## 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.