POWER SYSTEM SECURITY ASSESSMENT FOR FAULTS USING DIRECT METHODS By Ahmad Sadeghi Yazdankhah A DISSERTATION Submitted to Michigan State University in partial fuifiiiment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of E1ectrica1 Engineering and Systems Science 1984 ABSTRACT POWER SYSTEM SECURITY ASSESSMENT FOR FAULTS USING DIRECT METHODS By Ahmad Sadeghi Yazdankhah Power system operators and planners have long desired to be able to simulate the transients due to electrical faults on-line and without the extensive computation and time required to solve the non- linear system differential equations. Extensive research has been devoted to developing Lyapunov methods that would eliminate the com- putation and thus permit (a) on-line assessment of stability by operators and (b) much more extensive evaluation of the security of the system for different faults by system planners. The Lyapunov methods for assessing the stability of power system for a particular fault have been unable to predict whether the system will or will not be stable. Moreover, the procedures developed for eliminating the need to solve the system differential equations require either approximately the same computation as the solution of the dif- ferential equations themselves or are not accurate. Two algorithms have recently been proposed for assessing the stability by (a) identifying a critical machine that determines whether the system will be stable and (b) determining whether the deceleration . . C on. I; a O I . a a. 69" O - - 3 . :a- ':Q: .vvi do. O'-o 0 O T; :I‘ :0 0A . . " "l III 3'- . o p s _ I c ‘r C Ahmad Sadeghi Yazdankhah energy of the transmission network is sufficient to maintain stability for the particular fault and clearing time. The first contribution of this thesis is to show that these two algorithms are extremely accurate and can easily identify the critical fault clearing time at which the system just loses stability. Moreover, both methods were shown to predict identical fault clearing times. The second contribution is to develop an accurate method of f directly predicting the maximum angular deviation during the transient ’ for a particular fault, fault clearing time, and operating condition. This prediction of peak angular deviation is required to detennine whether the system is stable or unstable using the above algorithms. The computation required for this angular prediction and thus the resulting direct stability assessment method is shown to be extremely small and less than one-hundredth of that required for solution of the differential equations. The accuracy of the direct stability assessment method using the peak angle prediction method is shown to be Quite good based on the extensive computational results on the Reduced Iowa System. For my parents Homay and Rahim and my sisters and brothers ii ACKNOWLEDGMENTS The longer a man lives the more he realizes that his success is mostly due to the help and support received from his family and friends. I am forever thankful to my family for their unconditional love and sacrifices, and for getting me started right. I sincerely wish to express my thanks and appreciation to Professor R.A. Schlueter, my major advisor, for his invaluable assis- tance in the planning and preparation of this dissertation. I would like to extend my appreciation to the members of the guidance com- mittee, Professors H. Khalil, R.0. Barr, w. Symes, and R.A. Schlueter, for sharing their thoughtful and constructive comments. I am also indebted to Professor J.B. Kreer, the Chairman of the Department of Electrical Engineering and Systems Science, for support of the exten- sive computer cost of the thesis. Finally, my special thanks to Mrs. Carol Cole, whose fast and ’ excellent typing of this dissertation is greatly appreciated. v». " U... .- \ . - \ .a- fil \ '~~' . u. A. ... -'- \ . . I .0‘ In. v~ .‘ ~- I Ozoo: 'irvn ‘ ‘3 D. n '- V c. \'. 6s .. ' - ‘ e ' l ' "’0- '00 ~ ". 9....--Ifl. ; a. _, . .' ‘ Iv- P’t-. \ ‘ I.-.) a -..‘ I . . K... *9 I (7 c. '5. v'r" ’F. I - 1‘ L0 .’. (I) I?! (4:) ¢ Lu 1 (h J P 3 [II In t 7 I TABLE OF CONTENTS LIST OF TABLES ........................ LIST OF FIGURES ....................... Chapter 1. POWER SYSTEM STABILITY ................ 1.1. Introduction .................. THEORETICAL DEVELOPMENT ON POWER SYSTEM TRANSIENT STABILITY ....................... 2.1. The Dynamics of Power System During a Transient . 2.2. Correspondence of the Equal Area Criterion and the Transient Energy Method ........... 2.2.1. Introduction .............. 2.2.2. Definition of Transient Energy ..... 2.2.3. The Conservative Nature of Classical Lyapunov Methods ............ INDIVIDUAL ENERGY FUNCTION AND ITS APPLICATION IN TRANSIENT STABILITY ASSESSMENT ............ 3.1. Introduction .................. 3.2. Transient Energy of Individual Machines ..... 3.3. Analytical Justification of Using the Critical Energy of Individual Machines for Transient Stability Assessment .............. 3.3.1. Potential Energy Boundary Surface (PEBS) Method ................. 3.3.2. Local Equal Area Criterion ....... 3.4. Critical Group, Generator, and Boundary ..... iv vii ix 10 10 12 12 16 24 30 3O 32 36 42 46 53 ”'CO- .- ‘ -:... 9633' ' h . U I. .00: . 1.2. C: I! a: O... -_ “-‘W‘nu. "‘"'-5U ' o N v__ C... .- ' i I do. ’ a . JD. 0 A - . I >321 "bo --‘ ’ A . / v.~. - ‘ u- ‘1 .::‘ ‘0“. ‘v‘ ' A ‘ . U.-. ‘0 ':' "v-. Uh “- “ lh'_ -- ‘.\.-. ‘ kw" 5' I 5‘ . T... .‘I I. . 5 l; '2' on 61 ,. ‘P- (a. a Chapter 4. STABILITY SIMULATION STUDIES USING INDIVIDUAL ENERGY FUNCTION .................... 4.1. Introduction .................. 4.2. Cooper Case ................... 4.3. Fort Calhoun Case ................ 4.4. Raun Case .................... PROPOSED DIRECT STABILITY ASSESSMENT ALGORITHMS . . . . 5.1. Modified Transient Energy Method ........ 5.1.1. Fault Trajectory Approximation by Cosine Series ................. 5.1.2. Trajectory Approximations by Taylor Series ................. 5.2. Fast PEBS Method ................ 5.2.1. Efficient PEBS Algorithm for Predicting Stability ................ 5.3. Fast Equal Area Method ............. 5.3.1. Efficient Equal Area Algorithm for Predicting Stability .......... DERIVATION, JUSTIFICATION, AND VERIFICATION OF SECURITY MEASURE, PREDICTION OF PEAK ANGLES .......... 6.1. Introduction .................. 6.2. Linearized Power System Model .......... 6.3. Disturbance Model ................ 6.3.1. Step Disturbance ............ 6.3.2. Electrical Faults ............ 6O 6O 63 76 85 97 100 100 102 105 110 112 115 117 117 124 125 126 . O 9:. :p I .U U. C" ‘7‘ (I. ‘- . . C I Y! .. RA VI ) ‘I l P In 9 In vi Chapter 6.4. Linear RMS Coherency Measure .......... 128 6.4.1. RMS Coherency Measure for Impulse Input Disturbance ............... 132 6.4.2. RMS Coherency Measure for Pulse Input Disturbances .............. 137 6.4.3. Justification of Nonlinear RMS Coherency Measure ................. 138 6.4.4. Theoretical Justification of Fault Security Measure for Second Order System 142 6.5. Computational Algorithm for Infinite Interval Pulse Coherency Measure ............. 147 7. STABILITY ANALYSIS USING FAST DIRECT METHODS AND COMPUTATIONAL RESULTS ................. 153 7.1. Introduction .................. 153 7.2. Cooper Case ................... 153 7.3. Fort Calhoun Case ................ 161 7.4. Raun Case .................... 161 8. REVIEW, CONCLUSION, AND TOPICS FOR FUTURE INQUIRY . . . 168 8.1. Chapter Review ................. 168 8.2. Topics for Future Research ........... 173 APPENDIX A .......................... 175 A.1. Computation of Vk+1 ............... 175 A.2. Computation of v0 ................ 177 BIBLIOGRAPHY ......................... 179 .— In. I- 0‘ J- (I. I... '\I A .‘n n:‘ "Udv~‘ “r‘90. .- hv ‘ . .00 ' p. _'O-; A P'Q.‘.-‘I \ . ~H Table 4.1. 4.2. LIST OF TABLES Reduced Iowa System generator data and initial conditions ...................... Potential energy boundary surface energy as a function of tc for determining tcc ............... Minimum of AE2(tB,tC )= A1(tC ) + A2(tB,tC t) for critical generator 2 using stability simulation program Minimum of AE16(tB = for critical generator 16) using 1(sEabiliEyt8 simulation program ........................ Potential energy boundary surface energy as a function of tc for determining tCC ............... Comparison of fault trajectory angles at t= 0.4 sec. (fault on bus 15, New England system) ......... Generator peak angles( - ‘91 ) for Cooper using the RMS coherency measure and simu ation programs. Clearing time = .192 seconds ............. Generator peak angles( - '91 ) for Cooper using the RMS coherency measure and simu ation programs. Clearing time = 0.210 seconds ............. Determination of maximum potential energy (PEBS) for Cooper using the fast direct method .......... Determination of minimum energy margin (EAC) for Cooper using the fast direct method ............. . Detennination of maximum potential energy (PEBS) for Fort Calhoun using the fast direct method ....... . Determination of minimum energy margin (EAC) for Fort Calhoun using the fast direct method ......... vii 62 69 75 82 83 101 155 156 157 159 162 162 O. c"- v4 C viii Table 7.6a. Determination of maximum potential energy (PEBS) for Raun using the fast direct method ........... 164 7.6b. Determination of minimum energy margin (EAC) for Raun using the fast direct method ............. I64 h '\u . o . )- v F. a . r l A: a. n .4. a 2' a: r ' O .Q Q h\- AU C .Q 3. .u» an o. . u h x‘ I. . . P u 1 . u v 6 6 )5 an 3. RU O. \Au or o\v In. gr 1 .1 S on Al. F I I O I .I O 0 Al.‘ O 0 ‘/b n I- I o I I . Al. I]. QI' Ill . .u. r r) 1.- :1 r an. s ' \flu .‘ S F c» A v H ‘4‘ F s . up 15y V .t \c N ‘v d . f u A ‘ «ll K b V I .55 I . v. 1 Q I ad'- '1!- a... a. .. I V I I .t d at by F F\ r F .t ah ah» H NU H Va A 1‘4 A a F TRIU 1' a .‘U . (U I. l. I p .1 .n\ .FI .u.‘ uf- an.‘ m: .e. h P1 ~ A 3 P. r a. o 5 6L n $1» 6.» .le .uu ,fv‘ a 3 n = z a: P = = P .3 I in QC 5: h v P-.. C. C» alu A. u n u - - -. 1/ h. s , - I rsCuMht‘u pus A1» .MN‘ n‘J. :3. -‘v “I Figure 2.1. 2.2. LIST OF FIGURES Power angle curves for one-machine infinite-bus system Stable, unstable, and critical trajectory for a three- machine system .................... Partial energy analysis. Clearing time = .1922 seconds (a) Sum of partial energies for generators 5 and 6 (b) Partial energy for generators 5 and 6 ....... Stable, unstable, and critical trajectory for a three- machine system .................... Equal area criterion (a) Power angle representation of one—machine infinite- bus system (b) Variation of energy margin vs. time ........ 17-generator system (Reduced Iowa System) [13] 17-generator system (Reduced Iowa System) [13] Swing curves for Cooper. Clearing time .218 seconds (a) Generators 2, 17 (b) Generators 12, 16 ................. Swing curves for C00per. Clearing time .220 seconds (a) Generators 2, 17 (b) Generators 12, 16 ................. Swing curves for C00per ................ Energy margin A1(t + A2(t,tc) for Cooper case. ) 218 seconds. Critical generator Clearing time = = 2 W/C = transfer conductance included N/C = transfer conductance excluded .......... Energy margin A1(t ) + A2(t,tc) for C00per case. Clearing time = .220 seconds. Critical generator = 2 W/C = transfer conductance included N/C = transfer conductance excluded .......... ix 15 28 41 44 48 55 61 64 65 66 72 73 I II | . 4.4 .u .4. ‘4. a o. u u b h. - n. I. f F D a ‘1 f P a . p . u o . . rx. r n r o u o . r o c O u a» .v; i; a a .u. .r» In ,a .9- ~ a -. r . .u a .o a v. a n \a A .a a - P o. -. a s: P -. .. n . . a . u r 3. F 2- .. p s a. . C .9» at. p v A u :- .F~ P v A b C .P. 3. s v . .u. 3' Ha I w r v !\. ’\a. . -\v u) . no. I 5 .' \. ’55 n1v ' M“ Av. . r 1 1 1 A v I Q and I“. u 0 a .0 I o n I '0 '0 'O ’.. r F r d C. C. by r n .hy .us Aachen no I ' In hi n\vl .1. . P- F\- I.) Q‘)‘ C. A. s 1 a. ‘4.- .J .t. (d ‘5 P ; a a N! 1-. . -‘I I ‘ C. s . C. L. C. . a. p: r .u u. .u r A‘.$.. r» 2.4 a n 3‘ p: e n . F C. r .V P1 a» .4 -.§ ‘4... r .1 u 1 .6, s - F s 1 r- 1 g 6... In G. a n1 ‘4- :- nN .\ .NU: ...U as 111 1 / o a i Q e 1.1 cl. 6 ,1. .55 Aliv 1. 1. I U Phi. Phi. ! C I Q . lb. Figure 4.7. 4.8. 4.9. 4.10. 4.11. 4.12. 4.13. 4.14. 4.15. Swing curves for Fort Calhoun. Clearing time = .354 seconds (a) Generators 16, 17 (b) Generators 10, 12 ................. Swing curves for Fort Calhoun. Clearing time = .356 seconds (a) Generators 16, 17 (b) Generators 10, 12 ................. Energy margin A1(tc) + A2(t,tc) for Fort Calhoun case. Clearing time = .354 seconds. Critical generator = 16 W/C transfer conductance included N/C transfer conductance excluded .......... Energy margin A1(tc) + A2(t,tc) for Fort Calhoun case. Clearing time = .356 seconds. Critical generator = 16 W/C transfer conductance included ' N/C transfer conductance excluded .......... Swing curves. Clearing time .1900 seconds (a) Generators 10, 13, 16 (b) Generators 5, 6 ............ , ...... Swing curves. Clearing time .1925 seconds (a) Generators 2, 10, 13, 16 (b) Generators 5, 6 .................. .1922 seconds ..... Swing curves. Clearing time Partial energy analysis. Clearing time = .1922 seconds (a) Sum of partial energies for generators 5 and 6 (b) Partial energy for generators 5 and 6 ....... Equal area analysis AE(t) = A1(t) + A2(t). Raun case, 6-infinite bus. Clearing time = .18 seconds. (a), (b) Transfer conductance excluded and included, respectively ................... . Equal area analysis. Raun case, 6-infinite bus. Clearing time = .1925 seconds (a) AE(t) (t) + A2(t ) vs. time (transfer conduc- tance echuded) )Areas A( )and A2(t )(transfer conductance bexcludeé)fi 77 78 80 81 86 87 89 9O 92 u“ 1 -‘C— A- .2:- u u h a U a 1 1 1 g I ~ I 1 9 1 ~ ~ ’ 1- .‘i- I. n. In" 6 .. a 1 l I (I xi Figure 4.17. Equal area analysis. Raun case, 5-infinite bus. Clearing time = .1922 seconds (a) AE(t) = A (t) + A2(t) vs. time (transfer conduc- tance incIuded) (b) Areas A1(t) and A2(t) vs. time (transfer conduc- tance included) .................. 94 4.18. Equal area analysis. Raun case, S-infinite bus. Clearing time = .1925 seconds (a) AE(t) = A (t) + A (t) vs. time (transfer conduc- tance included). (b) Areas A1(t) and A2(t) vs. time (transfer conduc- tance included) .................. 95 5.1. Determination of critical boundary for critical gen- erator using maximum potential energy VPE(tc) method . 108 5.2. Determination of critical boundary for critical gen- erator using minimum energy margin AE(tB,tC) method . . 114 CHAPTER 1 POWER SYSTEM STABILITY 1.1. Introduction Since the industrial revolution1man's demand for and consump- tion of energy has increased steadily. A major portion of the energy needs of a modern society is supplied in the form of electrical energy. A The ever-increasing dependence of societies on electrical energy requires not only the production of a continuous electric supply but also energy within acceptable Quality limits. Very complex power systems have been built to satisfy this increasing demand. The trend in electric power production is toward an interconnected network of transmission lines linking generators and loads into large integrated systems. In the interconnected power system, the ability to provide reliable and uninterrupted srevice to the loads is the main concern for both planning and operating engineers in their decision making. 1" Practical terms this means that both voltage and frequency must be held‘within close tolerances so that the consumer's equipment may OPerate satisfactorily. The concept of stability arises when the Power system is subjected to the occurrence of a disturbance. If the disturbance does not involve any net change in power, the power n . . o 2’“. f :153'1! ‘. 55. . C '1 - .‘IA ‘ 1.1 .1913. ,F r. O. o‘;;' Ar;~‘.':'l , 111.. v- ‘ ‘ \ 0 'er n. u, . I.~-b S» Ea_:‘ _ ...‘Q.g " O . .1. . 4,551.11: n; .'-':":‘\ ‘ Op.- .. .’: .1 .T j . 1. ."‘-na 1 ‘ . . ..‘ 9'.P“‘ Id“ “ 1‘6 .~.. 5: fl'fin ‘. ‘ ~». 3 ‘_.I 9 A u; " U ' a .2: o ‘ u no ‘he ‘ah‘!‘ ‘ - ‘u b V ‘a. it *6. 'T v a? a ' 1 F I ”E 9’51er 1 ‘a .PI . 'v" I" Ff N 5.3 E'l‘e n ‘. 1'. 1 ' 9 . 1 : u . F 1 T 1 ;, ‘ 1.1 “He ’0'. 1‘; . :Q‘. Q s, \ ‘~ I4” U11 9 _ S :3“? a - ~' 53'9. u-Q C \. .Ffir‘ ‘ VII We .0 f‘ :I‘f’ANr l ‘ '2‘:~| I V! v I. Nmkc. 'v -f‘ '11 :M o 5. 'Fat SQ" s..;e § 1P1“. ' :n" A‘- 5 55 jcnl ‘fi “ \‘V :3“: ‘1 3.. fl: system variables, such as the rotor angles, powers, etc., undergo a small deviation from their nominal value and then return very shortly to their original state. If an unbalance between the supply and demand is created by a change in load, in generation, or in network conditions, a transition from one operating state to another results. The behavior of the system response to occurrence of a large dis- turbance (electrical fault, loss of a generator, etc.) is called the transient stability analysis of the power system. During this state transition some of the severely disturbed generators will "swing" far enough from their equilibrium positions to lose synchronism in the process. Usually the severe disturbance under which transient stability is tested is a short circuit in the high-voltage trans- mission network. In power system terminology this is referred to as a fault. When a fault occurs, certain generators which are electrically close to the fault location are disturbed to a greater extent than the other generators which are remote from the fault. These generators tend to accelerate or decelerate, depending on the nature of the fault, from the rest of the generators in the system. If the fault lasts long enough, eventually one machine or a group of machines Separates from the rest of the system, causing instability (loss of Qynchronism). However, the power network is equipped with automatic devices that sense the existence of the faults in the network and l'I'Iitiate action to "clear" the fault, i.e., isolate the faulted Sfiiction of the network. A matter of great practical importance, therefore, is that the time required to clear the fault should be nor .h‘f .fi' J p :21: U C' b and ' o ;,.; :pQ a“ a o I» a oi. id 1 o 4 . Q. - 1 'c '5 <-.‘ I o O 4» u -p 0.. ‘10. 4 b .0" f I 5 .5 .d o. ‘ ' -~ ‘< O.;' - ~ ‘ IO .4 I‘ 10".. .n-,.. ', ' c». c. .‘ '51:“Sllfia1, NI 5‘ ' it... ‘- 0‘...‘ . :, _.: 3M4 . u.’ I a . ‘11:. ' 1: 15 <~-----. '1 h an» ,.~ I on . - . ‘ u: '..‘n‘- 0.. "vi. \- '- n .9 ‘0. . : P:n...r .- “v. v. v ‘ E0 "ftb T “on ‘ I. ~9. . 0‘ "e .‘.e .‘ I ”'J 0 It . h '5 \nl‘w ~ '53:“ '0. q 1 n - .,;".-o S“: n ‘ s ’91 ‘4 I S‘b::‘ Q P 'u' an ‘NA . mp: "R ‘b (2“ 1 1'- 1) .h'e k era. ‘3: S‘\‘ u,:.'A' 1 “a”! e“ :‘u ‘-'.' PM ~. 2 ' e‘rC‘EFf II. it: ‘p "‘93". 3‘ .1 .. 3L :I'I ‘ t‘ ' “5ch AC 1‘ | less than the duration of the fault that would create a disturbance large enough to cause one or more machines to lose synchronism. This is the so-called "critical clearing thne (tcc)’ commonly used in the literature on transient stability of power systems. This term is often quoted by researchers on power system transient stability by direct mechods as a figure of merit to be used (a) to compare results obtained by time simulation with those obtained by direct methods, and (b) to determine how "robust" a power network may be as it is subjected to disturbance. (The state vector, evaluated at the critical clearing time, provides a means of estimating the region of attraction of the pdst-fault systems.) The most widely used transient stability analysis is obtained by the time solution of the machine's rotor angles. Then, based on the observation of the swing curves and engineering judgment, the stability or instability of the power system is decided. However, there are some disadvantages of this technique such as: (1) Stability (or instability) depends on network configura- tion and the type of disturbance (2) The computation is cumbersome and time consuming for a large system. The drawbacks of the time solution and the need for fast, com- 13utationally efficient and approximate transient stability analysis ITIiade researchers inquire into an alternatiVe approach. As a result, tlie concept of direct methods of stability was pursued. From the I early stages of development, the direct methods of Lyapunov and the o a..- :I;I" F u - I J v- u “ - \Ozu- o. '=-‘ . .l a Q 'I‘ ‘4' I. O n -1 '1 3 ". ~1 0 “ "3":"H- ~- -0. .. . 'z. O\ ‘P .n. v . .A. 5. ..SI energy function analysis showed promise of assessing transient stability rapidly without the computation required to integrate the many system differential equations even though the method remained far from implementation. The use of such a method for contingency analysis in expansion planning, operation planning, and on-line operation was exciting. It is clear, however, that such approxima- tion methods would never replace time solution for accurate stability assessment. The historical development of the direct methods for transient stability in this area is divided into the following distinct but continuous phases. (1) The work (Hi Magnusson in 1947 [30] considers a classical model representation of the power system. In this representation the transfer conductance is omitted and an energy function for the system is evaluated. Then the critical energy, by which the region of stability is identified, is determined by the energy of the lowest _§§ddle point, V15. The work of Aylett [31] is devoted to finding an lenergy integral. The kinetic (KE) and potential (PE) components of energy are identified and the stability of the power system is decided QY'determining whether KE < PE. (2) The work in phase (1), although outstanding in the elabora- titon of the concept, was far from implementation. The main issues rEflnaining were R. 0" .91.... 1 O. u. 1- " ‘~I the. .fi. - l U ' " t u 11"“ g 9.. v.~ l A (a) to resolve the difficulty in obtaining all singular points, (b) to be able to identify the correct singular point, (c) to be able to include the transfer conductances in the model, and (d) to identify the critical value of energy which if exceeded would result in loss of stability. The work of El-Abiad et al. [32] and Prabhakara et al. [33] was devoted to finding the appropriate saddle points and hence the critical energy. The work of Uyemura et al. [34] was devoted to approximating the transfer conductance term, while Smith and Tavora [35] initiated the first work toward considering the critical energy which was related to the faulted trajectory. (3) The work in references [4, 9] in the development of the Potential Energy Boundary Surface (PEBS) and [13, 14, 28] in the development of energy accounting using the transient energy function tnark the latest advances of algorithms for direct assessment of transient stability. A main point of this work is that the critical energy evaluated is directly related to the fault trajectory and hence a larger region of stability is obtained. The work in this Phase will be discussed in detail in Chapter 2. However, in spite of these encouraging results, the work in [29] showed that the true region of stability is identified by con- S‘ideration of local kinetic and potential energy of an individual l"achine rather than global kinetic and potential energies of all 'z':'=”'3 I“ I~ ‘ -'. . ‘3 3m“ ' U u add p.‘ :- F‘s vi ‘: .310 ,1 r" IJ‘Dg : ~. :11: generators in the system. This investigation attempted to identify the particular individual machine whose behavior dictates the stability of the entire system. Furthermore, it was shown that the region of stability obtained by the individual machine energy function concepts is more accurate than total system energy function concepts. In this research, it is believed that the efficiency and relia- bility of algorithms for direct assessment of transient stability using individual machine energy function could be improved by further investigating the following concepts: (1) development of a method for determining the accelerated group and the critical generator without simulating the system for the particular fault and analyzing the individual generator energy function in time frame, (2) testing of the Local Equal Area and Local Potential Energy Boundary methods on several fault cases to show the extreme accuracy and the ease in determining the critical clearing times when applied to transient stability simulations of the faults, (3) development of very fast and computationally efficient algorithms for implementing the algorithms based on the Local Equal Area Condition or Potential Energy Boundary Condition using individual machine energy function without simulation of the system and thus integration of the differential equations. (4) extensive verification of the algorithms developed on a test system and extensive fault cases to detennine if there are spe- cial cases for which these algorithms fail. \ .-;-,o 5‘ o».- p.\- O D on. .3 . To achieve the goals, the contents of Chapter 2 are devoted to describing the behavior of the power system and the concept of tran- sient stability analysis. The historical development of the direct methods for transient stability assessment is revisited in further detail. The concepts and algorithms based on potential energy bound- ary surface [4, 9] and on equal area with global energy accounting [13, 15, 28] are explained. Chapter 3 proposes and justifies two hypotheses, (a) that the stability of a group of machines and thus the system is dictated by a region of stability for one machine in that group, and (b) that this region of stability is reflected in the kinetic and potential energy of this machine. The individual machine energy function is then presented and shown to violate the conditions for it to be a Lyapunov function. A PEBS algorithm [16] is then justified that utilizes the maximum individual generator potential energy as a function of time for a fault-on trajectory (t: >> tcc) as a critical energy threshold for deciding whether the system is or is not stable. This threshold energy value is compared with the maximum of the individual generator potential energy as a function of time for some tc S t: to decide retention or loss of stability. The second algorithm, Equal Area Criterion (EAC), is then justified that utilizes the minimum energy margin AE* = A1 + A2 as a function of time to decide retention or loss of stability. A1 is the accelerating energy produced during the fault period and A2 is the decelerating energy after the fault is cleared of the individual generator with respect to the rest of "a art—r?" ,. I. i C'- H . ’1‘ 5.1: A L D s 1 u I. v VJ C v i ' a -:.:':O-' a, 'I - C-U ’ I l . . ’2’ :r I CD.» ~ Q o. .._ - x.:', ‘ 5's- . Q 9- U- ‘ . P," "'F Is. 'I I, \ p ‘l d u C :." ‘.§.: uu'_ -“:. g. F .0. a _, C... ' l 6 '1' .A. . .' ~ A . 5 D . "v .-" ‘o 3 u I. ' ‘9? RA“. ' ‘1 ‘: pF;‘ 1 ‘ “ "11:" 1." a w I.' E‘ A: ‘ by ‘ l Far. y... , ”3".5111‘ r. 'l ‘ NJ; the :J' ‘ . 5“. "'v Fl v- ' j ay:ha . 2‘9" 61., ' 'I.::e ; h .. the generators in the system. This minimum value of energy differ- ence should be less than zero if stability is to be retained for any clearing time. Then the concepts of critical group, critical generator, and critical boundary are also defined. Chapter 4 presents two algorithms based on the kinetic and potential energy conditions discussed above. Simulation results are then presented that indicate these algorithms are extremely accurate and hold significant promise for the development of both accurate and computationally efficient procedures. Chapter 5 proposes two fast and computationally efficient algorithms (Fast PEBS and Fast EAC). Both algorithms are based on the potential energy of the individual machine with respect to the other generators in the system. Computation of this potential energy requires calculation of the initial operating states, the final operating states (peak values of generator rotor angles), and the post-fault network conditions. It is shown that this information about the system state trajectory can be obtained by one of two tra- jectory approximations, the Taylor series or Cosine series, to imple- ment these algorithms without simulation. However, it is shown that the first requires extensive computation and the second is in general not sufficiently accurate. Chapter 6 derives the Root Mean Square (RMS) coherency measure for different disturbances (step, impulse, and pulse). A nonlinear RMS coherency measure based on the critical unstable equilibrium point is also derived. The impulse RMS coherency measure is then o.‘ p 6. or ‘ . u' D than A ‘- w' E. 7' .- o-wv U u .. on. o.- :¢U Sh:. a -":':F‘ 1 q I. l o F‘ ‘. P‘ C I up. ‘ . .m- shown to predict the peak angle deviation of the generator for a second order linear single machine infinite bus system model. As last step a computational algorithm for the finite interval pulse coherency measure is derived. Chapter 7 applies the two fast algorithms (PEBS, EAC), dis— cussed in Chapter 5, directly to the Reduced Iowa system. The results of these applications for different fault cases are then presented that indicate the algorithms are extremely accurate. Finally in Chapter 8 the contribution of this investigation and the avenues for further inquiry are considered. CHAPTER 2 THEORETICAL DEVELOPMENT ON POWER SYSTEM TRANSIENT STABILITY 2.1. The Dynamics of Power System During a Transient When the power system is operating in normal state, all the equality and inequality constraints are satisfied, all the generators are operating at synchronous speed, and its dynamics are defined by a nonlinear vector differential equation §= E(§.E) (2.1) where F(§,P) is the transient stability model of the power system, and 5 and P are states and parameters of the system, respectively. Of special interest in the analysis of the system (2.1) is the equi- librium, which is the state of the system where the rate 2.is zero and the system is "in balance." The most significant interpretation attached to the equilibrium in a wide variety of application in engineering is stability. Stability is commonly understood as a situation where the system is in equilibrium, and if perturbed, returns in time to equilibrium. The equilibrium operating point 351 (s.e.p.) of the system (2.1) is defined by X= E0318) =9 (22) 10 11 Upon the occurrence of an electrical fault, the power system under- goes two new phases [5]: (1) during the fault, where the dynamic behavior of the power system governed by O < t < tc (2.3) (2) post-fault (after clearance of the fault); once the fault is cleared, the system will assume a new configuration and thus its behavior will be governed by another set of nonlinear differential equations of the form where tc is called the clearing time. If, after the transition from the fault phase to the post- fault phase, synchronism for all the generators in the system is maintained, then transient stability results and the system trajec- tory will converge toward a post-fault s.e.p. 52(352’Epf) = 0 (2'5) If, after this transition, synchronism of all the generators is lost, the trajectory will pass close to an unstable equilibrium point (u.e.p.) that satisfies Ez1x“.epf) = o (2.6) One of the basic problems of stability analysis is to find (necessary and sufficient) conditions on the system parameters so 00;. ”a - ,. at O i 9" ID. . .j;'a-‘ -3. vi. ‘ ' ".1: :x" D h .‘ V Y‘\ r .. :f n\v .\ .12. ea. Q. :\ H- C .11.. Axu \ Fm 12 that the convergence of solutions to a post fault s.e.p. takes place. The most powerful method of solving such a problem is the Lyapunov direct method [6]. The method answers questions of stability without the explicit solution of the related differential equations, such as (2.1). The underlying idea of the Lyapunov method is to find a posi- tive scalar function V(§) with the rate of change V(§) negative for every possible state 3 belonging to some region of stability R except for a single equilibrium state 3* where V(§) attains its minimum V(§*). Then the function V(§) will continuously decrease along the solutions of the system until it assumes its minimum V(X*) and the system reaches the equilibrium 5*. As a candidate for Lyapunov's function V(§), the energy func- tion of the power system will be chosen. The analysis of kinetic and potential energy components at two different instants of time has been used to conclude transient stability or instability of the system. The next section is devoted to the transient stability analysis from the energy point of view. 2.2. Correspondence of the Equal 1Area Criterion and the Transient Energy Method 2.2.1. Introduction Before the occurrence of the fault, the power system is operat- ing at the nominal state (pre-fault s.e.p.) and the machine veloci- ties with respect to a synchronous reference are zero. The fault changes the network configuration of the system and the machine .I. g “A t.' I" >91 - 1 .g . 1,...” .in ‘ N i . C I d :"'F4"pn .1 a UV. ' = " 3.? ‘qu . I. 9" hrn‘ ‘. rH 5 o . h 1 “ 1 i s; u o: ‘A I .h-V'I'S ' O 1»: 'u U“: A: 9 . '1 ~91“. 1,.) a {n O \ ‘1’“? ., “r s I .~€ A; . (‘11QP‘ 5“ ‘1' ~ -. ' '1.” FA“ ‘ “ -: :!\‘ 1-. A :h§ . '\ 3‘ 1 'b. n... s 1 ' ‘71s V ‘l : H1 ‘ \ AU“; ‘1‘ 1“ h,CIF1 13 velocities will increase, thus increasing kinetic energy. Obviously, the acceleration that increases kinetic energy also moves the system from its pre-fault s.e.p. After the fault is removed, a new network configuration results and the excess kinetic energy produced during the fault period is distributed in the post-fault network according to the network requirements. If the motion of the acceler- ated generators with respect to the rest of the system's center of inertia is reversed due to the kinetic energy distribution, then the system converges toward the post-fault s.e.p. where again the machine velocities are zero. If the motion of the accelerated generators with respect to the center of inertia is not reversed, a loss of stability occurs. The capability of the post-fault network for pro- ducing restoring forces is measured by the potential energy of the network elements. The transient energy function contains both kinetic and poten- tial terms. The system kinetic energy, associated with the relative motion of machine rotors, is independent of the network. The system potential energy, associated with the potential energy of network elements and machine rotors, is always defined for the post-fault system, whose stability is to be analyzed. The principal idea of the direct methods is that a system's transient stability can, for a given contingency, be determined directly by comparing the total system energy which is gained during the fault on period, with a certain critical potential energy. For a two-machine system this critical energy is uniquely defined and the direct analysis is equivalent to the equal area criterion. a, o :a;p ' to; I e o n b- N C 1 . i O I. n u v n V . o ';P"f (n "‘ '5 do A 1‘ :. ‘1 .: AP..:'A‘ ' —- .. :'. ~01. r. ‘ P v‘¢-. V. ‘.‘ ‘H 1 ' C . 0A 0 'P -I\ T :1 I e 5 V ._ . Go: | I ‘ pl :4 5 .I' - car 1 v. a“? 's‘ :S a- I Q I, a C‘ “C. F a :F:‘ . .‘jls'. 11' §' . 'P. :5'« ‘. ~ ‘ S 1 .1. I "" :9‘ho Hi.“ 'h "-9 '11-; V1 4‘ 14 The correspondence of the equal-area criterion and the transient energy method for a two-machine system is illustrated for the equiva- lent single machine infinite bus system in Figure 2.1. In this figure the kinetic energy KE(6C) = A1 gained during the fault-on period (using equation (2.3)) c a V(5°) = KE(6C) = [ F1(X,Pf)dX 551 is compared to the critical potential energy PE(6C,6U) = A2 (using equation (2.4)) a” f v(a”,6c) = PE(aC.s”> = f Few“)?dx <5c For A2 = PE(5U,6c) ; KE(6C) = Al, the system remains stable and for A2 = PE(6u,6c) < KE(6C) = Al, the system loses stability. For a system with three or more machines, the direct analysis becomes more difficult. In this case the critical energy is not uniquely defined and its detennination becomes the key step in the analysis. According to the Lyapunov-based theorem [7], the critical energy is chosen to be the potential energy at the unstable equilib- rium point. This unstable equilibrium point is called the lowest saddle point [10]. This critical energy frequently yields results that are very conservative, especially for large systems. 15 PREFAULT POSTFAULT /A" FAULTED z . ‘ \ 651 652 6c 6U Point 0: prefault operating point; 6 = 651, t = t; Point a: electrical power at t = t3, 6 = 651 Point b: electrical power at t = t2, 6 = 6c Point c: electrical power at t = t2, 6 = 6c Point d: operaggng point when transient subsides, t + a, 6‘6 Figure 2.1. Power angle curves for one-machine infinite-bus system. 16 2.2.2. Definition of Transient Energy The previous section describes the sequence of functions (2.1- 2.6) that characterize stability or loss of stability and the energy transfers associated with this transient stability problem. The equal area criterion, which precisely describes the energy transfers for a single machine-infinite bus (equivalent infinite machine that represents the rest of the system) model, is discussed based on Figure 2.1. Although the equal area criterion has motivated the methods used for multimachine power system, these methods have not yet been shown to achieve the desired accuracy. llfis subsection will develop the nmltimachine classical tran- sient stability model and present the "total" energy function for this model. Then the areas A1 and A2 for the equal area criterion will be defined using this energy function with the appropriate model (the fault model for A1 and post-fault model for A2), the appropriate terms in this expression (kinetic for A1 and potential for A2), and the proper limits of integration (651 and 6C for A1 and 5c and 6" for A2). The PEBS and UEP methods, which are both based on the definition of energy Vcl and Vcr for post-fault network and related to A1 and A2 respectively, are then discussed in subsection 2.2.3. The importance of the discussion of the equal area criterion is (1) its use to justify the PEBS and UEP methods for the total energy function in this chapter and the PEBS and equal area methods based on an individual machine energy function in the next chapter; and (2) the results in Chapter 4 that show the equal area criterion ...-¢ doc-- a!- a\d I... .4 'V . Frl-r . u"‘~ ‘ C -.\v is . ‘v ass as VI! 17 based methods for the individual energy function are extremely pre- cise and accurate for determining region of stability. These results using the individual energy function are much more accurate than the results obtained using the total energy function. The dynamic behavior of the power system is described by two sets of differential equations. One set describes the internal structure of machine quantities and their mutual relationships, and the other set relates the tenninal voltage and current of each machine to those of the other machines [2]. Because of the fact that a syn- chronous machine has several coupled circuits, inclusion of one or more of the coupled circuits within the machine increases the com- plexity of the power system model. The complexity will become more apparent if a multimachine power system is considered. For the pur- pose of investigation of stability for approximate and easily computed transient security assessment, a simplified classical model will be used to determine the dynamic behavior of the power system. The classical model is characterized by [1] (1) Mechanical input power is constant. (2) Damping coefficient, both mechanical and electrical, is neglected. (3) The voltage behind transient reactance of the synchronous machine is assumed to be constant. (4) Loads are represented by constant impedances. For the system model being considered, the equations of motion are: P‘p NE 1'- ‘9 h Y) rr‘ [It 18 where n i=1 , #i _ 2 pi ‘ Pmi ' EiGii Dij = ElElej and, for unit i, Pmi = mechanical power input Gii = driving point conductance E1 = constant voltage behind the direct axis transient reactance mi,61 = generator rotor speed and angle deviations, respectively Mi = moment of inertia Bij(Gij) = transfer susceptance (conductance) in the reduced bus admittance matrix The transformation of equations (2.7) into the center of angle coordinates provides a concise framework for the analysis of systems with transfer conductances. Define: ECTE b o; N I in 3t E. a-.. Gk 19 n 1 so 9—1? M16] n . A MT = "1 i=1 then n n n-l n . = - = - A ”Two 2”? Pei) Pi 2: Z Dij C05 513' = PCOA i=1 1=l i=1 j=i+1 50 = mo (2.9) By defining new anlges and speeds relative to the center of angle reference, 9i Q 61 - 60 and Di é mi — mo, the system equations of motion become {1:113 i=1, 2, ....,n (2.10) 2111.91. = . Mia, = o (2.11) The transient energy function V is obtained from equation (2.7) by first establishing the n(n - 1)/2 relative acceleration equations, multiplying each of these by the corresponding relative velocity and 20 integrating the sum of the resulting equations from a fixed s.e.p. (as) to a variable upper limit [4] n-l n 2 2 Wm.- wine». w.» i=1 j=i+1 n-l n =2 (MJP1 - M1PJ)(w1 - wJ) i=1 J=l+1 n-l n i=1 j=i+1 n-l n a __1_ 2 _l_ s V Z [ MT M1MJ(‘”1 ‘ “3) 'MT (PiMj'PJMiH‘Sij'Gu) i=1 j=i+1 s - Cij(cos 513 - cos Sij) 61+6j-250 + s s s Dij cos Sijd(61 + SJ - 250)] (2.13) 6 +6j-26 The system transient energy components in equations (2.13) are identifiable. The first term is the kinetic energy. The second term is related directly to the rotor angle position of generators, so it is called position energy. The third term is magnetic energy and the fourth term is the dissipation energy, which is the energy dissipated in the network transfer conductances. It is common to use the term "potential energy" to indicate the last three terms. Tc :1" te °. .. n- I'ecc :. . 5 ‘-. = u " ' Q |v ] ‘1. q I Ie -h'(‘r u' § ' "Ar ‘ s 5“. q S En‘ q. .1» I!“ yd- “ ,,$ 3'9 ”Wu v ~(I: 7'- M“ lie!- ‘ ‘ e - " ‘ '\ U ‘Q I H's'c- H1 ‘9'. ~ . A. 'i. ”a 5": 0 ‘u ) I‘e . . 1 re .4; . ~‘. “In“; 21 To write the energy function in a more convenient form using the center of angle variables, apply the above steps to the n center of angle acceleration equations (2.10) n n V =§ZM15€ 2 PM ‘ 6?) i=1 i=1 n-1 n -2 Z [Cij(cos 91.3. - cos 6%) i=1 j=i+1 [6 +62 6s 6s Dij cos eijd(ei + Gj) ] (2.14) where eij = e. - e The physical significance of the center of angle reference in the transient stability problem formulation is illustrated by the fact that, for systems without transfer conductances, the equilibrium points are obtained by solving n - 1 real power equations (2.10, 2.11 where $1 = 0, i = 1, 2, ...., n) for an n machine system, given appropriate initial angles [11]. This works satisfactorily for such systems because there is no change in load. For systems with transfer conductances, however, the total load will differ from one operating point to another and therefore a closed form expression for the total system energy cannot be obtained. Many previous researchers have neglected transfer conduc- tances, i.e., real part of the off-diagonal elements of the reduced bus admittance matrix, which depend not onlyCNithe transmission line these 3'3 :mg v~j (d ‘5 . U a. q ... l. ... ‘ u v G 9:04». - . C -\ :vr' ’- l 'h D 22 resistances but also on loads modeled as constants impedances. There- fore, these terms can be large and in general cannot be neglected. For transient stability assessment using the transient energy function as a direct method, an approximation to the conductance energy term is necessary since the conductance term is path dependent and the entire trajectory must be known. A linear trajectory in the angle space is assumed to express the integral term as ei+ej Iij = Dij cos eijd(ei + ej) 6 - e I (.4 m-I-m mt—Jom . . s [Sln 6.. - Sln e..]~ D.. (2.15) 6i '9j-9.+e. 1J 1J 1:] ...; L; The transient energy function (2.13) or (2.14) for a two- machine system with the assumption of zero transfer conductance is analogous to the equal-area criterion. Considering the situation at clearing time, the transient 1 energy function, using the post-fault network and as as reference, is given by [13]: 6 2 = = ~c _ c _ $1 _ c _ 51 V V '2 M(w ) C12(cos 6 cos 6 ) P1(e e ) 9 (2.16) The first term in the right hand side of (2.16) is the kinetic energy produced during the fault period and is proportional to the area oabf of Figure 2.1. The second and third terms add up to the n-t‘ Ivo< .0. ed. 5. We 0. C. .u. u 23 potential energy of the system during the fault which is associated with the (area cdfc- area oed). The clearing energy is thus the area oabf + (cdfc- oed) . 52 The critical energy is evaluated from post-fault s.e.p. e to u 2 the u.e.p., e . The velocity of the system at both 65 and e” is zero and thus the critical energy consists of only potential energy, V = V (2.17) The right-hand side of (2.17) corresponds to the area cgf + cdf. 2 as a reference [4], the transient energy 5 When vcr use 6 function is not analogous to the equal area criteria. However, Fouad et al. in [13] argue that the critical energy shouldtxeevaluated. from 651 and thus a correction term of 952 vcor=v (2.18) 51 9 must be added to the critical energy, Vcr' eu B52 Su Vér = Vcr + Vcor=v +V =V (2.19) e52 e51 651 Vér then corresponds to the area dgcd - oed and contains the area cdfc - oed in addition to A2. The VC1 in (2.16) contains area .1. v.. .1. ~-.' *5 ... I. Ann P ‘4. nhi :u n‘. \h\ h 1. D I 24 cdfc - oed in addition to A1. Thus, the condition VC] 5_Vér with the correction Vcor in Vér corresponds to the equal area criterion (A1 5_A2). The critical energy can be found for multimachine power systems by computing the proper u.e.p. a” and knowing initial operat- ing point 651. 2.2.3. The Conservative Nature of Classical Lyapunov Methods Unstable Equilibrium Point (UEP) Method. In a large inter- connected network a fault followed with or without line switching will result in a mode of stability which is different for different fault locations in the system even though the post-fault configura- tion may be the same. For a given post-fault configuration, there are several singular points among which one is s.e.p. and others are u.e.p. or saddle points. Depending on the location of fault, severity, and type of fault, the post-fault trajectory if cleared at t = tcc will pass in the vicinity of one among the saddle points. It is not possible to find a unique closed surface (boundary) in the state space separating stable and unstable regions which will give accurate results for all faults. In the past, the critical value was obtained as Min V(§) evaluated at all the saddle points. Fouad [13] and Athey [14] showed that for t = t the post-fault cc’ trajectory just becomes unstable and theoretically passes through but practically approaches very close to the u.e.p. for a specific fault. The critical energy (boundary) is defined as the energy of the system when the post-fault trajectory passes very close to the u.e.p. For different fault conditions, there is a different fault 'n‘5r’aw 2" ......v j u a N J- a” n #5:. . . 3‘53 ft” *7- ,H 31 L s ., id. Sta: S r- .U- ' ~ s l U - . me SE: A: w gal Sgess J ~ (I) ‘H 2.3n, u :e 1- '” 1°21 i‘Dr . 513+ “i‘ U 25 trajectory and a different u.e.p., and therefore the computation of critical energy for each fault is a difficult problem in the multi— machine case. A method of selecting the proper u.e.p. from the possible 2"'1 (for n machine system) has been suggested in [14]. For systems without transfer conductances, the calculation of the unstable equilibrium points using a Newton-Raphson or Gauss Siedel technique can be used successfully. The recommended procedure consists of solving n - 1 real power equations (2.10) having $i==0, i= 1, ...., for an n machine system with one reference machine. An initial guess for the u.e.p. for machine i going unstable given the post- fault stable equilibrium point 652 is . 9? iii e‘!= (2.20) J n- 952 j ,1 i j K If the set of generators {ik} are assumed to go unstable, the k=1 initial guess for the unstable equilibrium point is 52 . . . . u ej J # 11, 12, ...., 1k e. = (2.21) J - 952 ° = 1' 1° 1' , 17 j J 1, 2, ..... , k This works satisfactorily for systems without transfer conductances since the starting values are close to the solution, and there is no change in load. For systems with transfer conductances, the total load at a u.e.p. will differ from that at the s.e.p. The difference will be n ..--A. ‘. \ 9' o'c- O 6: ~ 1. § '- . C“ p ..‘d ‘ Q .I."--‘ 26 allocated to the reference generator bus resulting in a substantial mismatch. The scalar function J, defined by .1 é: (P. - MT PCOA) (2.22) is the objective function whose minimization provides an alternative approach to the calculation of the u.e.p. Even this algorithm does not readily converge based on the initial guess given above and in this case the angle from the transient stability simulation when the potential energy is maximum is used. Having selected the proper u.e.p. and computed it, the pro- cedure used to test for whether the system is stable or unstable is based on n-1 n +2 2 [Cij(cos 8% - cos 9:93.) i=1 j=i+1 u u 51 51 e. + e- - e. - e. 1 j 1 j . u _ $1 + 013 a” e” 651 + 851 (s1n eij sin 9. J.)] (2.23) i ’ j ' i j A-§ . u \ ‘5- .5 A b «n u 27 BC 11 n V -V(ec)=V =IZM"C2 ZP(eC-651) cl ‘ $1 2 1‘”: " 1' 1' 1' 6 i=1 i=1 c +2 Z[C1.J.( cos 61.11- cos 61.1.) 1j=-1+1 c c $1 51 e. + e. - e. -e . 1 J 1 J - c _ $1 + Dij 8c - 6c - 651 + 951 (s1n eij sin 611)] (2.24) i j i . _ 1 where D.j and C.j are the parameters of the post-fault network. For V" >V cl system is stable and for V“ < Vcl system loses stability. Potential Energy Boundary Surface (PEBS) Method. The potential energy boundary surface method [3] is an abstraction which examines the potential energy function VPE(6) in the angle space for a multi- machine power system. For a three machine system, the corresponding potential energy surface is shown in Figure 2.2 as are the actual stable and unstable system trajectories and the corresponding saddle points. The line joining the saddle points and orthogonal to the constant Vp contours is known as the potential energy boundary sur- face (PEBS) (dotted line). A faulted trajectory if cleared at t > t will cross the PEBS at some point on the curve. The critical CC trajectory (fault cleared at t = t will just touch the boundary cc) but does not cross it. The value of Vcr can be taken to be the value of VPE(gB) (2.23) for a fault-on trajectory at the point 68 ‘. A: 28 (0.0) 9 Figure 2.2. Stable, unstable, and critical trajectory for a three- machine‘system. 12 ... 29 when it crosses the PEBS. At the PEBS crossing, VPE(eB) has a rela- tive maximum and Vk(&), kinetic energy, has a relative minimum which is near zero and assumed to be zero in this method. This is the Kakimoto method [3] of detecting PEBS crossing. The stability of the system is determined by the following comparisons: (1) If V(6C) 5-Vcr = VPE/max’ then the system is stable. (2) If V(ec) > vcr = VPE/max’ then the system is unstable. Computational experience [8] indicates that the PEBS method of computing Vcr is extremely fast and results are reliable and accurate for only first swing stability cases. In the latest effort on transient stability analysis, it is claimed that the kinetic and potential energies of the "individual" machines (and not the "total" energy) must be considered for accu- rately estimating the stability boundary (critical clearing time). The next chapter is denoted to derivation, discussion, and compari- son of the individual energy function derived by two different methods. Ii 1 f k /\ ‘1 ‘II‘ c If Q . l '0: l P o\ I V I "4' Au» a ”1‘ Au\ F1 411‘ 1‘ . I. 3. -.I at a 3 ‘4. a: \J u . .I n. F q\~ F as. s - Ea.- an. LP. .- J A V \f- :u 1.11, :- c . .\ . D I i g. 1.51.“ \ . A 5 Pi: -\§ A\V -\e CHAPTER 3 INDIVIDUAL ENERGY FUNCTION AND ITS APPLICATION IN TRANSIENT STABILITY ASSESSMENT 3.1. Introduction In Chapter 2 an attempt was made to investigate the recent development in assessment of transient stability by Lyapunov's direct methods. The concept of potential energy boundary surface [4] and energy accounting of the total system energy [13] was introduced. Several stability criteria based on "total" system energy were iden- tified, and the boundary energy Vcr was calculated alternatively by (1) computing the energy at the lowest saddle point (u.e.p. with minimum enerQY) a”; (2) computing the energy at the u.e.p. e” closest to trajectory; (3) computing the value of the potential energy of the system for a fault-on trajectory when it crosses the PEBS boundary. It was pointed out that the region of stability evaluated based on total system energy produces conservative results. For a larger and more accurate region of stability and boundary (critical clearing time), the stability of the power system must be investi- gated in terms of the energy components that truly reflect loss of stability for a particular fault and post-fault network. As a first step, the separation of the total system energy into "within" and 30 ‘.' .h; "1 5|. ‘6 “Joh;p J a. acgwm \ mate i‘ ‘ ad 9‘ ’u «a. M. 31 "between" coherent group energies was considered. The kinetic and potential energy between the accelerated group and the stationary group of generators, which can be considered an equivalent single machine or infinite bus, was considered as the “between" group energy. The "within" group energy was defined as the kinetic and potential energy within both the stationary and accelerated groups. Although the PEBS and UEP methods described in the previous chapter are more accurate if the between group and within group kinetic and potential energy are accounted for at clearing time and at the potential energy boundary surface, the results are still conservative. The most recent work [16,29] indicates that a much more precise determination of the region of stability and the boundary (defined by the critical clearing time) can be made if (1) an individual machine is identified as the critical generator upon which the sta- bility of the system depends for a particular fault; (2) an indi- vidual machine energy function is.defined and then developed for the generator determined as critical for the particular fault; and (3) appropriate methods are developed for defining the boundary of the region of stability. These methods are developed based on the equal area criterion which’precisely determines the boundary of the region of stability for the single machine infinite bus model. The methodology based on (1)-(3) above is shown via computational results in Chapter 4 to precisely detennine the boundary of the region of stability for a particular fault for several fault cases on a multi- machine power system model. ,. [‘3 oc'npaa :- hr‘d‘c u F":P:.R ’- ». ‘ _ h. ' 32 The individual machine energy function is derived and the methods for detennining the boundary of the region of stability for a particular fault are developed in sections 3.2 and 3.3, respec- tively, assuming knowledge of how to identify the critical generator that detennines stability or loss of stability for a particular fault and for which the individual machine energy function is written. Section 3.4 then develops a procedure for identifying the critical generator. 3.2. Transient Energy of Individual Machines For the classical model described in section (2.2.2) of Chap- ter 2, an expression will be derived for the individual machine transient energy. Some energy functions describe the system tran- sient energy using a synchronous frame of reference [9, 12]. Others have used a center of inertia (COI) formulation [13]-[15]. The fol- lowing two subsections describe the derivation of the individual energy function with these two different reference frames: (1) Center of Inertia (COI) Reference. Consider the mathe- matical model described by equations (2.10) in Chapter 2. Multiply the ith post-fault swing equation (equation of motion of critical generator by 61 to obtain 2 . ( M1 . M.w.6. = P. - P ---P )6. 1 e1 T C01 1 Integrating (3.1) with respect to time, using as a lower limit 51 51 t = t$1 where 6 = 0 and 6(tsl) = e is the s.e.p., yields Owl/b 1' y. .11 g . “f\ I. \1‘ v4 “ n 61 1 2 _ $1 -2M1w -P1(e1-e )-Zf51 C1J s1n e1Jde1 j=1 e1 in n 61 M1 6i '=1 6S 6? i141 ‘ ‘ n e. _ 1 ..2 51 Z 1 . ._ sl 3"]. 61 jfi " 91 M1 91 ._ 51 T 51 in 1 = 1, 2, , n This integral is evaluated using the values of e1, 61, 61, and using the values C1j, D1j, and 9:1 for the post-fault network. The first term in the right-hand side of (3.3) is the kinetic energy of machine 1 with respect to the system inertial center. It is customary to consider the remaining terms as a potential energy. Thus (3.3) can be expressed as V1 3 VKE. + VPE. (3-4) 1 1 Since (3.3) contains path dependent integrals, the verifica- tion of V1 as being Lyapunov function cannot readily be detennined analytically in closed form, and the transient stability analysis '- law ..I D. Av (“F 34 using this energy function requires simulation of system trajectory for critical generator. The above derivation of the individual energy function can be found in [16]. (2) Synchronous Reference. Consider the mathematical model described by equations (2.7) in Chapter 2. For generator i (being known as critical generator) and an arbitrary generator j (j = 1, 2, . i-l, i+1, ...., n) the equations of motions of these generators are written as 3 E II .0 1 .0 (3.5) 3 E II .U 1 1:) Multiplying first the equation of generator i(j) by moment of inertia of the other generator Mj(M1) and then subtracting these equations from each other results in (3.6) j = 1, ...., n i f i Equation (3.6) represents the equation of motion of the relative velocity of generator i with generator j. Multiplying (3.6) by (61 - wj) and adding these new n - 1 equations to each other yields 35 n n 2 M1MJ(w1- wJ)(w1- (113) =Z(MjP1 - M‘inHwi - wJ) 1=1 i=1 in in n - 26111161 - M1PeJ)(w1 - 1.1) (3 7) 3:1 3f1 Integrating (3.7) with respect to time, using as a lower limit t = t51 where w51 = D and 5(tsl) = 651, results in the energy func- tion of critical generator 1 n n 1 :2 2 2 1 :2 : 51 J: J: jfi 3fi n 51 - ZC1j(cos 61.3. - cos 51.1) i=1 ifi + 0.. cos 5..d(a. + a. - 25 ) 3.8 551+651-2651 13 13 1 J o i j o This representation of the individual energy function, derived in [29], contains only one path dependent integral and requires trajectory simulation for the transient stability analysis of the system. However, in order to obtain the critical energy Vcr’ which is the energy at the u.e.p. corresponding to the actual boundary of separation, a linear trajectory in the angle space is assumed. This ,‘_.‘O( a. rd nu‘ p\o . .l q 1 Nu ..l I G h v At F, .4,,- a VW ‘5 Fl A... 9 r I I O I u Ah- - I V 1' ... .. . .r. .. . .r: OK. 36 allows the term in (3.8) to be analytically evaluated between the limits 651 and 6” with the expression given in (3.8a) U U U 6 . ~260 1.+6J u u $1 $1 60 +60 - 6. - 5. = 1 j 1 j . u _ . 51 013 6” _ u - 651 + 651 (s1n 61j s1n 61j) (3.8a) i j i j Testing the above technique of transient stability assessment on several moderately large power systems showed that the approxima- tion is quite acceptable. The stability criteria based on the total system kinetic and potential energy for assessing the region of stability of the system could be applied to the individual generator kinetic and potential energy function with more accurate results. The following section justifies the idea of using the individual energy function in tran- sient stability assessment of the power system. 3.3. Analytical Justification of Using the Critical Energy of Individual Machines for Transient Stability Assessment One way of assessing the transient stability of a power system is by comparison of the critical energy, the energy at the lowest saddle point, with the clearing energy V A second method is cl' based on the concept of potential energy boundary surface (PEBS) as discussed in Chapter 2. In this method the critical energy is con- sidered to be the potential energy at the crossing of the total 'a*;ertna.1 ’. Twas: sache 5639C Or; 51"?“ ’E'§T'C."S 0‘ 35° ’hocra-- ... -. 'sgiar. R = {1 (I be the :5- .-=0 := 1r Miter) (1) its Rd :2) 915-- . Fe gr'.: 3'” 35' ' To ottai. 139 of the the: “gazetheses of 1 3313111 that tie Hp“. H 9‘9 Edi”? to t 37 system trajectory and the PEBS. It was pointed out previously that the region of stability obtained by the application of the PEBS is larger than the one resulting by consideration of the energy at the lowest saddle point. Theoretically, these different approaches are based on similar Lyapunov stability theorems but with different regions of definiteness. This is clarified by the following theorem. Theorem--Let V(§) be a scalar function. Suppose that the region R = {§|V(§) < k} which contains the origin is bounded. Let V(§) be the derivative of V(§) along the solutions of X = f(§); i(g) = 0. IF V(x) is positive definite and 1(5) negative definite in R, then (1) the origin is an asymptotically stable equilibrium state, and (2) every solution of 8 = f(§) starting in R converges to the origin as t + m. To obtain the best estimate for the domain of attraction by the use of the theorem, one chooses the largest value k = k for which the hypotheses of this theorem are satisfied. In the following, it is shown that the individual machine energy function of the critical group of machines does satisfy the hypotheses of the theorem. But before proceeding to the proof for the critical groupcnimachines, it is appro- priate to show how the conditions of the theorem can be shown to be satisfied for the total system energy function. The sign definiteness of the total system energy function V and its derivative V cannot be proven when the transfer conductance term is included in V [14, 18]. 38 However, the total system energy function restricted to a suitably small region of interest can be shown to be positive. The derivative of this total system energy function without mechanical damping is zero for t1; tc based on simulation results for all system trajec- tories. This observation is then coupled with a proof that the effect of the mechanical damping is negative definite to suggest that V(§) ;,O. This same "proof" has been attempted for the indi- vidual energy function. This proof is now outlined and then shown to be incorrect. This suggests that the individual energy may not be a Lyapunov function and thus that one may not be able to apply normal Lyapunov theory to establish a region of convergence. The excellent computational results obtained using the individual energy function and the physical understanding of how loss of stability occurs in relation to the individual energy function are the basis for using it at this point in its development. Consider the mathematical model including the mechanical damping where D1 presents the mechanical damping for i = 1, 2, ...., n. Writing the equation of motion in terms of center of angle results in (3.9) , M1. M1. " Mi‘*’i = (Pi ' Pei ‘ Diwi) ‘ "M‘T‘ PCOA I if}: Djwj 1‘1 M. n - I 1 3:1 ' I. he :er‘vat‘ ' :2”: i=2 4 r“*—1 ‘~"0 PEQ-fi u. n n; ’3‘ sq In ‘h“ . e 1 n We: tive (1) n 551 . a] Suoreg1 / .2) ,1 1fl : S'mUIEti Uflci 'i70n . IS 21 I (3 J to 5'rfd4‘. LADDS : ‘3 If t; 39 From (3.9) the contribution of the mechanical damping to the time derivative of the energy function is n M n 1 ~ ' Z [Diwi ‘ TM; 01% I “’1 1=1 j=1 n n n _ .. _ 2 _ -[Z 01,,1m1 - 0] - {2 011,1. - Z 01w1w0] (3.10) i=1 i=1 i=1 The right-hand side of (3.10) is negative, if no t A method for determining stability cc' requires the determination of the maximum potential energy along the trajectory for a particular fault as a function of clearing time. The clearing time at which this function is maximum is the critical clearing time tcc and the system is stable if tC1; tcc' A second method, which is not as accurate, uses the potential energy at the PEBS for a fault on trajectory as the critical energy Vcr based on the incorrect assumption that the energy along the PEBS is constant. This method may not be accurate in determining tcc or determining whether the system is stable or not if the peak potential energy along the trajectory is greater than this value of Vcr’ How- ever, the method will determine whether the system is stable if the peak potential energy along the trajectory is less than Vcr since at this value the trajectory never reaches the potential energy boundary surface. Figure 3.2 indicates selecting vcr as the value of the potential energy at the PEBS for a fault on trajectory results in an interval around tcc where stability or loss of stability cannot be determined. However, for development of fast on-line stability assessment by operators, this interval of uncertainty appears to be 44 (D (0.0) 12 Figure 3.2. Stable, unstable, and critical trajectory for a three- machine system. arpak' 'uu-yi Q TCV‘EI frzrsi A1... 116. '. C .51 ,t a": I!) 1, O 1. >'_ 13‘185 45 acceptable in terms of the accuracy required in determining tCC or more likely the stability for a particular fault. The procedure for transient stability assessment using the transient energy of individual machines (or groups of machines) is outlined below: (1) For the post-fault network, the stable equilibrium point 651, the admittance matrix YBUS’ and the parameters C11 and D1j for all i, j = 1, ...., n are determined. (2) For the particular fault and a fault on trajectory where tC > tcc’ using the special computer program compute the energy values M n n = i _ £2_1_L_:E: _ _ sl i=1 i=1 in jfi ” 51 a1. + 53 - 5131- 631 - :5: C1j(cos 61j - cos 613) + D1j 6 - 6 - 651 + 651 j=1 i j i . j #1 , . . sl (Sln 611 - s1n 61j) (3.12) 46 for i = 1, 2, ...., n along the faulted trajectory at every time interval At. (3) By examining the value of V1 = VPE + VKEi and its com- 1 ponents at each time step, determine the value of V1 where VPE- has 1 maximum value and the value of VKE- is near zero (this particular 1 value of energy V1 is called the critical energy and defined as vi/critical = VPEi/max) and store for each fault location. (4) Repeat step (3) for the same fault for the particular clearing time tc < tcc to be investigated and determine the maximum potential energy along this trajectory and let this value of maximum potential energy be denoted as Vi/t=tc' (5) Check for the stability of' the machine i. If Vi/t=tc5=vi/critical’ mach1ne 1 1s stable, and 1f Vi/t=tc> Vi/critical machine i would be unstable. 3.3.2. Local Equal Area Criterion The second method for determining the region of stability boundary of stability for a multimachine power system is based on the well-known "equal-area criterion" of one machine infinite bus systems. The local equal-area method is an extension of equal-area criterion in the sense that a particular machine of the critical group is considered against the rest of the generators in the power system. Then by comparison of the energy transfers between this particular generator and the rest of the system during and after the fault period, a decision on the stability of the entire power system is made for a fault at or near this generator. The discussion FF "P' w 1.. .1 '1 I" z" 47 concerning determination of the critical generator is left to the next section. In order to clarify the concept of this equal-area criterion applied to a multimachine power system, the equal-area criterion applied to a single machine infinite bus is reviewed. For the one-machine infinite bus model, Mm = Pm - Pe (3.13) where E1E2 P = ——-— sin 6 = C sin 6 (3-14) e X12 Consider the power angle representation Pe of Figure 3.3 illustrat- ing the behavior of the single machine against the infinite bus during the transition from one state to another where Po sin 6, P2 sin 6, and P1 sin 6 represent the electrical real power of the system before the occurrence of the fault (pre-fault), during the fault period and after the fault is cleared (post-fault), respec- tively. The area A1 in Figure 3.3a which is obtained from the mis- match of power existing between the mechanical input and the faulted electrical output, represents the kinetic energy of the generator's inertia that resulted from the fault that reduced the electrical power Pe below mechanical input power Pm. The area A1 is compared with a critical energy A2, which is shown in Figure 2.1 and represents the energy capacity of the transmission network for a particular mechanical power Pm, network configuration and fault clearing time. The critical energy A3 shown in Figure 3.3a is the amount of .18.”. 48 ‘d 6 AE(t) 1 / / unstable //// / stable / / z ” tctCC tB tB time Figure 3.3. Equal area criterion (a) Power angle representation of one-machine infinite-bus system. (b) Variation of energy margin vs. time. l I ‘wl P. A 39:2 91.», 1:"; fer the 3:333:r 'ctar angle w 'Vu CEC'EéSE d’C 35. wine and ca 3‘9" the sewer *1 ‘.0. (L) C 1 (D A) cSz'Ji-Te the DOS {- 49 decelerating energy produced by the post-fault network to compensate for the acceleration energy A1. Note that for any tC < tCC the rotor angle position 6(t) peaks when [A1] = |A3| and starts to decrease and oscillate afterwards. If the system is damped then the rotor angle also damps out and assumes the post—fault steady-state angle. If the post-fault transmission network cannot decelerate the machine and cause a reversal of direction of motion at some time t8, then the generator will lose stability. This can be understood by defining A1 and A3 and AE(t). AE(t) = A1(tc) + A3(t) is defined to be a function of rotor angle position which in turn is a function of time as follows: ( 6(tc)=6c .jr 1 (PM - stin 6)d6 6(t) < 6C 5 6 A1(tc) =« O 0') A H . V "V O? O A3(t) =1 6(t) 1jf (PM - Plsin 5)d5 6(t) 3_6. (3'15) 6 Figure 3.3b depicts the quantity AE(t) = A3 + A1 as a,function of time. In the fault period AE(t) > O, and reaches its maximum value at the clearing time tc. At tc, the network is switched to assume the post-fault network and hence for tc=é t AE(t) decreases CC’ 50 until it becomes zero at tB (decelerating energy A3(tB) < 0, provided by the post fault network is capable of capturing the accelerating (kinetic) energy A1(tc) produced by the fault). For tc = tCC the minimum of AE(t) takesa longer time to become zero. Note that tBl is also the time at which the rotor angle position is maximum (6(tBl) = 6 For tc > t the quantity AE(t) = A1(t) increases with time max)' cc’ for t < tc and then decreases (AE(t) = A1(tc) + A3(t)) for t > tc. However, in this case 0E(tB)==Min (AE(t)) > O and occurs where 6(t) = 6u at t = tBZ. Thus, the decelerating energy capability of the post-fault network A3(t32) is less than the acceleration energy A1(tc). Since AE(tBZ) = A1(tc) + A3(th) > D is a measure of the net decelerating energy and the kinetic energy remaining in the machine's inertia and since AE(t) remains positive and never reaches zero for all t > O, the machine angle 6(t) never changes its direc- tion of motion and continues to increase for t > tBZ as shown in Figure 3.2. Thus AE(t) = A1(tc) + A3(th) + A4(t) for t > th where A4(t) =‘ 6(t) f (PM - Plsin 6)d6 5(t) > a“ L 6 where A4(t) > 0 It should be noted that increasing tc for tc < t tB where 1 AE(tB ) = 0, increases since more accelerating energy A1(tc) is put 1 . CC’ Iii‘ 51 into the system and thus a longer time for AE(tB )==A1(t )i-A t )==O. ( 1 c 3 B For tc> t decreases for increasing tC since there is a larger , t cc B2 excess accelerating and thus kinetic energy AE(tB ) = A1(t ) + 2 A3(tB ) > D, which permits the trajectory to reach the PEBS 2 faster. Thus the maximum t8 and t8 occurs when tB = t8 for 1 2 1 2 tc = tcc‘ Therefore, the boundary of stability (tcc) can be pre- dicted by the following two different ways: (2) the maximum value of tc for which the minimum value of AE(t) over t is zero; or (2) the max1mum t1me tBl for tC é=tcc or t82 for tc ggt at which AE(t) reaches its minimum value which satisfies tB = t and 1 B2 AE(tBl) = AE(th) for tC = tcc' This equal area criterion is now extended to multimachine systems represented as Mi M1‘”1 = Pi ' pe1 "M; PCOA 91 = mi I = 1, ..oo, n (3.17) where n Pei =2: [C1j sin 91.1.4: 01.1 cos e1j] i=1 ifi by attempting to apply a similar equal area analysis to the energy associated with accelerating and decelerating torques between critical generator i and the rest of the generators j # i in the system. The potential energy measure is in part contributed by the torques on ra:tf¢e i “ECOenp n 3 ‘: Cur an; “‘31 rec- i 'eVEY‘SC Tfiar'uu' '1 \ES acceTera. Ja! engra- '3 52 machine i and all j # i from the equivalent transmission lines con- necting generator i to the rest of the generators in the system. By observation of the behavior of AE(t) = A1 + A3, in Chapter 4, the equal area criterion based on either (1) or (2) above apply to the multimachine model (3.17) just as in the single-machine infinite bus case. Before going to the next chapter, the following remarks and limitations are in order. (1) The concept of equal area is justified only for the loss- less systems. (2) The loss of stability in a multimachine system will not occur when the generator i (critical generator) reverses direction with respect to a synchronous reference but rather when the generator i reverses direction with respect to the inertial center of the other machines in the power system model. Since these generators also accelerate with respect to a synchronous reference the minimum resid- ual energy AEi(tB) of generator i at the boundary, where generator i reverses direction with respect to the inertial center, may be positive. (3) The generators in a multimachine power system do not act as an infinite bus or even as a single equivalent machine. The deceleration energy A3(tB) may exceed the acceleration kinetic energy A1(t and thus AE(tB) could be negative. C) It will be shown in Chapter 4 that if one generator is acceler- ated by a fault the minimum residual energy value AE(tB) will be negative for any clearing time tC < tCC for that fault, since the 53 other generators do not act as a single machine and the network has more deceleration energy than acceleration energy due to that fault. If, however, a fault accelerates a large number of generators the minimum residual energy AE(tB) for this fault will be positive since generator 1 reverses direction with respect to the other generators when it still has some_kinetic energy with respect to a synchronous reference. The last and possibly most important point of interest in direct stability analysis of the power system based on the individual energy function is to determine which of the accelerated generators will dictate the stability of the group and in general the stability of the system. The following section proposes a procedure for iden- tifying this individual generator (critical generator) that dictates stability of the system. 3.4. Critical Group, Generator, and Boundary It was argued previously that once the individual machine energies are related to the entire system rotor angle position and angular velocity,it could be used to estimate the boundary of the region of stability of the entire system. Knowing that it is possible to predict the critical clearing time by an individual machine raises the argument that one has to identify a particular individual machine whose behavior dictates most accurate the stability of the entire power system. In reSponse to occurrence of a fault, the group of generators which are most affected and disturbed by the fault energy (those 54 generators which are electrically closer to the fault location) is called the accelerated group. The longer the fault remains on the system, the larger will be the number of generators in this group. For example, in the Cooper case where the fault is applied to the high side of the transformer connected to generator 2 and the fault is cleared at tc = .220 seconds, only generator 2 is contained in the accelerated group. On the other hand, if the fault is kept on for a longer time and cleared at tC = .24 seconds, then generator 17 also joins to the accelerated group. Note that from Figure 3.4 generator 17 (Neb. CT, bus 774) is close to fault location and thus would logically enter the accelerated group as the fault clearing time increases. The rest of the generators in the system constitute the stationary group (generators which are least affected by the fault energy and remain relatively close to their pre-fault condi- tions). Based on the fact that the behavior of the generators of the accelerated group is very different from that of their pre-fault condition, it is believed that the specific generator dictating the transient stability of the entire system is contained in the acceler- ated group. It is worth noting that the generators initially forming the accelerated group do not necessarily remain in this group, and some of them may decelerate and join the less accelerated or "sta- tionary" group at a later time. The stationary group plays the role of the infinite bus in the classical equal area criterion for the single machine infinite bus model. 55 .mmS Emumxm ~33 “.8255 53m? Louacmcmmuz .¢.m $53... _ 82528 6 $2: .338 l.l.l ... m. ...n ....... \\._\\, >x man ....fi |\.. Ex \ \ \ \ \ a. . or... A}: I \\ \ x \ x \ :21: \ \\ urine" in.” 25.: \W\ \\ c.0107 . . {ea-w... _ 3.2.9.. 2' 51.1.: .9. If“ U 0“ genera Hy‘ Q ne:“S.y "nu“, ' VI \- T3] also "CCE 3f deter“ n 9'0ub ‘n. 2 an‘ 17 56 For a given small clearing time a generator or a group of generators which accelerates and pulls out of the system simulta- neously is the one which causes instability and it is called the "critical group." For a longer fault clearing time, other generators may also pull out from the system and thus result in a different mode of instability, but these generators do not play any role in determining the stability of the system. For example, the critical group for the Cooper case consists of generator 2 and not generators 2 and 17. As another example, the critical group for the Raun case consists of both generators 5 and 6. These two generators pull out of the system simultaneously. A more detailed discussion for prac- tically identifying the critical group will be presented in Chapter 4 where the simulation results are considered. Once the critical group is identified, the dynamic behavior and energy transfers between the individual machines in this group must be investigated. Although all of the generators of the critical group pull out of synchronism with the system, there is only one particular generator whose stability or instability accurately indi- cates the stability or loss of stability of the critical group and thus the system. This particular generator in the critical group, which dictates the stability or instability of the critical group, is called the "critical generator." The appropriate boundary encircling the critical generator is called the "critical boundary." The critical boundary determines a potential or kinetic energy bound- ary whose violation results in instability. The kinetic energy 57 boundary is evidenced by a minimum of energy Min (A1 + A3) in equal- area criterion. The crossing of a potential energy boundary for the critical generator is evidenced by a maximum in potential energy of the critical generator after the fault is cleared (PEBS). Generators of the critical group will each cross their own potential or kinetic energy boundary with respect to the generators of the stationary group one at a time and the critical generator is the last one in the critical group which crosses a potential or kinetic energy boundary. If the critical generator crosses its potential or kinetic energy boundary, then the entire critical group loses synchronism with the stationary group. If this critical gen- erator never crosses its potential or kinetic energy boundary, the critical group will remain stable. To clarify the loss of synchronism between the critical generator and the generators of the stationary group, a very simplified example is in order. Consider the real power transmitted between two generators i and j connected by a lossless line with reactance Xi J" lE-HE-l .. =——‘—J—sina.. (3.18) 13 Xij lJ where 5ij = 61 - 5j and E1, Ej are the magnitude of voltage at buses i and j. If E1 and Ej are kept constant, then Pij = Pmax sin aij (3.19) E. E. Where P = iii—[J]. max Xij 58 The real power transmitted from bus i to bus j through line ij clearly depends on the phase angle difference between buses i and j. When the phase angle difference (due to load increase or a change in generation due to a fault) is forced to attain a value near 90°, the power transmitted will reach Pmax’ the maximum value, and any additional phase angle difference (beyond 90°) will decrease the transmitted power. At the point where 5 = 90° (the static stability limit), the system "pulls apart electrically" and the synchronism between bus i and j is lost [20] if buses i and j are only connected through this one path. If buses i and j are operating in such a way that the phase angle difference is small, then these two generators are said to be operating in synchronism or strongly coupled. In contrast, if the angle difference exceeds 90°, buses i and j are weakly coupled. If there are several paths connecting two sets of buses I and 0, then all buses ikeI and jzed must be weakly coupled for I and J to lose synchronism. One can now argue that once the potential energy of the line connecting bus ik to bus jfl achieves its maximum capacity, then generators ik and j£ become weakly coupled. In a dynamic sense, if all of the generators ik belonging to the critical group and all generators j belonging to the stationary group exceed the potential energy capacity of the equivalent line connecting them, the two groups lose synchronism and thus the critical group goes unstable. The last generator in the critical group which approaches its potential energy boundary of the lines connecting it to the stationary group decides the stability of the critical group 59 and hence the entire system. In Chapter 5, two boundary conditions are investigated and the accuracy of the critical clearing time estimates which are based on these stability boundaries will be discussed. CHAPTER 4 STABILITY SIMULATION STUDIES USING INDIVIDUAL ENERGY FUNCTION 4.1. Introduction The power network used in the validation studies of this research is an equivalent of the real Iowa system (referred to here as the Reduced Iowa System) [28]. This network consists of 17 gen- erators and 163 buses. Figure 4.1 shows a one-line diagram of the Reduced Iowa System. The study done at Iowa State University confirms that this reduction preserves the dynamic behavior of the system for "first swing" stability. The generator data, together with the initial conditions including the generator internal voltages, are given in Table 4.1. The Reduced Iowa System model was tested by running stability studies for three different fault cases: the Cooper case (BUS 6), the Fort Calhoun case (BUS 773), and the Raun case (Bus 372). The objectives of this chapter are: (1) To establish the critical generator concept on several fault cases as well as on the Raun case where the critical group consists of more than one generator. The critical generator must be identified 60 61 .mmdu Asmpmxm mon nmuzummv Empmxm coumcmcmm-ufi .H.¢ mczmwg . mopHacm III?! . ° ..ll.ll Ill-lull— >¥ mp— M“ WNW IIIIIII . \\.§ 3 . >x mi” ... \\ \ c \\ \ :3... ..\\ .\\ 6...."0NOlhdmu ,- . JUN-... . \ \\ z... _ . _ 3:26.: 2. O 3...”... _ a. on" . 23.1.. .3qu; ‘ x . \q‘ -.5_Ar\fll \ \ .03... I; \ \I o it. :4— . a \ ..// 3.6.3:: .. 62 Table 4.1. Reduced Iowa System generator data and initial conditions. Initial Conditions Generator Parametersa Internal Voltage Generator H Xd Pmoa E Number (MW/MVA) (p.u.) (p.u.) (p.u.) (degrees) 1 100.00 0.004 20.000 1.0032 -27.92 2 34.56 0.043 7.940 1.1333 - 1.37 3 80.00 0.0100 15.000 1.0301 ~16.28 4 80.00 0.0050 15.000 1.0008 —26.09 5 16.79 0.0507 4.470 1.0678 - 6.24 6 32.49 0.0206 10.550 1.0505 - 4.56 7 6.65 - 0.1131 1.309 1.0163 -23.02 8 2.66 0.3115 0.820 1.1235 -26.95 9 29.60 0.0535 5.517 1.1195 -12.41 10 5.00 0.1770 1.310 1.0652 -11.12 11 11.31 0.1049 1.730 1.0777 -24.30 12 19.79 0.0297 6.200 1.0609 ' -10.10 13 200.00 0.0020 25.709 1.0103 - 8.10 14 200.00 0.0020 23.875 1.0206 -26.76 15 100.00 0.0040 24.670 1.0182 -21.09 16 28.60 0.0559 4.550 1.1243 - 6.70 17 0.0544 5.750 1.116 - 4.35 20.66 aon 100-MVA base l , prgceriy 1; function T0: bility using ir extrerfe‘s'. meet! to methods in t1 (4} 7 0f the P533 or excluded The f: the rESU‘iIS far 7, they testing Us: A AESCr‘] bed 1 63 properly to determine the machine for which the individual energy function must be constructed. (2) To show that the PEBS method for assessing transient sta- bility using the transient energy of the critical generator results in extremely accurate determination of the critical clearing times compared to similar total transient energy methods. (3) To compare the accuracy of PEBS and Local equal area methods in terms of determining critical clearing time. (4) To determine the effect of conductances on the accuracy of the PEBS method by comparing the results when they are included or excluded in individual transient energy function. The following sections discuss these objectives by showing the results of simulation runs for each fault case. Later in Chap- ter 7, these results will be compared to the results obtained by testing the proposed fast stability algorithms (which will be described in Chapter 5) on each fault case. 4.2. Cooper Case A three-phase fault is applied to generator 2 (Cooper) and is removed by clearing line 6-439. The system trajectory was simulated for different fault clear- ing times and it was observed that although the generators 1, 17, 12, and 16 are electrically close to the fault location (see Figure 4.1), their behavior is different than that of generator 2. The fault energy separated the system into two groups; one consisting of the most accelerated (critical) generator 2, and the second of the rest of the generators in the system. Figures 4.2, 4.3, and 4.4 depict 64 90; 702 so: 302 10‘ 17 HNGLEI DEGREE) .910 ' ' .650 Time (seconds) (a) '310 Time (seconds) '620 (b) Figure 4.2. Swing curves for Cooper. Clearing time = .218 seconds. (a) Generators 2, 17 (b) Generators 12, 16 - .52: .2— a .22: E 4 2 2 . «h ~12 .2 ~.~ ~01: ”guns 4 65 3 so: 2%” 70: 17 § 601 :2: ° .33 .66 .99 1.2 Time (seconds) (a) O) O .66 .99 . < Time (seconds) (b) Figure 4.3. Swing curves for Cooper. Clearing time = .220 seconds. (a) Generators 2, 17 (b) Generators 12, 16 I N o L—A- (A) w HNGLEI DEGREE) N o 66 .330 SECONDS. 1r nu nu 2 1 . 0°00 000 983 mucouo.uaoz¢ (A) TIflE (SECONDS) 0. o 1 a 0000 52:. mucouo.uaoa¢ Swing curves for Cooper. Figure 4.4. 67 the swing curves for generators 2, 17, 16, and 12 for fault clearing times of tC = .218, .220, and .330 seconds, reSpectively. When the fault is cleared at tc = .218 all the generators stay stable, and although the peak of the swing curves of generator 2 reaches approxi- mately 150° (62 > 90°), it ultimately decelerates and remains stable ‘with the rest of the system. Figure 4.3, however, illustrates that for a longer clearing time (t = .220 seconds) generator 2 acceler- c ates and pulls away from the rest of the system and hence by defini- tion is the critical generator and the critical group. As a further step, Figure 4.4 illustrates that for tC = .330 seconds both gener- ators 2 and 17 lose synchronism with respect to the rest of the system, but the group consisting of these two generators, which did not lose synchronism simultaneously, is not considered as the critical group. Once the critical generator or group is determined, it is appropriate to calculate the critical energy of the individual energy function for this critical generator using the transient stability simulation program. The critical boundary or the PEBS for Cooper is based on the maximum of potential energy as a function of time for t > tC n _ 1 VPE2(t)-Kr-Z: (PZMj' -.P M2“ - 9237+ HZ C2j( cos 92j( (t) J=1 ifz 372 1 51 e (t)+-e (t)- 0S - e. - cos 0:1) 023 2 :1 :1 [Sin 02j(t)- sin 031] 68 This critical generator maximum potential energy is calculated for different clearing times tC and the results are summarized in Table 4.2. The maximum potential energy (4.1) as a function of time is determined for each clearing time tc' The time tB(tc) occurs at the point in time when the trajectory most closely approaches the potential energy boundary surface for the individual energy function of the critical generator for the case where the clearing time tC is less than the critical clearing time tcc‘ For clearing times tc greater than the critical clearing time, the maximum potential energy occurs at the point in time tB(tc) when the trajectory crosses the potential energy boundary surface. The peak potential energy Vp52(t8(tc)) should clearly increase for increasing clearing time tc when the clearing time tC is less than the critical clearing time tcc' The peak potential energy function VPE2( t3(tc )) has been claimed to be nearly constant for all tc greater than the critical clearing time in the previous literature [16]. It is clear from Table 4.2 that VPE 2(tB(tc )) decreases slowly for tC greater than critical clearing time since VPEZ(tB(tC )) decreases for tC > 0.220 seconds, where it is clear that the system is unstable for tc = .220 seconds from Figure 4.3. There is a very significant peak in VPE2(tB(tC)) when tc is close to tcc’ making it easy to accurately identify tcc The decrease in the peak potential energy for increasing tc beyond the critical clearing time is due to the fact that the generator 17 and ultimately 16 and 12 will lose stability with generator 2 and thus the angle differences between generator 2 and these generators 69 Table 4.2. Potential energy boundary surface energy as a function of tC for determining tcc' with conductance without conductance * * tc VPE(tB) t3 VPE(tB) ‘0 .192 6.546 .368 4.961 .368 .208 8.105 .448 6.586 .448 .210 8.362 .480 6.879 .480 .217 10.173 .644 8.876 .637 .218 10.843 .687 9.549 .687 .219 12.897 .814 11.042 .821 .220 15.133 .935 13.924 .946 .222 13.672 .848 11.388 .848 .224 12.021 .672 11.241 .672 .240 9.403 .432 8.180 .464 tB - time at which the maximum of potential energy occurs tc - clearing time 70 at tB(tc) will be smaller making VPEZ(tB(tc)) decrease with clearing time tC > tcc‘ From the above discussion the critical clearing time should occur precisely at the time that VPE2(tB(t )) is maximum. c For the Cooper case it is estimated that the actual critical clearing time is tce:(.219-.220) seconds based on the above criteria for selecting tcc' This predicted critical clearing time is very accurate compared to the results in [16] and very well comparable to simula- tion results that indicate the critical clearing time lies in (.219-.220). The local equal-area critierion for Cooper is based on the minimum overtime of the energy margin AE2(t;tc) = A1(tc) + A2(t,tc) (4.2) where 0(tc) A1(tc) ‘ V962 951 is evaluated using the local potential energy function VpEz(t) with the faulted network admittance matrix and 0(t) A2(t;tc) = VPE (4.3) 2 0(tc) is the local potential energy function with the post-fault network admittance matrix. The local EAC (tc) is thus 71 EAC (tc) = min 0E2(t;tc) = AE2(tB;tC) (4.4) t > tC Figures 4.5 and 4.6 show the simulation of AE2(t;tc) performed with small integration step for two different clearing times, .218 and .220 seconds. In both figures AE2(t;tC) starts from zero value at time t = 0 and increases with time until it peaks at clearing time tc = .218 and .220 seconds. This increase of energy margin is due to the excess kinetic energy (accelerating energy) A1 of generator 2 during the fault period where A1 is positive energy and A2 = 0. After the fault is cleared t > tc, the post-fault network starts to absorb the kinetic energy at clearing time and the minimum AE2(tB;tc) is reached at some time tB after the clearing time. For tC = .218 seconds, the minimum of AE2(tB;tc) occurs at t8 = .676 and tB = .687 with and without conductance term included, respectively, in equation (4.1). The value of EAC(tC) being negative for tC < tCC indicates that the post-fault network is completely capable of absorbing the faulted energy and decelerates the critical generator to a point where its direction of motion is changed and the system remains stable. However, for large clearing time tc > tcc’ more kinetic energy is produced during the fault-on period which causes the post-fault network in: abSOrb more kinetic energy but no longer all of it. As a result, the minimum of AE2(tB,tC) for tC > tcc never reaches zero and remains positive |A1| > [AZ], indicating the fact that the critical generator does not change its direction of 72 .umczpuxm mocmuusucou gmymcmcu U\z .umuzpocw mocmuozucoo commence Q\2 .N u go acmcmm quwu_co .mvcoomm mHN. u mswu newcmmpu .mmmu Lmaoou cow Auu.uvm< + Auuv < :_mcae Amcmcm .m.¢ mczmwd ADHBNE 1°n°a1 73 .mncoomm omm. u we?“ mcwcmmFU N.H .umuzpuxm mucouozucou commcmcu .vmnzpoc_ mocmuozucou cmwmcacu .mmmu cmaoou co; Auu.uvm< + Amazouum. mzah U\z U\3 .N u cowmcmcmm pmu_u_cu Auuv < :_mgme Amcmcm .0 mm. hwfll ma. u\3 Q\z mo. #111. r0 Hm“ rum .e wczm_m ('n'd) ADUBNB 74 motion and the system loses stability. In contrast to the case of the equal-area criterion of one-machine infinite bus, the minimum AE2(tB;tcc) in a multimachine power system is negative and not zero when the system is stable. As was discussed earlier, this phenomenon was expected because of the fact that the non-critical machines do not behave as an infinite bus and thus require more deceleration energy A2 to reverse the direction of motion of the critical genera- tor than if they acted in unison as a single bus. The stability simulation program was run for different clearing times to predict the critical boundary using Minimum Energy Equal- Area Criteria and the results are summarized in Table 4.3. The minimum of AE2(tB;tc) occurs at tB = .946 for both cases where the transfer conductances are and are not included. This observation certainly shows that for the qualitative analysis the concept of the equal-area criterion can be extended for a multimachine case. From Table 4.3 it is clear that the estimated critical clearing time is tc e (.219-.220) seconds which is comparable to simulation results tc e (.219-.220). Note that the very sharp narrow minimum for AE2(tB,tc) for tC near tcc makes accurate identification of tcc quite easy. The results of both algorithms PEBS and EAC in Tables 4.2 and 4.3 show that increasing tC for tC < tCC causes tB(t to increase c) since the angle deviation of the accelerated generators from their nominal (initial operating) state becomes larger and it takes more time for generators to reverse their directions and return to the 75 Table 4.3. Minimum of AE2(tB,tC) = A1(tc) + A2(tB,tc) for critical generator 2 using stability simulation program. with conductance without conductance tC AE2(tB,tc) t8 AE2(tB,tC) tB .192 .423 .368 -2.199 .368 .208 .011 .464 -2.578 .480 .210 - .036 .495 -2.695 .495 .217 -1.522 .637 -4.067 .637 .218 -2.111 .676 -4.647 .687 .219 -3.873 .821 -5.910 .821 .220 -6.358 .946 -8.826 .946 .222 -5.422 .848 -7.266 .848 .224 -4.032 .688 -5.909 .672 .240 1.399 .496 -1.041 .464 tB - time at which the minimum of AE2(t,tC) occurs t C - clearing time 76 previous stable position. The maximum tB(tC) occurs when tC = tcc' Thus there are two indicators of tCC by observing the minimum value of AE2(tB): (1) the maximum value of tc for which the minimum value of AE2(t,tc) over time is the smaller than some value a . (2) the maximum time tB(tC) at which the minimum of AE2(t,tC) is less than c. 4.3. Fort Calhoun Case A three-phase fault is applied to generator 16 (Ft. Calhoun) in Figure 4.1 and is removed by clearing line 773-779. From Figure 4.1 the generators electrically close to the fault location are generators 17, 12,11L 5, and 6. For different fault clearing times the system trajectory was simulated and generator 16 was found to be the first generator which was accelerated and separated from the rest of the generators. Therefore, generator 16 constitutes the critical group and the critical generator. Based on simulation, it is observed that for clearing time of tC = .354 seconds the system is stable but that for tC = .356 seconds the system becomes critically unstable. Figures4u7 and 4.8 illustrate the swing curves of some of the generators of the stationary group and the critical generator for clearing times of .354 and .356 seconds, respectively. Note that the peak of the swing curves of the stationary group is some- where around 80°-100° while that of the critical generator is about 170°-180° for the stable case, confirming the fact that the critical generator is initially pulling away from the stationary group but at 77 16 1______‘__~____- E 901 17 g 60: b so: .42 .84 1.12 Time (seconds) (a) E S 8 171 d E Figure 4.7. .32 .84 1.12 Time (seconds) (b) Swing curves for Fort Calhoun. Clearing time = .354 seconds. (a) Generators 16, 17. (b) Generators 10, 12. 80: 40. RNGLEIDEGREE) -401 H (D . N D O D 78 16 / 17 .5 1.0 \ Time (seconds) (a) 12 10 RNGLEIDEGREE) ‘ c3 I L I b D Figure 4.8. 1 6 ‘ 1 10 § Time (seconds) (b) Swing curves for Fort Calhoun. Clearing time = .356 seconds. (a) Generators 16, 17. (b) Generators 10, 12. 79 a later time all of the generators decay and resume a relatively small angle indicating stability. For clearing times tC = .354 and tc = .356 seconds, Figures 4.9 and 4.10 illustrate the energy margin AE16(t,tC) = A1(tc) + A2(t,tc) produced between critical generator 16 and all of the generators of the stationary group. From Figure 4.9, it is clearly seen that the minimum of AE16(t,tc) occurs at t8 = 1.2 seconds and it has negative value, i.e., IAll < IAZI, indicating that the post-fault network captures the total kinetic (accelerating) energy produced during the fault period and hence changes the direction of motion of the critical generator; therefore the system is stable. For the larger clearing time tC = .356 seconds, Figure 4.10 shows that the minimum of AE(t,.356) = EAC(.356) occurs at earlier time tB = .997 seconds (because the post-fault network does not have the capability of absorbing the faulted energy completely and it takes less time to reach the minimum of (A11-A2). Table 4.4 shows the minimum of AE16(t,tc) for different clearing times. From the entries of this table it is clear that the lowest possible value occurs close to tc = .354 seconds, indicating critical clearing time. Note that AE16(tB,tC) has a sharp narrow minimum, making the accurate identi- fication of tcc easy. Similar analysis was done for different clearing times to cal- culate the boundary energy (Maximum Potential Energy) of the critical generator using the PEBS method. The Stability simulation program was run and the maximum of potential energy was calculated at each clearing 80 .umuapuxm mocmuusucou commence u\z .umnzpucw mucmuuzucou gammcmcu Q\2 .cH u coumcmcmm Pmu_ FL .mncoumm «mm. H mew» mcwcmmPU .mmmu caogpmu “com com Auu.pv~< + A av < cwmcme zmcocm .m.¢ mczmwu ('D'd) ADUENB 81 .cmczpoxm mocmuozucou commence U\z .umuapocw mocmuosucou cmwmcmcu u\z .oH u Loumcmcmm Pmu_u_co .mucoomm 0mm. u warp mcwcmmFU .mmmo czospwo “com com Auu.wvm< + Auuvfl< cwmcms xmcmcu .oH.¢ mczmwu 3 ”8.7% 948%. ME: . 8 c1 .0 IA . n H? mm 0;. m2 ..n u; 8 1 82 Table 4.4. Minimum of 0615(t8,tc) = A1(tc) + A2(tB,tC) for critical generator 16 using stability simulation program. with conductance without conductance tc AE16(tB,tc) tB AE16(tB,tC) tB .320 .486 .624 - 3.287 .624 .345 - .064 .885 - 4.107 .885 .352 -3.440 .928 - 7.836 .944 .354 -6.220 .200 -11.149 .200 .356 -5.880 .997 - 9.982 .997 .357 -5.442 .959 - 9.400 .959 .360 -4.315 .900 - 8.011 .900 .368 -2.005 .816 - 5.428 .816 tB - time at which the minimum of AE16(t,tc) occurs tc - clearing time 83 Table 4.5. Potential energy boundary surface energy as a function of tc for determining tcc' with conductance without conductance * * tc VPE16(tB) t8 Vpel6(t8) t8 .320 7.681 .624 5.672 .624 .345 12.561 .885 9.825 .885 .352 14.402 .960 11.888 .944 .354 18.267 1.200 15.315 1.200 .356 16.382 .997 14.265 .997 .357 15.708 .959 13.741 .959 .360 14.230 .888 12.528 .900 .368 11.848 .800 ‘ 10.428 .816 tB - time at which the maximum of potential energy occurs tC - clearing time 84 time for two different cases, one with transfer conductances included and the other with transfer conductancesnot.included. The results of this analysis are summarized in Table 4.5. The entries in columns 2 and 4 of Table 4.5 indicate that for clearing time tC = .354 seconds the maximum potential energy of the critical generator has its largest value. This potential energy value represents the boundary energy and the corresponding clearing time (close to .354 seconds in this case) is called the critical clearing time tcc' The accurate identification of tcc is again possibly due to the sharp narrow peak of Vp516(t8). It should also be noted that the critical generator trajectory for this particular ) clearing time achieves the potential energy maximum at a time tB(tCC which is larger than all the trajectories for which the fault was cleared at tC # tcc' This phenomenon happens because of the fact that for tc = tCC the angle deviation of the accelerated generators from their initial operating state has the largest value and it takes the longest time for generators to reverse their directions and return to their previous stable position. The results obtained for the Fort Calhoun case indicate that the concepts of the selection of the critical generator and both EAC and PEBS are valid for multimachine power systems and both pre- dict the critical clearing time very accurately' tC e (.354, .356). This investigation of the accuracy of assessing critical clear- ing time by (a) identifying a critical generator, and (b) using the PEBS or equal area method with the critical generator's individual 85 rnachine energy function is extremely accurate and indicates these methods truly capture the energy conditions that cause loss of sta- taility. This attempt to determine the accuracy of the PEBS and annual area criterion methods using the critical generator's individual energy function has shown that there are very large changes in ‘VPE(tB,tc) for PEBS and AE(tB;tc) for equal area when tC is near tcc' “This makes very accurate assessment of tCC quite easy. The previous research using the PEBS method on an individual energy function [16] did not attempt to determine how accurate the rnethod was and only indicated the critical clearing time belonged to intervals of .011 and .008 seconds rather than .002 and .001 for the Fort Calhoun and Cooper cases, respectively, as in this research. 4.4. Raun Case A three-phase fault is applied to the high side of the trans- ‘former’connected to generator 6 (Raun) and is removed by clearing line 372-193. Stability run for Raun case was done earlier in Reference [29]. 'To complete the thesis, the analysis and the results obtained are summarized here. For different fault clearing times the system trajectory was simulated and it was observed that generator 5 was electrically closest to the fault location and thus possesses similar behavior to that of generator 6. The fault energy separated the System into two groups, one consisting of the accelerated generators (5 and 6) and the other by the rest of the system. Figures 4.11 and 4.12 show the swing curves of the generators for fault clearing 86 (I) .504 Figure 4.11. Swing curves. 1 v “ fi v Tun (second) v 0) Clearing time = .1900 seconds. (a) Generators 10, 13, 16. b) Generators 5, 6. m 8.1.-10601.0) 87 16 ‘0 )5 10 16 10 'rj' V T ‘ Y V V V t T Y r ‘7 " V f U 1' " V V U r I I' r r V I O, ' ‘ o. 13 (a) T'VT‘IFiftitftt‘j'UT‘Tfft‘Terl .9 1.- Tm (mend) (b) Figure 4.12. Swing curves. Clearing time = .1925 seconds. (a) Generators 2, 10, 13, 16. (b) Generators 5, 6. 88 times tC of 0.19 and 0.1925 seconds, respectively. For tc = 0.19 seconds all the rotor positions do n0t exceed the stability limit and do not accelerate indefinitely. However, it is clear that the behavior of generators 5 and 6 is different from that of the other generators. Figure 4.12, where the fault was cleared at tC = 0.1925 (sec.), also indicates the similarity in behavior of generators 5 and 6, but here they are both accelerated and thus pull out of step from the rest of the system, causing instability. Based on simula- tion, it was observed that for clearing time of tc = 0.1922 seconds the system was critically stable as shown in Figures 4.13a and 4.13b. Note that the peak of the swing curves of the stationary group is somewhere around 90°-100°, while that of the critical group is about 160°-170°, confirming the fact that the critical group is initially pulling away from the stationary group but at a later time all of the generators resume £1 relatively small angle indicating stability. For the same clearing time, i.e., t = .1922 seconds, Figure 4.14a 0 illustrates the sum of the potential energy produced between gen- erators 5 and 6 and all of the generators of the stationary group. Figure 4.14b shows the plot for the potential energy produced between generator 5 and the stationary group and a similar one for the partial potential energy between generator 6 and the stationary group. The peak of the partial potential energy of generator 6 indicates the maximum energy capacity of the transmission network connecting gen- erator 6 to the stationary group. Before reaching the peak of poten- tial energy. there is a strong coupling between generator 6 and the 89 16 t”. ..q 16 2 10 rfi’TiTij‘jWfijTTT—rj'I‘T' Yrrj .31 L24 ,6 “(I 0.10.10.101001 «a. 6 00.1 “(I IJCKO) 1 U V jfi T I j Y I I j 1 T T—Y—V 1—T 7 1 1 ‘j‘ T T w .31 124 \\3<:::;7 1 Time (second) Figure 4.13. Swing curves. Clearing time = .1922 seconds. 90 (5+6) 12., "I!" JV. ‘4 I r 1 I I T V —r T I T T j V I Trfi' 1 ‘U T j Tfi‘a T '1'“ .31 V1.55\/ (cl 1 j 11111 1 I r I T I' I ‘5 1 1 T f 1.55 U‘I.*O‘ o PU . U H Time (second) (b) 49.1 l’igure 4.14. Partial energy analysis. Clearing time = .1922 seconds. (a) Sum of partial energies for generators 5 and 6. (b) Partial energy for generators 5 and 6. 91 stationary group, and after the energy exceeds the maximum potential energy capacity of all lines connecting 6 and the stationary group, the magnetic coupling between generator 6 and the stationary group becomes weakly coupled. If generator 6 was the only machine in the critical group, one could conclude that generator 6 would pull away from the system and thus lose synchronism. However, for this case, where generator 5 is also in the critical group, one cannot yet make any decision on the loss of stability. Now consideration of the energy behavior of generator 5 reveals that the peak of the partial pcrtential energy of generator 5 is reached at a later time, indicating the fact that although generator 6 is trying to pull away from the system, generator 5 maintains a strong coupling to the stationary group. Thus, among the generators of the critical group (5 and 6), generator 5 is the last generator to exceed its potential energy boundary capacity and therefore is by definition the Mai _ggnerator. Now that the critical generator is identified, it still remains to identify a boundary of stability. For both the inclusion and exclusion of the transfer conductances, several simulation runs for (Tifferent clearing times were performed by Rastgoufard [29]. Fig- ures 4.15-4.18 illustrate some of these results. Figure 4.15 depicts the partial potential energy across the boundary of generator 6, cleared at tc = .18 seconds. It is clearly seen that for this case the minimum of AE6(t,tc) is negative (for both cases with and without transfer conductances), confirming the stability of the system. However, in contrast to the case of equal area criterion of one 92 6. ’T 2« 3 S M /\/\ E” ) .30 ’ T U 1.8 g '21 LLJ -4.532 -6* (a) 6) 5 2~ "5 é a. T ' ‘ L .30 1.8 3 1.1.1 'ZJ .1 -6« Time (seconds) -7.465 (b) Figure 4.15. Equal area analysis AE(t) = A1(t) + A2(t). Raun case, 6-infinite bus. Clearing time = .18 seconds. (a), (b) Transfer conductance excluded and included, respectively. 93 45— =3 154 (1 V 0 ' ' r f fl § 15 .35 1.75 m - d ,5 J (a) -45. 60. 30« 5 A1 a. V 0 61 ’ 1.75 S. Q) .5 '301 l -60j Time (seconds) (b) Figure 4.16. Equal area analysis. Raun case, 6-infinite bus. Clearing time = .1925 seconds. (a) AE(t) = A1(t) + A2(t) vs. time (transfer conductance excluded). (b) Areas A1(t) and A2(t) (transfer conductance excluded). 94 4. 2. 5 . d. " 0 >5 8’ Q) E. -2. -4.1 5n 3'1 3. 14 3 >5 8‘ q, -1. C LIJ -3.. -5. Figure 4.17. (b) Time (seconds) Equal area analysis. Raun case, 5-infinite bus. Clearing time = .1922 seconds. . (a) AE(t) = A (t) + A1(t) vs. time (transfer conduc- tance inc uded). (b) Areas A1(t) and A2(t) vs. time (transfer conduc- tance included). 20. H O A n O 95 A1 + A2 Energy (p.u.) -10. '20 u 20 . 10 . W ' 1.75 Energy (p.u.) O -10 . -20 . W ' 1375 Time (seconds) (b) Figure 4.18. Equal area analysis. Raun case, 5-infinite bus. Clearing time = .1925 seconds. (a) AE(t) = A1(t) + A2(t) vs. time (transfer conductance included). (b) Areas A (t) and A2(t) vs. time (transfer conductance include ). 96 machine infinite bus, the minimum of AE6(t,.18) is not zero. As was discussed earlier; ‘this phenomenon was expected. In Figure 4.16, where the fault is cleared at tC = 0.1925, the minimum of energy margin AE6(t,tC) has positive value (lAll > IAZI), indicating loss of stability. Figures 4.17 and 4.18 depict the generator 5 potential energy boundary for tc = .1922 and .1925 seconds, respectively. In comparing the behavior of generator 5 with that of generator 6, it is seen that the minimum of AE(t,tC) for generator 5 takes place at a later time than that of generator 6. Although the energy boundary of both generators predicts tcc e (.1922, .1925), the fact that the minimum of AE(t,tc) for generator 5 occurs at a later time confirms that generator 5 is the critical generator. Both the PEBS and EAC methods using the individual energy function accurately determined the critical clearing time for the cases studied. However, these results were based on step-by-step integration of system differential equations to evaluate the indi- vidual potential energy required at each integration step. The major point of interest here is to test the proposed direct methods (to be described in Chapter 5) on the same Reduced Iowa System and compare the results of predicting the boundary of stability with that of simulation results. CHAPTER 5 PROPOSED DIRECT STABILITY ASSESSMENT ALGORITHMS The purpose of this chapter is to develop a fast PEBS method and a fast equal area method using the individual machine energy functions. The results of the previous chapter indicated that both the PEBS method and equal area method using the individual machine energy function are extremely accurate in determining the critical clearing time for a particular fault, which is a measure of its ability to test whether a system is stable for that fault and the clearing time and the margin of stability. However, these results were based on simulation of the system differential equations to evaluate the individual potential energy: 7 n 1 2i: 1 J- 171 n 51 +2 Cij(cos 61.j(t) - cos 81.3.) i=1 in 01(t) + ej(t) - 0:1 - 031 $1 01(t) - ej(t) 81 ej (5.1) 97 98 required for the PEBS and equal area methods at each integration step. Thus the results of Chapter 4 suggest that the equal area and PEBS methods are extremely accurate in confinning the conclusion concerning whether the system is stable or unstable, which can also be determined by observing the simulated system trajectory. Thus the PEBS and equal area methods are totally dependent on the tran- sient stability simulation and only provide a quantitative measure in terms of energy of the relative margin of stability or instability. The objective in developing the PEBS and equal area methods was to develop fast stand alone algorithms that could determine whether a system is stable or unstable; the margin of stability for a particular fault, clearing time, and operating condition; and the sensitivity of the stability to such operating conditions as network configuration, load level, generation dispatch, etc. These stand alone algorithms obviously should not require time step inte- gration of the transient stability model but should provide the same results and conclusions concerning the stability of the system; the margin of stability for the fault, clearing time, and operating conditions; and sensitivity to operating conditions. The fast PEBS and fast equal area methods require methods for: (1) Obtaining the total energy which the system gains during the fault-on period. This has previously been accomplished by per- forming a step-by-step integration of the faulted system equations and simultaneously calculating the appropriate energy; 99 (2) Determining the critical generator and thus the generator for which the individual energy function is evaluated; (3) Determining the actual critical energy by calculating the peak potential energy using the PEBS method; (4) Computing the energy at the proper unstable equilibrium point (u.e.p.) using the UEP method. Two algorithms have been proposed for producing system trajec- tory approximations that could provide the above information required by fast PEBS and fast equal area methods. These Taylor series and cosine series approximations algorithms are discussed in this chapter and shown to either be inaccurate or require computational require- ments comparable to simulating the system trajectory. A fast PEBS and a fast equal area method are then proposed based on using an RMS coherency measure to accurately predict the state when the system trajectory either most closely approaches or crosses the PEBS. These fast PEBS and equal area methods do not require approximating the faulted system trajectory over time but rather only require predicting the trajectory when it comes closest to or crosses the PEBS for a specific fault and fault clearing time. The cosine and Taylor series approximations of the system trajectory are discussed first before the fast PEBS and equal area methods based on the RMS coherency measure [21]. The RMS coherency measure is then developed in Chapter 6. 100 5.1. Modified Transient Energy Method 5.1.1. Fault Trajectory Approximation by Cosine Series From the concepts of the previous section it is understood that a knowledge of the faulted system trajectory is necessary or very useful. A simple approximation of the system fault trajectory developed by Athay et al. has been proven to be sufficiently accurate in some cases for the four purposes mentioned above. Representing the center of angle referenced accelerating powers of the faulted system by f. (i.e., f. 1 1 = the right hand side of equation (2.10)), the form of the approximation is fi = aI + bl COS wt i = 1: 2: ----, n (5.2) The The method for determining the unknown constants ai, bi’ i = 1, 2, ...., n and the frequency w are quickly summarized; basically, two power flow solutions are utilized [17]. The first, at the instant of fault application, determines the parameters ai’bi’ i = 1, 2, ...., n for a given frequency w. The second, along an approximate trajectory shortly after the fault, is used to compute m. Angles obtained from this fault trajectory approximation are given in Table 5.1 for a particular case on the ten machine New England System. The comparison of the actual angles and those obtained by this approximation indicates that the approach can be quite accurate in some cases. The approach will not be accurate at all for long faults where the post-fault network determines the 101 Table 5.1. Comparison of fault trajectory angles at t = 0.4 sec. (fault on bus 15, New England system). cosine unit actual approximations 1 - 38.0 - 38.1 2 55.7 55.4 3 63.2 63.2 4 98.5 98.8 5 91.2 91.0 6 95.0 95.7 7 100.7 101.2 8 43.9 42.4 9 71.7 71.5 10 8.2 9.4 102 .critical generator that loses stability and not the initial acceler- ation during the fault. Thus, the cosine approximation can only be used in cases where the fault clearing time is small and stability is assured. 5.1.2. Trajectory Approximations by Taylor Series The Taylor series method proposed in Ref. [19] is conceptually attractive. The absolute rotor angles 61(t), i = 1, 2, ...., n are approximated by Taylor series expansion and the coefficients of the series are computed from the prefault operating point 551 using the faulted admittance matrix. Because of the fact that all 81's are zero, the computation of the Taylor series coefficients would be simplified because all the odd coefficients are zero. A more accurate form of this method would be to update the Taylor series coefficients at successively smaller time intervals until the desire accuracy is achieved. This procedure is in con- trast with the alternative of increasing the order of the Taylor series until the desired accuracy is achieved. This updating pro- cedure involves more terms since the 01's are no longer zero at each update. The form for updating at t = t0 for the fourth order series is + i o o i o o (5 3) 103 where + 2! + i o o (5.4) where to is the time of last update n (2 _ 1 . 51. )(t) 'M;[Pi -JZ=:1(C1-jsm 61.j(t) + Dij cos 61.j(t))] , if” n (4) _ 1 61 (t) - ET; {[013 $111 613(t) m 2 - C13“ cos 6ij(t)][dgl)(t) - 59%)] 104 To implement this Taylor series updated given t0 = 0 and {61(0)}?=1 n - . and {wi(0)}i=1 "' 0. (1) Set m = 0. (2) Compute the new {5gk)(to)}?=1 for k = 0, 1, 2, 3, 4 at to = m- A . (3) Set the new t = (m + 1)- A (time of update). (4) Utilizing dgk)(to), to, and t, compute {61(t)}9 1:1 and {wi(t)}?=1 using (5.3) and (5.4). (5) If (m + 1)- A gatf, stop; otherwise return to step (2) with m = m + 1. These approximated trajectories produce very nearly the same energy functions as the exact trajectories. From numerical results of the fault on cases studied, a single update at t = 0.2 was enough to give the same results as the exact trajectory. Although it is found that the Taylor series method of Ribbens Pavella [19] avoids integration of the differential equations and gives acceptable accuracy, updating this Taylor series which cal- culates the rotor angles {614”}?=1 and their successive derivatives agk)(to) for k = 0, 1, 2, 3, 4 at each updating time using equa- tions (5.4) and (5.5) requires a large number of algebraic operations (additions, multiplications, and divisions) which is comparable to the number of operations needed to solve load flow equations using a Gauss Seidel method at each integration step for a transient sta- bility simulation. 105 Thus the computation for each update of the Taylor series algorithm at time step m- A is comparable to that for solving the network equations using a Gauss Seidel algorithm at each integration step. Since the time step for the Taylor series update and the integration step sizes are comparable, the Taylor series algorithm does not appear to significantly reduce computation compared to a conventional transient stability simulation even though the Taylor sieres can be quite accurate. Having presented the two methods that have been proposed for fast transient stability simulation, the PEBS and equal area methods using the individual machine energy function are now reviewed in terms of the information required to produce a fast transient sta- bility assessment procedure. The results of this review of these methods is that one could produce an extremely fast method for tran- sient stability assessment if a direct'method for predicting the state of the system when the system trajectory most closely approaches the boundary energy (stable case) or crosses it (unstable case) for a particular fault and clearing time could be developed. A method for direct prediction of this state using the RMS coherency measure is then proposed in the next chapter. 5.2. Fast PEBS Method Using the individual machine energy function, the boundary of the region of the stability is determined by the potential energy boundary surface (PEBS) as discussed in Chapters 2 and 3. At the PEBS which is defined for the post-fault network, the potential 106 energy of the critical generator with respect to the rest of the generators in the system is maximum, and on the boundary of the region of stability the total energy of the critical generator with respect to the rest of the system is equal in magnitude to the potential energy at PEBS. Since at the PEBS the potential energy is maximum, stability is maintained if the system kinetic energy is totally converted to potential energy (single machine infinite bus system) or if the system kinetic energy is minimum (multiple machine system) before reaching the PEBS. This maximum of potential energy as a function of time (5.1) varies with Clearing time tc < tcc since the maximum ei(t) depends on tc. Thus, one can write Max VPE(t,tc) = VPE(tB,tc) = v;E(tC) where tB(tc) is the time where VPE(t,tc) is maximum for clearing time tc. Increasing tc for tc < tCC will increase this maximum value of the potential energy of the critical generator v;E(tc) and it reaches its highest value for t t . This particular maximum value of potential energy for C CC tc to the rest of the system generators and it is denoted by Vcr' For tcc is called the critical energy of generator i with respect large clearing times tc > tcc’ some of the generators in the sta- tionary group may start accelerating and also become separated from the rest of the generators in the stationary group. As a result of this phenomenon, the system kinetic energy is not totally converted to potential energy (i.e., the power system does not behave as a single machine and infinite bus system) and even for large system the system kinetic energy is not minimum at the PEBS based on the 107 individual machine energy function. Therefore as tC keeps increasing for tc > t the maximum of potential energy VPE(tc) at PEBS decreases. cc’ In order to determine the precise value of the critical energy as well as the critical clearing time, one has to compute the maximum of potential energy for different clearing times and search for the highest value among these maximum potential energies. Based on the fact that the rate of change of the maximum of potential energy for tC > tcc may be small, one could use the maximum potential energy v;E(tc) of the critical generator or group for some arbitrary large clearing time tC > t 1 c ' potential energy of the critical generator or group at clearing time c’ and compare it with the tc < tcc < tC where the stability is desired to be tested. If 1 v;E(tcc) is much greater than VPE(tc1)’ as shown in Figure 5.1., one * I * O I could use VPE(tc1) as Vcr if VPE(tc) rises very rapidly for values . O O I * - of tc slightly less than tcc' Utilizing VPE(tC1) for tc1 - .240 would indicate the system is stable for tc < .217 rather than for o o o o * tc éltcc = .220 in the Cooper case from Table 4.2. UtiliZing VPE(tC1) for tc = .368 would indicate the system is stable for tC < .345 ratherlthan tcc z .354 in the Fort Calhoun case from Table 4.5. As tc1 were to increase the error in being able to determine tcc or judging values of tc for which the system is stable would increase. The error does not appear unacceptable for the applications of a fast transient stability assessment method such as a screening tool for assessing fault contingencies in operation or expansion planning or for use by operators in control centers. 108 r v .208 .212 .216 .152 14. 12. .220 .224 Sitcc Determination of critical boundary for critical generator using maximum potential energy VPE(tc) method. Figure 5.1. 109 Determination of the critical generator or group as well as the states (rotor angles of the generators) of the system at time tB is the critical requirement for calculating the critical energy e(tB,tC) e$1 for either tc1 or tc using equation (5.1). Transient stability simulation study shows that the rotor angles of the generators of the system at time tB are at their peak values. The previous algo- rithms discussed earlier in determining the generator angles at any specific time dealt with the simulation of the trajectories and thus required a significant amount of computations. The RMS coherency measure (linear/nonlinear) technique devel- Oped by Schlueter in 1978 [21] seems to be appropriate to use as a ) ma direct method of predicting the peak angles 8(t t = e x and also 8’ c the critical generator or group of the system. If one used the RMS coherency measure to predict the state 9(tB;tc) at which the trajec- tory most closely approaches or crosses the potential energy boundary surface for any clearing time in order to evaluate v;E(tc), the error in determining tcc by maximizing the function v;E(tc) is only the error in predicting 6(tB;tC) for each tc‘ However, if one evaluates v;E(tc1) for some tC1 > tCC using the RMS coherency measure, then the error in determining tCC and assessing whether the system is stable depends on both the difference VPE(tcc) - v;E(tC1) and the error in predicting e(tB;tC) for some tc and tC . The elimination 1 110 'k of the computation to maximize VPE(tc) is justified if this error in determining tCC due to VPE(tcc) - VPE(tc1) is not large. Chapter 6 of this thesis defines this RMS coherency measure and outlines the procedure of obtaining the peak angles 6(tB;tc) and identifying the critical generator. Finally, it provides the theoretical justification and verification for the use of the RMS coherency measure as a fault security measure based on analysis on a second order system. 5.2.1. Efficient PEBS Algorithm for Predicting Stability The procedure for computing the boundary (critical) energy Vcr and using it as a stability limit consists of the following steps: (1) Solve base case load flow equation (5.6) to obtain gen- erator angles at prefault (initial) operating point 651. n Mimi = P1. - Z [Cij sin 81.3. - 01.1 cos 61.3.] = 0 (5.6) i=1 jfi where 913- = 6i " 6:)- 6 =6 - -'- 3 n :5 —-i IH H H. 3 II ...n —-l .1 60:; center of inertial angle (2) For an arbitrary large clearing time (t >> tcc) use c the linear or nonlinear RMS coherency measure for tie particular fault to obtain the peak angles emax as well as the critical group or critical generator. (3) Having determined the critical generator i which is neces- sary to determine the proper generator to write the individual 1 machine energy function, the prefault operating point as and the peak angles 6(tBl,tc1) = emax, calculate the potential energy at time tBl, V;5i(tcl) = vPE(t81;tc1) emax vcr T VPEi 951 = VPE(tBl’tc1) 112 max max $1 $1 9. +6. -6- -6. - _ i j i j . max _ . sl Dij max max $1 $1 [51” eij 51” eij] (5'7) 61- +8J. -01. +6J. (4) For a specific clearing time tC < tc , where the stability 1 of the system is desired to be tested, repeat steps 2 and 3, and 'k then calculate potential energy of the critical generator VPE.(tc) . _ 'k and compare it to Vcr - VPEi(tc ). 1 (5) If VPEi(tc) §=Vcr=:vPEi(tc1) the system remains stable, and if VPEi(tc) > Vcr = VPEi(tc1) the system would be unstable. The applications of this algorithm to different fault cases on the Reduced Iowa system are discussed in Chapter 7 and the results obtained from these applications verify the accuracy of the algorithm. 5.3. Fast Equal Area Method From review of Chapter 3 it is understood that for multi- machine power systems the equal area criterion considers a par- ticular generator (critical generator) against the rest of the gen- erators in the system. The amount of energy produced during the fault period (accelerating energy) A1(t is added to the energy C) after the fault is cleared (decelerating energy) A3(t) and the quantity AE(tB,tC) = A1(tc) + A3(tB) at time t3, the time at which the system generators peak, determines the stability of the system as shown in Figure 3.2 in Chapter 3 for single machine infinite bus system. In order to determine the stability boundary of the system tCC very accurately, one has to compute the minimum energy 113 difference AE(tB,tC) = AE*(tC) for different clearing times and search for the smallest value among these minimum energy differences. This search technique, of course, requires more computations and hence it is time consuming and requires significant computation. An approximation method is proposed that does not require determining the minimum of AE*(tC). If AE*(tc) is near zero for t §=tcc for any fault case, it may be possible to test for sta- c bility of the system by determining whether AE*(tc) is less than some small a. The selection of 8 appears to be fault dependent based on the results for the Cooper and Fort Calhoun cases in Tables 4.3 and 4.4, respectively. Furthermore, the assumption that if AE*(tc) < a stability is assured is not exactly correct since the value of AE*(tC) for some interval tC above tCC is less than a as shown in Figure 5.2. If this interval is small, and if one could find an e for all fault cases, the algorithm presented in the next subsection could be used for fast stability assessment. Determination of the critical generator and the generator max) and clearing time angle positions of the system at time t8 (6 tc(ec) is the key point in calculating the minimum of energy dif- ference AE(tB,tc) = AE*(tc). Analogous to the "fast PEBS method," one could apply the same RMS coherency measure (linear/nonlinear) technique referenced in section 5.2 for post-fault network and X directly predict the peak angles ema as well as the critical gen- erator of the system. 114 1- tcc A? T I j .216 .220 . . ‘ i 4 1L.“ N N .5 ‘l .192 . .2653" .212 N O r!- »—-———-—-————4 h-_-—~__--—-1 *~_—--———-——-—--—-‘ Figure 5.2. Determination of critical boundary for critical generator using minimum energy margin AE(tB,tc) method. 115 The angles at clearing time tC(eC) can be approximated with enough accuracy by using the Taylor series approximation algorithm as described in subsection 5.1.2 for the faulted network. The fol- lowing subsection summarizes the algorithm and outlines the pro- cedure in a suitable order. 5.3.1. Efficient Equal-Area Algorithm for Predicting Stability The procedure for computing the transient energy margin AE*(tC) and predicting the stability of the power system involves the following computations: (1) Solve the base case load flow equation (5.6) to obtain generator angles at prefault (initial) operating point 951. (2) Apply Taylor series approximation algorithm for faulted network to obtain the generator angles at clearing time tc,e(tc) = 9c. (3) Use the linear pulse coherency measure to obtain angles at time tB,e(tB) = emax and the critical generator. (4) Having determined the critical generator and thus the generator fOr which the individual machine potential energy is to be evaluated, compute energy A1(951,ec) for faulted network using equation (5.1) A 951,6C) = V 1< PE. i 651 116 (5) Compute energy A2(ec,emax) for post-fault network using equation (5.1) max (6) Compute the sum of A1 + A2. (7) If A1 + A2:; a, the system is stable. If A1 + A2 > e, the system is unstable. This algorithm was tested on the 17-Generator Reduced Iowa System for different fault cases and the results are shown in Chapter 7. Comparison of the two algorithms, the fast equal-area method and the fast PEBS method, determines that the latter is more effi- cient and computationally faster because: (a) it is not necessary to calculate the system parameters for the faulted network, and (b) there is no need to calculate the generator angles at clearing time 6(tc) = 6C using the Taylor series approximation which in turn requires a reasonable amount of calculations. The equal-area method requires both (a) and (b) to compute accel- erating energy A1. CHAPTER 6 DERIVATION, JUSTIFICATION, AND VERIFICATION 0F SECURITY MEASURE, PREDICTION OF PEAK ANGLES 6.1. Introduction The concepts of this chapter deal with the development of a linearized power system state model, a generalized disturbance model, and the root mean square (RMS) coherency measure. These models and the generalized coherency measure are used to derive algebraic expres- sions which relate the RMS coherency measure, evaluated over an infinite observation interval for step, impulse, and pulse distur- bances in mechanical input power, to the parameters of the power system state model and probabilistic description of the disturbance vector. Finally, the theoretical justification of the fault security measure for a second order system is discussed and the prediction of the peak angles of the power system by RMS pulse coherency measure is verified. The last section derives the computational algorithm of pulse coherency measure for a multimachine power system. 6.2. Linearized Power System Model A system of linearized state equations is derived for a power system which is composed of classical synchronous machine models, voltage dependent load models, and a transmission network model. 117 118 A linear model can be derived from the nonlinear differential equations for the electromechanical motion of the classical syn- chronous generators plus a set of algebraic equations for the power 'Flows between the generators and the load buses of the system. The eelectromechanical equations for the motion of each synchronous gen- erator are: where d _ 3? Sim ‘ “i(t) (6.1a) d M,- 'a'f w,(t) = PM (t) - PGi(t) - 01...].(t) (6.1b) l = 1, 2, , n M. = inertia constant of generator i (in p.u.) D. = damping constant of generator i (in p.u.) 6. = rotor angle of generator i (in radians) w. = speed of generator i (in rad/sec) PM. = mechanical input power of generator i (in p.u.) PG. = electrical output power of generator i (in p.u.) n 8 total number of generators in the system In some papers, equation (6.1a) is given as: 34t— aim = wow) = 2nf0mi(t) (6.2) where fo is the synchronous frequency of the system in Hertz, and (L) i '18 in per unit (p.u.) instead of rad/sec. Equations (6 1) are 119 nonlinear because of the nonlinear relationship between P01 and the bus angles in the interconnected network. Equations (6.1) can be linearized around the nominal operating conditions 6:1, (0:1, PG?1, and PM?1 by introducing the following deviations: _ $1 061. - <51. - 61. _ 51 A101 "' (Di - 0.11- _ sl APM = PM - PM51 i i i The resulting linear model has the form 39%. A61.(t) = min) (6.3a) Mi é%-Awi(t) = APM1(t) - APGi(t) - 0166i (6.36) i =1, 2, ...., n Wher‘e A indicates that the variable represents a small deviation From a specified steady state operating point. The changes in the complex voltages and power injections at the network generator and load buses may be expressed using a JaCObian matrix as [22] where 3g = [261, P62, .... l8 P .g = [51, £2, .... §_ = [61, 62, 1 = [V1, v2, g = [61, 82, 399/39. ”ABE. Egg/36 Afl 335/36 AQ§_ - _AQ_|_.__ _39L/3§ [QG1, QGZ’ .... PL = [PL1, PL2, .... [QL1, 0L2, .... 120 322/31». egg/a]: egg/3y 300/831 pa 1 06an PLk]T T QLkJ BBQ/BE egg/61' [66. 33/3; egg/a! A]; aQ_G/6_E_ aQ_G/a_V_ Ag aQL/ag egg/ail _AL (6.4) real power injections at internal generator buses (p.u.) reactive power injections at internal generator buses (p.u.) real power residuals at load buses (p.u.) reactive power residuals at load buses (p.u.) voltages behind transient reac- tances at generator internal buses (p.u.) angles at internal generator buses (rad) voltages at load buses (p.u.) angles at load buses (rad) For low loss systems (resistances in the transmission network are close to zero) equations (6.4) can be simplified by accounting for the decoupling which exists between the real and reactive power flows [23]. The real power flows are largely dependent upon the voltage angles and as a first approximation, the effect of variations in load bus voltage magnitude may be neglected by setting the terms QEQ/a! and aft/ay_in equations (6.4) to zero. The voltages behind 121 the generator transient reactances are constant, thus Ag =10. There- fore, in equations (6.4) real power and phase angles are decoupled from reactive power and voltage magnitude. These linearized decoupled equations for real power flows can be written in polar form as: agg BEE/86 3333/33 66 43L agL/ag egg/62 go The partial derivatives corresponding to the above four terms are most precisely calculated using the voltages and angles at the post- fault steady state operating point 652, 652. The power angle Jacobian matrix in the network equations (6.5) is a sparse, symmetric, and singular matrix. Therefore, a unique solution for ag and A9: given aE§_and 63L, cannot be obtained. This minor problem can be solved by an angle referencing scheme [24]. Equations (6.3) and (6.5) are said to be synchronous frame model since the deviations in generator and bus angles and generator speeds are measured with respect to an external reference rotating at the nominal system speed (fo = 60 Hertz). The deviations in generator angles in response to a step disturbance in mechanical input power will appear as ramp function. Therefore, the synchronous frame modelhasian eigenvalue at the origin (step input, ramp output). Since a linear state model is desired that has all non-zero eigenvalues, an arbitrary reference is chosen for the angles and speeds of the generators. Selecting generator N of the system as 122 the reference is a common practice in power system analysis, and in a practical power system generator N is usually designated to be one of the generators with a comparatively large inertia. This fact suggests a modified version of the generator N reference known as the "Nth machine reference" frame. The resulting linear model with N-machine reference frame has the form mm D CD do A fl v H D E c-J A d v —J H -1 M1 [APMi(t) - APGi(t)] 53 D E ) a. A fl v II - M&1[APMN(t) - APGN(t)] - GAQi where o = Di/Mi i = 1, 2, , 81(t) = 61(t) - 5N(t) i = 1, 2, , 01(t) = 61(t) - 6N(t) 1 =1, 2, , win) = wJ-(t) - aw) i =1. 2. . The power equations in terms of the new variables written as: AE§_ egg/36 agg/ag 66 63!: egg/39 all/33g A! where 1, 2, ...., n - 1 can (6.6a) (6.6b) be 123 ]T - _ T §-[61,62, ....,6 ,i‘[w1,ng ...oa Wk‘l] n-l The network equations can be used to express 636 in terms of 66 and AEL- This can be done by solving the second equation in (6.7) for A? and substituting it in the first equation in (6.7) to obtain 63g agg aP_L -1331: afi 23L ’1 ‘39-" ‘53"??? 3519* “5'1“ ‘31" (.11 0?“ AE§_= Tag - LafiL (6.8) where 3g); 33g 33!; “163): 3% a_P_L “1 “first"? 33¢“?fo ‘5'” ‘1 is called the synchronizing torque coeffient matrix and L is called the load reflection matrix. Now, a linearized state space model can be derived by substi- tuting BE in (6.8) into equation (6.6b) and writing the 2(n - 1) equations in (6.6) in vector form to obtain 3(t) = Ax + §g(t) (6.10) where §= m. ,3: ..... (6.10a) . . -(n-1) (n-l): -n-1 9 I 9 l E = ............. 1 ---------- , 8 = ---T---- (5-10b) l _ I I. “.11. | Oin-1 J b M | ...".J -1 -1- LMI : ‘MN -1 ' M2 I o I I I E = : (6.10C) I 0 | . I -1 I -MN -1 I -1 _ M n-1 I “MN J The next task in this chapter is to derive the disturbance model which can be used for deterministic as well as probabilistic system disturbances. The disturbance model has been developed in [21] and the presentation here follows that development. 6.3. Disturbance Model The input g(t), composed of the deviations in the mechanical input power AB! on the generators and the deviations in load power 43L, can be used to model: (1) loss of generation due to generator dropping (2) loss of load due to load shedding (3) changes in load injections due to line switching (4) electrical faults 125 These contingencies can be modeled by an input u(t) that has the following form u(t) = u(t) + u2(t) (6.11) 6.3.1. Step Disturbance The vector function I 91 t ;=0 91(t) =1 (6°12) t < O ‘0 where 91(t) is a vector step function with amplitude 91‘ Thus, the non-zero entries in 91(t) can model the first three types of disturbances. The uncertainty due to a generator dropping, load shedding, and line switching disturbance could be modelled by r. ABM -l P.E11 . EIul} = E < ----- I = ----- =‘m1 (6.13a) . ABE-.j .-m12 ] P311591 “Ml-mHh-mflT= ----- i ----- =31 (6mm .. 9 5322. where (1) @11 and 311 can describe the uncertainty in the location and magnitude of generation changes due to generator dropping when 126 the particular station, the generator in the station, and the power produced on the generator are unknown. (2) @12 and 322 describe the uncertainty in the location and magnitude of the load being dropped by any manual or automatic load shedding operation. (3) @12 and 322 can describe the uncertainty in the location and the change in injections on buses due to any line switching operation. The uncertain model of 91 can handle the case of a specific deterministic disturbance by setting 81 = 0 and m1 = 91 for the particular disturbance. The function ul(t) can only model disturbances that resemble step changes. 6.3.2. Electrical Faults To model electrical faults, first define the vector function I Q t > T2 92(t) = 1 92 0 ;3t ;,T2 (6.14) I O t < O that represents a pulse of duration T2 and amplitude 92' This vector function can represent the effects of elettrical faults where T2 represents the fault clearing time and [ AEM_] 127 represents the step change in generation output equivalent to the accelerating powers due to a particular fault. This change of mech- anical powers, ABM, which is equal to the accelerating powers on generators due to a particular fault is calculated by an ACCEL pro- gram [22], and has been shown to adequately model the effects of that fault when a linearized model based on pre-fault load flow con- ditions is used. The above model can be generalized to model the uncertainty of any particular disturbance and yet handle specific deterministic disturbance as a special case. If the size and location of an elec- trical fault are not known and if the clearing time T2 for this fault is known, then a probabilistic description of this fault is '1‘21- E{92} = “““ = T2 (6.158) 19. . 321 E 9 . E{[y2 ‘ T2][92 ‘ T21T} = """ é"' = 82 (6.15b) (050 - -.I where m21 and 321 describe the uncertainty in accelerating power on all generators due to this electircal fault. This mean and variance should be determined based on observed historical records or hypothe- sized based on the present network conditions. If R = O, and m21 = AEH for a specific fault, this generalized model then reverts to the detenninistic model of a specific electrical fault. 128 It should be noted that 63M and ABE are assumed to be uncor- related because this model is to represent only one specific type of contingency at a time. For the same reason 91 and 92 are assumed uncorrelated with initial conditions, i.e. E{§(0)u{} = g (6.16) E{§(O)u;} = 9 where the initial conditions of the linear differential equations (6.10) are assumed random with 515(0)} = g (6.17a) E{§(0)§T(0)} = yx(0) (6.17b) 6.4. Linear RMS Coherency Measure The RMS measure of coherency between generator internal buses k and £ based on the uncertain description of disturbances is [25] Ck£(T1)=\/T%§ E{~/;T1[(15k(t)..A5N(t))- (Aa£(t)..AaN(t))]2 dt = J§l£§x(T1)§—‘k£ (6.18) where T1 -X(T1)=?15[ E{>_<(t)>_ T2 (6.22) Ftn" a specific step input disturbance (load shedding, loss of genera- ‘tion, or line switching) with the following specification P = 1, Tl = 91, m2 = O, 81 = R2 = Q, and V (0) = Q 131 the matrix §x(T1) has the form 1' Aq T (T)=T1—f:1{[fAeqqu]u1u1[[ e dQE] }dT (6-23) 0 If the specific deterministic disturbance is an electrical fault sinceirithis case = 82 = 9. and !x(0) = 9 . P=0s@1=0,m2=92,81 the matrix §x(T1) becomes T T T T = j w WJII no] I .. O 0 O T ,T_ T +[ 1{[e/M T2)] 2eAqqu] uzu u2|:eA (I Tsz szquIBJTdr} T 2 o o (6.24) This RMS coherency measure can handle both deterministic and probabilistic descriptions of power system disturbances. It is shown in [26] that the RMS coherency measure evaluated over an infi- nite interval (T1 = m) can be analytically related to generator inertias. synchronizing power coefficients tkz of equivalent lines connecting internal generator buses k, 1., and the statistics of the disturbances . 132 In the following subsections an infinite interval-RMS coherency measure will be derived for step, impulse, and pulse disturbances in the mechanical input power of the generators. 6.4.1. RMS Coherengy Measure for Impulse Input Disturbance The RMS coherency measure matrix for the impulse input dis- turbance can be obtained from the general equation (6.21) provided §x(w) for impulse is derived first, using -x T” T 1 s (o) =.2nn IL. E{§(t)§T(t)}dt (6.25) 1 where §(t) is the solution of 5(t) = A§(t) + 881(t) (6.26) for the impulse input 91(t) = g- 6 (t) . Assuming zero initial conditions, i.e., at t = 0, 3(0) = 0, the state of the system at time t can be found as t 8(t-t) 8t 5(t) = e 896(t)dt = e 89 (6.27) 0 Substituting (6.27) into (6.26), 8x (m) for impulse becomes 1 T1 At T T th 8 (m) = Kim E{ e 899 8 e J-dt xI T1+00 0 T1 At T T 5T: = Kim (e 8E{gg }B e )dt (6.28) T17“ o 133 where represents the step change in generation output equivalent to the accelerating powers due to this impulse. The statistics of this disturbance are defined to be I m -al -a = E{g} = TTTTT 91 T 8,1;9 B, = E{[9 - 0,1[9 - ma] 1 = ----- f-- o E o 1-.- thus I- T' B l + Talmal E I EIaaT} = R + m mT = .............. + -- -a -a-a ' 9 I b I where god is the variance matrix and Tel is (6.29a) (6.29b) (6.30) the mean vector of the uncertainty in accelerating power on all generators due to this impulse (electrical fault). Defining v = B(R + m mT) T -a - -a -a-a 8 equation (6.28) becomes S (m) = £im (e' V e )dt (6.31) -x I T1+0° In order to obtain a closed form solution for §XI(m) it is appropriate to approximate the impulse as a pulse with very short duration. To clarify this let us start with the pulse and its sta- tistics and try to relate it to the impulse and its statistics approximately. Define the pulse of duration T2 as I8 t>T2 920:) = < 92 o 9.- ;i2 (6.32) LQ t>0 where 92 - [ABM E 8]T , and its statistics are . r r1‘21 I112 = ELL-‘2} = “““ (6.33&) 9 . : . 321 : 9 I 32 = 61192 - mZJIQZ - 621T} = ----- I"‘ (6.33b) 9 59 'The impulse now can be approximated by the above pulse as follows r 8 t > T2 91(t) = 86(t) = fimo 82(t) z ( 1/T2 a 0:; t é=T2 2 19 t<0 135 and the statistics of pulse and impulse may be related as a .1. T2 T2 ma (6.34a) R =-l— R (6 34b) -2 T2 -a ' 2 . It has been shown in [26] that for a pulse of very short duration (impulse), 8x (m) can be determined from the following equation . g (6.35) where 8 is the solution of the following Lyapunov equation 6! + ggT = - Y2 (6.366) and 12 = ms + TZTEJET (6.368 T1 5t th W = Kim (e V e )dt _ T +m -2 1 o Tl'Tz 8t ATt = Kim (e 82a )dt (6.36c) T1+00 0 Comparing equations (6.31) and (6.36c) and knowing that 82 =-l§ ya T2 it is obvious that 8 =-l> 8x (m). Substituting matrices 8 and 82 TS I into equation (6.368), the following Lyapunov equation is obtained 136 6§x (.) + 5x (.)A = - ya (6.37) I I This Lyapunov equation (6.37) can be easily solved by considering the symmetric property of the matrix 8x (w). Thus, partitioning I S (w) as ..xI , . .S.1:§2 §XI(~) = ----f---- (6.38) sT's no-2 ‘34 and calculating ..g : 9 . _ TT_ ' -. - 9(8, + n._,)§ - "'I ------------------- I T T _.9 I ”(8&1 + Ta19a1)N along with some algebraic manipulation 81 becomes - ~ . ..1. -1 T T §1 - §XI( ) 40 [(fl) ”(F-{011 + r-I-IOLIU-‘odfl:4 T T -T + («(60,1 + malma1)M (MI) 1 (6.39) The expression (6.39) shows that for the impulse input disturbance the matrix 8x (m), which defines the infinite interval RMS coherency I measure, is related algebraically to the parameters of the linear system model and the disturbance statistics. 137 6.4.2. RMS Coherency Measure for Pulse Input Disturbances The RMS coherency measure for a pulse input disturbance 8x (a) may be obtained as the limit of 8x (T1) when T1 approaches infingty. 8x (w) for a pulse disturbance of duration T2 has been derived in P [24] and is shown to be Q _.l n T §x (...) = 3‘."- 15” )1 + («(504) l (6.40) p n=2 where 8 is the solution of the Lyapunov equation (6.36a). The solu- tion of this Lyapunov equation is similar to the solution of the Lyapunov equation for the impulse disturbance. The solution of 8 is exactly the same as the solution of 8x (m), except that we have to I use the statistics of the pulse rather than the impulse statistics. This solution has the form £E-[(MI)-1MAEM°APMTMT4-MAEMAPMTMTIMI)-TJ 9 o i; [8AfflAPMTMT] (6.40a) where 8 and Afl are the inertia matrix and the accelerating power on all generators due to the pulse input disturbance, respectively (electrical fault). If the pulse duration time T2 is very short,only the first term in the series (6.40) will be required, and under this assumption 8x (as) = TE 8. I 138 Equation (6.40) shows that for the pulse disturbance, 8 (m), which P defines infinite interval RMS coherency measure, is related alge- braically to system structure and disturbance statistics. 6.4.3. Justification of Nonlinear RMS Coherency Measure The nonlinear RMS coherency measure is now derived based on the linear RMS coherency measure derived in the previous section by showing the term [81]"18888_in 8 is a linearized inertial load- flow. The inertial load-flow equation is first proven to be defined as a singular point of the global energy function. An expression for a linearized inertial load-flow equation is then obtained. Finally, the angle changes for this inertial load-flow are shown to satisfy A? = [Mil-18' 42!. (6.41) which appears in 8. This justifies replacing expression (6.41) in W by §u _ ész. The inertial load-flow equation that defines the singular point of the global energy function is now derived. Consider the energy function of the system represented by n-1 n V = Z Z [L‘Mj (... - o )2 - (PIMJ' ' PJM‘M .- 652) ZMT i j MT 13 13 i=1 j=i+1 $2 - Cij(cos Sij - cos Gij)] (5-42) 139 Take partial derivative of V with respect to 5i and set equal to zero, aV/aoi = 0. n n P M. - P.M. - l J J I = 2 MT +2 C13 SH] 6 J 0 (6.43) i=1 i=1 jfi n Pi 25::Mj M n n i=1 _ _1 = . MT MT Pj Cij Sln Gij (6.44) i=1 i=1 Mi n n . i=1 i=1 Assume that Pi = PMi; therefore M. n n .1 = - A i=1 i=1 The equation (6.46) is an inertial load-flow equation and is used to solve for an unstable equilibrium point of the system. This equation can be linearized to form the following: 8PM1. - M71: APM. =ZU T. .86. (6.47) where 140 —.' II I O n O U) 0) —Jo _g _l —I II I ..1 .4. c... M. n-1 n-1 l _ 8PM. --5; 8PM, - Tijaei (6.48) i=1 i=1 where A0 = A61 - ooN or [ . [‘1 1‘1 11 MT MT MT MT ”T MT IA2=II- )AEM. E in . MT ' MT A (6.50) n where ZMJ. = MT' Multiplying both sides of equation (6.50) from i=1 the left by 8, where: 141 'L __1_1 M1 Mn N = _1____1_ _ Mn-1 Mn . and noting that: '_1_ ._1_1 11.141. ”LII M1 Mn MT MT MT 1212 112.0 MT MT MT '. -1-_1_ Flair: in. L Mn-l Mn . L MT MT MT ‘ then (6.50) may be written as: MAPM = MT AB (6.51) Thus, the inertial angle changes for step input disturbance (loss of generation i and generator buses are: [MU-1863M (6.52) A0 As a result of this discussion one could use the UEP angles eu _ 952 AG, calculated by a special program, instead of [8:]'18.88 to calculate the fault coherency measures (non-linear). This leads us to the following equation 0) + 66(0)(e” - 852) (6.53) In the next subsection the method of obtaining the peak angles from matrix 8x (6) is described. 1 6.4.4. Theoretical Justification of Fault Security Measure for Second Order System Consider a second order system such as single machine infinite bus power system; its behavior can be represented by the following linear second order differential equation M166(c) + DA6(t) + TA6(t) = u(t) (6.54) where 3 C) H II II inertia of the machine damping coefficient and u(t), A6(t) are defined as input and output of the system. The point of concern here is to study the response of this second order system to (1) step input disturbance (2) impulse input disturbance and then show that these responses are related to each other. System Response to Step Input Disturbance. Defining the step input function as 143 one could solve the linear differential equation I". > O (6.55) M1865(t) + DA6S(t) + TA6S(t) = ARM and obtain the closed form solution 865(t) -D/2M t 1 cos (\/4M1T - DZ/ZMl) t APM{1_e -D/2M t - D e 1 sin ( ./4M1i - 82/2141) t} (6.56) ,/4M1T - D2 Steady-state response of the system is obtained when t + w _ APM A6S( ) - ‘T” (6.57) To find initial acceleration 865(0), set D = 0 in equation (6.56): (6.58) _ APM A6S(t) - ‘T‘ {1 - cos (v/4M1T/2M1 ) t and then take the second derivative of equation (6.58) t) = fl cos (/4M1T/2M1)t 'A6( 5 M1 By setting t = 0, the initial acceleration will be obtained 144 665(0) = 434 (6.59) Multiplying equation (6.57) by equation (6.59), -8442! AOS(@) AOS(O) - —TP' M1 (6.60) Rearrange equation (6.60) and define M 94%; to obtain the result in 1 the compact form A6S(m)° 665(0) (MT)'1MAPM- APM- M (6.61) System Response to Impulse Input Disturbance. Consider the same second order model and apply an impulse function u(t) = APM-6(t) where 6(t) is defined as a delta function. M166(t) + DA6(t) + TA6(t) = APM- 6(t) (6.62) The solution of this differential equation can be expressed as ‘2fi‘ t A6 (t) = ZAP” e 1 sin (,I4M T - DZ/ZM ) t (6.63) 1 4M T Dz 1 1 I 1 - The maximum value of A6I(t) can be obtained by taking the first derivative of A6I(t) with respect to t. 1(1)/o: (t) = ZAPM e‘(tan 0° sin (tan'1 a) (6.64) AG {2 Imax where 145 o g\/ 41411 - Dz/D For the case where damping is zero (0 = 0), (t) = AP” = (M i)‘* APM (6.65) Imax \I MlT 1 Let M Q-£L and rewrite equation (6.65) in desired fonn 1 (t) = (MT)'* M- APM (6.66) By comparing equations (6.66) and (6.61) with each other, the rela- tionship between the response of the system to impulse disturbance and step disturbance is represented by (t) = L86 (w)A6 (0) (6.67) For multimachine power systems, the security measure for impulse input disturbance can be written as 2 . T s (a) = -3(}Mi)‘1MoPMoPMTMT + MAPMAPMTMT(MT)'T] (6.68) .% 4o— »——- -———-— substitute (Mll'ltIABUAPMTMT by 146 T 86 (t)- 86 _—Jmax _"{max(t) in (6.68) and write the resulted expression in vector form 5,6) =kztoel ~86} +82} 82., 1 (6.69) I max max max max where k2 2 k2 - T2 "16 Having calculated the matrix 8x (m), the diagonal elements of this I matrix are proportional to the square of the angle changes of the - 2 generators (86} ) for i = 1, 2, ...., n due to impulse disturbance max and the post-fault system generator angles at time tB are 9} = 652 + koe} , i = 1, 2, ...., n (6.70) max 7 max The above development of the expression for the peak angle deviation for the second order system (6.54) is based on the assump- tion of zero damping. However, the RMS coherency measure depends on the damping to inertia ratio 6 = Di/Mi which it should because the peak angle deviation should decrease for increased damping. Thus, 0 should be set experimentally to reflect the effect of turbine damping Di as well as the effective damping caused by the load impedance reflected in the conductances in the transmission network. Thus, the square root of the diagonal elements of the matrix SI(w) can be added to the base case load-flow angles to predict the 147 peak angles for the multimachine system. This prediction (6.70) of the peak angles using the linear or nonlinear RMS coherency measure has not been justified based on analysis of the multimachine model but solely based on the analysis on the special case of the second order system model. The accuracy of the prediction of the peak angles (6.70) using the RMS coherency measure from experimental results in the next chapter justifies the use of this prediction model. 6.5. Computational Algorithm for Infinite IntervéT'PuTSe Coherency Measure From the discussion of subsection (6.4.2) it is understood that the coherency measure for pulse input disturbance, which was derived in [24], has the following expression - 2 n-2 n-2 §x (...) - 2;; [A 21+ («A ) 1 (6.71) p n=2 where 8 is the solution of the Lyapunov equation (6.36a) and has the form shown in (6.40a). The objective of this section is to derive an algorithm for computing the coefficient matrices En-Z 8 in the above series expan- sion for the pulse coherency measure. Consider the matrix 8 shown in (6.40a) and define the sub- matrices 8 and 8 as Ix so that I: (6.72) For the power system model with zero damping (o = D/M = 0), the matrix 8 has the following form |J> form AN l>< IO '0 l-< l-< 1 1 fi IE IS (6.73) l-< '0 IO 149 13> I: II II (MT)"'18J — L MT 0 '0 Since the pulse coherency measure is computed from upper diago- nal submatrix, it is clear that the matrices 2k (82kg)11 = upper diagonal submatrix of 5 8 = (81)k8 (6.756) need to be computed since (82k'18)11 = upper diagonal submatrix of 52k'18 = g (6.75b) for k = 1, 2, If we define vectors 0 - —— 2’5 (6.76) v1 9 —l—-8888 2/3 then matrix 8 has the form T T vov1 + vlvo (6.77) |>< II 150 and the series (81)k8 could be written as a function of these vectors v0 and v1 as the following _ T T _ T T (M )8 - (MT)vov1 + (MT)v1vO - vlv1 + V2Vo 2 _ T T _ T T MT.) ’3 ' (M—T-IV1V1 T (wvzvo ‘ V2V1 I v3"o (ET-I33 ' (EIVZVI I (EIV3V6 = V3V1 I W; (111)"! = (_i)vk_1v{ + (mvkv; = vkq + vk+1vg (6.78) . where vk+1 MT vk , k = o, 1, 2, The series .. k T 2 (5k 25)11 TI k-2 can then be written as .. k 4 T T T 2 k-2 _ 2 T T 2 T 121“} ”11 ‘ 2T ”1% + Vovl) I 71T("2Vo T V1V1) + k=2 and 151 (MGR-2) I = 2T(V1Vo+" V1)”“‘I‘(" k=2 11 (6.79) 5' NHX' As a result of this analysis, one has to compute the vectors Vi i = 0, 1, 2, 3 to compute the pulse coherency measure, which pre- dicts the peak angles of the system for a particular fault. The computational algorithm of the pulse coherency measure for predicting peak angles of the generators in the system can be summarized in the following order. (1) Compute generator accelerating power 888 using ACCEL pro- gram, which requires one linear matrix equation to be solved. Then compute v1 = 8838_directly. /‘ (2) CofipZte vo from inertial load-flow equation for post- fault network (see Appendix A). (3) Compute Vk+1’ k = 1, 2, ... from iterative equation involv- ing pre-fault network Jacobian network matrix (see Appendix A). (4) Compute post-fault stable equilibrium points 652 from DC load-flow equation. (5) Substitute vés in equations (6.79) to obtain the pulse coherency measure matrix 8x (m). Then take the square root of diago- nal elements of this matrix and add them to the post-fault stable equilibrium point angles as was shown in equation (6.70). The results would predict the generator peak angles for this fault disturbance. This method of finding the peak angles of the system generators is very fast and computationally efficient in comparison with trajec- tory simulation method. Simulating the» systan trajectory 152 requires iterative solution of n + m (where n is the number of gen- erator buses and m is the number of load buses) dimensional set of nonlinear equations every integral step size At = .01 seconds during fault and every At = .05 seconds after fault clearing. This approxi- mation algorithm requires solution of only a few sparse linear matrix equations such as: (1) one linear equation solution for computing 838 (2) one solution of a nonlinear load flow for post-fault stable equilibrium point 6S2 (3) one linear equation solution for computing v0 (4) computational results showed that after four iterations terms (_MI)k 8, for k = 1, 2, 3, 4, the measure (6.71) converged. Therefore the algorithm requires in total six sparse linear equation solutions and one nonlinear load flow solution. CHAPTER 7 STABILITY ANALYSIS USING FAST DIRECT METHODS AND COMPUTATIONAL RESULTS 7.1. Introduction The same power network (Reduced Iowa System) described in Chapter 4 was used to study the stability of the system by fast direct methods for the same three fault cases, Cooper, Fort Calhoun, and Raun, as described in Chapter 4. The objective of this chapter is to apply the fast direct method stability algorithms (the PEBS and equal-area) and show the results of the direct stability run for each fault case and dichss the accuracy of these results. 7.2. Cooper Case A three-phase fault is applied to generator 2 (Cooper) and is removed by clearing line 6-439. The point of interest is to study the behavior of the system due to the fault and directly determine whether the system is stable or unstable without simulating the system trajectory. Recall the efficient PEBS algorithm described in subsection 5.2.1. To compute the boundary energy VPET(tB’tc) = vcr from equa- 51 tion (5.1) having the prefault Operating state 6 , it is understood max that one needs to determine the peak angles 6 (using linear or 153 154 nonlinear RMS coherency measure) as well as the critical generator or the critical group. The linear and nonlinear RMS coherency pro- gram was run for different clearing times and generator 2 was found to be the generator with the highest peak angle. Therefore, gener- ator 2 is predicted as the critical generator. Tables 7.1 and 7.2 show the peak angles of the generators with respect to generator 13 as a reference generator for two different clearing times tC = 0.192 and 0.210 seconds. Although the predicted peak angles by RMS coher- ency program (entries of columns 3 and 4) do not match with those obtained from simulation program (entries of column 2) in these tables, the results of computing the maximum potential energy of the critical generator using these approximated peak angles shows that the fast PEBS algorithm works accurately and predicts the stability boundary of the system. For different clearing times, the proposed PEBS method with both linear and nonlinear RMS coherency as peak angle predictors were tested on the Reduced Iowa Test system and the results are summarized in Table 7.3. In Table 7.3 columns 2 and 3 represent the boundary energies v;E(tc) calculated for different clearing times using the linear RMS coherency measure as predictor, for the energy function (5.1) with conductance term included and excluded, respectively. In both cases the largest peak value of the boundary energies occurs for clearing time within the interval tce (.210, .224) indicating that the critical clearing time is in this interval. 155 Table 7.1. Generator peak angles( (6 -81 for COOper using the RMS coherency measure and simulaEion programs. Clearing time = .192 seconds RMS coherency peak angles generator simulation number peak angles linear nonlinear 1 20.33 50.68 12.91 2 130.20 176.44 119.26 3 28.67 29.94 14.60 4 11.51 22.84 8.43 5 57.28 56.54 25.18 6 58.53 43.11 25.85 7 9.84 49.71 17.29 8 13.27 70.96 20.11 9 19.68 39.28 20.17 10 69.17 101.22 42.58 11 26.98 71.65 21.68 12 64.45 135.99 52.16 13 --- --- --- 14 - 0.26 6.49 2.12 15 13.66 16.52 9.40 16 74.12 113.28 46.51 17 92.78 138.14 56.29 156 Table 7.2. Generator peak angles (81-81 ) for Cooper using the RMS coherency measure and simula ion programs. Clearing time = 0.210 seconds. RMS coherency peak angles generator simulation number peak angles linear nonlinear 1 20.33 50.68 12.91 2 130.20 176.44 119.26 3 28.67 29.94 14.60 4 11.51 22.84 8.43 5 57.28 56.54 25.18 6 58.53 43.11 25.85 7 9.84 49.71 17.29 8 13.27 70.96 20.11 9 19.68 39.28 20.17 10 69.17 101.22 42.58 11 26.98 71.65 21.68 12 64.45 135.99 52.16 13 --- --- --- 14 - 0.26 6.49 2.12 15 13.66 16.52 9.40 16 74.12 113.28 46.51 17 92.78 138.14 56.29 Table 7. clearing times 0.192 0.208 0.210 0.216 0.224 0.24 \ 157 Table 7.3. Determination of maximum potential energy (PEBS) for Cooper using the fast direct method. linear nonlinear clearing with without with without times conductance conductance conductance conductance 0.192 9.907 4.915 9.937 6.886 0.208 9.941 5.156 10.896 7.928 0.210 9.947 5.174 11.009 8.057 0.216 9.949 5.197 11.336 8.187 0.224 9.773 5.112 11.744 8.930 0.24 9.516 5.062 10.917 8.735 158 Comparison of these results with the actual tcc s .220 seconds obtained from simulation program (Table 4.2) confirms the accuracy of the algorithm. It should be noted that in this research there was no attempt to determine the stability boundary very accurately. However, one could calculate the boundary energies v;E(tc) for addi- tional clearing times to precisely determine the highest possible value of V;E(tc). The corresponding clearing time for which this maximum energy value occurs was the critical clearing time from the results of Chapter 4. The results in columns 4 and 5 of Table 7.3 show that the boundary energy v;E(tc), where the nonlinear RMS coher- ency measure was used as the predictor, peaks for clearing time tC e (.216, .240), therefore the critical clearing time is within this interval. Although the critical clearing time tCC for the nonlinear case appears to occur at a later time compared to the critical clear- ing time of the linear case, the results are still in agreement with actual simulation results. For the same fault location (Cooper) the fast equal area algorithm was applied to the same test system and stability of the system was tested by computing the energy margin 8E(tc) = A1(tc) + A2(tc,tB) for differentclearing times. The results of this analysis are summarized in Table 7.4. From Table 7.4, where the local equal area energy margin for generator 2 is considered, it is observed that when the linear RMS coherency measure is used as the predictor of generator peak angles, 159 Table 7.4. Determination of minimum energy margin (EAC) for Cooper using the fast direct method. nonlinear clearing with without with without times conductance conductance conductance conductance .192 - .372 1.487 1.548 .791 .208 1.559 2.709 .486 .071 .210 1.832 2.886 .374 .004 .216 2.661 3.021 .018 .0038 .224 3.955 4.287 - .509 -.569 .24 6.244 6.215 1.573 .583 160 the minimum of 8E2(t8) increases as the clearing time increases. The critical clearing time based on the second and third columns of this table is not predictable because there is no minimum point for 8E2(tB) versus clearing time tC and hence the algorithm based on the linear RMS coherency measure does not work properly. The entries of the fourth and fifth columns with the nonlinear RMS coherency measure show that the energy margin 8E2(tB) decreases as the clearing time approaches the critical clearing time for tC<:tCC and then 8E2(tB) starts to increase as clearing time increases for tc > tcc’ The critical clearing time based on the fourth and fifth columns of Table 7.4 is again estimated to be in the interval tcc e (.216, .240). These results confirm that the approximation algorithm based on the nonlinear RMS coherency measure for predicting critical clearing time is very compatible to the results obtained from simulation. The nonlinear RMS coherency measure requires very significant computation to obtain the unstable equilibrium point. Since the linear RMS coherency measure appears to perform quite well for the PEBS method and since it can be computed at the cost of a decoupled load flow, the nonlinear RMS coherency measure was not applied to either the Fort Calhoun or Raun cases. However, the nonlinear RMS coherency measure performed satisfactorily for both the equal area and the PEBS methods where the linear RMS coherency measure only performed adequately for the PEBS method. 161 7.3. Fort Calhoun Case For the critical generator 16 of the Fort Calhoun case, a similar analysis to that of generator 2 of the Cooper case is per- formed. Generator 16 is correctly selected as critical from the maximum of peak angles predicted by the RMS coherency measure. Tables 7.5a and 7.5b summarize the results obtained by monitoring the individual machine potential energy (PEBS) and energy margin 8E(tB,tC) for generator 16, respectively. Note that in these analy- ses only the linear RMS coherency is used to predict the critical generator 16 and the peak angles of the system generators. From Table 7.5a it is clearly observed that as the clearing time increases toward the actual critical clearing time (which is not known at this time), the maximum of the potential energy increases until it reaches its highest value for tC c (.360, .368). However, this result is off by .008 seconds from the result obtained from simulation tc e (.352, .356) in Table 4.5. From Table 7.5b, where the minimum energy margin for generator 16 is considered, it is observed that again the minimum of 8E16(tB) increases as the clearing time increases. Since there is no minimum point for 8E16(tB) versus clearing time tc, the critical clearing time once again cannot be predicted using the fast equal-area algorithm and the linear RMS coherency measure. 7.4. Raun Case For the Raun case, for both the inclusion and exclusion of the transfer conductances the proposed fast PEBS and EAC algorithms for different clearing times were applied. In both cases the linear 162 Table 7.5a. Determination of maximum potential energy (PEBS) for Fort Calhoun using the fast direct method. linear clearing with without times conductance conductance .320 11.442 4.232 .336 11.584 4.451 .345 11.678 4.627 .352 11.714 4.721 .360 11.737 4.819 .364 11.748 4.936 .368 11.740 4.907 .384 11.690 4.749 Table 7.5b. Determination of minimum energy margin (EAC) for Fort Calhoun using the fast direct method. ‘ linear clearing with without times conductance conductance .320 -3.647 -1.504 .336 -2.831 -1.125 .345 -2.137 - .844 .352 -1.681 - .601 .360 -1.131 - .294 .364 - .883 - .082 .368 - .549 .046 .384 - .700 .822 163 RMS coherency measure was used to predict the critical generators and the generator peak angles. This coherency measure indicates gener- ator 6 is the most accelerated generator in the system. Therefore, by definition, generator 6 would determine the stability or insta- bility of the system. Thus, generator 6 is the critical generator in this case. However, in Chapter 4 it was shown that both gener- ators 5 and 6 were accelerated and separated from the rest of the system simultaneously, and it was also shown that more time was needed to drain out the excess clearing energy of generator 5 than that of 6. In other words, although both generators 5 and 6 lose synchronism with respect to the stationary group, generator 6 becomes weakly decoupled where generator 5 is still strongly coupled and becomes weakly decoupled at a later time if the system loses stability. This confinms the fact that the true mechanism of stability is dictated by generator 5 rather than 6. The linear RMS coherency measure therefore did not correctly identify the critical generator in this Raun case. However, in both algorithms (PEBS and EAC), the energy of gen- erator 6 was used to predict the stability boundary of the system. Tables 7.6a and 7.6b show the results obtained from the direct appli- cation of both the PEBS and EAC algorithms on the Reduced Iowa Test System, respectively. From the entries of Table 7.6a, for both the inclusion and exclusion of the transfer conductances, it is clear that the potential energy maximum of generator 6 with respect to the rest of the system has the highest value for clearing time around tc = .1922 seconds. This phenomenon indicates that the 164 Table 7.6a. Determination of maximum potential energy (PEBS) for Raun using the fast direct method. linear clearing with without times conductance conductance .160 10.384 7.205 .176 10.672 7.711 .1922 10.691 7.858 .208 10.184 7.608 .224 9.420 6.946 Tkable 7.6b. Determination of minimum energy margin (EAC) for Raun using the fast direct method. linear clearing with without times conductance conductance .160 .117 . 1.251 .176 2.785 3.216 .1922 6.294 5.941 .208 10.623 9.434 .224 15.686 13.640 x 165 estimated critical clearing time is approximately equal to tcc = .1922 seconds, which is very compatible to the actual boundary tc c (.1922, .1925) obtained from the simulation results. Table 7.6b shows the minimum energy margin of generator 6 with respect to the other generators 8E6(t in the system for dif- c) ferent clearing times for both the inclusion and exclusion of the transfer conductances. In both cases, the minimum energy margin 8E6(tc) varies with clearing time, i.e., for increasing clearing time tc, 8E6(tc) increases and this relationship holds even for large clearing times. Thus,function18E6(t ) does not have a minimum c point with respect to clearing time tc and hence one cannot predict the critical boundary tCC using the fast equal area prediction method with the linear RMS coherency measure. The PEBS method for the individual machine energy function with the linear RMS coherency measure used to predict the peak angles is quite accurate compared to results obtained: (1) Using the PEBS method for the individual machine energy function based on the actual simulation [16]. The results in Chap- ter 4 indicate that the critical clearing time could be determined much more accurately but the accuracy obtained in [16] was deemed adequate to indicate accuracy of the method. (2) Using the PEBS [17] or critical UEP methods [2] based on the total energy function. These methods have limitations on their accuracy due to use of the total energy function. 166 The fast PEBS method is quite acceptable in its level of accuracy because: (1) the results using the PEBS method for the individual energy function with the linear RMS coherency measure are as accu- rate as these previous results, and (2) for applications of this fast PEBS method for on line simulation at utility control centers and for screening contingen- cies for operation planning, and expansion planning it is more than sufficient. The fast PEBS method with the linear RMS coherency measure does not require the significant computation required for computing the nonlinear RMS coherency measure. Thus, since the linear RMS coherency measure appears to give results with adequate accuracy, the fast PEBS method with the nonlinear RMS coherency measure was not explored in depth. I The linear RMS coherency measure did not perform properly for the equal area criterion method for the individual machine energy function. The nonlinear RMS coherency measure did perform adequately for the equal area method but was not explored in depth due to the significant additional computation required for the nonlinear RMS coherency measure. The equal area method also requires computing the angles at clearing time using a Taylor series which is additional computation. 167 In conclusion, the fast PEBS method with the linear RMS coherency measure thus appears to provide adequate accuracy at a minimum computational requirement. CHAPTER 8 REVIEW, CONCLUSION, AND TOPICS FOR FUTURE INQUIRY “‘6’ 8.1. Chapter Review A fast accurate direct method for assessing whether the power I‘?‘ T“ system will or will not lose stability for a particular fault, line clearing action, and fault clearing time has been sought for over thirty-five years. The development of such a direct method should not require simulation of the fault and the clearing action for a particular fault clearing time but should require approximately the computation associated with a fast decoupled AC load flow. If this direct method of assessing stability for faults had the computa- tion requirements of an AC decoupled load flow, it could be applied in the following applications for fault contingencies as the decoupled load flow is presently used for line outage and loss of generation contingencies. (1) A screening tool for operation planning where all first and second contingencies are evaluated to (a) assess whether the operation plan for any day, week, or season is vulnerable to a loss of stability, (b) determine and rank contingencies for which the system is vulnerable, and (c) select the contingencies to be evalu- ated on-line at the utility control center. Theoperationpflan would 168 169 be modified if the particular operation plan was vulnerable to any credible first or second contingency. The fault contingencies that are found to be most severe can be evaluated using more accurate models to assess the cause and severity of the particular contingency. (2) As a check for system operators at control centers. The operators could assess the severity and whether the system would lose stability for any fault contingency or set of contingencies that appear of great concern because of loss of generation; line outages or fault contingencies haveoccurred that were not anticipated in the operation plan. The set of contingencies selected off line as part of the operation plan could also be simulated to determine if any of these contingencies would cause loss of stability or security due to changes in operating conditions. The operator will attempt to modify the operation based on the results of the con- tingency simulation to eliminate the security or stability problems . identified by the contingency simqlations. (3) A screening tool for expansion planning where all first and second contingencies are evaluated for each alternate expansion plan for several operating conditions. The relative security or reliability of the expansion plan will be used to help decide which expansion plan should be implemented. Detailed simulation of fault contingencies found to make the system vulnerable would be under- taken to accurately assess the cause and severity of the security or stability problem. The proposed contributions of this thesis are: 170 (1) To further refine the potential energy boundary surface and equal area methods developed previously [16, 19] using an indi- vidual machine energy function. The method for identifying the critical generator in [29] is used to select the machine for which the individual machine energy function is written. The potential energy boundary surface method, which determines the maximum poten- tial energy of the individual machine energy function as a function of clearing time, is shown to have a very sharp narrow peak at the critical clearing time. It was thought, based on [16], to be nearly flat for all clearing times greater than critical clearing time. This very sharp narrow peak for the maximum potential energy as a function of clearing time at the critical clearing time (8) makes extremely accurate identification of the critical clearing time quite easy, and (b) indicates that the combination of the method for identifying the critical generator and the method for determin- ing the maximum potential energy produces an energy metric that truly captures the structural conditions that cause the loss of stability for a particular fault. The equal area method measures the accelerating energy and decelerating energy for the individual machine energy function of the critical generator for a particular clearing time. The maximum decelerating energy is determined by noting the maximum of decelerating energy as a function of time. The difference between the accelerating energy as a function of clearing time is shown to be close to zero for small clearing times, have a very sharp negative peak at the critical clearing time, and II 171 have positive values for clearing times greater than the critical clearing time. This result was not known previously and (a) shows that the equal area condition of single machine infinite bus system does apply to the multimachine system if properly applied to the individual machine energy function for the critical generator, (b) makes extremely accurate identification of the critical clearing time quite easy, and (c) shows this difference in accelerating and maximum decelerating energy as a function of time is another energy metric that captures the structural conditions that cause loss of stability for a particular fault, clearing action, operating condi- tions, and clearing time. The individual machine energy function for the critical generator and the PEBS or equal area method appear to finally have been established experimentally as properly captur- ing the structural conditions that cause loss of stability in power systems. These methods still require theoretical justification, which is a topic of further research. (2) The second major contribution of this thesis is the development of a fast accurate method for determining loss of sta- bility without requiring transient stability simulation of the fault, the clearing action, and the fault clearing time as was required in the previous results to date [16, 17, 29]. This method is shown to require computation that is approximately that of an AC decoupled load flow and thus could be utilized in the applications mentioned earlier. The practical contribution of this method cannot be under- estimated because the previous literature on direct methods were 172 principally concerned with showing that the retention or loss of stability obtained from observing angle differences on a transient stability simulation could also be obtained via observing a measure of energy on that same transient stability simulation. The research performed to eliminate the need to perform the transient stability simulation was quite limited, never thoroughly investigated, and either required very extensive computation or was not very accurate. Thus, the development of the fast accurate method is an important contribution because the objective of the research on direct methods of stability assessment for faults is the elimination of the need to simulate the fault contingency. Such fast algorithms are based on the potential energy of the individual machine with respect to the rest of the generators in the system. Computation of this potential energy requires calcula- tion of the initial operating state, the final operating state (gen- erator peak angles), and the post-fault network conditions. The initial operating state and the network conditions are easily obtained by solving a load flow equation. However, the calculation of the generator peak angles is the key point to implement these algorithms. A linear RMS coherency measure for pulse input disturbance is a proper fast method to predict the generator peak angles of the system. A nonlinear RMS coherency measure based on the critical unstable equilibrium point is an alternative approach for predicting the peak angles, but it requires significant additional computation compared to that required by the linear RMS coherency measure. As 173 a theoretical basis, the RMS coherency measure is justified for a second order linear single machine infinite bus system model. The theoretical justification of RMS coherency measure, for a multi- machine power system, is a subject for further research. The algorithms are implemented and tested on the Reduced Iowa System consisting of 17 generators and 163 buses. The results in Chapter 7 indicate the accuracy of the algorithms and their signifi- cant promise for further improvement. 8.2. Topics for Future Research Based on the development of the first seven chapters, it is concluded that for direct stability assessment, the energy behavior of a particular individual machine (critical generator) is the deter- mining factor in accurately estimating the region of stability (critical clearing time). The results obtained from application of the fast direct methods for different fault cases on a test system are extremely promising and could be further investigated as follows: (1) development ofaitopological energy function that does not aggregate the network back to internal generator buses, allows arbitrary nonlinear real and reactive load models, and allows arbi- trarily complex generator models; (2) development of an individual machine energy function for this topological energy function; (3) showing that the PEBS and equal area criterion can easily and accurately identify the critical clearing time for this 174 topological individual machine energy function based on time simula- tion of the fault; . (4) development of an RMS coherency measure for the more complex power system model; (5) development of a fast computational method for predicting the state at which the trajectory most closely approaches or crosses the PEBS using this modified RMS coherency measure; (6) testing this fast computational method for predicting the state at which the trajectory most closely approaches or crosses the PEBS on several fault cases to show that it is a fast efficient and accurate computational method for direct prediction of stability on this more accurate power system model. APPENDIX A APPENDIX A A.1. Computation of Vk+1 In section 6.5 of Chapter 6 it was shown that Vk+1 = (81Wk for every k = 1, 2, .... Here it will be shown that vk+1 can be computed by solving the linear equation (9‘ ”111 lel .9l = (A.1) .AEQ.) L921 L122.. .Aé. where 88 = [861, 862, ...., 86m]T; angle changes at load buses 86 = [861, 862, ...., 86 1]T; angle changes at generator ' n- buses 888 = [APG1, APGZ, ...., APGn_1]T; real power changes at internal generator buses First solve equation (A.1) for A8 Oi“ . _ -1 . 86 - - 011 012 88 (A 3) Then substitute this A8 value in the second equation of (A.1) to 175 176 obtain the expression of 439 ABE = ‘ J21 J11 J12 44 + J22 44 (A°4) 4E9 = 1°46 (4.5) where I = 922 ‘ 921 911 912 (A‘6) Define vk Q 88 and multiply both sides of equation (A.5) by matrix 8 8838 = MT v _ k (A.7) By comparing vk+1 = 8: vk with equation (A.7) it is obvious that vk+1 = 8888 (A.8) This implies that the vk+1 is the solution of linear equation I’ ‘ "A“ 0 I011 012 86 v M0 Md v T k+1J _ --21 "221 _ kl for each k = 1, 2, .... ...—- ..n; T “‘8- 177 A.2. Computation of V0 Recall v0 and v1 defined in section 6.5: v = _1__ (fl) U41“. (11.10) ° 25 v1 =—l—8831_4_ . (A.11) 2/5 v1 = (141W, (4.12) Recall equation (A.6) and multiply both sides by matrix 8 and rewrite it in the following form: M8PG = (MT)86 (A.13) Again define v ==88 and compare equation (A.12) with equation (A.13). 0 As a result v1 = UAEE- Substitute this value of v1 in equation (A.11) and write the result as 88PM = M8_P_6_ (A. 14) NIH 31 Based on this analysis, one could obtain vo as the solution of linear equation (- 0 . )- . s - _ 911 212 L49 = (A.15) 1 . -——-M8Ph MJ MJ v _2/; --_:I _--21 --22] _ 01 178 Therefore computation of V0 requires one linear equation to be solved. BIBLIOGRAPHY 10. BIBLIOGRAPHY P.M. Anderson and A.A. Fouad, "Power System Controland.Stability," Iowa State University Press, Ames, Iowa, 1977. A.A. Fouad, "Stability Theory--Criteria for Transient Stability," Conference Paper, presented at "Systems Engineering for Power: Status and Prospects." Heniker, New Hampshire, Aug. 17-22, 1975. N. 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