Strategic Generation Investment and equilibria in oligopolistic electricity markets

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(1)UNIVERSIDAD DE CASTILLA-LA MANCHA. DEPARTAMENTO DE INGENIERÍA ELÉCTRICA, ELECTRÓNICA, AUTOMÁTICA Y COMUNICACIONES. STRATEGIC GENERATION INVESTMENT AND EQUILIBRIA IN OLIGOPOLISTIC ELECTRICITY MARKETS. TESIS DOCTORAL. AUTHOR: S. JALAL KAZEMPOUR DIRECTOR: ANTONIO J. CONEJO NAVARRO Ciudad Real, Mayo de 2013.

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(3) UNIVERSIDAD DE CASTILLA-LA MANCHA. DEPARTMENT OF ELECTRICAL ENGINEERING. STRATEGIC GENERATION INVESTMENT AND EQUILIBRIA IN OLIGOPOLISTIC ELECTRICITY MARKETS. PhD THESIS. AUTHOR: S. JALAL KAZEMPOUR SUPERVISOR: ANTONIO J. CONEJO NAVARRO Ciudad Real, May 2013.

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(5) To my parents, brothers and sister.

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(7) Acknowledgements I am deeply grateful to Prof. Antonio J. Conejo for his expert guidance, wise advise, support and help over the last four years. Working with him has been an outstanding, fruitful and enjoyable experience in all aspects of academic life. I would like to thank my friend, Dr. Carlos Ruiz, for his efficient advice regarding technical and mathematical aspects of this work. I am truly grateful to the Universidad de Castilla - La Mancha for providing an outstanding research environment. In addition, I wish to thank the Junta de Comunidades de Castilla - La Mancha for its partial financial support through project PCI-08-0102. Additionally, I am grateful to the Ministry of Economy and Competitiveness of Spain for its partial financial support through CICYT project DPI2009-09573. Many thanks to Prof. Hamidreza Zareipour for providing me an outstanding research atmosphere and financial support during six months (February 1st 2012-July 31st 2012) when I visited him at the University of Calgary, Calgary, AB, Canada. His comments were very helpful and efficient. I also appreciate the technical support and advice I gained from the research group of Prof. Lennart Söder at the Royal Institute of Technology (KTH), Stockholm, Sweden, from April 15th 2011 to July 31st 2011. Prof. Afzal Siddiqui from the University College London, U.K., and Prof. Maria Teresa Vespucci from the Università degli studi di Bergamo, Italy, read the last version of this report and made valuable and pertinent observations. Thanks to them. Special thanks to Dr. Kazem Zare for his help and advice, which facilitated the start of my PhD. vii.

(8) viii Thanks to my friends in Ciudad Real, Carlos, Luis, David, Edu, Salva, Juanmi, Alberto, Rafa, Ali, Marco, Morteza, Ricardo and many others, for their help in many practical aspects and for their friendship. I would also like to thank them for their help when I arrived in Spain. Thanks to my friends in Madrid, Stockholm and Calgary, Mehdi, Behnam, Amin, Behnaz, Hamid, Ebrahim, Ali, Eissa and many others. I will never forget our travel adventures. Thanks finally, from the bottom of my heart, to my family for their unconditional love, support, and everything. Ciudad Real, Spain May 2013.

(9) Contents Contents. ix. List of Figures. xvii. List of Tables. xxi. Notation. xxiii. 1 Generation Investment: Introduction. 1. 1.1 Electricity Markets . . . . . . . . . . . . . . . . . . . . . . . . .. 1. 1.2 Chapter Organization . . . . . . . . . . . . . . . . . . . . . . . .. 3. 1.3 Thesis Motivation . . . . . . . . . . . . . . . . . . . . . . . . . .. 3. 1.4 Problem Description . . . . . . . . . . . . . . . . . . . . . . . .. 6. 1.5 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . .. 9. 1.5.1. Literature Review: Generation Investment for a Strategic Producer . . . . . . . . . . . . . . . . . . . . . . . . .. 9. 1.5.2. Literature Review: Generation Investment Equilibria . . 10. 1.5.3. Other Related Works in the Literature . . . . . . . . . . 14. 1.6 Modeling Assumptions . . . . . . . . . . . . . . . . . . . . . . . 16 1.6.1. Investment Study Approach . . . . . . . . . . . . . . . . 16. 1.6.2. Load-Duration Curve . . . . . . . . . . . . . . . . . . . . 18. 1.6.3. Network Representation . . . . . . . . . . . . . . . . . . 20. 1.6.4. Market Competition Modeling . . . . . . . . . . . . . . . 21. 1.6.5. Additional Modeling Assumptions . . . . . . . . . . . . . 22. 1.7 Solution Approach . . . . . . . . . . . . . . . . . . . . . . . . . 25 ix.

(10) x. CONTENTS 1.8. Thesis Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . 31. 1.9. Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . 33. 2 Strategic Generation Investment. 37. 2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37. 2.2. Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37. 2.3. 2.2.1. Modeling Assumptions . . . . . . . . . . . . . . . . . . . 38. 2.2.2. Structure of the Proposed Model . . . . . . . . . . . . . 40. Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.3.1. Notational Assumptions . . . . . . . . . . . . . . . . . . 42. 2.3.2. Bilevel Model . . . . . . . . . . . . . . . . . . . . . . . . 42. 2.3.3. Optimality Conditions Associated with the Lower-Level Problems (2.2) . . . . . . . . . . . . . . . . . . . . . . . 45 2.3.3.1. KKT Conditions Associated with the LowerLevel Problems (2.2) . . . . . . . . . . . . . . . 46. 2.3.3.2. Strong Duality Equality Associated with the Lower-Level Problems (2.2) . . . . . . . . . . . 50. 2.3.4. MPEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51. 2.3.5. MPEC Linearization . . . . . . . . . . . . . . . . . . . . 52 2.3.5.1. 2.3.6 2.4. Linearizing Ztw . . . . . . . . . . . . . . . . . . 53. MILP Formulation . . . . . . . . . . . . . . . . . . . . . 55. Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . 60 2.4.1. Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61. 2.4.2. Deterministic Solution . . . . . . . . . . . . . . . . . . . 64. 2.4.3. 2.4.2.1. Uncongested and Congested Network . . . . . . 64. 2.4.2.2. Strategic and Non-Strategic Offering . . . . . . 65. Stochastic Solution . . . . . . . . . . . . . . . . . . . . . 67. 2.5. Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70. 2.6. Computational Considerations . . . . . . . . . . . . . . . . . . . 72. 2.7. Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . 72. 3 Strategic Generation Investment: Tackling Computational Burden via Benders’ Decomposition. 75.

(11) CONTENTS. xi. 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.2 Benders’ Approach . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.2.1. Complicating Variables . . . . . . . . . . . . . . . . . . . 76. 3.2.2. Proposed Algorithm . . . . . . . . . . . . . . . . . . . . 77. 3.3 Convexity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.3.1. Illustrative Example for Convexity Analysis . . . . . . . 79 3.3.1.1. Data . . . . . . . . . . . . . . . . . . . . . . . . 79. 3.3.1.2. Cases Considered . . . . . . . . . . . . . . . . . 81. 3.3.1.3. Convexity Analysis . . . . . . . . . . . . . . . . 82. 3.4 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.4.1. Decomposed Problems . . . . . . . . . . . . . . . . . . . 84. 3.4.2. MPEC Associated with the Decomposed Problems . . . 87. 3.4.3. Auxiliary Problems . . . . . . . . . . . . . . . . . . . . . 88. 3.4.4. Subproblems. 3.4.5. Master Problem . . . . . . . . . . . . . . . . . . . . . . . 95. . . . . . . . . . . . . . . . . . . . . . . . . 91. 3.5 Benders’ Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.6 Case Study of Section 2.5 . . . . . . . . . . . . . . . . . . . . . 97 3.7 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.7.1. Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99. 3.7.2. Investment Results . . . . . . . . . . . . . . . . . . . . . 101. 3.8 Computational Considerations . . . . . . . . . . . . . . . . . . . 103 3.9 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . 104 4 Strategic Generation Investment Considering the Futures Market and the Pool 107 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.2 Base and Peak Demand Blocks . . . . . . . . . . . . . . . . . . 108 4.3 Futures Market Auctions . . . . . . . . . . . . . . . . . . . . . . 109 4.4 Uncertainty Modeling . . . . . . . . . . . . . . . . . . . . . . . . 110 4.5 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.5.1. Hierarchical Structure . . . . . . . . . . . . . . . . . . . 111. 4.5.2. Modeling Assumptions . . . . . . . . . . . . . . . . . . . 113. 4.6 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.

(12) xii. CONTENTS 4.6.1. Notational Assumptions . . . . . . . . . . . . . . . . . . 115. 4.6.2. Bilevel Model . . . . . . . . . . . . . . . . . . . . . . . . 115 4.6.2.1. Futures Base Auction Clearing: Lower-level Problem (4.2) . . . . . . . . . . . . . . . . . . . . . 118. 4.6.2.2. Futures Peak Auction Clearing: Lower-level Problem (4.3) . . . . . . . . . . . . . . . . . . . . . 120. 4.6.2.3. Pool Clearing: Lower-level Problems (4.4) . . . 121. 4.6.3. Factor Ω . . . . . . . . . . . . . . . . . . . . . . . . . . . 124. 4.6.4. Optimality Conditions Associated with the Lower-Level Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.6.4.1. KKT Conditions Associated with the LowerLevel Problem (4.2) . . . . . . . . . . . . . . . 125. 4.6.4.2. Strong Duality Equality Associated with the Lower-Level Problem (4.2) . . . . . . . . . . . . 127. 4.6.4.3. KKT Conditions Associated with the LowerLevel Problem (4.3) . . . . . . . . . . . . . . . 129. 4.6.4.4. Strong Duality Equality Associated with the Lower-Level Problem (4.3) . . . . . . . . . . . . 132. 4.6.4.5. KKT Conditions Associated with the LowerLevel Problems (4.4) . . . . . . . . . . . . . . . 134. 4.6.4.6 4.6.5. MPEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139. 4.6.6. MPEC Linearization . . . . . . . . . . . . . . . . . . . . 140 4.6.6.1 4.6.6.2 4.6.6.3. 4.6.7 4.7. Strong Duality Equality Associated with each Lower-Level Problem (4.4) . . . . . . . . . . . . 137. Exact Linearization of Λ . . . . . . . . . . . . . 140 b . . . . . . . . . . . . . 143 Exact Linearization of Λ. Approximate Linearization of Λtw . . . . . . . . 143. MILP Formulation . . . . . . . . . . . . . . . . . . . . . 150. Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 4.7.1. Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163. 4.7.2. Cases Considered . . . . . . . . . . . . . . . . . . . . . . 169. 4.7.3. Investment Results . . . . . . . . . . . . . . . . . . . . . 171 4.7.3.1. Only Pool . . . . . . . . . . . . . . . . . . . . . 173.

(13) CONTENTS. xiii 4.7.3.2. Pool and Futures Base Auction Without Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . 173. 4.7.3.3. Pool and Futures Base Auction With Arbitrage 178. 4.7.3.4. Pool, Futures Base and Futures Peak Auctions 180. 4.7.3.5. Impact of Factor Υ on Generation Investment Decisions . . . . . . . . . . . . . . . . . . . . . 181. 4.8 Computational Considerations . . . . . . . . . . . . . . . . . . . 182 4.9 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . 182 5 Generation Investment Equilibria. 185. 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 5.2 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 5.3 Modeling Assumptions . . . . . . . . . . . . . . . . . . . . . . . 188 5.4 Single-Producer Problem . . . . . . . . . . . . . . . . . . . . . . 190 5.4.1. Structure of the Hierarchical (bilevel) Model . . . . . . . 190. 5.4.2. Formulation of the Bilevel Model . . . . . . . . . . . . . 192. 5.4.3. Optimality Conditions of the Lower-Level Problems . . . 195 5.4.3.1. KKT Optimality Conditions Associated with Lower-level Problems (5.1g)-(5.1n) . . . . . . . 196. 5.4.3.2. Optimality Conditions Associated with Lowerlevel Problems (5.1g)-(5.1n) Resulting from the Primal-Dual Transformation . . . . . . . . . . . 199. 5.4.4. MPEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201. 5.5 Multiple Producers Problem . . . . . . . . . . . . . . . . . . . . 205 5.5.1. EPEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205. 5.5.2. Optimality Conditions associated with the EPEC . . . . 206 5.5.2.1. Primal Equality Constraints . . . . . . . . . . . 206. 5.5.2.2. Equality Constraints Obtained From Differentiating the Corresponding Lagrangian with Respect to the Variables in ΞUL . . . . . . . . . . 208. 5.5.2.3 5.5.3. Complementarity Conditions . . . . . . . . . . 211. EPEC Linearization . . . . . . . . . . . . . . . . . . . . 214 5.5.3.1. Linearizing the Strong Duality Equalities (5.8g) 214.

(14) xiv. CONTENTS 5.5.3.2. Linearizing the Non-linear Terms Involving φSD yt 215. 5.6. MILP Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 215. 5.7. Searching For Investment Equilibria . . . . . . . . . . . . . . . . 230 5.7.1. Objective Function (5.29a): Total Profit . . . . . . . . . 231. 5.7.2. Objective Function (5.29a): Annual True Social Welfare 232. 5.8. Algorithm for the Diagonalization Checking . . . . . . . . . . . 232. 5.9. Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . 233 5.9.1. Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234. 5.9.2. Cases Considered . . . . . . . . . . . . . . . . . . . . . . 237. 5.9.3. Demand Bid and Stepwise Supply Offer Curves . . . . . 238. 5.9.4. General Equilibrium Results . . . . . . . . . . . . . . . . 240. 5.9.5. Triopoly Cases Maximizing Total Profit . . . . . . . . . . 245. 5.9.6. Triopoly Cases Maximizing Annual True Social Welfare . 246. 5.9.7. Monopoly Cases . . . . . . . . . . . . . . . . . . . . . . . 247. 5.9.8. Investment Results for Each Producer. 5.9.9. Diagonalization Checking. . . . . . . . . . . 248. . . . . . . . . . . . . . . . . . 250. 5.9.10 Impact of Factor Υ on Generation Investment Equilibria 252 5.9.11 Impact of the Available Budget on Generation Investment Equilibria . . . . . . . . . . . . . . . . . . . . . . . 253 5.10 Case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 5.10.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 5.10.2 Results of Generation Investment Equilibria . . . . . . . 258 5.11 Computational Considerations . . . . . . . . . . . . . . . . . . . 261 5.11.1 Computational Conclusions . . . . . . . . . . . . . . . . 261 5.11.2 Selection of values for φSD . . . . . . . . . . . . . . . . . 262 yt 5.11.3 Suggestions to Reduce the Computational Burden . . . . 262 5.12 Ex-post Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 263 5.13 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . 263 5.13.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 263 5.13.2 General Conclusions . . . . . . . . . . . . . . . . . . . . 265 5.13.3 Regulatory Conclusions . . . . . . . . . . . . . . . . . . . 267.

(15) CONTENTS. xv. 6 Summary, Conclusions, Contributions and Future Research 269 6.1 Thesis Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 269 6.1.1 Strategic Producer Investment . . . . . . . . . . . . . . . 270 6.1.2 Investment Equilibria . . . . . . . . . . . . . . . . . . . . 272 6.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 6.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 6.4 Suggestions for Future Research . . . . . . . . . . . . . . . . . . 280 A IEEE Reliability Test System: Transmission Data. 283. B Mathematical Background 287 B.1 Bilevel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 B.2 MPEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 B.2.1 MPEC Obtained from the KKT Conditions . . . . . . . 290 B.2.2 MPEC Obtained from the Primal-Dual Transformation . 293 B.2.2.1 B.2.2.2. Linear Form of the Lower-Level Problems (B.2) 293 Dual Optimization Problems Pertaining to Lower-. B.2.2.3. Level Problems (B.4) . . . . . . . . . . . . . . . 295 Optimality Conditions Associated with LowerLevel Problems (B.4) Resulting from the Primal-. B.2.2.4. Dual Transformation . . . . . . . . . . . . . . . 296 Resulting MPEC from the Primal-Dual Trans-. formation . . . . . . . . . . . . . . . . . . . . . 298 B.2.3 Equivalence Between the MPECs Obtained from the KKT Conditions and the Primal-Dual Transformation . . . . . 300 B.3 EPEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 B.4 Benders’ Decomposition . . . . . . . . . . . . . . . . . . . . . . 307 B.5 Linearization Techniques . . . . . . . . . . . . . . . . . . . . . . 311 B.5.1 Complementarity Linearization . . . . . . . . . . . . . . 312 B.5.2 Binary Expansion Approach . . . . . . . . . . . . . . . . 313 Bibliography. 317. Index. 332.

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(17) List of Figures 1.1 Introduction: Piecewise approximation of the load-duration curve for the target year. . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.2 Introduction: Demand blocks and demand-bid blocks (included in demand block t = t1 ) corresponding to the example given in Table 1.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.3 Introduction: General hierarchical (bilevel) structure of any single producer model. . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.4 Introduction: Interrelation between the upper-level and lowerlevel problems considering the futures market and the pool auctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.5 Introduction: Transformation of the bilevel model of a strategic producer into its corresponding MPEC. . . . . . . . . . . . . . . 30 1.6 Introduction: EPEC and its optimality conditions. . . . . . . . . 31 2.1 Direct solution: Bilevel structure of the proposed strategic generation investment model. . . . . . . . . . . . . . . . . . . . . . 40 2.2 Direct solution: Interrelation between the upper-level and lowerlevel problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.3 Direct solution: Six-bus test system (illustrative example). . . . 60 2.4 Direct solution: Locational marginal prices in (a) Cases A and B, and (b) Case C. . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.1 Benders’ approach: Flowchart of the proposed Benders’ algorithm. 78 xvii.

(18) xviii 3.2. LIST OF FIGURES Benders’ approach: Minus-producer’s profit as a function of capacity investment considering (a) non-strategic offering and one scenario, (b) strategic offering and one scenario, (c) nonstrategic offering and all scenarios, and (d) strategic offering and all scenarios. . . . . . . . . . . . . . . . . . . . . . . . . . . 83. 3.3. Benders’ approach: Evolution of the expected profit, the profit standard deviation and the CPU time with the number of scenarios (case study). . . . . . . . . . . . . . . . . . . . . . . . . . 102. 3.4. Benders’ approach: Evolution of Benders’ algorithm in case study involving 60 scenarios. . . . . . . . . . . . . . . . . . . . . 103. 4.1. Futures market and pool: Piecewise approximation of the loadduration curve for the target year, including peak and base demand blocks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108. 4.2. Futures market and pool: Demand blocks supplied through different markets, i.e., futures base auction, futures peak auction and pool. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109. 4.3. Futures market and pool: Hierarchical structure of the proposed strategic generation investment model. . . . . . . . . . . . . . . 112. 4.4. Futures market and pool: Maximum load level of a given demand supplied through the futures base auction, futures the peak auction and the pool. . . . . . . . . . . . . . . . . . . . . . 164. 4.5. Futures market and pool: Market outcomes as a function of factor Γ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177. 4.6. Futures market and pool: Expected profit of the strategic producer as a function of Γ and ∆1 . . . . . . . . . . . . . . . . . . . 179. 4.7. Futures market and pool: Strategic producer’s expected profit and its total investment as a function of factor Υ (Case 3). . . . 181. 5.1. EPEC problem: Hierarchical (bilevel) structure of the model solved by each strategic producer. . . . . . . . . . . . . . . . . . 191. 5.2. EPEC problem: Transformation of the bilevel model of a strategic producer into its corresponding MPEC (primal-dual transformation). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.

(19) LIST OF FIGURES. xix. 5.3 EPEC problem: Network of the illustrative example. . . . . . . 234 5.4 EPEC problem: Piecewise approximation of the load-duration curve for the target year (illustrative example). . . . . . . . . . 234 5.5 EPEC problem: Demand bid curve and stepwise supply offer curve corresponding to the first demand block t = t1 (Case 1 maximizing total profit). . . . . . . . . . . . . . . . . . . . . . . 239 5.6 EPEC problem: Total newly built capacity, total profit and annual true social welfare as a function of factor Υ (Case 1 maximizing total profit). . . . . . . . . . . . . . . . . . . . . . . 251 5.7 EPEC problem: Total newly built capacity, total profit and annual true social welfare as a function of the available investment budget (Case 1 maximizing annual true social welfare). . . . . . 254 5.8 EPEC problem: The simplified version of the IEEE RTS network (case study).. . . . . . . . . . . . . . . . . . . . . . . . . . 256. A.1 IEEE reliability test system: Network. . . . . . . . . . . . . . . 284.

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(21) List of Tables 1.1 Introduction: Relevant features of some models reported in literature and the model proposed in this dissertation (generation investment of a given strategic producer). . . . . . . . . . . . . . 11 1.2 Introduction: Relevant features of some models reported in the literature and the model proposed in this dissertation (generation investment equilibria). . . . . . . . . . . . . . . . . . . . . . 13 1.3 Introduction: Example for clarifying the demand-bid blocks. . . 18 2.1 Direct solution: Type and data for the existing generating units (illustrative example). . . . . . . . . . . . . . . . . . . . . . . . 61 2.2 Direct solution: Location and type of the existing units (illustrative example). . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.3 Direct solution: Type and data for investment options (illustrative example). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.4 Direct solution: Demand-bid blocks including maximum loads [MW] and bid prices [e/MWh] (illustrative example). . . . . . . 63 2.5 Direct solution: Investment results pertaining to the uncongested and congested cases (illustrative example). . . . . . . . . 65 2.6 Direct solution: Investment results considering and not considering strategic offering (illustrative example). . . . . . . . . . . . 67 2.7 Direct solution: Rival producer scenarios (illustrative example).. 68. 2.8 Direct solution: Investment results pertaining to the stochastic cases (illustrative example). . . . . . . . . . . . . . . . . . . . . 69 2.9 Direct solution: Location and type of existing units (case study). 70 2.10 Direct solution: Investment results (case study). . . . . . . . . . 71 xxi.

(22) xxii. LIST OF TABLES. 3.1. Benders’ approach: Type and data for the existing generating units (illustrative example for convexity analysis). . . . . . . . . 80. 3.2. Benders’ approach: Demand-bid blocks including maximum loads [MW] and bid prices [e/MWh] (illustrative example for convexity analysis). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80. 3.3. Benders’ approach: Investment options of the rival producers (illustrative example for convexity analysis). . . . . . . . . . . . 81. 3.4. Benders’ approach: Investment results of the case study presented in Section 2.5 obtained by the direct solution approach presented in Chapter 2. . . . . . . . . . . . . . . . . . . . . . . . 98. 3.5. Benders’ approach: Investment results of the case study presented in Section 2.5 obtained by the proposed Benders’ approach. 98. 3.6. Benders’ approach: Investment options. . . . . . . . . . . . . . . 100. 3.7. Benders’ approach: Investment options for rival producers (case study). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101. 3.8. Benders’ approach: Investment results (case study). . . . . . . . 102. 4.1. Futures market and pool: Data for the existing units of the strategic producer. . . . . . . . . . . . . . . . . . . . . . . . . . 167. 4.2. Futures market and pool: Data for rival units. . . . . . . . . . . 167. 4.3. Futures market and pool: Type and data for the investment options. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168. 4.4. Futures market and pool: Cases considered. . . . . . . . . . . . 170. 4.5. Futures market and pool: Investment results. . . . . . . . . . . . 172. 4.6. Futures market and pool: Market clearing prices pertaining to all cases considered. . . . . . . . . . . . . . . . . . . . . . . . . . 174. 4.7. Futures market and pool: Yearly production results pertaining to all cases considered. . . . . . . . . . . . . . . . . . . . . . . . 175. 5.1. EPEC problem: Data pertaining to the existing units (illustrative example). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235. 5.2. EPEC problem: Data pertaining to demands and their price bids (illustrative example). . . . . . . . . . . . . . . . . . . . . . 235.

(23) LIST OF TABLES. xxiii. 5.3 EPEC problem: Type and data for the candidate units (illustrative example). . . . . . . . . . . . . . . . . . . . . . . . . . . 236 5.4 EPEC problem: Cases considered for the illustrative example. . 237 5.5 EPEC problem: General results of generation investment equilibria (illustrative example). . . . . . . . . . . . . . . . . . . . . 241 5.6 EPEC problem: Three equilibria for Case 1 maximizing total profit (illustrative example). . . . . . . . . . . . . . . . . . . . . 248 5.7 EPEC problem: Data pertaining to the existing units (case study).257 5.8 EPEC problem: Location of the existing units (case study). . . 257 5.9 EPEC problem: results of generation investment equilibria (case study). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 A.1 IEEE Reliability Test System: Reactance (p.u. on a 100 MW base) and capacity of transmission lines. . . . . . . . . . . . . . 285.

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(25) Notation The notation used in this dissertation is listed below. The following observations are in order: 1) All symbols including index t pertain to a pool, e.g., symbols PtiS and θtn . These symbols are used in Chapters 2 to 5. 2) All symbols without index t, but including overlining refer to the futures S O base auction, e.g., symbols P i and C j . These symbols are only used in Chapter 4. 3) All symbols without index t, but including a hat refer to the futures bO . These symbols are only used in peak auction, e.g., symbols PbiS and C j Chapter 4.. 4) Symbols λ, µ, ν and ξ with subscripts and/or superscripts and/or overlining and/or hat are dual variables associated with lower-level problems, max max max e.g., dual variables λ, µ bSi , νtnm and ξtn . These symbols are used in Chapters 2 to 5.. 5) Symbol λ with subscripts and/or superscripts and/or overlining and/or b are market clearing prices, i.e., dual varihat, i.e., symbols λtn , λ and λ ables associated with energy balance constraints.. 6) Symbols χ, η, β, γ, τ , δ, ρ and φ with subscripts and/or superscripts and/or overlining and/or hat are dual variables associated with the MPEC of a strategic producer. These symbols are only used in Chapter 5. xxv.

(26) xxvi. NOTATION. Indices: t. Index for demand blocks running from 1 to T .. y. Index for producers from 1 to Y .. i. Index for candidate generating units of the strategic producer running from 1 to I.. k. Index for existing generating units of the strategic producer running from 1 to K.. j. Index for other generating units owned by rival producers running from 1 to J.. d. Index for demands running from 1 to D.. h. Index for available investment capacities running from 1 to H.. w. Index for scenarios running from 1 to W .. n, m. Indices for buses running from 1 to N, and from 1 to M, respectively.. Sets: ΨS. Set of indices of candidate units of the strategic producer.. ΨSn. Subset of set ΨS containing indices of candidate units located at bus n.. ΨES. Set of indices of existing units of the strategic producer.. ΨES n. Subset of set ΨES containing indices of existing units located at bus n.. ΨO. Set of indices of other units owned by rival producers..

(27) NOTATION. xxvii. ΨO n. Subset of set ΨO containing indices of rival units located at bus n.. ΨD. Set of indices of demands.. ΨD n. Subset of set ΨD containing indices of demands located at bus n.. Ωy. Set of indices of units owned by producer y.. Φn. Set of indices of buses adjacent to bus n.. Tb. Set of indices of base demand blocks.. Tp. Set of indices of peak demand blocks.. Constants: There are four types of constants in this dissertation, namely: 1) General constants regarding demand blocks, investment options, investment budget and transmission lines. These constants are used in Chapters 2 to 5. 2) Constants pertaining to the pool. These constants include index w if they refer to scenario w, and are used in Chapters 2 to 5. 3) Constants pertaining to the futures base auction. These constants are overlined, and are only used in Chapter 4. 4) Constants pertaining to the futures peak auction. These constants include a hat, and are only used in Chapter 4. All constants are defined as follows..

(28) xxviii. NOTATION. General Constants: φw. Probability of scenario w.. σt. Weighting factor of demand block t [hour].. Ki. Annualized capital cost of candidate unit i ∈ ΨS of the strategic producer [e/MW].. K max. Available investment budget [e].. Ximax. Maximum power of candidate unit i ∈ ΨS of the strategic producer [MW]. This constant is used if a set of continuous investment options is considered (Chapter 5). Option h for investment capacity of candidate unit i ∈ ΨS of. Xih. the strategic producer [MW]. This constant is used if a set of discrete investment options is considered (Chapters 2 to 4). Bnm max. Fnm. Susceptance of transmission line (n, m) [p.u.]. Capacity of transmission line (n, m) [MW].. Constants Pertaining to the Pool: max. PkES. max. Capacity of unit j ∈ ΨO of rival producers [MW].. max. Maximum load of demand d ∈ ΨD in demand block t [MW].. PjO PtdD. Capacity of existing unit k ∈ ΨE of the strategic producer [MW].. CiS. Marginal cost of candidate unit i ∈ ΨS of the strategic producer [e/MWh].. CkES. Marginal cost of existing unit k ∈ ΨE of the strategic producer [e/MWh].. CtjO. Price offer of unit j ∈ ΨO of rival producers in demand block t [e/MWh].. D Utd. Price bid of demand d ∈ ΨD in demand block t [e/MWh]..

(29) NOTATION. xxix. Constants Pertaining to the Futures Base Auction: O. Price offer of unit j ∈ ΨO of rival producers [e/MWh].. D. Price bid of demand d ∈ ΨD [e/MWh].. Cj. Ud. Dmax. Pd. Maximum load of demand d ∈ ΨD [MW].. Constants Pertaining to the Futures Peak Auction: bO C j. bD U d. max PbdD. Price offer of unit j ∈ ΨO of rival producers [e/MWh]. Price bid of demand d ∈ ΨD [e/MWh]. Maximum load of demand d ∈ ΨD [MW].. Primal Variables: There are four types of primal variables in this dissertation, namely: 1) General primal variables including variables on capacity of candidate units and voltage angles. These variables are used in Chapters 2 to 5. 2) Primal variables pertaining to the pool. These variables include index w if they refer to scenario w, and are used in Chapters 2 to 5. 3) Primal variables pertaining to the futures base auction. These variables are overlined, and are only used in Chapter 4. 4) Primal variables pertaining to the futures peak auction. These variables include a hat, and are only used in Chapter 4. All primal variables are defined below.. General Primal Variables: Xi. Capacity of candidate unit i ∈ ΨS of the strategic producer [MW]..

(30) xxx. NOTATION θtn. Voltage angle of bus n in demand block t [rad].. Variable θtn includes subscript w if it refers to scenario w.. Primal Variables Pertaining to the Pool: αtiS. Price offer by candidate unit i ∈ ΨS of the strategic producer in demand block t [e/MWh].. ES αtk. Price offer by existing unit k ∈ ΨES of the strategic producer in demand block t [e/MWh].. PtiS. Power produced by candidate unit i ∈ ΨS of the strategic producer in demand block t [MW].. PtkES. Power produced by existing unit k ∈ ΨES of the strategic producer in demand block t [MW].. PtjO. Power produced by unit j ∈ ΨO of rival producers in demand block t [MW].. PtdD. Power consumed by demand d ∈ ΨD in demand block t [MW].. Primal Variables Pertaining to the Futures Base Auction: αSi. Price offer by candidate unit i ∈ ΨS of the strategic producer [e/MWh].. αES k. Price offer by existing unit k ∈ ΨES of the strategic producer [e/MWh].. S. Pi. Power produced by candidate unit i ∈ ΨS of the strategic producer [MW].. ES. Power produced by existing unit k ∈ ΨES of the strategic producer [MW].. O. Power produced by unit j ∈ ΨO of rival producers [MW].. D. Power consumed by demand d ∈ ΨD [MW].. Pk. Pj. Pd.

(31) NOTATION. xxxi. Primal Variables Pertaining to the Futures Peak Auction: α biS. Price offer by candidate unit i ∈ ΨS of the strategic producer [e/MWh].. α bkES. Price offer by existing unit k ∈ ΨES of the strategic producer. PbiS. Power produced by candidate unit i ∈ ΨS of the strategic producer [MW].. PbkES. Power produced by existing unit k ∈ ΨES of the strategic producer [MW].. PbjO. Power produced by unit j ∈ ΨO of rival producers [MW].. PbdD. [e/MWh].. Power consumed by demand d ∈ ΨD [MW].. Dual Variables: There are four types of dual variables in this dissertation, namely: 1) Dual variables associated with the set of lower-level problems representing the clearing of the pool. These variables include index w if they refer to scenario w, and are used in Chapters 2 to 5. 2) Dual variables associated with the lower-level problem representing the clearing of the futures base auction. These variables are overlined, and are only used in Chapter 4. 3) Dual variables associated with the lower-level problem representing the clearing of the futures peak auction. These variables include a hat, and are only used in Chapter 4. 4) Dual variables associated with the upper-level, primal, dual and strong duality constraints included in the MPEC of a strategic producer. These variables are only used in Chapter 5. All dual variables are defined as follows..

(32) xxxii. NOTATION. Dual Variables Associated with the Clearing of the Pool: The dual variables below are associated with the following constraints: λtn. Pool energy balance in demand block t at bus n. These dual variables provide the pool locational marginal prices (LMPs) [e/MWh].. max. µSti. Capacity of candidate unit i ∈ ΨS of the strategic producer in demand block t.. min. µSti. Non-negativity of the production level of candidate unit i ∈ ΨS of the strategic producer in demand block t.. max. µES tk. Capacity of existing unit k ∈ ΨES of the strategic producer in demand block t.. min. µES tk. Non-negativity of the production level of existing unit k ∈ ΨES of the strategic producer in demand block t.. µO tj. max. Capacity of unit j ∈ ΨO of rival producers in demand block t.. µO tj. min. Non-negativity of the production level of unit j ∈ ΨO of rival producers in demand block t.. max. Maximum load of demand d ∈ ΨD in demand block t.. min. Non-negativity of demand d ∈ ΨD in demand block t.. µD td µD td. max. Transmission capacity of line (n, m) in demand block t and direction (n, m).. min. Transmission capacity of line (n, m) in demand block t and direction (m, n).. max. Upper bound of the voltage angle in demand block t at bus n.. min. Lower bound of the voltage angle in demand block t at bus n.. 1. Voltage angle in demand block t at the reference bus n = 1.. νtnm. νtnm. ξtn. ξtn ξt.

(33) NOTATION. xxxiii. Dual Variables Associated with the Clearing of the Futures Base Auction: The dual variables below are associated with the following constraints: λ. Energy balance in the futures base auction. This dual variable provides the clearing price of this auction [e/MWh]. max. Capacity of candidate unit i ∈ ΨS of the strategic producer.. min. Non-negativity of the production level of candidate unit i ∈ ΨS. µSi µSi. of the strategic producer. max. Capacity of existing unit k ∈ ΨES of the strategic producer.. min. Non-negativity of the production level of existing unit k ∈ ΨES of the strategic producer.. µES k µES k. µO j. max. Capacity of unit j ∈ ΨO of rival producers.. µO j. min. Non-negativity of the production level of unit j ∈ ΨO of rival producers.. µD d. max. Maximum load of demand d ∈ ΨD .. µD d. min. Non-negativity of demand d ∈ ΨD .. Dual Variables Associated with the Clearing of the Futures Peak Auction: The dual variables below are associated with the following constraints: b λ. Energy balance in the futures peak auction. This dual variable provides the clearing price of this auction [e/MWh]. max. Capacity of candidate unit i ∈ ΨS of the strategic producer.. min. Non-negativity of the production level of candidate unit i ∈ ΨS. µ bSi. µ bSi. of the strategic producer..

(34) xxxiv. NOTATION max. Capacity of existing unit k ∈ ΨES of the strategic producer.. min. Non-negativity of the production level of existing unit k ∈ ΨES of the strategic producer.. µ bES k. µ bES k µ bO j. µ bO j µ bD d. µ bD d. max. Capacity of unit j ∈ ΨO of rival producers.. min. Non-negativity of the production level of unit j ∈ ΨO of rival producers.. max. Maximum load of demand d ∈ ΨD .. min. Non-negativity of demand d ∈ ΨD .. Dual Variables Associated with the Upper-Level Constraints Included in the MPEC of Producer y: The dual variables below are associated with the following upper-level constraints: χmax yi. Upper-level constraint: Maximum capacity of candidate unit i ∈ ΨS to be built.. χmin yi. Upper-level constraint: Non-negativity of the production level of candidate unit i ∈ ΨS to be built.. χIB y. Upper-level constraint: Maximum available investment budget.. χSS y. Upper-level constraint: Supply security constraint imposed by the market regulator.. S. Upper-level constraint: Non-negativity of the strategic price offers of candidate unit i ∈ ΨS in demand block t.. ES. Upper-level constraint: Non-negativity of the strategic price offers of existing unit k ∈ ΨES in demand block t.. α ηyti. α ηytk.

(35) NOTATION. xxxv. Dual Variables Associated with the Primal Constraints Included in the MPEC of Producer y: The dual variables below are associated with the following primal constraints: βytn. Primal constraint: Energy balance in demand block t at bus n.. max. Primal constraint: Capacity of candidate unit i ∈ ΨS of the strategic producer in demand block t.. min. Primal constraint: Non-negativity of the production level of candidate unit i ∈ ΨS of the strategic producer in demand block t.. S γyti. S γyti. max. ES γytk. Primal constraint: Capacity of existing unit k ∈ ΨES of the strategic producer in demand block t.. min. ES γytk. Primal constraint: Non-negativity of the production level of existing unit k ∈ ΨES of the strategic producer in demand block t.. max. Primal constraint: Maximum load of demand d ∈ ΨD in demand block t.. min. Primal constraint: Non-negativity of demand d ∈ ΨD in demand block t.. D γytd. D γytd. τytnm. max. Primal constraint: Transmission capacity of line (n, m) in demand block t and direction (n, m).. min. Primal constraint: Transmission capacity of line (n, m) in de-. τytnm. mand block t and direction (m, n). max. δytn. Primal constraint: Upper bound of the voltage angle in demand block t at bus n.. min. δytn. Primal constraint: Lower bound of the voltage angle in demand block t at bus n.. 1. δyt. Primal constraint: Voltage angle fixed value in demand block t at the reference bus n = 1..

(36) xxxvi. NOTATION. Dual Variables Associated with the Dual Constraints Included in the MPEC of Producer y: The dual variables below are associated with the following dual constraints: ρD ytd. Dual constraint: Equality in the dual problem associated with consumption variable PtdD of demand d ∈ ΨD in demand block t.. ρSyti. Dual constraint: Equality in the dual problem associated with production variable PtiS of candidate unit i ∈ ΨS in demand block t.. ρES ytk. Dual constraint: Equality in the dual problem associated with production variable PtkES of existing unit k ∈ ΨES in demand block t.. ρθytn. Dual constraint: Equality in the dual problem associated with voltage angle variable θtn in demand block t at bus n.. max. Dual constraint: Non-negativity of dual variable µSti with candidate unit i ∈ ΨS in demand block t.. min. Dual constraint: Non-negativity of dual variable µSti with candidate unit i ∈ ΨS in demand block t.. S ηyti. S ηyti. max. associated. min. associated. max. Dual constraint: Non-negativity of dual variable µES tk ated with existing unit k ∈ ΨES in demand block t.. min. Dual constraint: Non-negativity of dual variable µES tk ES ated with existing unit k ∈ Ψ in demand block t.. ES ηytk. ES ηytk. max. D ηytd. max. associ-. min. associ-. max. associ-. Dual constraint: Non-negativity of dual variable µD td ated with demand d ∈ ΨD in demand block t.. min. D ηytd. min. Dual constraint: Non-negativity of dual variable µD td D. with demand d ∈ Ψ in demand block t.. associated.

(37) NOTATION. xxxvii. max. Dual constraint: Non-negativity of dual variable νtnm associated with transmission line (n, m) in demand block t and direction (n, m).. min. Dual constraint: Non-negativity of dual variable νtnm associated with transmission line (n, m) in demand block t and direction. ν ηytnm. ν ηytnm. max. min. (m, n). max. ξ ηytn. max. Dual constraint: Non-negativity of dual variable ξtn associated with voltage angle in demand block t at bus n.. min. ξ ηytn. min. Dual constraint: Non-negativity of dual variable ξtn associated with voltage angle in demand block t at bus n.. Dual Variable Associated with the Strong Duality Equality Included in the MPEC of Producer y: The dual variable below is associated with the following equality: φSD yt. Strong duality equality associated with the lower-level problem of producer y in demand block t.. Acronyms: ATSW. Annual True Social Welfare.. dc. Direct Current.. EPEC. Equilibrium Problem with Equilibrium Constraints.. GNE. Generalized Nash Equilibrium.. KKT. Karush-Kuhn-Tucker.. LCP. Linear Complementarity Problem.. LMP. Locational Marginal Price..

(38) xxxviii. NOTATION. MCP. Mixed Complementarity Problem.. MFCQ. Mangasarian-Fromovitz Constraint Qualification.. MILP. Mixed-Integer Linear Programming.. MPCC. Mathematical Program with Complementarity Constraints.. MPEC. Mathematical Program with Equilibrium Constraints.. RTS. Reliability Test System.. TP. Total Profit..

(39) Chapter 1 Generation Investment: Introduction 1.1. Electricity Markets. The restructuring of the electric power industry started in the 80’s to create competitive electricity markets [64, 76, 125, 132]. In an electricity market, each power producer submits its production offers, with the objective of maximizing its profit. On the other hand, each consumer submits its demand bids with the aim of maximizing its utility. Then, a non-profit entity, the market operator, clears the market. The objective of the market clearing procedure is to maximize the social welfare of the market. In addition to the market operator, there is another independent entity, the market regulator, that is in charge of the competitive functioning of the market. The market regulator supervises the market operation, and enforces a number of operation and planning policies in order to ensure that the market operates as close as possible to perfect competition. Observing diverse restructuring experiences throughout the world, one concludes that instead of perfectly competitive markets, oligopolistic markets have mostly been formed. In an oligopolistic market, which is the one considered in this dissertation, some producers denominated “strategic producers” are able to alter the market clearing outcomes through their decisions, including: 1.

(40) 2. 1. Generation Investment: Introduction. • Strategic decisions on production offers (operation issue). • Strategic decisions on generation investment (planning issue). Note that “strategic offering” and “strategic investment” refer to the offering and investment decisions of a strategic producer, respectively. Regarding electricity trading floors, there are several alternatives to trade electric energy. One prevalent market is the pool, cleared by the market operator once a day, one day ahead, and on an hourly basis. This is the case of OMIE [127], EEX [34], Nord Pool [128], ISO-New England [65] and PJM [103]. The market operator seeks to maximize the social welfare considering the production offers submitted by the producers and the demand bids submitted by the consumers. The market clearing results are hourly productions, consumptions and clearing prices. If the transmission network is modeled, a clearing price at each bus of the network is obtained, the so-called locational marginal price (LMP) of that bus. The LMP of a given bus represents the social welfare increment in the market as a result of a marginal demand increment at that bus. Note that in the case of congested transmission lines, LMPs vary across buses for any given hour. The futures market is another trading floor that has become increasingly relevant during the last decade. This is the case of OMIP [126], EEX [34] and Nord Pool [128], which include dedicated futures market auctions cleared by their respective market operators. Other markets include future derivatives traded in general stock exchanges. This is the case of ISO-New England [65] and PJM [103]. The futures market includes auctions encompassing a medium- or longterm horizon, e.g., one week to one year. Similarly to the pool, the market operator considers the production offers submitted by the producers and the demand bids submitted by the consumers, and then clears each futures market auction maximizing the corresponding social welfare. The futures market is generally cleared prior to the clearing of the pool, and thus it becomes possible for producers to engage in arbitrage, i.e., to purchase energy from the futures market and then to sell it in the pool..

(41) 1.2. Chapter Organization. 3. Further details on restructuring, diverse trading floors and market functioning are presented throughout the dissertation as required. The investment problems addressed in this dissertation consider the electricity market framework described above.. 1.2. Chapter Organization. The rest of this chapter provides an introduction to this dissertation and includes the sections below: • Section 1.3 presents the motivation for the approaches and models developed in this thesis. In other words, this section states why the subject matter of this dissertation deserves attention. • Section 1.4 describes in detail the problems that are addressed in this dissertation. • Section 1.5 presents a literature review. In addition, this section provides a comparison among the models/approaches proposed in this thesis work and others reported in the literature. • Section 1.6 describes the modeling assumptions considered throughout the dissertation. • Section 1.7 explains the general structure of the models proposed in this thesis and then explains the solution approaches for these models. • Section 1.8 presents the main objectives of this thesis. • Section 1.9 provides the outline of this document.. 1.3. Thesis Motivation. Since a reliable electricity supply is crucial for the functioning of modern societies, investment in electricity production is most important to guarantee supply security. However, within an electricity market framework, generation.

(42) 4. 1. Generation Investment: Introduction. investment decision-making problems for a power producer are complex, because such problems require the proper modeling of the following elements: 1) The market functioning, which leads to complementarity models. 2) The uncertainties plaguing markets, which leads to using stochastic programming models. 3) The behavior of rival producers, i.e., their operation and investment strategies, which has an effect on the producer’s own decisions. In addition to the need of such stochastic complementarity models, investment decision making is risky due to the long-term consequences of the decisions involved. Additionally, generation investment decision-making problems become particularly complex within an oligopolistic electricity market, where several strategic producers compete. The reason for such complexity is that each strategic producer is able to alter the formation of the market outcomes (e.g., clearing prices and production quantities) through its operation and planning strategies. Thus, the decision-making processes of all producers need to be jointly considered. The considerations above motivate the development of an appropriate mathematical tool to assist a strategic producer competing with its strategic rivals in an electricity market for making informed investment decisions. The objective of this mathematical tool is to maximize the expected profit of such strategic producer through its decisions in i) operations (offering), and ii) planning (generation investment). To develop such mathematical tool, several important issues need to be considered, namely: 1. Uncertainties: In an investment problem, the strategic producer faces a number of uncertainties, e.g., demand growth, behavior of rival producers, investment cost of different technologies, fuel price, regulatory policies and others. On one hand, modeling such uncertainties properly is important. On the other hand, a detailed description of such.

(43) 1.3. Thesis Motivation. 5. uncertainties may result in high computational burden and eventual intractability. Thus, the mathematical tool to address the generation investment decision-making problem of a strategic producer should be able to consider the most important uncertainties, while being computationally tractable. 2. Diverse trading markets: The strategic producer may participate in diverse trading markets. The prevalent market is the electricity pool; however, to obtain a higher expected profit or a lower risk, the strategic producer may trade in other markets as well, e.g., in the futures market. Thus, the mathematical tool to address the generation investment decision-making problem of a strategic producer should be able to model the functioning of all trading markets in which such producer may get involved. 3. Different investment technologies: The mathematical tool to address the generation investment decision-making problem of a strategic producer should be able to select the units to be built among the available investment options such as base technologies, e.g., nuclear units and peak technologies, e.g., CCGT units. 4. Locations for building the candidate generation units: The mathematical tool to address the generation investment decision-making problem of a strategic producer should be able to optimally allocate the units to be built throughout the network. To this end, a proper representation of the network is required. Additionally, generation investment equilibria need to be studied and analyzed in detail. Since the operation and planning strategies of any strategic producer are interrelated with those of other strategic producers, decisions made by one strategic producer may influence the strategies of other strategic producers. Thus, a number of investment equilibria may exist, where each strategic producer cannot increase its profit by changing its strategies unilaterally. Thus, it is important to identify such investment equilibria. This equilibrium analysis is useful for the market regulator to gain insight into the.

(44) 6. 1. Generation Investment: Introduction. investment behavior of the strategic producers and the generation investment evolution. Such insight may allow the market regulator to better design market rules, which in turn may contribute to increase the competitiveness of the market and to stimulate optimal investment in generation capacity. In addition, the market regulator may use an equilibrium analysis to assess the impact of certain policies on the investment evolution.. 1.4. Problem Description. Considering the thesis motivation presented in Section 1.3, this dissertation specifically addresses the four problems below: 1. Development of an optimization tool for a strategic producer trading in a pool to optimally solve its generation investment decision-making problem (direct solution approach): The aim of this tool is to identify the most beneficial investment strategy for a strategic producer competing in an electricity pool. This strategy includes the technology, the capacity and the network allocation of each new unit to be built. The strategies of the rival producers, i.e., their operation and planning decisions, are uncertain parameters represented through scenarios. That is, we use scenarios to describe uncertainties pertaining to i) rival production offers and ii) rival investments. To solve this problem, we propose a hierarchical (bilevel) model whose upper-level problem represents the investment and offering actions of the strategic producer, and whose multiple lower-level problems represent the clearing of the pool under different operating scenarios. Such model renders a mathematical program with equilibrium constraints (MPEC) by replacing each lower-level problem with its Karush-Kuhn-Tucker (KKT) conditions. In turn, this MPEC can be recast as a mixed-integer linear programming problem (MILP) solvable with commercially available software. Further details on the structure of this bilevel model and the resulting MPEC are provided in Section 1.7. In addition, the bilevel.

(45) 1.4. Problem Description. 7. model, the MPEC and the KKT conditions are described in Sections B.1 and B.2 of Appendix B. In the proposed model, all scenarios involved are considered simultaneously (direct solution approach). Thus, this direct approach generally suffers from high computational burden and eventual intractability in cases with many scenarios. This is the main drawback of this approach, which is presented in Chapter 2. 2. Development of an alternative approach for solving the generation investment decision-making problem of a strategic producer for cases with many scenarios (Benders’ decomposition approach): To tackle the computational burden of the direct solution approach in cases with many scenarios, an alternative approach based on Bender’s decomposition is proposed. This approach allows decomposing the considered bilevel model per scenario by fixing the investment variables. Therefore, unlike the direct solution approach, the scenarios are considered separately and thus the model can be solved even if a large number of scenarios is considered. A Benders’ approach is possible since exhaustive computational analysis indicates that if the producer behaves strategically and a sufficiently large number of scenarios is considered, the expected profit of the strategic producer as a function of the complicating investment decisions has a convex enough envelope. The proposed Benders’ approach is presented in Chapter 3. 3. Development of an optimization tool for a strategic producer trading in both the futures market and the pool to optimally solve its generation investment decision-making problem: Since the futures market has become increasingly relevant for trading electricity, it is important to analyze the effects of such market on the investment decisions of a strategic producer. To this end, we consider a hierarchical (bilevel) model, whose upper-level problem represents the investment and offering actions of the producer,.

(46) 8. 1. Generation Investment: Introduction. and whose multiple lower-level problems represent the clearing of the futures market and the pool under different operating conditions. Then, an MPEC is derived by replacing the lower-level problems with their respective KKT optimality conditions. Finally, the MPEC problem is linearized and recast as an MILP problem. This model is presented in Chapter 4.. 4. Identification of potential generation investment equilibria in an oligopolistic pool with strategic producers: The fourth problem considered in this dissertation is to mathematically identify the potential investment equilibria, where each producer cannot increase its profit by unilaterally changing its investment strategies. To this end, the investment and offering decisions of each strategic producer are represented through a hierarchical (bilevel) model, whose upperlevel problem decides on the optimal investment and the offering curves for maximizing the profit of the producer, and whose lower-level problems represent different market clearing scenarios. Replacing the lower-level problems by their primal-dual optimality conditions (Section B.2 of Appendix B) in the single-producer model renders an MPEC. The joint consideration of all producer MPECs, one per producer, constitutes an equilibrium problem with equilibrium constraints (EPEC). Further details on the EPEC are provided in Section 1.7 and in Section B.3 of Appendix B. To identify the solutions of the EPEC, each MPEC problem is replaced by its KKT conditions, which are in turn linearized. The resulting mixed-integer linear system of equalities and inequalities allows determining the EPEC equilibria through an auxiliary MILP problem. Finally, all equilibria identified are verified by a diagonalization checking. This problem is analyzed in Chapter 5..

(47) 1.5. Literature Review. 1.5. 9. Literature Review. This section reviews relevant works in technical literature regarding i) the generation investment problem of a strategic producer within an electricity market, and ii) the generation investment equilibrium problem in an oligopolistic electricity market. Finally, some other relevant works are also reviewed.. 1.5.1. Literature Review: Generation Investment for a Strategic Producer. Several works can be found in literature referring to the generation investment decision-making problem of a given strategic producer within an electricity market. In [14], a stochastic dynamic optimization model is used to evaluate generation investments under both centralized and decentralized frameworks, but not modeling the network. Long-term uncertainty in demand growth and its effect on future prices are modeled via discrete Markov chains. Reference [23] is a relevant paper that considers the generation expansion planning problem in an oligopolistic environment using a Cournot model and including no network constraint. The solution is found using an iterative search procedure, which assumes complete information of the rivals. The effects of both competition and transmission congestion on generation expansion are specifically considered in [69], where a Cournot model is used. In [94], a noncooperative game for generation investment is modeled using two tiers. In the first tier, the generation investment game is examined, and in the second tier the energy supply game is considered. The solution procedure is based on a reinforcement learning algorithm. In [97], the value of information pertaining to rival producers such as their marginal costs and conjectures on their behavior as well as demand levels are analyzed for making decision on generation investment. The model uses a Cournot approach and includes no network constraint. In [130], two different approaches pertaining to generation expansion in a electricity market are presented. Both of them consider a Cournot model.

(48) 10. 1. Generation Investment: Introduction. although they differ in how the producer determines its optimal capacity. In the first approach a mixed complementarity problem (MCP) is used, while for the second one an MPEC approach is considered based on a Stackelberg model [124]. In [131], the strategic generation capacity expansion of a producer considering incomplete information of rival producers is modeled using a two-level optimization problem. A genetic algorithm approach is used to solve such problem. Reference [134] proposes a bilevel model to assist a producer in making multi-stage generation investment decisions considering the investment uncertainty of rival producers. In the upper-level problem, the producer maximizes its expected profit, while the lower-level problem represents the market clearing characterized by a conjectural variations model and including no network constraint. The model proposed in [134] renders an MPEC. For clarity, we summarize in Table 1.1 the relevant features of the generation investment model proposed in papers [71–73], developed as part of this thesis, and other works reported in technical literature.. 1.5.2. Literature Review: Generation Investment Equilibria. There are several papers in the literature pertaining to generation investment equilibria as explained below. In pioneering reference [92], three models of the generation capacity expansion game are considered. The first model assumes perfect competition, thus being similar to a centralized capacity expansion model. The second model (open-loop Cournot duopoly) extends the well-known Cournot model to include investments in new generation capacity. The third model (closed-loop Cournot duopoly) separates the investment and production decisions considering investment in the first stage and sales in the second stage. All three models are static, and uncertainties as well as the network constraints are not modeled. Reference [35] considers a static but stochastic capacity expansion equilib-.

(49) Reference. Model. Transmission Static/ Stochastic constraints Dynamic. Uncertainty. model. Different investment. Bilevel. technologies. Strategic. Approach. offering. [14]. Stochastic optimization. No. Dynamic. Yes. Demand growth. Yes. No. No. Discrete Markov chain. [23]. Cournot. No. Static. No. -. Yes. No. No. Iterative. [69]. Cournot. Yes. Static. No. -. No. No. No. Quadratic programming. [94]. Supply function. Yes. Dynamic. Yes. Yes. No. No. Heuristic. Yes. No. No. Quadratic programming. Demand growth. 1.5. Literature Review. Table 1.1: Introduction: Relevant features of some models reported in literature and the model proposed in this dissertation (generation investment of a given strategic producer).. and line outages Demand growth, [97]. Cournot. No. Static. Yes. rival marginal cost and rival behavior. [130]. Cournot (Stackelberg). No. Static. No. -. Yes. No. No. Complementarity. [131]. Bilevel. Yes. Static. Yes. Unit outages. Yes. Yes. No. Genetic algorithm. [134]. Conjectural variation. No. Dynamic. Yes. Rival investment. Yes. Yes. No. MPEC. This. Stepwise. dissertation. supply. ( [71–73]). function. Rival offering, Yes. Static. Yes. rival investment and demand growth. MPEC (direct Yes. Yes. Yes. solution and Benders’ decomposition approaches). 11.

(50) 12. 1. Generation Investment: Introduction. ria in an electricity market. The objective of this work is to investigate the impact of four key parameters on the generation investment equilibria and on the supply security. Such parameters are: 1) profit risk, 2) investment incentives, 3) market organization (energy-only or energy and capacity) and price caps, and 4) carbon trading issues. In the capacity expansion equilibria reported in [35], fuel prices and climate change policies are considered uncertain parameters, while the demand is assumed known. The producers are considered price-takers, but subject to regulatory imperfections. The network is disregarded, and a small case study is analyzed. A multi-stage generation investment equilibrium considering uncertain demand is addressed in [45]. This reference proposes a Markov chain to model the strategic interaction between a short-run capacity-constrained Cournot generation game and a long-run generation investment game. It is concluded that a socially optimal level of capacity is not built, and that the distance between the capacity built and the optimal capacity is largely dependent on investment profitability. Investment incentives are studied in [46] using a simple strategic dynamic model with random demand growth. This model is based on a non-collusive Markovian equilibrium in which the investment decisions of each producer depend on its existing capacity. Similarly to [46], reference [49] formulates a dynamic capacity investment equilibrium considering a hydrothermal duopoly under uncertain demand. Both Markov perfect and open-loop equilibria are modeled in this reference, and then the incentives needed to promote investment are studied. In [66], the capacity expansion decisions are made by a leader on behalf of the market agents. The aim of this model is to maximize the total welfare of all market participants, which renders a bilevel model, whose upper-level problem determines the investment decisions, and whose lower-level problems represent the operation decisions of each market participant. Such bilevel model is transformed into a mathematical program with complementarity constraints (MPCC). The equilibrium of generation investment considering both futures and spot markets is studied in [93], where a Cournot model is used. This reference shows.

(51) Reference. Model. Bilevel. Strategic Transmission Static/ offering. Uncertainty. constraints dynamic. Different investment technologies. [35]. Complementarity. No. No. No. Static. Yes. Yes. [45]. Cournot-Marcov chain. No. No. No. Dynamic. Yes. No. [46]. Non-collusive Markov. No. No. No. Dynamic. Yes. No. [49]. Open-loop and Markov perfect. No. No. No. Dynamic. Yes. Yes. [66]. Cournot (MPCC). Yes. No. Yes. Static. No. Yes. [92]. Cournot (MPEC). Yes. No. No. Static. No. Yes. [93]. Cournot (EPEC). No. No. No. Static. Yes. Yes. [133]. Conjectural variation (EPEC). Yes. No. No. Dynamic. No. Yes. This dissertation. Stepwise supply. Yes. Yes. Yes. Static. No. Yes. ( [74] and [75]). function (EPEC). 1.5. Literature Review. Table 1.2: Introduction: Relevant features of some models reported in the literature and the model proposed in this dissertation (generation investment equilibria).. 13.

(52) 14. 1. Generation Investment: Introduction. that forward contracts may not mitigate market power in the spot market, in case that the production capacities of the producers are endogenous and constrain the production level. In [133], an EPEC is proposed to identify the generation capacity investment equilibria. To this end, a bilevel optimization problem is formulated so that each producer selects capacities in an upper-level problem maximizing its profit and anticipating the equilibrium outcomes of the lower-level problems, in which production quantities and prices are determined by a conjectured-price response approach. The work reported in the two-part paper [74, 75], developed as part of this thesis work, mainly differs from [35, 45, 46, 49, 66, 92, 93, 133] in that it uses a stepwise supply function model. For the sake of clarity, the relevant features of the equilibrium model proposed in this dissertation and other works reported in the literature are summarized in Table 1.2. The approach for modeling the generation investment equilibria used in this dissertation is similar to that used in [63] and [117]. Such references analyze the equilibria reached by strategic producers in a network-constrained pool in which the behavior of each producer is represented by an MPEC. However, references [63] and [117] address an operation, not an investment problem. Similarly to the model developed in [117], the EPEC is characterized in this dissertation by solving the optimality conditions of all MPECs, which are formulated as an MILP, and diverse linear objective functions are used to obtain different equilibria. Finally, note that the methodology presented in this dissertation extends the model in [117] by incorporating generation investment decisions and analyzing the impact that these decisions have on the competitiveness of the market.. 1.5.3. Other Related Works in the Literature. This subsection reviews some relevant works in the literature regarding 1) MPECs, EPECs, complementarity models, stochastic programming models and Benders’ decomposition, 2) other applications of bilevel models in elec-.

(53) 1.5. Literature Review. 15. tricity markets, 3) futures market auctions, 4) generation expansion planning in centralized power systems, and 5) operation equilibria in electricity markets. Such references are reviewed below: • The first mathematical model used in this dissertation, an MPEC, was first proposed in [55]. Further mathematical details on such model can be found in [82]. • The EPEC model was first proposed in [102] and then used in [108] to model the interaction among strategic producers of an electricity market. In addition, EPEC models and their applications to the equilibrium analysis of electricity markets are considered in [137]. • References [6,12,24,31–33] provide mathematical details on bilevel models. • Mathematical details on complementarity models and their applications in electricity markets are available in [41]. • Reference [25] provides mathematical details on stochastic programming models and their applications in electricity markets. • Mathematical details on Benders’ decomposition and their applications in electricity markets can be found in [26]. • In the technical literature associated with electricity markets, a number of works use a bilevel approach similar to the one proposed in this dissertation. Such works pertain to offering strategies [9, 60, 81, 113], transmission expansion [19, 44, 107, 120, 121], policy incentives for producers to invest in renewable facilities [138], transmission cost allocation [50], maintenance scheduling [100, 101], security analysis [18, 90, 119], estimation of the amount of market-integrable renewable resources [88] and retailer trading [20], among others. • Futures electricity market has been well analyzed in the literature as effective tool for power producers to hedge against the risk of pool price.

(54) 16. 1. Generation Investment: Introduction. volatility. For instance, pioneering reference [70] investigates the use of forward contracts to hedge profit risk. In addition, studies pertaining to the futures market, especially their impact on enhancing competitiveness, are reported in [1,38,78,83,118]. For example, reference [83] compares the market prices in a market with Bertrand competition with and without considering forward trading. Additionally, there are some works in the literature considering offering strategies of electricity producers if both futures and spot markets are considered, e.g., references [27, 48, 96]. • A large number of works can be found in the literature addressing the generation expansion planning problem in centralized power systems, e.g., references [85, 109, 135, 139]. • Finally, it is relevant to note that a number of works are available in the literature pertaining to electricity market equilibria from the operational point of view, e.g., references [4, 30, 61, 63, 104, 117, 136].. 1.6. Modeling Assumptions. Pursuing clarity, the main modeling assumptions considered throughout this dissertation are listed in this section.. 1.6.1. Investment Study Approach. As indicated in Tables 1.1 and 1.2, two different approaches are common in production investment studies as stated below: a) Static approach. b) Multi-stage or dynamic approach. The characteristics of these approaches are described below: In the static approach, the expansion exercise considers a single future target year - e.g., a single year 20 years into the future - and the optimal investment is established for that year. Once known the optimal generation.

(55) 17. 1.6. Modeling Assumptions. ı tT. Figure 1.1: Introduction: Piecewise approximation of the load-duration curve for the target year.. mix for the target year (e.g., year 20) and considering the generation mix of the initial year (year 0), it is possible to derive an appropriate building schedule to “move” from the generation mix of the initial year to that of the target year. In this approach, the building path from the initial year to the target year is not explicitly represented. In the multi-stage or dynamic approach, investments are considered at several steps throughout the planning horizon; and thus, the building path from the initial year to the target year is derived. This approach provides higher accuracy but at the cost of potential intractability. As it is customary in large-scale generation investment studies, e.g., [35,92, 93], and pursuing an appropriate tradeoff between accuracy and computational tractability, the static approach is used in this dissertation. It is important to note that since the static approach is used, only the available (not decommissioned) production facilities in the target year are considered. Existing production units being decommissioned along the planning horizon should not be included in the analysis..

(56) 18. 1. Generation Investment: Introduction. Table 1.3: Introduction: Example for clarifying the demand-bid blocks. Demand block (t) t = t1 t = t2 t = t3 t = t4. 1.6.2. Demand D1 Maximum Demand Bid load quantity price [MW] [MW] [e/MWh] 400 40 500 100 38 320 37 400 80 36 300 35 350 50 34 270 33 300 30 32. Demand D2 Maximum Demand Bid load quantity price [MW] [MW] [e/MWh] 600 36 800 200 35 500 33 650 150 32 400 30 500 100 29 350 27 400 50 26. Load-Duration Curve. As an input of the static approach, Figure 1.1 depicts the load-duration curve of the system for the target year of the planning horizon, approximated through a number of stepwise demand blocks. The number of steps used to discretize the load-duration curve needs to be carefully selected. A large number of steps may result in intractability while a small number of steps may affect the accuracy of the solution attained. In any case, it is important to check that the optimal solution does not significantly change by incorporating an additional step to describe the load-duration curve. Note that the weighting factor corresponding to demand block t (σt ) refers to a portion of the hours in the year for which the load of the system is approximated through the demand of that block. Clearly, the summation of the weighting factors of all demand blocks equals the number of hours in a P year, i.e., t σt = 8760.. For the sake of clarity, the terms demand blocks, demand-bid blocks, and production-offer blocks used throughout this dissertation are explained below. 1. Demand blocks: These blocks are obtained from a stepwise approximation of the load-.