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E-raamat: Complementarity Modeling in Energy Markets

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Complementarity Modeling in Energy MarketsSuurem pilt
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In a straightforward and approachable manner, this book introduces complementarity models, and uses them to carry out an in-depth analysis of energy markets, including formulation issues and solution techniques.

This addition to the ISOR series introduces complementarity models in a straightforward and approachable manner and uses them to carry out an in-depth analysis of energy markets, including formulation issues and solution techniques. In a nutshell, complementarity models generalize: a. optimization problems via their Karush-Kuhn-Tucker conditions b. on-cooperative games in which each player may be solving a separate but related optimization problem with potentially overall system constraints (e.g., market-clearing conditions) c. conomic and engineering problems that aren’t specifically derived from optimization problems (e.g., spatial price equilibria) d. roblems in which both primal and dual variables (prices) appear in the original formulation (e.g., The National Energy Modeling System (NEMS) or its precursor, PIES). As such, complementarity models are a very general and flexible modeling format. A natural question is why concentrate on energy markets for this complementarity approach? s it turns out, energy or other markets that have game theoretic aspects are best modeled by complementarity problems. The reason is that the traditional perfect competition approach no longer applies due to deregulation and restructuring of these markets and thus the corresponding optimization problems may no longer hold. Also, in some instances it is important in the original model formulation to involve both primal variables (e.g., production) as well as dual variables (e.g., market prices) for public and private sector energy planning. Traditional optimization problems can not directly handle this mixing of primal and dual variables but complementarity models can and this makes them all that more effective for decision-makers.

Arvustused

From the reviews:

This book is to address the topic of complementarity modeling to make these modeling approaches widely accessible to a larger audience. I strongly recommend it for energy practitioners and researchers in energy market modeling . I am convinced that they are a fruitful and challenging area of research, which will serve as a guide to real-world applications. Therefore, I strongly recommend this book to all modelers in energy systems and markets. (Dominik Mst, Interfaces, Vol. 43 (6), November-December, 2013)

This monograph represents the first comprehensive overview of electricity, environmental, and natural gas market models through the use of complementarity techniques. it provides a wonderful set of examples that are crucial in conveying the material. this monograph represents an indispensable and enduring resource for any serious researcher in energy markets. (Uday V. Shanbhag, Mathematical Reviews, October, 2013)

1 Introduction and Motivation
1(30)
1.1 Introduction
1(1)
1.2 Complementarity Models: Motivation and Description
1(17)
1.2.1 Illustrative Example. Three-Variable MCP
4(1)
1.2.2 Illustrative Example. Nonlinear Program Expressed as an MCP
5(2)
1.2.3 Illustrative Example. PIES Model
7(1)
1.2.4 Illustrative Example. Nash-Cournot Duopoly in Energy Production, Two Simultaneous Optimization Problems
8(2)
1.2.5 Illustrative Example. Generalized Nash Equilibria, Energy Production Duopoly
10(1)
1.2.6 Illustrative Example. Nash-Cournot Duopoly Expressed as a Variational Inequality
11(1)
1.2.7 Illustrative Example. Energy Network with Multiple Players
12(4)
1.2.8 Illustrative Example. MPEC
16(2)
1.3 Summary
18(1)
1.4 Appendix: Computational Issues for Selected Problems
19(12)
1.4.1 Illustrative Example 1.2.1
19(1)
1.4.2 Illustrative Example 1.2.4
20(1)
1.4.3 Illustrative Example 1.2.5
21(2)
1.4.4 Illustrative Example 1.2.7
23(4)
References
27(4)
2 Optimality and Complementarity
31(40)
2.1 Introduction
31(1)
2.2 Optimization Problems
32(7)
2.2.1 Illustrative Example. Optimization Problem: Only Equality Constraints
33(2)
2.2.2 Illustrative Example. Optimization Problem: Unconstrained
35(1)
2.2.3 Illustrative Example. Optimization Problem: Equality and Inequality Constraints
35(2)
2.2.4 Linear Optimization Problems
37(1)
2.2.5 Illustrative Example. LP Problem: Primal-Dual Formulation
38(1)
2.3 Karush-Kuhn-Tucker Conditions
39(3)
2.3.1 Illustrative Example. KKT Conditions: Equality Constraints
40(1)
2.3.2 Illustrative Example. KKT Conditions: Equality and Inequality Constraints
41(1)
2.4 Constraint Qualifications
42(2)
2.4.1 Illustrative Example. Constraint Qualification: Regular Solution
43(1)
2.4.2 Illustrative Example. Constraint Qualification: Non-Regular Solution
43(1)
2.5 Sufficiency Conditions
44(2)
2.5.1 Illustrative Example. Sufficiency Conditions
45(1)
2.6 Mixed Linear Complementarity Problem, MLCP
46(1)
2.6.1 Illustrative Example. MLCP
46(1)
2.7 Equilibrium Problems, EP
47(6)
2.7.1 Illustrative Example. Equilibrium Conditions: No Constraints
49(1)
2.7.2 Illustrative Example. Equilibrium Conditions: Only Equality Constraints
50(1)
2.7.3 Illustrative Example. Equilibrium Conditions: Equality and Inequality Constraints
50(2)
2.7.4 Illustrative Example. Linear Equilibrium Problem
52(1)
2.8 Mathematical Programs with Equilibrium Constraints, MPEC
53(7)
2.8.1 Illustrative Example. MPEC: Only Equality Constraints
56(2)
2.8.2 Illustrative Example. MPEC: Both Equality and Inequality Constraints
58(2)
2.9 Equilibrium Problems with Equilibrium Constraints, EPEC
60(6)
2.9.1 Illustrative Example. EPEC: Only Equality Constraints
62(2)
2.9.2 Illustrative Example. EPEC: Both Equality and Inequality Constraints
64(2)
2.10 Non-Convexity and Non-Regularity Issues
66(1)
2.11 Summary
67(1)
2.12 Exercises
68(3)
References
69(2)
3 Some Microeconomic Principles
71(56)
3.1 Introduction
71(1)
3.2 Basics of Supply and Demand
72(16)
3.2.1 Supply Curves
72(3)
3.2.2 Demand Curves
75(3)
3.2.3 Notion of Equilibrium as Intersection of Supply and Demand Curves
78(1)
3.2.3.1 Illustrative Example. Equilibrium in the Coal Market
79(1)
3.2.3.2 Illustrative Example. Changes in Consumers' and Producers' Surpluses Due to a Cartel
80(1)
3.2.4 Non-Price Influences: Shifting Supply and Demand Curves
81(2)
3.2.5 Multicommodity Equilibrium
83(1)
3.2.5.1 Illustrative Example. Simultaneous Equilibrium of Coal and Wood Markets
84(1)
3.2.6 Estimation of Parameters of Demand and Supply Functions
84(1)
3.2.6.1 Top-Down or Statistical Estimation on Observations
84(2)
3.2.6.2 Bottom-Up or Process-Based Estimation
86(1)
3.2.6.3 Auctions
87(1)
3.3 Social Welfare Maximization
88(6)
3.3.1 Definition of Social Welfare in Single Commodity Models: Consumers' Plus Producers' Surpluses
88(1)
3.3.2 Equilibrium as Maximization of Social Welfare in Single Commodity Models
89(1)
3.3.2.1 Illustrative Example. Equilibrium in Coal Market as Social Welfare Maximization
90(1)
3.3.3 Pareto Efficiency Versus Social Welfare Optimization
90(1)
3.3.4 Social Welfare in Multicommodity Models
91(1)
3.3.4.1 Possible Difficulty to Integrate Inverse Demand Functions in Multicommodity Models
91(1)
3.3.4.2 Illustrative Example. Impossibility of Integrating Inverse Demand Functions for Coal and Wood
92(1)
3.3.4.3 Measuring Changes in Social Welfare in Multicommodity Models
93(1)
3.3.4.4 Illustrative Example. Changes in Consumers' Surplus, for Wood and Coal, Due to a Tax on Coal
93(1)
3.4 Modeling Individual Players in Single Commodity Markets
94(24)
3.4.1 Profit-Maximization Problem for Price-Taking Firms, and Form of Equilibrium Problem
94(3)
3.4.2 Perfect Versus Imperfect Competition
97(1)
3.4.2.1 Illustrative Example. Three Price-Taking Firms: Social Welfare Maximization Model
98(2)
3.4.2.2 Illustrative Example. Three Price-Taking Firms: Complementarity Model
100(1)
3.4.2.3 Monopoly Model
101(1)
3.4.2.4 Illustrative Example. Three Firms Merged as One Firm: Monopoly Model
102(1)
3.4.2.5 Nash-Cournot Model
103(1)
3.4.2.6 Illustrative Example. Nash-Cournot Model of Three Firms: Complementarity Model
104(3)
3.4.2.7 Illustrative Example. Nash-Cournot Model of Three Firms: Optimization Model if Demand is Linear
107(1)
3.4.2.8 Illustrative Example. Mixed Behaviors: Firm 1 as Cournot, Firms 2 and 3 as Price-Takers
108(1)
3.4.2.9 Illustrative Example. Mixed Behaviors: Firms 1 and 2 as Cournot, Firm 3 as Price-Taker
108(1)
3.4.2.10 Bertrand Game
109(1)
3.4.2.11 Illustrative Example. Bertrand Model of Coal Market
109(1)
3.4.2.12 Cartels
110(1)
3.4.3 Nash Versus Generalized Nash Equilibria
111(3)
3.4.3.1 Illustrative Example. Generalized Nash Model for Coal Market: Limit on Coal Yard, with Government Allocation of Coal Yard Shares
114(1)
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(2)
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(1)
3.5 Multi-Level Games
118(3)
3.5.1 Stackelberg Leader-Follower Games (MPECs)
118(1)
3.5.1.1 Illustrative Example. Stackelberg MPEC with Firm 2 as Leader
119(1)
3.5.1.2 Illustrative Example. Stackelberg MPEC with Firms 1 and 2 Merged as One Leader
119(1)
3.5.2 Multi-Leader Games (EPECs)
120(1)
3.6 Summary
121(1)
3.7 Exercises
122(5)
References
125(2)
4 Equilibria and Complementarity Problems
127(54)
4.1 Introduction
127(2)
4.2 Economics and Engineering Equilibria
129(8)
4.2.1 Equilibria in Dominant Actions
129(1)
4.2.1.1 Illustrative Example. Energy Production Duopoly
129(1)
4.2.1.2 Illustrative Example. Energy Production Duopoly, β Changed from 1.5 to 2)
130(1)
4.2.2 Nash Equilibria
131(1)
4.2.2.1 Illustrative Example. Energy Production Duopoly, Nash Equilibrium
131(1)
4.2.2.2 Illustrative Example. Energy Production Duopoly, β = 1, Additional Costs
132(1)
4.2.3 Types of Game Theory Problems Considered
132(1)
4.2.4 Mixed Versus Pure Equilibria
133(2)
4.2.4.1 Illustrative Example. Energy Production Bimatrix Game, Version 1
135(1)
4.2.4.2 Illustrative Example. Energy Production Bimatrix Game, Version 2
136(1)
4.3 Duality in Optimization Versus Equilibria
137(4)
4.3.1 Linear Programs as Equilibrium Problems
137(1)
4.3.1.1 Illustrative Example. Energy Production Optimization Problem, One Player
138(2)
4.3.2 Nonlinear Programs as Equilibrium Problems
140(1)
4.4 More About the Connection Between Optimization and Equilibrium Problems
141(14)
4.4.1 Spatial Price Equilibrium Problem
142(1)
4.4.1.1 Illustrative Example. Spatial Price Equilibrium for Energy Products
143(3)
4.4.2 Optimization Problems from Equilibrium Conditions?
146(2)
4.4.2.1 Illustrative Example. Extended Energy Production Optimization Problem
148(2)
4.4.2.2 Illustrative Example. Extended Energy Production Optimization Derived from MCP
150(1)
4.4.3 Equilibria with No Corresponding KKT-Based Optimization Problem
151(2)
4.4.3.1 Illustrative Example. Spatial Price Equilibrium, Version 2
153(2)
4.5 Selected Existence/Uniqueness Results for Equilibrium Problems
155(6)
4.6 Extensions to Equilibrium Problems
161(9)
4.6.1 Overview
161(1)
4.6.1.1 Illustrative Example. Integer-Constrained Spatial Price Equilibrium
162(1)
4.6.2 Discretely-Constrained Mixed Linear Complementarity Problem
162(2)
4.6.2.1 Illustrative Example. Integer-Constrained Network Equilibrium
164(2)
4.6.3 Stochastic Equilibria
166(1)
4.6.3.1 Generator f's Problem
167(2)
4.6.3.2 Grid Owner's Problem
169(1)
4.6.3.3 Market Clearing
169(1)
4.7 Summary
170(1)
4.8 Appendix: Computational Issues for Selected Problems
170(4)
4.8.1 Computation of Nash Equilibrium Based on the Range for the Parameters
170(2)
4.8.2 Computations for Price Functions in Spatial Price Equilibrium-Version 2
172(1)
4.8.3 Uniqueness of Spatial Price Equilibrium Version 2 Solution
173(1)
4.9 Exercises
174(7)
References
177(4)
5 Variational Inequality Problems
181(40)
5.1 Introduction
181(1)
5.2 Formulation of Variational Inequality Problems
182(15)
5.2.1 Optimization Problem as a VI Problem
182(1)
5.2.2 VI Formulation of Nash Equilibrium: No Linking Constraints
183(2)
5.2.2.1 Illustrative Example. Nash-Cournot Model of Coal Market from
Chapter 3
185(1)
5.2.3 VI Formulation of Generalized Nash Equilibrium With Linking Constraints: A Special Case
186(3)
5.2.3.1 Illustrative Example. Nash-Cournot Model of Coal Market with Coal Yard Limit from
Chapter 3
189(1)
5.2.3.2 Illustrative Example. Competitive Equilibrium of Two Related Markets: Coal and Wood from
Chapter 3
190(3)
5.2.3.3 Illustrative Example. PIES Multicommodity Competitive Equilibrium Model from
Chapter 1
193(1)
5.2.3.4 Illustrative Example. Stochastic Equilibrium Model from
Chapter 4
194(3)
5.3 Relations between Variational Inequality and Complementarity Problems
197(9)
5.3.1 Any Complementarity Problem Has an Equivalent Variational Inequality Problem
198(1)
5.3.1.1 Illustrative Example. NCP and Two VI Forms for Coal Yard Model
198(2)
5.3.2 Any Variational Inequality Problem Has an Equivalent Complementarity Problem
200(1)
5.3.2.1 Illustrative Example. Comparison of MCP and VI Forms of Coal Market Model with Coal Yard Limits
201(1)
5.3.3 Alternative Equivalent Forms of Variational Inequality Problems
202(2)
5.3.3.1 Alternative Form of VI for Nash Equilibrium with Linking Constraints
204(1)
5.3.3.2 Illustrative Example. Alternative VI for Nash-Cournot Model of Coal Market with Yard Limit
205(1)
5.4 Generalized Nash Equilibrium as Quasi-variational Inequality Problem
206(9)
5.4.1 Some Important Properties of Quasi-variational Inequality Problems
209(1)
5.4.1.1 The VI Solution is a QVI Solution: Linking Duals are Equal
209(1)
5.4.1.2 Illustrative Example. Simple Electric Capacity Market Model with High Cost Green Energy and Equal Prices for All
210(1)
5.4.1.3 Modified VI: First Price-Directed Search for QVI Solutions
211(1)
5.4.1.4 Illustrative Example. Electric Capacity Market Model with Subsidized Green Energy
212(1)
5.4.1.5 Modified VI: Second Price-Directed Search for QVI Solutions
212(1)
5.4.1.6 Illustrative Example. Electric Capacity Market Model with Green Price a Multiple of Conventional Price
213(1)
5.4.1.7 Modified VI: Resource-Directed Search for QVI Solutions
214(1)
5.4.1.8 Illustrative Example. Electric Capacity Market Model with Quotas for Green and Conventional
214(1)
5.5 Summary
215(1)
5.6 Exercises
216(5)
References
219(2)
6 Optimization Problems Constrained by Complementarity and Other Optimization Problems
221(42)
6.1 Introduction
221(2)
6.1.1 Practical Interest
221(1)
6.1.2 Structure and Basic Classification
222(1)
6.2 Optimization Problems Constrained by Other Optimization Problems, OPcOP
223(18)
6.2.1 General Formulation
223(3)
6.2.2 Illustrative Example. Strategic Offering, OPcOP
226(3)
6.2.3 Illustrative Example. Vulnerability Assessment, OPcOP
229(2)
6.2.4 Illustrative Example. Transmission Investment, OPcOP
231(4)
6.2.5 Basic Assumption: Constraining Problems are Convex
235(1)
6.2.6 Mathematical Program with Complementarity Constraints, MPCC
235(1)
6.2.7 Illustrative Example. Vulnerability Assessment, MPCC
236(1)
6.2.8 Mathematical Program with Equilibrium Constraints, MPEC
237(1)
6.2.9 Illustrative Example. Strategic Offering, MPEC
237(1)
6.2.10 Illustrative Example. Transmission Investment, MPEC
238(1)
6.2.11 Stochastic OPcOPs
239(1)
6.2.12 Illustrative Example. Strategic Offering, sOPcOP
240(1)
6.3 Optimization Problems Constrained by Linear Problems, OPcLP
241(9)
6.3.1 Mathematical Program with Primal and Dual Constraints, MPPDC
242(1)
6.3.2 Illustrative Example. Strategic Offering, MPPDC
243(1)
6.3.3 Illustrative Example. Vulnerability Assessment, MPPDC
244(2)
6.3.4 Illustrative Example. Transmission Investment, MPPDC
246(1)
6.3.5 Mathematical Program with Complementarity Constraints, MPCC
247(1)
6.3.6 Stochastic OPcLPs
248(1)
6.3.7 Illustrative Example. Transmission Investment, sOPcLP
249(1)
6.4 Transforming an MPCC/MPEC/MPPDC into a MILP
250(3)
6.4.1 Fortuny-Amat McCarl Linearization
250(1)
6.4.2 SOS1 and Penalty Function Linearization
251(1)
6.4.3 Other Linearizations
251(1)
6.4.3.1 Illustrative Example. Strategic Offering: Exact Linear Transformation
252(1)
6.5 Writing and Solving the KKTs of an MPPDC
253(5)
6.5.1 KKTs of an MPPDC
253(1)
6.5.2 Illustrative Example. Strategic Offering, KKTs
254(2)
6.5.3 Reformulating an MCP as an Optimization Problem
256(1)
6.5.4 Illustrative Example. Strategic Offering: MCP Optimization Problem
257(1)
6.6 Summary
258(1)
6.7 Exercises
258(5)
References
261(2)
7 Equilibrium Problems with Equilibrium Constraints
263(60)
7.1 Introduction
263(1)
7.2 The EPEC Problem
264(7)
7.2.1 Problem Statement and Diagonalization Algorithm
264(3)
7.2.2 Diagonalization Applied to EPEC
267(4)
7.3 Energy Applications of EPECs
271(3)
7.4 EPEC Power Market Model 1: Strategic Quantity Decisions by Generators
274(17)
7.4.1 Model Formulation
274(1)
7.4.1.1 Model Structural Assumptions
274(2)
7.4.1.2 Consumer
276(1)
7.4.1.3 Transmission Provider
277(1)
7.4.1.4 Follower Equilibrium
277(1)
7.4.1.5 Generator (Leader) MPEC
278(1)
7.4.1.6 EPEC
279(1)
7.4.2 Illustrative Example
279(1)
7.4.2.1 Assumptions
279(1)
7.4.2.2 Follower Problem
280(1)
7.4.2.3 Leader Problems
281(1)
7.4.2.4 EPEC Statement and Analysis
282(3)
7.4.2.5 Attempted Solution by Diagonalization
285(1)
7.4.2.6 Mixed Strategy Solution
286(1)
7.4.2.7 Comparison of Outcomes of Alternative Game Formulations
287(1)
7.4.2.8 Sensitivity Case: Single Oligopolist
288(1)
7.4.2.9 Sensitivity Case: Transmission Expansion
289(1)
7.4.2.10 Summary of Cournot EPEC Example
290(1)
7.5 EPEC Power Market Model 2: Strategic Offering by Generators
291(11)
7.5.1 Model Formulation
291(1)
7.5.1.1 Structural Assumptions
291(1)
7.5.1.2 Auctioneer (Transmission Provider)
292(1)
7.5.1.3 Producer MPEC
293(1)
7.5.1.4 EPEC
294(1)
7.5.2 Illustrative Example
295(1)
7.5.2.1 Assumptions
295(1)
7.5.2.2 EPEC Formulation
295(1)
7.5.2.3 Application of Diagonalization
296(3)
7.5.2.4 Mixed Strategy Equilibrium Computation
299(2)
7.5.2.5 Comparison of Average MPEC Results, EPEC Mixed Equilibrium, and Competitive Equilibrium
301(1)
7.5.2.6 Pure Strategy Equilibrium
301(1)
7.6 Closed Loop Multistage Nash Equilibrium: Capacity Expansion
302(13)
7.6.1 Introduction
302(1)
7.6.2 Stage 2 Equilibrium: The Commodity Market
303(1)
7.6.2.1 Perfect Competition
303(2)
7.6.2.2 Cournot Competition
305(1)
7.6.3 Stage 1 EPEC Problem
306(2)
7.6.4 Illustrative Example: Consumers prefer Cournot to Bertrand Competition
308(7)
7.7 Summary
315(1)
7.8 Exercises
315(8)
References
319(4)
8 Algorithms for LCPs, NCPs and VIs
323(62)
8.1 Introduction
323(1)
8.2 Algorithms for LCP Models
324(16)
8.2.1 Lemke's Pivoting Method for LCPs
325(1)
8.2.1.1 General Background on Pivoting
325(1)
8.2.1.2 Illustrative Example. Pivoting in Simplex Method for a Linear Program
326(2)
8.2.1.3 Lemke's Method
328(1)
8.2.1.3.1 Illustrative Example. Simple LCP from
Chapter 1
328(1)
8.2.1.3.2 General Statement of Lemke's Method
329(2)
8.2.1.3.3 Illustrative Example. Equilibrium of two commodities
331(2)
8.2.1.3.4 Convergence of Lemke's Method
333(1)
8.2.2 Iterative Methods for LCPs
333(1)
8.2.2.1 General Background on Matrix Splitting
334(1)
8.2.2.2 Matrix Splitting for the LCP
335(1)
8.2.2.2.1 Illustrative Example. Matrix splitting for two-commodity model: B = diagonal part of M
336(1)
8.2.2.2.2 Illustrative Example. Matrix splitting for two-commodity model: symmetric B
337(1)
8.2.2.2.3 Convergence of Matrix Splitting Algorithms for the LCP
338(2)
8.2.2.3 Other Iterative Methods for LCPs
340(1)
8.3 Algorithms for NCP Models
340(19)
8.3.1 Newton's Method for Systems of Smooth Equations
342(1)
8.3.1.1 Undamped Newton Method for Smooth Equations
343(1)
8.3.1.1.1 Illustrative Example. Solving two equations in two unknowns by Newton's method
343(1)
8.3.1.1.2 Convergence of the Undamped Newton Method for Smooth Equations
344(1)
8.3.1.2 Damped Newton Methods for Smooth Equations
345(1)
8.3.1.2.1 Illustrative Example. Damping Procedures to Accelerate Convergence
345(2)
8.3.1.2.2 Convergence of the Damped Newton Method for Smooth Equations
347(2)
8.3.2 Newton's Method for the NCP
349(1)
8.3.2.1 Constructing an Approximate LCP
349(1)
8.3.2.2 Solving the Approximate LCP
350(1)
8.3.2.3 Getting Started: Solving the First Approximate LCP
351(1)
8.3.2.4 Two Examples Without Damping
352(1)
8.3.2.4.1 Illustrative Example. PATH method for two-commodity LCP
352(1)
8.3.2.4.2 Illustrative Example. PATH method for two-commodity NCP
353(2)
8.3.2.5 Damping in the Newton Method for NCPs
355(1)
8.3.2.5.1 Illustrative Example. Min-based merit function for two-commodity NCP
356(1)
8.3.2.5.2 Path Search between Previous Iterate and Newton Point
356(2)
8.3.2.6 Summary and Overview of Other Features of the PATH Algorithm
358(1)
8.4 Algorithms for VI Models
359(9)
8.4.1 Solve Equivalent KKT System as MCP
359(1)
8.4.2 Iterative Methods: Sequential Optimization
360(1)
8.4.2.1 Project Independence Evaluation System (PIES)
360(2)
8.4.2.1.1 Illustrative Example. Simple PIES model and algorithm
362(2)
8.4.2.1.2 PIES-q Algorithm
364(1)
8.4.2.1.3 Convergence of PIES and PIES-q Algorithms
364(1)
8.4.2.2 A Nonlinear Approximation of G - Diagonalization Method
365(1)
8.4.2.2.1 Illustrative Example. The PIES-q algorithm as diagonalization method on a VI
366(1)
8.4.2.3 Symmetric Linear Approximations of G
367(1)
8.4.2.4 Convergence of Diagonalization and Symmetric Linear Approximation
367(1)
8.5 Summary
368(1)
8.6 Appendix: Introduction to Theory for PATH and Other NCP Algorithms
369(9)
8.6.1 Projection Mappings
370(1)
8.6.1.1 Illustrative Example. Projection Mapping for B = Rn+
370(1)
8.6.1.2 Illustrative Example. Projection Mapping for B as a Rectangular Box
371(1)
8.6.2 NCP Reformulated as Nonsmooth Equation Using Projection Mapping
371(1)
8.6.2.1 Illustrative Example. Illustration of Theorem 8.3 with x ≠ z
372(1)
8.6.2.2 Illustrative Example. Illustration of Theorem 8.3 with x = z
372(1)
8.6.3 Some Useful Merit Functions and Corresponding Nonsmooth Equations
373(1)
8.6.3.1 Merit Function Based on Min Function
373(1)
8.6.3.2 Merit Function Based on Norm of the Normal Map
374(1)
8.6.3.3 Merit Function Based on Fischer-Burmeister Function
374(1)
8.6.3.3.1 Illustrative Example. Fischer-Burmeister-based merit function for two-commodity NCP
375(1)
8.6.3.4 Merit Function Based on Plus Function
375(1)
8.6.4 Damped Newton Method for NCP as Nonsmooth Equation
376(1)
8.6.5 Convergence of the PATH Algorithm
377(1)
8.6.6 Other Methods to Solve NCPs
377(1)
8.7 Exercises
378(7)
References
383(2)
9 Some Advanced Algorithms for VI Decomposition, MPCCs and EPECs
385(48)
9.1 Introduction
385(1)
9.2 Decomposition Algorithms for Vis
386(26)
9.2.1 Illustrative Example. Dantzig-Wolfe Decomposition of a Simple LP
386(6)
9.2.2 Illustrative Example. Simplified Stochastic Power Model from
Chapters 4 and 5
392(2)
9.2.3 Dantzig-Wolfe Decomposition of VIs
394(3)
9.2.3.1 Some Computational Enhancements to Dantzig-Wolfe Decomposition of VIs
397(1)
9.2.4 Illustrative Example. Dantzig-Wolfe Decomposition of Simplified Stochastic Power Model
398(2)
9.2.5 Simplicial Decomposition of VIs
400(2)
9.2.6 Illustrative Example. Simplicial Decomposition of Simplified Stochastic Power Model
402(1)
9.2.7 Benders Decomposition of VIs
403(1)
9.2.7.1 Illustrative Example. Benders Decomposition of a Simple LP
403(1)
9.2.7.2 General Development of Benders Decomposition for VIs
404(2)
9.2.7.3 Illustrative Example. Benders Decomposition of Simplified Stochastic Power Model
406(5)
9.2.8 Cobweb Decomposition Method - No Master Problem
411(1)
9.3 Algorithms for Mathematical Programs with Complementarity Constraints
412(10)
9.3.1 Why Are MPCCs Difficult to Solve?
414(1)
9.3.2 Applying Standard NLP Algorithms to MPCCs
415(1)
9.3.2.1 Regularization of Complementarity Constraints
415(1)
9.3.2.2 Illustrative Example. Regularization Applied to the Strategic Offer MPCC
416(1)
9.3.2.3 Penalization of Complementarity Constraints
417(1)
9.3.2.4 Illustrative Example. Penalization Applied to the Strategic Offer MPCC
418(1)
9.3.2.5 Sequential Quadratic Programming
419(1)
9.3.2.6 Illustrative Example. SQP Applied to the Strategic Offer MPCC
419(1)
9.3.2.7 Some Practical Advice
420(1)
9.3.3 Some Other Methods for MPCCs
420(2)
9.4 Algorithms for Equilibrium Programs with Equilibrium Constraints (EPECs)
422(7)
9.4.1 Diagonalization Method for EPECs
423(1)
9.4.2 NLP Reformulation of EPECs
424(1)
9.4.3 Illustrative Example. A simple 2-Leader, 1-Follower EPEC
424(5)
9.5 Summary
429(1)
9.6 Exercises
429(4)
References
431(2)
10 Natural Gas Market Modeling
433(44)
10.1 Introduction
433(2)
10.2 Natural Gas Market Models
435(4)
10.3 Engineering Considerations
439(1)
10.4 The Natural Gas Supply Chain and the Various Market Agents
440(29)
10.4.1 Sectoral and Seasonal Aspects and Gas Storage Operator
441(2)
10.4.2 Capacity Expansion and Multi-Year Perspective
443(1)
10.4.3 Representation of Consumers and Strategic Versus Non-Strategic Players
443(2)
10.4.4 Additional Players and Engineering Aspects
445(1)
10.4.5 Suppliers
445(1)
10.4.5.1 Production
445(3)
10.4.5.2 Delivering Gas to the Market
448(2)
10.4.5.3 Supplier's Problem (Version 1: Production and Export Functions)
450(2)
10.4.5.4 Storage Operations
452(2)
10.4.5.5 Supplier's Problem (Version 2: Production, Export and Storage Functions)
454(2)
10.4.6 Transportation
456(1)
10.4.7 A Model for the Whole Market
457(2)
10.4.8 Illustrative Example. Small Natural Gas Network Equilibrium
459(1)
10.4.8.1 Overview
459(2)
10.4.8.2 Base Case
461(4)
10.4.8.3 Analysis of Storage
465(1)
10.4.8.4 Analysis of Total Gas Reserves Constraint
466(2)
10.4.8.5 Analysis of Contract Sales
468(1)
10.5 Summary
469(1)
10.6 Exercises
470(7)
References
473(4)
11 Electricity and Environmental Markets
477(38)
11.1 Introduction
477(2)
11.2 Transmission-Constrained Electricity Markets
479(18)
11.2.1 Short-Run, Perfectly Competitive Market
480(5)
11.2.2 Illustrative Example. Transmission-Constrained Perfect Competition Equilibrium
485(4)
11.2.3 Oligopolistic Market: A Cournot Model
489(6)
11.2.4 Illustrative Example. Transmission-Constrained Cournot Equilibrium
495(2)
11.3 Environmental Markets: Emissions Trading
497(10)
11.3.1 A Simple Model of Emissions Trading among Producers
499(2)
11.3.2 Illustrative Example. Simple Source-Based Emissions Trading Equilibrium
501(2)
11.3.3 A Simple Model of Emissions Trading among Load-Serving Entities
503(1)
11.3.4 Illustrative Example. Simple Load-Based Market Equilibrium
504(2)
11.3.5 Model Analysis: Equivalence of Source-Based and Load Based Trading
506(1)
11.4 Summary
507(1)
11.5 Exercises
508(7)
References
511(4)
12 Multicommodity Equilibrium Models: Accounting for Demand-Side Linkages
515(104)
12.1 Introduction
515(1)
12.2 Linkages among Multiple Energy Markets
516(4)
12.3 Demand Relations over Time
520(10)
12.3.1 Regulated Vertically Integrated Utility Model
521(4)
12.3.2 Unbundled Power Market with and without Cross-Price Elasticities
525(5)
12.4 Multi-Sector Models with Demand Linkages
530(26)
12.4.1 The Project Independence Evaluation System
530(2)
12.4.2 PIES Model Components
532(1)
12.4.2.1 Consumers
533(1)
12.4.2.2 Fuel Producers
534(2)
12.4.2.3 Oil Refiners
536(2)
12.4.2.4 Shippers
538(1)
12.4.2.5 Market Clearing
539(1)
12.4.3 Assembling and Solving the PIES Model
540(1)
12.4.3.1 Market Equilibrium LCP
540(2)
12.4.3.2 Solution Approaches
542(1)
12.4.4 PIES Equilibrium: Interpreting the Solutions of a Multicommodity Model with Demand Linkages
543(1)
12.4.4.1 Interpreting Solutions: Where Do Prices Come Prom?
544(6)
12.4.4.2 Interpreting Solutions: Effects of Policy
550(2)
12.4.4.3 Comparison with Own-Elasticity Only Results
552(4)
12.5 Summary
556(1)
12.6 Exercises
557(4)
References
559(2)
A Convex Sets and Functions
561(10)
References
569(2)
B GAMS codes
571(36)
References
605(2)
C DC Power Flow
607(6)
References
611(2)
D Natural Gas Engineering Considerations
613(6)
References
617(2)
List of Tables 619(4)
List of Figures 623(2)
Index 625
Steven A. Gabriel received his M.A. and Ph.D. degrees in Mathematical Sciences from Johns Hopkins University in 1989 and 1992, respectively, and his M.S. in Operations Research from Stanford University in 1984. He is currently Associate Professor, Civil Systems Program, Department of Civil and Environmental Engineering, University of Maryland.

Antonio J. Conejo received the M.S. degree from Massachusetts Institute of Technology, Cambridge, MA, in 1987 and the Ph.D. degree from the Royal Institute of Technology, Stockholm, Sweden, in 1990. He is currently Professor of Electrical Engineering at the Universidad de Castilla La Mancha, Ciudad Real, Spain.

J. David Fuller received his Ph.D. in Interdisciplinary Studies from the University of British Columbia in 1980. His research interests focus on Energy Economics and Operations Research; Mathematical Programming Models of Economic Equilibrium with Applications to Energy Markets Forecasting and Electricity Market Design; and Decomposition of Linear, Nonlinear and Equilibrium Programs. He is currently a Professor of Management Sciences, in the Faculty of Engineering at the University of Waterloo, in Waterloo, Ontario, Canada.

Benjamin F. Hobbs received his Ph.D. in Environmental Systems Engineering from Cornell University in 1983; his MS in Resource Management and Policy from Syracuse University in 1978, and his BS in Mathematics and Environmental Sciences from North Dakota State University in 1976. He has served as Chair of the JHU Presidents Climate Change Task Force since 2008.

Carlos Ruiz is currently a Ph.D. candidate under Dr. Conejo at the University de Castilla.
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