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BOOKS IN THE IEEE PRESS SERIES ON POWER ENGINEERING
Power System Protection P. M. Anderson 1999 Hardcover 1344pp 0-7803-3472-2
Understanding Power Quality Problems: Voltage Sags and Interruptions Math H. 1.Bollen 2000 Hardcover 576pp 0-7803-4713-7
Electric Power Applicationsof Fuzzy Systems Edited by M. E. El-Hawary 1998 Hardcover 384pp 0-7803-1197-3
Principlesof Electric Machineswith Power ElectronicApplications, Second Edition M. E. El-Hawary 2002 Hardcover 496pp 0-471-20812-4
Analysis of Electric Machineryand Drive Systems,SecondEdition Paul C. Krause, Oleg Wasynczuk, and Scott D. Sudhoff 2002 Hardcover 624pp 0-471-14326-X
Power System Control and Stability
Second Edition
IEEE Power Engineering Society, Sponsor
IEEE PressPower Engineering Series Mohamed E. El-Hawary, Series Editor
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ISBN 0-471-23862-7
10 9 8 7 6 5 4 3 2 1
To Our Families
Part I Introduction P. M. Anderson and A. A. Fouad
Chapter 1. Power System Stability
1.1 Introduction 1.2 Requirements of a Reliable Electrical Power Service 1.3 Statement of the Problem 1.4 Effect of an Impact upon System Components 1.5 Methods of Simulation
Problems
Chapter 2. The Elementary Mathematical Model
2.1 Swing Equation 2.2 Units 2.3 Mechanical Torque 2.4 Electrical Torque 2.5 Power-Angle Curve of a Synchronous Machine 2.6 Natural Frequencies of Oscillation of a Synchronous Machine 2.7 System of One Machine against an Infinite Bus-The Classical Model 2.8 Equal Area Criterion 2.9 Classical Model of a Multimachine System 2.10 Classical Stability Study of a Nine-Bus System 2.11 Shortcomings of the Classical Model 2.12 Block Diagram of One Machine
Problems References
Chapter 3. System Response to Small Disturbances
3.1 Introduction 3.2 Types of Problems Studied 3.3 The Unregulated Synchronous Machine 3.4 Modes of Oscillation of an Unregulated Multimachine System 3.5 Regulated Synchronous Machine
Contents
xiii
10 11
13 15 16 20 21 24 26 31 35 37 45 47 48 52
53 54 55 59 66
vii
VIII
3.6 Distributionof Power impacts Problems References
Part II The Electromagnetic Torque P. M. Andersonand A. A. Fouad
Contents
4.1 Introduction 4.2 Park's Transformation 4.3 Flux Linkage Equations 4.4 Voltage Equations 4.5 Formulationof State-SpaceEquations 4.6 Current Formulation 4.7 Per Unit Conversion 4.8 Normalizingthe Voltage Equations 4.9 Normalizingthe Torque Equations 4.10 Torque and Power 4.11 EquivalentCircuit of a Synchronous Machine 4.12 The Flux Linkage State-SpaceModel 4.13 Load Equations 4.14 Subtransientand Transient Inductancesand Time Constants 4.15 SimplifiedModels of the Synchronous Machine 4.16 Turbine GeneratorDynamic Models
Problems References
5.1 Introduction 5.2 Steady-StateEquationsand Phasor Diagrams 5.3 Machine Connectedto an Infinite Bus througha TransmissionLine 5.4 Machine Connectedto an Infinite Bus with Local Load at Machine
Terminal 5.5 DeterminingSteady-State Conditions 5.6 Examples 5.7 Initial Conditions for a Multimachine System 5.8 Determinationof MachineParametersfrom Manufacturers' Data 5.9 Analog Computer Simulationof the Synchronous Machine 5.10 Digital Simulationof Synchronous Machines
Problems References
6.1 Introduction 6.2 Linearization of the GeneratorState-SpaceCurrentModel 6.3 Linearization of the Load Equation for the One-Machine Problem 6.4 Linearization of the Flux LinkageModel 6.5 SimplifiedLinear Model
83 83 85 88 91 91 92 99
103 105 107 109 114 122 127 143 146 148
150 150 153
208 209 213 217 222
Contents IX
6.6 Block Diagrams 231 6.7 State-Space Representation of Simplified Model 231
Problems 232 References 232
Chapter 7. Excitation Systems
7.1 Simplified View of Excitation Control 233 7.2 Control Configurations 235 7.3 Typical Excitation Configurations 236 7.4 Excitation Control System Definitions 243 7.5 Voltage Regulator 250 7.6 Exciter Buildup 254 7.7 Excitation System Response 268 7.8 State-Space Description of the Excitation System 285 7.9 Computer Representation of Excitation Systems 292 7.10 Typical System Constants 299 7.11 The Effect of Excitation on Generator Performance 304
Problems 304 References 307
Chapter 8. Effect ofExcitation on Stability
8.1 Introduction 309 8.2 Effect of Excitation on Generator Power Limits 311 8.3 Effect of the Excitation System on Transient Stability 315 8.4 Effect of Excitation on Dynamic Stability 321 8.5 Root-Locus Analysis of a Regulated Machine Connected to
an Infinite Bus 327 8.6 Approximate System Representation 333 8.7 Supplementary Stabilizing Signals 338 8.8 Linear Analysis of the Stabilized Generator 344 8.9 Analog Computer Studies 347 8.10 Digital Computer Transient Stability Studies 353 8.11 Some General Comments on the Effect of Excitation on Stability 363
Problems 365 References 366
Chapter 9. Multimachine Systems with Constant Impedance Loads
9.1 Introduction 368 9.2 Statement of the Problem 368 9.3 Matrix Representation of a Passive Network 369 9.4 Converting Machine Coordinates to System Reference 373 9.5 Relation Between Machine Currents and Voltages 374 9.6 System Order 377 9.7 Machines Represented by Classical Methods 378 9.8 Linearized Model for the Network 381 9.9 Hybrid Formulation 386 9.10 Network Equations with Flux Linkage Model 388 9.11 Total System Equations 390
x
Contents
392 396 397
Part III The Mechanical Torque Power System Control and Stability P. M. Anderson
Chapter 10. Speed Governing
10.1 The Flyball Governor 10.2 The Isochronous Governor 10.3 Incremental Equations of the Turbine 10.4 The Speed Droop Governor 10.5 The Floating-Lever Speed Droop Governor 10.6 The Compensated Governor
Problems References
Chapter 11. Steam Turbine PrimeMovers
11.1 Introduction 11.2 Power Plant Control Modes 11.3 Thermal Generation 11.4 A Steam Power Plant Model 11.5 Steam Turbines 11.6 Steam Turbine Control Operations 11.7 Steam Turbine Control Functions 11.8 Steam Generator Control 11.9 Fossil-Fuel Boilers 11.10 Nuclear Steam Supply Systems
Problems References
Chapter12. Hydraulic Turbine PrimeMovers
12.1 Introduction 12.2 The Impulse Turbine 12.3 The Reaction Turbine 12.4 Propeller-Type Turbines 12.5 The Deriaz Turbine 12.6 Conduits, Surge Tanks, and Penstocks 12.7 Hydraulic System Equations 12.8 Hydraulic System Transfer Function 12.9 Simplifying Assumptions 12.10 Block Diagram for a Hydro System 12.11 Pumped Storage Hydro Systems Problems
References
402 408 410 413 419 421 428 428
430 432 435 436 437 444 446 458 461 476 480 481
484 484 486 489 489 489 498 503 506 509 510 511 512
Contents
13.1 Introduction 13.2 The Combustion Turbine Prime Mover 13.3 The Combined-Cycle Prime Mover
Problems References
513 513 518 527 527
Appendix A. Appendix B. Appendix C. Appendix D. Appendix E. Appendix F. Appendix G. Appendix H. Appendix I. Appendix J.
Index
529 531 545 555 582 590 614 622 631 640
651
Preface
It is well over thirty years since some of the early versions of this book were used in our classes, and it is more than a quarter of a century since the first edition appeared in print. Nor­ mally, one would have expected users of the book to almost give it up as old-fashioned. Yet, un­ til very recently the questions the authors were frequently asked explained the rationale for the added material in this edition, especially by new users: When will the Second Edition be out?
Over these past thirty years the size of the systems analyzed in stability studies, the scope of the studies (including the kind of answers sought), the duration of the transients analyzed, and the methods of solution may have varied, but central to all is that the proper system model must be used. Such a model must be based on description of the physical system and on its be­ havior during the transient being analyzed.
This book has focused on modeling the power system components for analysis of the electromechanical transient, perhaps with emphasis on the inertial transient. The one possible exception reflects the concern of the time the book came into being, namely analysis of the lin­ ear system model for detection and mitigation of possible poorly damped operating conditions.
Since the 1970s, several trends made stability of greater concern to power system engi­ neers. Because of higher cost of money and delay of transmission construction because of envi­ ronmental litigations, the bulk power system has experienced more congestion in transmission, more interdependence among networks, and so on. To maintain stability, there has been more dependence on discreet supplementary controls, greater need for studying larger systems, and analysis of longer transients. Since then, additional models were needed for inclusion in stabili­ ty studies: turbine governors, power plants, discrete supplementary controls, etc. Thus, the need for modeling the power system components that make up mechanical torque has become more important than ever. The authors think it is time to meet this need, as was originally planned.
Now that the electric utility industry is undergoing major restructuring, the question arises as to whether the trend that started in the 1970s is likely to continue, at least into the near future. Many power system analysts believe that the answer to this question is yes.
Since the revised printing of this book appeared, the electric utility industry has undergone a significant restructuring, resulting in heavier use of the bulk power transmission for interre­ gional transactions. It is expected that new engineering emphasis will be given to what engi­ neers refer to as mid-term or long-term analysis. We believe that in the restructured environ­ ment, this type of analysis will continue be needed because there will be greater emphasis on providing answers about system limitations to all parties involved in the various activities as well as in the interregional transactions. Modeling of mechanical torque will be important in conducting these studies.
The material on the "mechanical torque" presented in Chapters 10 through 13 and in Ap­ pendices F through J are the work of author Paul Anderson and he should be contacted regard­ ing any questions, corrections, or other information regarding these portions of the book. This material is a bit unusual to include in a book on power system stability and control, but we have recognized that a complete picture of stability and the supporting mathematical models cannot
xiii
xiv Preface
be consideredcompletewithouta discussionof these importantsystem components. The mod­ els presented here can be described as "low-order" models that we consider appropriate addi­ tions to studiesof powersystemstability. This limits the modelsto a short time span of a minute or so, and purposely avoids the modeling of power plant behavior for the long term, for exam­ ple, in the study of economics or energy dispatch.
P. M. ANDERSON
A. A. FaUAD
Part I Introduction
chapter 1
1.1 Introduction
Since the industrial revolution man's demand for and consumption of energy has increased steadily. The invention of the induction motor by Nikola Tesla in 1888 sig­ naled the growing importance of electrical energy in the industrial world as well as its use for artificial lighting. A major portion of the energy needs of a modern society is supplied in the form of electrical energy.
Industrially developed societies need an ever-increasing supply of electrical power, and the demand on the North American continent has been doubling every ten years. Very complex power systems have been built to satisfy this increasing demand. The trend in electric power production is toward an interconnected network of transmission lines linking generators and loads into large integrated systems, some of which span en­ tire continents. Indeed, in the United States and Canada, generators located thousands of miles apart operate in parallel.
This vast enterprise of supplying electrical energy presents many engineering prob­ lems that provide the engineer with a variety of challenges. The planning, construction, and operation of such systems become exceedingly complex. Some of the problems stimulate the engineer's managerial talents; others tax his knowledge and experience in system design. The entire design must be predicated on automatic control and not on the slow response of human operators. To be able to predict the performance of such complex systems, the engineer is forced to seek ever more powerful tools of analysis and synthesis.
This book is concerned with some aspects of the design problem, particularly the dynamic performance, of interconnected power systems. Characteristics of the various components of a power system during normal operating conditions and during dis­ turbances will be examined, and effects on the overall system performance will be analyzed. Emphasis will be given to the transient behavior in which the system is de­ scribed mathematically by ordinary differential equations.
1.2 Requirements of a Reliable Electrical Power Service
Successful operation of a power system depends largely on the engineer's ability to provide reliable and uninterrupted service to the loads. The reliability of the power supply implies much more than merely being available. Ideally, the loads must be fed at constant voltage and frequency at all times. In practical terms this means that both voltage and frequency must be held within close tolerances so that the consumer's
3
4 Chapter 1
equipment may operate satisfactorily. For example, a drop in voltage of 10-15% or a reduction of the system frequency of only a few hertz may lead to stalling of the motor loads on the system. Thus it can be accurately stated that the power system operator must maintain a very high standard of continuous electrical service.
The first requirement of reliable service is to keep the synchronous generators running in parallel and with adequate capacity to meet the load demand. If at any time a generator loses synchronism with the rest of the system, significant voltage and current fluctuations may occur and transmission lines may be automatically tripped by their relays at undesired locations. If a generator is separated from the system, it must be re­ synchronized and then loaded, assuming it has not been damaged and its prime mover has not been shut down due to the disturbance that caused the loss of synchronism.
Synchronous machines do not easily fall out of step under normal conditions. If a machine tends to speed up or slow down, synchronizing forces tend to keep it in step. Conditions do arise, however, in which operation is such that the synchronizing forces for one or more machines may not be adequate, and small impacts in the system may cause these machines to lose synchronism. A major shock to the system may also lead to a loss of synchronism for one or more machines.
A second requirement of reliable electrical service is to maintain the integrity of the power network. The high-voltage transmisssion system connects the generating stations and the load centers. Interruptions in this network may hinder the flow of power to the load. This usually requires a study of large geographical areas since almost all power systems are interconnected with neighboring systems. Economic power as well as emergency power may flow over interconnecting tie lines to help maintain continuity of service. Therefore, successful operation of the system means that these lines must re­ main in service if firm power is to be exchanged between the areas of the system.
While it is frequently convenient to talk about the power system in the "steady state," such a state never exists in the true sense. Random changes in load are taking place at all times, with subsequent adjustments of generation. Furthermore, major changes do take place at times, e.g., a fault on the network, failure in a piece of equip­ ment, sudden application of a major load such as a steel mill, or loss of a line or gen­ erating unit. We may look at any of these as a change from one equilibrium state to another. It might be tempting to say that successful operation requires only that the new state be a "stable" state (whatever that means). For example, if a generator is lost, the remaining connected generators must be capable of meeting the load demand; or if a line is lost, the power it was carrying must be obtainable from another source. Unfortunately, this view is erroneous in one important aspect: it neglects the dynamics of the transition from one equilibrium state to another. Synchronism frequently may be lost in that transition period, or growing oscillations may occur over a transmission line, eventually leading to its tripping. These problems must be studied by the power sys­ tem engineer and fall under the heading "power system stability."
1.3 Statement of the Problem
The stability problem is concerned with the behavior of the synchronous machines after they have been perturbed. If the perturbation does not involve any net change in power, the machines should return to their original state. If an unbalance between the supply and demand is created by a change in load, in generation, or in network condi­ tions, a new operating state is necessary. In any case all interconnected synchronous machines should remain in synchronism if the system is stable; i.e., they should all re­ main operating in parallel and at the same speed.
Power System Stability 5
The transient following a system perturbation is oscillatory in nature; but if the sys­ tem is stable, these oscillations will be damped toward a new quiescent operating con­ dition. These oscillations, however, are reflected as fluctuations in the power flow over the transmission lines. If a certain line connecting two groups of machines undergoes excessive power fluctuations, it may be tripped out by its protective equipment thereby disconnecting the two groups of machines. This problem is termed the stability of the tie line, even though in reality it reflects the stability of the two groups of machines.
A statement declaring a power system to be "stable" is rather ambiguous unless the conditions under which this stability has been examined are clearly stated. This in­ cludes the operating conditions as well as the type of perturbation given to the system. The same thing can be said about tie-line stability. Since we are concerned here with the tripping of the line, the power fluctuation that can be tolerated depends on the initial operating condition of the system, including the line loading and the nature of the impacts to which it is subjected. These questions have become vitally important with the advent of large-scale interconnections. In fact, a severe (but improbable) distur­ bance can always be found that will cause instability. Therefore, the disturbances for which the system should be designed to maintain stability must be deliberately selected.
1.3.1 Primitive definition of stability
Having introduced the…