Stochastic complementarity models for investment in wind-power and transmission facilities.

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(1)UNIVERSIDAD DE CASTILLA-LA MANCHA. DEPARTAMENTO DE INGENIERÍA ELÉCTRICA, ELECTRÓNICA, AUTOMÁTICA Y COMUNICACIONES. STOCHASTIC COMPLEMENTARITY MODELS FOR INVESTMENT IN WIND-POWER AND TRANSMISSION FACILITIES. PhD THESIS. AUTHOR: LUIS BARINGO SUPERVISOR: ANTONIO J. CONEJO Ciudad Real, October 2013.

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(3) Contents Contents. iii. 1 Introduction. 1. 1.1 Electricity Markets . . . . . . . . . . . . . . . . . . . . . . . . .. 1. 1.2 Thesis Motivation . . . . . . . . . . . . . . . . . . . . . . . . . .. 4. 1.3 Problem Description . . . . . . . . . . . . . . . . . . . . . . . .. 7. 1.3.1. Stochastic Complementarity Models . . . . . . . . . . . . 1.3.1.1. The Perspective of a Wind Power Investor: UpperLevel Problem . . . . . . . . . . . . . . . . . .. 1.3.2. 1.3.3. 7 9. 1.3.1.2. The Perspective of the TSO: Upper-Level Problem . . . . . . . . . . . . . . . . . . . . . . . . 10. 1.3.1.3. Market Clearing: Lower-Level Problems . . . . 11. Solution Procedure . . . . . . . . . . . . . . . . . . . . . 13 1.3.2.1. MPECs . . . . . . . . . . . . . . . . . . . . . . 13. 1.3.2.2. MILP Problems . . . . . . . . . . . . . . . . . . 13. 1.3.2.3. Benders’ Decomposition . . . . . . . . . . . . . 13. 1.3.2.4. Computational Tool . . . . . . . . . . . . . . . 14. Model Assumptions . . . . . . . . . . . . . . . . . . . . . 14. 1.4 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4.1. Wind Energy . . . . . . . . . . . . . . . . . . . . . . . . 15. 1.4.2. Investment in Generation Facilities . . . . . . . . . . . . 16. 1.4.3. 1.4.2.1. Investment in Conventional Generation Units . 16. 1.4.2.2. Investment in Wind Power Facilities . . . . . . 17. Offering Strategies . . . . . . . . . . . . . . . . . . . . . 18 iii.

(4) iv. CONTENTS 1.4.4. Investment in Transmission Facilities . . . . . . . . . . . 19. 1.4.5. Mathematical Tools . . . . . . . . . . . . . . . . . . . . . 21 1.4.5.1. Uncertainty Modeling . . . . . . . . . . . . . . 21. 1.4.5.2. Bilevel Models . . . . . . . . . . . . . . . . . . 22. 1.4.5.3. MPECs . . . . . . . . . . . . . . . . . . . . . . 22. 1.4.5.4. Benders’ Decomposition . . . . . . . . . . . . . 22. 1.4.5.5. Risk Modeling . . . . . . . . . . . . . . . . . . 23. 1.5. Thesis Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . 23. 1.6. Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 26. 2 Uncertainty Modeling 2.1. 2.2. 29. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.1.1. Sources of Uncertainty . . . . . . . . . . . . . . . . . . . 29. 2.1.2. Chapter Organization . . . . . . . . . . . . . . . . . . . . 31. Modeling of the Uncertainty in Demand, Wind Power Production, and Balancing Market Price . . . . . . . . . . . . . . . . . 31 2.2.1. Modeling of the Uncertainty in Demand and Wind Power Production . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2.1.1. Method based on the Load- and Wind-Duration Curves . . . . . . . . . . . . . . . . . . . . . . . 32. 2.2.1.2. K-Means Clustering Method . . . . . . . . . . . 37. 2.2.2. Modeling of the Uncertainty in Demand Growth . . . . . 39. 2.2.3. Modeling of the Uncertainty in Wind Power Production and Balancing Market Price . . . . . . . . . . . . . . . . 40. 2.3. Modeling of the Uncertainty in Investment Cost . . . . . . . . . 42. 2.4. Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.4.1. Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44. 2.4.2. Results: Demand, Wind Power Production, and Balancing Market Price . . . . . . . . . . . . . . . . . . . . . . 46 2.4.2.1. Results: Demand and Wind Power Production. 46. 2.4.2.2. Results: Demand Growth . . . . . . . . . . . . 55. 2.4.2.3. Results: Wind Power Production and Balancing Market Price . . . . . . . . . . . . . . . . . 56.

(5) CONTENTS. v. 2.4.3. Results: Investment Cost . . . . . . . . . . . . . . . . . . 58. 2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3 Wind Power Investment: A Static Approach. 63. 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.1.1. Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 63. 3.1.2. Model Assumptions . . . . . . . . . . . . . . . . . . . . . 65. 3.1.3. Chapter Organization . . . . . . . . . . . . . . . . . . . . 66. 3.2 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.2.1. Bilevel Model Formulation . . . . . . . . . . . . . . . . . 67. 3.2.2. MPEC Formulation . . . . . . . . . . . . . . . . . . . . . 70. 3.2.3. MILP Formulation . . . . . . . . . . . . . . . . . . . . . 71. 3.3 Solution Procedure: A Benders’ Decomposition Approach . . . . 74 3.3.1. 3.3.2. Benders’ Decomposition Algorithm . . . . . . . . . . . . 75 3.3.1.1. Step 0: Initialization . . . . . . . . . . . . . . . 75. 3.3.1.2. Step 1: Subproblem solution . . . . . . . . . . . 76. 3.3.1.3. Step 2: Convergence checking . . . . . . . . . . 77. 3.3.1.4. Step 3: Master problem solution . . . . . . . . 77. On Convexity and Convergence . . . . . . . . . . . . . . 78. 3.4 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . 81 3.4.1. Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81. 3.4.2. Results: Uncongested Network . . . . . . . . . . . . . . . 84. 3.4.3. Results: Congested Network . . . . . . . . . . . . . . . . 86. 3.4.4. Computational Issues . . . . . . . . . . . . . . . . . . . . 88. 3.5 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.5.1. 3.5.2. IEEE 24-Node RTS . . . . . . . . . . . . . . . . . . . . . 89 3.5.1.1. Data . . . . . . . . . . . . . . . . . . . . . . . . 89. 3.5.1.2. Results . . . . . . . . . . . . . . . . . . . . . . 90. IEEE 118-Node TS . . . . . . . . . . . . . . . . . . . . . 96 3.5.2.1. Data . . . . . . . . . . . . . . . . . . . . . . . . 96. 3.5.2.2. Results . . . . . . . . . . . . . . . . . . . . . . 96. 3.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . 99.

(6) vi. CONTENTS. 4 Wind Power Investment: A Risk-Constrained Multi-Stage Approach 101 4.1. 4.2. 4.3. 4.4. 4.5. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.1.1. Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 101. 4.1.2. Chapter Organization . . . . . . . . . . . . . . . . . . . . 104. Problem Description . . . . . . . . . . . . . . . . . . . . . . . . 104 4.2.1. Time Framework . . . . . . . . . . . . . . . . . . . . . . 104. 4.2.2. Sources of Uncertainty: Scenario Tree . . . . . . . . . . . 105. 4.2.3. Decision Sequence . . . . . . . . . . . . . . . . . . . . . . 107. 4.2.4. Risk Management . . . . . . . . . . . . . . . . . . . . . . 108. 4.2.5. Model Assumptions . . . . . . . . . . . . . . . . . . . . . 109. Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.3.1. Bilevel Model Formulation . . . . . . . . . . . . . . . . . 109. 4.3.2. MPEC Formulation . . . . . . . . . . . . . . . . . . . . . 114. 4.3.3. MILP Formulation . . . . . . . . . . . . . . . . . . . . . 115. 4.3.4. Solution Procedure: Benders’ Decomposition . . . . . . . 117 4.3.4.1. Subproblem Formulation . . . . . . . . . . . . . 118. 4.3.4.2. Master Problem Formulation . . . . . . . . . . 119. Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . 120 4.4.1. System Data . . . . . . . . . . . . . . . . . . . . . . . . 120. 4.4.2. Planning Horizon and Scenario Data . . . . . . . . . . . 122. 4.4.3. Results: Uncertainty in Demand Growth . . . . . . . . . 123. 4.4.4. Results: Uncertainty in Wind Power Investment Cost . . 125. 4.4.5. Results: Uncertainty in both Demand Growth and Wind Power Investment Cost . . . . . . . . . . . . . . . . . . . 127. 4.4.6. Computational Issues . . . . . . . . . . . . . . . . . . . . 129. Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.5.1. 4.5.2. IEEE 24-Node RTS . . . . . . . . . . . . . . . . . . . . . 130 4.5.1.1. Data . . . . . . . . . . . . . . . . . . . . . . . . 130. 4.5.1.2. Results . . . . . . . . . . . . . . . . . . . . . . 130. IEEE 118-Node TS . . . . . . . . . . . . . . . . . . . . . 133 4.5.2.1. Data . . . . . . . . . . . . . . . . . . . . . . . . 133. 4.5.2.2. Results . . . . . . . . . . . . . . . . . . . . . . 134.

(7) CONTENTS. vii. 4.5.3. Computational Issues . . . . . . . . . . . . . . . . . . . . 138. 4.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . 139 5 Wind Power Investment: A Strategic Approach. 141. 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.1.1. Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 142. 5.1.2. Chapter Organization . . . . . . . . . . . . . . . . . . . . 143. 5.2 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . 143 5.2.1. Uncertainty Characterization . . . . . . . . . . . . . . . 143. 5.2.2. Decision Sequence . . . . . . . . . . . . . . . . . . . . . . 144. 5.2.3. Model Assumptions . . . . . . . . . . . . . . . . . . . . . 145. 5.3 Wind Power Offering Strategy: Model Formulation . . . . . . . 146 5.3.1. Bilevel Model Formulation . . . . . . . . . . . . . . . . . 146. 5.3.2. MPEC Formulation . . . . . . . . . . . . . . . . . . . . . 149. 5.3.3. MILP Formulation . . . . . . . . . . . . . . . . . . . . . 151. 5.4 Strategic Wind Power Investment: Model Formulation . . . . . 154 5.4.1. Bilevel Model Formulation . . . . . . . . . . . . . . . . . 155. 5.4.2. MPEC Formulation . . . . . . . . . . . . . . . . . . . . . 158. 5.4.3. MILP Formulation . . . . . . . . . . . . . . . . . . . . . 159. 5.5 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . 162 5.5.1. 5.5.2. Strategic Offering for a Wind Power Producer . . . . . . 162 5.5.1.1. System Data . . . . . . . . . . . . . . . . . . . 162. 5.5.1.2. Wind Power Production and Balancing Market Price Scenario Data . . . . . . . . . . . . . . . 165. 5.5.1.3. Results . . . . . . . . . . . . . . . . . . . . . . 165. Strategic Wind Power Investment . . . . . . . . . . . . . 169 5.5.2.1. System Data . . . . . . . . . . . . . . . . . . . 169. 5.5.2.2. Demand, Wind Power Production, and Balancing Market Price Data . . . . . . . . . . . . . . 172. 5.5.2.3 5.5.3. Results . . . . . . . . . . . . . . . . . . . . . . 172. Computational Issues . . . . . . . . . . . . . . . . . . . . 175. 5.6 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 5.6.1. IEEE 24-Node RTS . . . . . . . . . . . . . . . . . . . . . 176.

(8) viii. CONTENTS. 5.6.2. 5.6.3 5.7. 5.6.1.1. Strategic Offering for a Wind Power Producer . 176. 5.6.1.2. Strategic Wind Power Investment . . . . . . . . 180. IEEE 118-Node TS . . . . . . . . . . . . . . . . . . . . . 183 5.6.2.1. Strategic Offering for a Wind Power Producer . 183. 5.6.2.2. Strategic Wind Power Investment . . . . . . . . 190. Computational Issues . . . . . . . . . . . . . . . . . . . . 191. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193. 6 Transmission and Wind Power Investment 6.1. 6.2. 6.3. 6.4. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 6.1.1. Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 195. 6.1.2. Model Assumptions . . . . . . . . . . . . . . . . . . . . . 198. 6.1.3. Chapter Organization . . . . . . . . . . . . . . . . . . . . 199. Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 199 6.2.1. Bilevel Model Formulation . . . . . . . . . . . . . . . . . 199. 6.2.2. MPEC Formulation . . . . . . . . . . . . . . . . . . . . . 203. 6.2.3. MILP Formulation . . . . . . . . . . . . . . . . . . . . . 205. Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . 207 6.3.1. Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207. 6.3.2. Results: Effect of Subsidies. 6.3.3. Results: Effect of Demand Growth . . . . . . . . . . . . 212. 6.3.4. Computational Issues . . . . . . . . . . . . . . . . . . . . 213. . . . . . . . . . . . . . . . . 211. Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 6.4.1. 6.4.2. 6.4.3 6.5. 195. IEEE 24-Node RTS . . . . . . . . . . . . . . . . . . . . . 214 6.4.1.1. Data . . . . . . . . . . . . . . . . . . . . . . . . 214. 6.4.1.2. Results . . . . . . . . . . . . . . . . . . . . . . 215. IEEE 118-Node TS . . . . . . . . . . . . . . . . . . . . . 217 6.4.2.1. Data . . . . . . . . . . . . . . . . . . . . . . . . 217. 6.4.2.2. Results . . . . . . . . . . . . . . . . . . . . . . 218. Computational Issues . . . . . . . . . . . . . . . . . . . . 221. Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . 221.

(9) ix. CONTENTS 7 Closure. 223. 7.1 Thesis Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 223 7.1.1. Uncertainty Modeling . . . . . . . . . . . . . . . . . . . . 223. 7.1.2. Wind Power Investment: A Static Approach . . . . . . . 225. 7.1.3. Wind Power Investment: A Risk-Constrained MultiStage Approach . . . . . . . . . . . . . . . . . . . . . . . 226. 7.1.4. Wind Power Investment: A Strategic Approach . . . . . 227. 7.1.5. Transmission and Wind Power Investment . . . . . . . . 228. 7.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 7.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 7.4 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 A Wind Power Production and Balancing Market Price Scenario Data 237 B Mathematical Background. 249. B.1 Bilevel Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 B.1.1 Bilevel Formulation . . . . . . . . . . . . . . . . . . . . . 249 B.1.2 MPEC Formulation . . . . . . . . . . . . . . . . . . . . . 251 B.1.2.1. KKT Formulation . . . . . . . . . . . . . . . . 252. B.1.2.2. Primal-Dual Formulation . . . . . . . . . . . . 253. B.1.2.3. KKT vs. Primal-Dual Formulations . . . . . . . 255. B.2 Linearization Procedures . . . . . . . . . . . . . . . . . . . . . . 256 B.2.1 Fortuny-Amat Transformation . . . . . . . . . . . . . . . 256 B.2.2 Product of a Continuous Variable and a Binary One . . . 257 B.3 Benders’ Decomposition . . . . . . . . . . . . . . . . . . . . . . 258 B.3.1 Problem Structure . . . . . . . . . . . . . . . . . . . . . 258 B.3.2 Subproblems. . . . . . . . . . . . . . . . . . . . . . . . . 260. B.3.3 Master Problem . . . . . . . . . . . . . . . . . . . . . . . 261 B.3.4 Upper and Lower Bounds . . . . . . . . . . . . . . . . . 262 B.3.5 Benders’ Decomposition Algorithm . . . . . . . . . . . . 263 B.3.6 On Convexity and Convergence . . . . . . . . . . . . . . 264.

(10) x. CONTENTS. C IEEE 24-Node Reliability Test System 267 C.1 Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 C.2 Generation Unit and Demand Data . . . . . . . . . . . . . . . . 269 C.3 Transmission Line Data . . . . . . . . . . . . . . . . . . . . . . 269 D IEEE 118-Node Test System 273 D.1 Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 D.2 Generation Unit and Demand Data . . . . . . . . . . . . . . . . 273 D.3 Transmission Line Data . . . . . . . . . . . . . . . . . . . . . . 278 List of Figures. 287. List of Tables. 291. Notation. 297. Bibliography. 305. Index. 323.

(11) Chapter 1 Introduction. T. he main purpose of the thesis work reported in this document is to develop procedures to assist a wind power producer and the system operator to decide on the optimal investment in wind power and in. transmission facilities, respectively. This first chapter provides an introduction to the work reported in this document. Section 1.1 provides an overview of electricity markets. Section 1.2 gives the thesis motivation. Section 1.3 describes the main features of the investment problems considered in this thesis. Section 1.4 provides a detailed literature review of the existing works related to the topics addressed in this dissertation. The main thesis objectives are stated in Section 1.5. Finally, Section 1.6 outlines the structure of this document.. 1.1. Electricity Markets. Traditionally, the electric power sector was organized in such a way that a public or a publicly controlled entity was in charge of the generation, transmission, distribution, and supply of electric energy throughout a certain region. With the restructuring of the electric power industry in the 80’s, electricity markets [77,90,142] started to appear in different regions throughout the world. These electricity markets enable the electric energy trading among different market participants [79, 136], namely, producers, consumers, and retailers: 1.

(12) 2. 1. Introduction. 1. Producers own the generation units used to generate electricity and aim at maximizing their respective profits. 2. Consumers make use of the electric energy produced by the producers and aim at maximizing their utilities (or at minimizing their purchasing costs). 3. Retailers act as intermediaries among producers and consumers, purchasing electricity from the producers and selling it to their clients. Besides these market participants, there are, among others, four agents responsible for the adequate working of the market, namely, the market operator, the independent system operator (ISO), the transmission system operator (TSO), and the market regulator: 1. The market operator is the entity in charge of the financial management of the system. 2. The ISO is the entity in charge of the technical management of the system. In some jurisdictions, both the technical and the economic management of the system are performed by the ISO, e.g., in PJM [121] and ISO-New England [81]. In other markets, the market operator and the ISO are two independent entities, e.g., in the Iberian Peninsula [139] and in the New Zealand [110] markets. 3. The TSO is the organization responsible for the maintenance and expansion of the transmission infrastructure used to transport the electric energy. 4. The market regulator is an independent entity responsible for supervising the market and guaranteeing that it works appropriately. The energy trading among the different market participants occurs on different trading floors. The most important are the day-ahead, balancing, and futures markets:.

(13) 1.1. Electricity Markets. 3. 1. The day-ahead market is cleared once a day, one day in advance, and on an hourly basis. 2. The balancing market is used to compensate for the deviations between generation and demand in close time (e.g., half an hour) to the actual power delivery. 3. The futures market allows for medium and long-term transactions. Among these trading floors, the day-ahead market is generally the market with the largest volume of trading. The working of a pool-based electricity market can be summarized in the three steps below: 1. Producers submit their offers to the pool. These offers comprise a set of energy production blocks and the minimum prices that producers are willing to accept for the production of each of these energy blocks. 2. Consumers and retailers submit their bids to the pool. These bids comprise a set of energy demand blocks and the maximum prices they are willing to pay for the purchase of each of these energy blocks. 3. The market operator/ISO collects offers and bids from producers and consumers, respectively, and settles the production and consumption schedules, as well as the market clearing prices. To do so, an appropriate market clearing procedure is used. This market clearing procedure is generally formulated as an optimization problem that aims at maximizing the social welfare of the market (or at minimizing the generation cost). The complexity of electricity markets has stimulated the research on a wide range of topics. Additionally, in recent years, the implementation of new technologies, the integration of a large amount of renewable sources, and the development of efficient operation methods have resulted in important research challenges. Considering the above framework, in this thesis we focus on a particular but relevant problem: the investment in wind power and transmission facilities..

(14) 4. 1. Introduction. 1.2. Thesis Motivation. Nowadays, there is an increasing emphasis on limiting the greenhouse gas emissions that contribute to global warming. Although the electric power sector is far from being the main contributor to climate change, reducing the greenhouse gas emissions from electricity generation is important. Among the actions carried out in this sector to mitigate the greenhouse gas emissions, the integration of renewable energy sources is one of the most important. Among all available renewable sources, wind energy is the most mature. Wind energy and the production of electricity out of it have important advantages, namely: 1. It can be produced in many locations throughout the world. 2. It is emission free. 3. It requires no fuel/water consumption. 4. It has low forced outage rates. 5. It is generally well accepted by the public. This thesis focuses on wind power investment, but the proposed methodologies are general enough to be applied to investment in alternative renewable sources such as solar power. Many governments throughout the world have or have had special programs to promote the investment in wind power technology. This has motivated a great wind power expansion in some regions including Europe, North America, and Asia. According to the Global Wind Energy Council [70], the installed wind power capacity throughout the world was 282.6 GW at the end of 2012. The evolution of the total installed capacity in the world during the period 19962012 is depicted in Figure 1.1. The European Union, in particular, is currently pursuing the 20/20/20 objective that aims, among other goals, at increasing the amount of renewable energy up to 20% of the energy supply by 2020 [58]. To achieve these goals,.

(15) 1.2. Thesis Motivation. 5. Installed wind power capacity [GW]. 300 250 200 150 100 50 0. 1996 1998 2000 2002 2004 2006 2008 2010 2012 Year. Figure 1.1: Total installed wind power capacity in the world during the period 1996-2012 [70].. national governments have usually subsidized wind power producers through fixed feed-in tariffs or minimum prices for their productions. However, as the amount of installed capacity based on wind power becomes an important share of the total installed capacity, it is expected that these incentives significantly decrease or even disappear. If this happens, wind power investors would need to recover their investment costs by selling their productions in the pool with a profit-oriented vision, in a similar way as the conventional producers do. However, wind power units have an important drawback if compared with conventional generation units. Wind power production is uncertain since it depends on the availability of a natural source: the wind. Nevertheless, wind power units have the advantage of a very low production cost. It is important to note that the uncertain production of wind power units has to be accu-.

(16) 6. 1. Introduction. rately represented, which leads to rather different models to those used for conventional units. The above considerations motivate the work developed in this thesis. This work is intended to develop different models to determine the optimal investment in wind power facilities within a market environment. The investment decisions are carried out under a profit-maximization perspective. To do so, these models need to explicitly represent the pool-based electricity markets in which the energy trading occurs. Additionally, the proposed models need to represent the uncertainties in wind power production, as well as in other uncertain parameters. On the other hand, an important issue is the availability of enough transmission capacity to accommodate the power flows due to wind power production. This is a source of concern since wind power production centers are usually located away from and poorly-connected to demand centers. Thus, the investment in renewable units is generally conditioned by the transmission capacity of the system. To take into account network constraints, the proposed investment models need to represent the network topology of the electric energy system under study. Additionally, this thesis also seeks to develop a model for the investment in both transmission and wind power facilities. To develop these investment models, there are several issues that need to be considered, namely: 1. Uncertainty. Investment decisions are influenced by a number of uncertain parameters, e.g., demand (and its growth), investment cost, fuel price, or market participants’ behavior. Additionally, the wind power investor faces the uncertainty in the production of wind power units. 2. Trading markets. The day-ahead market is usually the market with the largest volume of trading and, thus, once the newly-built wind power units are ready to operate, the wind power producer participates in this market to sell the energy produced by these units. Generally, it also participates in the balancing market as the actual production of wind power units is not known in advance and the wind power producer may need to sell/purchase its production deviations. The working of these.

(17) 1.3. Problem Description. 7. trading markets needs to be represented. 3. Optimal siting. The investment model should determine the optimal siting of the wind power units to be built based not only on the wind power conditions but also on the characteristics of the transmission network. 4. Optimal sizing. The investment model should determine the optimal sizing of the wind power units to be built. 5. Optimal timing. Since the conditions that determine the investment decisions may change, the investment model should determine the optimal point in time to make the investments. 6. Offering strategy. The wind power investor has to decide its offering strategy to participate in the day-ahead and balancing markets. The investment model should represent different offering strategies.. 1.3. Problem Description. 1.3.1. Stochastic Complementarity Models. The aim of this thesis is to develop models to decide the optimal investment in wind power units and, to a lesser extent, in transmission facilities. These investment models can be generally formulated as optimization problems, i.e., problems with an objective function to be maximized (or to be minimized) and a set of constraints that must be satisfied. The outcomes of these optimization problems are the investment decisions, e.g., the sizing and siting of the wind power units to be built and the transmission capacity reinforcements to be carried out. Additionally, note that these investment decisions are made within a market environment and, thus, the following observations need to be taken into account: 1. The investment decisions are conditioned by the market in which investors sell their productions. For example, profit-oriented wind power.

(18) 8. 1. Introduction. investors mainly determine their investment decisions based on the market prices they get paid. 2. The market clearing is in turn influenced by the investment decisions. For example, the scheduling of producers and consumers depends on the production of the newly built wind power units. Thus, the proposed investment models must represent the clearing of the market as an additional constraint. Moreover, such a representation should include the market clearing under different operating conditions, e.g., under different demand and wind power production conditions. It is important to note that the market clearing is itself an optimization problem that seeks to maximize social welfare or to minimize generation cost, and is subject to a set of constraints, e.g., generation-demand balances at different nodes, power flow limits through transmission lines, or voltage angle limits. Thus, the proposed investment models become optimization problems constrained by other optimization problems, i.e., complementarity models (also known as hierarchical or bilevel models). Figure 1.2 illustrates the structure of these complementarity models that comprise an upper-level problem and a set of lower-level problems, i.e., a bilevel model. On one hand, the upper-level problem is an optimization problem to decide the optimal investment decisions. On the other hand, the lower-level problems represent the market clearing under different operating conditions. Note that the investment decisions influence the market clearing (through the production of the newly built wind power units) and that the outputs of the market influence in turn the investment decisions (through the market clearing prices). Additionally, the investment models need to represent the uncertainties in the parameters that influence the investment decisions, e.g., demand (and its growth), wind power production, investment cost, etc. This is done through a set of scenarios that represent possible realizations of these uncertain parameters. Thus, the investment models are finally formulated as stochastic complementarity models..

(19) 1.3. Problem Description. 9. Figure 1.2: Structure of complementarity models.. 1.3.1.1. The Perspective of a Wind Power Investor: Upper-Level Problem. We consider a wind power investor interested in building wind power units throughout an existing electric energy system. This investor seeks to determine both the siting and the sizing of each wind power unit to build, with the aim of maximizing the expected profit obtained from selling its production in the electricity market. To solve this wind power investment problem, different approaches can be considered, namely: 1. Static or dynamic. The static approach assumes that the investment decisions are made at a single point in time and for a future target year. On the other hand, a dynamic approach allows us to make the investment decisions at different points in time. 2. Risk-neutral or risk-averse. A risk-neutral approach considers the maximization of the expected profit and neglects the profit volatility. On the other hand, the risk-averse approach allows us to consider the risk of the profit volatility associated with the investment decisions..

(20) 10. 1. Introduction. 3. Non-strategic or strategic approach. Wind power production is generally offered to the market at zero or at a low price since its production cost is usually small. However, as wind power producers attain market power as their wind power capacities increase, they may behave strategically and offer prices different from zero. The approaches in items 1.-3. above, their advantages, as well as their disadvantages are explained and analyzed in Chapters 3-5 of this thesis. Besides this, a set of constraints limit and condition the wind power investment, i.e.: 1. Wind power units can only be built in selected locations. 2. The production of a newly-built wind power unit is limited by the wind availability in its location. 3. The budget for wind power investment is generally limited. These constraints need to be properly represented in the investment models. 1.3.1.2. The Perspective of the TSO: Upper-Level Problem. Wind power investment is highly conditioned by the transmission characteristics of the network. However, when deciding on the investment in both transmission and wind power facilities, the problem is conveniently formulated under the perspective of the TSO, which is the entity responsible for the transmission reinforcements to be carried out. Additionally, if the wind power investment is subsidized with public funds, it is reasonable to assume that a public entity such as the TSO is responsible for identifying the wind power projects that are beneficial for society, e.g., those wind power projects that minimize a measure of social welfare, such as the overall consumers’ payments. This approach is adopted in Chapter 6. As in the wind power investment problem of a profit-oriented wind power investor, a number of constraints needs to be properly represented, namely: 1. Wind power units can only be built in selected locations..

(21) 1.3. Problem Description. 11. 2. The number of prospective transmission lines to be built is limited. 3. The production of a newly built wind power unit is limited by the wind availability in its location. 4. Budgets for wind power investment and transmission investment are usually limited. 1.3.1.3. Market Clearing: Lower-Level Problems. As explained in Section 1.1 at the beginning of this chapter, there are different trading floors to enable energy trading among market participants, e.g, the day-ahead, balancing, and futures markets. Since wind power production is uncertain and difficult to forecast, futures markets are generally not of interest to wind power producers. Thus, throughout this dissertation we consider the participation of wind power producers only in the day-ahead and balancing markets. On one hand, the day-ahead market is cleared on an hourly basis and one day in advance. The clearing of this market determines the scheduled electric energy to be produced by each producer, the energy consumption of each consumer/retailer, and the market clearing prices. These prices are obtained as the dual variables associated with the energy balance equations. In a market clearing procedure considering the maximization of social welfare, the market clearing prices represent the social welfare increment in the market that results from a marginal increment of the demand. Market clearing prices can be considered uniform throughout the network or locational, i.e., different at the different nodes of the network. In the latter, market clearing prices are referred to as locational marginal prices (LMPs). The LMP differences across nodes are due to network congestion and line losses. The LMP approach is used in different electricity markets throughout the world, e.g., in PJM [121], ISO-New England [81], or New Zealand [110], and is of particular interest in systems with a high penetration of wind power since wind power units are generally located away from demand centers and transmission congestion occurs often..

(22) 12. 1. Introduction. In this thesis, we consider a market clearing procedure for the day-ahead market with the following characteristics: 1. The day-ahead market clearing is formulated as an optimization problem that aims at maximizing the overall social welfare. 2. Network constraints are modeled using a dc power flow representing both Kirchhoff’s laws [68]. Losses are neglected for the sake of simplicity. However, losses can be easily considered using a piecewise linear representation. 3. Constraints include the energy balance at each node, production limits of each generation unit, demand limits of each load, transmission capacity limits, and voltage angle limits. 4. Since we model the network, LMPs are the day-ahead market clearing prices. The above assumptions allow us to formulate the day-ahead market clearing procedure as a continuous and linear (and thus, convex) optimization problem. In particular, the day-ahead market clearing problem under different operating conditions is represented in the lower-level problems developed in this thesis. Since the day-ahead market is cleared one day in advance, there are generally energy imbalances between generation and demand that arise in real time, especially in systems with a high penetration of renewable sources. These imbalances are minimized in the balancing market. For the sake of simplicity, throughout this dissertation the network constraints are not modeled in the balancing market. Thus, we consider that balancing market prices are uniform across the system, i.e., not locational. Moreover, a single price is considered for buying/selling in the balancing market. Some markets, e.g., the Iberian Peninsula market [139], consider a two balancing price scheme. This approach penalizes the energy deviations that contribute to the overall system imbalance and is neutral with respect to those that contribute to restore the system balance..

(23) 1.3. Problem Description. 1.3.2. Solution Procedure. 1.3.2.1. MPECs. 13. The upper-level and the lower-level problems comprising a bilevel model have to be jointly solved. To do so, the lower-level problems representing the dayahead market clearing under different operating conditions are replaced by their optimality conditions. There are two manners of stating the optimality conditions (provided that the lower-level problems are convex), namely: 1. To replace each lower-level problem by its Karush-Kuhn-Tucker (KKT) optimality conditions. 2. To replace each lower-level problem by its primal constraints, its dual constraints, and its strong duality equality. The KKT optimality conditions, or the primal constraints, the dual constraints, and the strong duality equality, of each lower-level problem, are then included as additional constraints in the upper-level problem, rendering a mathematical program with equilibrium constraints (MPEC). 1.3.2.2. MILP Problems. The resulting MPEC is single-level but generally nonlinear, especially if it is obtained using the KKT optimality conditions. However, these nonlinearities can be generally linearized using exact mixed-integer linear equivalent expressions, and the MPEC can finally be formulated as a mixed-integer linear programming (MILP) problem. Thus, the stochastic complementarity models for the investment in wind power and transmission facilities are finally recast as MILP problems that can be efficiently solved using branch-and-cut techniques. 1.3.2.3. Benders’ Decomposition. Some of the considered MILP problems are large scale, especially if the system under study is large and/or a large number of operating conditions and sce-.

(24) 14. 1. Introduction. narios are considered. This is so because the linearization of MPECs requires using a large number of binary variables. However, if the investment decisions, i.e., the wind power capacity to be built throughout the system and the transmission lines to be built, are fixed to given values, the MILP problems considered can be decomposed into different subproblems. Moreover, the objective functions of these MILP problems expressed as a function of the investment variables have convex enough envelopes. Thus, Benders’ decomposition [23, 44] can be applied to reduce the computational burden in the case that the resulting MILP problems become intractable. 1.3.2.4. Computational Tool. As indicated in Subsection 1.3.2.2 above, the investment problems developed in this thesis are finally recast as MILP problems. Moreover, if these problems are tackled via Benders’ decomposition, the resulting subproblems are also MILP problems. Throughout this dissertation, MILP problems are solved using CPLEX 12.2.0.1 [47] under GAMS [64] on a Linux-based server with 4 processors clocking at 2.9 GHz and 250 GB of RAM.. 1.3.3. Model Assumptions. Subsections 1.3.1 and 1.3.2 above describe the main features of the investment problems addressed in this thesis. Additional model assumptions considered throughout this dissertation are listed below for the sake of clarity: 1. Producers (both wind power and conventional producers) participating in the day-ahead market get paid for their productions at the LMP corresponding to the node at which they are located. These LMPs are obtained as the dual variables associated with the generation/demand balance constraints [135]. 2. Demands participating in the day-ahead market pay for their consumptions the LMP corresponding to the node at which they are located..

(25) 1.4. Literature Review. 15. 3. Producers other than the wind power investor are assumed perfectly competitive and offer their productions at marginal costs. 4. Investment is only allowed at a selected subset of the network nodes. 5. Investment is only allowed in a selected subset of transmission lines. 6. The on/off status of generation units is not modeled for the sake of simplicity and to sidestep non-convexities [114, 131]. This is consistent with the current practice in most European markets. 7. This thesis focuses on the investment in wind power facilities. Thus, the investment in conventional generation units is not considered. However, the models developed in this thesis can be extended to also consider conventional units as investment alternatives. References [86, 87, 151] describe bilevel models for investment solely in conventional generation units. 8. The investments of rival producers are not considered for the sake of simplicity. This can be done using equilibrium problem with equilibrium constraints (EPEC) approaches [88, 152]. 9. For simplicity, security constraints are not considered in the market clearing problems. However, these constraints can be included as described in references [26, 35].. 1.4. Literature Review. This section provides an overview of the state-of-the-art of the topics related to this thesis.. 1.4.1. Wind Energy. The installed wind power capacity in the world has rapidly increased in the last decade (see Figure 1.1). Most of this wind power capacity has been installed in the last few years. For example, in the last four years, the installed wind.

(26) 16. 1. Introduction. power capacity has grown from 120.6 GW at the end of 2008 to 282.6 GW at the end of 2012, which represents an increase of 134%. This growth has stimulated the research in a number of wind power related issues, including operations [39,48,53,55,71,74,101,112,117], security [1,34,116,148], forecasting [46, 93, 153], emission control [53, 75], and economics [21, 54, 59, 102, 105, 133]. Among this wide range of topics, this thesis focuses on the economic aspects of investing in wind power facilities.. 1.4.2. Investment in Generation Facilities. The investment in generation facilities constitutes a relevant problem in electricity markets. This subsection provides a review of the broad literature addressing this topic. 1.4.2.1. Investment in Conventional Generation Units. The investment in conventional generation units has generally been tackled through two different approaches: a centralized framework [108] and a market one [86]. On the one hand, the centralized approach usually determines the generation capacity expansion plan based on a worst case scenario and considering the whole electric energy system. This approach is relevant for the efficient operation of the system as a whole. However, note that the owners of conventional generation units and the potential investors are generally independent profitoriented entities that aim at maximizing their own expected profits. Thus, the centralized approach is not of interest for these profit-oriented agents. On the other hand, the market approach allows us to represent the working of the market in which producers participate and sell their power productions. Such an approach allows us to represent the perspective of a profit-oriented investor. References addressing investment in conventional generation units under a market framework are numerous and consider different approaches, namely: 1. Static [40, 84, 86, 149] and dynamic [32, 109, 151]..

(27) 17. 1.4. Literature Review. 2. Stochastic [32, 86, 87, 113, 149] and deterministic [40, 84]. 3. Strategic [85–87] and competitive [113]. 4. With [84–87] and without [32, 40, 113, 151] network representation. The above references consider conventional generation units. This implies that the investors in these units have knowledge about and control over the productions of these generation units. However, it is important to note that investing in wind power facilities requires rather different models since wind power production is uncertain and this uncertainty must be accurately represented within the investment problem. 1.4.2.2. Investment in Wind Power Facilities. References on investment in wind power facilities are rather scarce. Table 1.1 summarizes the approaches used in technical literature to address the investment in wind power facilities. Table 1.1: Literature review: Relevant approaches used in technical literature for the wind power investment problem. Reference [91] [31, 92] [36, 154] [147] [3] [156] [31] [154] [49, 61, 99, 155]. Approach Screening and ranking Feed-in tariff scheme Security and reliability Stochastic dynamic programming Investment experience Design of incentives Renewable energy certificates Minimizing system social cost Wind and other technologies. Reference [36] provides an optimization approach to establish the maximum wind power penetration in a system with fixed transmission capacity, preserving network security levels. Reference [92] proposes a feed-in-tariff scheme.

(28) 18. 1. Introduction. to promote wind investment under a regulated environment. Reference [3] analyzes wind power investment in Turkey. In [147], wind power investment under uncertainty is analyzed using a stochastic dynamic programming approach. Reference [156] provides a methodology to design incentive policies for stimulating investment in renewable units. Reference [31] proposes a real options approach to analyze the investment in renewable sources under different support schemes including feed-in tariffs and renewable energy certificate tradings. In [154], a chance constrained approach is proposed to determine the optimal wind power capacity plan to minimize the system’s social cost and to satisfy operation constraints. Other references [49,61,99,155] consider the joint investment in wind power and other technologies. References [61,99] provide optimal expansion planning models for a wind-diesel energy system and for a photovoltaic-wind-fuel hybrid system, respectively. On the other hand, reference [49] proposes a linear programming model to determine the optimal technology mix, considering wind power production as a negative load. Finally, reference [155] analyzes different policy options to promote the investment in energy storage facilities to stimulate wind power investment.. 1.4.3. Offering Strategies. Chapter 5 of this thesis is devoted to the development of a decision-making model to determine the optimal investment of a strategic wind power producer. To do so, we consider that this producer is able to exercise market power and to behave strategically in the day-ahead market. There is a large number of references in technical literature proposing different offering strategies for producers. However, most of them are developed for conventional generation units, e.g., [8, 9, 12, 65, 130]. Reference [8] proposes a methodology to derive the optimal offers of a producer operating in the day-ahead market and considering the uncertainties in its own production and in its rivals’ behavior. In references [9] and [130], bilevel optimization models are used to develop offering strategies. Reference [9] develops an offering strategy for electricity producers in the day-ahead market.

(29) 1.4. Literature Review. 19. with step-wise offer curves. Reference [130] proposes a multi-period networkconstrained market clearing procedure, in which uncertainties in demand bids and in offering strategies of rival producers are represented. Reference [65] uses a risk-constrained stochastic programming approach to determine the self-scheduling of production units, the weekly forward contracts, and the offering strategy for a producer. Finally, references [12] and [56] propose robust optimization approaches for the optimal offering strategy of a conventional producer and a concentrating solar power plant that face uncertain but bounded energy prices, respectively. Regarding offering strategies for wind power producers participating in electricity markets, it is worthwhile to mention references [6, 20, 97, 102, 120, 157]. References [20] and [97] propose models to reduce the imbalance costs, using a Markov model and stochastic programming, respectively. In [6], an offering strategy considering wind power and hydro units is developed. Reference [120] uses probabilistic forecasts of wind power production to develop an offering strategy, considering the sensitivity of a wind power producer to regulation costs. On the other hand, reference [102] uses a stochastic approach to derive the offering strategy for a wind power producer considering different trading floors. Finally, reference [157] proposes a bilevel model to determine the pool strategy of a wind power producer that participates in the day-ahead market as a price-taker and in the balancing market as a price-maker.. 1.4.4. Investment in Transmission Facilities. Transmission lines constitute a fundamental part of an electric energy system. Therefore, the problem of investing in transmission facilites has been discussed in detail in technical literature. Techniques used to deal with the transmission expansion planning problem include heuristic [67, 118] and optimization [4, 50, 66] approaches. Reference [67] provides a method that uses linear flow estimates to derive network designs, which are then used for the transmission expansion planning. Reference [118] applies sensitivity methods as a tool for long-term transmission.

(30) 20. 1. Introduction. expansion planning, ranking different investment plans according to their effectiveness in increasing the system load supplying capability, and in reducing the system load curtailment. On the other hand, references [4, 50] propose MILP models for the longterm transmission expansion planning, considering losses [4] and using a set of scenarios to model the future demand and aiming at maximizing the overall social welfare [50]. Additionally, reference [66] proposes a bilevel model to solve the transmission expansion problem under the perspective of a transmission planner whose goal is to minimize the investment cost while facilitating the energy trading. The investment in transmission facilities becomes especially important in systems with high wind power penetration [60, 105, 115, 140, 146]. In [105], a bilevel model is proposed to decide on the transmission lines to be built and to estimate the amount of market-integrable wind resources. Reference [60] analyzes the expansion of the interconnection capacity required to integrate wind and solar power in the Iberian Peninsula, modeling the tradeoff between the investment cost in transmission capacity and the costs of renewable energy curtailment. Reference [146] proposes a method to decide on the optimal expansion of the transmission network to supply 50% of the electricity demand using wind power. In [115], a probabilistic approach for the transmission expansion planning considering uncertainties in demand and wind power production is developed. Finally, reference [140] evaluates the benefits of interconnection investments in the presence of high wind power penetration and other power plant investments. There are other references that consider the investment in both generation and transmission facilities, e.g., [5, 82, 106, 128, 129]. Reference [5] uses a stochastic programming approach that models as random events demands, availability of power units, and transmission capacity factors of lines. Reference [128] proposes an iterative procedure to coordinate the generation and transmission expansion, using incentives for investors to recover their investment costs. In [129], the interactions among producers, transmission companies, and the TSO are simulated in a competitive market in which a capacity payment mechanism is considered to promote agents’ investments. The model.

(31) 1.4. Literature Review. 21. proposed in [129] simulates the outages of generation units and transmission lines using the Monte Carlo method. Reference [106] proposes a four-level optimization model to determine the optimal transmission expansion anticipating generation expansion plans. Recently, reference [82] proposes a bilevel model to decide on the investment in the transmission network of a transmission operator, considering the investment of generation companies and a market environment.. 1.4.5. Mathematical Tools. This subsection provides a brief survey of the main mathematical tools used in the models developed in this dissertation. 1.4.5.1. Uncertainty Modeling. One key issue if dealing with investment problems is the modeling of the uncertain parameters that influence the investment decisions, e.g., demand growth, investment cost, fuel price, etc. There are two main techniques to tackle these uncertain parameters in optimization problems, namely, robust optimization [22, 25, 27] and stochastic programming [30, 43]. On one hand, robust optimization is a mathematical tool to tackle optimization problems that include uncertain but bounded parameters [25,27]. On the other hand, stochastic programming represents the uncertainty in the input data via scenarios [30]. Since the uncertainties in demand growth, investment cost, wind power production, etc., can be efficiently modeled through a set of scenarios, this second approach is the one used in this thesis. There are two important issues to be considered for developing stochastic programming models, namely: 1. Scenarios must accurately represent the uncertain data. Thus, appropriate scenario generation methods [103] should be considered. 2. Scenarios must represent all possible realizations of the uncertain data. This usually results in a very large number of scenarios that, if considering a large-scale problem, may result in intractability. Thus, appropriate.

(32) 22. 1. Introduction. scenario reduction techniques [104, 119] should be used to reduce the number of scenarios while maintaining an adequate representation of the uncertain data. 1.4.5.2. Bilevel Models. Bilevel models constitute the basis of the investment models developed in this dissertation. References [10, 28, 42, 52, 63] provide rigorous mathematical descriptions of such models. Additionally, references [2, 11, 41, 62, 63, 83, 145] propose different solution techniques for solving bilevel problems based on barrier methods [2], branch-and-bound algorithms [11,62], descent methods [145], sequential methods [83], and trust-region methods [41]. In recent years, bilevel models have been used in a wide variety of problems related to electricity markets [63], including generation capacity investment [86], transmission expansion planning [66, 82], offering strategy [73, 130], security analysis [134], or retailer trading [38], among others. 1.4.5.3. MPECs. The bilevel models proposed in this thesis are transformed into MPECs. MPECs were first reported in [72]. Detail mathematical descriptions of MPECs can be found in [63, 96]. One of the main drawbacks of MPECs is that they are generally nonconvex. However, several methods have been reported in technical literature to effectively solve MPECs. These methods include MILP reformulations of MPECs [9, 62, 63, 130], interior methods [94], relaxation methods [51], and sequential quadratic programming [7]. 1.4.5.4. Benders’ Decomposition. Benders’ decomposition constitutes a useful technique to address problems with complicating variables [23,44]. In a mathematical programming problem, the complicating variables are those variables that prevent the decomposition of the problem into different subproblems with the consequent reduction in the computation burden. Benders’ decomposition has been applied in a wide range.

(33) 1.5. Thesis Objectives. 23. of problems related to electric energy systems, including generation capacity expansion [85, 128], hydrothermal scheduling [137], or voltage security [124]. 1.4.5.5. Risk Modeling. A decision-making problem is generally formulated as an optimization problem whose objective is the maximization of profit or the minimization of cost. If the decision variables of this problem are influenced by uncertain data, the profit/cost becomes a random variable. For example, if the uncertain data is modeled via stochastic programming, the profit/cost can be characterized by a probability distribution. In these cases in which the objective function is a random variable, it is necessary to optimize a real-valued function characterizing the probability distribution of this random variable, e.g., its expected value. Maximizing expected profit or minimizing expected cost is generally appropriate for some applications. However, these approaches disregard the variability of the profit/cost distribution. In some cases, it is important to include a risk metric to control the profit/cos variability. There are multiple risk measures, e.g., variance, shortfall probability, expected shortage, value-at-risk (VaR), and conditional value-at-risk (CVaR) [43]. Among these risk measures, in Chapter 4 of this thesis we use the CVaR as a risk metric. The CVaR metric has been used in different problems, e.g., generation capacity expansion [109], participation in futures markets by producers [45], wind power trading [33], retailer decision-making [37], or scheduling and offering of a hydro producer [122].. 1.5. Thesis Objectives. Considering the context presented in the previous sections, the main objectives of this dissertation are fourfold: 1. To develop a stochastic complementarity model to determine the optimal investment for a wind power investor considering a static approach. This investor aims at maximizing its expected profit. Specific objectives are:.

(34) 24. 1. Introduction. 1.1 To model through a set of representative operating conditions the uncertainties in and the correlation between demand and wind power production. 1.2 To model the clearing of the day-ahead market under different operating conditions. 1.3 To tackle via Benders’ decomposition the potential intractability that may appear if a large number of operating conditions and a large system are considered. 1.4 To analyze the impact of the number of operating conditions on the convexity of the proposed model. 1.5 To analyze the impact of transmission congestion on the investment decisions. 1.6 To analyze the impact of the investment decisions on the resulting market clearing prices. 2. To develop a stochastic complementarity model to determine the optimal investment for a wind power investor considering a risk-constrained multi-stage approach. This investor aims at maximizing its expected profit while minimizing the risk pertaining to its profit volatility. Specific objectives are: 2.1 To model through a set of operating conditions and scenarios the uncertainties in demand, wind power production, future demand growth, and future wind power investment cost in different time periods. 2.2 To model the clearing of the day-ahead market under different operating conditions, scenarios, and time periods. 2.3 To represent the risk associated with the investment decisions through the CVaR metric. 2.4 To tackle via Benders’ decomposition the potential intractability that may appear if a large number of scenarios and a large system are considered..

(35) 1.5. Thesis Objectives. 25. 2.5 To analyze the impact of different risk-aversion levels on the investment decisions. 3. To develop a stochastic complementarity model to determine the optimal investment for a strategic wind power investor with market power considering a static approach. This investor aims at maximizing its expected profit. Specific objectives are: 3.1 To model through a set of operating conditions and scenarios the uncertainties in demand, wind power production, and balancing market price. 3.2 To model the clearing of the day-ahead and balancing markets under different operating conditions and scenarios. 3.3 To represent the strategic behavior of the wind power producer in the day-ahead market. 3.4 To analyze the impact of the strategic behavior of the investor on the investment decisions. 3.5 To analyze the impact of transmission congestion on the investment decisions. 4. To develop a stochastic complementarity model to determine the optimal investment in wind power and transmission facilities considering a static approach. The aim of this model is to minimize the overall consumers’ payments. Specific objectives are: 4.1 To model through a set of operating conditions the uncertainties in demand and wind power production. 4.2 To model the clearing of the day-ahead market under different operating conditions. 4.3 To determine the optimal transmission lines to be built by the TSO. 4.4 To determine the optimal wind power projects to be promoted among profit-oriented wind power investors..

(36) 26. 1. Introduction. 4.5 To analyze the impact of wind power investment subsidies on the investment decisions. 4.6 To analyze the impact of demand growth on the investment decisions.. 1.6. Thesis Structure. This document is organized as follows. Chapter 1 provides an introduction to the thesis work. First, an overview of electricity markets is provided. Second, the motivation of the work developed in this dissertation is stated. Third, the main features of the problems addressed in this thesis are described. Fourth, a literature review concerning the works related to this dissertation is carried out. Finally, the chapter concludes with a list of the objectives of this thesis. Chapter 2 provides models to characterize the different sources of uncertainty that affect the decision-making problems addressed in this thesis, including demand, wind power production, balancing market price, future demand growth, and future investment cost. Two different methods are proposed based on the load- and wind-duration curves, and the K-means clustering technique. A case study is provided to illustrate the modeling of these uncertain parameters. Chapter 3 provides a stochastic bilevel model for the wind power investment problem considering a static approach. The proposed model aims at maximizing the expected profit of a wind power investor. This chapter includes the reformulation of the proposed bilevel model as an MPEC that can be recast as a MILP problem. To tackle the computational burden of the resulting MILP problem, an approach based on Benders’ decomposition is provided. Finally, we numerically justify that the minus expected profit of the wind power investor as a function of the investment decision variables has a convex enough envelope if a large number of operating conditions is considered, and thus, Benders’ decomposition is effective. Chapter 4 proposes a stochastic bilevel model for the wind power invest-.

(37) 1.6. Thesis Structure. 27. ment problem considering a risk-constrained multi-stage approach. Instead of an investor that simply maximizes its expected profit as in Chapter 3, Chapter 4 considers a wind power investor that aims at maximizing its expected profit while minimizing its profit volatility. To to so, the CVaR metric is used. Uncertainties in demand, wind power production, future demand growth, and future investment cost are considered. A weighting parameter in the objective function is used to materialize different risk-aversion levels that lead to different investment strategies. The bilevel model is recast as a MILP problem that can be efficiently solved using Benders’ decomposition. Chapter 5 provides a stochastic complementarity model for the optimal wind power investment of a strategic investor with market power. This investor participates and exercises market power in the day-ahead market, while it buys/sells its production deviations in the balancing market, in which it behaves as a deviator. First, a strategic offering for a wind power producer is developed to determine the optimal production level and price to be offered to the day-ahead market. Then, this model is extended to consider investment in wind power facilities. Both the strategic offering and the strategic investment models are recast as MILP problems. Chapter 6 proposes a stochastic bilevel model to determine the optimal transmission lines to be built by the TSO and the optimal wind power projects to be promoted among private profit-oriented wind power investors with the aim of minimizing the overall consumers’ payments. Subsidies in wind power investment are considered as fixed percentages of the investment cost. The model is recast as a tractable MILP problem. Chapters 3-5 and 6 include clarifying examples based on a three-node system and the Garver’s six-node system, respectively. Additionally, Chapters 3-6 include two realistic case studies based on the IEEE 24-node Reliability Test System (RTS) and the IEEE 118-node Test System (TS) to illustrate the features and applicability of the proposed models. Chapter 7 concludes this dissertation and provides a summary of the thesis contents, a number of relevant conclusions, the main contributions of the work carried out, and some suggestions for future research. Appendix A provides the wind power production and balancing market.

(38) 28. 1. Introduction. price scenario data used in the case studies of Chapters 2 and 5. Appendix B provides an overview of the mathematical tools used in this thesis, including the bilevel model used in Chapters 3-6, two linearization methods used in Chapters 3-6, and the Benders’ decomposition algorithm used in Chapters 3-4. Appendices C and D provide the data of the IEEE 24-node RTS and the IEEE 118-node TS, respectively, used in the case studies of Chapters 3-6..

(39) Chapter 2 Uncertainty Modeling. A. key point when dealing with long-term decision-making problems is. the modeling of the uncertain parameters that influence the decisions. This thesis addresses different models to decide on the optimal investment in wind power units and in transmission lines. In this type of problems, the uncertain parameters include demand, demand growth, investment cost, fuel price, wind power production, market participants’ behavior, balancing market price, etc. This chapter describes the modeling of the different sources of uncertainty considered throughout this dissertation.. 2.1 2.1.1. Introduction Sources of Uncertainty. The approach used in Chapters 3-5 of this thesis is based on the maximization of the profit of a wind power investor interested in building new wind power units in an existing electric energy system. The profit of the wind power investor is computed as the revenue achieved by participating in both the dayahead and balancing markets, minus the investment cost incurred in building new wind power capacity. The revenue that the wind power investor obtains by selling its production in the day-ahead market is computed as the wind power production scheduled 29.

(40) 30. 2. Uncertainty Modeling. to be produced in this market by the investor times the locational marginal price (LMP) of the node at which it is produced. This revenue depends on two uncertain parameters. First, on the demand of the system: the higher the demand is, the higher the LMPs are. Second, on the available wind power production. Thus, the uncertainties in both demand and wind power production have to be adequately modeled. Additionally, in most electric energy systems, demand and wind power production are not statistically independent magnitudes since high wind power production (usually during the night) generally corresponds to low demand, and low wind power production (usually during the day) generally corresponds to high demand. This is the case, for example, in the power system operated by the Electric Reliability Council of Texas (ERCOT) [143]. In this system, the wind power production from the west zone is generally higher in winter, spring, and autumn than in summer, when the energy demand is higher due to the use of air-conditioning. Thus, it is necessary to properly represent this negative correlation between demand and wind power production. It is important to take into account that if the wind power scheduled to be produced in the day-ahead market is higher/lower than the actual wind power production as a result of its uncertain character, wind power producers need to buy/sell their production deviations in the balancing market. The balancing market prices that wind power producers pay/get paid for the production bought/sold in this market are also subject to uncertainty. Note that the balancing market price is usually anti-correlated with the wind power production because if the wind power production is high, the system usually experiences a surplus of generation and the balancing market price is usually low; on the other hand, if the wind power production is low, the system may have a deficit of generation and the balancing market price is expected to be high. Thus, this negative correlation needs to be considered. The profit achieved by a wind power investor also depends on the investment cost incurred in building new wind power capacity, which is usually uncertain, especially if considering a long term planning horizon. On the other hand, the approach used in Chapter 6 of this thesis is based on the minimization of consumers’ payments and investment costs. As in the.

(41) 2.2. Modeling of the Uncertainty in Dem., Wind Prod., and Bal. Market Price 31. profit maximization problem, demand and wind power production influence these payments, and thus, the uncertainties in both demand and wind power production need to be modeled. Considering the above context, this chapter provides different methods to represent both the uncertainties in and the correlation between important paraments of the proposed models. The outputs provided by these methods are used as input data for the investment models provided in the following chapters of this thesis.. 2.1.2. Chapter Organization. The remainder of this chapter is organized as follows. Section 2.2 provides the uncertainty modeling of demand, wind power production, and balancing market price using two different methods based on the load- and wind-duration curves, and the K-means clustering technique. Section 2.3 describes the uncertainty modeling of the investment cost. Section 2.4 provides a case study that shows the working of the uncertainty modeling methods. Finally, Section 2.5 summarizes the chapter.. 2.2. Modeling of the Uncertainty in Demand, Wind Power Production, and Balancing Market Price. In this section we describe the techniques used in this thesis to model the uncertainties in demand, wind power production, and balancing market price. These techniques are based on two methods that model both the uncertainties in and the correlation between these parameters, namely, the method based on the load- and wind-duration curves [13, 16, 86], and the K-means clustering technique [16, 76, 125, 144]..

(42) 32. 2.2.1. 2. Uncertainty Modeling. Modeling of the Uncertainty in Demand and Wind Power Production. Both the method based on the load- and wind-duration curves, and the Kmeans clustering technique use hourly historical data of demand and wind power production in different locations (e.g., at different nodes) as input, and provide a reduced data set that maintains the uncertainty information of and the correlation between the historical data as output. This reduced data set consists of a set of day-ahead market operating conditions, each one comprising a value for the demand in each demand location and a value for the wind power production in each wind location. These operating conditions represent both the uncertainties in and the correlation between demand and wind power production.. 2.2.1.1. Method based on the Load- and Wind-Duration Curves. The first method to model the uncertainties in and the correlation between the demand and the wind power production in each location of an existing electric energy system is based on the load- and wind-duration curves [13, 16, 86]. The working of this method can be summarized in the seven steps below:. Step 1: Using historical demand and wind power production hourly data, we first compute the demand factors (defined as the demands divided by the peak demand) and the wind power capacity factors (defined as the wind power productions divided by the installed wind power capacity). In the case of no wind power production historical data in a given location, historical wind power capacity factors are obtained considering historical data of wind speeds, and transforming these wind-speed data into wind power capacity factors through appropriate wind-speed/wind-power production curves..

(43) 2.2. Modeling of the Uncertainty in Dem., Wind Prod., and Bal. Market Price 33. Step 2: Using hourly historical data of demand factors, we build a load-duration curve. Wind power capacity factor [p.u.]. as represented by the continuous curve depicted in the lower plot of Figure 2.1.. Wind power capacity factor levels 1 0.8 0.6 0.4 0.2 0 0. 2000. 4000. 6000. 8000 Time [h]. 6000. 8000 Time [h]. 1. Demand factor [p.u.]. 0.9. Demand factor levels. 0.8 0.7 0.6 0.5 0.4 0. 2000. 4000. Demand blocks. Figure 2.1: Load and wind-duration curves..

(44) 34. 2. Uncertainty Modeling. Step 3: The load-duration curve is approximated using a set of demand blocks. Note that in terms of hour span, the first demand block is narrower than the others in order to represent the peak demand, which usually has a great impact on system-wide decisions and must be adequately represented. Step 4: The uncertainties in the demand factors within each demand block is represented by considering different demand factor levels. To do so, we build the cumulative distribution function (cdf) of the demand factors within each demand block, as depicted in Figure 2.2. Then, this cdf is divided into a selected number of segments (three in the example depicted in Figure 2.2), each one with an associated probability. The average values of the demand factors within each segment give the demand factor levels represented in the lower plot of Figure 2.1.. 1. Cumulative probability. 0.8. 0.6. 0.4. 0.2. 0 0.72. 0.74. 0.76. 0.78 0.8 0.82 Demand factor [p.u.]. 0.84. 0.86. Figure 2.2: Cumulative distribution function of demand factors..

(45) 2.2. Modeling of the Uncertainty in Dem., Wind Prod., and Bal. Market Price 35. Step 5: Next, we model the wind power production that is usually anti-correlated with the demand of the system, and thus, both the demand and the wind power production have to be jointly represented. We use historical data of wind power capacity factors throughout the same period considered for the demand. Then, for all the hours allocated to each demand block, we consider the corresponding wind power capacity factors. With these data, we build, for each demand block, a wind-duration curve as depicted in the upper plot of Figure 2.1 (continuous curves). Step 6: In order to account for the uncertainty in the wind power production within each demand block, the wind-duration curves are approximated by a set of wind power capacity factor levels. The procedure to select the wind power capacity factor levels is identical to that explained for the demand factor levels, i.e., we build the corresponding cdf of each wind-duration curve as depicted in Figure 2.3; then we divide this cdf plot into a selected number of segments (four in the case of the example in Figure 2.3); and finally we compute the average values of each segment, which represent the wind power capacity factor levels used in the study and represented in the upper plot Figure 2.1. Step 7: Within each demand block, we consider all possible combinations of demand factor levels and wind power capacity factor levels, which constitute the demand and wind power production conditions for each demand block. Each condition is assigned a probability within the demand block equal to the probability of the demand factor level times the probability of the wind power capacity factor level. For the example in Figure 2.1, we consider four demand blocks, three demand factor levels, and four wind power capacity factor levels, i.e., a total of 48 operating conditions (4 × 3 × 4). The weight of each operating condition is.

(46) 36. 2. Uncertainty Modeling. 1. Cumulative probability. 0.8. 0.6. 0.4. 0.2. 0. 0. 0.2 0.4 0.6 0.8 Wind power capacity factor [p.u.]. 1. Figure 2.3: Cumulative distribution function of wind power capacity factors.. computed as the number of hours in the corresponding demand block times the probability of each operating condition within this demand block. The number of demand blocks used to adjust the load-duration curve, as well as the number of demand factor levels and wind power capacity factor levels used to represent the uncertainties in demand and wind power production should be selected taking into account the nature of the study to be carried out. A large number of blocks and levels may result in intractability, while a small number may lead to a poor representation of the uncertainty in the demand and the wind power production of the system. This technique is simple and easy to implement. However, it has the disadvantage that it can only be used if the same correlation between demand and wind power production is considered in all locations (i.e., at all nodes) of the system. The K-means clustering technique provided in the next section does not have this limitation..