Steven A. Gabriel • Antonio J. ConejoJ. David Fuller • Benjamin F. HobbsCarlos Ruiz

Complementarity Modelingin Energy Markets

^J Springer

Contents

1 Introduction and Motivation 11.1 Introduction 11.2 Complementarity Models: Motivation and Description 1

1.2.1 Illustrative Example. Three-Variable MCP 41.2.2 Illustrative Example. Nonlinear Program Expressed

as an MCP 51.2.3 Illustrative Example. PIES Model 71.2.4 Illustrative Example. Nash-Cournot Duopoly in

Energy Production, Two Simultaneous OptimizationProblems 8

1.2.5 Illustrative Example. Generalized Nash Equilibria,Energy Production Duopoly 10

1.2.6 Illustrative Example. Nash-Cournot DuopolyExpressed as a Variational Inequality 11

1.2.7 Illustrative Example. Energy Network with MultiplePlayers 12

1.2.8 Illustrative Example. MPEC .. ..' 161.3 Summary 181.4 . Appendix: Computational Issues for Selected Problems 19

1.4.1 Illustrative Example 1.2.1 191.4.2 Illustrative Example 1.2.4 201.4.3 Illustrative Example 1.2.5 211.4.4 Illustrative Example 1.2.7 23

References 27

2 Optimality and Complementarity ." 312.1 Introduction 312.2 Optimization Problems 32

2.2.1 Illustrative Example. Optimization Problem: OnlyEquality Constraints 33

xii Contents

2.2.2 Illustrative Example. Optimization Problem:Unconstrained 35

2.2.3 Illustrative Example. Optimization Problem:Equality and Inequality Constraints . . • 35

2.2.4 Linear Optimization Problems 372.2.5 Illustrative Example. LP Problem: Primal-Dual

Formulation 382.3 Karush-Kuhn-Tucker Conditions 39

2.3.1 Illustrative Example. KKT Conditions: EqualityConstraints 40

2.3.2 Illustrative Example. KKT Conditions: Equalityand Inequality Constraints 41

2.4 Constraint Qualifications 422.4.1 Illustrative Example. Constraint Qualification:

Regular Solution 432.4.2 Illustrative Example. Constraint Qualification:

Non-Regular Solution 432.5 Sufficiency Conditions 44

2.5.1 Illustrative Example. Sufficiency Conditions 452.6 Mixed Linear Complementarity Problem, MLCP 46

2.6.1 Illustrative Example. MLCP 462.7 Equilibrium Problems, EP 47

2.7.1 Illustrative Example. Equilibrium Conditions: NoConstraints 49

2.7.2 Illustrative Example. Equilibrium Conditions: OnlyEquality Constraints 50

2.7.3 Illustrative Example. Equilibrium Conditions:Equality and Inequality Constraints 50

2.7.4 Illustrative Example. Linear Equilibrium Problem . . . 522.8 Mathematical Programs with Equilibrium Constraints, MPEC 53

2.8.1 ' Illustrative Example. MPEC: Only EqualityConstraints 56

2.8.2 Illustrative Example. MPEC: Both Equality andInequality Constraints 58

2.9 Equilibrium Problems with Equilibrium Constraints, EPEC.. 602.9.1 Illustrative Example. EPEC: Only Equality

Constraints 622.9.2 Illustrative Example. EPEC: Both Equality and

Inequality Constraints 642.10 Non-Convexity and Non-Regularity Issues •>..... 662.11 Summary 672.12 Exercises 68

References 69

Contents

Some Microeconomic Principles 713.1 Introduction 713.2 Basics of Supply and Demand 72

3.2.1 Supply Curves *; 723.2.2 Demand Curves 753.2.3 Notion of Equilibrium as Intersection of Supply and

Demand Curves 783.2.3.1 Illustrative Example. Equilibrium in the

Coal Market 793.2.3.2 Illustrative Example. Changes in

Consumers' and Producers' Surpluses Dueto a Cartel 80

3.2.4 Non-Price Influences: Shifting Supply and DemandCurves 81

3.2.5 Multicommodity Equilibrium 833.2.5.1 Illustrative Example. Simultaneous

Equilibrium of Coal and Wood Markets . . . . 843.2.6 Estimation of Parameters of Demand and Supply

Functions 843.2.6.1 Top-Down or Statistical Estimation on

Observations 843.2.6.2 Bottom-Up or Process-Based Estimation.... 863.2.6.3 Auctions 87

3.3 Social Welfare Maximization 883.3.1 Definition of Social Welfare in Single Commodity

Models: Consumers' Plus Producers' Surpluses 883.3.2 Equilibrium as Maximization of Social Welfare in

Single Commodity Models 893.3.2.1 Illustrative Example. Equilibrium in Coal

Market as Social Welfare Maximization 903.3"3 Pareto Efficiency Versus Social Welfare Optimization 903.3.4 Social Welfare in Multicommodity Models 91

3.3.4.1 Possible Difficulty to Integrate InverseDemand Functions in MulticommodityModels 91

3.3.4.2 Illustrative Example. Impossibility ofIntegrating Inverse Demand Functions forCoal and Wood 92

3.3.4.3 Measuring Changes in Social Welfare inMulticommodity Models j 93

3.3.4.4 Illustrative Example. Changes inConsumers' Surplus, for Wood and Coal,Due to a Tax on Coal 93

3.4 Modeling Individual Players in Single Commodity Markets .. 94

Contents

3.4.1 Profit-Maximization Problem for Price-TakingFirms, and Form of Equilibrium Problem 94

3.4.2 Perfect Versus Imperfect Competition 973.4.2.1 Illustrative Example. Three Price-Taking

Firms: Social Welfare Maximization Model.. 983.4.2.2 Illustrative Example. Three Price-Taking

Firms: Complementarity Model 1003.4.2.3 Monopoly Model 1013.4.2.4 Illustrative Example. Three Firms Merged

as One Firm: Monopoly Model 1023.4.2.5 Nash-Cournot Model .? . . . , 1033.4.2.6 Illustrative Example. Nash-Cournot Model

of Three Firms: Complementarity Model.... 1043.4.2.7 Illustrative Example. Nash-Cournot Model

of Three Firms: Optimization Model ifDemand is Linear 107

3.4.2.8 Illustrative Example. Mixed Behaviors:Firm 1 as Cournot, Firms 2 and 3 asPrice-Takers 108

3.4.2.9 Illustrative Example. Mixed Behaviors:Firms 1 and 2 as Cournot, Firm 3 asPrice-Taker 108

3.4.2.10 Bertrand Game 1093.4.2.11 Illustrative Example. Bertrand Model of

Coal Market 1093.4.2.12 Cartels 110

3.4.3 Nash Versus Generalized Nash Equilibria I l l3.4.3.1 Illustrative Example. Generalized Nash

Model for Coal Market: Limit on CoalYard, with Government Allocation of CoalYard Shares ' 114

3.4.3.2' Illustrative Example. Generalized NashModel for Coal Market: Limit on CoalYard, with Trading of Shares and EqualMarginal Utilities of Yard Shares 115

3.4.3.3 Illustrative Example. Generalized NashModel for Coal Market: Limit on CoalYard, with Auctioning of Shares andUnequal Marginal Utilities 117

3.5 Multi-Level Games f 1183.5.1 Stackelberg Leader-Follower Games (MPECs).'. 118

3.5.1.1 Illustrative Example. Stackelberg MPECwith Firm 2 as Leader 119

3.5.1.2 Illustrative Example. Stackelberg MPECwith Firms 1 and 2 Merged as One Leader .. 119

'. Contents xv

; 3.5.2 Multi-Leader Games (EPECs) 120\ 3.6 Summary 121i 3.7 Exercises 122i \

; References 125

I 4 Equilibria and Complementarity Problems 127; 4.1 Introduction 127i 4.2 Economics and Engineering Equilibria 129! 4.2.1 Equilibria in Dominant Actions 129i 4.2.1.1 Illustrative Example. Energy Productioni Duopoly ". 129', 4.2.1.2 Illustrative Example. Energy Production; Duopoly, j3 Changed from 1.5 to 2) 130i 4.2.2 Nash Equilibria 131! 4.2.2.1 Illustrative Example. Energy Production| Duopoly, Nash Equilibrium 131I 4.2.2.2 Illustrative Example. Energy Production! Duopoly, (3 = 1, Additional Costs 132I 4.2.3 Types of Game Theory Problems Considered 132[ 4.2.4 Mixed Versus Pure.Equilibria 133i 4.2.4.1 Illustrative Example. Energy Productioni Bimatrix Game, Version 1 135I 4.2.4.2 Illustrative Example. Energy Production[ Bimatrix Game, Version 2 136| 4.3 Duality in Optimization Versus Equilibria 137I 4.3.1 Linear Programs as Equilibrium Problems 137| 4.3.1.1 Illustrative Example. Energy Productioni ° Optimization Problem, One Player 138[ 4.3.2 Nonlinear Programs as Equilibrium Problems 140

4.4 More About the Connection Between Optimization andEquilibrium Problems 141

| 4.4.1 Spatial Price Equilibrium Problem 1424.4.1.1 Illustrative Example. Spatial Price

[ Equilibrium for Energy Products 143! 4.4.2 Optimization Problems from Equilibrium Conditions? 146\ 4.4.2.1 Illustrative Example. Extended Energy

Production Optimization Problem 1484.4.2.2 Illustrative Example. Extended Energy

| Production Optimization Derived from MCP 150I 4.4.3 Equilibria with No Corresponding KKT-Based

Optimization Problem 1514.4.3.1 Illustrative Example. Spatial Price

Equilibrium, Version 2 153

xvi Contents

4.5 Selected Existence/Uniqueness Results for EquilibriumProblems 155

4.6 Extensions to Equilibrium Problems 1614.6.1 Overview • 161

4.6.1.1 Illustrative Example. Integer-ConstrainedSpatial Price Equilibrium 162

4.6.2 Discretely-Constrained Mixed LinearComplementarity Problem 1624.6.2.1 Illustrative Example. Integer-Constrained

Network Equilibrium 1644.6.3 Stochastic Equilibria , 166

4.6.3.1 Generator / ' s Problem 1674.6.3.2 Grid Owner's Problem 1694.6.3.3 Market Clearing 169

4.7 Summary 1704.8 Appendix: Computational Issues for Selected Problems 170

4.8.1 Computation of Nash Equilibrium Based on theRange for the Parameters 170

4.8.2 Computations for Price Functions in Spatial PriceEquilibrium-Version 2 172

4.8.3 Uniqueness of Spatial Price Equilibrium Version 2Solution 173

4.9 Exercises 174

References 177

5 Variational Inequality Problems 1815.1 Introduction 1815.2 Formulation of Variational Inequality Problems 182

5.2.1 Optimization Problem as a VI Problem 1825.2.2 .. VI Formulation of Nash Equilibrium: No Linking

Constraints 1835.2.2.1 Illustrative Example. Nash-Cournot Model

of Coal Market from Chapter 3 1855.2.3 VI Formulation of Generalized Nash Equilibrium

With Linking Constraints: A Special Case 1865.2.3.1 Illustrative Example. Nash-Cournot Model

of Coal Market with Coal Yard Limit fromChapter 3 189

5.2.3.2 Illustrative Example. CompetitiveEquilibrium of Two Related Markets: Coaland Wood from Chapter 3 190

5.2.3.3 Illustrative Example. PIES MulticommodityCompetitive Equilibrium Model fromChapter 1 193

Contents

5.2.3.4 Illustrative Example. StochasticEquilibrium Model from Chapter 4 194

5.3 Relations between Variational Inequality and ^Complementarity Problems * 1975.3.1 Any Complementarity Problem Has an Equivalent

Variational Inequality Problem 1985.3.1.1 Illustrative Example. NCP and Two VI

Forms for Coal Yard Model 1985.3.2 Any Variational Inequality Problem Has an

Equivalent Complementarity Problem 2005.3.2.1 Illustrative Example. Comparison of MCP

and VI Forms of Coal Market Model withCoal Yard Limits 201

5.3.3 Alternative Equivalent Forms of VariationalInequality Problems 2025.3.3.1 Alternative Form of VI for Nash

Equilibrium with Linking Constraints 2045.3.3.2 Illustrative Example. Alternative VI for

Nash-Cournot Model of Coal Market withYard Limit 205

5.4 Generalized Nash Equilibrium as Quasi-variationalInequality Problem 2065.4.1 Some Important Properties of Quasi-variational

Inequality Problems 2095.4.1.1 The VI Solution is a QVI Solution: Linking

Duals are Equal 2095.4.1.2 Illustrative Example. Simple Electric

Capacity Market Model with High CostGreen Energy and Equal Prices for All 210

5.4.1.3 Modified VI: First Price-Directed Searchfor QVI Solutions ' 211

5.4.1.4 • Illustrative Example. Electric CapacityMarket Model with Subsidized Green Energy 212

5.4.1.5 Modified VI: Second Price-Directed Searchfor QVI Solutions 212

5.4.1.6 Illustrative Example. Electric CapacityMarket Model with Green Price a Multipleof Conventional Price 213

5.4.1.7 Modified VI: Resource-Directed Search forQVI Solutions j 214

5.4.1.8 Illustrative Example. Electric CapacityMarket Model with Quotas for Green andConventional 214

5.5 Summary 2155.6 Exercises 216

xviii Contents

References 219

6 Optimization Problems Constrained by Complementarityand Other Optimization Problems ^ 2216.1 Introduction 221

6.1.1 Practical Interest 2216.1.2 Structure and Basic Classification 222

6.2 Optimization Problems Constrained by Other OptimizationProblems, OPcOP 2236.2.1 General Formulation 2236.2.2 Illustrative Example. Strategic Offering, OPcOP 2266.2.3 Illustrative Example. Vulnerability Assessment,

OPcOP 2296.2.4 Illustrative Example. Transmission Investment,

OPcOP 2316.2.5 Basic Assumption: Constraining Problems are Convex 2356.2.6 Mathematical Program with Complementarity

Constraints, MPCC 2356.2.7 Illustrative Example. Vulnerability Assessment,

MPCC 2366.2.8 Mathematical Program with Equilibrium

Constraints, MPEC 2376.2.9 Illustrative Example. Strategic Offering, MPEC 2376.2.10 Illustrative Example. Transmission Investment, MPEC 2386.2.11 Stochastic OPcOPs 2396.2.12 Illustrative Example. Strategic Offering, sOPcOP 240

6.3 Optimization Problems Constrained by Linear Problems,OPcLP 2416.3.1 Mathematical Program with Primal and Dual

Constraints, MPPDC 2426.3.2- Illustrative Example. Strategic Offering, MPPDC 2436.3.3 Illustrative Example. Vulnerability Assessment,

MPPDC 2446.3.4 Illustrative Example. Transmission Investment,

MPPDC 2466.3.5 Mathematical Program with Complementarity

Constraints, MPCC 2476.3.6 Stochastic OPcLPs 248

: 6.3.7 Illustrative Example. Transmission Investment,sOPcLP .j 249

6.4 Transforming an MPCC/MPEC/MPPDC into a MILP 2506.4.1 Fortuny-Amat McCarl Linearization 2506.4.2 S0S1 and Penalty Function Linearization 2516.4.3 Other Linearizations 251

Contents xix

6.4.3.1 Illustrative Example. Strategic Offering:Exact Linear Transformation 252

6.5 Writing and Solving the KKTs of an MPPDC 2536.5.1 KKTs of an MPPDC .• 2536.5.2 Illustrative Example. Strategic Offering, KKTs 2546.5.3 Reformulating an MCP as an Optimization Problem . 2566.5.4 Illustrative Example. Strategic Offering: MCP

Optimization Problem 2576.6 Summary 2586.7 Exercises 258

References 261

7 Equilibrium Problems with Equilibrium Constraints 2637.1 Introduction 2637.2 The EPEC Problem 264

7.2.1 Problem Statement and Diagonalization Algorithm . . 2647.2.2 Diagonalization Applied to EPEC 267

7.3 Energy Applications of EPECs 2717.4 EPEC Power Market Model 1: Strategic Quantity Decisions

by Generators 2747.4.1 Model Formulation 274

7.4.1.1 Model Structural Assumptions 2747.4.1.2 Consumer 2767.4.1.3 Transmission Provider 2777.4.1.4 Follower Equilibrium 2777.4.1.5 Generator (Leader) MPEC 2787.4.1.6 EPEC 279

7.4.2 Illustrative Example 2797.4.2.1 Assumptions 279

>7.4.2.2 Follower Problem : 2807.4.2.3 .Leader Problems 2817.4.2.4 EPEC Statement and Analysis 2827.4.2.5 Attempted Solution by Diagonalization 2857.4.2.6 Mixed Strategy Solution 2867.4.2.7 Comparison of Outcomes of Alternative

Game Formulations 2877.4.2.8 Sensitivity Case: Single Oligopolist 2887.4.2.9 Sensitivity Case: Transmission Expansion . . . 2897.4.2.10 Summary of Cournot EPEC Example . . . . . . 290

7.5 EPEC Power Market Model 2: Strategic Offering by *Generators 2917.5.1 Model Formulation 291

7.5.1.1 Structural Assumptions 2917.5.1.2 Auctioneer (Transmission Provider) 292

xx Contents

7.5.1.3 Producer MPEC 2937.5.1.4 EPEC 294

7.5.2 Illustrative Example 2957.5.2.1 Assumptions l 2957.5.2.2 EPEC Formulation 2957.5.2.3 Application of Diagonalization 2967.5.2.4 Mixed Strategy Equilibrium Computation . . 2997.5.2.5 Comparison of Average MPEC Results,

EPEC Mixed Equilibrium, and CompetitiveEquilibrium 301

7.5.2.6 Pure Strategy Equilibrium 3017.6 Closed Loop Multistage Nash Equilibrium: Capacity

Expansion 3027.6.1 Introduction '. 3027.6.2 Stage 2 Equilibrium: The Commodity Market 303

7.6.2.1 Perfect Competition 3037.6.2.2 Cournot Competition 305

7.6.3 Stage 1 EPEC Problem 3067.6.4 Illustrative Example: Consumers prefer Cournot to

Bertrand Competition 3087.7 Summary : 3157.8 Exercises 315

References 319

8 Algorithms for LCPs, NCPs and Vis 3238.1 Introduction 3238.2 Algorithms for LCP Models 324

8.2.1 Lemke's Pivoting Method for LCPs 3258.2.1.1 General Background on Pivoting 325

<• 8.2.1.2 Illustrative Example. Pivoting in Simplex, Method for a Linear Program 326

8.2.1.3 Lemke's Method 3288.2.1.3.1 Illustrative Example. Simple

LCP from Chapter 1 3288.2.1.3.2 General Statement of Lemke's

Method 3298.2.1.3.3 Illustrative Example. Equilibrium

of two commodities 3318.2.1.3.4 Convergence of Lemke's Method .. 333

8.2.2 Iterative Methods for LCPs ? 3338.2.2.1 General Background on Matrix Splitting. . . . 3348.2.2.2 Matrix Splitting for the LCP 335

Contents

8.2.2.2.1 Illustrative Example. Matrixsplitting for two-commoditymodel: B = diagonal part of M .. 336

8.2.2.2.2 Illustrative Example. Matrixsplitting for two-commoditymodel: symmetric B 337

8.2.2.2.3 Convergence of Matrix SplittingAlgorithms for the LCP 338

8.2.2.3 Other Iterative Methods for LCPs 3408.3 Algorithms for NCP Models 340

8.3.1 Newton's Method for Systems of Smooth Equations . 3428.3.1.1 Undamped Newton Method for Smooth

Equations 3438.3.1.1.1 Illustrative Example. Solving two

equations in two unknowns byNewton's method 343

8.3.1.1.2 Convergence of the UndampedNewton Method for SmoothEquations 344

8.3.1.2 Damped Newton Methods for SmoothEquations 3458.3.1.2.1 Illustrative Example. Damping

Procedures to AccelerateConvergence 345

8.3.1.2.2 Convergence of the DampedNewton Method for SmoothEquations 347

8.3.2 Newton's Method for the NCP 3498.3.2.1 Constructing an Approximate LCP 3498.3.2.2 Solving the Approximate LCP 350

' 8.3.2.3 Getting Started: Solving the First• Approximate LCP 351

8.3.2.4 Two Examples Without Damping 3528.3.2.4.1 Illustrative Example. PATH

method for two-commodity LCP .. 3528.3.2.4.2 Illustrative Example. PATH

method for two-commodity NCP . 3538.3.2.5 Damping in the Newton Method for NCPs .. 355

8.3.2.5.1 Illustrative Example. Min-basedmerit function for two-commodityNCP V 356

8.3.2.5.2 Path Search between PreviousIterate and Newton Point 356

8.3.2.6 Summary and Overview of Other Featuresof the PATH Algorithm 358

i Contents

8.4 Algorithms for VI Models 3598.4.1 Solve Equivalent KKT System as MCP 3598.4.2 Iterative Methods: Sequential Optimization 360

8.4.2.1 Project Independence Evaluation System(PIES) 3608.4.2.1.1 Illustrative Example. Simple

PIES model and algorithm 3628.4.2.1.2 PIES-g Algorithm 3648.4.2.1.3 Convergence of PIES and PIES-g

Algorithms 3648.4.2.2 A Nonlinear Approximation of G -

Diagonalization Method 3658.4.2.2.1 Illustrative Example. The PIES-?

algorithm as diagonalizationmethod on a VI 366

8.4.2.3 Symmetric Linear Approximations of G . . . . 3678.4.2.4 Convergence of Diagonalization and

Symmetric Linear Approximation 3678.5 Summary 3688.6 Appendix: Introduction to Theory for PATH and Other

NCP Algorithms 3698.6.1 Projection Mappings 370

8.6.1.1 Illustrative Example. Projection Mappingfor B = Wl 370

8.6.1.2 Illustrative Example. Projection Mappingfor 5 a s a Rectangular Box 371

8.6.2 NCP Reformulated as Nonsmooth Equation UsingProjection Mapping 3718.6.2.1 Illustrative Example. Illustration of

Theorem 8.3 with x ^ z 372* 8.6.2.2 Illustrative Example. Illustration of

• Theorem 8.3 with x = z 3728.6.3 Some Useful Merit Functions and Corresponding

Nonsmooth Equations 3738.6.3.1 Merit Function Based on Min Function 3738.6.3.2 Merit Function Based on Norm of the

Normal Map 3748.6.3.3 Merit Function Based on Fischer-

Burmeister Function 3748.6.3.3.1 Illustrative Example. Fischer-j

Burmeister-based merit functionfor two-commodity NCP 375

8.6.3.4 Merit Function Based on Plus Function 3758.6.4 Damped Newton Method for NCP as Nonsmooth

Equation 376

Contents xxiii

8.6.5 Convergence of the PATH Algorithm 3778.6.6 Other Methods to Solve NCPs 377

8.7 Exercises -j 378

References 383

9 Some Advanced Algorithms for VI Decomposition,MPCCs and EPECs 3859.1 Introduction 3859.2 Decomposition Algorithms for Vis 386

9.2.1 Illustrative Example. Dantzig-Wolfe Decompositionof a Simple LP '. 386

9.2.2 Illustrative Example. Simplified Stochastic PowerModel from Chapters 4 and 5 392

9.2.3 Dantzig-Wolfe Decomposition of Vis 3949.2.3.1 Some Computational Enhancements to

Dantzig-Wolfe Decomposition of Vis 3979.2.4 Illustrative Example. Dantzig-Wolfe Decomposition

of Simplified Stochastic Power Model 3989.2.5 Simplicial Decomposition of Vis 4009.2.6 Illustrative Example. Simplicial Decomposition of

Simplified Stochastic Power Model 4029.2.7 Benders Decomposition of Vis 403

9.2.7.1 Illustrative Example. BendersDecomposition of a Simple LP 403

9.2.7.2 General Development of BendersDecomposition for Vis 404

9.2.7.3 Illustrative Example. BendersDecomposition of Simplified StochasticPower Model 406

9.2.8 . Cobweb Decomposition Method - No Master Problem 4119.3 Algorithms for _ Mathematical Programs with

Complementarity Constraints 4129.3.1 Why Are MPCCs Difficult to Solve? 4149.3.2 Applying Standard NLP Algorithms to MPCCs 415

9.3.2.1 Regularization of ComplementarityConstraints 415

9.3.2.2 Illustrative Example. RegularizationApplied to the Strategic Offer MPCC 416

9.3.2.3 Penalization of Complementarity Constraints 4179.3.2.4 Illustrative Example. Penalization Applied

to the Strategic Offer MPCC 4189.3.2.5 Sequential Quadratic Programming 4199.3.2.6 Illustrative Example. SQP Applied to the

Strategic Offer MPCC 419

xxiv Contents

9.3.2.7 Some Practical Advice 4209.3.3 Some Other Methods for MPCCs 420

9.4 Algorithms for Equilibrium Programs with EquilibriumConstraints (EPECs) ] 4229.4.1 Diagonalization Method for EPECs 4239.4.2 NLP Reformulation of EPECs 4249.4.3 Illustrative Example. A simple 2-Leader, 1-Follower

EPEC 4249.5 Summary 4299.6 Exercises 429

References 431

10 Natural Gas Market Modeling 43310.1 Introduction 43310.2 Natural Gas Market Models 43510.3 Engineering Considerations 43910.4 The Natural Gas Supply Chain and the Various Market Agents 440

10.4.1 Sectoral and Seasonal Aspects and Gas StorageOperator 441

10.4.2 Capacity Expansion.and Multi-Year Perspective 44310.4.3 Representation of Consumers and Strategic Versus

Non-Strategic Players 44310.4.4 Additional Players and Engineering Aspects 44510.4.5 Suppliers 445

10.4.5.1 Production 44510.4.5.2 Delivering Gas to the Market 44810.4.5.3 Supplier's Problem (Version 1: Production

and Export Functions) 45010.4.5.4 Storage Operations 452

k 10.4.5.5 Supplier's Problem (Version 2: Production,_ Export and Storage Functions) 454

10.4.6 Transportation 45610.4.7 A Model for the Whole Market 45710.4.8 Illustrative Example. Small Natural Gas Network

Equilibrium 45910.4.8.1 Overview 45910.4.8.2 Base Case 46110.4.8.3 Analysis of Storage 46510.4.8.4 Analysis of Total Gas Reserves Constraint . . 46610.4.8.5 Analysis of Contract Sales ? 468

10.5 Summary 46910.6 Exercises 470

References 473

Contents xxv

11 Electricity and Environmental Markets 47711.1 Introduction 47711.2 Transmission-Constrained Electricity Markets . .̂ 479

11.2.1 Short-Run, Perfectly Competitive Market 48011.2.2 Illustrative Example. Transmission-Constrained

Perfect Competition Equilibrium 48511.2.3 Oligopolistic Market: A Cournot Model 48911.2.4 Illustrative Example. Transmission-Constrained

Cournot Equilibrium 49511.3 Environmental Markets: Emissions Trading 497

11.3.1 A Simple Model of Emissions Trading amongProducers 499

11.3.2 Illustrative Example. Simple Source-BasedEmissions Trading Equilibrium 501

11.3.3 A Simple Model of Emissions Trading amongLoad-Serving Entities 503

11.3.4 Illustrative Example. Simple Load-Based MarketEquilibrium 504

11.3.5 Model Analysis: Equivalence of Source-Based andLoad Based Trading 506

11.4 Summary 50711.5 Exercises 508

References 511

12 Multicommodity Equilibrium Models: Accounting forDemand-Side Linkages 51512.1 Introduction 51512.2 Linkages among Multiple Energy Markets 51612.3 Demand Relations over Time 520

12.3.1 • Regulated Vertically Integrated Utility Model 52112.3.2 Unbundled Power Market with and without

Cross-Price Elasticities 52512.4 Multi-Sector Models with Demand Linkages 530

12.4.1 The Project Independence Evaluation System 53012.4.2 PIES Model Components 532

12.4.2.1 Consumers 53312.4.2.2 Fuel Producers 53412.4.2.3 Oil Refiners 53612.4.2.4 Shippers 53812.4.2.5 Market Clearing -. / . . . . 539

12.4.3 Assembling and Solving the PIES Model 54012.4.3.1 Market Equilibrium LCP 54012.4.3.2 Solution Approaches 542

xxvi Contents

12.4.4 PIES Equilibrium: Interpreting the Solutions of aMulticommodity Model with Demand Linkages 54312.4.4.1 Interpreting Solutions: Where Do Prices

Come From? .• 54412.4.4.2 Interpreting Solutions: Effects of Policy 55012.4.4.3 Comparison with Own-Elasticity Only

Results 55212.5 Summary 55612.6 Exercises 557

References 559

A Convex Sets and Functions 561

References 569

B GAMS codes 571

References 605

C DC Power Flow 607

References 611

D Natural Gas Engineering Considerations 613

References 617

List of Tables 619

List of Figures 623

Index ' : 625