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ОбложкаComplementarity modeling in energy markets / S.A.Gabriel et al. - New York: Springer, 2013. - xxvi, 629 p.: ill. - (International series in operations research & management science; vol.180). - Incl. bibl. ref. and index. - ISBN 978-1-4419-6122-8
Шифр: (И/У5-C73) 02
 

Место хранения: 02 | Отделение ГПНТБ СО РАН | Новосибирск

Оглавление / Contents
 
1    Introduction and Motivation ................................ 1
1.1  Introduction ............................................... 1
1.2  Complementarity Models: Motivation and Description ......... 1
     1.2.1  Illustrative Example. Three-Variable MCP ............ 4
     1.2.2  Illustrative Example. Nonlinear Program Expressed
            as an MCP ........................................... 5
     1.2.3  Illustrative Example. PIES Model .................... 7
     1.2.4  Illustrative Example. Nash-Cournot Duopoly in
            Energy Production, Two Simultaneous Optimization
            Problems ............................................ 8
     1.2.5  Illustrative Example. Generalized Nash Equilibria,
            Energy Production Duopoly .......................... 10
     1.2.6  Illustrative Example. Nash-Cournot Duopoly
            Expressed as a Variational Inequality .............. 11
     1.2.7  Illustrative Example. Energy Network with
            Multiple Players ................................... 12
     1.2.8  Illustrative Example. MPEC ......................... 16
1.3  Summary ................................................... 18
1.4  Appendix: Computational Issues for Selected Problems ...... 19
     1.4.1  Illustrative Example 1.2.1 ......................... 19
     1.4.2  Illustrative Example 1.2.4 ......................... 20
     1.4.3  Illustrative Example 1.2.5 ......................... 21
     1.4.4  Illustrative Example 1.2.7 ......................... 23
     References ................................................ 27

2    Optimality and Complementarity ............................ 31
2.1  Introduction .............................................. 31
2.2  Optimization Problems ..................................... 32
     2.2.1  Illustrative Example. Optimization Problem: Only
            Equality Constraints ............................... 33
     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 ....................... 37
     2.2.5  Illustrative Example. LP Problem: Primal-Dual
            Formulation ........................................ 38
2.3  Karush-Kuhn-Tucker Conditions ............................. 39
     2.3.1  Illustrative Example. KKT Conditions: Equality
            Constraints ........................................ 40
     2.3.2  Illustrative Example. KKT Conditions: Equality
            and Inequality Constraints ......................... 41
2.4  Constraint Qualifications ................................. 42
     2.4.1  Illustrative Example. Constraint Qualification:
            Regular Solution ................................... 43
     2.4.2  Illustrative Example. Constraint Qualification:
            Non-Regular Solution ............................... 43
2.5  Sufficiency Conditions .................................... 44
     2.5.1  Illustrative Example. Sufficiency Conditions ....... 45
2.6  Mixed Linear Complementarity Problem, MLCP ................ 46
     2.6.1  Illustrative Example. MLCP ......................... 46
2.7  Equilibrium Problems, EP .................................. 47
     2.7.1  Illustrative Example. Equilibrium Conditions: No
            Constraints ........................................ 49
     2.7.2  Illustrative Example. Equilibrium Conditions:
            Only  Equality Constraints ......................... 50
     2.7.3  Illustrative Example. Equilibrium Conditions:
            Equality and Inequality Constraints ................ 50
     2.7.4  Illustrative Example. Linear Equilibrium Problem ... 52
2.8  Mathematical Programs with Equilibrium Constraints, MPEC .. 53
     2.8.1  Illustrative Example. MPEC: Only Equality
            Constraints ........................................ 56
     2.8.2  Illustrative Example. MPEC: Both Equality and
            Inequality Constraints ............................. 58
2.9  Equilibrium Problems with Equilibrium Constraints, EPEC ... 60
     2.9.1  Illustrative Example. EPEC: Only Equality
            Constraints ........................................ 62
     2.9.2  Illustrative Example. EPEC: Both Equality and
            Inequality Constraints ............................. 64
2.10 Non-Convexity and Non-Regularity Issues ................... 66
2.11 Summary ................................................... 67
2.12 Exercises ................................................. 68
     References ................................................ 69

3    Some Microeconomic Principles ............................. 71
3.1  Introduction .............................................. 71
3.2  Basics of Supply and Demand ............................... 72
     3.2.1  Supply Curves ...................................... 72
     3.2.2  Demand Curves ...................................... 75
     3.2.3  Notion of Equilibrium as Intersection of Supply
            and Demand Curves .................................. 78
       3.2.3.1  Illustrative Example. Equilibrium in the
                Coal Market .................................... 79
       3.2.3.2  Illustrative Example. Changes in Consumers'
                and Producers' Surpluses Due to a Cartel ....... 80
     3.2.4  Non-Price Influences: Shifting Supply and Demand
            Curves ............................................. 81
     3.2.5  Multicommodity Equilibrium ......................... 83
       3.2.5.1  Illustrative Example. Simultaneous Equilibrium
                of Coal and Wood Markets ....................... 84
     3.2.6  Estimation of Parameters of Demand and Supply
            Functions .......................................... 84
       3.2.6.1  Тор-Down or Statistical Estimation on
                Observations ................................... 84
       3.2.6.2  Bottom-Up or Process-Based Estimation .......... 86
       3.2.6.3  Auctions ....................................... 87
3.3  Social Welfare Maximization ............................... 88
     3.3.1  Definition of Social Welfare in Single Commodity
            Models: Consumers' Plus Producers' Surpluses ....... 88
     3.3.2  Equilibrium as Maximization of Social Welfare in
            Single Commodity Models ............................ 89
       3.3.2.1  Illustrative Example. Equilibrium in Coal
                Market as Social Welfare Maximization .......... 90
     3.3.3  Pareto Efficiency Versus Social Welfare
            Optimization ....................................... 90
     3.3.4  Social Welfare in Multicommodity Models ............ 91
       3.3.4.1  Possible Difficulty to Integrate Inverse
                Demand Functions in Multicommodity Models ...... 91
       3.3.4.2  Illustrative Example. Impossibility of
                Integrating Inverse Demand Functions for
                Coal and Wood .................................. 92
       3.3.4.3  Measuring Changes in Social Welfare in
                Multicommodity Models .......................... 93
       3.3.4.4  Illustrative Example. Changes in Consumers'
                Surplus, for Wood and Coal, Due to a Tax on
                Coal ........................................... 93
3.4  Modeling Individual Players in Single Commodity Markets ... 94
     3.4.1  Profit-Maximization Problem for Price-Taking
            Firms, and Form of Equilibrium Problem ............. 94
     3.4.2  Perfect Versus Imperfect Competition ............... 97
       3.4.2.1  Illustrative Example. Three Price-Taking
                Firms: Social Welfare Maximization Model ....... 98
       3.4.2.2  Illustrative Example. Three Price-Taking
                Firms: Complementarity Model .................. 100
       3.4.2.3  Monopoly Model ................................ 101
       3.4.2.4  Illustrative Example. Three Firms Merged
                as One Firm: Monopoly Model ................... 102
       3.4.2.5  Nash-Cournot Model ............................ 103
       3.4.2.6  Illustrative Example. Nash-Cournot Model
                of Three Firms: Complementarity Model ......... 104
       3.4.2.7  Illustrative Example. Nash-Cournot Model
                of Three Firms: Optimization Model if
                Demand is Linear .............................. 107
       3.4.2.8  Illustrative Example. Mixed Behaviors:
                Firm 1 as Cournot, Firms 2 and 3 as Price-
                Takers ........................................ 108
       3.4.2.9  Illustrative Example. Mixed Behaviors:
                Firms 1 and 2 as Cournot, Firm 3 as Price-
                Taker ......................................... 108
       3.4.2.10 Bertrand Game ................................. 109
       3.4.2.11 Illustrative Example. Bertrand Model of
                Coal Market ................................... 109
       3.4.2.12 Cartels ....................................... 110
     3.4.3  Nash Versus Generalized Nash Equilibria ........... 111
       3.4.3.1  Illustrative Example. Generalized Nash
                Model for Coal Market: Limit on Coal Yard,
                with Government Allocation of Coal Yard
                Shares ........................................ 114
       3.4.3.2  Illustrative Example. Generalized Nash
                Model for Coal Market: Limit on Coal
                Yard, with Trading of Shares and Equal
                Marginal Utilities of Yard Shares ............. 115
       3.4.3.3  Illustrative Example. Generalized Nash
                Model for Coal Market: Limit on Coal
                Yard, with Auctioning of Shares and
                Unequal Marginal Utilities .................... 117
3.5  Multi-Level Games ........................................ 118
     3.5.1  Stackelberg Leader-Follower Games (MPECs) ......... 118
       3.5.1.1  Illustrative Example. Stackelberg MPEC
                with Firm 2 as Leader ......................... 119
       3.5.1.2  Illustrative Example. Stackelberg MPEC
                with Firms 1 and 2 Merged as One Leader ....... 119
     3.5.2  Multi-Leader Games (EPECs) ........................ 120
     Summary .................................................. 121
     Exercises ................................................ 122
     References ............................................... 125

4    Equilibria and Complementarity Problems .................. 127
4.1  Introduction ............................................. 127
4.2  Economics and Engineering Equilibria ..................... 129
     4.2.1  Equilibria in Dominant Actions .................... 129
       4.2.1.1  Illustrative Example. Energy Production
                Duopoly ....................................... 129
       4.2.1.2  Illustrative Example. Energy Production
                Duopoly, Я Changed from 1.5 to 2) ............. 130
     4.2.2  Nash Equilibria ................................... 131
       4.2.2.1  Illustrative Example. Energy Production
                Duopoly, Nash Equilibrium ..................... 131
       4.2.2.2  Illustrative Example. Energy Production
                Duopoly, /3 = 1, Additional Costs ............. 132
     4.2.3  Types of Game Theory Problems Considered .......... 132
     4.2.4  Mixed Versus Pure Equilibria ...................... 133
       4.2.4.1  Illustrative Example. Energy Production
                Bimatrix Game, Version 1 ...................... 135
       4.2.4.2  Illustrative Example. Energy Production
                Bimatrix Game, Version 2 ...................... 136
4.3  Duality in Optimization Versus Equilibria ................ 137
     4.3.1  Linear Programs as Equilibrium Problems ........... 137
       4.3.1.1  Illustrative Example. Energy Production
                Optimization Problem, One Player .............. 138
     4.3.2  Nonlinear Programs as Equilibrium Problems ........ 140
4.4  More About the Connection Between Optimization and
     Equilibrium Problems ..................................... 141
     4.4.1  Spatial Price Equilibrium Problem ................. 142
       4.4.1.1  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 ............... 148
       4.4.2.2  Illustrative Example. Extended Energy
                Production Optimization Derived from MCP ...... 150
     4.4.3  Equilibria with No Corresponding KKT-Based
            Optimization Problem .............................. 151
       4.4.3.1  Illustrative Example. Spatial Price
                Equilibrium, Version 2 ........................ 153
4.5  Selected Existence/Uniqueness Results for Equilibrium
     Problems ................................................. 155
4.6  Extensions to Equilibrium Problems ....................... 161
     4.6.1  Overview .......................................... 161
       4.6.1.1  Illustrative Example. Integer-Constrained
                Spatial Price Equilibrium ..................... 162
     4.6.2  Discretely-Constrained Mixed Linear
            Complementarity Problem ........................... 162
       4.6.2.1  Illustrative Example. Integer-Constrained
                Network Equilibrium ........................... 164
     4.6.3  Stochastic Equilibria  ............................ 166
       4.6.3.1  Generator/'s Problem .......................... 167
       4.6.3.2  Grid Owner's Problem .......................... 169
       4.6.3.3  Market Clearing ............................... 169
4.7  Summary .................................................. 170
4.8  Appendix: Computational Issues for Selected Problems ..... 170
     4.8.1  Computation of Nash Equilibrium Based on the
            Range for the Parameters .......................... 170
     4.8.2  Computations for Price Functions in Spatial
            Price Equilibrium-Version 2 ....................... 172
     4.8.3  Uniqueness of Spatial Price Equilibrium Version 2
            Solution .......................................... 173
4.9  Exercises ................................................ 174
     References ............................................... 177

5    Variational Inequality Problems .......................... 181
5.1  Introduction ............................................. 181
5.2  Formulation of Variational Inequality Problems ........... 182
     5.2.1  Optimization Problem as a VI Problem .............. 182
     5.2.2  VI Formulation of Nash Equilibrium: No Linking
            Constraints ....................................... 183
       5.2.2.1  Illustrative Example. Nash-Cournot Model
                of Coal Market from Chapter 3 ................. 185
     5.2.3  VI Formulation of Generalized Nash Equilibrium
            With Linking Constraints: A Special Case .......... 186
       5.2.3.1  Illustrative Example. Nash-Cournot Model
                of Coal Market with Coal Yard Limit from
                Chapter 3 ..................................... 189
       5.2.3.2  Illustrative Example. Competitive
                Equilibrium of Two Related Markets: Coal
                and Wood from Chapter 3 ....................... 190
       5.2.3.3  Illustrative Example. PIES Multicommodity
                Competitive Equilibrium Model from Chapter 1 .. 193
       5.2.3.4  Illustrative Example. Stochastic Equilibrium
                Model from Chapter 4 .......................... 194
5.3  Relations between Variational Inequality and
     Complementarity Problems ................................. 197
     5.3.1  Any Complementarity Problem Has an Equivalent
            Variational Inequality Problem .................... 198
       5.3.1.1  Illustrative Example. NCP and Two VI
                Forms for Coal Yard Model ..................... 198
     5.3.2  Any Variational Inequality Problem Has an
            Equivalent Complementarity Problem ................ 200
       5.3.2.1  Illustrative Example. Comparison of MCP
                and VI Forms of Coal Market Model with
                Coal Yard Limits .............................. 201
     5.3.3  Alternative Equivalent Forms of Variational
            Inequality Problems ............................... 202
       5.3.3.1  Alternative Form of VI for Nash Equilibrium
                with Linking Constraints ...................... 204
       5.3.3.2  Illustrative Example. Alternative VI for
                Nash-Cournot Model of Coal Market with
                Yard Limit .................................... 205
5.4  Generalized Nash Equilibrium as Quasi-variational
     Inequality Problem ....................................... 206
     5.4.1  Some Important Properties of Quasi-variational
            Inequality Problems ............................... 209
       5.4.1.1  The VI Solution is a QVI Solution: Linking
                Duals are Equal ............................... 209
       5.4.1.2  Illustrative Example. Simple Electric
                Capacity Market Model with High Cost
                Green Energy and Equal Prices for All ......... 210
       5.4.1.3  Modified VI: First Price-Directed Search
                for QVI Solutions ............................. 211
       5.4.1.4  Illustrative Example. Electric Capacity
                Market Model with Subsidized Green Energy ..... 212
       5.4.1.5  Modified VI: Second Price-Directed Search
                for QVI Solutions ............................. 212
       5.4.1.6  Illustrative Example. Electric Capacity
                Market Model with Green Price a Multiple
                of Conventional Price ......................... 213
       5.4.1.7  Modified VI: Resource-Directed Search for
                QVI Solutions ................................. 214
       5.4.1.8  Illustrative Example. Electric Capacity
                Market Model with Quotas for Green and
                Conventional .................................. 214
5.5  Summary .................................................. 215
5.6  Exercises ................................................ 216
     References ............................................... 219

6    Optimization Problems Constrained by Complementarity
     and Other Optimization Problems .......................... 221
6.1  Introduction ............................................. 221
     6.1.1  Practical Interest ................................ 221
     6.1.2  Structure and Basic Classification ................ 222
6.2  Optimization Problems Constrained by Other Optimization
     Problems, OPcOP .......................................... 223
     6.2.1  General Formulation ............................... 223
     6.2.2  Illustrative Example. Strategic Offering, OPcOP ... 226
     6.2.3  Illustrative Example. Vulnerability Assessment,
            OPcOP ............................................. 229
     6.2.4  Illustrative Example. Transmission Investment,
            OPcOP ............................................. 231
     6.2.5  Basic Assumption: Constraining Problems are
            Convex ............................................ 235
     6.2.6  Mathematical Program with Complementarity
            Constraints, MPCC ................................. 235
     6.2.7  Illustrative Example. Vulnerability Assessment,
            MPCC .............................................. 236
     6.2.8  Mathematical Program with Equilibrium
            Constraints, MPEC ................................. 237
     6.2.9  Illustrative Example. Strategic Offering, MPEC .... 237
     6.2.10 Illustrative Example. Transmission Investment,
            MPEC .............................................. 238
     6.2.11 Stochastic OPcOPs ................................. 239
     6.2.12 Illustrative Example. Strategic Offering, sOPcOP .. 240
6.3  Optimization Problems Constrained by Linear Problems,
     OPcLP .................................................... 241
     6.3.1  Mathematical Program with Primal and Dual
            Constraints, MPPDC ................................ 242
     6.3.2  Illustrative Example. Strategic Offering, MPPDC ... 243
     6.3.3  Illustrative Example. Vulnerability Assessment,
            MPPDC ............................................. 244
     6.3.4  Illustrative Example. Transmission Investment,
            MPPDC ............................................. 246
     6.3.5  Mathematical Program with Complementarity
            Constraints, MPCC ................................. 247
     6.3.6  Stochastic OPcLPs ................................. 248
     6.3.7  Illustrative Example. Transmission Investment,
            sOPcLP ............................................ 249
6.4  Transforming an MPCC/MPEC/MPPDC into a MILP .............. 250
     6.4.1  Fortuny-Amat McCarl Linearization ................. 250
     6.4.2  SOS1 and Penalty Function Linearization ........... 251
     6.4.3  Other Linearizations .............................. 251
       6.4.3.1  Illustrative Example. Strategic Offering:
                Exact Linear Transformation ................... 252
6.5  Writing and Solving the KKTs of an MPPDC ................. 253
     6.5.1  KKTs of an MPPDC .................................. 253
     6.5.2  Illustrative Example. Strategic Offering, KKTs .... 254
     6.5.3  Reformulating an MCP as an Optimization Problem ... 256
     6.5.4  Illustrative Example. Strategic Offering: MCP
            Optimization Problem .............................. 257
6.6  Summary .................................................. 258
6.7  Exercises ................................................ 258
     References ............................................... 261

7    Equilibrium Problems with Equilibrium Constraints ........ 263
7.1  Introduction ............................................. 263
7.2  The EPEC Problem ......................................... 264
     7.2.1  Problem Statement and Diagonalization Algorithm ... 264
     7.2.2  Diagonalization Applied to EPEC ................... 267
7.3  Energy Applications of EPECs ............................. 271
7.4  EPEC Power Market Model 1: Strategic Quantity Decisions
     by Generators ............................................ 274
     7.4.1  Model Formulation ................................. 274
       7.4.1.1  Model Structural Assumptions .................. 274
       7.4.1.2  Consumer ...................................... 276
       7.4.1.3  Transmission Provider ......................... 277
       7.4.1.4  Follower Equilibrium .......................... 277
       7.4.1.5  Generator (Leader) MPEC ....................... 278
       7.4.1.6  EPEC .......................................... 279
     7.4.2  Illustrative Example .............................. 279
       7.4.2.1  Assumptions ................................... 279
       7.4.2.2  Follower Problem .............................. 280
       7.4.2.3  Leader Problems ............................... 281
       7.4.2.4  EPEC Statement and Analysis ................... 282
       7.4.2.5  Attempted Solution by Diagonalization ......... 285
       7.4.2.6  Mixed Strategy Solution ....................... 286
       7.4.2.7  Comparison of Outcomes of Alternative
                Game Formulations ............................. 287
       7.4.2.8  Sensitivity Case: Single Oligopolist .......... 288
       7.4.2.9  Sensitivity Case: Transmission Expansion ...... 289
       7.4.2.10 Summary of Cournot EPEC Example ............... 290
7.5  EPEC Power Market Model 2: Strategic Offering by
     Generators ............................................... 291
     7.5.1  Model Formulation ................................. 291
       7.5.1.1  Structural Assumptions ........................ 291
       7.5.1.2  Auctioneer (Transmission Provider) ............ 292
       7.5.1.3  Producer MPEC ................................. 293
       7.5.1.4  EPEC .......................................... 294
     7.5.2  Illustrative Example .............................. 295
       7.5.2.1  Assumptions ................................... 295
       7.5.2.2  EPEC Formulation .............................. 295
       7.5.2.3  Application of Diagonalization ................ 296
       7.5.2.4  Mixed Strategy Equilibrium Computation ........ 299
       7.5.2.5  Comparison of Average MPEC Results,
                EPEC Mixed Equilibrium, and Competitive
                Equilibrium ................................... 301
       7.5.2.6  Pure Strategy Equilibrium ..................... 301
7.6  Closed Loop Multistage Nash Equilibrium: Capacity
     Expansion ................................................ 302
     7.6.1  Introduction ...................................... 302
     7.6.2  Stage 2 Equilibrium: The Commodity Market ......... 303
       7.6.2.1  Perfect Competition ........................... 303
       7.6.2.2  Cournot Competition ........................... 305
     7.6.3  Stage 1 EPEC Problem .............................. 306
     7.6.4  Illustrative Example: Consumers prefer Cournot
            to Bertrand Competition ........................... 308
7.7  Summary .................................................. 315
7.8  Exercises ................................................ 315
     References ............................................... 319

8    Algorithms for LCPs, NCPs and Vis ........................ 323
8.1  Introduction ............................................. 323
8.2  Algorithms for LCP Models ................................ 324
     8.2.1  Lemke's Pivoting Method for LCPs .................. 325
       8.2.1.1  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 ................................ 328
         8.2.1.3.1  Illustrative Example. Simple
                    LCP from Chapter 1 ........................ 328
         8.2.1.3.2  General Statement of Lemke's Method ....... 329
         8.2.1.3.3  Illustrative Example. Equilibrium
                    of two commodities ........................ 331
         8.2.1.3.4  Convergence of Lemke's Method ............. 333
     8.2.2  Iterative Methods for LCPs ........................ 333
       8.2.2.1  General Background on Matrix Splitting ........ 334
       8.2.2.2  Matrix Splitting for the LCP .................. 335
         8.2.2.2.1  Illustrative Example. Matrix splitting
                    for two-commodity model: В = diagonal
                    part of M ................................. 336
         8.2.2.2.2  Illustrative Example. Matrix splitting
                    for two-commodity model: symmetric В ...... 337
         8.2.2.2.3  Convergence of Matrix Splitting
                    Algorithms for the LCP .................... 338
       8.2.2.3  Other Iterative Methods for LCPs .............. 340
8.3  Algorithms for NCP Models ................................ 340
     8.3.1  Newton's Method for Systems of Smooth Equations ... 342
       8.3.1.1  Undamped Newton Method for Smooth Equations ... 343
         8.3.1.1.1  Illustrative Example. Solving two
                    equations in two unknowns by Newton's
                    method .................................... 343
         8.3.1.1.2  Convergence of the Undamped Newton
                    Method for Smooth Equations ............... 344
       8.3.1.2  Damped Newton Methods for Smooth Equations .... 345
         8.3.1.2.1  Illustrative Example. Damping Procedures
                    to Accelerate Convergence ................. 345
         8.3.1.2.2  Convergence of the Damped Newton Method
                    for Smooth Equations ...................... 347
     8.3.2  Newton's Method for the NCP ....................... 349
       8.3.2.1  Constructing an Approximate LCP ............... 349
       8.3.2.2  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 .................. 352
         8.3.2.4.1  Illustrative Example. PATH method for
                    two-commodity LCP ......................... 352
         8.3.2.4.2  Illustrative Example. PATH method for
                    two-commodity NCP ......................... 353
       8.3.2.5  Damping in the Newton Method for NCPs ......... 355
       8.3.2.5.1  Illustrative Example. Min-based merit
                  function for two-commodity NCP .............. 356
         8.3.2.5.2  Path Search between Previous Iterate and
                    Newton Point .............................. 356
       8.3.2.6  Summary and Overview of Other Features
                of the PATH Algorithm ......................... 358
8.4  Algorithms for VI Models ................................. 359
     8.4.1  Solve Equivalent KKT System as MCP ................ 359
     8.4.2  Iterative Methods: Sequential Optimization ........ 360
       8.4.2.1  Project Independence Evaluation System
                (PIES) ........................................ 360
         8.4.2.1.1  Illustrative Example. Simple PIES model
                    and algorithm ............................. 362
         8.4.2.1.2  PIES-g Algorithm .......................... 364
         8.4.2.1.3  Convergence of PIES and PIES-g
                    Algorithms ................................ 364
       8.4.2.2  A Nonlinear Approximation of G -
                Diagonalization Method ........................ 365
         8.4.2.2.1  Illustrative Example. The PIES-g
                    algorithm as diagonalization method on
                    a VI ...................................... 366
       8.4.2.3  Symmetric Linear Approximations of G .......... 367
       8.4.2.4  Convergence of Diagonalization and Symmetric
                Linear Approximation .......................... 367
8.5  Summary .................................................. 368
8.6  Appendix: Introduction to Theory for PATH and Other
     NCP Algorithms ........................................... 369
     8.6.1  Projection Mappings ............................... 370
       8.6.1.1  Illustrative Example. Projection Mapping
                for В = Ж1 .................................... 370
       8.6.1.2  Illustrative Example. Projection Mapping
                for В as a Rectangular Box .................... 371
     8.6.2  NCP Reformulated as Nonsmooth Equation Using
            Projection Mapping ................................ 371
       8.6.2.1  Illustrative Example. Illustration of
                Theorem 8.3 with хф z ......................... 372
       8.6.2.2  Illustrative Example. Illustration of
                Theorem 8.3 with x = z ........................ 372
     8.6.3  Some Useful Merit Functions and Corresponding
            Nonsmooth Equations ............................... 373
       8.6.3.1  Merit Function Based on Min Function .......... 373
       8.6.3.2  Merit Function Based on Norm of the
                Normal Map .................................... 374
       8.6.3.3  Merit Function Based on Fischer-Burmeister
                Function ...................................... 374
         8.6.3.3.1  Illustrative Example. Fischer-
                    Burmeister-based merit function for two-
                    commodity NCP ............................. 375
       8.6.3.4  Merit Function Based on Plus Function ......... 375
     8.6.4  Damped Newton Method for NCP as Nonsmooth
            Equation .......................................... 376
     8.6.5  Convergence of the PATH Algorithm ................. 377
     8.6.6  Other Methods to Solve NCPs ....................... 377
8.7  Exercises ................................................ 378
     References ............................................... 383

9    Some Advanced Algorithms for VI Decomposition,
     MPCCs and EPECs .......................................... 385
9.1  Introduction ............................................. 385
9.2  Decomposition Algorithms for Vis ......................... 386
     9.2.1  Illustrative Example. Dantzig-Wolfe
            Decomposition of a Simple LP ...................... 386
     9.2.2  Illustrative Example. Simplified Stochastic
            Power Model from Chapters 4 and 5 ................. 392
     9.2.3  Dantzig-Wolfe Decomposition of Vis ................ 394
       9.2.3.1  Some Computational Enhancements to
                Dantzig-Wolfe Decomposition of Vis ............ 397
     9.2.4  Illustrative Example. Dantzig-Wolfe
            Decomposition of Simplified Stochastic Power
            Model ............................................. 398
     9.2.5  Simplicial Decomposition of Vis ................... 400
     9.2.6  Illustrative Example. Simplicial Decomposition
            of Simplified Stochastic Power Model .............. 402
     9.2.7  Benders Decomposition of Vis ...................... 403
       9.2.7.1  Illustrative Example. Benders Decomposition
                of a Simple LP ................................ 403
       9.2.7.2  General Development of Benders Decomposition
                for Vis ....................................... 404
       9.2.7.3  Illustrative Example. Benders Decomposition
                of Simplified Stochastic Power Model .......... 406
     9.2.8  Cobweb Decomposition Method - No Master Problem ... 411
9.3  Algorithms for Mathematical Programs with
     Complementarity Constraints .............................. 412
     9.3.1  Why Are MPCCs Difficult to Solve? ................. 414
     9.3.2  Applying Standard NLP Algorithms to MPCCs ......... 415
       9.3.2.1  Regularization of Complementarity
                Constraints ................................... 415
       9.3.2.2  Illustrative Example. Regularization
                Applied to the Strategic Offer MPCC ........... 416
       9.3.2.3  Penalization of Complementarity Constraints ... 417
       9.3.2.4  Illustrative Example. Penalization Applied
                to the Strategic Offer MPCC ................... 418
       9.3.2.5  Sequential Quadratic Programming .............. 419
       9.3.2.6  Illustrative Example. SQP Applied to the
                Strategic Offer MPCC .......................... 419
       9.3.2.7  Some Practical Advice ......................... 420
     9.3.3  Some Other Methods for MPCCs ...................... 420
9.4  Algorithms for Equilibrium Programs with Equilibrium
     Constraints (EPECs) ...................................... 422
     9.4.1  Diagonalization Method for EPECs .................. 423
     9.4.2  NLP Reformulation of EPECs ........................ 424
     9.4.3  Illustrative Example. A simple 2-Leader,
            1-Follower EPEC ................................... 424
9.5  Summary .................................................. 429
9.6  Exercises ................................................ 429
     References ............................................... 431

10   Natural Gas Market Modeling .............................. 433
10.1 Introduction ............................................. 433
10.2 Natural Gas Market Models ................................ 435
10.3 Engineering Considerations ............................... 439
10.4 The Natural Gas Supply Chain and the Various Market
     Agents ................................................... 440
     10.4.1 Sectoral and Seasonal Aspects and Gas Storage
            Operator .......................................... 441
     10.4.2 Capacity Expansion and Multi-Year Perspective ..... 443
     10.4.3 Representation of Consumers and Strategic Versus
            Non-Strategic Players ............................. 443
     10.4.4 Additional Players and Engineering Aspects ........ 445
     10.4.5 Suppliers ......................................... 445
       10.4.5.1 Production .................................... 445
       10.4.5.2 Delivering Gas to the Market .................. 448
       10.4.5.3 Supplier's Problem (Version 1: Production
                and Export Functions) ......................... 450
         10.4.5.4 Storage Operations .......................... 452
         10.4.5.5 Supplier's Problem (Version 2: Production,
                  Export and Storage Functions) ............... 454
     10.4.6 Transportation .................................... 456
     10.4.7 A Model for the Whole Market ...................... 457
     10.4.8 Illustrative Example. Small Natural Gas Network
            Equilibrium ....................................... 459
       10.4.8.1  Overview ..................................... 459
       10.4.8.2  Base Case .................................... 461
       10.4.8.3  Analysis of Storage .......................... 465
       10.4.8.4  Analysis of Total Gas Reserves Constraint .... 466
       10.4.8.5  Analysis of Contract Sales ................... 468
10.5 Summary .................................................. 469
10.6 Exercises ................................................ 470
     References ............................................... 473

11   Electricity and Environmental Markets .................... 477
11.1 Introduction ............................................. 477
11.2 Transmission-Constrained Electricity Markets ............. 479
     11.2.1 Sfiort-Run, Perfectly Competitive Market .......... 480
     11.2.2 Illustrative Example. Transmission-Constrained
            Perfect Competition Equilibrium ................... 485
     11.2.3 Oligopolistic Market: A Cournot Model ............. 489
     11.2.4 Illustrative Example. Transmission-Constrained
            Cournot Equilibrium ............................... 495
11.3 Environmental Markets: Emissions Trading ................. 497
     11.3.1 A Simple Model of Emissions Trading among
            Producers ......................................... 499
     11.3.2 Illustrative Example. Simple Source-Based
            Emissions Trading Equilibrium ..................... 501
     11.3.3 A Simple Model of Emissions Trading among
            Load-Serving Entities ............................. 503
     11.3.4 Illustrative Example. Simple Load-Based Market
            Equilibrium ....................................... 504
     11.3.5 Model Analysis: Equivalence of Source-Based and
            Load Based Trading ................................ 506
11.4 Summary .................................................. 507
11.5 Exercises ................................................ 508
     References ............................................... 511

12   Multicommodity Equilibrium Models: Accounting for
     Demand-Side Linkages ..................................... 515
12.1 Introduction ............................................. 515
12.2 Linkages among Multiple Energy Markets ................... 516
12.3 Demand Relations over Time ............................... 520
     12.3.1 Regulated Vertically Integrated Utility Model ..... 521
     12.3.2 Unbundled Power Market with and without
            Cross-Price Elasticities .......................... 525
12.4 Multi-Sector Models with Demand Linkages ................. 530
     12.4.1 The Project Independence Evaluation System ........ 530
     12.4.2 PIES Model Components ............................. 532
       12.4.2.1 Consumers ..................................... 533
       12.4.2.2 Fuel Producers ................................ 534
       12.4.2.3 Oil Refiners .................................. 536
       12.4.2.4 Shippers ...................................... 538
       12.4.2.5 Market Clearing ............................... 539
     12.4.3 Assembling and Solving the PIES Model ............. 540
       12.4.3.1 Market Equilibrium LCP ........................ 540
       12.4.3.2 Solution Approaches ........................... 542
     12.4.4  PIES Equilibrium: Interpreting the Solutions of
             a Multicommodity Model with Demand Linkages ...... 543
       12.4.4.1 Interpreting Solutions: Where Do Prices
                Come From? .................................... 544
       12.4.4.2 Interpreting Solutions: Effects of Policy ..... 550
       12.4.4.3 Comparison with Own-Elasticity Only Results ... 552
12.5 Summary .................................................. 556
12.6 Exercises ................................................ 557
     References ............................................... 559

A    Convex Sets and Functions ................................ 561
     References ............................................... 569
В    GAMS codes ............................................... 571
     References ............................................... 605
С    DC Power Flow ............................................ 607
     References ............................................... 611
D    Natural Gas Engineering Considerations ................... 613
     References ............................................... 617

List of Tables ................................................ 619
List of Figures ............................................... 623
Index ......................................................... 625

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