The Next Generation of Electric Power Unit Commitment Models (International Series in Operations Research & Management Science)

THE NEXT GENERATION OF ELECTRIC POWER UNIT COMMITMENT MODELS

INTERNATIONAL SERIES IN OPERATIONS RESEARCH & MANAGEMENT SCIENCE Frederick S. Hillier, Series Editor Stanford University Saigal, R. / LINEAR PROGRAMMING: A Modern Integrated Analysis Nagurney, A. & Zhang, D. / PROJECTED DYNAMICAL SYSTEMS AND VARIATIONAL INEQUALITIES WITH APPLICATIONS Padberg, M. & Rijal, M. / LOCATION, SCHEDULING, DESIGN AND INTEGER PROGRAMMING Vanderbei, R. / LINEAR PROGRAMMING: Foundations and Extensions Jaiswal, N.K. / MILITARY OPERATIONS RESEARCH: Quantitative Decision Making Gal, T. & Greenberg, H. / ADVANCES IN SENSITIVITY ANALYSIS AND PARAMETRIC PROGRAMMING Prabhu, N.U. / FOUNDATIONS OF QUEUEING THEORY Fang, S.-C., Rajasekera, J.R. & Tsao, H.-S.J. / ENTROPY OPTIMIZATION AND MATHEMATICAL PROGRAMMING Yu, G. / OPERATIONS RESEARCH IN THE AIRLINE INDUSTRY Ho, T.-H. & Tang, C. S. / PRODUCT VARIETY MANAGEMENT El-Taha, M. & Stidham , S. / SAMPLE-PATH ANALYSIS OF QUEUEING SYSTEMS Miettinen, K. M. / NONLINEAR MULTIOBJECTIVE OPTIMIZATION Chao, H. & Huntington, H. G. / DESIGNING COMPETITIVE ELECTRICITY MARKETS Weglarz, J. / PROJECT SCHEDULING: Recent Models, Algorithms & Applications Sahin, I. & Polatoglu, H. / QUALITY, WARRANTY AND PREVENTIVE MAINTENANCE Tavares, L. V. / ADVANCED MODELS FOR PROJECT MANAGEMENT Tayur, S., Ganeshan, R. & Magazine, M. / QUANTITATIVE MODELING FOR SUPPLY CHAIN MANAGEMENT Weyant, J./ ENERGY AND ENVIRONMENTAL POLICY MODELING Shanthikumar, J.G. & Sumita, U./APPLIED PROBABILITY AND STOCHASTIC PROCESSES Liu, B. & Esogbue, A.O. / DECISION CRITERIA AND OPTIMAL INVENTORY PROCESSES Gal, T., Stewart, T.J., Hanne, T./ MULTICRITERIA DECISION MAKING: Advances in MCDM Models, Algorithms, Theory, and Applications Fox, B. L./ STRATEGIES FOR QUASI-MONTE CARLO Hall, R.W. / HANDBOOK OF TRANSPORTATION SCIENCE Grassman, W.K./ COMPUTATIONAL PROBABILITY Pomerol, J-C. & Barba-Romero, S. / MULT1CRITERION DECISION IN MANAGEMENT Axsäter, S. / INVENTORY CONTROL Wolkowicz, H., Saigal, R., Vandenberghe, L./ HANDBOOK OF SEMI-DEFINITE PROGRAMMING: Theory, Algorithms, and Applications Hobbs, B. F. & Meier, P. / ENERGY DECISIONS AND THE ENVIRONMENT: A Guide to the Use of Multicriteria Methods Dar-El, E./ HUMAN LEARNING: From Learning Curves to Learning Organizations Armstrong, J. S./ PRINCIPLES OF FORECASTING: A Handbook for Researchers and Practitioners Balsamo, S., Personé, V., Onvural, R./ ANALYSIS OF QUEUEING NETWORKS WITH BLOCKING Bouyssou, D. et al/ EVALUATION AND DECISION MODELS: A Critical Perspective Hanne, T./ INTELLIGENT STRATEGIES FOR META MULTIPLE CRITERIA DECISION MAKING Saaty, T. & Vargas, L./ MODELS, METHODS, CONCEPTS & APPLICATIONS OF THE ANALYTIC HIERARCHY PROCESS Chatterjee, K. & Samuelson, W./ GAME THEORY AND BUSINESS APPLICATIONS

THE NEXT GENERATION OF ELECTRIC POWER UNIT COMMITMENT MODELS

Editors Benjamin F. Hobbs The Johns Hopkins University Michael H. Rothkopf Rutgers University Richard P. O’Neill Federal Energy Regulatory Commission

Hung-po Chao Electric Power Research Institute

KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

eBook ISBN: Print ISBN:

0-306-47663-0 0-7923-7334-0

©2002 Kluwer Academic Publishers New York, Boston, Dordrecht, London, Moscow Print ©2001 Kluwer Academic Publishers Dordrecht All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Kluwer Online at: and Kluwer's eBookstore at:

http://kluweronline.com http://ebooks.kluweronline.com

CONTENTS Acknowledgments

vii

I. The Evolving Context for Unit Commitment Decisions

1. Why This Book?: New Capabilities and New Needs for Unit Commitment Modeling B. F. Hobbs, W. R. Stewart Jr., R. E. Bixby, M. H. Rothkopf, R. P. O'Neill, H.-p. Chao

1

2. Regulatory Evolution, Market Design and Unit Commitment R. P. O'Neill, U. Helman, P. M. Sotkiewicz, M. H. Rothkopf, W. R. Stewart Jr.

15

3. Development of an Electric Energy Market Simulator A. Debs, C. Hansen, Y.-C. Wu

39

II. New Features in Unit Commitment Models 4. Auctions with Explicit Demand-Side Bidding in Competitive Electricity Markets A. Borghetti, G. Gross, C. A. Nucci

53

5. Thermal Unit Commitment with a Nonlinear AC Power Flow Network Model C. E. Murillo-Sánchez, R. J. Thomas

75

6. Optimal Self-Commitment under Uncertain Energy and Reserve Prices R. Rajaraman, L. Kirsch, F. L. Alvarado, C. Clark

93

7. A Stochastic Model for a Price-Based Unit Commitment Problem and Its Application to Short-Term Generation Asset Valuation C.-L. Tseng

117

8. Probabilistic Unit Commitment under a Deregulated Market J. Valenzuela, M. Mazumdar

139

III. Algorithmic Advances

9. Solving Hard Mixed-Integer Programs for Electricity Generation S. Ceria

153

vi

The Next Generation of Unit Commitment Models

10. An Interior-Point/Cutting-Plane Algorithm to Solve the Dual Unit Commitment Problem -- On Dual Variables, Duality Gap, and Cost Recovery M. Madrigal, V. H. Quintana

167

11. Building and Evaluating Genco Bidding Strategies and Unit Commitment Schedules with Genetic Algorithms C. W. Richter, Jr., G. B. Sheblé

185

12. An Equivalencing Technique for Solving the Large-Scale Thermal Unit Commitment Problem S. Sen, D.P. Kothari

211

IV. Decentralized Decision Making 13. Strategic Unit Commitment for Generation in Deregulated Electricity Markets A. Baíllo, M. Ventosa, A. Ramos, M. Rivier, A. Canseco

227

14. Optimization-Based Bidding Strategies for Deregulated Electric Power Markets X. Guan, E. Ni, P. B. Luh, Y.-C. Ho

249

15. Decentralized Nodal-Price Self-Dispatch and Unit Commitment F. D. Galiana. A. L. Motto, A. J. Conejo, M. Huneault

271

16. Decentralized Unit Commitment in Competitive Energy Markets J. Xu, R. D. Christie

293

Index

317

ACKNOWLEDGMENTS This volume contains papers that were presented at a workshop entitled "The Next Generation of Unit Commitment Models", held September 2728, 1999 at the Center for Discrete Mathematics and Theoretical Computer Science (DIMACS), Rutgers University, Piscataway, NJ. The editors gratefully acknowledge the financial support of the co-sponsors of the workshop: DIMACS (funded by the National Science Foundation under grant NSF STC 91-19999); and the Electric Power Research Institute (EPRI), which supported the publication and distribution of this book. The editors also thank Sarah Donnelly of DIMACS for her organizational support of the workshop and the subsequent book. The editors would also like the many speakers and other participants of the workshop for their ideas and hard work. All the papers were subjected to anonymous peer review by at least two referees and two editors. The referees included authors of other papers in this volume, along with Paul Sotkiewicz and Judith Cardell of the Office of Economic Policy of the Federal Energy Regulatory Commission. Technical editing for the volume was ably provided by Debi Rager of The White Cottage Company. Liz Austin of Johns Hopkins Unviersity compiled the index. Funding for B. Hobbs' involvement in the workshop and book came from NSF Grants ECS 96-96014 and 00-80577. Partial support for M. Rothkopf was provided by NSF grant SBR 97-09861.

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Chapter 1 WHY THIS BOOK? NEW CAPABILITIES AND NEW NEEDS FOR UNIT COMMITMENT MODELING Benjamin F. Hobbs The Johns Hopkins University

William R. Stewart Jr. The College of William & Mary

Robert E. Bixby Rice University and ILOG

Michael H. Rothkopf Rutgers University

Richard P. O'Neill Federal Energy Regulatory Commission

Hung-po Chao Electric Power Research Institute

Abstract:

This book presents recent developments in the functionality of generation unit commitment (UC) models and algorithms for solving those models. These developments, the subject of a September 1999 workshop, are driven by institutional changes that increase the importance of efficient and market responsive operation. We illustrate these developments by demonstrating the use of mixed integer programming (MIP) to solve a UC problem. The dramatically lower solution times of modem MIP software indicates that it is now a practical algorithm for UC. Participants in the workshop also prioritized the features that need to be considered by UC models, along with topics for research and development. Among the highest research priorities are: market simulation; bid selection; reliability and reserve constraints; and fair processes for choosing from alternative near-optimal solutions. The chapter closes with an overview of the contributions of the other chapters.

2

1.

The Next Generation of Unit Commitment Models

PURPOSE OF THE BOOK

The unit commitment problem can be defined as the scheduling of production of electric power generating units over a daily to weekly time horizon in order to accomplish some objective. The problem solution must respect both generator constraints (such as ramp rate limits and minimum up or down times) and system constraints (reserve and energy requirements and, potentially, transmission constraints). The objective function should account for costs associated with energy production, ramping, start-ups and shutdowns decisions, along with possible effects upon revenues or customer costs of those decisions. The resulting problem is a large scale nonlinear mixed integer program. For many years, the electric power industry has been using optimization methods to help them solve the unit commitment problem. The result has been savings of tens and perhaps hundreds of millions of dollars in fuel costs. Things are changing, however. Optimization technology is improving, and the industry is undergoing radical restructuring. Consequently, the role of commitment models is changing, and the value of the improved solutions that better algorithms might yield is increasing. The purpose of this book is to explore the technology and needs of the next generation of computer models for aiding unit commitment decisions. Because of the unit commitment problem's size and complexity and because of the large economic benefits that could result from its improved solution, considerable attention has been devoted to algorithm development [1,2]. Heuristics such as priority lists have long been used by industry; but in the last three decades, more systematic procedures based on a variety of algorithms have been proposed and tested. These techniques have include dynamic programming, branch-and-bound mixed integer programming (MIP), linear and network programming approaches, and Benders decomposition methods, among others. Recently, metaheuristic methods have been tested, such as genetic programming and simulated annealing, along with expert systems and neural networks. The solution approach that has been most successful, and which is most widely used at present, is Lagrangian relaxation. This procedure decomposes the problem by multiplying constraints that couple different generators (such as energy demand and reserve constraints) by Lagrange multipliers and placing them in the objective function. Given a set of multiplier values, the problem is then separable in the generating units, and a dynamic program of low dimension can be used to obtain a trial schedule for each unit. A process of multiplier adjustment is used to search for feasible near-optimal solutions. Lagrangian relaxation has proven useful for quick development of good, if not optimal, generator schedules. Recent improvements in integer pro-

Introduction

3

gramming codes and other algorithms suggest, however, that it may be possible to find better solutions more rapidly. Further, such codes can more readily incorporate additional coupling constraints, such as transmission limits and emissions caps. Meanwhile, restructuring has sharpened the appetite of generation owners for more efficient operation. In the past, utilities sold power on a regulated cost-plus basis and so may not have put as much priority on squeezing out the last few percent improvements in the objective function. Furthermore, system operators realized that cost functions were approximate, so the operators were perhaps more likely to be satisfied with good solutions or marginal improvements that were technically suboptimal. Now, with restructuring, we have schedule coordinators making commitment decisions in a market environment, and independent system operators (ISOs) dealing with bids. Bids are precise, and small improvements in solutions can result in significant changes in payments to bidders. Further, the fact that optimization models are, in some cases, being used to determine which generators will be operated and thus paid implies that there is a greater incentive to get exact answers to make the bidding process fair and legitimate (and to disempower the bid-taker). In other words, electric markets are changing rapidly, as is the role of unit commitment models. How UC models are solved and what purposes they serve deserve reconsideration. The goal of the workshop that led to this book was to bring together people who understand the problem and people who know what improvements in algorithms are really possible. The papers in this book summarize the participants' assessments of industry needs together with new formulations and computational approaches that promise to make unit commitment models more responsive to those needs. In Section 2 of this chapter, we present an example to show how the capabilities of commercially available integer programming software to solve large unit commitment problems to optimality have dramatically improved in recent years. This example illustrates how improvements in software may make it possible to solve bigger problems in less time, while simultaneously including more of the complications that users want to represent. Section 3 then summarizes the results of a survey of workshop participants in which they were asked to identify what issues concerning unit commitment modeling are most in need of further research and development. Finally, in Section 4 we give an overview of the other chapters in this book. There, we describe how the papers contribute to our two goals of articulating the emerging needs of the restructured power industry and describing model developments that can make unit commitment models more responsive to those needs.

4

2.

The Next Generation of Unit Commitment Models

EXAMPLE OF NEW CAPABILITIES: SOLVING UNIT COMMITMENT PROBLEMS USING MIP

When formulated as a mixed integer program (MIP), the unit commitment (UC) problem is a large and complex mathematical programming problem. As a result, optimal solutions have been hard to obtain for practically sized problems due to the exponential behavior of solution times as problems grow larger. As a result, exact solutions to large UC problems have been unavailable, and approximate techniques are employed to produce solutions that are near optimal (within 0.5%-2%). In practice, Lagrangian relaxation methods have performed well, but the non-convexity and overall size of most practical problems have prevented solving such problems to optimality or even providing a bound on the optimal solution. Approximate solutions have two problems. First, since an approximate solution will be dispatching units that are slightly more costly than the ones that would have been dispatched had an optimal solution been available, the approximate solution almost certainly will be inefficient economically and a deserving facility may have been passed over by the approximate solution. The second, related problem consists of the political implications of passing over cost-effective units in a world in which generating units are owned by different competing entities, as opposed to the historical situation where all generation was centrally owned by a single, large regulated utility. The purpose of this section is to report on how the technology for solving MIPs to optimality (branch-and-bound, cutting planes, etc.) has improved computational times in the past several years, and the implications of those improvements for solving UC problems. Historically, MIPs have been notoriously difficult to solve due to the presence of multiple near-optimal solutions such as would occur when a system contained several generators with similar operating characteristics and costs. Traditional branch-andbound techniques would have to find all of these optimal solutions explicitly before an optimal solution could be verified. For this reason and because of the many near-optimal solutions in a typical large MIP, solution times grow very quickly with the size of the problem because of the large number of nodes in the branch-and-bound tree that must be fathomed. However, a number of theoretical improvements have been incorporated into commercial MIP codes, and further improvements are anticipated. Consequently, these codes can now solve to optimality problems thought impossible just a few years ago. As a concrete example of these improvements, we summarize the results of applying several recent generations of a widely used MIP code, CPLEX®, to a test problem. The problem comes from a paper by Johnson et al. [3]. It requires the scheduling of 17 generators to meet electricity demand over a

Introduction

5

seven-day (168-hour) period. The generators have different operating characteristics and the resulting MIP contains 25,755 variables (including 2,856 integer variables) and 48,939 constraints. While this may be a small problem by unit commitment standards, until recently it would have been considered intractable for direct solution by commercial MIP solvers using a linear programming/branch-and-bound/cutting plane technology. Recent advances have substantially accelerated these approaches, however; as a result, solution times for problems such as the one below are approaching the range where the MIP approach can be considered viable. Test UC Mixed Integer Linear Program: Minimize subject to:

where: is the MW of energy produced by generator i in period t, is a binary variable that is 1 if generator i is dispatched during t, is 1 if generator i is started at the beginning of period t, is 1 if generator i is shut down at the beginning of period t, is the MW of spinning reserves available from generator i in t, and are the fixed cost of operating ($/period), the cost of generation ($/MW/period), and the cost of start-ups ($), respectively, for generator i during t, and are the minimum and maximum MW capacities of plant i, and its maximum reserve contribution, respectively, and

6

The Next Generation of Unit Commitment Models and are the maximum ramp rates (in MW/period) for increasing and decreasing output, respectively, from unit i.

The seven-day MIP problem is non-convex, with one integer variable for each generator for each hour of the week. The rest of the variables are continuous. In addition to the formulation as shown in (1)-(10), the objective function coefficients for some of the generators grow with output level. These are approximated by piecewise linear cost functions. There is also a set of constraints not shown in (1)-(10) that controls for the minimum number of periods a generator can be up and running and the minimum number of periods that a generator must be shut down. The objective of UC is to choose a set of generators and operating levels so as to minimize the total cost of the dispatch (1) subject to meeting hourly demand (2), having sufficient spinning reserves (3) (arbitrarily set to 3% of demand in this problem), generation at each operating unit at or above its minimum run level and below its maximum output level (4)-(5), and spinning reserves below the maximum level for each generator (6). Constraints (7)-(8) force hour-to-hour changes in generation to respect ramp rate limits. Constraint (9) requires that a generator be started if it was not dispatched the previous period and will be this period, while (10) mandates that a unit be shut down this period if it is was dispatched the previous period and will not be used this period. The results for solving two related test problems are reported in Tables 1 and 2. The first problem is a one-day (24 time periods) problem and the other is the full seven-day problem. The one-day problem has one-seventh the number of continuous variables, discrete variables, and constraints that the week-long problem has. The main computational comparison is how the solution times for these problems improve as they are solved by later versions of a specific solver (CPLEX). In addition, we show the number of times a linear program is solved in the branch-and-bound portion of the algorithm (number of nodes).

Introduction

7

The marked improvement in times seen in Version 6.5 (Table 1) can be attributed to several factors: the inclusion of cutting planes eliminates a lot of extraneous fractional solutions that were previously explicitly considered; later versions of the dual simplex algorithm in CPLEX just solve the linear program faster (a couple of orders of magnitude faster on the seven day problem, see below); and the inclusion of pre-solve reductions substantially tightens the problem before branch-and-cut starts. The improvement of times from CPLEX 6.0 to CPLEX 6.5 of about an order of magnitude gives promise that this approach can be useful for solving UCPs in the future. The seven-day (168 time periods) problem was unsolvable in a reasonable amount of time in CPLEX 4.0; indeed, it took an hour just to get a solution to the linear programming relaxation. Yet, as Table 2 shows, that problem was solved in less than two hours using CPLEX 6.5 and under 30 minutes using a developmental version of CPLEX on a 500 MHz DEC Alpha. These results show that innovations in algorithms and their implementations will continue to have a strong impact on solution times for MIPs and that realistic unit commitment problems can be solved to optimality by off-theshelf software.

3.

NEW NEEDS: RESEARCH PRIORITIES

The potential of improved algorithms was one important focus of the workshop. The other was the functionality of unit commitment models. The workshop attendees participated in an afternoon meeting to evaluate the functions that unit commitment models should perform in restructured markets. The meeting was structured as a nominal group [4] to maximize the efficiency of information exchange and to ensure that all forty or so participants had a chance to voice their ideas. The steps of a nominal group exercise are as follows: participants silently write down ideas in response to one or more questions; the ideas are posted without attribution; the ideas are discussed in round robin fashion; and participants numerically rate the ideas either at the meeting or in a questionnaire after the meeting. The workshop participants performed the following tasks:

8

The Next Generation of Unit Commitment Models Identify features, characteristics, or requirements of system operation and bidding that are desirable to include in unit commitment models, and issues involved in representing these issues in those models. Describe the extent to which additional research is needed to incorporate those features in unit commitment models. This was done by grouping the features into three categories: I This feature/characteristic/requirement is modeled by existing models and software; II Models have been proposed for capturing this feature/ characteristic/requirement, but require development and implementation; and III There remain fundamental disputes or uncertainties as to how this feature/characteristic/requirement should be modeled. In this case, fundamental research is needed. Rate the importance of including each feature/characteristic/requirement on the following Likert scale: 1 Minor importance 3 Somewhat important 5 Very important 7 Crucial

Approximately 40 of the workshop attendees addressed the first two tasks during the afternoon session. Eighteen participants returned the follow-up questionnaire on importance rating. The latter group included nine industry representatives (primarily consultants, but also including generating company and EPRI representatives); eight university researchers; and a regulator. From the results of these tasks, a set of research priorities can be distilled. The following two tables summarize the results. Table 3 ranks the features by the mean importance rating given by the group, while Table 4 categorizes them in terms of whether further research is needed. Where participants disagreed concerning the category, a range is shown. Table 3 indicates near unanimity concerning the crucial importance of including the commodities of energy and ancillary services along with dynamic constraints and decision sequencing. The next most important issue was market prediction, interpreted as the projection of prices, and how the market might respond to alternative bidding strategies. Some importance ratings were mildly surprising. In particular, transmission issues were rated relatively low. It would seem that research on transmission is

Introduction

9

10

The Next Generation of Unit Commitment Models

Introduction

11

not a high priority with this group, as inclusion of AC representations fell in category II and received a moderate importance rating, while DC approximations were in I and received one of the lowest average importance ratings. Other features that received low ratings included distributed

12

The Next Generation of Unit Commitment Models

generation; emissions modeling; and particular stochastic methods for modeling reliability, uncertainty, and variability. Table 4 reveals that the group unanimously identified 11 features as requiring fundamental research (category III), of which nine received average importance ratings of 4.0 or more. These features are extremely diverse, ranging from risk and reliability modeling to strategic bidding, new product valuation, and dealing with the fairness and institutional implications of multiple near-optimal solutions. Another six features were thought by some but not all group members to be in category III and so are identified as being in category “II-III.” Several facets of market prediction were highly rated in that category. These facets include price prediction, bidding strategy assessment, and inclusion of priceresponsive demand. Meanwhile, in the “requires development” category (II), there are two very important topics: inclusion of ancillary services in unit commitment models and explicit accounting for the sequential nature of decisions (such as multi-settlement systems). Research of a more applied nature would seem to be justified in these cases.

4.

BOOK OVERVIEW

The chapters of this book are grouped into four sections. The first section includes this introduction and two other chapters that summarize the evolving institutional context that has motivated the functional and algorithmic developments described in the rest of the book. O'Neill, Helman, Sotkiewicz, Rothkopf, and Stewart (Chapter 2) review the recent history of short-term electricity markets in the U.S., focussing on alternative market designs and the implications for unit commitment modeling. They also suggest some principles for designing the next generation of UC market models. An alternative approach to presenting the evolving context of unit commitment is presented in Chapter 3. There, Debs, Hansen, and Wu present a general modeling framework that encompasses all the functions of short-term energy markets, including commitment, with a focus on whether the market participant is an ISO/RTO, generating company, market administrator, load serving entity, or even an energy service company. The other three sections of the book describe novel applications and features in UC models, new algorithms for solving those models, and modeling approaches that represent decentralized commitment by independent generating firms. Chapter 4 is the first chapter in the second section; there, Borghetti, Gross, and Nucci show how demand-side bidding can be included in Lagrangian-relaxation-based unit commitment models, and how such bidding can dampen price volatility and mitigate market power. Their formula-

Introduction

13

tion represents load recovery, hour-by-hour bids for demand reduction, and multiple bidders. Chapter 5 by Murillo-Sanchez and Thomas is the second chapter on new model features. They describe how a nonlinear AC power flow representation can be incorporated, with both active and reactive power sources. They also discuss a parallel processing implementation. The last three chapters of the second section concern the inclusion of price uncertainty in UC models. All use variations of a stochastic dynamic programming-based commitment model; when generators cannot individually influence price, the models neatly decompose into a single optimization model for each generator representing their self-commitment problem. In Chapter 6, Rajaraman, Kirsch, Alvarado, and Clark describe how uncertainties in both reserve and energy prices can be considered in such models. In Chapter 7, Tseng shows how such models can be used to quantify rigorously the worth of operating flexibility (“option value”) for a single generating asset. Finally, in Chapter 8, Valenzuela and Mazumdar present a model for single generator optimization that uses probability distributions of market prices directly derived from assumptions concerning demand variability and generator availability in the whole market. In the third section of the book, we turn our attention to improved algorithms for solving the UC problem. Four distinct approaches are represented in this section: mixed integer programming, Lagrangian relaxation, genetic algorithms, and aggregation approaches. Chapter 9 by Ceria, like Section 3 of this introduction, addresses the usefulness of MIP for UC, along with recent developments in MIP technology that have drastically improved solution times. He also briefly reviews two actual applications by utilities in Europe. Chapter 10, authored by Madrigal and Quintana, proposes an interior-point/cutting-plane algorithm to solve the Lagrangian relaxation problem and demonstrates its computational advantages over subgradient and other methods traditionally used to update Lagrange multipliers. They also offer some observations on several issues involved in using UC models to clear power markets, including duality gaps, cost recovery, and the existence of multiple solutions. Chapter 11, contributed by Richter and Sheble, reviews a range of considerations involved in creating bidding and commitment strategies. They then propose genetic algorithms and finite state automata-based simulations for strategy development and testing. Chapter 12 by Sen and Kothari shows how aggregation of generating units into a few sets of similar units can be exploited to improve solution times. The algorithm involves three basic steps: aggregation; solution of the simplified UC problem using dynamic programming or another optimization method; and disaggregation to create schedules for individual units.

14

The Next Generation of Unit Commitment Models

The fourth and final section of the book contains four chapters that address modeling and algorithmic issues associated with decentralized commitment decision processes. Important questions include opportunities for strategic manipulation of prices by market participants, coordination algorithms, and the ability of decentralized processes to approach optimality. The first two chapters of the fourth section focus on decision-making by individual firms. Baillo, Ventosa, Ramos, Rivier, and Canseco devote Chapter 13 to a model for committing a firm’s units while recognizing how commitment and dispatch decisions may affect market prices. Rival firms are assumed to behave according to price-elastic supply functions, which allows for derivation of total firm revenue as a function of its output. They use MIP to solve their model. In Chapter 14, Guan, Ni, Luh, and Ho describe two general approaches to bid development in decentralized markets. One is based on “ordinal optimization” for obtaining satisficing bidding strategies, and a second uses stochastic optimization to self-schedule and manage risks while considering interactions among different markets. The last two chapters of the fourth section turn to the issue of coordination of decentralized decisions. Galiana, Motto, Conejo, and Huneault (Chapter 15) propose a coordination process in which locational prices are announced, generating firms self-dispatch to maximize their individual profits, and prices are adjusted to ensure that demands and network constraints are satisfied. The authors use a Newton algorithm to update prices, and impose a “convexifying rule” to facilitate convergence. Case studies illustrate the process. In Chapter 16, Xu and Christie consider the interactions of strategic behavior by individual generating firms with a price-based coordinating mechanism. Firms optimize bidding strategies with the help of a simple price prediction model. The combined effects of multiple firms using that approach is explored with a market simulator, which reveals that convergence, feasibility, and price stability can be difficult to achieve, but that price cycling can be dampened with alternative price prediction models.

REFERENCES 1. 2. 3. 4.

G. Sheblé and G. Fahd. Unit commitment literature synopsis. IEEE Trans. Power Syst., 9(1): 128-135, 1994. S. Sen and D.P. Kothari. Optimal thermal generating unit commitment: a review. Elec. Power Energy Syst., 20(7): 443-451, 1998. R.B. Johnson, S.S. Oren, and A.J. Svoboda. Equity and efficiency of unit commitment in competitive electricity markets. Utilities Policy, 6(1): 9-20, 1997. A. Delbecq, A. Van de Ven, and D. Gustafson. Group Techniques for Program Planning — A Guide to Nominal Group and Delphi Processes. Glenview, IL: Scott Foresman and Co., 1975.

Chapter 2 REGULATORY EVOLUTION, MARKET DESIGN AND UNIT COMMITMENT

Richard P. O’Neill, Udi Helman, and Paul M. Sotkiewicz Federal Energy Regulatory Commission

Michael H. Rothkopf Rutgers University William R. Stewart Jr. The College of William and Mary

Abstract:

1.

In the context of competitive wholesale electricity markets, the unit commitment problem has shifted from a firm level optimization problem to a market level problem. Some centralized market designs use it to ensure reliability and determine day-ahead market prices. This chapter reviews the recent history of short-term electricity markets in the United States to evaluate the experience with alternative market designs and the implications for unit commitment modeling. It presents principles for the design of the next generation of unit commitment-based markets.

INTRODUCTION

Competitive wholesale electricity markets now operate in several major U.S. markets, confirming the analysis and recommendations of prescient economists, electrical engineers, and others over the past two decades.1 Since generation comprises approximately 75 percent of all electricity costs, com1

Seminal contributions on competitive wholesale electricity markets include [1,2]. As of January 1, 2000, regional markets with centralized wholesale electricity exchanges are operational in California, the Pennsylvania-New Jersey-Maryland (PJM) interconnection, New England, and New York.

16

The Next Generation of Unit Commitment Models

petition in generation promises large efficiency gains and cost savings to consumers. The unit commitment problem, the traditional method by which regulated utilities and power pools conducted internal scheduling of generation to meet demand at least cost over a multi-hour to multi-day time frame, is now embedded, in various ways, in competitive markets. Potentially, unit commitment models will be used by different market participants and institutions: individual firms, centralized auctioneers, decentralized aggregators of generation schedules, and transmission system operators. This environment presents a new set of modeling requirements and market design challenges.2 The market level unit commitment problem is typically much larger in scale than the firm level problem. Speed and accuracy are important if an auctioneer uses the solution by an auctioneer to determine market prices. Evaluating the characteristics of the solution, such as the presence of duality gaps (implying a lack of market clearing prices) and alternative optima, becomes of direct financial interest to market participants. The new electricity markets, and hence new applications of the unit commitment problem, are being developed within an evolving regulatory context. Indeed, an important driver of market designs is the guidance given by the regulator. The Federal Energy Regulatory Commission (henceforth “the Commission”) initiated regulatory reform of transmission in 1996, with the objective of encouraging competitive regional electricity markets that promote economic efficiency without compromising system reliability. The regulatory approach, embodied in a series of orders described below, has been to provide an open market architecture where alternative market designs are implemented, evaluated, and changed when necessary. Research into the unit commitment problem has largely been reactive to the new regulatory environment and the emerging issues in market design. A more proactive approach is needed. Among the issues that need consideration and research are the choices between simultaneous and sequential optimization of several energy and ancillary service products, alternative bidding rules for different products, different mechanisms for congestion pricing, and inter-regional coordination. In addition, unit commitment modeling now has to confront the issue of economic incentives in various market settings, which requires a more extensive familiarity with economics and game theory. In response to the regulatory evolution it has set in motion, the regulator also needs to adapt institutionally and develop its technical capabilities. This is imperative because the Commission is taking an oversight role in market design decisions across the United States. Several wholesale markets operate centralized unit commitment auction markets (e.g., PJM, New England, and 2

The literature on market design in electricity markets is extensive; for a survey, see the articles in [3].

Regulatory Evolution, Market Design, and Unit Commitment

17

New York), the market design of which is the focus of this chapter. These markets also allow bilateral trading and types of self-scheduling. Following approval of the basic design, the Commission provides oversight for a flood of subsequent adjustments and refinements in the search for well functioning markets. The underlying unit commitment model is often either an implicit or explicit matter in these market rule decisions. The objective of this chapter is to describe regulatory evolution and the market design challenges for unit commitment modeling. The chapter focuses on day-ahead markets, but much of the discussion is also applicable to realtime markets. Section 2 of the chapter describes the key regulatory developments and the design and recent experience of the major regional wholesale electricity markets. Section 3 focuses on principles that should guide the design of day-ahead energy and ancillary service markets. Finally, Section 4 offers conclusions.

2.

REGULATORY EVOLUTION AND THE ORGANIZATION OF ELECTRICITY MARKETS

The recent history of electricity regulatory reform in the United States began when the Commission issued Orders 888 and 889 in 1996 [4,5]. These orders required an open access transmission regime, based on nondiscriminatory transmission rates and transparent posting of available transmission capacity (ATC). Order 888 also included fairly broad organizational principles for an independent system operator (ISO), an institution which separates ownership from control of the grid and can perform market functions. Between 1997 and 2000, ISOs and power exchanges (PXs) were formed in California and in the three tight power pools of the eastern United States. These ISOs established day-ahead and real-time markets for energy, ancillary services, and transmission (in California, the day-ahead energy market is conducted by several separate scheduling coordinators, including the California Power Exchange). An ISO has also been established in the Midwest, but is not yet operational. Other regions of the country have been less successful or unwilling to centralize grid operations, and electricity trading remains bilateral, with a vertically integrated utility performing the balancing and reliability functions. By early 1999, a certain amount of inertia was evident in the development of wholesale markets. Electricity traders expressed dissatisfaction with the traditional methods of transmission grid management still employed in large parts of the United States. Specifically, there was substantial concern about

18

The Next Generation of Unit Commitment Models

frequent curtailments of transactions, justified on the basis of reliability but often questioned by parties to the transactions.3 On December 15, 1999, the Commission took a step toward clarifying the appropriate transmission access and market institutions with Order 2000, which requires the formation of regional transmission organizations (RTOs) [6]. The order establishes in the RTO many of the features that had emerged in the ISO markets as well as additional characteristics and functions that address unresolved issues in both ISO and non-ISO electricity markets. The RTO is required to serve a region of sufficient scope and configuration to provide for a reliable, efficient electricity market. With respect to the unit commitment problem, some of the important features of the RTO are that it must have exclusive authority for maintaining short term reliability, act as provider of last resort of ancillary services, address parallel path flows, provide real time energy balancing, and ensure development of market mechanisms for congestion management. In many ways, the functions assigned to the RTO are based on the principles of market design embodied in the better functioning ISOs that have emerged (in fact, RTOs will subsume existing ISOs). Section 2.1 surveys some of these ISO market design lessons; Section 2.2 discusses further the objectives of Order 2000 and some market design issues. Before considering the details of open access and market design, an important question is: Why should the regulator remain involved in the design and oversight of the emerging competitive markets? Recent experience has made clear that, in the near term, the Commission has a continuing role for at least three reasons: 1. There is the ongoing development of open access itself, including the provision of short-term reliability services linked to transmission and future expansion of the grid. There are public good aspects to reliability and transmission expansion.4 2. Where not currently available, there must be an efficient pricing mechanism for transmission congestion (i.e., pricing of the externalities created by parallel path flows). 3. There must be mitigation of market power in markets for electricity and ancillary services and in provision of transmission capacity. Market power is the ability of firms to raise prices above competitive lev3

Such curtailments are supposed to follow the North American Electricity Reliability Council’s (NERC) Transmission Loading Relief (TLR) procedures, which provide criteria for the management of congested transmission facilities.

4

A public good is a good that is non-rivalrous and non-excludable. In the case of reliability, a load’s or generator’s consumption of reliability in no way prevents others from enjoying the same reliability, and if all of the loads and generators are interconnected on the same system, they cannot be prevented from enjoying the benefits of reliability.

Regulatory Evolution, Market Design, and Unit Commitment

19

els. Firms can exercise market power in electricity markets because of both structural factors (e.g., firm concentration or transmission constraints) and opportunities offered by the market design (see [7,8] for a survey). The Commission’s policy heretofore has been to approve implementation of the markets (through more liberal standards for granting market-based rates) while at the same time evaluating the markets’ operational experience and providing guidance on market designs as a means to promote efficiency and competition. In addition, most of the responsibility for day-to-day monitoring and mitigation of market power has shifted to the ISOs and the future RTOs. The regulatory approach, then, seeks to balance the current reality some firms can exercise a degree of market power generally or under certain system conditions – with the expectation that entry of new firms and more efficient market designs will substantially mitigate future market power.5

2.1

Experience with ISO and Bilateral Markets

Order 888 outlined principles for, but did not require, a particular structure for competitive wholesale energy markets. Two broad types of market structures have developed. The first type consists of markets with ISOs, which may or may not include one or more scheduling coordinators or PXs.6 The ISO markets with PXs take the form of either a centralized ISO/PX or a decentralized ISO and PX(s). The centralized ISO/PX markets use unit commitment models for creating the day-ahead schedule (which incorporates bilateral transactions and self-schedules) while the decentralized ISO markets require self-commitment by the PXs. The second type of market has no ISO; rather, the transmission system and much of the generation continues to be operated by vertically integrated utilities. Power is traded bilaterally. This section focuses on the various market designs and performance of the existing ISO markets, as well as offer some conclusions and recommendations on what designs will work best. It also includes a brief review of the performance of the bilateral, non-ISO markets. Market Functions and Design of ISO Markets. Competitive wholesale 5

There is a large body of literature on market power due to both structural and market design characteristics of electricity markets. For analysis of regional energy and ancillary service markets in the United States, see [8-12].

6

Scheduling coordinators or power exchanges (PXs) are power trading operations functionally separate from the ISO. These terms will be used interchangeably.

20

The Next Generation of Unit Commitment Models

electricity markets can be complex, with multiple interdependent products sold on different time frames and differentially priced at different locations. Existing ISO markets can be characterized by the (1) number and types of different products (energy, ancillary services, capacity), (2) bidding and scheduling process, (3) relationship of temporal (forward and real-time) markets, (4) market clearing and settlement rules, and (5) type of congestion management and transmission rights. Each of the existing ISOs has also established market power monitoring and mitigation, but this function will not be examined here. The ISO carries out the basic function of assessing the feasibility of proposed generation schedules. The ISO also serves as the buyer, through contracted rates and bid-based auctions, of reliability services, including shortterm ancillary services (voltage support, operating reserves, and automatic generation control) and possibly longer-term capacity products. The PX facilitates and conducts a forward auction market for electric energy. PX functions can either be carried out by the ISO itself, as in New York, New England, and PJM, or by one or more separate, unaffiliated PXs, as in California. With the exception of the California ISO, the ISOs run a unit commitment model to determine which units will be scheduled to provide energy and ancillary services during the following day. In California, the ISO and PX (which is one of several scheduling coordinators) are separate, a decision intended to keep the transmission system operators (who may have been affiliated with an incumbent utility) functionally distinct from the market. In the California PX market, generation owners selfcommit their units through scheduling coordinators. This market structure has experienced certain disadvantages. One problem is that the scheduling coordinators’ submissions can be physically infeasible. The ISO must then engage in a time consuming iterative process with the scheduling coordinators to resolve the infeasibilities. The PX auction algorithm is such, however, that selfcommitted units are often asked to start-up and stop, disregarding minimum run and down times, with potentially adverse results.7 Another problem is that the separation of the ISO and PX raises the transaction costs for market participants. In addition, because the ISO’s decisions about congestion and purchases of ancillary services cannot be remedied by the PX, market participants may bid strategically into the ISO’s congestion market to ensure that profitable transactions are not curtailed. Bidding and Scheduling. The ISO markets began, largely, with only supply-side bidding for energy and certain ancillary services. While demandside bidding for energy is allowed in some markets (currently the California ISO and PX and New York, but planned for the other ISOs), the energy de7

Conversations with California ISO staff confirm this problem.

Regulatory Evolution, Market Design, and Unit Commitment

21

mand function is largely inelastic – that is, not price responsive – due to both technical limitations and historic rate designs. As more price responsiveness is introduced into demand bid functions (through installation of metering equipment and technological advances in distributed generation and information technology), there should be a reduction in both price volatility and the potential for exercise of market power, particularly during peak hours. The structure of bids is another market design issue that has attracted attention. Bids in current ISO energy markets vary in the number of cost components and required technical parameters, such as ramp rates, high and low operating limits, and so on. In the so-called “one-part” incremental energy bid, the bidder must factor its start-up, no load, and other costs into its dayahead energy bid for each megawatt-hour (MWh) offered.8 Even so, generators face the risk that they may not cover all of their costs in the auction. Onepart bids require generation owners to internalize this risk in some fashion, which in turn increases their costs (use of the real-time market to make adjustments can eliminate part of this risk).9 One-part bids also result in inefficiency if they are the only costs the dispatcher considers in commitment. In a three-part bid, the start-up and no-load costs can be separated out, allowing generators to bid actual operating costs more precisely and allowing for a more efficient unit commitment. In PJM and New York, generators are guaranteed to at least recover all of their bid costs if they are committed to run.10 This mechanism eliminates the uncertainty of whether a generator will be committed and dispatched only to lose money, and it allows for a more efficient dispatch. Market-Clearing and Settlement Systems. Market-clearing rules and settlement systems are the procedures that determine quantities produced and consumed, who pays, and who gets paid. As discussed above, ISOs typically operate multiple markets, including energy, several types of ancillary services, and transmission products. There are two basic ways to clear these different markets, sequentially or simultaneously, with variations on each method. In general, sequential auction markets clear each product separately 8

Currently, the California PX requires one-part energy bids without technical parameters, such as minimum run times, ramp rates, and so on. This could be called a pure one-part bid. New England currently requires one-part bids with technical parameters.

9

At its worst, the need to internalize risk in the one-part bidding system could lead to a greater incentive to internalize via bilateral contract or merger to avoid higher transactions costs. High market concentrations lead to market power concerns. Further, not being allowed to bid marginal cost is an easy defense to an inquiry on market power abuse.

10

Generators will receive an “uplift” payment to recover their costs only if the revenues they receive from the energy and ancillary services markets are less than their total bid costs.

22

The Next Generation of Unit Commitment Models

in a sequence, even though several of the products may be alternative (substitute) uses of the same generator. Variations of this approach were adopted initially in California and in the interim New England market. In contrast, a simultaneous auction, adopted in PJM and New York, clears the relevant markets at the same time, minimizing the joint bid cost of providing energy and ancillary services. This method explicitly takes into account that some products are substitutes. One of the clearest market design lessons from ISOs is that sequential market clearing without product and/or quantity substitution is economically inefficient and offers opportunities for strategic behavior. In California, energy, regulation, 10-minute spinning reserves, 30-minute non-spinning reserves, and replacement reserves are cleared in the order given.11 It has sometimes been the case that the price for ancillary services with lower production costs exceeds the price of ancillary services (and energy) with higher production costs (for example, providing spinning reserve requires using some fuel, whereas providing non-spinning reserve requires simply being on stand-by). The reason for some of these “price inversions” is that generators can strategically bid high prices in the last markets knowing that there will not be much capacity remaining after the other markets clear. Hence, the ISO must take these high price bids. To combat this design flaw, the California ISO has instituted a pre-processing algorithm, called the Rational Buyer Protocol, which will allow it to substitute higher quality services for lower quality services if and only if it reduces its ancillary service procurement cost. New England also initially used sequential market clearing and experienced problems similar to California; temporary solutions have included rolling over of bids between substitute services, as in California, as well as price caps. Experience with simultaneous market clearing is limited. The simultaneous market clearing method will largely, but not entirely, avoid the price inversions seen in the California ISO and New England markets because the software used to clear the markets automatically clears all remaining arbitrage opportunities. In addition, generators which bid strategically, as in California, would be far less likely to be selected to provide ancillary services at the higher price due to the greater substitution possibilities in the simultaneous market clearing. While more complex computationally, simultaneous marketclearing appears to be emerging as the better system from an efficiency standpoint. 11

In California, regulation refers to automatic generation control, but is defined in terms of whether generation output is increased (regulation up) or decreased (regulation down). Spinning reserves is reserve capacity available in a specified time period from a generator synchronized with the grid. Non-spinning reserves are reserve capacity available in a specified time period from a generator not synchronized with the grid. Replacement reserves are reserves that can be available within 60 minutes.

Regulatory Evolution, Market Design, and Unit Commitment

23

The settlement systems can be characterized as either single settlement or multi-settlement based on the number of temporal markets the ISO runs. In California, there are three temporal markets: day-ahead, hour-ahead, and realtime with a financial settlement for each. New York and PJM run two temporal markets: day-ahead and real-time. New England currently has only realtime markets (bids are due day-ahead but financial settlement only takes place at real-time prices), but is scheduled to implement day-ahead markets with financial settlement. A clear market design lesson is that single-settlement systems, which require generators to submit bids and stand-by day-ahead while awaiting financial settlement at real-time prices, create problems in scheduling and often require additional rules to constrain generator incentives to change their bids. The multi-settlement system has been adopted by all the ISOs in recognition of the value of the forward market as a financial hedge for real-time conditions. Also, the forward market should facilitate demand-side responses by giving demand that has bid to reduce load more time to react to price signals. Congestion Management and Pricing. In the ISO context, there are two general ways to manage congestion: locational pricing and non-locational pricing. Locational pricing can be sub-divided into approaches defined by the level of aggregation used to calculate the price. In the typical “nodal pricing” method, an energy price is calculated at each generation and load bus (node). The transmission congestion price between any two busses is the difference in energy prices at the busses. At higher levels of aggregation, the busses in the system operated by an ISO can be gathered into one or more congestion zones. Zones are intended to be indicative of the historical pattern of congestion in the system on the presumption that congestion will take place between the zones with little or no congestion within a zone. The price of congestion between zones is the difference in energy prices between the zones. If congestion occurs within a zone, the costs of managing it (typically through generator re-dispatch) are shared by all market participants inside the zone using a system of subsidies.12 This intrazonal congestion management method could be considered a type of nonlocational pricing.

12

Several ISOs have attempted to operate as single zones (PJM, New England), but have subsequently made the transition to locational pricing. If the single zone system has consistent congestion between sub-regions (that is, should be at least two zones), this can create opportunities for generators to leave the spot market and use bilateral contracts to take advantage of the system price. This was the experience in PJM before it implemented a nodal system; the ISO was required to adopt administrative measures to curtail the bilateral transactions.

24

The Next Generation of Unit Commitment Models

The zonal approach has been adopted in California, which currently has two (soon to be three) congestion zones but is experiencing congestion that should trigger new zones (also, each import point effectively creates a new zone). The California ISO manages inter-zonal congestion through adjustment bids submitted by generation and load. These bids indicate the price at which the market participant is willing to be ramped up or down in order to alleviate congested lines. If this fails to relieve the congested lines, then the ISO must call on generators with cost-based contracts to relieve congestion.13 California manages intra-zonal congestion by re-dispatch (which incorporates the transmission constraints into the original, transmission unconstrained dispatch) with the resulting costs averaged over load in the zone. Persistent intrazonal congestion indicates that the zones are not properly defined; in addition, the averaging of congestion costs within the zone is inefficient, since the congestion costs are also paid by participants not causing the congestion.14 In the long run, zonal pricing as practiced in California can lead to price signals that distort decisions on siting new generation and transmission assets.15 Neither maintaining fixed zones in the face of intrazonal congestion nor continuous re-zoning are efficient methods of congestion management. Zonal market design in California has been instituted in part under the rationale that it lowers market power. Both in theory and practice this assumptions has been proved wrong. Market power cannot be reduced by the declarations of large zones. If this were so, there would be no market power problem. Transmission constraints and generation costs determine the size of the market, not the declaration of zones. The California ISO rules recognize this by providing for dispatch orders and out-of-market payments to generators in the same zone separated by constraints. In contrast, despite opposition from some generators and marketers, nodal pricing has been adopted in New York and PJM and approved for New England (these systems actually use nodal prices for generators and zonal averages for loads). Nodal pricing eliminates the problem of properly defining 13

These are called Reliability-Must-Run (RMR) contracts. RMR contracts are intended to ensure that the ISO has sufficient generation capacity to meet various system contingencies, such as congestion relief and voltage support.

14

The California ISO can create a new congestion management zone if the cost to alleviate congestion over the previous 12 months exceeds 5 percent of the approximate annual revenue requirement of the transmission operators. In order to be considered an active congestion zone, the markets on either side of the congested interface must be “workably competitive” for significant portions of the year.

15

For example, the Commission has rejected a California ISO proposal (Tariff Amendment No. 19, filed June 23, 1999) that new generators upgrade transmission capacity to alleviate intrazonal congestion which might arise from their entry on the grounds that it could create further barriers to entry and market distortions. A similar New England proposal was also rejected.

Regulatory Evolution, Market Design, and Unit Commitment

25

zones and the need to average the costs of any intrazonal congestion. In the short run, load receives the proper price signals about how much to consume, and the long-run decisions can be made much more easily. Even though loads pay a zonal price, the nodal price information remains available for decision making. For financial markets, nodal prices for a region can be aggregated into fewer “hub” prices, which are weighted averages of the underlying nodal prices. For example, PJM has two hub prices. Transmission Rights. Transmission rights have traditionally been used to reserve access to the transmission system and to ensure that energy transactions would be curtailed only in extreme circumstances. These rights were physical rights – the right to transmit physically a specific amount of power over the system for the access charge paid. With the advent of congestion pricing (whether zonal or nodal), most ISOs have provided both physical rights and financial rights that can be used as a hedge against congestion costs (the stochastic nature and potentially high cost of congestion makes financial hedging necessary).16 In all the markets with locational congestion pricing, payment of congestion prices is essentially a physical right to transmit between nodes or zones (although not a right that is bought in advance). On the other hand, financial rights are typically purely financial mechanisms that provide revenues but confer no physical priority. They can be traded on a secondary market. For example, in New York and PJM, financial transmission rights give the holder the right to collect congestion rents between a designated point of injection and point of withdrawal, so that if a transaction incurs congestion costs, those costs would be offset by the revenues from the financial right. Auctions for these rights are typically held regularly. The California ISO has implemented a similar type of zone-to-zone right, but which also confers some physical priority.17 Performance of ISO Markets. As discussed above, none of the ISO markets has reached a stable point in terms of market design; some are undertaking major market re-designs while others are in the process of implementing major components of their market design. There is a convergence in market design in many areas: all the ISOs have implemented either sequential auctions with substitutions or simultaneous auctions for energy and ancillary services; most ISOs have established multi-settlement systems or will shortly. Most ISOs offer some form of financial transmission right; in the East coast 16

Transmission rights can take the form of either options or obligations.

17

If the California energy markets fail to clear, the holder of a transmission right usually gets a better position in the curtailment queue than a generator not holding a right.

26

The Next Generation of Unit Commitment Models

ISO markets, nodal pricing is used for generation or is planned for future implementation. Given these ongoing changes, the preliminary performance of the markets varies by product and time period. In transmission, the ISOs have recorded few curtailments. There has been some concern, however, that the number of bilateral transactions has decreased in nodal congestion management systems (because point-to-point congestion may be difficult to hedge with the available transmission rights). The energy markets seem to be functioning fairly well, although prices under certain system conditions reflect varying levels of market power [8-13]. Entry of generation, transmission capacity expansion, and demand-side bidding should lower prices and lessen volatility. The ancillary service markets have been more problematic. Reserve markets in particular have experienced price spikes and price inversions, reflecting the greater vulnerability of these markets to market power and to market design flaws that exacerbate strategic behavior [9,12]. Temporary price and bid caps and more permanent market re-designs should help solve some of these problems. Other general market problems include limitations in software implementation and technical capabilities (such as using telephone rather than electronic communications for dispatch), and conflicts that emerge when system operators depend on rules of thumb to dispatch the system rather than the outcomes of the auction. In general, however, many market design or implementation problems are amenable to satisfactory resolution, some through admittedly short-term “band-aid” solutions, but most with a longer term fix available. Business confidence is not equally robust in each ISO market (PJM appears to be the market with the fewest problems to date), but should increase as the markets mature. Performance of Bilateral (Non-ISO) Markets. The largely bilateral markets, especially those in the Midwest, have experienced many potential reliability problems as evidenced by the frequency of curtailments under Transmission Loading Relief (TLR) procedures. These may also be attributable to the lack of independence of the system operator and market participants.18 Market participants have complained that they could not get access to the transmission system even when capacity appeared to have been available. As described below, Order 2000 requires the implementation of more efficient congestion management practices.

18

The curtailment of transactions in the presence of prices 10 to 100 times the annual average, due to TLRs and voltage reductions concurrent with power outages, indicate markets are not working in harmony with reliability constraints. For example, in the summer of 1999 the ECAR region with bilateral trading called 87 TLRs and the adjacent PJM ISO called three. For a general review of these complaints, see [6].

Regulatory Evolution, Market Design, and Unit Commitment

2.2

27

Order 2000 and RTOs

A Regional Transmission Organization (RTO) is a transmission system operator that is independent of market participants, controls transmission facilities within a region of appropriate scope and configuration, and is responsible for operating those facilities to provide reliable, efficient, and nondiscriminatory service. All transmission owners must file a proposal to participate in a RTO or provide reasons for delaying or avoiding participation. Order 2000 explicitly notes that the designs for bid-based markets in the four ISOs operational before the year 2000 should form a basis for the design of RTO markets. Yet the open architecture adopted in Order 2000 does not propose a single market model and offers sufficient leeway for further experimentation within the RTO design principles. With respect to the unit commitment problem, the RTO has certain relevant functions. The RTO must have exclusive authority for maintaining shortterm reliability. To fulfill this function, Order 2000 makes clear that the RTO requires knowledge of the operational status of generators and load.19 This includes control over interchange schedules, the authority to require redispatch of generation connected to the grid, and approval over scheduled outages. The RTO will determine the required amount of each ancillary service and the location where the service is to be provided. It will also act as provider of last resort of ancillary services. That is, market participants can selfsupply or purchase ancillary services from third parties, but the RTO must have the capability to provide any residual. The RTO or a third party unaffiliated with market participants must provide real time energy balancing. With regard to transmission, the RTO must ensure development of market mechanisms for congestion management and must develop procedures to take into account parallel path flows. The RTO will sell physically feasible, shortand long-term, tradable transmission rights. The RTO may choose to expand the transmission system and/or invest in advanced technology to increase capacity.20 19

Such knowledge includes technical information supplied by generators such as ramp rates, upper and lower operating limits, whether the unit is running or not, start-up times and time between start-ups. In real-time and for day-ahead planning, the RTO must have information on generator injections and load withdrawals of energy in order to balance the system. 20 In the comments on Order 2000 [6], various policy suggestions were made regarding increasing transmission capacity, including overbuilding the transmission system (see Joskow comments) and/or investing in the high tech Flexible AC Transmission System (FACTS) and Wide-Area Measurement System (WAMS) to allow more robust competition to develop.

28

The Next Generation of Unit Commitment Models

Finally, the RTO is required to monitor for market power abuses and market design flaws. It should also evaluate and implement potential efficiency improvements in the markets it operates. Beyond these requirements and guidelines, specific market designs are left to the RTO market developers (subject to the proviso that they not limit the RTO’s ability to improve efficiency further). The remainder of this section discusses some issues about the conceptualization of the role of the RTO with respect to financial and physical transactions as well as the relationship of RTO-operated auction markets in relation to other energy markets. In addition, some pressing market design issues are reviewed, including pricing of reserves and inter-regional coordination. Section 3 then draws on the ISO experience and other sources to outline some principles for the design of dayahead RTO markets. Relationship Between Physical and Financial Transactions. An issue that has remained contentious in the preliminary design and operation of ISO markets is the relationship between physical and financial markets – specifically, the concern that the centralized ISO markets and nodal congestion pricing would inhibit development of the decentralized financial markets.21 An important principle underlying the future RTO markets is that wellfunctioning physical markets promote robust financial markets. For our purposes, physical trades are trades that the RTO has registered as feasible, considering all other physical trades and required ancillary services. This includes bids into the ISO markets, bilateral transactions and self-schedules that have been cleared in the ISO day-ahead schedule (even though these day-ahead transactions are actually financial contracts until physical delivery). Financial trades are trades that are not physical trades, but take the form of forward contracts, futures contracts, or options contracts. They are not considered physical trades until they are confirmed as physically feasible by the RTO. Indeed, the RTO should be concerned primarily with physical market transactions; it would not operate purely financial markets and need not be involved in any financial markets unless the transaction goes to delivery. Financial markets can and must exist in harmony and equilibrium with physical reliability markets. If not, the financial markets’ ability to reduce risk is diminished. Multiple PXs and bilateral trading can fit easily into this market velop. The latter appears more promising because it promises not only more capacity and less greenfield construction, but also better system control (see EPRI and EEI comments). 21

One argument has been that uncertainty over nodal congestion prices, calculated hourly in real-time, increases the risk of bilateral deals concluded prior to the hour. Another is that some rules regarding three-part bids, in which the start-up payments made by the ISO are averaged over all electricity load in the system (e.g., in New York), effectively amounts to a subsidy to generators in the ISO auction market.

Regulatory Evolution, Market Design, and Unit Commitment

29

framework. Each PX would act as a single scheduler, submitting schedules and technical information for generation and load to the RTO. The market design of the physical market should allow full, but optional interaction with financial markets. To become physical transactions, a market participant need only self-schedule, that is, submit quantities to the day-ahead market. If the transaction includes the necessary transmission rights and ancillary services, no charges will be assessed. This allows for fully hedged financial transactions. In addition, payment would be received for any additional service provided. Otherwise, the market participant will be billed for congestion, losses, and ancillary services caused by the bilateral transaction. If not self-supplied, a bilateral trader can place price limits on what it is willing to pay for transmission and ancillary services. If the price limits are not met, the transaction will not be scheduled. If all voluntary adjustments are tried and reliability constraints are still not met, the transaction will be canceled in the day-ahead market. This gives ample time for parties to make adjustments. This cancellation avoids a potential TLR and the resulting schedule is very likely to be physically feasible. Physical markets provide real-time price signals and additional liquidity. Without good price signals from the physical markets, the financial markets can become unstable and encourage more speculation and less hedging. Even though bilateral trading may be highly discriminatory (that is, sellers may charge different prices for the same delivered product to different buyers), the opportunity for buyers to participate in an efficient, nondiscriminatory RTO auction market will tend to discipline the bilateral market. RTO auction markets create options for all buyers and sellers and thereby allow for a light-handed approach to the regulation of these transactions. In sum, the benefits of well-designed RTO markets include lighter-handed regulation of financial markets, more liquidity, less gaming and risk, more visible prices, lower transactions costs, fewer curtailments, and compatibility with financial markets. Reserves Markets. The establishment of efficient ancillary service markets is an ongoing market design challenge. As the cost of reliability increases and in the absence of a way to represent willingness to pay for ancillary services, the RTO system operator can relax reserve margins and transmission constraints. This is, in fact, written into the market rules in some ISOs; it remains a contentious issue, largely because it has involved system operator discretion that results in changes in market prices. The relaxation of these constraints increases the probability of a system failure; as such, it should be part of the operational parameters of the auction decided in advance of the day-ahead auction so that actions of the system operator are not seen as arbi-

30

The Next Generation of Unit Commitment Models

trary. To be effective, reserves must be able to respond to loads that need them. If transmission is congested between generation and load, reserves on the generation side are of little help to the load side. Both transmission and generation can be used to meet reserve requirements. They have substitute characteristics (strengthening transmission connections within a region can lower the total generation reserves needed in a region) and complementary characteristics (if the reserves for a load are not located at the same node, transmission capacity between the reserves and load will also need to be set aside). This set-aside is similar to the capacity benefit margin (CBM) concept. The auction algorithm would set aside transmission capacity to allow reserves to respond. The modeling of this process is very similar to the modeling of the energy market itself. Prices for reserves would have a locational component and the transmission price would reflect the set aside transmission capacity. Using this locational method of allocating reserves, it could appear that transmission capacity is being withheld. One method to deal with this is to have a “use or lose” requirement of transmission rights. The set-aside “use” of transmission for reserves markets would be considered a “use,” not withholding, and is under the control of the auction process and the operator. Dealing with the combination of congestion constraints and soft reserve constraints simultaneously requires operator independence and transparency in the market environment, as a means to promote trust in the market. Over time, as demand becomes more responsive to price, generation reserve margins can decline as offers to reduce demand can substitute for reserves. Currently the costs of reserves are averaged among all end users. Some of these costs can be more directly assigned to specific generating units and individual customers. For example, a unit with a good reliability record should be responsible for a lower reserve margin or be charged less for reserves based on size and the historic probability of unit failure. Payments or discounts that differentiate more reliable from less reliable generators should be handled in the pre-day-ahead markets. RTO and Expanded Inter-regional Coordination. Spatial boundary conditions – also called the “seam” or interface problem – are becoming an important design issue as trading between ISOs increases. ISO coordination efforts on this matter are in the nascent stages. In terms of system representation, some ISOs have included a reasonably detailed representation of the interconnection with control areas outside the ISO as part of the boundary conditions. Approaches to the seam problem have included proposed interface auctions for inter-control area exchanges [16,17]. In day-ahead markets, there is some time to coordinate these interfaces offline via iterative trading rules. Research into this problem generally is in its infancy. PJM, New York, New England, and Ontario are examining a broad set of inter-regional market de-

Regulatory Evolution, Market Design, and Unit Commitment

31

sign and procedural issues, and these markets have signed a memorandum of understanding on inter-regional coordination (on the need for such coordination generally, see [6]). Order 2000 anticipates that RTOs will assist in creating larger regional markets in which seams issues are resolved.

3.

DESIGN PRINCIPLES FOR DAY-AHEAD RTO AUCTIONS

In the early phases of electricity market design and implementation, the various disciplines – notably economics and electrical engineering – have not undertaken adequate inter-disciplinary research or sufficient professional exchanges. For example, ISOs have several times misunderstood the incentive issues in electricity and ancillary service auction designs, as evidenced in the remedial actions and market re-designs described in Section 2, above. At the same time, market design has sometimes proceeded with the economics basically correct but without an adequate consideration of reliability and technical constraints. As a result, there is sometimes little understanding of what basic principles ought to underlie these complex markets. This section attempts to elaborate such principles for day-ahead markets. These principles can be compared with the market design principles in [14]. This section addresses the prospective RTO day-ahead market, which is defined as the market in which the initial bidding to provide energy and ancillary services for reliability, congestion management, and energy balancing takes place. As is done currently in ISO markets, this market would be conducted on the day prior to the dispatch day. The dual objectives of the dayahead market are to achieve economic efficiency and ensure system reliability. The day-ahead market is a physical market where all expected balancing/ancillary services are scheduled. The design for the day-ahead market is discussed below and it assumes that there is a real-time market, in which adjustments are made to energy and ancillary services reflecting the differences between day-ahead expectations and real-time conditions.22 22

The real-time market will not be examined in depth here. Efficient market design requires, however, that most principles be adhered to with respect to the relationship of the real-time and day-ahead markets. Bids should be submitted separately into the real-time market, and market prices based just on those bids. Deviations from the day-ahead market should pay the real-time price unless there is a reliability problem. If a bidder does not deviate from the day-ahead schedule, there are no additional costs to pay based on the real-time market. Finally, the market operator needs to keep the system in balance at the nodal level using bids to the extent possible.

32

The Next Generation of Unit Commitment Models

The following is a list of recommended design principles derived from the foregoing analysis and experience to date, with short explanations and clarifications. Principle 1: Maximize economic efficiency. The RTO auction objective is to maximize economic efficiency through voluntary market bids, bilateral transactions, and self-scheduling, given the physical and reliability constraints. Not all current ISOs adopt this principle. For example, the California ISO cannot adjust power schedules submitted to it by the California PX to improve economic efficiency. This principle requires that if market participants use the RTO markets, the resulting prices are efficient. The market-clearing procedure will balance the system and the purchase of ancillary services using the bids it has received. If the financial markets are efficient, there may be few additional trade gains in this market and this market becomes a reliability check. Efficiency requires that prices be consistent (e.g., no price inversions due to market design flaws) and that arbitrage opportunities reflected in the bids be exhausted. Principle 2: Ensure physical feasibility of market transactions and system reliability. Without physical feasibility, reliability problems cannot be fully addressed. For ancillary services, the market design should require that generators committed to provide these services are located so that their capacity is available when and where they are needed. Principle 3: Remove disincentives to market participation. Participation in an RTO market should involve low transaction costs and create minimal additional risks. Minimal participation in the market is the submission of generation and consumption quantities (that is, bids which are taken at any price, or “at market”). Any unit dispatched should be guaranteed bid-cost recovery. Principle 4: Bidding protocols should promote flexibility of participation. All market participants should be allowed, but not required, to submit multi-part bids that reflect short-term marginal costs. Market participants should be allowed to self-schedule, that is, allowed to submit quantity-only bids. This principle requires that all resources have the option to bid a reasonable approximation of their short-term marginal cost function, including start-up, no load, and energy costs (in addition to technical parameters, such as minimum and maximum load limits, ramp rates, and minimum shutdown time). Although a bid function will seldom serve as a perfect match for

Regulatory Evolution, Market Design, and Unit Commitment

33

actual marginal (incremental or going forward) costs, a good approximation should be available. This principle will require changes in some current unit commitment auctions, which allow only one-part energy bids (of course, in a multi-part bidding rule, generators can still submit one-part bids, by bidding zero for start up and no load). As discussed above, there are technical, financial, and economic reasons for adopting multi-part bidding. While the multi-part bid allows for more accurate representation of marginal costs and thus, in the absence of market power, should result in a more efficient solution, it also results in a non-convex supply function. In turn, this makes the market equilibrium and prices harder to derive. This complication can be addressed and made manageable with some simplifying assumptions about which generators are allowed to bid non-convex costs, whether some parameters should be fixed for a specified period (such as ramp rates, maximum and minimum output), and what should be fixed in the bid function. Demand bid functions are essentially the mirror images of generator bid functions but will not be discussed in detail here. Consumers need more explicitly defined contracts to participate properly in the market and should be allowed bid functions similar to the generators. Principle 5: Make clear the distinction between financial and physical commitments. If accepted, bids are financially binding. If needed for reliability, bids are physically binding. The RTO schedules are physical and financial commitments subject to liquidated damages. That is, if market participants deviate from their commitments, they are liable for making the affected market participants whole. Under emergency conditions, the RTO may exact other penalties for nonperformance and/or issue perform-to-contract orders. Principle 6: Minimize opportunities for arbitrage between different product markets. This principle addresses generally the problems that arise when generators can bid into different product markets (energy and ancillary services) that are sequentially cleared. As discussed above, simultaneous markets eliminate opportunities for arbitrage. If the market design allows opportunities for lower quality products to be priced higher than high quality products, then there may be cases where generating units are paid more for not generating than for generating. Preferably, this should not be the case. The New England ISO attempted to establish this principle administratively by proposing that the energy price always act as a cap for operating reserves prices. This proposal was rejected by the Commis-

34

The Next Generation of Unit Commitment Models

sion; administrative measures should be at best transitions to market designs which can efficiently reach the same outcome. As discussed above, the simultaneous auction design will automatically allocate energy and reserves efficiently, but cannot guarantee the elimination of price inversions in the presence of market power. Principle 7. Prices should be unbundled where possible to minimize averaging (i.e., socializing) of costs. Averaged prices (whether for congestion, reserves, or other costs) do not send the correct price signals for the entry of new generation. Uplift charges, the mechanism used for passing through costs of energy and ancillary services not covered by market prices, should be coupled with incentives for the RTO to minimize the use of such charges. Principle 8. Market-clearing information should be made available as soon as possible. Fuller information on bids should follow with a suitable delay.23 Market-clearing prices and quantities are the basic results of the bid acceptance process. They enable market participants and potential future market participants to assess the market and plan their businesses efficiently. They also allow market participants to spot and correct obviously erroneous bid acceptance and rejection decisions. Disclosure of individual bids should be made eventually, but not immediately. Such disclosure will allow detection of subtle bid acceptance errors and it will also allow study of the market by independent analysts and market participants. It may lead to the exposure of the exercise of market power. Immediate disclosure of individual bids is undesirable because it might facilitate collusion by the market participants. Immediate disclosure might reveal information about market participants who wish to keep their costs confidential. After 6 months or a year, the information on individual bids has essentially no value for collusion and discloses little new about any bidder’s current costs, but the information would have high value for auditing and independent analysis. The auction software should be available to the market participant or public at a reasonable cost. Improvements to the software are desirable, and the best way to accomplish this is by making the software available with a set of test problems. 23

The California ISO, PJM, and New England appear to set the benchmark with updated prices every five minutes. At a minimum, prices and aggregate quantities should be available before the next round of bids with enough lead time to allow a reasonable response to the new information.

Regulatory Evolution, Market Design, and Unit Commitment

35

Principle 9: Minimize the incentives for market participants to engage in strategic behavior. The design should not favor market participants with market power. Market designs can only imperfectly address structural sources of market power. An auction does not eliminate the ability of a large firm to withhold capacity profitably. Auctions have been devised to “pay-off” market power, but these require significant computation and may not be revenue sufficient.24 Hence, absent a market design solution, the basic problem of structural market power has to be addressed using both structural remedies (vertical and horizontal dis-integration, encouragement of entry) and regulatory remedies, such as market power monitoring and mitigation. Of course, structural remedies may be wrapped up in market design issues. For example, an ISO may seek to promote rules on transmission interconnection for new generation which appear to favor incumbent generators. While market design cannot necessarily mitigate structural market power, it can certainly exacerbate it; market design can also create opportunities for strategic behavior by generators other than the obvious large players. An example, discussed above, is the sequential clearing of energy and ancillary service markets without substitution in both the initial California and New England markets. Even a small generator can try to take advantage of shortages of certain types of reserves in this type of market to raise prices.

4.

CONCLUSIONS

The considerable developments in the design of electricity markets over the past few years have provided the groundwork for the next generation of short-term markets. This chapter has emphasized that unit commitment models are now embedded in a variety of market contexts governed by an evolving regulatory framework that presents new requirements for modeling, including incorporation and understanding of market design issues. The open architecture promoted by the Commission allows for continued experimentation with RTO market design, but within parameters reflecting lessons learned heretofore from the ISO markets as well as the non-ISO bilateral markets. The market design principles presented in the paper are intended to reflect those lessons.

24

The Vickrey-Clark-Groves (VCG) mechanism, which has been applied to electricity auctions by [15], is a method to elicit truthful bids from players with market power.

36

The Next Generation of Unit Commitment Models

The other primary argument in the chapter is that a well-designed RTO day-ahead auction market with unit commitment complements “decentralized” financial markets. The computer does the work of resolving reliability constraints and will ensure that no offered trade gains are missed. Financial markets are free to create whatever innovative deals they can. The only restriction is that if they go to delivery they must satisfy any reliability constraints. Also, the opportunity for buyers to participate in an RTO auction market will tend to discipline the financial market. This will allow for lighterhanded regulation of financial transactions. The paper identified several other benefits of well-designed RTO markets: fewer curtailments, more visible prices, lower transactions costs, and less gaming and risk. The open architecture also allows for continued progress in efficient pricing, so that causality has financial consequences – prime candidates are transmission rights and reliability. Over time, as price signals are sent and acted on in real-time, accurate pricing can allow the public good aspects of these markets to shrink in importance and the private good aspects to grow.

ACKNOWLEDGEMENTS This paper reflects ongoing discussions among the authors and Carolyn Berry, Judith Cardell, Benjamin Hobbs, Thanh Luong, David Mead, William Meroney, and Roland Wentworth.

REFERENCES 1. 2. 3. 4.

5.

6. 7. 8.

P. Joskow and R.M. Schmalansee. Markets for Power. Boston: MIT Press, 1983. F.C. Schweppe, M.C. Caramanis, R.E. Tabors, and R.E. Bonn. Spot Pricing of Electricity. Norwell, MA: Kluwer Academic Press, 1988. H. Chao and H.G. Huntington, eds. Designing Competitive Electricity Markets. Boston: Kluwer Academic Press, 1998. Federal Energy Regulatory Commission. “Promoting Wholesale Competition Through Open Access Non-discriminatory Transmission Services by Public Utilities and Recovery of Stranded Costs by Public Utilities and Transmitting Utilities.” Order No. 888, 61 FR 21, 540, May 10, 1996. Federal Energy Regulatory Commission. “Open Access Same-Time Information System (formerly Real-Time Information Networks) and Standards of Conduct.” Order No. 889, 61 FR 21,737, May 10, 1996. Federal Energy Regulatory Commission. Regional Transmission Organizations. Order No. 2000, 89 FERC 61,285, December 20, 1999. C.A. Berry, B.F. Hobbs, W.A. Meroney, R.P. O’Neill, and W.R. Stewart. Analyzing Strategic Bidding Behavior in Transmission Networks. Utilities Policy, 8(3): 139-158, 1999. S. Borenstein, J. Bushnell, and C.R. Knittel. Market Power in Electricity Markets: Beyond Concentration Measures. Energy J., 20(4): 65-88, 1999.

Regulatory Evolution, Market Design, and Unit Commitment 9. 10 .

11. 12. 13. 14. 15. 16 . 17.

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California ISO. “Annual Report on Market Issues and Performance.” Prepared by the Market Surveillance Unit, California Independent System Operator, June 6, 1999. R.E. Bohn, A.K. Klevorick, and C.G. Stalon. “Second Report on Market Issues in the California Power Exchange Energy Markets.” Prepared for the Federal Energy Regulatory Commission by The Market Monitoring Committee of the California Power Exchange, March 9, 1999. S. Borenstein, J. Bushnell, and F. Wolak. “Diagnosing Market Power in California’s Deregulated Wholesale Electricity Market.” PWP-064, Univ. of Cal. Energy Institute, Berkeley, CA, Revised, March 2000. P. Cramton and J. Lien. “Eliminating the Flaws in New England’s Reserve Markets.” Working Paper, Department of Economics, University of Maryland, College Park, MD, March 2, 2000. J.E. Bowring, W.T. Flynn, R.E. Gramlich, M.P. Mclaughlin, D.M. Picarelli, and S. Stoft. “Monitoring the PJM Energy Market: Summer 1999.” PJM Market Monitoring Unit, undated draft. R.D. Wilson. “Design Principles.” In [3]. B.F. Hobbs, M. Rothkopf, L. Hyde, and R.P. O’Neill. Evaluation of a truthful revelation auction in the context of energy markets with non-concave benefits. J. Regulatory Econ., 18(1): 5-32, 2000. M.D. Cadwalader, S.M. Harvey, W.W. Hogan, S.L. Pope. “Coordinating Congestion Relief Across Multiple Regions.” PHB Hagler Bailly Inc., Navigant Consulting Inc., and J.F.K. School of Government, Harvard University, Cambridge, MA, October 7, 1999. B.H. Kim and R. Baldick. Coarse-grained distributed optimal power flow. IEEE Trans. Power Syst., 12(2): 932-939, 1997.

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Chapter 3 DEVELOPMENT OF AN ELECTRIC ENERGY MARKET SIMULATOR

Atif Debs and Charles Hansen Decision Systems International

Yu-Chi Wu National Lien-Ho Institute of Technology and Commerce

Abstract:

1.

The paper outlines the development of an electricity market operation simulator (EMOS) that can be used by competing market participants as well as independent system operators and independent market operators such as power exchanges. The uses include training personnel to make control and operating decisions and to assess bidding strategies by energy trading entities for energy and ancillary service products. We base the simulator is based on the power system model used in the operator training simulator. In the EMOS, market participants use unit commitment to assess bidding strategies and/or to schedule generation in a pool-type electricity grid. The EMOS provides a framework for users to implement their own applications with minimal effort and to model their competitors in a generic manner.

INTRODUCTION

This chapter reports on efforts to develop a realistic electricity market operation simulator (EMOS) for the emerging electric energy market. Several key features of the market are: (a) An independent system operator (ISO) coordinates supplies and demands with a focus on system security and reliability; (b) A market clearing entity determines which bids by market participants are accepted. In some cases this entity is an independent power exchange (PX), such as the California Power Exchange. In other

40

The Next Generation of Unit Commitment Models

cases, the ISO and the market clearing entity are integrated into one organization (e.g., the PJM Interconnection); and (c) Market participants (MPs) consist of a mix of unregulated and average-cost regulated entities. The unregulated entities compete against each other and submit daily, hourly, and even real-time bids to the market clearing entities and/or the ISO. They are generally generation companies (GENCOs), energy service companies (ESCOs), or scheduling coordinators (SCs). The regulated entities are mainly distribution companies whose aim is to purchase the energy through the market at competitive prices. In order to compete, each MP has to follow the rules of the market area involved and make appropriate “bids” to meets its goals. The bidding is normally based on the MP’s estimates of market prices and other factors associated with various risks involved. The purpose of the EMOS is to simulate the overall market composed of the entities just mentioned for very short-term operations (one day ahead, up to real-time). The working tool is the power system model (PSM) of the EPRI-operator training simulator (OTS) [1-7]. Each MP may have its own model of the system, but with outputs consisting of its bids into the market. The market clearing entity generates market clearing prices (MCP) and also most of the information needed to schedule system operation. The ISO does the final balancing of the market and the provision of needed “ancillary services” for system balancing, security, reliability, and real-time control [8-10]. Through the use of many existing software packages, standard application program interfaces, and communication protocols, the EMOS can be realized in a variety of configurations. It is, therefore, usable by any of the entities mentioned above. How do unit commitment (UC) and other forms of system optimization fit into the overall scheme of things? The approach given here uses UC at both the MP level and the market clearing entity level, whenever applicable. The EMOS also recognizes the fact that network optimization through the optimal power flow (OPF) is essential for such market considerations as: congestion management, reactive power/voltage scheduling, maximization of transfers, and others. Section 2 of this chapter provides a background summary of the emerging electricity market with a focus on short-term scheduling. Section 3 provides a conceptual framework for the development of MP models. Section 4 describes the overall market model, and Section 5 contains concluding remarks.

An Electric Energy Market Simulator

2.

ELEMENTS OF EMERGING ELECTRICITY MARKETS

2.1

Generation Scheduling in the Competitive Electricity Market

41

The traditional approaches of UC scheduling, hydro scheduling (HS), or a combination thereof (so-called hydro-thermal coordination (HTC)) [11] are being replaced rapidly by various forms of short-term competitive bidding schemes [8]. The majority of such schemes are based on a 24-hour, day-ahead scheduling process – a Forward Day-Ahead Market in the new jargon. In all the new schemes of electric utility restructuring, the generation component, as compared with the transmission and distribution components, is strictly market-driven and competitive. As a result, the central dispatcher plays the role of implementing the rules of generation scheduling as obtained through a market auction (bidding) mechanism. The main differences between the traditional and the “market-driven” approaches for generation scheduling consist of the following [11-13]: (a) Market prices are the result of the market design and associated rules and regulations under consideration. These rules may vary from one system to another. What complicates the picture is that the “products” being traded are many – starting with MWH energy and continuing through a host of ancillary services (A/S) and interconnected operation services (IOS). Some bidding schemes require full “multi-part” bids. Others require single-part bids for energy, and either simultaneous or sequential bids for the ancillary and IOS services. (b) Individual for-profit MPs aim to improve their short- and long-term market positions through various strategies for increasing their profits, controlling their market share, and/or other objectives. In the area of short-term (e.g., day-ahead and same-day) scheduling their main focus is on profit maximization. For the long-term strategies, they tend to use different mixes of tools including financial ones (futures, options, and swaps), or strategic ones (mergers and acquisitions, investments in other markets, etc.). (c) The new markets can become more or less effective through the existence of catalysts such as a power exchange, scheduling coordinators, energy brokers, service providers, e-commerce, and independent system operators.

42

2.2

The Next Generation of Unit Commitment Models

Network and Other “System” Issues in the New Environment

Principles that have been followed in restructuring transmission include: 1. In order to allow generating companies – as well as customers and distribution companies – to compete effectively, an open-access requirement has been suggested and implemented. The functions of generation scheduling have been “separated” from those of network access and control. 2. Certain markets (e.g., the PJM Interconnection, New Zealand, NYISO) use locational marginal prices (LMP). These depend on a solution of both a resource scheduling program (unit commitment, hydro-scheduling, or hydro-thermal coordination) and a securityconstrained optimal power flow (SCOPF). The resulting locational prices reflect a combination of “charges,” such as transmission losses, congestion, reactive support, reserves and, others [14,15]. (See Figure 1.) Even in markets that do not use LMPs (or equivalent), network information and the ancillary services protocols can be of great value for all market players, e.g., the California ISO [8].

An Electric Energy Market Simulator

43

In short, different market players will have to adapt to the operating rules for the systems served by different types of ISOs. In many cases, knowledge of real-time and forecasted network conditions can be critical in assessing various risk exposures and in evaluating hedging instruments.

2.3

Market Prediction is a “Learning Process”

Competitive markets do not behave like centrally dispatched systems. Price discovery is attained through the auction mechanism, which has its own dynamics [12]. Prices tend to be uncertain with a random component and a systematic component. The random component stems from external uncertainties (unpredictable weather, forced outage of a unit, system disturbance, fluctuating fuel prices and interest rates, etc.). The systematic component is related to physical constraints, market rules, auction clearing mechanisms, etc. It is also related to “gaming” and market strategies by various players (e.g., to manipulate prices and increase prices). Within this systematic component of market price prediction, a “learning process” becomes necessary to predict the behavior of the competition. Since energy markets clear prices very frequently, the learning process can be formally analyzed [13]. In most cases, however, it becomes the purview of “experts,” like skilled traders who earn their income from being very good learners.

2.4

Rationale of a Simulation-Based Approach

The foregoing discussion points to the fact that various analytical models may not provide the full answer to all the questions. Given the complexity of the power system and hence the electric energy market, there is a need in our opinion for a simulation tool to help in the learning process. Such a tool must by necessity represent both the systematic and random parts of the process. From a modeling viewpoint, we have considered the following to be critical [1-8,13]: The power system under consideration should be modeled realistically. Control and decision mechanisms by the main dispatcher (ISO, TSO, etc.) should be customized in accordance with the rules of the system under consideration. The market bidding and price clearing mechanisms should also be modeled for each respective market.

44

The Next Generation of Unit Commitment Models

The transitions from one market regime to the next (i.e., day ahead to hour ahead to real-time) should be clearly modeled and accommodated, The various tools used by individual market participants to play the market should be modeled. The simulation should have a growing library of such tools as one learns about the bidding strategies of various players. The simulation should explicitly model random and systematic risk exposures.

3.

MARKET-PARTICIPANT MODELS

The simulation model recognizes that the power system is run as a result of decisions made by groups of various market participants (MPs). At a minimum, these MPs would represent the following entities: Generating companies (GENCOs) who are competing against each other, Demand-side entities, Independent System Operators (ISOs) Energy Service Providers (ESCOs) Figure 1 gives the functional structure of MP models. This structure is in turn specialized for each of the categories of market participants as given in Figure 2. Figure 3 in Section 3.6 displays the ESCO functional structure.

3.1

Generating Companies (GENCOs)

GENCOs comprise a variety of entities: (a) the generation component of an integrated utility that is required to have functional separation of generation trading from other services, (b) a pure generating company with resources scattered over several geographic areas, or (c) an independent power producer (IPP). All these entities share similar needs and requirements. As a result, we designed the GENCO model to consist of three basic modules: (1) input module, (2) bid optimization module and (3) post-mortem analysis module. These are now presented in some detail as they form the core of the overall model.

3.1.1

Input Module

The GENCO will not be able to compete successfully unless it is able to forecast energy and ancillary service prices. Other inputs are related to spe-

An Electric Energy Market Simulator

45

cific market rules for the clearing of bids. We discuss these in some detail below. The GENCO should be able to forecast market-clearing prices based on historical information and its relative market position [12]. The approach used in our modeling is a regression model, which relates market-clearing prices to: (a) overall system demand and (b) available generation by the GENCO. Given the system demand forecast, the GENCO can then predict the corresponding price forecast. Several price-forecast scenarios are provided based on uncertainties in the demand forecast. At a minimum, these are expected, pessimistic, and optimistic scenarios with associated probabilities of occurrence.1 Price forecasts are done for both energy (MWh) prices as well as ancillary service prices. A given GENCO sits somewhere between the two extremes of being (a) a pure price leader or (b) a pure price follower [12]. Furthermore, the GENCO may be constrained by socio/economic/regulatory factors, which would limit its ability to manipulate price, even if it has market dominance.2 The output 1 2

An alternative to this is fuzzy modeling of the uncertainty. For example, the GENCO may be concerned with anti-monopoly laws or excess profit regulation, which may arise out of aggressive practices.

46

The Next Generation of Unit Commitment Models

of this sub-module consists of: (1) limits on market share and (2) sensitivity of market prices and energy supply to GENCOs bid prices. Both of these are to be used in the bid optimization module.

3.1.2

Bid Optimization Module (BOM)

The BOM consists of two components, described below. Component 1: Deterministic bid optimization based on specific input scenarios The optimization is performed for two specific markets: (a) day ahead and (b) hour ahead markets.3 The market rules vary from market to market. Our approach models initially two types of market rules – (1) those which are based on LMP considerations, e.g., the PJM Interconnection, and (2) those based on bidding by scheduling coordinators4 using a PX, e.g., the California ISO. Thus, there are four basic sub-modules to consider. Day Ahead/LMP-Based. The bidding system for LMP-based day-ahead markets consists of a multi-part bidding process whereby the generation (and load) bidders are required to submit typically the following: Energy multi-block price curves which are piecewise constant No-load price bids Operating range (min and max MW) Minimum up-time Minimum down-time The ISO control center as a result performs a combination of unit commitment and contingency-constrained optimal power flow computations to yield an optimal solution. The performance criterion is the minimization of overall price to meet customer demand. When combined with elastic demand bids, it is possible to use the same software to maximize social welfare. Ideally, if every generation bidder submits his marginal cost data (plus a specified mark-up for profit), the result would be almost the same as that of a fully integrated utility system. The constrained OPF would minimize system losses. In case the ISO would pay for reactive power production, then the OPF would attempt to maximize profit from the use of reactive power production. From the perspective of the GENCO, the objective is to submit a bid that maximizes profit – at least at this initial deterministic level. Thus for each 3

4

It is possible to extend the functionality of this module to cover both real-time bids and longer-term forward market bids. Scheduling coordinators (SCs) are allowed to submit strictly balanced generation/load bids for the energy part of the market.

An Electric Energy Market Simulator

47

forecasted scenario, the control variables consist of the above bid parameters, as long as they do not violate technical limitations (such as bidding minimum up-time which is always greater than or equal to the technical minimum uptime). One of the key issues here is to ensure that the GENCO MarketParticipant is able to model its own cost information, including their nonconvex price incremental cost curves. Hour-Ahead/LMP-Based. Under this regime, the basic bidding process consists of incremental/decremental bids to allow for adjustments of schedules based on deviations from the day-ahead schedules. The main tool is strictly the OPF with an objective function to maximize market surplus. From the bidder’s perspective, the goal is to maximize profit. Day-Ahead/SC/PX5-Based. In the SC/PX-based market, the typical bidding process is decomposed into hourly energy price bid curves and other bids for congestion management and ancillary services. However, there is a distinction between a scheduling coordinator (SC) and the power exchange (PX). In California, the SCs are exposed to network congestion and ancillary service risks (and opportunities). Their main bid parameters to improve profitability consist of incremental/decremental (INC/DEC) bids for congestion management and ancillary service bids. In the simulation model the focus is strictly on congestion management bids. For GENCOs bidding through the PX, the main objective is to optimize hourly energy price bid curves. Furthermore, there is a question here whether portfolio bidding should be allowed. If portfolio bidding is permitted, then the GENCO has to perform optimization at two levels: one that uses a single curve for the entire company and a second that provides generation schedules at the cleared market prices. Again, the GENCO has the opportunity to bid into the congestion management and ancillary services market. Hour-Ahead/SC/PX-Based. This is similar to the day-ahead bidding system with the exception that portfolio bidding may not be allowed. Component 2: Bid optimization through risk management For each of the four bidding regimes, the GENCO optimization is performed for a set of scenarios. In the input module, there is a probability associated with each scenario. The key output here is a profit probability profile. The analysis would select a scenario that yields bid prices for the best profit/risk trade-off.

5

PX = Power exchange, whether independent or part of the central market clearing organization.

48

3.2

The Next Generation of Unit Commitment Models

Demand-Side Entities

For the purposes of the model under consideration, we distinguish the following demand-side entities: (a) Major Load Serving Entities (LSE) (e.g., distribution company, or commercial or industrial loads): An LSE may be able to bid a declining multi-block purchase price curves based on the composition of its load and the demand-side management systems in place (typically interruptible loads). The objective here for the LSE is to reduce the cost of purchases based on its price forecasts (and contracts already in place). (b) Pumped Storage Power Plants: These plants try to combine their generation bids with their demand bids (for pumping). The result is a composite optimization.

3.3

Independent System Operators (ISOs)

In our model, the ISO represents the component that implements the results of the bidding process to create a full schedule of system operation that meets regulatory reliability and security requirements. Normally, the ISO controls only its own area of responsibility (control area). Again, we distinguish between two types of ISOs: LMP-based and SC/PX-based.

3.3.1

LMP-Based ISO

The model of the LMP-based ISO permits the following functions to be performed: Issuance of ISO-demand forecast for the day-ahead and hour-ahead markets: The demand forecast is a net forecast for all demands to be met by participants in the day-ahead and hour-ahead markets. In essence, the forecast is adjusted by subtracting any bilateral contracts and export/import contracts, which are also bilateral in nature. Hourly optimization of the generating system and the network through a combination of unit commitment and contingencyconstrained OPF, as described above, based on the totality of the bids by the market participants. The optimization would yield LMPs, which are published and released to the market. Computation of congestion charges and firm transmission rights (FTR) payments.

An Electric Energy Market Simulator

49

Similar functions for (1,2, and 3) the hour-ahead market (no unit commitment is involved, however). Real-time system control through automatic generation control (AGC). The AGC utilizes an economic dispatch component which is based on the hour-ahead bid prices, or alternative bid prices for system regulation as an ancillary service

3.3.2

SC/PX-Based ISO

In this case, the following functions are supported: Demand forecast similar to the one in the previous section for the LMP-based ISO Congestion management for interzonal congestion as well as intrazonal congestion for both the day-ahead and hour-ahead markets Simultaneous and sequential ancillary services bid clearing Automatic generation control based on ancillary service bids for system regulation and load following

3.4

Power Exchange (PX) Model

There is no need for a separate PX model for the LMP-based system, simply because the ISO in this case clears the market using the various multipart bids. The PX model for the SC/PX-based market is quite simple as it just matches supply and demand curves. The PX, however, may engage in ancillary service markets and options/futures markets and these have to be taken into account.

3.5

Energy Service Provider (ESCO)

For the purposes of the market simulation model, the ESCO is a combination of a GENCOs and LSEs. A large ESCO may be bidding in multiple markets and these have to be modeled separately. The trading performed by the ESCO may combine a portfolio of contracts/bids to manage risk.

3.6

Overall Market Model

Figure 3 shows the resulting EMOS overall market model. We have designed the model for the following uses:

50

The Next Generation of Unit Commitment Models

(a) Studies and training by each market participant entity described above to improve its performance and meet its own needs. (b) Studies at regional or national levels to evaluate different market designs. (c) On-line activities by MPs, including energy bidding strategies, ancillary services bidding, market surveillance, etc., whenever the system is linked to real-time data sources. Again, a given market participant should be able to model its own activities using its own proprietary systems.

3.7

Note on Implementation

The EMOS is implemented using standard software applications developed primarily by EPRI and other developers. These include:

An Electric Energy Market Simulator

51

EPRI-OTS by EPRI for the primary PSM modeling EPRI-DYNAMICS unit commitment program for various market participant modules EPRI ANN-STLF for short-term load forecasting DSI-OPF for the multi-area optimal power flow interfaced with EPRI-DYNAMICS Other components of the EMOS are still under development. The overall system uses an effective database system which is compliant with EPRI’s common information model [1].

4.

CONCLUDING REMARKS

The development of an overall market model for the electric energy industry under restructuring is a major challenge mainly because the electric physical system is highly complex, changes constantly, and creates a significant amount of physical risk to market participants (MPs). The overall market model consists of an interacting population of MPs. Market rules and designs are aimed at coordinating all efforts through ISO’s. The model implementation is based on such developments to allow for effective interoperability and plug-compatibility among various applications. We expect that the developments reported here can be used in a variety of ways by each MP type: GENCOs, LSEs, ISOs, PXs and ESCOs. The main uses consist of: Just-In-Time training of personnel – traders, dispatchers, operations engineers and managers Enhanced mechanisms for improving competitive positions by traders Ability of ISOs and market designers to analyze the behavior of market participants and test improvements in market rules and overall market design Unit commitment as such is used as a tool whenever applicable. The main users of UC are the GENCOs, ESCOs, and some ISOs with multi-part bidding market rules.

ACKNOWLEDGEMENTS The developments reported in this chapter are under the sponsorship of the National Science Foundation (NSF) under SBIR Phase II Grant, DMI-980 10161. The EPRI-OTS developments and related enabling technologies have been partially sponsored by EPRI, Palo Alto, California.

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The Next Generation of Unit Commitment Models

REFERENCES 1.

2.

3.

4. 5.

6.

7. 8.

9.

10. 11. 12. 13.

14.

15. 16.

17. 18.

A. Debs, Y.-C. Wu, and C. Hansen. “Enhancement of the EPRI-OTS for the Restructured Electric Utility,” Project Reports, Decision Systems International, Atlanta, Georgia, under Small Business Innovations Program of the National Science Foundation. A. Debs and C. Hansen. “The Total Power System Simulator: A Comprehensive Tool for Operation, Control and Planning.” In Proc. Arab Electricity ’97 Conference & Exhibition, PennWell Europe and DSI, Bahrain, 1997. A. Debs and C. Hansen. “The EPRI-OTS as the Standard for Training and Studies in the New Era: Strategy for Global Application.” Presentation at the First Asia Pacific Conference on Operation and Planning Issues in the Emerging Electric Utility Environment, sponsored by EPRI, Kuala Lumpur, Malaysia, 1997. M.K. Enns, et al. Considerations in designing and using power system operator training simulators. EPRI EL-3192: 1984. V. Calovic and A. Debs. “The Fully Integrated DTS for ESB, Dublin. ” In Proc. EPRI First European Workshop on Power System Operation and Planning Tools, Amsterdam, The Netherlands, 1995. C. Hansen and M. Foley. “Power System Model Enhancement at ESB, Dublin.” In Proc. EPRI First European Workshop on Power System Operation and Planning Tools, Amsterdam, The Netherlands, 1995. S. Lutterodt and A. Debs. “Issues in Use of Operator Training Simulators.” Paper presented at the CEPSI Conference, Kuala Lumpur, Malaysia, 1996. A. Debs and F. Rahimi. “Modern Power Systems Control and Operation in the Restructured Environment.” Class notes for intensive short course by Decision Systems International, San Francisco, CA, 1999. A. Debs, P. Gupta, C. Hansen, A. Papalexopoulos, and F. Rahimi. “System Planning in the Context of Competition and Restructuring.” Class notes for intensive short course by Decision Systems International, San Francisco, CA, 1996. M. Ilic, F. Graves, L, Fink, and A. DiCaprio. Operating in the open access environment. Elec. J., 9(3): 61-69,1996. A. Debs. Modern Power Systems Control and Operation. Norwell, MA: Kluwer Academic Publishers, 1988. B. Sheblé. Computational Auction Mechanisms for Restructured Power Industry Operations. Boston, MA: Kluwer Academic Publishers, 1999. A. Papalexopoulos. “Design of the Wholesale Market in the USA.” In Proc. EPRI Second European Conference – Enabling Technologies and Systems for the Business-Driven Electric Utility Industry, Vienna, Austria, 1999. Y.C. Wu, A.S. Debs, and R.E. Marsten. A direct nonlinear predictor-corrector primal-dual interior point algorithm for optimal power flows. IEEE Trans. Power Syst., 9(2): 776-883, 1994. H. Kim and R. Baldick. Coarse-grained distributed optimal power flows. IEEE Trans. Power Syst., 12(2): 932-939,1997. A. Debs. “The OPF in the Deregulated Environment.” Lecture notes for intensive short course by Decision Systems International: Modern Power Systems Control and Operation, San Francisco, CA, 1996. A. Papalexopoulos, et al. Cost/benefits analysis of an optimal power flow: The PG&E experience. IEEE Trans. Power Syst. 9(2): 796-804, 1994. Y.C. Li and A.K. David. Optimal multi-area wheeling. IEEE Trans. Power Syst., 9(1): 288-294, 1994.

Chapter 4 AUCTIONS WITH EXPLICIT DEMAND-SIDE BIDDING IN COMPETITIVE ELECTRICITY MARKETS A. Borghetti University of Bologna, Italy

G. Gross University of Illinois at Urbana-Champaign

C. A. Nucci University of Bologna, Italy

Abstract:

1.

This paper focuses on the development of a model for and the simulation of electricity auctions with demand-side bidding (DSB) explicitly considered. We generalize the competitive power pool (CPP) framework developed in [1] to include DSB. In order to allow customers to play a proactive role in the price determination process, the DSB provides the opportunity for them to submit bids for load reductions in specific periods. We study the behavior of DSB inclusion in electricity auctions simulations obtained with a specially developed Lagrangian relaxation scheme that effectively takes advantage of the structure of the problem. We present some numerical results for a 24-hour simulation on a small system. This case study is effective in illustrating the various economic impacts of DSB including system efficiency effects, changes in the system marginal price and the load recovery effects.

INTRODUCTION

This paper deals with the development of a model for the simulation of electricity auctions with demand-side bidding (DSB) explicitly considered. The model is a generalization of the competitive power pool (CPP) framework developed in [1] to include DSB. Such a mechanism has been used at times in the England & Wales Power Pool by large industrial consumers who can offer their ability to reduce load directly to the Pool and receive a payment for actually making such a reduction possible [2,3].

54

The Next Generation of Unit Commitment Models

In the CPP with only supply-side offers, the point of intersection of the aggregate supply curve and the fixed forecasted load determines the marketclearing price. To improve competition, the consumers may be allowed to participate in the market price definition process by providing them the opportunity to submit bids. Without DSB, the system marginal price (SMP) is determined by constructing the supply curve from the offers submitted by the supply-side bidders and determining the uniform price paid to all bidders who are selected to serve the fixed forecasted load demand. The basic idea in the application of DSB is to use the demand profile to lower SMP by cutting load during peak periods. DSB may take advantage of large industrial customers who can cut load or shift load to benefit from lower electricity prices. Since industrial customers are not expected to reduce significantly their daily energy consumption, the overall effectiveness of DSB depends on the how, the how much, and the when of the energy recovery. Clearly, the more flexibility such a customer has, the more possible it becomes for him to participate in DSB. The demand recovery is typically a function of the weather, of the economic conditions as well as of the number of reduction periods, and the amount of load cut in each period. In [4] an analysis has been presented of data relevant to a four-year period for a regional electricity company (REC) in the United Kingdom that quantifies the extent of intertemporal substitution in electricity consumption across pricing periods within the day for five groups of different industries. Every industry shows electricity substitution possibilities with its own firm-specific characteristics. We have constructed an econometric model from the results of this analysis. This model is for the use of the REC to formulate its demand-side bids and as such takes into account the specific pricing scheme and rules adopted in the England & Wales Power Pool. There are additional data available on the characteristics of the load recovery from experiences gained with the application of commercial/industrial and residential load control programs [5-7]. In this case, there is electricity substitution only across adjacent load periods, i.e., the energy is recovered in the periods that follow right after the reduction periods. In this paper, we simulate and study the inclusion of DSB in electricity auctions to examine various economic and policy aspects in DSB. The simulation tool implements a Lagrangian relaxation-based scheme, which takes advantage of the problem structure with the inclusion of DSB. The sufficiently detailed representation of the supply-side bidders allows the modeling of the unique physical characteristics of the power generation system. In particular, it takes into account the main operating considerations including the operational limits and up- and down-time constraints. The supply-side bidders are also allowed to include in their bids a separate price for start-up in addition to the price for the MWh commodity. In the simulation tool, the impact of DSB is taken into account by incorporating specific energy con-

Auctions with Explicit Demand-Side Bidding

55

straints and load recovery into the Lagrangian relaxation algorithm. Researchers have applied Lagrangian relaxation methods widely in recent years for the solution of unit commitment problems for large-scale systems due to their ability to include a more detailed system representation than would be possible with other techniques [8]. Some aspects of DSB have been analyzed in [9,10]. The studies indicate that the specific characteristics of the demand-side bidders have a substantial impact on the system economies. However, additional work is required to better quantify such impacts. This chapter aims to bring additional insight to the study of DSB. The principal combinations are the development of a generalized framework for the analysis of various aspects of DSB and the illustration of the salient characteristics through the simulations obtained with the tool developed for the implementation of this framework. The chapter is organized as follows. In section 2, we present the generalization of the CPP structure with the inclusion of DSB into the CPP framework and a description of the Lagrangian relaxation-based algorithm for its solution. In section 3 we present some numerical results obtained for a 24hour simulation on a small system. We compare costs and price signals obtained from DSB in addition to supply-side bidding (SSB) to costs and price signals obtained from the use of only supply-side bidders. The paper provides an analysis of the numerical results. Section 4 offers final thoughts on this research, and the Appendix presents details concerning the implemented code and the power system data.

2.

GENERALIZATION OF THE COMPETITIVE POWER POOL FRAMEWORK FOR INCLUSION OF DEMAND SIDE BIDDING

We start with the extension of the CPP framework developed in [1] to include DSB in addition to SSB. With the addition of DSB, the CPP dispatcher problem is to select the winning bids and offers from the set of both supply-side offers and demand-side bids. We use henceforth the term bid to refer to both the supply-side offer and the demand-side bid and we refer to each player as a bidder. Each bid has three components: the bid variable price which describes the per hour price as a function of MW provided/reduced. The function is here assumed to be a quadratic or piece-wise linear function; the bid start-up price which is charged whenever the bidder is committed; and the bid offered capacity

which is a vector whose

56

The Next Generation of Unit Commitment Models

component is the maximum capacity offered by the bidder for use in period t, expressed as a fraction of the maximum output. The supply- and demand-side bids have different characteristics. In particular, for the case of SSB, we have made the assumption that the operational data1, including the minimum and maximum output and and minimum up- and down-times and need to be provided by each bidding plant i. Moreover, the bid start-up price which is charged whenever unit i is started in period t, is assumed to be a function of the down time of unit i at period t-1

where is the thermal time constant of unit i, I is the number of supply side bids, is a constant related to the cold start charge, and is a constant related to the fixed operating and maintenance charges [11]. The characteristics of the demand-side schemes require the specification of the following information for each demand-side bid j,j=1, ..., J: the bid start-up price that is considered a constant value the subset of the operator-designated load-reduction periods that is the subset of the periods dertake load reduction the minimum and maximum demand

in which bidder j may unand

that can be re-

2

duced by bid j the subset of the operator-designated load-recovery periods

that

is the subset of the periods in which bidder j may undertake a load recovery load recovery at period h, that is related to all the load reductions in each reduction period of by the following expression

1

2

We do not take ramp and network constraints into account in the model discussed in this chapter. Note that a requirement is introduced in the problem formulation to ensure that the sum of the maximum load reductions of all the demand-side bids in a period t cannot exceed the load forecasted for that period.

Auctions with Explicit Demand-Side Bidding

where

57

represents the load recovery ratio in period h of the load

reduction

in period t.

The above representation of load recovery is similar to that used in [9]. It allows the representation in a flexible and simple way of the electricity substitution characteristics of consumers who may participate in load modification. It allows, moreover, load recovery to occur both in pre- and post-load reduction periods. In [9] it is pointed out that the beneficial effects of DSB are greatly attenuated if some bidders recover their load in the same periods in which others are reducing it. Therefore, we shall assume that there are no overlapping periods in which the load reductions and the load recoveries may cancel each other out. By taking into account both load reductions and load recoveries, the actual load in reduction period t is given by

and the actual load

where

and

in recovery period h is given by

are the forecasted loads in period t and h, respectively.

The additions of the relations for the DSB participation do not change the basic structure of the framework in [1]. Consequently, we can extend the Lagrangian relaxation framework developed in [1] to the more general model described here. For this purpose, we use the following notation:

I: J: T: D:

the number of supply-side bidders the number of demand-side bidders the number of time periods the T-dimensional vector of the load demands in each period t in the scheduling horizon the T-dimensional vector of the zero-one variables indicating whether supply-side bidder i is committed in period t or not the T-dimensional vector of the zero-one variables indicating whether demand-side bidder j is committed in period t or not the T-dimensional vector of the amount of power that the supplyside bidder i is producing in period t the T-dimensional vector of the amount of power that the demand-side bidder j is reducing in period t.

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The Next Generation of Unit Commitment Models

The CPP dispatcher problem determines the most economic commitment that satisfies the forecasted demands D without violating physical and operating constraints of the generation equipment and demand specifications.

subject to initial conditions and, by taking into consideration equations (2) and (3), subject to

We refer to the optimization problem in equations (4)-(7) as P. While the demand-side bids are defined similarly to supply-side offers, problem P differs from problem with only supply-side bidding. In fact, the incorporation of DSB adds the two decision variables w and y that are characterized by specific constraints specified in (7). The solution approaches with and without DSB are similar. Also with the introduction of new DSB decision variables w and y, the Lagrangian relaxation approach allows the decomposition of P into I+J sub-problems associated with each supply-side and demand-side bid. With the inclusion of DSB, we use the following Lagrangian dual function

or, more explicitly, taking into account (4),

Auctions with Explicit Demand-Side Bidding

59

subject to the specified initial conditions and to constraints (6) and (7). t= l,...,T, are non-negative Lagrange multipliers. The dual function is rearranged as

where, for supply-side bid i

and, for demand-side bid j, by explicitly taking into consideration equations (2) and (3),

subject to the specified initial conditions and to the constraints (6) and (7). Problems (10) and (11) are the I + J sub-problems, one for each bidder, that need to be solved. In order to find the optimal values of the multipliers to use in equations (10) and (11) the following dual problem is solved

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The Next Generation of Unit Commitment Models

In fact, L * provides a lower bound of the final solution of the primal problem P [12]. As a by-product of the process of maximizing L we obtain the optimal Lagrange multipliers and a system schedule {u, w, p, y} resulting from the solution of the Lagrangian relaxation for In certain cases, this system schedule does not satisfy the conditions imposed by the demand constraint and, therefore, a practical approach for computing a near-optimal schedule has to be implemented [13]. It may be useful to note that the optimal schedule, obtained with the solution of the proposed framework, is the one that minimizes the total “production” costs, i.e., the costs paid to satisfy the forecasted demand, on the basis of the bids of the supply-side and demand-side bidders. The solution can differ, in general, from the schedule that minimizes the consumer payments, for the various reasons cited in [14].

3.

NUMERICAL RESULTS

We implemented the solution approach based on the extended framework for the inclusion of DSB as a software package. Some of the salient aspects of this package are summarized in Appendix A. This package has been used to perform many numerical studies to assess the characteristics and the impacts of DSB in a competitive power pool. To illustrate the application of the generalized framework, we present some typical results of our simulation studies. We use the 10-unit system in [13] as the supply side of the resources. The operational data and the bid functions used for the supply-side bidders are given in Appendix B. The behavior of the power pool is simulated for the 24-hour period load profile of Table B.1 in Appendix B. Figure 1 plots the hourly loads. The load reductions are dispatched on the basis of the demand-side bids that represent the prices at which the bidders wish to reduce their consumption by the specified amounts. The bidder that is scheduled to reduce its consumption in a certain period is also allowed to recover its load partially or totally in other periods. The model specifies the periods in which load reduction and load recovery are allowed. These periods have been defined in the previous section as the operator-designated load-reduction periods and load-recovery periods To show the influence of such a constraint, the results of a sensitivity study are presented. The study was carried out by varying only the following two triggering levels: the load-reduction triggering level as a fraction of the peak load, specifies the load level above which the load reductions are allowed; and

Auctions with Explicit Demand-Side Bidding

61

the load-recovery triggering level as a fraction of the difference between the peak and the base load, specifies the load level below which the load recoveries are allowed. For each given and values, the corresponding permitted reduction and recovery periods, and respectively, are then established. The load recovery is considered uniform throughout the recovery period. We discuss the results of several simulations on this system with a single demand-side bidder in addition to the ten supply-side bidders. The demandside bidder limits are and The demand-side bid is 1 $/MWh with 1 $/h being the no-load bid. These values were selected specifically to provide a lower cost per unit of energy to the CPP dispatcher on the demand side than the supply-side. We start out with a case in which, the DSB is allowed whenever the load exceeds For this case is set at 0.5. The periods when load reductions are allowed, are The corresponding recovery periods

are

and the recovery is implemented to spread the recovery of load in equal amounts in each of the periods in

62

The Next Generation of Unit Commitment Models

To assess the impact of DSB, we use the no-DSB case as the reference basis. We show in Figure 2 the modified loads as a result of DSB under different levels of load recovery. These levels of load recovery are indicated by the recovery fraction value, that is the total amount of load recovery over the total amount of reduced load. In Figure 3, we depict the corresponding impact on the system marginal prices (SMP). For the supply-side bidding strategy used, the system marginal prices track closely the modified system demands Furthermore, DSB, in effect, acts to “smooth” out the price variations across periods. We now evaluate the impacts of varying some of the parameters in DSB implementation. One parameter that has a major impact on deployment of DSB is the load reduction recovery factor. This parameter is the ratio of the recovery energy to the cut energy. As this factor decreases from 1 to 0, the savings increase with respect to the costs for the case without DSB (what we shall call reference costs). Figure 4 shows results for the test system. The assessment of DSB on the variability of the SMP is important. DSB tends to impact prices in different periods lowering the prices in periods with higher loads and increasing them in periods with lower loads. We evaluate the hourly average SMP, as well as the volatility impacts in terms of the standard deviation as a function of recovery fraction. Figure 5 gives these results with the no-DSB case as reference. While there is a reduction in the volatility, a reduction in the average value of the SMP is not obtained for all

Auctions with Explicit Demand-Side Bidding

63

recovery fractions. The reason is that the schedule is obtained without a minimization of consumer payments. In the case of DSB without load recovery, the reduction of the volatility is only due to the reduction of the SMP in the peak periods. Therefore, the reduction in the volatility is lower in this case than for the case in which there are both the reduction in the price in the peak periods and the increase of the price in load recovery periods. We note that at the higher fraction of load recovery, DSB is committed in only a limited number of periods, and as such both the savings and the reduction in price volatility are limited. An important aspect to examine is the impact of different triggering levels for the definition of the operator-designated load-reduction and recovery periods. For the test system we evaluate the savings with respect to the reference DSB case for sixteen different triggering levels under the fixed load recovery fraction of 0.8. Table 1 gives the data, and Figure 6 presents the plots as a function of the reduction triggering levels for different values of Due to the fewer number of periods in which the demand-side bidder may be committed, higher reduction triggering levels result in lower savings. On the other hand, the effect of the recovery triggering level is less uniform: as the recovery triggering level increases, the number of recovery periods increases and the recovery of the load decreases in each recovery period. In the case of the higher recovery triggering levels, the recovery periods are in the shoulder load periods resulting in a decrease in the savings compared to the lower values of

64

The Next Generation of Unit Commitment Models

Auctions with Explicit Demand-Side Bidding

65

We examine the impact on loads and SMPs for as a function of Figure 7 shows the load modifications, and Figure 8 depicts the impact on the SMPs. These plots make more concrete the observations in the previous paragraph on the impacts of different We next discuss the impacts of a single-supply unit’s bidding strategy on the market. We focus on the unit 9, and we obtain the results in Figures 2-8 with the bid of unit 9 declaring unavailability in periods 10 to 21. A supplyside bidder for various reasons or market objectives may select such “strategic” behavior. When unit 9 changes its bid and declares availability for the periods 10-21, the total costs without DSB change $537,271. This change

66

The Next Generation of Unit Commitment Models

Auctions with Explicit Demand-Side Bidding

67

represents a saving of $3,186 with respect to the reference case of no-DSB without the participation of unit 9 for the periods 10-21. From Figure 4, we can see that such a saving is lower than the saving obtained with DSB recovering the entire energy. We may, therefore, interpret the impacts of DSB to include its ability to mitigate the market power on the supply-side. Thus far, we have concentrated on displaying the salient characteristics of DSB through a single demand-side bidder. The rationale for this was to provide a good insight into the interaction of the DSB with the electricity market. We next study the case with multiple DSB bidders. To illustrate the behavior of DSB with multiple demand-side bidders, we consider two cases: (i) five DSB bidders and (ii) nine DSB bidders. We compare the behavior in these two cases to that in the case of the single bidder discussed above. For these cases, the demand-side bidders have and the load recovery fraction is 0.9. Recall that for the single DSB bidder, the bid is 1 $/MWh with 1 $/h being the no-load amount and In case (i) the bids are 1, 2, 3, 4, and 5 $/MWh, respectively, with 1 $/h the no-load amount for each bidder. In case (ii) the bids are 1, 1.25, 1.5, 1.75, 2., 2.25, 2.5, 2.75, and 3 $/MWh with 1 $/h the no-load amount for each bidder. In the two cases is 200 MW for each DSB bidder. Figures 9 and 10 provide plots of the results obtained with the five and nine demand-side bidders together with the results obtained with the single DSB bidder. Figure 9 gives the impacts on the load modification, and Figure 10 displays the impacts on the hourly SMPs. The number of demand-side bidders has been chosen to exceed the number that is committed in the multiple-bidder cases: in (i) only two are committed, and in (ii) only five are committed with the fifth being committed only for a single period in the hour 10.

4.

CONCLUSIONS

This chapter focused on the integration of DSB into electricity markets. We discussed the development of a general framework for the inclusion of DSB in addition to supply-side bidders. We used the Lagrangian relaxation based solution approach to examine the impacts of DSB. The tool we developed for simulation is very useful for investigating a wide spectrum of policy issues. In our case, we have used this tool to examine the ability of DSB to mitigate the potential for exercise of market power by the supply-side bidders. In addition, the impacts of DSB on smoothing system marginal prices and on mitigating price volatility are important attributes as indicated by the numerical results.

68

The Next Generation of Unit Commitment Models

Auctions with Explicit Demand-Side Bidding

69

The numerical studies of the paper provide good insights into various aspects of DSB. In particular, it is important that load reductions and load recoveries do not cancel each other out. Therefore, the specification of permitted load reduction periods and permitted load recovery periods has been added to the model. The results of a sensitivity analysis show that the specification of appropriate triggering levels, that define the permitted load reduction periods and permitted load recovery periods, is particularly critical to make effective use of DSB. In the chapter, we provide a simple example to study multiple DSB players. The impacts of competition and possible collusion among DSB players need to be investigated in more depth. Also, there are many additional policy issues that remain to be examined. One key issue for further study is that of DSB remuneration, including the needs for such incentives and the levels at which to set them. The changes in supply-side bidder behavior brought about by the inclusion of DSB are another key issue that requires investigation.

ACKNOWLEDGMENTS We are grateful for the financial support for the research reported here from the Italian National Research Council. George Gross received support from the Power System Engineering Research Center administered by Cornell University.

REFERENCES 1. 2. 3. 4. 5. 6. 7.

G. Gross and D.J. Finlay. “An Optimization Framework for Competitive Electricity Power Pools.” In Proc. PSCC, 815-823, 1996. F.A. Wolak and R.H. Patrick. The impact of market rules and market structure on the price determination process in the England and Wales electricity market. Working Paper PWP-047, University of California Energy Institute, 1997. D.W. Bunn. Demand-side participation in the electricity pool of England and Wales. Decision Technology Centre report, London Business School, 1997. R.H. Patrick and F.A. Wolak. Estimating the customer level demand for electricity under real time pricing. Working Paper, Department of Economics, Stanford University, 1997. D.V. Stocker. Load management study of simulated control of residential central air conditioner on the Detroit Edison Company system. IEEE Trans. Power Apparatus Syst., 4: 1616-1624, 1980. A.I. Cohen, D.H. Oglevee, and L.H. Ayers. An integrated system for residential load control. IEEE Trans. Power Syst., 3: 645-651, 1987. C.N. Kurucz, D. Brandt, and S. Sim. A linear programming model for reducing system peak through customer load control programs. IEEE Trans. Power Syst., (11)4: 18171824, 1996.

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70 8. 9.

10. 11.

12. 13. 14. 15. 16.

A.J. Svoboda and S.S. Oren. Integrating price-based resources in short-term scheduling of electric power systems. IEEE Trans. Energy Conversion, (9)4: 760-769, 1994. G. Strbac, E.D. Farmer, and B.J. Cory. Framework for the incorporation of demand-side in a competitive electricity market. In IEE Proc. – Generation, Transmission and Distribution, (143)3: 232-237, 1996. G. Strbac and D. Kirschen. Assessing the competitiveness of demand side bidding. IEEE Trans. Power Syst., (14)1: 120-125, 1999. J. Gruhl, F. Schweppe and M. Ruane. “Unit Commitment Scheduling of Electric Power Systems.” In System Engineering for Power: Status and Prospects, LH Fink and K. Carlsen eds., Henniker, NH, 1975. A. Geoffrion. Lagrangian relaxation for integer programming. Math. Prog. Study, (2): 82-114, 1974. J.F. Bard. Short-term scheduling of thermal-electric generators using Lagrangian relaxation. Oper. Res., (36)5: 756-766, 1988. S. Hao, G.A. Angelidis, H. Singh, and A.D. Papalexopoulos. Consumer payment minimization in power pool auctions. IEEE Trans. Power Syst., (13)3: 986-991, 1999. R. Fourer, D.M. Gay and B.W. Kernighan. A modeling language for mathematical programming. Manage. Sci., (36): 519-554, 1990 B.A. Murtagh and M.A. Saunders. “MINOS 5.1 User’s Guide.” Technical Report SOL 83-20R, Systems Optimization Laboratory, Department of Operations Research, Stanford University, 1987.

APPENDICES A.

Development of a Computer Program for Validation of the Proposed Approach

We have developed a computer program incorporating both supply-side and demand-side bidding. The program allows multiple demand-side bidders and supply-side bidders to interact for meeting a load profile over a specified period. The generality of the program allows us, in particular, to study the characteristics of the demand-side bidders and the influence of their parameters on the results. For the Lagrangian relaxation implementation we have followed the procedure proposed in [13]. To solve subproblems (10) and (11), the values of are, at first, considered assigned as well as the values of the zero-one decision variables and It follows that these subproblems, for supply-side bid i and demand-side bid j, respectively, become

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subject to upper/lower limits specified in equation (6) (without the minimum up/down constraints) and in equation (7). These problems are piece-wise or quadratic problems (it depends on how the bid variable price function is defined), and can be solved with a linear or quadratic programming technique. These optimizations give the values of power productions and load reductions To find the optimal values for the decision variables and we solve subproblems (10) and (11) using the values of and in a forward dynamic programming, taking into account the start-up prices and the minimum up-/down-time constraints. In the case of demand-side bidder problem (11) the dynamic programming approach differs from that of supply-side bidders. In particular, to take into account the load recoveries, we add the following term to the values corresponding to the states in which demand-side bid j is committed at period t

This term represents the contribution to the function of all the load recoveries due to the scheduled load reduction at period t that has to be minimized in equation (11). Because of the presence of this additional term, the solution of the problem (A.2) and the dynamic programming procedure are included in a loop that ends when two consecutive solutions result in the same commitment, i.e., in the same values of the zero-one decision variables To find the maximum value L * of equation (12), we use a standard subgradient technique [12]. With this technique, the solution process is iterated, starting from a set of tentative values of multipliers and at each iteration k the multipliers are updated using

where, taking into consideration equations (2) and (3),

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and denote the current solution (iteration k) of the dual problem and the lowest available solution of the primal one, which represent the lower and upper bound of the final solution at iteration k, respectively. is a scalar parameter that is progressively reduced whenever the solution of the dual problem is not improved. To implement such a technique, we need to find a feasible solution for the primal problem P at each iteration. This is accomplished by adopting the priority-list heuristic procedure proposed in [13], based on the calculation of the marginal costs associated with each supply-side bidder and each period. The heuristic procedure is implemented in such a way that it does not change w, i.e,. the commitment of the demand-side bidders. The procedure here described is stopped when the relative duality gap (defined as is lower than a specified value. We implement the program in the algebraic modeling language for mathematical programming called AMPL [15]. This language is particularly useful for the piecewise linear representation of the cost functions. A major advantage of the modular structure resulting with the AMPL implementation is the ability to use a library of solvers. In this paper the MINOS [16] solver has been used to solve the problems (A.1) and (A.2).

B.

Case Study System Data Table B.1 gives the load profile.

Ten generators, considered as supply-side bidders, compose the supply system. Tables B.2, B.3, and B.4 show the supply-side bidder data and have been adapted from those used in [13]. Table B.1 shows the values of the minimum and maximum power outputs the minimum up- and down-times and the initial status of the units i.e., the number of

Auctions with Explicit Demand-Side Bidding periods the unit had been up

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or down

In Table B.3 the quadratic expression of the cost data of [13] is converted into piecewise linear form so that the bid variable price is specified by 6 parameters: the no-load price three incremental prices two elbow points respectively.

and

Table B.4 displays the values of the components of the bid start-up price defined in equation (1).

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Chapter 5

THERMAL UNIT COMMITMENT WITH A NONLINEAR AC POWER FLOW NETWORK MODEL Carlos E. Murillo-Sánchez and Robert J. Thomas Cornell University

Abstract: This chapter presents a formulation of the thermal unit commitment

problem that includes nonlinear power flow constraints, thus allowing a more accurate representation of the network than is possible with DC flow models. This also permits potential VAr production to be used as a criterion for commitment of otherwise expensive generators in strategic locations. We use a Lagrangian relaxation framework with duplicated variables for each active and reactive source, permitting the exploitation of the separable structure of the dual cost. Results for medium-sized systems in a parallel processing environment are available.

1. INTRODUCTION The central theme surrounding the use of Lagrangian relaxation for the unit commitment problem is that of separation. Ever since the early papers [1, 2] this separability was the objective and for a good reason: the unit commitment problem, as a mixed-integer mathematical program, suffers from combinatoric complexity as the number of generators increases. This is what dooms other algorithms intended for solving it, such as dynamic programming: the combined state space of several generators in a dynamic program is too large to be able to attack many realistic problems, even with limited memory schemes. Classical Lagrangian relaxation permits the decomposition of the problem into several one-machine problems at each iteration; the coupling to other constraints involving more machines is achieved by sharing price information corresponding to the relaxed constraints, which is updated from one iteration to another. The complexity of a given iteration becomes linear in the number of discrete variables. The price to pay is that of switching to an iterative method

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rather than a direct method, as well as giving up certainty of global optimality if there is a duality gap. This last drawback is not as bad as it seems, however; very small duality gaps are routinely obtained using Lagrangian relaxation. The unit commitment problem can be formulated generally as:

where

Length of the planning horizon Number of generators to schedule Real power output for generator at time Reactive power output for generator at time On/off status (one or zero) for generator at time

The total production cost The sum of any startup costs A set of dynamic generator-wise constraints A set of static instantaneous system-wide constraints A set of nonseparable constraints. The production cost function F is assumed to be convex (in fact, quadratic) and separable over each generator and time period:

For our purposes, the constraints in the problem have been classified into three kinds. Category groups constraints that pertain to a single generator, but may span several time periods. These include minimum up or down times and ramping constraints. Category groups constraints that span the complete system but involve only one time period, such as load/demand matching, voltage limits, reserve constraints, and generation upper/lower limits. Finally, category groups constraints that involve more than one generator and more than one time period. A typical example is the infeasibility of turning on more than one unit at a time in a given location because of crew constraints. To show the specific type of separation achieved by means of Lagrangian relaxation, we take a look at the work of Muckstadt and Koenig [2]. They considered a lumped one-node network with losses modeled as fixed penalty factors. They also considered reserve constraints. We write an example formulation including demand and reserve constraints, the relaxation of which yields the Lagrangian

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where is the real power demand in period is the desired minimum total committed capacity for the same period, and is the upper operating limit for the ith generator. Consider the dual objective

and the corresponding dual problem

which can be written explicitly in the following form after collecting terms on a per generator basis:

Thus, for fixed and evaluation of amounts to solving single-generator dynamic programs of the form

separate,

The dynamic programs can readily accommodate constraints such as minimal up or down times. Transition costs can easily include cold start or warm start costs, derived from either banked or total shut-downs. Ramp-rate constraints can also be introduced by discretizing the generation domain for the unit, though the dynamic program grows considerably. For a detailed description of a dynamic programming graph including most of these constraints, see reference [3]. After confirming that the evaluation of the dual functional is simple, a dual maximization algorithm can be readily applied to since the sub-gradient is easy to compute from Equation (3). A method as simple as a Poljak’s subgradient ascent or as sophisticated as a bundle method [4] can be utilized. A proper algebraic constraint structure is the key to the separability of the dual functional. Note that the step from Equation (3) to Equation (6) was possible due to the fact that the constraints are affine in the optimization variables. We want to separate the variables on a per-generator basis, so, failing affinity, we need at least separability into sets of variables related to a single generator each. The unit commitment problem has such separability as long as the system-wide constraints are separable by generator. Thus, general nonlinear constraints cannot be readily included in the model, at least not directly. We will show a way around this later.

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2. THE CASE FOR AN AC POWER FLOW MODEL In the past 20 years, many advances have been made, enhancing the number and type of constraints that can be treated, addressing convergence issues when the production costs are not strongly convex, and so on. In particular, transmission line constraints have been included by using the so-called DC power flow network models. These constraints, being linear, can be relaxed with appropriate multipliers and the dual functional will still be separable by generator. There are, however, genuine engineering considerations for including a nonlinear AC flow model. These considerations stem from the fact that some very important system constraints can only be modeled accurately with a full-blown AC power flow model. For example, in an actual transmission line the limit is best expressed as a limit in the current, whereas a transformer’s capacity is best described by its MVA rating. Both actual MVA loading and current depend on its orthogonal active and reactive components. The DC flow model can only predict (and even then, only for relatively small angle deviations and under a nominal voltage assumption) the active component, and therefore it is easy to find situations in which it does a poor job of modeling important constraints. Active branch limits inextricably tie together active and reactive dispatch restrictions; if the reactive component is large, the dispatch obtained by the DC flow-based unit commitment algorithm may have to be altered to comply with actual line limits, or conservative limits may have to be used. Other limits that cannot be modeled accurately via linear approximations are voltage limits. Appropriate voltage levels are crucial to the operation of most electrical apparatus, and it is not uncommon to find so-called “must run” generators that are needed not so much because of their actual power generation capacity, but because of their reactive power capacity. In order to raise the voltage to levels that are adequate for consumers, sometimes large amounts of reactive power are needed at specific locations in the network, even if the only sources are generators that are costly to operate. Granted, common practice to this day is to use linear network models for constrained economic dispatch, and in many cases accuracy is good enough. The reason why they work, however is that usually the only branches that are modeled are tielines, which tend to have controllable reactive compensation in order to keep power factors near unity. Linear congestion models work best if unity power factor is maintained. They become inaccurate if used to approximate thermal limits over broad ranges of voltage and reactive and active injections, especially if the branch being represented does not have dispatchable reactive compensation. In summary, current Lagrangian relaxation algorithms using DC flow-based network constraints or other linear approximations may do poorly enforcing voltage limits and true current or MVA limits in grids requiring large reactive injections. As a result of the limited accuracy of this network representation, it is quite possible that a solution obtained by such an algorithm is actually infeasible, requiring additional generators to be committed in order to meet voltage or congestion constraints.

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We find some of the works that have followed the DC flow formulation in their incorporation of line limits to the dual maximization in [5, 6, 7, 8, 9, 10, 11]. In [12], the authors use linear approximations to include the voltage constraints in the formulation, but it is not clear that a linear approximation will work under large excursions in the reactive dispatch. Baldick [8] uses a general formulation that could in principle be used to address AC flow constraints, and he elaborates a little more in the speculative paper [13], although the authors do not seem to have actually implemented their scheme. The formulation proposed in this work was first reported in [14]. It uses fewer multipliers and, by explicitly employing an augmented Lagrangian, should have improved convergence properties.

3.

FORMULATION AND ALGORITHM

Our approach has its origins in the variable duplication technique credited to Guy Cohen in [6] by Batut and Renaud. Baldick [8] later used this technique in his more general formulation of the unit commitment problem. The principal achievement of this work is the inclusion of reactive power output variables to the formulation, so that better loss management may be performed and generators that are necessary because of their VAr output but not their real power are actually committed. This is the logical next step in the development of Lagrangian relaxation techniques. At this point, when typical algorithms reduce the duality gap to figures close to 1% [15, 10], it is important to recognize that a better handling of the reactive power considerations at the unit commitment stage may have a payoff that is higher than those last few percentage points in the duality gap. The variable duplication technique exploits the fact that the problem

is equivalent to

Notice that the last constraint is linear and therefore amenable to relaxation. We start by defining two sets of variables. The dynamic constraints will be posed in terms of the dynamic variables, whereas the static or system-wide constraints will be posed in terms of the static ones: Dynamic variables: Commitment status {0,1} for generator at time Real power output for generator at time VAr output for generator at time

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Static variables: Real power output for generator at time VAr output for generator at time

We define the optimization problem as:

subject to: (1.) constraints

(2.)

constraints

(3.) and the following additional constraints

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where is the minimum combined capacity that is acceptable for the zone in the period and is the set of indices of generators in the zone. A word about the representation of the minimum up-time or down-time constraints is in order. While it is perfectly possible to state these restrictions in terms of additional state variables obeying state transition rules that reflect these constraints (see [10]), we have chosen not to include any more variables because of the already overburdened notation. These constraints can readily be incorporated in the structure of the dynamic programming graph used to solve the resulting subproblems, as in [3]. Thus, it will be assumed that we can enforce the constraints (9–11) on the D variables and the constraints (12– 14) on the S variables, so that we only relax the three last constraints (15–17), which leads to the following Lagrangian:

where are multipliers on the relaxed equalities of the two kinds of variables, is the multiplier associated to the zone’s reserve requirement at the period, and returns the index of the zone to which generator belongs.

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The separation structure of the Lagrangian is obvious upon looking at equations (19) and (20). It makes it possible to write the dual objective as

By looking again at (19) and (21), we see that the first term can be computed by solving dynamic programs again; the second term separates into optimal power flow (OPF) problems with all generators committed but with special cost curves for generator at time Notice that also has a price. We assume that the solutions of the dynamic programs meet the constraints and that the solutions of the optimal power flows meet the constraints. It would be tempting to apply a dual maximization procedure to the dual objective as stated, but there are some issues that prevent us from doing that without some modification of the Lagrangian. The first issue is that the cost of reflected in the dynamic programs, being linear, is not strongly convex; this can cause unwanted oscillations in the prescribed by the dynamic program [6]. Therefore, we set out to fix this before addressing any other problems by augmenting the Lagrangian with quadratic functions of the equality constraints. This will introduce nonseparable terms. At this stage, we will invoke the Auxiliary Problem Principle described by Cohen in [16] and [17]. This principle is a rather general formalism characterizing optimality in an implicit manner for convex programming, and an algorithm of the fixed point type can be inferred from it. Convergence for the convex case with affine constraints has been proved by Cohen. We proceed to write the new augmented Lagrangian as

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Within the context of the auxiliary problem principle, it is possible to substitute the augmentation terms by the following at iteration (see [17] for the general technique and [6, 7] for the first documented application to unit commitment):

where and are the values obtained at the iteration. Since (22) is separable, we can collect terms of the augmented Lagrangian on a per-generator basis, so that at the iteration we are faced with

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Notice that (24) has the same separation structure of (19). Now that the separability issue has been resolved, we propose the following Algorithm: AC Augmented Lagrangian relaxation Step 0: Step 1: Initialize to the values of the multipliers on the power flow equality constraints at generator buses when running an OPF with all units committed. Initialize to zeros. Initialize a database of tested commitments to be empty; it will later be filled with the commitment schedules obtained in the course of the algorithm and their corresponding feasibility status. Step 2a: Compute

by solving

one-generator dynamic programs.

Step 2b: Compute by solving OPFs in which all generators are committed, their generation range has been expanded to include and the special cost is used. Note: all tasks in steps 2a and 2b can be solved in parallel. Step 3: If the commitment schedule Û is not yet in the database of tested commitments, perform a cheap primal feasibility test. If the results are not encouraging (i.e., not enough committed capacity), store the schedule in the database and label it as infeasible, then go to Step 6. Step 4: Perform a more serious primal feasibility test by actually attempting to run OPFs with the original constraints. If all OPF’s are successful in finding a feasible dispatch, store the commitment in the database, together with the primal cost including startup costs. Else label the commitment as infeasible, store it in the database, and go to Step 6. Step 5: If the mismatch between the two sets of variables is small enough, and there is already a set of feasible commitment schedules, stop. Step 6: Update all multipliers using sub-gradient techniques, and

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Step 7: Go to Step 2. The proposed algorithm is very OPF-intensive: the major computational cost is that of computing OPFs for every iteration in order to solve the static subproblems, plus extra OPFs in selected iterations when a given commitment is promising in terms of primal feasibility. Thus, every effort possible must be made to try to alleviate the burden of OPF computation. The first thing that can be done is to use as a starting point for the OPF the result of the previous iteration for the same time period. Most of the times, the only difference in the data for the OPF would be a small change in the costs (reflected by the change in from one iteration to another). This, in theory, should result in fewer iterations needed for the OPF. Another drawback of the algorithm is that a different set of OPF computations must be performed to compute the value of the dual objective and to compute the value of the primal. Thus, before even trying to compute the value of the primal objective, one should make sure that such a costly computation is worth doing. Some of the cheap tests include verifying that the reserve constraint is met and that the mismatch between the S and the D variables is small, since feasibility is a given if the mismatch is zero. With respect to the latter, we have found that if a smaller mismatch should be specified as requisite to feasibility than if More costly feasibility tests would involve power flow problems starting from appropriate initial values. Currently a constrained power flow is being performed at this stage.

4.

COMPUTATIONAL RESULTS

An implementation of the algorithm has been written in the MATLAB™ environment. The dynamic subproblems can accommodate minimal up or down times, warm start and cold start-up costs and are solved using forward dynamic programming. We solve the static subproblems using a version of MINOS [18] that has been incorporated in the MATPOWER package [19] by the first author. It incorporates box constraints on the generator’s active and reactive output, piecewise-linear or polynomial cost functions for both P and voltage constraints, line MVA limits, and, of course, the power flow equations. Additionally, any linear constraint on the optimization variables can be imposed. A preconditioner for MINOS that performs a constrained power flow is used if necessary. It implements a Levenberg-Marquardt-like minimization of the sum of squares of the power flow constraints, with penalty functions on some other box contraints and constrained variables in the case of voltage limits. Thus, each iteration involves the solution of a problem rather than a Newton step. We solve the problem using a MEX-file version of BPMPD [20], an interior method solver. The specific sub-gradient technique being used at this point is a simple Poljac step size schedule, with an iteration-dependent weight that is inversely proportional to iteration number. We chose simplicity initially because convergence conditions for Poljac’s method are “mild” and well documented: it is advanta-

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geous to be able to blame any convergence problems on the formulation itself and not on the particular kind of sub-gradient update. The obvious drawback to this decision is that the convergence of the dual iteration process may not be very fast. More sophisticated gradient updates will be investigated once the basic features of the algorithm are well understood. The program was originally tested on a modified IEEE 30-bus system [21] with six generators and a planning horizon of length six. We modified the test case so that generator #4, located at bus #27, is needed for voltage support for many load levels even though it is most uneconomical to operate. For comparison purposes, we also wrote a version of the Lagrangian relaxation algorithm with DC flow-based relaxed line limits. The AC-based algorithm correctly identified this unit as a must-run for those time periods, even providing some price information on the MVArs that this unit produced by means of the corresponding The number of iterations required was usually in the vicinity of one hundred. In contrast, the DC flow-based algorithm failed to commit unit #4 for any period, producing a commitment schedule that was infeasible in light of the AC power flow constraints. The importance of proper selection of the parameters was apparent from the beginning. After several trial runs, we obtained good results with and Values very different from these, however, tended to produce somewhat smooth, damped oscillations in the values of some of the To highlight one of the new features found in the algorithm, we show the evolution of versus iteration number for a typical run in Figure 1. The multipliers with the higher values are all P-type multipliers. Those with the smaller values correspond to the Most of them settle to zero, indicating that is essentially free almost always. Yet, a few of them actually have high prices: these belong to generators and time periods where the OPF tries to use their MVArs in order to force feasibility or guided by economic considerations, but the generators are not actually committed. In the course of the algorithm, these may grow so large that they trigger the respective unit on. Once this happens, such multipliers tend to approach zero again, since is now plentiful. In Figure 1 there are two clear examples of this behavior, corresponding to unit 4 being committed for certain time periods. As the multiplier approaches zero, the static copy will approach the dynamic

A slightly more ambitious test has been performed using the IEEE 118-bus system, with 54 generators and a time horizon of 24 periods, corresponding to two “weekdays” and one day of the weekend, each with 8 three–hour periods. The total variation of the load relative to the base case is –50% and +40%. Figure 2 shows the behavior of the norm of the active and reactive mismatches between the two sets of variables. Figure 3 depicts the evolution of the multipliers in this case.

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88

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The Next Generation of Unit Commitment Models

PARALLEL IMPLEMENTATION

One of the purported advantages of separation by Lagrangian relaxation is that it allows for simultaneous solution of all of the static and dynamic subproblems in a given dual iteration, making the computation amenable to parallelization. Profiling of the algorithm has indicated that more than 95% of the computation time in this particular implementation is allocated to the solution of the optimal power flows. In order to test larger systems, a parallel implementation was deemed not just convenient, but in fact necessary since running times were already several hours long. Fortunately, MultiMATLAB, a parallel processing toolbox for MATLAB is being developed at Cornell University by John Zollweg (see [22] and the original work of Trefethen et. al. [23]); this allowed us to take advantage of much of the existing code. Although the dynamic programs are also parallelizable, their time-granularity is much smaller than that of the OPFs, so communications overhead is likely to reduce the efficiency of their parallelization. We have opted to parallelize only the OPF computation at this time. MultiMATLAB works by having several copies of MATLAB, each running in a different node, communicate by means of a subset of the Message Passing Interface (MPI) library [24]. The calls to the MPI functions are implemented by means of MEX (MATLAB-Executable) files. A master node performs the main algorithm and the dynamic programming subproblems and when a set of OPFs need to be carried out it sends the OPF data to the other nodes,

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working in a master/slaves configuration. The workers’ only task is to receive OPF input data, run the OPF solver and report the results back to the master. Two parallel scheduling strategies have been programmed so far. The first one is a straightforward round-robin scheme in which the master cycles through the workers, receiving the results of any previously assigned OPF by means of a blocking receive (MPI’s Recv function), which ties the master until data does arrive. Then the master sends the worker data for the next OPF and turns its attention to the next worker. Clearly, having the master wait for the worker is not optimal, but it has the advantage of employing only MPI’s Send and Recv calls, whose implementation is stable in MultiMATLAB. A second, more sophisticated parallel scheduling strategy makes use of MPI’s Irecv non-blocking receive function. Immediately after sending data to a worker, the master node posts a non-blocking receive, setting aside an incoming message reception area. The master can then turn its attention to other workers and assign further work. Every now and then the master calls MPI’s Testany function, which informs the master if any pending Irecv’s have been completed. If so, the master reads the data from the corresponding buffer and assigns more work if needed. This strategy promises the most efficient use of the workers. In limited experiments with up to seven workers, the master is actually free most of the time, testing for incoming data. This means that it is efficiently keeping the workers busy. The efficacy of this scheduling strategy can be further improved by performing the OPFs in order of decreasing expected execution time. Unfortunately, as of now the Testany implementation is not sufficiently stable to allow running a complete problem. The round-robin strategy has been tested using the 118-bus system with a much longer time horizon (168 periods) and longer start-up and shut-down times on the generators. Figure 4 shows a plot of the evolution of the mismatches.

6. FUTURE WORK Historically, the best justification for using Lagrangian relaxation has been the regular achievement of small relative duality gaps in larger problems. So far, our algorithm seems to behave correctly with proper tuning of parameters, but there is a need to test much larger (i.e., more than 300 generators) systems to verify whether small duality gaps are also routine. At that point, comparison to the results obtained using a DC-flow or linear network model formulation should also be performed, with special attention to the cost of any commitment corrections needed to make the schedule generated by the linear network model algorithm to be AC-feasible. It would be worth it to incorporate other kinds of constraints and continue improving the robustness of the optimal power flow subsystem, which has been the weakest link so far. Three major efforts are being undertaken. First, John Zollweg is implementing several changes that should make the non-blocking receive implementation

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in MultiMATLAB more stable, allowing the use of improved parallel scheduling strategies. Secondly, work is being done to improve the robustness of the OPF solver. While MINOS has been found to do a good job of finding optima given a good starting point, the early stages of the Lagrangian relaxation algorithm, when prices are being adjusted with larger steps, result in OPF problems where the solution lies far away from the starting point. We have found MINOS not to behave as well in these cases, especially for larger (i.e., 3000 buses) systems, even with the constrained power flow preconditioner. So the preconditioner is being turned into a first-stage optimizer intended to locate binding constraints and provide a better, feasible starting point. Finally, work is being done on how to include ramping constraints into the formulation. There are two basic variations in the literature: the first one involves discretizing the generation range for the dynamic copy of the active sources and disallowing transitions that violate ramping constraints in the dynamic program. While relatively straightforward, this approach would result in much larger dynamic programs. The second method involves relaxing the linear ramping constraints and adding them to the Lagrangian. There is, however, some concern about the speed of convergence of the corresponding multipliers, and special updating strategies may be part of the solution. Finally, there remains the question about whether such an algorithm is useful anymore. Lagrangian relaxation methods have not been used to clear energy markets that also produce commitment schedules due to fairness problems: since the cheapest commitment schedule obtained may depend on parameter tuning, a Lagrangian relaxation solution may lack the clarity and uncontestability required in a method used to clear a market. In addition, the issue of computation time also arises, since the algorithm requires the solution of many

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complex AC OPFs; a method used to clear a market should be expeditious due to practical considerations. This second consideration is actually an issue that will become moot in time, given the exponential growth of typical computing capacity and the algorithm’s amenability to parallel computation. The first issue is more serious, and inherent in Lagrangian relaxation methods for unit commitment. So at this point it is difficult to advocate the use of the algorithm for market clearing. However, it is still a tool that can be used by ISOs for location-based market power studies as well as transmission capability assessment and expansion. Lastly, from an experimental economics viewpoint, the best way to assess the performance of a market is to compare its efficiency to that of the maximum social welfare solution. This algorithm can provide this solution (the modifications for elastic demand are trivial) and help to make market efficiency comparisons.

ACKNOWLEDGEMENTS We wish to thank Ray Zimmerman and Deqiang Gan for providing us with their package MATPOWER [19] for the initial tests of the algorithm. We would also like to thank Csaba Mészáros, whose QP program [20] BPMPD we use in several of our programs, and finally, John Zollweg, principal developer of MultiMATLAB.

REFERENCES 1. J.A. Muckstadt and R.C. Wilson. An application of mixed-integer programming duality to scheduling thermal generating systems. IEEE Trans. Power Apparatus Syst., 87(12):1968–1978, 1968. 2. J. A. Muckstadt and S.A. Koenig. An application of Lagrange relaxation to scheduling in power-generation systems. Oper. Res., 25(3):387–403, 1977. 3. G.S. Lauer, D.P. Bertsekas, N.R. Sandell, and T.A. Posbergh. Solution of large– scale optimal unit commitment problems. IEEE Trans. Power Apparatus Syst., 101(1):79–85, 1982. 4. C. Lemarechal and J. Zowe. A condensed introduction to bundle methods in nonsmooth optimization, in Algorithms for Continuous Optimization, E. Spedicato, Ed., Kluwer Academic Pub., 1994. 5. and A new approach for solving extended unit commitment problem. IEEE Trans. Power Syst., 6(l):269–277, 1991. 6. J. Batut and A. Renaud. Daily generation scheduling optimization with transmission constraints: a new class of algorithms. IEEE Trans. Power Syst., 7(3):982–989, 1992. 7. A. Renaud. Daily generation management at Electricité de France: from planning towards real time. IEEE Trans. Autom. Cont., 38(7):1080–1093, 1993. R. Baldick. The generalized unit commitment problem. IEEE Trans. Power Syst., 8. 10(l):465–475, 1995. 9. J.J. Shaw. A direct method for security-constrained unit commitment. IEEE Trans. Power Syst., 10(3): 1329–1339, 1995.

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10. S.J. Wang, S.M. Shahidehpour, D.S. Kirschen, S. Mokhtari, and G.D. Irisarri. Short-term generation scheduling with transmission and environmental constraints using an augmented Lagrangian relaxation. IEEE Trans. Power Syst., 10(3):1294–1301, 1995. 11. K.H. Abdul-Rahman, S.M. Shahidehpour, M. Aganagic, and S. Mokhtari. A practical resource scheduling with OPF constraints. IEEE Trans. Power Syst., 11(1):254–259, 1996. 12. H. Ma and S.M. Shahidehpour. Unit commitment with transmission security and voltage constraints. IEEE Trans. Power Syst., 14(2):757-764, 1999. 13. X. Guan, R. Baldick, and W.H. Liu. Integrating power system scheduling and optimal power flow. In Proc. 12th Power Systems Computation Conference, Dresden, Germany, August 19–23, 1996, pp. 717–723. 14. C.E. Murillo-Sánchez and R.J. Thomas. Thermal unit commitment including optimal AC power flow constraints. In Proc. 31st HICSS Conference, Kona, Hawaii, Jan. 6–9 1998. 15. D.P. Bertsekas, G.S. Lauer, N.R. Sandell, and T.A. Posbergh. Optimal shortterm scheduling of large-scale power systems. IEEE Trans. Autom. Cont., 28(1):1–11, 1983. 16. G. Cohen. Auxiliary problem principle and decomposition of optimization problems. Optim. Theory Appl., 32(3):277-305, 1980. 17. G. Cohen and D.L. Zhu. Decomposition coordination methods in large scale optimization problems: the nondifferentiable case and the use of augmented Lagrangians. In Advances in Large Scale Systems, Vol. 1; J.B. Cruz, Ed., JAI Press Inc, 1984, pp. 203–266. 18. B.A. Murtagh and M.A. Saunders. MINOS 5.5 user’s guide. Stanford University Systems Optimization Laboratory Technical Report SOL 83-20R. 19. R. Zimmerman and D. Gan. MATPOWER: A Matlab power system simulation package. http://www.pserc. Cornell.edu/matpower/. 20. Cs. Mészáros. The Efficient Implementation of Interior point Methods for Linear Programming and Their Applications. Ph.D. Thesis, Eötvös Loránd University of Sciences, 1996. 21. O. Alsac and B. Stott. Optimal load flow with steady-state security. IEEE Trans. Power Apparatus Syst., 93(3):745–751, 1974. 22. J. Zollweg. MultiMATLAB for NT Clusters. Cornell Theory Center Software Documentation, http://www. tc.Cornell. edu/UserDoc/Intel/Software/ multimatlab/. 23. A.E. Trefethen, V.S. Menon, C.C. Chang, G.J. Czajkowski, C. Myers, and L.N. Trefethen. MultiMatlab: MATLAB on Multiple Processors. Cornell University Computer Science Technical Report 96TR239. 24. W. Gropp, E. Lusk, and A. Skjellum. Using MPI: Portable parallel programming with the Message Passing Interface. MIT Press, 1994. 25. A.I. Cohen and V.R. Sherkat. Optimization-based methods for operations scheduling. Proc. IEEE, 75(12):1574–1591, 1987. 26. G.B. Sheblé and G.N. Fahd. Unit commitment literature synopsis. IEEE Trans. Power Syst., 9(1):128–135, 1994.

Chapter 6 OPTIMAL SELF-COMMITMENT UNDER UNCERTAIN ENERGY AND RESERVE PRICES

R. Rajaraman and L. Kirsch Laurits R. Christensen Associates

F.L. Alvarado University of Wisconsin

C. Clark Electric Power Research Institute

Abstract:

1.

This paper describes and solves the problem of finding the optimal self-commitment policy in the presence of exogenous price uncertainty, inter-product substitution options (energy versus reserves sales), and different markets (real-time versus day-ahead), while taking into consideration intertemporal effects. The generator models consider minimum and maximum output levels for energy and different kinds of reserves, ramping rate limits, minimum up- and down-times, incremental energy costs and start-up and shut-down costs. Finding the optimal market-responsive generator commitment and dispatch policy in response to exogenous uncertain prices for energy and reserves is analogous to exercising a sequence of financial options. The method can be used to develop bids for energy and reserve services in competitive power markets. The method can also be used to determine the optimal policy of physically allocating generating and reserve output among different markets (e.g., hour-ahead versus day-ahead).

INTRODUCTION

Unit commitment refers to the problem of deciding when to start and when to shut-down generators in anticipation of changing demand [1]. In traditional utility systems, the problem of unit commitment was formulated and solved as a multi-period optimization problem. In the traditional problem formulation, the anticipated demand was an input variable. The problem was solved for multiple generators, generally owned by the same entity (a utility). The start-up, shut-down and operating costs of the generators were assumed known. The standard way to analyze and solve this problem was by dynamic programming, and within this category of problems, the most popular solution

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method in recent years has been the use of Lagrangian relaxation [2,3]. Recently, a new method of decommitment has also been proposed [4,5]. Several things change in a deregulated market. Generators generally have to self-commit their units optimally. Since in most power pools, no single merchant owns all the generating assets, the need to meet system load is replaced with the need to optimize profits of the merchant’s generating plants based on the uncertain market prices at the locations where the generators are located. Of course, the forecasts of markets prices depend upon a number of factors, the most important of which include demand, system-wide generation availability and cost characteristics, and transmission constraints. We pose the problem of finding the profit-maximizing commitment policy of a generating plant that has elected to self-commit in response to exogenous but uncertain energy and reserve price forecasts. Typically, one generator’s output does not physically constrain the output of a different generator1, so this policy can be applied to each generator in the merchant’s portfolio separately and independently. Therefore, for ease of exposition, we assume the case of a single generator. Generator characteristics such as start-up and shut-down costs, minimum and maximum up- and down-times, ramping rates, etc., of this generator are assumed known. The variation of prices for energy and reserves in future time frames is known only statistically. In particular, the prices follow a stochastic rather than deterministic process. We model the process using a Markov chain. The method is applicable to multiple markets (e.g., day-ahead, hour-ahead) and multiple products (energy, reserves). Other researchers have modeled the effect of energy price uncertainty on generator valuation. In [6], the author models the effect of the spark spread on short-term generator valuation. In [7], the authors propose mean reverting price processes and use financial options theory [9] to value a generating plant. In both [6] and [7], however, the authors neglect the effect of realistic operating constraints such as minimum start-up and shut-down times. In [8], the authors improve upon this work to more realistically include the effect of operating constraints to find the short-term value of a generating asset; yet they neglect the effect of ramping rates. All these papers make assumptions about the risk-neutral price process in order to value the power plant. This chapter focuses not on generator valuation, but on finding the general principles for generator self-commitment in the presence of exogenous price uncertainty and market multiplicity. We derive the basic mathematical principles from dynamic programming theory [10]. We consider energy and reserve markets, although the method can be extended to include additional market choices, such as day-ahead versus real-time markets. Section 2 of this 1

Exceptions include multiple hydroelectric units connected in series and restrictions on aggregate emission levels from multiple generators within an area.

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chapter defines the problem. Section 3 gives the dynamic programming solution to the problem, and Sections 4 and 5 illustrate the features of the optimal commitment policy using simple illustrative examples. Section 6 gives a more detailed numerical result for a peaking generator, and Section 7 concludes the paper. Appendix A is a technical section that solves the single-period optimal generator dispatch problem given exogenous prices of energy and reserves.

2.

THE PROBLEM

We begin by describing the exogenous inputs to the problem. Generator capability and cost characteristics. At any given time t, generator G is assumed to be in state where is a member of a discrete set X={state 1, state 2, ..., state K} of possible states. Intertemporal constraints are represented by state transition rules that specify the possible states that the generator can move to in time period t+1, given that the generator is in a state at time t. Generally, there is a cost associated in moving between different states. In a simple representation, two states are sufficient: “in service” (or “up”) and “out of service” (or “down”). In general, however, many more states may be needed to represent the various conditions of the start-up and shut-down process. The degree to which a generator can participate in providing reserves depends on its ability to respond to the reserve needs in a timely manner. For regulating and spinning reserves, the generator must be already be in service; the amount of MW of reserves that a generator can offer must be consistent with its ramping rate. Generators that are already at a maximum in terms of energy provision are unable to also participate in the reserves market. Thus, to participate in the reserves markets, the generator cannot simultaneously sell all of its capability in the energy market. The parameters that describe a generator include: Minimum and maximum output levels Ramping rates Minimum up- and down-times for the generator Incremental energy costs and no-load costs Start-up, shut-down, and banking costs. Generator states and state transitions. Generators can be in any of a number of several possible UP, DOWN, or transitional states. For example, for a generator with total capacity of 200 MW, and ramp rate of 100 MW/hour, we could define two UP states: and The state would cover the operating range [0 MW, 100 MW], while the state would cover the operating range [101 MW, 200 MW]. Likewise, minimum down times

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can be enforced by defining multiple DOWN states. Only certain transitions among these states are permissible. Furthermore, transition between states generally involves a cost. For example, going from a cold shutdown to an online state will involve a start-up cost. Generator dispatch constraints. A generator may have additional dispatch constraints that restrict its operation. For example, the generator may need to be offline (in the “off” state) during certain periods for scheduled repairs. These restrictions are modeled as time-dependent constraints on generator states. Exogenous price forecast of energy. A discrete Markov process models the exogenous price for energy. In each time period, a discrete price state represents a price range. The price at time is probabilistically related to the price at time t via a price transition matrix. That is, is a known quantity. One can think of a price forecast at any time t to be a baseline price point plus a random uncertainty around the baseline forecast. Since the exogenous price forecast is an important input of the problem, we digress a little to discuss how one may obtain estimates of this input. We consider two ways in which price forecasts can be made: One possible method is to use historical data. For example, to obtain a price forecast for next week, one could use past week data and data from other weeks with similar load/weather patterns as that predicted for next week. This would be a statistical data-mining problem. Another way to forecast prices would be to use numerous Monte Carlo iterations of structural computer models (such as optimal power flow models and production cost models) to model the uncertainty of prices. The physical spot markets for energy include real-time, hour-ahead, dayahead, and possibly week-ahead markets2. Each of these spot markets is a different market, e.g., energy prices for a particular hour could be different in day-ahead and real-time markets and could have very different characteristics in terms of price volatility. A generator will often have a choice as to which market to use to sell its energy. Exogenous price forecast of reserves. Operating reserves are distinguished by the speed with which they become available and the length of time that they remain available. In the nomenclature of the Federal Energy Regulatory Commission, the primary types of reserves (from fastest to slowest) are regulating, spinning, supplemental, and backup reserves. For example, regulating reserves need to be available for following moment to moment fluctua2

It is doubtful whether forward prices quoted month-ahead (or more) will influence an individual generator’s commitment and dispatch decisions.

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tions in system demand and can generally be offered by generators on automatic generation control (AGC). As another example, to offer 10 MW of spinning reserves, a generator must be online and must be capable of producing 10 MW within 10 minutes. Supplemental reserves and backup reserves are slower reserve types. For generators who provide reserve services, there are two types of reserve costs (see [11] for more details). These are reserve availability costs, which are the costs of making reserves available even if they are not actually used, and reserve use costs, which are the costs incurred when the reserves are actually used. Generally, reserve use costs are compensated at the spot price of energy3. Reserve availability costs are the opportunity costs of generators, i.e., they include off-economic dispatch costs, and costs of starting up or shutting down generators. In California, New York, New England, and the Pennsylvania-Jersey-Maryland (PJM) system, there is currently a competitive market-clearing process for setting the reserve availability costs of some or all of the above reserve types. Reserve availability prices are modeled similarly to energy prices, i.e., as a discrete Markov chain. We allow, however, the reserve availability prices to be correlated with the energy prices. One particularly simple way to model reserve uncertainty is to assume perfect correlation between energy prices and reserve availability prices. Exogenous fuel price forecasts. Fuel prices can be modeled similarly to reserve availability prices. Since most practical commitment periods are less than a week and fuel prices typically show much less volatility than energy prices over this interval, it is a not a bad approximation to keep these inputs constant. For ease of exposition, therefore, we assume fixed fuel prices throughout the paper. It is fairly straightforward to also include uncertainty in fuel prices; see, for example, [8]. Start and end time periods. We will assume that the start time period is at time 1 and that the end time period is at time T. For most practical problems, T will be between one day and one week. Next, we define some notation; explicit functional dependencies are often omitted for clarity. Given generator state and a vector of energy and reserve availability prices, at time t, 1. (or simply defines the commitment policy for time t, i.e., it signifies a particular valid rule to move the generator from state at time t to a new state at time t+1.

3

In general, reserve use costs could also include the wear-and-tear costs of ramping up and down to follow system demand. The current custom is that, in competitive markets for reserve services, these costs are not directly compensated and must somehow be internalized by the generators.

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(or simply defines the dispatch policy for time t, i.e., it represents a particular dispatch of energy and reserves for the generator at time t, given that the generator is in state 3. denotes single-period profits at time i.e., the profits realized by the dispatch 4. denotes the cost of transitioning (e.g., start-up or shut-down costs) from state at time t to state at time t+1 due to the commitment policy used in time period t. The problem can now be posed as: [PROBLEM] Find the best commitment and dispatch policy (u*, d*) that maximizes expected total profits over all possible commitment policies and all possible dispatch policies where E denotes the expected value over the uncertain price forecasts5. Before we proceed further, it is useful to summarize the essential features of the problem: Inter-product substitutability. Market participants have a choice between sales in the energy versus sales in the reserve markets. These markets operate simultaneously (though the markets for energy and reserves may clear sequentially, as they currently do in California). Moreover, market participants have a choice of offering their products in different markets (e.g., day-ahead versus hour-ahead markets). Price uncertainty. The future prices of energy and of reserves at the location of the generator of interest are unknown but follow a known random process. The general characteristics of the random process are estimated by the generator wishing to self-commit. Intertemporal ffects. Intertemporal constraints affect the generator’s operations. This may lead to situations when a market participant can elect to remain on during certain periods when operation will be at a loss in return for likely (but not certain) profits in later periods. We are interested in finding both the optimal commitment and the optimal dispatch policy. We stress that the problem is complicated by the fact that at the time the commitment decision is made, future prices are uncertain. The next section addresses this problem. 2.

4 5

Current period profits do not include transition costs. This objective function assumes that the generator is risk-neutral. If the generator is riskaverse, the objective function should reduce the expected outcome according to some measure of risk. For example, the objective might be to maximize where “V” is the variance of net profit and “a” is a risk aversion coefficient. As another example, the objective function could be an exponential utility function with constant relative risk aversion [12]. Now the objective function would be multiplicative in nature, but we can take natural logarithms to convert the objective function to the form shown in this chapter.

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3.

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OPTIMAL COMMITMENT AND DISPATCH POLICY

We now present the optimal commitment and dispatch policy. The optimal dispatch policy is fairly straightforward: given an exogenous price forecast for time period t, the generator takes its current state as given and dispatches energy and reserves in an optimal manner for time period t, without regard to other time periods. Appendix A describes the single-period optimal dispatch. The profit-maximizing commitment decision for transitioning to the next time period is more complicated, however, because actions taken now affect future time periods. The backward dynamic programming (DP) method for solving this problem starts at the final time period T and works backward using the following steps. The backward DP method [3,10] is as follows6: Step 1. Let over all possible commitment policies and dispatch policies Let the optimal dispatch policy be denoted by and the optimal commitment policy by must be computed for each possible state and each possible price level Step 2. Let over all possible commitment policies and dispatch policies The expectation E is taken over all possible price levels given that the price in time T–l is is probabilistically related to via the Markov chain. The state at time T, is related to by the commitment policy Let the optimal dispatch policy be and the optimal commitment be must be computed for each possible state and each possible price level Step T. Let over all possible commitment policies dispatch policies and price levels The expectation E is taken over all possible price levels given that the price in time 1 is The state at time 2, is related to the previous state by the commitment policy i.e., Let the optimal dispatch policy be and the optimal commitment policy be must be computed for each possible state and each price level If the generator is in state and sees price level at time t=l, the profit-maximizing schedules are represented by the commitment actions and the dispatch decisions where for t=1,2,...,T-l, and is related probabilistically to via the Markov chain, and the maximum expected profits are Actual profits and actual 6

For ease of exposition, we have ignored the boundary condition at time T+l.

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schedules depend upon actual price levels encountered in the different time periods. The algorithm for finding the optimal commitment policy is similar to the problem of determining the value of an option using a tree approach [9]. Therefore, we can characterize the problem of finding an optimal commitment policy as a generalized tree approach that values and exercises a sequence of complicated options in each time period. The options involve decisions such as whether to commit or not, whether to ramp up or ramp down, whether to participate in the energy or reserves markets, etc.

4.

ILLUSTRATIVE EXAMPLE 1

This section illustrates the concept. For simplicity we assume only one product, energy, and one three-period market, i.e., T=3. Suppose that the generator parameters are as shown in Table 1. For this example, the generator at time t can be in one of two states, “UP” or “DOWN," i.e., X={UP,DOWN}. There are no intertemporal constraints. The generator can move from any state at time t to any state at time t+1.

A shut-down cost is incurred when the generator moves from an UP state to the DOWN state. A start-up cost is incurred when the generator moves from the DOWN state to the UP state. All other transitions result in zero costs. In each time period, an exogenous price forecast may be described by two possibilities: is HIGH or is LOW, each having a specified probability of occurrence. The HIGH and LOW prices in each time period are allowed to vary, as Table 2 shows. Further assume that the prices at time t+1 depend probabilistically upon the prices at time t. The probability of a HIGH-HIGH transition is and the probability of a LOW-LOW transition is These are exogenous, with assumed values and

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The backward DP algorithm finds the profit-maximizing commitment and dispatch policy. Tables 3(a)-(c) depict the solution. Columns in these tables correspond to time periods. The entry in Table 3(a) that corresponds to a “time period t” column and a “state” row represents the maximum total expected profits for time periods t to T, given that the state at time t is and the price is

Similarly, given state and price at time t, the corresponding entries in Tables 3(b)-(e) show respectively: 1. The optimal dispatch for time t. 2. The optimal commitment policy for time t, i.e., the next state to move to at time t+1. 3. The maximum profits obtained from the optimal dispatch at time t. 4. The cost of the optimal commitment policy, or the cost to move from the current state to the new state at time t+1. Table 3 illustrates the results of a standard Backward DP computation. These results are obtained, as is standard practice, by solving for the respective entries in the table from right to left.

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103

Suppose that, at time t=l, the price level is HIGH and the generator is UP. The maximum expected profits are $415.4 over the three time periods. Since the current price level is HIGH and the generator is UP, the optimal commitment policy is to stay UP at t=2 (from Table 3(c)), in spite of the possibility of net losses over the three periods. The actual profits and the commitment policies at other times would, however, depend upon the actual price levels that occur in those time periods. For example, if the price level stays HIGH for both t=2 and t=3, the optimal commitment policy is to stay UP at t=3, realizing total profits of –5+500+50=$545. If, on the other hand, the price level becomes LOW for t=2, and LOW for t=3, the optimal commitment policy is to go DOWN at t=3, realizing profits of –5–10–22= –$37. In other words, the generator loses money under some price patterns, even with the optimal policy. The expected profits are maximized, however. The optimal schedules given by the backward DP are not static. Instead, they depend upon the exogenous prices in each time period. Thus the DP method does not merely give an optimal schedule. Rather, it gives an optimal scheduling policy corresponding to different conditions.

5.

ILLUSTRATIVE EXAMPLE 2

In this section, we describe the “optionality” features of the generator self-commitment problem, and show that it has features analogous to financial options. We also make additional three points: 1. Assuming a single average price forecast generally understates the value of the optionality, and could severely understate expected generator profits. 2. Running Monte Carlo methods without taking care to ensure that future prices are always uncertain (at the time the commitment is made) generally overstates the value of the optionality and overstates expected generator profits. 3. Not considering reserve products (and multiple markets) tends to lower expected generator profits, because these additional products increase generator optionality. First, we consider the optionality due to price uncertainty. Assume a single time-period horizon and a single product – the energy service. Assume that the generator for which we want to find the optimal commitment and dispatch policy has no start-up or shut-down costs and no intertemporal constraints. Assume that generator has a capacity of 100 MW, no minimum generation constraint, and constant incremental costs of $30/MWh over this range.

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Table 4 shows five different price forecast scenarios. Two possible price states, HIGH and LOW, each with a 50% probability of occurrence, represent each price forecast. The expected value of the prices for all price forecasts is $30/MWh. The optimal policy for the generator is to produce 100 MW whenever the energy price exceeds its incremental costs and to produce 0 MW whenever the energy price is below its incremental costs. For example, for forecast #3, the generator will produce 0 MW when the price is LOW (or $20/MWh), and will make no profits. When the price is HIGH ($40/MWh), the generator will produce 100 MW and make a profit of 100*(40-30) = $1000. Since both price scenarios are equally likely for this forecast, expected profits are 0*0.5+1000*0.5 = $500. Table 4 shows that expected generator profits increase with increasing price volatility8. This is analogous to the value of a financial option that increases in value when price volatility increases [9]. On the other hand, if one uses an average price of $30/MWh to find a commitment policy for the generators, then one will estimate that the generator will make no profit for any price forecast because the generator incremental costs will not exceed the expected energy price. Therefore average price forecasts fail to calculate the value of optionality and understate generator profits. Next, we examine a potential pitfall associated with Monte Carlo methods. One possible way of finding the expected generator profits to capture the optionality value could be to generate a large number of random price scenarios for the time interval [0,T] by Monte Carlo methods. Using the ensemble 7 8

Price volatility here is defined as the ratio of the standard deviation to the expected price, expressed in percent (and rounded). All other factors (e.g., expected energy price) remaining constant.

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of all generated price scenarios one could then use a deterministic model to solve for the optimal commitment and dispatch over this period for each member of the ensemble and then average over different Monte Carlo runs. This is, however, not always correct. To see this, consider the following example. Assume a two-period system, with each period having a 50% chance of HIGH price ($35/MWh) and a 50% chance of having a LOW price ($10/MWh), regardless of the previous period. The generator has the same characteristics as the above example, except that there is a minimum generation limit of 90 MW, and an additional intertemporal constraint that, once online, the generator has to stay online for two consecutive periods. The boundary condition is that the generator is offline initially and must be offline at the end of two periods. It can be verified that the optimal policy is not to commit the generators regardless of what the period 1 price is9. Therefore, the expected profit under the optimal commitment policy is zero. On the other hand, suppose that we first generate all the price scenarios (using a Monte Carlo method), and then run a deterministic optimal unit commitment on each possible price sequence. The four equally possible price sequences in the two periods are {HIGH, HIGH}, {HIGH, LOW}, {LOW, LOW}, and {LOW, HIGH}. If we make four deterministic unit commitment runs on these four price sequences, the deterministic unit commitment will only run the generator at maximum output (100 MW) for both time periods when the price sequence is {HIGH, HIGH}. The profit for this price sequence is $1000. For all other price sequences, the generator will not run, and the profit will be zero. Hence, expected profits using this method will be 0.25*1000+0.75*0 = $250, which clearly overstates the expected profits of zero under the true optimal commitment policy. The reason for this is that in each Monte Carlo run, the generator “peeped ahead” and “knew” the future prices and therefore chose the commitment accordingly. Models for commitment based on the traditional approach are likely to follow a variant of this deterministic optimization method. This approach would result in overestimation of generator profits. Monte Carlo methods are very efficient when one needs to simulate a large number of different random outcomes and find the expected value (or some other statistic) of some function based on these random outcomes. They

9

If period 1 price is HIGH, and the generator decided to commit, then the generator would produce 100 MW in period 1 to make a profit of $500. There is a 50/50 chance, however, that the period 2 price is HIGH or LOW. If the period 2 price is HIGH, the generator’s twoperiod profit will be $1000. If period 2 had a LOW price, the generator would produce the minimum 90 MW and lose 90*30=$2700 in period 2 for a net two-period loss of $2200. Therefore, if the generator commits to be online when the period 1 price is HIGH, the expected two-period payoff is 1000*0.5 – $2200*0.5 = ($600), for an expected loss of $600.

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are much more complicated to implement10, and prohibitively expensive, when the value of a function at any given time t itself depends on what may happen in the future, as in optimal commitment policy problems that have intertemporal constraints11. For example, using the finance analogy, Monte Carlo methods are used for European style options and for those other types of options when one does not have to worry about when it is optimal to exercise the option. American style options, however, are much more difficult to solve for using Monte Carlo methods [12, pp. 685]. Consider a final example to show the optionality value of multiple products (or multiple markets). Assume a single time-period problem and a single reserve product. Consider a generator whose incremental cost is $30/MWh, maximum capacity is 100 MW, minimum capacity is 0 MW, and maximum reserve capacity of 30 MW. Assume that the price of energy is $45/MWh and reserve availability costs of $20/MW/h. If the generator offers 100 MW of energy only, it will make profits of 100*(45–30) = $1500, on revenues of 100*45=$4500. On the other hand, if it maximizes its profits12 and offers 30 MW of reserve and 70 MW of energy, its profits are 30*20+ 70*(45–30)=$1650 on revenues of 30*20+70*45=$3750. (Shifting more of the generator output to reserves increases profits, even though total revenues decrease relative to the energy-only sale. This is typical.) In summary, the more optionality that a generator has, the higher its expected profits will be. Conversely, the more the operational constraints reduce this optionality (e.g., intertemporal constraints), the lesser will be its expected profits, all other factors being equal.

6.

ILLUSTRATIVE EXAMPLE 3

We now consider a slightly more realistic example.13 Assume a peaking generator with the characteristics illustrated in Table 5. The start-up and shutdown times for the generator are assumed to be zero. Figure 1 shows the base10

See reference [8] for the correct way to implement Monte Carlo methods for such problems. In [8], however, the Monte Carlo method becomes prohibitively expensive as the number of generator states and price uncertainty states increase. See also Section 6 for how to use Monte Carlo methods once the optimal self-commitment policy is known. 11 The Monte Carlo method discussed in this section will work correctly on the problem described in Table 4, because the optimal self-commitment policy does not have intertemporal features. 12 Appendix A shows how one may approach the problem when there are more than one reserve type. In this example, while there is profit to be made on the sale of both energy and reserves, the generator sees a higher profit margin in reserves and so maximizes the sale of reserves (30 MW). The remainder is offered as energy (100-30=70 MW). 13 The results of this section were derived using EPRI’s PROFITMAX model.

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line energy price forecast. Figures 2 and 3 illustrate the anticipated baseline prices for 4 types of reserves: regulating, spinning, supplemental, and backup.14 The total number of time periods is one week (168 hours).

14

Spinning reserves are defined to be the capability that can be offered in 10 minutes (if online). Supplemental reserves are the amount of MW available in 20 minutes, and backup reserves are the amount available in 60 minutes.

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We model price uncertainty using a three-node Markov process. We assume that prices can be at one of three levels: HIGH, BASELINE, or LOW. The BASELINE prices are as illustrated. The HIGH price for energy and re-

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serves is 115% of the corresponding baseline price, and the LOW price is 85% of the baseline price. We assume that there is perfect correlation between energy and reserve availability prices; e.g., when the energy price is HIGH, so are the reserve availability prices. For simplicity, we assume that the probability of transition between any one price state to any other price state is 1/3. That is, it is equally likely for the price to change states regardless of the present state of prices. Using the backward DP methods described in Section 3, we derived the optimal commitment and dispatch policy. The optimal commitment and dispatch policy at any time t is a function of the state that the generator is in, and a function of the uncertain price forecast for future time periods as observed at time t. Future prices are always considered uncertain. We then applied the optimal policy15 in numerous Monte Carlo runs to simulate different profit outcomes. From the Monte Carlo prices, we then calculated the actual eventual profits and generator outputs. Using these outcomes, we then illustrate the probability distribution of different generator outputs: generator profits, optimal generator dispatch of energy and reserves, etc. When we use the optimal policy found by the DP to simulate a number of possible states, we obtain a distribution of possible outcomes. Figure 4 illustrates this distribution. There is no assurance that the distribution will be neatly “bell-shaped” as in this example. In other examples it is possible to have distributions that are skewed or bimodal, particularly when we consider start-up and shut-down costs. The effects of the optimal commitment policy on the state transitions are shown in Figure 5. For a large number of scenarios, all transitions between states that result from the optimal policy are recorded. For some times, the state is unique (either UP or DOWN), but for other times the system could end up in either state, depending on the price sequence. This is because it is not known which state the generator will be at a future time t. Figure 5 shows that, depending on the prices that are actually realized, the generator could be in any of the different states at a future time. The ball “size” represents the probability of ending up in a particular state. The “thickness” of a transition line indicates the likelihood that the particular transition will take place.

15

We stress that the optimal policy was an input (not an output) in the Monte-Carlo runs.

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During the interval depicted in Figure 5, the optimal policy has the option to take several transitions, depending on the actual price realized. Only hours 93, 94, and 104 to 106 have a certain state (OFF in this case). For other periods the relative probability of being in either state is represented by the size of the “ball” and the relative transition probability is represented by the thickness of the transition “line.” During certain periods there is both an up transition probability and a down transition probability in the optimal policy.

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For comparison, Figure 6 illustrates the optimal commitment policy when there is no uncertainty, i.e., the baseline price forecast is the perfect forecast. It can be seen that the effect of price uncertainty recognizes the possibility that the generator will either begin producing output (periods 104 through 108) or will turn off (periods 94 through 104) during certain periods according to what prices are actually realized. That is, a quick-responding generator has the luxury of producing when the prices are high, and going offline when the prices are low. Thus high price volatility tends to be beneficial for the expected profits of the peaking generator. When prices are higher than its incremental costs, the peaking generator will maximize its output (and increase profits), while when prices are below its incremental costs, the generator will shut down (and have zero profits). Since profits are bounded from below at zero, and monotonically increase as a function of price (above the generator’s incremental costs) the generator’s expected profits will increase. Figure 7 illustrates this notion. It shows that when expected prices are held constant among scenarios, expected profits increase as price uncertainty increases. This again shows that committing and dispatching a generator is analogous to exercising a financial option. The option value generally increases when uncertainty increases.16 This is because a peaking generator is able to follow changes more readily than other plant types. Thus, price volatility is beneficial to peaking units, a result that may be familiar to many readers. In Figure 11, we are able to precisely quantify this benefit. 16

An intermediate or cycling generator has less optionality features because inter-temporal constraints affect its profits, and it has features similar to Asian options [9]. A baseload generator has even less optionality features (excepting for the important question of which market to sell into), usually because its incremental costs are generally well below the market prices and is analogous to forward contracts [9].

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Figure 8 illustrates the expected sales of energy and reserves as a function of time. Note the striking fact that the generator in question offers energy only during certain periods, but derives income from making reserves available during many more periods.

7.

CONCLUSIONS

The contribution of this chapter is to describe the multi-period multimarket uncertainty framework within which decisions for unit commitment and dispatch will have to take place for many units that operate in a deregulated market. The chapter applies directly to the problem of optimal generator self-commitment. It describes a method for finding the most profitable market-responsive commitment and dispatch policy that takes into full account the optionality available to a generator: reserve market opportunities, multiple markets, price uncertainty, and intertemporal constraints. The model uses backward dynamic programming, and the algorithm in the model can be thought of as a generalized tree that values and exercises a sequence of complicated options. This algorithm can be used to obtain optimal power market bids for energy and reserve services in markets that integrate both these needs. The method can also be used to profitably allocate output in different physical forward markets, e.g., hour-ahead versus day-ahead.

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ACKNOWLEDGMENTS We thank EPRI PM&RM for support of this work, with special thanks to Victor Niemeyer of EPRI for his support and encouragement. In particular, the work on the PROFITMAX model developed by Christensen Associates was sponsored by EPRI and initiated in 1997. We thank Blagoy Borissov of Christensen Associates for his help in simulating some of the results in this paper. We also thank Fritz Schulz for his help with the original implementation of PROFITMAX.

REFERENCES 1. A.J. Wood and B.F. Wollenberg. Power Generation Operation and Control, Wiley 1984. 2. A. Merlin and P. Sandrin. A new method for unit commitment at Electricite De France. IEEE Trans. Power Apparatus Syst., PAS-102(5): 1983. 3. D.P. Bertsekas et al. Optimal short-term scheduling of large-scale power systems. IEEE Trans. Autom. Cont., AC-28(1): 1983. 4. C.A. Li, R.B. Johnson and A.J. Svoboda. A New Unit Commitment Method. IEEE Trans.. Power Syst., 12(1): 1997. 5. C.L. Tseng, S.S. Oren, A.J. Svoboda, and R.B. Johnson. A unit decommitment method in power system scheduling. Elec. Power Energy Syst., 19(6): 1997. 6. M. Hsu. Spark spread options are hot! Electricity J., 11(2): 1998.

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7. S. Deng, B. Johnson, A. Sogomonian. “Exotic Electricity Options and the Valuation of

8. 9. 10. 11. 12. 13.

Electricity Generation and Transmission Assets.” Research Report #PSERC 98-13, 1998 (also available in the limited access http://www.pserc.wisc.edu/index_publications.html). C.L. Tseng and G. Bartz. “Short Term Generation Asset Valuation.” Research Report #PSERC 98-20, 1998 (also available in the limited access website http://www.pserc. wisc.edu/index_publications.html). J.C. Hull. Options, Futures, and Other Derivatives, Prentice Hall, 1997. D.P. Bersekas. Dynamic Programming and Optimal Control, Volumes 1 and 2, Athena Scientific, June 1995. L. Kirsch and R. Rajaraman. Profiting from operating reserves. Electricity J., March: 1998. H.R. Varian. Microeconomic Analysis, W.W. Norton & Company, Edition, 1992. P. Wilmott. Derivatives: The Theory and Practice of Financial Engineering, John Wiley & Sons, 1998.

APPENDIX A This appendix describes how a profit-maximizing generator would dispatch energy and reserve availability services given exogenous market prices for a given hour, and given that it is committed to be online. Assumptions Consider the output choice faced by a generator that can offer, in any given hour, energy and four reserve services. Assume that the energy price is PE, and that the availability prices for the four reserves are PR1, PR2, PR3, PR4, respectively. Suppose that the generator has maximum output level ME, and that, because of ramping limitations, the generator can provide the maximum quantities of the reserve XR1 for reserve 1, XR2 for reserve 2, XR3 for reserve 3, and XR4 for reserve 4. Further suppose that the generator’s production cost function is: where a, b, and c are constants and XE is the generator’s quantity of offered energy. What quantities of energy and reserves should the generator offer if it is maximizing profits? The Optimization Problem The generators problem is to maximize profits17:

subject to the constraints that all energy and reserve quantities must respect maximum limits: 17

When a generator offers reserves, there is a certain probability of these reserves being called. When reserves are called to produce energy, they will receive the market energy price. One can easily include this effect in the objective function (A-2).

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For simplicity and without loss of generality, we ignore the constraints that all reserve quantities must be non-negative, and that the energy quantity have a minimum limit. Although equations (A-2) and (A-3) make the reserves all appear to be mathematically identical, they are not because we assume (reasonably) that reserve prices have a particular order18: Because production costs (A-l) depend only upon energy output, the generator will prefer to sell Reserve 1 first and Reserve 4 last. The Solution The Lagrangian for the optimization problem is:

The solution to the foregoing problem is:

Note that the shadow value of capacity use the generator to its full capacity, and that The solution to the above problem is:

18

only if energy and reserves only if XRj=Mj for j=1,2,3.

Theoretically, the reserve availability prices must be highest for the “highest” quality reserve (regulation) and lowest for the “lowest” quality reserves (backup). The reasoning is that generators that can offer “higher” quality reserves can always offer “lower” quality reserves, but not necessarily vice-versa. Therefore, the availability prices for “higher” quality reserves must be higher than the “lower” quality reserves. Because of market imperfections, however, this relationship is not always obeyed; e.g. see historical ancillary service prices from the California ISO website (http://www.caiso.com). Note that in equation (A-4), we do not necessarily assume that the highest quality reserves will have the highest price, i.e., we allow market imperfections.

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If Reserve 1 is (optimally) offered in a positive quantity that is less than its limit, then But if Reserve 1 is (optimally) offered to its limit, then In general: if Reserve j is offered at all, then all reserves <j are at their limits; if Reserve j is offered in a positive quantity that is less than its limit, then and all reserves >j are not offered at all.

Chapter 7 A STOCHASTIC MODEL FOR A PRICE-BASED UNIT COMMITMENT PROBLEM AND ITS APPLICATION TO SHORT-TERM GENERATION ASSET VALUATION

Chung-Li Tseng University of Maryland

Abstract:

1.

In this paper, we model the unit commitment problem as a multi-stage stochastic programming problem under price and load uncertainties. We assume that there are hourly spot markets for both electricity and fuel consumed by the generators. In each time period, the operator needs to determine which units are to be scheduled so as to maximize the profit while meeting the demand. Assuming that the price and load uncertainties can be represented by a scenario tree, we develop a unit decommitment method using dynamic programming to solve this problem. When there is only one unit under consideration, we show that a scenario tree can be converted to a lattice that allows branch recombination, which may greatly reduce the size of state space. This one-unit problem can be used to value a generation asset over a short-term period. In conclusion, we present our numerical results.

INTRODUCTION

Unit commitment is a problem to schedule generating units economically to meet forecasted demand and operating constraints, such as spinning reserve requirements, over a short time horizon. The unit commitment decision determines which units will be used in each time period. It is a mixedinteger programming problem and is in the class of NP-hard problems (e.g. [1]). Because of its problem size and the NP-hardness, the optimal solution of the unit commitment problem is normally difficult to obtain. Many optimization methods have been proposed to solve the unit commitment problem. These methods include the priority list method [2], the dynamic programming approaches (e.g. [3-5]), the branch-and-bound methods [6-8], and

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the Lagrangian relaxation methods (e.g. [9-11]). Among them, the Lagrangian relaxation methods are the most advanced and widely used approaches. Although the unit commitment problem has been widely studied during the past decades, most of the approaches do not consider uncertainties. The traditional unit commitment problem aims to schedule generation units to meet the forecasted demand. When the actual demand is not equal to the forecasted value, the discrepancy can be handled by system spinning reserve to some extent. In [12], we present a stochastic model for the unit commitment problem in which the demands are not known with certainty. We then use a scenario tree to capture the demand uncertainty and the apply Lagrangian relaxation to decompose the problem into scenario subproblems using progressive hedging [13]. Carpentier et al. in [14] propose another decomposition scheme using the augmented Lagrangian method. In these two approaches, the unit commitment decision is modeled as a multi-stage problem. Carøe and Schultz develop a two-stage stochastic programming model for the unit commitment problem, in which the authors emphasize the planning decision over the entire planning horizon rather than multi-stage implementation of the operating decision [15]. With the evolution of deregulation in the electricity industry and the introduction of spot markets for both electricity output and fuel input, incorporating uncertainties to the unit commitment problem becomes a necessity for utilities or power generators. In this chapter, from the perspective of a firm owning generating units, we model the unit commitment problem as a multistage stochastic programming problem under price and load uncertainties. We assume that there are hourly spot markets for electricity and the fuel consumed by the generators. Our research uses a scenario tree to represent the uncertainties as in [12]. However, we develop a new method using unit decommitment to solve this problem. Li et al. [16] and Tseng et al. [17] proposed independently the method of unit decommitment for the traditional unit commitment problem. In [16], the authors propose a solution procedure, which initially turns on all available units at all hours and then performs only decommitment. The authors view their method as a Lagrangian relaxation-like method and take the multipliers from economic dispatch instead of sub-gradients. On the other hand, in [17] the authors propose using unit decommitment as a post-processing tool for existing solution procedures for solving the unit commitment problem. They consolidate these two approaches as a unified unit commitment algorithm [18], in which they also conduct extensive numerical tests. Their results show that the unit decommitment method on average obtains solutions almost as good as the Lagrangian relaxation approaches, but with much less CPU times. In this chapter, we extend the unified approach presented in [18] to solve the stochastic unit commitment problem. Assume that the price and

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119

load uncertainties can be represented by a scenario tree. Initially, all generating units are committed in all scenarios and in all time periods, to the best extent. Each unit’s schedule (in all scenarios) is then tentatively improved in turn with other units’ schedules fixed. At each iteration, only one unit’s tentative schedule is selected and updated. The iteration proceeds until no improvement can be made. Initial numerical tests seem to suggest that the unit decommitment approach applied to the stochastic unit commitment problem retains the properties when applied to the traditional deterministic case, as reported in [18]. When there is only one generating unit under consideration, the problem can be used to value a generation asset over a short-term period (e.g. [19]). This problem appraises the flexibility of a power plant’s real options, e.g. committing or decommitting a unit, in a competitive environment (e.g. [20]). We will show that in this special case we can convert a scenario tree to a lattice that allows branch recombination, which may greatly reduce the size of state space. In Section 2, we establish the mathematical model for the stochastic unit commitment problem. Section 3 presents a special case in which the uncertainties of the problem are perfectly known. Through this deterministic special case, we derive the method of unit decommitment. Section 4 extends the unit decommitment method to the stochastic case. We then discuss a shortterm generation asset valuation problem in Section 5. Section 6 gives numerical test results, and we present future directions in Section 7.

2.

THE MATHEMATICAL MODEL

In the model development, we introduce the following standard notation, with additional symbols introduced when necessary. : index for the number of units : index for time : zero-one decision variable indicating whether unit is up or down in time period : state variable indicating the length of time that unit has been up or down in time period : the minimum number of periods unit must remain on after it has been turned on : the minimum number of periods unit must remain off after it has been turned off

120

The Next Generation of Unit Commitment Models : the number of periods required for the boiler of unit to cool down

: decision variable indicating the amount of power unit in time period : minimum rated capacity of unit

is generating

: maximum rated capacity of unit : reserve available from unit in time period : system demand (MW) in time period : system reserve (MW) in time period (assumed to be a function of ) : amount of power transaction in time period (power input) or negative (power output). : electricity price ($/MWh) in time period

Its value can be positive

: fuel price ($/MMBtu) in time period : the heat (MMBtu) required for unit to generate (MW) of power (assumed strictly convex, increasing, and smooth) : start-up cost associated with turning on unit at state : shut-down cost associated with turning off unit

2.1

Operational Constraints

The operational constraints to be discussed in this paper can be described in terms of state transition. Let be the set of unit commitment state space for unit the on-line states, and

is composed of two subsets of the states: for the off-line states.

The operational constraints include: Minimum up-time/down-time constraints, for

and

for

A Stochastic Model for a Price-Based Unit Commitment Problem

State transition constraints, for

and

Equation (4) also implies the following relation between

Capacity constraints for Initial conditions on

2.2

121

and

and for

at

A Multi-Stage Stochastic Programming Formulation

This research considers the problem from the perspective of a firm that owns generating units. We assume that there are hourly spot markets for both electricity and fuel consumed by the generators. Each generating unit is viewed as a cross-commodity instrument. That is, each generator buys fuel from the fuel market, converts it to electricity, and sells the electricity to the electricity market. The uncertainties considered in this model are the prices for both electricity and fuel and the quantity of the electricity sold. The quantity of electricity sold in the market , called load or demand in this paper, may be contingent upon time (e.g. peak or non-peak hours) or may be price-elastic. The firm may also involve option-type transactions, such that the amount of transactions may depend on the price of electricity. For example, a customer may tend to call an option of quantity (e.g. swing options) when the price of electricity is high and vice versa. The firm would like to maximize its expected profit while meeting its load and transactions. In the proposed stochastic model, the timing of the event occurrence is as follows. In time period , the uncertainties are revealed. Based on the states of all units from the previous hour, , the operator needs to schedule the units to achieve maximum profit while satisfying the demand. The operator’s decision is not only based on the information obtained in the current period, but also the expectation of future return. Any commitment decision made in the current period will then become the

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The Next Generation of Unit Commitment Models

initial condition for decision-making in the following hour and will constrain the availability or flexibility of the unit commitment in the subsequent hour(s). In addition, there are start-up and shut-down costs associated with the commitment decision. In this paper, both costs are generalized in the following function. Let

be the decision problem to be made in time pe-

riod , given the initial conditions

and

and the observed elec-

tricity and fuel prices The dynamic programming type recurrence equations can be formulated as follows:

where denotes the expectation operator, and the subscript indicates that the expectation is based on the price information available at time . The last term in (8) with the expectation operator defines another stochastic program to be considered in the subsequent time period, which is also called a recourse function. Since the fist term on the right-hand side of (8) is a constant that can be pulled out from the maximization, can be decomposed to two terms as follows.

Equation (10b) is subject to the physical constraints (3) to (6) and the following constraints. Demand constraints for

Reserve constraints for

A Stochastic Model for a Price-Based Unit Commitment Problem

where and unit i. In (9), accounts for revenue, and The terminal condition of (9) is as follows

123

is the maximum reserve for represents cost minimization.

subject to (11) and (12) at time The optimal value representing the maximum expected profit that the operator can make over the period can be obtained from the last step of the recursive relation as Finally, note that in the formulation the fuel price

directly affects a

generator’s cost characteristics, and the electricity price is correlated to the transaction amount and the load , which are uncertain per se. All of these uncertainties influence unit commitments from different perspectives.

2.3

Economic Dispatch

The minimization problem on the right-hand side of (10b) represents scheduling generating units over a single period with generating costs minimized. The scheduling decisions include the determination of what units will be on (i.e. the commitment) and the generation levels for on-line units. Given a commitment, the economic dispatch problem is to allocate electricity generation economically to on-line units while satisfying the demand and system reserve constraints. At time given state variable and for each unit, let be the index set of on-line units at time (Note that in this chapter a variable with a tilde hat will denote a realization of the variable.) The economic dispatch problem at time denoted by rced is defined below.

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The Next Generation of Unit Commitment Models

Note that rced stands for “reserve-constrained economic dispatch.” In [18] the optimality condition of rced is interpreted as follows: the units in are divided into two sets, one set of units with “cheap reserve” but “expensive generation” (relatively), and one is the counterpart. The units with cheap reserve but expensive generation are operated with the same marginal costs (or as close as possible), which are the Lagrange multipliers corresponding to (15b). The units with expensive reserve but cheap generation are operated such that their marginal costs for reserve equal (or are as close as possible) to the Lagrange multipliers corresponding to (15c). Proposition 1. The solution of rced following conditions hold.

3.

exists if and only if the

THE DETERMINISTIC CASE

In this section we discuss a special case with certainty. An algorithm will be developed, and will be extended to handle stochastics in a later section. Assume that the future prices for electricity and fuel as well as the load and transaction , are fully and perfectly known. Equation (8) reduces to the following deterministic price-based unit commitment formulation.

subject to constraints (3) to (6), (11) to (12), and initial conditions on at for The first term in (17b) is a constant, and the minimization term is a traditional unit commitment problem, which can be solved using the methods reviewed in the introduction of this paper, such as the Lagrangian relaxation method. In this section, we focus on the unit decommitment method proposed in [17, 18] for solving the cost minimization problem in (17b). First we review the unit decommitment method.

A Stochastic Model for a Price-Based Unit Commitment Problem

3.1

125

Unit Decommitment

The unit decommitment method was proposed in [17] as a postprocessing tool to improve solution quality for methods solving the traditional unit commitment problem. Given and a feasible schedule consider the problem of optimally improving one unit’s schedule by decommitment. That is, in the time periods when the unit under consideration is already off-line, the unit remains off-line. The unit may be turned off in some on-line periods only if doing so is cost-saving and would not cause infeasibility. While we are improving a unit’s schedule, say determining for unit by decommitment, the commitments of units other than are kept fixed. However, their generation levels in some time period may change in order to balance demand and reserve if unit is decommitted in the same time period. The decommitment rule is summarized below.

Constrained by the decommitment rule, the optimal decommitment problem for a unit, say is formulated below.

subject to additional constraints such as (3) to (6) and (11) to (12), where captures the total cost changed from units other than if unit were turned off in time period

The exact value of

tained by solving the dispatch problem twice, one with unit time and the other without.

can be obcommitted at

is a 0-1 integer programming problem, and can be solved using the following dynamic programming recursive equation.

with boundary condition

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The Next Generation of Unit Commitment Models

where

can be interpreted as the minimum cost for op-

erating unit

over a period starting from time to

The optimal solution of

is obtained from the last step of the dynamic

programming algorithm as tion of

with initial state

In this research, we call the solu-

the tentative commitment of unit

In the following algorithm, superscript denotes the the algorithm. Let be the total cost for unit the schedule

and

iteration of associated with

be the optimal objective

value of solved with respect to bold faces are vectors. For example,

(Note the variables in

The unit decommitment algorithm Prices given. Set Step 0:

for

Data:

and a feasible solution

and evaluate

Step 1: Solve Step 2: Select a unit

are

for

with respect to

and obtain

such that

for

If there is no such unit,

stop; otherwise update the commitment of unit the tentative commitment obtained in

in

by

The resultant unit

commitment is assigned to be Step 3: Perform dispatch on for go to Step 1. Step 4: Set

to obtain

and evaluate

Although the unit decommitment method starts with an initial feasible solution of the unit commitment problem, it can be used as a complete algorithm for solving the unit commitment problem. Initially, as many units as possible are turned on in all hours without violating the minimum downtime constraints at the initial hours. This resultant unit commitment serves as an initial commitment, to which the unit decommitment algorithm is then applied. There are, however, minor modifications required in order to make this approach work. First, the commitments in the early iterations tend to be

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127

seriously overcommitted such that the first inequality in (16a), the minimum load condition, is violated. In other words, the dispatch phase in Step 3 of the unit decommitment algorithm may not be feasible. When the infeasible situation occurs, the algorithm would dispatch the units to satisfy the optimality condition to the best extent, in the sense of matching the units’ marginal costs for both fuel cost and reserve to the corresponding multipliers and , respectively, as closely as possible (see [18]). By doing this, as the decommitment procedure proceeds, the commitment obtained eventually satisfies the minimum load condition, and the algorithm starts to produce feasible schedules. Instead of obtaining the exact value as in (20), may also be approximated. In [18] the first order approximation of

is derived.

where and are the multipliers associated with rced With (23) plugged into (19), it is shown in [18] that starting from an economically dispatched schedule performing either the unit decommitment step with respect to or the Lagrangian relaxation subproblem with respect to to any unit would result in the same (tentative) commitment. However, these two approaches differ in the commitment updating (the unit decommitment updates one unit at a time, while the Lagrangian relaxation approach updates all units at once) and the multiplier updating (the unit decommitment performs economic dispatch, while the Lagrangian relaxation approach uses sub-gradients). In [18], based on their numerical testing results on randomly generated instances, the authors report that the error between the solutions obtained by the unit decommitment method and Lagrangian relaxation approach is within 0.2%, and the unit decommitment method takes at least 50% less CPU time than the Lagrangian relaxation approach on average. In the following section, we extend and apply the unit decommitment method to the stochastic unit commitment problem.

4.

SOLVING THE MULTI-STAGE STOCHASTIC MODEL USING UNIT DECOMMITMENT

The multi-stage stochastic model formulated in (8) appears to be intractable generally. A popular approach is to summarize the future realization of uncertainties by a finite number of possible scenarios. We represent scenarios by the nodes of a tree such that given any node in the tree, there exists a

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The Next Generation of Unit Commitment Models

unique path leading to it from the starting node. A (directed) arc in a tree connecting two nodes represents the evolution from one scenario to the other. Consider a scenario tree T(N, A), where N is the set of nodes, and A is the set of arcs in the tree. Let

where

is an ordered set

of nodes in the tree corresponding to time period t, and notes the set of the descendents for node at time t such that Assume the scenario at node and a descendent

de-

is . For each node there is a branching (conditional) prob-

ability associated with this arc denoted by . Given a scenario tree T(N, A) , the stochastic unit commitment formulated in (10) reduces to the following equations for a node

and

subject to the demand constraints for

and the reserve constraints for

and (3) to (5) with an appropriate superscript in each variable to denote the node in the tree. Next, we will show that we can extend the unit decommitment method to solve (25). Given a scenario tree T(N, A) , and a feasible schedule for similar to the development in Section 3.1, we discuss what the best strategy is to improve the generating cost from unit with other units’ strategies fixed in all scenarios (cf. (21)).

with boundary condition (cf. (22))

A Stochastic Model for a Price-Based Unit Commitment Problem

for

Again

129

is subject to the decommitment

rule,

and the minimum up-time and down-time constraints between in (3), and the state transition constraint for the

5.

last

step

of

and

defined in (4), for

We obtain the optimal solution of from the dynamic programming algorithm as (Note contains only the start node.)

SHORT-TERM GENERATION ASSET VALUATION

In this section, we consider a special case in which there is only one generating unit (say unit Solving this problem can determine the expected profit for a generation asset over a short-term period (e.g. [19]). This problem appraises the flexibility of a power plant’s real options, e.g. committing or decommitting a unit, in a competitive environment (e.g. [20]). We will show that, in this special case, a scenario tree can be converted to a lattice that allows branch recombination, which may greatly reduce the size of state spaces. Recall the general formulation of the stochastic unit commitment in (8) to (10). With employment of a scenario tree, these equations are reduced to (24) and (25). In (28), the unit subproblem (for unit using unit decommitment is presented. Although each unit subproblem (28) is solved independently, the commitment status of units other than are also required, which are implicitly accounted through That is, the value of depends on the states of all other units. However, at each node

to evaluate

at

all possible unit commitments is virtually impossible. The approach suggested here is to incorporate a scenario tree such that for each node there is one corresponding commitment state for each unit i to be determined,

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The Next Generation of Unit Commitment Models

as described in Section 4. In a tree there exists a unique path leading to any given node from the starting node, and the commitment of all units can be tracked in each path. By doing this, the tree size is inevitably large, an exponential function of the number of times that the tree branches. For example, if we would like to design a scenario tree such that each node branches into two new nodes at each hour, for a 24-hour period the tree would contain ,more than 16 millions paths. When there is only one generating unit (say j) under consideration, the situation changes because Equation (28) reduces to

It is now possible to evaluate all possible states at each node are only

states of

If

is evaluated at all possible

since there we

would not rely on the unique path leading to to track the history of unit j’s commitment. Therefore, we can merge a scenario tree to a lattice that allows branch recombination without losing any information. The following section gives numerical results, along with an example of constructing a scenario lattice.

6.

NUMERICAL RESULTS

6.1

General Case with Multiple Units

The presented algorithm has been implemented in Fortran and applied to a test problem on a Pentium II PC (400MHz). In the test problem, 10 generating units are considered. The heat requirement of each generator is modeled as a quadratic function. The start-up cost for

is modeled as follows.

Table 1 summarizes the parameters of the generators.

A Stochastic Model for a Price-Based Unit Commitment Problem

131

Starting from a current electricity price at 25 ($/MWh), a scenario tree is used to represent the variability of the electricity prices over the period [1,24]. This 24-hour period is equally divided into 4 subperiods (1 to 6, 7 to 12, 13 to 18, and 19 to 24). For the first and the fourth subperiods, there are two price scenarios indicating high (H) and low (L) price cases; for the second and the third subperiods, there are three price scenarios indicating high (H), medium (M) and low (L) price cases. Notations and i= 1,2,3,4 and i=2,3 are used to represent the high, low, or medium price scenarios in the i-th subperiod. Figure 1 illustrates the price tree, and Table 2 gives the data of scenarios. Table 3 presents the conditional probabilities between scenarios. For simplicity, assume that hourly demand follows the following relation (including the effect of transaction where

can be considered as some base load, and

represents some

“mean” prices for the electricity. When is deviated from , assume the demand will also deviate from . The load deviation is proportional to . This may be due to some options the firm has sold, which may be exercised at different price levels. Table 4 summarizes the data of and

,

,

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The Next Generation of Unit Commitment Models

We apply the proposed unit decommitment method to the test problem and obtain a near-optimal solution within 2.3 seconds of CPU time. Table 5 reviews the algorithm performance. In all, we performed eight iterations. Initially, at iteration k=0, all units in all scenarios are turned on at all hours, to the most extent. The economic dispatch is then performed, and then the expected profit of each unit is summarized under the column of Note that at this point the system may be overcommitted, and the dispatch may not exactly match the demand. In the first iteration (k=1), we perform subproblem to each unit and obtain the tentative schedule. One unit m whose tentative schedule can yield the most improvement on the expected profit, is selected. This value is recorded under the column of for each unit. In the first iteration, unit 1 yields the most improvement for the expected profit. Its tentative commitment that completely shuts down the unit in all periods replaces its original one. Economic dispatch is performed to the new unit commitment, and the expected profit of each unit is recorded under The iteration goes on in a similar manner until no improvement can be made. We can observe the following: 1. The expected total profit is strictly increasing as the iteration increases. 2. Once a unit has been selected for improvement at some iteration, its (new) commitment remains “optimal” in the subsequent iterations. For example, we select unit 1 at the first iteration. The improvement that can be made through decommitment (under the column of remains 0 for unit 1 in the subsequent iterations. We can adopt this observation to improve the algorithm performance, because once a unit has been selected at some iteration, it can be exempted from further consideration. Therefore, the number of iterations required by the algorithm is bounded by the number of units.

A Stochastic Model for a Price-Based Unit Commitment Problem

3.

133

It can be verified that the following relation:

starts to hold when That is, after iteration 4, the algorithm has produced a feasible unit schedule. Therefore, solution feasibility is maintained after the 4-th iteration. Finally, to verify how good the solution obtained from the proposed method is, the dual problem using Lagrangian relaxation has been established. We created a simplistic version of dual optimization to obtain an estimate of the

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The Next Generation of Unit Commitment Models

dual optimum, which can serve as an upper bound of the primal optimum. Without much effort spent to fine-tune dual iterations, we obtain an approximate dual optimum of $1,100,339. Compared with the primal optimum obtained from the proposed method of $1,089,819, the duality gap is within 0.96%. This implies that the obtained solution is fairly close to the true optimum.

A Stochastic Model for a Price-Based Unit Commitment Problem

6.2

135

Short-Term Generation Asset Valuation

Assume that the electricity and fuel prices follow some Ito processes. As an example, consider the following two processes advocated in [21].

and where and are two Wiener processes with instantaneous correlation In the price models above, captures the seasonal average price. The parameter captures reverting speed when the price deviates from and is the price volatility. We characterize the above commodity price models by mean reversion and log normally distributed, seasonal prices. Let equivalently We devised a two-dimensional price lattice to encompass both electricity and gas prices. Each price node in the plane is designed to branch out into a 3×3 grid of 9 price nodes in the plane corresponding to the following time period (see Figure 2). To form a lattice, the plane of each time period is divided into a predetermined grid such that branching is only allowed from grid nodes to grid nodes of the following time period. How to select the nine nodes to which to map, as well as their corresponding (conditional) probabilities, is discussed in [22]. Basically each branching must match (discrete-time) price movement characteristics (mean, variance, and correlation) implied by (36a) and (36b). We apply the branching process to each mapped grid node and repeat the process until time period T is reached. Figure 2 illustrates one such lattice. Note that the size of the lattice grows quadratically (linearly in each variable) as time increases. Furthermore, at each node we assume

Equation (37) essentially assumes that the operator would sell to the market at the amount of electricity that optimizes profit, and that the market has infinite capacity. Note that (37) ignores the reserve constraint (27), and the transaction is set to zero. Remember that Table 1 give the proposed method as applied to unit 1. Consider a 7-week (168-hour) period. Assume that the electricity prices and fuel prices both follow the processes described by (36a) and (36b). We obtain the parameters of the price processes are by fitting the historical price data series of Nymex natural gas prices and electricity prices from the California Power Exchange, taking the logarithm of these prices as our basic

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The Next Generation of Unit Commitment Models

data series. For gas we obtain ity we obtain

and

and

For electric-

Refer to [22] for detailed

val-

ues. At time 0 suppose prices and are observed. We assume that the instantaneous correlation coefficient between electricity and natural gas is Figures 3 and 4 depict the relation between the expected profit of the power plant and the length of planning horizon T and the volatility respectively. We see that the expected profit for the power plant increases as either T or increases. The expected profit is extremely sensitive to the price volatility. From Figure 4, we can estimate that a 1% increase in the value of would result in roughly a 1% increase of the expected profit for the power plant. An intuitive interpretation of these results is to view owning a power plant as holding a series of spark spread call options [20]. The value of these options increases when (i) the number of options increases (as T increases) or (ii) the price volatility increases.

A Stochastic Model for a Price-Based Unit Commitment Problem

7.

137

CONCLUSIONS AND FUTURE DIRECTIONS

In this chapter, we present and model a general formulation of a pricebased unit commitment problem as a multi-stage stochastic programming problem under price and load uncertainties. We extend the method of unit decommitment using dynamic programming to solve this problem. The unit decommitment method is very efficient in computation and does not require a fine-tuning process for algorithm parameters like in the Lagrangian relaxation method. Preliminary numerical test results are promising. When we use a scenario tree to describe the evolution of uncertainties as presented in this research, it can be fully and efficiently integrated with a dynamic programming approach such as the proposed unit decommitment method. The establishment of scenario trees remains an open problem. Statistical methods can be used to measure the correlation of random variables such as prices and loads at different hours. Also, the scenario trees seem to suffer the curse of dimensionality. We demonstrate, using the example of short-term generation asset valuation, that under some problem structure the dimensionality curse can be relieved. How to model the stochastic unit commitment problem such that it can provide a nice structure to cope with the dimensionality problem, and can still reflect real operational restrictions, seems to be a new direction of research.

REFERENCES 1.

C.L. Tseng. On Power System Generation Unit Commitment Problems, Ph.D. Thesis, University of California at Berkeley, 1996.

138 2.

3. 4.

5. 6. 7. 8.

9. 10.

11. 12. 13. 14.

15.

16. 17. 18. 19. 20. 21. 22.

The Next Generation of Unit Commitment Models W.G. Chandler, P.L. Dandeno, A.F. Gilmn, and L.K. Kirchmayer. Short-range operation of a combined thermal and hydroelectric power system. AIEE Trans., 733: 1057-1065, 1953. C.K. Pang and H.C. Chen. Optimal short-term thermal unit commitment. IEEE Trans. Power Apparatus Syst., 95(4): 1336-1342, 1976. C.K. Pang, G.B. Sheble, and F. Albuyeh. Evaluation of dynamic programming based methods and multiple area representation for thermal unit commitments. IEEE Trans. Power Apparatus Syst., 100(3): 1212-1218, 1981. A.J. Wood and B. F. Wollenberg. Power Generation, Operation and Control. New York: Wiley, 1984. A. Turgeon. Optimal unit commitment. IEEE Trans. Autom. Control, 23(2): 223-227, 1977. A. Turgeon. Optimal scheduling of thermal generating units. IEEE Trans. Autom. Control, 23(6): 1000-1005, 1978. T.S. Dillon. Integer programming approach to the problem of optimal unit commitment with probabilistic reserve determination. IEEE Trans. Power Apparatus Syst., 97(6): 2154-2164, 1978. J.A. Muckstadt and S. A. Koenig. An application of Lagrangian relaxation to scheduling in power-generation systems. Oper. Res., 25(3): 387-403, 1977. D.P. Bertsekas, G.S. Lauer, N.R. Sandell Jr., and T.A. Posbergh. Optimal short-term scheduling of large-scale power systems. IEEE Trans. Autom. Control, 28(1): 1-11, 1983. J.F. Bard. Short-term scheduling of thermal-electric generators using Lagrangian relaxation. Oper. Res., 36(5): 756-766, 1998. S. Takriti, J.R. Birge, and E. Long. A stochastic model for the unit commitment problem. IEEE Trans. Power Syst., 11(3): 1497-1508, 1996. R.T. Rockafellar and R.J.-B. Wets. Scenarios and policy aggregation in optimization under uncertainty. Math. Oper. Res., 16(1): 119-147, 1991. P. Carpentier, G. Cohen, J.-C. Culioli, and A. Renaud. Stochastic optimization of unit commitment: a new decomposition framework. IEEE Trans. Power Syst., 11(2): 10671073, 1996. C.C. Carøe and R. Schultz. A Two-Stage Stochastic Program for Unit Commitment under Uncertainty in a Hydro-Thermal Power System. Working paper, University of Copenhagen, Institute of Mathematics, Denmark, 1998. C.A. Li, R.B. Johnson, and A.J. Svoboda. A new unit commitment method. IEEE Trans. Power Syst., 12(1): 113-119, 1997. C.L. Tseng, S.S. Oren, A.J. Svoboda, and R.B. Johnson. A unit decommitment method in power system scheduling. Elec. Power Energy Syst., 19(6): 357-365, 1997. C.L. Tseng, C.A. Li, and S.S. Oren. Solving the unit commitment problem by a unit decommitment method. J. Optimization Theory Appl., 105(3), 2000 (in press). C.L. Tseng and G. Barz. “Short-term generation asset valuation.” In Proc. 32nd Hawaii International Conf. Syst. Sci. (CD-ROM), Maui, Hawaii, Jan 5-8, 1999. M. Hsu. Spark spread options are hot! Electricity J., 11(2): 28-39, 1998. G. Barz. Stochastic Financial Models for Electricity Derivatives. Ph.D. Thesis, Stanford University, 1999. C.L. Tseng. Valuing the operational real options of a power plant using discrete-time price trees. Submitted to Elect. Power Energy Syst.

Chapter 8 PROBABILISTIC UNIT COMMITMENT UNDER A DEREGULATED MARKET Jorge Valenzuela and Mainak Mazumdar University of Pittsburgh

Abstract:

1.

In this paper, we propose a new formulation of the unit commitment problem that is suitable for the deregulated electricity market. Under these conditions, an electric power generation company will have the option to buy or sell from a power pool in addition to producing electricity on its own. We express the unit commitment problem as a stochastic optimization problem in which the objective is to maximize expected profits and the decisions are required to meet the standard operating constraints. Under the assumption of competitive market and price-taking, we show that the unit commitment problem for a collection of M generation units can be solved by considering each unit separately. The volatility of the spot market price of electricity is represented by a stochastic model. We use probabilistic dynamic programming to solve the stochastic optimization problem pertaining to unit commitment. We show that for a market of 150 units the proposed unit commitment can be accurately solved in a reasonable time by using the normal, Edgeworth, or Monte Carlo approximation methods.

INTRODUCTION

In the short-term, typically considered to run from twenty-four hours to one week, the solution of the unit commitment problem (UCP) is used to assist decisions regarding generating unit operations [1]. In a regulated market, a power generating utility solves the UCP to obtain an optimal schedule of its units in order to have enough capacity to supply the electricity demanded by its customers. The optimal schedule is found by minimizing production cost over a time interval while satisfying demand and operating constraints. Minimization of the production costs assures maximum profits because the power generating utility has no option but to supply the prevailing load reliably. The price of electricity over this period is predetermined and unchanging; therefore, operating decisions have no effect on the firm’s revenues. As various regions of the United States implement deregulation [2], the traditional unit commitment problem continues to remain applicable for the

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The Next Generation of Unit Commitment Models

commitment decisions made by the Independent System Operator (ISO). The ISO resembles very much the operation of a power generating utility under regulation. The ISO manages the transmission grid, controls the dispatch of generation, oversees the reliability of the system, and administers congestion protocols [3,4,5]. The ISO is a non-profit organization. Its economic objective is to maximize social welfare, which is obtained by minimizing the costs of reliably supplying the aggregate load. Under deregulation, the UCP for an electric power producer will require a new formulation that includes the electricity market in the model. The main difficulty here is that the spot price of electricity is no longer predetermined but set by open competition. Thus far, the hourly spot prices of electricity have shown evidence of being highly volatile. The unit commitment decisions are now harder and the modeling of spot prices becomes very important in this new operating environment. Different approaches can be found in the literature in this regard. Takriti et al. [6] have introduced a stochastic model for the UCP in which the uncertainty in the load and prices of fuel and electricity are modeled using a set of possible scenarios. The challenge here is to generate representative scenarios and assign them appropriate probabilities. Allen and Ilic [7] have proposed a stochastic model for the unit commitment of a single generator. They assume that the hourly prices at which electricity is sold are uncorrelated and normally distributed. In [8] Tseng uses Ito processes to model the prices of electricity and fuel in the unit commitment formulation. The purpose of this chapter is to present a new formulation to the UCP suitable for an electric power producer in a deregulated market and consider computationally efficient procedures to solve it. We express the UCP as a stochastic optimization problem in which the objective is to maximize expected profits and to meet operating constraints such as capacity limits and minimum up- and down-times. We show that when the spot market of electricity is included, the optimal solution of a UCP with M units can be found by solving M uncoupled sub-problems. We obtain a subproblem by replacing the values of the Lagrange multipliers by spot power prices. The volatility of the spot market price of electricity is taken into account by using a variation of the stochastic model proposed by Ryan and Mazumdar [9], The model, which is referred to as the probabilistic production-costing model, incorporates the stochastic features of load and generator availabilities. It is often used to obtain approximate estimates of production costs [10,11,12]. This model ignores the unit commitment constraints and assumes that a strict predetermined merit order of loading prevails. This implies that a generator will be dispatched only when the available unit immediately preceding it in the loading order is working at its full capacity. We believe that this model provides a good approximation to the operation of an electricity market, such as the California market, in which no centralized unit commitment decisions are taken. The model cap-

Probabilistic Unit Commitment

141

tures the fundamental stochastic characteristics of the system. At any moment, a power producer may not be fully aware of the exact characteristics of the units comprising the market at that particular time. But it is likely to possess information about the steady state statistical characteristics of the units participating in the market. Ryan and Mazumdar’s probabilistic production costing model can be used to provide a steady-state picture of the market. We determine the hourly spot market price of electricity by the marketclearing prices. The market-clearing price can be shown to be the variable cost or bid of the last unit used to meet the aggregate load prevailing at a particular hour. This unit is called the marginal unit. We determine the probability distribution of the hourly market-clearing price based on the stochastic process governing the marginal unit, which depends on the aggregate load and the generating unit availabilities. We model the aggregate load as a GaussMarkov stochastic process and use continuous-time Markov chains to model the generating unit availabilities [10,13]. We assume that the information is available on mean time to repair, mean time to failure, capacity, and variable operating cost of each unit participating in the market required to characterize these processes. We use probabilistic dynamic programming to solve the stochastic optimization problem pertaining to unit commitment. We also report results on the accuracy and computational efficiency of several analytical approximations as compared to Monte Carlo simulation in estimating probability distributions of the spot market price for electric power.

2.

FORMULATION

We consider the situation in which an electric power producer owns a set of M generating units and needs to determine an optimal commitment schedule of its units such that the profit over a short period of length T is maximized. Revenues are obtained from fulfilling bilateral contracts and selling electric power, at spot market prices, to the power pool. It is assumed that the electric power company is a price-taker. If at a particular hour the power supplier decides to switch on one of its generating units, it will be willing to take the price that will prevail at this hour. We also assume that the power supplier has no control over the market prices and the M generating units will remain available during the short time interval of interest. In determining an optimal commitment schedule, there are two decision variables, denoted and The first variable denotes the amount of power to be generated by unit at time and the latter is a control variable, whose value is “1” if the generating unit is committed at hour and “0” otherwise. The cost of the power produced by the generating unit depends on the amount of fuel consumed and is given by a known cost function

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The Next Generation of Unit Commitment Models

where is the amount of power generated. The start-up cost, which for thermal units depends on the prevailing temperature of the boilers, is given by a known function The value of specifies the consecutive time that the unit has been on (+) or off (-) at the end of the hour In addition, a generating unit must satisfy operating constraints. The power produced by a generating unit must be within certain limits. When the generating unit is running, it must produce an amount of power between and If the generating unit is off, it must stay off for at least hours, and if it is on, it must stay on for at least hours.

2.1

Problem Formulation

The objective function is given by the sum over the hours in the interval [0,T] of the revenue minus the cost. The revenue is obtained from supplying the bilateral contracts and by selling to the power pool at a price of per MWH the surplus energy produced in each hour t. The cost includes the cost of producing the energy, buying shortfalls (if needed) from the power pool, and the start-up costs. Defining the supply amount stipulated under the bilateral contract by and by R ($/MWH) the price, the objective function (maximum total profit) is given by:

A positive value of indicates that megawatts hour are bought from the power pool and a negative value indicates that megawatts hour are sold to the pool. Since the quantity is a constant, the optimization problem reduces to:

subject to the following constraints(for Load: Capacity limits:

and

Probabilistic Unit Commitment

143

Minimum down-time: Minimum up time: where and

unrestricted in sign

After substituting in the objective function the value of obtained from Equation 4, we re-write Equation 3 as follows

which after removing constant terms is equivalent to:

subject to the operating constraints. Because the constraints (5) to (7) refer to individual units only, the advantage of Equation 9 is that the objective function is now separable by individual units. The optimal solution can be found by solving M de-coupled subproblems. Thus, the subproblem for the unit

subject to operating constraints of the unit. Equation 10 is similar to the subproblem obtained in the standard version of the UCP [14] excepting that the value of the Lagrange multipliers are now replaced by the spot market price of electricity.

2.2 Stochastic Formulation of the Subproblem We next consider the spot market price of electricity, which is determined by the market-clearing price, as a random variable. When the optimization subproblem is being solved for a particular unit, we assume that the mar-

144

The Next Generation of Unit Commitment Models

ket, which includes the M-1 units owned by the power producer solving the problem, consists of N generating units (N>>M). The generating unit for which the subproblem is solved is excluded from the market. We assume that the unit commitment decisions for any one unit have a negligible effect on the determination of the marginal unit of the market for a given hour. To model the market-clearing price, we assume that the generators participating in the market are brought into operation in an economic merit order of loading. The unit in the loading order has a capacity (MW), variable energy cost ($/MWH), and a forced outage rate Under the assumption of economic merit order of loading, the market-clearing price at a specific hour is equal to the operating cost (S/MWH) of the last unit used to meet the load prevailing at this hour. The last unit in the loading order is called the marginal unit and is denoted by The market-clearing price, is thus equal to The values of and depend on the prevailing aggregate load and the operating states of the generating units in the loading order. We write the objective function of the subproblem for one of the M generating units as follows:

subject to the operating constraints: capacity limits, minimum up-time, and minimum down-time.

2.3

Probabilistic Dynamic Programming Solution

The maximum profit over the period T (Equation 11) is a random variable because the hourly market-clearing price is a random variable. We assume that at the time of the decision, hour zero, the marginal unit and the load for all the hours before hour zero are known. We denote the marginal unit at time zero by and solve the subproblem by maximizing the conditional expected profit over the period T. We express the objective function as:

This equation is subject to the same operating constraints described earlier. We use probabilistic dynamic programming to solve this optimization problem. We define the function by the following equation:

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145

This function denotes the maximum profit at hour given that at this hour the unit is determining the market-clearing price and the generator to be scheduled is in the operating state We also define the recursive function to be the optimum expected profit from hour to hour T of operating the generator that is in state at time Thus, the expression for hour zero is

For hour

the expression is given by the following recursive relation:

Setting the expected incoming profit at time T+1 to be zero we obtain the boundary condition for the last stage

The initial conditions are given by the initial state of the generator and and the marginal unit at hour zero Consequently, the optimal schedule is given by the solution of To solve the problem, the following conditional probabilities need to be computed:

Thus, the joint probability distribution of probability distribution of are needed.

3.

and

and the marginal

STOCHASTIC MODEL FOR THE MARKETCLEARING PRICE

The stochastic model of the market-clearing price uses the productioncosting model proposed by Ryan and Mazumdar [9]. This model has been used in estimating the mean and variance of production cost [12] and marginal cost [15] of a power generating system.

146

3.1 1.

2.

3.

4.

5.

The Next Generation of Unit Commitment Models

Stochastic Model of the Market

The assumptions of a market model with N generating units are: The generators are dispatched at each hour in a fixed, pre-assigned loading order, which depends only on the load and the availability of the generating units. Operating constraints such as minimum up- and down-times, spinning reserve, and scheduled maintenance are not considered. The unit in the loading order has capacity (MW), variable energy cost (S/MWH), mean time to failure mean time to repair and forced outage rate, After adjusting for the variations in the ambient temperature and periodicity, the load at time is assumed to follow a Gauss-Markov process [16,17] with and where and are assumed to be known. (An analysis given in [13] validates this assumption.) The operating state of each generating unit follows a two-state continuous-time Markov chain, with failure rate and repair rate The forced outage rate equals For and are statistically independent for all values of and Each is independent of for all values of

3.2

Probability Distribution of the Marginal Unit

To derive an analytical expression for the probability mass function of the marginal unit at time first note that and that the events

and

are equivalent. Thus,

the following equality holds:

3.3

Thus, to obtain the probability mass function of

the probability that

is greater than zero for all values of

needs to be computed.

Bivariate Probability Distribution of the Marginal Unit

An analytical approximation for the bivariate probability mass function of and needed for evaluating Equation 17, requires the following development. Writing

Probabilistic Unit Commitment

147

and observing that the events

are equivalent, we obtain the following equality:

Therefore, to compute the bivariate probability mass function of and

the probability that

needs to be evaluated for all values of

and

The computational effort in evaluating equations 18 and 20 depends on the many values that the expression

can take, which in the worst

case is (when Thus, the computational time increases exponentially as N increases and it would make an exact computational procedure prohibitive for large N. In our numerical examples, we have used three approximation methods: the normal, Edgeworth, and Monte Carlo approximations. The Edgeworth approximation [18] is known in the power system literature as the method of cumulants. We also attempted the use of the large deviation or equivalently, the saddlepoint approximation method [19], but it turned out to be prohibitively time-consuming for very large systems.

4.

SOLUTION OF THE PROBABILISTIC UNIT COMMITMENT PROBLEM: A NUMERICAL EXAMPLE

For our purpose, we assume that a complete description of the electricity market is given by the data concerning the N power generators that comprise

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The Next Generation of Unit Commitment Models

the market, historical data of the aggregate load, and the hourly temperature forecast for the day of trading. The description of the power generators includes the order in which they will be loaded by the ISO, their capacities, energy costs, mean times to failure, and mean times to repair. The data for the aggregate load gives the historically forecast ambient temperature and the corresponding load for each hour in the region served by the marketplace. In this example, a data set that gave the actual ambient temperature and the corresponding load for each hour in a region covering the northeast United States during the calendar years 1995 and 1996 was used. We chose the last day of this data set, September 20, 1996, as the trading day. Table 1 gives the actual temperatures on this day, which were assumed to be the forecast temperatures.

Example: The market is described by the aggregate load and a power generation system consisting of generators participating in the market. Using statistical time series analysis on the data at hand, we found that an ARIMA provided a very good description of the actual load observed. The model [13] used is as follows:

where

and

is the average ambient temperature at hour and

is defined as:

Probabilistic Unit Commitment where variance

149

is a Gaussian white noise process with mean zero and estimated Table 2 provides the estimates of the least-square re-

gression coefficients,

relating load to temperature.

We assume the market consists of 150 power generating units. This system was obtained by repeating ten times each unit of a 15-unit system, which in its turn is a smaller version of the IEEE Reliability Test System (RTS) [20]. The load data from [13] was also multiplied by a factor of ten. Defining as the cumulative capacity of the first i units,

we assume that the variable cost of each unit is given by the function:

This assumption allows for the unit operating costs to increase in order of the position of the units in the loading order. Table 3 gives the relevant characteristics of the units, in their loading order. The problem is to schedule one of the generators of the power producer for the next 23 hours given the information about the electricity market and the known initial operating conditions for the generating unit. The characteristics of this generator were taken from [1], and they are reproduced in Table 4. We modified the fuel-cost function of the unit to be consistent with the range of the individual units’ energy costs. The objective is to maximize the expected profit over this period. We assumed the generator to have been on for eight consecutive hours. As mentioned in Section 2.3, this generator is not

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The Next Generation of Unit Commitment Models

included in the set of generators that comprise the market. We also assumed that the variable cost of the unit is currently determining the marketclearing price, which is $19.73 per MWH.

Table 5 summarizes the unit commitment solutions obtained using the different algorithms. The optimal schedule produced by the Monte Carlo simulation (200,000 replicates used in estimating the probability distributions) is to turn the generating unit off during the first four hours. Then, the unit is turned back on for the next nineteen hours. The Monte Carlo procedure estimates that the execution of this schedule will generate an expected profit of $37,509. The normal and Edgeworth approximation methods provide this schedule as well. However, they estimate expected profits of $37,483 and $37,351, respectively. Details of these computations are given in [21].

Probabilistic Unit Commitment

5.

151

DISCUSSION

In this chapter, we have proposed a new formulation of the unit commitment problem that is valid under deregulation. We have shown that, when we assume a competitive market and price-taking, the unit commitment problem can be solved separately by each individual generating unit. The solution method for the new formulation requires the computation of the probability distribution of the spot market price of electricity. The power generation system of the marketplace has been modeled using a variation of the RyanMazumdar model. This model takes into account the uncertainty on the load and the generating unit availabilities. The probability distribution of the spot market price, which is determined by the market-clearing price, is based on the probability distribution of the marginal unit. The exact computation of the probability distribution is prohibitive for large systems. We evaluated three approximation methods. From the computational experience, it appears that the proposed unit commitment can be accurately solved in a reasonable time by using the normal, Edgeworth, or Monte Carlo approximations.

ACKNOWLEDGMENTS The authors are indebted to the editors and the two reviewers for their very helpful comments. This research was supported by the National Science Foundation under grant ECS-9632702.

REFERENCES 1.

A. Wood and B. Wollenberg. Power Generation Operation and Control. New York: Wiley & Sons, Inc., 1996.

152 2. 3. 4. 5. 6.

7. 8.

9. 10. 11. 12.

13. 14. 15. 16.

17.

18.

19. 20. 21.

The Next Generation of Unit Commitment Models “Promoting Wholesale Competition Through Open Access Non-discriminatory Transmission Services by Public Utilities.” FERC Order No. 888, 1996. F.D. Galiana and M.D. Ilic. A mathematical framework for the analysis and management of power transactions under open access. IEEE Trans. Power Syst., 13: 681-687, 1998. “Introduction to the New California Power Market.” Draft/work document posted on the Internet, http://www.calpx.com, July 1997. “Zonal Market Clearing Prices: A Tutorial California Power Exchange.” Document posted on the Internet, http://www.calpx.com. S. Takriti, B. Krasenbrink, and L. Wu. “Incorporating Fuel Constraints and Electricity Spot Prices into the Stochastic Unit Commitment Problem.” IBM Research Report, RC 21066, Subject Area Computer Science/Mathematics, 1997. E.H. Allen and M.D. Ilic. “Stochastic Unit Commitment in a Deregulated Utility industry.” In Proc. North American Power Conference, Laramie, WY: 105-112, 1997. C. Tseng. “A Stochastic Model for the Price-based Unit Commitment Problem.” Paper presented at the Workshop on Next Generation of Unit Commitment Models, Cosponsored by DIMACS and EPRI, September 27-28, 1999. S.M. Ryan and M. Mazumdar. Effect of frequency and duration of generating unit outages on distribution of system production costs. IEEE Trans. Power Syst., 5: 191-197, 1990. M. Mazumdar and A. Kapoor. Stochastic models for power generation system production costs. Electric Power Syst. Res., 35: 93-100, 1995. S.M. Ryan and M. Mazumdar. Chronological influences on the variance of electric power production costs. Oper. Res., 40: S284-S292, 1992. F. Shih, M. Mazumdar, and J.A. Bloom. Asymptotic mean and variance of electric power generation system production costs via recursive computation of the fundamental matrix of a Markov chain. Oper. Res., 47: 703-712, 1999. J. Valenzuela and M. Mazumdar. Statistical analysis of electric power production costs. Accepted for publication in IIE Trans. J.F. Bard. Short-term scheduling of thermal-electric generators using Lagrangian Relaxation. Oper. Res., 36: 756-766, 1988. F. Shih and M. Mazumdar. An analytical formula for the mean and variance of marginal costs for a power generation system. IEEE Trans. Power Syst., 13: 731-737, 1998. A.M. Breipohl, F.N. Lee, D. Zhai, and R. Adapa. A Gauss-Markov load model for application in risk evaluation and production simulation. IEEE Trans. Power Syst., 7: 1493-1499, 1992. J. Valenzuela, M. Mazumdar, and A. Kapoor. Influence of temperature and load forecast uncertainty on estimates of power generation production costs. Accepted for publication in IEEE Trans. Power Syst. M. Mazumdar and Y. Wang. On the application of Esscher's approximation to computation of generating system reliability and production costing indexes. IEEE Trans. Power Apparatus Syst., 104: 3029-3036, 1985. S. lyengar and M. Mazumdar. A saddle point approximation for certain multivariate tail probabilities. SIAM J. Scient. Comput, 19: 1234-1244, 1998. APM Subcommittee. IEEE reliability test system. IEEE Trans. Power Apparatus Syst., 98: 2047-2054, 1979. J. Valenzuela. Stochastic Optimization of Electric Power Generation in a Deregulated Market. Ph.D. Dissertation, School of Engin., University of Pittsburgh, 1999.

Chapter 9 SOLVING HARD MIXED-INTEGER PROGRAMS FOR ELECTRICITY GENERATION Sebastian Ceria Columbia University and Dash Optimization, Inc.

Abstract:

1.

In this chapter, we describe the most recent advances in the solution of mixedinteger programming problems. The last ten years have seen enormous improvements in the solution of the most difficult mixed-integer programs. The trend towards integration of modeling and optimization now makes it possible to solve the hardest optimization problems arising from electricity generation, such as the unit commitment problem. We report results with a leading software package that was used successfully to solve unit commitment problems in two European utility companies.

INTRODUCTION

Electricity generating companies and power systems face the daily problem of deciding how best to meet the varying demand for electricity. In the short-term, electric utilities need to optimize the scheduling of generating units so as to minimize the total fuel cost or to maximize the total profit over a study period of typically a day, subject to a large number of constraints that must be satisfied. This problem, which is of crucial importance for electric generation companies, is referred to as the “unit commitment problem.” This problem can be best solved by using state-of-the-art optimization technology. Over the last twenty years, we have been witnesses to a revolution in computational optimization. The availability of powerful computers and the improvements in algorithmic development now make it possible to solve problems that only a few years ago were thought to be unsolvable. Indeed, modern commercial linear and integer programming codes, such as CPLEX1, LINDO2, OSL3, and XPRESS-MP4 and research linear and integer program1

CPLEX is a trademark of ILOG, Inc. LINDO is a trademark of LINDO Systems, Inc 3 OSL is a trademark of IBM Corporation 4 XPRESS-MP is a trademark of Dash Associates, Ltd. 2

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The Next Generation of Unit Commitment Models

ming codes, such as MINTO [28, 34], ABACUS [24], MIPO [8], and bc-opt [9], are several orders of magnitude more powerful than the most efficient implementations of linear and integer programming algorithms that were available only ten years ago (when compared the present generation of computers). In this chapter, we will focus on the solution of basic mixed-integer (binary) programming problems of the form shown in Figure 1.

In Figure 1, is an A is a matrix of rows, is a vector of components, and is an The first variables are constrained to taking only 0-1 values; thus, the problem in Figure 1 is called a mixed 0-1 program. Traditionally, integer programming problems have been solved by using one of two algorithms, branch-and-bound and cutting planes, both based on linear programming technology (see [23] for a complete survey of the area). For the last ten years, several researchers have been studying and implementing a combination of the above, which is commonly referred to as branch-andcut. The term was originally coined by Padberg and Rinaldi [31, 32] who were the first researchers to demonstrate the computational efficiency of the algorithm for the solution of traveling salesman problems. But Padberg and Rinaldi’s branch-and-cut code goes beyond a simple combination of branch-and-bound and cutting planes. It adds heuristics to speed up the identification of feasible (almost optimal) solutions. It uses preprocessing to reduce the size of the problem and hence improve its solvability. Finally, Padberg and Rinaldi exploit the algebraic structure of the problem in order to generate cutting planes. Hence, large-scale traveling salesman problems, involving thousands of cities, which translates into millions of variables and an exponential number of constraints, can now be solved efficiently in a few minutes. The success of their algorithm can be largely attributed to exploiting the mathematical knowledge of the problem. Researchers have solved other hard optimization problems using similar techniques. The caveat of the approach is that the algorithm has to be specifically tailored for a particular problem, with the added difficulty that any

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changes to the model cannot be easily accommodated. It is for this reason that commercial software companies have recently concentrated their efforts on the development of programming environments that facilitate the development of such algorithms (see for example [11]). The main idea behind such products is to break the separation that has existed for decades between modeling environments and optimization algorithms, thus making it easy for the developer to prototype rapidly sophisticated algorithms for the solution of the hardest optimization problems. In Section 4 we briefly explore an application of this technology to the solution of unit commitment problems. The resurgence of mixed-integer programming as a viable technology for solving hard optimization problems is due to significant improvements in the quality of the basic optimizers. Section 2 lists some of these developments and includes several references. In Section 3, we study integrated tools for modeling and optimization and discuss this paradigm shift. The integration of modeling and optimization has been made possible by new software tools available in the market and has recently been embraced by practitioners. Finally, let us add that it is not the intent of this chapter to serve as a full reference for the state-of-the-art in integer programming, nor to provide a complete list of commercial or research software available for solving such problems. Interested readers should also consult references, such as [7] and [23]. Classic references for integer programming include [15], [29], and [35].

2.

INTEGER PROGRAMMING: RECENT ADVANCES

Research in the algorithmic solution of mixed-integer programming problems started in the early sixties with the development of two classes of methods. The first one is that of cutting plane algorithms. In a cutting plane algorithm, valid inequalities that cut off the current solution of the linear programming relaxation are used to tighten the initial formulation until a feasible integer solution can be found (see Figure 2). Their origins can be traced as far back as the mid-1950s, with the landmark paper of Dantzig, Fulkerson, and Johnson [12, 13] where they show how cutting planes can be used to solve a 48-city traveling salesman problem. Gomory [16, 17, 18], was the first one to propose cutting planes as a general solution procedure. Branch-and-bound is the second class of methods, which are used to solve mixed-integer programs. The seminal papers of Land and Doig [25] for general mixed integer programs and of Balas [1] for the case of pure 0-1 programs introduced the basic ideas that would later become the foundation for the area. Branch-and-bound methods are of the enumerative type, since they solve the problem by enumerating a certain number, hopefully as small as possible, of feasible solutions. Branch-and-bound has been for long the prevalent way chosen by practitioners to solve mixed 0-1 programs.

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In the last 10 to 15 years there have been several important innovations in the practical solution of mixed-integer programs. The first is one of incorporating combinatorial cutting planes, as opposed to general cutting planes for tightening the linear programming relaxation of certain classes of pure integer programming problems. The main drawback of these methods is that there has to be an underlying combinatorial structure in order to generate the cutting planes. Crowder, Johnson, and Padberg [10] propose a method that circumvents this difficulty and that applies to pure 0-1 programs. In order to obtain deep cuts they generate facets of the knapsack polyhedra obtained by considering each constraint in the constraint set separately. Van Roy and Wolsey [37, 38] use similar ideas for mixed 0-1 programs. In both cases, after the linear programming relaxation had been tightened with the newly added cuts, the authors apply branch-and-boun, and show that a significant reduction in the number of nodes in the branch-and-bound tree can be obtained. Recently, researchers have explored other ways of tightening the linear programming relaxation of an integer program that involves reformulating the problem into a higher dimensional space (one that uses more variables), where a more convenient formulation may give a tighter relaxation. Such procedures are based on the work of Sherali and Adams [36]; Lovász and Schrijver [27]; and Balas, Ceria, and Comuejols [2], [3], [4]. Cutting planes can also be used in conjunction with branch-and-bound as a method for strengthening the formulations at the nodes of the search tree (see Figure 3). Padberg and Rinaldi [31] were the first to propose such a method – that they baptized branch-and-cut – for the traveling salesman prob-

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lem and obtained impressive results with it. There are two important keys to the success of their approach. First, the combinatorial cutting planes they generate are deep cuts, and often facets, of the underlying integer polyhedron. Second, because of the combinatorial nature of their cutting planes, the cuts they generate at one part of the tree can be easily made valid throughout the enumeration tree. There has recently been considerable interest in the development of a branch-and-cut method that can be applied to general mixed 0-1 programs. Hoffman and Padberg [21, 22] obtained impressive results by using the cutting planes of Crowder, Johnson, and Padberg in a branch-and-cut context for a class of sparse problems arising from applications.

Recent implementations of branch-and-cut have been proven it to be an effective method for tackling some classes of mixed 0-1 programs, but there is still an important need for a procedure that can solve general mixed 0-1 programs, independently of any underlying combinatorial structure. In [2, 3, 4] we show that a branch-and-cut method based on lift-and-project cutting planes is able to find optimal solutions to some instances of mixed 0-1 pro-

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grams that were previously unsolved, and it is faster than competing methods on other instances. Most of these recent advances have been implemented in commercial software. These improvements have been combined with preprocessing, heuristics, and fancy data structures. A modern MIP code starts by preprocessing the problem, thus reducing the size of the coefficients and eliminating unneeded variables; then running a heuristic to improve the quality of the bounds; and finally applying a combination of cutting planes and branch-andbound. It is now possible to solve problems that only a few years ago seemed unsolvable. Most commercial packages incorporate features that are particularly helpful when modeling some realistic problems in electricity generation. For example, XPRESS-MP allows the user to define semi-continuous variables (variables that take either the value 0 or a value between a lower-bound (greater than 0) and an upper-bound). This feature can be used to model electricity generation or consumption processes naturally.

We include computational results with XPRESS-MP5 are included in Figure 5 (for linear programming) and Figure 6 (for mixed-integer program5

XPRESS-MP is one of several commercial LP/MIP packages commercially available.

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ming). The code is compared with its earlier version that came on the market only one year before the current release. The significant improvements in solution times are due to improved preprocessing, faster LP engines, and the utilization of state-of-the-art cutting planes. The data sets are publicly available through either NETLIB or MIPLIB.

3.

INTEGRATED MODELING AND OPTIMIZATION

Traditionally, mathematical programmers that use modeling languages to express their problems have “artificially” separated modeling from optimization. Practitioners and researchers have to sacrifice either performance by constantly switching between a modeling and optimization environments, or they have to design specific algorithms and implementations that completely ignore modeling environments and work directly with an internal representation of the problem. In the first case, knowledge about the structured algebraic model either missing and thus difficult to exploit. In the second, it is so embedded into the implementation that small alterations to the model may render the implementation completely useless.

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There have been several attempts to integrate modeling environments and optimization. The first such environment that has been implemented for the solution of mixed-integer programming problems is EMOSL6 (Entity Modeling Optimization Subroutine Library) [6, 14]. EMOSL is a combined modeling and optimization subroutine library that overcomes the separation between modeling and optimization by allowing the problem to be manipulated using the notation of the model language. By integrating modeler and optimizer, the scope and ease of algorithm development is greatly improved, without any observable degradation of performance. Figure 7 provides an illustration of the principle. EMOSL was used in [11] for four applications: a block structure extraction tool and the implementations of three MIP heuristics and cutting planes techniques. The result of their research effort is now a commercial product that is also used in the electricity industry (see Section 4). This does not mean that researchers have not been integrating modeling and optimization in the implementation of their mixed-integer programming algorithms. If this integration is done in the modeling environment then the drawbacks in terms of speed are so great that these platforms cannot be considered for serious application of efficient algorithms. On the other hand, if researchers use a low-level language, the original model becomes very hard to maintain. For mixed-integer programming, the majority of current algorithmic 6

EMOSL is a trademark of Dash Associates Ltd.

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implementations and development work uses conventional optimizer subroutine libraries (for example: ABACUS [24], bc-opt [9], MINTO [28, 34] and MIPO [8]). EMOSL is essentially a unification of an optimizer subroutine library and a callable modeler. It provides all of the standard functions of an optimizer library and many advanced features; we mention several later. It provides functions to parse a model and to regenerate it after changes have been made. The key feature is the new functionality to access the attributes of the variables and constraints of the problem, i.e., functions to inspect and change the coefficients of the variables in constraints, the sense of constraints, the upper and lower bounds on variables, the type of variables (continuous, binary, integer), and functions to inspect the primal and dual solution values of the variables and constraints. Other modeling languages, such as AMPL,7 GAMS,8 MPL,9 and OPL,10 are increasingly blurring the dividing line between modeler and optimizer. For example, MPL has a feature that allows access to entities in the modeling language from inside a C program, and OPL allows C/C++ model generation, effectively allowing the combination of model and algorithm.

7

AMPL is a trademark of AT&T. GAMS is a trademark of Gams Development Corp. 9 MPL is a trademark of Maximal Software, Inc. 10 OPL is a trademark of ILOG, Inc. 8

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EXAMPLES IN ELECTRICITY GENERATION

There are several ways of formulating the UC problem. Most formulations make use of mixed-integer and nonlinear mixed-integer models. Unit commitment problems are intrinsically difficult because in many cases operating constraints are best represented by logical conditions, whereas in other cases they are represented by non-convex functions. The UC problem has been tackled by using a variety of methods. Heuristic procedures were first studied in [20], while most of the current research has focused on a combination of dynamic programming and Lagrangian relaxation [5, 19, 30, 33, 39]. In [26], Lekane and Gheury, from the Belgian electricity company Tractebel, consider a system including nuclear units, fossil-fired units, and pumped storage plants. Their model takes into account operating considerations such as the operating reserve constraint, the minimum operating power and ramping rate constraints, the commitment constraints of the fossil-fired units, the operation of the nuclear plants at constant output levels, and the energy constraints associated with the pumped storage plants. Their paper first presents a solution approach based on the Lagrangian relaxation method and discusses the difficulties associated with discovering a primal feasible solution. Then the authors describe a new solution system, which was investigated in the frame of the ESPRIT project called MEMIPS. The approach presented there is based on a column generation method that integrates dynamic programming to generate feasible power profiles for the resources and linear programming using EMOSL to find a set of profiles minimizing the total operating cost while respecting the system constraints. Heuristics are then applied to determine integer solutions. The system heavily exploits the integration of modeler and optimizer available through EMOSL. By using EMOSL, the authors have the flexibility to change the model by adding new constraints without necessarily affecting the algorithm implementation. The approach presented in [26] is based on a reformulation of the UC problem as a problem of determining the set of feasible thermal unit profiles which minimizes the total operation cost over the studied period while respecting the demand and reserve constraints as well as the pumped storage plant constraints. A feasible profile for a thermal unit is defined by both power and reserve profiles, which specify its hourly generation and reserve contribution over time. The principle of the proposed approach consists in generating a set of feasible thermal profiles and solving the MIP problem over the specified set of profiles. Since a huge number of thermal profiles exist, an approximate procedure is used to solve the MIP problem over a limited set of profiles. This procedure comprises four main steps, namely:

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Generation of feasible thermal profiles: An initial set of feasible thermal profiles is generated by using the solutions to the sub-problems found when solving the dual problem in the Lagrangian relaxation approach. Column Generation: The LP relaxation of the MIP that contains only these thermal profiles is solved using column generation. Relax and Fix: The resulting MIP problem is solved approximately by using a relax and fix method. Solution of reduced MIP: The MIP problem obtained by fixing the binary variables associated with the thermal units at the values found as a result of the relax and fix heuristic is solved to optimality. In this case, the advantages of using a framework that combines modeling and optimization for such an implementation are clear: The combined nature of the modeling and optimization subroutine library facilitates the integration. Access to variables and constraints in the modeler’s language allows for ease of algorithm implementation. An implementation that uses both modeling and optimization constructs is extremely efficient since no overhead is incurred while communicating with the modeling system. The integrated modeling and optimization implementation (in EMOSL) results in easy-to-maintain code. Another example solving unit commitment problems using mixed-integer programming comes from Power Optimisation, Ltd11. One version of the unit commitment software, developed for Northern Ireland Electricity (NIE), has been used every day since December 1996 to schedule the generating units in the Northern Ireland power system. The users at NIE report that the schedules produced by the software are consistently of a very high quality. The software uses a novel multi-stage solution method, which drastically reduces the computer run-time required to find an excellent feasible schedule to just a few minutes. A great advantage of using and MIP method is that it has proved to be easy and quick to introduce new constraints and features into the unit commitment software. Hence the software can easily be adapted to model the changes in the plant and operating rules of the power system that occur from time to time. 11

More information on this application can be found at www.poweroptimisation.com. The application was implemented using the XPRESS-MP software package.

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CONCLUSIONS

In this chapter, we reviewed the latest developments in integer programming and the integration of modeling and optimization. We illustrated the principles in this integration through a several examples in the area of electricity generation problems. Even though mixed-integer programming techniques are not yet widely used in the electricity industry, we believe that the recent developments in theoretical and practical mixed-integer programming will enable practitioners to use these techniques more effectively. Undoubtedly, the biggest challenge will be the efficient solution of large-scale instances of these problems.

REFERENCES 1. 2. 3. 4.

5. 6. 7.

8. 9.

10. 11. 12. 13. 14. 15.

E. Balas. Disjunctive programming. Ann. Discrete Math., 5: 3-51, 1979. E. Balas, S. Ceria, and G. Cornuéjols. Mixed 0-1 programming by lift-and-project in a branch-and-cut framework. Manage. Sci., 42(9): 1229-1246, 1996. E. Balas, S. Ceria, and G. Cornuéjols. A lift-and-project cutting plane algorithm for mixed 0-1 programs. Math. Prog., 58: 295-324, 1993a. E. Balas, S. Ceria, and G. Cornuéjols. “Solving Mixed 0-1 Programs by a Lift-and-Project Method.” In Proc. Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, 232242, 1993b. J. Batut and R.P. Sandrin. “New Software for the Generation Rescheduling in the Future EDF National Control Center.” In Proc. PSCC, Graz, Austria, 1990. Belvaux, G., Solving lot-sizing problems by branch & cut: Basic guide to use of mgEMOSL. Working paper, CORE, Université catholique de Louvain, February 1998. A. Caprara and M. Fischetti. Branch-and-Cut Algorithms, in Annotated Bibliographies in Combinatorial Optimization. M. Dell’Amico, F. Maffioli, and S. Martello, eds. Chichester: John Wiley & Sons, Chichester, 45-63, 1997. S. Ceria. “MIPO: A Mixed-Integer Program Optimizer.” Presented at: International Symposium on Mathematical Programming, Lausanne, August 1996. C. Cordier, H. Marchand, R.S. Laundy, and L.A. Wolsey. “bc-opt: A branch-and-cut code for mixed integer programs.” Discussion paper DP 9778, CORE, Universite catholique de Louvain, 1997. H. Crowder, E. Johnson, and M. Padberg. Solving large-scale zero-one linear programming problems. Op. Res., 31: 803-834, 1983. R. Daniel and J. Tebboth. A tightly integrated modelling and optimisation library: A new framework for rapid algorithm development. Ann. Op. Res., 0: 1-21, 1998. G.B. Dantzig, D.R. Fulkerson, and S.M. Johnson. Solution of a large-scale traveling salesman problem. Op. Res., 2: 393-410, 1954. G.B. Dantzig, D.R. Fulkerson, and S.M. Johnson. On a linear programming, combinatorial approach to the traveling salesman problem. Op. Res., 7: 58-66, 1959. Dash Associates. XPRESS-MP EMOSL Reference Manual, Release 10. Blisworth, Northants NN7 3BX, UK: Blisworth House Church Lane, 1997. R.S. Garfinkel and G.L. Nemhauser. Merger Programming. New York: John Wiley & Sons, Inc., 1972.

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16. R. Gormory. “An Algorithm for the Mixed-Integer Problem. Technical Report RM-2597, The Rand Corporation, 1960. 17. R. Gormory. 1958. Outline of an algorithm for integer solutions to linear programs. Bull. Am. Math. Soc., 64: 275-278, 1958. 18. R. Gormory. “Solving Linear Programming Problems in Integers. In Combinatorial Analysis R.E. Bellman, M. Hall Jr., eds. Providence, RI: American Mathematical Society, 211-216, 1960. 19. X. Guan, P.B. Luh, and L. Zhang. Non-linear approximation method in Lagrangian relaxation-based algorithms for hydrothermal scheduling. IEEE Trans. Power Syst., 10(2): 772778, 1995. 20. H.H. Happ, R.C. Johnson, and W.J. Wright. Large-scale unit commitment method and results. IEEE Trans. PAS, PAS-90: 1373-1383, 1971. 21. K.L. Hoffman and M. Padberg. Techniques for improving the LP-representation of zeroone linear programming problems. ORSA J. Computing, 3: 121-134, 1991. 22. K.L. Hoffman and M. Padberg. Solving airline crew scheduling problems by branch-andcut. Manage. Sci, 39: 657-682, 1993. 23. E.L. Johnson, G.L. Nemhauser, and M.W.P. Savelsbergh. “Progress in Linear Programming Based Algorithms for Integer Programming: An Exposition.” School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia, August 1999. 24. M. Jünger and S. Thienel. Introduction to ABACUS – A branch-and-cut system. Op. Res. Lett., 22: 83-95, 1998. 25. H.A. Land and A.G. Doig. An automatic method for solving discrete programming problems. Econometrica 28: 497-520, 1960. 26. T. Lekane and J. Ghuery. “Short Term Operation of an Electric Power System.” Tractebel Energy Engineering, Brussels, Belgium. 27. L. Lovasz and A. Schrijver. Cones of matrices and set-functions and 0-1 optimization. SIAM J. Optimization, 1: 166-190, 1990. 28. G.L. Nemhauser, M.W.P. Savelsbergh, and G.C. Sigismondi. MINTO, a Mixed INTeger Optimizer. Op. Res. Lett., 15: 47-58, 1993. 29. G.L. Nemhauser and L.A. Wolsey. Integer and Combinatorial Optimization. New York: John Wiley & Sons, 1988. 30. S.O. Orero and M.R. Irving. Large-scale unit commitment using a hybrid genetic algorithm. Elec. Power Energy Syst., 19(1): 45-55, 1997. 31. M.W. Padberg and G. Rinaldi. A branch-and-cut algorithm for the resolution of largescale symmetric traveling salesman problems. SIAM Rev., 33: 60-100, 1991. 32. M.W. Padberg and G. Rinaldi. Optimization of a 532 city symmetric traveling salesman problem by branch-and-cut. Op. Res. Lett., 6: 1-7, 1987. 33. C.K. Pang, G.B. Sheblé, and F. Albuyeh. Evaluation of dynamic programming based methods and multiple area representation for thermal unit commitments. IEEE Trans. PAS, PAS 100(3): 1212-1218, 1981. 34. M.W.P. Savelsbergh and G.L. Nemhauser. Functional description of MINTO, a mixed integer optimizer, version 2.3. School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia, November 1996. 35. A. Schrijver. Theory of Linear and Integer Programming. Chicester: John Wiley & Sons, 1986. 36. H. Sherali and W. Adams. A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM J. Disc. Math., 3: 411430, 1990. 37. T.J. Van Roy and L.A. Wolsey. Solving mixed-integer programming problems using automatic reformulation. Op. Res., 35: 45-57, 1987.

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38. T.J. Van Roy and L.A. Wolsey, Solving mixed 0-1 programs by automatic reformulation. Op. Res. 35: 145-163, 1987. 39. S.J. Wand, S.M. Shahidehpour, D.S. Kirschen, S. Mokhtari, and G.D. Irisarri. Short-term generation scheduling with transmission and environmental constraints using an augmented Lagrangian relaxation. IEEE Trans. Power Syst., 10(3): 1294-1301, 1995.

Chapter 10 AN INTERIOR-POINT/CUTTING-PLANE ALGORITHM TO SOLVE THE DUAL UNIT COMMITMENT PROBLEM – ON DUAL VARIABLES, DUALITY GAP, AND COST RECOVERY

Marcelino Madrigal and Victor H. Quintana University of Waterloo

Abstract:

1.

In this chapter, we use an interior-point/cutting-plane (IP/CP) method for nondifferentiable optimization to solve the dual to a unit commitment (UC) problem. The IP/CP method has two advantages over previous approaches, such as the sub-gradient and bundle methods: first, it has proved to have better convergence characteristics in an actual implementation; and second, it does not suffer from the parameter-tuning drawback. The results of performance tests using systems with up to 104 units confirm the superiority of the IP/CP method over previous approaches to solve the dual UC problem. We discuss issues that have influenced whether or not UC models are used as the clearing mechanism in electricity markets; these issues include duality gap, cost recovery, and the existence of multiple solutions.

INTRODUCTION

Lagrangian relaxation (LR) has become one of the most practical and accepted approaches to solve unit commitment (UC) problems of real-life dimensions [1-3]. The key idea in LR-based approaches is to solve the dual problem instead of the primal; the dual function has a separable structure, i.e., in a per thermal-unit basis, which permits its easy evaluation and, at the same time, provides a primal solution. The dual function is concave but not differentiable [16]; therefore, non-differentiable optimization techniques are required to solve the dual problem. Pioneering work on LR-based UC solution approaches has used sub-gradient (SG) methods as the dual maximization engine [2, 4]. Despite their bad convergence characteristics, they are still be-

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ing used due to its simplicity and low per-iteration computer effort. Several cutting-plane (CP) variants to solve non-differentiable optimization have been employed to solve the dual to unit commitment or other power scheduling problems. For instance, in [5] a penalty-bundle (PB) method is used to solve the UC problem. In [6], a CP with dynamically adjusted constraints is used to solve the hydrothermal coordination problem. In [7], a reduced complexity bundle method is introduced to solve the dual of a power-scheduling problem. All these cutting-plane variants still have the disadvantage that parameters need be carefully tuned; these parameters define a stabilization scheme that prevents unboundedness in the maximization of the dual function and help improve convergence [8]. Interior-point/cutting-plane (IP/CP) methods have been used to solve nondifferentiable problems in engineering applications, such as the lot sizing [9] and multi-commodity flow problems [10]. Recently, the authors have proposed the use of an IP/CP method to solve the UC problem [11]. This paper describes the use of such a non-differentiable optimization technique to solve the dual to a unit commitment problem. IP/CP methods have two advantages over previous approaches, such as the sub-gradient and penalty-bundle methods: first, they have better convergence characteristics; and second, they do not suffer from the parameter-tuning drawback. The first mechanism used by an electricity market to select power among competing generators was a unit commitment model. In this Pool marketmodel, generators act as independent entities and the pool operator, based on the costs submitted by generators and their physical limitations, decides the schedules for all generators and sets a market price. UC models may naturally have multiple solutions that, from the Pool point of view, are equally good (all minimize the cost) but, for generators, it means different schedules and, therefore, different revenues that create a clear conflict of interest. LR algorithms are still dependent on heuristics and are not able to identify or distinguish multiple solutions [12-13]. The IP/CP method presented in this chapter has eliminated one of the drawbacks (tuning in the dual maximization) of LR algorithms. We also present some other findings that relate duality gap and cost recovery when dual variables set the market price. It has been shown that UC cost-minimization models and artificial price definitions, derived from their solutions, do not necessarily lead to lower prices for consumers [14]. For a simplified UC model, we show that, in the absence of duality gap, the dual variables are the minimum uniform market price that recovers participants’ cost. In Sections 2 and 3, we describe the unit commitment problem and its dual. Section 4 presents the solution of the dual problem by the IP/CP method. Section 5 discusses the issues of duality gap, stopping criterion, cost recovery,

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and multiple solutions. Section 6 contains numerical results on the performance of the IP/CP method, and we give some conclusions in Section 7.

2.

THE UNIT COMMITMENT PROBLEM

The unit commitment problem consists on determining the generators that need to be committed and the production levels that are required for supplying the forecasted short-term (24 or, at most, 168 hours) demand and spinning reserve requirements (2-3); all of this at the minimum cost (1). The operation of the units is subject to several constraints (4), to be described. The primal unit commitment (PUC) is a very large, non-linear, mixed-integer problem; therefore, the non-convex optimization problem:

T is the set of time periods in the optimization horizon (subsequently | T | denotes the number of periods); I is the set of thermal units. The objective function is comprised of production, and start-up cost, These costs are represented, respectively, using the classic [2] quadratic-convex and exponential models:

The commitment state of unit at time is defined by the binary variable (1: committed, 0: decommitted); represents the power output of the unit. Equations (2) and (3) are the constraints to satisfy the system-wide power, and reserve, requirements. The reserve contribution of every thermal plant is given by where is the maximum allowable power output. In (4), the vector contains the commitment states and production levels of the particular unit that is, Each thermal unit operation constraint set, contains: (i) minimum and maximum power

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output constraints; (ii) ramp-constraints; and, (iii) minimum up- and downtime constraints [2]. In general, the sets contain non-linear and mixedinteger restrictions. The UC has been proven to be NP-hard [15].

3.

THE DUAL UNIT COMMITMENT PROBLEM The (Lagrangian) dual problem to PUC is

We obtain the dual objective function, from the Lagrangian that is formed by adding to the objective function (1), through Lagrange multipliers, the system-wide demand constraints (2), and the reserve constraints (3). The dual variables of the power-balance equality constraints are not necessarily constrained to be positive, but, since the objective function, see (5-6), always increases with power production, they will always take positive values. After arranging some terms, the dual objective function has the following separable structure:

where

and

is the dual variable vector. The dual func-

tion (8), as proven in [16], is concave but not differentiable. The separable nature of the dual function is exploited in a two-stage solution process (Lagrangian Relaxation) of the DUC problem.

3.1

The Lagrangian Relaxation Algorithm The LR approach to solve the UC problem is outlined below:

Initialization. Obtain an initial dual vector and set Obtain a priority-list UC and afterwards perform a simplified economic dispatch for each time

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The dual variables of the power demand constraints are used to initialize the respective Spinning reserve constraints’ dual variables are initialized to zero, i.e., Phase 1. DUC Maximization. 1. Dual objective function evaluation. From (8), evaluate the dual objective function by solving the | I | individual unit commitment subproblems (9) using a forward dynamic programming approach. A primal vector, (not necessarily feasible) is obtained after the subproblems are solved. 2. Convergence test. If a convergence criterion is satisfied, then go to Phase 2. Otherwise, continue. 3. Dual Maximization. Find an improved dual solution vector, using a non-differentiable optimization technique. Set go back to 1. Phase 2. Feasibility search. Use a heuristic procedure to map the non primal-feasible solution, (obtained in Phase 1) to a feasible one, say Usually, feasibility is achieved with little modifications on We follow the procedure described in [2] to perform such feasibility search. The structure of the non-linear, mixed-integer subproblems (9) is such that they can be easily solved using a dynamic programming [2-3] or branch and bound techniques. Semi-definite programming has also been used to solve the UC subproblems [17].

4.

AN INTERIOR-POINT/CUTTING-PLANE METHOD TO SOLVE THE DUC PROBLEM

Classic methods for non-differentiable optimization, such as penaltybundle and Kelley’s cutting plane methods [8], maximize a cutting plane approximation of the objective function over a set of restrictions (stabilization region) that encloses the optimal solution and helps improve convergence by properly setting parameters. Instead of maximizing a cutting plane approximation over a stabilization region, IP/CP method takes the analytic center of a localization set as the new improved dual solution [9, 18]. The localization set, is a convex closed region defined by

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The components of are described as follows: (i) a cutting-plane approximation of the UC dual function; (ii) the known dual variables’ lower bounds (iii) a box constraint and (iv) a lower bound to the dual objective function value. Constraints (ii) and (iii) limit the localization set from the “left” and “right,” respectively; and the constraints (i) and (iv) limit the localization set from “above” and “below,” respectively. The cuttingplane approximation of the dual function is constructed using a bundle of information from all previous iterations of the maximization process. A bundle is a collection of: (i) dual vectors (ii) their corresponding dual objective function values

and (iii) the sub-gradients

A sub-gradient at iteration is readily available by computing the mismatches of the power and reserve constrains,

This vector always belongs to the sub-differential set

Therefore, it qualifies as a sub-gradient for the concave function (8) at As pointed in [18], the selection of the box constraint, in (10), has a limited influence on the convergence characteristics of the IP/CP method; in practice, any large enough number based on knowledge of the problem can be chosen. For the UC problem, similar convergence characteristic is achieved with any large value as presented in the results section. This value is obtained based on units’ cost coefficients. A lower bound of the dual objective function is readily available from previous iterations, as The localization set in (10) can be rewritten as

where

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with

and

and

Note that of

where

has iteration-increasing size. The analytic center is defined as the point solving the potential problem

defines a potential function whose maximum is achieved at

a point that is centered in the localization set. For example, a non-centered point touching hyperplane has and the associated potential compoFrom nent is taken as an updated dual solution vector, which is used to evaluate the dual function. If a stopping criterion is not satisfied, the localization set (10) is updated to by adding a new cut, and by replacing the dual-function lower bound approximation by The analytic center of the updated localization set is obtained and the process in repeated again. We depict the third, fourth, and fifth iterations of an IP/CP method for a dual function in schematically in Figure 1. The dot inside each updated localization set (shaded region) represents the analytic center; the horizontal dotted line represents the lower bound and the bold curved line represents the dual function. This figure depicts classical behaviour of the IP/CP method; cuts generated from the analytic center are deeper and the localization set rapidly shrinks towards a single point corresponding to the optimal value.

Problem (14) can be efficiently solved using a primal, dual, or primaldual interior-point method [19]. In [11], the authors use a semi-definite pro-

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gramming formulation to solve the potential problem. In this chapter, we employ a primal-dual logarithmic-barrier interior-point method to solve the potential problem (14) and describe it briefly in the following paragraphs. We derive the primal-dual interior-point method from Kojima, Megiddo, and Mizuno’s primal-dual infeasible-interior-point algorithm [20]. The barrier-Lagrangian to (14) is given by

where is an iteration-decreasing barrier parameter order necessary conditions for the optimality of (15) are

In these equations, the complementarity

The first-

primal

and dual

residuals are defined. From an initial strictly positive point, primal-dual interior-point method generates new points of the form

the

We compute the search directions by a one-iteration Newton’s method that is applied to solve the optimality conditions (16)-(18). The x and y Newton’s directions can be computed by solving the reduced system

The s -search direction is obtained from

The step length in (19) is computed as

where

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and is a safety factor that prevents variables touching zero. A barrier parameter reduction scheme, derived from (16), is given by

The algorithm can be stopped when the optimality conditions (16)-(17) are satisfied up to an specified tolerance.

5.

ON DUALITY GAP, COST RECOVERY, AND MULTIPLE OPTIMAL SOLUTIONS

From well-known results on duality theory [16], we know that, for any feasible primal dual pair the dual objective value is always a lower bound to the primal objective value Duality gap is defined as the difference between the optimal primal and optimal dual objective function values, i.e.,

If a feasible primal dual pair satisfies then the pair is optimal, and there is no duality gap. It may happen, as in several non-convex programming problems, that duality gap exists; that is, the optimal primal and dual objective function values are not equal, and therefore,

5.1

Duality Gap And Stopping Criterion

At every iteration of the LR algorithm, a primal and dual feasible solutions can be obtained. The dual solution is directly obtained from the dual maximization. We achieve a primal solution (not necessarily feasible) when the dual function is evaluated; a feasibility search (based on heuristics) can yield a feasible one. Let us define the complementarity and relative complementarity gaps as

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Through the iterative process, the values CG or RCG can be used as optimality indicators. The LR algorithm can be executed until DG is as small as possible. If after a number of iterations it is found that CG = 0, then an optimal solution has been found and there is no duality gap. If CG cannot be reduced below some value there are two possibilities: (i) a duality gap exists, and the solution at hand is optimal; or (ii) the LR approach fails to further improve the solution. Experience on solving UC problems [2, 15, 21] shows that RCG can be reduced to “small” values, about 1-2%, especially for large problems. These experimental results can be explained by theoretical results [1, 16] that show that, as the number of separable components (generators) in the dual function increases, the DG decreases.

5.2

Duality Gap and Cost Recovery: Do Dual Variables Recover the Participants’ Cost ?

Using a simplified unit commitment problem, the authors have been able to show that there is a direct relationship between duality gap and participants’ cost recovery when dual variables are used to set the market price when the unit commitment is used as an optimization-based auctioning device [22]. The simplified (static, 1-hour) UC problem considers cost functions that contain a linear and a constant start-up cost. The reserve constraint is dropped and the operation of units is only constrained by the upper production limit; that is,

Let be the uniform price set by a market to pay participants for their power output (optimal solution to (24)). The profit (revenue minus production cost) for each participant is given by

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where has been introduced for convenience. The market price has to be the minimum possible that leads to the recovery of participants’ cost; i.e., the profits are positive The dual problem to (24) has the form of (7), and the dual function is given by

The dual function is piece-wise concave, and has slope-changing points at every where as shown in Figure 2.

The optimal value of the dual function (26) can be found in a closed form; it just suffices to determine the point at which the dual-function slope becomes zero or negative. Without loss of generality, let us assume that the units are ordered so that An optimal dual solution is given by

where

is the smallest index such that

In [22], the

conditions under which duality gap exists for this problem are derived. In the absence of duality gap, if the optimal dual variable (or any of its multiple values) is used to set the market price, then all the units will recover their cost. Since for all scheduled units, then If a duality gap exists, there is a cost that is not recovered and its magnitude equals the gap.

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Table 1 shows the data from an illustrative example, where we consider three different demand conditions and 290 MW). In the first case, the dual problem has multiple dual solutions, but there is no duality gap; and, therefore, as Table 3 shows, all the units’ costs are recovered. For this case, there is only one optimal (global) primal solution. Notice that the profit of the last scheduled unit is zero; hence, there is no other smaller value (that is used as an uniform price) which can recover participants’ cost. This means that in the absence of duality gap, the optimal dual variables from the cost minimization problem are also the minimum possible prices. In the second case, the dual problem has a unique solution. Since the optimal primal objective function is different from the optimal dual objective function, however, there is duality gap. The cost not recovered is equal to the duality gap; see Table 3. In the last case, there is also a duality gap of $3 and, therefore, the same quantity is not recovered. In this case, there are two multiple primal solution; 40 MW can be supplied either by unit 4 or 5.

Some other results in [22] show that for systems with a larger number of units, the size of the duality gap dramatically decreases. This suggests that small (or no) adjustments to the optimal dual variables need to be made in order to use them to set the market price. Even though the relation between duality gap and cost recovery has not been proved for the general UC problem (1)-(4), the results in the next section suggest that the relation may hold. Multiple solutions are more likely to exist for this type of combinatorial problem, as happened in the last example. An approach that can identify all the alternative optimal solutions is, obviously, the use of enumeration. This

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would require the enumeration of possible unit commitment combinations; from these combinations, we take the ones that can satisfy the demand and solve them using a simple linear programming model. If different combinations have the same objective function, then there are multiple (combinatorial) solutions. For a particular combination, multiple solutions (continuous multiple solutions) may also exist. The identification of multiple solutions for the PUC problem is a much more complex task.

6.

RESULTS OF THE IP/CP FOR THE DUC PROBLEM

The LR procedure to solve the UC problem has been coded in C; test runs have been performed on a Pentium 200Mhz computer under the LINUX operative system. We solve the potential problem (14) using the primal-dual interior-point method, described in Subsection 4.1.1; it is also coded in C. The Newton’s system (20) is solved using an efficient sparse-matrix processing and factorization techniques. We use the following four UC test systems: (i) a 26-unit system [23]; (ii) a 32-unit system [15]; (iii) two larger systems with 76 and 104 units, which are created by using the data from the systems in (i) and (ii). The data for the last two systems is selected in such a way that large, medium, and small units, as well as expensive, moderate, and cheap units coexist. All tests are performed over a | T |= 24 hour optimization horizon.

6.1

Comparative Performance

Table 4 exhibits the number of iterations and time required by each of the methods to solve the dual problem up to a point where the RCG is less than 2% for the 26-units case, and less than 1% for the other cases. Notice that the IP/CP method requires far less iterations than the SG and PB methods, and its solution time is competitive, especially for larger systems. The results obtained with the SG and PB methods, shown in Table 4, are achieved after extensive individual parameters tuning for each system; whereas no parameter tuning is required for the IP/CP method. The box constraint is easily set up and left unchanged in all the tests performed. Figure 3 presents the evolution of the dual function for each method. The results shown correspond to the 104-unit system; we observe a similar behaviour with the other test systems. In the first four to five iterations, the IP/CP method is able to reduce the RCG to about 3%.

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The Next Generation of Unit Commitment Models

Effect of the Box Constraint

We established and left fixed box constraints in all test systems to the maximum incremental cost (evaluated at maximum power) of a system unit; that is, It is likely that the incremental cost at the solution point will never be larger than this value. If the incremental cost in the system at a particular time reaches this value, unfeasibility is likely to occur since the most expensive unit is being put at maximum generation. It has been observed that if the box constraint is set up to any other large value, similar convergence occurs. This claim is supported by the results shown in Table 5, where the box constraint has been changed to 75%, 200% and even 300% of the base value Notice that for larger systems the set up of the box constraint has less effect; i.e., in the UC-104 system, only 10 more iterations are required for convergence if the set up is 300% of

Unit Commitment By Interior-Point/Cutting-Planes

6.3

181

Duality Gap and Cost Recovery

As shown in Section 5.2 for a simplified UC problem, the magnitude of the duality gap is equal to the cost not recovered by the units when the optimal dual variables are used to set the market price. For the PUC (1)-(4) the same relationship seems to hold as shown by the results presented in Table 6. For all the UC test cases, the dual variables to the power balance constraints (obtained in the last iteration of the LR algorithm) are used to set the market price The profit for each unit is and the total non-recovered cost is As we can see, the CG and NRC are very similar; this suggests that the relation between duality gap and cost recovery may still hold for the UC problem. Small modifications could be made to the dual variables in order to use them as the market price. For instance, the revenues of participants whose profits are positive can be proportionally (to the total positive profits) adjusted (reduced) so that the reduction is equal the duality gap and, therefore, is enough to recover all participants costs. This is equivalent to define a non-uniform market price; each supplier received a price computed from the dual variable, but scaled to proportionally recover the duality gap. Modification of prices to achieve desirable revenue levels (revenue reconciliation) has been proposed for spot markets [24]. It is worthwhile to point out that DG is a characteristic of the model and its parameters; it cannot be reduced by the use of one or other algorithm used to solve the problem. An algorithm does better than the other if it reduces CG closer to DG. Changes in the model can lead to different DG, but now dual variables have different meanings.

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CONCLUSIONS

Interior-point methods are among the best alternatives to solve a wide class of very large linear and even non-linear programming problems. Research on combinatorial and non-differentiable optimization has again put interior-point methods as one of the promising approaches to most efficiently solve such problems. In this chapter, we proposed an interior-point/cutting-plane method to solve the dual to UC problems. Comparisons to previously used approaches, such as sub-gradient and penalty-bundle methods, demonstrate the advantages of the IP/CP method. The IP/CP method has better convergence characteristics and completely eliminates the need for parameter tuning. The chapter also presents results that relate duality gap and cost recovery if the dual variables are used as market prices. We showed, using a simplified UC model, that when duality gap does not exist, the dual variable is the minimum market price that can be used to set a uniform market price that recovers participants’ costs. When duality gap exists, the optimal dual variables do not recover the costs; the un-recovered cost equals the magnitude of the duality gap. The same relation seems to hold for the general UC problem, as shown by the numerical results. This suggests that for large unit commitment problems, small (or none) modification can be made to the dual variables in order to be used as market prices. More research into this problem and on the effective detection of multiple optimal solutions may lead to better understanding of the failures encountered in pioneering markets, as well as help design better auctioning optimization models.

ACKNOWLEDGEMENTS The first author gratefully acknowledges CONACyT and Instituto Tecnológico de Morelia in México, for providing financial support to pursue his Ph.D. studies at University of Waterloo.

REFERENCES 1. 2. 3.

D.P. Bertsekas, G.S. Lauer, N.R. Sandell, and T.A. Posberg. Optimal short-term scheduling of large scale power systems. IEEE Trans. Autom. Control, AC-28(1): 1-11, 1983. F. Zhuang and F.D. Galiana. Towards a more rigorous and practical unit commitment by Lagrangian relaxation. IEEE Trans. Power Syst., 3(2): 763-773, 1988. X. Guang, P.B. Luh, and H. Yan. An optimization-based method for unit commitment. Electrical Power Energy Syst., 14(1): 9-17, 1992.

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6. 7.

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13. 14. 15. 16. 17.

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J.A. Muckstadt and S.A. Koenig. An application of Lagrangian relaxation to scheduling power-generation systems. Oper. Res., 25(3): 387-403, 1977. F. Pellegrino, A. Renaud, and T. Socroun. “Bundle and augmented Lagrangian methods for short-term unit commitment.” In Power Systems Computation Conference Proc., pp. 730-739, 1996. N. Jiménez and A.J. Conejo. Short-term hydro-thermal coordination by Lagrangian relaxation: solution to the dual problem. IEEE Trans. Power Syst., 14(1): 89-95, 1999. P.B. Luh, D. Zhang, and R.N. Tomastik. An algorithm for solving the dual problem of hydrothermal scheduling. In IEEE-PES, Winter Meeting paper PE-333-PWRS-0-12-1997, New York, 1997. C. Lemarechal and J Zowe. “A Condensed Introduction to Bundle Methods in Nonsmooth Optimization.” In Algorithms for Continuous Optimization, pp. 357-382, ed. E. Spedicato. Kluwer Academic Publishers, 1994. O. du Merle, J.L. Goffin, C. Trouiller, and J.P. Vial. A Lagrangian relaxation of the capacitated multi-item lot sizing problem solved with an interior point cutting plane method. Technical report, Faculty of Management, McGill University, 1997. J.L. Goffin, J. Gondzio, R. Sarkissian and J.P. Vial. Solving nonlinear multi-commodity flow problems by the analytic center cutting plane method. Math. Prog., (76): 131-154, 1996. M. Madrigal and V.H. Quintana. “An Interior-point/Cutting-plane Algorithm to Solve Unit Commitment Problems.” In IEEE-PES Power Industry Computer Applications Conference Proc., pages 179-185, Santa Clara, California, 1999. To appear in IEEE Trans. Power Syst. R.B. Johnson, S.S. Oren, and A.J. Svodoba. Equity and efficiency of unit commitment in competitive electricity markets. Technical Report PWP-039, POWER-series, The University of California Energy Institute, 1996. S. Dekrajangpetch, G.B. Sheble, and A.J. Conejo. Auction implementation problems using Lagrangian relaxation. IEEE Trans. Power Syst., 14(1): 82-88, 1999. J.M. Jacobs. Artificial power markets and unintended consequences. IEEE Trans. Power Syst, 12(2): 968-972, 1997. C.L. Tseng. On Power Systems Generation Unit Commitment Problems. Ph.D. Thesis, University of California, Berkeley, 1996. D.P. Bertsekas. Nonlinear Programming. Athena Scientific, 1997. M. Madrigal and V.H. Quintana. “Semi-definite Programming Relaxations for {0,1}Power Dispatch Problems.” In IEEE-PES, 1999 Summer Meeting Conference Proc., pp. 697-702, Edmonton, Alberta, Canada, 1999. O. Bahn, J.L. Goffin, J.P. Vial, and O. Du Merle. Experimental behaviour of an interior point cutting plane algorithm for convex programming: an application to geometric programming. Discrete Appl. Math, 49: 3-23, 1994. J.L. Goffin and J.P. Vial. Interior point methods for nondifferentiable optimization. Technical reports, Faculty of Management, McGill University, 1997. M. Kojima, N. Megiddo, and S. Mizuno. A primal-dual infeasible-interior-point algorithm for linear programming. Math. Prog, (61): 263-280, 1993. S. Sen and D.P. Kothari. Optimal thermal generating unit commitment: a review. Electrical Power Energy Syst., 20(7): 443-451, 1998. M. Madrigal and V.H. Quintana. “Using Optimization Models and Techniques to Implement Electricity Auctions.” In IEEE-PES, 2000 Winter Meeting Conference Proc., Singapore, 2000.

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23. S.J. Wang, S.M. Shahidehpour, D.S. Kirschen, S. Mokhtari, and G.D. Irisarri. Short-term generation scheduling with transmission and environmental constraints using augmented Lagrangian relaxation. IEEE Trans. Power Syst., 10(3): 1294-1301, 1994. 24. F.C. Schweppe, M.C. Caramanis, R.D. Tabors and R.E. Bohn. Spot Pricing of Electricity. Kluwer Academic Publishers, 1987.

Chapter 11 BUILDING AND EVALUATING GENCO BIDDING STRATEGIES AND UNIT COMMITMENT SCHEDULES WITH GENETIC ALGORITHMS

Charles W. Richter Jr. and Gerald B. Sheblé Iowa State University

Abstract:

1.

Far from being an artifact of the past, the unit commitment (UC) algorithm is essential to making economical decisions in today’s competitive electricity industry. Increasing competition; decreasing obligations-to-serve; and enhanced futures, forwards, and spot market trading in electricity and other related markets make the decision of which units to operate more complex than ever before. Decentralized auction markets currently being implemented in countries like Spain use UC-type models, which should encourage researchers to continue working on finding better and faster solution techniques. UC schedules may be developed for a generation company, a system operator, etc. The need for many flavors of UC algorithms, each considering different inputs and objective functions, is growing. Factors such as historical reliability of units should be considered in designing the UC algorithm. Although a particular schedule may result in the lowest cost, or highest profit, it may depend on generators that have varying availabilities. Traditionally, consumers had very reliable electricity whether they needed it or not. Given a choice in a market-based electricity system, many consumers might choose to pay for a slightly lower level of power availability if it would result in sufficient savings. As the number of inputs and options grows in UC problems, the genetic algorithm (GA) becomes an important tool for searching the large solution space. GA times-to-solution often scale up linearly with the number of units, or hours being considered. Another benefit of using the GA to generate UC schedules is that an entire population of schedules is developed, some of which may be well suited to situations that may arise quickly due to unexpected contingencies.

INTRODUCTION

Electric generators and energy service companies around the world are presently embracing competition. Generation companies and energy service companies can negotiate profitable electricity contract prices bilaterally or via auction markets, which will likely play a vital role in price discovery. Auction

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market bidding strategies will be important for participants’ profitability and survival. No longer guaranteed consumers nor a rate-of-return by a public utilities commission (PUC), it is the responsibility of each generation company (GENCO) to operate units profitably. Good bidding strategies must allow the trader to negotiate profitable contracts, not only in the short-term, but also in the mid- to long-term. Providing a basic foundation for effective bidding strategies, the unit commitment (UC) algorithm will remain one of the central tools in the competitive electricity industry. The business of GENCO is ultimately to generate electricity. The changes brought about by deregulation (e.g., increases in competition, decreases in the utility's obligations-to-serve, enhanced futures, forwards, and spot market trading in electricity and other related markets) make the decision of which units to operate more complex than ever before. The need for many varieties of UC algorithms is growing. UC schedules may be developed and optimized for a particular generation company or for a system operator. When choosing a UC schedule to the objective function may consider a host of factors, such as historical availability of units or transmission lines, the credit risk of dealing with a particular consumer, etc. Although a particular schedule may yield the lowest cost or greatest profit, consideration of historical generator availability may reduce expected profits or increase expected costs. Historically, consumers were provided with (and paid for) very reliable electricity regardless of their requirements. Given a choice in a market-based electricity system, many consumers would choose to pay for a slightly lower level of power availability if it resulted in substantial savings. Allowing the trade of electricity with varying reliability levels in a market environment will require improvements in allocation algorithms including UC algorithms. As the number of choices and contract-types grow, the UC schedule search space becomes large. The genetic algorithm (GA) is an important tool in searching large solution spaces. GA times-to-solution often scale up linearly with the number of units or the number of hours being considered. Another strength that GAs have over other solution techniques is that entire populations of solutions are developed in parallel. While GAs work well for searching large spaces, there is no proof that they are converging to the optimal solution. Because entire populations are being evolved during each run, complex fitness evaluations may require much computation. If a trader is participating in a market, he wants to take as much profit as he can. The auction market may be viewed as a game for which various strategies exist. How does the GENCO use its UC schedules and bidding strategies in a manner that will result in long term profit? The procedure begins by developing a working model of the competition. Since generating unit characteristics don’t change fundamentally in the move from monopolistic

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operation to competitive operation, information made public during regulated operation remains valid and can help develop a basic competitor model. Additional information influencing the competitor is also public, e.g., price of fuel. These basic competitor models can be used to simulate future (e.g., for T hours into the future) trading. The models allow study of strategic behavior that may affect how close the price is to the price achieved under perfect competition. Any additional useful information should be considered in determining the best forward price and demand forecasts. A price-based UC algorithm uses these forecasts when searching for the weekly schedule that maximizes expected profit, subject to some risk level. The schedule(s) should be analyzed under various contingencies. Candidate UC schedules should be used as an input to the bidding strategy development process. Figure 1 shows an outline of this procedure in block diagram form.

This chapter gives a brief introduction to a basic market framework to experimental economics. (The reader should be cautioned that this is a representative market and is not identical to existing markets.) We discuss issues related to determining competitor models, and we present methods used to simulate the various markets (spot and forward). Our work investigates the models’ impact on GENCO operations and planning on daily and weekly operations. We highlight creation of bidding strategies for single and multiperiod auctions and discuss how forwards and futures markets provide a

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means to manage risk. We present a price-based unit commitment formulation, followed by a unit commitment genetic algorithm (UC-GA) formulation. Additional criteria (e.g., considering contingencies such as non-availability of generators, network problems, unforeseen market disturbances) may highlight differences in UC schedules that have equivalent cost and profit. The authors discuss each procedure to compare UC schedules with each other based on their expected monetary value under likely scenarios or contingencies. Adaptive agents modelling buyers and sellers test the developed strategies by trading electricity in a simulated deregulated electricity market.

2.

MARKETS AND EXPERIMENTAL ECONOMICS

2.1

A Basic Market Framework

Many countries and some regions of the United States (e.g., California, Pennsylvania-Jersey-Maryland (PJM), New England, etc.) have deregulated. There does not yet appear to be a standardized final market structure that will be adopted for all areas, but each market developed should be an improvement over those previously developed. The framework described here is one in which electric energy is produced by GENCOs, sold to energy service companies (ESCOs), and delivered on wires owned by distribution companies (DISTCOs) and transmission companies (TRANSCOs). An entity such as the North American Electric Reliability Council (NERC) sets the reliability standards. The contract prices are discovered in an auction. Buyers and sellers of electricity make bids and offers that are matched subject to the approval of the independent contract administrator (ICA), who ensures that the contracts will result in a system operating safely within limits. The ICA submits information to an independent system operator (ISO) for implementation. The ISO is responsible for physically controlling the system to maintain its security and reliability. Participants are provided with average contract price and volume, last accepted bid or offer, and whether the participant’s previous bid was accepted or rejected. Most of the market framework assumed here has been developed in previous publications [1,2,3,4]. It allows for cash (spot and forward), futures, and planning markets as shown in Figure 2. The spot market is a market where sellers and buyers negotiate (either bilaterally or multilaterally through an exchange) a price for a certain number of MWh to be delivered in the near future (e.g., 30 MWh – 10 MW from 1:00 p.m. to 4:00 p.m. tomorrow). The buyer and seller must arrange a transmission path for the electrons. The dayahead market in California is an example of a spot market. The forward market allows trading in a manner similar to the spot market but is further into the

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future than is the spot market. In both forward and spot markets, the buyer and seller intend to exchange the physical good (i.e., the electrical energy). In contrast, the futures market is primarily financial, allowing traders to reduce uncertainty by locking in a price for a commodity in some future month. The provision for physical delivery exists, but since it is not normally intended, other entities (e.g., TRANSCOs, ICAs, and ISOs) need not be informed of futures trading. Buying a futures contract is akin to purchasing insurance. It allows the traders to manage their risk by limiting potential losses or gains. To ensure sufficient interest for price discovery, futures contracts are generally standardized such that it is not possible to tell one unit of the good from another (e.g., 1 MWh of electricity of a certain quality, voltage level, etc.). Although provisions for delivery exist, they are generally not convenient. The trader ultimately cancels his position in the futures market either with a gain or loss. The physical goods are then purchased on the spot market to meet demand with the profit or loss having been locked-in via the futures contract. The planning market aids in financing long-term projects like transmission lines and power plants.

2.2

Experimental Economics

Economic theories are often based on several assumptions that tend to be true when looking at society as a whole, but the theories may not necessarily ring true for a particular agent or group of agents. The data used to substantiate these theories often comes from indicators observed in the real economy. It is hard to isolate the effect of a change in one input on one output in the real world economy, because many things are changing simultaneously. Experimental economics seeks to advance the theories of economics through ex-

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perimentation. Pioneers such as Charles Plott and Vernon Smith popularized the field through their work using the auction as an allocation mechanism. In general, researchers study agents in a laboratory setting where their behavior is monitored as they take part in experiments. Although initial experiments were with humans, intelligent computerized agents are faster, cheaper, and more consistent than human agents in the laboratory environment. Drawing from the field of complex adaptive systems/artificial intelligence, computerized agents can be given the ability to learn and to develop sensible GENCO bidding strategies. (Figure 1 shows that simulating auction markets is one of the first steps to validate and fine-tune the forward price and demand curves.) The simulated markets (described later) are populated with computerized trading agents (seeded with GENCOs and ESCOs models) to obtain forecasts that account for the strategic behaviors of the competitors.

3.

DETERMINING THE COMPETITOR MODEL

If the auction simulations are going to produce any results that are to be used in actual GENCO operation, the models of the competing ESCOs and GENCOs need to be reasonably accurate. Developing competitor models is an involved process. Under the market framework assumed for this research, the results of the auctions are public information, similar to the Australian electric power market. As indicated in Figure 3, if it is available, a database of auctions from previous periods contains bidding information and can be intelligently mined to determine the general rules that the competitors are using. This information can be combined with any additional information known about the system or about the competitors to develop a fine-tuned model of the competing computerized agents in the simulated markets [5]. The competitor models populate computerized agents participating in simulated auctions for each period of interest. The competitor models may represent a single aggregated GENCO competitor and a single aggregated ESCO, or many independent GENCOs and ESCOs. Important information (e.g., weather, unit outage information, status of transmission system, and time of day) that will influence the bidding process is used to fine-tune the competitor model as shown in Figure 4.

Using Genetic Algorithms in GENCO Strategies and Schedules

4.

191

SIMULATING AN HOURLY AUCTION MARKET

The procedure described here is loosely based on the auctions seen at the Chicago Board of Trade. To prevent an infinite loop, we select a maximum number of cycles per round and maximum number of rounds per period. Buyers and sellers determine how much they would like to buy and sell. They determine their bids and offers and submit them to the auctioneer. All transactions are subject to the approval of the ICA. Price discovery occurs when there are a sufficient number of buy bids and sell offers to allow a predetermined portion of the total participants to be satisfied with the resulting contract. Submission of the bids marks the beginning of the bidding cycle. When

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the auctioneer reports the results of the auction to the market participants the cycle is complete. If, after the present cycle, the price has not been discovered, the auctioneer reports that price discovery did not occur and asks for new bids and offers. Market rules (funnel rules) force subsequent cycles closer to price discovery by requiring the buyers to increase their bids and sellers to decrease their offers. The cycles continue until price discovery occurs, or until the auctioneer decides to match whatever valid matches exist and continue to the next round or hour of bidding. Figure 5 shows this process.

5.

FORECASTING PRICES AND DEMANDS

For each auction market period, it is possible to determine a curve relating the price to the quantity demanded and a curve relating the price to the quantity supplied. In the cost-based world of the past, the forward price curve for an individual hourly market was simply a horizontal summing of GEN-

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COs’ cost curves, which were public information. The forward price curve in the competitive world will look fairly similar. In the price-based competitive world, however, GENCOs no longer have to reveal their true cost. This information can be approximated from the forward and futures trading. Models for generating unit output costs are relatively consistent from year-to-year; so much of the information is publicly available from historical filings made with regulators. Some uncertainty comes with not knowing the consumer demand, which is highly dependent on the weather. We can use neural networks and statistical techniques to obtain demand forecasts. These price and demand forecasts are given to the unit commitment scheduler, which attempts to find the schedule of generating units that maximizes utility. For simplification, the UC algorithm described in the next section uses a single expected price and quantity for each period. A UC schedule is typically developed for each hour of the following week. Each period the latest forecasts and market price are given to the algorithm and updated schedule is developed.

6.

PRICE-BASED UNIT COMMITMENT

In an environment with vertically integrated monopolies, the scheduling of generating units to be on, off, or in stand-by/banking mode is done in a manner that minimizes costs. Consideration must be given to factors like varying fuel costs, start-up and shut-down parameters/constraints of each power plant, and crew constraints. In order to determine the cost associated with a given schedule, an economic dispatch calculation (EDC), in which all unconstrained operating units are set so that their marginal costs are equal, must be performed for each hour under consideration. One possible way to determine the optimal schedule is to do an exhaustive search. Exhaustively considering all possible ways that units can be switched on or off for a small system can be done, but for a reasonably sized system the amount of time it would take is too long. Solving the problem generally involves using methods like Lagrangian relaxation, dynamic programming, genetic algorithms, or other methods using heuristic search techniques. Many references for the traditional UC can be found in Sheblé and Fahd [6] and in Wood and Wollenberg [7]. Without competition, demand forecasts of total consumer demand in the service territory advised power system operators of the amount of power that needed to be generated. Under competition, bilateral spot and forward contracts make part of the total demand known a priori. Predicting the GENCO’s share of the remaining demand may be difficult, however, since it is dependent on its prices compare to that of other suppliers, and the condition of the

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transmission network, among other factors. Since the number of units switched on or in banking mode affects the average cost of production, the UC schedule indirectly affects the price, making it an essential input to any successful bidding strategy. Utilities operating in regulated monopoly fashion have an obligation to serve their customers within a designated service territory. This translates into a demand constraint that ensures all demand is met. For the UC problem, this might mean switching on an additional unit to meet a remaining MW or two. Without an obligation to serve, the competitive GENCO can consider a schedule that produces less than the predicted demand. It can allow other suppliers to provide that one or two MWs that might have increased average costs (they might not have secured that contract for which they would have had to compete). Demand forecasts and expected market prices are an important input to the profit-based UC algorithm; they are used to determine the expected revenue, which affects the expected profit. If a GENCO comes up with two potential UC schedules each having different expected costs and different expected profits, it should take the one that provides for the greatest expected benefit to cost ratio, which will not necessarily cost least. Since price and demand are so important in determining the optimal UC schedule, price prediction and demand forecasts become crucial. Mathematically the traditional cost-based UC problem has been formulated as follows [8]:

Subject to the following constraints:

The cost based UC algorithm for the regulated monopolist has been well researched. Recently, research has included the use of markets in developing UC schedules. Takriti, Krasenbrink, and Wu [9] present a good description and a stochastic solution of the UC problem that considers spot markets. Their research differs from that presented here in that it minimizes costs rather than maximizes profits. Here, we redefine the UC problem for the GENCO operating in the competitive environment by changing the demand

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constraint from an equality to less than or equal (assume buyers are required to purchase reserves per contract) and changing the objective function from cost minimization to profit maximization:

Subject to:

Where individual terms are defined as follows: (Capacity limits) (Ramp rate limits) up-/down-time status of unit n at time period power generation of unit n during time period t load level in time period t forecasted demand w/ reserves for period t forecasted price for period t system reserve requirements in time period t production cost of unit n in time period t start-up cost for unit n, time period t shut-down cost for unit n, time period t maintenance cost for unit n, time period t number of units number of time periods generation low limit of unit n generation high limit of unit n maximum contribution to reserve for unit n Since a GENCO is not obligated to serve under competition, it may choose to generate less than the total consumer demand. Therefore, maximizing the profit is not the same as minimizing the cost. Choosing the amount of demand to serve allows the GENCO more flexibility in developing the UC schedules. Because prices fluctuate with supply and demand, engineers often assume that making the load curve level minimizes cost and should be a reasonable operating strategy. When maximizing profit, the GENCO may find that during certain conditions greater profit may be possible under a non-level load curve. If revenue increases more than the cost does, the profit will increase. Figure 6 shows a block diagram of the UC solution process. EDC is an important part of UC. Formerly used to minimize costs, it is necessary to redefine EDC for price-based operation. Where the costminimizing EDC ignored transition and fixed costs to adjust the power level

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of the units until they each had the same incremental cost our new EDC attempts to set equal to a pseudo price (i.e., produce until the marginal cost equal the price). This pseudo price accounts for transition and fixed costs as shown in the following formula:

Transition costs include start-up, shut-down, and banking costs. Fixed costs (present for each hour that the unit is on), are represented by the constant term in the typical quadratic cost curve approximation. During each period, the fixed and transition costs are accounted for by adding an average value to the incremental cost. Note that this is one method of allocating the transition and fixed costs, but there are many others that could be used. For instance, if GENCO A’s generators are able to produce electricity far less expensively than the competing GENCOs during the night, but don’t have that advantage during the daytime, GENCO A could shift some of its daytime costs to be recovered through bids from the night-time periods.

7.

A GENETIC-BASED UC ALGORITHM

7.1

The Basics of Genetic Algorithms

A genetic algorithm is a search algorithm often used in non-linear discrete optimization problems. A population of data structures appropriate for the problem solution is initialized randomly, evolves over time, and finds a suitable answer (or answers) to the problem. The data structures often consist of strings of binary numbers, which are mapped onto the solution space for evaluation. Each solution (often termed a creature) is assigned a fitness - a heuristic measure of its quality. During the evolutionary process, those creatures having higher fitness are favored in the parent selection process and are allowed to procreate. The parent selection is essentially a random selection with a fitness bias. The parent selection method determines the type of fitness bias. Following the parent selection process, the processes of crossover and mutation are utilized and new creatures are developed which ideally explore a different area of the solution space. These new creatures replace less fit creatures from the existing population.

Using Genetic Algorithms in GENCO Strategies and Schedules

7.2

197

GA Formulation for Price-Based UC

The algorithm presented here solves the UC problem for the profitmaximizing GENCO operating in the competitive environment [10]. Various GA formulations for finding optimal cost-based UC schedules have been proposed by researchers [11, 12]. The profit or price-based algorithm presented here is a modification of a genetic-based UC algorithm for the costminimizing monopolist that was described by Maifeld and Sheblé [13]. The fitness function now rewards schedules that maximize profit. The intelligent mutation operators are preserved in their original form. The schedule format is the same. Figure 7 shows the algorithm. The algorithm first reads in the contract demand and prices, the forecast of remaining demand, and forecasted spot prices. During the initialization step, a population of UC schedules is randomly initialized (see Figure 8). For each UC period of each member of the population, EDC is called to set the units’

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generation levels. The cost of each schedule is determined from the parametric generator data and the demand and price data read at the beginning of the program. Next, the fitness (i.e., the profit) of each schedule in the population is calculated. “Done?” checks to see whether the algorithm has either cycled through for the maximum number of generations allowed, or whether other stopping criteria have been met. If the algorithm is done, the results are written to a file; if it is not done, the algorithm proceeds to the reproduction process.

During reproduction, the algorithm creates new schedules. The first step of reproduction is to select parents from the population. After selecting parents, children are created using two-point crossover as shown in Figure 9. Following crossover, standard mutation is applied. Standard mutation involves turning a randomly selected unit on or off within a given schedule.

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An important feature of this UC-GA is that it spends as little time as possible doing EDC. After standard mutation, EDC is called to update the profit only for the mutated hour(s). An hourly profit number is maintained and stored during the reproduction process, which dramatically reduces the amount of time required to calculate the profit over what it would be if EDC had to work from scratch at each fitness evaluation. In addition to the standard mutation, the algorithm uses two “intelligent” mutation operators that work by recognizing that, because of transition costs and minimum up-time and down-time constraints, 101 or 010 combinations are undesirable. The first

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of these operators purges these undesirable combinations by randomly changing 1s to 0s or vice versa. The second of these intelligent mutation operators purges the undesirable combination by changing 1 to 0 or 0 to 1 based on which of these is more helpful to the profit objective.

7.3

Price-Based UC-GA Results

We tested the UC-GA on some small systems and the results were compared to the solutions found by exhaustive search. In all of the trials for which known optima existed, the GA was successful in locating the optimal UC schedule. Figure 10 shows the costs and average costs (without transition costs) of the 10 generators, as well as the hourly price and load forecasts for the 48 hours. We chose the data so that the optimal solution was known a priori. The dashed line in the load forecast represents the maximum output of the 10 units. In addition to the information shown in the figure, the UC-GA program requires the start-up and shut-down costs, the minimum up and down times, and the cost to bank each generator. The generators are modelled with a quadratic cost curve (e.g., , where P is the power level of the unit). Though the GA took 730 seconds to find a population of solutions containing the best possible solution for a 10-unit, 48-hour case (see Figure 11). Prior to initiating the UC-GA, the control parameters shown in Table 1 are specified, including the “Number of Units” and Number of Hours” to consider. “Popsize” is the size of the GA population. Increasing the population size increases the amount of space searched during each generation of the GA. This must be balanced with the consideration that execution time for each generation of the algorithm varies approximately linearly with the size of the popsize. The number of “Generations” indicates the number of times the GA will undergo the selection and reproduction. “System reserve” is the percentage of reserves that the buyer must maintain for each contract. “Children per generation” tells us how much of the population will be replaced each generation. Changing this can affect the convergence rate. If there are multiple optima, faster convergence may trap the GA in local sub-optimal solution. “UC schedules to keep” indicates the number of evolved schedules to write to file. There is also a “random number seed” that may be set between 0 and 1.

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The algorithm accurately calculates the cost of schedules in which minimum up- and down-time constraints appear to be violated by considering a zero surrounded by ones to be a banked unit, and a one surrounded by zeros is ignored (unit remains off) if it violates the minimum up constraint. An advantage of using the GA is that its solution time scales up only linearly with the number of hours and units, while dynamic programming quickly becomes too computationally expensive to solve. The existence of additional

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valid solutions, which may be only slightly sub-optimal in terms of profit, is another main advantage of using the GA. It gives the system operator the flexibility to choose the best schedule from a group of schedules to accommodate things like forced maintenance.

8.

COMPARING AND SELECTING UC SCHEDULES

Even though a large percentage of the UC schedules encountered by the genetic algorithm (or other search technique) may satisfy (within some small tolerance) the primary objective function, they may not be equal. A set of UC schedules may initially have indistinguishable costs or profits, but when we consider additional criteria, differences between the schedules may be revealed. A few examples of additional criteria might be: Impact of units or transmission system availability Ability to respond to spot-market price fluctuations Schedule’s profitability during network contingencies Ability to accommodate maintenance activities A unit that is unavailable likely reduces GENCO profitability. A unit may be unavailable due to a unit outage (scheduled maintenance or forced), the transmission line connecting it to the load may be congested, or other unforeseen circumstances may exist that prevent the GENCO from selling electricity. The amount of time that a generating unit is forced out of the market is often unpredictable and variable. If a unit undergoes a forced outage or is taken off-line for other reasons, costs such as shut-down (and subsequent start-up) must be recovered. These unanticipated costs may have a large impact on the profitability of the UC schedule. Although reserves may mitigate

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the consequences of a single unit outage, a possibility exists that many units may be inoperable or at reduced capacity simultaneously, since independent unit outages may occur, as well as contingencies promoting system-wide disturbances. Therefore, the possibility that more than one unit is forced off-line in a given period of time must be considered. We can use historical availability of generating units and of the transmission system itself to differentiate between the UC schedules under consideration. The spot market prices may undergo short-term unanticipated changes that could be quite profitable for the GENCO having a UC schedule allowing the amount of power to be increased or decreased easily (i.e., without turning on or off additional units). A schedule’s performance under various contingencies may distinguish it from others. While searching for the optimal UC schedule, certain network conditions, unit availabilities, load and price forecasts may have been assumed. Contingencies will impact some of the candidate schedules more than others. The ability to schedule maintenance activities may be a characteristic of schedules that distinguishes them from each other. Perhaps two schedules result in roughly the same amount of profit, but one of the schedules allows for preventive maintenance activities on some key units.

9.

TESTING THE UC SCHEDULE IN SIMULATED COMPETITION

9.1

Intelligent Bidding Strategies

Once the tentative UC schedule(s) is developed, the GENCO should have bidding strategies that guide it in placing bids and in taking market positions. These strategies might be designed to limit risk, maximize profit, a combination of both, or something entirely different. Intelligent strategies can detect various market scenarios and respond appropriately (i.e., profitably). Because the number of scenarios that the agent might encounter is extremely large, discrete, and non-linear, finding an optimal bidding strategy is a problem naturally suited for genetic algorithms. Genetic algorithms and genetic programming have been used to evolve bidding strategies that maximize profit for individual hourly spot markets [2, 14]. Representing the agent’s strategy in a easy-to-use and easy-to-evolve format is sometimes the most difficult aspect of GA application. Fixed binary strings may be the simplest method of encoding a problem. Finite state machines/automata offer a more powerful means of encoding the strategies, but might be rather large to encode an independent response for each scenario

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encountered. The authors use GP-Automata to circumvent this problem. We test the evolved bidding strategies are tested by repeatedly using them in simulated trades against competing strategies. Before we present the basics of GP-Automata, here is a brief introduction of genetic programming.

9.2

The Basics of Genetic Programming

A sub-class of genetic algorithms, genetic programming (GP) is a new discipline attributed to John Koza [15, 16]. Although not for the electricity market, Andrews and Prager published research indicating that GP works for representing simple double auction market strategies [19]. The evolving data structures in GP are “parse trees” which allow complex relations to be described. Genetic programs (GPs) contain nodes and branches, with branches connecting the nodes. Nodes can be either operational nodes, having arguments and performing operations involving those arguments, or terminal nodes. Figure 12 provides examples of randomly generated GP-trees. The tree on the right side of the figure would return the average of five plus the average buy bid from the previous round of bidding. The tree on the left would multiply 10 by the absolute value of the high buy bid, and then (inefficiently) take the absolute value of the result.

The designer specifies the set of valid operators and terminals suitable to the problem being investigated. For instance, in developing bidding strategies, suitable operators and terminals might be those described in Table 2. In designing GPs for the GP-Automata, it is desirable to give the trees an opportunity to return numbers in the range of competitive bids.

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Valid GP trees are initialized randomly and then evolved in a standard genetic algorithm (as described in the previous section) with the following modifications. The crossover of two parents involves randomly selecting a node from each parent and swapping the sub-trees rooted at those nodes. Mutation involves randomly selecting a node in the candidate child and throwing away its sub-tree. In its place a new sub-tree is generated randomly.

9.3

The Basics of GP-Automata

GP-Automata are a combination of finite state automata and GP. They were first described as such by Ashlock [17] and were used by Ashlock and Richter [18]. The typical finite state automaton specifies an action and “next state” transition for a given input or inputs. With only one or two binary inputs to work with, it can be fairly simple to develop a finite state diagram to cover the possible input/output relations. When the number of inputs is large the task is much harder. The number of transitions needed to cover all possible combinations of inputs grows exponentially (e.g., 10 inputs, each having five possible values would require transitions). This is where genetic programming comes in. The GP-trees perform bandwidth compression for the GP-Automata by selecting which inputs to consider and performing computations involving these inputs. The four-state GP-Automaton in Figure 12 begins by bidding the number in the “Initial Action” field (in this case, 24). The “Initial State” tells us which state is used next (in this case, 2). Coupled with each of these states is a GP-

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The Next Generation of Unit Commitment Models

tree, termed a “Decider.” When executed, the decider returns a value in the valid range of bidding. Following the decider evaluation, one of the following two things will happen: (a) if the value is even after truncation, the action listed under “IF EVEN” is taken and the current state becomes the one listed under the “IF EVEN” next state; or (b) if the returned value is odd after truncation, then the action and next state listed under “IF ODD” is used. The “Action” is the number listed in the action field of the automaton, with two exceptions. The first exception is the “U” that indicates that the value returned by the decider should be taken directly as the action. The second exception is a “*” indicating further computation is necessary – the GPautomaton immediately moves to the next state. This gives rise to the possibility of complex (multi-state) computation. To prevent infinite loops, after an externally specified maximum number of *s have been processed, an action is selected at random from the valid actions.

A population of GP-Automata bidding strategies evolves in a GA. Fitness is dependent on the goal of the strategies. Children/offspring are produced using crossover and mutation. Crossover for the GP-Automata involves selecting (with a uniform probability) a crossover point ranging from zero to the number of states. Then parentl’s states from zero to the crossover point are copied to childl and parent2’s states are copied to child2. Following the crossover point, childl gets parent2’s state information and child2 gets parentl’s state information (including the associated decider). Before replacing less fit members of the population, each child is subjected to mutation. Mutation may be standard GA mutation that selects a state or action at random and replaces it with a valid entry. Other forms of mutation are acceptable as well. The goal is to introduce new combinations of genetic material into the population.

Using Genetic Algorithms in GENCO Strategies and Schedules

9.4

207

Auction Bidding with the GP-Automata Strategies

While they are evolving in the genetic algorithm, each GP-Automaton’s strategy competes against several other GP-Automata every generation (see Figure 14). This helps to ensure that the resulting GP-Automata strategies will be robust. Based on the competition, some fitness measure (e.g., the expected amount of profit that results from using the strategies to win contracts) is assigned to each GP-Automaton. Through the natural selection process, the population evolves, finding strategies that are more likely to achieve good fitness.

State information is supplied to the GP-Automata via the terminals. The GP-trees use both the state information accessed by the terminals as well as constants in the valid bidding range. Bids are taken from the action cell of the automata, except in the cases where the action is listed as a * or a U, as described previously. The bids are submitted, along with the bids from the competing sellers and buyers, to the auctioneer for evaluation. The bids and offers are matched and a would-be price is reported, completing one cycle of the auction. The cycles continue until price discovery occurs or until some maximum number of cycles (maxcycles) has passed. There is a maxcycles parameter, which is selected uniformly over a range to prevent the strategies from

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falling in a local optima where the strategies only work well when the number of cycles never changes over the trials in a given generation. An evolved GP-Automaton contains trained rules, which may be quite complex. These rules may be used directly in a real auction just as they were used in the simulated auction during evolution.

10.

SUMMARY

The GENCO’s business is still one of generating electricity, and it must ultimately determine a profitable schedule to operate its generating units. Thus, the UC algorithm will continue to be an important tool in the evolving industry. While the ICA may minimize total costs when matching bids, the GENCO must maximize its profit. GENCOs must make decisions based on market projections. Customers and demand may no longer be guaranteed, but bilateral and multilateral forward contracts will ensure that the GENCO knows much of its load ahead of time. Accurate forecasts of the quantity demanded and prices are crucial when solving the UC problem. If a GENCO’s market projections are incorrect, the UC schedule may no longer be optimal. Flexible schedules and bidding strategies are important. With huge potential losses/profit at stake, UC schedules should be tested before use. Intelligent agents with evolvable strategies provide realistic competitive behavior and are thus ideal for robustness testing. GAs have demonstrated an ability to learn and to build and adapt UC and bidding strategies for given scenarios.

REFERENCES 1.

2.

3. 4. 5. 6. 7.

J. Kumar and G. Shebté. “Framework for Energy Brokerage System with Reserve Margin and Transmission Losses.” In Proc. 1996 IEEE/PES Winter Meeting, 96 WM 190-9 PWRS, NY: IEEE. C. Richter and G. Sheblé. “Genetic Algorithm Evolution of Utility Bidding Strategies for the Competitive Marketplace.” In Proc. 1997 IEEE/PES Summer Meeting, PE-752PWRS-1-05-1997. New York: IEEE. G. Sheblé. “Electric energy in a fully evolved marketplace.” Paper presented at the 1994 North American Power Symposium, Kansas State University, KS, 1994. G. Sheblé. “Priced based operation in an auction market structure.” Paper presented at the 1996 IEEE/PES Winter Meeting. Baltimore, MD, 1996. C. Richter. Profiting from Competition: Financial Tools for Competitive Electric Generation Companies. Ph.D. dissertation, Iowa State University, Ames, IA, 1998. G. Sheblé and G. Fahd. Unit commitment literature synopsis. IEEE Trans. Power Syst., 9(1): 128-135, 1994. A. Wood and B. Wollenberg. Power Generation, Operation, and Control. New York: John Wiley & Sons, 1996.

Using Genetic Algorithms in GENCO Strategies and Schedules 8. 9.

10. 11. 12.

13. 14.

15. 16. 17. 18.

19.

209

G. Sheblé. Unit Commitment for Operations. Ph.D. Dissertation, Virginia Polytechnic Institute and State University, 1985. S. Takriti, B. Krasenbrink, and L.S.-Y. Wu. “Incorporating Fuel Constraints and Electricity Spot Prices into the Stochastic Unit Commitment Problem,” IBM Research Report: RC 21066, Mathematical Sciences Department, T.J. Watson Research Center, Yorktown Heights, NY, 1997. C. Richter and G. Sheblé. “A Price-Based Unit Commitment GA for Uncertain Price and Demand Forecasts.” In Proc. 1998 North American Power Symposium, 1998. S. Kondragunta. Genetic algorithm unit commitment program, M.S. Thesis, Iowa State University, Ames, IA, 1997. S. A. Kazarlis, A. G. Bakirtzis, and V. Petridis. “A Genetic Algorithm Solution to the Unit Commitment Problem.” In Proc. 1995 IEEE/PES Winter Meeting 152-9 PWRS, New York: IEEE, 1995. T. Maifeld and G. Sheblé. Genetic-based unit commitment. IEEE Trans. Power Syst., 11(3): 1359, 1996. C. Richter, D. Ashlock, and G. Sheblé. “Effects of Tree Size and State Number on GPAutomata Bidding Strategies.” In Proc. 1998 Conference on Genetic Programming, Denver, CO: Morgan Kaufmann, 1998. J. Koza. Genetic Programming. Cambridge, Massachusetts: The MIT Press, 1992. J. Koza. Genetic Programming II. Cambridge, Massachusetts: The MIT Press,1994. D. Ashlock. “GP-automata for Dividing the Dollar.” Mathematics Department, Iowa State University, Ames, IA 1995. D. Ashlock and C. Richter. “The Effects of Splitting Populations on Bidding Strategies.” In Proc. 1997 Conference on Genetic Programming, Denver, CO: Morgan Kaufmann, 1997. M. Andrews and R. Prager. “Genetic programming for the acquisition of double auction market strategies.” In Advances in Genetic Programming, K. Kinnear Jr., ed. Cambridge, MA: The MIT Press, 1994.

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Chapter 12 AN EQUIVALENCING TECHNIQUE FOR SOLVING THE LARGE-SCALE THERMAL UNIT COMMITMENT PROBLEM Subir Sen Power Grid Corporation of India, Ltd.

D.P. Kothari Indian Institute of Technology at Delhi

Abstract:

1.

This chapter presents a new efficient solution approach for solving the unit commitment schedule of thermal generation units of a realistic large scale power system. We base the approach on cardinality reduction by the generator equivalencing concept. This concept reduces the number of units in the large-scale power system to the lowest possible number based on the units’ fuel/generation cost and other physical characteristics, such as minimum up and down time, etc. with units having similar (almost the same) characteristics form one group. The reduced system consists of only each group of representative units and is first solved by the modified dynamic programming technique (one of the new solution methods developed by the authors). Another option is to use any of the standard unit commitment solution techniques. We obtain the overall solution to the original unit commitment problem of the entire system by un-crunching the solved reduced system based on certain rules. This chapter also presents test results for real-life systems of up to 79 units and comparisons with results obtained using Lagrangian relaxation and truncated dynamic programming (DP-TC).

INTRODUCTION

One of the most important problems in operational scheduling of electrical power generation is the unit commitment (UC) problem. It involves determining the start-up and shut-down schedules of thermal units to be used to meet forecasted demand over a future short term (24-168 hour) period. The objective is to minimize total production cost while observing a large set of operating constraints. The unit commitment problem (UCP) is a complex mathematical

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optimisation problem having both integer and continuous variables. One obtains the exact solution to the problem by complete enumeration, which cannot be applied to realistic power systems due to its excessive computation time requirements [1,2]. In solving the unit commitment problem of a large system, the main cause of difficulty is the involvement of large number of units for commitment. The problem cannot be solved easily if all units are involved in the search for the optimal solution, since computational facilities could be exhausted. Research efforts have concentrated, therefore, on efficient, sub-optimal UC algorithms which can be applied to realistic power systems and have reasonable storage and computation time requirements. The basic UC methods reported in the literature can broadly be classified in six categories [3]: Priority list Dynamic programming Lagrangian relaxation Augmented Lagrangian relaxation Branch-and-Bound Benders decomposition Since improved UC schedules may save the electric utilities substantial resources per year in production costs, the search for closer to optimal commitment schedules continues. Recent efforts include application of simulated annealing, expert systems, Hopfield neural networks and genetic algorithms to solve the UCP. References [4,5] give a survey of various approaches and their merits and demerits in this field. Some of these methods achieved a reduction of the computation requirement for large power systems. Researchers have yet to obtain an optimal solution to the problem for such systems. There have been some past attempts in other areas such as coal modeling [6] to reduce a large-scale system to a smaller system. This chapter proposes a new, efficient solution approach to the UCP of a large-scale power system. The approach is based on cardinality reduction by generator equivalencing (hereafter called “equivalencing”), which reduces the number of units in the largescale power system to the lowest possible number according to their similar fuel/generation cost characteristics and minimum up- and down-time characteristics. We first solve the reduced system using a modified dynamic programming technique [7] and then obtain an overall solution to the original unit commitment problem of the entire system by un-crunching the reduced solved system and using certain rules. The Appendix explains the modified dynamic programming (MDP) technique.

An Equivalencing Method

2. N : T :

213

NOTATION number of thermal generation units total scheduling period load demand, in MW power generation by nth unit, in MW minimum generation capacity limit for n-th unit maximum generation capacity limit for n-th unit cost of power generation by nth unit, in Rs/hour start-up and shut-down cost for nth unit, in Rs/hour system reserve requirement in time period t up-/down-time status of nth unit unit on; unit off duration of unit n on and off, in hour minimum up- and down-time for nth unit, in hour number of units in group i number of groups in the system power output of the equivalent system minimum and maximum generation limit of group g

3.

DESCRIPTION OF THE UNIT COMMITMENT PROBLEM

The objective of the UC problem is to minimize the total production cost over the scheduling horizon. The total production cost consists of: fuel costs, start-up costs, and shut-down costs. We calculate fuel costs by using unit heat rate and fuel price information. We express the start-up cost as a function of the number of hours the unit has been down (exponential when cooling and linear when banking). The shutdown cost is given by fixed amount for each unit per shut down. The following must be satisfied during the optimization process: (a) system power balance (demand plus loss plus exports) (b) system reserve requirement (c) unit initial condition (d) unit high and low MW limits (economic, operating)

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(e) unit minimum up time (f) unit minimum down time (g) unit status restrictions (must-run, fixed-MW, unavailable, available) (h) unit rate limits (i) unit start up ramps (j) plant crew constraints Constraints (a) and (b) concern all the units of the system and are called system, or coupling, constraints. For multi-area unit commitment, the system constraints must be modified to take into account the interchange schedules and the tie-line limitations. In general, the system constraints must take into account possible transmission bottlenecks in the allocation of the demand and the reserves to the generating units. Constraints (c) through (i) concern individual units and are called local constraints. Plant crew constraints can also be classified along with local constraints, but they involve all the units in a plant.

4.

THE EQUIVALENCING METHOD

The “equivalencing” method for solving large scale thermal unit commitment problem consider the particular reference to units generation cost (inputoutput characteristic), minimum up and down time, and ramp rate criterion. A large-scale power system consists of a large number of generation units, in which all the units are not of same size and similar characteristics. Yet there are many clusters of units throughout the entire power system which have almost similar cost coefficients and other physical characteristics. Therefore, units in the large scale system are re-grouped/ partitioned into various groups based on their fuel cost, having cost coefficients of almost similar value (maximum 1% variation), and other physical characteristics, such as same capacity, minimum up and down time, ramp rate, etc. so that identical/similar characteristic units form one group. Each such group is then represented by any one unit of this group and is called a representative unit. Consequently, an equivalent smaller system of units consisting only of representative units is generated. The basic concept of this method is that the large-scale power system is represented by an equivalent smaller system with a smaller number of generation units, such that the unit commitment problem of the equivalent system would be easier to handle and solve than that of the original large size system. Once the solution of the unit commitment problem of the equivalent small system is obtained, then the solution to the original unit commitment problem can be determined accordingly by un-crunching the equivalent problem solution.

An Equivalencing Method

5.

215

PROBLEM FORMULATION

The function to be minimized for the unit commitment problem can be expressed in mathematical form as follows:

subject to the following major constraints: i) demand constraint:

ii) capacity constraint:

iii) unit’s minimum up- and down-time constraints:

iv) unit’s generation capacity constraint:

Formulate the large-scale unit commitment problem by using the “equivalencing” method as given below: v) Generate the equivalent system (based on the generator cost and other physical characteristics) with the lowest possible number of representative units only. vi) Determine the number of groups in the system and the number of units in a group, such that identical units form one group. The total number of units in the system would be

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vii) Represent each group by one representative unit to form equivalent system. The total output of the equivalent system would be:

viii) Maximum and minimum capacity limit of each group (i.e., representative unit) would be

ix) Minimum up and down time, ramp rate, etc. of the representative unit of a group is set as any individual unit’s characteristics of that group.

5.1

Solution Technique

The large-scale unit commitment problem is solved by performing the commitment (0-1 status of each unit) of the equivalent system using the modified dynamic programming technique [7] developed by the authors. It can be solved, however, by any other standard unit commitment solution technique. If, during the solution of the equivalent system, the minimum output of a particular representative unit of a group is less than the equivalent minimum run level of that group, perform the sub-unit commitment of the individual units of that particular group. Once the reduced system is solved, the units in each group in the equivalent system are treated according to the status of its representative unit, based on the following strategy: If the representative unit of group i at interval t is “OFF,” then all units of the group should be off. If the representative unit of group i at interval t is “ON” and operated at its equivalent minimum/maximum output, all the units in the group should be operated at its individual minimum/maximum output. If the representative unit of group i at interval t is “ON” and operated at some percentage of its maximum/minimum output, carry out further commitment scheduling within the loop among the units in that group. Finally, obtain the total cost over the commitment period using Equation (1).

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5.2

217

The New Algorithm

Step 1: Read input data, i.e., unit characteristics, load demand profile, etc. Step 2: Categorize the units having similar/identical fuel costs, minimum up and down time, and ramp rate characteristics into different groups. Step 3: Generate the equivalent smaller system of the representative units. Step 4: Perform the commitment schedule of the equivalent system by modified dynamic programming or by any standard technique with certain rules, i.e., if during the solution of the equivalent system, the minimum output of a particular representative unit of a group comes out less than the equivalent minimum run level of that group, perform the sub-unit commitment of the individual units of the group. Step 5: Generate the unit commitment schedule of the original system using the rules as specified below: if representative unit is “OFF,” all units in that group would be off. if representative unit is “ON,” and generate up to its equivalent maximum or minimum capacity output, all units in that group would generate their minimum/ maximum output. if the representative unit is “ON” and generate a percentage of its maximum/minimum capacity output, schedule the units in that group using modified dynamic programming to generate the optimal scheduling of the units in that group. Step 6: Calculate the total cost for scheduling the normal way by Equation (1). Figure 1 presents the outline of the new algorithm for solving large scale unit commitment problem.

6.

TEST SYSTEMS AND RESULTS

The authors have tested the new “equivalencing” method for solving short term thermal unit commitment problem solution for two large power systems consisting of the 26 and 79 thermal units, respectively, with a particular daily load demand profile [8,9]. We base the system spinning reserve on the capacity of the largest on-line unit. The 79-unit system represents Eastern Regional grid of the Indian power system. We implemented the new algorithm in FORTRAN 77 code on a PC-486 computer. In the equivalent system representation, 8 units represent the complete 26unit system, and 19 units represent the 79-unit system. Therefore, we can see that the size of the problem has been reduced drastically. The comparison of the proposed method with other traditional techniques like Lagrangian relaxation and truncated dynamic programming [10,11] are presented in Tables 1 and 2 for 26-unit and 79-unit systems, respectively.

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An Equivalencing Method

219

In the Lagrangian relaxation technique, form the Lagrangian dual by appending the relaxed constraints to the primal objective. Find the minimum of the primal objective by maximizing the dual objective. Update the Lagrangian multipliers using a sub-gradient method that drives the solution towards feasibility [12]. In the DP-TC technique, the ordering of the units are made based on an average full load cost of each unit, and the size of the search range of four units (16 combinations) has been considered.

7.

DISCUSSION

The results given in Tables 1 and 2 clearly show that the new equivalencing method for solving the large scale unit commitment problem is able to provide solutions very close to the best solutions found by other approaches. The variation in cost is within 0.05% and 0.19% respectively for the 26-unit system compared to truncated dynamic programming and the Lagrangian relaxation approach, respectively. For the 79-unit system, the cost obtained using the new

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method is 0.7% more compared to the DP-TC-based method while it is 1.27% less compared to Lagrangian relaxation-based method. The solution time in the new approach, however, is smaller as compared to other techniques. This is one of the main requirements to solve the unit commitment problem for a large-scale power system. In addition, the computer space requirement for the new method is comparatively less than other standard methods of UCP solution. Also, the new method simplifies the unit commitment problem in terms of dimensionality of the problem. As a result, the new method of solving the unit commitment problem for large-scale systems turns out to be a promising one. We tested the algorithm for one sector of Indian power system, however, it is not presently implemented by Indian power utility. In fact, due to restrictions on the available generation capacity, presently India uses just merit order scheduling. Further, the algorithm neglects to consider the transmission constraint. If it must be considered, however, then the concept presented in this work would have to be applied to a cluster of units which are relatively close to each other, and then the algorithm would not violate the transmission constraint.

8.

CONCLUSION

In this chapter, we developed the “Equivalencing” method based on cardinality reduction by generator equivalencing for estimating the short-term thermal unit commitment schedule for large scale power systems. In this method, we reduce a large-scale power system to an equivalent reasonably small-sized power system. We first solve the equivalent system by using a modified dynamic programming approach (or it can be solved by any other standard unit commitment solution technique). Finally, we return to the original system to calculate the unit commitment schedule cost and determine the units’ final status. This method simplifies the unit commitment problem in terms of dimensionality, and consequently, computer space as well as the computer time required for solution can be reduced remarkably with an acceptable overall solution of the large-scale power system. We have tested the model on a practical large and complex Indian power system. The results obtained from the above model are highly impressive and encouraging for implementation of real-life large scale power systems having large numbers of units. In short, the proposed unit commitment model yields a promising approach to solve the short-term thermal unit commitment problem and offers good performance which provides fast solutions for large-scale power systems.

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ACKNOWLEDGEMENTS The chapter’s presentation was greatly improved by the comments and suggestions of the editors and three anonymous referees. The authors also wish to thank B. Hobbs for suggestions, which improved many aspects of this chapter.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.

10.

11.

12.

A.J. Wood and B.F. Wollenberg. Power Generation Operation and Control. New York: John Wiley, 1996. I.J. Nagrath and D.P. Kothari. Power System Engineering. New Delhi: Tata McGraw-Hill, 1994. S.A. Kazarils, A.G. Bakirtzis and V. Petridis. A genetic algorithm solution to the unit commitment problem. IEEE Trans. Power Syst., 11(1): 83-92, 1996. G.B. Sheble and G.N. Fahd. Unit commitment literature synopsis. IEEE Trans. Power Syst., 9(1): 128-135, 1994. S. Sen and D.P. Kothari. Optimal thermal generating unit commitment: A review. Elec. Power Energy Syst., 20(7): 443-451, 1998. S. Bullard and R.E. Wiggans. Intelligent data compression in a coal model. Oper. Res., 36: 521-531, 1988. D.P. Kothari and S. Sen. Optimal thermal generating unit commitment – a novel approach. In Proc. International Seminar on Modelling & Simulation, Australia, 331-336, 1997. C. Wang and S.M. Shahidehpour. Effects of ramp-rate limits on unit commitment and economic dispatch. IEEE Trans. Power Syst., 8(3): 1341-1350, 1993. S. Sen and D.P. Kothari. Evaluation of benefit of inter-area energy exchange of Indian power system based on multi area unit commitment approach. Elec. Machines Power Syst., 26(8): 801-813, 1998. S.J. Wang, S.M. Shahidehpour, D.S. Kirschen, and G.D. Irisarri. Short-term generation scheduling with transmission and environmental constraints using an augmented Lagrangian relaxation. IEEE Trans. Power Syst., 10(3): 1294-1301, 1995. C.K. Pang, G.B. Sheble and F. Albuyeh. Evaluation of dynamic programming-based methods and multiple area representation for thermal unit commitments. IEEE Trans. Power Syst. PAS-100, 3: 1212-1218, 1993. W.L. Peterson and S.R. Brammer. A capacity-based Lagrangian relaxation unit commitment with ramp rate constraints. IEEE Trans. Power Syst., 10(2): 1077-1084, 1995.

APPENDIX: Modified DP Technique for Solving UCP Theory Formation of Unit Commitment Schedule Table Let the cost function of the n-th unit at a plant be [2]:

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For minimization of cost, using well-known dynamic programming, a simple recursive expression can be obtained as given below:

Where

is the cost of generation at the n-th unit with as power dispatch. is the minimum cost of generating by the remaining (N1) units of the plant. Re-writing equation (10) as

where,

For minimum of

or

where

From equations (11) and (12), obtain the composite cost function of units #1 and #2 for a demand of as shown below:

where

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Equation (13) represents the most economical cost of the two units for total generation allocation over two units with as generation allocation on the second unit. In general form,

where

Equation (14) can also be expressed as

We obtain the critical value of generator loading/generation dispatch by equating (14) and (15) above, in which the combination of n-number of unit commitment will be economical as compared to number of units combination. Therefore, a “loading range” of operation can be obtained for units for which the cost is minimum as compared to n-units combination. In this chapter, we use this principle to prepare a sequential economic order of units for commitment and the economic “loading range” of operation for those combinations. Generation Limits Constraints Fixing To satisfy the maximum and minimum generation limits during generation scheduling, we apply the following limits constraint fixing technique. i)

Solve unconstrained generation dispatch problem.

ii) If there is no limit violation, the solution is optimal, otherwise compute as follows:

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for all generation limit upper-bound violation. for all generation limit lower bound violation. (iii)If ¨ fix all upper bound violations units to the upper limits, if fix all lower bound violations units to the lower limits, otherwise fix both upper and lower bound violations to upper and lower limits respectively. iv) Determine the new demand which is the original demand minus the sum of fixed generation levels. v) Set aside fixed generation level units for scheduling of new demand. vi) Reschedule new demand among remaining committed units and return to step (ii). UP- and DOWN-Time Constraint Fixing The purpose of this postprocessor algorithm (rules) is to detect the violation of the minimum up- and down-time constraint of units. Minimum up and down times are particularly difficult to model and cannot be incorporated directly in the main program routine. So to detect a violation, we have established certain rules: Check Constraint (up time/down time violated) Condition (unit on/off) Condition (unit on/off time < min. up/down time) If a unit is committed, then de-committed, and the duration between on and off state is less than its minimum up time, then the unit up time is violated. Therefore, the unit is charged as if it was on stand-by for those hours. In the original unit commitment schedule, the stand-by hours are set to 1 (unit “on”) instead of 0 (unit “off”). The additional number of hours needed to satisfy the minimum up-time constraint are multiplied by the banking cost and then added to the UC schedule cost. On the other hand, if a unit is de-committed, then committed, and the duration between off and on states is less than its minimum down time, then the unit down time is violated. Therefore, the unit is charged as if it was on stand-by for the additional number of hours needed to satisfy the constraint. The original unit commitment schedule has those hours set to 1 instead of 0. UC schedule is “costed” as if it was banking for those hours.

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Algorithm Step 1: Read number of units, unit parameters, hourly demands, etc. Step 2: Select any unit as a first unit from the list of the total number of units and using equations (14) and (15), the composite cost function of the two units, taking all units one by one is formed. Step 3: We find the critical loading value for the combination of two units by equating the cost function of the first unit with the composite cost function of two units or equivalent two units taking sequentially. The accepted combination of two units is that having minimum critical “loading” value Step 4: Repeat Steps 1 and 2, taking all units sequentially as the first unit and find the N-number of minimum critical loading are found from each combination. Among these, the maximum of minimum critical loading is the best combination and sequential order of the first two units. If is higher than the sum total of maximum capacity of two or equivalent two units, then is reset to the sum of the maximum capacity. For the equivalent two units combination case, the minimum loading will be the just slightly higher than the maximum loading of the previous combination. Step 5: The sequence of the two units formed in Step 3 can be used as a basic combination as the equivalent first unit to search the third most economic unit. Repeat the procedure in Steps 1 to 3 to find the best combination of three units and the corresponding third unit in sequence. The algorithm uses the combination of three units as basic combination to search the next economic combination of four units. In this way, for any addition of units in basic combination of j units, follow the procedure mentioned in Steps 1 to 3 until all units are considered to form a sequential order/ combinations of units and a corresponding order of combination and loading range of operation. Step 6: Execute limits constraints fixing, up- and down-time violation rules. Step 7: Based on the general unit commitment schedule table and Step 6, perform the unit commitment schedule for the particular load profile.

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Chapter 13 STRATEGIC UNIT COMMITMENT FOR GENERATION IN DEREGULATED ELECTRICITY MARKETS

A. Baíllo, M. Ventosa, A. Ramos, M. Rivier Universidad Pontificia Comillas de Madrid.

A. Canseco IBERDROLA S.A.

Abstract:

1.

In this chapter we address some of the new short-term problems that are faced by a generation company in a deregulated electricity market, and we propose a decision procedure to address them. Additionally, we propose a strategic unit commitment model, which deals with the weekly operation of the firm’s generating facilities. In it we combine traditional cost-evaluation techniques and technical constraints that grant a feasible schedule with new market-modeling equations. We suggest strategic constraints that allow the accomplishment of the firm’s medium-term objectives. We have formulated the model as a mixedinteger-programming problem and solved it by means of a commercial algorithm, instead of using the traditional Lagrangian relaxation approach. Results of the application of the method to a numerical example are presented. The procedure is a simplified version of one of several tools currently being used by a leading Spanish generation company, Iberdrola, for the weekly operation of its generation assets in the Spanish wholesale electricity market.

INTRODUCTION

The electricity industry is in the midst of a profound restructuring process in an increasing number of countries. These changes are intended to bring about competition in some of the electricity business activities, so as to promote a higher level of efficiency in the provision of electric services. Although the details of the deregulated marketplace may vary from one case to

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another, it is generally assumed that electricity should be traded in a similar fashion to other energy commodities. Generation companies have traditionally been subject to regulatory policies that guaranteed the full recovery of their costs. In the new framework, electricity generation is a deregulated activity and firms have to compete to sell the electric services provided by their facilities. Therefore, generation companies are now fully responsible not only for the efficient operation of their units, but also for selling their output. The particular design of the marketplace where the electric services are traded is of great importance. Two mechanisms coexist in recently deregulated electric industries throughout the world. The most common is a centralized power exchange based on auctions where the price for each electric service during a certain time period is determined by the intersection of the aggregate supply and demand bid curves. Additionally, bilateral trading between buyers and sellers is usually permitted. The fact that generation companies’ revenues depend on the market forces leads to a higher degree of uncertainty and risk. New procedures and tools devoted to the maximization of the firm’s profit, taking into account the different market mechanisms available and keeping an upper bound on the degree of risk exposure, are needed. However, introducing in a model all the complexity of electricity trading does not necessarily result in a better understanding of the environment. A gradual implementation of solutions for the open issues should lead to a deeper and more solid knowledge of their implications. In this chapter, we address some of the new problems that are faced by a generation company in the short term (one day to a week). Assuming that the majority of energy is traded in an energy exchange based on 24 day-ahead hourly uniform-price auctions, we outline a decision procedure. One of the tools incorporated in this procedure is a strategic unit-commitment model, which deals with the weekly operation of the firm’s generating facilities. It includes traditional cost evaluation techniques and takes into account technical constraints that grant a feasible schedule. Its major contribution lies in the inclusion of a set of market-modeling equations intended to express the relationship between the firm’s output and the revenue obtained in the successive auctions. Additionally, we show the need to consider a set of strategic constraints so as to direct the solution towards the firm’s medium-term objectives. We have formulated the model as a MIP optimization problem and solved it by means of a commercial algorithm instead of using the traditional Lagrangian relaxation approach. We present results of the application of the method to a numerical example.

Strategic Unit Commitment for Generation Companies

2.

229

DECISION PROCEDURE

In the new competitive framework, a generation company not only has to determine how to operate its generation facilities in the most efficient manner, but also must decide on the amount of each electric service that should be supplied, at which moment it should be produced, at what price it should be sold, and with which units it should be provided. These new challenges require decision procedures and tools specifically oriented to the maximization of the firm’s profit and the hedging of its risk. The short-term decision procedure outlined in this section is represented in Figure 1.

2.1

Medium Term Guidelines

As in the past, generation firms have to decompose the problem of planning the operation of their units into different time scopes to make it tractable. The traditional hierarchy used to classify the decision support tools into long-, medium-, and short-term models is still completely in force. The results obtained from a model with a longer time scope must affect all the inferior models. Consequently, in a short-term decision procedure the generation firm’s medium-term goals play an important role. In its medium term analysis (one month to a year) the generation firm still has to make traditional decisions related to the maintenance of the groups, the annual management of water reserves, or the fuel consumption of thermal plants. New issues include determining the expected equilibrium between

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generation companies and estimating the prices that are likely to appear during the following months. Two very important results of the medium term, which the firm must consider in the short term (one day to a week), are the water value assigned to the water reserves and the medium-term market position that allows an equilibrium with the rest of generation firms. Firms must assign a certain value to their stored energy. Otherwise, shortterm tools will interpret that producing with these reserves has no associated costs and the result will be that hydro units must permanently produce at their maximum capacity. Bushnell [1] analyzed the strategic management of hydro resources in a competitive environment and established a relationship between the value of the available hydro energy and the marginal cost of thermal units. Scott and Read [2] used Dual Dynamic Programming methods to build up a weekly curve giving the optimal output for a certain water value. Similarly, if a short-term model is not aware of the market position defended in the medium term by the firm, it will tend to follow blindly the shortterm signals transmitted by the competitors through their supply curves. We will analyze in detail how the slope of the bid curve presented both by other generators and by the demand exerts an influence on the results given by a short-term model that tries to maximize the short-term profit of a generation firm. This influence must be limited and controlled so that the firm is able to keep a steady pace towards its medium-term objectives, which include defending its market position. A firm may lose its market position by systematically producing less than the market share it should have according to the cost and size of its generating assets relative to those of its competitors. Several approaches have been proposed for approximating the medium-term equilibrium that a number of generating firms should reach in a competitive electricity market. Some of these consider the generation firms as Cournot agents whose decision variables are the quantities produced in each time period. Otero-Novas et al. [3] developed a simulation platform to evaluate the medium-term evolution of the Spanish electricity market. Ramos et al. [4] and Ventosa et al. [5] successfully combined a detailed representation of the generation operational costs with the Cournot-equilibrium conditions in a costminimization framework. Rivier et al. [6] used the complementarity problem approach with satisfying results to determine the expected medium-term equilibrium reached in the recently deregulated Spanish electricity industry. Hobbs [7] and Wei and Smeers [8] extended the usage of the complementarity problem to predict the outcome of an electricity market with significant transmission constraints. An important feature of all these models is that they reflect the objective of profit maximization of all generation firms so that the medium-term market equilibrium is accurately represented.

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2.2

231

Forecasting Techniques

A major challenge for a generation company in the new framework is the development of forecasting techniques devoted to the estimation of the opponents’ expected behavior. Information concerning the amount of each service traded at different prices is extremely relevant. These data can be exploited to estimate future competitors’ hourly offer curves. The expected level of demand is also a decisive variable, whereas the elasticity of the demand curve is a parameter whose importance is expected to grow as the agents gain experience in the new regulatory scheme.

2.3

Short-term Generation Scheduling

Given a certain scenario for the competitors’ offers and the demand bids for the different electric services markets, the firm has to determine the energy that it should offer as well as the capacity that should be reserved for ancillary services. When the firm’s revenue is based on administrative and centralized decisions, the amount of energy that each unit must produce and the precise moments when this unit should start up and shut down are tackled with weekly cost-minimization unit-commitment models. However, deregulation has shifted focus from obligation of supply and cost-minimization to competition and profit maximization. Therefore, the competitive environment requires unit-commitment models that take into account the expected price series for each of the electric services. Moreover, if the firm has a significant market share, the response of prices to the output of the generation firm must be considered [9]. Unit-commitment models provide a commitment schedule. However, they usually give only approximate generation levels for the generators [10]. Once the commitment decisions are taken, the best hourly output of each individual generator can be determined with a daily model. This model will include a more detailed representation of the generating equipment, such as an accurate description of hydro generation resources.

2.4

Strategic bidding

The final stage of the generation firm’s decision procedure is the design of the hourly offer curves that must be submitted to the different auctions. The results of the previous short-term decision support tools include the firm’s expected hourly optimal productions together with expected hourly prices.

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However, the competitors’ sell offers as well as the demand-side buy bids are estimated with a certain degree of uncertainty, since the short-term volatility of electricity prices is higher than those of other energy commodities. In this context, instead of sticking to a single hourly quantity, the firm can submit an hourly offer curve for each of the electric services. The bigger and more flexible its generation portfolio is, the wider the variety of hourly offer curves the firm can design. In this chapter, we assume that offer curves consist of a set of quantity-price pairs and that no additional information, such as fixedcosts, is submitted. Given a probability distribution for the last accepted bid (Figure 2), the generation firm can derive the offer curve that maximizes its objective function. This can be a combination of the expected short-term profit and other targets such as market share goals.

3.

MODEL DESCRIPTION

In the previous section, we suggested a short-term decision procedure for a generation firm participating in a day-ahead auction-based energy exchange. The unit-commitment model that we will develop henceforth, however, is a simplified part of the complex combination of tools that a generation firm should use to face the short-term problems that will arise in the new competitive framework. We will only consider the day-ahead hourly energy market. Our aim is to gain insight into certain modeling features such as the management of hydro reserves or the influence of strategic constraints.

3.1

Objective Function

A major difference between the strategic unit commitment model and a traditional unit commitment model lies in the objective function. The aim of a

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traditional model is the minimization of the overall system costs. In spite of their complexity, cost functions of thermal units have frequently been modeled as convex piecewise linear functions. In contrast, the strategic unit commitment model guides a generation firm to its maximum profit objective, which is a non-linear (and frequently non-convex) function of the firm’s energy output.

3.2

Generation System Representation

Thermal units’ costs representation includes fuel costs (which, for simplicity, can be defined as a convex piecewise linear function of the unit’s output), start-up costs and shut-down costs. Thermal units’ most relevant constraints are the minimum stable load, maximum output, and upwards and downwards ramp limits. Hydro units produce with nearly zero variable costs. Water reserves have associated opportunity costs, however, as they can be used to substitute thermal units. Therefore, we can define a cost function known as water value. The water value function gives the hydro energy that must be produced if the marginal revenue of the firm exceeds a certain value. In the strategic unitcommitment model, we divide hydro reserves into several reserve levels and assign a different water value to each one of them. Depending on the market circumstances, the model will decide to use a certain amount of each one of these levels. To keep track of the contents of these reserve levels we will include hourly reserve balance equations. Hydro units also have a minimum and a maximum power output. The operation of pumped-storage power plants is subject to the same constraints as regular hydro power plants, except that the water balance constraints are modified to include the pumping mode of operation.

3.3

Market Representation

The strategies followed by the firm’s competitors are expressed by means of their expected hourly offer curves (known also as hourly supply functions). Similarly, instead of using a fixed level of demand for each time period, energy buyers submit hourly demand curves.

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From the generation firm’s point of view, the competitors’ hourly offer curves and the hourly demand curve exert a very similar influence on the firm’s profit (see Figure 3). If the firm increases its energy output, lower prices will result. This is due to the combined effect of a decrease in the competitors’ output and an increase in the energy consumption. Therefore, to a certain extent, the firm is able to adjust its hourly revenue by varying its energy output. In microeconomic theory, this is modeled by means of the residual demand function. This gives the energy the firm is able to sell at each price. The firm should try to estimate this function for each hour. Going a little further, a change in the firm’s production also modifies the competitors’ revenue. It must not be forgotten that a decrease in the firm’s production may or may not increase its profit, but it will surely benefit the competitors as they are able to produce more and at a higher price (Figure 4).

The objective function whose maximum is sought is the firm’s profit, defined as the difference between the obtained revenue and the incurred costs. Owing to the fact that the most powerful available solvers are those designed

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for mixed integer linear problems, such as CPLEX and OSL, a linearization procedure for the firm’s revenue function is of great interest. An intuitive method is to divide the firm’s hourly revenue function into convex sections and approximate each one by a piecewise linear function. The slope obtained for each linear segment is the firm’s marginal revenue at the corresponding energy output (Figure 5).

A group of consecutive segments with strictly decreasing marginal revenues defines a convex section in the revenue function. When we seek the optimum we select a specific convex section by switching its binary variable from zero to one. Once we have chosen a convex section we fill its segments with continuous bounded variables. In other words, we obtain the hourly revenue by integrating the marginal-revenue function. With this approach, we replace the set of constraints that represent the competitors’ generation facilities in a traditional unit commitment by a set of hourly constraints, which define the firm’s hourly revenue as a function of its energy output. Hourly prices do not appear explicitly in our model.

3.4

Medium-term Guidelines

The results obtained from medium-term models add information to shortterm decision-support tools. Good examples are the water value assigned to hydro reserves or the position that the firm must defend if it wants to maximize its profit in the long run. Our model decides the amount of hydro energy to be used depending on the water value received from a hydrothermal co-ordination model. It also distributes this energy along the time scope of the model, trying to obtain a uniform marginal revenue. The total hourly energy production that the model suggests not only depends on the expected price and on the firm’s marginal production costs. We also consider the slope of the residual demand curve, which exerts a major

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influence on the clearing price (Figure 6). If this curve is very steep and the firm’s output is high (on-peak hours) the model will blindly tend to reduce the firm’s production. This causes a rise of the energy price and an increase of the firm’s profit. Another result is that competitors are able to produce more at a higher price. Taking into account that the price of electricity usually behaves like a mean-reverting process, if the firm gives up its position repeatedly during on-peak hours, competitors will increase their market shares and, in the long run, prices will return to the original level.

An alternative is to define a set of hourly minimum-market-share constraints. In this fashion, the short-term model maximises the firm’s short-term profit while following the right medium-term strategy. We will analyze the influence of minimum-market-share constraints in the case study. Many other constraints can be designed ad hoc to fulfill the firm’s requirements such as existing physical or financial contracts. In our model, they must be expressed as linear equations. The strategic UC solution will be different in each particular case. A great effort must be made to interpret correctly the influence of each factor both in the short and long term.

4.

MATHEMATICAL FORMULATION

4.1

Notation

In this section, we identify the symbols used in this chapter and classify them according to their use. Table 1 shows the indices and sets considered, capitals being used for sets and lower-case for indices. Table 2 includes the decision variables. Table 3 lists the auxiliary variables. Table 4 defines the information given to the model as fixed data.

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238

4.2

The Next Generation of Unit Commitment Models

Model formulation

We formulate the model as an MIP optimization problem. The objective function to be maximized is the firm’s total profit for the scope of the model. We have classified operating constraints into thermal and hydro constraints. Additional market constraints model the behavior of the competitors and the demand side in the day-ahead power market.

Strategic Unit Commitment for Generation Companies

4.2.1

239

Objective Function

The objective function represents the firm’s profit defined as the difference between the firm’s revenue and the firm’s operating costs for all load levels within the scope of the model:

4.2.2 Thermal Generation Constraints Total thermal operating costs include fuel costs, O&M costs, start-up costs and shut-down costs:

For each committed thermal unit, the maximum generation is less than the maximum available capacity, and the minimum generation is greater than the minimum stable load:

The hourly change in the output of each thermal unit is limited by the ramp rates:

A logical relationship exists between the start-up, shut-down, and commitment variables: Since the commitment decision variables are binary, both the start-up and the shut-down decision variables can be continuous but must have upper and lower bounds:

4.2.3

Hydro Generation Constraints

The water reserves not used by the model have a value for the future:

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Each unit has an upper and a lower limit for its power output:

The contents of the reservoirs depend on the energy produced or stored during each time period and have upper and lower bounds:

4.2.4

Market Constraints

Each segment of the firm’s net hourly energy output is valued at a different marginal revenue. The sum of all the segments must equal the sum of the power produced by thermal and hydro units minus the power consumed by pumped-storage units:

Each segment has an upper and a lower bound and the convex sections must be chosen in order:

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241

We calculate the total revenue by valuing the different segments of the net energy output at their corresponding marginal revenues. In other words, the revenue is obtained by integrating the marginal revenue function:

The hourly price of energy is not obtained explicitly with this formulation. It must be calculated after the execution of the model. To do so, we simply divide the firm’s hourly revenue by the firm’s hourly production. 4.2.5

Strategic Constraints

We define a set of hourly minimum-market-share constraints. In the numerical example, we investigate the influence of this constraint on hourly prices and on the firm’s short-term benefit. In our formulation, we suppose that demand is perfectly inelastic. Consequently, the only variations of demand we allow are those introduced by pumping:

5.

NUMERICAL EXAMPLE

The strategic unit commitment model has been implemented in GAMS [11]. A case study has been solved with the optimizer CPLEX 6.5.

5.1

Case Study

We include the results of the application of the model to a case study. Our aim is to highlight the influence of the different modeling decisions previously explained. Computational and convergence issues are secondary and should be treated only when the researcher is sure that the model adequately represents the problem he is trying to solve. Consequently, we will only give the essential information to define the case study. (The authors may be contacted for details if the reader wishes to reproduce the results.) The firm’s generating equipment is described in Table 5.

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Hydro and reserves have been classified into the levels shown in Table 6 according to the results of a hydrothermal coordination model.

We have estimated a piecewise linear residual demand function for each hour. Figure 7 shows an example of a residual demand curve with its corresponding revenue function, formed by two convex sections. Each of these sections has been divided into five segments. We have assigned a constant marginal revenue to each segment.

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In this example, we include a set of hourly minimum-market-share constraints to obtain a generation schedule similar to the traditional one. If no strategic constraints were used, the model would blindly follow all the shortterm opportunities and the resulting operation would require extremely inefficient dynamic performance of the generating units. The weekly problem is formed by 19446 equations, 24555 continuous variables and 4692 binary variables. It is solved in 89 seconds on a PC Pentium II 350 MHz 64 MB.

5.2

Results

In this section, we describe the results of the model when a minimummarket-share constraint of 29% is used. Table 7 states the differences between the solution given as optimal by CPLEX and the first feasible solution.

The model decides the hourly power output for each ofthe firm’s generating units. The hourly energy that each kind of unit should produce to achieve this hourly market share has been represented in Figure 8.

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Each hourly energy output determines an hourly revenue by means of the estimated marginal revenues. Additionally, the production of that energy leads to an hourly cost. The difference gives the firm’s expected hourly profit. These three variables have been represented in Figure 9.

A subproduct of the problem is an estimation of the hourly price of energy. We calculate it after the execution of the model, dividing the firm’s hourly revenue by its hourly production. As we observe in Figure 10, due to the differences among hourly revenue functions, the series of energy prices is only partially correlated to the series of the firm’s energy outputs.

The model also decides the optimum strategic management of the existing water reserves. The usage of hydro energy depends on the assigned water val-

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ues. The model will use a certain amount of water reserves if their value is lower than the maximum marginal revenue reached during the week. In this example, all reserve levels are used except for the one valued at 37.5 $/MWh. This indicates that the marginal revenue never reaches that value. Similarly, the pumped-storage unit consumes energy when the value of its reserve is times higher than the weekly minimum marginal revenue. Conversely, this water will be released if the weekly maximum marginal revenue reaches the water value of the pumped-storage (Figure 11). In this case the final contents of the pumped-storage reservoir have been set equal to the initial ones.

5.3

Influence of the Strategic Constraints

We now analyze the influence of introducing a minimum-market-share constraint. In our case study, the firm faces such a steep residual demand curve that the incentive to withdraw production from the market is very strong. Although this leads to high prices in the short term, the firm should administrate its market power with care to guarantee a solid long-term market position. We have solved the case study with ten levels of minimum market share, ranging from 26% to 30.5%. Figure 12 shows part of the price series obtained for three of these market-share levels.

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As we can see, lower market shares produce higher prices. In this case, the rise of prices overcomes the market share reduction. Consequently, by withdrawing, the firm achieves both higher revenues and lower costs (Figure 13).

As stated by Viscusi et al. [12], one pricing strategy is for an incumbent firm always to set price so as to maximize current profit. Typically, setting such a high price will cause the fringe to invest in capacity and expand. Therefore, this can be called myopic pricing. The polar opposite case is for the incumbent to set price so as to prevent all fringe expansion (limit pricing). Myopic pricing gives higher profits today, while limit pricing gives higher profits in the future. Pricing at a level to exclude from the market less efficient competitors is, of course, what competition is supposed to do. Pricing to exclude equally or more efficient competitors is known as predatory pricing and constitutes an intent to acquire the monopoly position. To determine the position that must be defended in the market, a generation firm can assign a cost to the deviations from its medium-term marketshare objective. The firm must keep an eye on the, say, one-month moving

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average of its market share. If its market share remains for more than one month below the objective, then its competitors may understand this as a change in the medium-term equilibrium conditions. Therefore, the firm can expect prices to stay high for a month. After this period, prices will revert to their medium-term mean. If this happens, the firm will be forced to suffer low prices to recover the lost position. With this approach, each time the firm’s one-month market-share moving average lies below the medium-term objective it accounts for a loss. On the other hand, the firm should increase its market share cautiously, as this can lead to a price war. The cost function looks like the one sketched in Figure 14.

Depending on the short-term market conditions and on the accumulated market-share, there will be weeks when the model will suggest producing above or below the market-share objective. An additional consideration is that the cost of market-share deviations will probably be higher in on-peak hours than in off-peak hours.

6.

CONCLUSION

In the new deregulated electric marketplace, generation companies have to compete to sell the electric services provided by their facilities. They must develop new procedures and tools devoted to maximize profit and hedge risks. In this chapter, we addressed several new short-term problems that are faced by a generation company. Stemming from the medium-term objectives of the firm, our aim has been to determine the optimal combination of offers and contracts for the supply of electric services through the different market mechanisms available. We have tried to emphasize the importance of an adequate design and use of the decision-support tools.

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In this context, we presented a strategic unit-commitment model specifically devoted to the profit maximization of an electric generation firm. It incorporates new market-modeling equations intended to express the relationship between the firm’s output and the obtained revenue. We have made a special effort to interpret the effect of these new equations on the management of hydro reserves. Additionally, we demonstrated how the position that the firm must defend in the market must guide the short-term operation towards the medium-term equilibrium. The procedure is a simplified version of the one currently being used by a leading Spanish generation company, Iberdrola, for the weekly operation of its generation assets in the Spanish wholesale electricity market.

REFERENCES 1. J. Bushnell. Water and Power: Hydroelectric Resources in the Era of Competition in the Western US. POWER Conference on Electricity Restructuring. University of California Energy Institute, 1998. 2. T.J. Scott, and E.G. Read. Modeling hydro reservoir operation in a deregulated electricity market Int. Trans. Oper. Res., 3: 243-253, 1996. 3. I. Otero-Novas, C. Meseguer, and J.J. Alba. A Simulation Model for a Competitive Generation Market. IEEE Power Engr. Soc., Paper PE-380-PWRS-1-09-1998. 4. A. Ramos, M. Ventosa, and M. Rivier. Modeling competition in electric energy markets by equilibrium constraints. Utilities Policy, 7(4): 233-242, 1998. 5. M. Ventosa, A. Ramos, and M. Rivier. “Modeling Profit Maximization in Deregulated Power Markets by Equilibrium Constraints.” PSCC Conference, Norway, 1: 231-237, 1999. 6. M. Rivier, M. Ventosa, and A. Ramos. A generation operation planning model in deregulated electricity markets based on the complementarity problem. ICCP99 Conference, Wisconsin, 1999. 7. B.F. Hobbs. “LCP Models of Nash – Cournot Competition in Bilateral and POOLCO– Based Power Markets.” In Proc. IEEE Winter Meeting, New York, 1999. 8. J.Y. Wei and Y. Smeers. Spatial oligopolistic electricity models with Cournot generators and regulated transmission prices. Oper. Res., 47(1): 102-112, 1999. 9. J. Garcia, J. Roman, J. Barquín, and A. Gonzalez. “Strategic Bidding in Deregulated Power Systems.” PSCC Conference, Norway, 1: 258-264, 1999. 10. R. Baldick. The generalized unit commitment problem. IEEE Trans. Power Syst., 10(1): 465-475, 1995. 11. A. Brooke, D. Kendrick, and A. Meeraus. GAMS A User’s Guide. Boyd and Fraser, 1992. 12. W.K. Viscusi, J.M. Vernon, and J.E. Hamington. In Economics of Regulation and Antitrust, ed, The MIT Press, 1998.

Chapter 14 OPTIMIZATION-BASED BIDDING STRATEGIES FOR DEREGULATED ELECTRIC POWER MARKETS

Xiaohong Guan Harvard University, on leave from Xian Jiaotong University, China

Ernan Ni and Peter B. Luh University of Connecticut

Yu-Chi Ho Harvard University

Abstract:

1.

Deregulation of the electric power industry worldwide raises many challenging issues. Aiming at these challenging issues and using California and New England power markets as background, this chapter focuses on the methodologies for integrated generation scheduling and bidding strategies for deregulated electric power markets. We present a systematic bid selection method based on ordinal optimization for obtaining “good enough” bidding strategies for generation suppliers. A stochastic optimization method for integrated bidding and scheduling is developed with consideration of risk management, self-scheduling requirements, and the interaction between different markets.

INTRODUCTION

The electric power industry worldwide is experiencing an unprecedented restructuring. In the United States, California was the first state to establish a deregulated power market starting in April 1998. Since then, almost every state has or is deregulating its power industry [1-3]. This chapter summarizes the methodologies developed by the authors and the results achieved so far in dealing with some challenging issues on integrated resource bidding and scheduling in deregulated electric power markets. In terms of the structure of resource allocation and scheduling, current power markets can be classified into two types: individual and pool. The Cali-

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fornia market belongs to the first kind. It includes a Power Exchange (PX) managing the “day-ahead” and “hour-ahead” energy markets, an Independent System Operator (ISO) handling real-time balancing, reserve and other ancillary markets, and various energy and service suppliers and demanders. The structure and functions of ISO and PX have been described in [3-4]. In the day-ahead energy market, a power supplier submits to the PX piece-wise linear and monotonically increasing power-price “supply bid curves” for each generator or for a portfolio of generating units, one for each hour of the next day. On the other hand, an energy service company submits to the PX an hourly power-price “demand bid curve” reflecting its forecasted demand. The PX aggregates supply and demand bid curves to determine a “Market Clearing Price” (MCP) and “Market Clearing Quantity” (MCQ) as shown in Figure 1. The power to be awarded to each bidder is then determined based on the individual bid curves and the MCP. All the power awards will be compensated at the MCP. After the auction closes, each supply bidder aggregates its power awards as its system demand, and performs a traditional unit commitment or hydrothermal scheduling to meet its obligations at minimum cost over the bidding horizon. The ISO will check if the schedules submitted by suppliers can be implemented through the transmission grid by performing a power flow calculation and modify the schedules by calling some must-run units as ancillary service. In this case, the constrained MCPs will be re-determined and would be different for different areas. As pointed out in [1], suppliers’ bidding decisions are coupled with generation scheduling since generator characteristics and how they will be used to meet the accepted bids in the future have to be considered before bids are submitted. Therefore, bidding decisions must consider the anticipated MCP, generation award and costs, and competitors’ decisions and other complicating factors, such as transmission constraints of the power grid.

The power market in the United States New England region is formed based on the New England Power Pool and belongs to the second type [5-6].

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Organizationally, the functions of PX and ISO are combined under the auspices of a single ISO. Unlike the separate and sequential energy and ancillary service markets in California, the energy bids are integrated with other service bids such as reserve and AGC. In addition, since the ISO has all the system operational parameters of each generator, it clears the MCP and other market prices by performing unit commitment or generation scheduling for the whole power system in the market based on the power-price bid curves received. Although the capacity of each unit has to be bid, an energy supplier may “withhold” or self-schedule some capacity to meet some percentage of its own load or to fulfill bilateral transactions with other market participators by bidding zero or negative prices. Another difference is that a bid curve in the New England market is a staircase or piece-wise constant function rather than a piece-wise linear function. Many challenging issues arise under the new competitive market structure. Instead of centralized decision-making in a monopoly environment as in the past, many parties with different goals are now involved and competing in the market. The information available to a party may be limited, regulated, and received with time delay, and decisions made by one party may influence the decision space and well-being of others. These difficulties are compounded by the underlying uncertainties inherent in the system, such as the demand for electricity, fuel prices, outages of generators and transmission lines, tactics by certain market participants, etc. Consequently, the market is full of uncertainty and risk. The recent experience learned from the many markets has shown that MCPs are volatile and often out of the range of bidders’ expectation. How to handle MCP volatilities, how to manage uncertainties and risks, and how to allocate the generating capacity into different markets have become extremely important under the new market environment. Aiming at these challenging problems and using the California and New England power markets as background, we focus on the methodologies for optimizing bidding strategies. Since bidding problems are multi-person, game theoretic problems generally associated with inherent uncertainties and computational difficulties, it is more desirable to ask which decision is better as opposed to seeking an optimal solution. In our research, we develop two bidding methods based on the structures and rules of two actual power markets: the United States California and New England markets. We first concentrate on a systematic bid selection method based on ordinal optimization to obtain “good enough bidding” strategies for generation suppliers. We then present a stochastic optimization method for integrated bidding and scheduling with self-scheduling constraints. The risks in supply bidding are managed in a systematic way. We also explore the interactions between energy and other ancillary service markets. Numerical testing shows that the algorithm is efficient for daily bidding and scheduling.

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2.

The Next Generation of Unit Commitment Models

LITERATURE REVIEW

Many approaches have been reported in the literature to address the structures and mechanism of deregulated power markets. Prior to the deregulation in the United States, the market structure model discussed most is the “British Model” [2]. The California model is presented in [3]. The structure of the New England market is described in [5] and its ISO’s energy and ancillary service dispatch problem is presented in [6]. Some recent studies on market mechanism are primarily concerned with market analysis and market power issues [7-8]. Game theory is a natural platform to model a gaming environment where each participant is determined to maximize its profit [2, 9-12]. Optimal bidding strategies to maximize a bidder’s profit based on the pool model of England and Wales were presented in [2] under the assumption that any particular bid has no effect on the MCP. For a market where a bid consists of start-up cost, variable price, and generator capacity, it was demonstrated that profit is maximized by bidding each generator at its physical cost curve and maximum capacity. This is done by showing that such a strategy is a “Nash equilibrium” for the market. The perfect conditions assumed in [2], however, may not be true. Matrix games have been reported in [11] and [12]. Bidding strategies are discretized, such as “bidding high,” “bidding low,” or “bidding medium,” and an ”equilibrium” of the “matrix bidding game” can be obtained. The strategic gaming behaviors and how the market structure affect the competition is analyzed in [9]. It is shown that the strategic behavior on electric network may produce unexpected results from the traditional economic theory. Game theory is used in [13] to minimize the risk in bidding problems. Various other methods for solving bid selection problems at different levels of the market have also been discussed. In [14], a bid clearing system in New Zealand is presented. Detailed models are used, including network constraints, reserve constraints, and ramp-rate constraints, and linear programming are used to solve the problem. Other approaches addressing various aspects of generation and ancillary service bidding can be found in [15-16], where Lagrangian relaxation, and decision trees were used to analyze and support the bidding process. For example, a bidding strategy considering revenue adequacy was presented in [17] based on Lagrangian relaxation and an iterative bid adjustment process. However this process may not be available for the current California PX market. A bidding method considering the uncertainties of other bidders, the ISO’s bid selection process and selfscheduling in New England power market is presented in [18]. The problem is solved within a simplified game theoretical framework. The exact “gaming” phenomenon among bidders however is not captured. Bidding behaviors under a simple auction market are studied in [19]. The results show that power suppliers would tend to bid above their production costs to hedge against the

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possibility of winning on the margin. The factors affecting bidding strategies in Australia power market are analyzed in [7]. More recent work on bidding strategies concentrates on the adaptive method, i.e., reacting to inputs, based on genetic programming and finite state automata [31]. It should be noted that Lagrangian relaxation is a very successful price based method for hydrothermal scheduling [20], and the framework is also very useful in dealing with the new integrated bidding and scheduling issues [4]. Recently, an intelligent computational method – Ordinal Optimization (OO) has been developed to solve complicated optimization problems possibly with uncertainties [21-22]. Ordinal optimization is based on the following two tenets. First, it is much easier to determine “order” than “value.” To determine whether A is larger or smaller than B is a simpler task than to determine the value of (A-B) especially when uncertainties exist. Second, instead of asking the “best for sure,” we seek the “good enough with high probability.” Softening the goal of optimization should make the problem easier. A bid selection strategy is developed to generate good enough bids based on the framework of ordinal optimization [23]. It can be seen from above that tools to support integrated bidding and scheduling process of deregulated power markets are far from satisfactory in view of the inherent complexity (multiple participants with their own objectives in a dynamic and uncertain environment) and the sizes of practical problems (tens or hundreds of generators with various constraints). High quality and computationally efficient approaches are critically needed to address the new challenges.

3.

ORDINAL OPTIMIZATION-BASED BIDDING STRATEGIES

Assume the bidding strategy is developed for an energy or generation supply company E. Suppose there are I generating units in E. Supply bids can be submitted for individual units or an aggregated bid can be submitted in the PX market. The bidding objective for E is to select its bid curves to maximize its profit over a time horizon T, i.e.,

where Aggregated energy (generation)-price supply bid curve of E for hour t; Generation and demand bids of other bidders unknown

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to E for hour t; Generation cost for delivering generation award Power generation award for E depending on the bid of E and the bids of other bidders; T: Bidding time horizon (24 hours for the day-ahead market); and Market clearing price (MCP) determined by the aggregated supply bid curve and the aggregated demand bid curve of all market participants as shown in Figure 1. The bid curves of other participants are assumed to be fixed, and we are only interested in the influence of E’s own bids on the MCP, which will be modeled later. According to the PX rules, if a supplier bidder awarded it will be compensated by the dollar amount regardless of the original bid submitted by that supplier. Start-up costs should be accounted for in bid curves since there is no direct start-up compensation. The problem described in (1) is thus an optimization problem to determine the optimal supply bid curves to maximize the profit subject to relevant operating constraints, such as the minimum down-/up-time, ramp-rate constraints, etc. Since the PX rules require bid curves to be piece-wise linear and monotonically increasing, searching the optimal bidding strategy is to determine the corner points of the bid curves. Note that MCPs are determined by the bids submitted by all the bidders, and when submitting the bids, a bidder does not know the bid curves submitted by others. Once the PX determines the MCP, the generation award for individual units are determined as the intersection from their bid curves or for the generation supply company E from its aggregated bid curve with as the given price. Although E can submit bids for each individual unit, the PX view total generation award to E as obligation. Given a total generation award, it is desirable for E to schedule or to reallocate all generating units across the bidding horizon to deliver its total award at the minimal cost. This can be formulated as a traditional unit commitment or hydrothermal scheduling problem with the total generation award as system demand as follows:

subject to

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and other individual operating constraints such as minimum down-/up-times described in [23-26], where C: Total generation cost over the entire bidding horizon; Cost for generating for unit i at hour t; Start-up or shut-down cost associated with the up-/downstate transitions for unit i to generate at hour t; Power generation of unit i at hour t; and Generation award for unit i at hour t according to unit’s individual bid curve. If we know the market-clearing price (MCP) and consequently know the generation award, we can re-write the profit calculation of (1) as

To select a good bidding strategy, it is necessary to identify two situations: 1) the participant is small and has no significant influence on the MCP; 2) the participant has market power and can influence the MCP. In the first case, it is desirable to find a nominal bid curve that would maximize its profit at any given MCP. For the second case, it is necessary to establish the relationship between one’s bidding strategy and the MCP. One approach is to obtain the MCP by simulating market participants’ bidding strategies. By changing one’s own bidding strategies, it is possible to establish the influence of bidding strategies on the MCP. Another method is to establish an influence function by experience or by regression through one’s historical bids and the MCPs. Either way, assume the following influence functions:

where

is the nominal MCP forecasted by the historical

MCPs the forecasted system demand and other forecasted factors such as fuel prices, etc. The nominal MCP forecasting using an ANN model has been reported in our recent work [30]. The influence function is determined by the difference between one’s bid curve and the nominal bid curve and can also be established by regression method based on the historical MCPs and records of one’s own bids plus bidders’ experience. The advantage the ordinal optimization method is that it has no restriction on any model used to describe bidding strategy influence on the MCP. In

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fact (5) can be replaced by a Monte Carlo simulation procedure or other game-theoretic models. The basic idea of the ordinal optimization method includes the following steps: 1. Generating a nominal bid curve by using Lagrangian relaxation method for hydrothermal scheduling; 2. Generating N bid curves by perturbing the nominal bid curve and obtaining the MCPs associated with these bids; 3. Generating a “good enough” select bid set by evaluating these N bids using a very crude model with little computation effort, and ranking and selecting them based on ordinal optimization; and 4. Evaluating the bids in the select bid set using an accurate model and solving the computationally time-consuming hydrothermal scheduling problem (2), and then selecting the best one. The above four steps are briefly described next (see [23] for details). It should be noted that reflecting the transmission constraints of the power grid in bids is very difficult and has not been considered in the above bid selection procedure. In fact, a unit can provide must-run service when the transmission grid is limited by its transfer capability versus providing energy and other services. Resource allocation or asset allocation among different energy and service markets is an important issue that will be partially addressed in the next section.

3.1

Generating the Nominal Bid Curve

A nominal bid curve should serve two purposes: 1) it is a basis in (5) to define the relationship between bid curves and the MCP; and 2) it is an optimal bid curve if bidding strategy has no influence on the MCP. That is, if E is a “price-taker,” the nominal bid curve should maximize E’s profit for any MCP determined by the market. To achieve this goal, let a set of MCPs be given as bid prices

and the optimal generation for an indi-

vidual unit to maximize its profit can be obtained based on (6):

subject to the individual operating constraints, where k is the point index on a bid curve. For the given series of

(6) is a subprob-

lem within the unit commitment or hydrothermal scheduling context when solved using the Lagrangian relaxation technique, where can be viewed as the Lagrange multipliers given by the high level problem. The problem can

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thus be efficiently solved by dynamic programming as in our previous work [24-25]. The results are the optimal generation levels for all units at each hour. The nominal bid curves for individual units and the aggregated bid curve for E are thus generated as

and

Based on the procedure where the nominal bid curve is created, we see that if the MCP determined by the market is equal to the generation award would maximize the profit of an individual unit as in (6).

3.2

Perturbing the Nominal Bid Curve The nominal bid curve given above is perturbed to generate N bid curves

as

where

is a perturbation function. A simple way to implement is to keep the power generation of a bidding generation point

same and uniformly sample

the

in the neighborhood

so that

always specifies a monotonically

increasing piece-wise linear bid curve as required by the PX. Based on a perturbed bid curve the corresponding estimated by using (5) can be obtained.

3.3

Selecting Good Bids

The N bid curves obtained in (7) can be evaluated and ranked by ordinal optimization. The estimated profit of each set of bid curves is calculated as T

I

Note (9) is just a rough profit evaluation with the MCP given but without

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solving scheduling problems. It is assumed that the generation award to a unit based on its bid curves will also be delivered by that unit. This may not be true since the generation award to a unit may not even satisfy the intertemporal individual operating constraints. A generation company will schedule or re-allocate all its units to meet its total generation award with the minimal total generation cost for delivering its total obligation and satisfying every individual operating constraint of its units. The true profit can be obtained by solving (2)-(4), and there may be significant error due to the above approximation where is error. The advantage of the ordinal optimization method is its capability to separate the good from bad even with a very crude model, namely, the performance “order” is relatively immune to large approximation errors. Even if the rough estimation is used to rank N bid curves, some good enough bids will be kept within the select set with high probability. The major task in applying ordinal optimization is to construct the selected subset S containing “good enough” bids with high probability, including the determination of its size The quantitative measure of the “good enough” is the alignment probability defined as where G is the “good enough” bid set and a is called the alignment level. Intuitively the alignment probability is the probability of the event that there are at least a elements in the good enough set G matched in the select set S. To select s good ones from the N perturbed bids generated by (7), the profits are estimated by (9) and ranked. The top s bids are then selected as S and its size s is determined by a regressed nonlinear equation to satisfy certain confidence requirement ([27]). The value of s can be estimated by where is the size of G and and are coefficients or parameters obtained by nonlinear regression. Evaluating N bidding strategies using (9) is computationally efficient and the ordinal optimization method can guarantee that good enough bids will be among the s selected strategies. The result of [27] tells us how large s should be. We apply a more accurate, but time-consuming, evaluation that requires solving generation scheduling problems to evaluate the s selected bids. For each bid with the generation award, a traditional generation scheduling or unit commitment problem described by (2) and (3) is solved to calculate E’s profit using the Lagrangian relaxation-based algorithm in ([24-25]). The best bid is then selected from by evaluating those bids in the subset S. Since is much smaller than N, the ordinal optimization method is extremely efficient

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in comparison with the brute force method of solving the N scheduling problems.

3.4

Numerical Testing

We perform numerical testing is performed for the California day-ahead energy market based on a 10-unit generation company and historical PX market clearing prices as the nominal MCPs. For testing purposes, we use the published MCPs for May 1, 1998, and January 4, 1999, on the California PX day-ahead energy market as the nominal MCPs in (5). The bidding influence function in (5) is simplified as a linear function just to demonstrate the effectiveness of the ordinal optimization method. The parameters for ordinal optimization are selected as follows. Search space size: N= 1000; Alignment probability: Good enough set: G = top 50 bids among N bids, i.e., g = 50; Alignment level: To observe “order is relatively immune from error” claimed by the ordinal optimization method, the size of intersection of S and G, is listed in Table 1. The ordinal optimization method tells us that at least 5 bids in S should be also be in G. It is observed from Table 1 that sizes of the intersections of G and S are greater than 5 for all cases. Therefore, the good enough bids will not “slip away” from select set S because of very crude approximation and the ordinal optimization method is very effective. The testing results also show that for the four cases tested the best bid is selected among the N perturbed bids. The computational time is reduced from 11 hours to about 40 minutes in comparison with the brute-and-force evaluation method. For details see [14].

4.

INTEGRATED BIDDING AND SCHEDULING WITH RISK MANAGEMENT

This section presents an integrated bidding and scheduling problem for maximizing the profit of a power supplier while reducing bidding risks. The method is generic, but the problem considered here is for the New England

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market. As mentioned in the introduction, the MCP may be volatile in view of the underlying uncertainties in the market. It is, therefore, important to deal with the volatility and to manage the risks. Another issue is how to allocate the generating capacity among different markets such as the energy and reserve markets. First we present a formulation of integrated bidding and scheduling. We assume the “perfect market” where the MCP is not affected by any single bid and assume the MCP series to be a Markov chain. The risk management is explicitly modeled, and the reserve market and the self-scheduling requirements are also considered. The problem is formulated as a mixed-integer stochastic optimization problem with separable structure. To solve this problem, we develop a Lagrangian relaxation-based method to decompose the problem into a number of subproblems, one for each unit. Then we apply stochastic dynamic programming to solve individual unit subproblems, and generate a set of generation strategies: how much power should each unit provide for each possible market price value at what probability? We then create the bid curves based on these generation strategies. Numerical testing base on an 11unit system including a large pumped-storage unit shows the algorithm can generate good energy and reserve bid curves in 4-5 minutes on a P-III/600 PC. The results also demonstrate how a pumped-storage unit affects the bidding of thermal units and how the risk management and the reserve market affect the bidding in the energy market.

4.1

Problem Formulation

Consider a utility company in New England with I generators of thermal, hydro, and pumped-storage units. The problem is to select bid curves for individual units to maximize the profit while reducing risks. The formulation involves market assumptions, the objective function to be minimized, and the constraints to be satisfied. In this section, the “perfect market” assumption is assumed, i.e., the MCP is not affected by a single market participant. Since there exist various timedependent operating constraints such as minimum up-/down-times and ramp rate limits, the bid curves and thus the MCPs between hours are coupled. To model time dependence of the MCPs, the MCP series is assumed to be a Markov chain. This means that the MCP at hour only depends on that of the previous The time independent MCP series can be considered as a special case of the Markov chain. In practice, a “perfect market” may not exist, and the MCP is usually affected by the bids. This issue will also be discussed in this section. As mentioned in several previous sections, the bid curve for unit

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i at time t is a power-price function indicating how much power the company wants the unit to generate at any given price. Based on the bid curve, if the MCP is then unit i will be awarded the amount of power Therefore, to determine the bid curve for unit i is to decide a generation level for any possible price The form of bid curves may be different in different markets. In the New England market, up to 10 power blocks, each with a price, form a bid. Similar to the procedure of generating the nominal bid curve presented in section 3.1, for each hour are given based on the MCP forecasting. The transition probabilities between and are also known and will not be affected by any particular bidder. The same is also assumed for the reserve prices We can construct the bid curves by obtaining the optimal generation for each given (k= 1, 2, ..., K), and the optimal reserve for each reserve price (j=1, 2, ..., K) for each unit i and any hour t. The utility may have its “own load” and the associated reserve requirement R(t) at hour t. We ignore the own load uncertainties since they are usually accurate to within 2% and are less significant than the MCP uncertainties. As a strategic decision for reducing the risks, the company may want to cover by itself, on an average, at least a certain percentage of its own load and reserve requirements. This “self-scheduling requirement” can be formulated as the following constraints on expected and reserve price

and

where and are the self-scheduling coefficients. The algorithm would be just simpler for the cases without considering the selfscheduling requirements. Currently there is no demand market in New England. Therefore, if the MCP at hour t is

and the reserve price is

then

is the amount of the energy that the company has to buy (positive) from or sell (negative) to the market at

and

the company has to buy (positive) or sell (negative) at

the amount of reserve The profit equals

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the revenue from the energy and reserve markets minus the operation costs. The objective is to maximize the expected profit or equivalently to minimize negative expected profit, which requires the utility to sell power at hours with high prices and buy power at times with low prices. With known transition probabilities of market prices, the MCP variances and reserve price and

reflecting the market uncertainties at hour t can be calculated. Ac-

cording to historical observation, the MCP may jump from its normal value of 20 or 30 dollars to over 1000 dollars per megawatt hour. A company usually prefers to sell power when the market has large uncertainty and buy power when the market has low uncertainty so as to avoid risks. Therefore, bidding risks can be reduced by penalizing the product of price variances and the purchased amount of energy. Combining the above analysis, the objective to be minimized is a weighted sum of the negative profit (the first part) and the risk terms (the second part), i.e.,

where is a weight to balance the profit versus risks. The problem is also subject to individual unit constraints such as minimum up-/down-time and ramp rate limits for thermal units, and reservoir dynamics and volume limits for hydro and pumped-storage units. The constraints for thermal and hydro units have been presented in [24, 25, 28], and those for pumped-storage units were presented in [26] and [29].

4.2

Solution Methodology

The above integrated bidding and scheduling problem is a mixed-integer stochastic optimization problem. It is very difficult to obtain the optimal solution in view of its computational complexity. One way to solve the problem is to perform scenario analysis. Since the number of scenarios may be very large, scenario analysis is inefficient. In view of the separable structure of the problem, Lagrangian relaxation is a very efficient approach, and we apply it to solve the problem. By taking (14) and (15) as the demand and reserve requirements and relaxing them using two sets of multipliers and respectively, a two-level optimization is formed. The high level is to update the multipliers so as to maximize the dual. It is formulated as

Optimization-Based Bidding Strategies

263

where

and

Given multipliers and the low level problem consists of I subproblems, one for each generator. The subproblem i is formulated as min

subject to individual unit constraints. These two levels iterate until a convergence criterion is achieved. Based on the subproblem solutions, the bid curves are constructed after the dual procedure terminates. We briefly describe these steps next. For details, refer to [29]. 4.2.1

Solving Subproblems

The above subproblem (20) is similar to that of a hydrothermal scheduling problem as in [24, 25, 28], where and can be interpreted as the energy and reserve marginal costs of the system at hour t, respectively. However, and are random variables depending on market prices and The subproblem solution is a set of strategies rather than a generation level for each hour. It establishes a relation of and the reserve price with a probability to generate a certain quantity. The subproblems for different type units are different. For example, there are usually no fuel cost and start-up cost for hydro and pumped-storage unit subproblems. In view of the combinatorial nature and the large number of possible MCP realizations, it is hard to solve the subproblem (20). Based on the transition probabilities of a Markov chain, a stochastic dynamic programming (SDP) is developed, in which a stage corresponds to an hour, and a state at hour t is associated with a pair of and (k, j= 1, 2, ..., K). Constraints that couple the decisions at different hours, such as ramp rate and pond limit constraints, are relaxed by using an additional set of Lagrangian

264

The Next Generation of Unit Commitment Models

multipliers so that the problems are stage-wise decomposable. We then obtain the optimal generation level and reserve level for each state by optimizing the stage-wise cost function, subject to feasible operation regions. Since the coefficients of the stage-wise cost function are determined by and depends on and and is generally an increasing function of

The probability

for each

state is calculated based on the transition probabilities of the Markov chain. The subproblem solution here is to provide a set of strategies: for each pair of and each unit should provide amount of power with probability

We will use these strategies in constructing

bid curves. 4.2.2

Solving the Dual Problem

After solving the subproblems, the sub-gradient associated with (14) can be obtained as

Then the multipliers can be iteratively updated by using a sub-gradient or bundle method. The updating of is similar. 4.2.3

Generating and Selecting Bid Curves

After the dual procedure converges, the bid curves for each unit are constructed based on the strategies obtained in the subproblem solutions. First, we calculate the amount of power to be submitted for each (k=1, 2, ..., K) as the expectation of over for j=1, 2, ..., K. Since the operation regions of a unit may be discontinuous, the result obtained is then projected into a feasible operation region to get the optimal power amount. This procedure is

Then we construct the bid curve for unit i at home t. As mentioned, the bid curve required by the New England market is a staircase function. The size of its power block equals and the price is as

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The reserve bid curves can also be constructed similarly. In view of the existence of integer decision variables such as the unit commitment decision, the maximum dual function does not mean the best primal solution. Therefore, the ordinal optimization described in Section 3 is applied to sort the bid curves generated at the last several iterations. The best one is selected as the final set of bid curves. 4.2.4

MCP Influenced by Bidding Strategies

In the above, we model the MCP series as a Markov chain under the “perfect market.” As mentioned in section 3, however, the MCP may be influenced by the bids to be determined. To estimate the impact of bidding behaviors on the MCP, a neural network model for predicting the MCP is under development. The idea is that in addition to the market information such as forecasted market demand and MCP lags [30], the aggregate bid curves are also part of the inputs to the neural network, and the sensitivity of the MCP versus the bid curves can thus be analyzed. This model is to be integrated with the above algorithm and the MCP distribution will be updated based on the bid curves generated at each iteration.

4.3

Self-scheduling, Market Interactions, and Risk Management

Since plays the role of marginal energy cost and the marginal reserve cost, their values determine the bid curves for each unit. From (18)-(20) and (22), one can see the impact of self-scheduling constraints and risk management on bidding curves, and the interaction of the energy and reserve markets. The self-scheduling constraints couple the different units via and and affect the marginal and as in (18) and (19). For example, if the self-scheduling requirement at hour t is high, will be large, causing large Consequently, the units have to bid the same amount of energy at a lower price to satisfy the self-scheduling constraints. The tradeoff in allocating the limited generation capacity in the energy and reserve markets depends on the market prices and In the normal situation, where is much higher than the profit largely depends on the MCPs for energy. Occasionally, when the reserve price is relatively high as observed in many markets, more capacity should be allocated to the reserve market. This may result in significant changes in the energy bidding strategies, especially for pumped-storage units. A pumped-storage unit pro-

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The Next Generation of Unit Commitment Models

vides large reserves at either pumping or generating. In view of the reservoir dynamics and limits, the bidding strategies are coupled across hours. That is, decisions in one hour will affect results of other hours. Furthermore, the pumped-storage bidding strategies also affect thermal bidding strategies in view of self-scheduling constraints. Therefore, the pumped-storage units play a key role in both the energy and reserve markets. The risk management affects bidding strategies through and related to the variances large and

and

If

is high, uncertainty on

is large. Then the optimal generation level

is ob-

tained in DP is large, resulting in large as in (22). As a result, the units will bid the same amount of energy at low prices so that large amounts of energy could be sold on the market or buying large amounts of power at potential high MCP could be avoided.

4.4

Numerical Testing

We perform numerical testing for the New England energy and reserve markets based on a system with 10 thermal units and a 4-identical-generator pumped-storage system with a large reservoir and the historical market prices. To investigate the influence of the pumped-storage units on thermal bidding strategy, we test the pure thermal system. To investigate the influence of the reserve market on the bidding strategies, we also perform testing where the reserve market is ignored. The testing results are summarized below. For detailed analysis refer to [29]. The results show that the pumped-storage unit plays important roles in both the energy and reserve markets and can significantly change the thermal bid curves. For example, they pump at hours with low MCPs, making the thermal units bid at a low price to satisfy the high self-scheduling requirements and generate at hours with high MCPs, reducing the self-scheduling requirement for the thermal units. Risk management proves to be an effective way to avoid buying large amounts of power from the market at potentially high MCPs and to reduce the profit variances. In section 4.3, we explained why the bid price should be low when the MCP variance is high. In this case, it is observed that the generation award is large. The significant influence of the reserve market on the bidding strategies is also observed. Figure 2 shows the comparison of pumped-storage bid curves for hour eight on July 12, 1999, with and without considering the reserve market. The reserve market is considered in Case 1, but ignored in Case 2. Figure 3 gives the MCPs and reserve prices. According to the market rules,

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267

the reserve contribution of a pumped-storage generator is same as the pumping level or the one-line generation capacity minus the current generation level. In view of the high reserve price at hour eight, each generator of the pumped-storage system should either pump at its capacity or generate at its minimum generation limit so as to provide maximum reserve for maximizing the profit. Figure 2 depicts these two bidding strategies, and the bid curves are significantly different. The CPU time of the algorithm is about four to five minutes on a 600 Hz Pentium III PC. Therefore, the algorithm is efficient for daily bidding and scheduling.

268

5.

The Next Generation of Unit Commitment Models

CONCLUSIONS AND ON-GOING WORK

In this research, we developed effective methods for dealing with the new challenging problems in deregulated electric power markets. We created an ordinal optimization based bidding strategy to select “good enough” bids for a power supplier. Numerical results show this method is efficient and can yield good bids. We developed a stochastic optimization approach for integrated bidding and scheduling with the consideration of risk management, selfscheduling requirements, and interaction between energy and reserve markets. Numerical testing results based on an 11-unit system of the New England market show that the algorithm is efficient for daily bidding and scheduling. The bidding strategies of pumped-storage units have significant influence on the bidding strategies for thermal units and play important roles in both the energy and reserve markets. On-going work includes simulation and forecasting of the market indicators such as the MCP, game theoretic modeling and analysis of the MCP price spikes and bidding strategies, and integration of forecasting and bidding methods.

ACKNOWLEDGEMENTS The research reported in this chapter is supported in part by EPRI/ARO Contract WO833-03, National Science Foundation under Grant ECS9726577; National Outstanding Young Investigator Grant 6970025; and a Key Project Grant 59937150, National Science Foundation of China and 863 Project of China. The authors would like to thank Dr. David Pepyne of Harvard University for his valuable insight and comments.

REFERENCES 1. 2. 3. 4.

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W. Dunn, M. Rossi, and B. Avaramovic. Impact of market restructuring on power systems operation. IEEE Comput. Appl. Power Engr., 8(1): 42-47,1995. G. Gross and D. J. Finlay. “Optimal Bidding Strategies in Competitive Electricity Markets.” In Proc. Power Systems Computation Conference, Dresden, 1996, pp. 815-822. A. Farrokh and Z. Alaywan. California ISO formation and implementation. IEEE Comput. Appl. Power Engr., 12(4): 30-34,1999. X. Guan and P.B. Luh. Integrated resource scheduling and bidding in the deregulated electric power market: New challenges. Special Issue, J. Discrete Event Dyn. Syst. 9(4): 331350, 1999. P.A. Fedora. “Development of New England Power Pool’s Proposed Markets.” In Proc. Hawaii International Conference on System Sciences, Maui, Hawaii. 1999. K.W. Chenug, P. Shamsollahi, and D. Sun. “Energy and Ancillary Service Dispatch for the Interim ISO New England Electricity Market.” In Proc. International Conference on Power Industry Computer Applications, Santa Clara, California, 1999, pp. 47-53.

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W. Mielczarski, G. Michalik, and M. Wildjaja. “Bidding Strategies in Electricity Markets.” In Proc. 21th International Conference on Power Industry Computer Applications, Santa Clara, California, 1999, pp. 71-76. T. J. Overbye, J.D. Weber, and K. J. Patten. “Analysis and Visualization of Market Power in Electric Power Systems.” In Proc. Hawaii International Conference on Systems Science, Maui, Hawaii, 1999. C.A. Berry, B.F. Hobbs, W.A. Meroney, R.P. O’Neill, and W.R. Stewart. “Analyzing Strategic Bidding Behavior in Transmission Networks.” In IEEE Tutorial on Game Theory Applications in Electric Power Markets, TP-136-0: 7-32, 1999. Owen G. Game Theory, Third Edition, Academic Press, 1995. V. Krishna and V.C. Ramesh. “Intelligent Agents for Negotiations in Market Games, Part I and Part II.” In Proc. International Conference on Power Industry Computer Applications, Columbus, Ohio, 1997, pp. 388-399. R.W. Ferrero, S.M. Shahidehpour, and V C. Ramesh. Transaction analysis in deregulated power systems using game theory. IEEE Trans. Power Syst., 12(3): 1340-1347, 1997. H.Y. Yamin and S.M. Shahidepour. “Risk Management Using Game Theory by Transmission Constraint Unit Commitment in a Deregulated Power Market.” In IEEE Tutorial on Game Theory Applications in Electric Power Markets, TP-136-0: 50-60, 1999. T. Alvey, D. Goodwin, X. Ma, D. Streiffert, and D. Sun. A security-constrained bidclearing system for the New Zealand wholesale electricity market. IEEE Trans. .Power Syst., 13(3): 986-991, 1998. S. Dekrajangpetch, G.B. Sheble, and A.J. Conejo. Auction implementation problem using Lagrangian relaxation. IEEE Trans. Power Syst., 14(1): 82-88, 1999. G.B. Sheble. “Decision Analysis Tools for GENCO Dispatchers.” IEEE/PES Summer Meeting, San Diego, California, 1998, PE-231-PWRS-0-06-1998. C. Li, A. Svoboda, X. Guan, and H. Singh. Revenue adequate bidding strategies in competitive electricity markets. IEEE Trans. Power Syst., 14(2): 492-497, 1999. D. Zhang, Y. Wang, and P.B. Luh, “Optimization-Based Bidding Strategies in the Deregulated Market.” In Proc. International Conference on Power Industry Computer Applications, Santa Clara, California, 1999, pp. 63-69. S. Hao. “A Study of Basic Bidding Strategy in Clearing Pricing Auctions.” In Proc. International Conference on Power Industry Computer Applications, San Clara, California, 1999, pp. 55-60. A. Cohen and V. Sherkat. “Optimization-Based Methods for Operations Scheduling.” In Proceedings of IEEE, 75(2): 1574-1591,1987. Y. C. Ho. “Soft Optimization for Hard Problems.” Web-distributed copy of a tutorial for ordinal optimization, http://www.hrl.harvard.edu/people/faculty/ho/DEDS, 1997. Y.C. Ho. On the numerical solution of stochastic optimization problems. IEEE Trans. Autom. Cont, 42(5): 727-729, 1997. X. Guan, Y.C. Ho, and F. Lai. An ordinal optimization-based bidding strategy for electric power suppliers in the daily energy market. Submitted, 1999. X. Guan, P.B. Luh, H. Yan, and J.A. Amalfi. An optimization-based method for unit commitment. Int. J. Electric Power Energy Syst., 14(1): 9-17, 1992. X. Guan, E. Ni, R. Li, and P.B. Luh. An algorithm for scheduling hydrothermal power systems with cascaded reservoirs and discrete hydro constraints. IEEE Trans. Power Syst., 12(4): 1775-1780, 1997. X. Guan, P.B. Luh, H. Yan and P.M. Rogan. Optimization-based scheduling of hydrothermal power systems with pumped-storage units. IEEE Trans. Power Syst., 9(2): 10231031, 1994. T.W.E. Lau and Y.C. Ho. Universal alignment probabilities and subset selection for ordinal optimization. J. Optim. Theory Appl., 39(3): 455-490, 1997. E. Ni, X. Guan, and R. Li. Scheduling hydrothermal power systems with cascaded and head-dependent reservoirs. IEEE Trans. Power Syst., 14(3): 1127-1132, 1999.

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29. E. Ni and P.B. Luh. “Optimal Integrated Bidding and Scheduling for Hydrothermal Power Systems with Risk Management and Self-Scheduling Requirements.” To appear in Proc. World Congress on Intelligent and Control, Hefei, Anhui, China, 2000. 30. F. Gao, X. Guan, X. Cao, and A. Papalexopoulos. “Forecasting Power Market Clearing Price and Quantity Using a Neural Network Method.” To appear in Proc. 2000 IEEE/PES Summer Meeting. 31. C. Richter, G. Sheble’, and D. Ashlock. Comprehensive bidding strategies with genetic programming/finite state automata. IEEE Trans. Power Syst., 14(4): 1207-1212, 1999.

Chapter 15 DECENTRALIZED NODAL-PRICE SELF-DISPATCH AND UNIT COMMITMENT

Francisco D. Galiana and Alexis L. Motto McGill University

Antonio J. Conejo University of Castilla-La-Mancha

Maurice Huneault Hydro-Quebec Research Institute (IREQ)

Abstract:

This chapter sets forth a scheme for self-scheduling independent market participants in a power pool. The approach, named DNSA for Decentralized NodalPrice Self-Scheduling Auction, is proposed as an alternative to centralized Pool auctions and operation. DNSA exploits the intrinsic parallelism of the dual unit commitment problem to decentralize the various scheduling and dispatch functions. Each competing participant (GENCO, DISTCO) maximizes its profit for any set of nodal prices by choosing its level of production or consumption. Similarly, the TRANSCO independently maximizes its merchandising surplus within the network security constraints. The price caller, a centralized entity without access to proprietary cost information, updates prices through an effective Newton algorithm until the power balance at each bus is satisfied. DNSA does not assume a perfect market and accounts for the AC load flow model including transmission losses and line congestion, in addition to integer variables, ramping rates, start-up costs, and minimum up and down times. The convergence of DNSA hinges on the notions of profit optimality and the convexifying market rule. We present several study cases to illustrate the characteristics of DNSA. We conclude that to achieve fairness of treatment for all competing participants, they should be allowed to optimize their profit by self-scheduling. Therefore, to the extent possible, the next generation of unit commitment models should include profit optimality.

272

1.

The Next Generation of Unit Commitment Models

INTRODUCTION

This chapter summarizes a number of research activities conducted at McGill University on the decentralized self-operation of electricity market participants, namely, GENCOs, DISTCOs, and TRANSCOs. A main goal of this research is to demonstrate that, under certain conditions, self-operation based on maximizing individual profits yields the same electrical and economic operating point as centralized maximum welfare operation. This equivalence is known for simple cases of economic dispatch, but is less obvious for more complex scheduling problems including unit commitment, intertemporal constraints, nonlinear transmission losses, flow congestion, and other network constraints. In particular, we will show that the equivalence between self- and centralized operation is also valid for conditions other than a perfect market. The proof of equivalence is based on purely mathematical arguments under a main assumption, here named “profit optimality,” by which we mean that all competing participants are free to maximize profit subject only to market prices. Without profit optimality, the centralized solution may require some individual participants to operate at less than maximum profit, an unavoidable consequence of the difficulty of defining a measure of social welfare.1 Therefore, we argue that market rules can be considered unfair if they can lead to solutions where some but not all participants are dispatched at maximum profit. Such a possibility must be precluded by the market rules since equality of treatment is of paramount concern among competing entities. Accordingly, in our view, profit optimality is not a theoretical issue but a desirable restriction of the next generation of unit commitment problems. Under profit optimality, the dual and primal solutions of the mixed integer unit commitment problem are shown to be identical. This then leads to the proposed decentralized, nodal-price self-dispatch and scheduling auction, DNSA, the essence of which is to optimize the dual function while simultaneously meeting the relaxed constraints of the primal problem. The DNSA comprises some key innovations centered on the application of an iterative auction. These are the introduction of a central price caller that updates and broadcasts trial prices, the delegation of the self-scheduling tasks to the participants who then respond to the trial prices, the use of a fast Newton priceupdating scheme, and the consideration of the full set of non-linear network constraints. This chapter is organized as follows: after the nomenclature, we compare the general features of centralized operation to those of decentralized selfoperation, in particular under the proposed DNSA scheme. Next, the unit 1

See [1] for a compact summary on Arrow’s impossibility theorem. Kenneth J. Arrow was a 1972 Nobel laureate in economic science.

Decentralized Nodal-Price Self-Dispatch and Commitment

273

commitment problem primal and dual forms are formulated and shown to yield identical solutions under the condition of profit optimality. The DNSA scheme is then described and compared to the classical centralized dual-based unit commitment. DNSA requires two conditions to converge, the previously defined profit optimality, together with the “convexifying” rule imposed to ensure rational participants’ behavior in response to nodal prices. We then describe the Newton-based price-updating component of DNSA. Finally, a number of numerical results illustrate the characteristics of DNSA.

2.

NOTATION

For quick reference, we classify below the main mathematical symbols used throughout this chapter. feasible space of demand i feasible space of generation i minimum down time of generator i minimum up time of generator i set of network nodes average real power injection at bus i during period average real power of load i during period average real power of generator i during period profit of load i over the scheduling horizon profit of generator i over the scheduling horizon Benefit function of load i ramp-down limit of generator i ramp-up limit of generator i shut-down cost of generator i occurring at the beginning of period start-up cost of generator i occurring at the beginning of period set of time intervals in scheduling horizon 0/1 variable, which is equal to 1 if generator i is committed during period or 0 otherwise 0/1 variable, which is equal to 1 if load i is committed during period or 0 otherwise 0/1 variable, which is equal to 1 if generator i is started up at the beginning of period or 0 otherwise

274

The Next Generation of Unit Commitment Models

0/1 variable, which is equal to 1 if generator i is shut down at the beginning of period or 0 otherwise average voltage phase angle at node i during period cumulative up/down time of generator i at the start of period average nodal price of active power at node i during period the vector over all time intervals for node i the vector over all nodes for time the vector all nodes and all time intervals

3.

MOTIVATION FOR SELF-OPERATION

3.1

Current Practices

The emerging restructured power industry is guided towards two generic trading models: pool and bilateral markets [3,4].2 In the pool context, a central agent collects the bids from sellers (GENCOs) and buyers (DISTCOs), determining the winners and the amount of power each is required to sell or buy as well as the market-clearing prices. Generally, this selection results from a centralized constrained optimization. In contrast, the bilateral model allows participants to arrange contracts among themselves. Though the transmission monopoly may create unforeseeable challenges to the implementation of bilateral transactions, some authors argue that such contracts offer participants more freedom, thereby achieving greater decentralization in decision-making [5]. In view of their widespread use, the scope of this chapter is limited to pure pool-operated markets. Figure 1 illustrates the state of the art of today’s Pool operation. The market operates with three independent entities, GENCOs, DISTCOs, and the ISO.3 Under this model, strictly speaking, there is no independent transmission provider, as the network is under the authority of the ISO. The latter sets the pool price (or nodal prices) by trying to maximize the overall social welfare, irrespective of the participants’ revenue or profit requirements. Clearly, the ISO has an important and often coercive decision-making authority, a practice inherited from the traditional centrally controlled utility industry.

2

The central buying entity for all suppliers of electricity, which in turn is the single agent for selling power to retail customers and their aggregators [2]. 3 We use the term ISO to refer to the combined power exchange and system operator.

Decentralized Nodal-Price Self-Dispatch and Commitment

3.2

275

On Decentralizing Pool Operation

Reduced reliance on the decisions of a centralized scheduling and dispatching entity is an alternative that some electricity market participants already actively seek [6,7,8]. Centralized authority has been brought into question for both technical and philosophical reasons. For example, it is accepted that solutions to optimal power flow [9] and unit commitment [10] programs can be highly sensitive to small variations in the input parameters and to the algorithm heuristics. Whereas the sensitivity of the objective function to parameter variations is usually insignificant, that of the individual participants’ responses can be more substantial, a potentially unfair result. Philosophically, it can also be argued that, in a competitive environment, the decision as to whether to trade or not and how much to trade must rest solely with each trader. Furthermore, such a choice should be guided by each participant’s private and confidential profitability expectations as well as on the market conditions. For this decision-making independence to function in the context of a power system, market prices alone must offer sufficient incentives to satisfy all network security and power balance requirements without the need for centralized intervention. One alternative to the centralized pool with the above characteristics is the DNSA scheme put forward in this chapter. DNSA is an auction mechanism [11,12] that allows each independent participant to self-commit and dispatch based on its own profit evaluation (see Figure 2).

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The Next Generation of Unit Commitment Models

It is instructive to compare the dual communication in DNSA between each participant and the auctioneer – here the Price Caller – to the one-way relation between participants and the ISO in the centralized auction of Figure 1. Under DNSA, convergence to equilibrium involves repeated interaction of all participants, with the Price Caller updating trial prices and the independent entities responding accordingly. Note that the DNSA differs from other attempts to decentralize the authority of the ISO. For example, in the approach of Griffes [13], GENCOs and DISTCOs are allowed to resubmit new bids to the ISO if the previous schedule is not to their satisfaction. In that approach, however, for each new set of bids the ISO has to repeat a fully centralized scheduling operation. The DNSA, in a similar fashion as an iterative Lagrangian relaxation algorithm [14,15], exploits the decomposability of the unit commitment problem in its dual form to achieve the goal of decentralization and selfscheduling. The next section analyzes the conditions under which dual and primal methods yield the same solutions.

4.

THE UNIT COMMITMENT PROBLEM

Traditionally, the unit commitment problem develops the on/off schedule of generating units (loads) over an operating horizon. Once committed, a generator (load) is synchronized to the grid and is ready to deliver (consume) power. The dispatch problem consists of determining the levels of production (consumption) of the committed generators (loads). In practice, however, these two problems are solved concurrently and often referred to as the unit commitment problem.

Decentralized Nodal-Price Self-Dispatch and Commitment

4.1

277

Primal Problem

The primal unit commitment problem [16], abbreviated here as PUC, can be formulated as the maximization of the social welfare,

subject to the constraints,

The objective function (1) includes generation costs, sumer benefit functions, The total operation cost, incurred by Genco(i) can be written as,

where,

is the fixed cost,

is the variable cost,

and conin ($/h)

is the

4

start-up cost, and is the shut-down cost. Note that cost and benefit data are bids. Constraints (2) implement the real power balance equation at each network node, line power flow limits are enforced by (3), consumption limitations such as budget constraints are represented by (4), while generator constraints are contained in (5). The symbols and represent the feasible regions of the variables and respectively. The set includes initial conditions, (7), output limits, (8), minimum up/down times, (11) –(12), and ramp-up/down limits, (9)–(10), that is,

4

Refer to [17] for a formal discussion of cost and consumer benefit functions.

5

A similar set of constraints could apply to

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The Next Generation of Unit Commitment Models

A few caveats must be made at this point. For the sake of clarity, we made the decision to simplify some of the models. We have attempted, however, to introduce the results in such a way that generalizations are apparent. Thus, the network optimization variables have been limited to the voltage phase angles, while still retaining the full nonlinear AC load flow. Reserve constraints and multiple-generator (-loads) GENCOs (DISTCOs) have not yet been incorporated in DNSA. The implementation of PUC for a typical power system involves a large number of 0/1 variables and numerous constraints. This problem is known to be NP-hard and, unless drastic simplifications are made, seldom tractable.6 Furthermore, the complexity of the models that are solved increases under competition since profit-driven agents have no choice but to model their systems with additional variables in order to achieve greater accuracy. In order to apply integer programming [18], in our simulations all variables are constant during the discrete time intervals and all functions are approximated by discrete piecewise-linear mappings [19,20,21].

4.2

Dual Problem

In contrast to the primal, experience demonstrates that dual methods such as Lagrangian Relaxation are quite successful at solving the unit commitment problem [22,23,15], notwithstanding the heuristic approximations that become necessary whenever the duality gap is non-zero.7 In the dual approach, relaxing the power balance constraints at every bus and for every time period yields the Lagrangian function,

6

A problem is said to be NP-hard if the zero-one integer programming problem can be mapped to it in polynomial time. 7 In such cases, the primal solution is sub-optimal and can dispatch some participants at less than the maximum possible profit for the market prices.

Decentralized Nodal-Price Self-Dispatch and Commitment

279

from which the dual unit commitment problem (DUC) can be written as

By the weak duality theorem, the dual value is a lower bound to the primal value defined by equation (1). Regardless of the structure of the objective and constraints of the primal problem, the domain of the dual function is convex and the function is concave over These elegant convexity properties combined with the problem decomposability allow a solution of the dual problem consisting of several reduced sub-problems that can be solved in parallel as outlined next.

4.3

Self-Operation Sub-Problems

The three components in the right-hand-side of equation (18) can be interpreted as monetary objectives to be maximized by each type of entity for the given prices. 4.3.1

The GENCO Profit Maximization Sub-Problem

Given any set of nodal prices GENCO(i) maximizes its profit while satisfying its generation constraints, that is,

The arguments of this sub-problem are functions of the prices and are denoted by and The above-defined maximum profit is non-negative, that is,

This result stems from the fact that since a GENCO can

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The Next Generation of Unit Commitment Models

control its commitment, it will always turn itself off over the entire time horizon rather than operate at a loss. 4.3.2

The DISTCO Profit Maximization Sub-Problem

Given any set of nodal prices, DlSTCO(i) also maximizes its profit while satisfying its load constraints, that is,

The arguments of this sub-problem are functions of the prices and are denoted by and The maximum profit of any self-committing DISTCO (with controllable 0/1 variable 4.3.3

is also non-negative, that is,

The TRANSCO Merchandising Surplus Maximization SubProblem

Assuming that a single Transco operates the network, this sub-problem consists of maximizing the merchandising surplus over the planning horizon, subject to the various network security constraints, Thus,

The argument of this sub-problem is a function of the prices and is denoted by a solution that is de facto security feasible, as the TRANSCO enforces security constraints on all network variables. We emphasize that in (24) the TRANSCO does not physically adjust the voltage phase angles or the power injections, it merely computes the injections that are then submitted to the Price Caller. What is perhaps surprising is that giving the TRANSCO the freedom to maximize its merchandising surplus is consistent with the primal problem objective of maximum social welfare under the set of network constraints, Even so, the abuse of monopoly power by a TRANSCO in the process of maximizing its merchandising surplus is a possibility that must be addressed. For example, manipulation of the line flow limits in could artificially create transmission congestion and nodal price differences, thereby increasing the merchandising surplus. We believe that this practice can be virtually ruled out by making the TRANSCO data and response to prices public and open to scrutiny by all market participants. This is consistent with the rulings of FERC [24,25]. Furthermore, because its revenues and/or profits are regulated, there is no incentive for a TRANSCO to create congestion artificially, unless it were in an anti-competitive collusion with a GENCO or a DISTCO.

Decentralized Nodal-Price Self-Dispatch and Commitment

4.4

281

Profit Optimality Let

and

be the main arguments of the pri-

mal problem (1) at its optimum. The corresponding profit of GENCO(i) is

Analogously, the profit of DISTCO(i) is

The TRANSCO surplus under the primal optimum conditions is given by

With these results, we can now define the notion of profit optimality. DEFINITION 1 (Profit Optimality): A primal solution is said to be profit optimum if,

Profit optimality states that the primal profit of GENCO(i), the maximum achievable profit,

is

A similar statement applies to the

profit of DISTCO(i) and to the merchandising surplus of the TRANSCO.

4.5

Equivalence between Primal and Dual

PROPOSITION: Assume that PUC and DUC are each unique. DUC and PUC yield identical solutions if and only if the solution of is profit optimum. PROOF: Necessity: Let PUC and DUC have identical solutions, that is, Since maximizes the individual profits in DUC for the prices then also maximizes the same profits for solution of PUC is profit optimum.

Thus, the

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Sufficiency: Let the solution to PUC, be profit optimum. We show that PUC and DUC yield identical solutions. First calculate the dual function at the primal solution prices, From (18)–(24),

Next, applying the profit optimality conditions in (28), and then refining with equations (25)–(27),

Since the power balance at all nodes and times is guaranteed by the primal solution, the second right-hand side term above disappears. Thus,

which is the value of the primal. This implies that at the primal prices, the dual function is maximized and the duality gap is nil. Making use of the generally accepted assumption that the solutions of DUC and PUC are unique, it then follows that these solutions are equal, that is, Q.E.D.

5.

BASIS OF THE DECENTRALIZED NODAL-PRICE SELF-SCHEDULING AUCTION (DNSA)

5.1

Profit Optimum Unit Commitment

We know (see Case C in Section 6) that the standard formulation of the unit commitment problem, (1)-(5), can yield solutions that are not profit optimum. Although mathematically feasible, such solutions may require some participants to operate at a profit below the maximum possible for the current prices, a result that we contend is unfair in the context of competition. In fact, the current practice of “uplift” charges in Pool operation is a mechanism for correcting such inequities, albeit in a heuristic and sub-optimal manner. To preclude the potential of unequal treatment of participants’ profits and to do so systematically, we assert that the next generation unit commitment formulation should include the condition of profit optimality. This is a subject of our ongoing research for both the centralized PUC and for DNSA.

Decentralized Nodal-Price Self-Dispatch and Commitment

5.2

283

Basic Steps of DNSA

DNSA differs from standard dual methods (SDM) in several respects: On the centralized nature of the method: all scheduling and price updating is centralized in SDM, whereas in DNSA, only price updating is centralized. GENCOs, DISTCOs, and TRANSCO self-schedule. On the requirements of price updating and commercial information: In SDM, as price updating is based on maximizing the central scheduler needs full knowledge of private cost and benefit functions. In DNSA, as price updating is based on solving the Price Caller needs only a pointwise price-response by GENCOs and DISTCOs.10 On the requirements of network information: In SDM, all TRANSCO data is required by a central scheduler to assure network security. In DNSA, the TRANSCO assures security. The Price Caller needs only a reduced set of network data and sensitivities. On the completeness of network modeling: In SDM, network constraints, particularly those describing nonlinear behavior, have been often simplified or ignored. In DNSA, most network constraints can be or have been included. On the evolution of bids through the auction process: In SDM, bid functions are fixed throughout the solution process. In DNSA, response to prices may evolve during the price-updating trials subject to the “convexifying” rule, which is defined next. 8 9

Here

The state-of-the-art methods for updating dual variables are the sub-gradient method [26], the bundle method and the cutting plane method. In the current implementation of DNSA, a Newton approach is used to update prices. 10 The Price Caller requires no explicit bid functions. To prove that the self-scheduling scheme yields the same solution as PUC, however, we assume the existence of such functions. This implies that GENCOs and DISTCOs behave according to some deliberate strategy.

284

5.3

The Next Generation of Unit Commitment Models

Convergence of DNSA

We now address the question of equilibrium attainment, that is, if a competitive equilibrium is feasible, in the sense that PUC is profit optimum, how does the market attain it? The difficulty of this question is old and well known to economists. Indeed, it was first contemplated by Leon Walras, one of the progenitors of modern equilibrium theory [12]. In his conjecture, known as Walrasian tatônnement, Walras states that attaining equilibrium would involve a process by which a market groped toward equilibrium with the help of a fictitious auctioneer who announced prices, then collected demands from consumers as to how much of each good they would wish to purchase at the announced prices. If demand exceeded supply, the price was adjusted upward. If supply exceeded demand, the price was adjusted downward. Walras conjectured that this procedure would cause the market to eventually settle into equilibrium. This simple and intuitively plausible idea was widely accepted until Herbert Scarf demonstrated, by his famous examples, the existence of an open set of economies having a unique equilibrium that was unstable, therefore unattainable, under basic tatônnement [27]. In this study, we came to the conclusion that it was possible to construct an improved tatônnement procedure in decentralized electricity market. We conjecture that this procedure will converge, of course assuming the existence of a competitive equilibrium, by imposing a restriction on the participants’ bidding here called the convexifying rule. This constraint is not a theoretical limitation of DNSA but a practical necessity for attaining a competitive equilibrium in both decentralized as well as centralized auctions. A consequence of the convexifying rule is that during the price-updating trials, participants are prevented from derailing the auction process or rendering it chaotic. DEFINITION 2 (Convexifying Market Rule): An electricity market is said to operate under the convexifying market rule if, for each time period, the collection of trial price-response pairs of GENCO(i), to the k-th auction iteration – can be extended to a monotone increasing incremental cost [monotone decreasing incremental benefit] function. 11

11

A real-valued function of a real variable x is said to be monotone increasing if the value of decrease as increases; that is,

Decentralized Nodal-Price Self-Dispatch and Commitment

285

CONJECTURE (Convergence of DNSA): Extensive simulations and related experience [28] indicates that DNSA converges well when PUC is profit optimum and the competing participants behave rationally according to the convexifying market rule.

5.4

Price Updating

In this subsection, a vector variable with the superscript k represents its value at the k-th price-updating trial. One important innovation of DNSA is the use by the Price Caller of a price-updating algorithm that uses the first and second derivatives of the dual function. This Newton algorithm [29] solves the nodal power balance equations, that is, via the iterative procedure,

The second derivative matrix or Hessian is

The first two terms in the right-hand side of (35) are based on confidential competing participant information and can only be estimated by the Price Caller. Any estimation method may be used, among others, first differences, and regression using all the past price-response pairs. In the simulations of this chapter, each generator responds myopically so that the first two terms are diagonal matrices whose elements can be estimated from the participant’s behavior during the price-updating trials. The convergence of Newton’s method does not however hinge on knowing the exact Hessian, a good approximation being sufficient. The third term in (35) is computed from public network data available to the Price Caller.

This Hessian term corresponds to the case with no active line flow constraints. Under congestion, the TRANSCO is required to disclose the Lagrange multipliers associated with the active line flow limits. These values may then be used by the Price Caller to refine the Hessian for faster convergence [30].

286

6.

The Next Generation of Unit Commitment Models

CASE STUDIES

To illustrate how DNSA works, we use a 5-bus system with network and generator data shown in Table 1 and 2. The resistance, R, reactance, X, and total line charging susceptance, Bcap, are per unit on a base of 100 MVA.

We named the generators and loads according to their location in the network. We chose identical generator costs because this case has been challenging for traditional centralized UC algorithms that do not adequately represent the transmission system. In this chapter, we report three simulations. In Cases A and C, line flow limits are large enough to rule out congestion, while in Case B some line capacities are reduced as indicated later to create congestion. The planning horizon consists of three equal length periods. The load patterns over time of DISTCO(2) and DISTCO(5) appear in Table 3 and are assumed unaffected by nodal prices. 12

Following the notation in (6), the variable cost of GENCO(i)

is given by

Decentralized Nodal-Price Self-Dispatch and Commitment

287

CASE A (Primal solution is profit optimal): Figure 4 and Figure 5 show that convergence to within is obtained in seven trials (or Newton price-updating iterations), at which point the nodal prices have also steadied. Plainly, the auction is stopped successfully when the maximum nodal power imbalance (defined in footnote 8, page 12) is less than 1 MW. This is common practice in actual power system operation. The generator output and profit profiles are reported in Table 4 and Table 5, respectively. All GENCOs make some profit over the planning horizon even though GENCO(l) and GENCO(3) must supply power at a loss during Period 3 owing to their minimum up-time constraint, In contrast, GENCO(4), with a lower minimum up time, elects to operate during Period 1 and Period 2 only, de-committing during Period 3 when there is no financial incentive because of the low demand and low nodal price. Table 6 shows the merchandising surplus of the TRANSCO that, without congestion, is due entirely to transmission losses.

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The Next Generation of Unit Commitment Models

CASE B (Effects of transmission congestion): Here, the data of Case A is modified by setting the flow limit of line 1–5 to 35 (MW), producing a light congestion during Period 1 and a heavier one during Period 2. As evidenced by Figure 6 and Figure 7, DNSA convergence is obtained within 11 trials. Price updating exploits the Lagrange multipliers of the active flow limits made available by the TRANSCO. Without this information, the DNSA still converges but at a slower rate (47 iterations).

Decentralized Nodal-Price Self-Dispatch and Commitment

289

The final generator output and profit profiles are shown in Tables 7 and 8, respectively. Comparing these results to those of Case A, we note a redistribution of the generation profits and a substantial increase in the Transco merchandising surplus (Table 6 versus Table 9). CASE C (Primal solution is profit sub-optimum): The same data as in case A are used except that the minimum up-times of GENCO(l) and GENCO(3) are decreased to two hours As the generating units can now switch off during Period 3, under the profit maximizing trial iterations of DNSA the participants do not come to a power balancing agreement at this time period. The resulting cycling behavior in prices and outputs during period 3 is shown in Figure 8 and Figure 9. This is a signal to the Price Caller that profit optimality is infeasible and that an alternative price-updating scheme must be initiated, as discussed in the Conclusions. The primal unit commitment solution without profit optimality does however exist as shown in Table 10, Table 11, and Table 12.13 Comparing the last two columns of Table 12, we see that each generator primal profit, is positive but less than the maximum possible,

The unfairness of primal solutions that are

profit sub-optimal can be observed from the relative differences between primal and maximum profits.

13

During those intervals where the DNSA and the primal solutions are theoretically equal, the numerical results may differ slightly due to convergence tolerance.

290

7.

The Next Generation of Unit Commitment Models

CONCLUSIONS

As an alternative to centralized pool auctions and operation, we propose a Decentralized Nodal-Price Self-Scheduling Auction (DNSA) for selfcommitment and dispatch of electricity market participants in a power pool. The scheme exploits the decomposability of the dual unit commitment problem. Each competing GENCO (DISTCO) self-computes its level of production (consumption) and the 0/1 commitment by maximizing profit for a given set of trial nodal prices. Similarly, the Transco independently maximizes its merchandising surplus while satisfying all network security constraints.

Decentralized Nodal-Price Self-Dispatch and Commitment

291

The Price Caller, a centralized entity without direct access to proprietary cost information, updates prices through an effective Newton algorithm until the power balance at each bus is satisfied. DNSA does not assume a perfect market and accounts for the AC load flow model including transmission losses and line congestion, in addition to integer variables, ramping rates, start-up costs, and minimum up and down times. The convergence of DNSA hinges on the notions of profit optimality and the convexifying market rule. The latter rule is imposed to ensure that the response to the Price Caller by market participants will not derail the price updating process. Interestingly, the convexifying rule, while preventing irrational and chaotic bidding, also offers the competing participants some latitude to refine their bids during the auction. In study cases A and B, we illustrate the convergence characteristics of DNSA with and without transmission congestion. In addition, we solve the traditionally difficult case of generators with identical cost functions. In Case C, we illustrate two important points as regards profit sub-optimal problems. First, the primal profits of individual participants are not the maximum possible for the primal nodal prices. We contend that under free competition this result is unfair to some participants. Second, price cycling is a clear signal that profit optimality is infeasible. To ensure fairness of treatment for all competing participants, the next generation of unit commitment models should allow for self-scheduling, and should therefore include profit optimality constraints. Ongoing research is considering various mathematical reformulations and computational solutions of the profit optimal primal problem, as well as DNSA derivations that solve this new problem in a decentralized manner [30]. We are looking at DNSA price updating schemes that will enforce profit optimality through the trial iterations for those cases that are not naturally profit optimum, that is for which the primal unit commitment formulation is not already profit optimum. The crucial point here is to enforce profit optimality whenever possible while achieving maximum social welfare.

REFERENCES 1. 2. 3. 4. 5.

K.J. Arrow. Social Choice and Individual Values, Second Edition. New Haven: Yale University Press, 1963 ed. 1951). R. Green. “The Political Economy of the Pool.” In Power Systems Restructuring: Engineering and Economics. Kluwer Academic Press, 1998. M. Ilic, and F.D. Galiana. “Power Systems Operation: Old vs. New.” In Power Systems Restructuring: Engineering and Economics. Kluwer Academic Press, 1998. M. Huneault, F.D. Galiana, and G. Gross. “A Review of Restructuring in the Electricity Business.” In Proc. 13th PSCC Conf., Tronheim, Norway, 1999. F. Wu, and P. Varaiya. Coordinated multilateral trades for electric power networks: Theory and implementation. Elec. Power Energy Syst., 21(2): 75-102, 1999.

292 6. 7. 8. 9.

10.

11. 12.

13. 14. 15. 16. 17. 18. 19.

20. 21. 22. 23. 24.

25. 26. 27. 28. 29. 30.

The Next Generation of Unit Commitment Models C. Imparato. “Market-Making Attributes of Alternative ISO Structures.” Comments in a Panel Session, In Proc. IEEE PES Winter Meeting, 1999. Federal Energy Regulatory Commission (U.S.). “Regional Transmission Organizations: Rulemaking.” Docket No. RM99-2-000, Order No. 2000, December 1999. R. Wilson. “Activity Rules for the Power Exchange.” Report to the California Trust for Power Industry Restructuring, March 3, 1997. G. Gross, E. Bompard, P. Marannino, and G. Demartini. “The Uses and Misuses of Optimal Power Flow in Unbundled Electricity Markets.” In Proc. of the IEEE PES, Summer Meeting, Edmonton, Alberta, Canada, July 1999. R. Johnson, S.S. Oren, and A. Svoboda. “Equity and Efficiency of Unit Commitment in Competitive Electricity Markets.” In Proc. POWER Conf. and paper PWP-039, Univ. of California Energy Institute, March 15, 1996. C.W. Smith. Auctions: The Social Construction of Value. London: The Free Press, 1989. M.-E.L. Walras. Éléments d'Économie Politique Pure; ou la Théorie de la Richesse Sociale. First edition, 1874. [English translation: Elements of Pure Economics, or; The Theory of Social Wealth. Homewood, I11.: Published for the American Economic Association and the Royal Economic Society, by R.D. Irwin, 1954. P.H. Griffes. “Iterative Bidding in the PX Market.” A report to the California Power Exchange, http://www.calpx.com/news/publications/index.htm., January 2000. H. Everett. Generalized Lagrange multiplier method for solving problems of optimal allocations of resources.” Op. Res. 11: 399-417, 1963. F. Zhuang, and F.D. Galiana. Towards a more rigorous and practical unit commitment by Lagrangian relaxation. IEEE Trans. Power Syst., 3(2): 763-773, 1988. A. Merlin, and P. Sandrin. A new method for unit commitment at Electricité de France. IEEE Trans. PAS -102, 5: 1218-1225, 1983. P.A. Samuelson. Foundations of Economic Analysis. Harvard University Press, 1947. L.A. Wolsey. Integer Programming. John Wiley & Sons, Inc., 1998. T.S. Dillon, K.W. Edwin, H.D. Kochs, and R.J. Tand. Integer programming approach to the problem of optimal unit commitment with probabilistic reserve determination.” IEEE Trans. PAS-97, 6:2154-2166, 1978. J.M. Arroyo and A.J. Conejo. Optimal response of a thermal unit to an electricity spot market. To appear in the IEEE Trans. Power Syst. Paper PE040PRS (10-99). A. Brooke, D. Kendrick, A. Meeraus, and R. Raman. GAMS: A User’s Guide. GAMS Development Corp., 1998. J.A. Muckstadt, and S.A. Koenig. An application of Lagrangian relaxation to scheduling in power generation systems. Op. Res., 25(3): 387-403, 1977. D.P. Bertsekas, G.S. Lauer, N.R. Sandel, and T.A. Posberg. Optimal short-term scheduling of large-scale power systems. IEEE Trans. Autom. Cont., 28(1): 1-11, 1983. N. Nadira, and A.S. Cook. “On the Availability of Data Required by Optimal Power Flows under Increased Competition,” IEEE PES, Tutorial Course – Optimal Power Flow: Solution Techniques, Requirements, and Challenges, 96 TP 111-0, 1996. Federal Energy Regulatory Commission (FERC–US). “Form 715: Annual Transmission Planning and Evaluation Report.” http://www.ferc.fed.us/electric/F715/, January 2000. B.T. Polyak. Introduction to Optimization. Optimization Software, 1987. H.E. Scarf. Some examples of global instability. Int. Econ. Rev., 1: 157-172, 1960. M. Huneault. “An Investigation of the Solution to the Optimal Power Flow Problem Incorporating Continuation Methods.” Doctoral thesis, McGill University, August 1988. J.M. Ortega and W.C. Rheinboldt. Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, 1970. A.L. Motto. “Equilibrium of Electricity Market Under Competition.” Ph.D. thesis in progress, McGill University.

Chapter 16 DECENTRALIZED UNIT COMMITMENT IN COMPETITIVE ENERGY MARKETS

Jinghai Xu and Richard D. Christie University of Washington

Abstract:

1.

In a competitive energy market, instead of, or in addition to, a centralized unit commitment, individual generation owners will make independent unit commitment decisions. They will seek to maximize their profits against the predicted market clearing price. Their unit commitment strategy will be expressed in their bids, so that they shut-down or start-up when the market price indicates such activity. In this chapter, we develop a unit commitment based price-taking (UCPT) bidding strategy with a simple price prediction mechanism and explore it using a market simulator. Simulation results show that an individual generator has higher profits with UCPT bidding than with simple price-taking bidding, and that the cost of supplying price-inelastic loads achieved by the market is lower when all generators use UCPT bidding. It appears that UCPT bidding gives results similar to those from a Lagrangian relaxation unit commitment (LRUC), without a fix-up step, and it has problems with convergence and feasibility similar to LRUC. We observe cyclic behavior in market prices with UCPT bidding, and we show that it depends on the price prediction mechanism. Alternative price prediction mechanisms can reduce cyclic behavior. Finally, we conceptually explore potential strategic behavior and market power arising from unit commitment constraints.

INTRODUCTION

The unit commitment problem – scheduling generator start-ups and shutdowns over a period of time to minimize the cost of serving expected loads has been applied by the power industry and studied by researchers for decades. Since unit commitment was typically performed for a set of generators all owned by one company—to meet load exclusively served by that company—it was natural for the algorithm to assume that one central authority controlled the status of every generator. This case is called centralized unit commitment. Deregulation has invalidated the assumption of centralized control. A number of different companies now own generators. Each company must make its own individual start-up and shut-down decisions, and cannot control

294

The Next Generation of Unit Commitment Models

the decisions made by other companies. This case is called decentralized unit commitment, because the commitment decision making is carried out in a decentralized control structure. Recent publications that consider deregulation and unit commitment deal mostly with centralized unit commitment. In some cases (such as the England and Wales Power Pool), the market structure requires generators to submit to centralized unit commitment [1]. In other cases, researchers have assumed the existence of a centralized unit commitment in decentralized markets [2, 3, 4]. Oren et al. [5], however, identify the problems inherent in the use of centralized unit commitment. Specifically, they point out that due to the nearoptimal nature of the solutions obtained by practical unit commitment algorithms, small changes in total cost can have large consequences for individual generators. When all generators are owned by one company, these differences are not important. When generators are owned by different companies, these differences are highly problematical. The problems with centralized unit commitment have been recognized in practice by various deregulated markets. California and the Nord Pool market in Norway have no centralized unit commitment in the market process. The PJM Interconnection has a voluntary centralized unit commitment, but allows market participants to self-commit. Based on economic simulations the California tariff proposed an iterative energy market bidding scheme [13] to account for start-up costs, but has so far decided not to implement it due to the cost of implementation and time constraints of the required communications. Even if centralized unit commitment is required by, for example, connection agreements, it seems unlikely that it can practically be enforced. If a generator is required to run, and thinks that it should be shut down, it may suffer an operating problem of some kind that forces it off-line. Generators are well known to be the least reliable components of the power system and separating intentional shut-downs from truly inadvertent ones is likely to be impractical. If a generator wants to run, but is required to shut down, it could, perhaps, claim restraint of trade, especially if its bid for the time period in question shows that it is willing to run at minimum power for any price. Note that we base this discussion of generator non-compliance on economic motivations over a relatively long period of time, and it does not address compliance during emergency conditions. In [6], Li et al. introduce a market model that uses generator selfcommitment to determine generator bids over a fixed time period, and then the usual market resolution process resolves the bids to determine market prices. Then generator bids are redetermined for the same time period using the new prices. The concept of individual self-commitment to maximize profits given future market prices and the decentralized nature of the commitment problem are new. But the idea that bidders would have several opportunities to bid for the same time period is unworkable, as mentioned above.

Decentralized Unit Commitment

295

In the same journal, Huse et al. [7] introduce a method of computing generator bids that includes self-commitment and generates one bid at a time. This is better-suited for most existing market resolution processes and for the general desire in most markets to move bidding as close to operation as possible. This technique, however, is introduced in the context of using a market simulation to solve a centralized unit commitment problem. Once market clearing prices have been obtained for the time periods of interest, the bidding process is repeated for the same time periods, so an iteration loop is present which is not present in real markets. In this chapter, we apply the bidding strategy developed in [7] to a somewhat more realistic market simulation. We then use the simulation to explore the following questions: Is there an incentive for individual generators to pay attention to unit commitment when bidding? What is the impact of unit selfcommitment on the market, in particular, in comparison to a centralized unit commitment? Section 2 gives the mathematical notation of this model. Section 3 discusses how generators can control their commitment through the form of their bids. Section 4 describes several different bidding strategies that pay more or less attention to unit commitment. Section 5 illustrates the market simulation used, including the generation and loads. Section 6 gives results from the use of the different strategies, identifies an interesting cycling behavior in the market, corrects it, and addresses the first question. Section 7 addresses the second question, and Section 8 discusses strategic bidding issues.

2.

NOTATION We used the following notation in our research: thermal time constant for the generator, in hours cold start cost, in $ fixed start cost, in $ C(P) : cost function, in $/hour modified cost function, in $/hour IC(P) :incremental cost function, in $/MWh M : unit self-commitment period profit, in $/hour P : power output, in MW high power output limit, in MW low power output limit, in MW p : market price, in $/MWh predicted market price, in $/MWh STC : start-up cost, in $

296

The Next Generation of Unit Commitment Models SDC: shut-down cost, in $/hour t: time that the generator was cooled, in hours T: cyclic period of price u : unit status, u = 1 if running, u = 0 if not running w : price prediction weight

3.

CONTROLLING COMMITMENT WITH BIDDING

In a Marshallian market where all bidders are paid the market clearing price (MCP), the price-taking bidding strategy sets bids equal to incremental cost for generators operating between minimum and maximum output power (segment CD in Figure 1a). What happens to the bid when these limits are approached? For the upper power limit, the answer is easy. The generator cannot produce more power no matter how much it is paid, so the bid curve is simply vertical (segment DE in Figure 1a). For the lower limit the answer is both more difficult and more interesting.

Decentralized Unit Commitment

297

If the generator is willing to stay on line at minimum power once MCP goes below its incremental cost at its low output power limit at then the curve simply extends downward, as shown by the segment BC in Figure la. When MCP reaches a level at which the generator is no longer willing to operate, (the minimum acceptable price, or MAP), it will shut down. The generator transitions from to zero output, as shown by the dotted line AB. Thus, by submitting the bid curve ABCDE, the generator informs the market about its willingness to operate, and the market tells the generator whether to operate or shut down via the MCP. It is certainly possible that the MAP for operation could be greater than In this case, the MAP supercedes the incremental cost. The bid curve for this case is shown in Figure 1b. The transition from operating to shutdown poses feasibility problems. The generator cannot actually operate below its minimum power but it is quite possible for a market solution to return an MCP equal to MAP, with a scheduled output between zero and (on section AB of Figure 1a). Resolution of this infeasibility depends on the market type. In an ex ante market, the generator has at least three possible ways to deal with the problem. There is some time before the bid period starts in which the generator may be able to adjust its schedule by making bilateral contracts. The generator also has an implicit contract with the ISO, since there must be a generation adjustment mechanism of some kind to account for real time variations in load and inadvertent generator outages. The generator could simply shut down at the start of the hour and accept any costs charged for deviating from its schedule. Finally, what is actually scheduled is the energy to be delivered over the bid period. If metering is done on an hourly basis, the generator could simply operate at minimum power for the amount of time required to deliver the scheduled energy, and then shut down. The time needed to reduce generator power to zero may cause the generator to supply some energy to the power system while it shuts down even if it is scheduled for zero output, aiding this strategy. The generator will choose the lowest cost strategy, and it can factor any associated costs into its bids, as described later. In an ex post market, a generator should simply shut down when assigned an output less than The real time nature of the ex post market will change bids in the next resolution period to account for changes in generator status. Thus, if a minimum acceptable price (MAP) can be computed, generators can use this to modify their ideal price-taking bids, and the interaction of the modified bid with the market will signal the generator whether to stay on-line or shut down in a given bid period.

298

4.

The Next Generation of Unit Commitment Models

BIDDING BEHAVIOR

Two bidding strategies with different approaches to finding the MAP are discussed in this section. In developing these strategies, we make four assumptions: (1) generators are price-takers; (2) generators try to maximize its profit in the market; (3) each generator bids independently in the market; and (4) the generator’s incremental cost function is monotonically increasing.

4.1

Finding Minimum Acceptable Price

If a generator has no shut-down and start-up costs, the minimum acceptable price at which it will choose to operate is, therefore, the MCP at which its profit is zero. It is well known from microeconomic theory that this occurs at the point where incremental cost equals average cost, which is also the minimum of average cost, i.e. [8]

The bid curve of the generator with this MAP is shown in Figure 2 (dark line). P* is the power output where average cost (AC) and incremental cost (IC) intersect.

Decentralized Unit Commitment

4.2

299

Applying Power Output Limits

The power output limits of a generator, and modify the MAP. When P* is within the power output limits, i.e., the MAP is unaffected, as shown in Figure 3a. With at the profit of the generator is and is positive (see Figure 3b). Since the profit is zero at

which is the minimum value of average cost in the range the bid curve is shown in Figure 3b. Similarly, when zero at as shown in Figure 3c. In general,

4.3

and the profit is

Including Shut-down and Start-up Costs

The MAP is also modified by non-zero shut-down cost or start-up cost. Consider a non-zero shut-down cost. The MAP is the price that sets the generator’s profit equal to the negative of shut-down cost (SDC), i.e., or

This is similar to equation (2) with a modified cost function: Then the MAP is:

The MAP given by equation (8) may even be negative if SDC is high enough, giving the bid curve in Figure 4.

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Decentralized Unit Commitment

301

If a generator is shut-down in hour n-1, then the MAP for hour n may have to include the start-up cost STC. STC here can be treated as an extra fixed cost of operation. The modified cost function can be further revised to

and

If the generator is operating in hour n-1, then value of STC in hour n is zero.

4.4

Shut-Down Price-Taking Strategy

The MAP given by equation (10) controls commitment decisions based on cost and income in hour n only. This strategy is referred to as shut-down price-taking (SDPT). The SDPT strategy is naive, since it considers only those costs incurred in a single period. A better approach to finding the MAP would include the effect of the commitment decision on future profit.

302

4.5

The Next Generation of Unit Commitment Models

Unit Commitment-Based Price-Taking Strategy

A bidding strategy that incorporates the future profitability of the generator was first suggested in the unit commitment literature by Huse et al. [7]. The future profitability here is evaluated by a unit self-commitment approach, so this strategy is referred to as unit commitment-based price-taking (UCPT). In the UCPT strategy, when preparing the bid curve for hour n, a generator first computes its future profits for the two cases of running and shutting down in hour n. Then the difference of the two projected future profits, or the lost profit due to shutting down, is treated as a extra cost and incorporated into the MAP computation. Suppose is the future profit if the generator is running in hour n, and is the future profit if the generator is shut down in hour n. The lost future profit (which is referred to as in [7]) is given by:

Then the MAP for UCPT is computed by extending equation (10) as follows:

4.6

Unit Self-Commitment for UCPT

We find the future profit by solving a unit self commitment problem, formulated as: Given a known operating state in hour n and predicted prices for hours n+1 to n+M, find the commitment schedule for a generator that maximizes its profit in the interval n+1 to n+M, subject to generator output limits and minimum shut-down time constraints. Profits are income (which is market price times power output) minus cost (fuel cost, and start-up cost) if operating, or the negative of shut-down cost if shut down. The commitment does not consider minimum run time, ramp rate, or spinning reserve constraints, in order to simplify the computation. Time varying start-up costs are considered. In mathematical form, Given Find

s.t.

and

to maximize

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303

and minimum down-time constraints Where

This problem is simple enough to be solved by forward dynamic programming. It resembles the single generator subproblem in Lagrangian relaxation unit commitment.

4.7

Price Prediction for UCPT

As indicated in equation (13), a future price profile is required when computing future profits. This price profile has to be predicted by each individual generator. In this paper, we use two simple price prediction methods based on historical prices. In real life, various sophisticated price prediction algorithms will be employed. The first method is called the single period prediction (SPP). It assumes that the market price is cyclic with period T, because the load profile used in the example is also cyclic with period T. Then the estimated future market price is just the historical market price for the preceding cycle. Suppose that the historical prices before hour n are known. Then predicted future prices

starting from hour n are as follows (and are illustrated in Figure 5):

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Note that is not computed because is unknown. The unit selfcommitment is therefore actually performed for m = l,...,T-1. Omitting the last hour of a 168-hour week should have little effect on the result of future profit differential If the historical price is not available (in the initial period), the predicted price is set as:

As discussed later, the use of SPP leads to a cycling market price pattern with period 2T. To correct this, we use a weighted average price (WAP). Similar to SPP, a generator using WAP also assumes that the market prices are cyclic with period T, but the predicted price here is a weighted combination of the last two historical prices, i.e.,

Setting of the price prediction weight is discussed in the next section.

5.

BID STRATEGIES IN MARKET SIMULATION

In this section, we test SDPT and UCPT in a simulated power market environment. In addition, we address questions regarding the incentive for using the UCPT and the impact on the market efficiency.

5.1

Simulated Power Market

We constructed a 15-generator test case was constructed on a market simulator platform. There are no transmission constraints or costs. The generator parameters, listed in Appendix 1, were chosen randomly from the 110generator system published in [9]. In the market simulation, we include startup and shut-down costs in the cost computation. The minimum shut down time constraint of the generator is also enforced. We do not enforce other constraints, such as ramp rate limits and minimum run time. We took a weekly load (energy demand) shape from published market loads [12] and scaled it to the generation in the example (Appendix 2). The same weekly load profile was applied in every week of the simulation. The load bid for each hour is price inelastic.

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Market resolution proceeds hour by hour for ten weeks. In each hour, each generator submits a bid. The market price is decided by crossing the aggregated generator and load bids.

5.2

SDPT Results

When all generators use SDPT, the weekly average price is $13.220/MWh. The hourly price trajectory is shown in Figure 6. As expected, the prices for each week are identical.

5.3

UCPT Results

When all generators use UCPT with SPP price prediction with a oneweek time horizon, we observe a weekly price oscillation with a period of two weeks, presented in Figure 7. After a high average price week, price estimates for the coming week are also high. With high predicted future prices in the UCPT computation, lower MAPs result. With lower MAPs in supply bids, more generators stay on-line, which decreases the actual market prices and vice versa. We show weekly average prices in Figure 8.

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When the price prediction horizon is extended to two weeks, we observe a four-week period of price oscillation (Figures 9, 10). This periodicity may be because near term predicted prices dominate the profit differential calculation and thus the value of MAP. The near term prices come from the initial hours of the preceding period and are unaffected by recent market results.

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To damp the weekly price oscillation, we replaced the SPP by WAP. Different price weights, i.e. w = 0, 0.1... 1.0, in the WAP were tested. We find the least price oscillation when w is 0.7, shown in Figure 11. The weekly average prices in this case exhibit only small changes after five weeks as shown in Figure 12. In general, oscillations are well-damped for Figure 13 shows the last four weeks of hourly prices from UCPT overlaid on each other. It is clear that the prices vary from week to week, indicating that there is not a clean convergence of the process.

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Results show that WAP is better suited to predict future market prices. In later sections we used only WAP (with w = 0.7) in UCPT.

5.4

Addressing Questions

1. If all generators use SDPT, is one generator motivated to use UCPT? To answer this question, we constructed a test case with Generator 27 using UCTP and all other generators using SDPT. We chose Generator 27 because it has the most status changes in the all SDPT case. In the all SDPT case, Generator 27’s profit in the last four weeks is $5,367. When the generator changes its bidding strategy to UCPT, its profit increases to $32,443. When Generator 27 uses UCPT in its bid, because of the long-term profitability, lower MAPs are used in its bids. Lower MAPs allow it to run continuously, saving the start-up and shut-down costs. Thus, Generator 27 is motivated to use UCPT by increased profits. 2. If all generators use UCPT, is a generator motivated to use SDPT? When all generators use UCPT, Generator 27’s profit for the last four weeks is $15,807. If Generator 27 changes to SDPT while the rest of the generators stay with UCPT, Generator 27’s profit falls to $5,614. Generator 27 is motivated to use UCPT. Profits of Generator 27 for the four scenarios are listed in Table 1.

3. Is the market better off with UCPT? With a price inelastic load, the total cost of generation is related to social welfare. The best market, with the highest social welfare, has the lowest cost of generation. We compare generation costs of the four scenarios in previous subsections. To avoid the initial unstable results, we use the data from the last four weeks. A minimum total generation cost is observed in the scenario where all generators use UCPT. The maximum total cost is found when all generators bid with SDPT. Total generation costs in the last four weeks of different scenarios are listed in Table 2.

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When all generators use UCPT, the lowest average weekly price also occurs. This indicates that the price inelastic load also benefits from the supplier’s UCPT bidding. The result with lower costs and lower average prices in the UCPT case indicates that UCPT improves the efficiency of the specific market. To broaden this observation, the 110-generator system and its associated daily load curve from [9] are simulated. In this case, the market process is shortened to 10 days but the daily load curves are identical. The price prediction period is 23 hours. Other assumptions made in Section 5.1 still apply. We test two cases. First, all generators use SDPT, then UCPT. In each day of the SDPT case – except the first day when no start-up cost applies – the daily average price is $15.955/MWh, and the daily average cost and profit are $3,793,107 and $1,313,937 respectively. In the UCPT case, the result is stabilized after four days when using WAP with w = 0.7. Table 3 shows the daily average prices, total cost and total profit for the UCPT case. In the 110-generator system, after the initial day, the market efficiency of UCPT is also higher than that of SDPT.

310

6.

The Next Generation of Unit Commitment Models

COMPARING UCPT WITH A LAGRANGIAN RELAXATION SOLUTION

With the same 110-generator system and the same daily load curve used in Section 5, we solve the unit commitment problem with a Lagrange Relaxation Unit Commitment (LRUC) algorithm. The initial unit state for LRUC is taken from the UCPT solution for the last hour before the final day of the simulation, allowing a reasonable comparison of LRUC results with the last day of UCPT. We iterate LRUC 20 times and present the results in Table 4. The minimum generation cost was found in iteration 11. The total cost of the LRUC solution for the 11th iteration was only 0.02% less than UCPT. The profit was 0.47% lower in LRUC.

Note that the total daily cost of LRUC is lower than those reported in [7] and [9], because constraints such as minimum up-time and spinning reserve are not observed and the initial state differs. LRUC and UCPT had very similar commitment patterns as shown in Appendix 3. The LRUC commitment pattern is shown, with UCPT differences highlighted. Only three of 110 generators had different commitment patterns, two with different run time durations and one with a three-hour shut-down.

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The differences in operating durations of generators 13 and 15 may be attributed to the way LRUC deals with infeasible solutions, such as generator power outputs between zero and These are allowed by UCPT, as discussed in Section 3, but not by LRUC. Overall, comparison of LRUC and UCPT results gives some assurance that centralized unit commitment solutions can be used as reasonable predictors of energy market prices, assuming no exercise of market power, and thus as benchmarks for market efficiency measures.

7.

STRATEGIC BIDDING

We assumed in previous sections that generators are price takers. This assumption may not hold in reality. In a real world power pool, a generator will have non-zero market power – the ability to affect the price of electricity. Instead of bidding at its incremental cost, generators are likely to explore other bidding strategies that maximize long term profit. This approach to bidding is often called strategic bidding. A common form of strategic bidding is to anticipate the response of other bidders to one’s own bid. By examining the expected commitment behavior of other generators, a strategic bidder can arrive at a computed profit differential far different fromt that in a price-taking calculation, changing its MAP. The technical constraints of unit commitment offer another opportunity for strategic bidding. If competitors have generators with long time constants, such as long minimum shut-down times or long minimum run times, a strategic bidder with market power can attempt to drive market price down far enough to force the slow generators off-line at a strategic time, for example, just as load starts to increase, and then recoup the cost of this by exercising market power while the slow units are off-line. This problem is reminiscent of entry problems from economics. In general, slow units are cheap ones, and expensive units have short response times. The potential for entry (start-up) of an expensive unit should control the effects of this sort of exercise of market power. This discussion has only scratched the surface of unit commitment-aware strategic bidding strategies. A lot of work can be done developing and evaluating such strategies, and developing methods to detect their employment, measure their effects on market efficiency and limit their negative effects.

8.

CONCLUSIONS AND FUTURE WORK

We have explored two methods of including start-up costs and shut-down constraints in market-based generation bidding. We found that prices depend

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on the price prediction mechanism, and we showed that a simple weighted price prediction methodology resulted in more stable market prices. Examples show that the Unit Commitment Price Taking (UCPT) strategy results in higher profits for generators using it and in improved market efficiency. A comparison between market based UCPT solutions and centralized Lagrange relaxation unit commitment shows that the UCPT solution is very close to the LRUC solution, indicating that centralized unit commitment can be used for price prediction in decentralized markets. The area of strategic bidding, taking unit commitment considerations into account, appears fruitful for future research. The possibility of a theoretical link between the decentralized market-based unit commitment process with UCPT bidding, and centralized Lagrangian relaxation unit commitment, is an intriguing theoretical topic.

ACKNOWLEDGEMENTS This work was funded by the Advanced Power Technologies Center at the University of Washington, Chen-Ching Liu, Director. The authors are grateful to Karl Seeley for discussions on economic issues.

REFERENCES 1. D.P. Mendes and D.S. Kirschen, “Modelling of a competitive electricity power pool.” 32nd Universities Power Engineering Conference, UPEC ’97, UK, 1: 387-390, 1997. 2. G. Gross and D.J. Finlay. “Optimal Bidding Strategies in Competitive Electricity Markets.” In Proc. Twelfth Power Systems Computation Conference, PSCC, Dresden, 2: 815-823, 1996. 3. J.J. Ancona. A bid solicitation and selection method for developing a competitive spot priced electric market. IEEE Trans. Power Syst., 12 (2): 743-748, 1997. 4. G. Huang and Q. Zhao. “A Multi-objective Formulation for Competitive Power Market.” In Proc. 1998 Large Engineering Systems Conference on Power Engineering, LESCOPE 98, Canada: 317-22, 1998. 5. S.S. Oren, A.J. Svoboda, and R.B. Johnson. “Volatility of Unit Commitment in Competitive Electricity Markets.” In Proc. Thirtieth Hawaii International Conference on System Sciences, 5: 594-601, 1997. 6. C. Li, A.J. Svoboda, X. Guan, and H. Singh. Revenue adequate bidding strategies in competitive electricity markets. IEEE Trans. Power Syst., 14(2): 492-497, 1999. 7. E.S. Huse, I. Wangensteen, and H.H. Faanes. Thermal power generation scheduling by simulated competition. IEEE Trans. Power Syst., 14(2): 472-477, 1999. 8. R.S. Pindyck and D.L. Rubinfeld. Microeconomics, Fourth Edition. Prentice Hall, 1998. 9. S.O. Orero and M.R. Irving. Large scale unit commitment using a hybrid genetic algorithm. Elec. Power Energy Syst, 19(1): 45-55, 1997. 10. A.J. Wood and B.F. Wollenberg. Power Generation, Operation, and Control, Second edition. John Wiley & Sons, Inc., 1996.

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11. 12. 13.

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S. Takriti, J.R. Birge, and E. Long. A stochastic model for the unit commitment problem. IEEE Trans. Power Syst., 11(3): 1497-1508, 1996. California ISO Web Site, http://www.caiso.com. P.H. Griffes. “A Report to the California Power Exchange: Iterative Bidding in the PX Market.” ANALYSIS GROUP/Economics, 1999, http://www.calpx.com/news/publications /index.htm.

APPENDIX 1. PARAMETERS OF 15 GENERATORS

MDT: minimum down time PUC: A, B, and C: coefficients of polynomial cost function Pmax / Pmin: maximum/minimum power output SCC: cold start-up cost SCF: fixed start-up cost ST: time constant of cold start-up cost SDC: shut-down cost

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APPENDIX 2. WEEKLY LOAD PROFILE

APPENDIX 3.

UNIT COMMITMENT PATTERN FROM LRUC FOR 110 GENERATOR CASE

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INDEX ABACUS 154, 161 AC transmission 9, 75, 78, 271, 278 AMPL 72, 161 ancillary services 6, 9, 16, 18, 20, 22, 26, 29, 33, 41, 76, 93, 95, 96, 103, 107, 120, 122, 169, 194, 251, 261, 266 arbitrage 33 ARIMA 148 artificial neural network 255, 265 auctions 35, 185, 228 Australia 190, 253 autonomous agent models 190, 203, 253 bc-opt 154, 161 bid optimization 46, 231 bidding 9, 16, 20, 21, 32, 34, 43, 55, 93, 185, 191, 207, 233, 256, 283 bidding strategies 251, 252, 255, 257, 259, 264, 267, 295, 296, 304, 311 bids, multipart 33 bilateral trading 17, 19, 26, 28, 29, 185, 188, 228, 274 BPMPD 85 branch-and-cut 154, 156, 157 bundle method 77, 264 California 17, 20, 22, 24, 32, 39, 46, 97, 135, 188, 249, 259, 294 Chicago Board of Trade 191 competitive market 15, 41, 53, 55, 94, 185, 227, 249, 274, 275, 293 complementarity problem 230 cost recovery 175, 176, 181 Cournot model 230 CPLEX 4, 6, 153, 235, 241, 243

cutting plane 154, 155, 167, 168, 171 DC power flow approximation 10, 75, 78, 79 decision trees 252 demand-side 23, 26, 30, 44, 48, 56, 242 demand-side bidding 53, 67, 276 dispatch 114, 123, 193, 195, 231 DISTCO 188, 271, 274, 276, 280 distributed energy 10 DSI-OPF 51 duality gap 72, 76, 89, 134, 167, 175, 176, 181,278 dynamic programming 2, 71, 75, 77, 81, 93, 125, 129, 162, 171, 193, 212, 219, 221, 230, 260, 303 dynamic programming, stochastic 99,109,117,122,141,144, 260, 263 economic efficiency 32 Edgeworth approximation 147, 150 emissions 10 EMOS 40, 50 EMOSL 160, 161, 162, 163 EPRI ANN-STLF 51 EPRI-DYNAMICS 51 EPRI-OTS 51 equivalencing method 211, 214 ESCO 40, 44, 49, 188, 190 experimental economics 91, 188, 189 fairness 90, 294 FERC 16, 96 fuel price 123, 135 fuel price forecasts 97 game theory 252, 268

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The Next Generation of Unit Commitment Models

GAMS 161, 241 GENCO 40, 44, 185, 188, 190, 193, 271, 274, 276, 279 genetic programming 2, 185, 196, 204, 212, 253 genetic programming automata 205 hydropower 9, 41, 230, 233, 235, 237, 239, 244, 260, 262, 263 Iberdrola 248 India 217 integer programming 2, 4, 125, 153, 155, 159, 171, 212, 235, 238, 262 interior point method 167, 168, 171, 174, 182 interruptible loads 9 ISO 3, 9, 17, 18, 19, 23, 24, 25, 26, 31, 39, 43, 44, 46, 48, 140, 188, 250, 274, 276 Ito processes 135 Lagrange multipliers 2, 9, 60, 70, 77, 81, 143, 170, 176, 263, 279, 285 Lagrangian relaxation 2, 4, 54, 57, 70, 75, 78, 79, 84, 88, 89, 90, 94, 118, 127, 133, 162, 167, 170, 176, 193, 212, 219, 252, 256, 258, 260, 262, 278, 293, 303, 310 limits constraints fixing technique 223 LINDO 153 linear programming 6, 153, 158 load recovery 54, 57, 62 locational pricing 23, 25, 42, 46, 48, 271, 287, 290 LSE 50 maintenance 203 market design 15, 18, 28, 31, 35 market power 9, 18, 20, 28, 35, 227, 231,293,311

market simulation 9, 39, 187, 189, 190, 203, 228, 250, 268, 293, 304 Markov process 94, 95, 96, 97, 108, 141, 146, 260, 262, 263, 265 Mathematical Programs with Equilibrium Constraints 9 MATLAB 85, 88 MATPOWER 85 MINOS 72, 85, 90 MINTO 154, 161 MIPO 154, 161 modeling environments 160 modeling languages 161 Monte Carlo 96, 103, 104, 105, 106, 109, 141, 147, 150 MPL 161 MultiMATLAB 88, 90 New England 16, 20, 22, 23, 24, 33, 97, 188, 250, 252, 260, 266 New York 20, 21, 23, 24, 25, 42, 97 New Zealand 42, 252 Newton algorithm 285 nominal group 7 Nord Pool 294 normal approximation 147, 150 Northern Ireland Electricity 163 numerical application 60, 85, 100, 106, 127, 130, 147, 162, 178, 179, 200, 217, 241, 259, 266, 286, 304 NYMEX 135 OPF 42, 84, 85, 90 OPL 161 optimal self-commitment 93 optimization 2 options 103, 121, 129, 139 Order 2000 18, 27 Order 888 17, 19 ordinal optimization 253, 258, 259 OSL 153, 235

Index

parallel implementation 88 penalty bundle method 168 PJM 16, 18, 20, 21, 23, 24, 25, 26, 40, 42, 46, 97, 188 POOLCO 16, 168, 274, 275, 282 price 54, 67, 120, 143, 193, 244, 250, 254, 261, 296, 304, 306, 307 price elasticity 9 price forecasting 9, 45, 96, 103, 107, 187, 192, 197, 231, 260, 268, 293, 303, 306, 311 price inversion 22 price uncertainty 93, 98, 135, 143, 145 price volatility 104, 111, 140, 251 priority lists 2, 72, 212 probabilistic production-costing model 140, 145 profit optimality 271, 281 PROFITMAX 106 programming 233 pumped storage 48, 233, 237, 245, 260, 262, 263, 265, 266 PX 17, 19, 20, 28, 39, 47, 49, 250, 259 ramp rate constraints 76, 90, 93, 95, 170, 195, 214, 216, 239, 262, 277 Rational Buyer Protocol 22 reactive power 78, 79 regression model 45, 258 regulation 15, 17, 186 reliability 9, 10, 33, 188, 202 reserves: See ancillary services revenue adequacy 252

319

risk management 47 RTO 18, 27, 30, 31 scenario analysis 262 scenario tree 117, 127, 129 schedule coordinators 3, 16, 40, 47 settlement systems 21, 23 simulated annealing 212 simulation 43, 54 Spain 248 start-up 21, 55, 93, 95, 100, 107, 120, 142, 193, 195, 233, 238, 239, 255, 263, 273, 299, 302 statistical data-mining 96, 190 statistical methods 137 stochastic dynamic programming: See dynamic programming, stochastic stochastic programming 118 strategic behavior 65 sub-gradient technique 71, 77, 127, 167, 172, 264 Tractebel 162 TRANSCO 188, 272, 280 transmission 18, 20, 23, 27, 30, 32, 42, 202, 272, 277, 280, 283, 285, 286, 288 transmission rights 25, 28 UK 53, 54, 252, 294 uncertainty 44, 251, 266 unit decommitment method 117, 125 variable duplication technique 79 voltage limits 76, 78, 80 Walrasian equilibrium 284 XPRESS-MP 153, 158, 163 zonal pricing 24