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Analyzes the dynamic performance of interconnected power systems.
* Examines the characteristics of the various components of a power system during normal operating conditions and during disturbances.
* Explores the detailed mathematical models of system components and analyzes the system behavior using the necessary computational tools.

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This is the revised printing of a text originally published in 1977 by the Iowa State University Press. Providing a comprehensive description of the dynamic condition of the power system, it presents critical information on synchronous generators and their excitation systems, as well as extensive material on mathematical modeling of these critical components. For practicing engineers in the electric power industry, as well as advanced students. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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Power System Control and Stability

John Wiley & Sons

Chapter One

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 signaled 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 entire 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 problems 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 disturbances 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 described 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 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 resynchronized 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 transmission 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 remain 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 equipment, sudden application of a major load such as a steel mill, or loss of a line or generating 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 system 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 conditions, 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 remain operating in parallel and at the same speed.

The transient following a system perturbation is oscillatory in nature; but if the system is stable, these oscillations will be damped toward a new quiescent operating condition. 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 includes 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) disturbance 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 term "stability," we now propose a simple nonmathematical definition of the term that will be satisfactory for elementary problems. Later, we will provide a more rigorous mathematical definition.

The problem of interest is one where a power system operating under a steady load condition is perturbed, causing the readjustment of the voltage angles of the synchronous machines. If such an occurrence creates an unbalance between the system generation and load, it results in the establishment of a new steady-state operating condition, with the subsequent adjustment of the voltage angles. The perturbation could be a major disturbance such as the loss of a generator, a fault or the loss of a line, or a combination of such events. It could also be a small load or random load changes occurring under normal operating conditions.

Adjustment to the new operating condition is called the transient period. The system behavior during this time is called the dynamic system performance, which is of concern in defining system stability. The main criterion for stability is that the synchronous machines maintain synchronism at the end of the transient period.

Definition: If the oscillatory response of a power system during the transient period following a disturbance is damped and the system settles in a finite time to a new steady operating condition, we say the system is stable. If the system is not stable, it is considered unstable.

This primitive definition of stability requires that the system oscillations be damped. This condition is sometimes called asymptotic stability and means that the system contains inherent forces that tend to reduce oscillations. This is a desirable feature in many systems and is considered necessary for power systems.

The definition also excludes continuous oscillation from the family of stable systems, although oscillators are stable in a mathematical sense. The reason is practical since a continually oscillating system would be undesirable for both the supplier and the user of electric power. Hence the definition describes a practical specification for an acceptable operating condition.

1.3.2 Other stability problems

While the stability of synchronous machines and tie lines is the most important and common problem, other stability problems may exist, particularly in power systems having appreciable capacitances. In such cases arrangements must be made to avoid excessive voltages during light load conditions, to avoid damage to equipment, and to prevent self-excitation of machines.

Some of these problems are discussed in Part III, while others are beyond the scope of this book.

1.3.3 Stability of synchronous machines

Distinction should be made between sudden and major changes, which we shall call large impacts, and smaller and more normal random impacts. A fault on the high-voltage transmission network or the loss of a major generating unit are examples of large impacts. If one of these large impacts occurs, the synchronous machines may lose synchronism. This problem is referred to in the literature as the transient stability problem. Without detailed discussion, some general comments are in order. First, these impacts have a finite probability of occurring. Those that the system should be designed to withstand must therefore be selected a priori. Second, the ability of the system to survive a certain disturbance depends on its precise operating condition at the time of the occurrence. A change in the system loading, generation schedule, network interconnections, or type of circuit protection may give completely different results in a stability study for the same disturbance. Thus the transient stability study is a very specific one, from which the engineer concludes that under given system conditions and for a given impact the synchronous machines will or will not remain in synchronism. Stability depends strongly upon the magnitude and location of the disturbance and to a lesser extent upon the initial state or operating condition of the system.

Let us now consider a situation where there are no major shocks or impacts, but rather a random occurrence of small changes in system loading. Here we would expect the system operator to have scheduled enough machine capacity to handle the load. We would also expect each synchronous machine to be operating on the stable portion of its power-angle curve, i.e., the portion in which the power increases with increased angle. In the dynamics of the transition from one operating point to another, to adjust for load changes, the stability of the machines will be determined by many factors, including the power-angle curve. It is sometimes incorrect to consider a single power-angle curve, since modern exciters will change the operating curve during the period under study. The problem of studying the stability of synchronous machines under the condition of small load changes has been called "steady-state" stability. A more recent and certainly more appropriate name is dynamic stability. In contrast to transient stability, dynamic stability tends to be a property of the state of the system.

Transient stability and dynamic stability are both questions that must be answered to the satisfaction of the engineer for successful planning and operation of the system. This attitude is adopted in spite of the fact that an artificial separation between the two problems has been made in the past. This was simply a convenience to accommodate the different approximations and assumptions made in the mathematical treatments of the two problems. In support of this viewpoint the following points are pertinent.

First, the availability of high-speed digital computers and modern modeling techniques makes it possible to represent any component of the power system in almost any degree of complexity required or desired. Thus questionable simplifications or assumptions are no longer needed and are often not justified.

Second, and perhaps more important, in a large interconnected system the full effect of a disturbance is felt at the remote parts some time after its occurrence, perhaps a few seconds. Thus different parts of the interconnected system will respond to localized disturbances at different times. Whether they will act to aid stability is difficult to predict beforehand. The problem is aggravated if the initial disturbance causes other disturbances in neighboring areas due to power swings. As these conditions spread, a chain reaction may result and large-scale interruptions of service may occur. However, in a large interconnected system, the effect of an impact must be studied over a relatively long period, usually several seconds and in some cases a few minutes. Performance of dynamic stability studies for such long periods will require the simulation of system components often neglected in the so-called transient stability studies.

1.3.4 Tie-line oscillations

As random power impacts occur during the normal operation of a system, this added power must be supplied by the generators. The portion supplied by the different generators under different conditions depends upon electrical proximity to the position of impact, energy stored in the rotating masses, governor characteristics, and other factors. The machines therefore are never truly at steady state except when at standstill. Each machine is in continuous oscillation with respect to the others due to the effect of these random stimuli. These oscillations are reflected in the flow of power in the transmission lines. If the power in any line is monitored, periodic oscillations are observed to be superimposed on the steady flow. Normally, these oscillations are not large and hence not objectionable.

The situation in a tie line is different in one sense since it connects one group of machines to another. These two groups are in continuous oscillation with respect to each other, and this is reflected in the power flow over the tie line. The situation may be further complicated by the fact that each machine group in turn is connected to other groups. Thus the tie line under study may in effect be connecting two huge systems. In this case the smallest oscillatory adjustments in the large systems are reflected as sizable power oscillations in the tie line. The question then becomes, To what degree can these oscillations be tolerated?

(Continues...)

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Excerpted from Power System Control and Stability by Paul M. Anderson A. A. Fouad Excerpted by permission.
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