Anderson, A. The third edition of the landmark book on power system stability and control, revised and updated with new material The revised third edition of Power System Control and Stability continues to offer a comprehensive text on the fundamental principles and concepts of power system stability and control as well as.
Power System Dynamics. Bialek, James R. An authoritative guide to the most up-to-date information on power system dynamics The revised third edition of Power System Dynamics and Stability contains a comprehensive, state-of-the-art review of information on the topic. The third edition continues the successful approach of the first and second editions by progressing from simplicity to. For many power systems this time is on the order of one second or less.
The classical model is the simplest model used in studies of power system dynamics and requires a minimum amount of data; hence, such studies can be conducted in a relatively short time and at minimum cost. Furthermore, these studies can provide useful information. For example, they may be used as preliminary studies to identify problem areas that require further study with more detailed modeling.
Thus a large number of cases for which the system exhibits a definitely stable dynamic response to the disturbances under study are eliminated from further consideration. A classical study will be presented here on a small nine-bus power system that has three generators and three loads. A one-line impedance diagram for the system is given in Figure 2. The prefault normal load-flow solution is given in Figure 2.
Generator data for the three machines are given in Table 2. This system, while small, is large enough to be nontrivial and thus permits the illustration of a number of stability concepts and results. In the performance of a transient stability study, the following data are needed: I. The equivalent impedances of the loads are obtained from the load bus data. The Elementary Mathematical Model Table 2. Note: Reactance values are in pu on a MVA base. All time constants are in s.
Several quantities are tabulated that are as yet undefined in this book. These quantities are derived and justified in Chapter 4 but are given here to provide complete data for the sample system. System data as follows: a. The inertia constant H and direct axis transient reactance x j for all generators.
Transmission network impedances for the initial network conditions and the subsequent switchings such as fault clearing and breaker reclosings. The type and location of disturbance, time of switchings, and the maximum time for which a solution is to be obtained. To prepare the system data for a stability study, the following preliminary calculations are made: 1. All system data are converted to a common base; a system base of M V A is. The loads are converted to equivalent impedances or admittances.
The needed data for this step are obtained from the load-flow study. These internal angles may be computed from the pretransient terminal voltages V k as follows.
Let the terminal voltage be used temporarily as a reference, as shown in Figure 2. V matrix for each network condition is calculated. The following steps are usually needed: a. The equivalent load impedances or admittances are connected between the load buses and the reference node; additional nodes are provided for the internal generator voltages nodes 1, 2,. Also, simulation of the fault impedance is added as required, and the admittance matrix is determined for each switching condition.
All impedance elements are converted to admittances. Elements of the matrix are identified as follows: ITi is the sum of all the adis the negative of the admittance between mittances connected to node i, and node i and node j. Finally, we eliminate all the nodes except for the internal generator nodes and obtain the k matrix for the reduced network.
The reduction can be achieved by matrix operation if we recall that all the nodes have zero injection currents except for the internal generator nodes. This property is used to obtain the network reduction as shown below.
Thus for the network in Figure 2. Expanding 2. It has the dimensions n x n where n is the number of the generators. The network reduction illustrated by 2. If the loads are not considered to be constant impedances, the identity of the load buses must be retained.
Network reduction can be applied only to those nodes that have zero injection current. The technique of solving a classical transient stability problem is illustrated by conducting a study of the nine-bus system, the data for which is given in Figures 2. The disturbance initiating the transient is a three-phase fault occurring near bus 7 at the end of line The fault is cleared in five cycles 0.
For the purpose of this study the generators are to be represented by the classical model and the loads by constant impedances. The damping torques are neglected. The system base is M V A. Make all the preliminary calculations needed for a transient stability study so that all coefficients in 2. Solution The objective of the study is to obtain time solutions for the rotor angles of the generators after the transient is introduced. These time solutions are called "swing curves.
Therefore, mathematically, we are to obtain a solution for the set of equations 2. All impedance data are given to this base. For convenience bus numbers I , 2, and 3 are used to denote the generator internal buses rather than the generator low-voltage terminal buses. Values for the generator x i are added to the reactance of the generator transformers. For example, for generator 2 bus 2 will be the internal bus for the voltage behind transient reactance; the reactance between.
The prefault network admittances including the load equivalents are given in Table 2. The P matrix for the faulted network and for the network with the fault cleared are similarly obtained. The results are shown in Tables 2. Elimination of the network nodes other than the generator internal nodes by network reduction as outlined in step 5 is done by digital computer. The resulting reduced Y matrices are shown in Table 2. We now have the values of the constant voltages behind transient reactances for all three generators and the reduced Y matrix for each network.
Thus all coefficients of 2. The fault is cleared in five cycles by opening line of Figure 2. Plot the angles a,, a2, and 4 and their difference versus time. S o ht ion. The problem is to solve the set of equations 2. All the coefficients for the faulted network and the network with the fault cleared have been determined in Example 2. Since the set 2. A brief survey of numerical integration of differential equations is given in Appendix B. For hand calculations see [ I ] for an excellent discussion of a numerical integration method of the swing equa-.
Also see Chapter IO of [8] for a more detailed discussion of several numerical schemes for solving the swing equation. The so-called transient stability digital computer programs available at many computer centers include subroutines for solving nonlinear differential equations. Discussion of these programs is beyond the scope of this book.
Numerical integration of the swing equations for the three-generator, nine-bus system is made by digital computer for 2. The maximum angle difference is about 8 5. To determine whether the system is stable or unstable for the pareither ticular transient under study, it is sufficient to carry out the time solution for one swing only. If the rotor angles or the angle differences reach maximum values and then decrease, the system is stable.. If any of the angle differences increase indefinitely, the system is unstable because at least one machine will lose synchronism.
System stability depends on the characteristics of all the components of the power system. This includes the response characteristics of the control equipment on the turbogenerators, on the dynamic characteristics of the loads, on the supplementary control equipment installed, and on the type and settings of protective equipment used. The machine dynamic response to any impact in the system is oscillatory. In the past the sizes of the power systems involved were such that the period of these oscillations was not much greater than one second.
Furthermore, the equipment used for excitation controls was relatively slow and simple. Thus the classical model was adequate. Today large system interconnections with the greater system inertias and relatively weaker ties result in longer periods of oscillations during transients. Generator control systems, particularly modern excitation systems, are extremely fast.
It is therefore. Indeed there have been recorded transients caused by large impacts, resulting in loss of synchronism after the system machines had undergone several oscillations. Another aspect is the dynamic instability problem, where growing oscillations have occurred on tie lines connecting different power pools or systems. As this situation has developed, it has also become increasingly important to ensure the security of the bulk power supply.
This has made many engineers realize it is time to reexamine the assumptions made in stability studies. This view is well stated by Ray and Shipley [ 14 : We have reached a time when it is appropriate that we appraise the state of the Art of Dynamic Stability Analysis.
In conjunction with this we must: 1. Expand our knowledge of the characteristictime response of our system loads to changes in. Re-examine old concepts and develop new ideas on changes in system networks to improve system stability. Update our knowledge of the response characteristics of the various components of energy systems and their controls boilers, reactors, turbine governors, generator regulators, field excitation, etc.
Reformulate our analytical techniques to adequately simulate the time variation of all of the foregoing factors in system response and accurately determine dynamic system response. Let us now make a critical appraisal of some of the assumptions made in the classical model: 1. Transient stability is decided in thefirst swing. A large system having many machines will have numerous natural frequencies of oscillations.
The capacities of most of the tie lines are comparatively small, with the result that some of these frequencies are quite low frequencies of periods in the order of s are not uncommon.
It is quite possible that the worst swing may occur at an instant in time when the peaks of some of these nodes coincide. It is therefore necessary in many cases to study the transient for a period longer than one second.
Constant generator mainfield-windingflux linkage. This assumption is suspect on two counts, the longer period that must now be considered and the speed of many modern voltage regulators. The longer period, which may be comparable to the field-winding time constant, means that the change in the main field-winding flux may be appreciable and should be accounted for so that a correct representation of the system voltage is realized.
Furthermore, the voltage regulator response could have a significant effect on the field-winding flux. We conclude from this discussion that the constant voltage behind transient reactance could be very inaccurate. Negfecting the damping powers. A large system will have relatively weak ties. In the spring-mass analogy used above, this is a rather poorly damped system.
It is important to account for the various components of the system damping to obtain a correct model that will accurately predict its dynamic performance, especially in loss of generation studies [8]. Constant mechanical power. If periods on the order of a few seconds or greater are of interest, it is unrealistic to assume that the mechanical power will not change.
The turbine-governor characteristics, and perhaps boiler characteristics should be included in the analysis. Representing loads by constant passive impedance. Let us illustrate in a qualitative manner the effect of such representation. Let the load be represented by the static ad-. During a transient the voltage magnitude V and the frequency will change. In the model used in Figure 2. This assumption is often on the pessimistic side. There are situations, however, where this assumption can lead to optimistic results.
This discussion is intended to illustrate the errors implied. To illustrate this, let us assume that the transient has been initiated by a fault in the transmission network. Initially, a fault causes a reduction of the output power of most of the synchronous generators.
Some excess generation results, causing the machines to accelerate, and the area frequency tends to increase. At the same time, a transmission network fault usually causes a reduction of the bus voltages near the fault location. In the passive impedance model the load power decreases considerably since PL a V2 ,and the increase in frequency does not cause an increase in load power. In real systems the decrease in power is not likely to be proportional to Y 2 but rather less than this.
A n increase in system frequency will result in an increase in the load power. Thus the model used gives a load power lower than expected during the fault and higher than normal after fault removal.
From the foregoing discussion we conclude that the classical model is inadequate for system representation beyond the first swing. Since the first swing is largely an inertial response to a given accelerating torque, the classical model does provide useful information as to system response during this brief period.
Block diagrams are useful for helping the control engineer visualize a problem. We will be considering the control system for synchronous generators and will do so by analyzing each control function in turn.
It may be helpful to present a general block diagram of the entire system without worrying about mathematical details as to what makes up the various blocks. Then as we proceed to analyze each system, we can fill in the blocks with the appropriate equations or transfer functions. Such a block diagram is shown in Figure 2. The basic equation of the dynamic system of Figure 2.
Three separate control systems are associated with the generator of Figure 2. The first is the excitation system that controls the terminal voltage. Note that the excitation system also plays an important role in the machines mechanical oscillations, since it affects the electrical power, P,.
The second control system is the speed control or governor that monitors the shaft speed and controls the mechanical power P,. Finally, in an interconnected system there is a master controller for each system.
This to sends a unit dispatch signal UDS each generator and adjusts this signal to meet the load demand or the scheduled tie-line power. It is designed to be quite slow so that it is usually not involved in a consideration of mechanical dynamics of the shaft. Thus in most of our work we can consider the speed reference or governor speed changer GSC position to be a constant. In an isolated system the speed reference is the desired system speed and is set mechanically in the governor mechanism, as will be shown later.
In addition to the three control systems, three transfer functions are of vital importance. The first of these is the generator transfer function. The generator equations are nonlinear and the transfer function is a linearized approximation of the behavior of the generator terminal voltage C: near a quiescent operating point or equilibrium state. The load equations are also nonlinear and reflect changes in the electrical output quantities due to changes in terminal voltage?.
Finally, the energy source equations are a description of the boiler and steam turbine or of the penstock and hydraulic turbine behavior as the governor output calls for changes in the energy input. These equations are very nonlinear and have several long time constants. To visualize the stability problem in terms of Figure 2. If a sudden change in w occurs, we have two ways of providing controlled responses to this change.
One is through the governor that controls the mechanical power P,,,. A second controlled response acts through the excitation system to control the electrical power P,. Time delays are involved here too, but they are smaller than those in the governor loop.
Hence much effort has been devoted to refinements in excitation control. Analyze 2. I dimensionally using a mass, length, time system and specify the units of each quantity see Kimbark [I].
Let r, be the accelerating time constant, Le. Then show that r, can also be related to H , the pu inertia constant. Solve the swing equation to find the time to reach full load speed wR starting from any initial speed uo with constant accelerating torque as in Problem 2.
Relate this time to rr and the slip at speed u ,. Write the equation of motion of the shaft for the following systems: a An electric generator driven by a dc motor, where in the region of interest the generator torque is proportional to the shaft angle and the motor torque decreases linearly with increased speed.
State any necessary assumptions. Will this system have a steady-state operating point? Is the system linear? What factor must be used to make the units consistent?
What are the units of 6? Compute the regulation R of this machine. What are the units of R? Compute the pu constant K that must be used with these machines in their governor simulations if the system base is MVA. Verify the expressions in a and b.
O pu power. Find the natural frequency of oscillation for this machine, assuming small perturbations from the operating point. The machine has the same characteristics and operating conditions as given in Problem 2. I 1 and is connected to an infinite bus. Find the natural frequency of oscillation and the damping coefficient, assuming small perturbations from the operating point.
For this condition find the expression for Pa and for the synchronizing power coefficient. I4 Derive an expression similar to that of 2. Show that the equations for , such a case are exactly equivalent to that of a single finite machine of inertia 2. I5 Derive linearized expressions similar to Example 2. M2, , Is there a simple expression for the natural frequency of oscillation in this case?
Designate synchronizing power between machines I and 2 as P S l 2etc. The reactance x includes the transmission line and the machine reactances. Write the swing equation for each machine, and show that this system can be reduced to an equivalent one machine against an infinite bus.
Give the inertia constant for the equivalent machine, the mechanical input power, and the amplitude of its power-angle curve. The inertia constants of the two machines are HI and H2s. Write the equations of motion for this system. Assume that the transmission networks are reactive. Generator and transmission line data are given below. The result of a load-flow study is also given.
A three-phase fault occurs near node 2 and is cleared in 0. Convert the system to a common MVA base, convert the loads to equivalent passive impedances, and calculate the generator internal voltages and initial angles. Write the swing equation for the faulted network and for the network after the fault is cleared. Apply the equal area criterion to the fault discussed in Problem 2. What is the critical clearing angle? This is more typical of the arcing re0 sistance commonly found in a fault.
Implementation will require computation of Y,,, , , , the initial conditions, and the potentiometer settings. Y c Devise a method of introducing additional damping on the analog computer by adding a term K d b in the swing equation. Estimate the value of Kd by assuming that a slip of 2. Repeat the analog simulation and determine the critical clearing time to the nearest cycle.
This will require a means of systematically changing from the fault condition to the postfault one line open condition after a measured time lapse. This can be accomplished by logical control on some analog computers or by careful hand switching where logical control is not available. Use the computer for this, writing the admittance matrices by inspection and reducing to find the two-port admittances.
Compare these results with similar plots with no local load present. References I. Kimbark, E. Power System Stability. Wiley, New York, Stevenson, W. Elements qfPower System Analysis. New York, Federal Power Commission. Narional Power Survey. Effects of future turbine-generator characteristics on transient stability. PAS First report of power system stability. Transient Phenomena in Electrical Power Systems.
Pergamon Press, Macmillan. New York. Crary, S. Cottipurer Me1hod. McGraw-Hill, New York. Erect of steam turbine reheat on speed-governor performance. Power -6, Kirchmayer, L. Economic Control oflnrerronnected Systels. A new stability program for predicting the dynamic performance of electric power systems. Byerly, R. Stability program data preparation manual.
Westinghouse Electric Corp. Concordia, C. Synchronous machine damping and synchronizing torques. Ray, J. Dynamic system performance. Anderson, P. A n analysis and comparison of certain low-order boiler models. ISA Trans. This chapter reviews the behavior of an electric power system when subjected to small disturbances.
It is assumed the system under study has been perturbed from a steady-state condition that prevailed prior to the application of the disturbance.
This small disturbance may be temporary or permanent. If the system is stable, we would expect that for a temporary disturbance the system would return to its initial state, while a permanent disturbance would cause the system to acquire a new operating state after a transient period. In either case synchronism should not be lost. Under normal operating conditions a power system is subjected to small disturbances at random. It is important that synchronism not be lost under these conditions.
Thus system behavior is a measure of dynamic stability as the system adjusts to small perturbations. We now define what is meant by a small disturbance. The criterion is simply that the perturbed system can be linearized about a quiescent operating state.
A n example of this linearization procedure was given in Section 2. While the power-angle relationship for a synchronous machine connected to an infinite bus obeys a sine law 2. In general, the response of a power system to impacts is oscillatory. If the oscillations are damped, so that after sufficient time has elapsed the deviation or the change in the state of the system due to the small impact is small or less than some prescribed finite amount , the system is stable.
If on the other hand the oscillations grow in magnitude or are sustained indefinitely, the system is unstable. For a linear system, modern linear systems theory provides a means of evaluation of its dynamic response once a good mathematical model is developed. The mathematical models for the various components of a power network will be developed in greater detail in later chapters.
Here a brief account is given of the various phenomena experienced in a power system subjected to small impacts, with emphasis on the qualitative description of the system behavior. The method of small changes, sometimes called the perturbation method [ 1.
If the power system is perturbed, it will acquire a new operating state. If the perturbation is small, the new operating state will not be appreciably different from the initial one. I n other words, the state variables or the system parameters will usually not change appreciably. Thus the operation is in the neighborhood of a certain quiescent state xo.
In this limited range of operation a nonlinear system can be described mathematically by linearized equations. This is advantageous, since linear systems are more convenient to work with. This procedure is particularly useful if the system contains control elements. The method of analysis used to linearize the differential equations describing the system behavior is to assume small changes in system quantities such as b,, u,, PA change in angle, voltage, and power respectively.
Equations for these variables are found by making a Taylor series expansion about xo and neglecting higher order terms [4,5,6]. The behavior or the motion of these changes is then examined. In examining the dynamic performance of the system, it is important to ascertain not only that growing oscillations do not result during normal operations but also that the oscillatory response to small impacts is well damped.
If the stability of the system is being investigated, it is often convenient to assume that the disturbances causing the changes disappear. The motion of the system is then free. Stability is then assured if the system returns to its original state. Such behavior can be determined in a linear system by examining the characteristic equation of the system.
If the mathematical description of the system is in state-space form, i. When a power impact occurs at some bus in the network, an unbalance between the power input to the system and the power output takes place, resulting in a transient. When this transient subsides and a steady-state condition is reached, the power impact is shared by the various synchronous machines according to their steady-state characteristics, which are determined by the steady-state droop characteristics of the various governors [5,7].
During the transient period, however, the power impact is shared by the machines according to different criteria. If these criteria differ appreciably among groups of machines, each impact is followed by oscillatory power swings among groups of machines to reflect the transition from the initial sharing of the impact to the final adjustment reached at steady state. Under normal operating conditions a power system is subjected to numerous random power impacts from sudden application or removal of loads.
As explained above, each impact will be followed by power swings among groups of machines that respond to the impact differently at different times. These power swings appear as power oscil-.
This gives rise to the term tie-line oscillations. In large interconnected power systems tie-line oscillations can become objectionable if their magnitude reaches a significant fraction of the tie-line loading, since they are superimposed upon the normal flow of power in the line. Furthermore, conditions may exist in which these oscillations grow in amplitude, causing instability.
This problem is similar to that discussed in Section 3. It can be analyzed if an adequate mathematical model of the various components of the system is developed and the dynamic response of this model is examined. If we are interested in seeking an approximate answer for the magnitude of the tie-line oscillations, however, such an answer can be reached by a qualitative discussion of the distribution of power impacts.
Such a discussion is offered here. We start with the simplest model possible, i. The equation of motion of a synchronous machine connected to an infinite bus and the electrical power output are given by 2.
The system described by 3. If the electrical torque is assumed to have a component proportional to the speed change, a damping term is added to 3. If either one of these quantities is negative, the system is unstable. Venikov [4] reports that a situation may occur where the machine described by 3.
This would be the case where there is appreciable series resistance see [4], Sec. A negative value of P leads to , unstable operation. The model of constant main field-winding flux linkage neglects some important effects, among them the demagnetizing influence of a change in the rotor angle 6. To account for this effect, another model of the synchronous machine is used. It is not our concern in this introductory discussion to develop the model or even discuss it in detail, as this will be accomplished in Chapter 6.
Rather, we will state the assumptions made in such a model and give some of the pertinent results applicable to this discussion. These results are found in de Mello and Concordia [8] and are based on a model previously used by Heffron and Phillips To account for the field conditions, equations for the direct and quadrature axis quantities are derived see Chapter 4.
Major simplifications are then made by neglecting saturation, stator resistance, and the damper windings. The transformer voltage terms in the stator voltage equations are considered negligible compared to the speed voltage terms. Linearized relations are then obtained between small changes in the electrical power Pea, the rotor angle ,a the field-winding voltage uFArand the voltage proportional to the main field-winding flux EA. For a machine connected to an infinite bus through a transmission network, the following s domain relations are obtained,.
The constants K I , K 2 , and K4 depend on the parameters of the machine, the external network, and the initial conditions. Note that K , is similar to the synchronizing power coefficient P, used in the simpler machine model of constant voltage behind.
Equations 3. Substituting in the linearized swing equation 3. The first of the above criteria states that the synchronizing power coefficient K, must be greater than the demagnetizing component of electrical power.
The second criterion is satisfied if the constants K2, K3, and K4 are positive. Venikov [4] points out that if the transmission network has an appreciable series capacitive reactance, it is possible that instability may occur.
This would happen because the impedance factor producing the constant K, would become negative. In the linearized version of 3. The change in power due to. In this case the new differential equation becomes 3. From Section 2. The system block diagram with speed regulation added is shown in Figure 3. Again Rouths criterion may be applied to determine the conditions for stability. This is left as an exercise see Problem 3. For a given initial condition sin Sijo and cos bij0 are known, and the term in parentheses in 3.
Thus we write n. It is a synchronizing power coefficient between nodes i and j and is identical to the coefficient discussed in Section 2. We also note that since 3. Thus the concept of the synchronizing power coefficients can be extended to mean the change in the electrical power of a given machine due to the change in the angle between its internal EMF and. An implied assumption is that the voltage at the remote bus is also held constant.
This expanded definition of the synchronizing power coefficient will be used in Section 3. Using the inertial model of the synchronous machines, we get the set of linearized differential equations,.
Research Feed. View 5 excerpts. Sliding Mode Power System Stabilizer for Fouad, A. Details of the composition of these mathematical models are given in references Anderson and Fouad, ; Kundur, ; Sauer and Pai, Upload a Thing!
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