THE UNIVERSITY OF MICHIGAN SYSTEMS ENGINEERING LABORATORY Department of Electrical Engineering SEL Technical Report no. 41 METHODS OF IMPROVING TRANSIENT STABILITY OF A SYNCHRONOUS GENERATOR Chaman N. Kashkari July 1969 sponsored by Consumers Power Company Jackson, Michigan

ACKNOWLEDGEMENTS The author wishes to express his appreciation to Professors J. J. Carey, C. L. Dolph, W. D. Getty, K. B. Irani, E. L. McMahon and R. F. Mosher for serving on his doctoral committee. In particular, the author wishes to record his debt of gratitude to: Professor J. J. Carey, Chairman of the committee, without whose inspiring influence and invaluable guidance, this work would not have been possible. Professor R. F. Mosher for his active help and close supervision at all stages of this work. Professor K. B. Irani for his encouragement, inspiration and assistance. Professor K. Chuang for many fruitful discussions during the course of this research. Sincere thanks are due to Professor A. J. Pennington, formerly of the University of Michigan, for his assistance in the early stages of the author's graduate work. Special thanks are due to Miss Joyce Doneth for her assistance in preparing the manuscript. The support given by Consumers Power Company for this research is appreciated. Finally, the author is indebted to his wife for her patience, understanding, and continued encouragements. ii

TO SHEILA AND MEERA iii

TABLE OF CONTENTS Page List of Symbols v List of Figures viii Abstract xi Chapter I Scope of Investigation 1 Chapter II Mathematical Representation of the System 6 Chapter III Excitation Systems - Phase Plane Analysis of Derivative Signals 20 Chapter IV Excitation Systems - Digital Computer Simulation 45 Chapter V Excitation Systems - Digital Computer Results 49 Chapter VI Normalized Stability Margin 58 Chapter VII Phase Shifting Transformers - Basic Analysis 64 Chapter VIII Application of Phase Shifting Transformers 83 Chapter IX Switched Series Capacitors - Approximate Analysis 87 Chapter X Switched Series Capacitors - Experimental Study 106 Chapter XI Switched Series Capacitors - Fault Study and Relaying 123 Chapter XII Conclusions 140 References 142 Appendix I Mathematical Representation of a Synchronous Generator 144 Appendix II Computer Programs 158 iv

List of Symbols A1 - A6 Gain in Excitation System Auxiliary Signal Channels Ed Direct-Axis Component of Excitation Voltage E'd Direct-Axis Component of Transient Internal Voltage E Exciter Voltage Referred to Armature Circuit ex E. Voltage Behind Transient Reactance E Voltage Behind Synchronous Impedance E' Voltage Proportional to Field Flux Linkage E Voltage Behind Quadrature-Axis Synchronous Reactance Eqo Steady State Value of Eq E Exciter Field Voltage E1 Voltage Behind Transient Reactance E2 Infinite Bus Voltage G Machine Rating H Inertia Constant id Direct-Axis Component of Armature Current if Current in Machine Field Winding i Quadrature Axis Component of Armature Current K1, K2 Design Parameters in Auxiliary Signals Channels Lf Inductance of Field Winding M Inertia Constant of the Machine N Slope of the Phase Trajectory v

List of Symbols (Continued) P1, P2, Pi' P Input Power Pm1' P m2 Pm3 Amplitudes of Prefault, During Fault and Post Fault Power Angle Curves Pu, Po Output Power r Armature Resistance a rf Resistance of Field Winding Sd Direct Axis Saturation Factor Sq Quadrature Axis Saturation Factor T'do Open Circuit Transient Time Constant T Exciter Field Constant e T'qo Quadrature Axis Transient Open-Circuit Time Constant Vb Busbar Voltage Vd Direct Axis Component of Terminal Voltage vdm Direct Axis Component of Machine Terminal Vdm v ~Voltage vdp Direct Axis Component of Potier Voltage Vm Machine Terminal Voltage v Quadrature Axis Component of Machine qm Terminal qp Quadrature Axis Component of Potier Voltage v Quadrature Axis Component of Terminal ~~q ~Voltage Vr Reference Voltage co Frequency in Radians per Second vi

List of Symbols (Continued) Xd Direct Axis Synchronous Reactance X'd Direct Axis Transient Reactance x External Reactance x Potier Reactance x Quadrature Axis Synchronous Reactance x' Quadrature Axis Transient Reactance q 6 Torque Angle 6 o Steady-State Torque Angle <^0+~ Magnetic Flux Jv Gain in the Voltage Channel vii

List of Figures Figure Page 2.1 Elementary Power System 7 2.2 Phasor Diagram of the System 11 2. 3 Phasor Diagram of the System with Saturation 13 2. 4 An Exciter for a Synchronous Generator 16 3. 1 Torque Angle vs. Rate of Change of Torque Angle Curve 24 3. 2 Effect of Excitation Control on Operating Point 26 3. 3 Effect of Excitation Control on Torque Angle/Rate of Change of Torque Angle Curve 28 3. 4 Effect of Excitation Control on Operating Point 30 3. 5 Construction of a Phase Trajectory using the Method of Isoclines 33 3. 6a Effect of K2 on Isoclines k =. 005 x 180,/T 34 3. 6b Effect of K2 on Isoclines k2=.01 x 180/7r 35 3. 6c Effect ofK2 on Isoclines k2=. 015 x 180/7 36 5. 1 Swing Curves 54 5.2 Sending End Terminal Voltage 54 5.3 Machine Field Voltage 55 5. 4 Swing Curve for input 1. 4 56 5. 5 Exciter Field Voltage 57 6. 1 Determination of Stability Margin 60 6. 2 Pre-fault, During-fault and Post-fault Power Angle Curves 62 viii

List of Figures (Continued) Figure Page 7. 1 Application of P. S. Transformers in a Typical System 65 7. 2 Effect of Phse Shift on Power Angle Curve 66 7. 3 Phase-Plane Analysis of Phase Shifting Transformers 69 7. 4 Determination of the Effect of Phase-Shift on Stability Margin 73 7. 5 0 for Dead-Beat Control 76 7. 6 Determination of Minimum 0 79 8.1 A Typical System Using a P. S. Transformer 84 8. 2 Effect of Phase Shift Transformer on Swing Curve 86 9.1 A Typical System Using Switched Series Capacitors 89 9. 2 Power Angle Curves for the Above System 89 9. 3 Effects of Switched Series Capacitors on Power Angle Curves 93 9. 4 Series Compensation for Maximum Transient Stability Limit 93 9. 5 Determination of Minimum Capacitive Reactance 99 9.6 Single Machine System 99 9,.7 Positive Seauence Networks 101 9. 8 Effect of Switched Series Capacitor on Swing Curves 105 10. 1 Schematic Diagram of the Laboratory System 109 10.2 Experimental Set Up 111 10.3 Experimental Set Up 111 ix

List of Figures (Continued) Figure Page 10. 4 Experimental Set Up 112 10. 5 Experimental Set Up112 10. 6 Principle of Torque Angle Measurement 114 10. 7 Schematic Diagram of the System 119 10. 8 Phasor Diagram 119 10. 9 Comparis on of Theoretical and Experimental Curves 121 10.10 Experimental Recordings 122 11.1 A Generalized Circuit 124 11.2 A Typical By-pass Circuit 124 11. 3 Swing Curve with Capacitor Insertion 130 11. 4(a) Circuit for Distance Relay Study 132 11. 4(b) A Simplified Circuit 132 11. 5 A Simple Two Machine System 134 11. 6 Impedance Seen by Relay during Swing 139 A-1. 1 Two Pole Synchronous Machine 146 A-1. 2 Y-Connected Synchronous Machine Windings 147 A- 1.3 Phasor Diagram 157 x

ABSTRACT This work is an investigation of three techniques for enhancing transient stability of a synchronous machine connected to an infinite bus through parallel transmission lines. These techniques are as follows: 1. Application of Auxiliary Signals in Excitation Systems This research includes an analytical study of the effects of auxiliary signals used in Excitation Systems. These signals are proportional to derivatives of torque angle. A very interesting conclusion of the study is that if the signals are of correct magnitude, then the oscillations will be strongly damped. The mathematical techniques used in this study can be applied while designing an excitation system. These techniques are based on Phase Plane Analysis. 2. Use of Phase-Shifting Transformers Another interesting method for controlling transients utilizes phase-shift windings in step-up transformers. This research is concerned with an analytical study of the effect of phase-shifting transformers on the transient stability of a Synchronous Generator. Since the power is carried by the phase-shift winding during transients only, it can be designed for short time rating with resultant economy. 3. Switched Series Capacitors An experimental and an analytical investigation has been made on switched series capacitors. The analytical study includes some application problems associated with the capacitors. The results indicate that it is technically feasible to use switched series capacitors in long transmission lines. xi

Chapter I SCOPE OF INVESTIGATION Introduction Since the early days of alternating current power generation and transmission, engineers have been faced with the problem of power system stability. During the past few years this problem has assumed new importance due to system interconnections, which have created complex networks having load areas far away from the generating plants. The Northeast blackout of November 9, 1965, created a new awareness of this problem in the minds of the public, the government and the electric utilities. The effects of the blackout were so profound that the utilities cannot afford to have another such occurence. Engineers must therefore search for new techniques to improve the reliability of electric power supply. Definitions According to the American Standards Association, stability with reference to power systems is defined as follows: "Stability when applied to a system of two or more synchronous machines through an electrical network, is the condition in which the difference of angular positions of the machines either remains constant while not subject to a disturbance or becomes constant following an aperiodic distrubance. "Transient Stability is a condition which exists in a power system if, after an aperiodic disturbance, the system regains its steady-state stability~ "Steady State Stability is a condition which exists in a power system if it operates with stability when not subjected to an aperiodic disturbance. "

2 It will be observed that depending upon the nature of the disturbance, stability can be classified in the following categories: 1. Steady-state-stability under slow or gradual load change. 2. Transient-stability-when the system is subjected to sudden and large magnitude disturbance. By means of an automatic voltage regulator, it is possible to create stability of a generator under conditions for which the generator is inherently unstable. Regulator induced generator stability is referred to as "Dynamic Stability". Methods of Improving Stability During system planning, a stability study is required in order to ensure that it will be able to ride through the dynamic oscillations following any system disturbance and maintain stability. The power capability of long, interregional tie lines is usually limited by the transient stability limits. Such economic factors as the high cost of long lines and the additional revenue obtainable from the delivery of additional power provides a strong incentive to explore all economically and technically feasible means of raising the stability limits. The methods of increasing the stability limits may be classified as follows: 1. Switching 2. Excitation systems 3. Compensation 4. Energy control.

3 1. Switching includes rapid clearing of faults, increasing the number of intermediate switching stations in the case of two or more parallel lines and the use of reclosing breakers. These methods have been traditionally used by power system engineers. 2. Excitation systems. Fast excitation systems have considerable effect on stability, particularly dynamic stabilityo There is however considerable interest in the development of faster excitation systems which would improve both dynamic and transient stability. 3. Compensation. By compensation is meant the introduction of series capacitors or shunt reactors. 4. Energy control. This includes braking resistors and rapid reduction of input power by fast valve action, on the prime mover. Scope of Investigation The aim of this research is to develop new techniques for improving transient stability of a synchronous generator, connected to an infinite bus by means of a double circuit transmission line. The following research was carried out: (1) Application of Auxiliary Signals The main function of an excitation system is to regulate the machine terminal voltage, A voltage signal proportional to the difference between the magnitudes of reference voltage and the

4 terminal voltage is applied to the field circuit of a control exciter, thereby correcting the terminal voltageo The possibility of using signals other than the terminal voltage as input signals to the 2-5 regulator has been mentioned recentlyo These investigators have reported that the transient performance of the synchronous generator is improved if additional signals are applied to the excitation system0 This research is concerned with an analytical study of the fundamental effects of auxiliary signals, proportional to derivative of torque angle 6, on the transient performance of a synchronous generator0 The study is based on phase plane techniques of nonlinear control systemso Previous investigations of excitation system problems involved either linearizing of the differential equations or use of empirical methodsO By the approach followed in this report, expressions for appropriate signals are developed in a straightforward manner. The gains in various feedback channels can be determined from the knowledge of system parameterso This research lays foundations for future research on excitation system designo (2) Use of Phase-Shifting Transformers An interesting method is developed which may have considerable application in long distance transmission of power. This involves the use of auxiliary windings in the sending-end

5 step-up transformer. This winding provides phase shift. During a transient, power flows through this winding, instead of the normal winding, with resultant damping of the oscillations and enhancement of the stability limit. Since the auxiliary winding carries power during the transient only, it can be short-time rated with resultant economy. (3) Use of Switched-Series Capacitors 6 12 Kimbark and Smith2 have suggested inserting capacitors in the lines to compensate for the increase in line reactance due to switching out of a line section. The author has extended this idea by an experimental and theoretical study of a typical installation using switched series capacitors. Some application problems, inherent in use of switched series capacitors have been taken into consideration, thus establishing the feasibility of using this technique in modern installations.

Chapter II MATHEMATICAL REPRESENTATION OF THE SYSTEM Introduction This study is based on the classical work of Park7 who developed a mathematical model for an ideal synchronous machine. Details of the assumptions on which the model is based and the derivation of the equations are given in Appendix I. A simplifying approximation in the model is that the magnetic saturation has been neglected. In this chapter the machine equations have been suitably modified to account for saturation. This chapter also gives a description of the system under consideration and develops mathematical models for the components so that the system can be represented by a set of differential equations. The equations derived here have been used in Chapters III to V, for digital simulation of the system, to investigate the effects of feeding auxiliary signals to the excitation system. Description of the System The elementary power system which has been considered for the purpose of analysis is shown in figure 2. 1. It consists of a salient-pole synchronous generator, connected to an infinite bus by a double circuit transmission line. This is a classical configuration for such a study. 6

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8 Machine Equations and Saturation The equations of an ideal synchronous machine have been developed in Appendix I. It will now be necessary to modify these equations so that saturation can be taken into account. Many methods, all approximate, have been given in literature to include saturation. 8 The method used here is that outlined by Olive and is based upon a correction of the flux linkages within the machine. It is approximate in that leakage flux has been assumed known for stator circuits only and has been assumed zero for rotor circuits. The saturation factor Sd or Sq at any specified voltage for the direct or quadrature axis, is defined as Field current to give that voltage on the magnetization curve Field current to give that voltage on the airgap line The direct axis saturation factor may be obtained directly from the machine open circuit curve corresponding to the value of the quadrature axis component of the Potier voltage or at =v +r i +i x qp q a q dp where Sd= (vqp) v is the quadrature axis component of Potier voltage, v is qp q the quadrature axis component of terminal voltage, ra is the armature resistance, i is the quadrature axis component of current, id is the q

9 direct axis component of current, x is the Potier reactance and f is a symbol for a certain function. The value of the quadrature axis saturation factor may also be obtained from the same open-circuit curve by multiplying the value found at the quadrature axis component of Potier voltage by the ratio xq/xd, where xq and xd are quadrature axis and direct axis synchronous reactances of the machine. Vdp = vd +r id -x i dp d a d p q where Vd is the direct axis component of Potier voltage and vd is the direct axis component of terminal voltage. x s =iq f(v q d The equations of a synchronous machine as developed in Appendix I are ~q= -'a aq - xd id + q (2.1) d= -a id + iq d (2.2) q d q q E = -(xd - 2d) id + E (2.3) Ed =(xq Xq) iq d (2.4) dE T [Ex - E (2.5) dt q q d d E d d't d d (2.6) dt d'-qo qo

10 In the above equations, E and Ed are the quadrature and direct axis components of excitation voltageo E'q and E' are the quadrature and direct-axis components of transient internal voltage, x'q and x' are the quadrature and direct axis transient reactanceso T'qo and T'd are the quadrature and direct axis open circuit time constants. Eex is the exciter voltage referred to the armature circuito Since the machine under consideration is a salient pole synchronous machine (xq = x' ), E d= Ed= 0 and if the resistance terms are neglected, the equations reduce to - id d (2. 7) q q dd vd x i (2. 8) d qq E'= E - (Xd- X'd) id (2.9) q q dxd) dEq _ Eex Eq] (2 10) dt Tt ^A (2. 10) dt - T' do If saturation is included, the equations modify to the following xd- x v =E _-[ d p +xid (2. 11) q q 1+Sd P d x -X 1+S q P +X ](2.12) Vd [ 1 +S + p q (xd x id) E =E - id (2.13) q q (1~S d) (3

~re~ ~~ ~~~~~~~~~~~~~11~~~~~~~~C Ip~~~~~~~~~~~~~~~~~~~~~~~~l C) tI2 P E~~~~~~~~~~~~C)'"a r,0 x~~~~~~~~~~~~~~~~~~~~~~~ b0 Ca~~~~U -9~~~~~~~rr Q)~ IC a~~~~C

12 dE' q [Ex (1+Sd) E] (2. 14) dt - T'do ex d q A phasor diagram showing the above situation is shown in figure 2. 3. Power Output Relationships (Neglecting Saturation) The power output from a salient pole machine during a transient is given by Vb E (x - x ) P X —- e sin 6 + Vb2 ( q sin26 (2.15) u x6 + b 2(x + xe) (x'd+ xe) where vb is the busbar voltage, 6 is the angle between the infinite bus and the quadrature axis of the rotor and xe is the external reactance. Since the rate of change of E' involves Eq, it would be convenient to relate E and E' by certain known quantities as follows. Refer to figure 2.2 qd - id(q + e) =vb cos6 (2. 16) where Eqd is the voltage behind the quadrature-axis synchronous reactance. Therefore Eqd - vb Cos 6 i qd e (2. 17) d (x +x) q e But E't - - (2.18) q = Eqd id(xq - Xd) (2.18) Substituting equation (2. 17) in equation (2. 18),

13 0 - + ct < ) W, / \.o (II C)! 7L ):,. i /.+ // CA) 0) -"% \ ^<^ ^/ ^C~

14 E - v cos 6 E' = qd b (x -x'd) q qd +x + x q q e (xd+ xe) (x - x) d e q d =E - -v+ v cos (2.19) qd (x q+ x) Vb c (x +x ) Again from the figure, Eq Eqd + id(xd - ) (2.20) and Eqd E =id(q -'d) (2.18) From the above equation, Eqd - E' id qd q d- X-Xd q d Substituting this expression for id in eq. (2.20), the result is (xd- x'd) (, - xq) Eq q=E dx E (2.21) q qd (Xq d)' From (2. 19) and (2.21), (xd+ Xe) (x -x d) E = E'q(d rx - Vb cos 6 x (2.22) q(x+x (X'd e ) During a stability study, it will be necessary to use the equation involving rate of change of E q, namely dE' dt -T'do [ex Eq](2.5)

15 Since E is known from eq. (2.22), it will be necessary to know E ex the machine field voltage, which is also the exciter armature voltage. It is therefore, necessary to develop a mathematical model for the exciter as will be shown in the next section. Exciter Representation Figure 2. 4 represents an exciter with a separately excited field winding. Ex is voltage applied to the field and Eex is its armature voltage. Usually the machine voltage is regulated by controlling the exciter field voltage Ex. If rf, Lf are the resistance and the inductance of the exciter field winding and if is the field current, then the following relationships hold. di rfif + Lf dt = Ex (2.23) Eex =k = k'if (2.24) where 0 is the magnetization flux and k and k' are constants. The flux 0 is proportional to if, as saturation in the exciter field is neglected. This is an acceptable approximation, since the ceiling voltage of modern exciters is many times the normal voltage. From eqs. (2.23) and (2.24) E x d if r + p is operator f- rf+PLf

16 0 I 0 o 0 bD v80 0 0 0 0 X o * g. O " ^~~~~~~I ^-^( &~~~ L/ vJ i *^ V ^ ^~~~~~~~G Cti bs^-^ *S C^~~~~~~~~~~~~~~~~r 2) y9 <D > F (S^~~ o)

17 k'E k'E E X X ex f f r+.rf.p Te e field winding If Eex and E are in per unit system, then E E - (2.25) ex 1 + p T or dE ex 1 dt = T (E -E) (2.26) Ex, the exciter field voltage will consist of a steady state component and a transient component proportional to the difference between terminal voltage and reference voltage, or Ex -o v(V - Vr) (2.27) where Jv is gain in voltage channel. From eq. (2.26) and eq. (2. 5) the steady state component of E is equal to Eq, the steady-state value of Eq. In any excitation control with auxiliary signals eq. (2. 27) will be Ex Eqo - (v m vr) + f[auxiliary signals] (2.28) where fis a certain function. Terminal Voltage In eqs. (2.27) and (2.28), vm is the machine terminal voltage and it is necessary to find an expression for vm in terms of the known quantities. Referring to figure 2.2, x v = v sin 6 q (2.29) dm b xin + x x e

18 Vqm = Vb Cos + id (2.30) But E -vb cos 6 id=-q b (2.31) d- x + x e d Therefore xd v = v cos 6 d-+E x (2.32) qm b o + xd q e ( Xe+ xd 2 2 2 v = v + v (2.33) m dm qm Machine Equations with Saturation Referring to figure 2. 3, the following equations hold (Xd - X'd) d Eq Eq 1 + Sd id b cos 6E - id(Xd + ) or E' - vb cos 6 q b d (x' d+x) E =E' + (-'dd) [q Vb cos6 q xd +xe Vb cos (xd - Cxtd d d. (x' d + (1 d) ) (2 34) v,~~ x - x' dm q p(X ) v b sin - 1 + b d d +X)(1+S) ~~~~~~~(2.34) d e d Vdm x q P X +X +X -X e P q p 1+S q

19 x S +x v - - q q sin 6 (2.35) dm (x +x ) S +x +x b e p q q e Similarly v = v cos 6 + i x qm b d e But E - v cos 6 q b d x +x Xp e p +Xd Xp 1 + Sd Therefore E - vb cos 6 qm Vb cos + + x e p d P 1 +Sd Ex x q e evbcos e Xe P d - p e p d p 1 + Sd 1 + Sd (2. 36) 2v = 2 2 (2.33) m Vdm (2 33) Similarly d dEq Tdo[E -(1+Sd) E] (2.37) dt q Td ex dq The equations derived in this chapter represent the mathematical model for digital simulation of the synchronous generator.

Chapter III EXCITATION SYSTEMS (PHASE PLANE ANALYSIS OF DERIVATIVE SIGNALS) Introduction The main function of an excitation system is to regulate the machine terminal voltageo A voltage signal proportional to the terminal voltage is compared with a reference voltage and the error is amplified and fed to the exciter field. Besides regulating the terminal voltage, an excitation system will have a beneficial effect on the stability of a synchronous machine. It will improve dynamic stability to an appreciable degree and transient stability to some extent by rapidly increasing the airgap flux following a fault which produces a depression of machine terminal voltage. During the last few years many researchers have worked on the problem of improving transient stability by making improvements in 2-5 the excitation systems. They have attempted to feed auxiliary signals, besides the voltage deviation, to the excitation system. Notable 3 among them are Langer et al who have used linearized equations on an analog computer, feeding first and second derivatives of load angle to the regulator. Dineley et al have used a digital computer to find the effect of varying gains in load angle channels. This research is concerned with an analysis of nonlinear equations of a synchronous machine by the phase plane method and lays the 20

21 foundations for a systematic study and design of amplification factors in derivative channels. The Phase Plane10 The phase plane approach is a very convenient method of analyzing some non-linear systems. Consider the equation n dn (3.1) A? dx A + d...+A'x + Ax =0 (3.1) n dtn n-1 dtn-1 1 0 * *e dn-l If x, x, x, d 1 are treated as co-ordinates of a certain phase dn-i dt space, then this equation can be represented by a single curve in the space. This curve is called a phase trajectory. In second order systems, it is fairly easy to obtain a phase trajectory. In the above equation A? is the coefficient of the nth derivative. n If the differential equation is of second order, the equation may be * * e x +Ax + Bx = F. A phase plane applied to this equation will have coordinates x and x. The Swing Equation The equation governing the oscillations of the salient-pole synchronous machine rotor is 2 E' v M P6 =P - [ s6-kin 6 -k 26] dt2 X

22 where M = inertia constant E'q = voltage proportional to field flux linkages 6 = torque angle vb = infinite bus voltage t = time X = x + x where d e Pi = input power x' = transient reactance d X = external reactance e x - x k = vb + X XT) =q a constant, if x? is assumed to be constant. d eq e d6 E' v dt q "'^ rt1' sin 5 +k16 sin 25] -dt [Pi- Xn ksin2 d6 wM or dA -E v sin6 (3.3) P. q + k sin 26 1 X If Et, vb X and M are constants, then the variables in eq. (3~ 3) can be separated to yield E' v M d = [Pi - q +ksin + s26 ] d6 Integrating both sides 2 E' v Mw2 q b 2 = Pi(6 - 6 o) + X (cos 6 - cos 6 o) - k [cos 26 - cos 26o]

23 where initial values of 6 and o are 6 o and zero respectively. Therefore 2 2 ^E'vb k - [=P(6 -6 o)+ b (cos 6 -cos 6o)- k (cos 26 -cos 26 o)] (3. 4) A phase plane plot was drawn for a synchronous machine system having the following constants and initial conditions. xd= 1.25 X? =.28 v =1.0 x =.70 x =.5 ~d d ~ b =q e H, the inertia constant = 3. 0 G, the machine rating = 1. 0 GH i. 0 x 3. 0 M, the inertia constant = H 180 x 30 =.000278 Input power = 1. 0 Power factor at infinite bus = 1. 0 60, the initial steady state torque angle = 50. 2~ With reference to figure 2. 2, the value of E' is calculated as follows: q Eq qd- (Xq X )id E =-+j(x +x)T qd b + e q = 1.00 + j(. 5 + 7) 1.00 = 1.00 + j 1.2 = 1. 57 id =i sin 6 =.765 Therefore E' = 1. 57 - (.7 -.28).765 = 1. 245 q The phase plane plot is shown in figure 3. 1 when the input power was suddenly increased from 1. 0 to 1. 3. From the plot it

24 C4 0\ >0 0 torque angle -— o. degrees Fig. 3. 1 Torque angle/Rate of change of torque angle curve

25 is observed that the speed of the machine increases rapidly at the beginning of the transient and it reaches a maximum at an angle of about 70', when it begins to decrease. The oscillations are from 50. 20 to 87. 0~ and since damping has been neglected, these are continuous. Auxiliary Signals During the preceding analysis, it was assumed that the voltage behind transient reactance, E' was constant and figure 3. 1 was drawn on the same basis. In this section, the influence of varying this voltage will be taken into account as follows: Signal Proportional to Torque Angle Assume that the voltage E' was regulated according to the following relationship E q =E qo + K1(6-) (3.5) where 60 is the steady state value of 6 before the transient occurred, E'qo is the initial value of Eq and K1 is a constant, the value of qo q 1 which will be determined. 6 and 6 are in radians. Substituting (3. 5) in (3. 3) gives d6 WM (3.6) (3.6) dw vb Pi X [E' + K1(6-60)] sin 6 + k sin 26 The effect of this signal is depicted in figure 3. 2 which shows that the amplitude of the power angle curves increases from instant to instant due to action of the auxiliary signal. The operating point

26 New input power Sg Input Power rb Before 0 Torque Angle —-- Fig. 3. 2 Effect of Excitation Control on operating point

27 moves from a to d through b and c. It is observed that the amplitude of the oscillation is decreased from 6, to 61. Again separating the variables in eq. (3. 6) Vb Mww= [Pi - [E qo + K1(6-6 ) ]sin 6 + k sin 26]d6 Integrating this equation, the result is 2 M E qo Vb 2 Pi(5-6) + qoX ( 6-cos 6 ) 1 0 X 0 K1 v + X [(6 - 6 ) cos 6 + (sin 65 - sin 6)] k 2 (cos 26-cos 26) (3.7) Comparing equations (3.4) and (3.7), the extra term in (3.7) is K1 v -- [?(6 - 56 ) cos 56 + (sin 6 - sin 6 ) For positive value of K1, this expression will have a negative value, hence this type of control will decrease the rotor speed and it will also decrease the angular swing. A phase plane plot for this condition is shown in figure 3. 3 for a value of K1 =. Olx From the above analysis it is clear that if a signal proportional to the rotor angle is utilized, such that the voltage behind the transient reactance is changed, according to eq. (3. 5), a substantial improvement in the reduction of transient oscillations will result. The improvement will depend on the value of K1 which will have to be carefully chosen.

28 rQ 0 T without excitation control Q,n a I y E I |i with excitation I /, 0 control 0) 0 0 U 1 2~40 40 Torque angle —> degrees o / to C-4'I Fig. 3. 3 Effect of Excitation Control on torque angle/rate of change of torque angle curve

29 Signal Proportional to Rate of Change of Torque Angle Assume now that a signal proportional to the rate of change of torque angle was used in such a way that E'q = E'o + K2 dt in q qo 2 dt which K2 is a design parameter. dt is in radians/sec. 2 dt A great advantage of this type of control is that is is possible to design the parameter K2 such that, for maximum value of transmitted power, the transient is controlled in a dead-beat nonoscillatory fashion. This can be readily observed from figure 3.4. At point a' the value of dt= 0 but very soon dt becomes dt dt positive, and the amplitude of the power angle curve continues to increase rapidly and the operating point a' moves along through b' c', d' and reaches e' which is the new operating point. It should be kept in mind that the maximum value of d6/dt is at d' and while above the Pi line, it steadily becomes less. Hence if the value of K2 is chosen properly, the movement from a' to e' can be non-oscillatory. Consider eq. (3.3) again and substitute the new value for E' d6 W dw _d c IM Pi - [Eqo+ K2 dt ] sin 6 + k sin 26 or Vb d6 [k sin 26 + [Eqo 2 d ]sin6]d6 = Md since d6 = c dt

30 area aad'= area do'e final input a d initial input dt torque angle Fig. 3. 4 Effect of Excitation Control on operating point

31 Vb [ksin26 +P. [E' K2W sin 6] d6 = Mwdw 1 X qo or vb, dw Pi- X [E qo+ K2] sin 6 +k sin 26 Mw =[A -B [C + K2w] sin 6 + k' sin 26] -1 (3. 8) where A, B, C and k' are constants. The variables of this equation cannot be separated and hence another method will be used to draw the phase trajectory. The Method of Isoclines0 The method of isoclines can be illustrated by considering the following equation dy Q (x,y) 3 P(x, y) dy There are three variables x, y and - in this equation. dx dy is the slope of the curve (x,y) in the x,y plane. Asume that dx the slope is constant and equal to N1, then Q (x, y) -- = N (3. 10) P(x,y) 1 which is a function of x and y only. This curve is called an "isocline". If a number of isoclines are drawn, the phase trajectory canbe constructed from the given initial conditions by drawing straight line segments directed according to the slope of the isoclines.

32 The method of drawing isoclines is illustrated in figure 3. 5. The initial operating point is A. The phase trajectory should cross isocline N1 at a slope N1. Therefore one dotted line is drawn at a slope N1. But the trajectory should cross isocline N2 at a slope N2. Accordingly another dotted line is drawn at a slope N2. Point B is therefore chosen as the mid point of the segment intercepted by the dotted lines. Similar procedure is followed in locating points C and D. It may be pointed out that this is an approximate procedure to draw the trajectory but the error is very small. A great advantage of this method is that a number of phase trajectories may be drawn with little more effort than is required for a single trajectory. Consider eq. (3. 8) again, dw={A-B(C+K w)sinS+ k sin 2S} / o d8 2 Let d= d- = N1 = a constant d S 1

33 I2 Initial 2 Point IN I Rate of B Change of N2 Torque/ / Angle Torque Angle ---- Fig.3.5 Construction of a Phase Trajectory using the method of Isoclines.

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36 ^ ^I'EEEEEEfill EE1I111il^ 18 0 0 1;,0j5 AL~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

37 If N1 is substituted for dwin eq. (3. 8),then the following d6 equation is obtained. w N1 = A-B(C + K2 w) sin 6 + k'sin 26 w(N1 + BK2sin6) = A-BC sin 6 + k' sin 26 or A-BC sin 6 + k'sin 26 ( N1 + B K2 sin 6 (311) Equation (3. 11) is the equation of the isocline. It would be possible to draw a set of isoclines for different values of N and to construct a phase trajectory from them. If K2 is varied its effect can be conveniently analyzed from the shape of the phase trajectories. Figure 3. 6a through 3. 6c show isoclines and trajectories drawn for the parameters under consideration. From these figures the best 180 value of K2 is found to be o 01 x 18 7T Exciter Input In the previous sections, expressions were derived for the voltage behind the transient reactance to produce maximum damping0 In a salient pole machine this voltage is equal to E'q which is proportional to field flux linkages. In a practical system, E'q is controlled by means of signals applied to the regulator. It is, therefore, necessary to derive expressions for voltage input to the regulator for the conditions of maximum damping.

38 As developed in Chapter II, the relationships between E' and q the voltage applied to the exciter is as follows: For Excitation Winding (Neglecting Saturation) dE' dt _ 1 (E - E) (3.12) where E = open-circuit excitation voltage in p. u. ex E = voltage behind synchronous reactance in p. u. T' = field time constant in seconds. do E' = voltage proportional to field flux linkages. q For Exciter dE ex 1 dt T (Ex E ) (3. 13) where Ex = exciter field voltage in per unit. T = exciter field time constant in seconds. e (a) Assume that a signal proportional to rotor angle deviation is applied to the excitation system such that E'q =E'qo +K(6-6) (3.4) From eq. (3. 12)E Eq + T' E E =Edex q do dt q E,,=E' q+id (xdxd)

39 Therefore Ee q d(Xd ) + T do d E (3. 14) ex qdd Tdo dt Substituting eq. (3.4) in eq. (3. 14), the result is ex = q' + K1(6-60) + id(Xd x'd) + T'do[K dt (3.15) From eq. (3. 13) E = exciter field voltage = E + T d E x ex e dt ex qoE + K (5-) + i (xd- x') + T' K dt qo 1 o d d Tdo 1 dt +Te K1 dt (X d) dt Tdo 1 d2 dt d26 d (Te T' K1) + dt (T K + T'd K) qeo e 1 d d do dt -+ E' qo + (6-6) K1 + (Xd- Xd)i did + (xd- Xid) Te d d2 die dt d26 d6 did =A- - + A.- A + dAid~ +E (3.16) 1 2 +A2 dt 3 dt q Therefore the voltage applied to the exciter field should have signals proportional to the second derivative of the torque angle, the first derivative of the torque angle and direct axis component of current.

40 (b) Assume that a signal is applied to the excitation system proportional to dt such that E' =ET + K d (3. 17) q qo 2 dt q q d(Xd ) Eqo + dt +d(xd Eex =Eq +id(xd-xd) +Tdo dt q(3.14) Substituting (3. 17) in (3. 14) d26 E E' +K + - )i + T' [ ex =qo 2 dt +(Xd - x d + T do[K2 dt2 E =E +T E x ex e dt ex 2 d6 __ =E Kqo (x x' + K2 qo +2 dt +(Xd d)id + Tdo[K2 d 21 + T [K + (xd - X d) dt + T doK2 d qdt 2 d2i 3 3 3(Te T do K2) + w(T K2 + T'd K2) dtO e doK2+ e 2 do2 dt dt di + (xd -d) e dt + q =A4-3 +A5 2 +A6 dt +Eq ( 18) dt d d (3.18) where A's are constants. Regulator Input (with saturation) (a) Assume that a signal is applied to the excitation system such that?'q=E'qo'1(-60)8' Ee = (1 + Sd + T'do

41 xd - x q Eq+id(l + Sd eq + d( d ) + T do dt q!d x ex e dt ex [E qo K1(6-6o)](1 + Sd) +id[xd- d]+T) [K + do K1{dt } Te{K1(1 + Sd) d-t }+ Te(xd - x'd)) } d T'do e 1 2 d TI +2 dt + A3 dt + (1+ Sd)E (3.19) (b) Assume that a signal proportional to -d- is applied to E=E +T d E ~~~x ex e ~dt ex ^E = {E' +i1 d) T do2di d q = q + 1 + Sd) + Ki -d do E d E fE d26 d d 1Sid TfAE -x A i d +A2 dt +3 dt

42 =(1 +S ) ( +K + ( + T? d qo 2 dt +(Xd- Xd)id+TddoK2 dt2 dt E =E +T E x ex e dt ex =+ ( +Sd) Eqo+K 2 + (Xd - d) d + Tdo K2 } dto e do 2 dtiL e d 2+do 2 6 dt 2 dt 3 d+ ( dd d6d + Te(l1 +Sd)K2-2 + (X d X'd) dt +T do K2 3 dt dt d6TeTdK d 6 e Tdo K2 + 2 T (1 + Sd) K2 + T'd K dt dt (1+ e d) K2 d + Te(Xd- x-d) qdt + (xd, Xd) id +(1 + Sd)E'q 3 2 d 6 d T dt dt did + T (xd- Xd)'d +(1 + Sd E E Amplification Factors (without saturation) Two modes of controlling transients have been developed. In one ~,2 ~2 mode, signals proportional to and are needed and dt 2 dt dt amplification factors in various channels are d26 t2 amplification factor A1= T T'do K1 dt2 dt amplification factor A2T itdo Ka

43 did dt amplification factor A3 = (xd - x d) Te. dt e In the second mode, which results in a dead-beat transient control, for maximum transmitted power the amplification factors are d35 d-3 amplification factor A4 = Te doK2 dt d2 d- 6amplification factor A5 = T K2 + T'do K2 dt di dt amplification factor A6 = (Xd - xd) Te. (with saturation) First Mode d2 d2 amplification factor A1 = Te T'do K1 dt dt amplification factor A2 = T K + K (1 + Sd) dt 2 do 1 e 1 d di dt amplification factor A3 = Te(xd - x'd) Second Mode d36 t3 amplification factor A4 Te Tdo K2 dt d26 t2- amplification factor A5 = T(1 + Sd) K2 + Tdo K2 di dt amplification factor A6 = (xd - x'd) Te.

44 Conclusion An analytical method has been developed to determine the correct magnitudes of feedback signals, in excitation systems, to produce the desired control of swings. This information will now be applied, in Chapters IV and V, to a typical synchronous machine system.

Chapter IV EXCITATION SYSTEMS (DIGITAL COMPUTER SIMULATION) Introduction The theory developed in Chapter III is applied to the system of figure 2. 1. The equations describing the system were developed in Chapter IL The excitation system utilized auxiliary derivative signals as developed in Chapter III. The system was simulated on a digital computer. For the investigation of the transient stability of the generator, several tests may be applied but all are concerned with the behavior of the system subsequent to a sudden disturbance. The latter may be a fault on a transmission line, the dropping of a line section, a sudden application of load or a change of input power. The change in input power is the method used in this study. The generator is adjusted initially to operate at rated power output and terminal voltage and is then subjected to a sudden increase in the power input. Transient stability is said to exist, if the machine regains a state of equilibrium after such a disturbance. Excitation Control In Chapter III two modes for input voltage to the exciter field were developed which are as follows: 45

46 Mode I d26 d6 d id Ex = A1 + A2 dt +A3 +dt Eq x 1 -, ^ 4 dd tt d t Mode II 3 2 di d6 d6 did Ex = A4 v + A5 dt +I A dt + Eq dt dt+ There are some practical considerations in applying the above theory to actual cases, as can be observed from the following: a) There are upper and lower limits to the voltage that can be applied to the exciter field. b) In actual cases it will be necessary to provide a signal proportional to the difference between the magnitudes of the terminal voltage and the desired terminal voltage. c) Circuits have been designed to measure quickly and with reasonable accuracy the phase angle1. 5 Researchers have experimented with signals based on speed, from which it appears that it is possible db d26 to obtain with reasonable accuracy -d and dt2 signals. It is not, however, certain as yet whether a d36 noise free 3 signal can be obtained in actual practice. dt At this stage it was therefore decided to restrict the dI d2d computer investigation to application of dr- and - 2 dt dt2 signals only, or Mode 1.

47 d) The influence of saturation depends largely on the shape of the saturation curve. In Chapter nI and III, machine equations were modified to take saturation into account. So that the true effects of derivative signals are not masked, it was decided to neglect saturation in the computer study. Other investigators have followed the 3-5 same procedure. It can, however, be taken into account if desired, by using the modified equations of Chapters II and III. System Equations v E' v 2( -x ) VbE q vb (X xq Pu = qsin 6 + sin 26 (2. 15) xd e q e de E' (Xd+ Xe) q(xd+X) (X -X') E = d - vb cos q d (2.22) q (xW+xe) (x d+e) dE ex = (E -E ) (2.26) dt Te x ex dt e The exciter field voltage is E and by equation 3. 16 d26 d6 did E= A1 dt2 2dt 3 dt +q (3.18) In the steady-state condition Exo = Eqo where subscript o denotes the steady-state value in the pre-fault state. On account of automatic voltage regulator requirement, a signal proportional to (v -v ) will be inserted into the exciter.

48 Therefore 2 did d26 d di =E+ A +A +11(v, -V ) Ex Eq 1 d2 +A2 dt +A3 dt v(vr vm) dtt where v is the gain in the voltage channel. The above equation can be written as follows: __ d2 did Ex Eqo + (E - Eo) + A1 d2 A2 dt + A dt + I(Vr-Vm) did In the above equation the terms (Eq - Eqo) and A3 dt are small d2 d5 as compared to A1 2 and A2 dt' If these are neglected, for simdt plification, the equation reduces to E =E +ALd6+A (V"- V x qo+ 1 d2 +A2 dt +vm The input signal to the exciter is, therefore, d2 d5 Ex = Exo + A 2 + A2 dt + v(vr - ), (4. 01) dt since E = E as shown in Chapter II. qo xo In eq. 4. 01 A2 = K1 (T'd, + Te) and A1 = Te T' doK as derived in Chapter III.

Chapter V EXCITATION SYSTEMS (DIGITAL COMPUTER RESULTS) Introduction It is desired to study the transient stability of a synchronous generator, connected to an infinite bus, by means of a double circuit transmission line, when the input power of the generator is suddenly increased, The results given in this chapter were obtained by simulating the system on a digital computer. Input power was increased suddenly until the system became unstable. The system was simulated with a conventional Automatic Voltage Regulator (A. Vo R ) to maintain the mrachine terminal voltage constant. Curves were drawn from the data abtained with and without the use of auxiliary signals to make a comparison between the two caseso In this particular study, transient instability occurred when the input power was suddenly increased by 40 percento System Equations and Constraints The investigation is based on the equations developed in previous chapters and for ready reference they are reproduced hereunder: v E' v2 (x' - X) bE' b d P sin 6 + sin 26 (2.15) u Xd+X2(x +X d e q e ae E' (Xd+xe) (x -X ) E = d - vbcos 6 (XqX') (2.22) ^(Xd+xe) c(X'+ x4) 49

50 dE' dt T'd (x -E) (2. 5) 0 dE et (Ex -E ) (2. 26) e d6 d6 r + ~v (v -)A Ex= Exo + jv, (Vr Vm)+ A1 d2 A dt (4. 01) dt A1Te Tdo K' A2 = (T'do + Te)K (3. 16) A typical machine was selected for the investigation with the constants as given below. All values are in per unit system. x =1.25 H =3.0 x =.70 T' =5.0 d q do x.i 28 T = 15 x =.5 d e e Exciter Field Ceiling voltage Max = 3. 5 Min = -1 The following operating conditions are assumed. Pu = 1., vb = 1. 0 vm = 1. 18 f = 60 c/s. Power Factor=l. 0. The value of K1 used in eq. (3. 16) was. Olx 180 This value was JL~~~~~~~ ~~7T selected by plotting eq. (3. 7) for this system as shown in figure 3. 3. By calculations, the initial values of E'q, 6, Eqd9 and E were found as follows: E' = 1.245 6 =50.2~ Eqd = 1. 562 E = 1.986 q qd. q A2 = 01(5 + o 15) = 0515x 18 A =.01( 5x. 15) =.0075x 180 2^~ 7T 1 IT The computer program is enclosed with the appendixO Computer Results: Figure 5. 1 shows the Swing Curve, when the input power is raised by 30 percent. It can be seen that the auxiliary signals have a significant

51 damping effect on the oscillations. Figure 5. 2 shows the variation of terminal voltage with time during a transient. Without the derivative signals, the terminal voltage is higher but it has large fluctuations. With the derivative signals, the fluctuations have almost disappeared although the magnitude of the voltage is slightly less. The terminal voltage begins to rise as the oscillations of the rotor disappear. It is, therefore, observed that the quick damping of rotor oscillations is achieved at the expense of slight drop in terminal voltage during the transient. Figure 5.3 shows the variation of field voltage with time. When auxiliary signals are used, the machine field voltage undergoes rapid changes, even reversing at a particular instant. This shows that to attain quick damping of rotor oscillations, the field voltage should be increased quickly to the maximum positive value reversed and again increased in the positive direction. This is an application of the classic "Bang-Bang" principle. Figure 5. 4 shows two swing curves drawn when the input power was 1. 4 p.u. Both curves are for a system which includes the normal Automatic Regulator Action. It is observed that the auxiliary signals have significant effect on the damping of the oscillations.

52 Figure 5. 5 is a graph of exciter field voltage and time. If there were no ceiling limit on the exciter field voltage, then the magnitude of this voltage would rise to as high a value as 9. 5 per unit. Dut to the ceiling limit, it does not go above 3. 5 per unit. The results of this limit is that the damping of the oscillations is not as rapid as it would be if there was no ceiling on the exciter field voltage.

53 Input Power 1. 3 Normal A. V. R in operation with auxiliary signals oo O O / so without auxiliary signals!' _, I I 1 0.2.4.6.8 1.0 1.2 1.4 1.6 1.8 2.0 Time-secs Fig. 5.1 Swing Curves

54 without auxiliary signals C4 bb with auxiliary signals Cd 4-> 00 -.2.4.6.8 1. 1.2 1.4 1.6 1.8 2. Time-sees Fig. 5. 2 Sending End Terminal Voltage

55 - a\ A \& \- - - - <S r4o U C) Cd >. 0 0 U)P~ ~ ~ ~' - ~...4 SN > - a I'-.. 0 7L) - - Z) \ ~ Z0'-4 / \ / ^ ^>.nd^ w1op~i "^. ~ ~~~~ I^> If

56 / 4!'UI~\, Sr l - R *.M +.oi o X SS C LO /.?. —)Q) PO., 0 s OOI 08 09 0> s. I1^-. — 11 / * b saaojap-g Q^at2u anbio 001 08 09 O0 O o

57 oo b.OO~~~D Z% ~ 4-. lop0. $. Q^ 1'~ - - - - - _ _ 6 8 1~ 9 S, ~ Z I ~ I n'd e41IA 3Xz.,

CHAPTER VI NORMALIZED STABILITY MARGIN Introduction Since digital computers are becoming commonplace in utility system operation, there is a strong possibility that during the next decade they will be used to make decisions and initiate corrective action during transient periods. It will therefore be convenientto have a numerical technique by which relative stability of a system can be conveniently determined. Baba et al suggested the use of phase -plane technique to develop an expression for stability margin of a synchronous generator. In this chapter the familiar equal-area criterion will be used to define stability margin and an expression will be developed for what the writer will call "Normalized Stability Margin", wherein the positive limit of stability margin will be unity. The concept of normalized stability margin will be used in Chapters VII and VIII to evaluate the effect of Phase-Shifting transformers. Stability Margin Figure 6-1 shows pre-fault and post-fault power angle curves. The amplitude of the post-fault curve is less due to the fact that postfault reactance is higher, due to switching out of a line section. For simplicity, fault clearing time, being very small, is neglected but it will be taken into account later on. 58

59 According to equal area criterion, if area A2 > area A1, the system is stable if area A2 < area Al, the system is unstable. 62 Area A = S (P1 - Pm2 sin 6 )d = P1(2 - 61) 6 + m2(cos6 2 - cos6 1) 7r -6 AreaA = 2 (P 2 sin 6 - P1 )d6 = Pm2(os62 + cos62) - P1(r -262) Therefore A2 - A1 = Pm2(os 62 + cos 61) + P(6 +62 - r) (6. 1) It will be convenient to normalize expression 6. 1 so that maximum stability margin will be unity which will occur at no load. The lesser the load, the larger will be the area A2 and smaller the area A1; hence, maximum value of (A2- A1) will occur when P1 = 0 It will be convenient to designate the maximum value as unity; therefore, if the expression 6. 1 is divided by maximum value of A2 (or total area under the post-fault curve), the expression is normalized. The area under post-fault curve is equal to 2P Therefore normalized stability margin 1 P1 2 {(cos 6 + cos 61) + P (6 + 2 - 7) (6.2) m2

60 pm. —_- _ Prefault Curve ml A i ~/ \^ ~ ~Post-fault Curve Pm2 Input P 1 — m 6.1 62 Fig.6. 1-Determination of Stability Margin

61 Derivation of Formula for Stability Margin when Fault Clearing Time is Taken into Account Let Pml' Pm2 and Pm3 be the maximum power transferrable ml m2 m3 under pre-fault, during-fault and post-fault conditions. Refer to, figure 6. 2. Let P be the input power in this case. 62 6 AreaA = 62 (P - Pm2 sin 6)d6 + (P - Pm3 sin 6)d6 61 62 = P(6 2-6 1) + P2(os 6 2-cos 6 1) + P(6 3-6 2) + Pm3(cos 63 - cos 62) - P(6 3-6 1) + P2(Cos 6 2 - cos 6 1) + Pm3(COs 6 3-cos 6 2) 7r-6 AreaA2 = S [P m sin - P]d5 m3 - P 3[cos(7-6 3) - cos 6 3] - P[ T - 6 - 6 3] =2P3 coS 63 - P[ - 263] Therefore A2 - A1 = P[6 3+6 1-7r] + Pm2[cos 6 -Cos 6 2] + Pm3[cos 6 3 + cos 6 2] Dividing it by area under post-fault curve equal to 2 Pm3 1 Pm2 Stability Margin = Pcos 6 2 + cos 6 3] + 2p-[cs 6 1 - os 6 2] r3

62 Pmi Prefault P Post-fault During-f ault 1-1 gt 1 62 Fig. 6.2 Pre-fault, During-fault, and Post-fault Power Angle Curves

63 Example 6.1 The following example illustrates the application of the concent of normalized stability margin. The equations describing the power flow from a generator are Before the fault P = 1. 735 sin 6 o During the fault P =.42 sin 6 After the fault P =1.25 sin 6 0 where P is the power output from the generator. 0 Assume fault removal when 6 = 45. Calculate the stability margin, if generator is delivering 1. 0 per unit power. -1. 10 P 1= 17356 = sin 1. = 35.2 =.615 radian ml' 1 1.735 0o Pm2 =. 42 6 = 45 =. 785 radian P3 = 1.25 6 =sin 1 3.0 = 53. 1 =.928 radian Pgm3 1. ^3 1.25 Stability Margin = {. 60 +.707}+ 4250 {. 818 -.70} 2 2.50 + 2.50 {.61 +.928- 3. 141} =.20 unit. If the fault is removed at an angle 6 = 51. 6, then stability margin 1.42 1.0 615928 = 2 {.60 +.622}+ {.615 +.928 - 2. 50 2. 50 - 3. 141} = 0.0 per unit Hence this corresponds to critical switching angle.

Chapter VII PHASE SHIFTING TRANSFORMERS (BASIC ANALYSIS) Introduction Another method of improving transient stability in a power system is investigated. This method involves using phase-shifting transformers to shift the power angle curve by a certain angle to produce the desired control of swings. The phase-shift windings of the transformers can be short time rated which will reduce their cost. Principle of the Method Figure 7. 1 shows a generator G supply power to an infinite bus through a double circuit transmission line. On the sending end there is a transformer T which has a phase shifting arrangement as shown in the figure. A is the main breaker, while B is an additional breaker. The two breakers have an interlocking device so that at any time when one breaker is closed, the other is open and vice versa. The generator is supplying power P1. The voltage behind transient reactance of the generator is E' and that of the infinite bus is vb. X q b is the sum of the line reactance, the transformer reactance and the transient reactance. Therefore E' vb P= q b sin 6 1 where 6 is phase angle between the two voltages. 64

65 Infinite Bus Circuit Breakers Line Reactances Circuit Breakers -Circuit Breaker A Circuit Breaker B T A^NAn^V^^^V Phase Shifting Transformer G X j? I Synchronous Generator Fig. 7. 1 Application of P. S. Transformers in a Typical System

66 Pmi Pm2 I I I I'4 62 7f-62 Phase-shift' 62 r62_ Phase-hift Torque Angle -- - Fig. 7. 2 Effect of Phase Shift on Power Angle Curve

67'he usual power angle curve is shown in figure 7.2. A fault occurs on one of the lines which is switched out due to the action of the fast-acting breakers. The loss of the line increases the reactance between the two voltages and operation moves from pre-fault to postfault power angle curve. Since the duration of the fault is very short, no "during-fault" curve is shown for the sake of simplicity. As expected, the generator rotor begins to accelerate. In the case illustrated in the figure, the system is unstable since P is greater than transient stability limit of the system. If it is desired to control the swings, breaker A opens and B closes at the same time that the line is switched out. This is not difficult, since the relays which switch out the line can open A and close B also. Due to the phase-shifting property of the windings, the post-fault power angle is shifted towards the left by an angle 0 as shown in figure 7.2. The result of this shift is the accelerating area is reduced while the decelerating area is increased, thereby increasing the stability margin. The magnitude of the swing is reduced too. If the angle 0 is of correct magnitude, it is possible to bring the rotor d6 to point o such that d- = Oat o. Under that condition, if the phaseshift is removed when the operating point reaches o, there will be no back swing. Thus the use of phase shifting transformers has resulted in bringing the operating point to o in a dead-beat non-oscillatory manner.

68 Phase-Plane Analysis of the Effect of Pnase Shifting To illustrate the basic effects of phase-shifting transformers, the phase- plane technique is used to analyze the swing equation. The following assumptions are made. 1. Damping is ignored. 2. The synchronous machine is represented by a fixed voltage behind the transient reactance. 3. Saturation is neglected. 4. The prime mover input remains constant during a swing. 5. The machine has a smooth rotor. Therefore d2 M 2 P 1-P P sin 6 (7.1) dt E' v where P = q b ml X The steady-state condition is given by E' v q b si (7. 2) X The representative point is shown in phase-plane figure 7.3, by A. A 3-phase fault occurs on the system which reduces the output to zero, hence d5 M 2 = P during the fault dt 1 M d =P t+C (7.3) dt 1

69 4^ -Ij 0 0 T~~~~~~~o~or ~,, 4 -) QP e0 5 E ap 9p a)' \ 11 o r j' 1 9P

70 t = time after occurrence of the fault d6 or Mw = P t+C, where = dt C is a constant. When t =0, = Q, therefore C=O. Therefore the phase-trajectory equation during the fault is M = P t Ilt d5 or = dt (7.4) Integrating again the above equation P. 2 = t + C1. C1 is a constant. When t =, = 60, therefore C1 =5 6. Hence P1 t2 2Mt 6 60 (7.5) Eliminating the variable t in equations (7. 4) and (7. 5), the result is P1 2 M 2 5 -6 0 2M 2 2P P1 1 2 2P o r (w M (6 -6 ) (7.6) w M In the phase-plane, equation (7. 6) is represented by a trajectory AB, which represents the "during fault" portion of the trajectory. After a time t1, the fault is removed and the swing equation becomes

71 M d26 Md2 =P1 - Pm sin 6 (7.7) dt2 = m2 E' v where P = q and X' is the post-fault reactance. m2 X' The initial conditions for equation 7. 7 are: when t = 0, 6 = 6 1' P1 M 1 w = 1where =M 1 1 2P1 + 6 f from equations 7.4 and 7.6. Assume the new steady-state point in the phase plane to be(6 2' 0). Then Pm2 sin62 =P1 Also, d6 W= dt 2 d6 _ dw 1 - sin 6 dt2 at M P1= m2si dt po 1 or dw [P - P2 sin6] o d6 2 P1 Pm2 Pm2 or wdw = [ - M sin 6 ]d 2 P P w 1 m2 Integrating, = -6 + cos 6 +C (7. 8) 2 - M M 2 C2 is a constant. For the transient stability limit o = 0 when 6 =7 -62, which follows from the application of equal area criterion to figure 7.2.

72 2 M ( -62) M [cos (r - 62)] P P P1 m2 - 6 2) + M cos 6 2 2 P P P P 2 p p m2 P1 or 16 +m cos6 + cos 6 ( - m2 - (6 - r6 + ) + (co +os +cos62) (7. 9) This is the equation of the Seperatrix. If the fault is removed while the representative point is within the area enclosed by the seperatrix, the system is stable, otherwise it is unstable. Suppose that the fault is removed when the representative point is at B, which has coordinates'l, 6 1 Substituting this condition in equation 7. 8. W1 P P - = 6 + m2 cos +C M + 2 2 61 M 1 2 w1 P P m2 ^C2 = -21 - -_- cos 6 2 P P 2 1~ (5-51)m2 Therefore -2 = -- ) (cos 6 - cos 6 ) 2 M M 2 + (7. 10) (7. 10)

73 Pmi B \^^ "\ Prefault \i p M Post-fault i I /!16 6 6 7T- 63 61 53 62 -63 Fig. 7. 4 Determination of the Effect of Phase-Shift on Stability Margin

74 This is shown in figure 7. 3 by post-fault trajectory. It will be seen that the representative point oscillates continuously around the new steady state point 62. Suppose that at the time of switching out of the line, the power angle is shifted towards the left by an angle 8. Then the swing equation becomes d2 M 2 = P P sin(6 + 6) dt2 1 m2 The apparent singular point is 63 = 62 - 0. The trajectory equation (7. 10) becomes 2 Pi P a)2 ct ~1 m2 1 2 = M ( - 6) + M (cos(6 + 0) - cos (61+ 0)) + 2 (7.11) The new trajectory is again oscillatory and is shown in the dotted lines. Figure 7. 3 shows qualitatively how the trajectories are shifted towards the left by the action of the phase shifters. If the phase shift is of such a magnitude that the point C passes through 62, then it would be possible to remove the phase shift at that instant and the representative point will stay there without further movement, so that the transient is controlled in a dead-beat manner. Effect of Phase Shift on Stability Margin In chapter VI, the concept of stability margin was introduced as it gives a quantitative idea of stability. Referring to figure 7.4 Stability Margin _ area b'd'c' - area o d _b' with phase shifting - Total area under post-fault curve (7.12)

75 Stability margin without phase shifting area bdc - area oab 1 Total area under post-fault curve Therefore increase in stability margin = (7. 12) - (7. 13) area oab - area oa'b' Total area under post-fault curve (since area b'd'c' = area bdc) area oab = g 2(P1 - Pm sin 6) d 1 P (62 61) + P 2(cos6 - cos 6) Similarly area oab' = P1(63 - 61) + P 2(cos 63 - cos 6) Total area under post-fault curve = 2 Pm2 approximately with and without phase shift. P1(62- 63) + Pm2(Cs 62- cos 63) Therefore increase in stability margin = 2 P2 m2 Since 63 = 62 - 0 P1 0 + Pm2(Cos 62 -cos(62 - 0)) Increase in stability margin= - cos( 2 Pm2

76 Pml P m2 0 Fig. 7. 5 0 for Dead-Beat Control TOrque Angle 6 Fig. 7. 5 0 for Dead-Beat Control

77 Determination of 0 for Dead-beat Transient Control Figure 7.5 shows the usual pre-fault and post-fault power angle curves. The apparent power angle curve, while the phase shifters are energized, is shown in the dotted curve. It is desired to find the value of 0 which moves the representative point from a to e in a dead-beat non-oscillatory fashion, if the phase-shift is removed when it reaches d. The criterion for the above is that the area abc = the area cde such that when the representative point reaches d, its velocity is zero. If the phase-shift is removed at d, it will occupy point e and stay there permanently, since power input equals output. Therefore, 6 [P1- Pm2 sin(6 + 0) ]d6 = 2 [Pm2 sin(6 + 0) Pl]d6 or P1(6 3 - 1) + Pm2 [cos(6 3 + 0) - cos (61 + )] = P C (62+) - cos( (63 +0)] - P1[62 63] or P [6 2 -6 1] = Pm2 [os (61 +0) - cos ( 2 + 0)] P [6 or [6 2 - 6 1 = cos 6 1 cos 0 - sin 6 1 sin 0 - cos 2 cos 0 p L 2 -1 1 m2 + sin6 2 sin 0 cos 0 [ cos 61 - cos 62] - sin 0[sin 6 - sin 62]

78 cos 6 - cos 6 = A sin a 12| cos 61 - cos 62 Let or tan a = 2 — and sin 61 - sin 62 sin 61- sin 6 A cos a A2 = (cos 6 - cos 62) + (sin 6 - sin 62) 2 6- 6 =4 sin2 ( 2 61 -6 or A = 2 sin ( 2 Then P P [[62 61] = [cos 0 sin a - sin cos a]A m2 = A sin (a - 0) - i1 a- 0= sin A P [62- 51] A m2 -1 P or 0 = a- sin A Pm [62- 61] - cos 61- cos2 61 P or 0tan sin 6 - sin 62 s A P [62 - ] (7 16) 1 2 m2 Increase in Power Transmitted Figure 7.6 represents a pre-fault, a post-fault without phase shift and a post-fault with phase shift power angle curves. Without phase shift control, the maximum power transmitted will be P1

79 P ~ ^ Pre -fault ml/~ ^\'Pm2 P; P "-1 _ Post-fault'/ b Id' \ \ / l i _ Phase-shifted 0~~ & Post-fault i //^i | 6163 6 -63 Torque Angle 6 Figure 7. 6 Determination of Minimum 0

80 which is determined graphically, satisfying the equal area criterion, such that area oab = area bcd. It is desired to transmit power greater than P1 over the system by employing phase shifting technique. Suppose it is desired to increase P1 to Pi, then the power angle curve needs to be shifted by an amount 0 such that area o'a"b' = area b'c'd' which is possible for some value of 0, as can be observed from the figure. Thus the phase shifting transformers have made it possible to raise the power transmitted to a level more than the transient stability limit P1, in the case under consideration. The maximum value of the power transmitted is Pm2' the steady-state limit in the post-fault condition. It will not be practicable to raise the power to this value. The maximum value of the transmitted power will depend upon the maximum angle 62 desired in the post-fault state, which should be less than 900. Hence if 62 be the maximum angle desired, then P1 = P sin 62 The phaseshifting transformer windings will be designed in such a way that power P1 can be transmitted. The minimum angle of phase shift will be necessary if area o'a"b' = area b'c'd' or that full areaabove the power line is used in decelerating the rotor. Referring again to figure 7.6, area o'a"b' = area b'c'd' or 6 ~rv-5 -8 f3 [P1 - Pm2 sin(6 + 0) ]d5 = P 3 [m2 sin(6 + 0) - P1]d6 This gives This gives

81 p1 pm [7i - 61- 3] = [cos(61 + 0) + cos 63] m2 -0 But Pm2 sin 6 = P sin 6 m2 2 ml 1 and 62 = 63 + Therefore p p2 [ - 61 62]= [cos(61 + 0) + cos(62- 0)] (7.17) P1 p [T - 61 - 6] = cos 61 cos 0 - sin 61 sin 0 + cos 62 cos 0 + sin 62 sin 0,cos= fc os 66 c os c + [sin 62-sin 6 ] sin 0. 1m^j 26 Let cos 61 + cos 62 = A sin a cos 61 + cos 62 or tan= sin6- sin sin 62 - sin 6 = A cos a 2 1 -1 cos 6 + cos 62 or ac tan sin 6 - sin 61 2 1 and (61+ 62) A= 2cos 2 Therefore p! p1 [ - 51 6 2] = A[sin(oa + 0)] m2

82 or Pi [i - 6 - 62] cos 6 1 + cos 6 6= sin~ (61+62) - tan sin 2 - sin 61 P 2 cos 2 Application to a Typical System Chapter VII gives details of an actual system and computer details when the technique of phase shift control was applied to a typical system,

Chapter VIII APPLICATION OF PHASE SHIFTING TRANSFORMERS Introduction This chapter gives details of an investigation of a typical system using the method of phase-shifting transformers as developed in the preceding chapter. The study which was done by digital simulation indicates that there is considerable potential in this technique and it is hoped that the electric companies will be encouraged to explore its application to their systems. System Studied The system consists of a synchronous generator supplying power to an infinite bus by means of a double circuit transmission line, as shown in Figure 8. 1. A fault occurs at point P which results in opening of the circuit breakers adjacent to P thereby raising transfer reactance between the generator and the infinite bus. The equations for the power output of the generator are: Before the fault Pml sin 6 = 1. 735 sin e During the fault Pm2 sin 6 = 0O 42 sin 6 After the fault Pm3 sin 6 = 1. 25 sin e -1 1. 15 Input Power = 1. 15, hence 6 1 = sin 1 = 41. 4~, r - 62 =sin - 1 =113~. The power angle curves are shown in figure 6. 2. 83

84 Synchronous Generator -' m, ~._ r Inf.Bus Phase-shifting Transformer Fig. 8. 1 A Typical System Using a P. S. Transformer

85 Modern beakers are capable of operating in 3 cycles (clearing time. 05 sec.). From standard curves, the clearing angle for this case is 45. 3. A computer program was written to calculate the data for the swing curve. The constants of the generator were as follows: H =3.0 G = Therefore M= GH = 2. 78 x 10 per unit 180 x f A time interval of 0.05 sec. was selected for the calculations. Figure 8. 2 shows the results of different values of phase shift on stability of the generator. The figure shows three curves obtained for phase shifts of 0, 5, and 10~. The effect of 10~ phase shift is to decrease the maximum torque angle from 85~ to 68~, a decrease of 17. Similarly, the phase shift of 5~ decreases the maximum torque angle by 10~. Therefore the amount of phase shift has considerable effect on the shape of the swing curve. The computer program and the flow chart are enclosed in the appendix. It may be mentioned here that there are many appli cation problems to be analyzed before the techniques of phase shifting transformers can be used in actual systems.

86 0o Lt",-. _ I l II,c: ) Q -. ~I Cai -) 0 4) go I, XI p/ 61 4) Cl, \/ / / /'~ 0 0 "06 08 L 09 0' 0v 0~ z 0 s- a a IIanb o, ~'~\ *' 0' =- I " ~"..''.'"I 06 080/ 09 Og 0i' Oo 01 0 ^ IL fs) CO 0$~~~~~~~~~~~~a saaejSaa ^ITaa ai~uvT anbjxoj

Chapter IX SWITCHED SERIES CAPACITORS (APPROXIMATF ANALYSIS) Introduction A very effective method of raising stability limits of long transmission lines is by means of fixed series capacitors. There are some excellent papers on this subject in the literature. In 1965 Kimbark6 suggested the use of switched series capacitors which involves switching in a capacitor in series with the line as soon as a line section is switched out. The idea behind this scheme is that by switching in a capacitor, the increase in line reactance, caused by switching out of a section, is minimized. Kimbark's paper was followed 12 by that of Smith. He suggested that capacitors be switched in and out at appropriate instants to control the transients, the exact instants would be determined by a centralized control computer. Scope of Investigation Chapters IX to XI of this report relate to switched series capacitors as shown below: Chapter IX. This chapter is concerned with an approximate analysis of the fundamental effects of switched series capacitors. As 6 12 in Kimbark and Smith's papers, equal area criterion is the method used in the investigation. It is shown that the capacitive reactance needed to maintain stability can be minimized if dead-beat operation is 87

88 not desired and formulas are derived for capacitor sizes under different modes of operation. A simple system is simulated on a digital computer to find the effect of switching in different sizes of capacitorso Chapter Xo This chapter is a detailed report on some experimental work performed in the Power Systems Laboratory of the University of Michigan. The experimental results are compared with the results obtained by digital simulation of the system0 Chapter XI. This chapter is concerned with the study of the following problems arising in application of switched series capacitors. i. Effect, on system stability, of a secondfault which may occur while the capacitor is in circuit and the machines are swinging. ii. Effect of Switching in a series capacitor on a circuit with distance relaying Approximate Analysis The aim of the approximate analysis is to investigate the fundamental effects of switched series capacitors. The investigation is restricted to a system which has two similar parallel lines such that the reactances of the two lines are equal. Figure 9. 1 represents a double circuit transmission line connecting a generator to an infinite bus. X is the reactance of each circuit. X is the reactance of the switched series capacitor. Circuit C

89 Syn. Gen. x B1 x1 C, X Inf. Bus Et -S 1CB4 E2 Fig. 9. 1 A Typical System Using Switched Series Capacitors I IP 1 ( ml ~~~~~~pm X I m2 P, 0a I l S6 ~6 Torque Angle 6 Fig. 9. 2 Power Angle Curves for the Above System

90 breaker S is normally closed and it opens when any of the circuit breakers CB-1 through CB-4 opens. It is assumed that the same relay initiates the tripping of the breakers S and any of the breakers CB-1 through CB-4. When a fault occurs at point F, circuit breakers CB-1 and CB-2 trip, raising the line impedance from Xl to x1. Since circuit breaker 2 S opens, the net reactance between A and B is raised from x1 to 2 x1-x. It is therefore clear that if the value of x is chosen to be 1, 1 ~~~~~~~~~~~~~~~c c 2 the line reactance will remain unchanged although one circuit has been relayed out. It will however be shown that it is not necessary to have the x1 capacitive reactance x equal to -, since a significant improvement in c 2 the transient stability limit will be achieved even when x is much lower xl than Without Switched Capacitor Figure 9. 2 is the power angle diagram for the above system. Sine-curve no. 1 represents the prefault curve and no. 2 the post fault curve. The during-fault curve is neglected since fault duration is very small. P. and P2 are the amplitudes of the power angle curves given by E' E E' E2 -- dq2 q 2 ml xd + 0. 5 x m2 + x1 where E' is the voltage behind the transient reactance of the generator E2 is the voltage of the infinite bus x'd is the transient reactance of the generator x1 is the reactance of each line.

91 Since it is an approximate method, the assumption will be made that x'd is small in comparison with x1 and hence may be neglected in the derivation to follow. Let Pm be the transient stability limit derived on the basis of the equal area criterion such that area abc = area cod. P can then be determined from the following equation Pm[7 62 - 6 1] = Pm2[Co 5 2 + cos 6 1] (3. 1) where 61 and 62 are given by the following equations -1 Pm -1 m P. 6 = sin Pml 5 = sin m2 Switched-Series Compensation The effect of switched series compensation can be seen from figure 9. 3. It is assumed that the circuit breaker S is operated by the same relay which operates either circuit breaker CB1 or CB2, thus the capacitor x is inserted as soon as the faulty circuit is taken out and reactance will c X1 to ( - Consequently, the operation will be increase from n to (xI -x ). Consequently, the operation will be transferred from curve 2 to 2'. The operating point moves from 1 to 2 and 2 to 3. The rotor starts to advance along the dashed curve from 3 to 4 at which point

92 d6 - is maximum. From 4, the operating point moves along the dashed curve to 5 at which point d6= 0o If the value of xc is chosen such that 5 and 6 lie on the same vertical line, it is sufficient to remove the capacitor at 5 and the operating point will move from 5 to 6 and stay there, since 6 is the steady-state point in 12 the post-fault condition. This method was originally suggested by Smith. The operation has therefore moved from 1 to 6 in a dead-beat, non-oscillatory fashion with the aid of a switched-series capacitoro The value of the capacitor which will be required for the process will be unique for the loading Pm It can be argued from the above that, by means of a switched series capacitor, it will be possible to increase the transient stability limit up to the value of Pm2, with a maximum value of 6 = 90~. This is indicated in figure 9. 4. However, it would not be practicable to operate at 6 = 90~ and, therefore, the transient stability limit will be less than m2' Selection of Capacitor Size There are two criteria for selecting the capacitor size as follows: (1) To produce dead-beat non-oscillatory transfer from prefault to post-fault state. (2) To maintain stability but with operation which is not deadbeat. (This will result in a smaller capacitor size. )

93 \Pt 2 Torque Angle 6 Fig. 9. 3 Effects of Switched Series on Power Angle Curves Pant P44 /- ^ / I I $~ 62sz=90 Torque Angle 6 Fig. 9. 4 Series Compensation for Maximum Transient Stability Limit 3~~S 1^9< Fig. 9. 4 Series Compensation for Maximum Transient Stability Limit

94 1. Dead-beat operation. Refer to figure 9. 3. Select 62 to be the maximum angle at which operation in the post-fault state may take place. Then sin 62 = Pm/ Pm2 or Pm = Pm2 sin 6. Thus P is the maximum amount of power which is required to be transmitted along the lines. -1 Pm 61 = sin Pml 622 6 P2- - Pm62-6!] =-^ 6 sin6 d6 =-PI 2[cos 6]6 =P'[cos 61-cos 62] m[62 -61] =[cos 1 -cos 62] P2 P [62 -61] [cos 6I - cos 62] Eq _ P ol 61] cx [cos 51-cos 62] where x' is the total reactance when the capacitor is in the circuit.

95 Eq 2 x1 Pm[62 61] xl X2 [c6 C6 - cos 62] Pml X1 Pm [62 - 61 T = X2 [cos6 -cos - cos 2] Pml x [cos 6 - cos 62] x2 2P m [62- 61] But x2 = x - xc pml x (cos 61 - cos 62] or - x = P 1 c 2Pm (62- 61) Pml (co-cos - cos x = X- iC Pml 2 i. c 1 2P 62 J m -P2 P 2 2 1 1- mi 1 2 - - 22 2 FD rP P ml P ml Pml m 4 m2 m11 m 2Ppm2 X1'2Pm 6 -1 m 2 - sin 2ml Pml m

96 14E E2 K m2 2 2 m 2 m E' E x xl x 1 q 2 1 X X 1I X1 Pm 2 E E' E E Px x 62- sin 2E E q 2 Since Eq, E2, x1, 62 and Pm are known, the value of capacitive reactance can be determined from the above mathematical relation. 2. Lowest Bound on the Capacitor Size. As before 6 2 is selected as the maximum operating angle in the post-fault state. Instead of being dead-beat the rotor is allowed to swing up to angle a < vT - 62, see figure 9. 5. The operating point moves from 1 through 2. 3. 4 to 5. At this stage the capacitor is taken out. The operation continues along curve 2 from 6 to 7 to 8 such that area 607 = area 789. The rotor oscillates around position 7 and finally settles down at 7 corresponding to angle 52. The value of x which corresponds to the case of the rotor swinging upto an angle a is calculated on the next page.

97 As before, with reference to figure 9. 5, using the equal area criterion. 6' J62 (m -Pm sin 6)d = (P2 sin 6 - P) d6 1 r m22 or Pm(6 I - 6 ) + P2 s (cos 6 1) P 2 cos2 cos -s 5m( 2 P (6 6 + a - ) = P2 {cos 6 cos a - cos 6 + cos 6 Pm(-6) = P'm2(Cos 6 1 - cos a) (a-6 1) m2 m cos 6 1-cos a E E2 (a -61) X2 cosm cos-cos a E qE2(cos 6 1 - cos a) x 2 (a -61) X Xc Therefore E qE2(cos 61 - cosa) xc 1 P ma - 6 ) If in the limit a is allowed to approach a value of r- 6 2, then the value of x in the above equation is the lowest bound on te se of te caacir fr s m s y. on the size of the capacitor for system stability.

98 Application to an Actual System Figure 9. 6 represents a single machine connected to an infinite bus through parallel transmission lines. This system is studied with respect to the influence of switching in a capacitor in series with the transmission line. The numbers on the diagram indicate the values of the reactances in per unit. The breakers adjacent to a fault on both sides are arranged to clear simultaneously. Resistance and shunt capacitance are neglected. The following data are provided: i. voltage behind transient reactance = 1. 25 pu ii. voltage of infinite bus = 1. 0 pu iii. power transferred = 1.0 pu. The transient starts due to a 3 phase fault at P which is cleared by opening of the circuit breakers adjacent to P. Positive sequence impedance diagram for the system is as shown in fig. 9.7. Amplitudes of the power angle curves are: (See figure 6. 2) 1. Ox 1.25 (A) Before fault = 72 = 1. 735 (B) During fault = 1. x 125- 0. 420 2.98 1. Ox.25 (C) After the fault = 1.. 2 1 25 1.00 Pm2.42 therefore m2. = 242 Pm3 _ 1.25 Pml 1.735=

99 on I -j Torque Angle 6 Fig. 9. 5 Determination of Minimum Capacitive Reactance.16.24.16 ~.28r""-0 —- —. i P 24 gt o16.16 Fig. 9. 6 Single Machine System

100 -si1 1. =35. 2=. 615 Radian 6. -1 1.(0 7r- 6 3sin1 1.0 = 126.9~= 2.22Radians ~1. 25' cos ( 17.5 ) (2. 22 - 0. 615) +.72 cos 126.9 -.242 cos 35.2 62 (critical) = 0.72 -.242 = 51.6~ Since 3 cycle breakers (0. 05 sec. clearing time) are commonly used, it would be interesting to find out if the power level can be raised above 1. 0 p. u. Initial angles for various power levels are as follows:? 6 0 (7T - o 0 0 ~P 61~ (n-6 3)=6 m Clearing angle 6 2 (front standard curves) 1.0 35.2 127.0 38.6 1. 05 37.2 122.8 40.7 1.10 39.3 118.0 43.9 1.15 41.4 113.0 45.3 1.20 43.7 106.5 47.7

101 o 16.24 16 o 56 1'-1-w- 11 —1 b C.16.24 16.28 16 16 409.16 1. 25 n.08 20 2,.98.,a d.28.16 7Sw-1.057 1o0 Figure 9. 7 Positive Sequence Networks

102 Knowing the clearing angle corresponding to the minimum clearing time of 3 cycles, it is necessary to draw swing curves to check whether power level can be raised from 1.0 pu. to 1. 20 pu. or an increase of 20%9 The data for the swing curves as obtained from digital calculations are given and swing curves plotted in figure 9. 8. Fault cleared in 3 cycles Power a-1.0 b-1.05 c-1.10 d-1.15 e-1.20 Time- sec 6~ 6 6~ 6 6~ 0.00 35.2 37.2 39.3 41.4 43.7 0.05 38.6 40.7 43.0 45.3 47.8 0.10 46.3 48.9 51.5 54.2 57.1 0.15 54.8 58.0 61.1 64.3 67.7 0.20 63.2 67.0 70.7 74.7 78.8 0.25 70.6 75.1 79.7 84.5 89.6 0.30 76.3 81.8 87.5 93.5 9.99 0.35 80.1 86.8 93.9 101.7 110.0 0.40 81.8 90.0 99.0 109.1 120.3 0.45 81.4 91.4 102.9 116.3 131.7 0.50 78.8 91.0 105.7 123.7 145.5 0.55 74.2 88.8 107.6 132.2 163.7 0.60 67.8 84.8 108.7 142.6 189.5 0.65 60.0 79.1 109.0 156.6 228.0 0.70 51.4 71.8 108.6 176.5 285.7 0.75 43.1 63.2 107.5 205.9 365.0

103 It is observed from the swing curves that cases d and e, corresponding to power transmitted of 1. 15 and 1. 20 respectively are unstable. So it has been possible to increase power by 10% by employing the fastest possible fault clearing. Consider the effect of adding a suitable capacitor in series as soon as the line is tripped out. From the data provided, the amplitude of power angle curve during the post-fault condition is 1. 25. Temporarily it is necessary to slightly increase this value to. create stability for power levels of 1. 15 and 1. 20. Assume that the amplitude of the power angle curve during the post- fault condition is x, and if Pm is the input power then from equal area criterion Pm(6 - 51) +.42 (Cos 62 -Cos 61) Cos 62 - Cos 6 2 m The value of x as calculated from this equation is: Pm = 15 pu x = 1.30 pu Pm 1.20 pu x = 1.36pu or ETE Eq 2 1.25x 1.0 Pm 1.15pu, - x 1.30 i _ L c c 1. 25 x =. 0- = 1.0 -.962=.038 pu c 1. 30 P= 1. 20 pu x =1.0 = 1.0 -.883=.117 pu c 1.36

104 The swing curves are again calculated after including the series capacitors in the circuit and the following results are obtained. Power = 1.15 (x =.038) 1.20 (x =.117) Time- sec 6~ o6 0.00 41.4 43.7 0.05 45.3 47.8 0.10 54.2 57.1 0.15 63.9 66.9 0.20 73.6 76.3 0.25 82.3 84.5 0.30 89.8 91.4 0.35 95.9 96.8 0.40 100.8 100.9 0.45 104.5 103.8 0.50 107.3 105.6 0.55 109.2 106.4 0.60 110.4 106.3 0.65 111.0 105.3 0.70 111.1 103.2 0.75 110.6 100.1 The swing curves are replotted on the same graph as broken lines from which the stability achieved by using capacitors can be observed.

105 00 I /o r,.,) i 0 ( QC.) P / /5'o < 0 0 -1v \ \\/3.-4 l)l \ \ ^\I c <D 3 c0: 0 CO C C \ \CD C soe:[Sep I~;)!.z3,Oee-c - s~~o~~~~ep reo1.I1~~~~~~io - Q~Q

Chapter X SWITCHED SERIES CAPACITORS (EXPERIMENTAL STUDY) Introduction This chapter describes some experimental work, on Switched Series Capacitors, which was performed in the Power Systems Laboratory of The University of Michigan. The purpose of the experiment was to make a comparison between the digital solution of the machine equations and the experimental results. It will be seen that there was fairly good agreement between the computer and the experimental results. Brief Description of the Experiment The experiment was performed on a synbhronous Generator/D. C. Motor set. The D. C. Motor which was a compound wound machine acted as the prime mover for the generator. The generator was synchronized to the A.C. Supply system through two parallel transmission lines, one consisting of reactors to simulate a long transmission system. The other a simple direct connection essentially zero reactance. Under steady state conditions, power was flowing through both lines. The direct connection was opened suddenly which created a transient and the generator began to oscillate. The oscillations were measured by a specially designed measuring apparatus. Different sizes of capacitors were introduced in the system and the oscillations were recorded. It was observed that for a certain value of the capacitive reatance, the oscillations were minimal. The experimental results were 106

107 compared with the digital solution of the machine equations. Machine Specifications Synchronous Generator Make Westinghouse KVA 11 Excitation volts 125 Volts 240 Excitation amps 5. 6 Amps 26. 4 R. P. M. 1200 Phases 3 Cycles 60 Power factor. 9 D. C. Motor Make Westinghouse Compound Wound HP 15 Volts 250 Amps 52 RPM 1200 Auxiliary Generator Make Dayton Electric Volts 115 Phase 1 Amps 3. 7 Cycles 60 Power Factor 1. 0 RPM 3600 VA 1000

108 Operation of the System Figure 10. 1 is a schematic diagram of the system which was set up in the laboratory. The synchronous generator was connected to the infinite bus, which was the Detroit Edison supply in this case, through two parallel transmission lines. As shown in the schematic, line A was a direct connection but line B was made of lumped reactors to simulate a long transmission line. Line A had a contact 1 in series with it, which was controlled by coil CR1. Since CR1 was energized, this contact was closed; hence power was flowing from the synchronous generator to the infinite bus through line A. The coil CR1 had a push button'a' in series with it which was normally closed. It was pushed by hand to de-energize the coil and open contact 1. Line A was thus switched out and power began to flow through line B. Since contact 3 was open, power was flowing through path CC' which was a series circuit comprising inductance and capacitance. The net reactance of path CC' was capacitive. It was necessary to keep a combination of inductance and capacitance in CC' since otherwise the value of capacitance required would have been very large. Since coil CR1 was de-energized, it closed contact 1', which energized the timer. After a pre-set time delay, coil CR2 was energized which closed contact 2'. Closing of this contact energized coil CR3. As soon as this coil was energized, contact 3 was closed, shorting out the capacitor circuit.

109 CO 1-4 e~ ~ ~ ~ ~ ~ ~ ~ ~ e k ~~~~~rz - r< E -Ok~~~ kS~ I.... * -s'- -.. ~^ — ^ ^ ^ < _, ~.. ^ ~? s~~~hi T-I ^-1 T-I < --- --- -------' o^J^~^H i ~~~~~~~~~~~~~~~~~~~~~~~~~~c^f ~r ------ 1 cQ <u __ __ _____________________ C __ __ __ __ __ _ __ __ __ __ __ _ __ ___ ~ ~ ~ P, 0 ---- --- ^~~ (g) —~ —'^11~~~~ c

110 After a certain time when oscillations had disappeared and steady state conditions realized, push button'a' was released which energized coil CR1, closing contact 1, thus reinserting line A and returning to the initial conditions. The capacitor insertion time was varied by varying the time setting on the timer. The above procedure was repeated for different loads, capacitors and the capacitor insertion times. Figures 10. 2 through 10. 5 are the photographs of the system as set up in the laboratory. Torque Angle Measurement The principle of the measurement of torque angle can be illustrated by reference to figure 10. 6. OB and OA are two phasors representing two voltage sources which are assumed to remain constant in magnitude. OB is the voltage of the remote' generating station and OA is the voltage of the receiving station. If 6 is the torque angle, then AB is the phase difference between the two voltages. If OB is equal in length to OA, then AB= 2 OB sin - = K sin 6 2 2 where K is a constant or 1 AB 6 =-2 sin1 (10. 1) K

Q~ Fig. 10. 2 Fig. 10. 3 C~~~~~Fg 0

Fig. 10. 5 14~ ~~~~l.1,

113 KEY TO FIGURES 10. 2, 10. 3, 1010. 5 1E Sanborn Recorder 2. D. Co Driving Motor 3. Synchronous Generator 4. Artificial Transmission Line 5. Switched Series Capacitors 6. Single Phase Reference Voltage Generator 7. Gear and Belt coupling 8. Control Contactor 9. Infinite Bus 10. Welding Timer 11. D.C. Motor Starter 12. Synchronous Generator Field Regulator 13. Phase Shifter 14. Variac

114 B 6 Fig. 10. 6 Principle of Torque Angle Measurement

115 If the phase difference AB vs. time is recorded during a transient, by means of a recorder, it is possible to draw a torque angle vs. time curve by the equation 10. 1. The above principle was used to measure the torque angle during the transient. The voltage of an auxiliary generator was used as a reference signal. The auxiliary generator was coupled to the main shaft by gear and toothed belt drive as shown in figure 10. 3, which was necessary since the auxiliary generator had a rated speed of 3600 rpm while the main shift was running at 1200 rpm. Since there was no load on this generator, its voltage remained constant during a transient, but its phase changed with the position of the rotor. This was the voltage OB in figure 10.6. Voltage signal OA was obtained from the Edison supply, through an auto transformer as shown in figure 10. 1. The voltage signal OB was adjusted by means of a variac such that OA = OB. A phase shifter was used to bring the two voltages in phase with each other when no power was transmitted across the line, so that 6 was zero under no load condition. Signal AB was connected across a Sanborn Recorder. As soon as the transient occurred, the fluctuations in the magnitude of this signal were recorded uy this recorder. The equation 10. 1 was then used to draw a torque angle/time curve from the recorder tracing.

116 Experimental Results During the experiment, a number of machine loadings, capacitor sizes and switching times were used. A typical case was selected for analysis and comparison with the digital computer results. The data obtained from the Sanborn Recorder were converted to torque angles by equation 10.1. Experimental torque angle curves were then superimposed on the theoretical curves as obtained from digital computer analysis, the details of which are given in the following sections. Theoretical Analysis A theoretical study of the system was made by digital computer simulation and results were obtained as follows: The digital computer investigation was to find the time solution of the swing equation d26 M 2 = Pi - P dt2 1 0 where P. and P are power input and power output of the machine. To find these quantities we proceed as follows: Machine and System Parameters Direct Axis Synchronous Reactance = 3.8 ohms per phase Quadrature Axis Synchronous Reactance = 2.46 ohms per phase Open Circuit Time Constant = 0.4 second Direct Axis Transient Reactance = 0. 89 ohm per phase Line Reactance = 4 x 0. 39 = 1. 56 ohms Additional Reactance = 0.48 ohm

117 Per Unit System The following calculations are done on 11 KVA and 240 volt base. 240 Per unit voltage = 240 138 V 1 11000 Per unit current = f3 240 = 26. 5 amp. 138 Per unit impedance = 26 = 5. 236 S2. On the above per unit system, the values of the parameters of the system are as follows: xd = 0.725 d x = 0.47 q xd = 0.17 xE(Line Reactance) = 0. 300 Additional Reactance =0. 092 Figure 10. 7 shows the reactances in per unit. Determination of Inertia Constant The inertia constant of the machine can be found from the following formula. 2 2 x 10 H =.231x WR x (rpm) KVA where WR2 is moment of inertia in lb-ft2, H is the inertia constant in kw-sec/KVA and rpm is the speed of the machine in revolutions per minute. The moment of inertia of the machine rotor and the coupled DC rotor was calculated experimentally and found equal to 32. 9 lb-ft2

118 Therefore 2 -6 0. 231x 32.9x (1200) x 106 H = = 0.995 kw/sec/KVA. Transient Stability Analysis The machine was operated at full load and at unity power factor and suddenly line 1 was opened by pushing the button a. Figure 10. 8 shows the phasor diagram of the system prior to the transient. OA is the voltage of the infinite bus as well as the terminal voltage of the machine, since line 1 is closed. OA is also the direction of the current since the power factor is unity. v b= 1. 00 where v is the machine terminal voltage and vb is the bus voltage. i = 1. 00 / 00 = power factor angle = 0 E = v + j x I qd q = 1.00 0+ L90 x 0.47x 1.00 /0 = 1.00 0+.047 /o0~ =1.00+j 0.47 = 1.105 /25 6=25~ i = 1.00 sin 6 = 1. 00 sin 25 = 0.423 E' = 1.105-(xq -Xd) id = 1.105 - (. 47 -.17). 423 = 0.9781

119 synchronous machine transmission line 1 4, 1 cb xd=' 725 392 infinite bus Xq 47 -2 transmission line 2 H P series capacitor xc Fig. 10. 7 Schematic Diagram of the System - / quadrature axis /// // direct axis Fig. 10. 8 Phasor Diagram

120 Eq Eqd + d(xd q) = 1. 105 + 0.423(. 725 -.47) = 1. 213 A transient stability program was written and the swing curve was calculated by the help of this program. Figure 10.9 shows both the theoretical and the experimental results as explained below. 1. Experimental swing curve without capacitor insertion. 2. Theoretical swing curve without capacitor insertion. 3. Experimental swing curve with. 1 capacitor inserted for 10 cycles. 4. Theoretical swing curve with. 1 capacitor inserted for 10 cycles. It will be observed from the figure that there is a reasonable amount of agreement between experimental and theoretical data.

121 0 o o C.) 0 ~ C.. C Cd I~~~~~~~~~~~~~~o 0~ I o I'^-/^^d, I I / I-f ID ~'~ *' "~0~ 0.-4 OI OL 09 O 0 0 0 (g 9l2Ue / ^S 0 08 0L 09 09 0T O OZ Ot 0

122 without capacitor with capacitor insertion Fig. 10. 10 Experimental Recordings with capacitor insertion Fig. 10. 10 Experimental Recordings

Chapter XI SWITCHED SERIES CAPACITORS (FAULT STUDY AND RELAYING) Introduction In Chapter IX, an approximate analysis of switched series capacitors was presented. Chapter X related to the experimental demonstration of this technique. In this chapter some problems which will arise in the application of switched capacitors to practical systems will be investigated. These problems include the effect of a second fault while the capacitor is in circuit and the effect of the capacitor insertion on distance relaying. System Description Refer to figure 11. 1 which is rather a generalized circuit although similar to the one set up in the laboratory. There are three lines in parallel which connect the synchronous machine to the infinite bus. Circuit no. 1 has negligible reactance as was the case in the laboratory set up, in order to get a good transient. Circuits no. 2 and 3, each consist of two sections. Each section has a reactance of 0. 16 p. u. There are two Y-A transformers at the two ends of the line, which have solidly grounded neutrals. A capacitor of 0.1 p.u. reactance, which has a circuit breaker in parallel with it, is located at the center of the line, as shown in the figure. Each section has two circuit breakers at its ends which have distance relays on them. It is admitted that since circuit no. 1 has been assigned a reactance of zero, this cannot be called a practical system. 123

124 x=O x =.725 1 = d --- x =.47 q 2 xd=. 17 \.16.16.16.16 ^ 2 F3 F4 YJ 3 ~Y Fig. 11.1 A Generalized Circuit x line c line current transformer gap resistance. electrically operated valves air bl:ast reservoir Fig. 11.2 A Typical By-pass Circuit

125 It was, however, selected in order to have as much similarity to the laboratory set up as possible. The reactances of the generator are shown in the figure. The total line reactance is the same as in the experiment. Since the parameters of the system are the same as the system in the laboratory had, the initial conditions for per unit power transfer are also the same. Initially all these lines are in operation. Due to a fault line no. 1 is relayed out and the capacitor is inserted into the circuit. While the system is oscillating, there is another fault which is removed by switching out the corresponding line section. The initial steady state conditions are the following. These are shown in figure 10. 8. I = 1. 00 L0 E' = 0.98 E = 1.21 Vb =V = 1.00 i0 vb =m 6 = 25~ Position and Type of Fault In figure 11. 1, Fj, F2, F3 and F4 are four different locations at which the fault will be assumed in order to get an overall picture of its effect. A three phase fault will be considered in the investigation since this results in the worst condition from the stability standpoint.

126 The following table shows the fault current through the capacitor for different locations. Fault Location Current through the Capacitor F1 7.70 pu F 20. 00 F3 4.45 F4 3.27 From the above table, it is obvious that as soon as a fault occurs at any of the locations F1 through F4, the capacitor will be bypassed due to excessive voltage across it. The fault will result in switching out of a line section by action of the associated relays and the breaker, which will take about 3 cycles. The capacitor will be fully reinserted in the transmission circuit within one half cycle after the overvoltage condition is removed. Capacitor Protection Should a second fault occur while the capacitor is in circuit, the current through the capacitor will be many times normal. It would, therefore, be necessary to bypass the capacitor to protect it against overvoltage. It is important that the capacitor bypass circuit be opened immediately after the capacitor voltage is reduced below the critical value.

127 A typical protective bypass circuit is shown in figure 11, 2. 22 This has been designed for use with fixed series capacitors and can equally well be utilized in the case of switched series capacitors. The breakdown voltage of the bypass circuit can be adjusted in the range of 2. 5 to 3. 0 times the rated voltage of the capacitor. To protect against moderate overvoltages, additional relay equipment has to be provided. Flow of current in the gap circuit causes operation of the electrically operated airvalves and the air-blast reservior discharges a stream of air into the gap conducting area. After establishment of sufficient air flow, the arc in the gap space is quenched at each current zero and gap flashover occurs again on the next half-cycle of high voltage. Thus when the cause of the capacitor overvoltage is removed the gap arc is quenched at the first current zero thereafter and no restrike occurs. The capacitor is fully reinserted in the transmission circuit within one-half cycle after the overvoltage condition is removed. Analysis of the Effect of Fault on Power Transfer Power output from the synchronous generator to the infinite bus is given by the following equation E'q vb vb (XdX P = sin 6 + sin 26 u x d e 2(Xq +Xe) ('d + e) E =.98 vb 1.00

128 6 = 250 0 x'=.17 d x =.47 q The transient proceeds as follows: Initially all three lines were in operation and therefore xe, the external reactance was equal to zero (note this is the same condition as in the laboratory). Therefore Pu.98x1.0 sin 25 1(.30) 1 sin 500 = 1.00 pu u.17 2x.47 x.17 As soon as a fault occurs on line no. 1, it is switched out and the capacitor is switched in. Therefore xe = 0. 07 + 0. 08 - 0. 10 + 0.08 + 0.07=.20 pu Assume that while the capacitor is in the circuit, a second fault occurs at any of the locations F1 through F4, then the power output during the period of the fault is reduced to zero. The faulted section will be relayed out in 3 cycles. For half a cycle, following the relaying out of the faulted section, the capacitor remains bypassed. Therefore the value of xe during this half cycle =.07 +.16 +.08 +.07 =.38. Next the capacitor is inserted and therefore x =.38 -. 1 =.28. The machine continues e to swing until the oscillations are damped out. d6 The capacitor is inserted when (it is positive and removed when it is negative. This technique requires a sophisticated control

129 equipment to switch the capacitor in and out of the circuit but this is a superior method of using a capacitor as compared to the method used in the laboratory where the capacitor was switched out after a certain number of cycles. Computer Study Acomputer program was written to simulate the above system. The value of H used in the study was. 995 as obtained in the experimental investigation. The program makes the following simulation: The generator is operating in the steady-state supplying a power equal to 1. 00. A fault occurs which results in switching out of line no. 1 and insertion of the capacitor. A second fault occurs immediately which drops the output to zero for a period of 3 cycles. At the end of 3 cycles, the fault is removed by switching out a line section. Since the capacitor will be out for another half cycle, the value of xe during the period immediately following the clearance of the fault =. 07 +. 08 +. 16 +.07 =. 38. Another half cycle afterwards, the capacitor is reinserted into the circuit and the value of xe = 38 -. 1 =.28. The capacitor iccth d6. d6 remains in circuit when t is positive and is out of circuit when dt is negative. The enclosed swing curve shows that the system is stable. It may be pointed out that the second fault was assumed to occur at the very start of the first swing which makes it rather a special case. In actual practctice, the second fault may occur any time during the swing which fact should be considered when designing real systems.

130 a 0 bc f Q) c, - m. Q CO'D IC Q?-*b~* m D/ o 0 bD, C i 0 C"' 06 08 OL 09 09 0' 0f 0E 01 0 S3oaJp-Q o~i Q onbo ~~~/ r,,O~~00 b.A ^ ^~~~ y *<0 l 5~~~~~~~~b.^ E-i b^~~~~~~~~' / ln'^~~~~~~~~~~~~~~~~1 ^ - * M~~~~~~~~~~~~~~~~~u "s^^ oo~~~~~~d ^s^ ^ ^~~~~~~~~~~~~~~~j "^' * r-< ~ 06 08 OL 09 0~~ Olti OE OZ b i saa~~~x~ap-g. ooSt anb4

131 Effect of Switched Series Capacitors on Distance Relaying Refer to figure 11.4(a) which shows a fault at the location F1. Since the capacitive reactance is subtracted from the total inductive reactance of the lines 1 and 2, the fault appears much closer to relay A which may cause operation of this relay. Thus instead of opening the faulty section 3, the unfaulted section 1 may be switched out. In practice, due to the action of the bypass circuit, the capacitor will generally remain shorted out during a fault. In most cases, therefore, the operation of the distance relays will not be affected by capacitor insertion during fault conditions. An exception will be a "line-to-ground" fault where the fault current may not be high enough to short out the capacitor. In such a case the relaying will have to be properly designed to take care of the effect of the switched capacitor. A capacitor insertion can have substantial effect on distance relaying during machine swinging. It would therefore be desirable to study the effect of capacitor insertion on the impedance "seen by the distance relay" during a swing. Consider figure 11.4 (a) again which can be simplified as shown in figure 11.4(b)y if the parallel direct connection is omitted. The following analysis is based on the assumption that the voltage behind the transient reactance remains constant, otherwise the calculations will become unduly complicated.

132 X=0 2 16 4 16.e 16 o 16 d o 07 1 A 3~ O07 F, Fig. 11. 4 (a) Circuit for Distance Relay Study ~32 Fig. 11.b A S pl Fig. 11. 4 (b) A Simplified Circuit

133 To investigate the problem, the following discussion will be 23 helpful. Consider figure 11.5, which shows a simple two machine system. Here VA and V are the voltages behind the transient reactances which are A 8B assumed constant in magnitude but varying in phase during swings or out-of-step conditions. VA leads VB by the variable angle 6. The current anywhere in the series circuit is VAL - VB z where Z is the impedance of the connecting circuit including the transient reactance of the two machines. The total impedance Z is divided by the relay location M into two parts, mZ and (l-m) Z where m is a real number less than 1. At point M the voltage V is then V = (l-m) VA + mVB The impedance "seen" by the relays at M is V = = (1-m) VA n+ mVB r VAL - VB

134 I >Ft-~e -- 0-~.I)i -? I W" ((I- r I v VB Fig. 11. 5 A Simple Two Machine System Fig. 11. 5 A Simple Two Machine System

135 V (1-m)_ /A +m r VB k(l-m) + m Z _ /; 1 k/6L- 1 z VA j V 13 where k is the magnitude of V A/'B k(l-m)A + m k(l-m) + mZ-6 k/L - 1 k -L6 In the system under study k = 1.01 and if it can be taken as equal to 1, the above equation simplifies to zr _ (l-m) + m 1_ _ -m(l- -_6 ) + 1 -m + 1 ( 1 6 -L-6 = - -m)- cot (1) This equation represents a vertical line, because the real part is constant while the imaginary part varies as a function of 6. If both sides are multiplied by Z, lengths are multiplied by the magnitude of Z and the line is rotated counterclockwise through the impedance angle 8. In the post-fault steady-state condition the capacitor will be out of the circuit and equation (1) will hold. However when the capacitor is inserted the value of m will change. The new value of m can be found as follows:

136 Let m = mI when the capacitor is inserted Total Impedance = Z - X Then m Z = mlZ1 =ml (Z -Xc) z or m= Z-X m C Thus the value of m1 increases by an amount ZX. C Applying the above analysis to the system of Figure 11. 5(a) the following impedance is "seen" by the relay at location A without capacitor insertion. zrF Zr I.16.17 +.07.24 m = =- =-.51 17 +.07 +.08 +.08+.07.47 r 1 Thus the locus of Z is a vertical line at a distance of. 51 = -.01 from the origin. The ordinates of points on this line representing various values of 6 are given by 1 6 y =- 2 cot 2 Table Table 1 shown calculation of y for 30~ increments, from 0~ to 180~.

137 Table 1 6 y 0 00 25~ -2. 320 30~ -1. 866 60~ -.866 90~ -. 500 120~ -.289 150~ -. 134 180~ 0 Since the resistance of the line is neglected, Z =. 47 90~. Hence the vertical line is turned counterclockwise by 900 and since only one half current flows in the protected line, the loci is multiplied by.47 x 2 =. 94. The loci is shown in Figure 11. 16. As soon as the capacitor is inserted, the value of m becomes m1 = m Z - X =.51x - -.51 x 4 =646 Z'X.47'. -.37 c and Z =. 47 -. 1 =.37. Therefore the loci jumps from A to B. Assume that the first zone impedance element of the relay at breaker 1 roaches to 80% of the length of the line which it protects or to. 80 x. 16 =. 128. Let the second zone be set at. 256 and the third at. 384. The tripping characteristics of these elements are concentric circles of radii. 128,. 256 and. 384 with centers at the origin. Let the directional

138 element have maximum torque at a phase angle of 45~. Its characteristic is a straight line through the origin and perpendicular to the line of maximum torque. The characteristic is shown in Figure 11.6. From the figure it is observed that although insertion of the capacitor adversely affects the performance characteristics of distance relay no. 1, for the assumed conditions of the problem, it creates no serious difficulty as can be seen from the following discussion. 0 It has been noted earlier that the value of 6 varies from 25 to o 90. The initial locus takes it into the zone Z2. The new locus keeps it in the same zone. Therefore under conditions of "capacitor in" or "capacitor out" the worst condition is the same. The diagram shows that instantaneous tripping of this particular relay will not occur during swinging or out-of-step operation, but that delayed tripping will occur if the angle 6 lies close enough to 900 for a sufficient time to close the contacts. The above study demonstrates that while planning systems involving switched series capacitors care must be exercised to ensure that the protective relaying is not adversely affected.

139 o.. t <t 0 dCt C;1) C 0 P4 r i N s 4 gV 1 E \\ ^, ^^' a -~~~~~C1 ~c^ e dQYb / ^-''^./ o / /^ {^^ \N (^~~~~b1 / E4N s-i ^ - - - ^ ^ ^^ \ /T 0, \( 0 ^ \ be~~~~~~~r /I \ *r3 0d \ p 4->'i \ 0 ^ ^ ^ * o \~ 5 \ ^ x

Chapter XII CONCLUSIONS Three methods of improving transient stability of a synchronous generator have been investigated. The first method involves the use of signals proportional to the derivatives of the torque angle 6. It has been demonstrated that the transient stability of a synchronous generator is substantially improved if the derivative signals of proper magnitudes are used in the excitation system. A mathematical method has been demonstrated which develops formulas for amplification factors in various feedback channels. The values of line constants of the excitation system can be substituted in appropriate formulas to determine the magnitudes of these signals. Formulas have also been derived which takes saturation into account. The second method involves the use of Phase-shifting transformers. A mathematical analysis of the effect of these transformers, on the transient stability, has been made. It has been demonstrated that this method should have considerable effect on improving the transient stability. The third method uses switched series capacitors. By experiment and by theory the feasibility of this method has been demonstrated. Many technical problems associated with the use of the series capacitors have been thoroughly investigated. 140

141 In conclusion, it appears that there is considerable scope for improving the system stability during transient conditions by all these methods. While switched series capacitors and phase-shifting transformers are strong measures, due to economy these may have limited applications. There will, however, be situations where these will be the most economical techniques of controlling transient instability. On the other hand, excitation control continuously contributes to damping. With the development of static excitation systems, which have low time constants and much higher voltage limits, there is great potential for improving transient stability by using derivative signals.

REFERENCES 1. American Institute of Electrical Engineers, American Standard Definitions of Electrical Terms, 35.20.200 and 35. 20. 203, New York, 1942. 2. Glavitsch, J., "Theoretical Investigations into the Steady-State Stability of Synchronous Machines", The Brown Boveri Review, vol. 49, no. 3/4, 1962. 3. Langer, P. and Johansson, K. E., tInfluence of Load-AngleDependent Signals on the Voltage Regulation of Synchronous Machines", Proc. Int. Conf. Large Electric Systems (C. I. G. R. E.), Paris, 18th Convention, 1960, vol. III, paper no. 315. 4. Dineley, J. L., Morris, A. J. and Preece, C., "Optimized Transient Stability from Excitation Control of Synchronous Generators", I.E.E. E. Trans. (Power Apparatus and Systems) vol. PAS-87, no. 8, August 1968. 5. Dandeno, P. L., Karas, A. N., McClymont, K. R. and Watson, W., "Effect of High Speed Rectifier Excitation Systems on Generator Stability Limits", I. E.E. E. Trans. (Power Apparatus and Systems) vol. PAS-87, no. 1, January 1968. 6. Kimbark, E. W., "Improvement of System Stability by Switched Series Capacitors", I. E. E.E. Trans. (Power Apparatus and Systems) vol. PAS-85, no. 2, February 1966. 7. Park, R. H., "Two-Reaction Theory of Synchronous Machines - Part I, Generalized Method of Analysis", A. I. E. E. Trans., vol., July 1929. 8. Olive, D. W., "New Techniques for the Calculation of Dynamic Stability", I. E.E.E P.I.C.A. Conference, May 1965. 9. Kron, G., "A Super Regulator Concelling the Transient Reactance of Synchronous Machines", Matrix and Tensor Quarterly 1955, 5, p. 71. 10. Thaler, G. J. and Pastel, M. P., "Analysis and Design of Nonlinear Feedback Control Systems", McGraw-Hill Book Co. Inc., 1962. 142

143 11. Baba, J., Hayashi, S., Yamada, I., Haneda, H. and Ishiguro, F., "Sensitivity Analysis of Power System Stability", I. E. E. E. P.I.C.A. Conference, May 1967. 12. Smith, O. J. M., "Power System Transient Control by Capacitor Switching", I.E. E. E. Trans. (Power Apparatus and Systems) vol. PAS-88, no. 1, January 1969. 13. Breuer, G. D., Rustebakke, H. M., Gibley, R. A. and Simmons, H. O. Jr., "The Use of Series Capacitors to Obtain Maximum EHV Transmission Capability", I.E.E.E. Trans. (Power Apparatus and Systems) vol. 83, no. 11, November 1964. 14. Mittelstadt, W. A., "Four Methods of Power System Damping", I. E. E. E. Trans. (Power Apparatus and Systems) vol. PAS-87, no. 5, May 1968. 15. Stagg, G. W., Gabrielle, A. F., Moore, D. R. and Hohenstein, F., "Calculation of Transient Stability Problems Using a High Speed Digital Computer", I. E. E. E. Trans. (Power Apparatus and Systems) vol. 72, August 1959. 16. Kimbark, E. W., "Power System Stability", vol. 1, John Wiley and Sons, Inc., 1967. 17. Kimbark, E. W., "Power System Stability - Synchronous Machines", Dover Publications, Inc., New York 1968. 18. Crary, S. B., "Power System Stability", vol. II, Transient Stability, New York: John Wiley and Sons, Inc., 1962. 19. Aldred, A. S., Shackshaft, G., "The Effect of a Voltage Regulator on the Steady State and Transient Stability of a Synchronous Generator", I.E.E.E. Trans., vol. 105, part A, August 1958. if 20 R. Fairfield, Posicast Switched Series Capacitors to stablize a transmission System:' M. S. thesis, Un. of California. June 1966 21. Richard, C. R., Stemler, G. E., "A Fast Response Instrument for Measurement of Power System Phase Angles, " IEEE Trans-PAS-87, no. 1, January 1968. 22. Harder, E. L., Barkle, J. E., Ferguson, R. W., "Series Capacitors During Faults and Reclosing, " AIEE Trans., vol. 70, 1951. 23. Kimbark, E. W., "Power System Stability, " vol. 2, John Wiley and Sons, Inc., 1967.

Appendix I Mathematical Representation of a Synchronous Generator The analysis to be presented here employs the concept of an ideal synchronous machine and is based on the classical work of Park. An ideal synchronous machine has no saturation, hysteresis or eddy currents and all of its fields are sinusoidally distributed. The assumption of sinusoidal field distribution removes the harmonics of the airgap mmf and flux waves, from consideration. Under most operating conditions, these harmonics have a secondary effect on machine behavior, hence this assumption is justified. The neglect of magnetic saturation is a serious approximation, particularly in steady state stability study. Hence the technique by which magnetic saturation can be taken into account will be discussed. Magnetic hysteresis is negligible for materials used in the construction of modern machines and will be ignored here. In the discussion to follow, a salient pole rotor is analyzed, rather than a nonsalient pole. The reason for this is that a hydroset is usually a slow speed set and hence has a salient pole rotor. Since 144

145 this research deals with the analysis of a single machine connected with a large system by means of a long transmission system, it is more representative of a remote hydro-station connected to a load center by means of a transmission line. It would be appropriate here to briefly describe the construction of a typical synchronous machine. The salient-pole rotor has two axes of symmetry. One, called the direct axis, is identical to the polar axis, the other is coincident with the interpolar axis and is called the quadrature axis. By definition, the positive quadrature axis is taken ahead of the positive direct axis in the direction of rotation. The stator has three, distributed phase windings, the positive axes of which are equally spaced about the stator periphery at intervals of 271/3 electrical radians. The three phases are lettered abc to indicate the order in which their positive axes are encountered in skirting the stator periphery in the direction of rotor rotation. A schematic diagram of an idealized, two-pole synchronous machine is given in figure Al-1. The angle 0 is the electrical angle by which the rotor reference axis is advanced on the stator reference axis in the direction of rotation. To develop the equations of motion for an idealized, three-phase, salient pole synchronous alternator, linear system of coupled networks will be used which is the usual method of treatment for such a study. Saturation will be taken into account later on by means of certain

146 factors known as saturation factors. A set of inductance coefficients will be defined and the principal of superposition used in determining the total flux linkages of each of the several machine windings. Double subscript notation will be used to identify the inductance coefficients, like subscipts designate a self-inductance coefficient, while unlike subscripts indicate the mutual inductance between the windings indicated by the subscripts. \/ Rotor Quadrature Axis *,* Axis of phase b / /\ A. ~/ \ b. \ ~//Rotor Direct Axis Axis of phase a ~igue... 1 — — Reference Axis Axis of phase\ Figure A-1. 1 Two Pole Synchronous Machine

147 vb 0 Va V vc i C i Fig. A 2 Y-Connected Synchronous Machine Windings Fig. A-1. 2 Y-Connected Synchronous Machine Windings

148 The induced voltage in a coil is calculated as v(t) d t) (1) where 4(t) is the coil's flux linkage. If current flows in the windings, d 4, (t) va(t) = r(t) + dt d 4b(t) vb(t) = rib(t) + dt (2) d ~c(t) vc(t) = ric(t) + dt d 4/(t) vf(t) = rf if(t) + dt Since saturation is not as yet taken into account, flux linkages can be assumed to be proportional to mmf's and superposition is applicable. Dropping t for convenience, =a- L i +L ib+L i +L a aa a a+ b + ac c af if = Lba ia+Lbbb + Lbc ic + Lbf f 4,=L i+L i+L i+L i + 3,c ca a Lcb b cc c cf (3) f = Lfa a Lfb ib+ Lf i + Lf if L, Lbb, L are self inductances, Lab, Lac etc. are mutual inducta n e s.ac inductances.

149 L will have a maximum value when the direct axis is aligned aa with the axis of coil a and Lab will have a maximum value when the ab rotor has moved 30 clockwise from the center line of coil a. L has aa a minimum value when the quadrature axis is aligned with the axis of coil a. The variation of L between its extreme values is sinusoidal aa and can be expressed as L = L + L cos 20 aa s m where 0 is measured from the center line of coil a. Lbb= L + L cos 2( - 1200) = L + L cos(20 + 120~) s mr Lcc = L + L cos(20 - 120~) The expressions for mutual inductances are Lab = Lba=- [M + L cos 2(0 + 300]) = -M + L cos(20 - 120~) s m Lb = L =-M +L cos 20 Lc cb s m L = L = -M + L cos(20 + 1200) ca ac s m Laf= Lfa = Mf cos Lbf = Lfb = Mf cos(O - 120~0) Lcf = Lfc = Mf cos(0 + 120~)

150 Since the rotor mmf is driving flux across a constant air gap, Lff is a constant. These expressions for the various inductances can be substituted in equation (3) to calculate flux linkages in terms of currents. The constant velocity of the rotor is taken into account by writing 0(t) =o t +.00 The equation for v is a Va = [r - 2wL sin 2(ct + )] ia -[2wLL sin(2o + 200 - 120~) ]ib - [2cwL sin(2wt + 20 + 120~)] i -[wMf Sin(ot +0 ) ]if di dib + [L + L( cos 2(wdt + ) d +Ms + Lm cos(2wtt + 20 - 120~)] dt diC dif + [Ms + Lm cos(2 t + 200 + 120~)] dt + [Mf cos(wt + ) ] dt Similar differential equations are obtained for v, v and vf. c It is obviously very difficult to solve these equations. d-q Transformations It will simplify the calculations if the flux linkages due to the stator coils are resolved along the direct and quadrature axes of the machine. See Figure Al. 1. Using the following Park's transformation /d = 3 [a2 cos 0 + b sin(0 - 120~) + 4c cos(0 + 120~)]'d 3 a c c -q = -3 [VNa sin 0 + Lb sin(0 - 120~) + 4/c sin(0 + 120~)] (4)'o 3 (oa +b + c) or

151,/a = 4/d cos 0- /q sin 0 + /o ib = /d cos(0 - 1200) - q sin(O - 120P) + q/ 4/c = id cos(8 + 120) - qq sin(O + 120~) + /o (5) Similarly v = vd os - v sin + v a d q o b = d cos(0 - 120~) - v sin(0 - 120) + V' V = V cos( + 120) - v sin(O + 120~) + v C vd q o i = i cos 0 - i sin 0 + i a d q o ib = i cos( - 120) - i sin ( - 120~)+ i ic = id cos( + 120~)- i sin(0 + 120~) + i Substituting expressions for 4/a',b' and qc in (4), the result is,d = (Ls + Ms + L) d + Mf f 4d=(L +M 3 Lm)i s + s 2 Lm) q (6) = (L -2M )i 0 S S 0 4f= 2 Mf id + Lff if 3 Let L= L + M + L This may be callei the direct axis d s s 2 m synchronous inductance. L =L +M - L q s s 2 m This is called the quadrature axis synchronous inductance.

152 L L -2M, o s s this is called the zero sequence inductance. Then d Ld id + Mf f - = L i q q q Uo-/ L i o o0 0 ~f =2 Mf id+ Lff if Transforming the voltage equations to the d-q reference frame. v= - [v cos 0 + V cos(0 - 12C) + v cos(0 + 12W0)] 2 rd d % [(ria + d ) cos 0 + (rib + ) cos(0 - 120~) d 4 + (ric + t cos(O + 12C0)] 2 = r[i cos 0 + ib cos(0 - 120~) + i cos(0 + 120~)] 3 a c d i d / d V d % +3[ co dt cos( 0 + dt) cos( - 1200)]~) d 0~ Vd=id + dt - dq (8) Similarly v ri+ (9) d% t s ae l e as cd ri and t ( Note that - w and C terms are large as compared to ri and d4 terms and correspond to the voltages induced as the air gap flux wave sweeps past the stationary stator coils.

153 The same equations can represent a generator if the sign of the currents is changed, which will give the following set of equations d 4 vd=-rid+ dt - q (12) =-riq + d + qd (13) q q d Vf = r if + -df f (14) d= -Ldid+Mf if (15) -Mi3Li (16) f 2 Mf id + f f(16) f=-L i (17) q qq For a balanced system, vd= v inm V (1) v =v cos 6 (19) q where S is the angle between the voltage v and the quadrature axis. Solving equations (15)-(17) for currents and substituting in (12)-(14) M f id - L d (20) ff L'd f d ff d 3 f2 where L'd= Ld 2 Lff is called the direct axis transient inductance.

154 _d r Mf rWd Vd dt Lff L'd Lfd q (21) d 4/ V q +d t + i (22) q dt L */Cq q d d Yf Ld 3 rf vf dt Lff Ld' 4'f 2 Lff' MfL d (23) If the derivatives of 4d and 4q are neglected, since these are small compared to the wo iterms, this reduces (21) - (23) to vd=- cWOq- rid v = w4d- ri d4Cj r Lrq d _f _rf d 3 f Mf dt Vf L Ld' I +2 L L d ff d ff Ld Lff Let = T = open circuit field time constant, then rf do d 4f Ld 3 Mf dt = f - L'T' T f + 2 L' T' d d do d do Let wo M if = E q OMf Multiply equation (14) by L ff ff wM Mf (d oMf Lff vf = L- rf if + dt ( -Lff f ) wMf Let Eq= Lf'f q- Lt ~Cf'

155 WMf V_ WMf rf Vf _ where Lff Lff rf T' where ff f do E is the open circuit armature voltage. do where E = v + rI +jw Ld id + j oL i from equation (2 0) id - f- L' dLhf 4'f Ld ff L'd d w Mf or w L'di f - d dd Lff f 4/d or L'd id= Eq- v -ri Hence E'= vq + WL'd id + ri (24) Since vq =-ri -wLd id +E Substituting in (24) gives Eq' -riq - (Ld') idE + ri But Eq = Eqd +(Ld- Lq) id Therefore Eq = Eq - (Lq - Ld')id q qd q di

156 or in phasor form Eq = qd - j(Lq - Ld')i If stator resistance is neglected, the equations can be summarized as follows: dE' qdt T1 ex -Eq) (25) dt T' ex q do Eq =Eqd+ (Ld Lq) id (26) Ed= v+jwL I (27) E'= Eqd -(L Ld)id (28) E v + jLd id + jLq i(29) The phasor diagram corresponding to above equations is shown on the next page. In the diagram, vb is the busbar voltage, vm the machine terminal voltage and xe the external reactance.

157 I COCC /, F / w l / ij_ J^_ ___;0- - sx ea 0 ~!rS~F37aXa

APPENDIX II Computer Programs Symbols Used- Program nos 1,2 and 3 G = Machine Rating H = Inertia Constant DLTT = Time Interval XD = Direct Axis Synchronous Reactance XQ = Quadrature Axis Synchronous Reactance XDP = Direct Axis Transient Reactance XE = External Reactance TDO = Open Circuit Time Constant TE = Exciter Time Constant VFO = Initial Value of Excitation Voltage VR = Reference Voltage MUV = Gain in Voltage Channel Av d6 AMUD = A = Gain in dt Channel 2 dt 2A AMU2D = A= Gain in - Channel dt VB = Busbar Voltage XC = Input Power in the First Interval YC = Input Power in the Second and Subsequent Intervals EEXO = Initial Value of Exciter Voltage Referred to Armature Circuit EQPO = Initial Value of Voltage Proportional to Field Flux Linkage PU = Output Power 158

159 DF dO dt d28 DS 2 dt D(N) = n V = Machine Terminal Voltage F - Frequency

160 Program No. 3 C = A Constant = M GH 180f DLTT = Time Interval 6T PAI = Accelerating Power in the First Interval DI = Initial Value of 6 PI = Input Power THETA = 0 the Angle by which the Power Angle Curve is Shifted

161 Computer Program No. 1 Chapter V: Excitation Systems (Digital Computer Results) Title: To find transient Stability of a System using auxiliary signals in the excitation system. DIMENSION D(50) NAMELIST/NAME1/F,G,H,DLTT,XD,XQ,XDP,XE,TDO,TE IVFO,VR,MUV,AMUD,AMU2D,VB,XC,YC,EEXO,EQPO PU=1.0 READ(5,NAME ) D(1)=50.2 PI=XC EEX=EEXO EQP=EQPO T-O. DF=O. N-1 I T=T+DLTT N-N+I DS-(PI -PU)*1O.*F/(G*H ) DDF=DS*DLTT DF=DDF+DF DD=DF*DLTT D(N )=D(N-I )+DD WRITE(6,3)T,D(N),EEX 3 FORMAT(2HT=,F6.3,2X,2HD=,F6.3,2X,44HEEX,F6.3) X=D (N)*3. 14 159/1 0. AA=VB*VB*(XQ-XDP)*SIN (X )*COS(X)/((XQ+XE)*(XDP+XE)) BB - (XDP-XQ )/ (XQ+XE) EQ =(EQP+BB*COS (X) )/( I.+BB) VD=XQ*VB*SI N(X) / (XQ+XE) CC= (XD-XDP )/(XQ-XDP) DD= (XD-XQ)/(XQ-XDP) VF CC*EQ-DD*EQP VQ (XD *VB *COS (X )+XE*VF ) / (XD+XE) V-SQRT (VD*VD+VQ*VQ) WRITE(6,10)V 10 FORMAT(2HV-,F6.3) EX VFO-MUV*(V -VR )+A MUD*DF+A MU 2D *D S IF(EX+I.)6,6,7 6 EX=-I. GO TO 4 continued

162 7 IF(EX-3.5)4,5,5 5 EX=3.5 4 CONTINUE AK 1DLTT*(EX-EEX)/TE AK2=DLTT*(EX-EEX+.5*AK1 )/TE AK3=DLTT*(EX-EEX+.5*AK2)/TE AK4-DLTT*(EX-EEX+AK3) /TE EEXXEEX+(AK 1+AK2+AK2+AK3+AK3+A K 4)/6. DNM- (EEX-VF)*DLTT DNMM- (VF/EQP )*DLTT/2. DEQP =DNM/ (TDO+DNMM) EQP =EQP+DEQP PU (EQP*VB*SI N (X)/(XDP+XE) ) -AA PI=YC IF(N-50)2,.,8 2 GO TO 1 8 CONTINUE END

163 Computer Program No. 2 Chapter VIII: Application of Phase Shifting Transformers. Title: To compute data for a Swing Curve for a system using Phase Shifting Transformers. >I DIMENSION D(25),DLTD(25),P0(25) 2 NAMELIST/NAMEI/C,DLTT,PAI,DI,PI DDI THETA 3 READ(5,NAMEI) 4 D(l)=DI > 5 PO(1)=PAI > 6 DLTD(1)-0. > 7 T=O. > 8 N-= 8.1 DDIO=DDI 9 1 T=T+DLTT 10 N=N+1 11 DLTD(N)=DLTD(N- )+PO(N-1 )*DLTT*DLTT/C 1 1. DLTD (N) =DLTD (N)+DDIO 12 D(N)=D(N-I)+DLTD(N) 13 WRITE(6,3)T,D(N) 14 3 FORMAT(2HT: F6.3,5X,2HD=,F10.5) 15 X=((D(N)+THiTA)*3.1416/180.) 16 PO(N)=PI-1.25*SIN(X) 16.1 DDIOO0. > 7 IF(N-24)1, 1,9 18 9 CONTINUE 19 END # END OF FILE

164 Computer Program No. 3 Chapter X: Switched Series Capacitors (Experimental Study) Title: To determine Transient Stability of a system using switched series capacitors. The capacitors are taken out after a fixed number of of cycles. DIMENSION D(30) NAELI ST/NAiE 1/F, GH H i.DLTT. XD X-0~.XDP, XEI 9 TD'3 TE 1 VF'. M3 VR MUVtAMUD AM? Dv, AML JI? D, V3 J XC. YC, rEEX EgP3J D 1 0 TF Pt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continued

165 6 EX=-1 G3 TO 4 7 IF(EX-3.5)4, 5 5 5 EX=3.5 4 CONTINlUE AK 1 =DLTT*(EX-EEX)/TE AK2=DLTT* (EX- EEX +. 5AK )/TE AK 3=DLTT*( EX-EEX+. 5*AK2) /TE AK4=DLTT*( E EX-EEX+AK3) /TE EEX=EEX+(AK1+AK2+AK2+AK3+AK3+AK4)/6. DNM=( EEX-VF)*DLTT DNMM= ( VF/EQP) *DLTT/2. DEOP=DNM/(TD0 +DNMM) EQP=EQP+DEQP PU=(EQP*VB*SIN(X)/(XDP+XE) )-AA IF(DF) 1, 1,2 11 XE=XEI GO TO 12 2 XE=XE2 12 IF(N-30)13,8,8 13 GO TO 1 8 CONTINUE END

166 Computer Program No. 4 Chapter XI: Switched Series Capacitors (Fault Study and Relaying) Title: To determine Transient Stability of a system using switched series capacitors. The capacitors are taken out when the torque angle is decreasing and inserted when the torque angle is increasing. DIMENSION D(30) NAMELI ST/NAMEI/F, G.H- DLTT, XD, XQ, XDP, XE I TDO, TE, 1 VF0 rVR, MUV AMUD, AMU2D. VB, XC, YC, EEX0, EQP0, XE2, D O PU= 1 READ( 5, NAME ) D( I )=D10 PI=YC EEX=EEXO EQP=EOPO V=VR T=0. DF=0. N= = XE=XEI-XC I T=T+DLTT N=N+ I DS= (PI-PU) *180.*F/(G*H) DDF=DS*DLTT DF=DDF+DF DD=DF*DLTT D(N)=D(N- I )+DD WRITE( 6, 3) T D(N), V 3 Fi0RMAT(2HT=F6.3 2X,2HD=,F6. I,2Xg2HV=,F6.2) X=D(N)*3. 14159/180. AA=VB*VB*(XQ-XDP)*SIN(X)*C0S(X) / ((XQ+XE)*(XDP+XE)) BB=(XDP-XO)/(XQ+XE) EQ=(EQP+BB*C0S(X))/( 1 +BB) VD=XQ*VB*SIN(X)/ (XO+XE) CC=(XD-XDP)/(XQ-XDP) DD=(XD-XQ)/(XQ-XDP) VF=CC*EQ-DD*EQP VQ=(XD*VB*CS(X) +XE*VF)/(XD+XE) V=SORT( VD*VD+VQ*VQ ) EX=VF- MUV*( V- VR) +AMUD*DF+AMU2 D*DS IF(EX+I. )6. 6,7 continued

167 4 C3NTINLUE AK 1 =DLTT*( EX-EEX)/TE AK2=DLTT*( EX-EEX+. 5*AK l )/TE AK3=DLTT*( EX-EEX+. 5*AK2)/TE AK4=DLTT*( EX-EEX+AK3)/TE EEX=EEX+(AK1 +AK2+AK2+AK3+AK3+AK4)/ 6 DNM=( EEX-VF)*DLTT DNMM=( VF/EQP)*DLTT/2. DEQP=DNM/(TD0+DNMM) EQP=EQP+DEOP PU=(EQP*VB*SIN(X)/(XDP+XE))-AA IF(T-TF)2, 11 11 11 XE=XEI G0 TO 12 2 XE=XEI-XC 12 IF(N-30)13,8,8 13 G6 TO 1 8 C3NTI NUJL. END

168 start Read System Constants TRANSIENT STABILITY STUDY Read Initial Vale PU=I,D(I)=50.2, PI=XC,EEX=EEXO, EQP=EQPO,T=O, DF=O.,N=I T=T+DLTT (STOPI) Calculate D(N) yes,,' / \^^^^^^~ ~ WRITE IF N <50 1> / r, no I-| TD(N),EEiX Calculate V PI=YI Write V Calculate J_ EX=3.5 Calculate EX E..EX, EQP, PIU no -o ^y^es * no ~^r7 —--— SFs —<'- ^EX ="-I -- EX=-~, I yes M>^ ^ ^ FLOW CHART I FOR THE COMPUTER Programs

169 Transient Stability Study using phase shift transformers in the system (START) READ System Constants Initial Values D(I)=DI,PO(I)=PAI, DLTD(I)=O,T=O,N=I, DDIO=DDI T=T+ DLTT N=N+I Calculate D(N) Write,T,D(N)J ICalculate PO(N) no N >24 yes Flow Chart II for Computer Programs