EFFECTS OF EHV POWER CIRCUIT BREAKER RECLOSING ON POWER SYSTEM STABILITY

Transcription

1 EFFECTS OF EHV POWER CIRCUIT BREAKER RECLOSING ON POWER SYSTEM STABILITY A THESIS Presented to The Faculty of the Division of Graduate Studies By U. C. Roumillat, Jr. In Partial Fulfillment of the Requirements for the Degree Master of Science in Electrical Engineering Georgia Institute of Technology May, 1978

2 EFFECTS OF EHV POWER CIRCUIT BREAKER RECLOSING ON POWER SYSTEM STABILITY Approved: Rogery Webb, Ch airman Chee-Yee Chong ^ Claytoh H. Griffin v "V Ed W. Kamen Date approved by Chairman S"/z3/ r y J S'

3 ii ACKNOWLEDGEMENTS I wish to express my appreciation to all those who have contributed to the successful completion of this thesis. The subject was suggested by Mr. Clayton Griffin, Georgia Power Company Chief Protection Engineer; and, his advice and encouragement is gratefully acknowledged. Thanks are also extended to my thesis advisor, Dr. Roger Webb, and to the reading committee members, Dr. C-Y Chong, Mr. C. H. Griffin, and Dr. E. W. Kamen, for their guidance. I am indebted to the Southern Company's Georgia Power Company and Southern Company Services, Incorporated, for the use of their digital computer facilities. Also, I appreciate the help of Mr. W. R. Startt of Georgia Power Company, Mr. N. J. Balu, and Mr. S. G. Burks of Southern Company Services, Incorporated, in using the computer facilities. Thanks also go to Mrs. W. A. Moncrief for typing the manuscript. I would like to dedicate this thesis to my wife, Lucie. Her sacrifices and understanding were essential for the completion of this work.

4 iii f TABLE OF CONTENTS ACKNOWLEDGEMENT., LIST OF TABLES LIST OF ILLUSTRATIONS LIST OF SYMBOLS AND ABBREVIATIONS SUMMARY Page ii V vi vii viii Chapter I. INTRODUCTION 1 II. FORMULATION OF THE STABILITY STUDY... 7 The Purpose and Approach of the Study 7 Methods Used to Calculate System Behavior Following a Disturbance 7 Cases Studied 8 Definition of Terms Used 13 III. ANALYSIS BY THE USE OF A DIGITAL COMPUTER Introduction 15 Data Preparation 15 Initial Conditions and Method of Analysis Analysis of Case 1 17 Analysis of Case 2 19 Analysis of Case 3 19 Analysis of Case 4 19 Analysis of Cases 5 and 6 22 Analysis of Cases 7 and 8 22 IV. ANALYSIS OF THE REDUCED SYSTEM 26 Approach and Assumptions 26 Analysis of Cases 9 and Analysis of Cases 11 and 12 32

5 iv TABLE OF CONTENTS Chapter Page V. EVALUATION OF REDUCED SYSTEM ANALYSIS AND LIAPUNOV METHOD Purpose 36 Comparison of the Computer Analysis with the Point-By-Point Analysis 36 Comparison of the Point-By-Point Analysis with the Liapunov Method 37 Evaluation of the Comparisons 39 VI. CONCLUSION 4 0 Effect of Reclosing on Stability 40 Basis for Conclusion 40 VII. RECOMMENDATIONS 42 APPENDIX 43 BIBLIOGRAPHY 56

6 V LIST OF TABLES Table Page 1. Load Flow System Parameters Generator Parameters Computer Computation of Case Computer Computation of Case Point-By-Point Computcition of Case Point-By-Point Computation of Case Liapunov Computation of Case 11 55

7 VI LIST OF ILLUSTRATIONS e Page Generator Shaft Torques 4 Reduced 500-KV System 9 Swing Curves of Cases 1 and 9 18 Swing Curves of Cases 2 and Swing Curves of Cases 3 and 4 21 Swing Curves of Cases 5 and 6 23 Swing Curves of Cases 7 and 8 24 Equivalent System 28 Power-Angle Curves of Cases 9 and Rotor Speed-Rotor Angle Curves of Cases 9 and Liapunov Method Cases 11 and 12 38

8 Vll LIST OF SYMBOLS AND ABBREVIATIONS Generator rotor angle relative to an infinite bus. th Change in 6 for the n time interval. Time in seconds. Maximum power transfer. Accelerating power. Electrical power input. Electrical power output. Angular rotor speed above synchronous speed. Power circuit breaker. Extra high voltage. Inertia constant. Per unit on a 100 MVA base. With respect to. Moment of inertia in pound-feet 2 of the entire unit. (Generator, Prime Mover and Exciter.)

9 viii SUMMARY Power circuit breakers have been closed as soon as possible to aid system stability. However, closing a breaker that places a fault near a generator can damage the rotor shaft of the generator due to the resulting torques impressed on the rotor. If the closing of the breakers is delayed ten seconds, the rotor oscillations have decayed sufficiently to prevent shaft damage when a fault is placed near the machine. The purpose of this thesis is to investigate the effects of closing a breaker as rapidly as the breaker manufacturers allow on generator stability. On modern EHV circuit breakers this limitation requires that closing should be delayed by about one-third second. The four large generators at Georgia Power Company's Plant Bowen were reduced to an equivalent generator feeding an infinite bus thirough an equivalent transmission system. This reduction made* it possible to solve the classical swing equation by the point-by-point method. Various fault-clearing and breaker-closing times were, used until the critical closing time was determined. The change in rotor angular velocity with respect to angular displacement was plotted to graphically determine a separatrix curve. A Liapunov function was plotted on the same graph for a comparison of the results. The curves

10 ix illustrate that reclosing into a fault that is not cleared by the critical time can result in generator instability. Digital computer methods were used to reduce the power system to an equivalent 124 bus network. Transient stability conditions, similar to the conditions used in the manual calculations, were studied for comparison. The swing curves resulting from various system conditions are also included. The results obtained by the computer were in close agreement with those obtained by manual calculations.

11 1 CHAPTER I INTRODUCTION The objective of this thesis was to study the stability of a large section of the southeastern electric utility power system. The effect of interconnections of this section with other power systems was included in the study. The power system is constructed by connecting electromechanical generators in a network of transmission lines. The transmission lines are used to supply consumer loads and to connect the generators in synchronous parallel operation. The power supplied by the generators must equal the system losses plus the system load for the generators to maintain synchronous parallel operation. The power system is in stable operation when the generation matches the load. A system disturbance such as a line opening or faulting will produce a power swing in the interconnected system. The effect of the disturbance will be most severe at points in the system that are electrically close to the disturbance. If the disturbance is a three phase fault, all of the generator rotors will accelerate causing a power swing or oscillation to begin. The resulting electrical angular difference between generator rotors during the swing will determine the stability of the system. Electromechanical restoring forces will

12 2 return the rotors of the machines to synchronous operation if the angular difference decreases with successive power swing oscillations after a fault is cleared. If the angular difference of the generator rotors continues to increase with successive oscillations, the generator or group of generators must be automatically separated from the system. This reduces the strength of the system and can cause other generator rotors to swing out of synchronism and collapse the system. A short circuit on a modern power system EHV bus must be cleared quickly by the opening of the power circuit breakers surrounding the short circuit. Protective relay operating time plus breaker clearing time is normally 0.05 second. If the short circuit occurs on a transmission line, the clearing of the fault will result in the loss of the transmission line. Since some short circuits are due to arc discharges to ground that are extinguished when the transmission line is de-energized, the line can possibly be restored to service automatically in 0.47 second. The reclosing time includes breaker opening time plus a preset closing time delay as suggested by the breaker manufacturer. Short circuits near a generator effect the generator and turbine shafts even more than the stator windings since the shafts become torsionally excited and are exposed to oscillatory stresses (l). A typical modern generator shaft system can have six natural resonant frequencies. Three of

13 3 these frequencies are above the power system frequency and three below. Simulated shaft system computer studies have shown that the resulting mechanical shaft torque can be as high as seven times normal after reclosing to a line that is not short circuited. (See Figure 1) To reduce the possibility of fatigue resulting in the generator shaft system one generator manufacturer has suggested that power circuit breaker reclosing near generators be delayed for at least ten seconds. The manufacturer believes the decay of torsional oscillation in the generator shaft will be sufficient to eliminate the risk of shaft fatigue after ten seconds. Manufacturers agree that shaft damage is also possible if an average generator power output change exceeding one-half pea: unit results from normal line switching. The literature on power system stability prior to 1945 included stability studies of interconnected power systems with high speed power circuit breaker reclosing. These stability studies provided the engineer with information on the maximum possible time for de-energization of the faulted circuit without loss of synchronism. The minimum permissible time for de-ionization of the fault arc was the determining factor in setting the reclosing time. The only risk considered in high speed reclosing was that of restriking the fault arc and that was considered to be negligible. In February, 1945, an article appeared in the AIEE

14 High Low Pressure Pressure 1 Intermediate Low Pressure Pressure 2 Generator \ Exciter HHZ] Coupling Mechanical Torque Electrical Torque Figure 1. Generator Shaft Torques ^

15 5 Transactions on Power Apparatus and Systems discussing the transient electrical torques of turbines during short circuits (10). However, the manufacturers of generators did not show much concern until fatigue of a generator shaft was experienced recently by a power company in the western part of the country. This fatigue was attributed to subsynchronous resonance. Subsynchronous resonance is a problem in the western part of the country because generators are frequently connected to the end of long lines with reactances approaching the subsynchronous reactance of the generator. This shaft fatigue experience caused the manufacturers to become concerned about the high speed reclosing of circuit breakers as well as synchronizing and line switching near generators. Generators manufactured recently have inertia constants more than twice the inertia constants of smaller units that were manufactured in the last decade. The subsynchronous, direct axis and quadrature reactances of these later units are about one-fourth the reactances of the earlier units. The accelerating power and available shaft torque during a short circuit condition has increased as a result. When a short circuit occurs on a utility network, a step function of shaft torque is applied to the generator shaft system. Another step function will be applied when the short circuit is cleared by the opening of surrounding circuit breakers. If another system disturbance occurs such as circuit breaker reclosing another step function will be applied to the shaft

16 6 system. These step functions will cause additive or subtractive torques to be applied to the shaft depending on the point of the oscillatory period they occur. The requirements of the manufacturers have resulted in power system operating changes that have created a problem for the power system protection engineer. Resulting power system stability and network line loading as a function of power circuit breaker reclosing time delay must be researched. The data accumulated in this research should help the protection engineer design protective schemes to minimize adverse effects on the power system. Since the protection engineer cannot risk the possibility of subjecting the generator shafts to excessive torques, he must delay reclosing by the time suggested by the manufacturer. He must then design the system protective relaying around the new operating conditions. Therefore, he needs to know the effects of power circuit breaker reclosing on power system stability as a function of reclosing time. He also needs to know the effect of not reclosing until the power swings have decayed.

17 7 CHAPTER II FORMULATION OF THE STABILITY STUDY The Purpose and Approach of the Study The purpose of this thesis was to study the effects of EHV power circuit breaker reclosing on the power system. The approach was to simulate the reclosing of breakers near a large generating facility after a fault was cleared by the opening of breakers. The ressulting power system swings were calculated at 0.17 second time intervals. The time for clearing the initial fault is normally three cycles in practice, and this initial clearing time was used throughout the study. The time taken to clear the fault after the breakers reclose was varied in the cases studied. The reason for varying the time for clearing the fault after reclosing was to determine the effects of reclosing on the power system under abnormal conditions. Abnormal conditions include those that occur due to a stuck breaker, failure of primary protective relays, incorrect operation of reclosing relays or incorrect operation of the control scheme. Methods Used to Calculate System Behavior Following a Disturbance A digital computer was used to reduce the Georgia Power Company system and its interconnections to other

18 8 systems to 124 buses. A portion of this system is shown in Figure 2. A load flow solution was used as the initial system conditions for the stability studies. Three phase faults were placed near the Plant Bowen 500-KV bus. Various reclosing times and fault clearing times were used to disturb the system. System conditions were calculated for every second following the disturbance by the use of a digital computer. Similar system disturbances were calculated by the classical swing equation using a point-by-point solution method, and also by the use of a Liapunov function developed by Gless (9). Cases Studied Case 1 In this case a three phase fault was placed on the Bowen-Sequoyah 500-KV line near the Bowen 500-KV bus. The fault was cleared in three cycles. The fault was replaced by reclosing the associated breakers after 2 8 cycles from the initial disturbance. The second fault remained on the system for 11 cycles before it was cleared. The three cycles clearing time of the initial fault is a normal operation. A closing delay setting of 25 cycles on the EHV breakers would cause a normal reclosing time of approximately 2 8 cycles. However, the 11 cycle clearing time is abnormal. This delayed clearing time will be experienced if a breaker associated with the faulted line fails to trip. In this case a back-up

19 V Bowen Sequoyah T Villa Rica I T Union City Norcross X Wansley Klondike 5HI I Figure 2. Reduced 500-KV System V >

20 10 relaying scheme is employed to trip back-up breakers. The breaker failure clearing time is 11 cycles in this case. This time was chosen because it is the critical switching time to maintain system stability. Case!2 A three phase fault was placed on the Bowen-Sequoyah 500-KV line near Bowen. The fault was cleared in three cycles. The fault was replaced by reclosing the associated breakers after 28 cycles. This time the second fault remained on the system for 12 cycles before it was cleared. The 12 cycle time was chosen since it is probable that breaker failure clearing time could require 12 cycles, and it is one cycle beyond the critical switching time. Case 2 A three phase fault was placed on the Bowen-Norcross 500-KV line near Norcross. The fault was cleared after three cycles and the system was left undisturbed for the remainder of the one second calculating period. This case was chosen to illustrate the system response to a remote fault being cleared in normal time without reclosing. Case A three phase fault was placed on the Bowen-Norcross 500-KV line near Bowen. The fault was cleared after three cycles. The associated breakers closed in 28 cycles placing the fault back on the system. The Bowen breaker tripped in three cycles, but the Norcross breaker remained closed for

21 30 cycles. This case illustrates the effect of leaving the fault connected to the system through the impedance of the Bowen-Norcross line. Case _5 A three phase fault was placed on the Bowen-Norcross 500-KV line near Norcross. The fault was cleared after three cycles. The Norcross breaker reclosed after 28 cycles and remained closed for 30 cycles. This case illustrates the effect of a remote terminal reclosing and failing to clear a fault in primary relaying time. Case S_ A three phase fault was placed on the Bowen-Sequoyah 500-KV line near Bowen. Reclosing occurred after 28 cycles. In this case the fault was removed before reclosing took place. This case was chosen to illustrate reclosing to an unfaulted line. Case ]_ A three phase fault was placed on the Bowen-Sequoyah 500-KV line near Eiowen. Reclosing occurred after 28 cycles with the fault present. The fault was cleared in 11 cycles and then the breakers reclosed again six cycles later. Even though this sequence of events would not likely occur during normal operation of the system, it was chosen to illustrate the resulting power swing. Case8: A three phase fault was placed on the Bowen-Sequoyah

22 500-KV line near Bowen. The fault was cleared at the Bowen terminal in three cycles. The fault was cleared from the Sequoyah terminal after 2 8 cycles. Reclosing of the Bowen- Sequoyah line took place after an additional 25 cycles into a fault that remained nine cycles before it was cleared. The sequence of events that could cause this operation in practice is the failure of primary protective relaying at the Sequoyah terminal. Nornicil reclosing took place in 25 cycles, but the Sequoyah breciker failed to clear the fault. Breaker failure back-up relaying at Sequoyah finally cleared the fault after nine cycles. Case This case was used to compare the results of a pointby-point solution of the swing equation with the results of a digital computer solution. A three phase fault was placed on the Bowen-Sequoyah 500-KV line near the Bowen 500-KV bus. The fault was clecired in three cycles. Reclosing into the fault occurred after 28 cycles. The fault was cleared after 11 cycles. Computations were ended after 1.2 seconds. Case 10 This case was also used to compare the results of a point-by-point solution with a digital computer solution. A three phase fault was placed on the Bowen-Sequoyah 500-KV line near the Bowen 500-KV bus. The fault was cleared in three cycles. Reclosing into the fault occurred after 28 cycles. The fault was cleared after 12 cycles and the Bowen

23 13 generators became unstable. Case 11 A Liapunov function developed by Gless (9) was used to calculate the maximum stable trajectory, in the rotor speed-rotor displacement plane, of the generator stability boundary. Rotor speed is a function of inertia, rotor displacement with respect to a synchronous reference bus, mechanical shaft input power and the power transfer between the generators and the infinite bus. Case 12 The same Liapunov function that was used in Case 11 was used in this case. However, the initial rotor angle is radians and switching into a permanent fault occurs when the rotor angular displacement reaches 1.22 radians. The system conditions simulated are similar to those in Cases 2 and 10. This case was included for comparison of the Liapunov method with the digital computer and point-by-point calculation methods. Definition of_ Terms Used The IEEE standard definition of power system stability is as follows: In a system of two or more synchronous machines connected through an eleictric network, the condition in which the difference of the angular positions of the rotors of the machines either remains constant while not subjected to a disturbance, or becomes constant following an aperiodic disturbance.

24 14 The IEEE standard definition of a Liapunov function is as follows: A scalar differentiable function V(x) defined in some open region including x e such that in that region (1) V(x) > 0 for x ^ x e (where x e = system equilibrium point) (2) V(x e ) = 0 (3) V (x) < 0. The "Critical Angle" is defined as follows: If a severe fault that is placed on a power system is removed before conditions are reached that would lead to instability, the system would remain stable. The maximum time the fault can remain with stable system operation maintained is known as the critical clearing time. The critical condition occurs when the sum of the potential and kinetic energy stored within the system at the instant immediately after clearing the fault does not exceed the maximum amount of potential energy which the system is able to accommodate. The relative rotor angle of the equivalent machine associated with this condition is known as the critical switching angle. The time taken to reach the*, critical switching angle is known as the critical switching time.

25 15 CHAPTER III ANALYSIS BY THE USE OF A DIGITAL COMPUTER Introduction Cases 1 through 8 were the cases studied by the use of a digital computer. In these cases the entire Georgia Power Company system with the interconnections to neighboring companies was reduced to 124 buses. A modified transient stability program by the Philadelphia Electric Company was used for the analysis (3). This program was also used for star-mesh reduction of the network (2]. Equivalent lines were created between buses, and loads were modified to represent the interconnected network by a reduced system. Data Preparation Data was pirepared for transformers and lines, loads, generators, governors and exciters. The impedances of the lines and trans formers were entered into the program in percent quantities on a 100 MVA base. The line charging MVA was also entered. This program assigns half of the charging MVA to each terminal bus. Transformers are distinguished from transmission lines by an entry in the transformer tap field. The transformer tap setting is the actual per-unit ratio of the transformer. Phase shifts introduced by the transformers were also included. Megawatt and megavar loads

26 16 were entered and represented as constant impedances. Initial Conditions and Method of Analysis The results of a load flow program were used as initial conditions for the transient stability study. These results for the generators and buses near Bowen were listed in Table 1. The generators were modeled in the stability study in more detail than the classical models used in load flow programs. The effects of field variations in the synchronous machines were represented. Generator damping was computed and inserted by the computer program on the generator bus for each generator near the fault. Generator saturation constants were entered into the program for calculation of the voltages behind the leakage reactances, which are proportional to the field currents. The excitation system was represented by entering the regulator, exciter, and amplifier time constants and other quantities for detailed modeling of the exciter. The speed governor control during transient periods was represented by transfer functions and time constants describing the steam and control systems. Solution of network performance equations was required to obtain system conditions at the instant after a fault occurs (0+ seconds). In this study, a three phase fault was placed on the network near Plant Bowen. The Gauss-Seidel iterative method was used to solve the network equations (2.).

27 17 Changes in genercitor internal voltage angles and generator speeds were calculated by solving first order differential equations by the modified Euler method (2). Network solutions were obtained for every second after the fault for at least one second. The initial conditions were not varied during the digital computer studies. Since the purpose of this research was to determine the effect of PCB reclosing on system stability, many cases were programmed for various switching conditions. The conditions that produced results of interest are included in the descriptions that follow with reference to the accompanying illustration. Analysis of Case1^ Case 1 was chosen to illustrate the effects of reclosing the Bowen-Sequoyah 50 0-KV line into a fault that remained on the system just long enough to allow the system to remain stable after the fault was cleared. The swing curve for this case is illustrated in Figure 3 as Curve 1. In this case the initial fault was cleared in primary clearing time, but the second fault placed on the system after reclosing took place was cleared in a breaker failure clearing time of 11 cycles. System stability is evident from the computer results at the end of one second. The results show that the relative rotor speed is approaching zero and the accelerating power is decreasing.

28 Case 1, Curve 1: Three phase fault near Bowen on Bowen- Sequoyah line. Fault cleared in three cycles. Reclose into fault at 28 cycles. Clear fault in 11 cycles. Calculation by computer. Case 9, Curve 9: Three phase fault near Bowen on Bowen- Sequoyah line. Fault cleared in three cycles. Reclose into fault at 2 8 cycles. Clear fault in 11 cycles. Manual point-by-point calculation. 160 h en Q) o u <D G <o I <u rh CP U O -P h 8o r 40 h o Time-t-Seconds Figure 3. Swing Curves of Cases 1 and 9 00

29 19 Analysis of Case 2_ Case 2 was chosen to illustrate the effects of reclosing the Bowen-Sequoyah 500-KV line into a fault that remained on the system just long enough to produce instability. The swing curve for this case is illustrated in Figure 4, Curve 2. In this case the initial fault was cleared in primary clearing time, but the second fault placed on the system after reclosing took place was cleared in a breaker failure clearing time of 12 cycles. System instability is evident since the computer results at the end of one second show a relative rotor position in excess of 180 degrees, an increasing accelerating power, and am increasing rotor speed. The one cycle variation in breaker failure clearing time can occur in practice due to variations in operating times of relays in the control circuits. Analysis of Case3^ The swing curve resulting from a fault being placed on the Bowen-Norcross 500-KV line near Norcross is shown in Figure 5 as Curve 3. This is Case 3 in which the initial fault was cleared in primary relaying time. Analysis of Case Case 4 is plotted in Figure 5. In this case the initial fault was cleared in three cycles; however, the line was reclosed after 2 8 cycles placing the fault back on the system. The Plant Bowen breaker tripped in three cycles but

30 Case 2, Curve 2: Three phase fault near Bowen on Bowen-Sequoyah line. Fault cleared in three cycles. Reclose into fault at 2 8 cycles. Clear fault in 12 cycles. Calculation by computer. 200 h 10 en a) (U U tn (U Q l <o i <D -H tp U O 4-> s 160 h 120 h 80I- 40 h Case 10, Curve 10: Three phase fault near Bowen on Bowen-Sequoyah line. Fault cleared in three cycles. Reclose into fault at 28 cycles Clear fault in 12 cycles. Point-by-point calculation Time-t-Seconds Figure 4. Swing Curves of Cases 2 and ro o

31 Case 3, Curve 3: Three phase fault near Norcross on Bowen-Norcross line. Fault cleared in three cycles. No reclosing. Calculation by computer. Case 4, Curve 4: Three phase fault near Bowen on Bowen-Norcross line. Fault cleared in three cycles. Reclose into fault at 28 cycles. Bowen tripped in three cycles. Calculation by computer. w Q) 80 U CP <D Q <o 1.H t^ 40 u 0 4J 0 « Time-t-Seconds Figure 5. Swing Curves of Cases 3 and 4

32 22 the Norcross breaker remained closed for 30 cycles due to failure of the primary protective relays at Norcross. If the fault is removed at this time, the system will remain stable. Stable operation was indicated by a decreasing accelerating power and a decreasing relative rotor angle in the computer results. Analysis of Cases _5 and 6_ Case 5 was used to investigate the effect of a remote breaker reclosing into a fault and failing to trip. The swing curve in Figure 6, Curve 5, resulted from a fault near Norcross which was cleared in three cycles by the breakers at. the Bowen and Norcross terminals. The breaker at Norcross reclosed after 2 8 cycles placing the fault on the system. The breaker at Bowen remained open. This case is stable provided the fault is removed after 30 cycles since the accelerating power and relative rotor angle is decreasing at this point. This case is plotted in the same figure with Case 6 for comparison. Case 6 resulted from reclosing Bowen-Sequoyah after 28 cycles to an unfaulted line. Analysis of Cases 7_ and8_ Cases 7 and 8 are shown in Figure 7. The Case 7 swing curve resulted from a fault near Bowen that was cleared from the system in three cycles. Reclosing into the fault occurred at 2 8 cycles after which the fault was cleared in 11

33 23 Case 5, Curve 5: Three phase fault near Norcross on Bowen-Norcross line. Fault cleared in three cycles. Norcross terminal reclosed into fault at 28 cycles. Fault remained 30 cycles. Calculation by computer Case 6, Curve 6: Three phase fault near Bowen on Bowen-Sequoyah line. Fault cleared in three cycles. Reclose unfaulted line at 28 cycles. Calculation by computer. w <u <D u 0) a i <o I <U rh c 0* < u o -p o « Time-t-Seconds Figure 6. Swing Curves of Cases 5 and 6

34 24 Case 7, Curve 7: Three phase fault near Bowen on Bowen-Sequoyah line. Reclose into fault at 28 cycles. Clear fault in 11 cycles. Reclose in 6 cycles. Calculation by computer. Case 8, Curve 8: Three phase fault near Bowen on Bowen-Sequoyah line. Trip Bowen terminal in three cycles. Trip Sequoyah terminal in 28 cycles. Reclose at 53 cycles into fault,. Fault cleared in nine cycles. Calculation by computer Time-t-Seconds Figure 7. Swing Curves of Cases 7 and 8

35 25 cycles. After six cycles, at 45 cycles, the line was reclosed into the fault. Even though impractical, this shows the effect of six cycle reclosing during a power swing. The resulting swing was large but stability was maintained. In Case 8 the Bowen terminal tripped for the initial fault in three cycles, but the Sequoyah terminal took 28 cycles to trip. Reclosing of the line occurred at 53 cycles into the fault, which was then cleared at 62 cycles. Cases 7 and 8 involved multiple reclosing and produced the largest rotor swings of the cases studied. Reclosing was not found to prevent system instability in any of the cases.

36 26 CHAPTER IV ANALYSIS OF THE REDUCED SYSTEM Approach and Assumptions Two methods of approach to the stability problem are presented in this chapter. Both methods require assumptions that simplify the problem to make it possible to solve without the use of a computer. The first method is referred to as the point-by-point or step-by-step solution. The pointby-point method was used in Cases 9 and 10. The second method involves the use of a Liapunov function. The Liapunov method was used in Cases 11 and 12. Values obtained for relative rotor displacement and speed were plotted to illustrate the comparison of the two methods. Cases 9, 10, 11 and 12 are covered in this chapter. The assumptions made were as follows: (a) a synchronous machine may be represented by a constant voltage back of the machine transient reactance; (b) flux linkages in the rotor circuit of a synchronous machine are constant; (c) torque input into the synchronous machine is constant; (d) angular momentum of the machine is constant; (e) the resistance and capacitive reactance of lines and transformers are neglected; (f) the resistance of machines is neglected; (g) all damping torques are neglected.

37 27 A partial 500-KV one-line diagram is shown in Figure 2. The double lines are 500-KV buses. The "G" at Plant Bowen represents the machine equivalent of the four steam generators connected to the bus. The buses are shown with interconnecting lines and equivalent sources from the interconnected network. were tabulated in the appendix. The line and generator parameters The system can be reduced to an equivalent machine connected to an infinite bus as shown in Figure 8. Since PCB 3 and PCB 4 have to open to clear the fault marked x, line Number 2 represents the impedance of the system with the faulted line removed. The network was reduced by making all of the possible series and parallel combinations of the line reactances surrounding the bus or buses to be retained. The remaining buses were eliminated by star-mesh and delta-wye conversions [8]. The equivalent impedances are shown in Figure 8. The four machines at Plant Bowen swing together during a system disturbance; therefore, the inertias of the machines should be added in deriving the equivalent machine (8) Ml = 180*f = x 10 ~ 3 (D Mi = 3^7-2.1 X 10 3 (2) M 3 = M = X 10" 3 (3) M = jmj = X 10" 3 F-u. power sec. 2 (4) e< 3 electrical degrees

38 28 Plant Bowen o A 1 HD L_]-X- Line 2 iy- B Infinite Bus Line 1 ^ 3 phase fault EHV Power Circuit Breaker o Transmitted Power = a & sin /ga - x-n x e q u-c* *-a Where x EA EI is the equivalent system reactance. P mabv Prefault: C. = 5^Ja = = max i Xl ou - /0 P- u - P max D» rin 9 Fault: C * = - ^ - OS554 = 17 ' 6 P ' U - 2 Pmax Postfault: C 3 = ^A_l = = 46>0Q p ^ Figure 8. Equivalent System

39 29 The solution of the classical swing equation was performed by successively calculating the changes in accelerating power (P ) and machine rotor angle (<5) over a successive second intervals. The Swing Equation: M -4 =: P * = p n " P m sin <5 = P. - P (5) 3t 2 a i m I u Where, P. = Combined shaft power input to the four 1 Bowen units P = Ci, prefault; C 2, during fault; C 3 m postfault P = Electrical power output Since, 51 = OJ = u 0 + 2at (6) u dt M Where, w = Angular rotor speed above synchronous speed Then, 6 = <5 0 + w 0 t + ^ - (7) u =u +^P,, (8) n n-i M a(n-i) «n " S n +1 + At«n _ l + - ^ P a(a. l) (9) A5 n = 5 n " «n-i = **» -! + If 1 *a(n-i) <10) (At) 2 = (0.017) * - = ~ 2 (11) M X 10-3 K X 10 U1) - These equations can be used to derive the following equations that were used to calculate the machine angle for each time interval.

40 30 C sin 6 = F (12) n-1 un P - P = P (13) m un an KP = A<5 (14) an n A<5 + 5 =5 (15) n n- I n a) - ^ (16) n At Analysis of Cases 9_ and 10 The solution of each interval is tabulated in Table 6 for switching conditions that produced stable operation. For Case 9 a fault was placed on the Bowen-Sequoyah line for three cycles and then removed. Since the system impedance changes at the switching point, two values of system impedance were inserted in column C at the switching points. The accelerating powei: resulting from each impedance was calculated and averaged to provide a smooth transition between states. The fault was again placed on the system after 28 cycles. In other words, PCB 3 and PCB 4 reclosed in 28 cycles The fault was cleared after 11 cycles and the resulting calculations were listed in Table 6. For Case 10, an unstable swing was produced by leaving the fault connected for one additional cycle. These values are listed in Table 7 and indicate that the critical switching time for a three phase fault on the Sequoyah 500-KV line is 11 cycles. Power-angle curves are shown in Figure 9. The power

41 Case 9, Curve 9: Three phase fault near Bowen on Bowen-Sequoyah line. Fault cleared in three cycles. Reclose into fault at 28 cycles. Clear fault in 11 cycles. Manual point-by-point calculation. Case 10, Curve 10: Three phase fault near Bowen on Bowen-Sequoyah line. Fault cleared in three cycles. Reclose into fault in 28 cycles. Clear fault in 12 cycles. Point-by-point calculation. «! Rotor Angle-6-Degrees Figure 9. Power-Angle Curves of Cases 9 and 10 CJ

42 32 input to the machine was plotted as a straight line. Three maximum electriccil power curves are shown representing the three system reactances resulting from the switching operations. The Case 9 stable trajectory is indicated by single arrows progressing from (<5 ;l) to (<5 2 ). The unstable trajectory (Case 10) is indicated by double arrows. The swing curves are shown in Figures 3 and 4 for the two cases. The computer solutions are also shown plotted on these curves for comparison of the results. Analysis of Cases 11 and 12 The method of Liapunov was applied to the problem to determine if this method is comparable with the point-bypoint method. The Liapunov function developed by Gless (9) was used. The Liapunov theorum states that the equilibrium of a system is stable if there exists a function of time and motion such that its total derivative with respect to time along the trajectories of the system is not positive. The function used is as follows;* V = T + 5!* ( _P i + P m sin (X " V) Where 6 = x + 6 d d(x + 62) _ dx _ 'dt " dt ~ dt " w The time derivative of equation (17) is dx < 17 > 3I = V=.f + i (-P i + P m sin <x )) H (13)

43 33 Since M 7- &0L = P. - P_ sin (x + <5 2 ) dt " i " m v = its ^pi - p m sin < x + 5 a>) + i (- P. + P sin (x + 6 )) f*- (19) 2 M 1 m ' dt Or V = 0 If the integration in equation (17) is performed, the result is 2 1 V = S- + ~ f- P.x - P cos (x + <SJ + P cos 5 2 ) (20) Substituting u> = x = (20) becomes (ffj 2 - I t p i (5 " V + P m (cos 6 - cos S 2 ] 1 + 2V (21) After adding and subtracting and rearranging P.5 + P cos 5 n 1 0 m 0 (H) 2 " I ( p i <«" V +P m < cos * * cos V + H ( P i (6 Ci " 6 2> + p m (cos 6 o " cos 5 2 } ) < 22 > If V = i fp. (6, 2-6 J + P (cos 26, - cos 5 )) M ^ 1 m 0 ; Then w 2 = 1 (p ± (6-6 0 ) + P m (cos 6 - cos <5 0 )) (23)

44 34 The maximum trajectory for stable operation was plotted by solving this equation for successive values of the machine angle (Table 8). The maximum value of the machine angle shown in Figure 11 is known as the saddle point. The w - 6 plot of the solution by the point-by-point method is illustrated in Figure 10 for comparison. Case 12 in Figure 11 illustrates an unstable trajectory comparable with Case 10.

45 Case 9, Curve 9: Three phase fault near Bowen on Bowen-Sequoyah line. Fault cleared in three cycles. Reclose into fault at 28 cycles. Clear fault in 11 cycles. Manual point-by-point calculation. Case 10, Curve 10: Three phase fault near Bowen on Bowen-Sequoyah line. Fault cleared in three cycles. Reclose into fault in 28 cycles. Clear fault in 12 cycles. Pointby-point calculation TJ c o <u en \ to Q) Q) U & Q) Q I 3 I Tf 0) 0) CU cn o -P o PC Rotor Angle-6-Degrees 160 Figure 10. Rotor Speed-Rotor Angle Curves of Cases 9 and

46 36 CHAPTER V EVALUATION OF REDUCED SYSTEM ANALYSIS AND LIAPUNOV METHOD Purpos e The purpose of this evaluation was to make comparisons of the various methods used to study the effects of power circuit breaker reclosing on the stability of the power system. The computer analysis was compared with the pointby-point analysis to determine the effects of making the assumptions that make manual computations practical. The reduced system analysis was then compared with the computation of a Liapunov function that made possible a prediction of the stability boundaries of the system by plotting the separatrix curve of Figure 11. The comparisons were used to make an evaluation of the validity of the reduced system analysis and the Liapunov method. Comparison of the Computer Analysis with the Point-By-Point Analysis The curves in Figure 3 illustrate the solution of the simulated system swing curve by a digital computer (Case 1) and the point-by-point calculation (Case 9). Reclosing took place at 28 cycles into a fault that was cleared in 11 cycles. Case 1 resulted in a larger swing of the Bowen generators.

47 37 However, stability was maintained as evidenced by the relative rotor speed approaching zero and the accelerating power decreasing in Table 4. The curves in Figure 4 illustrate the solution of the simulated system swing curve by a digital computer (Case 2) and the point-by-point calculation (Case 10). Reclosing took place at 28 cycles into a fault that was cleared in 12 cycles. The curves from either solution indicate generator instability at Plant Bowen. The slope of Case 2 is greater than the slope of Case 10 after one second, but the similarity of the solutions is evident. The critical clearing time of 11 cycles was obtained in each study. Therefore, the assumptions made in the reduced analysis did not effect the critical clearing time. Comparison of the Point-By-Point Analysis with the Liapunov Method The point-by-point solution was compared to the Liapunov solution by the use of Figures 10 and 11. The unstable trajectory, (Case 10) beginning at (6 0 ) in Figure 10, crosses the separatrix at 88 degrees rotor angle and 250 degrees per second rotor speed. In Case 12 of Figure 11 the unstable trajectory crosses the separatrix at 94 degrees rotor angle and 241 degrees per second rotor speed. This represents a difference of six percent in rotor angle and four percent in rotor speed. For all practical purposes

48 38 Case 11, Curve 11: Maximum stable trajectory, Bowen Generators, fault on Bowen-Sequoyah 5Q0-KV line. Calculated by Liapunov method. (<5 0 = radian) Case 12, Curve 12: Three phase fault near Bowen on Bowen-Sequoyah line. Reclose at 28 cycles into a 12 cycle fault. Unstable trajectory calculated by Liapunov method. (55» 0.43 radian) 6.0 h c o CD cn \ w <d H td «i 3 I d <u o en u o -P s 4.0 L 2.0 h -2.0 h -4.0 U -6.0 h Rotor Angle-<5-Radians Figure 11. Liapunov Method Cases 11 and 12

49 39 the results of Figures 10 and 11 are identical. Evaluation of the Comparisons In the comparison of the computer analysis with the point-by-point analysis the critical switching time to clear a fault before the units became unstable was found to be the same. Since the point-by-point analysis did not include the influence of machines remote from Plant Bowen, the resulting power swings in the digital computer analysis were greater than those calculated by the point-by-point method. Therefore, the results obtained by the point-by-point analysis are conservative. The Liapunov method was compared to the point-by-point solution. The results of the two methods were almost identical. The ease of solution of the Liapunov function used makes it the most desirable method used in the study. The ease of varying initial conditions in the Liapunov method makes it more useful.

50 40 CHAPTER VI CONCLUSION Effect of Reclosing on Stability System stability was not preserved because of reclosing a line in any of the cases studied. Reclosing did cause system instability when a faulted line was reclosed and left on the system in excess of 11 cycles as shown in Figure 4. The conclusion drawn from the system studied is that high speed reclosing cannot be used to aid generator stability. Basis for Conclusion When reclosing was used, the rotor displacement angle w.r.t. the infinite bus increased rapidly in cases where the fault was still present. The swing curve shown as Curve 6 in Figure 6 did not change appreciably when the unfaulted line was reclosed. The only way to keep the system stable after a three phase fault is placed near the generators is to remove the fault before the generator rotors accumulate more potential energy than the system is able to accommodate. In Figures 10 and 11 the separatrix curve gives a graphical illustration of the trajectory of the generator rotor speed following a three phase fault. Starting at (6 ) in Figure 11 the trajectory will be contained within the separatrix if the fault is cleared in time. If not, the

51 41 system becomes unstable. The OJ-6 curves gave better insight to the operation of the system than the swing curves plotted by a digital computer. In Figure 11 an infinite number of initial rotor angles and reclosing times can be plotted within the separatrix.

52 42 CHAPTER VII RECOMMENDATIONS High speed reclosing of EHV breakers near generating plants should not be used to improve system stability. The possibility of causing fatigue to the generator shaft system is reason enough to make this choice; furthermore, the possibility of protective relay or breaker malfunction causing delayed fault clearing and instability is also present. Further research is necessary to determine a safe system impedance between the generators and the fault to insure that the generator shaft would not be damaged after a reclosure or switching operation. Also, a simplified method for calculating system stability is needed to reduce computer time. Since the separatrix curve that was calculated from the Liapunov function used in this thesis is a conservative estimate of system, stability, this method could be used in an on-line computer to determine stability limits under actual system operating conditions. The weak points could then be further examined by a detailed transient stability analysis. Rotor swings resulting from actual system disturbances could be plotted to determine the accuracy of the results.

53 APPENDIX 43

54 44 Table 1. Load Flow Bus Bowen Bus Voltage Mag. p.u. Angle degrees Norcross Klondike Union City Wansley Sequoyah To Norcross To Union City To Villa Rica To Gen. No. 1 To Gen. No. 2 To Gen. No. 3 To Gen. No. 4 To Equivalent To Sequoyah To Bowen To Klondike To Equivalent To Norcross To Union City To Equivalent To Bowen To Union City To Wansley To Equivalent To Villa Rica To Equivalent To Gen. No. 1 To Bowen To Equivalent To Gen. No. 1 MW Power Flow MVAR

55 45 Table 2. System Parameters From Bus To Bus Reactance Charging Percent MVA Bowen Unit 1 Bowen 500-KV Bowen Unit 2 Bowen 500-KV Bowen Unit 3 Bowen 500-KV Bowen Unit 4 Bowen 500-KV Bowen 500-KV Norcross 500-KV Bowen 500-KV Sequoyah 500-KV Bowen 500-KV Union City 500-KV Bowen 500-KV Villa Rica 500-KV Klondike 500-KV Norcross 500-KV Klondike 500-KV Union City 500-KV Union City 500-KV Villa Rica 500-KV Villa Rica 500-KV Wansley 500-KV

56 Table 3. Plant Bowen Generator Parameters Unit 1 Unit 2 Unit 3 Manufacturer Westinghouse General General Electric Electric Rated Output (KW) 805, , ,000 Rated Output (KVA) 948, ,000 1,120,000 WR 2 738, , ,280 Rated Voltage 25,000 25,000 18,000 Rated Current (Amperes) 21,893 21,431 35,924 XI (p.u.)* x (p.u.)* X^ (p.u.)* H (MWS/MVA) * Each generator is a 3-phase, 60 Hertz, 3600 RPM unit. *On a 100 MVA base. Unit 4 General Electric 952,000 1,120, ,000 35, >» Ov

57 47 Table 4 Computer Computation of Case 1 Time P 6 0) Sec. 12 degrees degrees/sec p.u

58 Table 4. Continued Time P u <5 Sec. p.u. degrees

59 49 Table 5. Computer Computation of Case 2 Time P u 5 OJ Sec. p.u. degrees degrees/sec

60 50 Table 5. Continued Time p u 5 0) Sec. p.u. degrees degrees/sec ,