Cover Page

CONTENTS

PREFACE TO THE THIRD EDITION

PREFACE TO THE SECOND EDITION

PREFACE TO THE FIRST EDITION

ACKNOWLEDGMENT

1 INTRODUCTION

1.1 PURPOSE OF THE COURSE

1.2 COURSE SCOPE

1.3 ECONOMIC IMPORTANCE

1.4 DEREGULATION: VERTICAL TO HORIZONTAL

1.5 PROBLEMS: NEW AND OLD

1.6 CHARACTERISTICS OF STEAM UNITS

1.7 RENEWABLE ENERGY

APPENDIX 1A Typical Generation Data

APPENDIX 1B Fossil Fuel Prices

APPENDIX 1C Unit Statistics

REFERENCES FOR GENERATION SYSTEMS

FURTHER READING

2 INDUSTRIAL ORGANIZATION, MANAGERIAL ECONOMICS, AND FINANCE

2.1 INTRODUCTION

2.2 BUSINESS ENVIRONMENTS

2.3 THEORY OF THE FIRM

2.4 COMPETITIVE MARKET SOLUTIONS

2.5 SUPPLIER SOLUTIONS

2.6 COST OF ELECTRIC ENERGY PRODUCTION

2.7 EVOLVING MARKETS

2.8 MULTIPLE COMPANY ENVIRONMENTS

2.9 UNCERTAINTY AND RELIABILITY

PROBLEMS

REFERENCE

3 ECONOMIC DISPATCH OF THERMAL UNITS AND METHODS OF SOLUTION

3.1 THE ECONOMIC DISPATCH PROBLEM

3.2 ECONOMIC DISPATCH WITH PIECEWISE LINEAR COST FUNCTIONS

3.3 LP METHOD

3.4 THE LAMBDA ITERATION METHOD

3.5 ECONOMIC DISPATCH VIA BINARY SEARCH

3.6 ECONOMIC DISPATCH USING DYNAMIC PROGRAMMING

3.7 COMPOSITE GENERATION PRODUCTION COST FUNCTION

3.8 BASE POINT AND PARTICIPATION FACTORS

3.9 THERMAL SYSTEM DISPATCHING WITH NETWORK LOSSES CONSIDERED

3.10 THE CONCEPT OF LOCATIONAL MARGINAL PRICE (LMP)

3.11 AUCTION MECHANISMS

APPENDIX 3A Optimization Within Constraints

APPENDIX 3B LINEAR PROGRAMMING (LP)

APPENDIX 3C Non-Linear Programming

APPENDIX 3D Dynamic Programming (DP)

APPENDIX 3E Convex Optimization

PROBLEMS

REFERENCES

4 UNIT COMMITMENT

4.1 INTRODUCTION

4.2 UNIT COMMITMENT SOLUTION METHODS

4.3 SECURITY-CONSTRAINED UNIT COMMITMENT (SCUC)

4.4 DAILY AUCTIONS USING A UNIT COMMITMENT

APPENDIX 4A Dual Optimization on a Nonconvex Problem

APPENDIX 4B Dynamic-Programming Solution to Unit Commitment

PROBLEMS

5 GENERATION WITH LIMITED ENERGY SUPPLY

5.1 INTRODUCTION

5.2 FUEL SCHEDULING

5.3 TAKE-OR-PAY FUEL SUPPLY CONTRACT

5.4 COMPLEX TAKE-OR-PAY FUEL SUPPLY MODELS

5.5 FUEL SCHEDULING BY LINEAR PROGRAMMING

5.6 INTRODUCTION TO HYDROTHERMAL COORDINATION

5.7 HYDROELECTRIC PLANT MODELS

5.8 SCHEDULING PROBLEMS

5.9 THE HYDROTHERMAL SCHEDULING PROBLEM

5.10 HYDRO-SCHEDULING USING LINEAR PROGRAMMING

APPENDIX 5A Dynamic-Programming Solution to Hydrothermal Scheduling

PROBLEMS

6 TRANSMISSION SYSTEM EFFECTS

6.1 INTRODUCTION

6.2 CONVERSION OF EQUIPMENT DATA TO BUS AND BRANCH DATA

6.3 SUBSTATION BUS PROCESSING

6.4 EQUIPMENT MODELING

6.5 DISPATCHER POWER FLOW FOR OPERATIONAL PLANNING

6.6 CONSERVATION OF ENERGY (TELLEGEN’S THEOREM)

6.7 EXISTING POWER FLOW TECHNIQUES

6.8 THE NEWTON–RAPHSON METHOD USING THE AUGMENTED JACOBIAN MATRIX

6.9 MATHEMATICAL OVERVIEW

6.10 AC SYSTEM CONTROL MODELING

6.11 LOCAL VOLTAGE CONTROL

6.12 MODELING OF TRANSMISSION LINES AND TRANSFORMERS

6.13 HVDC LINKS

6.14 BRIEF REVIEW OF JACOBIAN MATRIX PROCESSING

6.15 EXAMPLE 6A: AC POWER FLOW CASE

6.16 THE DECOUPLED POWER FLOW

6.17 THE GAUSS–SEIDEL METHOD

6.18 THE “DC” OR LINEAR POWER FLOW

6.19 UNIFIED ELIMINATED VARIABLE HVDC METHOD

6.20 TRANSMISSION LOSSES

6.21 DISCUSSION OF REFERENCE BUS PENALTY FACTORS

6.22 BUS PENALTY FACTORS DIRECT FROM THE AC POWER FLOW

PROBLEMS

7 POWER SYSTEM SECURITY

7.1 INTRODUCTION

7.2 FACTORS AFFECTING POWER SYSTEM SECURITY

7.3 CONTINGENCY ANALYSIS: DETECTION OF NETWORK PROBLEMS

7.4 AN OVERVIEW OF SECURITY ANALYSIS

7.5 MONITORING POWER TRANSACTIONS USING “FLOWGATES”

7.6 VOLTAGE COLLAPSE

APPENDIX 7A AC Power Flow Sample Cases

APPENDIX 7B Calculation of Network Sensitivity Factors

PROBLEMS

REFERENCES

8 OPTIMAL POWER FLOW

8.1 INTRODUCTION

8.2 THE ECONOMIC DISPATCH FORMULATION

8.3 THE OPTIMAL POWER FLOW CALCULATION COMBINING ECONOMIC DISPATCH AND THE POWER FLOW

8.4 OPTIMAL POWER FLOW USING THE DC POWER FLOW

8.5 EXAMPLE 8A: SOLUTION OF THE DC POWER FLOW OPF

8.6 EXAMPLE 8B: DCOPF WITH TRANSMISSION LINE LIMIT IMPOSED

8.7 FORMAL SOLUTION OF THE DCOPF

8.8 ADDING LINE FLOW CONSTRAINTS TO THE LINEAR PROGRAMMING SOLUTION

8.9 SOLUTION OF THE ACOPF

8.10 ALGORITHMS FOR SOLUTION OF THE ACOPF

8.11 RELATIONSHIP BETWEEN LMP, INCREMENTAL LOSSES, AND LINE FLOW CONSTRAINTS

8.12 SECURITY-CONSTRAINED OPF

APPENDIX 8A Interior Point Method

APPENDIX 8B Data for the 12-Bus System

APPENDIX 8C Line Flow Sensitivity Factors

APPENDIX 8D Linear Sensitivity Analysis of the AC Power Flow

PROBLEMS

9 INTRODUCTION TO STATE ESTIMATION IN POWER SYSTEMS

9.1 INTRODUCTION

9.2 POWER SYSTEM STATE ESTIMATION

9.3 MAXIMUM LIKELIHOOD WEIGHTED LEAST-SQUARES ESTIMATION

9.4 STATE ESTIMATION OF AN AC NETWORK

9.5 STATE ESTIMATION BY ORTHOGONAL DECOMPOSITION

9.6 AN INTRODUCTION TO ADVANCED TOPICS IN STATE ESTIMATION

9.7 THE USE OF PHASOR MEASUREMENT UNITS (PMUS)

9.8 APPLICATION OF POWER SYSTEMS STATE ESTIMATION

9.9 IMPORTANCE OF DATA VERIFICATION AND VALIDATION

9.10 POWER SYSTEM CONTROL CENTERS

APPENDIX 9A Derivation of Least-Squares Equations

PROBLEMS

10 CONTROL OF GENERATION

10.1 INTRODUCTION

10.2 GENERATOR MODEL

10.3 LOAD MODEL

10.4 PRIME-MOVER MODEL

10.5 GOVERNOR MODEL

10.6 TIE-LINE MODEL

10.7 GENERATION CONTROL

PROBLEMS

REFERENCES

11 INTERCHANGE, POOLING, BROKERS, AND AUCTIONS

11.1 INTRODUCTION

11.2 INTERCHANGE CONTRACTS

11.3 ENERGY INTERCHANGE BETWEEN UTILITIES

11.4 INTERUTILITY ECONOMY ENERGY EVALUATION

11.5 INTERCHANGE EVALUATION WITH UNIT COMMITMENT

11.6 MULTIPLE UTILITY INTERCHANGE TRANSACTIONS—WHEELING

11.7 POWER POOLS

11.8 THE ENERGY-BROKER SYSTEM

11.9 TRANSMISSION CAPABILITY GENERAL ISSUES

11.10 AVAILABLE TRANSFER CAPABILITY AND FLOWGATES

11.11 SECURITY CONSTRAINED UNIT COMMITMENT (SCUC)

11.12 AUCTION EMULATION USING NETWORK LP

11.13 SEALED BID DISCRETE AUCTIONS

PROBLEMS

12 SHORT-TERM DEMAND FORECASTING

12.1 PERSPECTIVE

12.2 ANALYTIC METHODS

12.3 DEMAND MODELS

12.4 COMMODITY PRICE FORECASTING

12.5 FORECASTING ERRORS

12.6 SYSTEM IDENTIFICATION

12.7 ECONOMETRIC MODELS

12.8 TIME SERIES

12.9 TIME SERIES MODEL DEVELOPMENT

12.10 ARTIFICIAL NEURAL NETWORKS

12.11 MODEL INTEGRATION

12.12 DEMAND PREDICTION

12.13 CONCLUSION

PROBLEMS

REFERENCE

INDEX

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Allen Wood passed away on September 10, 2011, during the preparation of this edition. Al was my professor when I was a student in the Electric Power Engineering Program at Rensselaer Polytechnic Institute (RPI) in 1966. Allen Wood and other engineers founded Power Technologies Inc. (PTI) in Schenectady, NY, in 1969. I joined PTI in 1974, and Al recruited me to help teach the course at RPI in 1979. The original text was the outcome of student notes assembled over a 5 year period from 1979 to 1984 and then turned over to John Wiley & Sons. Allen Wood was my professor, my mentor, and my friend, and I dedicate this third edition to him.

BRUCE F. WOLLENBERG

I dedicate this work to my family, my wife Yvette Sheblé, my son Jason Sheblé, my daughter Laura Sheblé, and grandson Kiyan, as they helped me so much to complete this work.

GERALD B. SHEBLÉ

PREFACE TO THE THIRD EDITION

It has now been 17 years from the second edition (and a total of 28 years from the publishing of the first edition of this text). To say that much has changed is an understatement. As noted in the dedication, Allen Wood passed away during the preparation of this edition and a new coauthor, Gerald Sheblé, has joined Bruce Wollenberg in writing the text. Dr. Sheblé brings an expertise that is both similar and different from that of Dr. Wollenberg to this effort, and the text clearly shows a new breadth in topics covered.

The second edition was published in 1996, which was in the midst of the period of “deregulation” or more accurately “reregulation” of the electric industry both in the United States and worldwide. New concepts such as electric power spot markets, Independent System Operators (ISOs) in the United States, and independent generation, transmission, and distribution companies are now common. Power system control centers have become much larger and cover a much larger geographic area as markets have expanded. The U.S. government has partnered with the North American Electric Reliability Corporation (formerly the North American Electric Reliability Council) and has begun a much tighter governance of electric company practices as they affect the system’s reliability and security since the events of 9/11.

We have added several new chapters to the text to both reflect the increased importance of the topics covered and broaden the educational and engineering value of the book. Both Sheblé and Wollenberg are professors at major universities and have developed new examples, problems, and software for the text. Both Wollenberg and Sheblé are consultants and expert witnesses to the electric energy industry. We hope this effort is of value to the readers.

Today, students and working engineers have access to much more information directly through the Internet, and if they are IEEE members can access the very extensive IEEE Explore holdings directly from their home or office computers. Thus, we felt it best not to attempt to provide lists of references as was done in earlier editions.

We would like to extend our thanks to those students who provided excellent programming and development skills to difficult problems as they performed research tasks under our direction. Among them are Mohammad Alsaffar and Anthony Giacomoni at the University of Minnesota; George Fahd, Dan Richards, Thomas Smed, and David Walters at Auburn University; and Darwin Anwar, Somgiat Dekrajangpetch, Kah-Hoe Ng, Jayant Kumar, James Nicolaisen, Chuck Richter, Douglas Welch, Hao Wu, and Weiguo Yang at Iowa State University; Chin-Chuen Teoh, Mei P. Cheong, and Gregory Bingham at Portland State University; Zhenyu Wan at University of South Wales.

Last of all, we announce that we are planning to write a sequel to the third edition in which many of the business aspects of the electric power industry will be presented, along with major chapters on topics such as extended auction mechanisms and reliability.

BRUCEF. WOLLENBERG
GERALDB. SHEBLÉ

PREFACE TO THE SECOND EDITION

It has been 11 years since the first edition was published. Many developments have taken place in the area covered by this text and new techniques have been developed that have been applied to solve old problems. Computing power has increased dramatically, permitting the solution of problems that were previously left as being too expensive to tackle. Perhaps the most important development is the changes that are taking place in the electric power industry with new, nonutility participants playing a larger role in the operating decisions.

It is still the intent of the authors to provide an introduction to this field for senior or first-year graduate engineering students. The authors have used the text material in a one-semester (or two-quarter) program for many years. The same difficulties and required compromises keep occurring. Engineering students are very comfortable with computers but still do not usually have an appreciation of the interaction of human and economic factors in the decisions to be made to develop “optimal” schedules, whatever that may mean. In 1995, most of these students are concurrently being exposed to courses in advanced calculus and courses that explore methods for solving power flow equations. This requires some coordination. We have also found that very few of our students have been exposed to the techniques and concepts of operations research, necessitating a continuing effort to make them comfortable with the application of optimization methods. The subject area of this book is an excellent example of optimization applied in an important industrial system.

The topic areas and depth of coverage in this second edition are about the same as in the first, with one major change. Loss formulae are given less space and supplemented by a more complete treatment of the power-flow-based techniques in a new chapter that treats the optimal power flow (OPF). This chapter has been put at the end of the text. Various instructors may find it useful to introduce parts of this material earlier in the sequence; it is a matter of taste, plus the requirement to coordinate with other course coverage. (It is difficult to discuss the OPF when the students do not know the standard treatment for solving the power flow equations.)

The treatment of unit commitment has been expanded to include the Lagrange relaxation technique. The chapter on production costing has been revised to change the emphasis and introduce new methods. The market structures for bulk power transactions have undergone important changes throughout the world. The chapter on interchange transactions is a “progress report” intended to give the students an appreciation of the complications that may accompany a competitive market for the generation of electric energy. The sections on security analysis have been updated to incorporate an introduction to the use of bounding techniques and other contingency selection methods. Chapter 13 on the OPF includes a brief coverage of the security-constrained OPF and its use in security control.

The authors appreciate the suggestions and help offered by professors who have used the first edition, and our students. (Many of these suggestions have been incorporated; some have not, because of a lack of time, space, or knowledge.) Many of our students at Rensselaer Polytechnic Institute (RPI) and the University of Minnesota have contributed to the correction of the first edition and undertaken hours of calculations for homework solutions, checked old examples, and developed data for new examples for the second edition. The 1994 class at RPI deserves special and honorable mention. They were subjected to an early draft of the revision of Chapter 8 and required to proofread it as part of a tedious assignment. They did an outstanding job and found errors of 10 to 15 years standing. (A note of caution to any of you professors that think of trying this; it requires more work than you might believe. How would you like 20 critical editors for your lastest, glorious tome?)

Our thanks to Kuo Chang, of Power Technologies, Inc., who ran the computations for the bus marginal wheeling cost examples in Chapter 10. We would also like to thank Brian Stott, of Power Computer Applications, Corp., for running the OPF examples in Chapter 13.

ALLEN J. WOOD
BRUCE F. WOLLENBERG

PREFACE TO THE FIRST EDITION

The fundamental purpose of this text is to introduce and explore a number of engineering and economic matters involved in planning, operating, and controlling power generation and transmission systems in electric utilities. It is intended for first-year graduate students in electric power engineering. We believe that it will also serve as a suitable self-study text for anyone with an undergraduate electrical engineering education and an understanding of steady-state power circuit analysis.

This text brings together material that has evolved since 1966 in teaching a graduate-level course in the electric power engineering department at Rensselaer Polytechnic Institute (RPI). The topics included serve as an effective means to introduce graduate students to advanced mathematical and operations research methods applied to practical electric power engineering problems. Some areas of the text cover methods that are currently being applied in the control and operation of electric power generation systems. The overall selection of topics, undoubtedly, reflects the interests of the authors.

In a one-semester course it is, of course, impossible to consider all the problems and “current practices” in this field. We can only introduce the types of problems that arise, illustrate theoretical and practical computational approaches, and point the student in the direction of seeking more information and developing advanced skills as they are required.

The material has regularly been taught in the second semester of a first-year graduate course. Some acquaintance with both advanced calculus methods (e.g., Lagrange multipliers) and basic undergraduate control theory is needed. Optimization methods are introduced as they are needed to solve practical problems and used without recourse to extensive mathematical proofs. This material is intended for an engineering course: mathematical rigor is important but is more properly the province of an applied or theoretical mathematics course. With the exception of Chapter 12, the text is self-contained in the sense that the various applied mathematical techniques are presented and developed as they are utilized. Chapter 12, dealing with state estimation, may require more understanding of statistical and probabilistic methods than is provided in the text.

The first seven chapters of the text follow a natural sequence, with each succeeding chapter introducing further complications to the generation scheduling problem and new solution techniques. Chapter 8 treats methods used in generation system planning and introduces probabilistic techniques in the computation of fuel consumption and energy production costs. Chapter 8 stands alone and might be used in any position after the first seven chapters. Chapter 9 introduces generation control and discusses practices in modern U.S. utilities and pools. We have attempted to provide the “big picture” in this chapter to illustrate how the various pieces fit together in an electric power control system.

The topics of energy and power interchange between utilities and the economic and scheduling problems that may arise in coordinating the economic operation of interconnected utilities are discussed in Chapter 10. Chapters 11 and 12 are a unit. Chapter 11 is concerned with power system security and develops the analytical framework used to control bulk power systems in such a fashion that security is enhanced. Everything, including power systems, seems to have a propensity to fail. Power system security practices try to control and operate power systems in a defensive posture so that the effects of these inevitable failures are minimized. Finally, Chapter 12 is an introduction to the use of state estimation in electric power systems. We have chosen to use a maximum likelihood formulation since the quantitative measurement–weighting functions arise in a natural sense in the course of the development.

Each chapter is provided with a set of problems and an annotated reference list for further reading. Many (if not most) of these problems should be solved using a digital computer. At RPI, we are able to provide the students with some fundamental programs (e.g., a load flow, a routine for scheduling of thermal units). The engineering students of today are well prepared to utilize the computer effectively when access to one is provided. Real bulk power systems have problems that usually call forth Dr. Bellman’s curse of dimensionality—computers help and are essential to solve practical-sized problems.

The authors wish to express their appreciation to K. A. Clements, H. H. Happ, H. M. Merrill, C. K. Pang, M. A. Sager, and J. C. Westcott, who each reviewed portions of this text in draft form and offered suggestions. In addition, Dr. Clements used earlier versions of this text in graduate courses taught at Worcester Polytechnic Institute and in a course for utility engineers taught in Boston, Massachusetts.

Much of the material in this text originated from work done by our past and current associates at Power Technologies, Inc., the General Electric Company, and Leeds and Northrup Company. A number of IEEE papers have been used as primary sources and are cited where appropriate. It is not possible to avoid omitting, references and sources that are considered to be significant by one group or another. We make no apology for omissions and only ask for indulgence from those readers whose favorites have been left out. Those interested may easily trace the references back to original sources.

We would like to express our appreciation for the fine typing job done on the original manuscript by Liane Brown and Bonnalyne MacLean.

This book is dedicated in general to all of our teachers, both professors and associates, and in particular to Dr. E. T. B. Gross.

ALLEN J. WOOD
BRUCE F. WOLLENBERG

ACKNOWLEDGMENT

I am indebted to a number of mentors who have encouraged me and shown the path toward development: Homer Brown, Gerry Heydt, Pete Sauer, Ahmed El-Abiad, K Neal Stanton, Robin Podmore, Ralph Masiello, Anjan Bose, Jerry Russel, Leo Grigsby, Arun Phadke, Saifur Rahman, Aziz Fouad, Vijay Vittal, and Mani Venkata. They have often advised at just the right time with the right perspective on development. My coauthor, Bruce, has often provided mentorship and friendship over the last several decades. I have had the luxury of working with many collaborators and the good fortune of learning and of experiencing other viewpoints. I especially thank: Arnaud Renaud, Mark O’Malley, Walter Hobbs, João Abel Peças Lopes, Manuel Matos, Vladimiro Miranda, João Tomé Saraiva, and Vassilios G. Agelidis.

GERALD B. SHEBLÉ

1

INTRODUCTION

1.1 PURPOSE OF THE COURSE

The objectives of a first-year, one-semester graduate course in electric power generation, operation, and control include the desire to:

1. Acquaint electric power engineering students with power generation systems, their operation in an economic mode, and their control.
2. Introduce students to the important “terminal” characteristics for thermal and hydroelectric power generation systems.
3. Introduce mathematical optimization methods and apply them to practical operating problems.
4. Introduce methods for solving complicated problems involving both economic analysis and network analysis and illustrate these techniques with relatively simple problems.
5. Introduce methods that are used in modern control systems for power generation systems.
6. Introduce “current topics”: power system operation areas that are undergoing significant, evolutionary changes. This includes the discussion of new techniques for attacking old problems and new problem areas that are arising from changes in the system development patterns, regulatory structures, and economics.

1.2 COURSE SCOPE

Topics to be addressed include

1. Power generation characteristics
2. Electric power industry as a business
3. Economic dispatch and the general economic dispatch problem
4. Thermal unit economic dispatch and methods of solution
5. Optimization with constraints
6. Optimization methods such as linear programming, dynamic programming, nonlinear optimization, integer programming, and interior point optimization
7. Transmission system effects
a. Power flow equations and solutions
b. Transmission losses
c. Effects on scheduling
8. The unit commitment problem and solution methods
a. Dynamic programming
b. Lagrange relaxation
c. Integer programming
9. Generation scheduling in systems with limited energy supplies including fossil fuels and hydroelectric plants, need to transport energy supplies over networks such as pipelines, rail networks, and river/reservoir systems, and power system security techniques
10. Optimal power flow techniques
11. Power system state estimation
12. Automatic generation control
13. Interchange of power and energy, power pools and auction mechanisms, and modern power markets
14. Load forecasting techniques

In many cases, we can only provide an introduction to the topic area. Many additional problems and topics that represent important, practical problems would require more time and space than is available. Still others, such as light-water moderated reactors and cogeneration plants, could each require several chapters to lay a firm foundation. We can offer only a brief overview and introduce just enough information to discuss system problems.

1.3 ECONOMIC IMPORTANCE

The efficient and optimum economic operation and planning of electric power generation systems have always occupied an important position in the electric power industry. Prior to 1973 and the oil embargo that signaled the rapid escalation in fuel prices, electric utilities in the United States spent about 20% of their total revenues on fuel for the production of electrical energy. By 1980, that figure had risen to more than 40% of the total revenues. In the 5 years after 1973, U.S. electric utility fuel costs escalated at a rate that averaged 25% compounded on an annual basis. The efficient use of the available fuel is growing in importance, both monetarily and because most of the fuel used represents irreplaceable natural resources.

An idea of the magnitude of the amounts of money under consideration can be obtained by considering the annual operating expenses of a large utility for purchasing fuel. Assume the following parameters for a moderately large system:

Annual peak load: 10,000 MW

Annual load factor: 60%

Average annual heat rate for converting fuel to electric energy: 10,500 Btu/kWh

Average fuel cost: $3.00 per million Btu (MBtu), corresponding to oil priced at 18$/bbl

With these assumptions, the total annual fuel cost for this system is as follows:

Annual energy produced: 107 kW × 8760 h/year × 0.60 = 5.256 × 1010 kWh

Annual fuel consumption: 10, 500 Btu/kWh × 5.256 × 1010 kWh = 55.188 × 1013 Btu

Annual fuel cost: 55.188 × 1013 Btu × 3 × 10− 6 $/Btu = $1.66 billion

To put this cost in perspective, it represents a direct requirement for revenues from the average customer of this system of 3.15 cents/kWh just to recover the expense for fuel.

A savings in the operation of this system of a small percent represents a significant reduction in operating cost as well as in the quantities of fuel consumed. It is no wonder that this area has warranted a great deal of attention from engineers through the years.

Periodic changes in basic fuel price levels serve to accentuate the problem and increase its economic significance. Inflation also causes problems in developing and presenting methods, techniques, and examples of the economic operation of electric power generating systems.

1.4 DEREGULATION: VERTICAL TO HORIZONTAL

In the 1990s, many electric utilities including government-owned electric utilities, private investor–owned electric utilities were “deregulated.” This has had profound effects on the operation of electric systems where implemented. This topic is dealt with in an entire chapter of its own in this text as Chapter 2.

1.5 PROBLEMS: NEW AND OLD

This text represents a progress report in an engineering area that has been and is still undergoing rapid change. It concerns established engineering problem areas (i.e., economic dispatch and control of interconnected systems) that have taken on new importance in recent years. The original problem of economic dispatch for thermal systems was solved by numerous methods years ago. Recently there has been a rapid growth in applied mathematical methods and the availability of computational capability for solving problems of this nature so that more involved problems have been successfully solved.

The classic problem is the economic dispatch of fossil-fired generation systems to achieve minimum operating cost. This problem area has taken on a subtle twist as the public has become increasingly concerned with environmental matters, so “economic dispatch” now includes the dispatch of systems to minimize pollutants and conserve various forms of fuel, as well as to achieve minimum costs. In addition, there is a need to expand the limited economic optimization problem to incorporate constraints on system operation to ensure the “security” of the system, thereby preventing the collapse of the system due to unforeseen conditions. The hydrothermal coordination problem is another optimum operating problem area that has received a great deal of attention. Even so, there are difficult problems involving hydrothermal coordination that cannot be solved in a theoretically satisfying fashion in a rapid and efficient computational manner.

The post–World War II period saw the increasing installation of pumped-storage hydroelectric plants in the United States and a great deal of interest in energy storage systems. These storage systems involve another difficult aspect of the optimum economic operating problem. Methods are available for solving coordination of hydroelectric, thermal, and pumped-storage electric systems. However, closely associated with this economic dispatch problem is the problem of the proper commitment of an array of units out of a total array of units to serve the expected load demands in an “optimal” manner.

A great deal of progress and change has occurred in the 1985–1995 decade. Both the unit commitment and optimal economic maintenance scheduling problems have seen new methodologies and computer programs developed. Transmission losses and constraints are integrated with scheduling using methods based on the incorporation of power flow equations in the economic dispatch process. This permits the development of optimal economic dispatch conditions that do not result in overloading system elements or voltage magnitudes that are intolerable. These “optimal power flow” techniques are applied to scheduling both real and reactive power sources as well as establishing tap positions for transformers and phase shifters.

In recent years, the political climate in many countries has changed, resulting in the introduction of more privately owned electric power facilities and a reduction or elimination of governmentally sponsored generation and transmission organizations. In some countries, previously nationwide systems have been privatized. In both these countries and in countries such as the United States, where electric utilities have been owned by a variety of bodies (e.g., consumers, shareholders, as well as government agencies), there has been a movement to introduce both privately owned generation companies and larger cogeneration plants that may provide energy to utility customers. These two groups are referred to as independent power producers (IPPs). This trend is coupled with a movement to provide access to the transmission system for these nonutility power generators as well as to other interconnected utilities. The growth of an IPP industry brings with it a number of interesting operational problems. One example is the large cogeneration plant that provides steam to an industrial plant and electric energy to the power system. The industrial-plant steam demand schedule sets the operating pattern for the generating plant, and it may be necessary for a utility to modify its economic schedule to facilitate the industrial generation pattern.

Transmission access for nonutility entities (consumers as well as generators) sets the stage for the creation of new market structures and patterns for the interchange of electric energy. Previously, the major participants in the interchange markets in North America were electric utilities. Where nonutility, generation entities or large consumers of power were involved, local electric utilities acted as their agents in the marketplace. This pattern is changing. With the growth of nonutility participants and the increasing requirement for access to transmission has come a desire to introduce a degree of economic competition into the market for electric energy. Surely this is not a universally shared desire; many parties would prefer the status quo. On the other hand, some electric utility managements have actively supported the construction, financing, and operation of new generation plants by nonutility organizations and the introduction of less-restrictive market practices.

The introduction of nonutility generation can complicate the scheduling–dispatch problem. With only a single, integrated electric utility operating both the generation and transmission systems, the local utility could establish schedules that minimized its own operating costs while observing all of the necessary physical, reliability, security, and economic constraints. With multiple parties in the bulk power system (i.e., the generation and transmission system), new arrangements are required. The economic objectives of all of the parties are not identical, and, in fact, may even be in direct (economic) opposition. As this situation evolves, different patterns of operation may result in different regions. Some areas may see a continuation of past patterns where the local utility is the dominant participant and continues to make arrangements and schedules on the basis of minimization of the operating cost that is paid by its own customers. Centrally dispatched power pools could evolve that include nonutility generators, some of whom may be engaged in direct sales to large consumers. Other areas may have open market structures that permit and facilitate competition with local utilities. Both local and remote nonutility entities, as well as remote utilities, may compete with the local electric utility to supply large industrial electric energy consumers or distribution utilities. The transmission system may be combined with a regional control center in a separate entity. Transmission networks could have the legal status of “common carriers,” where any qualified party would be allowed access to the transmission system to deliver energy to its own customers, wherever they might be located. This very nearly describes the current situation in Great Britain.

What does this have to do with the problems discussed in this text? A great deal. In the extreme cases mentioned earlier, many of the dispatch and scheduling methods we are going to discuss will need to be rethought and perhaps drastically revised. Current practices in automatic generation control are based on tacit assumptions that the electric energy market is slow moving with only a few, more-or-less fixed, interchange contracts that are arranged between interconnected utilities. Current techniques for establishing optimal economic generation schedules are really based on the assumption of a single utility serving the electric energy needs of its own customers at minimum cost. Interconnected operations and energy interchange agreements are presently the result of interutility arrangements: all of the parties share common interests. In a world with a transmission-operation entity required to provide access to many parties, both utility and nonutility organizations, this entity has the task of developing operating schedules to accomplish the deliveries scheduled in some (as yet to be defined) “optimal” fashion within the physical constraints of the system, while maintaining system reliability and security. If all (or any) of this develops, it should be a fascinating time to be active in this field.

1.6 CHARACTERISTICS OF STEAM UNITS

In analyzing the problems associated with the controlled operation of power systems, there are many possible parameters of interest. Fundamental to the economic operating problem is the set of input–output characteristics of a thermal power generation unit. A typical boiler–turbine–generator unit is sketched in Figure 1.1. This unit consists of a single boiler that generates steam to drive a single turbine–generator set. The electrical output of this set is connected not only to the electric power system, but also to the auxiliary power system in the power plant. A typical steam turbine unit may require 2–6% of the gross output of the unit for the auxiliary power requirements necessary to drive boiler feed pumps, fans, condenser circulating water pumps, and so on. In defining the unit characteristics, we will talk about gross input versus net output. That is, gross input to the plant represents the total input, whether measured in terms of dollars per hour or tons of coal per hour or millions of cubic feet of gas per hour, or any other units. The net output of the plant is the electrical power output available to the electric utility system. Occasionally, engineers will develop gross input–gross output characteristics. In such situations, the data should be converted to net output to be more useful in scheduling the generation.

FIGURE 1.1 Boiler–turbine–generator unit.

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FIGURE 1.2 Input–output curve of a steam turbine generator.

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In defining the characteristics of steam turbine units, the following terms will be used:

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Occasionally, the $/h operating cost rate of a unit will include prorated operation and maintenance costs. That is, the labor cost for the operating crew will be included as part of the operating cost if this cost can be expressed directly as a function of the output of the unit. The output of the generation unit will be designated by P, the megawatt net output of the unit. Figure 1.2 shows the input–output characteristic of a steam unit in idealized form. The input to the unit shown on the ordinate may be either in terms of heat energy requirements [millions of Btu per hour (MBtu/h)] or in terms of total cost per hour ($/h). The output is normally the net electrical output of the unit. The characteristic shown is idealized in that it is presented as a smooth, convex curve.

These data may be obtained from design calculations or from heat rate tests. When heat rate test data are used, it will usually be found that the data points do not fall on a smooth curve. Steam turbine generating units have several critical operating constraints. Generally, the minimum load at which a unit can operate is influenced more by the steam generator and the regenerative cycle than by the turbine. The only critical parameters for the turbine are shell and rotor metal differential temperatures, exhaust hood temperature, and rotor and shell expansion. Minimum load limitations are generally caused by fuel combustion stability and inherent steam generator design constraints. For example, most supercritical units cannot operate below 30% of design capability. A minimum flow of 30% is required to cool the tubes in the furnace of the steam generator adequately. Turbines do not have any inherent overload capability, so the data shown on these curves normally do not extend much beyond 5% of the manufacturer’s stated valve-wide-open capability.

FIGURE 1.3 Incremental heat (cost) rate characteristic.

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FIGURE 1.4 Net heat rate characteristic of a steam turbine generator unit.

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The incremental heat rate characteristic for a unit of this type is shown in Figure 1.3 This incremental heat rate characteristic is the slope (the derivative) of the input–output characteristic (ΔHP or ΔFP). The data shown on this curve are in terms of Btu/kWh (or $/kWh) versus the net power output of the unit in megawatts. This characteristic is widely used in economic dispatching of the unit. It is converted to an incremental fuel cost characteristic by multiplying the incremental heat rate in Btu per kilowatt hour by the equivalent fuel cost in terms of $/Btu. Frequently, this characteristic is approximated by a sequence of straight-line segments.

The last important characteristic of a steam unit is the unit (net) heat rate characteristic shown in Figure 1.4. This characteristic is H/P versus P. It is proportional to the reciprocal of the usual efficiency characteristic developed for machinery. The unit heat rate characteristic shows the heat input per kilowatt hour of output versus the megawatt output of the unit. Typical conventional steam turbine units are between 30 and 35% efficient, so their unit heat rates range between approximately 11,400 Btu/kWh and 9,800 Btu/kWh. (A kilowatt hour has a thermal equivalent of approximately 3412 Btu.) Unit heat rate characteristics are a function of unit design parameters such as initial steam conditions, stages of reheat and the reheat temperatures, condenser pressure, and the complexity of the regenerative feed-water cycle. These are important considerations in the establishment of the unit’s efficiency. For purposes of estimation, a typical heat rate of 10,500 Btu/kWh may be used occasionally to approximate actual unit heat rate characteristics.

FIGURE 1.5 Approximate representations of the incremental heat rate curve.

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Many different formats are used to represent the input–output characteristic shown in Figure 1.2. The data obtained from heat rate tests or from the plant design engineers may be fitted by a polynomial curve. In many cases, quadratic characteristics have been fit to these data. A series of straight-line segments may also be used to represent the input–output characteristics. The different representations will, of course, result in different incremental heat rate characteristics. Figure 1.5 shows two such variations. The solid line shows the incremental heat rate characteristic that results when the input versus output characteristic is a quadratic curve or some other continuous, smooth, convex function. This incremental heat rate characteristic is monotonically increasing as a function of the power output of the unit. The dashed lines in Figure 1.5 show a stepped incremental characteristic that results when a series of straight-line segments are used to represent the input–output characteristics of the unit. The use of these different representations may require that different scheduling methods be used for establishing the optimum economic operation of a power system. Both formats are useful, and both may be represented by tables of data. Only the first, the solid line, may be represented by a continuous analytic function, and only the first has a derivative that is nonzero. (That is, d2F/d2P equals 0 if dF/dP is constant.)

At this point, it is necessary to take a brief detour to discuss the heating value of the fossil fuels used in power generation plants. Fuel heating values for coal, oil, and gas are expressed in terms of Btu/lb or joules per kilogram of fuel. The determination is made under standard, specified conditions using a bomb calorimeter.

This is all to the good except that there are two standard determinations specified:

The difference between the HHV and LHV for a fuel depends on the hydrogen content of the fuel. Coal fuels have a low hydrogen content with the result that the difference between the HHV and LHV for a fuel is fairly small. (A typical value of the difference for a bituminous coal would be of the order of 3%. The HHV might be 14,800 Btu/lb and the LHV 14,400 Btu/lb.) Gas and oil fuels have a much higher hydrogen content, with the result that the relative difference between the HHV and LHV is higher; typically on the order of 10 and 6%, respectively. This gives rise to the possibility of some confusion when considering unit efficiencies and cycle energy balances. (A more detailed discussion is contained in the book by El-Wakil, [reference 1].)

A uniform standard must be adopted so that everyone uses the same heating value standard. In the United States, the standard is to use the HHV except that engineers and manufacturers that are dealing with combustion turbines (i.e., gas turbines) normally use LHVs when quoting heat rates or efficiencies. In European practice, LHVs are used for all specifications of fuel consumption and unit efficiency. In this text, HHVs are used throughout the book to develop unit characteristics. Where combustion turbine data have been converted by the authors from LHVs to HHVs, a difference of 10% was normally used. When in doubt about which standard for the fuel heating value has been used to develop unit characteristics—ask!

1.6.1 Variations in Steam Unit Characteristics

A number of different steam unit characteristics exist. For large steam turbine generators the input–output characteristics shown in Figure 1.2 are not always as smooth as indicated there. Large steam turbine generators will have a number of steam admission valves that are opened in sequence to obtain ever-increasing output of the unit. Figure 1.6 shows both an input–output and an incremental heat rate characteristic for a unit with four valves. As the unit loading increases, the input to the unit increases and the incremental heat rate decreases between the opening points for any two valves. However, when a valve is first opened, the throttling losses increase rapidly and the incremental heat rate rises suddenly. This gives rise to the discontinuous type of incremental heat rate characteristic shown in Figure 1.6. It is possible to use this type of characteristic in order to schedule steam units, although it is usually not done. This type of input–output characteristic is nonconvex; hence, optimization techniques that require convex characteristics may not be used with impunity.

FIGURE 1.6 Characteristics of a steam turbine generator with four steam admission valves.

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Another type of steam unit that may be encountered is the common-header plant, which contains a number of different boilers connected to a common steam line (called a common header). Figure 1.7 is a sketch of a rather complex common-header plant. In this plant, there are not only a number of boilers and turbines, each connected to the common header, but also a “topping turbine” connected to the common header. A topping turbine is one in which steam is exhausted from the turbine and fed not to a condenser but to the common steam header.

A common-header plant will have a number of different input–output characteristics that result from different combinations of boilers and turbines connected to the header. Steinberg and Smith (reference 2) treat this type of plant quite extensively. Common-header plants were constructed originally not only to provide a large electrical output from a single plant but also to provide steam sendout for the heating and cooling of buildings in dense urban areas. After World War II, a number of these plants were modernized by the installation of the type of topping turbine shown in Figure 1.7. For a period of time during the 1960s, these common-header plants were being dismantled and replaced by modern, efficient plants. However, as urban areas began to reconstruct, a number of metropolitan utilities found that their steam loads were growing and that the common-header plants could not be dismantled but had to be expected to provide steam supplies to new buildings.

FIGURE 1.7 A common-header steam plant.

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Combustion turbines (gas turbines) are also used to drive electric generating units. Some types of power generation units have been derived from aircraft gas turbine units and others from industrial gas turbines that have been developed for applications like driving pipeline pumps. In their original applications, these two types of combustion turbines had dramatically different duty cycles. Aircraft engines see relatively short duty cycles where power requirements vary considerably over a flight profile. Gas turbines in pumping duty on pipelines would be expected to operate almost continuously throughout the year. Service in power generation may require both types of duty cycle.