phd thesis

UNIVERSIDAD DE CASTILLA-LA MANCHA

DEPARTAMENTO DE INGENIERIA ELECTRICA,

ELECTRONICA, AUTOMATICA Y COMUNICACIONES

STRATEGIC GENERATION INVESTMENT

AND EQUILIBRIA IN OLIGOPOLISTIC

ELECTRICITY MARKETS

TESIS DOCTORAL

AUTHOR: S. JALAL KAZEMPOUR

DIRECTOR: ANTONIO J. CONEJO NAVARRO

Ciudad Real, Mayo de 2013

UNIVERSIDAD DE CASTILLA-LA MANCHA

DEPARTMENT OF ELECTRICAL ENGINEERING

STRATEGIC GENERATION INVESTMENT

AND EQUILIBRIA IN OLIGOPOLISTIC

ELECTRICITY MARKETS

PhD THESIS

AUTHOR: S. JALAL KAZEMPOUR

SUPERVISOR: ANTONIO J. CONEJO NAVARRO

Ciudad Real, May 2013

To my parents, brothers and sister

Acknowledgements

I am deeply grateful to Prof. Antonio J. Conejo for his expert guidance, wise

advise, support and help over the last four years. Working with him has been

an outstanding, fruitful and enjoyable experience in all aspects of academic

life.

I would like to thank my friend, Dr. Carlos Ruiz, for his efficient advice

regarding technical and mathematical aspects of this work.

I am truly grateful to the Universidad de Castilla - La Mancha for providing

an outstanding research environment.

In addition, I wish to thank the Junta de Comunidades de Castilla - La

Mancha for its partial financial support through project PCI-08-0102. Addi-

tionally, I am grateful to the Ministry of Economy and Competitiveness of

Spain for its partial financial support through CICYT project DPI2009-09573.

Many thanks to Prof. Hamidreza Zareipour for providing me an outstand-

ing research atmosphere and financial support during six months (February 1st

2012-July 31st 2012) when I visited him at the University of Calgary, Calgary,

AB, Canada. His comments were very helpful and efficient.

I also appreciate the technical support and advice I gained from the research

group of Prof. Lennart Soder at the Royal Institute of Technology (KTH),

Stockholm, Sweden, from April 15th 2011 to July 31st 2011.

Prof. Afzal Siddiqui from the University College London, U.K., and Prof.

Maria Teresa Vespucci from the Universita degli studi di Bergamo, Italy, read

the last version of this report and made valuable and pertinent observations.

Thanks to them.

Special thanks to Dr. Kazem Zare for his help and advice, which facilitated

the start of my PhD.

vii

viii

Thanks to my friends in Ciudad Real, Carlos, Luis, David, Edu, Salva,

Juanmi, Alberto, Rafa, Ali, Marco, Morteza, Ricardo and many others, for

their help in many practical aspects and for their friendship. I would also like

to thank them for their help when I arrived in Spain.

Thanks to my friends in Madrid, Stockholm and Calgary, Mehdi, Behnam,

Amin, Behnaz, Hamid, Ebrahim, Ali, Eissa and many others. I will never

forget our travel adventures.

Thanks finally, from the bottom of my heart, to my family for their uncon-

ditional love, support, and everything.

Ciudad Real, Spain

May 2013

Contents

Contents ix

List of Figures xvii

List of Tables xxi

Notation xxiii

1 Generation Investment: Introduction 1

1.1 Electricity Markets . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Chapter Organization . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Thesis Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.5.1 Literature Review: Generation Investment for a Strate-

gic Producer . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.5.2 Literature Review: Generation Investment Equilibria . . 10

1.5.3 Other Related Works in the Literature . . . . . . . . . . 14

1.6 Modeling Assumptions . . . . . . . . . . . . . . . . . . . . . . . 16

1.6.1 Investment Study Approach . . . . . . . . . . . . . . . . 16

1.6.2 Load-Duration Curve . . . . . . . . . . . . . . . . . . . . 18

1.6.3 Network Representation . . . . . . . . . . . . . . . . . . 20

1.6.4 Market Competition Modeling . . . . . . . . . . . . . . . 21

1.6.5 Additional Modeling Assumptions . . . . . . . . . . . . . 22

1.7 Solution Approach . . . . . . . . . . . . . . . . . . . . . . . . . 25

ix

x CONTENTS

1.8 Thesis Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . 31

1.9 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . 33

2 Strategic Generation Investment 37

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.2 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.2.1 Modeling Assumptions . . . . . . . . . . . . . . . . . . . 38

2.2.2 Structure of the Proposed Model . . . . . . . . . . . . . 40

2.3 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.3.1 Notational Assumptions . . . . . . . . . . . . . . . . . . 42

2.3.2 Bilevel Model . . . . . . . . . . . . . . . . . . . . . . . . 42

2.3.3 Optimality Conditions Associated with the Lower-Level

Problems (2.2) . . . . . . . . . . . . . . . . . . . . . . . 45

2.3.3.1 KKT Conditions Associated with the Lower-

Level Problems (2.2) . . . . . . . . . . . . . . . 46

2.3.3.2 Strong Duality Equality Associated with the

Lower-Level Problems (2.2) . . . . . . . . . . . 50

2.3.4 MPEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.3.5 MPEC Linearization . . . . . . . . . . . . . . . . . . . . 52

2.3.5.1 Linearizing Ztw . . . . . . . . . . . . . . . . . . 53

2.3.6 MILP Formulation . . . . . . . . . . . . . . . . . . . . . 55

2.4 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . 60

2.4.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.4.2 Deterministic Solution . . . . . . . . . . . . . . . . . . . 64

2.4.2.1 Uncongested and Congested Network . . . . . . 64

2.4.2.2 Strategic and Non-Strategic Offering . . . . . . 65

2.4.3 Stochastic Solution . . . . . . . . . . . . . . . . . . . . . 67

2.5 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

2.6 Computational Considerations . . . . . . . . . . . . . . . . . . . 72

2.7 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . 72

3 Strategic Generation Investment: Tackling Computational Bur-

den via Benders’ Decomposition 75

CONTENTS xi

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.2 Benders’ Approach . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.2.1 Complicating Variables . . . . . . . . . . . . . . . . . . . 76

3.2.2 Proposed Algorithm . . . . . . . . . . . . . . . . . . . . 77

3.3 Convexity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.3.1 Illustrative Example for Convexity Analysis . . . . . . . 79

3.3.1.1 Data . . . . . . . . . . . . . . . . . . . . . . . . 79

3.3.1.2 Cases Considered . . . . . . . . . . . . . . . . . 81

3.3.1.3 Convexity Analysis . . . . . . . . . . . . . . . . 82

3.4 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.4.1 Decomposed Problems . . . . . . . . . . . . . . . . . . . 84

3.4.2 MPEC Associated with the Decomposed Problems . . . 87

3.4.3 Auxiliary Problems . . . . . . . . . . . . . . . . . . . . . 88

3.4.4 Subproblems . . . . . . . . . . . . . . . . . . . . . . . . 91

3.4.5 Master Problem . . . . . . . . . . . . . . . . . . . . . . . 95

3.5 Benders’ Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 96

3.6 Case Study of Section 2.5 . . . . . . . . . . . . . . . . . . . . . 97

3.7 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

3.7.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

3.7.2 Investment Results . . . . . . . . . . . . . . . . . . . . . 101



4 Strategic Generation Investment Considering the Futures Mar-

ket and the Pool 107

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.2 Base and Peak Demand Blocks . . . . . . . . . . . . . . . . . . 108

4.3 Futures Market Auctions . . . . . . . . . . . . . . . . . . . . . . 109

4.4 Uncertainty Modeling . . . . . . . . . . . . . . . . . . . . . . . . 110

4.5 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.5.1 Hierarchical Structure . . . . . . . . . . . . . . . . . . . 111

4.5.2 Modeling Assumptions . . . . . . . . . . . . . . . . . . . 113

4.6 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

xii CONTENTS

4.6.1 Notational Assumptions . . . . . . . . . . . . . . . . . . 115

4.6.2 Bilevel Model . . . . . . . . . . . . . . . . . . . . . . . . 115

4.6.2.1 Futures Base Auction Clearing: Lower-level Prob-

lem (4.2) . . . . . . . . . . . . . . . . . . . . . 118

4.6.2.2 Futures Peak Auction Clearing: Lower-level Prob-

lem (4.3) . . . . . . . . . . . . . . . . . . . . . 120

4.6.2.3 Pool Clearing: Lower-level Problems (4.4) . . . 121

4.6.3 Factor Ω . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

4.6.4 Optimality Conditions Associated with the Lower-Level

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 124


Level Problem (4.2) . . . . . . . . . . . . . . . 125


Lower-Level Problem (4.2) . . . . . . . . . . . . 127


Level Problem (4.3) . . . . . . . . . . . . . . . 129




Level Problems (4.4) . . . . . . . . . . . . . . . 134

4.6.4.6 Strong Duality Equality Associated with each


4.6.5 MPEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

4.6.6 MPEC Linearization . . . . . . . . . . . . . . . . . . . . 140

4.6.6.1 Exact Linearization of Λ . . . . . . . . . . . . . 140

4.6.6.2 Exact Linearization of Λ . . . . . . . . . . . . . 143

4.6.6.3 Approximate Linearization of Λtw . . . . . . . . 143

4.6.7 MILP Formulation . . . . . . . . . . . . . . . . . . . . . 150

4.7 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

4.7.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

4.7.2 Cases Considered . . . . . . . . . . . . . . . . . . . . . . 169

4.7.3 Investment Results . . . . . . . . . . . . . . . . . . . . . 171

4.7.3.1 Only Pool . . . . . . . . . . . . . . . . . . . . . 173

CONTENTS xiii

4.7.3.2 Pool and Futures Base Auction Without Arbi-

trage . . . . . . . . . . . . . . . . . . . . . . . . 173

4.7.3.3 Pool and Futures Base Auction With Arbitrage 178

4.7.3.4 Pool, Futures Base and Futures Peak Auctions 180

4.7.3.5 Impact of Factor Υ on Generation Investment

Decisions . . . . . . . . . . . . . . . . . . . . . 181



5 Generation Investment Equilibria 185

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

5.2 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

5.3 Modeling Assumptions . . . . . . . . . . . . . . . . . . . . . . . 188

5.4 Single-Producer Problem . . . . . . . . . . . . . . . . . . . . . . 190

5.4.1 Structure of the Hierarchical (bilevel) Model . . . . . . . 190

5.4.2 Formulation of the Bilevel Model . . . . . . . . . . . . . 192

5.4.3 Optimality Conditions of the Lower-Level Problems . . . 195

5.4.3.1 KKT Optimality Conditions Associated with

Lower-level Problems (5.1g)-(5.1n) . . . . . . . 196

5.4.3.2 Optimality Conditions Associated with Lower-

level Problems (5.1g)-(5.1n) Resulting from the

Primal-Dual Transformation . . . . . . . . . . . 199

5.4.4 MPEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

5.5 Multiple Producers Problem . . . . . . . . . . . . . . . . . . . . 205

5.5.1 EPEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

5.5.2 Optimality Conditions associated with the EPEC . . . . 206

5.5.2.1 Primal Equality Constraints . . . . . . . . . . . 206

5.5.2.2 Equality Constraints Obtained From Differen-

tiating the Corresponding Lagrangian with Re-

spect to the Variables in ΞUL . . . . . . . . . . 208

5.5.2.3 Complementarity Conditions . . . . . . . . . . 211

5.5.3 EPEC Linearization . . . . . . . . . . . . . . . . . . . . 214

5.5.3.1 Linearizing the Strong Duality Equalities (5.8g) 214

xiv CONTENTS

5.5.3.2 Linearizing the Non-linear Terms Involving φSDyt 215

5.6 MILP Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 215

5.7 Searching For Investment Equilibria . . . . . . . . . . . . . . . . 230

5.7.1 Objective Function (5.29a): Total Profit . . . . . . . . . 231

5.7.2 Objective Function (5.29a): Annual True Social Welfare 232

5.8 Algorithm for the Diagonalization Checking . . . . . . . . . . . 232

5.9 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . 233

5.9.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

5.9.2 Cases Considered . . . . . . . . . . . . . . . . . . . . . . 237

5.9.3 Demand Bid and Stepwise Supply Offer Curves . . . . . 238

5.9.4 General Equilibrium Results . . . . . . . . . . . . . . . . 240

5.9.5 Triopoly Cases Maximizing Total Profit . . . . . . . . . . 245

5.9.6 Triopoly Cases Maximizing Annual True Social Welfare . 246

5.9.7 Monopoly Cases . . . . . . . . . . . . . . . . . . . . . . . 247

5.9.8 Investment Results for Each Producer . . . . . . . . . . 248

5.9.9 Diagonalization Checking . . . . . . . . . . . . . . . . . 250

5.9.10 Impact of Factor Υ on Generation Investment Equilibria 252

5.9.11 Impact of the Available Budget on Generation Invest-

ment Equilibria . . . . . . . . . . . . . . . . . . . . . . . 253

5.10 Case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

5.10.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

5.10.2 Results of Generation Investment Equilibria . . . . . . . 258


5.11.1 Computational Conclusions . . . . . . . . . . . . . . . . 261

5.11.2 Selection of values for φSDyt . . . . . . . . . . . . . . . . . 262

5.11.3 Suggestions to Reduce the Computational Burden . . . . 262

5.12 Ex-post Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 263


5.13.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 263

5.13.2 General Conclusions . . . . . . . . . . . . . . . . . . . . 265

5.13.3 Regulatory Conclusions . . . . . . . . . . . . . . . . . . . 267

CONTENTS xv

6 Summary, Conclusions, Contributions and Future Research 269

6.1 Thesis Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 269

6.1.1 Strategic Producer Investment . . . . . . . . . . . . . . . 270

6.1.2 Investment Equilibria . . . . . . . . . . . . . . . . . . . . 272

6.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

6.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

6.4 Suggestions for Future Research . . . . . . . . . . . . . . . . . . 280

A IEEE Reliability Test System: Transmission Data 283

B Mathematical Background 287

B.1 Bilevel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

B.2 MPEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

B.2.1 MPEC Obtained from the KKT Conditions . . . . . . . 290

B.2.2 MPEC Obtained from the Primal-Dual Transformation . 293

B.2.2.1 Linear Form of the Lower-Level Problems (B.2) 293

B.2.2.2 Dual Optimization Problems Pertaining to Lower-

Level Problems (B.4) . . . . . . . . . . . . . . . 295

B.2.2.3 Optimality Conditions Associated with Lower-

Level Problems (B.4) Resulting from the Primal-

Dual Transformation . . . . . . . . . . . . . . . 296

B.2.2.4 Resulting MPEC from the Primal-Dual Trans-

formation . . . . . . . . . . . . . . . . . . . . . 298

B.2.3 Equivalence Between the MPECs Obtained from the KKT

Conditions and the Primal-Dual Transformation . . . . . 300

B.3 EPEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

B.4 Benders’ Decomposition . . . . . . . . . . . . . . . . . . . . . . 307

B.5 Linearization Techniques . . . . . . . . . . . . . . . . . . . . . . 311

B.5.1 Complementarity Linearization . . . . . . . . . . . . . . 312

B.5.2 Binary Expansion Approach . . . . . . . . . . . . . . . . 313

Bibliography 317

Index 332

List of Figures

1.1 Introduction: Piecewise approximation of the load-duration curve

for the target year. . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.2 Introduction: Demand blocks and demand-bid blocks (included

in demand block t = t1) corresponding to the example given in

Table 1.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.3 Introduction: General hierarchical (bilevel) structure of any sin-

gle producer model. . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.4 Introduction: Interrelation between the upper-level and lower-

level problems considering the futures market and the pool auc-

tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.5 Introduction: Transformation of the bilevel model of a strategic

producer into its corresponding MPEC. . . . . . . . . . . . . . . 30

1.6 Introduction: EPEC and its optimality conditions. . . . . . . . . 31

2.1 Direct solution: Bilevel structure of the proposed strategic gen-

eration investment model. . . . . . . . . . . . . . . . . . . . . . 40

2.2 Direct solution: Interrelation between the upper-level and lower-

level problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.3 Direct solution: Six-bus test system (illustrative example). . . . 60

2.4 Direct solution: Locational marginal prices in (a) Cases A and

B, and (b) Case C. . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.1 Benders’ approach: Flowchart of the proposed Benders’ algorithm. 78

xvii

xviii LIST OF FIGURES

3.2 Benders’ approach: Minus-producer’s profit as a function of

capacity investment considering (a) non-strategic offering and

one scenario, (b) strategic offering and one scenario, (c) non-

strategic offering and all scenarios, and (d) strategic offering

and all scenarios. . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.3 Benders’ approach: Evolution of the expected profit, the profit

standard deviation and the CPU time with the number of sce-

narios (case study). . . . . . . . . . . . . . . . . . . . . . . . . . 102

3.4 Benders’ approach: Evolution of Benders’ algorithm in case

study involving 60 scenarios. . . . . . . . . . . . . . . . . . . . . 103

4.1 Futures market and pool: Piecewise approximation of the load-

duration curve for the target year, including peak and base de-

mand blocks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

4.2 Futures market and pool: Demand blocks supplied through dif-

ferent markets, i.e., futures base auction, futures peak auction

and pool. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.3 Futures market and pool: Hierarchical structure of the proposed

strategic generation investment model. . . . . . . . . . . . . . . 112

4.4 Futures market and pool: Maximum load level of a given de-

mand supplied through the futures base auction, futures the

peak auction and the pool. . . . . . . . . . . . . . . . . . . . . . 164

4.5 Futures market and pool: Market outcomes as a function of

factor Γ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

4.6 Futures market and pool: Expected profit of the strategic pro-

ducer as a function of Γ and 1∆. . . . . . . . . . . . . . . . . . . 179

4.7 Futures market and pool: Strategic producer’s expected profit

and its total investment as a function of factor Υ (Case 3). . . . 181

5.1 EPEC problem: Hierarchical (bilevel) structure of the model

solved by each strategic producer. . . . . . . . . . . . . . . . . . 191

5.2 EPEC problem: Transformation of the bilevel model of a strate-

gic producer into its corresponding MPEC (primal-dual trans-

formation). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

LIST OF FIGURES xix

5.3 EPEC problem: Network of the illustrative example. . . . . . . 234

5.4 EPEC problem: Piecewise approximation of the load-duration

curve for the target year (illustrative example). . . . . . . . . . 234

5.5 EPEC problem: Demand bid curve and stepwise supply offer

curve corresponding to the first demand block t = t1 (Case 1

maximizing total profit). . . . . . . . . . . . . . . . . . . . . . . 239

5.6 EPEC problem: Total newly built capacity, total profit and

annual true social welfare as a function of factor Υ (Case 1

maximizing total profit). . . . . . . . . . . . . . . . . . . . . . . 251

5.7 EPEC problem: Total newly built capacity, total profit and an-

nual true social welfare as a function of the available investment

budget (Case 1 maximizing annual true social welfare). . . . . . 254

5.8 EPEC problem: The simplified version of the IEEE RTS net-

work (case study). . . . . . . . . . . . . . . . . . . . . . . . . . 256

A.1 IEEE reliability test system: Network. . . . . . . . . . . . . . . 284

List of Tables

1.1 Introduction: Relevant features of some models reported in lit-

erature and the model proposed in this dissertation (generation

investment of a given strategic producer). . . . . . . . . . . . . . 11

1.2 Introduction: Relevant features of some models reported in the

literature and the model proposed in this dissertation (genera-

tion investment equilibria). . . . . . . . . . . . . . . . . . . . . . 13

1.3 Introduction: Example for clarifying the demand-bid blocks. . . 18

2.1 Direct solution: Type and data for the existing generating units

(illustrative example). . . . . . . . . . . . . . . . . . . . . . . . 61

2.2 Direct solution: Location and type of the existing units (illus-

trative example). . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.3 Direct solution: Type and data for investment options (illustra-

tive example). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

2.4 Direct solution: Demand-bid blocks including maximum loads

[MW] and bid prices [e/MWh] (illustrative example). . . . . . . 63

2.5 Direct solution: Investment results pertaining to the uncon-

gested and congested cases (illustrative example). . . . . . . . . 65

2.6 Direct solution: Investment results considering and not consid-

ering strategic offering (illustrative example). . . . . . . . . . . . 67

2.7 Direct solution: Rival producer scenarios (illustrative example). 68

2.8 Direct solution: Investment results pertaining to the stochastic

cases (illustrative example). . . . . . . . . . . . . . . . . . . . . 69

2.9 Direct solution: Location and type of existing units (case study). 70

2.10 Direct solution: Investment results (case study). . . . . . . . . . 71

xxi

xxii LIST OF TABLES

3.1 Benders’ approach: Type and data for the existing generating

units (illustrative example for convexity analysis). . . . . . . . . 80

3.2 Benders’ approach: Demand-bid blocks including maximum loads

[MW] and bid prices [e/MWh] (illustrative example for convex-

ity analysis). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.3 Benders’ approach: Investment options of the rival producers

(illustrative example for convexity analysis). . . . . . . . . . . . 81

3.4 Benders’ approach: Investment results of the case study pre-

sented in Section 2.5 obtained by the direct solution approach

presented in Chapter 2. . . . . . . . . . . . . . . . . . . . . . . . 98

3.5 Benders’ approach: Investment results of the case study pre-

sented in Section 2.5 obtained by the proposed Benders’ approach. 98

3.6 Benders’ approach: Investment options. . . . . . . . . . . . . . . 100

3.7 Benders’ approach: Investment options for rival producers (case

study). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

3.8 Benders’ approach: Investment results (case study). . . . . . . . 102

4.1 Futures market and pool: Data for the existing units of the

strategic producer. . . . . . . . . . . . . . . . . . . . . . . . . . 167

4.2 Futures market and pool: Data for rival units. . . . . . . . . . . 167

4.3 Futures market and pool: Type and data for the investment

options. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

4.4 Futures market and pool: Cases considered. . . . . . . . . . . . 170

4.5 Futures market and pool: Investment results. . . . . . . . . . . . 172

4.6 Futures market and pool: Market clearing prices pertaining to

all cases considered. . . . . . . . . . . . . . . . . . . . . . . . . . 174

4.7 Futures market and pool: Yearly production results pertaining

to all cases considered. . . . . . . . . . . . . . . . . . . . . . . . 175

5.1 EPEC problem: Data pertaining to the existing units (illustra-

tive example). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

5.2 EPEC problem: Data pertaining to demands and their price

bids (illustrative example). . . . . . . . . . . . . . . . . . . . . . 235

LIST OF TABLES xxiii

5.3 EPEC problem: Type and data for the candidate units (illus-

trative example). . . . . . . . . . . . . . . . . . . . . . . . . . . 236

5.4 EPEC problem: Cases considered for the illustrative example. . 237

5.5 EPEC problem: General results of generation investment equi-

libria (illustrative example). . . . . . . . . . . . . . . . . . . . . 241

5.6 EPEC problem: Three equilibria for Case 1 maximizing total

profit (illustrative example). . . . . . . . . . . . . . . . . . . . . 248

5.7 EPEC problem: Data pertaining to the existing units (case study).257

5.8 EPEC problem: Location of the existing units (case study). . . 257

5.9 EPEC problem: results of generation investment equilibria (case

study). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

A.1 IEEE Reliability Test System: Reactance (p.u. on a 100 MW

base) and capacity of transmission lines. . . . . . . . . . . . . . 285

Notation

The notation used in this dissertation is listed below. The following observa-

tions are in order:

1) All symbols including index t pertain to a pool, e.g., symbols P Sti and θtn.

These symbols are used in Chapters 2 to 5.

2) All symbols without index t, but including overlining refer to the futures

base auction, e.g., symbols PS

i and CO

j . These symbols are only used in

Chapter 4.

3) All symbols without index t, but including a hat refer to the futures

peak auction, e.g., symbols P Si and CO

j . These symbols are only used in

Chapter 4.

4) Symbols λ, µ, ν and ξ with subscripts and/or superscripts and/or over-

lining and/or hat are dual variables associated with lower-level problems,

e.g., dual variables λ, µSmax

i , νmax

tnm and ξmax

tn . These symbols are used in

Chapters 2 to 5.

5) Symbol λ with subscripts and/or superscripts and/or overlining and/or

hat, i.e., symbols λtn, λ and λ are market clearing prices, i.e., dual vari-

ables associated with energy balance constraints.

6) Symbols χ, η, β, γ, τ , δ, ρ and φ with subscripts and/or superscripts

and/or overlining and/or hat are dual variables associated with the MPEC

of a strategic producer. These symbols are only used in Chapter 5.

xxv

xxvi NOTATION

Indices:

t Index for demand blocks running from 1 to T .

y Index for producers from 1 to Y .

i Index for candidate generating units of the strategic producer

running from 1 to I.

k Index for existing generating units of the strategic producer run-

ning from 1 to K.

j Index for other generating units owned by rival producers run-

ning from 1 to J .

d Index for demands running from 1 to D.

h Index for available investment capacities running from 1 to H .

w Index for scenarios running from 1 to W .

n,m Indices for buses running from 1 to N , and from 1 to M , respec-

tively.

Sets:

ΨS Set of indices of candidate units of the strategic producer.

ΨSn Subset of set ΨS containing indices of candidate units located at

bus n.

ΨES Set of indices of existing units of the strategic producer.

ΨESn Subset of set ΨES containing indices of existing units located at

bus n.

ΨO Set of indices of other units owned by rival producers.

NOTATION xxvii

ΨOn Subset of set ΨO containing indices of rival units located at bus

n.

ΨD Set of indices of demands.

ΨDn Subset of set ΨD containing indices of demands located at bus

n.

Ωy Set of indices of units owned by producer y.

Φn Set of indices of buses adjacent to bus n.

Tb Set of indices of base demand blocks.

Tp Set of indices of peak demand blocks.

Constants:

There are four types of constants in this dissertation, namely:

1) General constants regarding demand blocks, investment options, invest-

ment budget and transmission lines. These constants are used in Chap-

ters 2 to 5.

2) Constants pertaining to the pool. These constants include index w if

they refer to scenario w, and are used in Chapters 2 to 5.

3) Constants pertaining to the futures base auction. These constants are

overlined, and are only used in Chapter 4.

4) Constants pertaining to the futures peak auction. These constants in-

clude a hat, and are only used in Chapter 4.

All constants are defined as follows.

xxviii NOTATION

General Constants:

φw Probability of scenario w.

σt Weighting factor of demand block t [hour].

Ki Annualized capital cost of candidate unit i ∈ ΨS of the strategic

producer [e/MW].

Kmax Available investment budget [e].

Xmaxi Maximum power of candidate unit i ∈ ΨS of the strategic pro-

ducer [MW]. This constant is used if a set of continuous invest-

ment options is considered (Chapter 5).

Xih Option h for investment capacity of candidate unit i ∈ ΨS of

the strategic producer [MW]. This constant is used if a set of

discrete investment options is considered (Chapters 2 to 4).

Bnm Susceptance of transmission line (n,m) [p.u.].

Fmax

nm Capacity of transmission line (n,m) [MW].

Constants Pertaining to the Pool:

PESmax

k Capacity of existing unit k ∈ ΨE of the strategic producer [MW].

POmax

j Capacity of unit j ∈ ΨO of rival producers [MW].

PDmax

td Maximum load of demand d ∈ ΨD in demand block t [MW].

CSi Marginal cost of candidate unit i ∈ ΨS of the strategic producer

[e/MWh].

CESk Marginal cost of existing unit k ∈ ΨE of the strategic producer

[e/MWh].

COtj Price offer of unit j ∈ ΨO of rival producers in demand block t

[e/MWh].

UDtd Price bid of demand d ∈ ΨD in demand block t [e/MWh].

NOTATION xxix

Constants Pertaining to the Futures Base Auction:

CO

j Price offer of unit j ∈ ΨO of rival producers [e/MWh].

UD

d Price bid of demand d ∈ ΨD [e/MWh].

PDmax

d Maximum load of demand d ∈ ΨD [MW].

Constants Pertaining to the Futures Peak Auction:

COj Price offer of unit j ∈ ΨO of rival producers [e/MWh].

UDd Price bid of demand d ∈ ΨD [e/MWh].

PDmax

d Maximum load of demand d ∈ ΨD [MW].

Primal Variables:

There are four types of primal variables in this dissertation, namely:

1) General primal variables including variables on capacity of candidate

units and voltage angles. These variables are used in Chapters 2 to 5.

2) Primal variables pertaining to the pool. These variables include index w

if they refer to scenario w, and are used in Chapters 2 to 5.

3) Primal variables pertaining to the futures base auction. These variables

are overlined, and are only used in Chapter 4.

4) Primal variables pertaining to the futures peak auction. These variables

include a hat, and are only used in Chapter 4.

All primal variables are defined below.

General Primal Variables:

Xi Capacity of candidate unit i ∈ ΨS of the strategic producer

[MW].

xxx NOTATION

θtn Voltage angle of bus n in demand block t [rad].

Variable θtn includes subscript w if it refers to scenario w.

Primal Variables Pertaining to the Pool:

αSti Price offer by candidate unit i ∈ ΨS of the strategic producer in

demand block t [e/MWh].

αEStk Price offer by existing unit k ∈ ΨES of the strategic producer in

demand block t [e/MWh].

P Sti Power produced by candidate unit i ∈ ΨS of the strategic pro-

ducer in demand block t [MW].

PEStk Power produced by existing unit k ∈ ΨES of the strategic pro-

ducer in demand block t [MW].

POtj Power produced by unit j ∈ ΨO of rival producers in demand

block t [MW].

PDtd Power consumed by demand d ∈ ΨD in demand block t [MW].

Primal Variables Pertaining to the Futures Base Auction:

αSi Price offer by candidate unit i ∈ ΨS of the strategic producer

[e/MWh].

αESk Price offer by existing unit k ∈ ΨES of the strategic producer

[e/MWh].

PS

i Power produced by candidate unit i ∈ ΨS of the strategic pro-

ducer [MW].

PES

k Power produced by existing unit k ∈ ΨES of the strategic pro-

ducer [MW].

PO

j Power produced by unit j ∈ ΨO of rival producers [MW].

PD

d Power consumed by demand d ∈ ΨD [MW].

NOTATION xxxi

Primal Variables Pertaining to the Futures Peak Auction:

αSi Price offer by candidate unit i ∈ ΨS of the strategic producer

[e/MWh].

αESk Price offer by existing unit k ∈ ΨES of the strategic producer

[e/MWh].

P Si Power produced by candidate unit i ∈ ΨS of the strategic pro-

ducer [MW].

PESk Power produced by existing unit k ∈ ΨES of the strategic pro-

ducer [MW].

POj Power produced by unit j ∈ ΨO of rival producers [MW].

PDd Power consumed by demand d ∈ ΨD [MW].

Dual Variables:

There are four types of dual variables in this dissertation, namely:

1) Dual variables associated with the set of lower-level problems represent-

ing the clearing of the pool. These variables include index w if they refer

to scenario w, and are used in Chapters 2 to 5.

2) Dual variables associated with the lower-level problem representing the

clearing of the futures base auction. These variables are overlined, and

are only used in Chapter 4.

3) Dual variables associated with the lower-level problem representing the

clearing of the futures peak auction. These variables include a hat, and

are only used in Chapter 4.

4) Dual variables associated with the upper-level, primal, dual and strong

duality constraints included in the MPEC of a strategic producer. These

variables are only used in Chapter 5.

All dual variables are defined as follows.

xxxii NOTATION

Dual Variables Associated with the Clearing of the Pool:

The dual variables below are associated with the following constraints:

λtn Pool energy balance in demand block t at bus n. These dual

variables provide the pool locational marginal prices (LMPs)

[e/MWh].

µSmax

ti Capacity of candidate unit i ∈ ΨS of the strategic producer in

demand block t.

µSmin

ti Non-negativity of the production level of candidate unit i ∈ ΨS

of the strategic producer in demand block t.

µESmax

tk Capacity of existing unit k ∈ ΨES of the strategic producer in

demand block t.

µESmin

tk Non-negativity of the production level of existing unit k ∈ ΨES

of the strategic producer in demand block t.

µOmax

tj Capacity of unit j ∈ ΨO of rival producers in demand block t.

µOmin

tj Non-negativity of the production level of unit j ∈ ΨO of rival

producers in demand block t.

µDmax

td Maximum load of demand d ∈ ΨD in demand block t.

µDmin

td Non-negativity of demand d ∈ ΨD in demand block t.

νmax

tnm Transmission capacity of line (n,m) in demand block t and di-

rection (n,m).

νmin

tnm Transmission capacity of line (n,m) in demand block t and di-

rection (m,n).

ξmax

tn Upper bound of the voltage angle in demand block t at bus n.

ξmin

tn Lower bound of the voltage angle in demand block t at bus n.

ξ1

t Voltage angle in demand block t at the reference bus n = 1.

NOTATION xxxiii

Dual Variables Associated with the Clearing of the Fu-

tures Base Auction:


λ Energy balance in the futures base auction. This dual variable

provides the clearing price of this auction [e/MWh].

µSmax

i Capacity of candidate unit i ∈ ΨS of the strategic producer.

µSmin

i Non-negativity of the production level of candidate unit i ∈ ΨS

of the strategic producer.

µESmax

k Capacity of existing unit k ∈ ΨES of the strategic producer.

µESmin

k Non-negativity of the production level of existing unit k ∈ ΨES


µOmax

j Capacity of unit j ∈ ΨO of rival producers.

µOmin

j Non-negativity of the production level of unit j ∈ ΨO of rival

producers.

µDmax

d Maximum load of demand d ∈ ΨD.

µDmin

d Non-negativity of demand d ∈ ΨD.

Dual Variables Associated with the Clearing of the Fu-

tures Peak Auction:


λ Energy balance in the futures peak auction. This dual variable

provides the clearing price of this auction [e/MWh].

µSmax

i Capacity of candidate unit i ∈ ΨS of the strategic producer.

µSmin

i Non-negativity of the production level of candidate unit i ∈ ΨS


xxxiv NOTATION

µESmax

k Capacity of existing unit k ∈ ΨES of the strategic producer.

µESmin

k Non-negativity of the production level of existing unit k ∈ ΨES


µOmax

j Capacity of unit j ∈ ΨO of rival producers.

µOmin

j Non-negativity of the production level of unit j ∈ ΨO of rival

producers.

µDmax

d Maximum load of demand d ∈ ΨD.

µDmin

d Non-negativity of demand d ∈ ΨD.

Dual Variables Associated with the Upper-Level Con-

straints Included in the MPEC of Producer y:

The dual variables below are associated with the following upper-level con-

straints:

χmaxyi Upper-level constraint: Maximum capacity of candidate unit i ∈

ΨS to be built.

χminyi Upper-level constraint: Non-negativity of the production level of

candidate unit i ∈ ΨS to be built.

χIBy Upper-level constraint: Maximum available investment budget.

χSSy Upper-level constraint: Supply security constraint imposed by

the market regulator.

ηαS

yti Upper-level constraint: Non-negativity of the strategic price of-

fers of candidate unit i ∈ ΨS in demand block t.

ηαES

ytk Upper-level constraint: Non-negativity of the strategic price of-

fers of existing unit k ∈ ΨES in demand block t.

NOTATION xxxv

Dual Variables Associated with the Primal Constraints

Included in the MPEC of Producer y:

The dual variables below are associated with the following primal constraints:

βytn Primal constraint: Energy balance in demand block t at bus n.

γSmax

yti Primal constraint: Capacity of candidate unit i ∈ ΨS of the

strategic producer in demand block t.

γSmin

yti Primal constraint: Non-negativity of the production level of can-

didate unit i ∈ ΨS of the strategic producer in demand block t.

γESmax

ytk Primal constraint: Capacity of existing unit k ∈ ΨES of the

strategic producer in demand block t.

γESmin

ytk Primal constraint: Non-negativity of the production level of ex-

isting unit k ∈ ΨES of the strategic producer in demand block

t.

γDmax

ytd Primal constraint: Maximum load of demand d ∈ ΨD in demand

block t.

γDmin

ytd Primal constraint: Non-negativity of demand d ∈ ΨD in demand

block t.

τmax

ytnm Primal constraint: Transmission capacity of line (n,m) in de-

mand block t and direction (n,m).

τmin

ytnm Primal constraint: Transmission capacity of line (n,m) in de-

mand block t and direction (m,n).

δmax

ytn Primal constraint: Upper bound of the voltage angle in demand

block t at bus n.

δmin

ytn Primal constraint: Lower bound of the voltage angle in demand

block t at bus n.

δ1

yt Primal constraint: Voltage angle fixed value in demand block t

at the reference bus n = 1.

xxxvi NOTATION

Dual Variables Associated with the Dual Constraints In-

cluded in the MPEC of Producer y:

The dual variables below are associated with the following dual constraints:

ρDytd Dual constraint: Equality in the dual problem associated with

consumption variable PDtd of demand d ∈ ΨD in demand block t.

ρSyti Dual constraint: Equality in the dual problem associated with

production variable P Sti of candidate unit i ∈ ΨS in demand block

t.

ρESytk Dual constraint: Equality in the dual problem associated with

production variable PEStk of existing unit k ∈ ΨES in demand

block t.

ρθytn Dual constraint: Equality in the dual problem associated with

voltage angle variable θtn in demand block t at bus n.

ηSmax

yti Dual constraint: Non-negativity of dual variable µSmax

ti associated

with candidate unit i ∈ ΨS in demand block t.

ηSmin

yti Dual constraint: Non-negativity of dual variable µSmin

ti associated

with candidate unit i ∈ ΨS in demand block t.

ηESmax

ytk Dual constraint: Non-negativity of dual variable µESmax

tk associ-

ated with existing unit k ∈ ΨES in demand block t.

ηESmin

ytk Dual constraint: Non-negativity of dual variable µESmin

tk associ-

ated with existing unit k ∈ ΨES in demand block t.

ηDmax

ytd Dual constraint: Non-negativity of dual variable µDmax

td associ-

ated with demand d ∈ ΨD in demand block t.

ηDmin

ytd Dual constraint: Non-negativity of dual variable µDmin

td associated

with demand d ∈ ΨD in demand block t.

NOTATION xxxvii

ηνmax

ytnm Dual constraint: Non-negativity of dual variable νmax

tnm associated

with transmission line (n,m) in demand block t and direction

(n,m).

ηνmin

ytnm Dual constraint: Non-negativity of dual variable νmin

tnm associated

with transmission line (n,m) in demand block t and direction

(m,n).

ηξmax

ytn Dual constraint: Non-negativity of dual variable ξmax

tn associated

with voltage angle in demand block t at bus n.

ηξmin

ytn Dual constraint: Non-negativity of dual variable ξmin

tn associated

with voltage angle in demand block t at bus n.

Dual Variable Associated with the Strong Duality Equal-

ity Included in the MPEC of Producer y:

The dual variable below is associated with the following equality:

φSDyt Strong duality equality associated with the lower-level problem

of producer y in demand block t.

Acronyms:

ATSW Annual True Social Welfare.

dc Direct Current.

EPEC Equilibrium Problem with Equilibrium Constraints.

GNE Generalized Nash Equilibrium.

KKT Karush-Kuhn-Tucker.

LCP Linear Complementarity Problem.

LMP Locational Marginal Price.

xxxviii NOTATION

MCP Mixed Complementarity Problem.

MFCQ Mangasarian-Fromovitz Constraint Qualification.

MILP Mixed-Integer Linear Programming.

MPCC Mathematical Program with Complementarity Constraints.

MPEC Mathematical Program with Equilibrium Constraints.

RTS Reliability Test System.

TP Total Profit.

Chapter 1

Generation Investment:

Introduction

1.1 Electricity Markets

The restructuring of the electric power industry started in the 80’s to create

competitive electricity markets [64,76,125,132]. In an electricity market, each

power producer submits its production offers, with the objective of maximiz-

ing its profit. On the other hand, each consumer submits its demand bids

with the aim of maximizing its utility. Then, a non-profit entity, the market

operator, clears the market. The objective of the market clearing procedure is

to maximize the social welfare of the market.

In addition to the market operator, there is another independent entity,

the market regulator, that is in charge of the competitive functioning of the

market. The market regulator supervises the market operation, and enforces a

number of operation and planning policies in order to ensure that the market

operates as close as possible to perfect competition.

Observing diverse restructuring experiences throughout the world, one con-

cludes that instead of perfectly competitive markets, oligopolistic markets have

mostly been formed. In an oligopolistic market, which is the one considered in

this dissertation, some producers denominated “strategic producers” are able

to alter the market clearing outcomes through their decisions, including:

1

2 1. Generation Investment: Introduction

• Strategic decisions on production offers (operation issue).

• Strategic decisions on generation investment (planning issue).

Note that “strategic offering” and “strategic investment” refer to the offering

and investment decisions of a strategic producer, respectively.

Regarding electricity trading floors, there are several alternatives to trade

electric energy. One prevalent market is the pool, cleared by the market op-

erator once a day, one day ahead, and on an hourly basis. This is the case of

OMIE [127], EEX [34], Nord Pool [128], ISO-New England [65] and PJM [103].

The market operator seeks to maximize the social welfare considering the pro-

duction offers submitted by the producers and the demand bids submitted by

the consumers. The market clearing results are hourly productions, consump-

tions and clearing prices.

If the transmission network is modeled, a clearing price at each bus of the

network is obtained, the so-called locational marginal price (LMP) of that bus.

The LMP of a given bus represents the social welfare increment in the market

as a result of a marginal demand increment at that bus. Note that in the case

of congested transmission lines, LMPs vary across buses for any given hour.

The futures market is another trading floor that has become increasingly

relevant during the last decade. This is the case of OMIP [126], EEX [34] and

Nord Pool [128], which include dedicated futures market auctions cleared by

their respective market operators. Other markets include future derivatives

traded in general stock exchanges. This is the case of ISO-New England [65]

and PJM [103].

The futures market includes auctions encompassing a medium- or long-

term horizon, e.g., one week to one year. Similarly to the pool, the market

operator considers the production offers submitted by the producers and the

demand bids submitted by the consumers, and then clears each futures market

auction maximizing the corresponding social welfare.

The futures market is generally cleared prior to the clearing of the pool, and

thus it becomes possible for producers to engage in arbitrage, i.e., to purchase

energy from the futures market and then to sell it in the pool.

1.2. Chapter Organization 3

Further details on restructuring, diverse trading floors and market func-

tioning are presented throughout the dissertation as required.

The investment problems addressed in this dissertation consider the elec-

tricity market framework described above.

1.2 Chapter Organization

The rest of this chapter provides an introduction to this dissertation and in-

cludes the sections below:

• Section 1.3 presents the motivation for the approaches and models devel-

oped in this thesis. In other words, this section states why the subject

matter of this dissertation deserves attention.

• Section 1.4 describes in detail the problems that are addressed in this

dissertation.

• Section 1.5 presents a literature review. In addition, this section provides

a comparison among the models/approaches proposed in this thesis work

and others reported in the literature.

• Section 1.6 describes the modeling assumptions considered throughout

the dissertation.

• Section 1.7 explains the general structure of the models proposed in this

thesis and then explains the solution approaches for these models.

• Section 1.8 presents the main objectives of this thesis.

• Section 1.9 provides the outline of this document.

1.3 Thesis Motivation

Since a reliable electricity supply is crucial for the functioning of modern so-

cieties, investment in electricity production is most important to guarantee

supply security. However, within an electricity market framework, generation


investment decision-making problems for a power producer are complex, be-

cause such problems require the proper modeling of the following elements:

1) The market functioning, which leads to complementarity models.

2) The uncertainties plaguing markets, which leads to using stochastic pro-

gramming models.

3) The behavior of rival producers, i.e., their operation and investment

strategies, which has an effect on the producer’s own decisions.

In addition to the need of such stochastic complementarity models, in-

vestment decision making is risky due to the long-term consequences of the

decisions involved.

Additionally, generation investment decision-making problems become par-

ticularly complex within an oligopolistic electricity market, where several strate-

gic producers compete. The reason for such complexity is that each strategic

producer is able to alter the formation of the market outcomes (e.g., clearing

prices and production quantities) through its operation and planning strate-

gies. Thus, the decision-making processes of all producers need to be jointly

considered.

The considerations above motivate the development of an appropriate math-

ematical tool to assist a strategic producer competing with its strategic rivals in

an electricity market for making informed investment decisions. The objective

of this mathematical tool is to maximize the expected profit of such strate-

gic producer through its decisions in i) operations (offering), and ii) planning

(generation investment).

To develop such mathematical tool, several important issues need to be

considered, namely:

1. Uncertainties: In an investment problem, the strategic producer faces

a number of uncertainties, e.g., demand growth, behavior of rival pro-

ducers, investment cost of different technologies, fuel price, regulatory

policies and others. On one hand, modeling such uncertainties prop-

erly is important. On the other hand, a detailed description of such

1.3. Thesis Motivation 5

uncertainties may result in high computational burden and eventual in-

tractability. Thus, the mathematical tool to address the generation in-

vestment decision-making problem of a strategic producer should be able

to consider the most important uncertainties, while being computation-

ally tractable.

2. Diverse trading markets: The strategic producer may participate in di-

verse trading markets. The prevalent market is the electricity pool; how-

ever, to obtain a higher expected profit or a lower risk, the strategic

producer may trade in other markets as well, e.g., in the futures mar-

ket. Thus, the mathematical tool to address the generation investment

decision-making problem of a strategic producer should be able to model

the functioning of all trading markets in which such producer may get

involved.

3. Different investment technologies: The mathematical tool to address the

generation investment decision-making problem of a strategic producer

should be able to select the units to be built among the available in-

vestment options such as base technologies, e.g., nuclear units and peak

technologies, e.g., CCGT units.

4. Locations for building the candidate generation units: The mathematical

tool to address the generation investment decision-making problem of a

strategic producer should be able to optimally allocate the units to be

built throughout the network. To this end, a proper representation of

the network is required.

Additionally, generation investment equilibria need to be studied and an-

alyzed in detail. Since the operation and planning strategies of any strategic

producer are interrelated with those of other strategic producers, decisions

made by one strategic producer may influence the strategies of other strategic

producers. Thus, a number of investment equilibria may exist, where each

strategic producer cannot increase its profit by changing its strategies uni-

laterally. Thus, it is important to identify such investment equilibria. This

equilibrium analysis is useful for the market regulator to gain insight into the


investment behavior of the strategic producers and the generation investment

evolution. Such insight may allow the market regulator to better design mar-

ket rules, which in turn may contribute to increase the competitiveness of the

market and to stimulate optimal investment in generation capacity. In addi-

tion, the market regulator may use an equilibrium analysis to assess the impact

of certain policies on the investment evolution.

1.4 Problem Description

Considering the thesis motivation presented in Section 1.3, this dissertation

specifically addresses the four problems below:

1. Development of an optimization tool for a strategic producer trading

in a pool to optimally solve its generation investment decision-making

problem (direct solution approach):

The aim of this tool is to identify the most beneficial investment strategy

for a strategic producer competing in an electricity pool. This strategy

includes the technology, the capacity and the network allocation of each

new unit to be built. The strategies of the rival producers, i.e., their

operation and planning decisions, are uncertain parameters represented

through scenarios. That is, we use scenarios to describe uncertainties

pertaining to i) rival production offers and ii) rival investments.

To solve this problem, we propose a hierarchical (bilevel) model whose

upper-level problem represents the investment and offering actions of the

strategic producer, and whose multiple lower-level problems represent the

clearing of the pool under different operating scenarios. Such model ren-

ders a mathematical program with equilibrium constraints (MPEC) by

replacing each lower-level problem with its Karush-Kuhn-Tucker (KKT)

conditions. In turn, this MPEC can be recast as a mixed-integer lin-

ear programming problem (MILP) solvable with commercially available

software. Further details on the structure of this bilevel model and the

resulting MPEC are provided in Section 1.7. In addition, the bilevel

1.4. Problem Description 7

model, the MPEC and the KKT conditions are described in Sections B.1

and B.2 of Appendix B.

In the proposed model, all scenarios involved are considered simultane-

ously (direct solution approach). Thus, this direct approach generally

suffers from high computational burden and eventual intractability in

cases with many scenarios. This is the main drawback of this approach,

which is presented in Chapter 2.

2. Development of an alternative approach for solving the generation in-

vestment decision-making problem of a strategic producer for cases with

many scenarios (Benders’ decomposition approach):

To tackle the computational burden of the direct solution approach in

cases with many scenarios, an alternative approach based on Bender’s

decomposition is proposed. This approach allows decomposing the con-

sidered bilevel model per scenario by fixing the investment variables.

Therefore, unlike the direct solution approach, the scenarios are consid-

ered separately and thus the model can be solved even if a large number

of scenarios is considered.

A Benders’ approach is possible since exhaustive computational analysis

indicates that if the producer behaves strategically and a sufficiently large

number of scenarios is considered, the expected profit of the strategic

producer as a function of the complicating investment decisions has a

convex enough envelope.

The proposed Benders’ approach is presented in Chapter 3.

3. Development of an optimization tool for a strategic producer trading in

both the futures market and the pool to optimally solve its generation

investment decision-making problem:

Since the futures market has become increasingly relevant for trading

electricity, it is important to analyze the effects of such market on the

investment decisions of a strategic producer.

To this end, we consider a hierarchical (bilevel) model, whose upper-level

problem represents the investment and offering actions of the producer,


and whose multiple lower-level problems represent the clearing of the

futures market and the pool under different operating conditions. Then,

an MPEC is derived by replacing the lower-level problems with their

respective KKT optimality conditions. Finally, the MPEC problem is

linearized and recast as an MILP problem.

This model is presented in Chapter 4.

4. Identification of potential generation investment equilibria in an oligopolis-

tic pool with strategic producers:

The fourth problem considered in this dissertation is to mathematically

identify the potential investment equilibria, where each producer cannot

increase its profit by unilaterally changing its investment strategies.

To this end, the investment and offering decisions of each strategic pro-

ducer are represented through a hierarchical (bilevel) model, whose upper-

level problem decides on the optimal investment and the offering curves

for maximizing the profit of the producer, and whose lower-level problems

represent different market clearing scenarios. Replacing the lower-level

problems by their primal-dual optimality conditions (Section B.2 of Ap-

pendix B) in the single-producer model renders an MPEC. The joint

consideration of all producer MPECs, one per producer, constitutes an

equilibrium problem with equilibrium constraints (EPEC). Further de-

tails on the EPEC are provided in Section 1.7 and in Section B.3 of

Appendix B. To identify the solutions of the EPEC, each MPEC prob-

lem is replaced by its KKT conditions, which are in turn linearized. The

resulting mixed-integer linear system of equalities and inequalities allows

determining the EPEC equilibria through an auxiliary MILP problem.

Finally, all equilibria identified are verified by a diagonalization checking.

This problem is analyzed in Chapter 5.

1.5. Literature Review 9

1.5 Literature Review

This section reviews relevant works in technical literature regarding i) the gen-

eration investment problem of a strategic producer within an electricity mar-

ket, and ii) the generation investment equilibrium problem in an oligopolistic

electricity market. Finally, some other relevant works are also reviewed.

1.5.1 Literature Review: Generation Investment for a

Strategic Producer

Several works can be found in literature referring to the generation investment

decision-making problem of a given strategic producer within an electricity

market.

In [14], a stochastic dynamic optimization model is used to evaluate gen-

eration investments under both centralized and decentralized frameworks, but

not modeling the network. Long-term uncertainty in demand growth and its

effect on future prices are modeled via discrete Markov chains.

Reference [23] is a relevant paper that considers the generation expansion

planning problem in an oligopolistic environment using a Cournot model and

including no network constraint. The solution is found using an iterative search

procedure, which assumes complete information of the rivals.

The effects of both competition and transmission congestion on generation

expansion are specifically considered in [69], where a Cournot model is used.

In [94], a noncooperative game for generation investment is modeled using

two tiers. In the first tier, the generation investment game is examined, and in

the second tier the energy supply game is considered. The solution procedure

is based on a reinforcement learning algorithm.

In [97], the value of information pertaining to rival producers such as their

marginal costs and conjectures on their behavior as well as demand levels are

analyzed for making decision on generation investment. The model uses a

Cournot approach and includes no network constraint.

In [130], two different approaches pertaining to generation expansion in

a electricity market are presented. Both of them consider a Cournot model


although they differ in how the producer determines its optimal capacity. In

the first approach a mixed complementarity problem (MCP) is used, while

for the second one an MPEC approach is considered based on a Stackelberg

model [124].

In [131], the strategic generation capacity expansion of a producer consid-

ering incomplete information of rival producers is modeled using a two-level

optimization problem. A genetic algorithm approach is used to solve such

problem.

Reference [134] proposes a bilevel model to assist a producer in making

multi-stage generation investment decisions considering the investment uncer-

tainty of rival producers. In the upper-level problem, the producer maximizes

its expected profit, while the lower-level problem represents the market clear-

ing characterized by a conjectural variations model and including no network

constraint. The model proposed in [134] renders an MPEC.

For clarity, we summarize in Table 1.1 the relevant features of the gener-

ation investment model proposed in papers [71–73], developed as part of this

thesis, and other works reported in technical literature.

1.5.2 Literature Review: Generation Investment Equi-

libria

There are several papers in the literature pertaining to generation investment

equilibria as explained below.

In pioneering reference [92], three models of the generation capacity expan-

sion game are considered. The first model assumes perfect competition, thus

being similar to a centralized capacity expansion model. The second model

(open-loop Cournot duopoly) extends the well-known Cournot model to in-

clude investments in new generation capacity. The third model (closed-loop

Cournot duopoly) separates the investment and production decisions consid-

ering investment in the first stage and sales in the second stage. All three

models are static, and uncertainties as well as the network constraints are not

modeled.

Reference [35] considers a static but stochastic capacity expansion equilib-

1.5.Litera

ture

Review

11

Table 1.1: Introduction: Relevant features of some models reported in literature and the model proposed in thisdissertation (generation investment of a given strategic producer).

Reference ModelTransmission Static/ Stochastic

UncertaintyDifferent investment

BilevelStrategic

Approach

constraints Dynamic model technologies offering

[14] Stochastic optimization No Dynamic Yes Demand growth Yes No No Discrete Markov chain

[23] Cournot No Static No - Yes No No Iterative

[69] Cournot Yes Static No - No No No Quadratic programming

[94] Supply function Yes Dynamic YesDemand growth

Yes No No Heuristic

and line outages

[97] Cournot No Static Yes

Demand growth,

Yes No No Quadratic programmingrival marginal cost

and rival behavior

[130] Cournot (Stackelberg) No Static No - Yes No No Complementarity

[131] Bilevel Yes Static Yes Unit outages Yes Yes No Genetic algorithm

[134] Conjectural variation No Dynamic Yes Rival investment Yes Yes No MPEC

This Stepwise

Yes Static Yes

Rival offering,

Yes Yes Yes

MPEC (direct

dissertation supply rival investment solution and Benders’

( [71–73]) function and demand growth decomposition approaches)


ria in an electricity market. The objective of this work is to investigate the

impact of four key parameters on the generation investment equilibria and on

the supply security. Such parameters are: 1) profit risk, 2) investment incen-

tives, 3) market organization (energy-only or energy and capacity) and price

caps, and 4) carbon trading issues. In the capacity expansion equilibria re-

ported in [35], fuel prices and climate change policies are considered uncertain

parameters, while the demand is assumed known. The producers are con-

sidered price-takers, but subject to regulatory imperfections. The network is

disregarded, and a small case study is analyzed.

A multi-stage generation investment equilibrium considering uncertain de-

mand is addressed in [45]. This reference proposes a Markov chain to model the

strategic interaction between a short-run capacity-constrained Cournot gener-

ation game and a long-run generation investment game. It is concluded that

a socially optimal level of capacity is not built, and that the distance between

the capacity built and the optimal capacity is largely dependent on investment

profitability.

Investment incentives are studied in [46] using a simple strategic dynamic

model with random demand growth. This model is based on a non-collusive

Markovian equilibrium in which the investment decisions of each producer

depend on its existing capacity.

Similarly to [46], reference [49] formulates a dynamic capacity investment

equilibrium considering a hydrothermal duopoly under uncertain demand. Both

Markov perfect and open-loop equilibria are modeled in this reference, and then

the incentives needed to promote investment are studied.

In [66], the capacity expansion decisions are made by a leader on behalf of

the market agents. The aim of this model is to maximize the total welfare of

all market participants, which renders a bilevel model, whose upper-level prob-

lem determines the investment decisions, and whose lower-level problems rep-

resent the operation decisions of each market participant. Such bilevel model

is transformed into a mathematical program with complementarity constraints

(MPCC).

The equilibrium of generation investment considering both futures and spot

markets is studied in [93], where a Cournot model is used. This reference shows

1.5.Litera

ture

Review

13

Table 1.2: Introduction: Relevant features of some models reported in the literature and the model proposed in thisdissertation (generation investment equilibria).

Reference Model BilevelStrategic Transmission Static/

UncertaintyDifferent investment

offering constraints dynamic technologies

[35] Complementarity No No No Static Yes Yes

[45] Cournot-Marcov chain No No No Dynamic Yes No

[46] Non-collusive Markov No No No Dynamic Yes No

[49] Open-loop and Markov perfect No No No Dynamic Yes Yes

[66] Cournot (MPCC) Yes No Yes Static No Yes

[92] Cournot (MPEC) Yes No No Static No Yes

[93] Cournot (EPEC) No No No Static Yes Yes

[133] Conjectural variation (EPEC) Yes No No Dynamic No Yes

This dissertation Stepwise supplyYes Yes Yes Static No Yes

( [74] and [75]) function (EPEC)


that forward contracts may not mitigate market power in the spot market,

in case that the production capacities of the producers are endogenous and

constrain the production level.

In [133], an EPEC is proposed to identify the generation capacity invest-

ment equilibria. To this end, a bilevel optimization problem is formulated so

that each producer selects capacities in an upper-level problem maximizing its

profit and anticipating the equilibrium outcomes of the lower-level problems, in

which production quantities and prices are determined by a conjectured-price

response approach.

The work reported in the two-part paper [74,75], developed as part of this

thesis work, mainly differs from [35, 45, 46, 49, 66, 92, 93, 133] in that it uses a

stepwise supply function model.

For the sake of clarity, the relevant features of the equilibrium model pro-

posed in this dissertation and other works reported in the literature are sum-

marized in Table 1.2.

The approach for modeling the generation investment equilibria used in

this dissertation is similar to that used in [63] and [117]. Such references

analyze the equilibria reached by strategic producers in a network-constrained

pool in which the behavior of each producer is represented by an MPEC.

However, references [63] and [117] address an operation, not an investment

problem. Similarly to the model developed in [117], the EPEC is characterized

in this dissertation by solving the optimality conditions of all MPECs, which

are formulated as an MILP, and diverse linear objective functions are used

to obtain different equilibria. Finally, note that the methodology presented

in this dissertation extends the model in [117] by incorporating generation

investment decisions and analyzing the impact that these decisions have on

the competitiveness of the market.

1.5.3 Other Related Works in the Literature

This subsection reviews some relevant works in the literature regarding 1)

MPECs, EPECs, complementarity models, stochastic programming models

and Benders’ decomposition, 2) other applications of bilevel models in elec-

1.5. Literature Review 15

tricity markets, 3) futures market auctions, 4) generation expansion planning

in centralized power systems, and 5) operation equilibria in electricity markets.

Such references are reviewed below:

• The first mathematical model used in this dissertation, an MPEC, was

first proposed in [55]. Further mathematical details on such model can

be found in [82].

• The EPEC model was first proposed in [102] and then used in [108] to

model the interaction among strategic producers of an electricity mar-

ket. In addition, EPEC models and their applications to the equilibrium

analysis of electricity markets are considered in [137].

• References [6,12,24,31–33] provide mathematical details on bilevel mod-

els.

• Mathematical details on complementarity models and their applications

in electricity markets are available in [41].

• Reference [25] provides mathematical details on stochastic programming

models and their applications in electricity markets.

• Mathematical details on Benders’ decomposition and their applications

in electricity markets can be found in [26].

• In the technical literature associated with electricity markets, a number

of works use a bilevel approach similar to the one proposed in this dis-

sertation. Such works pertain to offering strategies [9,60,81,113], trans-

mission expansion [19, 44, 107, 120, 121], policy incentives for producers

to invest in renewable facilities [138], transmission cost allocation [50],

maintenance scheduling [100, 101], security analysis [18, 90, 119], estima-

tion of the amount of market-integrable renewable resources [88] and

retailer trading [20], among others.

• Futures electricity market has been well analyzed in the literature as

effective tool for power producers to hedge against the risk of pool price


volatility. For instance, pioneering reference [70] investigates the use of

forward contracts to hedge profit risk. In addition, studies pertaining to

the futures market, especially their impact on enhancing competitiveness,

are reported in [1,38,78,83,118]. For example, reference [83] compares the

market prices in a market with Bertrand competition with and without

considering forward trading. Additionally, there are some works in the

literature considering offering strategies of electricity producers if both

futures and spot markets are considered, e.g., references [27, 48, 96].

• A large number of works can be found in the literature addressing the

generation expansion planning problem in centralized power systems,

e.g., references [85, 109, 135, 139].

• Finally, it is relevant to note that a number of works are available in the

literature pertaining to electricity market equilibria from the operational

point of view, e.g., references [4, 30, 61, 63, 104, 117, 136].

1.6 Modeling Assumptions

Pursuing clarity, the main modeling assumptions considered throughout this

dissertation are listed in this section.

1.6.1 Investment Study Approach

As indicated in Tables 1.1 and 1.2, two different approaches are common in

production investment studies as stated below:

a) Static approach.

b) Multi-stage or dynamic approach.

The characteristics of these approaches are described below:

In the static approach, the expansion exercise considers a single future

target year - e.g., a single year 20 years into the future - and the optimal

investment is established for that year. Once known the optimal generation

1.6. Modeling Assumptions 17

Tt

Figure 1.1: Introduction: Piecewise approximation of the load-duration curvefor the target year.

mix for the target year (e.g., year 20) and considering the generation mix of the

initial year (year 0), it is possible to derive an appropriate building schedule to

“move” from the generation mix of the initial year to that of the target year.

In this approach, the building path from the initial year to the target year is

not explicitly represented.

In the multi-stage or dynamic approach, investments are considered at

several steps throughout the planning horizon; and thus, the building path

from the initial year to the target year is derived. This approach provides

higher accuracy but at the cost of potential intractability.

As it is customary in large-scale generation investment studies, e.g., [35,92,

93], and pursuing an appropriate tradeoff between accuracy and computational

tractability, the static approach is used in this dissertation.

It is important to note that since the static approach is used, only the

available (not decommissioned) production facilities in the target year are con-

sidered. Existing production units being decommissioned along the planning

horizon should not be included in the analysis.


Table 1.3: Introduction: Example for clarifying the demand-bid blocks.

Demand D1 Demand D2

Demand Maximum Demand Bid Maximum Demand Bid

block load quantity price load quantity price

(t) [MW] [MW] [e/MWh] [MW] [MW] [e/MWh]

t = t1 500400 40

800600 36

100 38 200 35

t = t2 400320 37

650500 33

80 36 150 32

t = t3 350300 35

500400 30

50 34 100 29

t = t4 300270 33

400350 27

30 32 50 26

1.6.2 Load-Duration Curve

As an input of the static approach, Figure 1.1 depicts the load-duration curve

of the system for the target year of the planning horizon, approximated through

a number of stepwise demand blocks. The number of steps used to discretize

the load-duration curve needs to be carefully selected. A large number of

steps may result in intractability while a small number of steps may affect the

accuracy of the solution attained. In any case, it is important to check that the

optimal solution does not significantly change by incorporating an additional

step to describe the load-duration curve.

Note that the weighting factor corresponding to demand block t (σt) refers

to a portion of the hours in the year for which the load of the system is

approximated through the demand of that block. Clearly, the summation of

the weighting factors of all demand blocks equals the number of hours in a

year, i.e.,∑

t σt = 8760.

For the sake of clarity, the terms demand blocks, demand-bid blocks, and

production-offer blocks used throughout this dissertation are explained below.

1. Demand blocks:

These blocks are obtained from a stepwise approximation of the load-


Figure 1.2: Introduction: Demand blocks and demand-bid blocks (included indemand block t = t1) corresponding to the example given in Table 1.3.

duration curve as illustrated in Figure 1.1. Each demand block may

include several demands located at different buses of the network.

2. Demand-bid blocks:

Each demand may contain several demand-bid blocks, each of them in-

cluding a demand quantity and its associated bid price.

As an example, we consider a load-duration curve of the target year that

is approximated through four demand blocks (i.e., t = t1, t = t2, t = t3


and t = t4) as provided in Table 1.3, whose weighting factors (σt) are

1095, 2190, 2190 and 3285, respectively. Additionally, we consider two

demands, D1 and D2, located at two different buses of the network. Each

of those demands contains two demand-bid blocks.

For the example considered, Table 1.3 includes the maximum load of

each demand (PDmax

td ) and its demand-bid blocks. The first column gives

the demand blocks. Columns 2 and 5 provide the maximum load of

demands D1 and D2, respectively. In addition, columns 3 and 4 provide

the demand-bid blocks of D1, and columns 6 and 7 those of D2. These

demand-bid blocks identify the actual values of the load bids (MW) and

corresponding bid prices (e/MWh).

As an instance derived from Table 1.3, the maximum load of demand

D1 in the first demand block (t = t1) is 500 MW. Such demand bids 400

MW at 40 e/MWh, and the remaining demand at 38 e/MWh.

Additionally, Figure 1.2 depicts the demand blocks corresponding to the

example considered as well as the demand-bid blocks included in the first

demand block (t = t1).

3. Production-offer blocks:

Each generator’s offer may consist of several production-offer blocks de-

rived from a stepwise linearization of its quadratic cost curve. Thus,

each production-offer block includes a production quantity and its corre-

sponding production cost. Note that since the generators are considered

strategic, they can offer at prices different than their actual production

costs.

1.6.3 Network Representation

In this work, a dc representation of the transmission system is embedded within

the investment model considered. This way the effect of locating new units at

different buses is adequately represented. Congestion cases are also easily rep-

resented. For simplicity, active power losses are neglected; however, they can


be easily incorporated using a piecewise approximation [86, 91]. The lossless

linearized dc model of the transmission network used in this thesis is similar

to the one used in [44, 59, 117]. Further details on the network modeling are

available in [51] and [52].

1.6.4 Market Competition Modeling

To accurately describe the functioning of real-world electricity markets, we use

a supply function model, in which each producer offers both prices and quanti-

ties to the market. This model reflects more accurately industry practice than

other models, such as Cournot, Bertrand or conjectural variations. For the

sake of comparison, these imperfect competition models are briefly explained

below:

1. Cournot model: In a Cournot model each producer assumes that it is able

to alter the market price through its production level [129]. However, the

production levels of the rival producers are considered to be fixed. The

objective of the producer is to maximize its profit.

There are a number of relevant references in the literature using Cournot

model within an electricity market framework, e.g., [22, 28, 53, 57, 59, 61,

67, 79, 84, 105]. Additionally, analytical expressions of price, production

quantities and profit are provided in [116] to gain insight into the behav-

ior of electricity market with Cournot producers.

2. Bertrand model: Unlike a Cournot model in which the producers com-

pete on production quantities, the producers compete on offer prices in

a Bertrand model [95]. Thus, each producer seeks to maximize its profit

through its price offering and, therefore, the production quantity of pro-

ducers is determined by the market.

There are several papers in the literature using Bertrand models, e.g.,

[37, 58, 79, 89].

3. Conjectural variations model: This model is an upgraded version of the

Cournot model. It is assumed that the production strategy of any pro-

ducer affects i) the market price, and ii) the production quantity of the


rival producers. The impact of the production level of each producer

on those of rival producers is modeled by a reaction parameters, whose

values represent the degree of market competition varying from perfect

competition to the formation of a cartel or a monopoly [17].

References [21, 29, 47, 133] are examples of applications of conjectural

variation models to electricity markets. In addition, an analytical study

characterizing the outcomes of an electricity market using a conjectural

variation model is provided in [114] and [118].

4. Supply function model: In this model, each producer submits its supply

function offer to the market, containing a price and a production quan-

tity offer [77]. This model constitutes a more accurate description of

the functioning of real-world electricity markets if compared with other

imperfect competition models such as Cournot, Bertrand or conjectural

variations.

Supply function models have been extensively applied to electricity mar-

kets, e.g., [2, 3, 5, 54, 56, 112].

In this dissertation, a stepwise supply function model [117] is used, in

which each producer submits a set of stepwise price-production quantity

offers to the market operator.

1.6.5 Additional Modeling Assumptions

Additional modeling assumptions are described below:

1) Demands are assumed to be elastic to prices, i.e., they submit stepwise

price-quantity bid curves. However, demands do not behave strategically.

In addition, since demands are considered elastic, they are not necessar-

ily supplied at their corresponding maximum levels. Additionally, no

constraint is included in the model to force the supply of a minimum

demand level.

2) For simplicity, to avoid non-convexities and to easily obtain clearing

prices, on-off commitment decisions [99] are not modeled. However, if


considered, clearing prices can still be derived through uplifts and other

techniques, e.g., [13, 62, 98, 115].

3) A pool-based electricity market is considered where a market operator

clears the pool once a day, one day ahead, on an hourly basis, using a dc

representation of the network. The market operator seeks to maximize

the social welfare of the market considering the stepwise supply function

offers and the stepwise demand function bids submitted by producers

and consumers, respectively. The market clearing results are hourly pro-

ductions/consumptions and LMPs.

4) As presented in Figure 1.1, the load-duration curve of the system for

the target year is approximated through a number of stepwise demand

blocks. Note that clearing outcomes are identical for those hours, in

which the load of the system is approximated through the same demand

block. Thus, it is enough to represent the clearing of the pool for each

demand block instead of for each hour.

5) Different types of futures market products can be considered, e.g.,

a) A product spanning all the hours of the target year, cleared by a

futures base auction.

b) A product spanning just the peak hours of the target year, cleared

by a futures peak auction.

c) Monthly products, cleared by futures monthly auctions.

d) Weekly products, cleared by futures weekly auctions.

Among those futures products, we select futures base and futures peak

products to be modeled in this dissertation. The detailed description of

such products and their auctions are presented in Chapter 4.

6) Similarly to the pool, the market operator clears each futures market

auction maximizing the corresponding social welfare and considering the

stepwise supply function offers submitted by the producers and the step-

wise demand function bids submitted by the consumers. The market


clearing results are productions/consumptions and clearing price of that

auction. Since the futures base auction spans all the hours of the target

year, its clearing outcomes span all hours (i.e., all demand blocks) of the

year. However, the clearing outcomes of the futures peak auction span

only the peak hours (i.e., peak demand blocks) of the target year.

7) The uncertainties considered are modeled through a set of plausible sce-

narios. For the sake of simplicity, the risk of profit variability is not

considered in this thesis work. However, since the conditional value at

risk (CVaR) can be linearly expressed [111], such risk measure can be

easily incorporated into the proposed formulations.

8) The marginal clearing prices corresponding to each market, i.e., the

LMPs, are obtained as the dual variables associated with the market

balance constraints of the corresponding market. That is, the marginal-

ist theory is considered [123].

9) Pursuing simplicity, it is assumed that the strategic producer considered

in this dissertation does not face uncertainties in the futures market,

while it does in the pool.

10) The transmission network is not explicitly modeled to clear the futures

market auctions. However, network constraints are considered to clear

the pool, a shorter term market.

11) Renewable facilities are not considered in this thesis work as investment

options. However, a number of works are available in the literature that

uses a hierarchical model for wind power investment, e.g., references [7]

and [8].

12) For the sake of simplicity, some features of real-world markets are not

included in the proposed models in this thesis, such as capacity pay-

ments and renewable credits. Note, however, that capacity payments

or renewable credits translate generally into a reduction/increase in the

annualized cost of investment of the candidate units, which can easily be

integrated into the proposed models.

1.7. Solution Approach 25

13) For the sake of simplicity, contingencies, i.e., generators and transmis-

sion line outages, are not considered. However, security constraints can

be incorporated into the market clearing (lower-level problems), as for

example in [11, 15, 16].

14) Transmission charges to producers are not considered in this thesis. How-

ever, a number of works are available in the literature considering such

charges, e.g., reference [68].

1.7 Solution Approach

A hierarchical (bilevel) model is proposed in this thesis work to model the

behavior of each single producer, whose solution determines its optimal invest-

ment and offering decisions. The general structure of such model is illustrated

in Figure 1.3 and described below.

Note that for the sake of clarity, a single futures market auction and the

pool are considered in Figure 1.3. However, this model can be easily expanded

for a case including several futures market auctions.

1) This bilevel model consists of an upper-level problem and a set of lower-

level problems.

2) Each single producer behaves strategically through its strategic invest-

ment and offering decisions made at the upper-level problem. The objec-

tive of the upper-level problem is to minimize the minus expected profit

(which is equivalent to maximize the expected profit) of the producer,

and is subject to i) the upper-level constraints, and ii) a set of lower-level

problems.

3) The upper-level constraints include bounds on generation investment op-

tions, the investment budget limit, market regulator policies and the

non-negativity conditions of the strategic offers. As a regulatory policy,

we impose a minimum available capacity (including existing and newly

built units) to ensure supply security.


Figure 1.3: Introduction: General hierarchical (bilevel) structure of any singleproducer model.


4) A set of lower-level problems are considered representing the clearing of

the markets under different conditions. As shown in Figure 1.3, this set

of lower-level problems includes: i) a lower-level problem representing the

clearing of the futures market, and ii) a collection of lower-level problems

representing the clearing of the pool, one per demand block and scenario.

5) Each strategic producer anticipates the outcomes of the futures market

and the pool, e.g., clearing prices and production quantities, versus its

strategic decisions made at the upper-level problem. To this end, con-

straining the upper-level problem, the futures market and pool auctions

per demand block and scenario are cleared at the corresponding lower-

level problems for given investment and offering decisions. This allows

each strategic producer to obtain feedback regarding how its offering and

investment actions affect the market. Thus, the investment and offering

actions are variables in the upper-level problem while they are parame-

ters in the lower-level problems.

6) The lower-level problems constrain the upper-level problem, and thus

the primal and dual variable sets of the lower-level problem are included

in the variable set of the upper-level problem as well.

7) Only one futures market auction is considered in this section to describe

the modeling framework and thus, only one lower-level problem is de-

picted in Figure 1.3 representing the clearing of that auction. However,

if both futures base and futures peak auctions are considered, two dif-

ferent lower-level problems are required, one representing the clearing of

the futures base auction, and another one representing the clearing of

the futures peak auction.

8) The upper-level and the lower-level problems are interrelated as illus-

trated in Figure 1.4. On one hand, the lower-level problem pertaining

to the futures market determines the clearing price and the production

quantities in that market. In addition, the lower-level problems related

to the pool provide the LMPs and production quantities in the pool for

each demand block and each scenario. All market clearing prices and

281.Gen

eratio

nInvestm

ent:

Intro

ductio

n

!

"

# $

# $

!

%#

# $

Figure 1.4: Introduction: Interrelation between the upper-level and lower-level problems considering the futuresmarket and the pool auctions.


production quantities obtained in the lower-level problems directly influ-

ence the producer’s expected profit in the upper-level problem. On the

other hand, the investment and offering decisions made by the strategic

producer at the upper-level problem affect the market clearing outcomes

in the lower-level problems.

9) The futures market is cleared prior to clearing the pool, and thus the pro-

duction quantities of the producers in the futures market are parameters

in the market clearing of the pool as shown in Figure 1.4.

To solve the bilevel model of each strategic producer, such model needs

to be transformed into a single level MPEC. This transformation is carried

out by replacing each lower-level problem with its optimality conditions as

illustrated in Figure 1.5. Note that the optimality conditions of each lower-level

problem can be derived using either the KKT conditions or the primal-dual

transformation. The details of these two approaches are described in Section

B.2 of Appendix B.

As explained in Section 1.4, the first decision-making problem addressed

in this dissertation pertains to the strategic investment decisions of a given

producer. To solve such problem, the resulting MPEC from the procedure

explained above and illustrated in Figure 1.5 is recast into an MILP problem

solvable using currently available branch-and-cut algorithms.

In addition to the single producer investment problems, as explained in

Section 1.4, the second problem addressed in this dissertation pertains to the

generation investment equilibrium. To model and solve such problem, the

following steps are proposed:

1) To consider a bilevel model identical to the one proposed in Figure 1.3

for each strategic producer.

2) To transform this bilevel model into a single level MPEC as illustrated

in Figure 1.5.

3) To concatenate all producer MPECs, one per strategic producer. The

joint consideration of all these MPECs characterizes an EPEC as shown

in the upper plot of Figure 1.6.

301.Gen

eratio

nInvestm

ent:

Intro

ductio

n

Figure 1.5: Introduction: Transformation of the bilevel model of a strategic producer into its corresponding MPEC.

1.8. Thesis Objectives 31

1y

2y

ny

1y

2y

ny

Figure 1.6: Introduction: EPEC and its optimality conditions.

4) To replace each MPEC by its KKT conditions as illustrated in Figure

1.6. This results in the optimality conditions associated with the EPEC,

whose solutions render market equilibria.

5) To identify the EPEC equilibria, the optimality conditions associated

with the EPEC are first linearized. Then, an auxiliary MILP problem

is formulated including as constraints the mixed-integer linear system of

equalities and inequalities characterizing the EPEC equilibria.

The details of the proposed EPEC approach to identify generation investment

equilibria are presented in Chapter 5.

1.8 Thesis Objectives

Considering the context presented in Sections 1.3-1.5 above, the main objec-

tives of this dissertation are twofold:

1. To develop a non-heuristic model based on mathematical programming

and complementarity to assist a strategic producer in making informed

investment decisions. Specific objectives are:


1.1 To model the functioning of the market considering both pool and

futures trading floors.

1.2 To represent the oligopolistic behavior of the strategic producer

through a hierarchical model.

1.3 To model uncertainties through a set of plausible scenarios, and to

handle a large number of scenarios without intractability by imple-

menting a Benders’ decomposition scheme.

1.4 To analyze the impact of transmission congestion on generation

investment.

1.5 To model the regulatory policies imposed by the market regulator.

1.6 To compare the investment actions of the strategic producer with

and without the futures market.

1.7 To represent the arbitrage between the futures market and the pool,

and to analyze its influence on the expected profit and the invest-

ment decisions of the strategic producer.

1.8 To select the technologies of the units to be built among the avail-

able investment options, to obtain the capacity of such units, and

to locate them throughout the network.

2. To propose a non-heuristic model based on optimization and complemen-

tarity to identify generation investment equilibria. Specific objectives

are:

2.1 To characterize all the potential generation investment equilibria

in a network-constrained electricity market, where the producers

behave strategically through their i) investment decisions, and ii)

production offers.

2.2 To represent the interactions between a number of strategic in-

vestors in a network-constrained market as a game-theoretic model.

2.3 To study the impact of the number of strategic producers trading

in the market on the investment outcomes.

1.9. Thesis Organization 33

2.4 To analyze the investment and operational outcomes of the market

if a number of producers offer at marginal costs.

2.5 To study the impact of a regulatory policy to ensure supply security

on generation investment.

2.6 To evaluate the influence of transmission congestion on generation

investment.

2.7 To analyze the generation investment outcomes versus the available

investment budget.

2.8 To study the impact of the available investment technologies on

general market measures, e.g., social welfare and total producer

profit.

2.9 To draw policy recommendations for the market regulator to pro-

mote a market that operates as close as possible to perfect compe-

tition.

1.9 Thesis Organization

This document is organized as follows:

Chapter 1 provides an introduction including the thesis motivation, the

description of the considered problems, the literature review, the modeling

assumptions, the proposed solution approaches, and the thesis objectives.

Chapter 2 proposes a bilevel model for the generation investment decision-

making of a strategic producer competing with rival producers in a network-

constrained pool. Such model is equivalent to an MPEC that can be recast

as a tractable MILP problem using an exact linearization approach. Uncer-

tainties of rival offering and rival investment are modeled via scenarios. All

involved scenarios are considered simultaneously (direct solution). A small-

scale illustrative example and a realistic case study are analyzed to illustrate

the usefulness of the proposed methodology and to study its scalability.

Chapter 3 also addresses the problem presented in Chapter 2. However,

to tackle the computational burden, an alternative approach based on Ben-

ders’ decomposition is developed. This alternative approach is tractable even


if a large number of scenarios is considered since the model decomposes by sce-

nario. First, it is numerically shown that the expected profit of the strategic

producer is convex enough with respect to investment decisions; thus, an effec-

tive implementation of Benders’ approach is possible. This chapter provides a

numerical comparison between the approach based on Benders’ decomposition

and the direct solution approach proposed in Chapter 2. In addition, a realis-

tic case study is analyzed considering a large number of scenarios representing

the uncertainties of rival offering, rival investment and demand growth.

Chapter 4 analyzes the effect of the futures market on the investment de-

cisions of a strategic producer competing with rival producers. To this end, a

bilevel model is proposed whose upper-level problem represents the investment

and offering actions of the considered strategic producer, and whose multiple

lower-level problems represent the clearing of the futures market auctions and

the pool under different operating conditions. The only uncertainty taken into

account pertains to rival offering in the pool. The proposed bilevel model ren-

ders an MPEC that can be recast as a tractable MILP problem using i) an ex-

act linearization approach, and ii) an approximate binary expansion approach

(Subsection B.5.2 of Appendix B). A realistic case study without network con-

straints is analyzed to illustrate the relevance of the proposed methodology.

Chapter 5 proposes a methodology to characterize generation investment

equilibria in a pool-based network-constrained electricity market, where the

producers behave strategically. To this end, the investment problem of each

strategic producer is represented using a bilevel model, whose upper-level prob-

lem determines the optimal investment and the supply offering curves to maxi-

mize its profit, and whose several lower-level problems represent different mar-

ket clearing scenarios. This model is transformed into an MPEC through

replacing the lower-level problems by their optimality conditions. The joint

consideration of all producer MPECs, one per producer, constitutes an EPEC.

To identify the solutions of this EPEC, each MPEC is replaced by its KKT

conditions, which are in turn linearized. The resulting mixed-integer linear

system of equalities and inequalities allows determining the EPEC equilibria

through an auxiliary MILP problem. To illustrate the ability of the proposed

approach to identify meaningful investment equilibria, a two-node illustrative

1.9. Thesis Organization 35

example and a realistic case study are examined and the results obtained are

reported and discussed.

Chapter 6 concludes this dissertation providing a summary, relevant con-

clusions drawn from the studies carried out throughout the thesis work, and

the main contributions of the thesis. Finally, some topics are suggested for

future research.

Appendix A provides the data of the 24-bus IEEE Reliability Test System

(RTS) used in Chapters 2 to 5.

Appendix B provides mathematical background including the bilevel model

used in Chapters 2 to 5, two alternative procedures used in Chapters 2 to 5

for deriving the optimality conditions associated with a linear optimization

problem, the MPEC used in Chapters 2 to 5, the EPEC used in Chapter 5,

Benders’ decomposition algorithm used in Chapter 3, the complementarity lin-

earization used in Chapters 2 to 5, and the binary expansion approximation

used in Chapter 4.

Chapter 2

Strategic Generation Investment

2.1 Introduction

This chapter proposes a hierarchical (bilevel) model for generation investment

decision-making of a strategic producer competing with rival producers in a

pool with supply function offers. The proposed model is transformed into a

stochastic mathematical program with equilibrium constraints (MPEC) that

can be recast as a large-scale mixed-integer linear programming (MILP) prob-

lem, which can be solved using commercially available branch-and-cut algo-

rithms. This model is able to optimally locate throughout the network the

generation units to be built and to select the best production technologies.

Note that the strategies of rival producers in the pool and their investment

decisions are uncertain parameters represented through scenarios. In other

words, we use scenarios to describe uncertainty pertaining to i) rival offers and

ii) rival investments.

2.2 Approach

The investment and offering decisions of the strategic producer under study

are described through a bilevel model. The general structure of bilevel models

is explained in Section 1.7 of Chapter 1. Additionally, mathematical details

on bilevel models are provided in Section B.1 of Appendix B.

37

38 2. Strategic Generation Investment

In the bilevel model proposed in this chapter, the upper-level problem rep-

resents both the investment decisions of the producer and its strategic offer-

ing corresponding to each demand block and scenario. Offering is carried out

through a stepwise supply function per demand block and scenario, which gen-

erally differs from the corresponding production cost function. This upper-level

problem is constrained by a collection of lower-level problems that represent

the clearing of the market for each demand block and scenario. The target

of each of these problems is to maximize the corresponding declared social

welfare.

2.2.1 Modeling Assumptions

For clarity, the main assumptions of the proposed model are summarized below:

1) A dc representation of the transmission network is embedded within the

considered investment model, as explained in detail in Subsection 1.6.3 of

Chapter 1. This way, the effect of locating new units at different buses is

adequately represented. Congestion cases are also easily represented. For

simplicity, active power losses are neglected.

2) A pool-based electricity market is considered in this chapter where a market

operator clears the pool once a day, one day ahead, and on an hourly basis.

The market operator seeks to maximize the social welfare considering the

stepwise supply function offers submitted by producers and the demand

function bids submitted by consumers. The market clearing results are

hourly productions, consumptions and locational marginal prices (LMPs).

3) The clearing prices corresponding to the pool, i.e., the pool LMPs, are ob-

tained as the dual variables associated with the balance constraints. Thus,

the marginalist price theory is adopted [123].

4) Pursuing simplicity, the futures market is not considered in this chapter.

5) The proposed investment model is static, i.e., a single target year is con-

sidered for decision-making, as explained in detail in Subsection 1.6.1 of

2.2. Approach 39

Chapter 1. Such target year represents the final stage of the planning hori-

zon, and the model uses annualized cost referred to that year.

6) The load-duration curve pertaining to the target year is approximated

through a number of demand blocks, as explained in detail in Subsection

1.6.2 of Chapter 1. Additionally, each demand block may include several

demands located at different buses of the network.

7) The producer under study is strategic, i.e., it can alter the market clearing

outcomes (e.g., LMPs and production quantities) through its investment

and offering strategies.

8) The strategic producer explicitly anticipates the impact of its investment

and offering actions on the market outcomes, e.g., LMPs and production

quantities. This is achieved through the lower-level market clearing prob-

lems, one per demand block and scenario.

9) The strategic generation investment problem is subject to several uncer-

tainties, e.g., the behavior of rival producers, demand growth, investment

costs for different technologies, regulatory policies, etc. Among such uncer-

tainties, the strategies of rival producers in the pool and their investment

decisions are considered in this chapter as uncertain parameters represented

through scenarios. Other parameters, e.g., demand growth, investment

costs for different technologies and regulatory policies are assumed to be

known.

10) We explicitly represent increasing stepwise offer curves (supply functions)

for producers and decreasing stepwise bidding curves for consumers.

11) Demands are assumed to be elastic to prices, i.e., demands submit stepwise

price-quantity bid curves to the market. However, they do not behave

strategically. In addition, since demands are considered elastic, they are

not necessarily supplied at their corresponding maximum levels (PDmax

td ).

Additionally, no constraint is included in the model to force the supply of

a minimum demand level.


Figure 2.1: Direct solution: Bilevel structure of the proposed strategic gener-ation investment model.

2.2.2 Structure of the Proposed Model

Figure 2.1 shows the bilevel structure of the proposed model. The upper-level

problem represents the expected profit maximization (or minus expected profit

minimization) of the strategic producer subject to investment constraints, and

to the lower-level problems. Each lower-level problem, one per demand block

and scenario, represents the market clearing with the target of maximizing

the social welfare (or minimizing the minus social welfare) and is subject to

the power balance at every bus, power limits for production and consump-

tion, transmission line capacity limits, voltage angle bounds and reference bus

identification.

Note that the upper-level and the lower-level problems are interrelated as

2.3. Formulation 41

Figure 2.2: Direct solution: Interrelation between the upper-level and lower-level problems.

illustrated in Figure 2.2. On one hand, the lower-level problems determine

the LMPs and the power production quantities, which directly influence the

producer’s expected profit in the upper-level problem. On the other hand, the

strategic offering and investment decisions made by the strategic producer in

the upper-level problem affect the market clearing outcomes in the lower-level

problems.

The optimality region of each lower-level problem is represented by its

Karush-Kuhn-Tucker (KKT) conditions. Considering the upper-level problem

and replacing the lower-level problems with the corresponding sets of the KKT

conditions results in an MPEC. This transformation is illustrated in Figure 1.5

of Chapter 1. Mathematical details are provided in Section B.2 of Appendix

B.

Finally, linearization techniques are used to convert the MPEC into an

MILP problem.

2.3 Formulation

The proposed bilevel model, the resulting MPEC and the final MILP problem

are formulated in this section.


2.3.1 Notational Assumptions

The following notational assumptions are considered in the formulation.

1) The supply offer curve of each unit may consist of several production-

offer blocks derived from a stepwise linearization of its quadratic cost

curve, as described in Subsection 1.6.2 of Chapter 1. To incorporate

such production-offer blocks into the formulation, an additional index

(e.g., index b) is required. For example, variable P Stib refers to the power

produced by generation block b of candidate unit i ∈ ΨS in demand

block t. Pursuing notational simplicity, such index is not included in

the formulation of this chapter, i.e., single-block production units are

considered. However, production-offer blocks are used in the case studies

of this chapter.

2) Each demand may contain several demand-bid blocks, as also described

in Subsection 1.6.2 of Chapter 1. To incorporate such demand-bid blocks

of the demands into the formulation, an additional index (e.g., index r)

is needed. For example, variable PDtdr refers to the power consumed by

demand-bid block r of demand d ∈ ΨD in demand block t. Pursuing

notational simplicity, such index is also not included in the formula-

tion of this chapter, i.e., single-block demands are considered. However,

demand-bid blocks are used in the case studies of this chapter.

2.3.2 Bilevel Model

The formulation of the proposed bilevel model is given by (2.1)-(2.2) below.

Note that the upper-level problem includes (2.1), and is constrained by the

lower-level problems (2.2), one per demand block t and scenario w. The dual

variables of each lower-level problem (2.2) are indicated at their corresponding

constraints following a colon.

2.3. Formulation 43

MinimizeΞUL

∑

i∈ΨS

KiXi

−∑

w

ϕw∑

t

σt

[(∑

i∈ΨS

P Stiwλt(n:i∈ΨS

n)w−∑

i∈ΨS

P StiwC

Si

)

+

(∑

k∈ΨES

PEStkwλt(n:k∈ΨES

n )w −∑

k∈ΨES

PEStkwC

ESk

)](2.1a)

subject to:

Xi =∑

h

uihXih ∀i ∈ ΨS (2.1b)

∑

h

uih = 1 ∀i (2.1c)

uih ∈ 0, 1 ∀i ∈ ΨS, ∀h (2.1d)

λtnw, PStiw, P

EStkw ∈ arg minimize

ΞPrimaltw ,∀t,∀w

∑

i∈ΨS

αStiwP

Stiw +

∑

k∈ΨES

αEStkwP

EStkw +

∑

j∈ΨO

COtjwP

Otjw −

∑

d∈ΨD

UDtdP

Dtdw (2.2a)

subject to:∑

d∈ΨDn

PDtdw +

∑

m∈Ωn

Bnm(θtnw − θtmw)−∑

i∈ΨSn

P Stiw

−∑

k∈ΨESn

PEStkw −

∑

j∈ΨOn

POtjw = 0 : λtnw ∀n (2.2b)

0 ≤ P Stiw ≤ Xi : µSmin

tiw , µSmax

tiw ∀i ∈ ΨS (2.2c)

0 ≤ PEStkw ≤ PESmax

k : µESmin

tkw , µESmax

tkw ∀k ∈ ΨES (2.2d)

0 ≤ POtjw ≤ POmax

jw : µOmin

tjw , µOmax

tjw ∀j ∈ ΨO (2.2e)

0 ≤ PDtdw ≤ PDmax

td : µDmin

tdw , µDmax

tdw ∀d ∈ ΨD (2.2f)

−Fmax

nm ≤ Bnm(θtnw − θtmw) ≤ Fmax

nm : νmin

tnmw, νmax

tnmw ∀n, ∀m ∈ Ωn (2.2g)

−π ≤ θtnw ≤ π : ξmin

tnw, ξmax

tnw ∀n (2.2h)

θtnw = 0 : ξ1

tw n = 1 (2.2i)


∀t, ∀w.

The primal optimization variables of each lower-level problem (2.2) are

included in set ΞPrimaltw below:

ΞPrimaltw = P S

tiw, PEStkw, P

Otjw, P

Dtdw, θtnw.

In addition, variable set ΞDualtw below contains the dual optimization vari-

ables of each lower-level problem (2.2):

ΞDualtw = λtnw, µ

Smin

tiw , µSmax

tiw , µESmin

tkw , µESmax

tkw , µOmin

tjw , µOmax

tjw , µDmin

tdw , µDmax

tdw , νmin

tnmw,

νmax

tnmw, ξmin

tnw, ξmax

tnw, ξ1

tw.

The producer under study behaves strategically through its following strate-

gic decisions made at the upper-level problem (2.1):

1) Strategic investment decisions, i.e., Xi ∀i ∈ ΨS.

2) Strategic offering decisions, i.e., αSti ∀t, ∀i ∈ ΨS and αES

tk ∀t, ∀k ∈ ΨES.

Note that the strategic producer anticipates the market outcomes, e.g.,

LMPs and production quantities, versus its strategic investment and offering

decisions. To this end, constraining the upper-level problem (2.1), the pool

is cleared in each lower-level problem (2.2) for given investment and offering

decisions. This allows each strategic producer to obtain feedback regarding

how its offering and investment actions affect the market.

Thus, αSti ∀t, ∀i ∈ ΨS, αES

tk ∀t, ∀k ∈ ΨES and Xi ∀i ∈ ΨS are variables in

the upper-level problem (2.1) while they are parameters in the lower-level

problems (2.2). Note that this makes the lower-level problems (2.2) linear and

thus convex.

In addition, since the lower-level problems (2.2) constrain the upper-level

problem (2.1), the lower-level variable sets ΞPrimaltw and ΞDual

tw are included in

the variable set of the upper-level problem as well. Thus, the primal variables

of the upper-level problem (2.1) are those in set ΞUL below:

ΞUL =ΞPrimaltw , ΞDual

tw , αStiw, α

EStkw, Xi, uih.

The objective function (2.1a) is the minus expected profit (investment cost

minus expected operations revenue) of the strategic producer where ϕw is

the probability associated with scenario w. LMP λtnw is the dual variable

2.3. Formulation 45

of the balance constraint at bus n, demand block t and scenario w, obtained

endogenously within the corresponding lower-level problem.

For each available investment technology i ∈ ΨS (e.g., nuclear, coal, gas,

CCGT, etc.), equations (2.1b)-(2.1d) allow the strategic producer to choose

among the available investment options, being one of the options no invest-

ment, e.g., investment options can be 0, 200, 500 or 1000 MW.

Note that the LMPs (λtnw) and the productions (P Stiw and PES

tkw) in the

upper-level problem (2.1) belong to the feasible region defined by lower-level

problems (2.2).

The minimization of the minus social welfare of each lower-level problem

is expressed by (2.2a).

The price offer of rival producers (COtjw) depends on w to model rival offering

uncertainty.

Equations (2.2b) enforce the power balance at every bus, being the associ-

ated dual variables LMPs.

Equations (2.2c), (2.2d) and (2.2e) enforce capacity limits for the new and

existing units of the strategic producer and the units of rival producers, re-

spectively.

Note that to model rival investment uncertainty, the upper bound of (2.2e),

i.e., POmax

jw , depends on w.

Constraints (2.2f) bound the power consumed by each demand.

Constraints (2.2g) enforce the transmission capacity limits of each line.

Constraints (2.2h) enforce angle bounds for each node, and constraints

(2.2i) impose n = 1 to be the reference bus.

To transform bilevel model (2.1)-(2.2) into a single-level MPEC, the opti-

mality conditions of lower-level problems (2.2) need to be derived. The next

subsection addresses this issue.

2.3.3 Optimality Conditions Associated with the Lower-

Level Problems (2.2)

To solve the proposed bilevel model (2.1)-(2.2), it is convenient to transform it

into a single-level optimization problem. To this end, each lower-level problem


(2.2) is replaced with its equivalent optimality conditions, which renders an

MPEC.

According to Section B.2 of Appendix B, the optimality conditions as-

sociated with the lower-level problems (2.2) can be formulated through two

alternative approaches: KKT conditions and primal-dual transformation. In

this respect, the following two observations are relevant:

a) The first approach (KKT conditions) includes a set of complementarity

conditions as a part of the optimality conditions. Such complementarity

conditions can be linearized as explained in Subsection B.5.1 of Appendix

B, but at the cost of adding a set of auxiliary binary variables to the variable

set ΞUL.

b) The second approach (primal-dual transformation) includes the non-linear

strong duality equality as a part of the optimality conditions. The source

of non-linearity is the product of continuous variables.

Considering the two observations above, the first approach (i.e., KKT con-

ditions) is used in this chapter to represent the optimality conditions associated

with the lower-level problems (2.2). However, the strong duality equality (ob-

tained from the primal-dual transformation) needs to be derived, because it

allows linearizing the resulting MPEC.

The KKT conditions associated with the lower-level problems (2.2) are

derived in the next subsection.

2.3.3.1 KKT Conditions Associated with the Lower-Level Prob-

lems (2.2)

To obtain the KKT conditions associated with the lower-level problems (2.2),

the corresponding Lagrangian function L is written below:

2.3. Formulation 47

L =∑

t(i∈ΨS)w

αStiwP

Stiw +

∑

t(k∈ΨES)w

αEStkwP

EStkw

+∑

t(j∈ΨO)w

COtjwP

Otjw −

∑

t(d∈ΨD)w

UDtdP

Dtdw

+∑

tnw

λtnw

∑

d∈ΨDn

PDtdw +

∑

m∈Ωn


i∈ΨSn

P Stiw

−∑

k∈ΨESn

PEStkw −

∑

j∈ΨOn

POtjw

+∑

t(i∈ΨS)w

µSmax

tiw

(P Stiw −Xi

)−

∑

t(i∈ΨS)w

µSmin

tiw P Stiw

+∑

t(k∈ΨES)w

µESmax

tkw

(PEStkw − P

ESmax

k

)−

∑

t(k∈ΨES)w

µESmin

tkw PEStkw

+∑

t(j∈ΨO)w

µOmax

tjw

(POtjw − P

Omax

jw

)−

∑

t(j∈ΨO)w

µOmin

tjw POtjw

+∑

t(d∈ΨD)w

µDmax

tdw

(PDtdw − P

Dmax

td

)−

∑

t(d∈ΨD)w

µDmin

tdw PDtdw

+∑

tn(m∈Ωn)w

νmax

tnmw

[Bnm(θtnw − θtmw)− F

max

nm

]

−∑

tn(m∈Ωn)w

νmin

tnmw

[Bnm(θtnw − θtmw) + F

max

nm

]

+∑

tnw

ξmax

tnw (θtnw − π)−∑

tnw

ξmin

tnw (θtnw + π)

+∑

tw

ξ1

twθt(n=1)w. (2.3)

Considering the Lagrangian function (2.3), the first-order KKT conditions

associated with the lower-level problems (2.2) are derived as follows:

∂L

∂P Stiw

=

αStiw − λt(n:i∈ΨS

n)w+ µSmax

tiw − µSmin

tiw = 0


∀t, ∀i ∈ ΨS, ∀w (2.4a)

∂L

∂PEStkw

=

αEStkw − λt(n:k∈ΨES

n )w + µESmax

tkw − µESmin

tkw = 0

∀t, ∀k ∈ ΨES, ∀w (2.4b)

∂L

∂POtjw

=

COtjw − λt(n:j∈ΨO

n )w+ µOmax

tjw − µOmin

tjw = 0

∀t, ∀j ∈ ΨO, ∀w (2.4c)

∂L

∂PDtdw

=

−UDtd + λt(n:d∈ΨD

n )w+ µDmax

tdw − µDmin

tdw = 0

∀t, ∀d ∈ ΨD, ∀w (2.4d)

∂L

∂θtnw=∑

m∈Ωn

Bnm(λtnw − λtmw)

+∑

m∈Ωn

Bnm(νmax

tnmw − νmax

tmnw)

−∑

m∈Ωn

Bnm(νmin

tnmw − νmin

tmnw)

+ξmax

tnw − ξmin

tnw + (ξ1

tw)n=1 = 0 ∀t, ∀n, ∀w (2.4e)

∑

d∈ΨDn

PDtdw +

∑

m∈Ωn


i∈ΨSn

P Stiw

−∑

k∈ΨESn

PEStkw −

∑

j∈ΨOn

POtjw = 0 ∀n, ∀w (2.4f)

θtnw = 0 n = 1, ∀t, ∀w (2.4g)

0 ≤ P Stiw ⊥ µSmin

tiw ≥ 0 ∀t, ∀i ∈ ΨS, ∀w (2.4h)

0 ≤ PEStkw ⊥ µESmin

tkw ≥ 0 ∀t, ∀k ∈ ΨES, ∀w (2.4i)

0 ≤ POtjw ⊥ µOmin

tjw ≥ 0 ∀t, ∀j ∈ ΨO, ∀w (2.4j)

2.3. Formulation 49

0 ≤ PDtdw ⊥ µDmin

tdw ≥ 0 ∀t, ∀d ∈ ΨD, ∀w (2.4k)

0 ≤ (Xi − PStiw) ⊥ µSmax

tiw ≥ 0 ∀t, ∀i ∈ ΨS, ∀w (2.4l)

0 ≤ (PESmax

k − PEStkw) ⊥ µESmax

tkw ≥ 0 ∀t, ∀k ∈ ΨES, ∀w (2.4m)

0 ≤ (POmax

jw − POtjw) ⊥ µOmax

tjw ≥ 0 ∀t, ∀j ∈ ΨO, ∀w (2.4n)

0 ≤ (PDmax

td − PDtdw) ⊥ µDmax

tdw ≥ 0 ∀t, ∀d ∈ ΨD, ∀w (2.4o)

0 ≤ [Fmax

nm +Bnm(θtnw − θtmw)] ⊥ νmin

tnmw ≥ 0 ∀t, ∀n, ∀m ∈ Ωn, ∀w(2.4p)

0 ≤ [Fmax

nm − Bnm(θtnw − θtmw)] ⊥ νmax

tnmw ≥ 0 ∀t, ∀n, ∀m ∈ Ωn, ∀w (2.4q)

0 ≤ (π − θtnw) ⊥ ξmax

tnw ≥ 0 ∀t, ∀n, ∀w (2.4r)

0 ≤ (π + θtnw) ⊥ ξmin

tnw ≥ 0 ∀t, ∀n, ∀w (2.4s)

λtnw : free ∀t, ∀n, ∀w (2.4t)

ξ1

tw : free ∀t, ∀w. (2.4u)

The structure of the KKT conditions (2.4) is explained below:

a) Equality constraints (2.4a)-(2.4e) are derived from differentiating the La-

grangian function L with respect to the primal variables included in the set

ΞPrimaltw .

b) Equality constraints (2.4f) and (2.4g) are the primal equality constraints

(2.2b) and (2.2i) in the lower-level problems (2.2).

c) Complementarity conditions (2.4h)-(2.4s) are related to the inequality con-

straints (2.2c)-(2.2h).

d) Conditions (2.4t) and (2.4u) state that the dual variables associated with

the equality constraints (2.2b) and (2.2i) are free.

Note that due to the linearity and thus convexity of the lower-level prob-

lems (2.2), KKT conditions (2.4) are necessary and sufficient conditions for

optimality.

As explained in Subsection 2.3.3, lower-level problems (2.2) are replaced by

the KKT conditions (2.4) rendering an MPEC. Additionally, the strong duality

equality obtained from the primal-dual transformation needs to be derived


since such equality is used to linearize the MPEC. The next subsection derives

the strong duality equality.

2.3.3.2 Strong Duality Equality Associated with the Lower-Level

Problems (2.2)

For clarity, the corresponding dual problem of each lower-level problem (2.2)

is derived. These dual problems are given by (2.5) below:

Maximize

ΞDualtw

−∑

i∈ΨS

µSmax

tiw Xi −∑

k∈ΨES

µESmax

tkw PESmax

k −∑

j∈ΨO

µOmax

tjw POmax

jw

−∑

d∈ΨD

µDmax

tdw PDmax

td −∑

n(m∈Ωn)

νmin

tnmwFmax

nm −∑

n(m∈Ωn)

νmax

tnmwFmax

nm

−∑

n

ξmin

tnwπ −∑

n

ξmax

tnwπ (2.5a)

subject to:

(2.4a)− (2.4e) (2.5b)

µSmin

tiw ≥ 0; µSmax

tiw ≥ 0 ∀i ∈ ΨS (2.5c)

µESmin

tkw ≥ 0; µESmax

tkw ≥ 0 ∀k ∈ ΨES (2.5d)

µOmin

tjw ≥ 0; µOmax

tjw ≥ 0 ∀j ∈ ΨO (2.5e)

µDmin

tdw ≥ 0; µDmax

tdw ≥ 0 ∀d ∈ ΨD (2.5f)

νmin

tnmw ≥ 0; νmax

tnmw ≥ 0 ∀n, ∀m ∈ Ωn (2.5g)

ξmin

tnw ≥ 0; ξmax

tnw ≥ 0 ∀n (2.5h)

(2.4t), (2.4u) (2.5i)∀t, ∀w.

2.3. Formulation 51

Considering each primal lower-level problem (2.2) and its corresponding

dual problem (2.5), the optimality conditions of each lower-level problem (2.2)

resulting from the primal-dual transformation can be derived as given by (2.6).

Note that the optimality conditions (2.6) are equivalent to the KKT conditions

(2.4).

(2.2b)− (2.2i) (2.6a)

(2.5b)− (2.5i) (2.6b)

∑

i∈ΨS

αStiwP

Stiw +

∑

k∈ΨES

αEStkwP

EStkw +

∑

j∈ΨO

COtjwP

Otjw −

∑

d∈ΨD

UDtdP

Dtdw =

−∑

i∈ΨS

µSmax

tiw Xi −∑

k∈ΨES

µESmax

tkw PESmax

k −∑

j∈ΨO

µOmax

tjw POmax

jw

−∑

d∈ΨD

µDmax

tdw PDmax

td −∑

n(m∈Ωn)

νmin

tnmwFmax

nm −∑

n(m∈Ωn)

νmax

tnmwFmax

nm

−∑

n

ξmin

tnwπ −∑

n

ξmax

tnwπ (2.6c)

∀t, ∀w,

where constraint (2.6c) represents the strong duality equality, i.e., it enforces

the equality of the values of primal objective function (2.2a) and dual objective

function (2.5a) at the optimal solution. This equality is used to linearize the

final MPEC.

2.3.4 MPEC

A single-level MPEC corresponding to the bilevel model (2.1)-(2.2) is obtained

by replacing each lower-level problem (2.2) with its KKT conditions (2.4). The

resulting MPEC is given by (2.7) below:


MinimizeΞUL

∑

i∈ΨS

KiXi

−∑

w

ϕw∑

t

σt

[(∑

i∈ΨS

P Stiwλt(n:i∈ΨS

n)w−∑

i∈ΨS

P StiwC

Si

)

+

(∑

k∈ΨES


n )w −∑

k∈ΨES

PEStkwC

ESk

)](2.7a)

subject to:

(2.1b)− (2.1d) (2.7b)

(2.4). (2.7c)

MPEC (2.7) includes some non-linearities, but it can be transformed into

the MILP problem as explained in the next subsection.

2.3.5 MPEC Linearization

MPEC (2.7) includes the following non-linearities:

a) A non-linear term in the objective function (2.7a). This term is denoted

as Ztw and given below:

Ztw =∑

i∈ΨS

P Stiwλt(n:i∈ΨS

n)w+∑

k∈ΨES


n )w ∀t, ∀w. (2.8)

The non-linearity in Ztw is the product of production quantity and price

variables. An exact linear expression of Ztw can be obtained as explained

in Subsection 2.3.5.1.

b) The complementarity conditions (2.4h)-(2.4s) are included in (2.7c). Such

conditions can be linearized through the approach explained in Subsec-

2.3. Formulation 53

tion B.5.1 of Appendix B, which relies on using the auxiliary binary

variables.

2.3.5.1 Linearizing Ztw

The exact linearization approach proposed in [113] is used here to find a linear

expression for the non-linear term Ztw that appears in the objective function

(2.7a).

To achieve this, the strong duality equality (2.6c) and some of the KKT

equalities (2.4) are employed as explained in the following. The derivation

below is valid for each demand block t and scenario w.

From complementarity conditions (2.4l) and (2.4m):

∑

i∈ΨS

µSmax

tiw Xi =∑

i∈ΨS

µSmax

tiw P Stiw (2.9a)

∑

k∈ΨES

µESmax

tkw PESmax

k =∑

k∈ΨES

µESmax

tkw PEStkw. (2.9b)

Substituting (2.9a) and (2.9b) in the strong duality equality (2.6c) renders

the equality below:

∑

i∈ΨS

P Stiw(α

Stiw + µSmax

tiw ) +∑

k∈ΨES

PEStkw(α

EStkw + µESmax

tkw ) =

−∑

j∈ΨO

COtjwP

Otjw +

∑

d∈ΨD

UDtdP

Dtdw − Ytw, (2.9c)

where

Ytw =∑

j∈ΨO

µOmax

tjw POmax

jw +∑

d∈ΨD

µDmax

tdw PDmax

td +∑

n(m∈Ωn)

νmin

tnmwFmax

nm

+∑

n(m∈Ωn)

νmax

tnmwFmax

nm +∑

n

ξmin

tnwπ +∑

n

ξmax

tnwπ. (2.9d)


On the other hand, from the KKT conditions (2.4c) and (2.4d):

λt(n:i∈ΨSn)w

= αStiw + µSmax

tiw − µSmin

tiw (2.9e)

λt(n:k∈ΨESn )w = αES

tkw + µESmax

tkw − µESmin

tkw . (2.9f)

Multiplying equalities (2.9e) and (2.9f) by production variables P Stiw and

PEStkw, respectively, renders the equalities below:

∑

i∈ΨS

P Stiwλt(n:i∈ΨS

n)w=∑

i∈ΨS

αStiwP

Stiw +

∑

i∈ΨS

µSmax

tiw P Stiw −

∑

i∈ΨS

µSmin

tiw P Stiw (2.9g)

∑

k∈ΨES


n )w =∑

k∈ΨES

αEStkwP

EStkw +

∑

k∈ΨES

µESmax

tkw PEStkw

−∑

k∈ΨES

µESmin

tkw PEStkw. (2.9h)

Additionally, from the complementarity conditions (2.4h) and (2.4i):

∑

i∈ΨS

µSmin

tiw P Stiw = 0 (2.9i)

∑

k∈ΨES

µESmin

tkw PEStkw = 0. (2.9j)

Using (2.9i) and (2.9j) to simplify (2.9g) and (2.9h) renders (2.9k) below:

∑

i∈ΨS

P Stiwλt(n:i∈ΨS

n)w+∑

k∈ΨES


n )w =

∑

i∈ΨS

P Stiw(α

Stiw + µSmax

tiw ) +∑

k∈ΨES

PEStkw(α

EStkw + µESmax

tkw ). (2.9k)

Finally, considering (2.9c) and (2.9k):

∑

i∈ΨS

P Stiwλt(n:i∈ΨS

n)w+∑

k∈ΨES

PESt(n:k∈ΨES

n )wλtnw =

−∑

j∈ΨO

COtjwP

Otjw +

∑

d∈ΨD

UDtdP

Dtdw − Ytw. (2.9l)

2.3. Formulation 55

Note that the equality (2.9l) provides a linear equivalent for the non-linear

term Ztw.

2.3.6 MILP Formulation

Considering the linearization techniques presented in Subsection 2.3.5 above,

MPEC (2.7) can be transformed into the MILP problem given by (2.10)-(2.18):

MinimizeΞUL, Ξψ

∑

i∈ΨS

KiXi −∑

w

ϕw∑

t

σt

(ZLintw −

∑

i∈ΨS

P StiwC

Si −

∑

k∈ΨES

PEStkwC

ESk

)(2.10)

subject to:

1) Exact linearization of the non-linear term Ztw that is included in the objec-

tive function (2.7a): This linear equivalent is denoted as ZLintw and derived

using the linearization procedure explained in Subsection 2.3.5.1.

ZLintw = −

∑

j∈ΨO

COtjwP

Otjw +

∑

d∈ΨD

UDtdP

Dtdw −

∑

j∈ΨO

µOmax

tjw POmax

jw

−∑

d∈ΨD

µDmax

tdw PDmax

td −∑

n(m∈Ωn)

νmin

tnmwFmax

nm

−∑

n(m∈Ωn)

νmax

tnmwFmax

nm −∑

n

ξmin

tnwπ −∑

n

ξmax

tnwπ. (2.11)

2) The upper-level constraints (2.1b)-(2.1d):

Xi =∑

h

uihXih ∀i ∈ ΨS (2.12a)

∑

h

uih = 1 ∀i ∈ ΨS (2.12b)

uih ∈ 0, 1 ∀i ∈ ΨS, ∀h. (2.12c)


3) Equality constraints included in the KKT conditions (2.4):

αStiw − λt(n:i∈ΨS

n)w+ µSmax

tiw − µSmin

tiw = 0 ∀t, ∀i, ∀w (2.13a)

αEStkw − λt(n:k∈ΨES

n )w + µESmax

tkw − µESmin

tkw = 0 ∀t, ∀k, ∀w (2.13b)

COtjw − λt(n:j∈ΨO

n )w+ µOmax

tjw − µOmin

tjw = 0 ∀t, ∀j, ∀w (2.13c)

−UDtd + λt(n:vd∈ΨD

n )w+ µDmax

tdw − µDmin

tdw = 0 ∀t, ∀d, ∀w (2.13d)

∑

m∈Ωn

Bnm(λtnw − λtmw)

+∑

m∈Ωn

Bnm(νmax

tnmw − νmax

tmnw)

−∑

m∈Ωn

Bnm(νmin

tnmw − νmin

tmnw)

+ξmax

tnw − ξmin

tnw + (ξ1

tw)n=1 = 0 ∀t, ∀n, ∀w (2.13e)

∑

d∈ΨDn

PDtdw +

∑

m∈Ωn


i∈ΨSn

P Stiw

−∑

k∈ΨESn

PEStkw −

∑

j∈ΨOn

POtjw = 0 ∀n, ∀w (2.13f)

θtnw = 0 n = 1, ∀t, ∀w.(2.13g)

4) Conditions (2.4t) and (2.4u) included in the KKT conditions (2.4):

λtnw : free ∀n (2.14a)

ξ1

tw : free n = 1. (2.14b)

5) Mixed-integer linear equivalents of the complementarity conditions (2.4h)-

(2.4k) included in the KKT conditions (2.4):

P Stiw ≥ 0 ∀t, ∀i ∈ ΨS, ∀w (2.15a)

2.3. Formulation 57

PEStkw ≥ 0 ∀t, ∀k ∈ ΨES, ∀w (2.15b)

POtjw ≥ 0 ∀t, ∀j ∈ ΨO, ∀w (2.15c)

PDtdw ≥ 0 ∀t, ∀d ∈ ΨD, ∀w (2.15d)

µSmin

tiw ≥ 0 ∀t, ∀i ∈ ΨS, ∀w (2.15e)

µESmin

tkw ≥ 0 ∀t, ∀k ∈ ΨES, ∀w (2.15f)

µOmin

tjw ≥ 0 ∀t, ∀j ∈ ΨO, ∀w (2.15g)

µDmin

tdw ≥ 0 ∀t, ∀d ∈ ΨD, ∀w (2.15h)

P Stiw ≤ ψSmin

tiw MP ∀t, ∀i ∈ ΨS, ∀w (2.15i)

PEStkw ≤ ψESmin

tkw MP ∀t, ∀k ∈ ΨES, ∀w (2.15j)

POtjw ≤ ψOmin

tjw MP ∀t, ∀j ∈ ΨO, ∀w (2.15k)

PDtdw ≤ ψDmin

tdw MP ∀t, ∀d ∈ ΨD, ∀w (2.15l)

µSmin

tiw ≤(1− ψSmin

tiw

)Mµ ∀t, ∀i ∈ ΨS, ∀w (2.15m)

µESmin

tkw ≤(1− ψESmin

tkw

)Mµ ∀t, ∀k ∈ ΨES, ∀w (2.15n)

µOmin

tjw ≤(1− ψOmin

tjw

)Mµ ∀t, ∀j ∈ ΨO, ∀w (2.15o)

µDmin

tdw ≤(1− ψDmin

tdw

)Mµ ∀t, ∀d ∈ ΨD, ∀w (2.15p)

ψSmin

tiw ∈ 0, 1 ∀t, ∀i ∈ ΨS, ∀w (2.15q)

ψESmin

tkw ∈ 0, 1 ∀t, ∀k ∈ ΨES, ∀w (2.15r)

ψOmin

tjw ∈ 0, 1 ∀t, ∀j ∈ ΨO, ∀w (2.15s)

ψDmin

tdw ∈ 0, 1 ∀t, ∀d ∈ ΨD, ∀w, (2.15t)

where MP and Mµ are large enough positive constants.

6) Mixed-integer linear equivalents of the complementarity conditions (2.4l)-

(2.4o) included in the KKT conditions (2.4):

Xi − PStiw≥ 0 ∀t, ∀i ∈ ΨS, ∀w (2.16a)

PESmax

k − PEStkw≥ 0 ∀t, ∀k ∈ ΨES, ∀w (2.16b)

POmax

jw − POtjw≥ 0 ∀t, ∀j ∈ ΨO, ∀w (2.16c)


PDmax

td − PDtdw≥ 0 ∀t, ∀d ∈ ΨD, ∀w (2.16d)

µSmax

tiw ≥ 0 ∀t, ∀i ∈ ΨS, ∀w (2.16e)

µESmax

tkw ≥ 0 ∀t, ∀k ∈ ΨES, ∀w (2.16f)

µOmax

tjw ≥ 0 ∀t, ∀j ∈ ΨO, ∀w (2.16g)

µDmax

tdw ≥ 0 ∀t, ∀d ∈ ΨD, ∀w (2.16h)

Xi − PStiw≤ ψSmax

tiw MP ∀t, ∀i ∈ ΨS, ∀w (2.16i)

PESmax

k − PEStkw≤ ψESmax

tkw MP ∀t, ∀k ∈ ΨES, ∀w (2.16j)

POmax

jw − POtjw≤ ψOmax

tjw MP ∀t, ∀j ∈ ΨO, ∀w (2.16k)

PDmax

td − PDtdw≤ ψDmax

tdw MP ∀t, ∀d ∈ ΨD, ∀w (2.16l)

µSmax

tiw ≤(1− ψSmax

tiw

)Mµ ∀t, ∀i ∈ ΨS, ∀w (2.16m)

µESmax

tkw ≤(1− ψESmax

tkw

)Mµ ∀t, ∀k ∈ ΨES, ∀w (2.16n)

µOmax

tjw ≤(1− ψOmax

tjw

)Mµ ∀t, ∀j ∈ ΨO, ∀w (2.16o)

µDmax

tdw ≤(1− ψDmax

tdw

)Mµ ∀t, ∀d ∈ ΨD, ∀w (2.16p)

ψSmax

tiw ∈ 0, 1 ∀t, ∀i ∈ ΨS, ∀w (2.16q)

ψESmax

tkw ∈ 0, 1 ∀t, ∀k ∈ ΨES, ∀w (2.16r)

ψOmax

tjw ∈ 0, 1 ∀t, ∀j ∈ ΨO, ∀w (2.16s)

ψDmax

tdw ∈ 0, 1 ∀t, ∀d ∈ ΨD, ∀w, (2.16t)

7) Mixed-integer linear equivalents of the complementarity conditions (2.4p)-

(2.4q) included in the KKT conditions (2.4):

Fmax

nm +Bnm(θtnw − θtmw) ≥ 0 ∀t, ∀n, ∀m ∈ Ωn, ∀w (2.17a)

Fmax

nm − Bnm(θtnw − θtmw) ≥ 0 ∀t, ∀n, ∀m ∈ Ωn, ∀w (2.17b)

νmin

tnmw ≥ 0 ∀t, ∀n, ∀m ∈ Ωn, ∀w (2.17c)

νmax

tnmw ≥ 0 ∀t, ∀n, ∀m ∈ Ωn, ∀w (2.17d)

Fmax

nm +Bnm(θtnw − θtmw) ≤ ψνmin

tnmwMF ∀t, ∀n, ∀m ∈ Ωn, ∀w (2.17e)

Fmax

nm − Bnm(θtnw − θtmw) ≤ ψνmax

tnmwMF ∀t, ∀n, ∀m ∈ Ωn, ∀w (2.17f)

νmin

tnmw ≤(1− ψν

min

tnmw

)Mν ∀t, ∀n, ∀m ∈ Ωn, ∀w (2.17g)

νmax

tnmw ≤(1− ψν

max

tnmw

)Mν ∀t, ∀n, ∀m ∈ Ωn, ∀w (2.17h)

2.3. Formulation 59

ψνmin

tnmw ∈ 0, 1 ∀t, ∀n, ∀m ∈ Ωn, ∀w (2.17i)

ψνmax

tnmw ∈ 0, 1 ∀t, ∀n, ∀m ∈ Ωn, ∀w, (2.17j)

where MF and Mν are large enough positive constants.

8) Mixed-integer linear equivalents of the complementarity conditions (2.4r)-

(2.4s) included in the KKT conditions (2.4):

π + θtnw≥ 0 ∀t, ∀n, ∀w (2.18a)

π − θtnw≥ 0 ∀t, ∀n, ∀w (2.18b)

ξmin

tnw≥ 0 ∀t, ∀n, ∀w (2.18c)

ξmax

tnw≥ 0 ∀t, ∀n, ∀w (2.18d)

π + θtnw≤ ψθmin

tnw Mθ ∀t, ∀n, ∀w (2.18e)

π − θtnw≤ ψθmax

tnw Mθ ∀t, ∀n, ∀w (2.18f)

ξmin

tnw≤(1− ψθ

min

tnw

)M ξ ∀t, ∀n, ∀w (2.18g)

ξmax

tnw≤(1− ψθ

max

tnw

)M ξ ∀t, ∀n, ∀w (2.18h)

ψθmin

tnw ∈ 0, 1 ∀t, ∀n, ∀w (2.18i)

ψθmax

tnw ∈ 0, 1 ∀t, ∀n, ∀w, (2.18j)

where Mθ and M ξ are large enough positive constants.

Variable set Ξψ below includes the auxiliary binary variables used to lin-

earize the complementarity conditions (2.4h)-(2.4s):

Ξψ = ψSmin

tiw , ψESmin

tkw , ψOmin

tjw , ψDmin

tdw , ψSmax

tiw , ψESmax

tkw , ψOmax

tjw , ψDmax

tdw , ψνmin

tnmw,

ψνmax

tnmw, ψθmin

tnw , ψθmax

tnw .

Thus, the variables of the MILP problem (2.10)-(2.18) are those included

in the variable set ΞUL (defined in Subsection 2.3.2) plus the binary variables

included in the set Ξψ.


Figure 2.3: Direct solution: Six-bus test system (illustrative example).

2.4 Illustrative Example

In this section, the strategic generation investment problem is analyzed using

a six-bus test system as illustrative example. The specific objectives of this

analysis are twofold:

a) To illustrate the interest of the proposed methodology to analyze the

impact of strategic generation investment decisions on market outcomes.

b) To illustrate the ability of the proposed methodology to assist a strategic

producer in its generation investment decisions.

2.4. Illustrative Example 61

Table 2.1: Direct solution: Type and data for the existing generating units(illustrative example).

Type of Capacity Capacity Capacity Production cost Production cost

existing[MW]

of block 1 of block 2 of block 1 of block 2

unit [MW] [MW] [e/MWh] [e/MWh]

Gas 12 5 7 23.41 23.78

Gas 20 15 5 11.09 11.42

Hydro 50 25 25 0 0

Coal 76 30 46 11.46 11.96

Gas 100 25 75 18.60 20.03

Coal 155 55 100 9.92 10.25

Gas 197 97 100 10.08 10.66

Coal 350 150 200 19.20 20.32

Nuclear 400 200 200 5.31 5.38

Table 2.2: Direct solution: Location and type of the existing units (illustrativeexample).

Strategic producer units Other units belonging to rival producers

k ∈ ΨES Unit CapacityBus j ∈ ΨO Unit Capacity

Bustype [MW] type [MW]

ES1 Coal 350 N1 O1 Coal 350 N1

ES2 Gas 100 N2 O2 Gas 197 N2

ES3 Coal 76 N3 O3 Coal 155 N3

ES4 Gas 20 N6 O4 Gas 100 N5

2.4.1 Data

The considered network in this illustrative example is depicted in Figure 2.3

and includes two separated areas (North and South) interconnected by two

tie-lines N2-N4 and N3-N6.

In the Northern area (buses N1, N2 and N3) generation prevails while in

the Southern one (buses N4, N5 and N6) the consumption does. In this figure,

“ES” identifies existing units belonging to the strategic producer under study

and “O” units belonging to rival producers.


Table 2.3: Direct solution: Type and data for investment options (illustrativeexample).

Candidate Annualized Options for Production Production

unit capital cost capacity of the cost of cost of

(i ∈ ΨSn)

(Ki) candidate units block 1 block 2

[e/MW] (Xih) [MW] [e/MWh] [e/MWh]

Base technology 75000 0, 500, 750, 1000 6.01 6.31

Peak technology 15000

0, 200, 250, 300,

14.72 15.20

350, 400, 450, 500,

550, 600, 650, 700,

750, 800, 850, 900,

950, 1000

Table 2.1 provides data for the existing units of the strategic producer and

other units of rival producers considered in this example. Each row refers to

a particular type of generation unit. The second column contains the power

capacity of each unit, which is divided in two generation blocks (columns 3

and 4) with associated production costs (columns 5 and 6).

Table 2.2 provides the location of the existing units throughout the network.

Note that each unit is defined by its type and capacity.

Table 2.3 gives investment options including two technologies:

1. Base technology (e.g., nuclear power plants) with high investment cost

but low production cost.

2. Peak technology (e.g., CCGTs) with low investment cost but high pro-

duction cost.

We consider that each new unit includes two production blocks. For the

sake of simplicity, note that the size of each of the two blocks is considered

equal to half of the capacity of the unit. Costs for these two generation blocks

are provided in the last two columns of Table 2.3.

The load-duration curve of the target year is approximated through seven

demand blocks. The considered weighting factors (σt) corresponding with

demand blocks are 0.5, 0.5, 1.0, 1.0, 1.0, 1.5 and 1.5 derived from the load


Table 2.4: Direct solution: Demand-bid blocks including maximum loads [MW]and bid prices [e/MWh] (illustrative example).

Demand D1∈ ΨD D2∈ ΨD D3∈ ΨD D4∈ ΨD

block Maximum Bid Maximum Bid Maximum Bid Maximum Bid

(t) load price load price load price load price

t = t1600.0 38.75 540.0 36.48 510.0 35.75 480.0 33.08

150.0 36.81 135.0 34.65 127.5 33.96 120.0 31.42

t = t2480.0 33.69 420.0 30.09 360.0 28.72 330.0 28.52

120.0 32.00 105.0 28.59 90.0 27.28 82.5 27.10

t = t3390.0 30.66 360.0 28.30 330.0 27.36 300.0 26.20

97.5 29.12 90.0 26.89 82.5 25.99 75.0 24.89

t = t4330.0 28.08 300.0 26.22 270.0 25.21 240.0 23.47

82.5 26.68 75.0 24.91 67.5 23.95 60.0 22.30

t = t5240.0 25.69 210.0 24.34 180.0 23.55 165.0 22.71

60.0 24.41 52.5 23.13 45.0 22.37 41.3 21.58

t = t6210.0 23.49 180.0 21.98 150.0 21.33 135.0 20.61

52.5 22.32 45.0 20.88 37.5 20.27 33.7 19.58

t = t7180.0 22.76 150.0 21.35 135.0 20.71 120.0 19.80

45.0 21.62 37.5 20.29 33.7 19.68 30.0 18.81

duration curve of the planning year, each multiplied by 87607, i.e., 1251.43.

That is, the total hours in a year (8760) divided by the number of considered

demand blocks (7 blocks).

Table 2.4 provides demand-bid blocks (maximum load and price) for each

demand block. Each column corresponds to a demand (D1 to D4), while each

row corresponds to a demand block. The cells of this table identify the actual

values of the load bids (MW) and the corresponding prices (e/MWh). Each

demand considers two bids per block with different sizes and prices.

Note that the demand blocks and demand-bid blocks have been explained

in detail in Subsection 1.6.2 of Chapter 1. In that subsection, a table with a

structure similar to the one of Table 2.4 is provided (Table 1.3). An illustrative

figure is also provided (Figure 1.2) in Subsection 1.6.2 of Chapter 1.

Finally, we consider that all lines have the same susceptance, Bnm = 10

p.u. (100 MW base).


2.4.2 Deterministic Solution

In this subsection, MILP problem (2.10)-(2.18) is solved considering one sce-

nario (deterministic solution) based on the data provided in the previous sub-

section.

2.4.2.1 Uncongested and Congested Network

Three single-scenario cases are considered below to analyze the impact of trans-

mission congestion on generation investment results.

Case A) The transmission capacity constraints (2.2g) are not enforced.

Case B) The total transmission capacity of both tie-lines N2-N4 and N3-N6

is limited to 450 MW.

Case C) The total transmission capacity of both tie-lines N2-N4 and N3-N6

is limited to 150 MW.

Table 2.5 provides results on generation investment and profit for the strate-

gic producer. Column 2 refers to Case A while columns 3 and 4 pertain to

Cases B and C, respectively. This table gives the investment in each bus (rows

2 to 7), the total investment (row 8), the profit (row 9), the investment cost

(row 10) and the operations profit (row 11).

Note that the investment and profit results for Cases A and B are identical.

However, due to transmission limits on the tie-lines in Case B, the newly built

base unit is located in the Southern area. In both of these cases, no tie-line

is congested, thus the clearing prices throughout the network are the same in

each demand block as shown in Figure 2.4(a).

In Case C, the tie-lines are congested, and the total investment is higher

than that in Cases A and B (row 8 of Table 2.5). Note that in Case C, all

investments are located in the Southern area and the profit of the strategic

producer is comparatively higher (47.64 Me).

The LMPs at each bus for each demand block in Case C are depicted in

Figure 2.4(b). Due to the prevailing demand in the Southern area, the tie-lines

are congested for most demand blocks. Observe that this fact makes LMPs


Table 2.5: Direct solution: Investment results pertaining to the uncongestedand congested cases (illustrative example).

Case A Case B Case C

Bus N1 [MW] No investment No investment No investment

Bus N2 [MW] 500 (base unit) No investment No investment

Bus N3 [MW] No investment No investment No investment

Bus N4 [MW] 200 (peak unit) 200 (peak unit) 600 (peak unit)

Bus N5 [MW] No investment 500 (base unit) No investment

Bus N6 [MW] No investment No investment 500 (base unit)

Total investment [MW] 700 700 1100

Profit [Me] 45.55 45.55 47.64

Investment cost [Me] 40.50 40.50 46.50

Operations profit [Me] 86.05 86.05 94.14

CPU time [second] 3.17 3.36 17.48

different throughout the network. This price behavior can be easily derived

from (2.4e) since νmin

tnmw or νmax

tnmw are not necessarily zero. In particular, for

demand blocks 1 to 5 congestion occurs and the Southern area exhibits higher

prices than the Northern area where generation prevails. On the contrary,

congestion does not occur at low demand blocks 6 and 7 and therefore prices

are identical throughout the network.

2.4.2.2 Strategic and Non-Strategic Offering

This subsection analyzes the impact of strategic offering of the producer under

study on generation investment results. Two cases, both including one single

scenario, are examined in the following:

Strategic offering) In this case, the producer under study behaves strategi-

cally.

Non-strategic offering) In this case, the producer under study offers at its

production cost. This is realized by replacing its strategic offering variables


1 2 3 4 5 6 718

20

22

24

26

28

30

32

34

36

Demand block

LMP

(a) Cases A and B (illustrative example).

1 2 3 4 5 6 7

16

18

20

22

24

26

28

30

32

34

36

38

Demand block

LMP

N1N2N3N4N5N6

(b) Case C (illustrative example).

Figure 2.4: Direct solution: Locational marginal prices in (a) Cases A and B,and (b) Case C.


Table 2.6: Direct solution: Investment results considering and not consideringstrategic offering (illustrative example).

Non-strategic offering Strategic offering

North (base unit) [MW] No investment No investment

North (peak unit) [MW] No investment No investment

South (base unit) [MW] No investment 500

South (peak unit) [MW] 350 200

Total investment [MW] 350 700

Profit [Me] 31.40 45.55

Investment cost [Me] 5.25 40.50

Operations profit [Me] 36.65 86.05

CPU time [second] 5.84 0.75

αStiw and αES

tkw in equation (2.2a) with cost parameters CSi and C

ESk , respectively.

For simplicity and for capturing just the impact of the strategic offering on

investment results, the considered network is reduced to two buses, North and

South, and no transmission limits on tie-lines are enforced.

The results are given in Table 2.6. Offering at marginal cost results in

comparatively lower market clearing prices with respect to a case with strategic

offers. Thus, non-strategic offering results in comparatively lower investment

and lower profit (rows 6 and 7 of Table 2.6).

2.4.3 Stochastic Solution

In this subsection, MILP problem (2.10)-(2.18) is first solved considering mul-

tiple scenarios and then the investment results obtained are analyzed. The

following observations are in order:

1) The strategies of rival producers in the pool and their investment decisions

are considered as uncertain parameters represented through scenarios.

2) Scenarios should be selected representing in the best possible manner the

real-world alternative values of the uncertain parameters as well as their

associated probabilities. Scenarios pertaining to rival offers need to be se-


Table 2.7: Direct solution: Rival producer scenarios (illustrative example).

Case Rival investment Bus Cost factor Probability

4 scenarios

No investment - 0.9 0.24


400 MW South 0.9 0.16

400 MW South 1.0 0.24

12 scenarios




400 MW South 0.9 0.05

400 MW South 1.0 0.15

400 MW South 1.1 0.05

197 MW North 0.9 0.05

197 MW North 1.0 0.15

197 MW North 1.1 0.05

400 MW and 197 MW South-North 0.9 0.02



lected covering all possible rival offering strategies, and scenarios pertaining

to rival investment should be based on the financial status and prospective

investments of rival producers.

In this subsection, the two stochastic cases below are analyzed:

4 scenarios) This case involves 4 scenarios including two rival offering scenarios

and two rival investment scenarios.

12 scenarios) This case involves 12 scenarios including three rival offering sce-

narios and four rival investment scenarios.

The details of rival producer scenarios (offering and investment) are given

in Table 2.7, whose structure is explained below:


Table 2.8: Direct solution: Investment results pertaining to the stochasticcases (illustrative example).

One scenario 4 scenarios 12 scenarios

North (base unit) [MW] No investment No investment No investment

North (peak unit) [MW] No investment 350 500

South (base unit) [MW] 500 No investment No investment

South (peak unit) [MW] 200 350 200


Expected profit [Me] 45.55 32.25 31.38


CPU time [second] 0.75 5.08 70.68

• Column 2 characterizes rival investment uncertainty considering alternative

investments consistent with the alternatives in Table 2.1 (Subsection 2.4.1).

Note that in the cases of no investment, only the rival offering uncertainty

is modeled.

• Column 3 identifies the location of rival investment throughout the network.

• Column 4 gives the cost factors pertaining to rival offers available in Table

2.1 (Subsection 2.4.1), i.e., a rival offering strategy is obtained multiplying

the production costs of all rival units by the corresponding cost factor.

• The last column presents the probabilities corresponding to each scenario.

Table 2.8 gives the generation investment results for cases involving one, 4

and 12 scenarios. The case involving one scenario corresponds with the one in

the third column of Table 2.6 (Subsection 2.4.2.2).

According to the results given in Table 2.8, although the total investment

in all cases is identical, the expected profit of the strategic producer decreases

as the rival uncertainty increases. The reason is that cases involving 4 and 12

scenarios consider a higher capacity for rival investment with respect to the

rival investment capacity in the deterministic case.


Table 2.9: Direct solution: Location and type of existing units (case study).

Strategic producer units Other units belonging to rival producers

k ∈ ΨES Unit CapacityBus j ∈ ΨO Unit Capacity

Bustype [MW] type [MW]

1 Gas 20 1 1,2 Gas 20 1

2 Coal 76 2 3,4 Gas 20 2

3 Gas 100 7 5 Coal 76 2

4 Coal 155 13 6 Gas 100 7

5 Gas 100 15 7 Gas 197 13

6 Gas 197 21 8-12 Gas 12 15

7 Coal 76 23 13 Coal 155 16

14 Gas 100 18

15-18 Hydro 50 22

19 Coal 155 23

20 Coal 155 23

2.5 Case Study

To highlight the capability of the proposed procedure to solve the strategic

generation investment problem in realistic systems, this section presents in-

vestment results for a case study based on the 24-bus IEEE one-area Reliability

Test System (RTS) [110], whose structure and data are presented in Appendix

A.

Similarly to the illustrative example (Section 2.4), we consider a load-

duration curve divided into seven demand blocks, whose weighting factors

ate those provided in Subsection 2.4.1.

Maximum load of demands and their bid prices are:

• The maximum load of each demand in the first block (PDmax

td t = t1, ∀d ∈ ΨD)

is identical to the one in [110].

• In demand blocks 2 to 7, the maximum load of each demand is the one in

the first demand block multiplied by 0.90, 0.75, 0.65, 0.60, 0.55 and 0.50,

respectively.

2.5. Case Study 71

Table 2.10: Direct solution: Investment results (case study).

One scenario 4 scenarios12 scenarios

(reduced version)

Base unit [MW] - - -

Peak unit [MW] 750 [bus 15] 550 [bus 11] 450 [bus 23]




CPU time 12.14 [second] 3.95 [hour] 3.76 [hour]

Optimality gap (%) 0.10 1.00 1.75

• Demands 11, 13 and 15 (located at buses 13, 15 and 18 in the original

system) bid in the first demand block at 40.00 e/MWh, other demands

located in the Northern area (buses 14-24) at 38.00 e/MWh, and other

demands located in the Southern area (buses 1-13) at 35.00 e/MWh.

• In demand blocks 2 to 7, each demand bids at a price identical to the one

in the first demand block multiplied by 0.95, 0.90, 0.85, 0.80, 0.75 and 0.70,

respectively.

As in Subsection 2.4.1 (illustrative example), Table 2.1 gives data for the

generating units.

Additionally, Table 2.9 provides the location of existing and rival units

throughout the RTS network.

The scenarios considered in the stochastic cases are the same as those in

Subsection 2.4.3 (illustrative example), available in Table 2.7. We assume that

the newly built units of rival producers are located at bus 15.

Since most transmission lines in a power network are designed to operate

at safe margins with respect to their capacities, congestion only occurs at

some critical lines that are generally well identified. Therefore, the buses

connected through lines that are not likely to suffer congestion are gathered

into a single bus without altering significantly the results of the study. This

simplification might be a computational requirement since network constraints


increase considerably the computational burden of the proposed model. Hence,

for the sake of simplicity and to decrease the computational burden in the case

of 12 scenarios, we reduce the number of buses in the system to nine, merging

buses 1 to 13 into a single one and buses 17 to 20 in another one.

Table 2.10 gives the generation investment results involving one and 4 sce-

narios considering 24 buses, 12 scenarios and 9 buses. The last row of this table

shows the optimality gap. Enforcing lower gaps may lead to higher accuracy,

but increases computational burden as well.

Similarly to the illustrative example (Section 2.4), higher uncertainty in-

volves higher capacity for rival investment, which results in lower expected

profit and investment in smaller units (Table 2.10).

2.6 Computational Considerations

MILP problem (2.10)-(2.18) is solved using CPLEX 12.1 [43] under GAMS [42]

on a Sun Fire X4600M2 with 8 Quad-Core processors clocking at 2.9 GHz and

256 GB of RAM.

The computational times required for solving the considered problems are

provided in Tables 2.5, 2.6, 2.8 and 2.10.

The CPU times given in the last row of Tables 2.5 and 2.6 show that the

required time for solving the considered problems increases with the size of the

problem and with the congestion of the lines.

Considering the CPU times reported in Tables 2.8 and 2.10, we can con-

clude that the computational time for solving linear, but stochastic MILP

problem (2.10)-(2.18) increases very significantly with the number of scenar-

ios, which is the main drawback of the proposed approach in this chapter.

2.7 Summary and Conclusions

In this chapter, we consider a strategic producer trading in a market through

supply function strategies. This producer seeks to derive its investment strat-

egy for a future target period. Thus, a methodology is developed to assist such

2.7. Summary and Conclusions 73

producer in making informed decisions on generation capacity investment. In

other words, the target of this chapter is to identify the investment strategy

most beneficial for the considered strategic producer. This strategy identifies

the technology, the capacity and the location of each production unit to be

built.

First, we propose a hierarchical (bilevel) model whose upper-level problem

represents the investment and offering actions of the strategic producer, and

whose multiple lower-level problems represent the clearing of the pool under

different operating conditions.

Then, the bilevel problem is transformed into a single-level MPEC by re-

placing each lower-level problem with its KKT conditions.

The resulting MPEC can be recast as an MILP solvable by commercially

available software. To linearize the MPEC, an exact linearization approach

based on the strong duality theorem and some of the KKT equalities is used.

Regarding generation investment options, we consider base and peak units.

Also, we make no simplification on the allocation of these units throughout

the network, i.e., any plant can be located at any bus throughout the network.

The strategies of rival producers in the pool and their investment decisions

are uncertain parameters represented in the model through scenarios. In other

words, we use scenarios to describe uncertainty pertaining to i) rival offers and

ii) rival investments.

The main conclusions that can be drawn from the work reported in this

chapter are listed below.

1) The strategic behavior of a producer in its offering and investment actions

is adequately characterized by an MPEC.

2) Congestion in the transmission system results in higher LMPs than those

in an uncongested case. This leads to an increase in the capacity of newly

built units by the strategic producer.

3) A non-strategic offering results in comparatively lower investment and

lower expected profit with respect to the case with strategic offers.


4) Higher uncertainty on rival producers (i.e., rival offering and rival invest-

ment uncertainties) results in comparatively lower expected profit, and

investment in smaller units.

5) In the proposed approach, all involved scenarios are considered simulta-

neously. Thus, the computational time for solving the proposed model

increases very significantly with the number of scenarios. Therefore, a

large number of scenarios may result in high computational burden and

eventual intractability, which is the main drawback of this proposed ap-

proach (direct solution).

To solve the drawback pointed out in this last conclusion, a methodology

based on Benders’ decomposition is presented in the next chapter to make the

proposed model tractable for cases with many scenarios.

Chapter 3

Strategic Generation

Investment: Tackling

Computational Burden via

Benders’ Decomposition

3.1 Introduction

Similarly to the previous chapter, in this chapter we address the generation

investment problem faced by a strategic power producer competing with other

rival producers in a pool. To identify optimal offering and investment decisions

for the strategic producer considered, a bilevel model identical to the one pro-

posed in Chapter 2 (Subsection 2.3.2) is considered. The upper-level problem

of this bilevel model determines investment and offering decisions to maxi-

mize expected profit, and its lower-level problems represent market clearing

per demand block and scenario.

As concluded in Chapter 2, the computational requirement for solving such

model (direct solution) increases significantly with the number of scenarios,

which is the main drawback of this direct solution approach. To overcome such

computational issue or even intractability, a Benders’ decomposition scheme is

proposed in this chapter that results in a tractable formulation even if hundred

75

76 3. Strategic Generation Investment via Benders’ Decomposition

of scenarios are used to describe uncertain parameters.

In other words, we seek to solve bilevel model (2.1)-(2.2) presented in Chap-

ter 2 (Subsection 2.3.2) through an alternative approach, i.e., Bender’s decom-

position. It is important to note that the proposed Benders’ decomposition

approach can be used since the total expected profit of the strategic producer

as a function of the investment decisions has a convex enough envelope.

In this chapter, uncertainties pertaining to i) rival producer investment,

ii) rival producer offering and iii) future demand are described in detail via a

large number of plausible scenarios.

To model the future demand uncertainty that has not been taken into ac-

count in model (2.1)-(2.2), presented in Chapter 2, the upper bound of equa-

tions (2.2f) is indexed by w to represent scenario dependency, i.e., it becomes

PDmax

tdw .

Finally, note that scenario reduction techniques (as described for instance

in [87] or [106]) can be used to trim down the number of scenarios while

keeping as intact as possible the description of the uncertain phenomena under

consideration.

3.2 Benders’ Approach

This section describes the structure of the proposed Benders’ algorithm and

discusses convexity issues. The theoretical foundations of Benders’ decompo-

sition are described in Section B.4 of Appendix B.

3.2.1 Complicating Variables

Solving model (2.1)-(2.2) presented in Chapter 2 (Subsection 2.3.2) requires

considering all involved scenarios simultaneously. Therefore, a high number of

scenarios may result in high computational burden and eventual intractability.

If investment decision variables Xi are considered to be complicating vari-

ables and the expected profit (objective function) is sufficiently convex with

respect to these investment variables, model (2.1)-(2.2) can be solved using

Benders’ decomposition. The reason for selecting Xi as complicating variables

3.3. Convexity Analysis 77

is that fixing these variables to given values renders a decomposable problem

per scenario.

3.2.2 Proposed Algorithm

The proposed Benders’ algorithm works as follows:

1. Given fixed investment decisions Xi (complicating variables), the result-

ing decomposed subproblems are solved and their solutions provide i)

offering and operating decisions for the producer and ii) sensitivities of

the producer expected profit with respect to the investment decisions.

2. The sensitivities obtained in step 1 above allow formulating a so-called

Benders’ master problem whose solution provides updated investment

decisions Xi.

3. Steps 1-2 above are repeated until no expected profit improvement is

achieved.

Figure 3.1 further illustrates the proposed Bender’s algorithm. Fixing the

complicating investment variables (Box B) renders a decomposed bilevel model

per scenario (Box C), which can be recast into two different but equivalent

MPECs: one is mixed-integer linear that renders the auxiliary problems (Box

D); and the other one that is continuous and generally non-linear and renders

the subproblems (Box E). These non-linear subproblems are made linear using

the optimal values of selected variables obtained from the auxiliary problems.

In turns, sensitivities obtained from the subproblems allow formulating the

master problem (Box F) to update the investment variables. The algorithm

continues until no expected profit improvement is achieved.

3.3 Convexity Analysis

If the minus expected profit of the strategic producer as a function of invest-

ment decisions has a convex envelope, an effective implementation of Benders’

decomposition is possible. Notwithstanding bilevel models do not generally


Figure 3.1: Benders’ approach: Flowchart of the proposed Benders’ algorithm.


meet such requirement, extensive numerical analysis shows that the consid-

ered MPEC does.

Two factors influence convexity:

1. The offering strategy of the producer.

2. The number of scenarios involved.

This convexity issue is analyzed in the next subsections using an illustrative

example.

3.3.1 Illustrative Example for Convexity Analysis

An illustrative example is considered in this section to analyze the convexity

issue. For the sake of simplicity, the transmission constraints are not explicitly

modeled in this example.

3.3.1.1 Data

Similarly to the illustrative example presented in Section 2.4 of Chapter 2,

seven demand blocks are considered. The corresponding weighting factors are

given in Subsection 2.4.1 of Chapter 2.

Four existing units with a single production block each are considered,

whose data are given in Table 3.1.

Additionally, seven identical gas units are considered as rival units. The

capacity of each rival unit is 100 MW with a production cost equal to 10.08

e/MWh.

A candidate unit (base technology) is considered, whose annual capital cost

is 75000 e/MW, and whose production cost is 6.01 e/MWh.

Four demands with different maximum loads and bid prices are considered.

Table 3.2 provides demand-bid blocks (maximum load and price) for each

demand block. Each column corresponds to a demand (D1 to D4), while each

row corresponds to a demand block. The cells of this table identify the actual

value of the load bids (MW) and corresponding prices (e/MWh).


Table 3.1: Benders’ approach: Type and data for the existing generating units(illustrative example for convexity analysis).

Type of existing unit Capacity [MW] Production cost [e/MWh]

Coal 350 19.20

Gas 100 18.60

Coal 76 11.46

Gas 20 11.09

Table 3.2: Benders’ approach: Demand-bid blocks including maximum loads[MW] and bid prices [e/MWh] (illustrative example for convexity analysis).

Demand D1 ∈ ΨD D2 ∈ ΨD D3 ∈ ΨD D4 ∈ ΨD

block Maximum Bid Maximum Bid Maximum Bid Maximum Bid

(t) load price load price load price load price

t = t1 750.0 38.75 675.0 36.48 637.5 35.75 600.0 33.08

t = t2 600.0 33.69 525.0 30.09 450.0 28.72 412.5 28.52

t = t3 487.5 30.66 450.0 28.30 412.5 27.36 375.0 26.20

t = t4 375.0 28.08 375.0 26.22 337.5 25.21 300.0 23.47

t = t5 300.0 25.69 262.5 24.34 225.0 23.55 225.0 22.71

t = t6 262.5 23.49 225.0 21.98 187.5 21.33 187.5 20.61

t = t7 225.0 22.76 187.5 21.35 150.0 20.71 150.0 19.80

Note that the structure of the demand blocks and the demand-bid blocks

is explained in detail in Subsection 1.6.2 of Chapter 1. In that subsection, a

table with a structure similar to the one in Table 3.2 is presented (Table 1.3),

and an illustrative figure is provided (Figure 1.2).

Three uncertain parameters are considered in this example, namely:

1) Rival investment.

2) Rival offering.

3) Future demand.

The uncertainties pertaining to rival producer investment (16 alternatives),

rival producer offering (5 alternatives) and demand (3 alternatives) render 240


Table 3.3: Benders’ approach: Investment options of the rival producers (il-lustrative example for convexity analysis).

Type of existing unit Capacity [MW] Production cost [e/MWh]

Coal 200 6.02

Gas 150 18.54

Gas 97 15.02

Coal 55 11.26

scenarios. The probability of each scenario is obtained by multiplying the

probabilities of the corresponding alternatives. These scenarios are described

below:

Regarding rival producer investment uncertainty, four options are taken

into account. These options are given in Table 3.3. Considering four rival

investment options results in 16 alternatives: 1 investing in all options (al-

ternative A), 4 investing in three of them (alternatives B), 6 investing in two

of them (alternatives C), 4 investing in one of them (alternatives D) and one

investing in none of them (alternative E). The probabilities of alternative A,

each alternative B, each alternative C, each alternative D and alternative E

are arbitrarily assumed to be 0.1, 0.025, 0.033, 0.075 and 0.3, respectively.

Rival producer offering uncertainty is represented by multiplying the price

offer (production cost) of rival units by a factor. Five alternatives for rival

producer offering are examined using factors 1.20, 1.15, 1.10, 1.05 and 1.00

with probabilities 0.10, 0.20, 0.20, 0.10 and 0.40, respectively.

Finally, three alternatives for demand uncertainty are modeled in a similar

way, using three different factors to multiply all maximum loads, 1.10, 1.00

and 0.90, with probabilities 0.25, 0.50 and 0.25, respectively.

3.3.1.2 Cases Considered

In this example, four cases are considered to analyze the convexity of the

considered problem, namely:

Case 1) The producer offers in a non-strategic way, and one single scenario is

considered.


Case 2) The producer offers strategically, and one single scenario is considered.

Case 3) The producer offers in a non-strategic way, and all scenarios are con-

sidered.

Case 4) The producer offers strategically, and all scenarios are considered.

In Cases 1 and 3, the producer offers in a non-strategic way, i.e., at its

marginal production cost, while the producer behaves strategically in Cases

2 and 4. Note that the non-strategic offering of the considered producer is

realized by replacing its strategic offering variables αStiw and αES

tkw in equation

(2.2a) of Chapter 2 with cost parameters CSi and CES

k , respectively.

Additionally, in Cases 1 and 2, one single scenario is considered, while

Cases 3 and 4 involve the 240 scenarios described in Subsection 3.3.1.1.

3.3.1.3 Convexity Analysis

To check convexity, mixed-integer linear programming (MILP) problem (2.10)-

(2.18) presented in Subsection 2.3.6 of Chapter 2 is solved, while the investment

decisions of the considered producer (Xi ∀i ∈ ΨS) are fixed to given values.

Each time that such MILP problem is solved, the investment decision term

(∑

i∈ΨS Xi) is fixed to a value within zero to 1000.

For each case considered, the expected profit of the producer as a function

of its total investment is obtained and depicted in Figure 3.2. Note that this

figure consists of four plots, one per case considered, and clarifies the convexity

issue.

Plots 3.2(a) and 3.2(b) of Figure 3.2 pertain to the case of a single scenario

(Cases 1 and 2). These two plots show that if the considered producer offers

strategically (Case 2), its expected profit is rather convex with respect to

investment decisions as shown in plot 3.2(b), while with non-strategic offering

(Case 1) it is not, as illustrated in plot 3.2(a).

A reason for this behavior is that newly built units may displace expensive

existing units from the market in a particular demand block. Once one of those

expensive units is displaced, if the strategic producer adapts the offer prices

of its available units to the new situation of demand-bid and production-offer

3.3.Convexity

Analysis

83

0 200 400 600 800 1000−40

−20

0

20

40

Investment (MW)

Min

us p

rofit

(m

illio

n eu

ro)

(a) Considering non-strategic offering and one scenario

0 200 400 600 800 1000−36

−32

−28

−24

Investment (MW)

Min

us p

rofit

(m

illio

n eu

ro)

(b) Considering strategic offering and one scenario

0 200 400 600 800 1000−40

−20

0

20

40

Investment (MW)

Min

us p

rofit

(m

illio

n eu

ro)

(c) Considering non-strategic offering and many scenarios

0 200 400 600 800 1000−34

−28

−22

−16

Investment (MW)M

inus

pro

fit (

mill

ion

euro

)

(d) Considering strategic offering and many scenarios

Figure 3.2: Benders’ approach: Minus-producer’s profit as a function of capacity investment considering (a) non-strategic offering and one scenario, (b) strategic offering and one scenario, (c) non-strategic offering and all scenarios,and (d) strategic offering and all scenarios.


blocks through its strategic offering (i.e., offering at the minimum bid price of

the demands), the expected profit profile becomes smooth. On the contrary,

the expected profit profile presents spikes with non-strategic offering.

Additionally, plots 3.2(c) and 3.2(d) show that an increasing number of

scenarios results in a smoother and rather convex expected profit function for

both strategic and non-strategic offerings (Cases 3 and 4). The theoretical

foundation of this behavior is provided in [10].

3.4 Formulation

This section presents the formulations of the decomposed problems, the aux-

iliary problems, the subproblems and the master problem.

3.4.1 Decomposed Problems

Consistent with Box C of Figure 3.1, fixing the complicating investment de-

cision variables Xi to given values XFixedi results in a problem, that can be

decomposed per scenario w. This problem is given by (3.1)-(3.2) below:

MinimizeΞUL

∑

i∈ΨS

KiXFixedi

−∑

w

ϕw∑

t

σt

[(∑

i∈ΨS

P Stiwλt(n:i∈ΨS

n)w−∑

i∈ΨS

P StiwC

Si

)

+

(∑

k∈ΨES


n )w −∑

k∈ΨES

PEStkwC

ESk

)](3.1a)

subject to:

3.4. Formulation 85

λtnw, PStiw, P



∑

i∈ΨS

αStiwP

Stiw +

∑

k∈ΨES

αEStkwP

EStkw +

∑

j∈ΨO

COtjwP

Otjw −

∑

d∈ΨD

UDtdP

Dtdw (3.2a)

subject to:∑

d∈ΨDn

PDtdw +

∑

m∈Ωn


i∈ΨSn

P Stiw

−∑

k∈ΨESn

PEStkw −

∑

j∈ΨOn


0 ≤ P Stiw ≤ XFixed

i : µSmin

tiw , µSmax

tiw ∀i ∈ ΨS (3.2c)


k : µESmin

tkw , µESmax



jw : µOmin

tjw , µOmax

tjw ∀j ∈ ΨO (3.2e)


td : µDmin

tdw , µDmax

tdw ∀d ∈ ΨD (3.2f)

−Fmax


nm : νmin

tnmw, νmax

tnmw ∀n, ∀m ∈ Ωn (3.2g)


tnw, ξmax

tnw ∀n (3.2h)

θtnw = 0 : ξ1

tw n = 1 (3.2i)∀t, ∀w.

Note that the term∑

i∈ΨS KiXFixedi appeared in the objective function

(3.1a) can be removed since it is fixed. Thus, problem (3.1)-(3.2) above de-

composes per scenario and demand block. This decomposed problem is still

a bilevel optimization problem given by (3.3)-(3.4) whose upper-level problem

seeks to minimize the minus operations revenues of the strategic producer.

The objective function of upper-level problem (3.3a) is subject to (3.3b), i.e.,

the condition that fixes the complicating variables to given values, and to the

lower-level problems (3.4):


MinimizeΞDecomposed

− ϕw∑

t

σt

[(∑

i∈ΨS

P Stiwλt(n:i∈ΨS

n)w−∑

i∈ΨS

P StiwC

Si

)

+

(∑

k∈ΨES


n )w −∑

k∈ΨES

PEStkwC

ESk

)]

(3.3a)

subject to:

Xi = XFixedi , ∀i ∈ ΨS (3.3b)

λtnw, PStiw, P



∑

i∈ΨS

αStiwP

Stiw +

∑

k∈ΨES

αEStkwP

EStkw +

∑

j∈ΨO

COtjwP

Otjw −

∑

d∈ΨD

UDtdP

Dtdw (3.4a)

subject to:∑

d∈ΨDn

PDtdw +

∑

m∈Ωn

Bnm (θtnw − θtmw)−∑

i∈ΨSn

P Stiw

−∑

k∈ΨESn

PEStkw −

∑

j∈ΨOn


0 ≤ P Stiw ≤ Xi : µSmin

tiw , µSmax

tiw ∀i ∈ ΨS (3.4c)


k : µESmin

tkw , µESmax



jw : µOmin

tjw , µOmax

tjw ∀j ∈ ΨO (3.4e)


tdw : µDmin

tdw , µDmax

tdw ∀d ∈ ΨD (3.4f)

−Fmax


nm : νmin

tnmw, νmax

tnmw ∀n, ∀m ∈ Ωn (3.4g)


tnw, ξmax

tnw ∀n (3.4h)

θtnw = 0 : ξ1

tw n = 1 (3.4i)∀t

∀w.

3.4. Formulation 87

The primal and dual optimization variables of each lower-level problem

(3.4) are identical to those of lower-level problems (2.2) presented in Chapter

2 (Subsection 2.3.2), i.e., the primal variables set ΞPrimaltw and the dual variables

set ΞDualtw .

Note that the market operator clears the pool for given fixed investment

decisions and given offering decisions made at the upper-level problem (3.3).

Thus, the upper-level variables Xi, αStiw and αES

tkw are parameters for the lower-

level problems (3.4). This makes the lower-level problems (3.4) linear and thus

convex.

Accordingly, the primal variables of the upper-level problem (3.3) are those

in the set below:

ΞDecomposed =ΞPrimaltw , ΞDual

tw , αStiw, α

EStkw, Xi.

3.4.2 MPEC Associated with the Decomposed Prob-

lems

Each decomposed subproblem (3.3)-(3.4) can be recast to an MPEC with re-

placing the lower-level problems (3.4) by its optimality conditions which are

obtained using either Karush-Kuhn-Tucker (KKT) conditions or the primal-

dual transformation (Section B.2 of Appendix B).

Both forms of the optimality conditions associated with the lower-level

problems (3.4) have already been derived in Chapter 2, i.e,

• KKT conditions (2.4) given in Subsections 2.3.3.1 of Chapter 2.

• Conditions (2.6) obtained from the primal-dual transformation and given in

Subsections 2.3.3.2 of Chapter 2.

Regarding the optimality conditions above, the following observations are

in order:

a) Replacing each lower-level problem (3.4) with its optimality conditions us-

ing the KKT format (2.4) results in an MPEC denoted in this chapter as

MPEC1.


b) The second MPEC (denoted in this chapter as MPEC2) can be obtained

by replacing each lower-level problem (3.4) with its equivalent optimality

conditions (2.6) derived from the primal-dual transformation.

c) Each MPEC provided by the KKT conditions (MPEC1) includes comple-

mentarity constraints (2.4h)-(2.4s). Although the complementarity con-

straints can be linearized using auxiliary binary variables (Subsection B.5.1

of Appendix B), and the corresponding mixed-integer MPEC1 solved, its

solution generally renders inaccurate sensitivities to build Benders’ cuts.

d) On the other hand, each MPEC obtained by the primal-dual transformation

(MPEC2) is continuous and non-linear and its solution provides accurate

sensitivities. The non-linearity is due to the non-linear terms αStiwP

Stiw,

αEStkwP

EStkw and µSmax

tiw Xi in the strong duality equality (2.6c). These non-

linearities may lead to convergence issues.

Considering the observations above, the two steps below are carried out:

Step a) MPEC1 is solved per scenario to obtain the optimal values of the vari-

ables involved in the non-linear terms of the corresponding MPEC2.

These variables in scenario w are:

αStiw, α

EStkw and µSmax

tiw .

Step b) Substituting the optimal values of variables αStiw, α

EStkw and µSmax

tiw in

MPEC2 for scenario w renders a version of MPEC2 continuous and

linear, whose solution provides accurate sensitivities.

Each MPEC1 is thus referred to as an auxiliary problem (Box D of Figure

3.1), while each MPEC2 is referred to as a subproblem (Box E of Figure 3.1).

3.4.3 Auxiliary Problems

The formulation of the auxiliary problem (MPEC1) pertaining to scenario w is

given by (3.5) below. Note that all variables are indexed by Benders’ iteration

index (v).

3.4. Formulation 89

MinimizeΞDecomposed(v)

− ϕw∑

t

σt

[(∑

i∈ΨS

PS(v)tiw λ

(v)

t(n:i∈ΨSn)w−∑

i∈ΨS

PS(v)tiw CS

i

)

+

(∑

k∈ΨES

PES(v)tkw λ

(v)

t(n:k∈ΨESn )w−∑

k∈ΨES

PES(v)tkw CES

k

)](3.5a)

subject to:

1) Fix the complicating variables to the given values:

X(v)i = XFixed

i ∀i ∈ ΨS. (3.5b)

2) The equality constraints included in the KKT conditions of lower-level prob-

lems (3.4):

αS(v)tiw − λ

(v)

t(n:i∈ΨSn)w

+ µSmax(v)tiw − µ

Smin(v)tiw = 0 ∀t, ∀i ∈ ΨS (3.5c)

αES(v)tkw − λ

(v)

t(n:k∈ΨESn )w

+ µESmax(v)tkw − µ

ESmin(v)tkw = 0 ∀t, ∀k ∈ ΨES (3.5d)

COtjw − λ

(v)

t(n:j∈ΨOn )w

+ µOmax(v)tjw − µ

Omin(v)tjw = 0 ∀t, ∀j ∈ ΨO (3.5e)

−UDtd + λ

(v)

t(n:d∈ΨDn )w

+ µDmax(v)tdw − µ

Dmin(v)tdw = 0 ∀t, ∀d ∈ ΨD (3.5f)

∑

m∈Ωn

Bnm

(λ(v)tnw − λ

(v)tmw

)

+∑

m∈Ωn

Bnm

(ν

max(v)tnmw − ν

max(v)tmnw

)

−∑

m∈Ωn

Bnm

(ν

min(v)tnmw − ν

min(v)tmnw

)

+ξmax(v)tnw − ξ

min(v)tnw +

(ξ1(v)tw

)n=1

= 0 ∀t, ∀n (3.5g)

∑

d∈ΨDn

PD(v)tdw +

∑

m∈Ωn

Bnm

(θ(v)tnw − θ

(v)tmw

)

−∑

i∈ΨSn

PS(v)tiw −

∑

k∈ΨESn

PES(v)tkw −

∑

j∈ΨOn

PO(v)tjw = 0 ∀n (3.5h)


θ(v)tnw = 0 n = 1, ∀t. (3.5i)

3) The complementarity constraints included in the KKT conditions of lower-

level problems (3.4):

0 ≤ PS(v)tiw ⊥ µ

Smin(v)tiw ≥ 0 ∀t, ∀i ∈ ΨS (3.5j)

0 ≤ PES(v)tkw ⊥ µ

ESmin(v)tkw ≥ 0 ∀t, ∀k ∈ ΨES (3.5k)

0 ≤ PO(v)tjw ⊥ µ

Omin(v)tjw ≥ 0 ∀t, ∀j ∈ ΨO (3.5l)

0 ≤ PD(v)tdw ⊥ µ

Dmin(v)tdw ≥ 0 ∀t, ∀d ∈ ΨD (3.5m)

0 ≤(X

(v)i − P

S(v)tiw

)⊥ µ

Smax(v)tiw ≥ 0 ∀t, ∀i ∈ ΨS (3.5n)

0 ≤(PESmax

k − PES(v)tkw

)⊥ µ

ESmax(v)tkw ≥ 0 ∀t, ∀k ∈ ΨES (3.5o)

0 ≤(POmax

jw − PO(v)tjw

)⊥ µ

Omax(v)tjw ≥ 0 ∀t, ∀j ∈ ΨO (3.5p)

0 ≤(PDmax

tdw − PD(v)tdw

)⊥ µ

Dmax(v)tdw ≥ 0 ∀t, ∀d ∈ ΨD (3.5q)

0 ≤[F

max

nm +Bnm

(θ(v)tnw − θ

(v)tmw

)]⊥ ν

min(v)tnmw ≥ 0 ∀t, ∀n, ∀m ∈ Ωn (3.5r)

0 ≤[F

max

nm − Bnm

(θ(v)tnw − θ

(v)tmw

)]⊥ ν

max(v)tnmw ≥ 0 ∀t, ∀n, ∀m ∈ Ωn (3.5s)

0 ≤(π + θ

(v)tnw

)⊥ ξ

min(v)tnw ≥ 0 ∀t, ∀n (3.5t)

0 ≤(π − θ

(v)tnw

)⊥ ξ

max(v)tnw ≥ 0 ∀t, ∀n. (3.5u)

The investment decision (complicating) variables are fixed in (3.5b). Sim-

ilarly to the KKT conditions (2.4) presented in Chapter 2, the set of equality

constraints (3.5c)-(3.5i) and complementarity constraints (3.5j)-(3.5u) is equiv-

alent to the lower-level problems (3.4).

Auxiliary problem (3.5) of scenario w can be transformed into an MILP

problem by linearizing the complementarity constraints (3.5j)-(3.5u) and the

3.4. Formulation 91

non-linear terms Z(v)tw ,

Z(v)tw =

∑

i∈ΨS

PS(v)tiw λ

(v)

t(n:i∈ΨSn)w

+∑

k∈ΨES

PES(v)tkw λ

(v)

t(n:k∈ΨESn )w

∀t,

that appears in the objective function (3.5a).

Such non-linearities are identical to those non-linear terms pertaining to

MPEC (2.7) in Chapter 2, whose linearization is explained in Subsection 2.3.5.

Thus, auxiliary problem (3.5) can be transformed into the MILP problem

similar to MILP (2.10)-(2.18) presented in Subsection 2.3.6 of Chapter 2, but

replacing (2.12) with (3.5b).

The solution of the MILP form of the auxiliary problem of scenario w gives

the optimal values of the variables below.

αS(v)tiw = α

S(v)tiw ∀t, ∀i ∈ ΨS (3.6a)

αES(v)tkw = α

ES(v)tkw ∀t, ∀k ∈ ΨES (3.6b)

µSmax(v)tiw = µ

Smax(v)tiw ∀t, ∀i ∈ ΨS. (3.6c)

3.4.4 Subproblems

The formulation of subproblem (MPEC2) for scenario w is given by problem

(3.7) below.

Note that the optimal values of the variables αS(v)tiw , α

ES(v)tkw and µ

Smax(v)tiw

obtained from solving the corresponding auxiliary problem (MPEC1) are in-

corporated in problem (3.7) through parameters αS(v)tiw , α

ES(v)tkw and µ

Smax(v)tiw .

Note also that all variables are indexed by Benders’ iteration index (v).


MinimizeΞSP(v)

G(v)w =

− ϕw∑

t

σt

(Z

Lin(v)tw −

∑

i∈ΨS

PS(v)tiw CS

i −∑

k∈ΨES

PES(v)tkw CES

k

)(3.7a)

subject to:

1) Exact linear equivalent for the non-linear term ZLin(v)tw in the objective

function (3.7a):

ZLin(v)tw = −

∑

j∈ΨO

COtjwP

O(v)tjw +

∑

d∈ΨD

UDtdP

D(v)tdw −

∑

j∈ΨO

µOmax(v)tjw POmax

jw

−∑

d∈ΨD

µDmax(v)tdw PDmax

tdw −∑

n(m∈Ωn)

(ν

min(v)tnmw + ν

max(v)tnmw

)F

max

nm

−∑

n

(ξmin(v)tnw + ξ

min(v)tnw

)π ∀t. (3.7b)

2) The complicating variables are fixed to given values:

X(v)i = XFixed

i : γ(v)iw ∀i ∈ ΨS. (3.7c)

3) The primal constraints:

∑

d∈ΨDn

PD(v)tdw +

∑

m∈Ωn

Bnm(θ(v)tnw − θ

(v)tmw)−

∑

i∈ΨSn

PS(v)tiw

−∑

k∈ΨESn

PES(v)tkw −

∑

j∈ΨOn

PO(v)tjw = 0 ∀t, ∀n (3.7d)

0 ≤ PS(v)tiw ≤ X

(v)i ∀t, ∀i ∈ ΨS (3.7e)

0 ≤ PES(v)tkw ≤ PESmax

k ∀t, ∀k ∈ ΨES (3.7f)

0 ≤ PO(v)tjw ≤ POmax

jw ∀t, ∀j ∈ ΨO (3.7g)

3.4. Formulation 93

0 ≤ PD(v)tdw ≤ PDmax

tdw ∀t, ∀d ∈ ΨD (3.7h)

−Fmax

nm ≤ Bnm

(θ(v)tnw − θ

(v)tmw

)≤ F

max

nm ∀t, ∀n, ∀m ∈ Ωn (3.7i)

−π ≤ θ(v)tnw ≤ π ∀t, ∀n (3.7j)

θ(v)tnw = 0 ∀t, n = 1. (3.7k)

4) The dual constraints:

αS(v)tiw − λ

(v)

t(n:i∈ΨSn)w

+ µSmax(v)tiw − µ

Smin(v)tiw = 0 ∀t, ∀i ∈ ΨS (3.7l)

αES(v)tkw − λ

(v)

t(n:k∈ΨESn )w

+ µESmax(v)tkw − µ

ESmin(v)tkw = 0 ∀t, ∀k ∈ ΨES (3.7m)

COtjw − λ

(v)

t(n:j∈ΨOn )w

+ µOmax(v)tjw − µ

Omin(v)tjw = 0 ∀t, ∀j ∈ ΨO (3.7n)

−UDtd + λ

(v)

t(n:d∈ΨDn )w

+ µDmax(v)tdw − µ

Dmin(v)tdw = 0 ∀t, ∀d ∈ ΨD (3.7o)

∑

m∈Ωn

Bnm

(λ(v)tnw − λ

(v)tmw

)

+∑

m∈Ωn

Bnm

(ν

max(v)tnmw − ν

max(v)tmnw

)

−∑

m∈Ωn

Bnm

(ν

min(v)tnmw − ν

min(v)tmnw

)

+ξmax(v)tnw − ξ

min(v)tnw +

(ξ1(v)tw

)n=1

= 0 ∀t, ∀n (3.7p)

µSmin(v)tiw ≥ 0 ∀t, ∀i ∈ ΨS (3.7q)

µESmin(v)tkw ≥ 0; µ

ESmax(v)tkw ≥ 0 ∀t, ∀k ∈ ΨES (3.7r)

µOmin(v)tjw ≥ 0; µ

Omax(v)tjw ≥ 0 ∀t, ∀j ∈ ΨO (3.7s)

µDmin(v)tdw ≥ 0; µ

Dmax(v)tdw ≥ 0 ∀t, ∀d ∈ ΨD (3.7t)

τmin(v)tnmw ≥ 0; τ

max(v)tnmw ≥ 0 ∀t, ∀n, ∀m ∈ Ωn (3.7u)

ξmin(v)tnw ≥ 0; ξ

max(v)tnw ≥ 0 ∀t, ∀n (3.7v)


5) The strong duality equalities:

∑

i∈ΨS

αS(v)tiw P

S(v)tiw +

∑

k∈ΨES

αES(v)tkw P

ES(v)tkw

+∑

j∈ΨO

COtjwP

O(v)tjw −

∑

d∈ΨD

UDtdP

D(v)tdw =

−∑

i∈ΨS

µSmax(v)tiw X

(v)i −

∑

k∈ΨES

µESmax(v)tkw PESmax

k

−∑

j∈ΨO

µOmax(v)tjw POmax

jw −∑

d∈ΨD

µDmax(v)tdw PDmax

td

−∑

n(m∈Ωn)

(ν

min(v)tnmw + ν

max(v)tnmw

)F

max

nm

−∑

n

(ξmin(v)tnw + ξ

max(v)tnw

)π ∀t. (3.7w)

The optimization variables of each subproblem (3.7) are those in set ΞSP(v)

as below:

ΞSP(v) = G(v)w ,X

(v)i , λ

(v)tnw, P

S(v)tiw , P

ES(v)tkw , P

O(v)tjw , P

D(v)tdw , θ

(v)tnw, µ

Smin(v)tiw , µ

ESmin(v)tkw ,

µESmax(v)tkw , µ

Omin(v)tjw , µ

Omax(v)tjw , µ

Dmin(v)tdw , µ

Dmax(v)tdw , τ

min(v)tnmw , τ

max(v)tnmw , ξ

min(v)tnw , ξ

max(v)tnw ,

ξ1(v)tw , γ

(v)iw .

Subproblem (3.7) pertaining to scenario w includes objective function (3.7a),

constraints to fix the complicating variables to given values (3.7c), primal con-

straints (3.7d)-(3.7k), dual constraints (3.7l)-(3.7v) and strong duality equali-

ties (3.7w).

Note that as in Subsection 2.3.5.1 of Chapter 2, the objective function

(3.7a) is linearized replacing its non-linear term by a linear equivalent, i.e.,

ZLin(v)tw , provided in (3.7b).

Note that subproblem (3.7) is continuous and linear.

Note also that dual variables γ(v)iw of equation (3.7c) are sensitivities.

The solutions of subproblems (3.7) for all scenarios provide G(v)w and γ

(v)iw ,

which allow computing the value of G(v)

to be used in the convergence check

and the values of G(v) and γ(v)i to be used in Bender’ master problem, i.e.,

3.4. Formulation 95

G(v)

=∑

w

G(v)w +

∑

i∈ΨS

KiXFixedi (3.8a)

G(v) =∑

w

G(v)w (3.8b)

γ(v)i =

∑

w

γ(v)iw ∀i ∈ ΨS. (3.8c)

Note that the subproblems are further restricted versions of the bilevel

optimization problem (2.1)-(2.2) presented in Chapter 2 (Subsection 2.3.2).

Therefore, G(v)

is an upper bound of the optimal value of the objective function

of the original problem, i.e., objective function (2.1a).

3.4.5 Master Problem

The formulation of Benders’ master problem (Box F of Figure 3.1) at iteration

(v) is as follows:

MinimizeΞMP(v)

G(v) =

(∑

i∈ΨS

KiX(v)i

)+ β(v) (3.9a)

subject to: (3.9b)

Xi =∑

h

uihXih ∀i ∈ ΨS (3.9c)

∑

h

uih = 1 ∀i ∈ ΨS (3.9d)

uih ∈ 0, 1 ∀i ∈ ΨS, ∀h (3.9e)

β(v) ≥ βmin (3.9f)

β(v) ≥ G(l) +∑

i∈ΨS

γ(l)i (X

(v)i −X

(l)i ) l = 1, ..., v − 1. (3.9g)


The optimization variables of master problem (3.9) at iteration v are those

in the set ΞMP(v) = G(v), X(v)i , β(v), u

(v)ih .

Note that master problem (3.9) is mixed-integer and linear.

The objective function (3.9a) corresponds to the objective function (2.1a)

of the upper-level problem (2.1)-(2.2) presented in Chapter 2, where β(v) rep-

resents the minus operations revenues.

Constraints (3.9c)-(3.9e) are the upper-level constraints (2.1b)-(2.1d) that

allow the strategic producer to select new units among the available investment

options.

Constraint (3.9f) imposes a lower bound on β(v) to accelerate convergence.

Finally, constraints (3.9g) are Benders’ cuts, which reconstruct from below

the objective function of the original problem, i.e., objective function (2.1a)

presented in Chapter 2.

Note that at every Benders’ iteration a new constraint is added to (3.9g),

i.e., a new Benders’ cut is incorporated into the master problem. Each solution

of the master problem updates the values of the complicating variables X(v)i ,

which are the investment decisions.

3.5 Benders’ Algorithm

A description of the proposed Benders’ algorithm to solve the strategic gener-

ation investment problem under uncertainty is presented in this section.

Input data for the proposed Benders’ algorithm include a tolerance ε, initial

guesses of the investment (complicating) variables X0i , and scenario data.

The initialization step includes setting v = 1, the expected profit (objec-

tive function) initial lower bound G(v) = −∞ and forcing the investment de-

cision (complicating) variables to be equal to the initial guesses, i.e., XFixedi =

X0i ∀i ∈ ΨS.

The steps of Benders’ algorithm are as follows:

Step 1 selects the first scenario.

Step 2 solves the auxiliary problem (3.5) per scenario w. Then, the optimal

values of variables αStiw, α

EStkw and µSmax

tiw are considered to be fixed to

3.6. Case Study of Section 2.5 97

solve subproblem (3.7).

Step 3 solves the subproblem (3.7) per scenario w.

Step 4 repeats the steps 2-3 for all involved scenarios.

Step 5 checks convergence by comparing the values of the upper bound profit

G(v)

obtained by (3.8a) and the lower bound profit G(v). If the differ-

ence between these two bounds is smaller than the tolerance ε, i.e.,

∣∣∣∣G(v)−G(v)

∣∣∣∣ ≤ ε (3.10)

the solution of iteration v is optimal with a level of accuracy ε. Other-

wise, the values of G(v) and γ(v)i are computed using (3.8b) and (3.8c)

and then the next iteration is considered, i.e., v ←− v + 1.

Step 6 solves the master problem (3.9) to update the values of XFixedi and of

the lower bound profit G(v).

The algorithm continues in step 1.

3.6 Case Study of Section 2.5

The objectives of this section is to illustrate the capability of the proposed

Benders’ algorithm to reduce the computational burden while obtaining opti-

mal results. In this section, the proposed Benders’ algorithm and the direct

MPEC solution approach proposed in Chapter 2 are applied to the case study

presented in Section 2.5 of Chapter 2. The results of those methods and the

computational time required are compared.

Tables 3.4 and 3.5 provide a comparison between the results obtained using

the direct MPEC solution approach proposed in Chapter 2 and the results

achieved using the proposed Benders’ algorithm.

Table 3.4 corresponds to the investment results obtained by the direct so-

lution approach, while Table 3.5 pertains to those results achieved by the

proposed Benders’ algorithm. The first row of both tables identifies the three


Table 3.4: Benders’ approach: Investment results of the case study presentedin Section 2.5 obtained by the direct solution approach presented in Chapter2.

Number of scenarios 1 scenario 4 scenarios12 scenarios

(network reduced to 9 buses)

Base unit [MW] No investment No investment No investment





Optimality gap (%) 0.10 1.00 1.75

CPU time 12.14 [second] 3.95 [hour] 3.76 [hour]

Table 3.5: Benders’ approach: Investment results of the case study presentedin Section 2.5 obtained by the proposed Benders’ approach.

Number of scenarios 1 scenario 4 scenarios12 scenarios

(network reduced to 9 buses)

Base unit [MW] No investment No investment No investment





Optimality gap (%) 0.10 1.00 1.00

CPU time 0.18 [hour] 0.22 [hour] 0.28 [hour]

cases considered in Section 2.5 of Chapter 2, i.e., cases with 1, 4 and 12 sce-

narios.

Note that in the case of 12 scenarios, the network is reduced to 9 buses due

to the computational burden of solving the MPEC directly. In both tables,

rows 2-4 pertain to investment results in base capacity, peak capacity and total

capacity. The next two rows give the expected profit of the strategic producer

and the investment cost, respectively. Rows 7 and 8 in those tables indicate the

optimality gap enforced in each case and the CPU time required, respectively.

3.7. Case Study 99

Note that lower optimality gaps may lead to higher accuracy but at the cost

of higher computational time.

A comparative analysis of the results shows that both methods provide

similar results, while computational burden is significantly smaller in the case

of the proposed Benders’ algorithm.

Note that contrary to the interconnecting tie-lines between the Northern

and Southern areas, the transmission lines within each area do not suffer from

congestion. Thus, the location of newly built units within each area is immate-

rial. Accordingly, in Table 3.4 (direct solution), newly built 550 MW and 450

MW peak units are located in bus 11 (Southern area) and bus 23 (Northern

area), respectively. However, based on the results obtained using the proposed

Benders’ approach (Table 3.5), such units are located in different buses, but at

the same areas, i.e., newly built 550 MW peak unit is located at Southern area

(in bus 10), and the newly built 450 MW peak unit is located at the Northern

area (in bus 15).

To highlight the ability of the proposed Benders’ algorithm to solve the

strategic generation investment problem in large systems and considering a

large number of scenarios, the next section considers a case study involving a

large number of scenarios.

3.7 Case Study

This section presents results from a case study based on the 24-bus IEEE

one-area Reliability Test System (RTS) [110], whose structure and data are

provided in Appendix A.

3.7.1 Data

The number of demand blocks and the corresponding weighting factors, demand-

bid blocks and generation-offer blocks are those provided in Section 2.5 of

Chapter 2.

The available investment options are given in Table 3.6. Note that each

investment option includes two production blocks. For the sake of simplicity,


Table 3.6: Benders’ approach: Investment options.



(i ∈ ΨSn)

(Ki) candidate units block 1 block 2


Base technology 66500 0, 500, 750, 1000 5.78 6.42


0, 100, 150, 200, 250

14.86 15.31300, 350, 400, 450, 500

550, 600, 650, 700, 750

800, 850, 900, 950, 1000

the size of each block is considered equal to half the installed capacity. In this

example, buses 9 and 14 are the two candidate locations to build new units.

To analyze the influence of the network in the investment results, the ca-

pacities of the North-South tie-lines 11-14, 12-23, 13-23 and 15-24 are limited

to 200 MW. Note that bus 9 is in the Southern area where demand prevails,

while bus 14 is in the Northern area where generation does.

A stochastic case involving 240 scenarios is analyzed. The uncertainties of

rival producer investment (16 alternatives), rival producer offering (5 alterna-

tives) and demand (3 alternatives) render 240 scenarios. The probability of

each scenario is obtained by multiplying the probabilities of the corresponding

alternatives. These scenarios are described below.

Regarding rival producer investment uncertainty, Table 3.7 gives the in-

vestment options for rival producers. Considering 4 rival producer investment

options results in 16 alternatives: 1 investing in all options (alternative A), 4

investing in three of them (alternatives B), 6 investing in two of them (alter-

natives C), 4 investing in one of them (alternatives D) and one investing in

none of them (alternative E).

The probabilities of alternative A, each alternative B, each alternative C,

each alternative D and alternative E are 0.1, 0.025, 0.033, 0.075 and 0.3, re-

spectively.

Rival producer offering uncertainty is characterized by multiplying the price

offer (production cost) of rival units by a factor. Five alternatives for rival

3.7. Case Study 101

Table 3.7: Benders’ approach: Investment options for rival producers (casestudy).

Rival Type of Capacity Capacity Production cost Capacity Production cost

Busunit rival[MW]


(j ∈ ΨO) unit [MW] [e/MWh] [MW] [e/MWh]

1 Nuclear 400 200 5.31 200 5.38 9

2 Gas 197 97 10.08 100 10.66 9

3 Coal 155 55 9.92 100 10.25 14

4 Coal 350 150 19.20 200 20.32 14

producer offering are considered using factors 1.20, 1.15, 1.10, 1.05 and 1.00

with probabilities 0.10, 0.20, 0.20, 0.10 and 0.40, respectively.

Finally, three alternatives for demand uncertainty are considered in a simi-

lar way, using three different factors to multiply all maximum loads, 1.10, 1.00

and 0.90, with probabilities 0.25, 0.50 and 0.25, respectively.

3.7.2 Investment Results

Unlike the direct solution proposed in Chapter 2, all scenarios are not simulta-

neously considered in the proposed Benders’ algorithm. Thus, this algorithm is

tractable even considering a large number of scenarios. Nevertheless, scenario

reduction techniques may reduce the computational burden significantly. To

check the effectiveness of scenario reduction technique, the investment prob-

lem for the stochastic case study described in the previous subsection is solved

with and without scenario reduction.

The considered scenarios are selected using the scenario reduction technique

reported in [87] and [106]. The working of this scenario reduction technique

is approximately as follows: first the model is solved for each individual sce-

nario and then scenarios with similar expected profits are merged and their

probabilities added.

For the stochastic case study considered, Figure 3.3 shows the evolution of

the expected profit, the profit standard deviation and the CPU time with the

number of scenarios.

According to Figure 3.3, both the expected profit and the profit standard


10 20 30 40 50 60146

50

54E

xpec

ted

prof

it(m

illio

n eu

ro)

10 20 30 40 50 6010

20

40

Pro

fitst

anda

rd d

evia

tion

(mill

ion

euro

)

10 20 30 40 50 6010

1

2

3

Number of scenarios

CP

U ti

me

(hou

r)

Figure 3.3: Benders’ approach: Evolution of the expected profit, the profitstandard deviation and the CPU time with the number of scenarios (casestudy).

Table 3.8: Benders’ approach: Investment results (case study).Number of scenarios 60 240

Expected profit [Me] 54.29 54.07

Base capacity [MW] No investment No investment

Peak capacity [MW] 150 (bus 9) 150 (bus 9)

Convergence error (|G−G|) [% of profit] 0.00 0.40

Number of iterations 10 10

Optimality gap [%] ≤1.00 ≤1.00

CPU time [hours] 2.86 23.63

deviation remain stable for a number of scenarios higher than 40, but to provide

a sufficient margin, 60 scenarios are considered for the study reported below.

Table 3.8 gives the investment results for a case involving 60 scenarios

(considering scenario reduction) and 240 scenarios (not considering scenario

reduction). This table provides the expected profit of the strategic producer

3.8. Computational Considerations 103

1 2 3 4 5 6 7 8 9 10−100

−80

−60

−40

−20

0

Iteration

Min

us e

xpec

ted

prof

it(m

illio

n eu

ro)

SubproblemMaster problem

Figure 3.4: Benders’ approach: Evolution of Benders’ algorithm in case studyinvolving 60 scenarios.

(row 2), the investment results (rows 3-4), the convergence error (row 5), the

number of iterations needed for convergence (row 6) and the optimality gap

enforced (row 7). Note that identical investment results for both cases confirms

the validity of the scenario reduction technique used.

According to the results of Table 3.8, a new unit is built at bus 9 in the

Southern area where demand prevails. Since the proposed Benders’ algorithm

does not consider all scenarios simultaneously, it is tractable even with a higher

number of scenarios. Nevertheless, scenario reduction techniques can reduce

the computational burden significantly.

Figure 3.4 illustrates the evolution of Benders’ algorithm in the case involv-

ing 60 scenarios. This algorithm converges in iteration 10, where the difference

between G(v)

(upper curve) and G(v) (lower curve) is smaller than the tolerance

ε = 0.01.


Each auxiliary problem (3.5), each subproblem (3.7) and the master problem

(3.9) in each iteration are solved using CPLEX 12.1 [43] under GAMS [42] on


a Sun Fire X4600M2 with 8 Quad-Core processors clocking at 2.9 GHz and

256 GB of RAM.

The computational times required for solving the proposed algorithm are

provided in Tables 3.4, 3.5, 3.8 and the last plot of Figure 3.3. As expected,

the required computational time increases with the size of the problem and

with the number of scenarios.

According to Table 3.8, the proposed Benders’ algorithm is tractable even

with a high number of scenarios, as all involved scenarios are not simultane-

ously considered in such model. However, scenario reduction techniques and

parallelization can reduce the computational burden significantly.

Note that the appropriate selection of parameter βmin in the master prob-

lem (3.9) and a suitable initialization accelerate the convergence of Benders’

procedure.


As in the previous chapter, in this chapter we address the generation invest-

ment problem faced by a strategic power producer and consider a detailed

description of the uncertainty parameters involved, namely, rival producer in-

vestment and market offering, and demand growth. Since the direct solution

approach proposed in Chapter 2 may suffer from high computational burden

and eventual intractability in cases with a high number of scenarios, the aim

of this chapter is to develop an alternative tractable approach even if a very

high number of scenarios is considered. To this end, an approach based on

Benders’ decomposition is proposed. If the strategic behavior of the producer

is modeled via supply functions and a sufficiently large number of scenarios

is considered, exhaustive computational analysis indicates that the expected

profit of the strategic producer is convex enough with respect to investment

decisions; thus, an effective implementation of Benders’ approach is possible.

We consider a bilevel model identical to model (2.1)-(2.2) proposed in Chap-

ter 2 (Subsection 2.3.2). The upper-level problem of this bilevel model deter-

mines investment and offering decisions to maximize expected profit, and its

lower-level problems represent market clearing per demand block and scenario.


This bilevel model can be transformed into two alternative MPECs, which

present structures exploitable by Benders’ decomposition. One MPEC is

mixed-integer linear and the other one is non-linear.

If investment decisions are fixed to given values, each of the two MPECs

representing the considered bilevel model decomposes by scenario. The mixed-

integer linear MPEC of each scenario (denoted auxiliary problem) is solved

to attain its optimal solution. Such optimal solution allows converting the

non-linear MPEC into a continuous linear programming problem (Benders’

subproblem) that provides the sensitivities of the expected profit with respect

to investment decision. In turn, these sensitivities are used to build Benders’

cut needed in Benders’ master problem.

Numerical simulations based on realistic case studies show the good perfor-

mance of the proposed decomposition approach. In addition, such numerical

studies illustrate that the proposed approach is tractable even if hundred of

scenarios are used to describe uncertain parameters.

The main conclusions that can be drawn from this chapter are:

1) If the considered number of scenarios is large enough, the expected profit

of the strategic producer as a function of its investment decisions has a

sufficiently convex envelope, which allows using Benders’ decomposition.

2) The efficient computation of the sensitivities of the expected profit of the

producer with respect to its investment decisions (needed for the Benders’

decomposition algorithm) requires sequentially solving two MPECs per

scenario, one mixed-integer linear and one linear.

3) The proposed Benders’ algorithm attaints the optimal solution in a mod-

erate number of iterations and behaves in a robust manner.

4) Two large-scale case studies illustrate the usefulness of the proposed

approach to solve realistic problems involving many scenarios.

Chapter 4

Strategic Generation

Investment Considering the

Futures Market and the Pool

4.1 Introduction

The futures market allows trading different derivatives, both financial and

physical, encompassing a medium- or long-term horizon, e.g., one week or one

year; while the pool is typically cleared on an hourly basis and one day in

advance, throughout the time horizon spanned by the futures market. Note

that the futures market is generally cleared prior to the clearing of the pool.

The objective of this chapter is to analyze the effect of the futures market

on the investment decisions of a strategic producer competing with other pro-

ducers. To this end, a hierarchical optimization model is proposed. Then, a

mathematical program with equilibrium constraints (MPEC) is derived, which

can be linearized and recast as a tractable mixed-integer linear programming

(MILP) problem.

Note that the approach used in this chapter is similar to one proposed in

Chapter 2 (i.e., direct MPEC solution). However, in cases with high computa-

tional burden or intractability, a similar approach to one proposed in Chapter

3 (i.e, Benders’ decomposition) can be used.

107

108 4. Strategic Generation Investment Considering Futures Market and Pool

Figure 4.1: Futures market and pool: Piecewise approximation of the load-duration curve for the target year, including peak and base demand blocks.

4.2 Base and Peak Demand Blocks

The load-duration curve of the system for the target year of the planning

horizon is approximated through a number of stepwise demand blocks, as

explained in Subsection 1.6.2 of Chapter 1. This section provides a further

elaboration on such approximation, which allows us to define different futures

market products.

Figure 4.1 illustrates a stepwise approximation of the load-duration curve

as well as the two types of demand blocks obtained, i.e., peak demand blocks

(t ∈ Tp) and base demand blocks (t ∈ Tb). Note that the base and peak

demand blocks encompass the base and the peak hours of the target year,

respectively.

Based on the base and peak demand blocks defined in this section, the next

section describes the two futures market auctions considered in this chapter.

4.3. Futures Market Auctions 109

Figure 4.2: Futures market and pool: Demand blocks supplied through differ-ent markets, i.e., futures base auction, futures peak auction and pool.

4.3 Futures Market Auctions

In this chapter, we consider the following two futures market auctions. Note

that such auctions are cleared independently.

a) Futures base auction spanning all the hours of the year (i.e., all base and

peak demand blocks);

b) Futures peak auction spanning just the peak hours of the year (i.e., only

the peak demand blocks).

Figure 4.2 illustrates the futures base and futures peak auctions. The

futures base auction encompasses the whole year (i.e., all four demand blocks),

while the futures peak auction spans the peak demand blocks (i.e., the first

two demand blocks).

The model developed in this chapter makes it possible to carry out a de-

tailed analysis of the impact of the two considered futures market auctions on

the investment decisions of the strategic producer under study. Particularly,

three cases are analyzed considering:


1) Only the pool.

2) Pool plus the futures base auction.

3) Pool plus the futures base and futures peak auctions.

For the sake of simplicity, note that only yearly futures market products

are considered in this chapter. Nevertheless, quarterly or monthly futures

products can be incorporated in the proposed model by replacing the yearly

load-duration curve in Figure 4.1 by quarterly or monthly load-duration curves.

4.4 Uncertainty Modeling

The strategic generation investment problem considering the futures market

and the pool is subject to the following uncertainties:

1) The offering of units owned by rival producers in the pool.

2) The offering of units owned by rival producers in the futures market.

3) The investment decisions of rival producers.

4) The demand level in the target year.

5) The investment cost of candidate units.

6) Regulatory policies.

7) Others.

In this chapter, for the sake of simplicity, only the first uncertainty, i.e,

the offering of units owned by rival producers in the pool is considered. Such

uncertainty is represented via a set of plausible scenarios, which can be built

based on historical data pertaining to rival offers.

Nevertheless, it is possible to consider all uncertainties above, but at the

cost of high computational burden and eventual intractability. In such cases,

a similar approach to one proposed in Chapter 3, i.e, Benders’ decomposition,

can be used.

4.5. Approach 111

We consider that each rival unit offers in both futures base and futures

peak auctions at its marginal cost (no uncertainty). In addition, no investment

action is considered for rival producers. Finally, the demand level in the target

year, the investment cost of candidate units and the regulatory policies are

considered known.

4.5 Approach

This section explains the structure of the proposed model and its mathematical

formulation.

4.5.1 Hierarchical Structure

As explained in Subsection 1.7 of Chapter 1, the model considered is hierarchi-

cal (bilevel) and includes an upper-level problem and a collection of lower-level

problems. The mathematical details on bilevel models are provided in Section

B.1 of Appendix B.

The upper-level problem represents the investment actions of the strategic

producer and its strategic offering, and is constrained by investment limits,

minimum available capacity imposed by the market regulator and the collection

of lower-level problems.

Lower-level problems include the clearing of considered markets:

1. Clearing of the futures base auction.

2. Clearing of the futures peak auction.

3. Clearing of the pool under different operating conditions that reflect pool

functioning throughout the target year.

Figure 4.3 illustrates the structure of the proposed hierarchical model. The

upper-level problem seeks to minimize the minus expected profit of the strate-

gic producer, and is subject to constraints pertaining to the investment op-

tions, minimum available capacity imposed by the market regulator and a set

of lower-level problems. The first lower-level problem represents the market


ω

Figure 4.3: Futures market and pool: Hierarchical structure of the proposedstrategic generation investment model.

4.5. Approach 113

clearing of the futures base auction, the second one the clearing of the futures

peak auction, and the remaining lower-level problems the clearing of the pool

for each demand block and scenario.

As explained in Subsection 1.7 of Chapter 1, and similarly to the bilevel

model (2.1)-(2.2) presented in Subsection 2.3.2 of Chapter 2, the upper-level

and the lower-level problems are interrelated. On one hand, the lower-level

problems determine the clearing prices of futures market auctions and pool as

well as the power production quantities in such markets, which directly influ-

ence the producer’s expected profit in the upper-level problem. On the other

hand, the strategic offering and investment decisions made by the producer at

the upper-level problem affect the market clearing outcomes in the lower-level

problems.

4.5.2 Modeling Assumptions


1) The model is to be used by a strategic producer to obtain its investment

and offering decisions considering the futures market and the pool. Two

futures market auctions are considered: futures base auction and futures

peak auction.

2) The producers are allowed to engage in arbitrage, i.e., to purchase energy

from the futures market and then to sell it in the pool.

3) The strategic producer under study explicitly anticipates the impact of

its actual investment and offering actions on the market outcomes, i.e.,

locational marginal prices (LMPs) and production quantities, as explained

in Subsection 1.7 of Chapter 1. This is achieved through modeling the

lower-level market clearing problems.

4) The proposed investment model is static, i.e., a single target year is con-

sidered for decision-making, as explained in Subsection 1.6.1 of Chapter 1,

and similarly to Chapters 2 and 3. Such target year represents the final

stage of the planning horizon, and the model uses annualized cost referred

to such year.


5) For the sake of simplicity, the transmission network is not explicitly modeled

in this chapter. However, transmission constraints can be incorporated into

the lower-level problems of the proposed model as they are modeled in

lower-level problems (2.2) presented in Subsection 2.3.2 of Chapter 2.

6) The offering of units owned by rival producers in the pool is represented

via scenarios, as described in Section 4.4, while we consider that each of

those rival producers offers in both futures base and futures peak auctions

at its marginal cost. Note that no investment action is considered for rival

producers. In addition, the demand level, the investment cost of candidate

units and the regulatory policies are assumed to be known.

7) The clearing price corresponding to each market are obtained as the dual

variable associated with the market balance constraint of that market. That

is, the marginalist theory is considered [123].

8) We explicitly represent stepwise increasing offer curves for producers and

stepwise decreasing bidding curves for consumers.




not necessarily supplied at their corresponding maximum levels, i.e., PDmax

d

(futures base auction), PDmax

d (futures peak auction) and PDmax

td (pool).

Additionally, no constraint is included in the model to force the supply of

a minimum demand level.

10) To achieve physically-based solutions, i.e., solutions consistent with futures

market auctions that are physically cleared, the total amount of power sold

by a unit in the futures base auction and futures peak auction is enforced

to be equal to or lower than its installed capacity.

4.6 Formulation

This section presents the formulation of the bilevel model.

4.6. Formulation 115

4.6.1 Notational Assumptions

The following notational assumptions are considered in the formulation:

1) All symbols including index t pertain to the pool. For example, variables

P Stiw refer to the power produced by candidate unit i ∈ ΨS of the strategic

producer and sold in the pool for demand block t and scenario w.

2) Symbols without index t, but including overlining refer to the futures base

auction. For example, variables PS

i refer to the power produced by can-

didate unit i ∈ ΨS of the strategic producer and sold in the futures base

auction.

3) Symbols without index t, but including a hat refer to the futures peak auc-

tion. For example, variables P Si refer to the power produced by candidate

unit i ∈ ΨS of the strategic producer and sold in the futures peak auction.

4) The uncertainty pertaining to the offer prices of each rival producer in

the pool is represented by considering different realization of the uncertain

parameters COtjw indexed by w.

5) Both notational assumptions explained in Subsection 2.3.1 of Chapter 2

regarding the production-offer blocks of generating units and the demand-

bid blocks of demands also apply to the formulation of this chapter.

6) Dual variables of each lower-level problem are indicated at their correspond-

ing constraints following a colon.

4.6.2 Bilevel Model

The proposed bilevel model is characterized as stated below. The upper-level

problem includes objective function (4.1a) and constraints (4.1b)-(4.1e), while

(4.1f) pertains to the lower-level problems.


MinimizeΞU

∑

i∈ΨS

KiXi

− 8760×

[∑

i∈ΨS

PS

i (λ− CSi ) +

∑

k∈ΨES

PES

k (λ− CESk )

]

−

∑

t∈Tp

σt

×

[∑

i∈ΨS

P Si (λ− C

Si ) +

∑

k∈ΨES

PESk (λ− CES

k )

]

−∑

w

φw∑

t

σt ×

[∑

i∈ΨS

P Stiw(λtw − C

Si ) +

∑

k∈ΨES

PEStkw(λtw − C

ESk )

](4.1a)

subject to:

Xi =∑

h

uihXih ∀i ∈ ΨS (4.1b)

∑

h

uih = 1 ∀i ∈ ΨS (4.1c)

uih ∈ 0, 1 ∀i ∈ ΨS, ∀h (4.1d)∑

i∈ΨS

Xi +∑

k∈ΨES

PESmax

k +∑

j∈ΨO

POmax

j

≥ Υ×

∑

d∈ΨD

(P

Dmax

d + PDmax

d + PDmax

td

)t = t1 (4.1e)

(4.2)− (4.4). (4.1f)

The primal optimization variables of the bilevel problem (4.1) are those in

set ΞU = Xi, uih, αSi , α

ESk , αS

i , αESk , αS

ti, αEStk plus all variables of the lower-

level problems (4.2), (4.3) and (4.4), which are defined after the formulation

of each of these problems through sets ΞBF, ΞPF and ΞS.

The producer considered behaves strategically through the following deci-

sions:

• Strategic investment decisions, Xi.


• Strategic offering decisions in the futures base auction, αSi and αES

k .

• Strategic offering decisions in the futures peak auction, αSi and αES

k .

• Strategic offering decisions in the pool, αStiw and αES

tkw.

Note that all decisions above are made by the strategic producer at the

upper-level problem (4.1a)-(4.1e).

The strategic producer anticipates the market outcomes, i.e., LMPs and

production quantities of futures market and pool versus its decisions stated

above. To this end, constraining the upper-level problem, lower-level problems

represent the clearing of the futures auctions and the pool for given investment

and offering decisions. This allows the strategic producer to obtain feedback

regarding how its offering and investment actions affect the markets. Thus,

αSi , α

ESk , αS

i , αESk , αS

tiw, αEStkw and Xi are variables in the upper-level problem

(4.1a)-(4.1e) while they are parameters in the lower-level problems (4.1f).

The objective function (4.1a) is the minus expected profit of the strategic

producer, i.e., investment costs (∑

i∈ΨS KiXi) minus operations profits. The

first row of (4.1a) pertains to the investment costs, while rows 2 and 3 give the

operations profits obtained from the futures base auction and the futures peak

auction, respectively. The last row of (4.1a) provides the expected operation

profit achieved from the pool.

Note that the probability (weight) of scenario w is φw. Variables λ, λ

and λtw appearing in (4.1a) are the market clearing prices of the futures base

auction, the futures peak auction and the pool, respectively, and are endoge-

nously generated within the lower-level problems (4.2), (4.3) and (4.4) included

in (4.1f).

As indicated in Figure 4.2, the futures base auction encompasses the whole

target year, thus the second row in (4.1a) is multiplied by the total number of

hours in a year, i.e., 8760. On the other hand, the futures peak auction spans

the peak demand blocks (t ∈ Tp), thus the third row of (4.1a) is multiplied by

the sum of the weighing factors of the peak demand blocks, i.e.,∑

t∈Tpσt.

Constraints (4.1b)-(4.1d) allow selecting the candidate units to be built

among available investment options, being no-investment one of such options

(e.g., the available options can be 0, 250, 500 and 1000 MW).


Constraint (4.1e) enforces a regulatory condition imposing a minimum pro-

duction capacity including rival and strategic (newly built and existing) units

to ensure supply security. The non-negative factor Υ adjusts the minimum

available capacity requirement as a function of the peak demand level, i.e.,

that of the first demand block, t = t1, of Figure 4.2. Note that the assumption

about the security of supply is important and it is motivated by regulatory

policies generally enforced by regulators in most electricity markets.

Constraint (4.1f) includes the set of lower-level problems (4.2), (4.3) and

(4.4) that represents the clearing of the futures base auction, the futures

peak auction and the pool, respectively. The formulation of such lower-level

problems are presented in the three following Subsections 4.6.2.1, 4.6.2.2 and

4.6.2.3.

4.6.2.1 Futures Base Auction Clearing: Lower-level Problem (4.2)

The formulation of the lower-level problem (4.2) included in (4.1f) to represent

the futures base auction clearing is stated below:

MinimizeΞPrimalBF

∑

i∈ΨS

αSi P

S

i +∑

k∈ΨES

αESk P

ES

k +∑

j∈ΨO

CO

j PO

j −∑

d∈ΨD

UD

d PD

d (4.2a)

subject to:

∑

d∈ΨD

PD

d −∑

i∈ΨS

PS

i

−∑

k∈ΨES

PES

k −∑

j∈ΨO

PO

j = 0 : λ (4.2b)

−Xi

∆≤ P

S

i ≤Xi

Ω: µSmin

i , µSmax

i ∀i ∈ ΨS (4.2c)

−PESmax

k

∆≤ P

ES

k ≤PESmax

k

Ω: µESmin

k , µESmax

k ∀k ∈ ΨES (4.2d)

−POmax

j

∆≤ P

O

j ≤POmax

j

Ω: µOmin

j , µOmax

j ∀j ∈ ΨO (4.2e)

0 ≤ PD

d ≤ PDmax

d : µDmin

d , µDmax

d ∀d ∈ ΨD. (4.2f)


The primal optimization variables of problem (4.2) are those in the follow-

ing set:

ΞPrimalBF = P

S

i , PES

k , PO

j , PD

d .

All optimization variables of problem (4.2) including its primal and dual

variables are included in the following set:

ΞBF = ΞPrimalBF , λ, µSmin

i , µSmax

i , µESmin

k , µESmax

k , µOmin

j , µOmax

j , µDmin

d , µDmax

d .

Since the lower-level problem (4.2) constrains the upper-level problem (4.1a)-

(4.1e), the variable set ΞBF is included in the variable set of upper-level problem

ΞU.

The market operator clears the futures base auction for given investment

and offering decisions made at the upper-level problem (4.1a)-(4.1e). Thus,

Xi, αSi and αES

k are considered as parameters in (4.2), while they are variables

in the upper-level problem (4.1a)-(4.1e). Therefore, problem (4.2) is linear and

thus convex.

The objective function (4.2a) is the minus social welfare of the futures base

auction.

Constraint (4.2b) enforces energy balance in the futures base auction, and

its dual variable λ provides the market clearing price of that auction.

Constraints (4.2c)-(4.2e) enforce lower and upper production bounds for

candidate and existing units of the strategic producer and rival units, respec-

tively.

Constraints (4.2f) bound the power consumed by each demand.

Note that the lower production bounds of constraints (4.2c)-(4.2e) are con-

sidered as the minus capacities of the units divided by a positive factor ∆

to allow producers to engage in arbitrage, i.e., to purchase energy from the

futures market and then to sell it in the pool.

It is important to note that bounds on arbitrage simplify the expansion

planning analysis. However, a variety of bound levels is used to analyze the

impact of such bounds on planning outcomes. In other words, we parameter-

ize the arbitrage level and express planning outcomes as a function of such

parameter.

In addition, the factor Ω ≥ 1 bounding the generation quantities sold in the

futures market is included to achieve physically-based solutions, i.e., solutions


consistent with the futures market that are physically cleared. The use of

this factor is further described in Subsection 4.6.3, after the description of all

lower-level problems (4.2)-(4.4).

4.6.2.2 Futures Peak Auction Clearing: Lower-level Problem (4.3)

Similarly to lower-level problem (4.2), the formulation of the lower-level prob-

lem (4.3) included in (4.1f) to represent the futures peak auction clearing is

stated below. Note that the factors ∆ and Ω are defined in lower-level problem

(4.2).

MinimizeΞPrimalPF

∑

i∈ΨS

αSi P

Si +

∑

k∈ΨES

αESk PES

k +∑

j∈ΨO

COj P

Oj −

∑

d∈ΨD

UDd P

Dd (4.3a)

subject to:

∑

d∈ΨD

PDd −

∑

i∈ΨS

P Si

−∑

k∈ΨES

PESk −

∑

j∈ΨO

POj = 0 : λ (4.3b)

−Xi

∆≤ P S

i ≤Xi

Ω: µSmin

i , µSmax

i ∀i ∈ ΨS (4.3c)

−PESmax

k

∆≤ PES

k ≤PESmax

k

Ω: µESmin

k , µESmax

k ∀k ∈ ΨES (4.3d)

−POmax

j

∆≤ PO

j ≤POmax

j

Ω: µOmin

j , µOmax

j ∀j ∈ ΨO (4.3e)

0 ≤ PDd ≤ PDmax

d : µDmin

d , µDmax

d ∀d ∈ ΨD. (4.3f)

The primal optimization variables of problem (4.3) are included in the set

below:

ΞPrimalPF = P S

i , PESk , PO

j , PDd .

All primal and dual optimization variables of problem (4.3) are those in

the following set:

ΞPF = ΞPrimalPF , λ, µSmin

i , µSmax

i , µESmin

k , µESmax

k , µOmin

j , µOmax

j , µDmin

d , µDmax

d .


Since the lower-level problem (4.3) constrains the upper-level problem (4.1a)-

(4.1e), the variable set ΞPF is included in the variable set of upper-level problem

ΞU.

The market operator clears the futures peak auction for given investment

and offering decisions made at the upper-level problem; thus, Xi, αSi and αES

k

are considered as parameters in (4.3), while they are variables in the upper-

level problem (4.1a)-(4.1e). This makes lower-level problem (4.3) linear and

thus convex.

The objective function (4.3a) is the minus social welfare of the futures peak

auction.

Constraint (4.3b) enforces energy balance in the futures peak auction, being

λ the market clearing price in that market.

Constraints (4.3c)-(4.3e) enforce lower and upper production bounds for

candidate and existing units of the strategic producer, and rival units.

Constraints (4.3f) enforce consumption bounds for demands.

4.6.2.3 Pool Clearing: Lower-level Problems (4.4)

The set of lower-level problems (4.4) included in (4.1f) represent the clearing

of the pool for each demand block and scenario:

Minimize

ΞPrimalS∑

i∈ΨS

αStiwP

Stiw +

∑

k∈ΨES

αEStkwP

EStkw +

∑

j∈ΨO

COtjwP

Otjw −

∑

d∈ΨD

UDtdP

Dtdw (4.4a)

subject to:

∑

d∈ΨD

PDtdw −

∑

i∈ΨS

P Stiw

−∑

k∈ΨES

PEStkw −

∑

j∈ΨO

POtjw = 0 : λtw (4.4b)

0 ≤(P Stiw + P

S

i + P Si

)≤ Xi : µSmin

tiw , µSmax

tiw ∀i ∈ ΨS (4.4c)


0 ≤(PEStkw + P

ES

k + PESk

)≤ PESmax

k : µESmin

tkw , µESmax


0 ≤(POtjw + P

O

j + POj

)≤ POmax

j : µOmin

tjw , µOmax

tjw ∀j ∈ ΨO (4.4e)


td : µDmin

tdw , µDmax

tdw ∀d ∈ ΨD (4.4f)∀t ∈ Tp, ∀w.

Note that problems (4.4a)-(4.4f) represent the clearing of pool for peak

demand blocks (t ∈ Tp). Since the futures peak auction does not span the

base demand blocks (see Figure 4.2), the value of parameters pertaining to

such market (i.e., P Si , P

ESk and PO

j ) are forced to be zero in the case of base

demand blocks. Thus, the lower-level problems representing the clearing of

the pool in the base demand blocks (t ∈ Tb) are formulated as follows:

Minimize

ΞPrimalS

(4.4a) (4.4g)

subject to:

(4.4b) (4.4h)

0 ≤(P Stiw + P

S

i

)≤ Xi : µSmin

tiw , µSmax

tiw ∀i ∈ ΨS (4.4i)

0 ≤(PEStkw + P

ES

k

)≤ PESmax

k : µESmin

tkw , µESmax

tkw ∀k ∈ ΨES (4.4j)

0 ≤(POtjw + P

O

j

)≤ POmax

j : µOmin

tjw , µOmax

tjw ∀j ∈ ΨO (4.4k)

(4.4f) (4.4l)∀t ∈ Tb, ∀w.

The set of primal variables of lower-level problems (4.4) either for peak or

for base demand blocks are included in the following set:

ΞPrimalS = P S

tiw, PEStkw, P

Otjw, P

Dtdw.


Thus, the primal and dual optimization variables of problem (4.4) are those

in the set below:

ΞS = ΞPrimalS , λtw, µ

Smin

tiw , µSmax

tiw , µESmin

tkw , µESmax

tkw , µOmin

tjw , µOmax

tjw , µDmin

tdw , µDmax

tdw .

Observe that since the lower-level problem (4.4) constrains the upper-level

problem (4.1a)-(4.1e), the variable set ΞS is included in the variable set of

upper-level problem ΞU.

For each demand block and scenario, the market operator clears the pool for

given investment and offering decisions made at the upper-level problem; thus,

Xi, αSti and α

EStk are considered as parameters in (4.4), while they are variables

in the upper-level problem (4.1a)-(4.1e). This makes lower-level problem (4.4)

linear and thus convex.

Moreover, since both futures base and futures peak auctions are cleared

prior to the pool, production variables in such auctions, i.e., PS

i , PES

k , PO

j , PSi ,

PESk and PO

j are considered as parameters in lower-level problem (4.4) that

represent the pool clearing.

Each objective function (4.4a) or (4.4g) is the minus social welfare of the

pool.

Each set of constraints (4.4b) or (4.4h) enforces the energy balance, and

its dual variable (λtw) corresponds to the pool clearing price at the demand

block t and scenario w.

The sets of constraints (4.4c)-(4.4e) and (4.4i)-(4.4k) represent for the peak

and base demand blocks, respectively, the production bounds of candidate and

existing units of the strategic producer and rival units. Note that the total

production of each unit in the futures base auction, the futures peak auction

and the pool are considered in (4.4c)-(4.4e), while the total production of each

unit in the futures base auction and the pool are considered in (4.4i)-(4.4k).

These sets of constraints link the productions of all markets.

Finally, constraints (4.4f) and (4.4l) bound the demand supplied in the pool

between zero and the maximum load.


4.6.3 Factor Ω

The factor Ω, Ω ≥ 1 is used to prevent any rival unit purchasing more energy

from the pool than what corresponds to its installed capacity. Note that if such

factor is not considered, based on (4.2e) and (4.3e), a rival unit can sell energy

in both futures base and futures peak auctions at its maximum capacity, and

considering (4.4e), this may result in purchasing an amount of energy from the

pool higher than that corresponding to the installed capacity of the unit. We

consider that this is not realistic in physically based markets.

Enforcing Ω ≥ 2 in both futures base and futures peak auctions results in a

correct functioning of the proposed formulation, while in the case of considering

just one of those futures market auctions, such factor is not considered (i.e.,

Ω = 1).

4.6.4 Optimality Conditions Associated with the Lower-

Level Problems

To solve the proposed bilevel model (4.1), it is convenient to transform it into a

single-level optimization problem. To this end, each lower-level problem (4.2)-

(4.4) included in (4.1f) is replaced with its equivalent optimality conditions.

This transformation renders an MPEC.

As Section B.2 of Appendix B explains, the optimality conditions associated

with each lower-level problem (4.2)-(4.4) can be obtained from two alternative

approaches: i) Karush-Kuhn-Tucker (KKT) conditions, and ii) primal-dual

transformation.

As stated in Subsection 2.3.3 of Chapter 2, the following two observations

are relevant:

a) The first approach (KKT conditions) requires the enforcement of comple-

mentarity conditions for each lower-level problem. Such complementarity

conditions can be linearized as explained in Subsection B.5.1 of Appendix

B, but at the cost of adding a set of auxiliary binary variables to the variable

set ΞU.


b) The second approach (primal-dual transformation) requires the enforce-

ment of a non-linear strong duality equality for each lower-level problem.

The source of non-linearity is the product of continuous variables.

Similarly to Chapter 2, the first approach (i.e., KKT conditions) is used in

this chapter to derive the optimality conditions associated with the lower-level

problems (4.2)-(4.4). However, the strong duality equalities (obtained from the

primal-dual transformation) need to be derived, because they allow linearizing

the resulting MPEC.

Subsections 4.6.4.1, 4.6.4.3 and 4.6.4.5 present the KKT conditions associ-

ated with the lower-level problems (4.2), (4.3) and (4.4), respectively.

Additionally, Subsections 4.6.4.2, 4.6.4.4 and 4.6.4.6 derive the strong du-

ality equality associated with the lower-level problems (4.2), (4.3) and (4.4),

respectively.

4.6.4.1 KKT Conditions Associated with the Lower-Level Problem

(4.2)

To obtain the KKT conditions associated with lower-level problem (4.2), the

corresponding Lagrangian function LBF below is needed.

LBF =∑

i∈ΨS

αSi P

S

i +∑

k∈ΨES

αESk P

ES

k +∑

j∈ΨO

CO

j PO

j −∑

d∈ΨD

UD

d PD

d

+ λ

∑

d∈ΨD

PD

d −∑

i∈ΨS

PS

i −∑

k∈ΨES

PES

k −∑

j∈ΨO

PO

j

+∑

i∈ΨS

µSmax

i

(P

S

i −Xi

Ω

)−∑

i∈ΨS

µSmin

i

(Xi

∆+ P

S

i

)

+∑

k∈ΨES

µESmax

k

(P

ES

k −PESmax

k

Ω

)−∑

k∈ΨES

µESmin

k

(PESmax

k

∆+ P

ES

k

)

+∑

j∈ΨO

µOmax

j

(P

O

j −POmax

j

Ω

)−∑

j∈ΨO

µOmin

j

(POmax

j

∆+ P

O

j

)


+∑

d∈ΨD

µDmax

d

(P

D

d − PDmax

d

)−∑

d∈ΨD

µDmin

d PD

d . (4.5)

Considering the Lagrangian function LBF given by (4.5), the KKT first-

order optimality conditions of the lower-level problem (4.2) are derived as

follows:

∂LBF

∂PS

i

= αSi − λ+ µSmax

i − µSmin

i = 0 ∀i ∈ ΨS (4.6a)

∂LBF

∂PES

k

= αESk − λ+ µESmax

k − µESmin

k = 0 ∀k ∈ ΨES (4.6b)

∂LBF

∂PO

j

= CO

j − λ + µOmax

j − µOmin

j = 0 ∀j ∈ ΨO (4.6c)

∂LBF

∂PD

d

= −UD

d + λ+ µDmax

d − µDmin

d = 0 ∀d ∈ ΨD (4.6d)

∑

d∈ΨD

PD

d −∑

i∈ΨS

PS

i

−∑

k∈ΨES

PES

k −∑

j∈ΨO

PO

j = 0 (4.6e)

0 ≤

(P

S

i +Xi

∆

)⊥ µSmin

i ≥ 0 ∀i ∈ ΨS (4.6f)

0 ≤

(P

ES

k +PESmax

k

∆

)⊥ µESmin

k ≥ 0 ∀k ∈ ΨES (4.6g)

0 ≤

(P

O

j +POmax

j

∆

)⊥ µOmin

j ≥ 0 ∀j ∈ ΨO (4.6h)

0 ≤ PD

d ⊥ µDmin

d ≥ 0 ∀d ∈ ΨD (4.6i)

0 ≤

(Xi

Ω− P

S

i

)⊥ µSmax

i ≥ 0 ∀i ∈ ΨS (4.6j)


0 ≤

(PESmax

k

Ω− P

ES

k

)⊥ µESmax

k ≥ 0 ∀k ∈ ΨES (4.6k)

0 ≤

(POmax

j

Ω− P

O

j

)⊥ µOmax

j ≥ 0 ∀j ∈ ΨO (4.6l)

0 ≤(P

Dmax

d − PD

d

)⊥ µDmax

d ≥ 0 ∀d ∈ ΨD (4.6m)

λ : free. (4.6n)


a) Equality constraints (4.6a)-(4.6d) are obtained from differentiating the La-

grangian function LBF with respect to the primal variables in set ΞPrimalBF .

b) Equality constraint (4.6e) is the primal equality constraint (4.2b) in the

lower-level problem (4.2).

c) Complementarity conditions (4.6f)-(4.6m) are related to the inequality con-

straints (4.2c)-(4.2f).

d) Condition (4.6n) states that the dual variable associated with the balance

equality (4.2b), i.e., the futures base auction price, is free.

Note also that because of the linearity and thus convexity of the lower-level

problem (4.2), the KKT conditions (4.6) are necessary and sufficient conditions

for optimality.

As explained in Subsection 4.6.4, lower-level problem (4.2) is replaced by

its KKT conditions (4.6). Additionally, the strong duality equality associated

with this lower-level problem needs to be derived since such equality is used

to linearize the final MPEC. The next subsection derives such equality.


Problem (4.2)

For clarity, the dual problem of lower-level problem (4.2) is given by problem

(4.7) below:


MaximizeΞDualBF

−∑

i∈ΨS

µSmax

i

Xi

Ω−∑

i∈ΨS

µSmin

i

Xi

∆

−∑

k∈ΨES

µESmax

k

PESmax

k

Ω−∑

k∈ΨES

µESmin

K

PESmax

k

∆

−∑

j∈ΨO

µOmax

j

POmax

j

Ω−∑

j∈ΨO

µOmin

j

POmax

j

∆

−∑

d∈ΨD

µDmax

d PDmax

d (4.7a)

subject to:

(4.6a)− (4.6d) (4.7b)

µSmin

i ≥ 0; µSmax

i ≥ 0 ∀i ∈ ΨS (4.7c)

µESmin

k ≥ 0; µESmax

k ≥ 0 ∀k ∈ ΨES (4.7d)

µOmin

j ≥ 0; µOmax

j ≥ 0 ∀j ∈ ΨO (4.7e)

µDmin

d ≥ 0; µDmax

d ≥ 0 ∀d ∈ ΨD (4.7f)

(4.6n). (4.7g)

The optimization variables of problem (4.7) are the dual optimization vari-

ables of the lower-level problem (4.2), i.e.,

ΞDualBF = λ, µSmin

i , µSmax

i , µESmin

k , µESmax

k , µOmin

j , µOmax

j , µDmin

d , µDmax

d .

Considering primal problem (4.2) and its corresponding dual problem (4.7),

the set of optimality conditions associated with the lower-level problem (4.2)

resulting from the primal-dual transformation is derived as given by (4.8) be-

low. Note that the optimality conditions (4.8) are equivalent to the KKT

conditions (4.6).


(4.2b)− (4.2f) (4.8a)

(4.7b)− (4.7g) (4.8b)

∑

i∈ΨS

αSi P

S

i +∑

k∈ΨES

αESk P

ES

k

+∑

j∈ΨO

CO

j PO

j −∑

d∈ΨD

UD

d PD

d =

−∑

i∈ΨS

µSmax

i

Xi

Ω−∑

i∈ΨS

µSmin

i

Xi

∆

−∑

k∈ΨES

µESmax

k

PESmax

k

Ω−∑

k∈ΨES

µESmin

K

PESmax

k

∆

−∑

j∈ΨO

µOmax

j

POmax

j

Ω−∑

j∈ΨO

µOmin

j

POmax

j

∆

−∑

d∈ΨD

µDmax

d PDmax

d , (4.8c)

where constraint (4.8c) enforces the strong duality equality, i.e., it enforces

the equality of the values of the primal objective function (4.2a) and the dual

objective function (4.7a) at the optimal solution. This equality is used to

linearize the MPEC.

4.6.4.3 KKT Conditions Associated with the Lower-Level Problem

(4.3)

Similarly to Subsection 4.6.4.1, the KKT conditions associated with the lower-

level problem (4.3) are derived in this subsection.

To this end, the corresponding Lagrangian function LPF below is needed.


LPF =∑

i∈ΨS

αSi P

Si +

∑

k∈ΨES

αESk PES

k +∑

j∈ΨO

COj P

Oj −

∑

d∈ΨD

UDd P

Dd

+ λ

∑

d∈ΨD

PDd −

∑

i∈ΨS

P Si −

∑

k∈ΨES

PESk −

∑

j∈ΨO

POj

+∑

i∈ΨS

µSmax

i

(P Si −

Xi

Ω

)−∑

i∈ΨS

µSmin

i

(Xi

∆+ P S

i

)

+∑

k∈ΨES

µESmax

k

(PESk −

PESmax

k

Ω

)−∑

k∈ΨES

µESmin

k

(PESmax

k

∆+ PES

k

)

+∑

j∈ΨO

µOmax

j

(POj −

POmax

j

Ω

)−∑

j∈ΨO

µOmin

j

(POmax

j

∆+ PO

j

)

+∑

d∈ΨD

µDmax

d

(PDd − P

Dmax

d

)−∑

d∈ΨD

µDmin

d PDd . (4.9)

Considering the Lagrangian function LPF given by (4.9), the KKT condi-

tions of the lower-level problem (4.3) are derived as follows:

∂LPF

∂P Si

= αSi − λ+ µSmax

i − µSmin

i = 0 ∀i ∈ ΨS (4.10a)

∂LPF

∂PESk

= αESk − λ+ µESmax

k − µESmin

k = 0 ∀k ∈ ΨES (4.10b)

∂LPF

∂POj

= COj − λ+ µOmax

j − µOmin

j = 0 ∀j ∈ ΨO (4.10c)

∂LPF

∂PDd

= −UDd + λ+ µDmax

d − µDmin

d = 0 ∀d ∈ ΨD (4.10d)

∑

d∈ΨD

PDd −

∑

i∈ΨS

P Si


−∑

k∈ΨES

PESk −

∑

j∈ΨO

POj = 0 (4.10e)

0 ≤

(P Si +

Xi

∆

)⊥ µSmin

i ≥ 0 ∀i ∈ ΨS (4.10f)

0 ≤

(PESk +

PESmax

k

∆

)⊥ µESmin

k ≥ 0 ∀k ∈ ΨES (4.10g)

0 ≤

(POj +

POmax

j

∆

)⊥ µOmin

j ≥ 0 ∀j ∈ ΨO (4.10h)

0 ≤ PDd ⊥ µDmin

d ≥ 0 ∀d ∈ ΨD (4.10i)

0 ≤

(Xi

Ω− P S

i

)⊥ µSmax

i ≥ 0 ∀i ∈ ΨS (4.10j)

0 ≤

(PESmax

k

Ω− PES

k

)⊥ µESmax

k ≥ 0 ∀k ∈ ΨES (4.10k)

0 ≤

(POmax

j

Ω− PO

j

)⊥ µOmax

j ≥ 0 ∀j ∈ ΨO (4.10l)

0 ≤(PDmax

d − PDd

)⊥ µDmax

d ≥ 0 ∀d ∈ ΨD (4.10m)

λ : free. (4.10n)


a) Equality constraints (4.10a)-(4.10d) are obtained from differentiating the

Lagrangian function LPF with respect to the primal variables in set ΞPrimalPF .

b) Equality constraint (4.10e) is the primal equality constraint (4.3b) in the

lower-level problem (4.3).

c) Complementarity conditions (4.10f)-(4.10m) are related to the inequality

constraints (4.3c)-(4.3f).

d) Condition (4.10n) states that the dual variable associated with the balance

equality (4.3b), i.e., the futures peak auction price, is free.


Note that because of the linearity and thus convexity of the lower-level

problem (4.3), the KKT conditions (4.10) are necessary and sufficient condi-

tions for optimality.

As explained in Subsection 4.6.4, lower-level problem (4.3) is replaced by

its KKT conditions (4.10). Additionally, the strong duality equality associated

with this lower-level problem needs to be derived since such equality is used

to linearize the final MPEC. The next subsection derives such equality.


Problem (4.3)

To derive the strong duality equality associated with the lower-level problem

(4.3), the procedure used in this subsection is similar to the one used in Sub-

section 4.6.4.2.

The dual problem of lower-level problem (4.3) is given by problem (4.11)

below:

MaximizeΞDualPF

−∑

i∈ΨS

µSmax

i

Xi

Ω−∑

i∈ΨS

µSmin

i

Xi

∆

−∑

k∈ΨES

µESmax

k

PESmax

k

Ω−∑

k∈ΨES

µESmin

K

PESmax

k

∆

−∑

j∈ΨO

µOmax

j

POmax

j

Ω−∑

j∈ΨO

µOmin

j

POmax

j

∆

−∑

d∈ΨD

µDmax

d PDmax

d (4.11a)

subject to:

(4.10a)− (4.10d) (4.11b)

µSmin

i ≥ 0; µSmax

i ≥ 0 ∀i ∈ ΨS (4.11c)


µESmin

k ≥ 0; µESmax

k ≥ 0 ∀k ∈ ΨES (4.11d)

µOmin

j ≥ 0; µOmax

j ≥ 0 ∀j ∈ ΨO (4.11e)

µDmin

d ≥ 0; µDmax

d ≥ 0 ∀d ∈ ΨD (4.11f)

(4.10n). (4.11g)

The optimization variables of problem (4.11) are the dual optimization

variables of the lower-level problem (4.3), i.e.,

ΞDualPF = λ, µSmin

i , µSmax

i , µESmin

k , µESmax

k , µOmin

j , µOmax

j , µDmin

d , µDmax

d .

Considering primal problem (4.3) and its corresponding dual problem (4.11),

the set of optimality conditions associated with the lower-level problem (4.3)

resulting from the primal-dual transformation is given by conditions (4.12)

below. Note that the optimality conditions (4.12) are equivalent to the KKT

conditions (4.10).

(4.3b)− (4.3f) (4.12a)

(4.11b)− (4.11g) (4.12b)

∑

i∈ΨS

αSi P

Si +

∑

k∈ΨES

αESk PES

k

+∑

j∈ΨO

COj P

Oj −

∑

d∈ΨD

UDd P

Dd =

−∑

i∈ΨS

µSmax

i

Xi

Ω−∑

i∈ΨS

µSmin

i

Xi

∆

−∑

k∈ΨES

µESmax

k

PESmax

k

Ω−∑

k∈ΨES

µESmin

K

PESmax

k

∆

−∑

j∈ΨO

µOmax

j

POmax

j

Ω−∑

j∈ΨO

µOmin

j

POmax

j

∆


−∑

d∈ΨD

µDmax

d PDmax

d , (4.12c)

where constraint (4.12c) enforces the strong duality equality, i.e., it enforces

the equality of the values of the primal objective function (4.3a) and the dual

objective function (4.11a) at the optimal solution. This equality is used to

linearize the final MPEC.

4.6.4.5 KKT Conditions Associated with the Lower-Level Prob-

lems (4.4)

In this subsection, we derive the KKT conditions associated with the pool in

the peak demand blocks, i.e., problem (4.4a)-(4.4f). In the case of base demand

blocks (t ∈ Tb), analogously to problem (4.4g)-(4.4l), the value of parameters

pertaining to the futures peak auction (i.e., P Si , P

ESk and PO

j ) are forced to be

zero.

To derive the KKT optimality conditions of lower-level problems (4.4) for

the peak demand blocks (t ∈ Tp), the corresponding Lagrangian function LS

below is needed.

LS =∑

(t∈Tp)iw

αStiwP

Stiw +

∑

(t∈Tp)kw

αEStkwP

EStkw

+∑

(t∈Tp)jw

COtjwP

Otjw −

∑

(t∈Tp)dw

UDtdP

Dtdw

+∑

(t∈Tp)w

λtw

∑

d∈ΨD

PDtdw −

∑

i∈ΨS

P Stiw −

∑

k∈ΨES

PEStkw −

∑

j∈ΨO

POtjw

+∑

(t∈Tp)iw

µSmax

tiw

(P Stiw + P

S

i + P Si −Xi

)

−∑

(t∈Tp)iw

µSmin

tiw

(P Stiw + P

S

i + P Si

)


+∑

(t∈Tp)kw

µESmax

tkw

(PEStkw + P

ES

k + PESk − P

ESmax

k

)

−∑

(t∈Tp)kw

µESmin

tkw

(PEStkw + P

ES

k + PESk

)

+∑

(t∈Tp)jw

µOmax

tjw

(POtjw + P

O

j + POj − P

Omax

j

)

−∑

(t∈Tp)jw

µOmin

tjw

(POtjw + P

O

j + POj

)

+∑

(t∈Tp)dw

µDmax

tdw

(PDtdw − P

Dmax

td

)−

∑

(t∈Tp)dw

µDmin

tdw PDtdw. (4.13)

Considering the Lagrangian function for t ∈ Tp given by (4.13), the KKT

conditions of the lower-level problems (4.4) for the peak demand blocks are

derived as follows.

∂LS

∂P Stiw

=

αStiw − λtw + µSmax

tiw − µSmin

tiw = 0 ∀t ∈ Tp, ∀i ∈ ΨS, ∀w (4.14a)

∂LS

∂PEStkw

=

αEStkw − λtw + µESmax

tkw − µESmin

tkw = 0 ∀t ∈ Tp, ∀k ∈ ΨES, ∀w (4.14b)

∂LS

∂POtjw

=

COtjw − λtw + µOmax

tjw − µOmin

tjw = 0 ∀t ∈ Tp, ∀j ∈ ΨO, ∀w (4.14c)

∂LS

∂PDtdw

=

−UDtd + λtw + µDmax

tdw − µDmin

tdw = 0 ∀t ∈ Tp, ∀d ∈ ΨD, ∀w (4.14d)


∑

d∈ΨD

PDtdw −

∑

i∈ΨS

P Stiw

−∑

k∈ΨES

PEStkw −

∑

j∈ΨO

POtjw = 0 ∀t ∈ Tp, ∀w (4.14e)

0 ≤(P Stiw + P

S

i + P Si

)

⊥ µSmin

tiw ≥ 0 ∀t ∈ Tp, ∀i ∈ ΨS, ∀w (4.14f)

0 ≤(PEStkw + P

ES

k + PESk

)

⊥ µESmin

tkw ≥ 0 ∀t ∈ Tp, ∀k ∈ ΨES, ∀w (4.14g)

0 ≤(POtjw + P

O

j + POj

)

⊥ µOmin

tjw ≥ 0 ∀t ∈ Tp, ∀j ∈ ΨO, ∀w (4.14h)

0 ≤ PDtdw ⊥ µDmin

tdw ≥ 0 ∀t ∈ Tp, ∀d ∈ ΨD, ∀w (4.14i)

0 ≤(Xi − P

Stiw − P

S

i − PSi

)

⊥ µSmax

tiw ≥ 0 ∀t ∈ Tp, ∀i ∈ ΨS, ∀w (4.14j)

0 ≤(PESmax

k − PEStkw − P

ES

k − PESk

)

⊥ µESmax

tkw ≥ 0 ∀t ∈ Tp, ∀k ∈ ΨES, ∀w (4.14k)

0 ≤(POmax

j − POtjw − P

O

j − POj

)

⊥ µOmax

tjw ≥ 0 ∀t ∈ Tp, ∀j ∈ ΨO, ∀w (4.14l)

0 ≤(PDmax

td − PDtdw

)⊥ µDmax

tdw ≥ 0 ∀t ∈ Tp, ∀d ∈ ΨD, ∀w (4.14m)

λtw : free ∀t ∈ Tp, ∀w. (4.14n)


a) Equality constraints (4.14a)-(4.14d) are obtained from differentiating the

Lagrangian function LS with respect to the primal variables in set ΞPrimalS .

b) Equality constraints (4.14e) are the primal equality constraints (4.4b) in-

cluded in the lower-level problems (4.4).


c) Complementarity conditions (4.14f)-(4.14m) are related to the inequality

constraints (4.4c)-(4.4f).

d) Conditions (4.14n) state that the dual variables associated with the balance

equalities (4.4b), i.e., the pool clearing prices, are free.

Observe that the KKT conditions (4.14) are necessary and sufficient con-

ditions for optimality due to the linearity and thus convexity of the lower-level

problems (4.4).

4.6.4.6 Strong Duality Equality Associated with each Lower-Level

Problem (4.4)

Similarly to Subsections 4.6.4.2 and 4.6.4.4, the strong duality equality asso-

ciated with each lower-level problem (4.4) is obtained in this subsection.

The dual problems of lower-level problems (4.4) related to the peak demand

blocks are formulated in (4.15) below:

Maximize

ΞDualS

∑

i∈ΨS

µSmax

tiw

(P

S

i + P Si −Xi

)−∑

i∈ΨS

µSmin

tiw

(P

S

i + P Si

)

+∑

k∈ΨES

µESmax

tkw

(P

ES

k + PESk − P

ESmax

k

)−∑

k∈ΨES

µESmin

tkw

(P

ES

k + PESk

)

+∑

j∈ΨO

µOmax

tjw

(P

O

j + POj − P

Omax

j

)−∑

j∈ΨO

µOmin

tjw

(P

O

j + POj

)

−∑

d∈ΨD

µDmax

tdw PDmax

td (4.15a)

subject to:


(4.14a)− (4.14d) (4.15b)

µSmin

tiw ≥ 0; µSmax

tiw ≥ 0 ∀i ∈ ΨS (4.15c)

µESmin

tkw ≥ 0; µESmax

tkw ≥ 0 ∀k ∈ ΨES (4.15d)

µOmin

tjw ≥ 0; µOmax

tjw ≥ 0 ∀j ∈ ΨO (4.15e)

µDmin

tdw ≥ 0; µDmax

tdw ≥ 0 ∀d ∈ ΨD (4.15f)

(4.14n) (4.15g)∀t ∈ Tp, ∀w.

The optimization variables of problem (4.15) are the dual variables of prob-

lems (4.4), i.e.,

ΞDualS = λtw, µ

Smin

tiw , µSmax

tiw , µESmin

tkw , µESmax

tkw , µOmin

tjw , µOmax

tjw , µDmin

tdw , µDmax

tdw .

Considering the primal problems (4.4) and the corresponding dual prob-

lems (4.15), the set of optimality conditions (4.16) associated with the lower-

level problems (4.4) is expressed as given by (4.16). Note that the optimality

conditions (4.16) are equivalent to the KKT conditions (4.14).

(4.4b)− (4.4f) (4.16a)

(4.15b)− (4.15g) (4.16b)

∑

i∈ΨS

αStiwP

Stiw +

∑

k∈ΨES

αEStkwP

EStkw +

∑

j∈ΨO

COtjwP

Otjw −

∑

d∈ΨD

UDtdP

Dtdw =

∑

i∈ΨS

µSmax

tiw

(P

S

i + P Si −Xi

)−∑

i∈ΨS

µSmin

tiw

(P

S

i + P Si

)

+∑

k∈ΨES

µESmax

tkw

(P

ES

k + PESk − P

ESmax

k

)−∑

k∈ΨES

µESmin

tkw

(P

ES

k + PESk

)


+∑

j∈ΨO

µOmax

tjw

(P

O

j + POj − P

Omax

j

)−∑

j∈ΨO

µOmin

tjw

(P

O

j + POj

)

−∑

d∈ΨD

µDmax

tdw PDmax

td . (4.16c)

∀t ∈ Tp, ∀w,

where constraints (4.16c) are the strong duality equalities related to the lower-

level problems (4.4), which for each problem enforce the equality of the values

of the primal objective function (4.4a) and the dual objective function (4.15a)

at the optimal solution. These equalities are used to linearize the final MPEC.

4.6.5 MPEC

A single-level MPEC corresponding to the bilevel model (4.1) can be obtained

by replacing the lower-level problems (4.2), (4.3) and (4.4) with either the KKT

optimality condition sets (4.6), (4.10) and (4.14), or the optimality condition

sets (4.8), (4.12) and (4.16) resulting from the primal-dual transformation.

Pursuing linearity, the KKT optimality condition sets, (4.6), (4.10) and

(4.14), are selected since the optimality condition sets (4.8), (4.12) and (4.16)

include non-linear terms due to the product of variables in the strong duality

equalities (4.8c), (4.12c) and (4.16c).

Thus, the MPEC below is obtained by replacing the lower-level problems

(4.2), (4.3) and (4.4) with KKT condition sets (4.6), (4.10) and (4.14), respec-

tively, i.e.,

MinimizeΞU

(4.1a) (4.17a)

subject to:

(4.1b)− (4.1e) (4.17b)

(4.6), (4.10), (4.14). (4.17c)


MPEC (4.17) includes non-linearities, but it can be transformed into an

MILP problem as explained in the next subsection.

4.6.6 MPEC Linearization

MPEC (4.17) includes the following non-linearities:

1) Complementarity conditions (4.6f)-(4.6m), (4.10f)-(4.10m) and (4.14f)-

(4.14m) included in (4.17c). Such conditions can be linearized using the

approach explained in Subsection B.5.1 of Appendix B using auxiliary

binary variables and large enough constants.

2) The term∑

i∈ΨS PS

i λ+∑

k∈ΨES PES

k λ in (4.1a) included in (4.17a). This

term is denoted as Λ.

3) The term∑

i∈ΨS P Si λ+

∑k∈ΨES PES

k λ in (4.1a) included in (4.17a). This

term is denoted as Λ.

4) The term∑

i∈ΨS P Stiwλtw+

∑k∈ΨES PES

tkwλtw in (4.1a) included in (4.17a)

for demand block t and scenario w. These terms are denoted as Λtw ∀t, ∀w.

The source of non-linearity in terms Λ, Λ and Λtw is the product of pro-

duction quantity and price variables.

Exact linear expressions for terms Λ and Λ can be obtained, as explained

in Subsections 4.6.6.1 and 4.6.6.2, respectively.

In addition, Subsection 4.6.6.3 describes an approximate linearization for

terms Λtw ∀t, ∀w using the binary expansion approach explained in Subsection

B.5.2 of Appendix B.

4.6.6.1 Exact Linearization of Λ

The exact linearization approach proposed in [113] is used in this subsection to

linearize Λ, similarly to Subsection 2.3.5.1 of Chapter 2. To this end, the strong

duality equality (4.8c) and some of the KKT equalities (4.6) are considered.

From the complementarity constraints (4.6f), (4.6g), (4.6j) and (4.6k):


∑

i∈ΨS

µSmin

i

Xi

∆= −

∑

i∈ΨS

µSmin

i PS

i (4.18a)

∑

k∈ΨES

µESmin

k

PESmax

k

∆= −

∑

k∈ΨES

µESmin

k PES

k (4.18b)

∑

i∈ΨS

µSmax

i

Xi

Ω=∑

i∈ΨS

µSmax

i PS

i (4.18c)

∑

k∈ΨES

µESmax

k

PESmax

k

Ω=∑

k∈ΨES

µESmax

k PES

k . (4.18d)

Substituting equalities (4.18a)-(4.18d) in the strong duality equality (4.8c)

renders the equality below:

∑

i∈ΨS

αSi P

S

i +∑

k∈ΨES

αESk P

ES

k

+∑

j∈ΨO

CO

j PO

j −∑

d∈ΨD

UD

d PD

d =

−∑

i∈ΨS

µSmax

i PS

i +∑

i∈ΨS

µSmin

i PS

i

−∑

k∈ΨES

µESmax

k PES

k +∑

k∈ΨES

µESmin

k PES

k

−∑

j∈ΨO

µOmax

j

POmax

j

Ω−∑

j∈ΨO

µOmin

j

POmax

j

∆

−∑

d∈ΨD

µDmax

d PDmax

d . (4.18e)

On the other hand, from the KKT equalities (4.6a) and (4.6b):

λ = αSi + µSmax

i − µSmin

i ∀i ∈ ΨS (4.18f)


λ = αESk + µESmax

k − µESmin

k ∀k ∈ ΨES. (4.18g)

Multiplying equalities (4.18f) and (4.18g) by production variables PS

i and

PES

k , respectively, renders the equalities below:

PS

i λ = αSi P

S

i + PS

i µSmax

i − PS

i µSmin

i ∀i ∈ ΨS (4.18h)

PES

k λ = αESk P

ES

k + PES

k µESmax

k − PES

k µESmin

k ∀k ∈ ΨES. (4.18i)

Adding equalities (4.18h) to equalities (4.18i) renders equality (4.18j) be-

low:

∑

i∈ΨS

PS

i λ+∑

k∈ΨES

PES

k λ =

∑

i∈ΨS

αSi P

S

i +∑

i∈ΨS

PS

i µSmax

i

−∑

i∈ΨS

PS

i µSmin

i +∑

k∈ΨES

αESk P

ES

k

+∑

k∈ΨES

PES

k µESmax

k −∑

k∈ΨES

PES

k µESmin

k . (4.18j)

Finally, considering equalities (4.18e) and (4.18j), an exact linear equivalent

for Λ is obtained and given by (4.18k) below:

∑

i∈ΨS

PS

i λ+∑

k∈ΨES

PES

k λ =

∑

d∈ΨD

UD

d PD

d −∑

j∈ΨO

CO

j PO

j

−∑

d∈ΨD

PDmax

d µDmax

d

−∑

j∈ΨO

POmax

j (µOmin

j

∆+µOmax

j

Ω). (4.18k)

Note that the equality (4.18k) provides a linear equivalent for non-linear


term Λ. We denote this exact linear term as ΛLin

.

4.6.6.2 Exact Linearization of Λ

An exact linear equivalent for Λ is obtained using an identical procedure to

the one presented in the previous subsection. To do this, the strong dual-

ity equality (4.12c) and some of the KKT equalities (4.10) are used. Note

that considering symbols with a hat instead of symbols with overlining in the

linearization process described through equations (4.18) renders the following

exact linear equivalent for Λ.

∑

i∈ΨS

P Si λ+

∑

k∈ΨES

PESk λ =

∑

d∈ΨD

UDd P

Dd −

∑

j∈ΨO

COj P

Oj

−∑

d∈ΨD

PDmax

d µDmax

d

−∑

j∈ΨO

POmax

j (µOmin

j

∆+µOmax

j

Ω). (4.19)

Note that the equality (4.19) provides a linear equivalent for non-linear

term Λ. We denote this exact linear term as ΛLin.

4.6.6.3 Approximate Linearization of Λtw

Using a similar approach to the one used in Subsections 4.6.6.1 and 4.6.6.2,

an exact equivalent for Λtw pertaining to the peak demand block t ∈ Tp and

scenario w is derived below.

Note that such exact equivalent for the base demand blocks (t ∈ Tb) can be

derived using a similar procedure, in which the value of parameters pertaining

to the futures peak auction (i.e., P Si , P

ESk and PO

j ) are forced to be zero.

From the complementarity conditions (4.14f), (4.14g), (4.14j) and (4.14k):


∑

i∈ΨS

µSmin

tiw (PS

i + P Si ) = −

∑

i∈ΨS

µSmin

tiw P Stiw ∀t ∈ Tp, ∀w (4.20a)

∑

k∈ΨES

µESmin

tkw (PES

k + PESk ) = −

∑

k∈ΨES

µESmin

tkw PEStkw ∀t ∈ Tp, ∀w (4.20b)

∑

i∈ΨS

µSmax

tiw (Xi − PS

i − PSi ) =

∑

i∈ΨS

µSmax

tiw P Stiw ∀t ∈ Tp, ∀w (4.20c)

∑

k∈ΨES

µESmax

tkw (PESmax

k − PES

k − PESk ) =

∑

k∈ΨES

µESmax

tkw PEStkw ∀t ∈ Tp, ∀w. (4.20d)

Substituting equalities (4.20a)-(4.20d) in the strong duality equalities (4.16c)

renders the equalities below:

∑

i∈ΨS

αStiwP

Stiw +

∑

k∈ΨES

αEStkwP

EStkw +

∑

j∈ΨO

COtjwP

Otjw −

∑

d∈ΨD

UDtdP

Dtdw =

−∑

i∈ΨS

µSmax

tiw P Stiw +

∑

i∈ΨS

µSmin

tiw P Stiw

−∑

k∈ΨES

µESmax

tkw PEStkw +

∑

k∈ΨES

µESmin

tkw PEStkw

+∑

j∈ΨO

µOmax

tjw

(P

O

j + POj − P

Omax

j

)

−∑

j∈ΨO

µOmin

tjw

(P

O

j + POj

)

−∑

d∈ΨD

µDmax

tdw PDmax

td ∀t ∈ Tp, ∀w. (4.20e)


On the other hand, from the KKT equalities (4.14a) and (4.14b):

λtw = αStiw + µSmax

tiw − µSmin

tiw ∀t ∈ Tp, ∀i ∈ ΨS, ∀w (4.20f)

λtw = αEStkw + µESmax

tkw − µESmin

tkw ∀t ∈ Tp, ∀k ∈ ΨES, ∀w. (4.20g)

Multiplying equalities (4.20f) and (4.20g) by production variables P Stiw and

PEStkw, respectively, renders the equalities below:

P Stiwλtw= αS

tiwPStiw + P S

tiwµSmax

tiw

−P Stiwµ

Smin

tiw ∀t ∈ Tp, ∀i ∈ ΨS, ∀w (4.20h)

PEStkwλtw= αES

tkwPEStkw + PES

tkwµESmax

tkw

−PEStkwµ

ESmin

tkw ∀t ∈ Tp, ∀k ∈ ΨES, ∀w. (4.20i)

Adding equalities (4.20h) to equalities (4.20i) renders (4.20j) below:

∑

i∈ΨS

P Stiwλtw +

∑

k∈ΨES

PEStkwλtw =

∑

i∈ΨS

αStiwP

Stiw +

∑

i∈ΨS

P Stiwµ

Smax

tiw

−∑

i∈ΨS

P Stiwµ

Smin

tiw +∑

k∈ΨES

αEStkwP

EStkw

+∑

k∈ΨES

PEStkwµ

ESmax

tkw −∑

k∈ΨES

PEStkwµ

ESmin

tkw

∀t ∈ Tp, ∀w. (4.20j)

Finally, considering (4.20e) and (4.20j), an exact equivalent for Λtw in the

peak demand blocks is obtained and given by (4.20k) below:

∑

i∈ΨS

P Stiwλtw +

∑

k∈ΨES

PEStkwλtw =


∑

j∈ΨO

µOmax

tjw

(P

O

j + POj − P

Omax

j

)

−∑

j∈ΨO

µOmin

tjw

(P

O

j + POj

)

+∑

d∈ΨD

UDtdP

Dtdw −

∑

j∈ΨO

COtjwP

Otjw

−∑

d∈ΨD

µDmax

tdw PDmax

td

∀t ∈ Tp, ∀w. (4.20k)

If a linearization procedure similar to (4.20) is carried out for driving an

exact equivalent for Λtw in the base demand block (t ∈ Tb), such equivalent

considering all demand blocks is obtained and given by (4.21) below:

Λtw =

∑j∈ΨO P

O

j (µOmax

tjw − µOmin

tjw ) + Ytw ∀t ∈ Tb, ∀w

∑j∈ΨO(P

O

j + POj )(µ

Omax

tjw − µOmin

tjw ) + Ytw ∀t ∈ Tp, ∀w(4.21a)

where,

Ytw =∑

d∈ΨD

UDtdP

Dtdw −

∑

j∈ΨO

COtjwP

Otjw

−∑

j∈ΨO

POmax

j µOmax

tjw −∑

d∈ΨD

PDmax

td µDmax

tdw ∀t, ∀w. (4.21b)

It is important to note that the exact equivalent (4.21) is non-linear due

to the product of primal and dual variables in the first terms of (4.21a), i.e.,

in the two terms below:

∑

j∈ΨO

PO

j (µOmax

tjw − µOmin

tjw ) ∀t ∈ Tb, ∀w, and

∑

j∈ΨO

(PO

j + POj )(µ

Omax

tjw − µOmin

tjw ) ∀t ∈ Tp, ∀w.


However, we can apply the approximate binary expansion approach explained

in Subsection B.5.2 of Appendix B to linearize either directly the terms Λtw∀t, ∀w

or the non-linear terms in (4.21a).

Regarding the binary expansion approach, note that the expansion of the

production (primal) variables PO

j and POj is more convenient than that of the

dual variables µOmin

tjw and µOmax

tjw because lower and upper bounds are available

for these primal variables in constraints (4.2e) and (4.3e).

On the other hand, applying the binary expansion approach directly to the

terms Λtw is not convenient because doing so requires expanding variables P Stiw

that are bounded by the investment decision variables Xi in (4.4c), and such

dependency may lead to infeasibility.

Therefore, the binary expansion approach explained in Subsection B.5.2 of

Appendix B is applied to the non-linear terms of (4.21a). Thus,

ΛLintw '

Ztw : ∀t ∈ Tb, ∀w

Ztw +∑

jq γOjq(ϕ

Omax

tjwq − ϕOmin

tjwq ) : ∀t ∈ Tp, ∀w(4.22a)

where,

Ztw = Ytw +∑

jq

γOjq(φOmax

tjwq − φOmin

tjwq ) ∀t, ∀w, (4.22b)

where ΛLintw are the approximate linear equivalent for non-linear terms Λtw given

in (4.21a).

The production (primal) variables PO

j and POj in (4.21) are substituted

in (4.22) by the discrete values∑Q

q=1 γOjq and

∑Q

q=1 γOjq, respectively. These

discrete values are as close as possible to the continuous ones, i.e.:

PO

j '

Q∑

q=1

γOjq ∀j ∈ ΨO (4.22c)

POj '

Q∑

q=1

γOjq ∀j ∈ ΨO. (4.22d)


Note that index q refers to the discretized productions of rival producers

in the futures market auctions running form 1 to Q. Note that considering

a higher number of discretization intervals results in higher accuracy in the

approximation of the production variables, but at the cost of higher computa-

tional burden.

In addition, φOmin

tjwq , φOmax

tjwq , ϕOmin

tjwq and ϕOmax

tjwq are auxiliary continuous vari-

ables.

For the binary expansion approach pertaining to the linearized terms in-

cluded in (4.22b) to work, i.e.,∑

jq γOjq(φ

Omax

tjwq − φOmin

tjwq ), ∀t, ∀w, the following

set of mixed-integer linear equations should be incorporated as constraints:

PO

j −δj2≤

Q∑

q=1

γOjq$jq ≤ PO

j +δj2

∀j ∈ ΨO (4.22e)

Q∑

q=1

$jq = 1 ∀j ∈ ΨO (4.22f)

$jq ∈ 0, 1 ∀j ∈ ΨO, ∀q (4.22g)

0 ≤ µOmax

tjw − φOmax

tjwq ≤ G(1−$jq) ∀t, ∀j ∈ ΨO, ∀w, ∀q (4.22h)

0 ≤ φOmax

tjwq ≤ G$jq ∀t, ∀j ∈ ΨO, ∀w, ∀q (4.22i)

0 ≤ µOmin

tjw − φOmin

tjwq ≤ G(1−$jq) ∀t, ∀j ∈ ΨO, ∀w, ∀q (4.22j)

0 ≤ φOmin

tjwq ≤ G$jq ∀t, ∀j ∈ ΨO, ∀w, ∀q. (4.22k)

Similarly to constraints set (4.22e)-(4.22k), for the binary expansion ap-

proach pertaining to the linearized terms∑

jq γOjq(ϕ

Omax

tjwq −ϕOmin

tjwq ), ∀t ∈ Tp, ∀w,

included in (4.22a) to work, the following set of mixed-integer linear equations

should be incorporated as constraints:


POj −

δj2≤

Q∑

q=1

γOjq$jq ≤ POj +

δj2

∀j ∈ ΨO (4.22l)

Q∑

q=1

$jq = 1 ∀j ∈ ΨO (4.22m)

$jq ∈ 0, 1 ∀j ∈ ΨO, ∀q (4.22n)

0 ≤ µOmax

tjw − ϕOmax

tjwq ≤ G(1− $jq) ∀t ∈ Tp, ∀j ∈ ΨO, ∀w, ∀q (4.22o)

0 ≤ ϕOmax

tjwq ≤ G$jq ∀t ∈ Tp, ∀j ∈ ΨO, ∀w, ∀q (4.22p)

0 ≤ µOmin

tjw − ϕOmin

tjwq ≤ G(1− $jq) ∀t ∈ Tp, ∀j ∈ ΨO, ∀w, ∀q (4.22q)

0 ≤ ϕOmin

tjwq ≤ G$jq ∀t ∈ Tp, ∀j ∈ ΨO, ∀w, ∀q, (4.22r)

where G is a large enough positive constant, and δj and δj are constants for

each rival producer j, defined as follows:

δj = γOj(q+1) − γOjq ∀j ∈ ΨO (4.22s)

δj = γOj(q+1) − γOjq ∀j ∈ ΨO. (4.22t)

The iterative procedure explained in Subsection B.5.2 of Appendix B is

used for selecting discrete values γOjq and γOjq for each rival unit j. This leads

to an increasing accuracy in the approximation.

The next subsection presents the mixed-integer linear form of MPEC (4.17)

according to the linearization techniques described in Subsection 4.6.6.


4.6.7 MILP Formulation

Using the linearization techniques presented in Subsection 4.6.6, MPEC (4.17)

is transformed into an MILP problem given by (4.23)-(4.41) below.

MinimizeΞU, Ξψ , ΞB

∑

i∈ΨS

KiXi

− 8760×

(Λ

Lin−∑

i∈ΨS

PS

iCSi −

∑

k∈ΨES

PES

k CESk

)

−

∑

t∈Tp

σt

×

(ΛLin −

∑

i∈ΨS

P Si C

Si −

∑

k∈ΨES

PESk CES

k

)

−∑

w

φw∑

t

σt ×

(ΛLintw −

∑

i∈ΨS

P StiwC

Si −

∑

k∈ΨES

PEStkwC

ESk

)(4.23)

subject to:

1) Exact linear expression for non-linear term ΛLin

appearing in the objective

function (4.23):

ΛLin

=∑

d∈ΨD

UD

d PD

d −∑

j∈ΨO

CO

j PO

j −∑

d∈ΨD

PDmax

d µDmax

d

−∑

j∈ΨO

POmax

j (µOmin

j

∆+µOmax

j

Ω). (4.24)

2) Exact linear expression for non-linear term ΛLin appearing in the objective

function (4.23):

ΛLin =∑

d∈ΨD

UDd P

Dd −

∑

j∈ΨO

COj P

Oj −

∑

d∈ΨD

PDmax

d µDmax

d

−∑

j∈ΨO

POmax

j (µOmin

j

∆+µOmax

j

Ω). (4.25)

3) Approximate linear expression for non-linear terms ΛLintw appearing in the


objective function (4.23):

ΛLintw '

Ztw : ∀t ∈ Tb, ∀w

Ztw +∑

jq γOjq(ϕ

Omax

tjwq − ϕOmin

tjwq ) : ∀t ∈ Tp, ∀w(4.26a)

where,

Ztw = Ytw +∑

jq

γOjq(φOmax

tjwq − φOmin

tjwq ) ∀t, ∀w. (4.26b)

4) Set of mixed-integer linear constraints required for the binary expansion

approach to work:

PO

j −δj2≤

Q∑

q=1

γOjq$jq ≤ PO

j +δj2∀j ∈ ΨO (4.27a)

Q∑

q=1

$jq = 1 ∀j ∈ ΨO (4.27b)

$jq ∈ 0, 1 ∀j ∈ ΨO, ∀q (4.27c)

0 ≤ µOmax

tjw − φOmax

tjwq ≤ G(1−$jq) ∀t, ∀j ∈ ΨO, ∀w, ∀q (4.27d)

0 ≤ φOmax

tjwq ≤ G$jq ∀t, ∀j ∈ ΨO, ∀w, ∀q (4.27e)

0 ≤ µOmin

tjw − φOmin

tjwq ≤ G(1−$jq) ∀t, ∀j ∈ ΨO, ∀w, ∀q (4.27f)

0 ≤ φOmin

tjwq ≤ G$jq ∀t, ∀j ∈ ΨO, ∀w, ∀q (4.27g)

POj −

δj2≤

Q∑

q=1

γOjq$jq ≤ POj +

δj2

∀j ∈ ΨO (4.27h)

Q∑

q=1

$jq = 1 ∀j ∈ ΨO (4.27i)

$jq ∈ 0, 1 ∀j ∈ ΨO, ∀q (4.27j)

0 ≤ µOmax

tjw − ϕOmax

tjwq ≤ G(1− $jq) ∀t ∈ Tp, ∀j ∈ ΨO, ∀w, ∀q (4.27k)


0 ≤ ϕOmax

tjwq ≤ G$jq ∀t ∈ Tp, ∀j ∈ ΨO, ∀w, ∀q (4.27l)

0 ≤ µOmin

tjw − ϕOmin

tjwq ≤ G(1− $jq) ∀t ∈ Tp, ∀j ∈ ΨO, ∀w, ∀q (4.27m)

0 ≤ ϕOmin

tjwq ≤ G$jq ∀t ∈ Tp, ∀j ∈ ΨO, ∀w, ∀q (4.27n)

δj = γOj(q+1) − γOjq ∀j ∈ ΨO (4.27o)

δj = γOj(q+1) − γOjq ∀j ∈ ΨO. (4.27p)

5) The upper-level constraints:

Xi =∑

h

uihXih ∀i ∈ ΨS (4.28a)

∑

h

uih = 1 ∀i ∈ ΨS (4.28b)

uih ∈ 0, 1 ∀i ∈ ΨS, ∀h (4.28c)∑

i∈ΨS

Xi +∑

k∈ΨES

PESmax

k +∑

j∈ΨO

POmax

j

≥ Υ×

∑

d∈ΨD

(P

Dmax

d + PDmax

d + PDmax

td

)t = t1. (4.28d)

6) KKT equalities related to the clearing of the futures base auction:

αSi − λ+ µSmax

i − µSmin

i = 0 ∀i ∈ ΨS (4.29a)

αESk − λ+ µESmax

k − µESmin

k = 0 ∀k ∈ ΨES (4.29b)

CO

j − λ+ µOmax

j − µOmin

j = 0 ∀j ∈ ΨO (4.29c)

−UD

d + λ+ µDmax

d − µDmin

d = 0 ∀d ∈ ΨD (4.29d)

∑

d∈ΨD

PD

d −∑

i∈ΨS

PS

i −∑

k∈ΨES

PES

k −∑

j∈ΨO

PO

j = 0. (4.29e)


7) KKT equalities related to the clearing of the futures peak auction:

αSi − λ+ µSmax

i − µSmin

i = 0 ∀i ∈ ΨS (4.30a)

αESk − λ+ µESmax

k − µESmin

k = 0 ∀k ∈ ΨES (4.30b)

COj − λ+ µOmax

j − µOmin

j = 0 ∀j ∈ ΨO (4.30c)

−UDd + λ+ µDmax

d − µDmin

d = 0 ∀d ∈ ΨD (4.30d)

∑

d∈ΨD

PDd −

∑

i∈ΨS

P Si −

∑

k∈ΨES

PESk −

∑

j∈ΨO

POj = 0. (4.30e)

8) KKT equalities related to the clearing of the pool (common to both base

and peak demand blocks):

αStiw − λtw + µSmax

tiw − µSmin

tiw = 0 ∀t, ∀i ∈ ΨS, ∀w (4.31a)

αEStkw − λtw + µESmax

tkw − µESmin

tkw = 0 ∀t, ∀k ∈ ΨES, ∀w (4.31b)

COtjw − λtw + µOmax

tjw − µOmin

tjw = 0 ∀t, ∀j ∈ ΨO, ∀w (4.31c)

−UDtd + λtw + µDmax

tdw − µDmin

tdw = 0 ∀t, ∀d ∈ ΨD, ∀w (4.31d)

∑

d∈ΨD

PDtdw −

∑

i∈ΨS

P Stiw

−∑

k∈ΨES

PEStkw −

∑

j∈ΨO

POtjw = 0 ∀t, ∀w. (4.31e)

9) Conditions enforcing that the dual variables of the market balance equali-

ties, i.e., the clearing prices, are free:

λ : free (4.32a)

λ : free (4.32b)

λtw : free ∀t, ∀w. (4.32c)


10) Linearization of complementarity conditions (4.6f)-(4.6m) related to the

clearing of the futures base auction:

(P

S

i +Xi

∆

)≥ 0 ∀i ∈ ΨS (4.33a)

(P

ES

k +PESmax

k

∆

)≥ 0 ∀k ∈ ΨES (4.33b)

(P

O

j +POmax

j

∆

)≥ 0 ∀j ∈ ΨO (4.33c)

PD

d ≥ 0 ∀d ∈ ΨD (4.33d)

µSmin

i ≥ 0 ∀i ∈ ΨS (4.33e)

µESmin

k ≥ 0 ∀k ∈ ΨES (4.33f)

µOmin

j ≥ 0 ∀j ∈ ΨO (4.33g)

µDmin

d ≥ 0 ∀d ∈ ΨD (4.33h)

(P

S

i +Xi

∆

)≤ ψ

Smin

i MP ∀i ∈ ΨS (4.33i)

(P

ES

k +PESmax

k

∆

)≤ ψ

ESmin

k MP ∀k ∈ ΨES (4.33j)

(P

O

j +POmax

j

∆

)≤ ψ

Omin

j MP ∀j ∈ ΨO (4.33k)

PD

d ≤ ψDmin

d MP ∀d ∈ ΨD (4.33l)

µSmin

i ≤(1− ψ

Smin

i

)Mµ ∀i ∈ ΨS (4.33m)

µESmin

k ≤(1− ψ

ESmin

k

)Mµ ∀k ∈ ΨES (4.33n)

µOmin

j ≤(1− ψ

Omin

j

)Mµ ∀j ∈ ΨO (4.33o)


µDmin

d ≤(1− ψ

Dmin

d

)Mµ ∀d ∈ ΨD (4.33p)

ψSmin

i ∈ 0, 1 ∀i ∈ ΨS (4.33q)

ψESmin

k ∈ 0, 1 ∀k ∈ ΨES (4.33r)

ψOmin

j ∈ 0, 1 ∀j ∈ ΨO (4.33s)

ψDmin

d ∈ 0, 1 ∀d ∈ ΨD (4.33t)

(Xi

Ω− P

S

i

)≥ 0 ∀i ∈ ΨS (4.33u)

(PESmax

k

Ω− P

ES

k

)≥ 0 ∀k ∈ ΨES (4.33v)

(POmax

j

Ω− P

O

j

)≥ 0 ∀j ∈ ΨO (4.33w)

(P

Dmax

d − PD

d

)≥ 0 ∀d ∈ ΨD (4.33x)

µSmax

i ≥ 0 ∀i ∈ ΨS (4.33y)

µESmax

k ≥ 0 ∀k ∈ ΨES (4.33z)

µOmax

j ≥ 0 ∀j ∈ ΨO (4.34a)

µDmax

d ≥ 0 ∀d ∈ ΨD (4.34b)

(Xi

Ω− P

S

i

)≤ ψ

Smax

i MP ∀i ∈ ΨS (4.34c)

(PESmax

k

Ω− P

ES

k

)≤ ψ

ESmax

k MP ∀k ∈ ΨES (4.34d)

(POmax

j

Ω− P

O

j

)≤ ψ

Omax

j MP ∀j ∈ ΨO (4.34e)

(P

Dmax

d − PD

d

)≤ ψ

Dmax

d MP ∀d ∈ ΨD (4.34f)


µSmax

i ≤(1− ψ

Smax

i

)Mµ ∀i ∈ ΨS (4.34g)

µESmax

k ≤(1− ψ

ESmax

k

)Mµ ∀k ∈ ΨES (4.34h)

µOmax

j ≤(1− ψ

Omax

j

)Mµ ∀j ∈ ΨO (4.34i)

µDmax

d ≤(1− ψ

Dmax

d

)Mµ ∀d ∈ ΨD (4.34j)

ψSmax

i ∈ 0, 1 ∀i ∈ ΨS (4.34k)

ψESmax

k ∈ 0, 1 ∀k ∈ ΨES (4.34l)

ψOmax

j ∈ 0, 1 ∀j ∈ ΨO (4.34m)

ψDmax

d ∈ 0, 1 ∀d ∈ ΨD, (4.34n)


11) Linearization of complementarity conditions (4.10f)-(4.10m) related to the

clearing of the futures peak auction:

(P Si +

Xi

∆

)≥ 0 ∀i ∈ ΨS (4.35a)

(PESk +

PESmax

k

∆

)≥ 0 ∀k ∈ ΨES (4.35b)

(POj +

POmax

j

∆

)≥ 0 ∀j ∈ ΨO (4.35c)

PDd ≥ 0 ∀d ∈ ΨD (4.35d)

µSmin

i ≥ 0 ∀i ∈ ΨS (4.35e)

µESmin

k ≥ 0 ∀k ∈ ΨES (4.35f)

µOmin

j ≥ 0 ∀j ∈ ΨO (4.35g)

µDmin

d ≥ 0 ∀d ∈ ΨD (4.35h)


(P Si +

Xi

∆

)≤ ψSmin

i MP ∀i ∈ ΨS (4.35i)

(PESk +

PESmax

k

∆

)≤ ψESmin

k MP ∀k ∈ ΨES (4.35j)

(POj +

POmax

j

∆

)≤ ψOmin

j MP ∀j ∈ ΨO (4.35k)

PDd ≤ ψDmin

d MP ∀d ∈ ΨD (4.35l)

µSmin

i ≤(1− ψSmin

i

)Mµ ∀i ∈ ΨS (4.35m)

µESmin

k ≤(1− ψESmin

k

)Mµ ∀k ∈ ΨES (4.35n)

µOmin

j ≤(1− ψOmin

j

)Mµ ∀j ∈ ΨO (4.35o)

µDmin

d ≤(1− ψDmin

d

)Mµ ∀d ∈ ΨD (4.35p)

ψSmin

i ∈ 0, 1 ∀i ∈ ΨS (4.35q)

ψESmin

k ∈ 0, 1 ∀k ∈ ΨES (4.35r)

ψOmin

j ∈ 0, 1 ∀j ∈ ΨO (4.35s)

ψDmin

d ∈ 0, 1 ∀d ∈ ΨD (4.35t)

(Xi

Ω− P S

i

)≥ 0 ∀i ∈ ΨS (4.35u)

(PESmax

k

Ω− PES

k

)≥ 0 ∀k ∈ ΨES (4.35v)

(POmax

j

Ω− PO

j

)≥ 0 ∀j ∈ ΨO (4.35w)

(PDmax

d − PDd

)≥ 0 ∀d ∈ ΨD (4.35x)

µSmax

i ≥ 0 ∀i ∈ ΨS (4.35y)


µESmax

k ≥ 0 ∀k ∈ ΨES (4.35z)

µOmax

j ≥ 0 ∀j ∈ ΨO (4.36a)

µDmax

d ≥ 0 ∀d ∈ ΨD (4.36b)

(Xi

Ω− P S

i

)≤ ψSmax

i MP ∀i ∈ ΨS (4.36c)

(PESmax

k

Ω− PES

k

)≤ ψESmax

k MP ∀k ∈ ΨES (4.36d)

(POmax

j

Ω− PO

j

)≤ ψOmax

j MP ∀j ∈ ΨO (4.36e)

(PDmax

d − PDd

)≤ ψDmax

d MP ∀d ∈ ΨD (4.36f)

µSmax

i ≤(1− ψSmax

i

)Mµ ∀i ∈ ΨS (4.36g)

µESmax

k ≤(1− ψESmax

k

)Mµ ∀k ∈ ΨES (4.36h)

µOmax

j ≤(1− ψOmax

j

)Mµ ∀j ∈ ΨO (4.36i)

µDmax

d ≤(1− ψDmax

d

)Mµ ∀d ∈ ΨD (4.36j)

ψSmax

i ∈ 0, 1 ∀i ∈ ΨS (4.36k)

ψESmax

k ∈ 0, 1 ∀k ∈ ΨES (4.36l)

ψOmax

j ∈ 0, 1 ∀j ∈ ΨO (4.36m)

ψDmax

d ∈ 0, 1 ∀d ∈ ΨD, (4.36n)


12) Linearization of complementarity conditions (4.14i) and (4.14m) related to

the clearing of the pool (common to the base and peak demand blocks):

PDtdw ≥ 0 ∀t, ∀d ∈ ΨD, ∀w (4.37a)


µDmin

tdw ≥ 0 ∀t, ∀d ∈ ΨD, ∀w (4.37b)

PDtdw ≤ ψDmin

tdw MP ∀t, ∀d ∈ ΨD, ∀w (4.37c)

µDmin

tdw ≤(1− ψDmin

tdw

)Mµ ∀t, ∀d ∈ ΨD, ∀w (4.37d)

(PDmax

td − PDtdw

)≥ 0 ∀t, ∀d ∈ ΨD, ∀w (4.37e)

µDmax

tdw ≥ 0 ∀t, ∀d ∈ ΨD, ∀w (4.37f)

(PDmax

td − PDtdw

)≤ ψDmax

tdw MP ∀t, ∀d ∈ ΨD, ∀w (4.37g)

µDmax

tdw ≤(1− ψDmax

tdw

)Mµ ∀t, ∀d ∈ ΨD, ∀w (4.37h)

ψDmin

tdw ∈ 0, 1 ∀t, ∀d ∈ ΨD, ∀w (4.37i)

ψDmax

tdw ∈ 0, 1 ∀t, ∀d ∈ ΨD, ∀w, (4.37j)


13) Linearization of complementarity conditions (4.14f)-(4.14h) and (4.14j)-

(4.14l) related to the clearing of the pool in the base demand blocks:

(P Stiw + P

S

i

)≥ 0 ∀t ∈ Tb, ∀i ∈ ΨS, ∀w (4.38a)

(PEStkw + P

ES

k

)≥ 0 ∀t ∈ Tb, ∀k ∈ ΨES, ∀w (4.38b)

(POtjw + P

O

j

)≥ 0 ∀t ∈ Tb, ∀j ∈ ΨO, ∀w (4.38c)

µSmin

tiw ≥ 0 ∀t ∈ Tb, ∀i ∈ ΨS, ∀w (4.38d)

µESmin

tkw ≥ 0 ∀t ∈ Tb, ∀k ∈ ΨES, ∀w (4.38e)

µOmin

tjw ≥ 0 ∀t ∈ Tb, ∀j ∈ ΨO, ∀w (4.38f)

(P Stiw + P

S

i

)≤ ψSmin

tiw MP ∀t ∈ Tb, ∀i ∈ ΨS, ∀w (4.38g)

(PEStkw + P

ES

k

)≤ ψESmin

tkw MP ∀t ∈ Tb, ∀k ∈ ΨES, ∀w (4.38h)


(POtjw + P

O

j

)≤ ψOmin

tjw MP ∀t ∈ Tb, ∀j ∈ ΨO, ∀w (4.38i)

µSmin

tiw ≤(1− ψSmin

tiw Mµ)

∀t ∈ Tb, ∀i ∈ ΨS, ∀w (4.38j)

µESmin


tkw Mµ)

∀t ∈ Tb, ∀k ∈ ΨES, ∀w (4.38k)

µOmin

tjw ≤(1− ψOmin

tjw Mµ)

∀t ∈ Tb, ∀j ∈ ΨO, ∀w (4.38l)

ψSmin

tiw ∈ 0, 1 ∀t ∈ Tb, ∀i ∈ ΨS, ∀w (4.38m)

ψESmin

tkw ∈ 0, 1 ∀t ∈ Tb, ∀k ∈ ΨES, ∀w (4.38n)

ψOmin

tjw ∈ 0, 1 ∀t ∈ Tb, ∀j ∈ ΨO, ∀w (4.38o)(Xi − P

Stiw − P

S

i

)≥ 0 ∀t ∈ Tb, ∀i ∈ ΨS, ∀w (4.39a)

(PESmax

k − PEStkw − P

ES

k

)≥ 0 ∀t ∈ Tb, ∀k ∈ ΨES, ∀w (4.39b)

(POmax

j − POtjw − P

O

j

)≥ 0 ∀t ∈ Tb, ∀j ∈ ΨO, ∀w (4.39c)

µSmax

tiw ≥ 0 ∀t ∈ Tb, ∀i ∈ ΨS, ∀w (4.39d)

µESmax

tkw ≥ 0 ∀t ∈ Tb, ∀k ∈ ΨES, ∀w (4.39e)

µOmax

tjw ≥ 0 ∀t ∈ Tb, ∀j ∈ ΨO, ∀w (4.39f)

(Xi − P

Stiw − P

S

i

)≤ ψSmax

tiw MP ∀t ∈ Tb, ∀i ∈ ΨS, ∀w (4.39g)

(PESmax

k − PEStkw − P

ES

k

)≤ ψESmax

tkw MP ∀t ∈ Tb, ∀k ∈ ΨES, ∀w (4.39h)

(POmax

j − POtjw − P

O

j

)≤ ψOmax

tjw MP ∀t ∈ Tb, ∀j ∈ ΨO, ∀w (4.39i)

µSmax

tiw ≤(1− ψSmax

tiw Mµ)

∀t ∈ Tb, ∀i ∈ ΨS, ∀w (4.39j)

µESmax


tkw Mµ)

∀t ∈ Tb, ∀k ∈ ΨES, ∀w (4.39k)

µOmax

tjw ≤(1− ψOmax

tjw Mµ)

∀t ∈ Tb, ∀j ∈ ΨO, ∀w (4.39l)


ψSmax

tiw ∈ 0, 1 ∀t ∈ Tb, ∀i ∈ ΨS, ∀w (4.39m)

ψESmax

tkw ∈ 0, 1 ∀t ∈ Tb, ∀k ∈ ΨES, ∀w (4.39n)

ψOmax

tjw ∈ 0, 1 ∀t ∈ Tb, ∀j ∈ ΨO, ∀w, (4.39o)


14) Linearization of complementarity conditions (4.14f)-(4.14h) and (4.14j)-

(4.14l) related to the clearing of the pool in the peak demand blocks:

(P Stiw + P

S

i + P Si

)≥ 0 ∀t ∈ Tp, ∀i ∈ ΨS, ∀w (4.40a)

(PEStkw + P

ES

k + PESk

)≥ 0 ∀t ∈ Tp, ∀k ∈ ΨES, ∀w (4.40b)

(POtjw + P

O

j + POj

)≥ 0 ∀t ∈ Tp, ∀j ∈ ΨO, ∀w (4.40c)

µSmin

tiw ≥ 0 ∀t ∈ Tp, ∀i ∈ ΨS, ∀w (4.40d)

µESmin

tkw ≥ 0 ∀t ∈ Tp, ∀k ∈ ΨES, ∀w (4.40e)

µOmin

tjw ≥ 0 ∀t ∈ Tp, ∀j ∈ ΨO, ∀w (4.40f)

(P Stiw + P

S

i + P Si

)

≤ ψSmin

tiw MP ∀t ∈ Tp, ∀i ∈ ΨS, ∀w (4.40g)

(PEStkw + P

ES

k + PESk

)

≤ ψESmin

tkw MP ∀t ∈ Tp, ∀k ∈ ΨES, ∀w (4.40h)

(POtjw + P

O

j + POj

)

≤ ψOmin

tjw MP ∀t ∈ Tp, ∀j ∈ ΨO, ∀w (4.40i)

µSmin

tiw ≤(1− ψSmin

tiw Mµ)

∀t ∈ Tp, ∀i ∈ ΨS, ∀w (4.40j)

µESmin


tkw Mµ)

∀t ∈ Tp, ∀k ∈ ΨES, ∀w (4.40k)


µOmin

tjw ≤(1− ψOmin

tjw Mµ)

∀t ∈ Tp, ∀j ∈ ΨO, ∀w (4.40l)

ψSmin

tiw ∈ 0, 1 ∀t ∈ Tp, ∀i ∈ ΨS, ∀w (4.40m)

ψESmin

tkw ∈ 0, 1 ∀t ∈ Tp, ∀k ∈ ΨES, ∀w (4.40n)

ψOmin

tjw ∈ 0, 1 ∀t ∈ Tp, ∀j ∈ ΨO, ∀w (4.40o)(Xi − P

Stiw − P

S

i − PSi

)≥ 0 ∀t ∈ Tp, ∀i ∈ ΨS, ∀w (4.41a)

(PESmax

k − PEStkw − P

ES

k − PESk

)≥ 0 ∀t ∈ Tp, ∀k ∈ ΨES, ∀w (4.41b)

(POmax

j − POtjw − P

O

j − POj

)≥ 0 ∀t ∈ Tp, ∀j ∈ ΨO, ∀w (4.41c)

µSmax

tiw ≥ 0 ∀t ∈ Tp, ∀i ∈ ΨS, ∀w (4.41d)

µESmax

tkw ≥ 0 ∀t ∈ Tp, ∀k ∈ ΨES, ∀w (4.41e)

µOmax

tjw ≥ 0 ∀t ∈ Tp, ∀j ∈ ΨO, ∀w (4.41f)

(Xi − P

Stiw − P

S

i − PSi

)

≤ ψSmax

tiw MP ∀t ∈ Tp, ∀i ∈ ΨS, ∀w (4.41g)

(PESmax

k − PEStkw − P

ES

k − PESk

)

≤ ψESmax

tkw MP ∀t ∈ Tp, ∀k ∈ ΨES, ∀w (4.41h)

(POmax

j − POtjw − P

O

j − POj

)

≤ ψOmax

tjw MP ∀t ∈ Tp, ∀j ∈ ΨO, ∀w (4.41i)

µSmax

tiw ≤(1− ψSmax

tiw Mµ)

∀t ∈ Tp, ∀i ∈ ΨS, ∀w (4.41j)

µESmax


tkw Mµ)

∀t ∈ Tp, ∀k ∈ ΨES, ∀w (4.41k)

µOmax

tjw ≤(1− ψOmax

tjw Mµ)

∀t ∈ Tp, ∀j ∈ ΨO, ∀w (4.41l)

ψSmax

tiw ∈ 0, 1 ∀t ∈ Tp, ∀i ∈ ΨS, ∀w (4.41m)

ψESmax

tkw ∈ 0, 1 ∀t ∈ Tp, ∀k ∈ ΨES, ∀w (4.41n)

4.7. Case Study 163

ψOmax

tjw ∈ 0, 1 ∀t ∈ Tp, ∀j ∈ ΨO, ∀w, (4.41o)


Note that set Ξψ below is the set of binary variables needed to linearize the

complementarity conditions (4.6f)-(4.6m), (4.10f)-(4.10m) and (4.14f)-(4.14m).

Ξψ = ψSmin

i , ψESmin

k , ψOmin

j , ψDmin

d , ψSmax

i , ψESmax

k , ψOmax

j , ψDmax

d , ψSmin

i ,

ψESmin

k , ψOmin

j , ψDmin

d , ψSmax

i , ψESmax

k , ψOmax

j , ψDmax

d , ψSmin

tiw , ψESmin

tkw , ψOmin

tjw , ψDmin

tdw ,

ψSmax

tiw , ψESmax

tkw , ψOmax

tjw , ψDmax

tdw .

In addition, the variable set ΞB below includes the auxiliary continuous

and binary variables introduced by the binary expansion approach to linearize

the terms Λtw ∀t, ∀w.

ΞB = γOjq, δj, $jq, φOmin

tjwq , φOmax

tjwq , γOjq, δj , $jq, ϕ

Omin

tjwq , ϕOmax

tjwq .

4.7 Case Study

This section presents results for a case study based on the IEEE one-area

Reliability Test System (RTS) [110], whose structure and data are presented

in Appendix A. The network is not modeled in this case study.

4.7.1 Data

The load duration curve of the target year is approximated by four demand

blocks with weighting factors (σt) 1095, 2190, 2190 and 3285, whose summation

renders the number of hours in a year (8760).

Note that the futures base auction encompasses the whole target year (i.e.,

all four demand blocks), while the futures peak auction spans the peak demand

blocks (i.e., the first two demand blocks). This is illustrated in Figure 4.4.

The maximum load level of each demand in block t is denoted Dmaxtd . The

maximum demand equals the summation of the maximum powers supplied

through the futures base auction, the futures peak auction and the pool, i.e.,


max

)1( dttD

=

max

)2( dttD

=

max

)3( dttD

=

max

)4( dttD

=

max

)2(

Dmaxˆdttd

DP=

×Θ=

max

)4(

Dmax

dttdDP

=×Γ=

Dmax

)4( dttP

=

Dmax

)3( dttP

=

Dmax

)1( dttP

=

Dmax

)2( dttP

=

Figure 4.4: Futures market and pool: Maximum load level of a given demandsupplied through the futures base auction, futures the peak auction and thepool.

Dmaxtd = P

Dmax

d + PDmax

d + PDmax

td ∀t, ∀d ∈ ΨD. (4.42)

Figure 4.4 illustrates the load level parameters Dmaxtd for a particular de-

mand. As shown in this figure, we use two non-negative factors Γ and Θ to

specify the percentage of each demand that can be supplied through the fu-

tures base auction and the futures peak auction, respectively. Factor Γ is fixed

based on the demand level of the fourth demand block Dmax(t=t4)d

, and Θ is fixed

based on the demand level of the second demand block Dmax(t=t2)d

.

Considering Figure 4.4, the maximum load of each demand in each consid-

ered market (i.e., futures base auction, futures peak auction and pool) is as

follows:

4.7. Case Study 165

a) The maximum level of each demand in the futures peak auction is

PDmax

d = Θ×Dmax(t=t2)d

.

b) The maximum level of each demand in the futures base auction is

PDmax

d = Γ×Dmax(t=t4)d

.

c) The maximum level of each demand in the pool for the first block (t = t1)

is PDmax

(t=t1)d= Dmax

(t=t1)d− PDmax

d − PDmax

d .

Note that t = t1 is a peak demand block.

d) The maximum level of each demand in the pool for the second block (t = t2)

is PDmax

(t=t2)d= Dmax

(t=t2)d− PDmax

d − PDmax

d .

Note that t = t2 is a peak demand block.

e) The maximum level of each demand in the pool for the third block (t = t3)

is PDmax

(t=t3)d= Dmax

(t=t3)d− P

Dmax

d .

Note that t = t3 is a base demand block.

f) The maximum level of each demand in the pool for the fourth block (t = t4)

is

PDmax

(t=t4)d= Dmax

(t=t4)d− P

Dmax

d .

Note that t = t4 is a base demand block.

In this case study, the following values for parameters Dmaxtd are considered:

• The maximum load of each demand in the first demand block, i.e., pa-

rameter Dmax(t=t1)d

, is the one reported in [110].

• The maximum load of each demand in the second, third and fourth

demand blocks is the one in the first demand block multiplied by factors

0.90, 0.75, and 0.65, respectively. Thus,

Dmax(t=t2)d

= 0.90×Dmax(t=t1)d

∀d.

Dmax(t=t3)d

= 0.75×Dmax(t=t1)d

∀d.

Dmax(t=t4)d

= 0.65×Dmax(t=t1)d

∀d.


Regarding the bid price of each demand in the futures base auction, the

futures peak auction and the pool, the corresponding values, i.e., parameters

UD

d , UDd and UD

td, are as follows:

• In the first demand block of the pool, demands 1-10 bid at 25.00 e/MWh,

demands 12, 14, 16 and 17 at 28.00 e/MWh, and the remaining demands

at 30.00 e/MWh, i.e.,

UDtd = 25.00 [e/MWh] t = t1, d = 1− 10.

UDtd = 28.00 [e/MWh] t = t1, d = 12, 14, 16, 17.

UDtd = 30.00 [e/MWh] t = t1, d = 11, 13, 15.

• In the next three demand blocks of the pool, each demand bids the

corresponding power in the first demand block multiplied by 0.90, 0.80

and 0.75, respectively. Thus,

UD(t=t2)d

= 0.90× UD(t=t1)d

∀d.

UD(t=t3)d

= 0.80× UD(t=t1)d

∀d.

UD(t=t4)d

= 0.75× UD(t=t1)d

∀d.

• Each demand bids in the futures base auction identically to its bid at

the fourth demand block of the pool, i.e.,

UD

d=UDtd t = t4, ∀d.

• Each demand bids in the futures peak auction its bid at the second

demand block of the pool, i.e.,

UDd =U

Dtd t = t2, ∀d.

Tables 4.1 and 4.2 give the data of the existing units of the strategic pro-

ducer and the rival units, respectively. In both tables, columns 2-3 contain the

4.7. Case Study 167

Table 4.1: Futures market and pool: Data for the existing units of the strategicproducer.

Existing Type of Capacity Capacity Capacity Production cost Production cost

unit existing[MW]


(k ∈ ΨES) unit [MW] [MW] [e/MWh] [e/MWh]

1-3 Coal 76 30 46 13.46 13.96

4 Coal 155 55 100 9.92 10.25

5-6 Gas 100 25 75 17.60 18.12

Table 4.2: Futures market and pool: Data for rival units.

Rival Type of Capacity Capacity Capacity Production cost Production cost

unit rival[MW]


(j ∈ ΨO) unit [MW] [MW] [e/MWh] [e/MWh]

1-2 Gas 197 97 100 10.08 10.66

3 Coal 76 30 46 13.46 13.96

4-6 Coal 155 55 100 9.92 10.25

7-8 Gas 120 40 80 18.60 19.03

9 Gas 100 25 75 17.60 18.12

type of each unit and its capacity. Each unit is characterized by two generation

blocks (columns 4-5) with corresponding marginal costs (columns 6-7).

Note that the total available capacity in the system is 1858 MW, 31.38%

of it (i.e., 583 MW) belonging to the strategic producer.

The available investment options are given in Table 4.3. Pursuing simplic-

ity, the size of each of the two production blocks of each unit is considered

equal to half of the installed capacity. Costs for these two generation blocks

are provided in the last two columns of Table 4.3.

All cases in this section take into account three scenarios that represent

the uncertainty of rival producer offering in the pool. We assume that each

rival unit offers in both futures base and futures peak auctions at its marginal

cost (no uncertainty), and in the pool at that cost multiplied by a factor. The

three scenarios considered are described below:


Table 4.3: Futures market and pool: Type and data for the investment options.



(i ∈ ΨS)(Ki) candidate units block 1 block 2


Base technology 600000, 100, 200, 300, 400, 500,

9.20 10.40600, 700, 800, 900, 1000


0, 100, 150, 200, 250,

15.42 16.90300, 350, 400, 450, 500,

550, 600, 650, 700, 750,

800, 850, 900, 950, 1000

Scenario 1) Each rival unit offers in the futures market and pool at its marginal

cost, i.e., the values for all cost parameters CO

j , COj and CO

tjw

are equal and identical to the costs provided in Table 4.2. The

probability of this scenario is arbitrarily fixed to 0.50.

Scenario 2) Each rival unit offers in both futures market auctions at its marginal

cost, and in the pool at that marginal cost multiplied by 1.15. This

means that the values for CO

j and COj are equal and identical to

the costs provided in Table 4.2, while the values for COtjw are those

marginal costs multiplied by 1.15. The probability of this scenario

is arbitrarily fixed to 0.30.

Scenario 3) Each rival unit offers in both futures market auctions at its marginal

cost, and in the pool at that marginal cost multiplied by 1.30. This

means that the values for CO

j and COj are equal and identical to

the costs provided in Table 4.2, while the values for COtjw are those

marginal costs multiplied by 1.30. The probability of this scenario

is arbitrarily fixed to 0.20.

Finally, we consider that the market regulator imposes that the available

capacity, including all units (newly built and existing), should be at least 10%

higher than the peak demand, i.e., Υ=1.10.

4.7. Case Study 169

4.7.2 Cases Considered

Table 4.4 characterizes the cases considered. Columns 2-4 provide the factors

Γ,Θ,∆ and Ω for each case. Additionally, the last column indicates the offering

behavior of the strategic producer (strategic or non-strategic). Considering

Table 4.4, the cases analyzed are described below:

Case 1) In this case, only the pool is considered, i.e., all demands are sup-

plied through the pool. In addition, all units offer at their marginal

costs (non-strategic offering). This is realized by replacing the offer-

ing variables αStiw and αES

tkw in the objective function (4.4a) with cost

parameters CSi and CES

k , respectively.

Case 2) Similarly to Case 1, only the pool is considered in this case; thus, all

demands are supplied through the pool. However, unlike Case 1, the

producer under study offers strategically.

Case 3) In this case, demands are supplied through two markets: futures base

auction and pool. In addition, the amount of demand that is supplied

through the futures base auction is at most 30% of the demand in the

fourth block (i.e., Γ = 30%). Moreover, considering ∆ = ∞ prevents

the producers from engaging in arbitrage. In this case, factor Ω is

considered to be 1, so that each producer can offer in the futures base

auction up to its maximum capacity.

Case 4) This case is similar to Case 3: i) the markets considered are the same,

i.e., futures base auction and pool, ii) parameter ∆ is equal to ∞

so that producers do not engage in arbitrage, and iii) Ω = 1, which

means that each producer can offer in the futures base auction up

to its maximum capacity. However, unlike Case 3, the amount of

demand that is supplied through the futures base auction is at most

75% of the demand in the fourth block (i.e., Γ = 75%).

Case 5) The similarities between this case and Cases 3 and 4 are as follows: i)

demands are supplied through the futures base auction and the pool,

ii) the value of factor Ω is 1. In this case, the amount of demand


Table 4.4: Futures market and pool: Cases considered.

Case Γ(%) Θ(%) ∆ Ω Offering

Case 1 No futures base auction No futures peak auction - - non-strategic

Case 2 No futures base auction No futures peak auction - - strategic

Case 3 30 No futures peak auction ∞ 1 strategic

Case 4 75 No futures peak auction ∞ 1 strategic

Case 5 30 No futures peak auction 10 1 strategic



Case 8 10 10 10 2 strategic

Case 9 45 45 10 2 strategic

that is supplied through the futures base auction is that of Case 3,

i.e., at most 30% of the demand in the fourth block (i.e., Γ = 30%).

However, unlike Cases 3 and 4 where arbitrage is not allowed, in this

case each producer can buy in the futures base auction at most 10%

of its capacity (∆ = 10), and then sell in the pool (arbitrage).

Case 6) Similarly to Cases 3-5, two markets (futures base auction and pool)

are considered to supply the demand in this case. Also, the value of

factor Ω is 1. In this case, the amount of demand that is supplied

through the futures base auction is at most 75% of the demand in the

fourth block (i.e., Γ = 75%). Similarly to Case 5, each producer can

buy in the futures base auction at most 10% of its capacity (∆ = 10),

and then sell in the pool (arbitrage).

Case 7) The markets considered (futures base auction and pool) and the value

of factor Ω (i.e., Ω=1) are similar to those in Cases 3-6. However, in

this case, the amount of demand that is supplied through the futures

base auction is comparatively low with respect to Cases 3-6, i.e., at

most 10% of the demand in the fourth block (i.e, Γ = 10%). Never-

theless, the arbitrage bound in this case is comparatively higher than

the one in Cases 3-6, i.e., each producer can buy in the futures base

4.7. Case Study 171

auction at most 50% of its capacity (∆ = 2), and then sell it in the

pool.

Case 8) In this case, the futures base auction, the futures peak auction and

the pool are considered. The amount of demand that is supplied

through the futures base auction is at most 10% of the demand in

the fourth block (i.e., Γ = 10%). Moreover, the amount of demand

that is supplied through the futures peak auction is at most 10% of

the demand in the second block (i.e., Θ = 10%). Each producer is

allowed to engage in arbitrage through buying energy at most 10%

of its capacity (∆ = 10) in each futures market auction. In addition,

factor Ω is considered to be 2, thus the maximum power that each

unit can sell in each futures market auction is half of its capacity.

Case 9) Similarly to Case 8, demands in this case are supplied through the

three markets (futures base auction, futures peak auction and pool).

However, in this case, the amount of demand that is supplied through

the futures base auction is at most 45% of the demand in the fourth

block (i.e., Γ = 45%). In addition, the amount of demand that is sup-

plied through the futures peak auction is at most 45% of the demand

in the second block (i.e., Θ = 45%). Similarly to Case 8, each pro-

ducer is allowed to engage in arbitrage through buying at most 10%

of its capacity (∆ = 10) in each futures market auction. In addition,

factor Ω is 2, and thus the maximum power that each unit can sell in

each futures market auction is half of its capacity.

4.7.3 Investment Results

The investment results are given in Table 4.5. The expected profit of the

strategic producer in each case is provided in the second column, while the

base capacity, peak capacity and total capacity to be built are provided in

columns 3, 4 and 5, respectively.

The results in Table 4.5 show that the strategic producer achieves compar-

atively higher expected profit in the following situations (ordered from higher


Table 4.5: Futures market and pool: Investment results.

Case

Expected Base capacity Peak capacity Total capacity

profit to be built to be built to be built

[Me] [MW] [MW] [MW]

Case 1 13.37 300 1000 1300

Cases 2, 3 53.62 500 800 1300

Cases 4, 6 48.98 500 800 1300

Case 5 67.75 400 900 1300

Case 7 100.26 400 900 1300

Case 8 73.97 400 900 1300

Case 9 47.84 300 1000 1300

to lower):

1) Being the marginal strategic producer in the futures market and pool,

provided that the amount of demands supplied in the futures market

auctions is comparatively low, and arbitrage is allowed. This occurs in

Cases 5, 7 and 8.

2) Being the marginal strategic producer just in the pool (Case 2), or be-

ing the marginal strategic producer in the futures market and the pool,


auctions is comparatively low, and arbitrage is not allowed (Case 3).

3) Being the marginal strategic producer in the futures market and the pool,


auctions is comparatively high. This occurs in Cases 4, 6 and 9.

4) Being the marginal non-strategic producer in every market (Case 1).

Note that in all the cases considered, the total capacity of candidate units to

be built by the strategic producer is 1300 MW, but with different configuration

of newly built units. The reason for this fixed investment level is constraint

(4.1e). Parameter Υ=1.10 through this constraint forces a capacity level higher

than the peak demand.

4.7. Case Study 173

The results for all cases considered are given in Tables 4.6 and 4.7 and

further analyzed in the following subsections.

In Table 4.6, rows 2 and 3 provide the futures base auction and the futures

peak auction prices. The pool prices per demand block pertaining to scenarios

1, 2 and 3 are provided in rows 4, 5 and 6, respectively.

In Table 4.7, yearly production of the strategic producer in the futures

base auction and futures peak auction are provided in rows 2 and 3, while its

production quantity in the pool pertaining to scenarios 1, 2 and 3 is provided

in rows 4, 5 and 6, respectively. Similar data for rival producers are given in

rows 7-11.

4.7.3.1 Only Pool

Cases 1 and 2 that correspond to a pool only market are analyzed below.

As expected, the pool prices in Case 2 (strategic offering) are comparatively

higher than the corresponding prices in Case 1 (non-strategic offering).

Moreover, the yearly production of the strategic producer in Case 2 is

comparatively lower than such production in Case 1.

Overall, the producer obtains a comparatively higher expected profit by

behaving strategically.

4.7.3.2 Pool and Futures Base Auction Without Arbitrage

In this subsection, both the futures base auction and the pool are considered.

Additionally, the strategic producer is not allowed to purchase energy from the

futures base auction (thus avoiding arbitrage). The cases considered are Cases

3 and 4.

In Case 3, since the demand bid prices in the futures base auction are

comparatively low, those demands are supplied by rival units. The strategic

producer participates in the pool as much as in Case 2; thus, its expected profit

and its newly built units do not change with respect to Case 2.

In Case 4, most of the demand is supplied through the futures base auction

(Γ = 75%); therefore, the strategic producer is forced to participate in this

1744.Stra

tegic

Gen

eratio

nInvestm

entConsid

eringFutures

Market

andPool

Table 4.6: Futures market and pool: Market clearing prices pertaining to all cases considered.

Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 Case 9

Futures base auction price [e/MWh] - - 10.25 18.75 10.66 18.75 10.66 10.66 18.75

Futures peak auction price [e/MWh] - - - - - - - 10.66 22.50

Pool prices per demand block (Scenario 1) [e/MWh]

λ1 18.12 25.00 25.00 25.00 25.00 25.00 25.00 25.00 25.00

λ2 16.90 22.50 22.50 22.50 22.50 22.50 22.50 22.50 22.50

λ3 16.90 20.00 20.00 19.03 20.00 19.03 20.00 20.00 19.03

λ4 15.42 18.60 18.60 18.60 18.60 18.60 18.60 18.60 18.12


λ1 20.83 25.00 25.00 25.00 25.00 25.00 25.00 25.00 25.00

λ2 16.90 21.89 21.89 21.89 22.50 21.89 22.50 22.50 21.40

λ3 16.90 20.00 20.00 20.00 20.00 20.00 20.00 20.00 20.00

λ4 15.42 18.75 18.75 18.75 18.75 18.75 18.75 18.75 18.75


λ1 23.56 24.74 24.74 24.74 24.74 24.74 24.74 24.74 24.74

λ2 17.50 22.50 22.50 22.50 22.50 22.50 22.50 22.50 22.50

λ3 16.90 20.00 20.00 20.00 20.00 20.00 20.00 20.00 20.00

λ4 15.42 18.75 18.75 21.00 18.75 21.00 18.75 18.75 18.75

4.7.Case

Study

175

Table 4.7: Futures market and pool: Yearly production results pertaining to all cases considered.

Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 Case 9

Yearly production by the SP∗ in the FBA [GWh] - - 0.0 2403.6 -1649.5 2403.6 -5902.0 -1649.5 2418.9

Yearly production by the SP in the FPA [GWh] - - - - - - - -568.3 1697.5

Yearly production by the SP in the pool (Scenario 1) [GWh] 11204.6 9124.1 9124.1 7071.0 10773.6 7071.0 15026.1 11341.9 5604.5



Yearly production by the RP† in the FBA [GWh] - - 4868.4 9767.4 6517.9 9767.4 7524.8 3272.3 4883.7

Yearly production by the RP in the FPA [GWh] - - - - - - - 1410.9 2094.2

Yearly production by the RP in the pool (Scenario 1) [GWh] 8300.1 10380.6 5512.2 262.8 3862.7 262.8 2855.8 5697.4 2805.9

Yearly production by the RP in the pool (Scenario 2) [GWh] 7976.0 8957.1 4088.7 -810.3 2789.6 -810.3 1782.7 4624.3 1804.0

Yearly production by the RP in the pool (Scenario 3) [GWh] 7693.5 8387.8 3519.3 -1379.7 1869.8 -1379.7 826.9 3704.5 1409.8

CPU time [s] 200.8 1.0 75.8 80.2 74.2 73.5 88.5 963.3 656.4

‡FBA: Futures base auction ; ?FPA: Futures peak auction ; ∗ SP: Strategic producer ; †RP: Rival producers


auction. Since the price in the futures base auction is lower than that in the

pool, the expected profit of the strategic producer decreases.

Market outcomes as a function of factor Γ are illustrated in Figure 4.5.

In this figure, the first plot pertains to the expected profit of the strategic

producer, the second plot shows the average pool prices regarding all three

scenarios, and the third plot illustrates the futures base auction price. The

average production of both strategic and rival units in the futures base auction

and pool are depicted in plots 4 and 5, respectively. The average demand

payment per MWh in the futures base auction and the pool is represented in

the last plot.

According to Figure 4.5, the pool prices do not change within the interval

Γ = 0% to Γ = 60% (plot 2), while the futures base auction price increases

with Γ (plot 3).

In addition, the production of the strategic producer in the futures base

auction is zero, while its production quantity in the pool does not change (plot

4); thus, its expected profit remains fixed (plot 1). Within the interval Γ = 0%

to Γ = 60%, all demands of the futures base auction are supplied by rival units

(plot 5).

A higher amount of demand to be supplied through the futures base auction

(Γ changing from 60% to 100%) forces the strategic producer to participate in

the futures base auction. Therefore, the production of the strategic producer

for the futures base auction increases with Γ while the production for the pool

decreases (plot 4), and its expected profit decreases too (plot 1).

Within the interval Γ = 60% to Γ = 100%, changing factor Γ is not

production-effective for rival producers as the total production does not change

(plot 5).

Finally, note that the best market configuration from the demands’ point

of view is Γ = 40%, where the demand payment per MWh is minimum (last

plot). Also, the most profitable market configuration from the producers’ point

of view is Γ = 0%, where the demands are only supplied through the pool. In

this case the demand payment per MWh is maximum (last plot).

To provide enough incentives for producers to invest in new capacity, it is

crucial for the market regulator to find an appropriate market configuration

4.7. Case Study 177

0 20 40 60 80 10040

45

50

55

Exp

ecte

d pr

ofit

(mill

ion

euro

s)

0 20 40 60 80 100

18

22

26

Ave

rage

poo

l pric

epe

r de

man

d bl

ock

(e

uro/

MW

h)

0 20 40 60 80 1000

10

20

Fut

ures

bas

eau

ctio

n pr

ice

(eur

o/M

Wh)

0 20 40 60 80 1000

5000

10000

Ave

rage

yea

rly

prod

uctio

n by

the

stra

tegi

c pr

oduc

er

(GW

h)

0 20 40 60 80 1000

5000

10000

Ave

rage

yea

rly

prod

uctio

n by

riv

al p

rodu

cers

(G

Wh)

0 20 40 60 80 100

18

20

22

Γ (%)

Ave

rage

pay

men

tpe

r M

Wh

for

dem

ands

(eur

o/M

Wh)

in pool in futures base auction

in pool in futures base auction

t1

t2

t3

t4

Figure 4.5: Futures market and pool: Market outcomes as a function of factorΓ.


through suitably selecting the value of factor Γ, so that (i) the producers are

motivated to invest, and (ii) the demand payment is as small as possible.

4.7.3.3 Pool and Futures Base Auction With Arbitrage

Here, the futures base auction and the pool are considered with the possibility

of arbitrage. Cases 5-7 are considered in this subsection.

Case 5 is similar to Case 3, but engaging in arbitrage is allowed for each

unit up to 10% of its capacity (∆ = 10). Hence, the strategic producer partic-

ipates in the futures base auction as a buyer. In fact, it purchases energy from

the futures base auction at λ = 10.66 e/MWh and then sells it in the pool at

comparatively higher prices; thus, its expected profit increases. Note that the

strategic producer purchases energy from the futures base auction at its max-

imum allowance, i.e., 10% of its candidate and existing capacity (1300+583)

multiplied by 8760.

Case 6 is similar to Case 4, but with the possibility of arbitrage. Since

in Case 4 (with no arbitrage), each rival unit with production cost smaller

than λ = 18.75 e/MWh sells energy in the futures base auction to its max-

imum capacity, in Case 6 (with possibility of arbitrage), purchasing energy

by the strategic producer from the futures base auction (arbitrage) results in

expensive rival units participating in that market. On the other hand, the

production cost of such expensive rival units are higher than the bids of most

demands in this market. Hence, in this situation, there is no potential for

the strategic producer to engage in arbitrage, i.e., engaging in arbitrage is not

profit effective and thus the investment results of Cases 4 and 6 are identical.

Case 7, involving a low level of demand supplied in the futures base auction

(Γ = 10%) and the possibility of each unit engaging in arbitrage up to 50%

of its capacity (∆ = 2) results in increasing expected profit for the strategic

producer (as in Case 5). However, note that this producer does not purchase

energy from the futures base auction (arbitrage) at its maximum allowance

(8247.5 GWh), i.e., 50% of its newly built and existing capacity (1300+583)

multiplied by 8760. The reason is that purchasing additional energy from the

futures base auction forces expensive rival units to participate in that market,

4.7. Case Study 179

1020

3040

50

1

1030

5070

100

40

50

60

70

80

90

100

110

Γ (%)1/∆ (%)

Exp

ecte

d pr

ofit

(mill

ion

euro

s)

Figure 4.6: Futures market and pool: Expected profit of the strategic produceras a function of Γ and 1

∆.

which results in a higher clearing price. Thus, although the strategic producer

can buy additional energy from the futures base auction, further arbitrage is

not profit effective, i.e., the profitability of engaging in arbitrage is saturated.

To illustrate this effect, Figure 4.6 depicts the strategic producer’s expected

profit as functions of factors Γ and 1∆. Four relevant conclusions can be drawn

from Figure 4.6:

1) If factor Γ is comparatively small (e.g., Γ = 10%), arbitrage is highly

profit effective.

2) If factor Γ is comparatively large (e.g., Γ = 100%), arbitrage is not

generally profit effective.


3) For each factor ∆, a comparatively smaller factor Γ renders a higher

expected profit and vice versa.

4) The profitability of engaging in arbitrage is limited. For example, for

Γ = 10%, the expected profit does not change if ∆ changes from 3 to 2.

4.7.3.4 Pool, Futures Base and Futures Peak Auctions

Next, the pool and the futures base and the futures peak auctions are consid-

ered with the possibility of arbitrage, which correspond to Cases 8 and 9.

In Case 8, similarly to Cases 5 and 7, the strategic producer engages in

arbitrage by purchasing energy from the futures market and then selling it

in the pool at higher prices. Since in Case 8 the strategic producer can buy

energy from both futures base and futures peak auctions, its profit in this case

is higher than in Case 5. Note that it purchases energy in the futures base

auction up to its maximum allowance (similarly to Case 5), while in the futures

peak auction it does not purchase at its maximum allowance (similarly to Case

7).

In Case 9, the strategic producer is forced to participate in the futures

market auctions due to the high amount of demand that is supplied in these

auctions. In this case, the production cost of all rival units is smaller than the

futures peak auction price (λ= 22.50 e/MWh), so all rival units sell energy

in the futures peak auction to their allowed capacities (half of their respec-

tive capacities). Thus, the rival units cannot sell further in the futures peak

auction. In addition, the rival units with production costs smaller than the

futures base auction price (λ= 18.75 e/MWh) sell energy in the futures base

auction to their allowed capacities (half of their respective capacities). On

the other hand, purchasing energy by the strategic producer from the futures

base auction (arbitrage) leads to the situation described in Case 6. Hence, the

strategic producer does not buy energy from the futures market to engage in

arbitrage.

4.7. Case Study 181

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2

52

54

56

58

Exp

ecte

d pr

ofit

(mill

ion

euro

)

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.20

500

1000

1500

2000

Factor ϒ

Tot

al in

vest

men

t (M

W)

Figure 4.7: Futures market and pool: Strategic producer’s expected profit andits total investment as a function of factor Υ (Case 3).

4.7.3.5 Impact of Factor Υ on Generation Investment Decisions

The impact of factor Υ on generation investment decisions of the strategic

producer is analyzed as follows.

As an example, Figure 4.7 illustrates the investment and expected profit

results in Case 3. The first plot pertains to the expected profit of the strategic

producer, while the second one depicts its total investment.

As shown in Figure 4.7, within the interval 0 ≤ Υ ≤ 1, the total investment

of the strategic producer is 1000 MW, and its expected profit does not change,

but enforcing a higher value for factor Υ leads to a comparatively higher total

investment and a comparatively smaller expected profit.

Finally, note that these trends are consistent for all Cases 1-9.



MILP problem (4.23)-(4.41) is solved using CPLEX 12.1 [43] under GAMS [42]

on a Sun Fire X4600M2 with 8 Quad-Core processors clocking at 2.9 GHz and

256 GB of RAM.

The computational times required for solving the proposed model are pro-

vided in the last row of Table 4.7 (Subsection 4.7.3). Note that the optimality

gap for all cases is enforced to be zero.

Among the cases considered in Table 4.7 (Subsection 4.7.3), the time re-

quired to solve Cases 8 and 9 is comparatively higher. Note that these are

the cases in which pool, futures base and futures peak auctions are considered,

with the possibility of engaging in arbitrage.

The approach used in this chapter is similar to one proposed in Chapter 2

(i.e., direct MPEC solution). However, considering stochastic cases with many

scenarios may lead to a high computational burden and eventual intractabil-

ity. Thus, a similar approach to one proposed in Chapter 3, i.e, Benders’

decomposition, can be used.


Futures markets are increasingly relevant for trading electric energy as they

help to hedge the volatility of the pool prices. In this chapter, we analyze the

effect of such market on the investment decisions of a strategic producer.

Two futures market auctions involving physical settlement are considered:

i) futures base auction spanning all the hours of the year, i.e., all base and peak

demand blocks, and ii) futures peak auction spanning just the peak hours of

the year, i.e., only the peak demand blocks.

The offering in the pool of units owned by the rival producers is represented

via scenarios. These scenarios can be built based on historical data pertaining

to rival offers. On the other hand, and for the sake of simplicity, we consider

that each of those rival producers offers in both futures market auctions at its

marginal cost.

To analyze the effect of the futures market on the investment decisions of


a strategic producer, we propose a bilevel model whose upper-level problem

represents the investment and offering actions of the producer, and whose

multiple lower-level problems represent the clearing of the futures base auction,

futures peak auction and pool under different operating conditions.

Such model is equivalent to an MPEC that can be recast as a tractable

MILP problem using i) an exact linearization approach, and ii) an approximate

binary expansion approach.

The approach used in this chapter is analogous to one proposed in Chap-

ter 2 (i.e., direct MPEC solution). However, considering stochastic cases with

many scenarios may result in a high computational burden and eventual in-

tractability. In such cases, a similar approach to one proposed in Chapter 3

using Benders’ decomposition can be used.

The studies carried out, intended to analyze the impact of the futures

market on the investment decisions by a strategic producer, allow drawing the

following relevant conclusions:

1) The futures market makes a difference for the strategic producer in both

expected profit and investment decisions.

2) The futures market with comparatively low energy trading, and with the

possibility of arbitrage results in higher expected profit for the strategic

producer. In the case of no arbitrage, the futures market is not profit

effective.

3) The futures market with comparatively high energy trading results in

lower expected profit for the strategic producer.

4) The total capacity of candidate units to be built by the strategic producer

directly depends on the value considered for parameter Υ in constraint

(4.1e) related to supply security. Enforcing a higher value for factor Υ

leads to a comparatively higher total investment and a comparatively

smaller expected profit.

5) The profitability of engaging in arbitrage is limited.

Chapter 5

Generation Investment

Equilibria

5.1 Introduction

The aim of a producer competing in an electricity market is to maximize its

profit through its investment and operation strategies. Since the strategies of

any producer are interrelated with those of other producers through the market

interaction, decisions made by one producer may influence the strategies of

other producers. Therefore, a number of investment equilibria may exist, where

each producer cannot increase its profit by changing unilaterally its strategies.

The objective of this chapter is to mathematically identify such investment

equilibria.

An equilibrium analysis is useful for a market regulator to gain insight

into the investment behavior of the producers and the generation investment

evolution. Such insight may allow the market regulator to better design market

rules, which in turn may contribute to increase the competitiveness of the

market and to stimulate optimal investment in generation capacity.

All producers considered in this chapter are “strategic”, i.e., they can alter

the market outcomes, including locational marginal prices (LMPs) and pro-

duction quantities, through their strategies. “Strategic offering” and “strategic

investment” refer to the offering and investment decisions of a strategic pro-

185

186 5. Generation Investment Equilibria

ducer, respectively.

The investment and offering decisions of each strategic producer are rep-

resented through a hierarchical (bilevel) model, whose upper-level problem

decides on the optimal investment and the supply offering curves for maximiz-

ing the profit of the producer, and whose several lower-level problems represent

different market clearing scenarios, one per demand block. Note that the gen-

eral structure of hierarchical (bilevel) models is explained in Section 1.7 of

Chapter 1. Additionally, mathematical details on bilevel models are provided

in Section B.1 of Appendix B.

Replacing the lower-level problems with their optimality conditions in the

single-producer model renders a mathematical program with equilibrium con-

straint (MPEC). Note that this transformation is explained in detail in Section

1.7 of Chapter 1. Additionally, mathematical details on MPEC are provided

in Section B.2 of Appendix B.

The joint consideration of all producer MPECs, one per producer, consti-

tutes an equilibrium problem with equilibrium constraints (EPEC). The struc-

ture of this problem is mathematically explained in Section B.3 of Appendix

B.

The specific details of the considered approach are described in the next

section.

5.2 Approach

The EPEC proposed in this chapter is analyzed by considering the optimality

conditions of all MPECs representing the strategies of all producers.

The detailed steps of this approach are listed below:

1) To formulate a hierarchical (bilevel) model for each strategic producer,

whose upper-level problem determines the optimal production invest-

ment (capacity and location) and the stepwise supply offering curves to

maximize its profit, and whose several lower-level problems represent the

market clearing, one per demand block.

2) To transform each hierarchical (bilevel) model into a single-level problem

5.2. Approach 187

by replacing the lower-level problems with their primal-dual optimality

conditions. The resulting model is an MPEC.

3) To simultaneously consider all producer MPECs, one per producer, and

thus to formulate an EPEC.

4) To derive the optimality conditions of the EPEC by replacing each MPEC

with its Karush-Kuhn-Tucker (KKT) conditions. This results in a set of

non-linear systems of equalities and inequalities.

5) To linearize the optimality conditions of the EPEC obtained in the pre-

vious step without approximation through i) a linearization of the com-

plementarity conditions (Subsection B.5.1 of Appendix B), and ii) a pa-

rameterization approach. The resulting conditions become mixed-integer

and linear.

6) To detect meaningful equilibria by formulating and solving an MILP

problem whose constraints are the system of equalities and inequalities

characterizing the EPEC and whose linear objective function is selected

targeting a particular equilibrium.

Note that the problem considered in this chapter can be classified as a

generalized Nash equilibrium (GNE) problem (see, for instance, [36] and [80])

since the feasibility regions of the producers’ problems are interrelated by their

strategies. Moreover, the proposed model is a GNE with shared constraints in

which the market clearing conditions are common to all producers.

We characterize the GNE problem by its associated optimality conditions

which are obtained by concatenating all the KKT conditions from all MPECs,

one per strategic producer (see item 1) to 6) above). However, due to the non-

convex nature of the MPECs, standard constraint qualifications are generally

not met and therefore the solution of the KKT system may include multiple

equilibrium points as well as saddle points. For this reason, in order to verify

that each solution attained is actually a Nash equilibrium, we perform an

ex-post analysis based on a one-step diagonalization algorithm (Section 5.8),

i.e., an iterative method in which each producer determines sequentially or in

parallel its investment decisions considering other producers’ strategies fixed.


It is important to note that diagonalization algorithms are inefficient with

respect to the proposed approach since they are iterative and heuristic, and

they provide, if convergence is achieved, at most one single equilibrium point.

5.3 Modeling Assumptions


1) The model is to be used by a market regulator to mathematically iden-

tify market equilibria. Additionally, an ex-post engineering and economic

analysis may be required to identify which of these equilibria are mean-

ingful and may actually occur in practice.

2) Similarly to Chapters 2 and 3, a dc representation of the transmission

network is embedded within the considered investment model. This way,

the effect of locating new units at different buses is adequately repre-

sented. Congestion cases are also easily represented. For simplicity,

active power losses are neglected.

3) Similarly to Chapters 2 and 3, a pool-based electricity market is con-

sidered in this chapter where a market operator clears the pool once a

day, one day ahead, and on an hourly basis. The market operator seeks

to maximize the social welfare considering the stepwise supply function

offers and the demand bids submitted by the producers and the con-

sumers, respectively. The market clearing results are hourly productions,

consumptions and LMPs.

4) Pursuing simplicity, the futures market is not considered. However, such

market can be incorporated into the proposed analysis using a model

similar to that presented in Section 4.5 of Chapter 4.

5) As explained in detail in Subsection 1.6.1 of Chapter 1, and similarly to

Chapters 2 to 4, the proposed investment model is static, i.e., a single

target year is considered for decision-making. Such target year represents


the final stage of the planning horizon, and the model uses annualized

cost referred to such year.

6) Similarly to Chapters 2 to 4, the load-duration curve pertaining to the

target year is approximated through a number of demand blocks. On

the other hand, each demand block may include several demands lo-

cated at different buses of the network. Further details on this stepwise

approximation are provided in Subsection 1.6.2 of Chapter 1.

7) All producers considered in this chapter are strategic, i.e., they can alter

the formation of the market clearing prices through their strategies.

8) Each strategic producer explicitly anticipates the impact of its invest-

ment and offering actions on the market outcomes, e.g., LMPs and pro-

duction quantities. This is achieved through the lower-level market clear-

ing problems, one per demand block.

9) The investment equilibrium problem is subject to several uncertainties,

e.g., demand growth, investment costs for different technologies, regula-

tory policies, etc. However, for the sake of simplicity, such uncertainties

are not considered.

10) The marginal clearing prices corresponding to each market, i.e., the

LMPs, are obtained as the dual variables associated with the market

balance constraints of that market. That is, the marginalist theory is

considered [123].

11) We explicitly represent stepwise increasing offer curves (supply functions)

for producers and stepwise decreasing bidding curves for consumers.




not necessarily supplied at their corresponding maximum levels (PDmax

td ).

Additionally, no constraint is included in the model to force the supply

of a minimum demand level.


5.4 Single-Producer Problem

The decision-making process by each strategic producer y to determine its

best investment and offering decisions is represented as a hierarchical (bilevel)

model.

5.4.1 Structure of the Hierarchical (bilevel) Model

The structure of the proposed bilevel model is illustrated in Figure 5.1 and

described below:

The upper-level problem represents the minus profit minimization for the

producer subject to i) the upper-level constraints, and ii) a set of lower-level

problems. Further details are provided below:

• The upper-level constraints include bounds on investment options, the

investment budget limit, the minimum available capacity imposed by the

market regulator and the non-negativity of strategic offers.

• Each lower-level problem per demand block represents the clearing of

the pool minimizing the minus social welfare and subject to the mar-

ket operation conditions, i.e., balance constraints at every bus, produc-

tion/consumption power limits, transmission line capacity limits, voltage

angle limits and the reference bus identification.

Note that the upper-level and the lower-level problems are interrelated as

illustrated in Figure 2.2 (Section 2.2.2 of Chapter 2). On one hand, the lower-

level problems determine the LMPs and the production quantities, which di-

rectly influence the producer’s profit in the upper-level problem. On the other

hand, the strategic offering and investment decisions made by the strategic

producer in the upper-level problem affect the market clearing outcomes in

the lower-level problems.

5.4. Single-Producer Problem 191

!!

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$($)'

$(*''

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$(-.',!& !!

($/',

0

/!

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Figure 5.1: EPEC problem: Hierarchical (bilevel) structure of the model solvedby each strategic producer.


5.4.2 Formulation of the Bilevel Model

The following notational assumptions are considered in the formulation below:

1) Both notational assumptions explained in Subsection 2.3.1 of Chapter 2

regarding the production-offer blocks of generators and the demand-bid

blocks of demands also apply to the formulation of this chapter.

2) Dual variables of each lower-level problem are indicated at their corre-

sponding constraints following a colon.

The formulation of the bilevel model corresponding to strategic producer

y is given below by (5.1).

The upper-level problem includes (5.1a)-(5.1f), while (5.1g)-(5.1n) pertain

to the lower-level problems, one per demand block t.

Objective function (5.1a) and constraints (5.1b) correspond to the strategic

producer y, while other upper-level constraints (5.1c)-(5.1f) and the lower-level

problems (5.1g)-(5.1n) are common to all strategic producers.

MinimizeΞUL

∑

i∈(ΨS∩Ωy)

KiXi −∑

t

σt

∑

i∈(ΨS∩Ωy)

P Sti

(λt(n:i∈ΨS

n)− CS

i

)

+∑

k∈(ΨES∩Ωy)

PEtk

(λt(n:k∈ΨES

n ) − CEk

) (5.1a)

subject to:

0 ≤ Xi ≤ Xmaxi ∀i ∈ (ΨS ∩ Ωy) (5.1b)

∑

i∈ΨS

KiXi ≤ Kmax (5.1c)

(∑

i∈ΨS

Xi +∑

K∈ΨES

PESmax

k

)≥ Υ×

∑

d∈ΨD

PDmax

td t = t1 (5.1d)


αSti ≥ 0 ∀t, ∀i ∈ ΨS (5.1e)

αEStk ≥ 0 ∀t, ∀k ∈ ΨES (5.1f)

λtn, PSti, P

EStk ∈ arg minimize

ΞPt ,∀t

∑

i∈ΨS

αStiP

Sti +

∑

k∈ΨES

αEStk P

EStk −

∑

d∈ΨD

UDtdP

Dtd (5.1g)

subject to:

∑

d∈ΨDn

PDtd +

∑

m∈Φn

Bnm(θtn − θtm)

−∑

i∈ΨSn

P Sti −

∑

k∈ΨESn

PEStk = 0 : λtn ∀n (5.1h)

0 ≤ P Sti ≤ Xi : µSmin

ti , µSmax

ti ∀i ∈ ΨS (5.1i)

0 ≤ PEStk ≤ PESmax

k : µESmin

tk , µESmax

tk ∀k ∈ ΨES (5.1j)

0 ≤ PDtd ≤ PDmax

td : µDmin

td , µDmax

td ∀d ∈ ΨD (5.1k)

−Fmax

nm ≤ Bnm (θtn − θtm) ≤ Fmax

nm : νmin

tnm, νmax

tnm ∀n, ∀m ∈ Φn (5.1l)

−π ≤ θtn ≤ π : ξmin

tn , ξmax

tn ∀n (5.1m)

θtn = 0 : ξ1

t n = 1 (5.1n)∀t.

The primal optimization variables of each lower-level problem (5.1g)-(5.1n)

are included in the set below:

ΞPt =P

Sti, P

EStk , PD

td , θtn.

In addition, the dual variable set of each lower-level problem (5.1g)-(5.1n)

is ΞDt below:


ΞDt =λtn, µ

Smin

ti , µSmax

ti , µESmin

tk , µESmax

tk , µDmin

td , µDmax

td , τmin

tnm, τmax

tnm, ξmin

tn , ξmax

tn ,

ξ1

t .

Producer y behaves strategically through its strategic decisions made at

the upper-level problem (5.1a)-(5.1f):

• Strategic investment decisions, i.e., Xi ∀i ∈ (ΨS ∩ Ωy).

• Strategic offering decisions in the pool, i.e., αSti ∀t, ∀i ∈ (ΨS ∩ Ωy) and

αEStk ∀t, ∀k ∈ (ΨES ∩ Ωy).

Note that strategic producer y anticipates the market outcomes, e.g., LMPs

and production quantities, versus its decisions stated above. To this end, con-

straining the upper-level problem (5.1a)-(5.1f), the pool clearing is represented

via each lower-level problem (5.1g)-(5.1n) for given investment and offering de-

cisions. This allows each strategic producer to obtain feedback regarding how

its offering and investment actions affect the pool.

Thus, αSti ∀t, ∀i ∈ (ΨS ∩ Ωy), α

EStk ∀t, ∀i ∈ (ΨES ∩ Ωy) andXi ∀i ∈ (ΨS ∩ Ωy)

are variables in the upper-level problem (5.1a)-(5.1f) while they are parameters

in the lower-level problems (5.1g)-(5.1n). Note that this makes the lower-level

problems (5.1g)-(5.1n) linear and thus convex.

In addition, since the lower-level problems (5.1g)-(5.1n) constrain the upper-

level problem (5.1a)-(5.1f), the lower-level variable sets ΞPt and ΞD

t are included

in the variable set of the upper-level problem as well. Thus, the primal vari-

ables of the upper-level problem (5.1a)-(5.1f) are those in the set:

ΞUL=ΞPt , Ξ

Dt , α

Sti, α

EStk , Xi.

Objective function (5.1a) of the upper-level problem is the minus profit,

i.e., the investment cost of strategic producer y (∑

i∈(ΨS∩Ωy)KiXi) minus its

operations revenue.

Note that the LMPs λtn are endogenously generated within the lower-level

problems (5.1g)-(5.1n) as dual variables.

Constraints (5.1b) bound the capacity of the candidate units.

Constraints (5.1c) and (5.1d) enforce the upper and lower bounds on the

total capacity to be built by all producers.


On one hand, the available investment budget of all producers considered

in (5.1c) imposes a cap on total investment by all producers, which reflects

the limited financial resources available to the market as a whole. On the

other hand, constraint (5.1d) enforces a minimum available capacity (including

existing and newly built units) to ensure supply security.

Similarly to the bilevel model (4.1) presented in Chapter 4, the minimum

available capacity condition is adjusted through the non-negative factor Υ that

multiplies the peak demand level (demand of the first block, t = t1).

In addition, constraints (5.1e) and (5.1f) enforce the non-negativity of the

offers of all strategic producers.

The objective function of each lower-level problem (5.1g) is the minus social

welfare.

Constraints (5.1h) represent the energy balance at each node, being the

associated dual variables LMPs.

Constraints (5.1i) and (5.1j) enforce lower and upper production bounds

on candidate and existing units, respectively.

Constraints (5.1k) impose lower and upper consumption bounds on de-

mands.

Limits on transmission line capacities and voltage angles of nodes are en-

forced through (5.1l) and (5.1m), respectively.

Finally, constraints (5.1n) identify n = 1 as the reference bus.

To transform the bilevel problem (5.1) into a single-level MPEC, the next

subsection provides the optimality conditions associated with each lower-level

problem (5.1g)-(5.1n).

5.4.3 Optimality Conditions of the Lower-Level Prob-

lems

According to Section B.2 of Appendix B, the optimality conditions associ-

ated with the lower-level problems (5.1g)-(5.1n) can be obtained from two

alternative approaches: i) the KKT conditions, and ii) the primal-dual trans-

formation.

Subsections 5.4.3.1 and 5.4.3.2 present two different, but equivalent sets


of optimality conditions associated with the lower-level problems (5.1g)-(5.1n)

resulting from the KKT conditions and the primal-dual transformation, re-

spectively.

To render an MPEC, the lower-level problems (5.1g)-(5.1n) are replaced

by the optimality condition set resulting from the primal-dual transformation,

while the KKT optimality condition set is used to derive an MILP formulation

of the MPEC. The reason for these choices is illustrated after describing such

optimality condition sets.

5.4.3.1 KKT Optimality Conditions Associated with Lower-level

Problems (5.1g)-(5.1n)

To obtain the KKT conditions associated with the lower-level problems (5.1g)-

(5.1n), the corresponding Lagrangian function L is considered:

L =∑

i∈ΨS

αStiP

Sti +

∑

k∈ΨES

αEStk P

EStk −

∑

d∈ΨD

UDtdP

Dtd

+∑

tn

λtn

∑

d∈ΨDn

PDtd +

∑

m∈Φn

Bnm(θtn − θtm) −∑

i∈ΨSn

P Sti −

∑

k∈ΨESn

PEStk

+∑

t(i∈ΨS)

µSmax

ti

(P Sti −Xi

)−∑

t(i∈ΨS)

µSmin

ti P Sti

+∑

t(k∈ΨES)

µESmax

tk

(PEStk − P

ESmax

k

)−

∑

t(k∈ΨES)

µESmin

tk PEStk

+∑

t(d∈ΨD)

µDmax

td

(PDtd − P

Dmax

td

)−

∑

t(d∈ΨD)

µDmin

td PDtd

+∑

tn(m∈Ωn)

νmax

tnm

[Bnm(θtn − θtm)− F

max

nm

]

−∑

tn(m∈Ωn)

νmin

tnm

[Bnm(θtn − θtm) + F

max

nm

]

+∑

tn

ξmax

tn (θtn − π)−∑

tn

ξmin

tn (θtnw + π) +∑

t

ξ1

t θt(n=1). (5.2)


Considering the Lagrangian function (5.2), the KKT first-order optimality

conditions of the lower-level problems (5.1g)-(5.1n) are derived as:

∂L

∂PDtd

= −UDtd + λt(n:d∈Ψn) + µDmax

td − µDmin

td = 0 ∀t, ∀d ∈ ΨD (5.3a)

∂L

∂P Sti

= αSti − λt(n:i∈Ψn) + µSmax

ti − µSmin

ti = 0 ∀t, ∀i ∈ ΨS (5.3b)

∂L

∂PEStk

= αEStk − λt(n:k∈Ψn) + µESmax

tk − µESmin

tk = 0 ∀t, ∀k ∈ ΨES (5.3c)

∂L

∂θtn=∑

m∈Φn

Bnm (λtn − λtm)

+∑

m∈Φn

Bnm

(ν

max

tnm − νmax

tmn

)

−∑

m∈Φn

Bnm

(ν

min

tnm − νmin

tmn

)

+ξmax

tn − ξmin

tn + (ξ1

t )n=1 = 0 ∀t, ∀n (5.3d)

∑

d∈ΨDn

PDtd +

∑

m∈Φn

Bnm (θtn − θtm)

−∑

i∈ΨSn

P Sti −

∑

K∈ΨESn

PEStk = 0 ∀t, ∀n (5.3e)

θtn = 0 ∀t, n = 1 (5.3f)

0 ≤ P Sti ⊥ µSmin

ti ≥ 0 ∀t, ∀i ∈ ΨS (5.3g)

0 ≤ PEStk ⊥ µESmin

tk ≥ 0 ∀t, ∀k ∈ ΨES (5.3h)

0 ≤ PDtd ⊥ µDmin

td ≥ 0 ∀t, ∀d ∈ ΨD (5.3i)

0 ≤(Xi − P

Sti

)⊥ µSmax

ti ≥ 0 ∀t, ∀i ∈ ΨS (5.3j)


0 ≤(PESmax

k − PEStk

)⊥ µESmax

tk ≥ 0 ∀t, ∀k ∈ ΨES (5.3k)

0 ≤(PDmax

td − PDtd

)⊥ µDmax

td ≥ 0 ∀t, ∀d ∈ ΨD (5.3l)

0 ≤[F

max

nm +Bnm (θtn − θtm)]⊥ ν

min

tnm ≥ 0 ∀t, ∀n, ∀m ∈ Φn(5.3m)

0 ≤[F

max

nm − Bnm (θtn − θtm)]⊥ ν

max

tnm ≥ 0 ∀t, ∀n, ∀m ∈ Φn (5.3n)

0 ≤ (π + θtn) ⊥ ξmin

tn ≥ 0 ∀t, ∀n (5.3o)

0 ≤ (π − θtn) ⊥ ξmax

tn ≥ 0 ∀t, ∀n (5.3p)

λtn : free ∀t, ∀n (5.3q)

ξ1

t : free ∀t. (5.3r)


a) Equality constraints (5.3a)-(5.3d) are obtained from differentiating the La-

grangian function L with respect to the primal variables included in the set

ΞPt .

b) Equality constraints (5.3e) and (5.3f) are the primal equality constraints

(5.1h) and (5.1n) in the lower-level problems (5.1g)-(5.1n).

c) Complementarity conditions (5.3g)-(5.3p) are related to the inequality con-

straints (5.1i)-(5.1m).

d) Conditions (5.3q) and (5.3r) state that the dual variables associated with

the equality constraints (5.1h) and (5.1n) are free.

Due to the linearity and thus convexity of the lower-level problems (5.1g)-

(5.1n), the KKT conditions (5.3) are necessary and sufficient conditions for

optimality.


5.4.3.2 Optimality Conditions Associated with Lower-level Prob-

lems (5.1g)-(5.1n) Resulting from the Primal-Dual Trans-

formation

For clarity, the corresponding dual optimization problems of lower-level prob-

lems (5.1g)-(5.1n) are first formulated as given by (5.4) below:

Maximize

ΞDt

−∑

i∈ΨS

µSmax

ti Xi −∑

k∈ΨES

µESmax

tk PESmax

k −∑

d∈ΨD

µDmax

td PDmax

td

−∑

n(m∈Ωn)

νmin

tnmFmax

nm −∑

n(m∈Ωn)

νmax

tnmFmax

nm −∑

n

ξmin

tn π −∑

n

ξmax

tn π (5.4a)

subject to:

(5.3a)− (5.3d) (5.4b)

µSmin

ti ≥ 0; µSmax

ti ≥ 0 ∀i ∈ ΨS (5.4c)

µESmin

tk ≥ 0; µESmax

tk ≥ 0 ∀k ∈ ΨES (5.4d)

µDmin

td ≥ 0; µDmax

td ≥ 0 ∀d ∈ ΨD (5.4e)

νmin

tnm ≥ 0; νmax

tnm ≥ 0 ∀n, ∀m ∈ Ωn (5.4f)

ξmin

tn ≥ 0; ξmax

tn ≥ 0 ∀n (5.4g)

(5.3q)− (5.3r) (5.4h)∀t.

Considering the lower-level primal problems (5.1g)-(5.1n) and their corre-

sponding dual problems (5.4), the following set of optimality conditions (5.5)

is obtained using the primal-dual transformation.


(5.1h)− (5.1n) (5.5a)

(5.4b)− (5.4h) (5.5b)

∑

i∈ΨS

αStiP

Sti +

∑

k∈ΨES

αEStk P

EStk −

∑

d∈ΨD

UDtdP

Dtd =

−∑

i∈ΨS

µSmax

ti Xi −∑

k∈ΨES

µESmax

tk PESmax

k −∑

d∈ΨD

µDmax

td PDmax

td

−∑

n(m∈Ωn)

νmin

tnmFmax

nm −∑

n(m∈Ωn)

νmax

tnmFmax

nm

−∑

n

ξmin

tn π −∑

n

ξmax

tn π (5.5c)

∀t,

where constraint (5.5c) is the strong duality equality related to each lower-level

problem (5.1g)-(5.1n), which enforces the equality of the values of the primal

objective function (5.1g) and the dual objective function (5.4a) at the optimal

solution.

Note that the set of optimality conditions (5.5) resulting from the primal-

dual transformation is equivalent to the set of KKT conditions (5.3). There-

fore, the lower-level problems (5.1g)-(5.1n) can be replaced either by system

(5.3) derived from KKT conditions or by system (5.5) resulting from the

primal-dual transformation.

A single-level MPEC corresponding to the bilevel model (5.1) is formulated

in the next subsection by replacing each lower-level problem (5.1g)-(5.1n) with

its optimality conditions.


Figure 5.2: EPEC problem: Transformation of the bilevel model of a strategicproducer into its corresponding MPEC (primal-dual transformation).

5.4.4 MPEC

In this subsection, the bilevel problem (5.1) of each strategic producer y is

transformed into a single-level problem by replacing the lower-level problems

(5.1g)-(5.1n) with their optimality conditions. The resulting problem is an

MPEC.

Unlike in Chapters 2 and 4 (direct MPEC solution) where the set of KKT

optimality conditions are used for deriving the corresponding MPEC, the op-

timality conditions resulting from the primal-dual transformation are used in

this chapter. To this end, the lower-level problems (5.1g)-(5.1n) are replaced

by its equivalent optimality condition set (5.5) resulting from the primal-dual

transformation. The reason for this selection is to avoid the use of non-convex

and difficult to handle complementarity conditions (5.3g)-(5.3p), but at the

cost of the non-linearities introduced by the strong duality equalities (5.5c).


Figure 5.2 depicts the transformation of bilevel problem (5.1) into its cor-

responding MPEC through replacing the lower-level problems (5.1g)-(5.1n) by

its equivalent optimality condition set (5.5).

The MPEC derived from bilevel problem (5.1), corresponding to strategic

producer y is given below by (5.6)-(5.7).

Dual variables of problem (5.6)-(5.7) are indicated at their corresponding

constraints following a colon.

MinimizeΞUL

∑

i∈(ΨS∩Ωy)

KiXi −∑

t

σt

∑

i∈(ΨS∩Ωy)

P Sti

(λt(n:i∈ΨS

n)− CS

i

)

+∑

k∈(ΨES∩Ωy)

PEtk

(λt(n:k∈ΨES

n ) − CEk

) (5.6a)

subject to:

0 ≤ Xi ≤ Xmaxi : χmin

yi , χmaxyi ∀i ∈ (ΨS ∩ Ωy) (5.6b)

∑

i∈ΨS

KiXi ≤ Kmax : χIBy (5.6c)

(∑

i∈ΨS

Xi +∑

k∈ΨES

PESmax

k

)

≥ Υ×∑

d∈ΨD

PDmax

td : χSSy t = t1 (5.6d)

αSti ≥ 0 : ηα

S

yti ∀t, ∀i ∈ ΨS (5.6e)

αEStk ≥ 0 : ηα

ES

ytk ∀t, ∀k ∈ ΨES (5.6f)


∑

d∈ΨDn

PDtd +

∑

m∈Φn

Bnm (θtn − θtm)

−∑

i∈ΨSn

P Sti −

∑

k∈ΨESn

PEStk = 0 : βytn ∀t, ∀n (5.6g)

0 ≤ P Sti ≤ Xi : γS

min

yti , γSmax

yti ∀t, ∀i ∈ ΨS (5.6h)

0 ≤ PEStk ≤ PESmax

k : γESmin

ytk , γESmax

ytk ∀t, ∀k ∈ ΨES (5.6i)

0 ≤ PDtd ≤ PDmax

td : γDmin

ytd , γDmax

ytd ∀t, ∀d ∈ ΨD (5.6j)

−Fmax

nm ≤ Bnm (θtn − θtm) ≤ Fmax

nm : τmin

ytnm, τmax

ytnm ∀t, ∀n, ∀m ∈ Φn(5.6k)

−π ≤ θtn ≤ π : δmin

ytn , δmax

ytn ∀t, ∀n (5.6l)

θtn = 0 : δ1

yt ∀t, n = 1 (5.6m)


n )+ µDmax

td

−µDmin

td = 0 : ρDytd ∀t, ∀d ∈ ΨD (5.6n)

αSti − λt(n:i∈ΨS

n)+ µSmax

ti

−µSmin

ti = 0 : ρSyti ∀t, ∀i ∈ ΨS (5.6o)

αEStk − λt(n:k∈ΨES

n ) + µESmax

tk

−µESmin

tk = 0 : ρESytk ∀t, ∀k ∈ ΨES (5.6p)

∑

m∈Φn

Bnm (λtn − λtm)

+∑

m∈Φn

Bnm

(ν

max

tnm − νmax

tmn

)

−∑

m∈Φn

Bnm

(ν

min

tnm − νmin

tmn

)

+ξmax

tn − ξmin

tn +(ξ1

t

)n=1

= 0 : ρθytn ∀t, ∀n (5.6q)


µSmin

ti ≥ 0 : ηSmin

yti ∀t, ∀i ∈ ΨS (5.6r)

µSmax

ti ≥ 0 : ηSmax

yti ∀t, ∀i ∈ ΨS (5.6s)

µESmin

tk ≥ 0 : ηESmin

ytk ∀t, ∀k ∈ ΨES (5.6t)

µESmax

tk ≥ 0 : ηESmax

ytk ∀t, ∀k ∈ ΨES (5.6u)

µDmin

td ≥ 0 : ηDmin

ytd ∀t, ∀d ∈ ΨD (5.6v)

µDmax

td ≥ 0 : ηDmax

ytd ∀t, ∀d ∈ ΨD (5.6w)

νmin

tnm ≥ 0 : ηνmin

ytnm ∀t, ∀n, ∀m ∈ Φn(5.6x)

νmax

tnm ≥ 0 : ηνmax

ytnm ∀t, ∀n, ∀m ∈ Φn(5.6y)

ξmin

tn ≥ 0 : ηξmin

ytn ∀t, ∀n (5.6z)

ξmax

tn ≥ 0 : ηξmax

ytn ∀t, ∀n (5.7a)

∑

i∈ΨS

αStiP

Sti +

∑

k∈ΨES

αEStk P

EStk

−∑

d∈ΨD

UDtdP

Dtd +

∑

i∈ΨS

µSmax

ti Xi

+∑

k∈ΨES

µESmax

tk PESmax

k

+∑

d∈ΨD

µDmax

td PDmax

td

+∑

n(m∈Φn)

(ν

min

tnm + νmax

tnm

)F

max

nm

+∑

n

(ξmin

tn + ξmax

tn

)π = 0 : φSD

yt ∀t, (5.7b)

where the primal optimization variable set of MPEC (5.6)-(5.7) is identical to

that of bilevel problem (5.1), i.e., set ΞUL.

5.5. Multiple Producers Problem 205

In addition, the dual variable set of MPEC (5.6)-(5.7) is the set:

ΞDual=χminyi , χmax

yi , χIBy , χSS

y , ηαS

yti, ηαES

ytk , βytn, γSmin

yti , γSmax

yti , γESmin

ytk , γESmax

ytk ,

γDmin

ytd , γDmax

ytd , τmin

ytnm, τmax

ytnm, δmin

ytn , δmax

ytn , δ1

yt, ρDytd, ρ

Syti, ρ

ESytk, ρ

θytn, η

Smin

yti , ηSmax

yti ,

ηESmin

ytk , ηESmax

ytk , ηDmin

ytd , ηDmax

ytd , ηνmin

ytnm, ηνmax

ytnm, ηξmin

ytn , ηξmax

ytn , φSDyt .

The structure of MPEC (5.6)-(5.7) is explained below:

• The objective function (5.6a) is identical to the objective function of

problem (5.1), i.e., (5.1a).

• The constraints (5.6b)-(5.6f) are the upper-level constraints of problem

(5.1), i.e., constraints (5.1b)-(5.1f).

• Each lower-level problem (5.1g)-(5.1n), one per demand block, is replaced

by the optimality conditions below:

i) Primal constraints (5.6g)-(5.6m) that are equivalent to (5.5a).

ii) Dual constraints (5.6n)-(5.6z) and (5.7a) that are equivalent to

(5.5b).

iii) Strong duality equality (5.7b) that is equivalent to (5.5c).

5.5 Multiple Producers Problem

5.5.1 EPEC

The joint consideration of all producer MPECs (5.6)-(5.7), one per producer,

constitutes an EPEC. This is depicted by the upper plot of Figure 1.6 presented

in Chapter 1.

The EPEC solution identifies the market equilibria. To attain such solu-

tion, the optimality conditions associated with the EPEC, i.e., the optimality

conditions of all producer MPECs, need to be derived.

To formulate the optimality conditions of all producer MPECs, it is impor-

tant to note that MPECs (5.6)-(5.7) are non-linear and thus the application

of the primal-dual transformation (second approach explained in Section B.2


of Appendix B) is not straightforward. Therefore, all MPECs (5.6)-(5.7) are

replaced by their corresponding KKT conditions (first approach explained in

Section B.2 of Appendix B) rendering the optimality conditions of the EPEC.

This transformation is illustrated in Figure 1.6 presented in Chapter 1.

5.5.2 Optimality Conditions associated with the EPEC

As described in the previous subsection and illustrated in Figure 1.6 presented

in Chapter 1, the optimality conditions associated with the EPEC are obtained

through replacing each MPEC (5.6)-(5.7), one per producer, with its KKT

optimality conditions.

The optimality conditions associated with the EPEC include the con-

straints below:

1) Primal equality constraints of MPECs (5.6)-(5.7).

2) Equality constraints obtained from differentiating the corresponding La-

grangian associated with each the MPEC (5.6)-(5.7) with respect to the

corresponding variables in ΞUL.

3) Complementarity conditions related to the inequality constraints of each

MPEC (5.6)-(5.7).

The system above is formulated in the next three subsections.

5.5.2.1 Primal Equality Constraints

The primal equality constraints of MPECs (5.6)-(5.7) are equalities (5.6g),

(5.6m)-(5.6q) and (5.7b) that are restated as (5.8) below:


∑

d∈ΨDn

PDtd +

∑

m∈Φn

Bnm (θtn − θtm)

−∑

i∈ΨSn

P Sti −

∑

k∈ΨESn

PEStk = 0 ∀y, ∀t, ∀n (5.8a)

θtn = 0 ∀y, ∀t, n = 1 (5.8b)


n )+ µDmax

td − µDmin

td = 0 ∀y, ∀t, ∀d ∈ ΨD (5.8c)


n)+ µSmax

ti − µSmin

ti = 0 ∀y, ∀t, ∀i ∈ ΨS (5.8d)


n ) + µESmax

tk − µESmin

tk = 0 ∀y, ∀t, ∀k ∈ ΨES (5.8e)

∑

m∈Φn

Bnm (λtn − λtm)

+∑

m∈Φn

Bnm

(ν

max

tnm − νmax

tmn

)

−∑

m∈Φn

Bnm

(ν

min

tnm − νmin

tmn

)

+ξmax

tn − ξmin

tn +(ξ1

t

)n=1

= 0 ∀y, ∀t, ∀n (5.8f)

∑

i∈ΨS

αStiP

Sti +

∑

k∈ΨES

αEStk P

EStk

−∑

d∈ΨD

UDtdP

Dtd +

∑

i∈ΨS

µSmax

ti Xi

+∑

k∈ΨES

µESmax

tk PESmax

k +∑

d∈ΨD

µDmax

td PDmax

td

+∑

n(m∈Φn)

(ν

min

tnm + νmax

tnm

)F

max

nm

+∑

n

(ξmin

tn + ξmax

tn

)π = 0 ∀t. (5.8g)


5.5.2.2 Equality Constraints Obtained From Differentiating the Cor-

responding Lagrangian with Respect to the Variables in ΞUL

The equality constraints (5.9) are obtained from differentiating with respect

to the variables in ΞUL the Lagrangian associated with each MPEC (5.6)-

(5.7). Note that LMPECy is the Lagrangian function of the MPEC (5.6)-(5.7)

pertaining to the strategic producer y.

∂LMPECy

∂P Sti

=

−σt(λt(n:i∈ΨS

n)− CS

i

)− βyt(n:i∈ΨS

n)

+γSmax

yti − γSmin

yti + φSDyt α

Sti = 0 ∀y, ∀t, ∀i ∈ (ΨS ∩ Ωy) (5.9a)

∂LMPECy

∂P Sti

=

−βyt(n:i∈ΨSn)+ γS

max

yti − γSmin

yti

+φSDyt α

Sti = 0 ∀y, ∀t, ∀i /∈ (ΨS ∩ Ωy) (5.9b)

∂LMPECy

∂PEStk

=

−σt(λt(n:k∈ΨES

n ) − CESk

)

−βyt(n:k∈ΨESn ) + γES

max

ytk

−γESmin

ytk + φSDyt α

EStk = 0 ∀y, ∀t, ∀k ∈ (ΨES ∩ Ωy) (5.9c)

∂LMPECy

∂PEStk

=


max

ytk − γESmin

ytk

+φSDyt α

EStk = 0 ∀y, ∀t, ∀k /∈ (ΨES ∩ Ωy) (5.9d)


∂LMPECy

∂PDtd

=

βyt(n:d∈ΨDn )

+ γDmax

ytd

−γDmin

ytd − φSDyt U

Dtd = 0 ∀y, ∀t, ∀d ∈ ΨD (5.9e)

∂LMPECy

∂Xi

=

Ki + χmaxyi − χ

minyi + χIB

y − χSSy

−∑

t

γSmax

yti +∑

t

φSDyt µ

Smax

ti = 0 ∀y, ∀i ∈ (ΨS ∩ Ωy) (5.9f)

∂LMPECy

∂Xi

=

χIBy − χ

SSy −

∑

t

γSmax

yti

+∑

t

φSDyt µ

Smax

ti = 0 ∀y, ∀i /∈ (ΨS ∩ Ωy) (5.9g)

∂LMPECy

∂αSti

=

−ηαS

yti + ρSyti + φSDyt P

Sti = 0 ∀y, ∀t, ∀i ∈ ΨS (5.9h)

∂LMPECy

∂αEStk

=

−ηαES

ytk + ρESytk + φSDyt P

EStk = 0 ∀y, ∀t, ∀k ∈ ΨES (5.9i)

∂LMPECy

∂θtn=

∑

m∈Φn

Bnm(βytn − βytm)

+∑

m∈Φn

Bnm(τmax

ytnm − τmax

ytmn)


−∑

m∈Φn

Bnm(τmin

ytnm − τmin

ytmn)

+δmax

ytn − δmin

ytn + (δ1

yt)n=1 = 0 ∀y, ∀t, ∀n (5.9j)

∂LMPECy

∂λtn=

−σt

∑

i∈(ΨSn∩Ωy)

P Sti +

∑

k∈(ΨESn ∩Ωy)

PEStk

+∑

d∈ΨDn

ρDytd −∑

i∈ΨSn

ρSyti −∑

k∈ΨESn

ρESytk

+∑

m∈Φn

Bnm(ρθytn − ρ

θytm) = 0 ∀y, ∀t, ∀n (5.9k)

∂LMPECy

∂µSmin

ti

=

−ρSyti − ηSmin

yti = 0 ∀y, ∀t, ∀i ∈ ΨS (5.9l)

∂LMPECy

∂µSmax

ti

=

ρSyti − ηSmax

yti + φSDyt Xi = 0 ∀y, ∀t, ∀i ∈ ΨS (5.9m)

∂LMPECy

∂µESmin

tk

=

−ρESytk − ηESmin

ytk = 0 ∀y, ∀t, ∀k ∈ ΨES (5.9n)

∂LMPECy

∂µESmax

tk

=

ρESytk − ηESmax

ytk + φSDyt P

ESmax

k = 0 ∀y, ∀t, ∀k ∈ ΨES (5.9o)

∂LMPECy

∂µDmin

td

=

−ρDytd − ηDmin

ytd = 0 ∀y, ∀t, ∀d ∈ ΨD (5.9p)


∂LMPECy

∂µDmax

td

=

ρDytd − ηDmax

ytd + φSDyt P

Dmax

td = 0 ∀y, ∀t, ∀d ∈ ΨD (5.9q)

∂LMPECy

∂νmin

tnm

=

−Bnm(ρθytn − ρ

θytm)− η

νmin

ytnm

+φSDyt F

max

nm = 0 ∀y, ∀t, ∀n, ∀m ∈ Φn (5.9r)

∂LMPECy

∂νmax

tnm

=

Bnm(ρθytn − ρ

θytm)− η

νmax

ytnm

+φSDyt F

max

nm = 0 ∀y, ∀t, ∀n, ∀m ∈ Φn (5.9s)

∂LMPECy

∂ξmin

tn

=

−ρθytn − ηξmin

ytn + φSDyt π = 0 ∀y, ∀t, ∀n (5.9t)

∂LMPECy

∂ξmax

tn

=

ρθytn − ηξmax

ytn + φSDyt π = 0 ∀y, ∀t, ∀n (5.9u)

∂LMPECy

∂ξ1

t

=

ρθyt(n=1) = 0 ∀y, ∀t. (5.9v)

5.5.2.3 Complementarity Conditions

The complementarity conditions included in the KKT conditions associated

with the EPEC are given by (5.10) below. These conditions are related to

the inequality constraints (5.6b)-(5.6f), (5.6h)-(5.6l), (5.6r)-(5.6z) and (5.7a)

included in MPECs (5.6)-(5.7).


0 ≤ Xi ⊥ χminyi ≥ 0 ∀y, ∀i ∈ (ΨS ∩ Ωy) (5.10a)

0 ≤ (Xmaxi −Xi) ⊥ χmax

yi ≥ 0 ∀y, ∀i ∈ (ΨS ∩ Ωy) (5.10b)

0 ≤

(Kmax −

∑

i∈ΨS

KiXi

)

⊥ χIBy ≥ 0 ∀y (5.10c)

0 ≤

[∑

i∈ΨS

Xi +∑

k∈ΨES

PESmax

k

−

(Υ×

∑

d∈ΨD

PDmax

td

)] ⊥ χSS

y ≥ 0 ∀y, t = t1 (5.10d)

0 ≤ αSti ⊥ ηα

S

yti ≥ 0 ∀y, ∀t, ∀i ∈ ΨS (5.10e)

0 ≤ αEStk ⊥ ηα

ES

ytk ≥ 0 ∀y, ∀t, ∀k ∈ ΨES (5.10f)

0 ≤ P Sti ⊥ γS

min

yti ≥ 0 ∀y, ∀t, ∀i ∈ ΨS (5.10g)

0 ≤(Xi − P

Sti

)⊥ γS

max

yti ≥ 0 ∀y, ∀t, ∀i ∈ ΨS (5.10h)

0 ≤ PEStk ⊥ γES

min

ytk ≥ 0 ∀y, ∀t, ∀k ∈ ΨES (5.10i)

0 ≤(PESmax

k − PEStk

)⊥ γES

max

ytk ≥ 0 ∀y, ∀t, ∀k ∈ ΨES (5.10j)

0 ≤ PDtd ⊥ γD

min

ytd ≥ 0 ∀y, ∀t, ∀d ∈ ΨD (5.10k)

0 ≤(PDmax

td − PDtd

)⊥ γD

max

ytd ≥ 0 ∀y, ∀t, ∀d ∈ ΨD (5.10l)

0 ≤[F

max

nm +Bnm (θtn − θtm)]

⊥ τmin

ytnm ≥ 0 ∀y, ∀t, ∀n, ∀m ∈ Φn (5.10m)


0 ≤[F

max

nm −Bnm (θtn − θtm)]

⊥ τmax

ytnm ≥ 0 ∀y, ∀t, ∀n, ∀m ∈ Φn (5.10n)

0 ≤ (π + θtn) ⊥ δmin

ytn ≥ 0 ∀y, ∀t, ∀n (5.10o)

0 ≤ (π − θtn) ⊥ δmax

ytn ≥ 0 ∀y, ∀t, ∀n (5.10p)

0 ≤ µSmin

ti ⊥ ηSmin

yti ≥ 0 ∀y, ∀t, ∀i ∈ ΨS (5.10q)

0 ≤ µSmax

ti ⊥ ηSmax

yti ≥ 0 ∀y, ∀t, ∀i ∈ ΨS (5.10r)

0 ≤ µESmin

tk ⊥ ηESmin

ytk ≥ 0 ∀y, ∀t, ∀k ∈ ΨES (5.10s)

0 ≤ µESmax

tk ⊥ ηESmax

ytk ≥ 0 ∀y, ∀t, ∀k ∈ ΨES (5.10t)

0 ≤ µDmin

td ⊥ ηDmin

ytd ≥ 0 ∀y, ∀t, ∀d ∈ ΨD (5.10u)

0 ≤ µDmax

td ⊥ ηDmax

ytd ≥ 0 ∀y, ∀t, ∀d ∈ ΨD (5.10v)

0 ≤ νmin

tnm ⊥ ηνmin

ytnm ≥ 0 ∀y, ∀t, ∀n, ∀m ∈ Φn (5.10w)

0 ≤ νmax

tnm ⊥ ηνmax

ytnm ≥ 0 ∀y, ∀t, ∀n, ∀m ∈ Φn (5.10x)

0 ≤ ξmin

tn ⊥ ηξmin

ytn ≥ 0 ∀y, ∀t, ∀n (5.10y)

0 ≤ ξmax

tn ⊥ ηξmax

ytn ≥ 0 ∀y, ∀t, ∀n. (5.10z)

The optimality conditions associated with the EPEC consist of the primal

equality constraints (5.8), the equality constraints (5.9) obtained from differ-

entiating the Lagrangian of the problem of each producer with respect to the

corresponding variables in ΞUL and the complementarity conditions (5.10).

It is important to note that the optimality conditions (5.8)-(5.10) associated

with the EPEC are non-linear and highly non-convex due to both products of

variables and complementarity conditions. Non-linearities are considered in


the next subsection.

5.5.3 EPEC Linearization

The optimality conditions (5.8)-(5.10) associated with the EPEC include the

following non-linearities:

a) The complementarity conditions (5.10). Such conditions can be lin-

earized through the approach explained in Subsection B.5.1 of Appendix

B using the auxiliary binary variables and large enough constants.

b) The products of variables involved in the strong duality equalities (5.8g)

included in (5.8). These non-linearities are linearized through the ap-

proach presented in Subsection 5.5.3.1 below.

c) The products of variables in (5.9a)-(5.9d), (5.9f)-(5.9i) and (5.9m). Ob-

serve that the common variables of such non-linear terms are the dual

variables φSDyt . Subsection 5.5.3.2 below presents a parameterization ap-

proach for linearizing such non-linear terms.

5.5.3.1 Linearizing the Strong Duality Equalities (5.8g)

Unlike complementarity conditions (5.10) that can be easily linearized by the

approach explained in Subsection B.5.1 of Appendix B through auxiliary bi-

nary variables, the strong duality equalities (5.8g) cannot be easily linearized

due to the nature of the non-linearities, i.e., the product of continuous vari-

ables.

However, as explained in Section B.2 of Appendix B, the strong duality

equality resulting from the primal-dual transformation is equivalent to the set

of complementarity conditions obtained from the KKT conditions.

Hence, pursuing linearity, the strong duality equalities (5.8g) are replaced

with their equivalent complementarity conditions (5.3g)-(5.3p) that have al-

ready been presented in Subsection 5.4.3.1.

Similarly to conditions (5.10), complementarity conditions (5.3g)-(5.3p)

can be linearized as explained in Subsection B.5.1 of Appendix B.

5.6. MILP Formulation 215

5.5.3.2 Linearizing the Non-linear Terms Involving φSDyt

From a mathematically point of view, we can parameterize the model in the

variables φSDyt because these are dual variables associated with MPECs (5.6)-

(5.7), whose constraints are non-regular, i.e., the Mangasarian-Fromovitz con-

straint qualification (MFCQ) is not satisfied at any feasible point [122].

In other words, the dual variables associated with MPECs (5.6)-(5.7) at

any solution are not unique and form a ray [39]. This redundancy allows the

parameterization of variables φSDyt .

Hence, non-linear terms of (5.9a)-(5.9d), (5.9f)-(5.9i) and (5.9m) become

linear if problem (5.8)-(5.10) is parameterized in dual variables φSDyt .

Regarding the selection of values to be assigned to dual variables φSDyt , note

that the combination of constraints (5.9l), (5.9m) and (5.10q)-(5.10r) requires

that dual variables φSDyt to be non-negative.

5.6 MILP Formulation

Using the linearization techniques presented in Subsection 5.5.3, the optimal-

ity conditions (5.8)-(5.10) associated with the EPEC are transformed into a

system of mixed-integer linear equalities and inequalities given by conditions

(5.11)-(5.28) below:

1) Equality constraint set (5.11) includes the primal equality constraints (5.8a)-

(5.8f) included in (5.8) as given below:

∑

d∈ΨDn

PDtd +

∑

m∈Φn

Bnm (θtn − θtm)

−∑

i∈ΨSn

P Sti −

∑

k∈ΨESn

PEStk = 0 ∀y, ∀t, ∀n (5.11a)

θtn = 0 ∀y, ∀t, n = 1 (5.11b)


n )+ µDmax

td − µDmin

td = 0 ∀y, ∀t, ∀d ∈ ΨD (5.11c)


n)+ µSmax

ti − µSmin

ti = 0 ∀y, ∀t, ∀i ∈ ΨS (5.11d)



n ) + µESmax

tk − µESmin

tk = 0 ∀y, ∀t, ∀k ∈ ΨES (5.11e)

∑

m∈Φn

Bnm (λtn − λtm)

+∑

m∈Φn

Bnm

(ν

max

tnm − νmax

tmn

)

−∑

m∈Φn

Bnm

(ν

min

tnm − νmin

tmn

)

+ξmax

tn − ξmin

tn +(ξ1

t

)n=1

= 0 ∀y, ∀t, ∀n. (5.11f)

2) As explained in Subsection 5.5.3.1, the strong duality equalities (5.8g) in-

cluded in (5.8) are replaced with the equivalent complementarity conditions

(5.3g)-(5.3p). The constraint sets (5.12)-(5.15) represent the mixed-integer

linear equivalent of such complementarity conditions.

The constraint set (5.12) below is the mixed-integer linear equivalent of the

complementarity conditions (5.3g)-(5.3i):

P Sti ≥ 0 ∀t, ∀i ∈ ΨS (5.12a)

PEStk ≥ 0 ∀t, ∀k ∈ ΨES (5.12b)

PDtd ≥ 0 ∀t, ∀d ∈ ΨD (5.12c)

µSmin

ti ≥ 0 ∀t, ∀i ∈ ΨS (5.12d)

µESmin

tk ≥ 0 ∀t, ∀k ∈ ΨES (5.12e)

µDmin

td ≥ 0 ∀t, ∀d ∈ ΨD (5.12f)

P Sti ≤ ψSmin

ti MP ∀t, ∀i ∈ ΨS (5.12g)

PEStk ≤ ψESmin

tk MP ∀t, ∀k ∈ ΨES (5.12h)

PDtd ≤ ψDmin

td MP ∀t, ∀d ∈ ΨD (5.12i)

µSmin

ti ≤(1− ψSmin

ti

)Mµ ∀t, ∀i ∈ ΨS (5.12j)


µESmin

tk ≤(1− ψESmin

tk

)Mµ ∀t, ∀k ∈ ΨES (5.12k)

µDmin

td ≤(1− ψDmin

td

)Mµ ∀t, ∀d ∈ ΨD (5.12l)

ψSmin

ti ∈ 0, 1 ∀t, ∀i ∈ ΨS (5.12m)

ψESmin

tk ∈ 0, 1 ∀t, ∀k ∈ ΨES (5.12n)

ψDmin

td ∈ 0, 1 ∀t, ∀d ∈ ΨD, (5.12o)


In addition, constraint set (5.13) below is the mixed-integer linear equiva-

lent of the complementarity conditions (5.3j)-(5.3l):

Xi − PSti ≥ 0 ∀t, ∀i ∈ ΨS (5.13a)

PESmax

k − PEStk ≥ 0 ∀t, ∀k ∈ ΨES (5.13b)

PDmax

td − PDtd ≥ 0 ∀t, ∀d ∈ ΨD (5.13c)

µSmax

ti ≥ 0 ∀t, ∀i ∈ ΨS (5.13d)

µESmax

tk ≥ 0 ∀t, ∀k ∈ ΨES (5.13e)

µDmax

td ≥ 0 ∀t, ∀d ∈ ΨD (5.13f)

Xi − PSti ≤ ψSmax

ti MP ∀t, ∀i ∈ ΨS (5.13g)

PESmax

k − PEStk ≤ ψESmax

tk MP ∀t, ∀k ∈ ΨES (5.13h)

PDmax

td − PDtd ≤ ψDmax

td MP ∀t, ∀d ∈ ΨD (5.13i)

µSmax

ti ≤(1− ψSmax

ti

)Mµ ∀t, ∀i ∈ ΨS (5.13j)

µESmax

tk ≤(1− ψESmax

tk

)Mµ ∀t, ∀k ∈ ΨES (5.13k)

µDmax

td ≤(1− ψDmax

td

)Mµ ∀t, ∀d ∈ ΨD (5.13l)

ψSmax

ti ∈ 0, 1 ∀t, ∀i ∈ ΨS (5.13m)

ψESmax

tk ∈ 0, 1 ∀t, ∀k ∈ ΨES (5.13n)

ψDmax

td ∈ 0, 1 ∀t, ∀d ∈ ΨD, (5.13o)



Additionally, constraint set (5.14) below is the mixed-integer linear equiv-

alent of the complementarity conditions (5.3m) and (5.3n):

Fmax

nm +Bnm (θtn − θtm)≥ 0 ∀t, ∀n, ∀m ∈ Ωn (5.14a)

Fmax

nm − Bnm (θtn − θtm)≥ 0 ∀t, ∀n, ∀m ∈ Ωn (5.14b)

νmin

tnm ≥ 0 ∀t, ∀n, ∀m ∈ Ωn (5.14c)

νmax

tnm ≥ 0 ∀t, ∀n, ∀m ∈ Ωn (5.14d)

Fmax

nm +Bnm (θtn − θtm)≤ ψνmin

tnmMF ∀t, ∀n, ∀m ∈ Ωn (5.14e)

Fmax

nm − Bnm (θtn − θtm)≤ ψνmax

tnm MF ∀t, ∀n, ∀m ∈ Ωn (5.14f)

νmin

tnm ≤(1− ψν

min

tnm

)Mν ∀t, ∀n, ∀m ∈ Ωn (5.14g)

νmax

tnm ≤(1− ψν

max

tnm

)Mν ∀t, ∀n, ∀m ∈ Ωn (5.14h)

ψνmin

tnm ∈ 0, 1 ∀t, ∀n, ∀m ∈ Ωn (5.14i)

ψνmax

tnm ∈ 0, 1 ∀t, ∀n, ∀m ∈ Ωn, (5.14j)

where MF and Mν are large enough positive constants.

Finally, constraint set (5.15) below is the mixed-integer linear equivalent of

the complementarity conditions (5.3o) and (5.3p):

π + θtn≥ 0 ∀t, ∀n (5.15a)

π − θtn≥ 0 ∀t, ∀n (5.15b)

ξmin

tn ≥ 0 ∀t, ∀n (5.15c)

ξmax

tn ≥ 0 ∀t, ∀n (5.15d)

π + θtn≤ ψθmin

tn Mθ ∀t, ∀n (5.15e)

π − θtn≤ ψθmax

tn Mθ ∀t, ∀n (5.15f)

ξmin

tn ≤(1− ψθ

min

tn

)M ξ ∀t, ∀n (5.15g)

ξmax

tn ≤(1− ψθ

max

tn

)M ξ ∀t, ∀n (5.15h)

ψθmin

tn ∈ 0, 1 ∀t, ∀n (5.15i)

ψθmax

tn ∈ 0, 1 ∀t, ∀n, (5.15j)

where Mθ and M ξ are large enough positive constants.


3) Equality constraint set (5.16) includes the KKT equality constraints (5.9),

in which dual variables φSDyt are parameterized as explained in Subsection

5.5.3.2. Thus, variables φSDyt are replaced by the corresponding parameters

φSDyt .

−σt(λt(n:i∈ΨS

n)− CS

i

)− βyt(n:i∈ΨS

n)

+γSmax

yti − γSmin

yti + φSDyt α

Sti = 0 ∀y, ∀t, ∀i ∈ (ΨS ∩ Ωy) (5.16a)

−βyt(n:i∈ΨSn)+ γS

max

yti

−γSmin

yti + φSDyt α

Sti = 0 ∀y, ∀t, ∀i /∈ (ΨS ∩ Ωy) (5.16b)

−σt(λt(n:k∈ΨES

n ) − CESk

)


max

ytk

−γESmin

ytk + φSDyt α

EStk = 0 ∀y, ∀t, ∀k ∈ (ΨES ∩ Ωy) (5.16c)


max

ytk

−γESmin

ytk + φSDyt α

EStk = 0 ∀y, ∀t, ∀k /∈ (ΨES ∩ Ωy) (5.16d)

βyt(n:d∈ΨDn )

+ γDmax

ytd

−γDmin

ytd − φSDyt U

Dtd = 0 ∀y, ∀t, ∀d ∈ ΨD (5.16e)

Ki + χmaxyi − χ

minyi + χIB

y − χSSy

−∑

t

γSmax

yti +∑

t

φSDyt µ

Smax

ti = 0 ∀y, ∀i ∈ (ΨS ∩ Ωy) (5.16f)

χIBy − χ

SSy −

∑

t

γSmax

yti

+∑

t

φSDyt µ

Smax

ti = 0 ∀y, ∀i /∈ (ΨS ∩ Ωy) (5.16g)


−ηαS

yti + ρSyti + φSDyt P

Sti = 0 ∀y, ∀t, ∀i ∈ ΨS (5.16h)

−ηαES

ytk + ρESytk + φSDyt P

EStk = 0 ∀y, ∀t, ∀k ∈ ΨES (5.16i)

∑

m∈Φn

Bnm(βytn − βytm)

+∑

m∈Φn

Bnm(τmax

ytnm − τmax

ytmn)

−∑

m∈Φn

Bnm(τmin

ytnm − τmin

ytmn)

+δmax

ytn − δmin

ytn + (δ1

yt)n=1 = 0 ∀y, ∀t, ∀n (5.16j)

−σt

∑

i∈(ΨSn∩Ωy)

P Sti +

∑

k∈(ΨESn ∩Ωy)

PEStk

+∑

d∈ΨDn

ρDytd −∑

i∈ΨSn

ρSyti −∑

k∈ΨESn

ρESytk

+∑

m∈Φn

Bnm(ρθytn − ρ

θytm) = 0 ∀y, ∀t, ∀n (5.16k)

−ρSyti − ηSmin

yti = 0 ∀y, ∀t, ∀i ∈ ΨS (5.16l)

ρSyti − ηSmax

yti + φSDyt Xi = 0 ∀y, ∀t, ∀i ∈ ΨS (5.16m)

−ρESytk − ηESmin

ytk = 0 ∀y, ∀t, ∀k ∈ ΨES (5.16n)

ρESytk − ηESmax

ytk + φSDyt P

ESmax

k = 0 ∀y, ∀t, ∀k ∈ ΨES (5.16o)

−ρDytd − ηDmin

ytd = 0 ∀y, ∀t, ∀d ∈ ΨD (5.16p)

ρDytd − ηDmax

ytd + φSDyt P

Dmax

td = 0 ∀y, ∀t, ∀d ∈ ΨD (5.16q)


−Bnm(ρθytn − ρ

θytm)

−ηνmin

ytnm + φSDyt F

max

nm = 0 ∀y, ∀t, ∀n, ∀m ∈ Φn (5.16r)

Bnm(ρθytn − ρ

θytm)

−ηνmax

ytnm + φSDyt F

max

nm = 0 ∀y, ∀t, ∀n, ∀m ∈ Φn (5.16s)

−ρθytn − ηξmin

ytn + φSDyt π = 0 ∀y, ∀t, ∀n (5.16t)

ρθytn − ηξmax

ytn + φSDyt π = 0 ∀y, ∀t, ∀n (5.16u)

ρθyt(n=1) = 0 ∀y, ∀t. (5.16v)

4) Condition set (5.17) below states that the dual variables associated with

the equality constraints included in the lower-level problems (5.1g)-(5.1n)

and in the MPECs (5.6)-(5.7) are free:

λtn : free ∀t, ∀n (5.17a)

ξ1

t : free ∀t (5.17b)

βytn : free ∀y, ∀t, ∀n (5.17c)

δ1

yt : free ∀y, ∀t (5.17d)

ρDytd : free ∀y, ∀t, ∀d ∈ ΨD (5.17e)

ρSyti : free ∀y, ∀t, ∀i ∈ ΨS (5.17f)

ρESytk : free ∀y, ∀t, ∀k ∈ ΨES (5.17g)

ρθytn : free ∀y, ∀t, ∀n. (5.17h)


5) The constraint sets (5.18)-(5.28) refer to the mixed-integer linear equivalent

of the complementarity conditions (5.10).

Constraint set (5.18) below is the mixed-integer linear equivalent of the

complementarity conditions (5.10a) and (5.10b):

Xi ≥ 0 ∀y, ∀i ∈ (ΨS ∩ Ωy) (5.18a)

(Xmaxi −Xi)≥ 0 ∀y, ∀i ∈ (ΨS ∩ Ωy) (5.18b)

χminyi ≥ 0 ∀y, ∀i ∈ (ΨS ∩ Ωy) (5.18c)

χmaxyi ≥ 0 ∀y, ∀i ∈ (ΨS ∩ Ωy) (5.18d)

Xi ≤ ψXmin

yi MP ∀y, ∀i ∈ (ΨS ∩ Ωy) (5.18e)

(Xmaxi −Xi)≤ ψXmax

yi MP ∀y, ∀i ∈ (ΨS ∩ Ωy) (5.18f)

χminyi ≤

(1− ψXmin

yi

)Mµ ∀y, ∀i ∈ (ΨS ∩ Ωy) (5.18g)

χmaxyi ≤

(1− ψXmax

yi

)Mµ ∀y, ∀i ∈ (ΨS ∩ Ωy) (5.18h)

ψXmin

yi ∈ 0, 1 ∀y, ∀i ∈ (ΨS ∩ Ωy) (5.18i)

ψXmax

yi ∈ 0, 1 ∀y, ∀i ∈ (ΨS ∩ Ωy), (5.18j)



alent of the complementarity conditions (5.10c) and (5.10d):

(Kmax −

∑

i∈ΨS

KiXi

)≥ 0 ∀y (5.19a)

[∑

i∈ΨS

Xi +∑

k∈ΨES

PESmax

k −

(Υ×

∑

d∈ΨD

PDmax

td

)]

≥ 0 ∀y, t = t1 (5.19b)

χIBy ≥ 0 ∀y (5.19c)


χSSy ≥ 0 ∀y (5.19d)(Kmax −

∑

i∈ΨS

KiXi

)≤ ψIB

y MK ∀y (5.19e)

[∑

i∈ΨS

Xi +∑

k∈ΨES

PESmax

k −

(Υ×

∑

d∈ΨD

PDmax

td

)]

≤ ψSSy M

P ∀y, t = t1 (5.19f)

χIBy ≤

(1− ψIB

y

)M IB ∀y (5.19g)

χSSy ≤

(1− ψSS

y

)MSS ∀y (5.19h)

ψIBy ∈ 0, 1 ∀y (5.19i)

ψSSy ∈ 0, 1 ∀y, (5.19j)

where MK, MP, M IB and MSS are large enough positive constants.

In addition, constraint set (5.20) below is the mixed-integer linear equiva-

lent of the complementarity conditions (5.10e) and (5.10f):

αSti ≥ 0 ∀y, ∀t, ∀i ∈ ΨS (5.20a)

αEStk ≥ 0 ∀y, ∀t, ∀k ∈ ΨES (5.20b)

ηαS

yti ≥ 0 ∀y, ∀t, ∀i ∈ ΨS (5.20c)

ηαES

ytk ≥ 0 ∀y, ∀t, ∀k ∈ ΨES (5.20d)

αSti ≤ ψα

S

ytiMα ∀y, ∀t, ∀i ∈ ΨS (5.20e)

αEStk ≤ ψα

ES

ytk Mα ∀y, ∀t, ∀k ∈ ΨES (5.20f)

ηαS

yti ≤(1− ψα

S

yti

)Mη ∀y, ∀t, ∀i ∈ ΨS (5.20g)

ηαES

ytk ≤(1− ψα

ES

ytk

)Mη ∀y, ∀t, ∀k ∈ ΨES (5.20h)

ψαS

yti ∈ 0, 1 ∀y, ∀t, ∀i ∈ ΨS (5.20i)


ψαES

ytk ∈ 0, 1 ∀y, ∀t, ∀k ∈ ΨES, (5.20j)

where Mα and Mη are large enough positive constants.


complementarity conditions (5.10g), (5.10i) and (5.10k). Note that some of

the constraints (5.21) are identical and thus redundant to those included

in (5.12). However, they are included in (5.21) to clarify the linearization

procedure used, while they are not repeated in the software implementation

of an equilibrium problem.

P Sti ≥ 0 ∀y, ∀t, ∀i ∈ ΨS (5.21a)

PEStk ≥ 0 ∀y, ∀t, ∀k ∈ ΨES (5.21b)

PDtd ≥ 0 ∀y, ∀t, ∀d ∈ ΨD (5.21c)

γSmin

yti ≥ 0 ∀y, ∀t, ∀i ∈ ΨS (5.21d)

γESmin

ytk ≥ 0 ∀y, ∀t, ∀k ∈ ΨES (5.21e)

γDmin

ytd ≥ 0 ∀y, ∀t, ∀d ∈ ΨD (5.21f)

P Sti ≤ ϑS

min

yti MP ∀y, ∀t, ∀i ∈ ΨS (5.21g)

PEStk ≤ ϑES

min

ytk MP ∀y, ∀t, ∀k ∈ ΨES (5.21h)

PDtd ≤ ϑD

min

ytd MP ∀y, ∀t, ∀d ∈ ΨD (5.21i)

γSmin

yti ≤(1− ϑS

min

yti

)Mγ ∀y, ∀t, ∀i ∈ ΨS (5.21j)

γESmin

ytk ≤(1− ϑES

min

ytk

)Mγ ∀y, ∀t, ∀k ∈ ΨES (5.21k)

γDmin

ytd ≤(1− ϑD

min

ytd

)Mγ ∀y, ∀t, ∀d ∈ ΨD (5.21l)

ϑSmin

yti ∈ 0, 1 ∀y, ∀t, ∀i ∈ ΨS (5.21m)

ϑESmin

ytk ∈ 0, 1 ∀y, ∀t, ∀k ∈ ΨES (5.21n)

ϑDmin

ytd ∈ 0, 1 ∀y, ∀t, ∀d ∈ ΨD, (5.21o)

where MP and Mγ are large enough positive constants.


In addition, constraint set (5.22) below is the mixed-integer linear equiv-

alent of the complementarity conditions (5.10h), (5.10j) and (5.10l). Note

that some of the constraints (5.22) are identical and thus redundant to

those included in (5.13). However, they are included in (5.22) to clarify the

linearization procedure used.

Xi − PSti ≥ 0 ∀y, ∀t, ∀i ∈ ΨS (5.22a)

PESmax

k − PEStk ≥ 0 ∀y, ∀t, ∀k ∈ ΨES (5.22b)

PDmax

td − PDtd ≥ 0 ∀y, ∀t, ∀d ∈ ΨD (5.22c)

γSmax

yti ≥ 0 ∀y, ∀t, ∀i ∈ ΨS (5.22d)

γESmax

ytk ≥ 0 ∀y, ∀t, ∀k ∈ ΨES (5.22e)

γDmax

ytd ≥ 0 ∀y, ∀t, ∀d ∈ ΨD (5.22f)

Xi − PSti ≤ ϑS

max

yti MP ∀y, ∀t, ∀i ∈ ΨS (5.22g)

PESmax

k − PEStk ≤ ϑES

max

ytk MP ∀y, ∀t, ∀k ∈ ΨES (5.22h)

PDmax

td − PDtd ≤ ϑD

max

ytd MP ∀y, ∀t, ∀d ∈ ΨD (5.22i)

γSmax

yti ≤(1− ϑS

max

yti

)Mγ ∀y, ∀t, ∀i ∈ ΨS (5.22j)

γESmax

ytk ≤(1− ϑES

max

ytk

)Mγ ∀y, ∀t, ∀k ∈ ΨES (5.22k)

γDmax

ytd ≤(1− ϑD

max

ytd

)Mγ ∀y, ∀t, ∀d ∈ ΨD (5.22l)

ϑSmax

yti ∈ 0, 1 ∀y, ∀t, ∀i ∈ ΨS (5.22m)

ϑESmax

ytk ∈ 0, 1 ∀y, ∀t, ∀k ∈ ΨES (5.22n)

ϑDmax

ytd ∈ 0, 1 ∀y, ∀t, ∀d ∈ ΨD, (5.22o)

where MP and Mγ are large enough positive constants.


alent of the complementarity conditions (5.10m) and (5.10n). Note that

some of the constraints (5.23) are identical and thus redundant to those

included in (5.14). However, they are included in (5.23) to clarify the lin-

earization procedure used.


Fmax

nm +Bnm (θtn − θtm)≥ 0 ∀y, ∀t, ∀n, ∀m ∈ Ωn (5.23a)

Fmax

nm −Bnm (θtn − θtm)≥ 0 ∀y, ∀t, ∀n, ∀m ∈ Ωn (5.23b)

τmin

ytnm ≥ 0 ∀y, ∀t, ∀n, ∀m ∈ Ωn (5.23c)

τmax

ytnm ≥ 0 ∀y, ∀t, ∀n, ∀m ∈ Ωn (5.23d)

Fmax

nm +Bnm (θtn − θtm)≤ ψτmin

ytnmMF ∀y, ∀t, ∀n, ∀m ∈ Ωn (5.23e)

Fmax

nm −Bnm (θtn − θtm)≤ ψτmax

ytnmMF ∀y, ∀t, ∀n, ∀m ∈ Ωn (5.23f)

τmin

ytnm ≤(1− ψτ

min

ytnm

)M τ ∀y, ∀t, ∀n, ∀m ∈ Ωn (5.23g)

τmax

ytnm ≤(1− ψτ

max

ytnm

)M τ ∀y, ∀t, ∀n, ∀m ∈ Ωn (5.23h)

ψτmin

ytnm ∈ 0, 1 ∀y, ∀t, ∀n, ∀m ∈ Ωn (5.23i)

ψτmax

ytnm ∈ 0, 1 ∀y, ∀t, ∀n, ∀m ∈ Ωn, (5.23j)

where MF and M τ are large enough positive constants.


complementarity conditions (5.10o) and (5.10p). Note that some of the

constraints (5.24) are identical and thus redundant to those included in

(5.15). However, they are included in (5.24) to clarify the linearization

procedure used.

π + θtn≥ 0 ∀y, ∀t, ∀n (5.24a)

π − θtn≥ 0 ∀y, ∀t, ∀n (5.24b)

δmin

ytn ≥ 0 ∀y, ∀t, ∀n (5.24c)

δmax

ytn ≥ 0 ∀y, ∀t, ∀n (5.24d)

π + θtn≤ ψδmin

ytn Mθ ∀y, ∀t, ∀n (5.24e)

π − θtn≤ ψδmax

ytn Mθ ∀y, ∀t, ∀n (5.24f)


δmin

ytn ≤(1− ψδ

min

ytn

)M δ ∀y, ∀t, ∀n (5.24g)

δmax

ytn ≤(1− ψδ

max

ytn

)M δ ∀y, ∀t, ∀n (5.24h)

ψδmin

ytn ∈ 0, 1 ∀y, ∀t, ∀n (5.24i)

ψδmax

ytn ∈ 0, 1 ∀y, ∀t, ∀n, (5.24j)

where Mθ and M δ are large enough positive constants.

Finally, constraint sets (5.25)-(5.28) below are the mixed-integer linear

equivalent of the complementarity conditions (5.10q) and (5.10z). Note

that some of the constraints (5.25)-(5.28) are identical and thus redundant

to those included in (5.12)-(5.15). However, they are included in (5.25)-

(5.28) to clarify the linearization procedure used.

µSmin

ti ≥ 0 ∀y, ∀t, ∀i ∈ ΨS (5.25a)

µSmax

ti ≥ 0 ∀y, ∀t, ∀i ∈ ΨS (5.25b)

µESmin

tk ≥ 0 ∀y, ∀t, ∀k ∈ ΨES (5.25c)

µESmax

tk ≥ 0 ∀y, ∀t, ∀k ∈ ΨES (5.25d)

µDmin

td ≥ 0 ∀y, ∀t, ∀d ∈ ΨD (5.25e)

µDmax

td ≥ 0 ∀y, ∀t, ∀d ∈ ΨD (5.25f)

νmin

tnm ≥ 0 ∀y, ∀t, ∀n, ∀m ∈ Φn (5.25g)

νmax

tnm ≥ 0 ∀y, ∀t, ∀n, ∀m ∈ Φn (5.25h)

ξmin

tn ≥ 0 ∀y, ∀t, ∀n (5.25i)

ξmax

tn ≥ 0 ∀y, ∀t, ∀n (5.25j)

ηSmin

yti ≥ 0 ∀y, ∀t, ∀i ∈ ΨS (5.25k)

ηSmax

yti ≥ 0 ∀y, ∀t, ∀i ∈ ΨS (5.25l)

ηESmin

ytk ≥ 0 ∀y, ∀t, ∀k ∈ ΨES (5.25m)

ηESmax

ytk ≥ 0 ∀y, ∀t, ∀k ∈ ΨES (5.25n)


ηDmin

ytd ≥ 0 ∀y, ∀t, ∀d ∈ ΨD (5.25o)

ηDmax

ytd ≥ 0 ∀y, ∀t, ∀d ∈ ΨD (5.25p)

ηνmin

ytnm ≥ 0 ∀y, ∀t, ∀n, ∀m ∈ Φn (5.25q)

ηνmax

ytnm ≥ 0 ∀y, ∀t, ∀n, ∀m ∈ Φn (5.25r)

ηξmin

ytn ≥ 0 ∀y, ∀t, ∀n (5.25s)

ηξmax

ytn ≥ 0 ∀y, ∀t, ∀n (5.25t)

µSmin

ti ≤ µSmin

ti Mµ ∀y, ∀t, ∀i ∈ ΨS (5.26a)

µSmax

ti ≤ µSmax

ti Mµ ∀y, ∀t, ∀i ∈ ΨS (5.26b)

µESmin

tk ≤ µESmin

tk Mµ ∀y, ∀t, ∀k ∈ ΨES (5.26c)

µESmax

tk ≤ µESmax

tk Mµ ∀y, ∀t, ∀k ∈ ΨES (5.26d)

µDmin

td ≤ µDmin

td Mµ ∀y, ∀t, ∀d ∈ ΨD (5.26e)

µDmax

td ≤ µDmax

td Mµ ∀y, ∀t, ∀d ∈ ΨD (5.26f)

νmin

tnm ≤ νmin

tnmMν ∀y, ∀t, ∀n, ∀m ∈ Φn (5.26g)

νmax

tnm ≤ νmax

tnmMν ∀y, ∀t, ∀n, ∀m ∈ Φn (5.26h)

ξmin

tn ≤ ξmin

tn Mξ ∀y, ∀t, ∀n (5.26i)

ξmax

tn ≤ ξmax

tn M ξ ∀y, ∀t, ∀n, (5.26j)

where Mµ, Mν and M ξ are large enough positive constants.

Also,

ηSmin

yti ≤(1− µSmin

ti

)Mη ∀y, ∀t, ∀i ∈ ΨS (5.27a)

ηSmax

yti ≤(1− µSmax

ti

)Mη ∀y, ∀t, ∀i ∈ ΨS (5.27b)

ηESmin

ytk ≤(1− µESmin

tk

)Mη ∀y, ∀t, ∀k ∈ ΨES (5.27c)

ηESmax

ytk ≤(1− µESmax

tk

)Mη ∀y, ∀t, ∀k ∈ ΨES (5.27d)

ηDmin

ytd ≤(1− µDmin

td

)Mη ∀y, ∀t, ∀d ∈ ΨD (5.27e)

ηDmax

ytd ≤(1− µDmax

td

)Mη ∀y, ∀t, ∀d ∈ ΨD (5.27f)

ηνmin

ytnm ≤(1− ν

min

tnm

)Mη ∀y, ∀t, ∀n, ∀m ∈ Φn (5.27g)

ηνmax

ytnm ≤(1− ν

max

tnm

)Mη ∀y, ∀t, ∀n, ∀m ∈ Φn (5.27h)


ηξmin

ytn ≤(1− ξ

min

tn

)Mη ∀y, ∀t, ∀n (5.27i)

ηξmax

ytn ≤(1− ξ

max

tn

)Mη ∀y, ∀t, ∀n, (5.27j)

where Mη is a large enough constant, and:

µSmin

ti ∈ 0, 1 ∀y, ∀t, ∀i ∈ ΨS (5.28a)

µSmax

ti ∈ 0, 1 ∀y, ∀t, ∀i ∈ ΨS (5.28b)

µESmin

tk ∈ 0, 1 ∀y, ∀t, ∀k ∈ ΨES (5.28c)

µESmax

tk ∈ 0, 1 ∀y, ∀t, ∀k ∈ ΨES (5.28d)

µDmin

td ∈ 0, 1 ∀y, ∀t, ∀d ∈ ΨD (5.28e)

µDmax

td ∈ 0, 1 ∀y, ∀t, ∀d ∈ ΨD (5.28f)

νmin

tnm ∈ 0, 1 ∀y, ∀t, ∀n, ∀m ∈ Φn (5.28g)

νmax

tnm ∈ 0, 1 ∀y, ∀t, ∀n, ∀m ∈ Φn (5.28h)

ξmin

tn ∈ 0, 1 ∀y, ∀t, ∀n (5.28i)

ξmax

tn ∈ 0, 1 ∀y, ∀t, ∀n. (5.28j)

The primal optimization variable set of the mixed-integer linear conditions

(5.11)-(5.28) includes the following variable sets:

a) Variable set ΞUL defined in Subsection 5.4.2.

b) Variable set ΞDual except dual variables φSDyt that have been parameter-

ized in Subsection 5.5.3.2. Note that the variable set ΞDual is defined in

Subsection 5.4.4.

c) The set of binary variables to linearize the complementarity conditions

(5.3g)-(5.3p) and (5.10) included in the following set:

ΞBin= ψSmin

ti , ψESmin

tk , ψDmin

td , ψSmax

ti , ψESmax

tk , ψDmax

td , ψνmin

tnm , ψνmax

tnm , ψθmin

tn ,

ψθmax

tn , ψXmin

yi , ψXmax

yi , ψ∆y , ψ

Γy , ψ

αS

yti, ψαES

ytk , ϑSmin

yti , ϑESmin

ytk , ϑDmin

ytd , ϑSmax

yti ,


ϑESmax

ytk , ϑDmax

ytd , ψτmin

ytnm, ψτmax

ytnm, ψδmin

ytn , ψδmax

ytn , µSmin

ti , µSmax

ti , µESmin

tk , µESmax

tk ,

µDmin

td , µDmax

td , νmin

tnm, νmax

tnm, ξmin

tn , ξmax

tn .

5.7 Searching For Investment Equilibria

The mixed-integer linear form of the optimality conditions of the EPEC, i.e.,

conditions (5.11)-(5.28) presented in Section 5.6 constitute a system of mixed-

integer linear equalities and inequalities that involves continuous and binary

variables and that generally has multiple solutions.

To explore such solutions, it is possible to formulate an optimization prob-

lem considering the mixed-integer linear conditions (5.11)-(5.28) as constraints.

In addition, several objective functions can be considered to identify different

equilibria [117], for example:

1) Total profit (TP).

2) Annual true social welfare (ATSW) considering the production costs of

the generation units.

3) Annual social welfare considering the strategic offer prices of the gener-

ation units.

4) Minus payment of the demands.

5) Profit of a given producer.

6) Minus payment of a given demand.

In this chapter, the first two objectives are selected because i) they can be

formulated linearly; and ii) they refer to general market measures. Thus, the

optimization problem to find equilibria is formulated as follows:

Maximize TP or ATSW (5.29a)

subject to: mixed-integer linear system (5.11)-(5.28) (5.29b)

Note that the primal variables of the optimization problem (5.29) are those

of the mixed-integer linear system (5.11)-(5.28) presented in Section 5.6.

5.7. Searching For Investment Equilibria 231

The two linear objective functions selected (i.e., total profit and annual

true social welfare) to be included in (5.29a) are described in the following two

subsections.

5.7.1 Objective Function (5.29a): Total Profit

The summation of objective function (5.1a) for all producers provides the

minus total profit of all producers, but such expression is non-linear due to the

product of continuous variables (productions and clearing prices) in the term

(5.30) below. We denote such non-linear term as Zt.

Zt =∑

i∈ΨS

P Stiλt(n:i∈ΨS

n)+∑

k∈ΨES

PEStk λt(n:k∈ΨES

n ) ∀t. (5.30)

An identical linearization approach to one presented in Subsection 2.3.5.1

of Chapter 2 is used to linearize Zt. Using such approach, the following exact

linear equivalent for Zt (denoted as ZLint ) is obtained:

ZLint =

∑

d∈ΨD

UDtdP

Dtd −

∑

d∈ΨD

µDmax

td PDmax

td

−∑

n(m∈Φn)

(νmin

tnm + νmax

tnm)Fmax

nm −∑

n

(ξmin

tn + ξmax

tn )π ∀t. (5.31a)

Therefore, the linear form of the total profit to be included in (5.29a) is

TP =∑

t

σt

(ZLint −

∑

i∈ΨS

P StiC

Si −

∑

k∈ΨES

PEStk C

ESk

)−∑

i∈ΨS

KiXi. (5.31b)


5.7.2 Objective Function (5.29a): Annual True Social

Welfare

The linear formulation of the annual true social welfare to be included in

(5.29a) is given by (5.32) below:

ATSW =∑

t

σt

(∑

d∈ΨD

UDtdP

Dtd −

∑

i∈ΨS

CSi P

Sti −

∑

k∈ΨES

CEStk P

EStk

). (5.32)

Note that to formulate the annual true social welfare in (5.32), instead of

the strategic offers of the generation units (αSti and α

EStk ), their true production

costs (CSi and CES

k ) are considered.

5.8 Algorithm for the Diagonalization Check-

ing

As explained in Section 5.2, we carry out a single step of the diagonalization

algorithm, because this allows us to check whether or not each stationary

point obtained by the approach proposed in this chapter is, in fact, a Nash

equilibrium.

Note that if under the diagonalization scheme no producer desires to deviate

from its current strategy, then the set of strategies of all producers satisfies

the definition of Nash equilibrium.

For example, assume a triopoly market with three strategic producers (Pro-

ducers 1, 2 and 3) with investment decisions X∗1 , X

∗2 and X∗

3 , respectively,

obtained by the proposed approach. In order to verify that this solution con-

stitutes a Nash equilibrium, the following steps are carried out:

a) Consider the mixed-integer linear form of MPEC (5.6)-(5.7), i.e., MILP

problem (2.10)-(2.18) presented in Subsection 2.3.6 of Chapter 2 pertain-

ing to Producer 1 (e.g., MPEC1).

b) Set the investment decisions of other producers (Producers 2 and 3)


to those obtained by the approach proposed in this chapter (i.e., X∗2

and X∗3 ), and then solve MPEC1. Note that its solution provides the

investment decisions of Producer 1, which we denote as X1.

c) Repeat the two steps above for every producer. For the example con-

sidered, this step results in the optimal investment decisions X2 and X3

pertaining to Producers 2 and 3, respectively.

d) Compare the results obtained from the above single-step diagonalization

algorithm (X1, X2 and X3) with those achieved from the approach pro-

posed in this chapter (X∗1 , X

∗2 and X∗

3 ). If the investment results of

each producer obtained from the single-step diagonalization algorithm

are identical to those attained by the approach proposed in this chapter

(i.e., X1=X∗1 , X2=X

∗2 and X3=X

∗3 ), then the solution of the algorithm

proposed in this chapter (X∗1 , X

∗2 and X∗

3 ) is a Nash equilibrium be-

cause each producer cannot increase its profit by changing its strategy

unilaterally.

5.9 Illustrative Example

To illustrate the numerical ability of the proposed approach to identify mean-

ingful generation investment equilibria, a two-node illustrative example is con-

sidered, where investment equilibria are identified considering annual true so-

cial welfare maximization and total profit maximization.

Additionally, the impact of the issues below on the generation investment

equilibria are analyzed in detail:

1) Strategic or non-strategic offering by the producers.

2) Transmission congestion.

3) Supply security.

4) Available investment budget.


Figure 5.3: EPEC problem: Network of the illustrative example.

1tt=σ

2tt =σ

3tt =σ

4tt =σ

Figure 5.4: EPEC problem: Piecewise approximation of the load-durationcurve for the target year (illustrative example).

5.9.1 Data

A two-node illustrative example is considered in this section and depicted in

Figure 5.3. The capacity of the transmission line is assumed to be 500 MW

for the uncongested cases and 100 MW for the congested ones. In addition,

its susceptance is considered equal to 10 p.u. (100 MW base). Note that both

nodes N1 and N2 are candidates to locate new units.

The load-duration curve of the target year is approximated through four

demand blocks (i.e, t = t1, t = t2, t = t3 and t = t4) as illustrated in Figure

5.4. The weighting factors (σt) of such demand blocks are 1095, 2190, 2190


Table 5.1: EPEC problem: Data pertaining to the existing units (illustrativeexample).

Existing Capacity Capacity Capacity Production cost Production cost

unit[MW]


(k ∈ ΨES) [MW] [MW] [e/MWh] [e/MWh]

G1 60 30 30 12.00 14.00

G2 60 30 30 12.00 14.00

G3 120 60 60 13.00 15.00

Table 5.2: EPEC problem: Data pertaining to demands and their price bids(illustrative example).

Demand Maximum load Price bid Price bid

block of each demand of demand D1 of demand D2

(t) [MW] [e/MWh] [e/MWh]

t = t1 200 20.00 22.00

t = t2 150 19.00 21.00

t = t3 125 18.00 20.00

t = t4 100 17.00 19.00

and 3285, respectively. Note that the summation of the weighting factors of

the demand blocks renders the number of hours in a year, i.e., 8760.

Three existing generation units (G1, G2 and G3) are considered and their

data given in Table 5.1. The second column provides the capacity of each

unit, which is composed of two generation blocks (columns 3-4), with their

corresponding production costs (columns 5-6). Regarding existing units, the

following observations are in order:

1) The capacities of units G1 and G2 are identical and equal to 60 MW,

while the capacity of unit G3 is 120 MW. Thus, the total capacity of

existing units is 240 MW.

2) The production costs of units G1 and G2 are identical, while the cost of


Table 5.3: EPEC problem: Type and data for the candidate units (illustrativeexample).

Candidate Annualized capital cost Production cost Maximum capacity

unit (Ki) (CSi ) to be built (Xmax

i )

(i ∈ ΨSn) [e/MW] [e/MWh] [MW]

Base technology 30000 10.00 250

Peak technology 6000 14.00 250

G3 is comparatively higher.

Two demands (D1 and D2) are considered in this example. Data for such

demands and their price bids per block are given in Table 5.2. The maximum

loads of demands D1 and D2 per block are considered identical and are given

in the second column of Table 5.2. Accordingly, the peak demand is 400 MW

corresponding to the first block, t = t1. The last two columns of Table 5.2

represent the price bids of each demand in each block.

Table 5.3 characterizes the candidate units for investment based on two

technologies: base technology (e.g., nuclear units) and peak technology (e.g.,

CCGT units). Columns 2 and 3 give annualized capital cost and production

cost for these candidate units, respectively. The maximum capacity of each

candidate unit to be built (Xmaxi ) is 250 MW.

Note that for the sake of simplicity, only one generation block is considered

for such units. Note also that the capital cost of each candidate base unit is

higher than that of each peak unit, while its operation cost is lower (Table

5.3).

We assume that the market regulator imposes that the available produc-

tion capacity after investment (i.e., summation of the capacities of existing

and newly built units) should be at least 10% higher than the peak demand

(Υ=1.10). Thus, since the capacity of existing units is 240 MW and the peak

demand is 400 MW, at least 200 additional MW of new capacity are required.

On the other hand, the maximum available investment budget (Kmax) is

considered to be e 7.5 million.


Table 5.4: EPEC problem: Cases considered for the illustrative example.

Case Competition type Strategic producers Non-strategic producers Congestion

Case 1 Triopoly Producers 1, 2 and 3 - ×

Case 2 Triopoly Producers 1, 2 and 3 - X

Case 3 Triopoly Producers 1 and 2 Producer 3 ×

Case 4 Triopoly - Producers 1, 2 and 3 ×

Case 5 Monopoly Producer 1 - ×

Case 6 Monopoly Producer 1 - X

In addition, the values of dual variables φSDyt ∀y, ∀t, are arbitrarily fixed to

5σt ∀t.

5.9.2 Cases Considered

The cases considered for this illustrative example are characterized in Table

5.4.

The second column of Table 5.4 gives the type of competition (triopoly or

monopoly) and columns 3 and 4 identify the strategic and the non-strategic

producers, respectively. In addition, the last column of Table 5.4 indicates

whether or not congestion occurs. The capacity of the transmission line N1-

N2 is 500 MW for the uncongested cases and 100 MW for the congested ones.

Regarding the cases considered in Table 5.4, the two following observations

are pertinent:

1) Each non-strategic producer offers at its production cost. This is realized

by replacing its strategic offering variables αSti and α

EStk in equation (5.1g)

with cost parameters CSi and CES

k , respectively.

2) Triopoly refers to a competition type whereas the generation units G1,

G2 and G3 are owned by three different Producers, 1, 2 and 3, respec-

tively. In addition, in the monopoly, all generation units belong to a

single producer, denominated Producer 1.

The cases considered are described in detail below:


Case 1) The type of competition is triopoly and all producers 1, 2 and 3 are

strategic. In addition, the capacity of the transmission line is assumed

to be 500 MW and thus line congestion does not occur.

Case 2) Similarly to Case 1, the type of competition is triopoly and all pro-

ducers are strategic. However, the capacity of the transmission line is

assumed to be 100 MW and thus the network suffers from congestion.

Case 3) Similarly to Cases 1 and 2, the type of competition is triopoly and

the capacity of the transmission line is assumed to be 500 MW (no

congestion). However, Producers 1 and 2 are strategic, while Producer

3 offers in a non-strategic way.

Case 4) Similarly to Cases 1, 2 and 4, the type of competition is triopoly and

the capacity of the transmission line is assumed to be 500 MW (no

congestion). Nevertheless, all producers 1, 2 and 3 offer in a non-

strategic way.

Case 5) Unlike Cases 1-4, the type of competition is monopoly, i.e., all units

belong to Producer 1. Note that such producer behaves strategically,

and the capacity of the transmission line is assumed to be 500 MW

(no congestion).

Case 6) Similarly to Case 5, all units belong to strategic Producer 1 (monopoly),

but the capacity of the transmission line is assumed to be 100 MW

and thus the network suffers from congestion.

Note that in Cases 1-4 (triopoly cases), the optimization problem (5.29) is

solved, while the mixed-integer linear form of MPEC (5.6)-(5.7), i.e., MILP

problem (2.10)-(2.18) presented in Subsection 2.3.6 of Chapter 2 is solved in

Cases 5 and 6 (monopoly cases).

5.9.3 Demand Bid and Stepwise Supply Offer Curves

In this subsection, the structure of the stepwise supply offers and the demand

bids is clarified.


0 40 80 120 160 200 240 280 320 360 400 440

12141618202224

Offe

r an

d bi

d pr

ices

(eur

o/M

Wh)

Unit G3 Newly built unit Unit G1 Unit G2

0 40 80 120 160 200 240 280 320 360 400 440

10111213141516

Production quantity (MW)

Pro

duct

ion

cost

(eur

o/M

Wh)

Market clearing point

Figure 5.5: EPEC problem: Demand bid curve and stepwise supply offer curvecorresponding to the first demand block t = t1 (Case 1 maximizing total profit).

As an example, Figure 5.5 depicts the demand bid and stepwise supply

offer curves pertaining to the first demand block, t = t1, of Case 1 maximizing

total profit. The upper plot of this figure illustrates the bid curve of demands

(dash line) and the strategic offer curve of existing units G1, G2, G3, and the

newly built 200 MW base unit. In addition, the lower plot depicts the actual

production cost of the generation units.

Note that unlike the case of using one single MPEC model, i.e., MILP

problem (2.10)-(2.18) presented in Subsection 2.3.6 of Chapter 2, all generation

units do not offer at the market clearing price (upper plot of Figure 5.5). This

is consistent with the results reported in [117].

In the equilibrium depicted in Figure 5.5, in order to be dispatched, the

comparatively expensive unit G3 offers its first production-offer block at 11

e/MWh (upper plot), i.e., at a price lower than its corresponding production

cost (13 e/MWh). As shown in the upper plot, all production-offer blocks of


other units as well as the second one of unit G3 are offered at prices higher

than the corresponding production costs (lower plot). Pursuing maximum

total profit, the most expensive production-offer block (second block of unit

G3) is offered at the minimum bid price of the demands, 20 e/MWh (upper

plot of Figure 5.5).

Note that other combinations of offers may result in different equilibria but

with the same market outcomes.

5.9.4 General Equilibrium Results

The general equilibrium results are given in Table 5.5 whose structure is de-

scribed below:

• Columns 2, 3 and 4 provide the total newly built base and peak capacity

and the total capacity built, respectively.

• Investment cost and the location of newly built units are given in columns

5 and 6, respectively.

• The market clearing prices per demand block are provided in columns

7-10.

• The total profit for all producers in the target year and the annual true

social welfare of the market are provided in the last two columns.

• Rows 2-5 pertain to equilibria in which the total profit is maximized,

while the next four rows give the results for equilibria in which the annual

true social welfare is maximized.

• The last two rows provide the investment results for the monopoly cases

(Cases 5 and 6).

According to the results presented in Table 5.5, several conclusions can be

drawn as stated below:

Conclusions pertaining to the newly built units, their investment costs and

locations (columns 2-6) are threefold:

5.9.Illu

strativ

eExample

241

Table 5.5: EPEC problem: General results of generation investment equilibria (illustrative example).

Case

Total newly Total newly Total newly Investment Location of λtn λtn λtn λtn Total Annual true

built base built peak built capacity cost new units for t = t1 for t = t2 for t = t3 for t = t4 profit social welfare

capacity [MW] capacity [MW] [MW] [Me] [node] [e/MWh] [e/MWh] [e/MWh] [e/MWh] [Me] [Me]

Case 1 (Max TP∗) 200 0 200 6.00 N1-N2 20 (N1-N2) 19 (N1-N2) 18 (N1-N2) 17 (N1-N2) 11.83 20.13

Case 2 (Max TP) 180 20 200 5.52 N1 20 (N1), 22 (N2) 19 (N1), 21 (N2) 18 (N1), 20 (N2) 17 (N1-N2) 12.34 19.61

Case 3 (Max TP) 80 120 200 3.12 N1-N2 20 (N1-N2) 19 (N1-N2) 18 (N1-N2) 17 (N1-N2) 10.37 15.79

Case 4 (Max TP) 80 120 200 3.12 N1-N2 15 (N1-N2) 14 (N1-N2) 14 (N1-N2) 14 (N1-N2) 1.67 16.73

Case 1 (Max ATSW†) 250 0 250 7.50 N1-N2 20 (N1-N2) 19 (N1-N2) 18 (N1-N2) 17 (N1-N2) 11.09 20.89

Case 2 (Max ATSW) 250 0 250 7.50 N1 20 (N1), 22 (N2) 19 (N1), 21 (N2) 18 (N1), 20 (N2) 17 (N1-N2) 11.52 20.89

Case 3 (Max ATSW) 250 0 250 7.50 N1-N2 15 (N1-N2) 13 (N1-N2) 10 (N1-N2) 10 (N1-N2) -4.02 20.89

Case 4 (Max ATSW) 250 0 250 7.50 N1-N2 14 (N1-N2) 12 (N1-N2) 10 (N1-N2) 10 (N1-N2) -5.08 20.89

Case 5 200 0 200 6.00 N1-N2 20 (N1-N2) 19 (N1-N2) 18 (N1-N2) 17 (N1-N2) 11.83 20.13

Case 6 180 20 200 5.52 N1 20 (N1), 22 (N2) 19 (N1), 21 (N2) 18 (N1), 20 (N2) 17 (N1-N2) 12.34 19.61

∗ TP: Total profit; †ATSW: Annual true social welfare


1.1) In all triopoly cases maximizing total profit (rows 2-5) and in the cases of

monopoly (last two rows), the total newly built capacity (base plus peak

units) is identical and equal to 200 MW. This result is due to constraint

(5.1d), because Υ=1.10 forces an available capacity level higher than the

peak demand. Nevertheless, the amounts of base and peak capacities are

different across cases.

1.2) In all triopoly cases maximizing annual true social welfare (rows 6-9),

the total newly built capacity is identical and equal to 250 MW, due to

constraint (5.1c) imposing a given investment budget.

1.3) In all congested cases (rows 3, 7 and 11), the newly built units are located

in node N1 where the comparatively cheaper demand (D1) is connected.

This allows congesting the network and driving the market to clear at

comparatively higher prices. Note that in all uncongested cases (rows 2,

4-6, 8-10), the location of newly built units is immaterial as new units

can be built at any node without altering the results.

Conclusions pertaining to the market clearing prices (columns 7-10) are

fourfold:

2.1) In all triopoly and monopoly cases in which congestion does not occur, if

all producers behave strategically (Cases 1 and 5), the market is cleared

in each demand block at the minimum bid price of the demands. Market

clearing prices in both nodes are identical.

2.2) In a similar situation as in 2.1), but with congestion (Cases 2 and 6), the

market clearing price of each node is identical to one of the bid prices

of the demand connected to that node. Note that in the fourth demand

block t = t4, the transmission line is not congested, and thus the market

prices at both nodes are equal.

2.3) If at least one producer behaves in a non-strategic way (Case 3), and the

total profit is maximized, the market clearing prices are identical to Case

1, while clearing prices comparatively decrease if the annual true social

welfare is maximized.


2.4) If all producer offer at their production costs (Case 4), and either total

profit or annual true social welfare is maximized, the market is cleared

at comparatively lower prices.

Conclusions pertaining to the total profit and the annual true social welfare

(last two columns) are fivefold:

3.1) As expected, each triopoly case maximizing total profit (rows 2-5) achieves

comparatively higher total profit than the corresponding case maximiz-

ing annual true social welfare (rows 6-9), but with comparatively lower

annual true social welfare.

3.2) In the triopoly cases maximizing annual true social welfare, only base

units are built, because they are cheaper from the operation point of

view, and thus they render a comparatively higher annual true social

welfare. In addition, since the same capacity of base units is built in

such cases (250 MW), their corresponding annual true social welfare are

equal (20.89 Me).

3.3) In the triopoly cases maximizing either total profit or annual true social

welfare, if all producers behave strategically (Case 1), a comparatively

higher total profit is obtained. In addition, congestion can increase this

total profit (Case 2).

3.4) In the triopoly cases maximizing total profit, if at least one producer

offers at its production cost (Case 3), total profit decreases (10.37 Me).

Behaving all producers in a non-strategic way (Case 4) results in a dra-

matic decrease of the total profit (1.67 Me).

3.5) In the triopoly cases maximizing annual true social welfare, a non-strategic

offer by one producer (Case 3) results in significantly lower total profit

(-4.02 Me). In this example, the total profit becomes negative for Cases

3 and 4, i.e., the producers cannot recover their investment costs. Note

that such negative profit results are obtained as a result of constraint

(5.1d) that enforces an installed capacity level higher than the peak de-

mand (Υ=1.10). These results are equilibria from a mathematical point

of view, but they are infeasible in practice.


In addition, one conclusion can be drawn pertaining to the monopoly cases

(last two rows) as stated below:

4.1) The investment results of the monopoly cases (Cases 5 and 6) are iden-

tical to those in the triopoly cases, if all producers behave strategically

(Cases 1 and 2) and the total profit is maximized.

Based on the specific conclusions 1-4 derived from Table 5.5, the main

conclusions of this subsection are summarized below:

• As expected, selecting the equilibria that maximize the annual true total

profit results in a comparatively higher total profit, and the capacity of

newly built units is conditioned by constraint (5.1d) related to supply

security. On the other hand, selecting the equilibria that maximize the

annual true social welfare leads to a comparatively higher annual true

social welfare, and the whole investment budget is spent in building base

units.

• If all producers behave strategically, comparatively higher prices and thus

a comparatively higher total profit are obtained for both maximizing

total profit or annual true social welfare.

• The market clearing prices and thus the total profit significantly decrease

if all producers behave in a non-strategic way and the equilibria that

maximize the total profit are selected, while one single non-strategic

producer may lead to a similar situation if the equilibria that maximize

the annual true social welfare are selected.

• Congestion leads to comparatively higher prices and thus total profit. In

addition, the newly built units are located at nodes in which compara-

tively cheaper demands are connected. The location of newly built units

is immaterial if the network does not suffer from congestion.

• If all producers are strategic and the equilibria that maximize the total

profit are selected, the outcomes of the investment equilibria are identical

to those obtained in a monopolistic market where the single producer

behaves strategically.


The results for all cases considered (Cases 1-6) are analyzed in detail in the

following three subsections.

5.9.5 Triopoly Cases Maximizing Total Profit

In this subsection, the triopoly cases maximizing total profit (i.e., Cases 1-4)

are analyzed in detail. The corresponding general results are provided in rows

2-5 of Table 5.5.

In Case 1 (row 2 of Table 5.5), all producers behave strategically, thus

the market is cleared in each demand block at the minimum bid price of the

demands. Since in this case the network does not suffer from congestion, the

location of candidate units is immaterial.

In Case 2 (row 3 of Table 5.5), the market prices at node N1 are identi-

cal to those in Case 1, but due to congestion, the market prices at node N2

corresponding to demand blocks t = t1, t = t2 and t = t3 are comparatively

higher. This leads to a higher total profit. The reason for obtaining compar-

atively higher market prices at node N2 is that demand D2 at node N2 bids

at comparatively higher prices. In the fourth demand block t = t4, the mar-

ket clearing prices in both nodes are identical, because the amount of energy

traded is comparatively low, and thus the transmission line is not congested.

Since the objective in this case is maximizing total profit, node N2 is not

selected to locate the newly built units, because building the newly built units

in that node may prevent congestion. Note also that in this case, a newly built

peak unit leads to a decrease in the annual true social welfare with respect to

Case 1, because its production cost is higher than that of base units.

In Case 3 (row 4 of Table 5.5), the most expensive producer (i.e., Producer

3) offers at its production cost, and thus it always produces and the total profit

decreases. Note that the newly built units do not belong to Producer 3. The

reason is that building those units by a non-strategic producer may decrease

the market clearing prices, thus rendering a comparatively lower total profit.

Note also that the higher capacity of newly built peak units results in lower

annual true social welfare with respect to Cases 1 and 2.

In Case 4 (row 5 of Table 5.5), all producers offer at their production cost,


thus the market prices and the total profit decrease dramatically. The annual

true social welfare in this case is lower than the corresponding to Cases 1 and 2

as a result of building a higher amount of peak capacity, while it is higher than

the annual true social welfare of Case 3, because the comparatively expensive

Producer 3 does not always produce.

Similarly to Case 1, the location of candidate units is immaterial in Cases

3 and 4, because the network does not suffer from congestion.

5.9.6 Triopoly Cases Maximizing Annual True Social

Welfare

In this subsection, the triopoly cases maximizing annual true social welfare

(i.e., Cases 1-4) are analyzed in detail. The corresponding general results are

provided in rows 6-9 of Table 5.5.

Observe that in a triopoly case in which the annual true social welfare is

maximized, the market may not clear at the highest possible price (e.g., the

minimum bid price of the demands). The reason is that in such a case, compar-

atively cheaper units are forced to produce, while may not offer strategically.

To obtain meaningful equilibria, we impose that the problem selects the

highest possible prices. This is achieved by adding in the objective function

(5.29a) the total profit expression (5.31b) multiplied by a small factor (e.g,

0.001) and the annual true social welfare expression (5.32). This way, on one

hand, the units offer strategically (i.e., to achieve the highest possible price),

and on the other hand, the objective (maximizing annual true social welfare)

is not altered due to the comparatively low weight of the annual true total

profit term.

In all triopoly cases maximizing annual true social welfare, the whole avail-

able investment budget (7.5 Me) is spent in building base units, and thus the

same capacity of base units is built, 250 MW. Thus, the annual true social

welfare of all triopoly cases maximizing annual true social welfare is identical

and equal to 20.89 Me.

In Cases 1 and 2 maximizing annual true social welfare (rows 6 and 7 of

Table 5.5), the location of newly built units and the market clearing prices per


demand block are identical to those in the corresponding cases maximizing

total profit (rows 2 and 3 of Table 5.5). The reason is the total profit term

added to achieve strategic offers. Nevertheless, the annual true social welfare

values in these cases are comparatively higher than those in the corresponding

cases maximizing total profit, but with comparatively lower total profit values.

In Case 3 (row 8 of Table 5.5), the most expensive producer (i.e., Producer

3) offers at its production cost. Although Producers 1 and 2 are strategic, their

comparatively cheaper units offer at prices lower than or equal to the offering

prices of Producer 3. Thus, those cheaper units produce and therefore a com-

paratively higher annual true social welfare is obtained. Note that the offers

of strategic units at comparatively lower prices results in clearing the market

at comparatively lower prices with respect to Cases 1 and 2 maximizing an-

nual true social welfare. Therefore, the total profit decreases dramatically, and

the producers cannot recover their investment costs in this case and thus the

equilibrium is unrealistic. Note also that in this case, unlike the corresponding

case maximizing total profit, the non-strategic producer may invest.

In Case 4 (row 9 of Table 5.5), since all producers offer at their production

cost, the market clearing prices and thus the total profit decrease drastically.

Note that the location of newly built units is immaterial in the uncongested

Cases 3 and 4.

5.9.7 Monopoly Cases

The results for the monopoly cases (Cases 5 and 6) given in the last two rows

of Table 5.5 show that the investment decisions made by the single producer

are identical to the investment decisions made by Producers 1, 2 and 3 in

the triopoly cases, provided that these producers behave strategically, and the

total profit is maximized (i.e., Cases 1 and 2 maximizing total profit).

Note that Subsections 5.9.4-5.9.7 discuss the global outcomes for the mar-

ket, i.e., total newly built base and peak units, total newly built capacity, total

profit and annual true social welfare, but the market outcomes for individual

producers are not specifically analyzed. The next subsection presents such

analysis.


Table 5.6: EPEC problem: Three equilibria for Case 1 maximizing total profit(illustrative example).

Results Equilibrium 1 Equilibrium 2 Equilibrium 3

Investment by Producer 1 [MW] 200 (base-N1) No investment No investment

Investment by Producer 2 [MW] No investment 200 (base-N2) 100 (base-N1)

Investment by Producer 3 [MW] No investment No investment 100 (base-N2)

Profit of Producer 1 [Me] 9.55 1.18 1.32



Total newly built capacity [MW] 200 (base) 200 (base) 200 (base)

Total profit [Me] 11.83 11.83 11.83

5.9.8 Investment Results for Each Producer

The investment results for each producer are analyzed in this subsection, in-

cluding the distribution of the total newly built capacity and the total profit

allocation among producers.

As an example, Table 5.6 provides three equilibria for the distribution of

the total newly built capacity and the total profit among producers in Case 1

(maximizing total profit). Columns 2, 3 and 4 characterize alternative equi-

libria. Rows 2-4 represent the investment decisions of producers, row 5-7 their

profit, and the last two rows the total newly built capacity and the total profit.

Although the total newly built capacity and the total profit (last two rows)

do not change across equilibria as shown in Table 5.6, their distributions among

producers may vary. For instance, in the equilibrium of the second column

(equilibrium 1), all newly built units belong Producer 1 (200 MW base capac-

ity), and thus its profit is comparatively higher than that of other producers,

while in the equilibrium in the last column (equilibrium 3), identical base

capacity (100 MW) is built by Producers 2 and 3, and thus they achieve com-

paratively higher profits.

Note that an infinite number of equilibria analogous to those provided in

Table 5.6 can be found through different allocation of 200 MW of base capacity


among producers.

Considering Table 5.6, the following three conclusions can be drawn re-

garding the investment results for each individual producer in the considered

triopoly competition:

5.1) If all producers behave identically (e.g., Cases 1, 2 and 4), the total

outcomes presented in Table 5.5 do not change regardless of which pro-

ducer/producers build the new units. In fact, the distribution of total

newly built capacity among producers and thus their profits can vary,

while the total newly built base/peak unit, the total capacity built, the

market clearing prices, the total profit and the annual true social welfare

do not change. This means that an infinite number of solutions exist.

5.2) This result suggests that the number of potential equilibria is infinite

since an infinite number of solutions exist.

5.3) If all producers do not behave identically (e.g., Case 3), the new units

are not built by the non-strategic producers if maximum total profit is

sought, while in the case of maximizing social welfare, the non-strategic

producers may invest (Subsection 5.9.6).

Summarizing, the main conclusions of this subsection are listed below:

• In a market in which all producers behave identically, the number of

investment equilibria can be infinite.

• In all equilibria obtained, the total newly built base/peak units, the total

capacity built, the market clearing prices, the total profit and the annual

true social welfare do not change. The distribution of the new units

and thus the total profit among producers distinguish the investment

equilibria.

• If both strategic and non-strategic producers participate in the market,

the new units are not build by non-strategic producers if the total profit

is maximized, while all producers may build new units if the annual true

social welfare is maximized (further details in Subsection 5.9.6).


• Based on the conclusions above, the number of potential equilibria can be

infinite for the cases in which i) all producers behave identically for both

total profit and annual true social welfare maximization, ii) all producers

do not behave identically and the annual true social welfare is maximized,

and iii) the total profit is maximized and at least two producers behave

strategically.

5.9.9 Diagonalization Checking

To check if each solution presented in columns 2-4 of Table 5.6 is a Nash

equilibrium, a single-iteration diagonalization checking as described in Section

5.8 is performed.

Through this checking mechanism, we have verified that the solutions ob-

tained by the proposed approach are Nash equilibria.

For example, Equilibrium 3 of Table 5.6 is verified through the following

single-iteration diagonalization checking:

Step 1) The investment decisions of Producers 1 and 2 are respectively fixed

to no investment and 100 MW base capacity.

Step 2) The mixed-integer linear form of MPEC (5.6)-(5.7), i.e., problem (2.10)-

(2.18) presented in Subsection 2.3.6 of Chapter 2 is solved for Producer

3 (maximizing Producer 3’s profit). The result obtained is to build

100 MW of base capacity by producer 3, whose profit is 5.21 Me.

Step 3) The result obtained from Step 2 is equal to the one obtained from the

proposed approach (rows 4 and 7 in the last column of Table 5.6).

Similar results are obtained solving the MPECs associated with Producers

1 and 2, which indicates that the solution presented in the last column of 5.6

is indeed a Nash equilibrium.


0.9 0.95 1 1.05 1.1 1.15 1.2140

180

220

250

Tot

al n

ewly

bui

lt ca

paci

ty(M

W)

0.9 0.95 1 1.05 1.1 1.15 1.2

11.7

11.8

11.9

12

Tot

al p

rofit

(mill

ion

euro

s)

0.9 0.95 1 1.05 1.1 1.15 1.2

19

19.5

20

20.5

Factor ϒ

Ann

ual t

rue

soci

al w

elfa

re(m

illio

n eu

ros)

Total newly built peak capacityTotal newly built base capacity

Figure 5.6: EPEC problem: Total newly built capacity, total profit and annualtrue social welfare as a function of factor Υ (Case 1 maximizing total profit).


5.9.10 Impact of Factor Υ on Generation Investment

Equilibria

Factor Υ enforces a minimum available total capacity to ensure supply security

through constraint (5.1d).

Based on conclusion 1.1) stated in Subsection 5.9.4, this factor is relevant

to the investment equilibria results of all triopoly cases maximizing total profit

as well as to all monopoly cases.

As an example, the impact of factor Υ on the generation investment equi-

libria for Case 1 maximizing total profit is analyzed below and illustrated in

Figure 5.6. This figure depicts the total capacity built (plot 1), the total profit

(plot 2) and the annual true social welfare (plot 3) as a function of factor Υ.

Within the interval 0 ≤ Υ ≤ 1.0, the total capacity built does not change,

thus the total profit and the annual true social welfare do not change. Imposing

a higher factor Υ leads to an increase in the total capacity built (plot 1)

while the total profit decreases (plot 2). In addition, the annual true social

welfare increases due to a higher capacity of newly built base units, but then

it decreases due to investment in peak units, as their production costs are

comparatively higher than those of base units.

Some conclusions are in order.

• Higher values of Υ bring more supply security, while rendering lower

total profit.

• Although more capacity would be built by imposing higher value of Υ,

the annual true social welfare depends on which technology is selected

to build new units. In fact, investment in base units leads to an increase

of the annual true social welfare, while investment in peak units renders

lower annual true social welfare.

• According to the two conclusions above, the market regulator need to

find a desirable value for factor Υ through a trade-off among i) supply

security, ii) total producer profit and iii) annual true social welfare. On

one hand, the market regulator aims to provide a specific level of supply

security. On other hand, a lower total profit may decrease the interest


of producers to invest. In addition, a higher annual true social welfare

indicates a more competitive market.

• Another pertinent conclusion is that providing incentives for producers

to build base units not only renders increasing supply security and annual

true social welfare, but also the total profit of producers might increase

due to lower investment costs.

5.9.11 Impact of the Available Budget on Generation

Investment Equilibria

The investment budget, Kmax, bounds the maximum total capacity built via

constraint (5.1c). According to the conclusion 1.2) stated in Subsection 5.9.4,

this parameter is relevant to the investment equilibria results of all triopoly

cases maximizing annual true social welfare.

In the cases where the annual true social welfare is maximized, the following

results are obtained if constraint (5.1c) is not enforced:

• Comparatively cheaper new base units with a capacity identical to the

peak demand level (i.e., demand of the first block t = t1) would be built.

• The highest demand (i.e., the demand of the first block t = t1) is entirely

supplied by newly built base units, and all comparatively expensive ex-

isting units are decommissioned. The reason for this result is that the

commitment of newly built base units renders the highest annual true

social welfare (lower production costs).

• Note that the results above are clearly not optimal from a profit maximiz-

ing point of view as investing in new base units requires a comparatively

higher investment budget.

Similarly to the results above, a higher investment budget enforced through

constraint (5.1c), results in a higher annual true social welfare, but a lower

total profit. This is shown in Figure 5.7 for Case 1 maximizing annual true

social welfare, which illustrates the total capacity built (plot 1), the total profit


6 7.5 9150

200

250

300T

otal

new

ly b

uilt

capa

city

(MW

)

6 7.5 99

10

11

12

Tot

al p

rofit

(mill

ion

euro

s)

6 7.5 920

20.5

21

21.5

Maximum investment budget(million euros)

Ann

ual t

rue

soci

al w

elfa

re(m

illio

n eu

ros)

Figure 5.7: EPEC problem: Total newly built capacity, total profit and annualtrue social welfare as a function of the available investment budget (Case 1maximizing annual true social welfare).

(plot 2) and the annual true social welfare (plot 3) as a function of available

investment budget.

Note that the results obtained are just valid for the example analyzed,

however the trends are generally valid.

5.10. Case study 255

5.10 Case study

In order to study the scalability of the investment equilibria approach proposed

in this chapter, this section presents the generation investment equilibrium

results for a case study based on the IEEE one-area Reliability Test System

(RTS) [110], presented in Appendix A.

To ease the computational burden, those transmission lines that operate

within a safe margin with respect to their capacities are not explicitly modeled.

Thus, buses 1 to 8 in the Southern area are merged into a single one and buses

17 to 20 in the Northern area into another one. This simplified version of the

IEEE RTS network is illustrated in Figure 5.8.

The flow capacity of transmission lines 11-14, 12-23, 13-23, and 3-24 be-

tween Northern and Southern areas is fixed to 1900 MW for the uncongested

cases, and to 500 MW for the congested ones.

In addition, two buses 9 and 12 in the Southern area as well as two buses

14 and 21 in the Northern area are candidate buses to locate the new units as

shown in Figure 5.8.

5.10.1 Data

In this case study, the demand blocks and the quantity and bid prices of each

demand are considered as stated below:

1) The number of demand blocks t and their weighting factors (σt) are

identical to those provided in the illustrative example (Subsection 5.9.1).

2) Demands for the first block t = t1 are those in [110].

3) Demands for blocks t = t2, t = t3 and t = t4 are those in the first block

t = t1 multiplied by 0.90, 0.75 and 0.65, respectively.

4) Each demand located in the Southern area bids at prices identical to

the prices bid by demand D1, provided in Table 5.2 of the illustrative

example (Subsection 5.9.1); while demands of the Northern area bid at

prices identical to those of demand D2, provided also in Table 5.2.


Figure 5.8: EPEC problem: The simplified version of the IEEE RTS network(case study).

5.10. Case study 257

Table 5.7: EPEC problem: Data pertaining to the existing units (case study).

Existing Type of Capacity Capacity Capacity Production cost Production cost

unit existing[MW]


(k ∈ ΨES) unit [MW] [MW] [e/MWh] [e/MWh]

1-4 Coal 76 30 46 13.46 13.96

5-7 Gas 100 25 75 17.60 18.12

8-9 Gas 120 40 80 18.60 19.03

10-13 Coal 155 55 100 9.92 10.25

14-15 Coal 197 97 100 10.08 10.66

Table 5.8: EPEC problem: Location of the existing units (case study).

ExistingLocation

ExistingLocation

ExistingLocation

unit unit unit

(k ∈ ΨES) [bus] (k ∈ ΨES) [bus] (k ∈ ΨES) [bus]

1 1 6 20 11 5

2 2 7 21 12 18

3 15 8 13 13 19

4 16 9 23 14 10

5 8 10 3 15 22

Table 5.7 gives the data of the existing units. Columns 2 and 3 provide the

type and the capacity of each existing unit. The next four columns characterize

the two production blocks of each unit. In addition, the locations of the existing

units are given in Table 5.8. Note that the total capacity of existing units is

1858 MW.

In this case study, two competition types are considered:

1) Duopoly with two strategic producers, i.e., Producers A and B.

2) Monopoly in which all units are owned by Producer A.

In the duopoly cases considered, Producers A owns all existing units in

the Southern area, while all existing units located in the Northern area belong

to Producer B. Thus, 47.31% of the total capacity of existing units, i.e., 879


MW, belong to Producer A, and the remaining capacity to Producer B, i.e.,

979 MW.

Other relevant data pertaining to this case study are indicated below:

1) The investment options are identical to those specified in Table 5.3 of

the illustrative example (Section 5.9).

2) The maximum capacity of each candidate unit (Xmaxi ) is fixed to 1500

MW.

3) Similarly to the illustrative example (Section 5.9), we consider Υ= 1.10.

Thus, since the total capacity of existing units is 1858 MW, and the peak

demand is 2850 MW, new capacity of at least 1277 MW is required.

4) The available investment budget (Kmax) is e 45 million.

5) The values of all dual variables φSDyt , ∀y, ∀t, are arbitrarily fixed to 15σt,

∀t.

5.10.2 Results of Generation Investment Equilibria

Table 5.9 gives the results of generation investment equilibria. The structure

of the columns of this table is similar to that of Table 5.5 (Section 5.9). Rows

3-5 pertain to uncongested network cases, while rows 7-9 refer to congested

network cases.

The results of generation investment equilibria for this case study are con-

sistent to those obtained for the illustrative example (Section 5.9). The main

conclusions that can be drawn from the results in Table 5.9 are stated below:

1) The total capacity built in both duopoly cases maximizing total profit

(rows 3 and 7) and in both monopoly cases (rows 5 and 9) are identical

and equal to 1277 MW. However, the quantities of base and peak ca-

pacities may be different across the uncongested and congested duopoly

cases.

2) A total available base capacity of 1500 MW is built in both duopoly

cases maximizing annual true social welfare (rows 4 and 8), while no

5.10.Case

study

259

Table 5.9: EPEC problem: results of generation investment equilibria (case study).

Uncongested network results

Case

Total newly Total newly Total newly Investment λtn λtn λtn λtn Total Annual true CPU

built base built peak built capacity cost for t = t1 for t = t2 for t = t3 for t = t4 profit social welfare time

units [MW] units [MW] [MW] [Me] [e/MWh] [e/MWh] [e/MWh] [e/MWh] [Me] [Me] [s]

Duopoly838.5 438.5 1277 27.79

20 (NA) 19 (NA) 18 (NA) 17 (NA)118.18 163.12 164

(Max TP∗) 20 (SA‡) 19 (SA) 18 (SA) 17 (SA)

Duopoly1500.0 0 1500 45.00

20 (NA) 19 (NA) 18 (NA) 17 (NA)113.82 175.97 945

(Max ATSW†) 20 (SA) 19 (SA) 18 (SA) 17 (SA)

Monopoly 838.5 438.5 1277 27.7920 (NA) 19 (NA) 18 (NA) 17 (NA)

118.18 163.12 0.8

20 (SA) 19 (SA) 18 (SA) 17 (SA)

Congested network results

Duopoly1130.0 147.0 1277 34.78

22 (NA) 21 (NA) 20 (NA) 19 (NA)125.85 169.43 3883

(Max TP) 20 (SA) 19 (SA) 18 (SA) 17 (SA)

Duopoly1500.0 0 1500 45.00

22 (NA) 21 (NA) 20 (NA) 19 (NA)120.53 175.97 15941

(Max ATSW) 20 (SA) 19 (SA) 18 (SA) 17 (SA)

Monopoly 1130.0 147.0 1277 34.7822 (NA) 21 (NA) 20 (NA) 19 (NA)

125.85 169.43 1.4

20 (SA) 19 (SA) 18 (SA) 17 (SA)

∗ TP: Total profit; †ATSW: Annual true social welfare

NA: Northern area; ‡SA: Southern area


peak capacity is built. Note that in such cases, the available budget

(45.00 Me) is fully spent.

3) In the uncongested cases (rows 3-5), the market is cleared in each demand

block at the minimum bid price of the demands, and market clearing

prices in both areas are identical.

4) In the congested cases (rows 7-9), the market is cleared at compara-

tively higher prices in the Northern area with respect to prices in the

uncongested cases (rows 3-5). The reason is that demands located at the

Northern area bid at comparatively higher prices. These comparatively

higher market prices result in a comparatively higher total profit.

5) As expected, each duopoly case maximizing total profit (rows 3 and 7)

achieves a comparatively higher total profit with respect to the corre-

sponding case maximizing annual true social welfare (rows 4 and 8), but

with comparatively lower annual true social welfare.

6) Since in every duopoly case maximizing annual true social welfare (rows

4 and 8) the whole investment budget is spent in building base units, the

same base capacity is built (1500 MW), and thus the annual true social

welfare in such cases is identical and equal to 175.97 Me.

7) The results of each monopoly case (rows 5 and 9) are identical to the

corresponding duopoly cases maximizing total profit (rows 3 and 7).

In the uncongested cases (rows 3-5), the location of new units is immaterial,

while in the congested ones (rows 7-9), the majority of newly built units are

located at either bus 9 or at bus 12 (in the Southern area). This is done to

create congestion in the network, because congestion leads to comparatively

higher market clearing prices and thus a comparatively higher total profit.

Similarly to the results of the illustrative example (Section 5.9), in every

case considered, there can be an infinite number of solutions resulting from

allocating the total capacity built and thus the total profit between Producers

A and B, while the total newly built base/peak units, the total capacity built,

5.11. Computational Considerations 261

the market clearing prices, the total profit and the annual true social welfare

do not change for such solutions.

Moreover, we have verified that the investment solutions of the proposed

approach are Nash equilibria using a single-iteration diagonalization procedure

as explained in Section 5.8.


Optimization problem (5.29) is solved for triopoly cases in the illustrative

example presented in Section 5.9 and for duopoly cases in the case study

presented in Section 5.10.

In addition, the mixed-integer linear form of MPEC (5.6)-(5.7), i.e., MILP

problem (2.10)-(2.18) presented in Subsection 2.3.6 of Chapter 2 is solved for

monopoly cases of both Sections 5.9 and 5.10.

All cases are solved using CPLEX 12.1 [43] under GAMS [42] on a Sun

Fire X4600M2 with 8 Quad-Core processors clocking at 2.9 GHz and 256 GB

of RAM.

The computational times required for solving the proposed model are given

in the last column of Table 5.9. Note that the optimality gap for all cases is

set to zero.

The next subsection presents some computational conclusions drawn.

5.11.1 Computational Conclusions

The following five conclusions regarding computational burden can be drawn:

1) Each oligopolistic case needs a significantly higher CPU time than any

monopoly case.

2) The CPU times needed for solving the duopoly cases maximizing annual

true social welfare are comparatively higher than the CPU times required

for solving the same duopoly cases but maximizing total profit.

3) Congested cases require higher computational time than uncongested

cases.


4) The proposed approach for identifying generation investment equilibria

is tractable and the computational times required are reasonable. For

example, the computational times needed to solve any case analyzed in

Section 5.10 (large-scale case study) are less than 4.43 hours.

5) Incorporating uncertainty into the proposed model may dramatically in-

crease the computational burden and thus other solution techniques e.g.,

decomposition and parallelization, would be required.

5.11.2 Selection of values for φSDyt

Pursuing linearity, dual variables φSDyt , ∀y, ∀t are parameterized as explained

in Subsection 5.5.3.2. To select appropriate values for φSDyt , numerical studies

show that unlike [117], the equilibria identified do not change with the values

of such variables, but the computational burden significantly does. Thus,

the main criterion for selecting values for these variables is to decrease the

computational burden, which is done by trial and error.

As described in Sections 5.9 and 5.10, the values of all dual variables φSDyt ,

∀y, ∀t are fixed to 5σt, ∀t in the illustrative example (Section 5.9), and they

are fixed to 15σt, ∀t in the case study (Section 5.10). The reason of such

selections is that the trial and error process show that such selections result in

the minimum computational time.

5.11.3 Suggestions to Reduce the Computational Bur-

den

To reduce the computational burden in comparatively larger case studies, the

following suggestions may prove effective.

1) To simplify the transmission network by merging those nodes connected

through transmission lines that operate within safe margins.

2) To appropriately select by trial and error the values for the parameterized

dual variables φSDyt , ∀y, ∀t as explained in Subsection 5.11.2.

5.12. Ex-post Analysis 263

3) To suitably initialize the investment equilibrium problem using the re-

sults obtained from a diagonalization algorithm.

5.12 Ex-post Analysis

Note that the main purpose of this chapter is to mathematically identify mar-

ket equilibria to characterize all the potential market outcomes. Additionally,

an ex-post engineering and economic analysis is required to identify which of

these equilibria are meaningful and may actually occur in practice.

Nevertheless, a subset of the equilibria obtained by the proposed approach

can be easily eliminated in the ex-post analysis. For example, infinitely many

equilibria are obtained in Subsection 5.9.8 corresponding to different alloca-

tions of the same quantity of base capacity among producers. In the ex-post

economic analysis, most of these equilibria may be eliminated considering the

available investment budget of each producer. In addition, the producers’

profitability and their investment actions during the recent years may provide

insights into which equilibria are most likely to be realized in practice.


5.13.1 Summary

This chapter proposes a methodology to characterize generation investment

equilibria in a pool-based network-constrained electricity market, where the

producers behave strategically. To this end, the following steps are carried

out:

Step 1) Similarly to Chapters 2 to 4, the investment problem of each strategic

producer is represented using a hierarchical (bilevel) model, whose

upper-level problem determines the optimal production investment

(capacity and location) and the supply offering curves to maximize

its profit, and whose several lower-level problems represent different

market clearing scenarios, one per demand block. This bilevel model


explicitly considers stepwise supply function offers, which constitutes

a more accurate description of the functioning of real-world electricity

markets if compared with other imperfect competition models such as

Cournot, Bertrand or conjectural variation.

Step 2) The single-producer bilevel model formulated in Step 1 is transformed

into a single-level MPEC by replacing the lower-level problems with

their optimality conditions resulting from the primal-dual transforma-

tion. The resulting MPEC is continuous, but non-linear due to the

product of variables in the strong duality equalities.

Step 3) The joint consideration of all producer MPECs, one per producer,

constitutes an EPEC, whose solution identifies the market equilibria.

Step 4) To identify EPEC solutions, the optimality conditions associated with

the EPEC, i.e., the optimality conditions of all producer MPECs, are

derived. To this end, each MPEC obtained in Step 2 is replaced by

its KKT optimality conditions. The resulting optimality conditions

of all MPECs, which are the optimality conditions of the EPEC, is a

collection of non-linear systems of equalities and inequalities.

Step 5) The optimality conditions associated with the EPEC obtained in Step

4 are linearized without approximation through three procedures: i)

linearizing the complementarity conditions, ii) parameterizing the op-

timality conditions in the dual variables corresponding to the strong

duality equalities, and iii) replacing the strong duality equalities with

their equivalent complementarity conditions. This linearization re-

sults in a mixed-integer and linear system of equalities and inequalities

characterizing the EPEC.

Step 6) To detect meaningful equilibria, an auxiliary mixed-integer linear op-

timization problem is formulated, whose constraints are the mixed-

integer linear conditions obtained in Step 5 and whose objective func-

tion is either a linear form of the total profit of all producers or a

linear form of the annual true social welfare.


Step 7) The auxiliary mixed-integer linear optimization problem formulated

in Step 6 is solved and a number of solutions (stationary points) are

obtained. To verify whether or not each solution obtained is a Nash

equilibrium, a single-step diagonalization checking is used.

To numerically validate the proposed methodology, a two-node illustrative

example and a realistic case study based on the IEEE reliability test system

(RTS) are examined and the results obtained are reported and discussed.

The main general conclusions that can be drawn from the study reported in

this chapter are listed in Subsection 5.13.2. In addition, the results reported al-

low drawing some regulatory observations included in Subsection 5.13.3, useful

for the market regulator to promote a market that operates as close as possible

to perfect competition.

5.13.2 General Conclusions

This subsection presents the main general conclusions that can be drawn from

the study reported in this chapter.

1) The approach developed in this chapter can be successfully implemented

to identify meaningful generation investment equilibria.

2) The number of equilibria can be infinite for the cases in which i) all

producers behave identically and total profit or annual true social welfare

is maximized, ii) all producers do not behave identically and annual true

social welfare is maximized, and iii) total profit is maximized and at least

two producers behave strategically.

3) As expected, a comparatively higher total profit is obtained if the total

profit is maximized. In addition, the capacity of newly built units is

affected by constraints (5.1d) related to supply security. A comparatively

higher factor Υ may lead to an increase in the total capacity built, while

the total profit may decrease.

4) As expected, a comparatively higher annual true social welfare, but a

comparatively lower total profit are obtained if the annual true social


welfare is maximized. In this case, the whole investment budget is spent

in building units with comparatively lower production costs (base units).

5) If all producers behave strategically, comparatively higher prices and

thus a comparatively higher total profit are obtained.

6) The investment results of a monopolistic market in which its single

producer behave strategically are identical to those pertaining to an

oligopolistic market with strategic producers if the total profit is maxi-

mized.

7) The total profit significantly decreases if all producers offer at their pro-

duction costs and the equilibria that maximize the total profit are se-

lected. In addition, one non-strategic producer may drive the market to

a similar situation (i.e., a significant decrease of the total profit) if the

equilibria that maximize the annual true social welfare are selected.

8) Congestion results in comparatively higher prices and thus higher total

profit. The location of newly built units is immaterial if the network

does not suffer from congestion. However, the new units are located at

nodes where comparatively cheaper demands are connected if congestion

occurs. This is done to achieve comparatively higher market clearing

prices.

9) The solutions of the proposed approach are Nash equilibria as verified

using a single-iteration diagonalization checking.

10) Although the distribution of new units among producers and thus the

profit of each producer may be different across equilibria, the total newly

built base/peak units, the total capacity built, the market clearing prices,

the total profit for all producers and the annual true social welfare do

not change.

11) If non-strategic and strategic producers compete in the market, the new

units are not built by non-strategic producers if the total profit is max-

imized, while non-strategic producers may decide to build new units if

the annual true social welfare is maximized.


5.13.3 Regulatory Conclusions

The analysis performed in the illustrative example (Section 5.9) and the case

study (Section 5.10) provide insights into which market configuration favors

increasing competitiveness in the market.

For instance, it is shown that network congestion generally diminishes the

competitiveness of the market and thus a well-designed transmission system

may contribute to increase the market competitiveness and thus the annual

true social welfare. Similarly, as the number of strategic producers increases,

market outcomes tend to be more competitive so that it is appropriate to find

ways to increase the number of producers. These type of findings can be of

interest for a market regulator.

Moreover, the numerical results of the studies reported in this chapter allow

drawing some policy observations for the market regulator to promote a market

that operates as close as possible to perfect competition. These observations

are:

1) To provide incentives for non-strategic producers to invest since such

producers may increase the competitiveness of the market.

2) To ensure that the network is adequately reinforced/expanded so that

congestion is unlikely. Congestion may decrease the competitiveness of

the market.

3) To provide incentives for all producers (either strategic or non-strategic)

to build comparatively cheaper base units since dispatching such units

leads to an increase in the true social welfare.

4) To make sure that enough investment funds are available for producers.

5) To select the minimum requirement for total investment through a trade-

off among i) supply security, ii) total profit for producers and iii) annual

true social welfare. On one hand, higher new capacity built brings more

supply security, while it may render lower total profit for producers.

On the other hand, the annual true social welfare highly depends on

which technology is selected to build new units. Thus, an appropriate


investment mix that results in high enough annual true social welfare

should be achieved.

6) To enforce mechanisms to avoid collusion among producers so that mo-

nopolistic behavior does not occur.

Chapter 6

Summary, Conclusions,

Contributions and Future

Research

In this closing chapter a summary of the thesis work is first provided. Then,

a list of relevant conclusions drawn from the thesis work is presented. Next,

the contributions of this thesis are stated. Finally, some proposals for future

research are suggested.

6.1 Thesis Summary

In this dissertation, we address the two topics below:

1. Development of a procedure based on optimization and complementarity

for a strategic producer to address its generation investment problem

within a network-constrained electricity market (Chapters 2-4).

2. Development of a non-heuristic methodology based on optimization and

complementarity to identify generation investment equilibria in a network-

constrained oligopolistic electricity market (Chapter 5).

The next two subsections summarize the thesis work pertaining to these

two topics.

269

270 6. Summary, Conclusions, Contributions and Future Research

6.1.1 Strategic Producer Investment

The first problem addressed in this thesis is the strategic investment of a power

producer.

We propose a bilevel model to represent the strategic behavior of a pro-

ducer competing with its rival producers in an electricity market. The purpose

of this model is investment decision-making. This bilevel model consists of an

upper-level problem and a set of lower-level problems. The upper-level problem

pursues maximizing the expected profit and determines the strategic decisions

of the producer, i.e., its strategic investment actions, and its strategic pro-

duction offers. The lower-level problems are considered below. In Chapters 2

and 3 only the pool is considered, thus, the set of lower-level problems repre-

sent just the pool clearing, one problem per demand block and scenario. In

Chapter 4 that also considers the futures market, the futures base auction, the

futures peak auction and the pool are represented through the set of lower-level

problems. One lower-level problem refers to the futures base auction, other

one refers to the futures peak auction, and the remaining lower-level problems

represent the pool, one per demand block and scenario.

To solve this bilevel model, two alternative approaches are developed in

this thesis work, namely:

1. Direct solution approach (Chapters 2 and 4).

2. Benders’ decomposition approach (Chapter 3).

In the first approach, the proposed bilevel model is solved considering si-

multaneously all involved scenarios; however, these scenarios are considered

separately in the second approach based on Benders’ decomposition. These

approaches are summarized in the following.

1. Direct Solution Approach (Chapters 2 and 4):

In this approach, the bilevel problem is transformed into a single-level

mathematical program with equilibrium constraints (MPEC) by replac-

ing each lower-level problem with its Karush-Kuhn-Tucker (KKT) con-

ditions. This transformation is illustrated in Figure 1.5 of Chapter 1,

and its mathematical details are provided in Section B.2 of Appendix B.

6.1. Thesis Summary 271

The resulting MPEC can be recast as a mixed-integer linear program-

ming problem (MILP) solvable by commercially available software. To

linearize the MPEC, an exact linearization approach based on the strong

duality theorem and some of the KKT equalities is used. Additionally,

a computationally efficient binary expansion approach explained in Sub-

section B.5.2 of Appendix B is used in Chapter 4.

One important observation regarding the direct solution approach is that

all involved scenarios are considered simultaneously. Therefore, a large

number of scenarios may result in high computational burden and even-

tual intractability.

To illustrate the performance of this direct approach, numerical results

pertaining to a small example and a realistic case study are reported and

discussed.

2. Benders’ Decomposition Approach (Chapter 3):

To tackle the computational problem of the direct solution approach in

cases with many scenarios, a methodology based on Benders’ decompo-

sition is proposed in Chapter 3. The objective is to make the proposed

bilevel model tractable even if many scenarios are used to describe un-

certain parameters.

To this end, we first perform a detailed numerical analysis to show that if

the strategic producer offers via supply functions and a sufficiently large

number of scenarios is considered, the expected profit of the strategic

producer as a function of investment decisions has a convex enough en-

velope. Thus, an effective implementation of Benders’ decomposition is

possible.

Next, the bilevel model considered is transformed into two alternative

MPECs:

• One MPEC is mixed-integer linear and is obtained by replacing the

lower-level problems with their corresponding KKT conditions. In

this case, the complementarity conditions are linearized using auxil-


iary binary variables (as explained in Subsection B.5.1 of Appendix

B).

• The second MPEC is continuous but non-linear as it is obtained

by replacing each lower-level problem with its primal-dual optimal-

ity conditions (Section B.2 of Appendix B). The non-linearities are

products of variables in the strong duality equality.

If investment decisions (complicating variables) are fixed to given values,

each of the two alternative MPECs decomposes by scenario. The mixed-

integer linear MPEC of each scenario (denoted as auxiliary problem)

is solved to attain its optimal solution. Such optimal solution allows

transforming the non-linear MPEC into a continuous linear program-

ming problem (Benders’ subproblem) that provides the sensitivities of

the expected profit with respect to investment decisions. In turn, these

sensitivities are used to build Benders’ cut needed in Benders’ master

problem.

A numerical comparison between the direct solution and the Benders’

decomposition approaches based on a realistic case study shows that the

proposed decomposition approach is accurate and efficient even if many

scenarios are considered.

6.1.2 Investment Equilibria

Chapter 5 of this dissertation proposes a non-heuristic methodology based on

optimization and complementarity to identify generation investment equilibria

in a network-constrained electricity pool. In each equilibrium, no producer can

increase its profit by changing unilaterally its strategies.

The importance of the proposed equilibrium identification methodology is

that it allows the market regulator to gain insight into the investment behav-

ior of the strategic producers and the generation investment evolution. Such

insight may allow the market regulator to better design market rules, which

in turn may contribute to increase the competitiveness of the market and to

stimulate optimal investment in generation capacity.

6.1. Thesis Summary 273

First, to model the strategic behavior of each single-producer, a similar

bilevel model to the one used in Chapters 2 and 3 is considered. The upper-

level problem of such model decides on the optimal investment and offering

functions to maximize the producer’s profit, while several lower-level problems

represent the pool operation per demand block. For the sake of simplicity,

uncertainties are not considered in this model and the futures market is not

modeled.

Second, each single-producer bilevel model is transformed into an MPEC by

replacing each lower-level problem with its primal-dual optimality conditions

(as explained in Section B.2 of Appendix B).

The joint consideration of all producer MPECs, one per producer, con-

stitutes an equilibrium problem with equilibrium constraints (EPEC), whose

solutions identify the investment equilibria. Mathematical details on the pro-

posed EPEC are provided in Section 1.7 of Chapter 1 and in Section B.3 of

Appendix B.

To search for the EPEC solutions, the optimality conditions associated with

the EPEC need to be derived. To this end, each producer MPEC is replaced

by its KKT optimality conditions. The joint consideration of the KKTs of all

MPECs is a collection of non-linear systems of equalities and inequalities that

constitutes the optimality conditions of the EPEC.

These optimality conditions are linearized without approximation using the

three techniques below:

1. Linearizing the complementarity conditions through auxiliary binary vari-

ables.

2. Parameterizing the optimality conditions using as parameters the dual

variables corresponding to the strong duality equalities, which is possible

since MPEC constraints are non-regular.

3. Replacing the strong duality equalities by their equivalent complemen-

tarity conditions, which are in turn linearized using auxiliary binary vari-

ables.


These linearizations render a mixed-integer and linear system of equalities and

inequalities characterizing the EPEC.

To search for meaningful equilibria, an auxiliary mixed-integer linear opti-

mization problem is formulated, whose constraints are the mixed-integer linear

conditions characterizing the optimality conditions of the EPEC and whose

objective function is either the total producer profit or the annual true social

welfare. These objective functions are both linear.

Finally, a single-iteration diagonalization procedure is used to check whether

or not each solution obtained is indeed a Nash equilibrium.

To numerically validate the proposed methodology, a small-scale illustrative

example and a realistic case study are considered. The obtained results are

reported and discussed.

6.2 Conclusions

At the end of each chapter of this thesis work, several conclusions are presented

that are specific to the analysis in that chapter. In this section, the most

important conclusions of this dissertation are listed.

Relevant conclusions related to the strategic generation investment problem

(Chapters 2-4) are:

1. The two proposed approaches (direct solution and Benders’ decompo-

sition) attain the optimal investment solution for a strategic producer

and present a robust computational behavior. The results obtained from

both approaches are identical; however, the computational time required

using Benders’ decomposition is significantly lower than that required by

the direct solution approach.

2. If many scenarios are taken into account and the considered producer

behaves strategically, its expected profit as a function of its investment

decisions has a convex enough envelope. This justifies the use of Benders’

decomposition.

3. The total capacity to be built by the strategic producer directly depends

6.2. Conclusions 275

on the market regulator policy regarding supply security. Imposing a

higher value for the minimum available capacity leads to a comparatively

higher investment level, but a comparatively smaller expected profit for

the investor.

4. Transmission congestion results in higher locational marginal prices (LMPs)

than those in an uncongested case. This leads generally to an increase

in the capacity built by the strategic producer.

5. Offering at marginal cost by the considered producer (non-strategic offer-

ing) results in comparatively lower investment and comparatively lower

expected profit with respect to a case with strategic offers.

6. Higher uncertainty on rival producers (i.e., rival offering and rival invest-

ment uncertainties) and demand growth results in lower expected profit,

and investment in smaller units.

7. The futures market makes a difference for the strategic producer in its

expected profit and both its offering and investment decisions as stated

below:

a) A futures market with comparatively low energy trading with re-

spect to the pool, and with the possibility of arbitrage results in

higher expected profit for the strategic producer. This stimulates

the strategic producer to build a comparatively larger number of

new units. In the case of no arbitrage, the futures market is not

profit/ investment effective.

b) A futures market with comparatively high energy trading with re-

spect to the pool results in lower expected profit for the strategic

producer. In this case, the strategic producer does not have incen-

tives to build new units. However, if the strategic producer is forced

to build new units due to supply security constraints imposed by

the market regulator, its expected profit decreases.

Relevant conclusions related to generation investment equilibria (Chapter

5) are:


8. The EPEC approach developed in this thesis can be successfully used to

identify meaningful generation investment equilibria.

9. The number of generation investment equilibria is infinite if:

a) All producers behave strategically and either the total profit or the

annual true social welfare is maximized.

b) All producers do not behave strategically, but the annual true social

welfare is maximized.

c) Total profit is maximized and at least two producers behave strategi-

cally.

10. Although the distribution of newly built units among producers and thus

the profit of each individual producer may be different across equilibria,

the global outcomes of the equilibria do not change. Such outcomes

are the total newly built base/peak units, the total capacity built, the

market clearing prices, the total profit for the producers and the annual

true social welfare.

11. If non-strategic and strategic producers compete in the market, new units

are not built by non-strategic producers if the total profit is maximized.

However, non-strategic producers may build new units if the annual true

social welfare is maximized.

12. As expected, a comparatively higher total profit is obtained if the total

profit is maximized. In addition, the capacity of newly built units is

affected by the market regulator’s policy related to supply security. A

comparatively higher value for the minimum available capacity may lead

to an increase in the total capacity built, while the total profit for all

producers may decrease.

13. As expected, a comparatively higher annual true social welfare, but a

comparatively lower total profit are obtained if the annual true social

welfare is maximized. In this case, the whole investment budget is spent

in building units with comparatively lower production costs.

6.3. Contributions 277

14. The investment results of a monopolistic market in which its single

producer behaves strategically are identical to those pertaining to an

oligopolistic market with strategic producers if the total profit is maxi-

mized.

15. Transmission congestion results in comparatively higher clearing prices

and thus higher total profit. The location of newly built units is im-

material if the network does not suffer from congestion. However, new

units are located at nodes where comparatively cheaper demands are con-

nected if congestion occurs. This is intended to achieve comparatively

higher market clearing prices.

6.3 Contributions

The main contributions of the thesis work are summarized below:

1. To propose a generation investment model for a strategic producer com-

peting in a network-constrained pool with supply function offers. This

model is able to optimally locate candidate units throughout the network,

and to select the best production technologies. The resulting model is

an MPEC.

2. To develop an investment model for a strategic producer competing in a

futures market and a pool with supply function offers, and to analyze the

impact of futures market auctions on investment decisions. This model

also renders an MPEC.

3. To represent the arbitrage between the futures market auctions and the

pool, and to analyze its influence on the expected profit and the invest-

ment decisions of the strategic producer.

4. To linearize the non-linear terms of the resulting MPECs using the pro-

cedures below:

a) An exact linearization technique based on the strong duality theorem

and some of the KKT equalities. This technique provides an exact


linear equivalent for the term “price”×“production” that appears in

the formulation of the strategic producer’s profit.

b) An approximate linearization technique: binary expansion. The ex-

act linearization procedure above (item a) cannot provide a linear

equivalent for the non-linear profit term “price”×“production” if the

futures market is considered. Therefore, the binary expansion ap-

proach is used in such case to approximately linearize the profit term.

c) An exact procedure for linearizing the complementarity conditions

using auxiliary binary variables.

5. To numerically show that the expected profit of a strategic producer as a

function of its investment decisions has a “sufficiently convex” envelope,

provided that (a) the producer behaves strategically, and (b) a large

enough number of scenarios is considered.

6. To develop a Benders’ decomposition approach as an alternative method-

ology for solving the strategic investment problem. This approach is com-

putationally efficient even if a large number of scenarios is considered to

describe uncertain data. In this approach, two MPECs are derived and

used, namely:

a) Auxiliary problem: A mixed-integer linear MPEC that is derived us-

ing the KKT conditions, and then linearized through auxiliary binary

variables.

b) Benders’ subproblem: A continuous and linear MPEC that is derived

using the primal-dual transformation, and then linearized using the

optimal solution of the auxiliary problem. This MPEC provides sen-

sitivities of the expected profit with respect to investment decisions.

7. To develop a methodology for representing the interactions among a num-

ber of strategic investors in a network-constrained oligopolistic market

as a game-theoretic model.

8. To identify generation investment equilibria through the formulation and

solution of an EPEC.

6.3. Contributions 279

9. The publication of five papers directly related to the thesis work in JCR

(Thompson Reuters) journals:

a) S. J. Kazempour, A. J. Conejo, and C. Ruiz. Strategic generation

investment using a complementarity approach. IEEE Transactions on

Power Systems, 26(2):940-948, May 2011. JCR 5-year impact factor:

3.258, position 27 of 245 (quartile Q1) in Engineering, Electrical and

Electronic.

b) S. J. Kazempour, and A. J. Conejo. Strategic generation investment

under uncertainty via Benders decomposition. IEEE Transactions on

Power Systems, 27(1):424-432, Feb. 2012. JCR 5-year impact factor:


Electronic.

c) S. J. Kazempour, A. J. Conejo, and C. Ruiz. Strategic generation

investment considering futures and spot markets. IEEE Transactions

on Power Systems, 27(3):1467-1476, Aug. 2012. JCR 5-year impact

factor: 3.258, position 27 of 245 (quartile Q1) in Engineering, Elec-

trical and Electronic.

d) S. J. Kazempour, A. J. Conejo, and C. Ruiz. Generation invest-

ment equilibria with strategic producers Part I: Formulation. IEEE

Transactions on Power Systems, In press. JCR 5-year impact factor:


Electronic.

e) S. J. Kazempour, A. J. Conejo, and C. Ruiz. Generation invest-

ment equilibria with strategic producers Part II: Case studies. IEEE

Transactions on Power Systems, In press. JCR 5-year impact factor:


Electronic.

10. The publication of an additional paper related to the thesis work:

f) C. Ruiz, A. J. Conejo and S. J. Kazempour. Equilibria in futures and

spot electricity markets. Electric Power Systems Research, 84(1):1-


9, Mar. 2012. JCR 5-year impact factor: 1.726, position 82 of 245

(quartile Q2) in Engineering, Electrical and Electronic.

6.4 Suggestions for Future Research

This concluding section suggests some relevant lines of future research. Sug-

gestions related to generation investment (Chapters 2-4) are:

1. To use a multi-stage investment decision-making approach instead of

the static one used in this thesis work. The multi-stage approach pro-

vides more accurate investment decisions, but at the cost of potential

intractability. Thus, a decomposition technique may be required.

2. To incorporate security constraints into the market clearing problem.

Such constraints ensure system security against a set of plausible contin-

gencies, i.e., generators and transmission line outages.

3. To consider consumers that behave strategically through their demand

function bids. The smart grids technology brings more flexibility for

consumers to become price sensitive and strategic.

4. To develop an analytical sensitivity analysis tool. Such sensitivity anal-

ysis tool allows the strategic producer to assess the impact of rival pro-

ducer parameters, demand parameters, investment costs and other pa-

rameters on investment decisions.

5. To consider other electricity trading floors other than those considered

in this thesis, e.g., bilateral contracts, ancillary services, and futures

monthly and weekly auctions. It is relevant to analyze the impact of

such trading floors on investment decisions.

6. To develop a robust generation investment model and to compare its

results with those obtained with a stochastic programming model.

7. To include in the objective function a risk term for the profit (e.g., con-

ditional value at risk, CVaR) and to analyze the impact of risk aversion

on investment decisions.

6.4. Suggestions for Future Research 281

8. To consider renewable sources and energy storage systems, e.g., pumped-

storage plants, as investment options.

Additionally, suggestions related to generation investment equilibria (Chap-

ter 5) are:

9. All suggestions 1-8 above related to the generation investment problem

of a strategic producer are also appropriate for the generation investment

equilibrium problem.

10. To evaluate the impact of uncertainties on generation investment equi-

libria using a stochastic framework. Since incorporating uncertainty may

dramatically increase the computational burden of the resulting model,

decomposition and parallelization may be required.

Appendix A

IEEE Reliability Test System:

Transmission Data

A description of a 24-node network based on the single-area IEEE Reliability

Test System (RTS) [110] is presented in this Appendix. This test system is

used in Chapters 2 to 5.

The considered network is depicted in Figure A.1, and includes two areas,

i.e., the Southern one (buses 1 to 13) and the Northern one (buses 14 to

24), interconnected by four tie-lines 3-24, 11-14, 12-23 and 13-23. This area-

splitting is used in the case studies of this dissertation to analyze the impact of

transmission congestion on investment decisions. Note that in the considered

system, the double-circuit transmission lines in the original reference [110] are

replaced with equivalent single-circuit ones.

Active power losses are not taken into account in this dissertation, and thus

the line resistances are ignored. Table A.1 provides the data for the reactance

and the transmission capacity of the 34 lines of the considered system (Figure

A.1).

Note that the technical data for the generating units and demands are

provided in the case study section of each chapter. Such data may vary across

chapters to illustrate different features of the proposed models.

283

284 A. IEEE Reliability Test System: Transmission Data

Figure A.1: IEEE reliability test system: Network.

285

Table A.1: IEEE Reliability Test System: Reactance (p.u. on a 100 MW base)and capacity of transmission lines.

From bus To busReactance Capacity

(p.u.) (MW)

1 2 0.0146 175

1 3 0.2253 175

1 5 0.0907 175

2 4 0.1356 175

2 6 0.2050 175

3 9 0.1271 175

3 24 0.0840 400

4 9 0.1110 175

5 10 0.0940 175

6 10 0.0642 175

7 8 0.0652 175

8 9 0.1762 175

8 10 0.1762 175

9 11 0.0840 400

9 12 0.0840 400

10 11 0.0840 400

10 12 0.0840 400

11 13 0.0488 500

11 14 0.0426 500

12 13 0.0488 500

12 23 0.0985 500

13 23 0.0884 500

14 16 0.0594 500

15 16 0.0172 500

15 21 0.0249 1000

15 24 0.0529 500

16 17 0.0263 500

16 19 0.0234 500

17 18 0.0143 500

17 22 0.1069 500

18 21 0.0132 1000

19 20 0.0203 1000

20 23 0.0112 1000

21 22 0.0692 500

Appendix B

Mathematical Background

This appendix provides mathematical background material relevant to this

thesis that includes:

1. Bilevel model used in Chapters 2 to 5.

2. Two alternative procedures used in Chapters 2 to 5 for deriving the

optimality conditions associated with a linear optimization problem.

3. Mathematical program with equilibrium constraints (MPEC) used in

Chapters 2 to 5.

4. Equilibrium problem with equilibrium constraints (EPEC) used in Chap-

ter 5.

5. Benders’ decomposition algorithm used in Chapter 3.

6. Complementarity linearization used in Chapters 2 to 5.

7. Binary expansion approximation used in Chapter 4.

B.1 Bilevel Model

This section presents the mathematical description for a hierarchical (bilevel)

optimization model [41].

287

288 B. Mathematical Background

Bilevel model (B.1)-(B.2) consists of an upper-level problem (B.1) and a

set of lower-level problems (B.2). Objective function (B.1a) of the upper-level

problem is constrained by the upper-level equality and inequality constraints

(B.1b)-(B.1c), and a set of n lower-level problems (B.2).

Regarding the notation used in the bilevel problem (B.1)-(B.2), the follow-

ing observations are relevant:

1. Symbols with superscript U pertain to the upper-level problem. For

example, variable vector xU includes the set of primal variables belonging

to the upper-level problem.

2. Symbols with superscript L refer to the lower-level problems. For exam-

ple, variable vector xLi includes the set of primal variables of lower-level

problem i.

3. Dual variable vectors associated with the lower-level problems are indi-

cated at the corresponding equations following a colon. For example,

vectors λLi and µLi are respectively the equality and inequality dual vari-

able vectors corresponding to the lower-level problem i.

MinimizeΞU

fU(xU, xL1 , ..., xLi , ..., x

Ln, λ

L1 , ..., λ

Li , ..., λ

Ln, µ

L1 , ..., µ

Li , ..., µ

Ln) (B.1a)

subject to:

0) Upper-level equality and inequality constraints:

hU(xU, xL1 , ..., xLi , ..., x

Ln, λ

L1 , ..., λ

Li , ..., λ

Ln, µ

L1 , ..., µ

Li , ..., µ

Ln) = 0 (B.1b)

gU(xU, xL1 , ..., xLi , ..., x

Ln, λ

L1 , ..., λ

Li , ..., λ

Ln, µ

L1 , ..., µ

Li , ..., µ

Ln) ≤ 0 (B.1c)

1) Lower-level problem 1:

MinimizexL1

fL1 (x

U, xL1 , ..., xLi , ..., x

Ln)

subject to:

hL1 (xU, xL1 , ..., x

Li , ..., x

Ln) = 0 : λL1

gL1 (xU, xL1 , ..., x

Li , ..., x

Ln) ≤ 0 : µL

1

(B.2a)

B.1. Bilevel Model 289

.

.

.

i) Lower-level problem i:

MinimizexLi

fLi (x

U, xL1 , ..., xLi , ..., x

Ln)

subject to:

hLi (xU, xL1 , ..., x

Li , ..., x

Ln) = 0 : λLi

gLi (xU, xL1 , ..., x

Li , ..., x

Ln) ≤ 0 : µL

i

(B.2b)

.

.

.

n) Lower-level problem n:

MinimizexLn

fLn (x

U, xL1 , ..., xLi , ..., x

Ln)

subject to:

hLn(xU, xL1 , ..., x

Li , ..., x

Ln) = 0 : λLn

gLn(xU, xL1 , ..., x

Li , ..., x

Ln) ≤ 0 : µL

n.

(B.2c)

Since the lower-level problems (B.2) constrain the upper-level problem

(B.1), all primal and dual variable vectors of the lower-level problems are

included in the variable set of the upper-level problem as well. Thus, the pri-

mal variable vectors of the upper-level problem (B.1) are those in the set ΞU

below:

ΞU=xU, xL1 , ..., xLi , ...x

Ln, λ

L1 , ..., λ

Li , ..., λ

Ln, µ

L1 , ..., µ

Li , ..., µ

Ln.

Note that in all bilevel problems proposed in this dissertation, the lower-

level problems are continuous, and linear, and thus convex.


B.2 MPEC

In this section, the bilevel problem (B.1)-(B.2) is transformed into a single-level

optimization problem. Since the lower-level problems of all bilevel problems

considered in this thesis are continuous and linear, and thus convex, each lower-

level problem (B.2) can be replaced by its first-order optimality conditions

rendering an MPEC.

The first-order optimality conditions associated with each lower-level prob-

lem (B.2) can be formulated through two alternative approaches:

1. Karush-Kuhn-Tucker (KKT) conditions.

2. Primal-dual transformation, i.e., enforcing primal constraints, dual con-

straints and the strong duality equality.

In the first approach (KKT conditions), a number of equalities are obtained

from differentiating the corresponding Lagrangian with respect to the primal

variables, and such equalities are equivalent to the set of primal and dual

constraints of the second approach (primal-dual transformation). In addition,

the set of complementarity conditions obtained by the first approach (KKT

conditions) is equivalent to the corresponding strong duality equality of the

primal-dual transformation [41].

In the following subsections, mathematical description for both alterna-

tive approaches and the resulting MPECs are presented. Then, further ex-

planations on the equivalence between the MPECs obtained from those two

approaches are provided.

B.2.1 MPEC Obtained from the KKT Conditions

In this subsection, a single-level MPEC is derived that is equivalent to the

bilevel problem (B.1)-(B.2). To this end, the lower-level problems (B.2) are

replaced by their first-order KKT conditions. The resulting MPEC is given by

(B.3):

B.2. MPEC 291

MinimizeΞU

fU(xU, xL1 , ..., xLi , ..., x

Ln, λ

L1 , ..., λ

Li , ..., λ

Ln, µ

L1 , ..., µ

Li , ..., µ

Ln) (B.3a)

subject to:

0) Upper-level equality and inequality constraints, which are identical to

constraints (B.1b)-(B.1c) of bilevel problem (B.1)-(B.2):

hU(xU, xL1 , ..., xLi , ..., x

Ln, λ

L1 , ..., λ

Li , ..., λ

Ln, µ

L1 , ..., µ

Li , ..., µ

Ln) = 0 (B.3b)

gU(xU, xL1 , ..., xLi , ..., x

Ln, λ

L1 , ..., λ

Li , ..., λ

Ln, µ

L1 , ..., µ

Li , ..., µ

Ln) ≤ 0 (B.3c)

1) KKT conditions associated with the lower-level problem 1:

∇xL1fL1 (x

U, xL1 , ..., xLi , ..., x

Ln) + λL1

T∇xL1

hL1 (xU, xL1 , ..., x

Li , ..., x

Ln)

+ µL1

T∇xL1

gL1 (xU, xL1 , ..., x

Li , ..., x

Ln) = 0 (B.3d)

hL1 (xU, xL1 , ..., x

Li , ..., x

Ln) = 0 (B.3e)

0 ≤ −gL1 (xU, xL1 , ..., x

Li , ..., x

Ln) ⊥ µL

1 ≥ 0 (B.3f)

λL1 : free (B.3g)

.

.

.

i) KKT conditions associated with the lower-level problem i:

∇xLifLi (x

U, xL1 , ..., xLi , ..., x

Ln) + λLi

T∇xLi

hLi (xU, xL1 , ..., x

Li , ..., x

Ln)

+ µLi

T∇xLi

gLi (xU, xL1 , ..., x

Li , ..., x

Ln) = 0 (B.3h)

hLi (xU, xL1 , ..., x

Li , ..., x

Ln) = 0 (B.3i)

0 ≤ −gLi (xU, xL1 , ..., x

Li , ..., x

Ln) ⊥ µL

i ≥ 0 (B.3j)

λLi : free (B.3k)


.

.

.

n) KKT conditions associated with the lower-level problem n:

∇xLnfLn (x

U, xL1 , ..., xLi , ..., x

Ln) + λLn

T∇xLn


Li , ..., x

Ln)

+ µLn

T∇xLn

gLn(xU, xL1 , ..., x

Li , ..., x

Ln) = 0 (B.3l)


Li , ..., x

Ln) = 0 (B.3m)

0 ≤ −gLn(xU, xL1 , ..., x

Li , ..., x

Ln) ⊥ µL

n ≥ 0 (B.3n)

λLn : free. (B.3o)

Note that each complementarity condition of the form 0 ≤ a ⊥ b ≥ 0 is

equivalent to a ≥ 0, b ≥ 0 and ab = 0.

To obtain MPEC (B.3), each lower-level problem (B.2) of the bilevel prob-

lem (B.1)-(B.2) is replaced by its KKT conditions. For example, lower-level

problem i given by (B.2b) in the bilevel problem (B.1)-(B.2) is replaced by its

KKT conditions including:

a) Equality (B.3h) obtained from differentiating the corresponding Lagrangian

of lower-level problem (B.2b) with respect to the variable vector xLi .

b) Equality (B.3i) that is identical to the one included in lower-level problem

(B.2b).

c) Complementarity condition (B.3j) related to the inequality constraint of

the lower-level problem (B.2b).

d) Condition (B.3k) stating that the dual variable vector λLi associated with

the equality constraint of the lower-level problem (B.2b) is free.

B.2. MPEC 293

B.2.2 MPEC Obtained from the Primal-Dual Transfor-

mation

In this subsection, another version of the MPEC associated with bilevel prob-

lem (B.1)-(B.2) is derived by replacing the lower-level problems (B.2) with

their primal-dual optimality conditions.

B.2.2.1 Linear Form of the Lower-Level Problems (B.2)

The primal-dual transformation is easily derived for linear optimization prob-

lems. Since the lower-level problems of all bilevel models proposed in this

dissertation are linear, the lower-level problems (B.2) are rewritten in a lin-

ear form as given by (B.4) below. Dual variable vectors associated with the

lower-level problems are indicated at the corresponding equations following a

colon.

1) Linear form of the lower-level problem 1:

MinimizexL1

kL1 (z1)TxL1

subject to:

AL1 (z1)x

L1 = bL1 (z1) : λL1

BL1 (z1)x

L1 ≥ cL1 (z1) : µL

1

xL1 ≥ 0 : ζL1

(B.4a)

.

.

.

i) Linear form of the lower-level problem i:

MinimizexLi

kLi (zi)TxLi

subject to:

ALi (zi)x

Li = bLi (zi) : λLi

BLi (zi)x

Li ≥ cLi (zi) : µL

i

xLi ≥ 0 : ζLi

(B.4b)


.

.

.

n) Linear form of the lower-level problem n:

MinimizexLn

kLn(zn)TxLn

subject to:

ALn(zn)x

Ln = bLn(zn) : λLn

BLn(zn)x

Ln ≥ cLn(zn) : µL

n

xLn ≥ 0 : ζLn .

(B.4c)

Regarding the linear form of the lower-level problems (B.4), the following

notational observations are relevant:

a) Primal variable vectors xU, xL1 , ..., xLi , ..., x

Ln are identical to those ones char-

acterized in the bilevel problem (B.1)-(B.2).

b) For example, zi =[xU

TxL1

T... xLi−1

TxLi+1

T... xLn

T]T

includes the pri-

mal variable vector corresponding to the upper-level problem and the ones

corresponding to all lower-level problems except the primal variable vec-

tor of lower-level problem i, i.e., xLi . Note that all primal variable vectors

included in zi are parameter vectors for lower-level problem i.

c) For example, vector kLi (zi), matrices ALi (zi) and B

Li (zi), and vectors bLi (zi)

and cLi (zi) are the cost vector, the constraint matrices and the right-handside

vectors, respectively, of the lower-level problem i.

d) Similarly to the bilevel problem (B.1)-(B.2), vectors λLi and µLi are respec-

tively the equality and inequality dual variable vectors corresponding to

the lower-level problem i. Additionally, dual variable vector ζLi associates

with the non-negativity of the primal variable vector xLi .

B.2. MPEC 295

B.2.2.2 Dual Optimization Problems Pertaining to Lower-Level Prob-

lems (B.4)

The dual optimization problem pertaining to each lower-level problem (B.4) is

given by (B.5) below:

1) Dual optimization problem pertaining to the lower-level problem 1:

MaximizeλL1 , µ

L1 , ζ

L1

bL1 (z1)TλL1 + cL1 (z1)

TµL1

subject to:

AL1 (z1)

TλL1 +BL

1 (z1)TµL1 + ζL1 = kL1 (z1)

µL1 ≥ 0 ; ζL1 ≥ 0

λL1 : free

(B.5a)

.

.

.

i) Dual optimization problem pertaining to the lower-level problem i:

MaximizeλLi , µ

Li , ζ

Li

bLi (zi)TλLi + cLi (zi)

TµLi

subject to:

ALi (zi)

TλLi +BL

i (zi)TµLi + ζLi = kLi (zi)

µLi ≥ 0 ; ζLi ≥ 0

λLi : free

(B.5b)

.

.

.

n) Dual optimization problem pertaining to the lower-level problem n:


MaximizeλLn, µ

Ln, ζ

Ln

bLn(zn)TλLn + cLn(zn)

TµLn

subject to:

ALn(zn)

TλLn +BL

n(zn)TµLn + ζLn = kLn(zn)

µLn ≥ 0 ; ζLn ≥ 0

λLn : free.

(B.5c)

B.2.2.3 Optimality Conditions Associated with Lower-Level Prob-

lems (B.4) Resulting from the Primal-Dual Transformation

Considering the lower-level primal problems (B.4) and their corresponding

lower-level dual problems (B.5), the following set of optimality conditions (B.6)

results from the primal-dual transformation.

1) Optimality conditions associated with the lower-level problem 1 resulting

from the primal-dual transformation:

AL1 (z1)x

L1 = bL1 (z1)

BL1 (z1)x

L1 ≥ cL1 (z1)

AL1 (z1)

TλL1 +BL

1 (z1)TµL1 + ζL1 = kL1 (z1)

kL1 (z1)TxL1 = bL1 (z1)

TλL1 + cL1 (z1)

TµL1

xL1 ≥ 0 ; µL1 ≥ 0 ; ζL1 ≥ 0

λL1 : free

(B.6a)

.

.

.

i) Optimality conditions associated with the lower-level problem i resulting


B.2. MPEC 297

ALi (zi)x

Li = bLi (zi)

BLi (zi)x

Li ≥ cLi (zi)

ALi (zi)

TλLi +BL

i (zi)TµLi + ζLi = kLi (zi)

kLi (zi)TxLi = bLi (zi)

TλLi + cLi (zi)

TµLi

xLi ≥ 0 ; µLi ≥ 0 ; ζLi ≥ 0

λLi : free

(B.6b)

.

.

.

n) Optimality conditions associated with the lower-level problem n resulting


ALn(zn)x

Ln = bLn(zn)

BLn(zn)x

Ln ≥ cLn(zn)

ALn(zn)

TλLn +BL

n(zn)TµLn + ζLn = kLn(zn)

kLn(zn)TxLn = bLn(zn)

TλLn + cLn(zn)

TµLn

xLn ≥ 0 ; µLn ≥ 0 ; ζLn ≥ 0

λLn : free.

(B.6c)

For example, optimality conditions (B.6b) associated with the lower-level

problem i consist of:

a) Primal constraints ALi (zi)x

Li = bLi (zi), B

Li (zi)x

Li ≥ cLi (zi) and xLi ≥ 0 in-

cluded in the primal problem (B.4b).

b) Dual constraints ALi (zi)

TλLi + BL

i (zi)TµLi + ζLi = kLi (zi), λ

Li : free, µL

i ≥ 0,

and ζLi ≥ 0 included in the dual problem (B.5b).


c) Strong duality equality kLi (zi)TxLi = bLi (zi)

TλLi + cLi (zi)

TµLi which enforces

the equality of the values of the primal objective function of (B.4b) and

the dual objective function of (B.5b) at the optimal solution.

B.2.2.4 Resulting MPEC from the Primal-Dual Transformation

MPEC (B.7) below associated with bilevel problem (B.1)-(B.2) is derived using

primal-dual optimality conditions (B.6). Dual variable vectors of MPEC (B.7)

are indicated at their corresponding constraints following a colon.

MinimizeΞU

fU(xU, xL1 , ..., xLi , ..., x

Ln, λ

L1 , ..., λ

Li , ..., λ

Ln, µ

L1 , ..., µ

Li , ..., µ

Ln) (B.7a)

subject to:

0) Upper-level equality and inequality constraints, which are identical to con-

straints (B.1b)-(B.1c) of bilevel problem (B.1)-(B.2):

hU(xU, xL1 , ..., xLi , ..., x

Ln, λ

L1 , ..., λ

Li , ..., λ

Ln, µ

L1 , ..., µ

Li , ..., µ

Ln) = 0 : αU (B.7b)

gU(xU, xL1 , ..., xLi , ..., x

Ln, λ

L1 , ..., λ

Li , ..., λ

Ln, µ

L1 , ..., µ

Li , ..., µ

Ln) ≤ 0 : βU (B.7c)

1) Optimality conditions associated with lower-level problem 1 resulting from

the primal-dual transformation:

AL1 (z1)x

L1 = bL1 (z1) : ρPC1 (B.7d)

BL1 (z1)x

L1 ≥ cL1 (z1) : ηPC1 (B.7e)

xL1 ≥ 0 : γx1 (B.7f)

AL1 (z1)

TλL1 +BL

1 (z1)TµL1 + ζL1 = kL1 (z1) : ρDC

1 (B.7g)

B.2. MPEC 299

µL1 ≥ 0 : γµ1 (B.7h)

ζL1 ≥ 0 : γζ1 (B.7i)

kL1 (z1)TxL1 = bL1 (z1)

TλL1 + cL1 (z1)

TµL1 : φSD

1 (B.7j)

.

.

.

i) Optimality conditions associated with lower-level problem i resulting from


ALi (zi)x

Li = bLi (zi) : ρPCi (B.7k)

BLi (zi)x

Li ≥ cLi (zi) : ηPCi (B.7l)

xLi ≥ 0 : γxi (B.7m)

ALi (zi)

TλLi +BL

i (zi)TµLi + ζLi = kLi (zi) : ρDC

i (B.7n)

µLi ≥ 0 : γµi (B.7o)

ζLi ≥ 0 : γζi (B.7p)

kLi (zi)TxLi = bLi (zi)

TλLi + cLi (zi)

TµLi : φSD

i (B.7q)

.

.

.

n) Optimality conditions associated with lower-level problem n resulting from



ALn(zn)x

Ln = bLn(zn) : ρPCn (B.7r)

BLn(zn)x

Ln ≥ cLn(zn) : ηPCn (B.7s)

xLn ≥ 0 : γxn (B.7t)

ALn(zn)

TλLn +BL

n(zn)TµLn + ζLn = kLn(zn) : ρDC

n (B.7u)

µLn ≥ 0 : γµn (B.7v)

ζLn ≥ 0 : γζn (B.7w)

kLn(zn)TxLn = bLn(zn)

TλLn + cLn(zn)

TµLn : φSD

n (B.7x)

Variable vectors λL1 ,...,λLi ,...,λ

Ln included in MPEC (B.7) are free. Note that

the dual variable vectors of MPEC (B.7) are indicated since these vectors are

used in Section B.3 to characterize EPEC.

B.2.3 Equivalence Between the MPECs Obtained from

the KKT Conditions and the Primal-Dual Trans-

formation

The equivalence between MPEC (B.3) obtained from the KKT conditions and

MPEC (B.7) resulting from the primal-dual transformation is explained below:

a) Constraints (B.3b)-(B.3c) included in MPEC (B.3) and constraints (B.7b)-

(B.7c) included in MPEC (B.7) are both identical to the upper-level con-

straints (B.1b)-(B.1c).

b) Primal equalities in MPEC (B.3) as well as the equalities obtained from

differentiating the corresponding Lagrangian with respect to the primal

variable vectors in such MPEC are equivalent to the collection of the pri-

mal and dual constraints included in MPEC (B.7). For example, equalities

B.3. EPEC 301

(B.3h) and (B.3i) of problem i in MPEC (B.3) are equivalent to the col-

lection of the primal and dual constraints of such problem, i.e., constraints

(B.7k)-(B.7p) included in MPEC (B.7).

c) Complementarity conditions included in MPEC (B.3) are equivalent to the

corresponding strong duality equalities included in MPEC (B.7). For ex-

ample, complementarity condition (B.3j) of problem i in MPEC (B.3) is

equivalent to the strong duality equality (B.7q) corresponding to problem

i and included in MPEC (B.7).

d) In both MPECs (B.3) and (B.7), the dual variable vectors associated with

the equalities, i.e, dual variable vectors λL1 ,...,λLi ,...,λ

Ln, are free.

B.3 EPEC

The joint consideration of a number of interrelated MPECs constitutes an

EPEC. To characterize the EPEC solutions, the optimality conditions of all

interrelated MPECs are jointly considered. In this process, two important

observations are in order:

1) MPECs obtained from the primal-dual transformation are preferably used

to avoid the use of non-convex and difficult to handle complementarity

conditions, but at the cost of the non-linearities introduced by the strong

duality equalities.

2) To derive the optimality conditions of the considered MPECs, it is impor-

tant to note that MPECs are generally non-linear and thus the application

of the primal-dual transformation is not generally possible. Therefore, to

obtain the optimality conditions associated with the EPEC, each MPEC is

replaced by its corresponding KKT conditions.

Considering the two observations above, KKT conditions of MPEC (B.7)

are derived as follows:


1) Equality (B.8a) below is obtained from differentiating the Lagrangian of

MPEC (B.7) with respect to variable vector xU. This vector appears in ob-

jective function (B.7a), upper-level constraints (B.7b)-(B.7c), and primal-

dual optimality conditions (B.7d)-(B.7x). Note that zj , j = 1, ..., i, ..., n

include variable vector xU.

∇xUfU(xU, xL1 , ..., x

Li , ..., x

Ln, λ

L1 , ..., λ

Li , ..., λ

Ln, µ

L1 , ..., µ

Li , ..., µ

Ln)

+αUT∇xUhU(xU, xL1 , ..., x

Li , ..., x

Ln, λ

L1 , ..., λ

Li , ..., λ

Ln, µ

L1 , ..., µ

Li , ..., µ

Ln)

+βUT∇xUgU(xU, xL1 , ..., x

Li , ..., x

Ln, λ

L1 , ..., λ

Li , ..., λ

Ln, µ

L1 , ..., µ

Li , ..., µ

Ln)

+n∑

j=1

ρPCjT∇xU

[ALj (zj)x

Lj − b

Lj (zj)

]

−

n∑

j=1

ηPCjT∇xU

[BLj (zj)x

Lj − c

Lj (zj)

]

+n∑

j=1

ρDCj

T∇xU

[ALj (zj)

TλLj +BL

j (zj)TµLj − k

Lj (zj)

]

+

n∑

j=1

φSDj

T∇xU

[kLj (zj)

TxLj − b

Lj (zj)

TλLj − c

Lj (zj)

TµLj

]= 0. (B.8a)

2) Equality (B.8b) below is obtained from differentiating the Lagrangian of

MPEC (B.7) with respect to variable vector xL1 . This vector appears in ob-

jective function (B.7a), upper-level constraints (B.7b)-(B.7c), primal con-

straints (B.7d)-(B.7f), strong duality equality (B.7j), and primal-dual op-

timality conditions (B.7k)-(B.7x). Note that zj , j = 2, ..., i, ..., n include

variable vector xL1 .

∇xL1fU(xU, xL1 , ..., x

Li , ..., x

Ln, λ

L1 , ..., λ

Li , ..., λ

Ln, µ

L1 , ..., µ

Li , ..., µ

Ln)

+αUT∇xL1hU(xU, xL1 , ..., x

Li , ..., x

Ln, λ

L1 , ..., λ

Li , ..., λ

Ln, µ

L1 , ..., µ

Li , ..., µ

Ln)

B.3. EPEC 303

+βUT∇xL1gU(xU, xL1 , ..., x

Li , ..., x

Ln, λ

L1 , ..., λ

Li , ..., λ

Ln, µ

L1 , ..., µ

Li , ..., µ

Ln)

+ρPC1TAL

1 (z1) − ηPC1

TBL

1 (z1) − γx1 + kL1 (z1)

TφSD1

+n∑

j=2

ρPCjT∇xL1

[ALj (zj)x

Lj − b

Lj (zj)

]

−

n∑

j=2

ηPCjT∇xL1

[BLj (zj)x

Lj − c

Lj (zj)

]

+n∑

j=2

ρDCj

T∇xL1

[ALj (zj)

TλLj +BL

j (zj)TµLj − k

Lj (zj)

]

+

n∑

j=2

φSDj

T∇xL1

[kLj (zj)

TxLj − b

Lj (zj)

TλLj − c

Lj (zj)

TµLj

]= 0. (B.8b)

3) Equality (B.8c) below is obtained from differentiating the Lagrangian of

MPEC (B.7) with respect to variable vector xLi . This vector appears in ob-


straints (B.7k)-(B.7m), strong duality equality (B.7q), and primal-dual op-

timality conditions (B.7d)-(B.7x) except conditions (B.7k)-(B.7q). Note

that zj , j = 1, ..., i− 1, i+ 1..., n include variable vector xLi .

∇xLifU(xU, xL1 , ..., x

Li , ..., x

Ln, λ

L1 , ..., λ

Li , ..., λ

Ln, µ

L1 , ..., µ

Li , ..., µ

Ln)

+αUT∇xLihU(xU, xL1 , ..., x

Li , ..., x

Ln, λ

L1 , ..., λ

Li , ..., λ

Ln, µ

L1 , ..., µ

Li , ..., µ

Ln)

+βUT∇xLigU(xU, xL1 , ..., x

Li , ..., x

Ln, λ

L1 , ..., λ

Li , ..., λ

Ln, µ

L1 , ..., µ

Li , ..., µ

Ln)

+ρPCiTALi (zi) − η

PCi

TBLi (zi) − γ

xi + kLi (zi)

TφSDi

+n∑

j=1j 6=i

ρPCjT∇xLi

[ALj (zj)x

Lj − b

Lj (zj)

]

−

n∑

j=1j 6=i

ηPCjT∇xLi

[BLj (zj)x

Lj − c

Lj (zj)

]

+n∑

j=1j 6=i

ρDCj

T∇xLi

[ALj (zj)

TλLj +BL

j (zj)TµLj − k

Lj (zj)

]


+n∑

j=1j 6=i

φSDj

T∇xLi

[kLj (zj)

TxLj − b

Lj (zj)

TλLj − c

Lj (zj)

TµLj

]= 0. (B.8c)

4) Equality (B.8d) below is obtained from differentiating the Lagrangian of

MPEC (B.7) with respect to variable vector xLn. This vector appears in ob-


straints (B.7r)-(B.7t), strong duality equality (B.7x), and primal-dual opti-

mality conditions (B.7d)-(B.7r). Note that zj , j = 1, ..., i, ..., n− 1 include

variable vector xLn.

∇xLnfU(xU, xL1 , ..., x

Li , ..., x

Ln, λ

L1 , ..., λ

Li , ..., λ

Ln, µ

L1 , ..., µ

Li , ..., µ

Ln)

+αUT∇xLnhU(xU, xL1 , ..., x

Li , ..., x

Ln, λ

L1 , ..., λ

Li , ..., λ

Ln, µ

L1 , ..., µ

Li , ..., µ

Ln)

+βUT∇xLngU(xU, xL1 , ..., x

Li , ..., x

Ln, λ

L1 , ..., λ

Li , ..., λ

Ln, µ

L1 , ..., µ

Li , ..., µ

Ln)

+ρPCnTALn(zn) − η

PCn

TBLn(zn) − γ

xn + kLn(zn)

TφSDn

+n−1∑

j=1

ρPCjT∇xLn

[ALj (zj)x

Lj − b

Lj (zj)

]

−n−1∑

j=1

ηPCjT∇xLn

[BLj (zj)x

Lj − c

Lj (zj)

]

+

n−1∑

j=1

ρDCj

T∇xLn

[ALj (zj)

TλLj +BL

j (zj)TµLj − k

Lj (zj)

]

+

n−1∑

j=1

φSDj

T∇xLn

[kLj (zj)

TxLj − b

Lj (zj)

TλLj − c

Lj (zj)

TµLj

]= 0. (B.8d)

5) Equalities (B.8e) below are obtained from differentiating the Lagrangian of

MPEC (B.7) with respect to variable vectors λLj , j = 1, ..., i, ..., n. Note

B.3. EPEC 305

that these variable vectors appear in objective function (B.7a), upper-level

constraints (B.7b)-(B.7c), dual constraints and strong duality equality of

lower-level problem j.

∇λLjfU(xU, xL1 , ..., x

Li , ..., x

Ln, λ

L1 , ..., λ

Li , ..., λ

Ln, µ

L1 , ..., µ

Li , ..., µ

Ln)

+αUT∇λLjhU(xU, xL1 , ..., x

Li , ..., x

Ln, λ

L1 , ..., λ

Li , ..., λ

Ln, µ

L1 , ..., µ

Li , ..., µ

Ln)

+βUT∇λLjgU(xU, xL1 , ..., x

Li , ..., x

Ln, λ

L1 , ..., λ

Li , ..., λ

Ln, µ

L1 , ..., µ

Li , ..., µ

Ln)

+ρDCj

TALj (zj)

T− φSD

j bLj (zj) = 0 j = 1, ..., i, ..., n. (B.8e)

6) Equalities (B.8f) below are obtained from differentiating the Lagrangian

of MPEC (B.7) with respect to variable vectors µLj , j = 1, ..., i, ..., n.

Note that these vectors appear in objective function (B.7a), upper-level

constraints (B.7b)-(B.7c), dual constraints and strong duality equality of

lower-level problem j.

∇µLjfU(xU, xL1 , ..., x

Li , ..., x

Ln, λ

L1 , ..., λ

Li , ..., λ

Ln, µ

L1 , ..., µ

Li , ..., µ

Ln)

+αUT∇µLjhU(xU, xL1 , ..., x

Li , ..., x

Ln, λ

L1 , ..., λ

Li , ..., λ

Ln, µ

L1 , ..., µ

Li , ..., µ

Ln)

+βUT∇µLjgU(xU, xL1 , ..., x

Li , ..., x

Ln, λ

L1 , ..., λ

Li , ..., λ

Ln, µ

L1 , ..., µ

Li , ..., µ

Ln)

+ρDCj

TBLj (zj)

T− γµj − φ

SDj cLj (zj) = 0 j = 1, ..., i, ..., n. (B.8f)

6) Equalities (B.8g) below are obtained from differentiating the Lagrangian

of MPEC (B.7) with respect to variable vector ζLj , j = 1, ..., i, ..., n. Note

that these vectors appear in dual constraints of problem j.


ρDCj − γ

ζj = 0 j = 1, ..., i, ..., n. (B.8g)

7) Primal equality constraints (B.8h)-(B.8k) below are included in MPEC

(B.7):

hU(xU, xL1 , ..., xLi , ..., x

Ln, λ

L1 , ..., λ

Li , ..., λ

Ln, µ

L1 , ..., µ

Li , ..., µ

Ln) = 0 (B.8h)

ALj (zj)x

Lj = bLj (zj) j = 1, ..., i, ..., n (B.8i)

ALj (zj)

TλLj +BL

j (zj)TµLj + ζLj = kLj (zj) j = 1, ..., i, ..., n (B.8j)

kLj (zj)TxLj = bLj (zj)

TλLj + cLj (zj)

TµLj j = 1, ..., i, ..., n. (B.8k)

8) Complementarity conditions (B.8l)-(B.8p) below are related to the inequal-

ities of MPEC (B.7):

0 ≤ −gU(xU, xL1 , ..., xLi , ..., x

Ln, λ

L1 , ..., λ

Li , ..., λ

Ln, µ

L1 , ..., µ

Li , ..., µ

Ln)

⊥ βU ≥ 0 (B.8l)

0 ≤[BLj (zj)x

Lj − c

Lj (zj)

]⊥ ηPCj ≥ 0 j = 1, ..., i, ..., n (B.8m)

0 ≤ xLj ⊥ γxj ≥ 0 j = 1, ..., i, ..., n (B.8n)

0 ≤ µLj ⊥ γµj ≥ 0 j = 1, ..., i, ..., n (B.8o)

0 ≤ ζLj ⊥ γζj ≥ 0 j = 1, ..., i, ..., n. (B.8p)

9) Conditions (B.8q)-(B.8t) state that the dual variable vectors associated

with the equalities of MPEC (B.7) are free:

B.4. Benders’ Decomposition 307

αU : free (B.8q)

ρPCj : free j = 1, ..., i, ..., n (B.8r)

ρDCj : free j = 1, ..., i, ..., n (B.8s)

φSDj : free j = 1, ..., i, ..., n. (B.8t)

Note that the joint consideration of the optimality conditions of all consid-

ered MPECs renders the optimality conditions of the EPEC, whose solution

provides the EPEC solution.

B.4 Benders’ Decomposition

The mathematical structure of the bilevel models proposed in this dissertation

is given by (B.9) below:

MinimizeX,xUw,x

Lw,λ

Lw,µ

Lw

aX +∑

w

φwfUw (x

Uw, x

Lw, λ

Lw, µ

Lw) (B.9a)

subject to:


Xmin ≤ X ≤ Xmax (B.9b)

hUw(X, xUw, x

Lw, λ

Lw, µ

Lw) = 0 ∀w (B.9c)

gUw(X, xUw, x

Lw, λ

Lw, µ

Lw) ≤ 0 ∀w (B.9d)


2) Lower-level problems:

MinimizexLw

fLw(X, x

Uw, x

Lw)

subject to:

hLw(X, xU, xL1 , ..., x

Ln) = 0 : λLw

gLw(X, xU, xL1 , ..., x

Ln) ≤ 0 : µL

w

∀w. (B.9e)

In the structure above, variable vector X belonging to the variable set of

the upper-level problem (B.9a)-(B.9d) is a complicating variable vector. The

reason is that if such vector fixed (X = XFixed), the bilevel problem (B.9)

decomposes to a number of smaller bilevel problems, one per w, as given by

(B.10) below. Note that each decomposed bilevel problem (B.10) is smaller and

thus easier to solve than the original bilevel problem (B.9). The decomposed

problem has the form as follows:

MinimizexUw,x

Lw,λ

Lw,µ

Lw

φwfUw (x

Uw, x

Lw, λ

Lw, µ

Lw) (B.10a)

subject to:


hUw(XFixed, xUw, x

Lw, λ

Lw, µ

Lw) = 0 (B.10b)

gUw(XFixed, xUw, x

Lw, λ

Lw, µ

Lw) ≤ 0 (B.10c)


MinimizexLw

fLw(X

Fixed, xUw, xLw)

subject to:

hLw(XFixed, xU, xL1 , ..., x

Ln) = 0 : λLw

gLw(XFixed, xU, xL1 , ..., x

Ln) ≤ 0 : µL

w

(B.10d)

B.4. Benders’ Decomposition 309

∀w.

This decomposable structure motivates applying a Benders’ decomposition

to the bilevel problem (B.9).

It is important to note that the Benders’ decomposition can be applied to

the bilevel problem (B.9) if the objective function of such problem expressed as

a function of the complicating variable vector X has a convex enough envelope.

The Benders’ decomposition algorithm works as follows [26]:

Input: A small tolerance ε to control convergence, and an initial guess of the

complicating variable vector X0.

Step 0) Initialization: Set v = 1, FU(v)= −∞ and XFixed = X0.

Step 1) Subproblem solution: Solve subproblem (B.11) below for each w.

Minimize

X(v),xUw(v),xLw

(v),λLw

(v),µLw

(v)φwf

Uw (x

Uw

(v), xLw

(v), λLw

(v), µL

w

(v)) (B.11a)

subject to:


X(v) = XFixed : Λ(v)w (B.11b)

hUw(X(v), xUw

(v), xLw

(v), λLw

(v), µL

w

(v)) = 0 (B.11c)

gUw(X(v), xUw

(v), xLw

(v), λLw

(v), µL

w

(v)) ≤ 0 (B.11d)


MinimizexLw

fLw(X

(v), xUw(v), xLw

(v), λLw

(v), µL

w

(v))

subject to:

hLw(X(v), xUw

(v), xLw

(v), λLw

(v), µL

w

(v)) = 0 : λLw

(v)

gLw(X(v), xUw

(v), xLw

(v), λLw

(v), µL

w

(v)) ≤ 0 : µL

w

(v)

(B.11e)


∀w,

where dual variable vectors Λ(v)w of (B.11b) are sensitivities.

Step 2) Convergence check: Calculate the objective function upper-bound,

FU(v)

, as follows:

FU(v)

=

[∑

w

φwfUw (x

Uw

(v), xLw

(v), λLw

(v), µL

w

(v))

]+ aXFixed. (B.12a)

If

∣∣∣∣FU(v)−FU(v)

∣∣∣∣ ≤ ε, the optimal solution with a level of precision ε

is X∗ = XFixed.

Otherwise, calculate Λ(v) by (B.12b) below and then set v ←− v + 1.

Λ(v) =∑

w

Λ(v)w . (B.12b)

Step 3) Master problem solution: Solve master problem (B.13) below:

MinimizeFU(v)

,X(v),Γ(v)

FU(v)= aX(v) + Γ(v) (B.13a)

subject to:

B.5. Linearization Techniques 311

Xmin ≤ X ≤ Xmax (B.13b)

Γ(v) ≥ Γmax (B.13c)

Γ(v) ≥

[∑

w

φwfUw (x

Uw

(l), xLw

(l), λLw

(l), µL

w

(l))

]

+ Λ(l)(X(v) −X(l)

)l = 1, ..., v − 1. (B.13d)

and then update XFixed and FU(v), and then continue the algorithm

in Step 1.

Note that each solution of the master problem updates the value of

the complicating variable vector X , i.e., XFixed ←− X(v).

The structure of master problem (B.13) is explained below:

a) Objective function (B.13a) corresponds to the upper-level objective

function (B.9a) in the bilevel problem (B.9), where Γ(v) represents∑

w φwfUw (x

Uw, x

Lw, λ

Lw, µ

Lw).

b) Constraint (B.13b) is identical to (B.9b) in the bilevel problem

(B.9).

c) Constraint (B.13c) imposes a lower bound on Γ(v) to accelerate

convergence.

d) Constraints (B.13d) are Benders cuts, i.e., inequality constraints

use to reconstruct the objective function. Note that at every iter-

ation, a new cut is added to (B.13d).

B.5 Linearization Techniques

This section provides the mathematical description of the linearization tech-

niques used in this thesis work.


B.5.1 Complementarity Linearization

The following exact linearization technique proposed in [40] is used in Chapters

2 to 5 to linearize the complementarity conditions.

Each complementarity condition of the form

0 ≤ a ⊥ b ≥ 0 (B.14)

can be replaced by conditions (B.15a)-(B.15e) below:

a ≥ 0 (B.15a)

b ≥ 0 (B.15b)

a ≤ ψM (B.15c)

b ≤ (1− ψ)M (B.15d)

ψ ∈ 0, 1, (B.15e)

where M is a large enough positive constant, and ψ is an auxiliary binary

variable.

For example, complementarity condition (B.3j), i.e.,

0 ≤ −gLi (xU, xL1 , ..., x

Li , ..., x

Ln) ⊥ µL

i ≥ 0

included in MPEC (B.3) is linearized as follows:

−gLi (xU, xL1 , ..., x

Li , ..., x

Ln) ≥ 0 (B.16a)

µLi ≥ 0 (B.16b)

−gLi (xU, xL1 , ..., x

Li , ..., x

Ln) ≤ ψL

i Mg (B.16c)

µLi ≤

(1− ψL

i

)Mµ (B.16d)

ψLi ∈ 0, 1, (B.16e)

where Mg and Mµ are both large enough positive constants.

Note that the complementarity conditions of each MPEC considered in this

dissertation linearized through the linearization technique above, at the cost


of adding auxiliary binary variables to the variable set of that MPEC.

B.5.2 Binary Expansion Approach

The binary expansion approach [115] is an approximate but computationally

efficient linearization technique that is used in Chapter 4 to linearize a product

of two continuous variables.

Based on this approach, each non-linear term P×µ where both P and µ are

continuous variables can be linearized by representing through discrete steps

the variable whose bounds are available (e.g., variable P ). This is clarified

below:

1) Variable P is substituted by an addition of discrete values∑Q

q=1 βq, i.e.,

P '

Q∑

q=1

βq. (B.17a)

Note that index q (1 to Q) refers to the discretized variable P .

2) Nonlinear term P × µ is replaced by a linear one, i.e.,

P × µ '

Q∑

q=1

βqϕq (B.17b)

where ϕq, ∀q, are auxiliary continuous variables.

3) Among the discrete values βq, the closest one to the variable P is selected

through the mixed-integer linear equations below:

P −∆β

2≤

Q∑

q=1

βqzq ≤ P +∆β

2(B.17c)

where ∆β is a constant defined as follows:

∆β = βq+1 − βq. (B.17d)


In addition, zq, ∀q, are binary variables. Note that among all those bi-

nary variables, the value of the one that makes shortest the discrete

and continuous values of P takes the value 1.0, while the other binary

variables are zero, i.e.,

Q∑

q=1

zq = 1. (B.17e)

4) Additionally, for the binary expansion to work, the following set of mixed-

integer linear inequalities should be incorporated as constraints:

0 ≤ µ− ϕq ≤ G× (1− zq) ∀q (B.17f)

0 ≤ ϕq ≤ G× zq ∀q (B.17g)

where G is a large enough positive constant.

Note that a higher number of discrete values βq renders higher accuracy in

the approximation, but at the cost of higher computational burden. To ease

the computational burden, an iterative algorithm for selecting the discrete

values is proposed as stated below:

a) Consider two discrete values β1 and β2 so that ∆β = β2 − β1.

b) Solve the optimization problem considering β1ϕ1+β2ϕ2 instead of P×µ,

obtain the optimal values of all variables (illustrated by superscript ∗),

and then compute Λ below:

Λ =

∣∣∣∣(β∗1ϕ

∗1 + β∗

2ϕ∗2)− (P ∗ × µ∗)

∣∣∣∣ (B.17h)

If Λ is smaller than tolerance ε, the approximation is accurate enough,

otherwise pick up the closest discrete value to P (e.g., β• = β1) and go

to step c.


c) Select two symmetrical new discrete values around β• so that their dif-

ference is δ×∆β, where 0 ≤ δ ≤ 1, and then return to step b. Note that

a higher value for δ generally results in increasing accuracy, but at the

cost of increasing the number of iterations.

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Index

Arbitrage, 119, 173, 178, 180

Auxiliary problem, 77, 88

Base demand block, 108, 164

Benders’ decomposition, 7, 74–77, 96,

97, 307

Bertrand model, 21

Bilevel model, 7, 25, 29, 37, 42, 75, 77,

111, 113, 115, 190, 192, 287

Binary expansion approach, 143, 313

Complementarity linearization, 311

Complementarity model, 4

Complicating variable, 76, 77

Computational burden, 72, 103, 182,

261

Conjectural variations model, 21

Convexity, 77

Cournot model, 21

dc power flow, 20

Demand block, 18, 20, 42, 62, 163, 234

Demand-bid block, 19, 42

Diagonalization Checking, 232, 250

Direct solution approach, 7, 74, 97

Dual optimization problem, 50, 127,

132, 137, 199, 294

Duopoly, 255

Electricity markets, 1

EPEC, 7, 8, 29, 186, 205, 206, 230,

301

Futures base auction, 25, 109, 118, 173,

178, 180

Futures market, 1, 7, 25, 107

Futures peak auction, 25, 109, 120,

180

Generalized Nash equilibrium, 187

Generation investment equilibria, 5, 7,

8, 29, 185, 186, 205, 230, 240

Investment technologies, 5, 62, 79, 100,

167, 236

KKT conditions, 7, 29, 41, 46, 87, 124,

125, 129, 134, 187, 195, 196,

290

Linearization, 52, 140, 214, 311

Load-duration curve, 18, 108

Locational marginal price, 1, 2, 25, 38,

40, 45, 64, 113, 185, 190, 194,

195

332

INDEX 333

Market operator, 1

Master problem, 95, 310

MILP, 7, 55, 150, 215

Monopoly, 247, 255

MPEC, 7, 29, 51, 87, 139, 186, 201,

289

Multi-stage approach, 16

Oligopolistic market, 1, 8

Optimality conditions, 7, 29, 45, 124,

195, 206, 290, 296

Peak demand block, 108, 164

Pool, 1, 8, 25, 37, 107, 121, 173, 180,

188

Primal-dual transformation, 91, 127,

132, 137, 195, 199, 290, 296,

298

Production-offer block, 20, 42

Scenario reduction, 101

Social welfare, 1, 2, 25, 38, 40, 45, 119,

121, 123, 195, 231, 246

Static approach, 16

Stochastic programming, 4

Strategic investment, 1, 25, 37, 44, 107,

116, 185, 194

Strategic offering, 1, 25, 37, 44, 65,

107, 116, 185, 194

Strategic producer, 1, 5, 25, 37, 44,

107, 116, 185, 194

Strong duality equality, 50, 127, 132,

137, 199

Subproblem, 77, 91, 309

Supply function model, 22

Supply security, 118, 180, 183, 195,

250

Triopoly, 232, 237, 245, 246

Uncertainty, 4, 5, 37, 45, 67, 68, 75,

80, 81, 96, 99, 110, 115, 167

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