major industry to be deregulated was the electric utility industry The start of this was the Electricity Act of 1983 which encouraged the creation of nonutility ower roducers, similar to PURPA (1978) and the Energy Policy Act (199) in the United States Finally, the restructuring was comletely imlemented in the Electricity Act of 1989 A new industry structure was hased in over several years The transmission system was restructured by requiring all comanies with ownershi in the electric transmission grid in the UK to divest their shares of the grid and a new ublicly-traded comany, the National Grid Comany (NGC), was created to manage this grid For generation, The nonnuclear generation was assigned to two comanies, National Power and PowerGen, accounting for 46% and 8% of the generation, resectively For distribution, twelve former regional area boards were rivatized to create regional electricity comanies (RECs) The UK ower ool was then develoed with the National Grid Comany acting as the ool oerator The NGC forecasts the demand each day and reresents this by 48 half-hour segments The suliers then submit block bids for each half-hour, and the NGC determines the disatch using the last acceted bid method discussed reviously The new market structure has been generally successful, but some concern has been exressed regarding the rice volatility as well as a concern over ossible unfair ractices of the two dominant generation comanies Some attemts have been made to remedy these concerns 9

22 In 1994, the regulatory authorities in the UK forced the two dominant comanies to sell 15% of their caacities (9% of the entire UK market) to try and reduce duooly behavior Also in late 1995, each comany attemted a takeover of an REC, romting the regulatory authorities to deny each request on the grounds that the takeovers would ose significant detriments to cometition [5] 13 South America South America is an examle of a region that has also restructured its state-owned monoolies In 198, Chile officially reorganized its electric ower industry Argentina followed in 199, Peru in 1993, Bolivia and Colombia in 1994, and then Brazil and Venezuela in 1996 [16] Examles of inefficiencies included ower rationing throughout Colombia in and an inability to raise caital for investment in the industry in Bolivia and Peru As a result of the reforms, a dramatic imrovement in economic efficiency as well as better service has been seen in many countries In Argentina for instance, availability of thermal generation went from an all-time low of 47% in 199 to 75% in 1996, while electricity rices droed from an average of more than $5 er MWh to $3 er MWh [16] As in the United States, the countries throughout South America have required oen-access schemes which ermit oen and nondiscriminatory use of their transmission systems [16] This oen-access has enabled the develoment of market structures that allow cometition to flourish Most of the market structures in South America have determined generation disatch and rices using sot ricing theory 1

23 133 Australia U until recently, the electricity in Australia has been rovided by ublicly-owned state utilities which met the needs of each individual state Transmission ties between the states were either nonexistent or extremely weak The state governments were resonsible for control and oeration of the industry as well as for lanning The national government was only eriherally involved in regulation through legal requirements such as foreign ownershi limits and taxation [5] Because Australia s oulation of 18 million eole is sread over an area aroximately the size if the 48 contiguous states of the United States, these searate ower systems made economic sense Because of the inefficiencies of government-owned industries, however, Australia has looked to restructure its electricity industry The reform effort has been under way since 1991 with discussions between the industry and government resulting in the National Electricity Code, which was comleted in Setember of 1996 The National Electricity Code established the rules and rocedures for oerating in the national electricity market (NEM) The electricity grid in Australia will remain under regulation of the states until early At this time, the Australia Cometition and Consumer Commission (ACCC) will take over regulation of the grid Methods for ricing transmission services have not been comletely determined at this stage Also, because of the geograhical limitations, initially the NEM will be limited to the southeastern art of Australia encomassing the states of New South Wales and Victoria, and the Australia Caital Territory In order to exand the market, new interstate transmission connections have been lanned to tie to Queensland to the North, South Australia to the West, and Tasmania via a sub-sea link 11

24 The Australia market will utilize a combination of bilateral contracts in the long-term (more than a month ahead) and short-term (days ahead) markets, and a sot ricing for the half-hour markets to match suly and demand All suliers with net exorts larger than 3 MW will be required to articiate in the market, while smaller suliers will be given an otion Initially, consumers with demand of greater than 1 MW will be given the otion to articiate in the market Eventually, however, all customers will be given this otion Full cometition is slated to begin in 1 14 The Need for Simulation Tools Regardless of what market structures are imlemented, there is a need for many new kinds of economic and engineering analysis tools In engineering analysis, tools such as available transfer caability (ATC) calculators are needed to determine the distance from which consumers can urchase their electricity roducts [17],[18] In economic analysis, tools that hel redict future rice trends are needed to hel create a stable market Other tools are needed to bridge the ga between engineering and economic analysis in order to analyze the imact of underlying engineering constraints on the market One such tool that bridges this ga is an individual welfare maximization algorithm which includes a full transmission system model This tool would allow an individual to determine its otimal bidding behavior in an electric ower ool This tool is develoed in this dissertation The alications of welfare maximization tool would be varied For examle, transmission system congestion can subdivide a market into several regions with vastly different rices By including the full transmission system model, market situations like this can be reresented For market articiants the resence of transmission congestion could resent oortunities to sell 1

25 generation into such a subdivided market in which local demand might be high and the number of otential sellers low Tools are needed to hel market articiants devise otimal bidding strategies Engineers at an ISO or other lanning agency could also use such a tool to hel with long-term system forecasting, enabling them to better determine where transmission system imrovements are needed Conversely, there is a need for regulators such as FERC, the Deartment of Justice (DOJ), and the state regulatory commissions to be vigilant against anticometitive acts by market articiants For examle, if a articular entity owns sufficient generation it may be ossible for them to maniulate the market in such a way as to deliberately induce congestion in order to raise rices [19],[] The recent wave of mergers and roosed mergers in the United States requires that regulators have access to tools to assess the otential for this tye of maniulation FERC s need for such a tool is described in its Order 59 Policy Statement on Utility Mergers in December of 1996 [1], and its formal adotion of the Deartment of Justice/Federal Trade Commission (DOJ/FTC) Horizontal Merger Guidelines [] Both the DOJ and the FERC, in a recent roosed rulemaking, have exlicitly stated a desire for comuter models The use of comuter models -- secifically, comuter rograms used to simulate the electric ower market -- has been raised in comments on the Policy Statement and also in secific cases In comments on the Policy Statement, DOJ recommended using comuter simulations to delineate markets and also noted that these simulations could be helful in gauging the market ower of the merged firm The Commission believes that use of a roerly structured comuter model could account for imortant hysical and economic effects in an analysis of mergers and may be a valuable tool to use in a horizontal screen analysis For examle, a comuter model might rove articularly useful in identifying the suliers in the geograhic market that are caable of cometing with the merged comany It could rovide a framework to hel ensure consistency in the treatment of the data used in identifying suliers in a geograhic market [3] 13

26 15 Overview This dissertation resents a new algorithm which allows a market articiant to determine the bidding strategy for maximizing its individual welfare in a sot market This algorithm must model the economics of the electricity sot market while also maintaining a full transmission system model Without modeling the transmission system, the algorithm would not be able to reroduce the rice volatility already seen in congested ower systems today [1] An overview of the sot market model and recent work in the area follows 151 The sot market model The starting oint of an individual welfare maximization algorithm is the develoment of a model which reresents the electricity market This dissertation studies the behavior of the electric ower ool which uses sot ricing This market is modeled by using the otimal ower flow (OPF) The OPF roblem was formulated in 196 [4] and has been an active area of research ever since Because the OPF can be a very large, nonlinear mathematical rogramming roblem, many numerical algorithms and techniques have been emloyed to hel solve it [5]- [3] An excellent literature survey of the different techniques is found in [31], while an overview of the challenges to solving the OPF is found in [3] This dissertation builds on the OPF written in [5] Chater introduces the concet of using an otimal ower flow to model the behavior of an electric ower ool Using an OPF, a sot rice market can be simulated with all engineering constraints enforced In this way, the imact of transmission constraints on the sot rices in the market can be directly assessed Much of the theory describing a sot market for electricity using the OPF is discussed in [8] Using the OPF as a software engine to simulate a sot market has been an 14

27 active area of research A good overview of how a sot market is modeled using the OPF is rovided in [33] Sot ricing in ower systems could also extend to reactive ower ricing Thus markets for real and reactive ower are conceivable and have been recommended by some [34] Reactive ower rices are tyically much smaller than the real ower rices; however, they can be extremely volatile For examle, in both [35] and [36], reactive ower sot rices are shown to vary by orders of magnitude when voltage limits are encountered in the ower system Also, caital investment for reactive ower equiment such as caacitor banks is much greater than oerating cost, so ricing must take into account this caital investment [35],[37] The markets of today have not yet included sot ricing of reactive ower This dissertation does not concentrate on studying reactive ower markets; however, a comlete model of such a market is investigated, and suly and demand models for reactive ower are introduced in Chater 3 Sot ricing ensures the efficient oeration of the transmission system by roviding rice differences across the system which reflect the social cost of enforcing limits The rice differences can become quite large during times of high system demand resulting in rice sikes in areas with a suly deficit Work has been done investigating the interaction of transmission limits with the sot market in [38]-[4] This dissertation will highlight this interaction throughout Throughout their history, electric utilities have concentrated on develoing the economics of the suly side of the industry Efforts have been made for decades to manage and minimize the costs of sulying electricity to customers Because of this, the extension to a free market with sulier bidding has been concetually straightforward Recently however, markets have begun to incororate behavior from the demand side through consumer bidding An examle in 15

28 Chater shows that this is necessary in order to achieve realistic market behavior Chater 3 shows that the OPF can easily be modified to include consumer bidding to cature the behavior of all market articiants [41],[4] A detailed exlanation and a comlete derivation of the incororation of consumer models into the OPF algorithm is rovided 15 Maximizing individual welfare in a sot market With a good market model, attention turns to determining how an individual can otimize its own outcome in the market Work done in [43]-[45] concentrated on otimal bidding strategies in markets which use the last acceted bid method for determining disatch and rice Genetic algorithms were alied to the roblem in [45] Game theoretic analysis is introduced as a ossible method for maximizing individual welfare in a sot market in [46]-[48] The introduction in [46]-[48] is useful and interesting, but in order to manage comlexity, otential bids by each market articiant are limited to three ossibilities: bid low, bid medium, or bid high A more general market model is resented in [49], which allows suliers and consumers to submit a general bid curve to the market In such a market, finding the individual welfare maximum resents a very difficult challenge Extending the algorithm resented in [49] to larger systems does not look romising Chater 4 develos a new algorithm based on Newton s method for use in maximizing an individual s welfare This algorithm is shown to be successful on several samle systems, and the rosects of scaling the algorithm u to systems of realistic size aear very good With the ability to model market articiant behavior more generally, the investigation of game theoretic concets such as market equilibrium oints can be investigated as was done in [5] Market equilibrium oints were found in [5] by iteratively solving the individual welfare maximization roblem for each market articiant until bids became constant This technique is 16

29 used in conjunction with the new individual welfare maximization algorithm develoed in this dissertation and encouraging results are resented in Section 44 The new individual welfare maximization technique develoed in Chater 4 is a calculusbased otimization routine Results in Chater 4, however, clearly demonstrate that maximizing the welfare of an individual is a highly nonconcave function with a otentially large number of local otima These local otima corresond to the ability of a market articiant to maniulate its bidding strategy around an engineering constraint in the system Any calculus-based method will not be able to generally solve for a global maximum when more than one local maximum exists As a result, the more global otimization aroach of genetic algorithms is considered Chater 5 develos a genetic algorithm (GA) for use in aiding the solution of welfare maximization 153 Contouring visualization The develoment of new electricity market structures also necessitates the exchange of huge amounts of information among the articiants in the market Power systems tyically involve information on tens of thousands of buses and transmission lines Analyzing all this information has been a daunting task for exerienced engineers for the last century With more information and new eole concerned with the oeration of the electricity markets, new ways to rocess this information more quickly and easily are needed As a result, this dissertation also develos several new techniques for the visualization of ower system data based on the use of contours Chater 6 illustrates these advancements 17

30 A MODEL OF BIDDING IN A POOL-BASED ELECTRICITY MARKET This chater introduces the electric ower ool structure used throughout this dissertation The modeling of this structure through the use of the otimal ower flow (OPF) is discussed followed by several examles demonstrating the use of the OPF in modeling sulier bidding 1 The Electric Power Pool Market Structure The ower ool structure is defined as a set of suliers bidding for the right to rovide energy service (MWh) to the consumers in the ower ool Suliers are then aid according to the disatch determined by the ool oerator at the sot rice for energy The disatch and sot rices are deendent on the system demand, transmission system structure, as well as the bids received from the suliers In order to kee the initial examles of this chater consistent with the history of ower system analysis, the market is initially confined to sulier bidding The examles in this chater will demonstrate the need to include consumer bidding behavior in the market A comlete discussion of the theory of sot markets and this imlementation of sulier and consumer bidding is reserved for discussion in Chater 3 Here the toics are only introduced to motivate their more thorough analysis later The Otimal Power Flow Over the last 3 years or so the otimal ower flow (OPF) algorithm has been an active area of research The OPF is defined as a static, nonlinear otimization roblem in which certain control variables are adjusted to otimize an objective function, while satisfying hysical and 18

31 oerational constraints Tyically the objective function has been to either minimize the cost of generation, or to minimize system losses Available controls have usually been ower system devices, such as generator real and reactive ower oututs, real ower transactions between oerating areas, transformer ta or hase ositions, and switched shunt devices To rovide some background, the traditional OPF with the objective of minimizing the total generation costs for the entire system is described In order to comare these equations with develoments later in the dissertation, maximizing the negative of the costs is resented instead of minimizing the costs max st s, x - C( s) h(x) ˆ sˆ + dˆ h(x,s, d) = = h(x) s min s g(x,s, d) = s s max g(x) (1) To solve this nonlinear rogram, form the Lagrange function for it: L = C( s) + T h C( s) + T = + gs min[ s T + g g( x) h(x,s, d) + T hˆ min T g g(x,s, d) [ ˆ + ˆ T h(x) sˆ d] + [ h(x) ] h T s] + [ ] gs max s smax () [51] The roblem can then be determined by solving for the necessary Kuhn-Tucker conditions 19

32 Stationarity Conditions L T h(x, s,d) = h + x x L C( s) ~ = hs ˆ s s L = h(x,s,d) = g(x,s,d) = ; g(x,s,d) gs min g(x, s,d) = x Comlementary Slackness Conditions T g g h T g + gs max = (3) The OPF roblem has been solved using a variety of different techniques Throughout this thesis, the Newton s method aroach as in [5] and [8] is used The software imlementation in Chaters and 3 are integrated into PowerWorld Simulator [5],[53] 3 A Market Model from the OPF The Lagrange multiliers determined from the solution of the otimal ower flow rovide imortant economic information regarding the ower system A Lagrange multilier can be interreted as the derivative of the objective function with resect to enforcing the resective constraint Therefore, the Lagrange multiliers associated with enforcing the ower flow equations of the OPF can be interreted as the marginal cost of roviding energy service ($/MWh) to that bus in the ower system This is what lies at the heart of sot ricing theory [8] and allows the OPF to model the ower ool

33 Market articiants submit bids to rovide energy services Market Particiants Power Pool Pool oerator determines the disatch for each sulier and a sot rice for each node in the system Figure 1 Power Pool Information Flow As is illustrated in Figure 1, a ower ool can be modeled in which suliers submit bid curves that define amounts of energy (MW), along with resective rices ($/MWh) Chater 3 will rigorously show these bids can be interreted as being directly related to the marginal cost curves of the suliers The ool oerator assumes the submitted bids are directly related to the true marginal cost curves of the suliers and solves the OPF of minimizing the total generation costs of the generators in order to determine the generation disatch as well as the sot rices for each node in the system Each sulier then generates the amount of ower secified by the ool oerator and is aid according to its bid Using the ideas just ut forth, an electricity market is constructed in which suliers submit linear bid curves, such as those seen in Figure Each sulier submits a minimum rice min at which they will sell ower along with a sloe m s defining the sloe of the linear curve 1 Using these bids, the ool oerator (such as a ower exchange or ossibly an ISO) solves the OPF 1 While only allowing single-segment linear bids may seem limiting, the analysis of the California ower market done in [54] shows that over time suly bid curves aear to be two-segment linear curves Two-segments could be added to this develoment in the future, but [54] shows that linear bids are reasonably accurate 1

34 under the assumtion that the articiants are submitting their true marginal The amount of disatch received is then awarded according to the solution of this OPF Generation Bid [MW] m s Price = min [$/MWh] Figure Linear Price-Deendent Suly Bids 4 Market Examles When erforming market analysis of the system, a market articiant is interested in what its rofit will be for various bidding strategies This rofit will deend not only on its bids, but also on the bids of the other articiants in the market In order to analyze the electricity market, the individual must make an estimate of the bids for each of the suliers which it does not control The individual then must determine the set of bids for the suliers it controls which are an otimal resonse to the cometitors bids 41 Examle market model for a four-sulier lossless system In order to demonstrate the market model, the six-bus, four-generator system shown in Figure 3 was used The OPF solution for the ool when all generators bid their true marginal cost is shown in Figure 3 Note that this is a lossless system As a result, the sot rices throughout the system are equal, as long as no transmission constraints are reached