CHAPTER I. the generator is represented by Eg and XQ) and the motor, by EM and XM- Upon combining the machine

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1 CHAPTER I THE STABILITY PROBLEM Definitions and illustrations of terms. Power-system stability is a term applied to alternating-current electric power systems, denoting a condition in which the various synchronous machines of the system remain in synchronism, or "in step," with each other. Conversely, instability denotes a condition involving loss of synchronism, or falling "out of step." Consider the very simple power system of Fig. 1, consisting of a synchronous generator supplying power to a synchronous motor over a circuit composed of series inductive reactance X&. Each of the synchronous machines may be represented, at least approximately, by a constant-voltage source in series with a constant reactance.* Thus Gen. Motor the generator is represented by Eg and XQ) and the motor, by EM and XM- Upon combining the machine I XG X X L X M reactances and the line re- actance into a single reactance, we E G E GT EMT E M have an electric circuit consisting of p IG< \ m Simple two-machine power two constant-voltage sources, Eg system, and EM, connected through reactance X =3 XG + XL + XM. It will be shown that the power transmitted from the generator to the motor depends upon the phase difference 8 of the two voltages E G and EM- Since these voltages are generated by the flux produced by the field windings of the machines, their phase difference is the same as the electrical angle between the machine rotors. The vector diagram of voltages is shown in Fig. 2. Vectorially, E G = EM + jxi [1] (The bold-face letters here and throughout the book denote com- *Either equivalent synchronous reactance or transient reactance is used, depending upon whether steady-state or transient conditions are assumed. These terms are defined and discussed in Chapters XII and XV, Vol. III. 1

2 2 THE STABILITY PROBLEM plex, or vector, quantities). Hence the current is JX jx ^^f The power output of the generator and like- ^^^^ * / wise the power input of the motor, since there ^^-<3]g /J xl is no resistance in the line is given by E M \^~~7/ P = R^(E(?I) [3] 90* I Ea EM ^ -Be(l E G - 0 5^) [4] FIG. 2. Vector diagram \ J jx / of the system of Fig. 1. where Re meang a^ ^ part of the system of Fig. 1. where Re meang ^ Qf, and g^ part Qf, and g^ means the conjugate of E(?. Now let means the conjugate of EQ. Now let EM = E M [0 [5] and E G = Eold [6] Then _ E G = Eotzl PI Substitution of eqs. 5, 6, and 7 into eq. 4 gives P = Re = Re EQI -6- 'E G 2 X EQEM X u TLQ "EM I = E G /±- -90 ^/90 EQEM X cos (-90-8) EM/0 - EGEM Z sin 5 m x This equation shows that the power P transmitted from the generator to the motor varies with the sine of the displacement angle 6 between the two rotors, as plotted in Fig. 3. The curve is known as a power" angle curve. The maximum power that can be transmitted in the steady state with the given reactance X and the given internal voltages E G and EM is _ E$GEM G E M Pm "~ X m

3 DEFINITIONS AND ILLUSTRATIONS OF TERMS 3 and occurs at a displacement angle 8 = 90. The value of maximum power may be increased by raising either of the internal voltages or by decreasing the circuit reactance. The system is stable only if the displacement angle 8 is in the range from 90 to +90, in which the slope dp/d8 is positive; that is, the range in which an increase in displacement angle results in an increase in transmitted power. Suppose that the system is operating in the steady state at point A, Fig. 3. The mechanical input of the generator and the mechanical output of the motor, if corrected for rotational losses, will be equal to the electric power P. Now suppose that a P c P m >A B 180* -90* FIG. 3. Power-angle curve of the system of Fig. 1. small increment of shaft load is added to the motor. Momentarily the angular position of the motor with respect to the generator, and therefore the power input to the motor, is unchanged; but the motor output has been increased. There is, therefore, a net torque on the motor tending to retard it, and its speed decreases temporarily. As a result of the decrease in motor speed, 5 is increased, and consequently the power input is increased, until finally the input and output are again in equilibrium, and steady operation ensues at a new point B, higher than A on the power-angle curve. (It has been tacitly assumed that the generator speed would remain constant. Actually the generator may have to slow down somewhat in order for the governor of its prime mover to operate and increase the generator input sufficiently to balance the increased output.)

4 4 THE STABILITY PROBLEM Suppose that the motor input is increased gradually until the point C of maximum power is reached. If now an additional increment of load is put on the motor, the displacement angle 8 will increase as before, but as it does so there will be no increase in input. Instead there will be a decrease in input, further increasing the difference between output and input, and retarding the motor more rapidly. The motor will pull out of step and will probably stall (unless it is kept going by induction-motor action resulting from damper circuits which may be present). P m is the steady-state stability limit of the system. It is the maximum power that can be transmitted, and synchronism will be lost if an attempt is made to transmit more power than this limit. If a large increment of load on the motor is added suddenly, instead of gradually, the motor may fall out of step even though the new load does not exceed the steady-state stability limit. The reason is as follows: When the large increment of load is added to the motor shaft, the mechanical power output of the motor greatly exceeds the electrical power input, and the deficiency of input is supplied by decrease of kinetic energy. The motor slows down, and an increase of the displacement angle 8 and a consequent increase of input result. In accordance with the assumption that the new load does not exceed the steady-state stability limit, 8 increases to the proper value for steadystate operation, a value such that the motor input equals the output and the retarding torque vanishes. When this value of 8 is reached, however, the motor is running too slowly. Its angular momentum prevents its speed from suddenly increasing to the normal value. Hence it continues to run too slowly, and the displacement angle increases beyond the proper value. After the angle has passed this value, the motor input exceeds the output, and the net torque is now an accelerating torque. The speed of the motor increases and approaches normal speed. Before normal speed is regained, however, the displacement angle may have increased to such an extent that the operating point on the power-angle curve (Fig. 3) not only goes over the hump (point C) but also goes so far over it that the motor input decreases to a value less than the output. If this happens, the net torque changes from an accelerating torque to a retarding torque. The speed, which is still below normal, now decreases again, and continues to decrease during all but a small part of each slip cycle. Synchronism is definitely lost. In other words, the system is unstable. If, however, the sudden increment in load is not too great, the motor will regain its normal speed before the displacement angle becomes too great. Then the net torque is still an accelerating torque and causes

5 DEFINITIONS AND ILLUSTRATIONS OF TERMS 5 the motor speed to continue to increase and thus to become greater than normal. The displacement angle then decreases and again approaches its proper value. Again it overshoots this value on account of inertia. The rotor of the motor thus oscillates about the new steady-state angular position. The oscillations finally die out because of damping torques, t which have been neglected in this elementary analysis. A damped oscillatory motion characterizes a stable system. With a given sudden increment in load, there is a definite upper limit to the load which the motor will carry without pulling out of step. This is the transient stability limit of the system for the given conditions. The transient stability limit is always below the steady-state stability limit, $ but, unlike the latter, it may have many different values, depending upon the nature and magnitude of the disturbance. The disturbance may be a sudden increase in load, as just discussed, or it may be a sudden increase in reactance of the circuit, caused, for example, by the disconnection of one of two or more parallel lines as a normal switching operation. The most severe type of disturbance to which a power system is subjected, however, is a short circuit. Therefore, the effect of short circuits (or "faults," as they are often called) must be determined in nearly all stability studies. A three-phase short circuit on the line connecting the generator and the motor entirely cuts off the flow of power between the machines. The generator output becomes zero in the pure-reactance circuits under consideration; the motor input also becomes zero. Because of the slowness of action of the governor of the prime mover driving the generator, the mechanical power input of the generator remains constant for perhaps f sec. Also, since the power and torque of the load on the motor are functions of speed, and since the speed cannot change instantly and changes by not more than a few per cent unless and until synchronism is lost, the mechanical power output of the motor may be assumed constant. As the electrical power of both machines is decreased by the short circuit, while the mechanical power of both remains constant, there is an accelerating torque on the generator and a retarding torque on the motor. Consequently, the generator speeds up, the motor slows down, and it is apparent that synchronism will be lost unless the short circuit is quickly removed so as to restore synchronizing power between the machines before they have drifted too fdiscussed in Chapter XIV, Vol. III. {Conventional methods of calculation, however, sometimes indicate that the transient stability limit is above the steady-state stability limit. This paradox is discussed in Chapter XV, Vol. III.

6 6 THE STABILITY PROBLEM far apart in angle and in speed. If the short circuit is on one of two parallel lines and is not at either end of the line, or if the short circuit is of another type than three-phase that is, one-line-to-ground, line-toline, or two-line-to-ground then some synchronizing power can still be transmitted past the fault, but the amplitude of the power-angle curve is reduced in comparison with that of the pre-fault condition. In some cases the system will be stable even with a sustained short circuit, whereas in others the system will be stable only if the short circuit is cleared with sufficient rapidity. Whether the system is stable during faults will depend not only on the system itself, but also on the type of fault, location of fault, rapidity of clearing, and method of clearing that is, whether cleared by the sequential opening of two or more breakers, or by simultaneous opening and whether or not the faulted line is redosed. For any constant set of these conditions, the question of whether the system is stable depends upon how much power it was carrying before the occurrence of the fault. Thus, for any specified disturbance, there is a value of transmitted power, called the transient stability limit, below which the system is stable and above which it is unstable. The stability limit is one kind of power limit, but the power limit of a system is not always determined by the question of stability. Even in a system consisting of a synchronous generator supplying power to a resistance load, there is a maximum power received by the load as the resistance of the load is varied. Clearly there is a power limit here with no question of stability. Multimachine systems. Few, if any, actual power systems consist of merely one generator and one synchronous motor. Most power systems have many generating stations, each with several generators, and many loads, most of which are combinations of synchronous motors, synchronous condensers, induction motors, lamps, heating devices, and others. The stability problem on such a power system usually concerns the transmission of power from one group of synchronous machines to another. As a rule, both groups consist predominantly of generators. During disturbances the machines of each group swing more or less together; that is, they retain approximately their relative angular positions, although these vary greatly with respect to the machines of the other group. For purposes of analysis the machines of each group can be replaced by one equivalent machine. If this is done, there is one equivalent generator and one equivalent synchronous motor ) even though the latter often represents machines that are actually generators. Because of uncertainty as to which machines will swing together, or

7 A MECHANICAL ANALOGUE OF SYSTEM STABILITY 7 in order to improve the accuracy of prediction, it is often desirable to represent the synchronous machines of a power system by more than two equivalent machines. Nevertheless, qualitatively the behavior of the machines of an actual system is usually like that of a two-machine system. If synchronism is lost, the machines of each group stay together, although they go out of step with the other group. Because the behavior of a two-machine system represents the behavior of a multimachine system, at least qualitatively, and because the two-machine system is very simple in comparison with the multimachine system which it represents, the two-machine system is extremely useful in describing the general concepts of power-system stability and the influence of various factors upon stability. Accordingly, the two-machine system plays a prominent role in this book. A mechanical analogue of system stability. 5 A simple mechanical model of the vector diagram of Fig. 2 may be built of two pivoted rigid arms representing the Eg and EM vectors, joined at their extremities by a spring representing the XI vector. (See Fig. 4.) Lengths represent voltages in the model, just 8 t( as they do in the vector diagram. The lengths of the arms, EQ and EM- EM, are fixed in accordance with the _,.,,. e,,.,. FIG. 4. A mechanical analogue of the assumption of constant internal system of Fig l voltages. The length of the spring XI is proportional to the applied tensile force (for simplicity, we assume an ideal spring which returns to zero length if the force is removed). Hence the tensile force can be considered to represent the current, and the compliance of the spring (its elongation per unit force), to represent the reactance. The torque exerted on an arm by the spring is equal to the product of the length of the arm, the tensile force of the spring, and the sine of the angle between the arm and the spring. (More torque is exerted by the spring when it is perpendicular to the arm than at any other angle for the same tensile force.) Obviously, the torques on the two arms are equal and opposite. The torque, multiplied by the speed of rotation, gives the mechanical power transmitted from one arm to the other. For convenience of inspection, the mechanical model will be regarded as stationary, rather than as rotating at synchronous speed, just as we regard the usual vector diagram as stationary. The formula for torque (or power) in the model is analogous to that for power in the vector Superior numerals refer to items in the list of References at end of chapter.

8 8 THE STABILITY PROBLEM diagram, namely: voltage X current X cosine of angle between them. (Since the XI vector is 90 ahead of the / vector, the cosine of the angle between E and / is equal to the sine of the angle between E and XI.) The shaft power of the machines may be represented by applying additional torques to the arms. A convenient method of applying Spring Arms Synchronous condenser constant equal and opposite torques to the two arms is to attach a drum to each arm and to suspend a weight pan from a pulley hanging on a cord, one end of which is wound on each drum, all as indicated in Fig. 5. As weights are added to the pan in small increments, the two arms of the model gradually move farther apart until the angle 8 between them reaches 90, at which position the spring exerts maximum torque. If further weights are added, the arms fly apart and continue to rotate in opposite directions until all the cord is unwound from the drums. The system is unstable. The steady-state power limit is reached at 8 = 90. Although from 90 to 180 the spring force (current) continues to increase, the angle between arm and spring changes in such a way that the torque decreases. The effect of changing the machine voltages can be shown by attaching the spring to clamps which slide along the arms. The effect of an intermediate synchronous-condenser station in in- -Axle- Drums. -Table Generator Motor Cord- Pullev -Weights- -Pan- FIG. 5. A mechanical analogue of FIG. 6. A mechanical analogue of a threethe system of Fig. 1, suitable for machine system consisting of generator, representing transient conditions. synchronous condenser, and synchronous motor.

9 BAD EFFECTS OF INSTABILITY 9 creasing the steady-state power limit can be shown by adding a third pivoted arm attached to an intermediate point of the spring (Fig. 6). The condenser maintains a fixed internal voltage. Since the condenser has no shaft input or output, no drum is provided on the third arm in the model. With the intermediate arm (representing the condenser) in place, the angle between the two outer arms (representing the generator and motor) may exceed 90 without instability, and the power limit is greater than before. The model can be used to illustrate transient stability by providing each arm with a flywheel such that the combined moment of inertia of the arm and flywheel is proportional to that of the corresponding synchronous machine together with its prime mover (or load). The drums can be made to serve this purpose. If not too great an increment of load is suddenly added to the pan, it will be found that the arms oscillate before settling down to their new steady-state positions. The angle between the arms may exceed 90 during these oscillations without loss of stability. If the increment of load is too large, the arms will fly apart and continue to rotate in opposite directions, indicating instability. This may happen even though the total load is less than the steadystate stability limit. The effect of switching out one of two parallel lines may be simulated by connecting the arms by two springs in parallel and then suddenly disconnecting one spring by burning the piece FlG - 7. A mechanical of string by which the spring is attached. analogue of the effect of a mv & > c r ii. J.U i- u I 1116 fault on th e power The effect of a fault on the line may be simu- system of Fi l lated by suddenly pushing a point on the spring toward the axle (Fig. 7). The arms will start to move apart, and stability will be lost unless the spring is quickly released. Models of this kind have been built to give a scale representation of actual power systems of three or four machines, and the oscillations of the arms have been recorded by moving-picture cameras. 6 There are practical difficulties, however, in applying the model representation to a complicated system. The chief value of the model is to illustrate the elementary concepts of stability. Other methods of analysis are used in practice. Bad effects of instability. When one machine falls out of step with the others in a system, it no longer serves its function. If it is a

10 10 THE STABILITY PROBLEM generator, it no longer constitutes a reliable source of electric power. If it is a motor, it no longer delivers mechanical power at the proper speed, if at all. If it is a condenser, it no longer maintains proper voltage at its terminals. An unstable two-machine system, consisting of motor and generator, may be compared to a slipping belt or clutch in a mechanical transmission system; instability means the failure of the system as a power-transmitting link. Moreover, a large synchronous machine out of step is not only useless; it is worse than useless it is injurious because it has a disturbing effect on voltages. Voltages will fluctuate up and down between wide limits. Thus instability has the same bad effect on service to customers' loads as does a fault, except that the effect of instability is likely to last longer. If instability occurs as a consequence of a fault, clearing of the fault itself may not restore stability. The disturbing voltage fluctuations then continue after the fault has been cleared. The machine, or group of machines, which is out of step with the rest of the system must either be brought back into step or else disconnected from the rest of the system. Either operation, if done manually, may take a long time compared with the time required to clear a fault automatically. As a rule, the best way to bring the machines back into step is to disconnect them and then re-synchronize them. Protective relays have been developed to open a breaker at a predetermined location when out-of-step conditions occur. Such relays, however, are not yet in wide use. Preferably the power system should be split up into such parts that each part will have adequate generating capacity connected to it to supply the load of that part. Some overload may have to be carried temporarily until the system is re-synchronized. Ordinary protective relays are likely to operate falsely during out-ofstep conditions, thereby tripping the circuit breakers of unfaulted lines. Such false tripping may unnecessarily interrupt service to tapped loads and may split the system apart at such points that the generating capacity of some parts is inadequate. The trend in power-system design has been toward increasing the reliability of electric power service. Since instability has a bad effect on the quality of service, a power system should be designed and operated so that instability is improbable and will occur only rarely. Scope of this book. This book will deal with two different phases of the problem of power-system stability: (1) methods of analysis and calculation to determine whether a given system is stable when subjected to a specified disturbance; (2) an examination of the effect of This aspect of relay operation is discussed fully in Chapter X, Vol. II.

11 HISTORICAL REVIEW 11 various factors on stability, and a consideration of measures for improving stability. In our discussion these two phases will be related: after a method of analysis is presented, it will be applied to show the effect of varying different factors. Among these factors are system layout, circuit impedances, loading of machines and circuits, type of fault, fault location, method of clearing, speed of clearing, inertia of machines, kind of excitation systems used with the machines, machine reactances, neutral grounding impedance, and damper windings on machines. Since transient power limits are lower than steady-state power limits, and since any power system will be subject to various shocks, the most severe of which are short circuits, the subject of transient stability is much more important than steady-state stability. Accordingly, the greater part of this book is devoted to transient stability. Chapter XV, Vol. Ill, deals with steady-state stability. Historical review. Since stability is a problem associated with the parallel operation of synchronous machines, it might be suspected that the problem appeared when synchronous machines were first operated in parallel. The first serious problem of parallel operation, however, was not stability, but hunting. When the necessity for parallel operation of a-c. generators became general, most of the generators were driven by direct-connected steam engines. The pulsating torque delivered by those engines gave rise to hunting, which was sometimes aggravated by resonance between the period of pulsation of primemover torque and the electromechanical period of the power system. In some cases improper design or functioning of the engine governors also aggravated the hunting. Hunting of synchronous motors and converters was sometimes due to another cause, namely, too high a resistance in the supply line. The seriousness of hunting was decreased by the introduction of the damper winding, invented by LeBlanc in France and by Lamme in America. Later, the problem largely disappeared on account of the general use of steam turbines, which have no torque pulsations. Nearly all the prime movers in use nowadays, both steam turbines and water wheels, give a steady torque. A few generators are still driven by steam engines or by internal combustion engines. These, as well as synchronous motors driving compressors, have a tendency to hunt, but, on the whole, hunting is no longer a serious difficulty. In the first ten or twenty years of this century, stability was not yet a significant problem. Before automatic voltage-controlling devices (generator-voltage regulators, induction feeder-voltage regulators, synchronous condensers, and the like) had been developed, the power

12 12 THE STABILITY PROBLEM systems had to be designed to have good inherent voltage regulation. This requirement called for low reactance in circuits and machines. As a consequence of the low reactances, the stability limits (both steady-state and transient) were well above the normally transmitted power. The development of automatic voltage regulators made it possible to increase generator reactances in order to obtain a more economical design and to limit short-circuit currents. By use of induction regulators to control feeder voltages, transmission lines of higher impedance became practicable. These factors, together with the increased use of generator and bus reactors to decrease short-circuit currents, led to a decrease in the inherent stability of metropolitan power systems. Stability first became an important problem, however, in connection with long-distance transmission, which is usually associated with remote hydroelectric stations feeding into metropolitan load centers. 1f The application of the automatic generator-voltage regulator to synchronous condensers made it possible to get good local voltage regulation from a hydroelectric station and a transmission line of high reactance and hence of low synchronizing power. The high investment in these long-distance projects made it desirable to transmit as much power as possible over a given line, and there was a temptation to transmit normal power approaching the steady-state stability limit. In a few cases instability occurred during steady-state operation, and more frequently it occurred because of short circuits. The stability problem is still more acute in connection with long-distance transmission from a generating station to a load center than it is in connection with metropolitan systems. It should not be inferred, however, that metropolitan systems have no stability problems. Another type of long-distance transmission which has frequently involved a stability problem is the interconnection between two large power systems for the purpose of exchanging power to obtain economies in generation or to provide reserve capacity. In many cases the connecting ties were designed to transmit an amount of power which was small in comparison with the generating capacity of either system. Consequently, the synchronizing power which the line could transmit was not enough to retain stability if a severe fault occurred on either system. There was also considerable danger of steady-state pull-out if the power on the tie line was not controlled carefully. From about 1920 the problem of power-system stability was the object of thorough investigation. Tests were made both on laboratory famong such hydroelectric stations are Big Creek, Bucks Creek, Pit River, Fifteen Mile Falls, Conowingo, and Boulder Dam.

13 HISTORICAL REVIEW 13 set-ups and on actual power systems, methods of analyses were developed and checked by tests, and measures for improving stability were developed. Some of the important steps in analytical development were the following: 1. Circle diagrams for showing the steady-state performance of transmission systems. These diagrams consist of a family of circles, each of which is the locus of the vector power for fixed voltages at both sending and receiving ends of the line. The circles are drawn on rectangular coordinates, the abscissas and ordinates of which are, respectively, active and reactive power at either end of the line. These diagrams show clearly the maximum power which a line will carry in the steady state for given terminal voltages, as well as the relation between the power transmitted and the angular displacement between the voltages at the two ends of the line. (Such diagrams are described in Chapter XV, Vol. III.) 2. Improvements in synchronous-machine theory, especially the extension of two-reaction theory to the transient performance of both salient-pole and nonsalient-pole machines. A number of new reactances were defined and used. (See Chapter XII, Vol. III.) More recently, the effect of saturation on these reactances has been investigated. (See Chapters XII and XV, Vol. III.) 3. The method of symmetrical components for calculating the effect of unsymmetrical short circuits. (See Chapter VI.) In this connection, methods of determining the sequence constants of apparatus by test and by calculation had to be devised. 4. Point-by-point methods of solving differential equations, particularly the swing equation (giving angular position of a machine versus time). (See Chapter II.) 5. The equal-area criterion for stability of two-machine systems, obviating the more laborious calculation of swing curves for such systems. (See Chapter IV.) 6. The a-c. calculating board or network analyzer for the solution of complicated a-c. networks. (See Chapter III.) The methods of analysis and calculation now in use are believed to be sufficiently accurate for determining whether any given power system in a given operating condition will be stable when subjected to a given disturbance. The calculations, however, are rather laborious when applied to a large number of different operating conditions of a complicated power system. Methods of analysis will be taken up in the following chapters. Calculated results have been checked in a number of instances by

14 14 THE STABILITY PROBLEM observations on actual power systems recorded with automatic oscillograph equipment. REFERENCES 1. R. D. BOOTH and G. C. DAHL, "Power System Stability a Non-mathematical Review," Gen. Elec. Rev., vol. 33, pp , December, 1930; and vol. 34, pp , February, A.I.E.E. Subcommittee on Interconnection and Stability Factors, "First Report of Power-System Stability," Elec. Eng., vol. 56, pp , February, O. G. C. DAHL, Electric Power Circuits, vol. II, Power-System Stability, New York, McGraw-Hill Book Co., Electrical Transmission and Distribution Reference Book, by Central Station Engineers of the Westinghouse Electric & Manufacturing Company, East Pittsburgh, Pa., 1st edition, a. Chapter 8, "Power System Stability Basic Elements of Theory and Application," by R. D. EVANS. b. Chapter 9, "System Stability Examples of Calculation," by H. N. MULLER, JR. 5. S. B. GRISCOM, "A Mechanical Analogy of the Problem of Transmission Stability," Elec. Jour., vol. 23, pp , May, R. C. BERGVALL and P. H. ROBINSON, "Quantitative Mechanical Analysis of Power System Transient Disturbances," A.I.E.E. Trans., vol. 47, pp , July, 1928; disc, pp Use of mechanical model with seven arms for investigating transient stability of Conowingo transmission system. PROBLEMS ON CHAPTER I 1. Two synchronous machines of equal rating, having internal voltages (voltages behind transient reactance) of 1.2 and 1.0 per unit, respectively, and transient reactances of 0.25 per unit each, are connected by a line having 0.50 per unit reactance and negligible resistance. Assume that the angle 5 between the two machines varies from 0 to 360 by 15 steps, and calculate for each step the current, the power, and the voltage at each of three points: at each end of the line and at its midpoint. Draw loci of the current and voltage vectors, marking the values of 5 thereon. Also plot in rectangular coordinates current, power, and voltage, all as functions of Draw the power-angle curve and discuss the condition for stability of two machines connected through series capacitive reactance which exceeds the internal inductive reactance of both machines.