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(1)UNIVERSIDAD DE CASTILLA-LA MANCHA UNIVERSIDAD DE CASTILLA-LA MANCHA. ESCUELA TÉCNICA SUPERIOR DE INGENIEROS INDUSTRIALES DE CIUDAD REAL. Ph.D. THESIS. Robust Generation Scheduling in Electricity Markets. AUTHOR NOEMI GONZÁLEZ COBOS. SUPERVISORS JOSÉ MANUEL ARROYO SÁNCHEZ NATALIA ALGUACIL CONDE. Ciudad Real, December 2018.

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(3) A mis padres, Filomena y Jesús.

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(5) Acknowledgements The realization of this thesis would not have been possible without the support and help in different ways of many people and institutions. Within these lines, I would like to express my sincere gratitude to all of them. First and foremost, I would like to express my deep gratitude to my thesis supervisors, José Manuel Arroyo and Natalia Alguacil, for their wise guidance, high availability, good advices, and patience over all these years. José Manuel, thank you for giving me the opportunity to undertake this research work. The realization of this thesis would not have been possible without your full support, tireless dedication, and good advices. Natalia, thank you for your useful and valuable suggestions and remarks, as well as for your personal and professional support during these four years. I am also very grateful to Alexandre Street for his constant encouragement and his contributions and comments which have greatly helped to improve the quality of this thesis. Thanks to my old and new colleagues at the Universidad de Castilla-La Mancha, Gregorio, Ricardo, Victoria, Pilar, Miguel, Agustı́n, Cristina, Ana, and Pablo, for their friendship, help in many technical and personal aspects, and for all good moments we have spent together at coffee breaks and conferences. I would also like to thank the rest of members composing the Power and Energy Analysis and Research Laboratory group, Raquel, Javier, José Ignacio, and Luis, for their help and support. I would like to acknowledge the Ministry of Education of Spain for its financial support through the grant Formación del Profesorado Universitario, FPU13/03725. This Ph.D. grant allowed me to conduct research through Ayudas a la movilidad para estancias breves y traslados temporales at two outstanding foreign research centers. In this regard, my gratitude goes to Dr. Jianhui Wang for hosting me with his research group at Argonne National Laboratory, Chicago, Illinois, USA. Furthermore, I am also indebted to Prof. Goran Strbac for giving me the opportunity of joining his research group at Imperial College London, United Kingdom, and for v.

(6) allowing me to work with him on a new research project different to the thesis topic. Moreover, thanks also to all people I met in Chicago and London. In particular, I would like to acknowledge Roberto Moreira for his technical support, and Darwin, Alexandre, Julio, Jochen, and Luis, among many others labmates, for making these research stays easier and more enjoyable. I also owe a big thank you to my close friends, especially Cris, Lola, and Vicky, for their valuable friendship and personal support. I would also like to thank my old friend and virtual labmate, Marı́a, for her help and support since we started our engineering studies. I would like to give special thanks to my parents, Jesús and Filomena, and my brothers, Jesús and Ignacio, for their endless patience, and their unconditional support and love, without which I would have never come this far. Last, but not least, I would like to thank to Jose for his love and endless understanding, and above all, for always being by my side supporting me.. vi.

(7) Contents List of Figures. xiii. List of Tables. xv. Nomenclature. xix. 1 Introduction. 1. 1.1. Electricity Markets Background . . . . . . . . . . . . . . . . . . . . .. 1. 1.2. Day-Ahead Generation Scheduling . . . . . . . . . . . . . . . . . . .. 3. 1.3. Thesis Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5. 1.4. Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 8. 1.4.1. Generation Scheduling under Uncertainty . . . . . . . . . . .. 8. 1.4.2. Robust Optimization . . . . . . . . . . . . . . . . . . . . . . .. 10. 1.4.3. Robust Generation Scheduling . . . . . . . . . . . . . . . . .. 12. 1.4.3.1 1.4.3.2 1.4.4. Generation Scheduling Disregarding the Base-Case Dispatch . . . . . . . . . . . . . . . . . . . . . . . .. 12. Generation Scheduling Considering the Base-Case Dispatch . . . . . . . . . . . . . . . . . . . . . . . .. 16. Limitations of Existing Robust Generation Scheduling Models 19 1.4.4.1. Consideration of Reserve Offers. . . . . . . . . . . .. 20. 1.4.4.2. Consideration of Dispatch Nonanticipativity . . . .. 20. 1.4.4.3. Consideration of Fast-Acting Dispatchable Generating Units . . . . . . . . . . . . . . . . . . . . . . . .. 21. Consideration of Bulk Energy Storage Units . . . .. 22. 1.5. Thesis Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 24. 1.6. Overview of the Remaining Chapters . . . . . . . . . . . . . . . . . .. 26. 1.4.4.4. vii.

(8) Contents 2 Robust Optimization 2.1 Uncertainty Characterization . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Ellipsoidal Uncertainty Set . . . . . . . . . . . . . . . . . . . 2.1.2 Polyhedral Uncertainty Set . . . . . . . . . . . . . . . . . . . 2.2 Types of Robust Optimization Problems . . . . . . . . . . . . . . . . 2.2.1 Robust Optimization without Recourse . . . . . . . . . . . . 2.2.2 Robust Optimization with Recourse . . . . . . . . . . . . . . 2.2.2.1 Two-Stage Robust Optimization . . . . . . . . . . . 2.2.2.2 Multistage Robust Optimization . . . . . . . . . . . 2.3 Solution Methodologies for Two-Stage Robust Optimization . . . . . 2.3.1 Two-Stage Robust Optimization without Discrete Recourse Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1.1 Benders Decomposition . . . . . . . . . . . . . . . . 2.3.1.1.1 Subproblem . . . . . . . . . . . . . . . . . 2.3.1.1.2 Master Problem . . . . . . . . . . . . . . . 2.3.1.1.3 Algorithm . . . . . . . . . . . . . . . . . . 2.3.1.2 Column-and-Constraint Generation Algorithm . . . 2.3.1.2.1 Subproblem . . . . . . . . . . . . . . . . . 2.3.1.2.2 Master Problem . . . . . . . . . . . . . . . 2.3.1.2.3 Algorithm . . . . . . . . . . . . . . . . . . 2.3.2 Two-Stage Robust Optimization with Discrete Recourse Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2.1 Subproblem . . . . . . . . . . . . . . . . . . . . . . . 2.3.2.1.1 Single-Level Approximation of the Subproblem . . . . . . . . . . . . . . . . . . . . . . 2.3.2.1.2 Lower-Level Optimization of the Subproblem 2.3.2.2 Master Problem . . . . . . . . . . . . . . . . . . . . 2.3.2.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . .. 29 30 30 31 33 34 35 35 37 39. 3 Robust Generation Scheduling with Reserve 3.1 Introduction . . . . . . . . . . . . . . . . . . . 3.2 Conventional Formulation . . . . . . . . . . . 3.3 Two-Stage Robust Optimization Approach . 3.3.1 Two-Factor Uncertainty Set . . . . . . 3.3.1.1 Nodal Net Injections . . . . . 3.3.1.2 System Component Outages 3.3.2 Two-Stage Robust Model . . . . . . . 3.4 Solution Methodology . . . . . . . . . . . . .. 53 54 58 59 60 60 61 62 65. viii. Offers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. 39 40 40 42 43 44 44 44 45 46 47 49 50 50 51.

(9) Contents. 3.5. 3.6. 3.4.1 Subproblem . . . . . . 3.4.2 Master Problem . . . 3.4.3 Acceleration Strategies 3.4.4 Algorithm . . . . . . . Numerical Results . . . . . . 3.5.1 Illustrative Example . 3.5.2 118-Bus System . . . . Summary and Conclusions . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. 4 Robust Generation Scheduling with an Enhanced Ramping 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Modeling Ramping under Nonanticipativity . . . . . . . . . . 4.3 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 4.4 Solution Methodology . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Subproblem . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Master Problem . . . . . . . . . . . . . . . . . . . . . 4.4.3 Computational Performance Enhancement . . . . . . . 4.4.4 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Solution Validation . . . . . . . . . . . . . . . . . . . . 4.5.2 Illustrative Example . . . . . . . . . . . . . . . . . . . 4.5.3 RTS-Based Case . . . . . . . . . . . . . . . . . . . . . 4.5.4 118-Bus System . . . . . . . . . . . . . . . . . . . . . . 4.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. 66 69 71 71 73 74 75 80. Model 83 . . . . 84 . . . . 86 . . . . 89 . . . . 93 . . . . 93 . . . . 99 . . . . 100 . . . . 101 . . . . 103 . . . . 103 . . . . 105 . . . . 109 . . . . 112 . . . . 114. 5 Robust Generation Scheduling with Fast-Acting Dispatchable Generating Units 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Solution Methodology . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Subproblem . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1.1 Single-Level Approximation of the Subproblem 5.3.1.2 Lower-Level Optimization of the Subproblem . 5.3.2 Master Problem . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Solution Validation . . . . . . . . . . . . . . . . . . . . . 5.4.2 Illustrative Example . . . . . . . . . . . . . . . . . . . . ix. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. 117 118 119 126 126 128 130 131 132 134 134 136.

(10) Contents. 5.5. 5.4.3 118-Bus System . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 140. 6 Robust Generation Scheduling with Bulk Energy Storage Units 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Solution Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Subproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1.1 Single-Level Approximation of the Subproblem . . . 6.3.1.2 Lower-Level Optimization of the Subproblem . . . . 6.3.2 Master Problem . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 RTS-Based Case . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 118-Bus System . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . .. 143 144 145 152 152 154 156 157 158 160 160 163 165 167. 7 Conclusions, Contributions, and Future Work 169 7.1 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 7.1.1 Robust Generation Scheduling with Reserve Offers . . . . . . 169 7.1.2 Robust Generation Scheduling with an Enhanced Ramping Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 7.1.3 Robust Generation Scheduling with Fast-Acting Dispatchable Generating Units . . . . . . . . . . . . . . . . . . . . . . . . . 171 7.1.4 Robust Generation Scheduling with Bulk Energy Storage Units172 7.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 7.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 7.4 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 A Inter-Temporal Generation Constraints. 179. B Linearization Schemes 181 B.1 Product of Two Binary Variables . . . . . . . . . . . . . . . . . . . . 181 B.2 Product of a Binary Variable and a Continuous Variable . . . . . . . 182 C Formulation of NS and NSUS 183 C.1 Formulation of NS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 C.2 Formulation of NSUS . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 x.

(11) Contents D Approximate Model. 187. E 24-Bus Test System E.1 Data for Dispatchable Generating Units E.2 Transmission Line Data . . . . . . . . . E.3 Demand Data . . . . . . . . . . . . . . . E.4 Wind Power Generation Data . . . . . . E.5 Storage Device Data . . . . . . . . . . .. . . . . .. 191 193 197 199 201 203. . . . . .. 205 207 213 218 222 223. F 118-Bus Test System F.1 Data for Dispatchable Generating Units F.2 Transmission Line Data . . . . . . . . . F.3 Demand Data . . . . . . . . . . . . . . . F.4 Wind Power Generation Data . . . . . . F.5 Storage Device Data . . . . . . . . . . . Bibliography. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. 225. xi.

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(13) List of Figures 3.1 3.2 3.3 3.4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8. 5.1. RGS with reserve offers. Flowchart of the solution methodology . RGS with reserve offers. Three-bus system . . . . . . . . . . . . . RGS with reserve offers. 118-bus system–Evolution of the values the objective functions: (a) master problem, (b) subproblem . . . RGS with reserve offers. 118-bus system–Economic impact of M. . . . . of . . . .. Dispatch decision tree neglecting nonanticipativity . . . . . . . . . . Dispatch decision tree considering nonanticipativity . . . . . . . . . Generation levels under uncertainty, generation bounds, and maximum generation ramps . . . . . . . . . . . . . . . . . . . . . . . . . . RGS with an enhanced ramping model. Flowchart of the solution methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . RGS with an enhanced ramping model. Three-bus system . . . . . . RGS with an enhanced ramping model. RTS-based case–Number of scheduled units and net demand . . . . . . . . . . . . . . . . . . . . RGS with an enhanced ramping model. RTS-based case–Dispatch for the base case at hour 10 . . . . . . . . . . . . . . . . . . . . . . . . . RGS with an enhanced ramping model. RTS-based case–Generation bounds for unit 15: (a) NS, (b) NSUS, (c) novel model . . . . . . . .. 72 74 77 78 87 87 88 102 106 110 110 111. 5.4. Examples of worst-case inter-period transitions for slow-acting dispatchable generating units . . . . . . . . . . . . . . . . . . . . . . . . 123 Examples of worst-case inter-period transitions for fast-acting dispatchable generating units . . . . . . . . . . . . . . . . . . . . . . . . 125 RGS with fast-acting dispatchable generating units. Flowchart of the solution methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 RGS with fast-acting dispatchable generating units. Three-bus system137. 6.1. Statuses of storage operation . . . . . . . . . . . . . . . . . . . . . . 149. 5.2 5.3. xiii.

(14) List of Figures 6.2 6.3 6.4. RGS with bulk energy storage units. Flowchart of the solution methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 RGS with bulk energy storage units. Three-bus system . . . . . . . . 161 RGS with bulk energy storage units. RTS-based case–Conventional generation and forecast wind power generation . . . . . . . . . . . . 165. E.1 24-bus test system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 F.1 118-bus test system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206. xiv.

(15) List of Tables 1.1. Comparison of robust generation scheduling models disregarding the base-case dispatch . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 15. Comparison of robust generation scheduling models considering the base-case dispatch . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 19. 3.1. Novel approach versus existing robust optimization approaches . . .. 57. 3.2. RGS with reserve offers. Illustrative example–Data for dispatchable generating units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 74. RGS with reserve offers. Illustrative example–Generation and reserve levels (MW) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 75. 3.4. RGS with reserve offers. 118-bus system–Computational results . . .. 76. 3.5. RGS with reserve offers. 118-bus system–Infeasibility of the conventional model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 79. RGS with reserve offers. 118-bus system–Results from the out-ofsample assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 79. 4.1. Novel approach versus the related literature . . . . . . . . . . . . . .. 85. 4.2. RGS with an enhanced ramping model. Illustrative example–Dispatchable unit data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106. 4.3. RGS with an enhanced ramping model. Illustrative example–Demand and wind power generation data . . . . . . . . . . . . . . . . . . . . 106. 4.4. RGS with an enhanced ramping model. Illustrative example–Economic results ($) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107. 4.5. RGS with an enhanced ramping model. Illustrative example–Basecase dispatch for the instance of maximum uncertainty (MW) . . . . 107. 4.6. RGS with an enhanced ramping model. Illustrative example–Generation redispatch in response to a wind ramp event at hour 3 (MW) . . . . 108. 1.2. 3.3. 3.6. xv.

(16) List of Tables 4.7. RGS with an enhanced ramping model. Illustrative example–Results from the out-of-sample assessment for the instance of maximum uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108. 4.8. RGS with an enhanced ramping model. RTS-based case–Results from the out-of-sample assessment . . . . . . . . . . . . . . . . . . . . . . 112. 4.9. RGS with an enhanced ramping model. 118-bus system–Optimal results from the novel approach . . . . . . . . . . . . . . . . . . . . . 113. 4.10 RGS with an enhanced ramping model. 118-bus system–Computational results with a cardinality-constrained uncertainty set . . . . . . . . . 114 5.1. RGS with fast-acting dispatchable generating units. Illustrative example–Economic data for dispatchable units . . . . . . . . . . . . 137. 5.2. RGS with fast-acting dispatchable generating units. Illustrative example–Technical data for dispatchable units . . . . . . . . . . . . . 137. 5.3. RGS with fast-acting dispatchable generating units. Illustrative example–Demand and wind generation data . . . . . . . . . . . . . . 138. 5.4. RGS with fast-acting dispatchable generating units. Illustrative example–Results for M = 2 (MW) . . . . . . . . . . . . . . . . . . . 138. 5.5. RGS with fast-acting dispatchable generating units. 118-bus system– Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140. 6.1. Effect of constraints (6.24)–(6.27) . . . . . . . . . . . . . . . . . . . . 151. 6.2. RGS with bulk energy storage units. Illustrative example–Conventional generating unit data . . . . . . . . . . . . . . . . . . . . . . . . . . . 161. 6.3. RGS with bulk energy storage units. Illustrative example–Demand and wind power generation data . . . . . . . . . . . . . . . . . . . . 162. 6.4. RGS with bulk energy storage units. Illustrative example–Results (MW) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162. 6.5. RGS with bulk energy storage units. RTS-based case–Economic impact of storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164. 6.6. RGS with bulk energy storage units. 118-bus system–Results . . . . 166. E.1 24-bus system–Economic data for dispatchable generating units . . . 193 E.2 24-bus system–Technical data for dispatchable generating units used in Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 E.3 24-bus system–Technical data for dispatchable generating units used in Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 E.4 24-bus system–Initial statuses and minimum up and down times of dispatchable generating units . . . . . . . . . . . . . . . . . . . . . . 196 xvi.

(17) List of Tables E.5 E.6 E.7 E.8 E.9. 24-bus system–Transmission line data . . . . . . . . . . . . . . . . . 24-bus system–System demands used in Chapter 4 . . . . . . . . . . 24-bus system–System demands used in Chapter 6 . . . . . . . . . . 24-bus system–Nodal consumptions . . . . . . . . . . . . . . . . . . . 24-bus system–Wind farm locations and maximum wind power generation used in Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . 24-bus system–Wind farm locations and maximum wind power generation used in Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . 24-bus system–Hourly scaling factors for wind power generation . . . 24-bus system–Economic data for bulk storage devices . . . . . . . . 24-bus system–Technical data for bulk storage devices . . . . . . . .. 197 199 199 200. F.1 118-bus system–Economic data for dispatchable generating units . . F.2 118-bus system–Technical data for dispatchable generating units . . F.3 118-bus system–Initial statuses and minimum up and down times of dispatchable generating units . . . . . . . . . . . . . . . . . . . . . . F.4 118-bus system–Transmission line data . . . . . . . . . . . . . . . . . F.5 118-bus system–Nodal peak demands used in Chapter 3 . . . . . . . F.6 118-bus system–Nodal peak demands used in Chapters 4–6 . . . . . F.7 118-bus system–Hourly scaling factors for nodal demands . . . . . . F.8 118-bus system–Wind farm locations and maximum wind power generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.9 118-bus system–Hourly scaling factors for wind power generation . . F.10 118-bus system–Economic data for bulk storage devices . . . . . . . F.11 118-bus system–Technical data for bulk storage devices . . . . . . .. 207 209. E.10 E.11 E.12 E.13. xvii. 201 201 202 203 203. 211 213 218 220 221 222 222 223 223.

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(19) NOMENCLATURE Sets and Indexes B. Set of bus indexes b and b0 .. Bu. Set of indexes b of the buses with uncertain wind power generation or uncertain net power injection.. Fi. Feasibility set for the decision variables associated with dispatchable generating unit i.. I. Set of indexes i of dispatchable generating units including both slow- and fast-acting generators.. IF A. Set of indexes i of fast-acting dispatchable generating units.. I SA. Set of indexes i of slow-acting dispatchable generating units.. Ib. Set of indexes i of the dispatchable generating units located at bus b.. K. Set of indexes k and k 0 of vectors of uncertainty realizations.. L. Set of transmission line indexes l.. r. Stage index.. S. Set of storage unit indexes s.. Sb. Set of indexes s of the storage units located at bus b.. T. Set of time period indexes t.. U. Uncertainty set.. UR. Uncertainty set corresponding to the Rth stage.. U2. Uncertainty set corresponding to the second stage. xix.

(20) NOMENCLATURE. Functions P (·) Cit. Production cost function offered by dispatchable generating unit i in period t.. f (·). Vector of linear functions defining the set of credible contingencies.. fR (·). Objective function term corresponding to the Rth stage.. fs (·). Objective function of the robust optimization problem without recourse.. f1 (·). Objective function term corresponding to the first stage.. f2 (·). Objective function term corresponding to the second stage.. gR (·). Set of constrained functions associated with inequality constraints corresponding to the Rth stage.. gs (·). Set of constrained functions associated with inequality constraints in the robust optimization problem without recourse.. g1 (·). Set of constrained functions associated with inequality constraints corresponding to the first stage.. g2 (·). Set of constrained functions associated with inequality constraints corresponding to the second stage.. hR (·). Set of constrained functions associated with equality constraints corresponding to the Rth stage.. hs (·). Set of constrained functions associated with equality constraints in the robust optimization problem without recourse.. h1 (·). Set of constrained functions associated with equality constraints corresponding to the first stage.. h2 (·). Set of constrained functions associated with equality constraints corresponding to the second stage. xx.

(21) NOMENCLATURE. Vectors λ. Vector of dual variables associated with the lower-level equality constraints.. λ(j). Vector of parameters representing the value of λ at iteration j.. λ(m). Vector of parameters representing the value of λ at iteration m.. λn. Vector of dual variables associated with the lower-level equality constraints at inner-loop iteration n.. π. Vector of dual variables associated with the lower-level inequality constraints.. π (j). Vector of parameters representing the value of π at iteration m.. π (m). Vector of parameters representing the value of π at iteration m.. πn. Vector of dual variables associated with the lower-level inequality constraints at inner-loop iteration n.. ã. Vector of the right-hand-side coefficients of the lower-level inequality constraints.. b̃, d̃. Vectors of the right-hand-side coefficients of the lower-level equality constraints.. c̃, ẽ. Coefficient vectors in the lower-level objective function.. x. Vector of decision variables in the robust optimization problem without recourse.. xR. Vector of decision variables of the Rth -stage problem.. x1. Vector of first-stage decision variables.. (j). Vector of parameters representing the value of x1 at iteration j.. x1. (m). Vector of parameters representing the value of x1 at iteration m.. x2. Vector of second-stage decision variables.. xm 2. Vector of recourse decision variables at iteration m.. u. Vector of variables modeling uncertainty realizations.. x1. xxi.

(22) NOMENCLATURE u(j). Vector of parameters representing the value of u at iteration j.. u(k). Vector of parameters representing the value of u at inner-loop iteration k.. u(m). Vector of parameters representing the value of u at iteration m.. u. Vector of parameters representing the nominal or expected values of uncertain parameters.. uR. Vector of variables modeling uncertainty realizations at the Rth stage.. u2. Vector of variables modeling uncertainty realizations at the second stage.. z. Vector of binary recourse decision variables of the second-stage problem.. zm. Vector of binary recourse decision variables at iteration m.. z(k). Vector of parameters representing the value of z at inner-loop iteration k.. z(n). Vector of parameters representing the value of z at inner-loop iteration n.. Matrices Σ. Variance-covariance matrix.. Ã, B̃, C̃, F̃. Coefficient matrices in the lower-level inequality constraints.. D̃, Ẽ, G̃. Coefficient matrices in the lower-level equality constraints.. Constants β. Uncertainty budget for the ellipsoidal uncertainty set.. β bt. Bounding parameter for βbt .. β bt. n. n Bounding parameter for βbt .. γ it. Bounding parameter for γit . xxii.

(23) NOMENCLATURE dn ∆Dbt. Maximum reduction in forecast net injection at bus b in period t.. up ∆Dbt. Maximum increase in forecast net injection at bus b in period t.. dn ∆Pbt. Maximum reduction in forecast wind power generation at bus b in period t.. up ∆Pbt. Maximum increase in forecast wind power generation at bus b in period t.. w ∆Pbt. Maximum fluctuation of wind power generation at bus b in period t.. ∆up. Maximum positive deviation of decision variable up from its expected value.. ∆udn p. Maximum downward deviation of uncertainty-related decision variable up from its nominal value.. ∆uup p. Maximum upward deviation of uncertainty-related decision variable up from its nominal value.. . Convergence parameter.. i. Convergence parameter of the inner loop of the nested columnand-constraint generation algorithm.. o. Convergence parameter of the outer loop of the nested columnand-constraint generation algorithm.. ηsc. Charging efficiency rate of storage unit s.. ηsd. Discharging efficiency rate of storage unit s.. Π. User-defined parameter in the polyhedral uncertainty set representing an upper bound for the sum of the weighted values for all uncertainty-related variables.. πp. Weighting coefficient associated with uncertainty-related variable up .. τ (j). Approximation of q wc at iteration j.. Φap(k). Power imbalance level resulting from the approximate subproblem at iteration k. xxiii.

(24) NOMENCLATURE Φ(j). Power imbalance level resulting from the subproblem at iteration j.. Φt. (j). Power imbalance level in period t resulting from the subproblem at iteration j.. Φ(k). Power imbalance level resulting from the lower-level optimization of the subproblem at iteration k.. χit. Bounding parameter for χit .. ω lt. Bounding parameter for ωlt .. (j). Value of ait at iteration j.. (m). Value of ait at iteration m.. (j). Value of alt at iteration j.. (m). Value of alt at iteration m.. dn(j). Value of adn bt at iteration j.. dn(k). Value of adn bt at iteration k.. dn(m). Value of adn bt at iteration m.. up(j). Value of aup bt at iteration j.. up(k). Value of aup bt at iteration k.. abt. up(m). Value of aup bt at iteration m.. b(i). Bus where dispatchable generating unit i is located.. c Cst. Cost rate offered by storage unit s in period t for charging power level.. c,dn Cst. Cost rate offered by storage unit s in period t for down-charging reserve.. c,up Cst. Cost rate offered by storage unit s in period t for up-charging reserve.. d Cst. Cost rate offered by storage unit s in period t for discharging power level.. ait ait alt alt. abt abt abt abt abt. xxiv.

(25) NOMENCLATURE dn Cit. Cost rate offered by dispatchable generating unit i in period t for down-spinning reserve.. d,dn Cst. Cost rate offered by storage unit s in period t for down-discharging reserve.. d,up Cst. Cost rate offered by storage unit s in period t for up-discharging reserve.. Cif. Fixed-cost coefficient of the production cost of dispatchable generating unit i.. CI. Penalty cost coefficient.. ns Cit. Cost rate offered by fast-acting dispatchable generating unit i in period t for nonspinning reserve.. sd Cit. Shut-down price offered by dispatchable generating unit i in period t.. su Cit. Start-up price offered by dispatchable generating unit i in period t.. up Cit. Cost rate offered by dispatchable generating unit i in period t for up-spinning reserve.. Civ. Variable-cost coefficient of the production cost of dispatchable generating unit i.. Dbt. Forecast demand level at bus b in period t.. n Dbt. Forecast net injection at bus b in period t.. DTi. Minimum down time of dispatchable generating unit i.. es0. Initial stored energy of storage unit s in period t.. E st. Maximum limit of stored energy of storage unit s in period t.. E st. Minimum limit of stored energy of storage unit s in period t.. Fl. Power flow capacity of line l.. f r(l). Origin bus of line l.. K. Number of unavailable system components. xxv.

(26) NOMENCLATURE KG. Number of unavailable dispatchable generating units.. KL. Number of unavailable transmission lines.. LB. Lower bound for the optimal value of the objective function.. LB i. Inner-loop lower bound.. LB o. Outer-loop lower bound.. M. Uncertainty budget for the polyhedral uncertainty set.. Mt. Uncertainty budget in period t for the polyhedral uncertainty set.. m̃. Dimension of vector u.. n. Number of system components.. nDT i. Number of periods during which dispatchable generating unit i must be initially scheduled off due to its minimum down time constraint.. T nU i. Number of periods during which dispatchable generating unit i must be initially scheduled on due to its minimum up time constraint.. f nof i0. Number of periods during which dispatchable generating unit i has been scheduled off prior to the first period of the time span (end of period 0).. non i0. Number of periods during which dispatchable generating unit i has been scheduled on prior to the first period of the time span (end of period 0).. ñ. Dimension of vector z.. Pi0. Initial production of dispatchable generating unit i.. k(∗). Optimal value of pkit provided by the single-period economic dispatch.. (j). Value of pit at iteration j.. (∗). Optimal or near-optimal value of pit .. pit. pit pit. xxvi.

(27) NOMENCLATURE 0. pmax it. Upper generation bound for dispatchable generating unit i in period t.. max(j). Value of pmax at iteration j. it. max(∗). Optimal or near-optimal value of pmax . it. pit pit. 0. pmin it. Lower generation bound for dispatchable generating unit i in period t.. min(j). Value of pmin at iteration j. it. min(∗). Optimal or near-optimal value of pmin it .. pit. mp(j). Value of pmp it at iteration j.. w Pbt. Forecast wind power generation level at bus b in period t.. pit. pit. w(j). Value of pw bt at iteration j.. pbt. w(m). Value of pw bt at iteration m.. w,k Pbt. Wind power generation level at bus b in period t under the vector of uncertainty realizations k.. P it. Upper production limit of dispatchable generating unit i in period t.. P it. Lower production limit of dispatchable generating unit i in period t.. pbt. (j). pcst. Value of pcst at iteration j.. P Cs. Rated charging capacity of storage unit s.. (j). pdst. Value of pdst at iteration j.. P Ds. Rated discharging capacity of storage unit s.. q ap(k). Recourse impact resulting from the single-level approximation of the subproblem at inner-loop iteration k.. q (j). Recourse impact at iteration j.. q (k). Recourse impact at inner-loop iteration k.. q (m). Recourse impact at iteration m. xxvii.

(28) NOMENCLATURE R. Number of stages.. dn(j). dn at iteration j. Value of rit. dn(∗). dn Optimal or near-optimal value of rit .. ns(j). ns Value of rit at iteration j.. ns(∗). ns Optimal or near-optimal value of rit .. up(j). up Value of rit at iteration j.. up(∗). up Optimal or near-optimal value of rit .. rit rit rit. rit rit rit. dn. Rit. Upper bound for the down-spinning reserve contribution of dispatchable generating unit i in period t.. Rdn t. Minimum system requirement of down-spinning reserve in period t.. ns. Rit. up. Upper bound for the nonspinning reserve contribution of fastacting dispatchable generating unit i in period t.. Rit. Upper bound for the up-spinning reserve contribution of dispatchable generating unit i in period t.. Rup t. Minimum system requirement of up-spinning reserve in period t.. dn(j). Value of rcdn st at iteration j.. up(j). Value of rcup st at iteration j.. dn(j). Value of rddn st at iteration j.. rdst. up(j). Value of rdup st at iteration j.. RDi. Downward ramp rate within two consecutive periods for dispatchable generating unit i.. RU i. Upward ramp rate within two consecutive periods for dispatchable generating unit i.. SDi. Ramp limit for the shut down of dispatchable generating unit i.. SU i. Ramp limit for the start up of dispatchable generating unit i.. to(l). Destination bus of line l.. rcst rcst. rdst. xxviii.

(29) NOMENCLATURE up. Nominal value of uncertainty-related decision variable up .. UB. Upper bound for the optimal value of the objective function.. U Bi. Inner-loop upper bound.. U Bo. Outer-loop upper bound.. UTi. Minimum up time of dispatchable generating unit i.. Vi0. Initial on/off status of dispatchable generating unit i.. k(∗). k Optimal value of vit .. u(k). u at inner-loop iteration k. Value of vit. u(n). u at inner-loop iteration n. Value of vit. (j). Value of vit at iteration j.. vit. (∗). Optimal or near-optimal value of vit .. xl. Reactance of line l.. vit vit. vit. vit. dn(j). Value of xdn bt at iteration j.. dn(m). Value of xdn bt at iteration m.. up(j). Value of xup bt at iteration j.. up(m). Value of xup bt at iteration m.. u(k). u at inner-loop iteration k. Value of zst. u(n). u Value of zst at inner-loop iteration n.. xbt xbt xbt xbt zst. zst. Decision Variables τ. Approximation of q wc in the master problem.. θbt. Phase angle at bus b in period t under the normal state.. k θbt. Phase angle at bus b in period t under the vector of uncertainty realizations k.. m θbt. Phase angle at bus b in period t under the extreme identified at iteration m. xxix.

(30) NOMENCLATURE u θbt. Phase angle at bus b in period t under uncertainty.. Φ. Level of system power imbalance resulting from the second stage.. Φt. Power imbalance level in period t resulting from the second stage.. Φap. Level of system power imbalance resulting from the single-level approximation of the subproblem.. Φwc. Worst-case system power imbalance.. +k Φ−k bt , Φbt. Slack variables in power balance equations used to characterize infeasibility under the vector of uncertainty realizations k.. +u Φ−u bt , Φbt. Variables representing the power imbalance at bus b in period t under uncertainty.. +k Ψ−k bt , Ψbt. Slack variables in ramping constraints used to characterize infeasibility under the vector of uncertainty realizations k.. ait. Binary variable that is equal to 1 if dispatchable generating unit i is available under the worst-case credible contingency in period t, being 0 otherwise.. alt. Binary variable that is equal to 1 if transmission line l is available under the worst-case credible contingency in period t, being 0 otherwise.. adn bt. Binary variable that is equal to 1 if the worst-case wind power generation at bus b in period t is equal to its lower bound, being 0 otherwise.. adn–f bt. Binary variables that is equal to 1 if the worst-case wind power generation at bus b in period t is less than its forecast level and greater than its lower bound, being 0 otherwise.. afbt–up. Binary variable that is equal to 1 if the worst-case wind power generation at bus b in period t is greater than its forecast value and less than its upper bound, being 0 otherwise.. aup bt. Binary variable that is equal to 1 if the worst-case wind power generation at bus b in period t is equal to its upper bound, being 0 otherwise. xxx.

(31) NOMENCLATURE csd it. Shut-down offer cost of dispatchable generating unit i in period t.. csu it. Start-up offer cost of dispatchable generating unit i in period t.. dnbt. Net injection at bus b in period t.. est. Level of stored energy of storage unit s in period t under the normal state.. emax st. Maximum level of stored energy of storage unit s in period t.. emin st. Minimum level of stored energy of storage unit s in period t.. flt. Power flow of line l in period t under the normal state.. dn fbb 0t. Binary variable equal to the product afbt–up adn b0 t .. fltk. Power flow of line l in period t under the vector of uncertainty realizations k.. fltm. Power flow of line l in period t under the extreme identified at iteration m.. fltu. Power flow of line l in period t under uncertainty.. up fbb 0t. Binary variable equal to the product afbt–up aup b0 t .. dn gbb 0t. Binary variable equal to the product adn–f adn b0 t . bt. up gbb 0t. Binary variable equal to the product adn–f aup bt b0 t .. hbt. Variable equal to the product βbt adn bt .. hdn–f bt. Variable equal to the product βbt adn–f . bt. hnbt. n dn Variable equal to the product βbt abt .. dn kbb 0t. dn Variable equal to the product βbt gbb 0t.. up kbb 0t. up Variable equal to the product βbt gbb 0t.. dn lbb 0t. dn Variable equal to the product βbt fbb 0t.. up lbb 0t. up Variable equal to the product βbt fbb 0t.. mbt. Variable equal to the product βbt xup bt .. nbt. Variable equal to the product βbt xdn bt . xxxi.

(32) NOMENCLATURE pit. Power output of dispatchable generating unit i in period t under the normal state.. pkit. Power output of dispatchable generating unit i in period t under the vector of uncertainty realizations k.. pm it. Power output of dispatchable generating unit i in period t under the extreme identified at iteration m.. pmax it. Upper generation bound for dispatchable generating unit i in period t.. pmin it. Lower generation bound for dispatchable generating unit i in period t.. pmp it. Auxiliary variable used to model the minimum attainable production of fast-acting dispatchable generating unit i in period t.. puit. Power output of dispatchable generating unit i in period t under uncertainty.. pw bt. Wind power generation level at bus b in period t.. pcst. Level of charging power of storage unit s in period t under the normal state.. pcm st. Level of charging power of storage unit s in period t under the extreme identified at iteration m.. pcust. Level of charging power of storage unit s in period t under uncertainty.. pdst. Level of discharging power of storage unit s in period t under the normal state.. pdm st. Level of discharging power of storage unit s in period t under the extreme identified at iteration m.. pdust. Level of discharging power of storage unit s in period t under uncertainty.. q. Recourse impact in the two-stage robust optimization problem.. qit. Variable equal to the product γit ait . xxxii.

(33) NOMENCLATURE qγ. Recourse impact involving only continuous variables.. q ap. Recourse impact resulting form the single-level approximation of the subproblem.. q wc. Worst-case recourse impact in the two-stage robust optimization problem.. wc qR. Worst-case recourse impact corresponding to the Rth stage.. q2wc. Worst-case recourse impact corresponding to the second stage.. dn rit. Down-spinning reserve contribution of dispatchable generating unit i in period t.. ns rit. Nonspinning reserve contribution of fast-acting dispatchable generating unit i in period t.. up rit. Up-spinning reserve contribution of dispatchable generating unit i in period t.. rcdn st. Down-charging reserve contribution of storage unit s in period t.. rcup st. Up-charging reserve contribution of storage unit s in period t.. rddn st. Down-discharging reserve contribution of storage unit s in period t.. rdup st. Up-discharging reserve contribution of storage unit s in period t.. sit. Variable equal to the product χit ait .. tbt. Variable equal to the product βbt aup bt .. tfbt–up. Variable equal to the product βbt afbt–up .. tnbt. n up Variable equal to the product βbt abt .. up. pth entry of vector u.. vit. Binary variable that is equal to 1 if dispatchable generating unit i is scheduled on in period t, being 0 otherwise.. c vst. Binary variable that is equal to 1 if storage unit s is scheduled to charge in period t, being 0 otherwise.. d vst. Binary variable that is equal to 1 if storage unit s is scheduled to discharge in period t, being 0 otherwise. xxxiii.

(34) NOMENCLATURE k vit. Binary variable that is equal to 1 if fast-acting dispatchable generating unit i is scheduled on in period t under the vector of uncertainty realizations k, being 0 otherwise.. m vit. Binary variable that is equal to 1 if fast-acting dispatchable generating unit i is scheduled on in period t under the extreme identified at iteration m, being 0 otherwise.. sd vit. Binary variable that is equal to 0 if fast-acting dispatchable generating unit i is not shut down under any of the plausible uncertainty realizations in period t, being 1 otherwise.. su vit. Binary variable that is equal to 0 if fast-acting dispatchable generating unit i is not started up under any of the plausible uncertainty realizations in period t, being 1 otherwise.. u vit. Binary variable that is equal to 1 if fast-acting dispatchable generating unit i is scheduled on in period t under uncertainty, being 0 otherwise.. xdn bt. Binary variable that is equal to 1 if the worst-case net injection at bus b in period t is equal to its lower bound, being 0 otherwise.. xup bt. Binary variable that is equal to 1 if the worst-case net injection at bus b in period t is equal to its upper bound, being 0 otherwise.. ylt. Variable equal to the product ωlt alt .. zst. Binary variable that avoids simultaneously charging and discharging storage unit s in period t under the normal state.. m zst. Binary variable that avoids simultaneously charging and discharging storage unit s in period t under the extreme identified at iteration m.. u zst. Binary variable that avoids simultaneously charging and discharging storage unit s in period t under uncertainty.. Dual Variables αit. Dual variable associated with the constraint imposing the lower bound for the generation dispatch under uncertainty of fastacting dispatchable generating unit i in period t in terms of the base-case generation and the down-spinning reserve level. xxxiv.

(35) NOMENCLATURE n αit. Dual variable associated with the constraint imposing the lower bound for the generation dispatch under uncertainty of fastacting dispatchable generating unit i in period t in terms of the base-case generation and the down-spinning reserve level for the values of lower-level binary variables identified at inner-loop iteration n.. βbt. Dual variable associated with the power balance equation at bus b in period t used in the lower-level problem.. n βbt. Dual variable associated with the power balance equation at bus b in period t under uncertainty for the values of lower-level binary variables identified at inner-loop iteration n.. γit. Dual variable associated with the constraint imposing the lower bound for the generation dispatch under uncertainty of slowacting dispatchable generating unit i in period t in terms of the base-case generation and the down-spinning reserve level.. n γit. Dual variable associated with the constraint imposing the lower bound for the generation dispatch under uncertainty of slowacting dispatchable generating unit i in period t in terms of the base-case generation and the down-spinning reserve level for the values of lower-level binary variables identified at inner-loop iteration n.. δit. Dual variable associated with the constraint imposing the lower bound for the generation dispatch under uncertainty of fastacting dispatchable generating unit i in period t in terms of u vit .. n δit. Dual variable associated with the constraint imposing the lower bound for the generation dispatch under uncertainty of fastacting dispatchable generating unit i in period t in terms of u vit for the values of lower-level binary variables identified at inner-loop iteration n.. ζst. Dual variable associated with the constraint imposing the upper bound for the discharging power level under uncertainty of (j) up(j) storage unit s in period t in terms of pdst and rdst .. n ζst. Dual variable associated with the constraint imposing the upper bound for the discharging power level under uncertainty of xxxv.

(36) NOMENCLATURE (j). up(j). storage unit s in period t in terms of pdst and rdst for the values of lower-level binary variables identified at inner-loop iteration n. κst. Dual variable associated with the constraint imposing the lower bound for the charging power level under uncertainty of storage (j) up(j) unit s in period t in terms of pcst and rcst .. κnst. Dual variable associated with the constraint imposing the lower bound for the charging power level under uncertainty of storage (j) up(j) unit s in period t in terms of pcst and rcst for the values of lower-level binary variables identified at inner-loop iteration n.. λit. Dual variable associated with the constraint relating pit and the generation dispatch under uncertainty of fast-acting dispatchable generating unit i in period t.. λnit. Dual variable associated with the constraint relating pit and the generation dispatch under uncertainty of fast-acting dispatchable generating unit i in period t for the values of lowerlevel binary variables identified at inner-loop iteration n.. µit. Dual variable associated with the constraint imposing the upper bound for the generation dispatch under uncertainty of fastacting dispatchable generating unit i in period t in terms of the base-case generation and the up-spinning reserve level.. µnit. Dual variable associated with the constraint imposing the upper bound for the generation dispatch under uncertainty of fastacting dispatchable generating unit i in period t in terms of the base-case generation and the up-spinning reserve level for the values of lower-level binary variables identified at inner-loop iteration n.. νst. Dual variable associated with the constraint relating pdust and u zst of storage unit s in period t.. n νst. u Dual variable associated with the constraint relating pdust and zst of storage unit s in period t for the values of lower-level binary variables identified at inner-loop iteration n.. ost. Dual variable associated with the constraint imposing the lower bound for the discharging power level under uncertainty of (j) dn(j) storage unit s in period t in terms of pdst and rdst .. mp(j). mp(j). xxxvi.

(37) NOMENCLATURE onst. Dual variable associated with the constraint imposing the lower bound for the discharging power level under uncertainty of (j) dn(j) storage unit s in period t in terms of pdst and rdst for the values of lower-level binary variables identified at inner-loop iteration n.. πlt. Dual variable associated with the constraint imposing the lower bound for fltu .. n πlt. Dual variable associated with the constraint imposing the lower bound for fltu for the values of lower-level binary variables identified at inner-loop iteration n.. $st. Dual variable associated with the constraint imposing the upper bound for the charging power level under uncertainty of storage (j) dn(j) unit s in period t in terms of pcst and rcst .. n $st. Dual variable associated with the constraint imposing the upper bound for the charging power level under uncertainty of storage (j) dn(j) unit s in period t in terms of pcst and rcst for the values of lower-level binary variables identified at inner-loop iteration n.. ρit. Dual variable associated with the constraint relating the lower generation bound, pmin it , and the generation dispatch under u uncertainty, pit , of slow-acting dispatchable generating unit i in period t.. σlt. Dual variable associated with the constraint imposing the upper bound for fltu .. n σlt. Dual variable associated with the constraint imposing the upper bound for fltu for the values of lower-level binary variables identified at inner-loop iteration n.. ςit. Dual variable associated with the constraint relating the upper generation bound, pmax , and the generation dispatch under it uncertainty, puit , of slow-acting dispatchable generating unit i in period t.. τst. Dual variable associated with the constraint relating pcust and u zst of storage unit s in period t. xxxvii.

(38) NOMENCLATURE n τst. u Dual variable associated with the constraint relating pcust and zst of storage unit s in period t for the values of lower-level binary variables identified at inner-loop iteration n.. χit. Dual variable associated with the constraint imposing the upper bound for the generation dispatch under uncertainty of slowacting dispatchable generating unit i in period t in terms of the base-case generation and the up-spinning reserve level.. χnit. Dual variable associated with the constraint imposing the upper bound for the generation dispatch under uncertainty of slowacting dispatchable generating unit i in period t in terms of the base-case generation and the up-spinning reserve level for the values of lower-level binary variables identified at inner-loop iteration n.. ψit. Dual variable associated with the constraint imposing the upper bound for the generation dispatch under uncertainty of fastacting dispatchable generating unit i in period t in terms of u vit .. n ψit. Dual variable associated with the constraint imposing the upper bound for the generation dispatch under uncertainty of fastacting dispatchable generating unit i in period t in terms of u vit for the values of lower-level binary variables identified at inner-loop iteration n.. ωlt. Dual variable associated with the equation relating power flow and phase angles under uncertainty for line l in period t.. n ωlt. Dual variable associated with the equation relating power flow and phase angles under uncertainty for line l in period t for the values of lower-level binary variables identified at inner-loop iteration n.. Acronyms ac. Alternating current.. ARO. Adjustable robust optimization.. BD. Benders decomposition. xxxviii.

(39) NOMENCLATURE CCGA. Column-and-constraint generation algorithm.. CNMC. National Commission on Markets and Competition.. CVaR. Conditional value-at-risk.. dc. Direct current.. FERC. Federal Energy Regulatory Commission.. GAMS. General Algebraic Modeling System.. ISO. Independent system operator.. ISO-NE. New England ISO.. KKT. Karush-Kuhn-Tucker.. MILP. Mixed-integer linear programming.. PJM. Pennsylvania-New Jersey-Maryland.. RGS. Robust generation scheduling.. RTO. Regional transmission organization.. RTS. Reliability Test System.. TSO. Transmission system operator.. U.S.. United States.. xxxix.

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(41) Chapter 1. Introduction The main purpose of this dissertation is to provide practical generation scheduling tools based on robust optimization to facilitate the integration of intermittent and nondispatchable renewable-based generation sources in the operation of restructured power systems. Section 1.1 provides a brief introduction to electricity markets, describing their basic concepts and main features. Section 1.2 focuses on day-ahead generation scheduling. Section 1.3 presents the thesis motivation. The state of the art is described in Section 1.4. Section 1.5 states the main thesis objectives. Finally, a summary of the remaining chapters is outlined in Section 1.6.. 1.1. Electricity Markets Background. The restructuring of the electrical power industry at the end of the last century led to the emergence of electricity markets across the world [34, 44, 82, 117, 119, 122]. Prior to this restructuring process, the power sector was traditionally organized as a natural monopoly wherein a conglomerate of vertically-integrated and state-owned companies was responsible for all activities associated with electrical power, including generation, transmission, distribution, and supply of electricity [138]. In other words, power systems were generally managed in a centralized manner. In contrast, current electricity markets feature a decentralized structure, wherein most of the activities are primarily carried out by private entities within a competitive environment. Generally, production and retail trading constitute competitive activities, whereas transmission and distribution are owned and operated as natural monopolies [34, 44, 82, 117, 119]. It is worth mentioning that the design of electricity markets varies widely across 1.

(42) 1. Introduction. countries and regions. However, some common aspects can be found, some of which will be discussed below. Most electricity markets are organized in wholesale and retail markets whereby electricity is traded among different market participants, namely producers, consumers, and retailers. Participants in wholesale markets include producers, retailers, and large consumers, whereas participants in the retail markets are retailers and consumers. These market participants can act directly in such markets as buyers or sellers of electricity. In particular, producers sell the electricity produced by their corresponding generating units with the aim of maximizing their own profits. Analogously, consumers purchase electricity with the aim of maximizing their profits. Finally, retailers are intermediary agents between producers and consumers purchasing electricity from the wholesale market and selling it to end users or customers at retail markets. Therefore, the aim of retailers is to maximize their profits obtained from selling electricity to consumers. Moreover, in order to guarantee a proper functioning of both wholesale and retail markets, there exist different entities in charge of their regulation and management. Such market entities include the market regulator, the market operator, the independent system operator (ISO), and the transmission system operator (TSO): • The market regulator is a nonprofit entity responsible for overseeing the proper functioning and the level of competition in the wholesale and retail markets, as well as the operation of the power system. For example, the regulator in the U.S. is the Federal Energy Regulatory Commission (FERC), while in Spain is the National Commission on Markets and Competition (CNMC in Spanish). • The market operator is a nonprofit entity in charge of the economic management of the electricity market. In particular, its main function is to clear the wholesale market and provide the prices and the traded quantities that maximize the overall social welfare. • The independent system operator is a nonprofit entity in charge of the technical management of the electricity market. In particular, its main role is to ensure at all time the security and continuity of balance between supply from producers and demand from consumers. • The transmission system operator (regional transmission organizations, RTOs, in the U.S.) is the organization in charge of the operation, maintenance, and planning of the transmission network. In some electricity markets, such as the Pennsylvania-New Jersey-Maryland (PJM) [129] and New England [128] systems, the market operator and the system operator 2.

(43) 1.2. Day-Ahead Generation Scheduling. are merged in a single entity. Conversely, in the case, for example, of the Iberian Electricity Market [104] and the Italian Electricity Market [65], both market agents operate separately and have their own functions. Furthermore, in some countries such as Spain, Italy, and New Zealand, the TSO and the ISO are usually the same entity [112, 127, 130]. In this dissertation, the market operator and the system operator are assumed to be the same entity, which will be referred to as the system operator. Thus, the system operator is responsible for both the economic and technical management of the electricity market. In wholesale electricity markets, the trading of electricity commodities such as energy and various ancillary services (e.g., frequency control, reserves, and voltage support) takes place in different organized markets which are sequentially cleared. The most important markets are the day-ahead, real-time, and futures markets. The day-ahead and real-time markets comprise short-time transactions, and both constitute what is known as the pool. The day-ahead market represents the backbone of the electricity trading [43]. Such a market allows the trading of electricity commodities one day ahead on an hourly or half-hourly basis. The real-time market, also called balancing market, constitutes the last market prior to the actual energy delivery. This market is used to make adjustments or manage deviations in order to ensure balance between supply and demand in real time. To that end, the real-time market is generally cleared 10–20 minutes prior to the actual energy delivery hour. On the contrary, the futures market involves mid-term and long-term transactions spanning from one year to several years. This market allows the trading of a certain amount of electricity commodities through forward contracts during a specific period at a fixed price. Finally, electricity commodities can also be traded outside an organized market through bilateral contracts, which are private trading agreements between different market participants. This dissertation focuses on the generation scheduling of day-ahead electricity markets. This practical problem is explained in detail in the next section.. 1.2. Day-Ahead Generation Scheduling. In day-ahead electricity markets, producers and consumers respectively submit offers and bids for the next 24 hours in the form of pairs quantity-price. Then, the system operator collects such offers and bids and clears the market through a market-clearing mechanism or auction model, which determines, for each time period, (i) the market-clearing price, (ii) the generation schedule, and (iii) the power production and consumption levels, while maximizing social welfare. 3.

(44) 1. Introduction. Generation offers submitted by producers can be either simple or complex. Simple offers comprise quantities and their corresponding prices. Complex offers contain the same information as simple offers with the addition of technical information, which is considered by the market-clearing mechanism in order to obtain a feasible generation schedule. This technical information may include, among others: (i) production limits, (ii) inter-temporal operational constraints such as ramp rates and minimum up and down times, and (iii) start-up and shut-down costs. Note that, in order to properly take into account time-related aspects, a multiperiod market-clearing procedure is required. Furthermore, the market-clearing mechanism may also embody a network representation as is the case in some electricity markets operating in the U.S., such as the PJM and New England markets. It should be noted that, when physical and operational aspects of the power system are neglected in the market-clearing mechanism, ex-post adjustments in the generation schedule and market-clearing prices are required, which may result in suboptimal social welfare. Moreover, during the last years, security has become a key issue for short-term system operation. The system operator must guarantee a perfect balance between demand and supply not only during normal operating conditions but also in case of unforeseen disturbances, such as the loss of system components or sudden deviations from the forecast demand levels. In this regard, the procurement of several types of reserve by dispatchable generating units constitutes an essential instrument to ensure the real-time implementation of the outcome of the day-ahead market in the event of an unforeseen disturbance. As such reserves must be available and ready to be deployed in real time, their procurement is scheduled in advance by the system operator. In this context, in most electricity markets, reserves are considered as tradable commodities in a similar way to energy. There exist mainly two current market implementations for the trading of reserves [45]. The first one relies on a sequential market-clearing procedure wherein various types of reserve are scheduled once the day-ahead energy schedule has been determined. This is the current practice in most countries in Europe [56], such as Spain and Portugal [103]. Such a procedure is simple to implement but it may lead to loss of social welfare and even infeasible solutions. On the contrary, in several electricity markets, such as those operating in the U.S. [35, 58, 59] and Greece [113], reserves are jointly scheduled with energy in the day-ahead market, thereby leveraging their strong interrelation. It has been widely demonstrated that the implementation of such a simultaneous or co-optimized market-clearing procedure results in a higher level of social welfare as compared with a sequential procedure. 4.

(45) 1.3. Thesis Motivation. In short, bearing in mind the above considerations, a desirable day-ahead marketclearing mechanism should accounted for: (i) the technical features of generators in a multiperiod setting, (ii) the network constraints, and (iii) the joint scheduling of energy and reserves. There are many electricity markets, such as those in the U.S., where the system operator clears the day-ahead market through a procedure very similar to the unit commitment problem traditionally solved in centralized power systems to determine the optimal generation schedule [68, 138]. The main difference between both generation scheduling procedures stems from the fact that, rather than minimizing the total operating cost, the market-clearing mechanism used in current electricity markets maximizes the declared social welfare based on the offers and bids submitted by market participants. Within a competitive framework and assuming an inelastic demand, the objective of the unit-commitment-based market-clearing procedure becomes the minimization of the total operating cost based on generation offers. Moreover, for every time period, the unit commitment solution provides the generation schedule and dispatch that meet (i) the forecast system demand, (ii) the technical information submitted by producers, and (iii) the system security requirements associated with the transmission network and the provision of several types of reserve. Thus, in the restructured power industry, the unit commitment problem also plays a crucial role in the generation scheduling of current day-ahead electricity markets and continues being a key decision-making component in the operation of power systems. Within this context, this dissertation focuses on the development of novel approaches based on two-stage adaptive robust optimization for unit-commitmentbased generation scheduling in day-ahead electricity markets.. 1.3. Thesis Motivation. Over the last years, power systems have experienced a steady increase in the use of intermittent renewable-based power generation technologies, in particular of wind power generators. A relevant example is the case of Spain, where the installed wind capacity grew from 16133 MW in 2008 to 23135 MW in 2017 [112], which represents an increase of 42.9%. Besides, it is expected that this figure continues growing since, according to the Renewable Energy Plan 2011–2020, the total installed wind capacity should amount to 35000 MW by 2020 [115]. The incorporation of wind-based generation resources in power systems brings well-known benefits including the reduction of greenhouse gas emissions and fuel costs. However, it poses serious technical and economic challenges to short-term 5.

(46) 1. Introduction. power system operation. In particular, wind power generation is considered as nondispatchable since it cannot be fully scheduled by system operators because of the intermittent and volatile nature of wind. Moreover, as opposed to demand, wind power generation cannot be forecasted with high accuracy within the time frames associated with the day-ahead generation scheduling process. As a result, wind power generation constitutes a significant source of uncertainty in the generation scheduling problem. Uncertainty is handled by system operators through the procurement of several types of reserve by dispatchable generating units. Consequently, with the growing penetration levels of wind-based generation, increased reserve levels are needed to cover possible sudden and unexpected wind power fluctuations, which entail additional operating costs. Hence, in order to guarantee not only system reliability but also economic efficiency, the optimal scheduling of reserves traditionally driven by the loss of system components should properly account for the intermittency and uncertainty of wind power generation. Current industry practice for energy and reserve scheduling in day-ahead cooptimized electricity markets relies on deterministic unit commitment models based on forecast values of nodal net power injections and pre-specified system reserve requirements [34, 59, 101, 117]. Note that nodal net power injection refers to the nodal demand supplied by conventional dispatchable generation. However, in the new context of increasing uncertainty, the ex-ante selection by the system operator of appropriate values for system reserve requirements becomes a critical issue since the required experience and engineering judgment may not be available. Moreover, the effect of the transmission network on the deployment of reserves is typically disregarded. As a consequence, the reserve schedule obtained from such models may be either insufficient or mislocated across the network, thereby leading to reserve undeliverability. Therefore, new effective methodologies for energy and reserve scheduling are required. In response to this requirement, the unit commitment problem under uncertainty has been extensively examined in the technical literature [44, 126, 152]. Among the recent relevant approaches, the focus has been mainly placed on robust optimization [125] due to its practical tradeoff between accuracy and tractability. However, available robust models neglect important aspects that make them unsuitable to optimally determine reserve levels in current unitcommitment-based day-ahead co-optimized electricity markets. Consequently, new robust methodologies for unit-commitment-based energy and reserve scheduling are yet to be developed. In addition, most existing robust optimization models for multiperiod generation scheduling in both energy-only electricity markets and co-optimized electricity markets for energy and reserves are instances of two-stage adaptive robust optimization. 6.

(47) 1.3. Thesis Motivation. Thus, the required nonanticipativity of dispatch decisions is not accounted for. In other words, due to the adoption of a two-stage setting, generation dispatch decisions in such models rely on the knowledge of future uncertain events. As a consequence, insufficient ramping capability may arise if the uncertain events do not materialize as anticipated. The significance of this information-related inconsistency is particularly stressed in the current context where recent issues involving wind power ramp events [101], i.e., sudden and unexpected wind power fluctuations, reveal the need for multistage generation scheduling approaches accounting for dispatch nonanticipativity. In order to manage the intermittency and volatility featured by wind-based generation, apart from increased levels of reserve, more flexible operational practices are required to allow the system operator to rapidly cater for sudden and unexpected deviations from forecast wind power generation levels. In this context, fast-acting dispatchable generating units, such as gas turbines and combined-cycle units, are becoming essential in some power systems with high levels of wind power generation. Fast-acting units feature high flexibility due to their short start-up and shut-down times, but they are generally more expensive to operate as compared with slow-acting dispatchable generating units such as coal-fired and nuclear power plants. As a consequence, such flexible resources have been traditionally used as occasional emergency or peaker units. Nevertheless, with the growing presence of intermittent and nondispatchable wind-based generation in current power systems, these generating units are increasingly called upon by system operators to guarantee real-time power balance. Moreover, as levels of wind power generation increase, so does the need for greater flexibility and reserve levels. Thus, in future decarbonized power systems, flexibility and reserves provided by conventional generating units might not be sufficient, thereby requiring the incorporation of more flexible generation resources. In this regard, bulk energy storage is widely deemed as one of the most promising flexible technologies. As stated in [47, 51, 111], the incorporation of energy storage devices can bring a wide range of potential economic and operational benefits to electricity markets that might facilitate the large-scale integration of wind power generation in a cost-efficient way. More specifically, because of their high flexibility and their short response time, these devices have the potential to (i) help system operators tackle sudden production and demand variations through the provision of charging and discharging reserves, (ii) defer investment on transmission network infrastructure by supporting network congestion management, and (iii) increase efficiency of system operation by improving the utilization of existing conventional generation assets. In addition, these devices can exploit exiting arbitrage opportunities in electricity 7.

(48) 1. Introduction. markets by charging during off-peak demand hours (i.e., at lower prices), and discharging during peak demand hours (i.e., at higher prices), which results in a smoothing effect for peak demand and, thus, in a potential increase in social welfare. However, in spite of the important role that fast-acting generating units and storage devices may play in future power systems, how to integrate their operation in the day-ahead generation scheduling problem under a robust optimization framework is still challenging. The above considerations constitute the main motivation of the work presented in this document. In particular, this dissertation tackles the following yet unresolved issues in two-stage robust generation scheduling: • The precise modeling of reserve offers in day-ahead co-optimized energy and reserve markets. • The nonanticipativity of generation dispatch decisions. • The operation of fast-acting dispatchable generating units in day-ahead cooptimized energy and reserve markets. • The operation of bulk energy storage units in day-ahead co-optimized energy and reserve markets.. 1.4. Literature Review. This section provides a review of the technical literature related to this thesis. First, relevant references addressing generation scheduling under uncertainty are described. Subsequently, fundamental references on the modeling framework adopted in this work, namely robust optimization, are presented. Then, references addressing generation scheduling based on robust optimization are summarized and categorized. Finally, some relevant practical limitations of the previously reported references on robust generation scheduling are discussed in detail.. 1.4.1. Generation Scheduling under Uncertainty. Generation scheduling is facing unprecedented levels of uncertainty due to the growing presence of intermittent renewable-based generation, in particular of wind energy. Under such increased uncertainty, power system operation is seriously challenged and effective generation scheduling methodologies are required. Over the last years, several optimization frameworks have been developed to deal with uncertainty in unit-commitment-based generation scheduling, being deterministic 8.

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