CONTENTS
References for Generation Systems / 31
Further Reading / 31
2 Industrial Organization, Managerial Economics, and Finance 35
2.1 Introduction / 35
2.2 Business Environments / 36
2.2.1 Regulated Environment / 37
2.2.2 Competitive Market Environment / 38
2.3 Theory of the Firm / 40
2.4 Competitive Market Solutions / 42
2.5 Supplier Solutions / 45
2.5.1 Supplier Costs / 46
2.5.2 Individual Supplier Curves / 46
2.5.3 Competitive Environments / 47
2.5.4 Imperfect Competition / 51
2.5.5 Other Factors / 52
2.6 Cost of Electric Energy Production / 53
2.7 Evolving Markets / 54
2.7.1 Energy Flow Diagram / 57
2.8 Multiple Company Environments / 58
2.8.1 Leontief Model: Input–Output Economics / 58
2.8.2 Scarce Fuel Resources / 60
2.9 Uncertainty and Reliability / 61
PROBLEMS / 61
Reference / 62
3 Economic Dispatch of Thermal Units and Methods of Solution 63
3.1 The Economic Dispatch Problem / 63
3.2 Economic Dispatch with Piecewise Linear Cost Functions / 68
3.3 LP Method / 69
3.3.1 Piecewise Linear Cost Functions / 69
3.3.2 Economic Dispatch with LP / 71
3.4 The Lambda Iteration Method / 73
3.5 Economic Dispatch Via Binary Search / 76
3.6 Economic Dispatch Using Dynamic Programming / 78
3.7 Composite Generation Production Cost Function / 81
3.8 Base Point and Participation Factors / 85
3.9 Thermal System Dispatching with Network Losses
Considered / 88
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3.10 The Concept of Locational Marginal Price (LMP) / 92
3.11 Auction Mechanisms / 95
3.11.1 PJM Incremental Price Auction as a
Graphical Solution / 95
3.11.2 Auction Theory Introduction / 98
3.11.3 Auction Mechanisms / 100
3.11.4 English (First-Price Open-Cry = Ascending) / 101
3.11.5 Dutch (Descending) / 103
3.11.6 First-Price Sealed Bid / 104
3.11.7 Vickrey (Second-Price Sealed Bid) / 105
3.11.8 All Pay (e.g., Lobbying Activity) / 105
APPENDIX 3A Optimization Within Constraints / 106
APPENDIX 3B Linear Programming (LP) / 117
APPENDIX 3C Non-Linear Programming / 128
APPENDIX 3D Dynamic Programming (DP) / 128
APPENDIX 3E Convex Optimization / 135
PROBLEMS / 138
References / 146
4 Unit Commitment 147
4.1 Introduction / 147
4.1.1 Economic Dispatch versus Unit Commitment / 147
4.1.2 Constraints in Unit Commitment / 152
4.1.3 Spinning Reserve / 152
4.1.4 Thermal Unit Constraints / 153
4.1.5 Other Constraints / 155
4.2 Unit Commitment Solution Methods / 155
4.2.1 Priority-List Methods / 156
4.2.2 Lagrange Relaxation Solution / 157
4.2.3 Mixed Integer Linear Programming / 166
4.3 Security-Constrained Unit Commitment (SCUC) / 167
4.4 Daily Auctions Using a Unit Commitment / 167
APPENDIX 4A Dual Optimization on a Nonconvex
Problem / 167
APPENDIX 4B Dynamic-Programming Solution to
Unit Commitment / 173
4B.1 Introduction / 173
4B.2 Forward DP Approach / 174
PROBLEMS / 182
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5 Generation with Limited Energy Supply 187
5.1 Introduction / 187
5.2 Fuel Scheduling / 188
5.3 Take-or-Pay Fuel Supply Contract / 188
5.4 Complex Take-or-Pay Fuel Supply Models / 194
5.4.1 Hard Limits and Slack Variables / 194
5.5 Fuel Scheduling by Linear Programming / 195
5.6 Introduction to Hydrothermal Coordination / 202
5.6.1 Long-Range Hydro-Scheduling / 203
5.6.2 Short-Range Hydro-Scheduling / 204
5.7 Hydroelectric Plant Models / 204
5.8 Scheduling Problems / 207
5.8.1 Types of Scheduling Problems / 207
5.8.2 Scheduling Energy / 207
5.9 The Hydrothermal Scheduling Problem / 211
5.9.1 Hydro-Scheduling with Storage Limitations / 211
5.9.2 Hydro-Units in Series (Hydraulically Coupled) / 216
5.9.3 Pumped-Storage Hydroplants / 218
5.10 Hydro-Scheduling using Linear Programming / 222
APPENDIX 5A Dynamic-Programming Solution to hydrothermal
Scheduling / 225
5.A.1 Dynamic Programming Example / 227
5.A.1.1 Procedure / 228
5.A.1.2 Extension to Other Cases / 231
5.A.1.3 Dynamic-Programming Solution to Multiple Hydroplant
Problem / 232
PROBLEMS / 234
6 Transmission System Effects 243
6.1 Introduction / 243
6.2 Conversion of Equipment Data to Bus and Branch Data / 247
6.3 Substation Bus Processing / 248
6.4 Equipment Modeling / 248
6.5 Dispatcher Power Flow for Operational Planning / 251
6.6 Conservation of Energy (Tellegen’s Theorem) / 252
6.7 Existing Power Flow Techniques / 253
6.8 The Newton–Raphson Method Using the Augmented
Jacobian Matrix / 254
6.8.1 Power Flow Statement / 254
6.9 Mathematical Overview / 257
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6.10 AC System Control Modeling / 259
6.11 Local Voltage Control / 259
6.12 Modeling of Transmission Lines and Transformers / 259
6.12.1 Transmission Line Flow Equations / 259
6.12.2 Transformer Flow Equations / 260
6.13 HVDC links / 261
6.13.1 Modeling of HVDC Converters
and FACT Devices / 264
6.13.2 Definition of Angular Relationships in
HVDC Converters / 264
6.13.3 Power Equations for a Six-Pole HVDC
Converter / 264
6.14 Brief Review of Jacobian Matrix Processing / 267
6.15 Example 6A: AC Power Flow Case / 269
6.16 The Decoupled Power Flow / 271
6.17 The Gauss–Seidel Method / 275
6.18 The “DC” or Linear Power Flow / 277
6.18.1 DC Power Flow Calculation / 277
6.18.2 Example 6B: DC Power Flow Example on the
Six-Bus Sample System / 278
6.19 Unified Eliminated Variable Hvdc Method / 278
6.19.1 Changes to Jacobian Matrix Reduced / 279
6.19.2 Control Modes / 280
6.19.3 Analytical Elimination / 280
6.19.4 Control Mode Switching / 283
6.19.5 Bipolar and 12-Pulse Converters / 283
6.20 Transmission Losses / 284
6.20.1 A Two-Generator System Example / 284
6.20.2 Coordination Equations, Incremental Losses,
and Penalty Factors / 286
6.21 Discussion of Reference Bus Penalty Factors / 288
6.22 Bus Penalty Factors Direct from the AC Power Flow / 289
PROBLEMS / 291
7 Power System Security 296
7.1 Introduction / 296
7.2 Factors Affecting Power System Security / 301
7.3 Contingency Analysis: Detection of Network Problems / 301
7.3.1 Generation Outages / 301
7.3.2 Transmission Outages / 302
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7.4 An Overview of Security Analysis / 306
7.4.1 Linear Sensitivity Factors / 307
7.5 Monitoring Power Transactions Using “Flowgates” / 313
7.6 Voltage Collapse / 315
7.6.1 AC Power Flow Methods / 317
7.6.2 Contingency Selection / 320
7.6.3 Concentric Relaxation / 323
7.6.4 Bounding / 325
7.6.5 Adaptive Localization / 325
APPENDIX 7A AC Power Flow Sample Cases / 327
APPENDIX 7B Calculation of Network Sensitivity Factors / 336
7B.1 Calculation of PTDF Factors / 336
7B.2 Calculation of LODF Factors / 339
7B.2.1 Special Cases / 341
7B.3 Compensated PTDF Factors / 343
Problems / 343
References / 349
8 Optimal Power Flow 350
8.1 Introduction / 350
8.2 The Economic Dispatch Formulation / 351
8.3 The Optimal Power Flow Calculation Combining
Economic Dispatch and the Power Flow / 352
8.4 Optimal Power Flow Using the DC Power Flow / 354
8.5 Example 8A: Solution of the DC Power Flow OPF / 356
8.6 Example 8B: DCOPF with Transmission Line
Limit Imposed / 361
8.7 Formal Solution of the DCOPF / 365
8.8 Adding Line Flow Constraints to the Linear
Programming Solution / 365
8.8.1 Solving the DCOPF Using Quadratic Programming / 367
8.9 Solution of the ACOPF / 368
8.10 Algorithms for Solution of the ACOPF / 369
8.11 Relationship Between LMP, Incremental Losses,
and Line Flow Constraints / 376
8.11.1 Locational Marginal Price at a Bus with No Lines
Being Held at Limit / 377
8.11.2 Locational Marginal Price with a Line Held at its Limit / 378
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8.12 Security-Constrained OPF / 382
8.12.1 Security Constrained OPF Using the DC Power Flow
and Quadratic Programming / 384
8.12.2 DC Power Flow / 385
8.12.3 Line Flow Limits / 385
8.12.4 Contingency Limits / 386
APPENDIX 8A Interior Point Method / 391
APPENDIX 8B Data for the 12-Bus System / 393
APPENDIX 8C Line Flow Sensitivity Factors / 395
APPENDIX 8D Linear Sensitivity Analysis of the
AC Power Flow / 397
PROBLEMS / 399
9 Introduction to State Estimation in Power Systems 403
9.1 Introduction / 403
9.2 Power System State Estimation / 404
9.3 Maximum Likelihood Weighted Least-Squares
Estimation / 408
9.3.1 Introduction / 408
9.3.2 Maximum Likelihood Concepts / 410
9.3.3 Matrix Formulation / 414
9.3.4 An Example of Weighted Least-Squares
State Estimation / 417
9.4 State Estimation of an Ac Network / 421
9.4.1 Development of Method / 421
9.4.2 Typical Results of State Estimation on an
AC Network / 424
9.5 State Estimation by Orthogonal Decomposition / 428
9.5.1 The Orthogonal Decomposition Algorithm / 431
9.6 An Introduction to Advanced Topics in State Estimation / 435
9.6.1 Sources of Error in State Estimation / 435
9.6.2 Detection and Identification of Bad Measurements / 436
9.6.3 Estimation of Quantities Not Being Measured / 443
9.6.4 Network Observability and Pseudo-measurements / 444
9.7 The Use of Phasor Measurement Units (PMUS) / 447
9.8 Application of Power Systems State Estimation / 451
9.9 Importance of Data Verification and Validation / 454
9.10 Power System Control Centers / 454
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APPENDIX 9A Derivation of Least-Squares Equations / 456
9A.1 The Overdetermined Case (Nm > Ns) / 457
9A.2 The Fully Determined Case (Nm = Ns) / 462
9A.3 The Underdetermined Case (Nm < Ns) / 462
PROBLEMS / 464
10 Control of Generation 468
10.1 Introduction / 468
10.2 Generator Model / 470
10.3 Load Model / 473
10.4 Prime-Mover Model / 475
10.5 Governor Model / 476
10.6 Tie-Line Model / 481
10.7 Generation Control / 485
10.7.1 Supplementary Control Action / 485
10.7.2 Tie-Line Control / 486
10.7.3 Generation Allocation / 489
10.7.4 Automatic Generation Control (AGC)
Implementation / 491
10.7.5 AGC Features / 495
10.7.6 NERC Generation Control Criteria / 496
PROBLEMS / 497
References / 500
11 Interchange, Pooling, Brokers, and Auctions 501
11.1 Introduction / 501
11.2 Interchange Contracts / 504
11.2.1 Energy / 504
11.2.2 Dynamic Energy / 506
11.2.3 Contingent / 506
11.2.4 Market Based / 507
11.2.5 Transmission Use / 508
11.2.6 Reliability / 517
11.3 Energy Interchange between Utilities / 517
11.4 Interutility Economy Energy Evaluation / 521
11.5 Interchange Evaluation with Unit Commitment / 522
11.6 Multiple Utility Interchange Transactions—Wheeling / 523
11.7 Power Pools / 526
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11.8 The Energy-Broker System / 529
11.9 Transmission Capability General Issues / 533
11.10 Available Transfer Capability and Flowgates / 535
11.10.1 Definitions / 536
11.10.2 Process / 539
11.10.3 Calculation ATC Methodology / 540
11.11 Security Constrained Unit Commitment (SCUC) / 550
11.11.1 Loads and Generation in a Spot Market Auction / 550
11.11.2 Shape of the Two Functions / 552
11.11.3 Meaning of the Lagrange Multipliers / 553
11.11.4 The Day-Ahead Market Dispatch / 554
11.12 Auction Emulation using Network LP / 555
11.13 Sealed Bid Discrete Auctions / 555
PROBLEMS / 560
12 Short-Term Demand Forecasting 566
12.1 Perspective / 566
12.2 Analytic Methods / 569
12.3 Demand Models / 571
12.4 Commodity Price Forecasting / 572
12.5 Forecasting Errors / 573
12.6 System Identification / 573
12.7 Econometric Models / 574
12.7.1 Linear Environmental Model / 574
12.7.2 Weather-Sensitive Models / 576
12.8 Time Series / 578
12.8.1 Time Series Models Seasonal Component / 578
12.8.2 Auto-Regressive (AR) / 580
12.8.3 Moving Average (MA) / 581
12.8.4 Auto-Regressive Moving Average (ARMA):
Box-Jenkins / 582
12.8.5 Auto-Regressive Integrated Moving-Average
(ARIMA): Box-Jenkins / 584
12.8.6 Others (ARMAX, ARIMAX, SARMAX, NARMA) / 585
12.9 Time Series Model Development / 585
12.9.1 Base Demand Models / 586
12.9.2 Trend Models / 586
12.9.3 Linear Regression Method / 586
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12.9.4 Seasonal Models / 588
12.9.5 Stationarity / 588
12.9.6 WLS Estimation Process / 590
12.9.7 Order and Variance Estimation / 591
12.9.8 Yule-Walker Equations / 592
12.9.9 Durbin-Levinson Algorithm / 595
12.9.10 Innovations Estimation for MA and ARMA
Processes / 598
12.9.11 ARIMA Overall Process / 600
12.10 Artificial Neural Networks / 603
12.10.1 Introduction to Artificial Neural Networks / 604
12.10.2 Artificial Neurons / 605
12.10.3 Neural network applications / 606
12.10.4 Hopfield Neural Networks / 606
12.10.5 Feed-Forward Networks / 607
12.10.6 Back-Propagation Algorithm / 610
12.10.7 Interior Point Linear Programming Algorithms / 613
12.11 Model Integration / 614
12.12 Demand Prediction / 614
12.12.1 Hourly System Demand Forecasts / 615
12.12.2 One-Step Ahead Forecasts / 615
12.12.3 Hourly Bus Demand Forecasts / 616
12.13 Conclusion / 616
PROBLEMS / 617
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.
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.
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 graduatelevel
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
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xxii preface to the first edition
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.
