THE’:‘l$ 4 *‘thma. «b—‘vw ; LluR/i}: VT?- A ! RIILID‘VW‘ Qt- {he “& dry ‘1 Lmdvc‘“ 'f " {:5 ‘mr‘. -mmwv- "\0- nm .m, This is to certify that the thesis entitled THE DETERMINATION OF REDUCED ORDER MODELS FOR LOCAL AND GLOBAL ANALYSIS OF POWER SYSTEMS presented by John Frederic Dorsey has been accepted towards fulfillment of the requirements for Electricai Engineering Doctoral degreein & Systems Science GMW Major professor Date AUQUSt 6, 1980 0-7 639 OVERDUE FINES: 25¢ per day per item RETURNING LIBRARY MATERIALS: Place in book return to renow charge from circulation recon I‘ /.-. - © 1981 'JOHN FREDERIC DORSEY All Rights Reserved .I' A ' - . rrrsv '.'. |L V_.._.., .1. . .nfla H" "A .Ulir-‘L F ‘J 3‘3 by . Ni ' DC: a": THE DETERMINATION OF REDUCED ORDER MODELS FOR LOCAL AND GLOBAL ANALYSIS OF POWER SYSTEMS By John Frederic Dorsey A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering and Systems Science 1980 i .. "m‘fi" 9‘1? HP""V'| " IID * ' - ‘u I .. ”LIL \- a ' I A."I '| H r. - 5;." l“ ‘u J5 .. " l UV .. 1-. c ”-3.”. fear "c g._ .3,- HOI . gnu ‘5. M- $.14. .. , u 'I ':d ..e S5JLI' 1"- V y T.” P" “F A“.' .. fl " ‘n u mus t b ‘- .., ‘r- .t;,.~“.° :3 1 .4 . ~u 1" ’rUJ: to r;- we Ian ' - ""'«-fSIS IS n3 find Lnjquar DEF-fur ’ . | HE S .4475: ‘7”: ‘r .v vie DresEn? t a '- "7531‘1 . . we SpECIijd Hr ABSTRACT THE DETERMINATION OF REDUCED ORDER MODELS FOR LOCAL AND GLOBAL ANALYSIS OF POWER SYSTEMS By John Frederic Dorsey The nonlinear model of a power system is divided into two parts called the study group (system) and the specified group. Structural conditions on the power system are determined which cause the specified group to remain strictly coherent and respond effectively as a single generator for disturbances within the stugy_group. Three conditions called strict synchronizing coherency (SSC), strict geometric coherence (SGC), and pseudo-coherenty (PC) are identified. The analysis is repeated for the linear model and it is shown that if the structural conditions for any of the conditions SSC, SGC, or PC hold in the linear model, then the coherent equivalent for the specified group is identical with the equivalent determined by apply- ing the rules of modal analysis. The singular perturbation model for the power system is then subsumed into the present theory by showing that the structural con- ditions on the specified group of n generators necessary to apply the singular perturbation method are exactly the structural conditions for strict synchronizing coherency. The concept of linear decoupling is introduced as the final structural condition under which the modal and coherency equivalents I" H can Ezesaecified group are :‘EZSCE'T‘TC. Two are C? ite';utaticna‘.ly e 1e :3-'~:"}‘.i:ns strict 5,1110“ 23233:: strict stmrg ‘. retire in cursor: £715: iIVII'u') disturtar; "5‘3! csniitions are i.» Tess-1:1: tiona‘. a 1 i=1”; two 2m dist. Lia-:2 of all generators ‘ 3:13"; St'lCt Sy'nchrt'fi; ”send, a ZVII'J disturb: I'ttges'etric coherency a ItiesaI-ocal miel for c Fir'iiirs used in the ZVII'. The theory and coca. R My testing on the 1 John Frederic Dorsey of the specified group are identical. Three types of linear decoupling are determined. Two are classified as weak and one as strong. A computationally efficient algorithm is found for detecting the conditions strict synchronizing coherency, strict geometric co- herency and strict strong linear decoupling, using an r.m.s. co- herency measure in consort with a Zero Mean, Independent, Inertially Weighted (ZMIIW) disturbance. Weak linear decoupling and pseudo— coherency conditions are intentionally not detected. The computational algorithm distinguishes two levels of models by applying two ZMIIW disturbances. The first, a general ZMIIW dis- turbance of all generators detects the principal groups of machines satisfying strict synchronizing coherency, and provides a global model. The second, a ZMIIW disturbance of a subset of generators detects strict geometric coherency and strict strong linear decoupling and provides a local model for disturbances confined to the subset of generators used in the ZMIIW disturbance. The theory and computational algorithm are satisfactorily verified by testing on the 39 Bus New England System. ‘A ‘ 2“" For John Gauw and Thelma Gladys ii Toe longer a ran 21:55; is die to his cw 313231, have come a‘:-c intraielers in life lane a great de: I'e'Tectnl-ly I can ne ,1 3"?! 155% of encourage ‘~':._ g 3 "Sid have been ' Tutti last four ,veé 'ce'ticnlueter, Gerald E", 3an Kreer, Herman Event for continuing Efren the old dog dic The preparation c "fat the excellent he? W "v friend David iii x 2'31' . ”as and flgures. ACKNOWLEDGEMENTS The longer a man lives the more he realizes how little of his success is due to his own efforts, and how much his victories, large and small, have come about from the help of his friends, family and fa? Tow travelers in life. I owe a great deal to my family for getting me started right. Irite‘l 'l ectually I can never repay Professor George H. Meyer, for his thEnty ,years of encouragement, guidance and friendship, without which "‘7 1 ‘3 fe would have been inmeasurably poorer. But for helping me t"Min-19h the last four years of labor, I would especially like to thank Robert Schlueter, Gerald Park, James Resh, Donald Reinhard, Robert 86"": John Kreer, Herman Koeniga Ronald Rosenberg, and Bruce McFar‘l and for continuing to believe an old dog could learn new tricks. eve... When the old dog didn't believe it himself. The preparation of the manuscript could not have been done without the excellent help of my typist Noralee Burkhardt. A Kudu is due my friend David Wilson for doing such a fine job helping with t he graphs and figures. \ 3‘. 'I‘. .q a : ‘ . ‘5 "“Hfifl" “Y t. ‘:“i;" A I”? V" ‘J FA - :Ar‘f .', Lyhi.‘: ‘LI :Flll ’I E‘- :‘IV:“‘-‘05 ARETIE'n l rm 1m. a- . | squIhLii ’hllfl-o -.. .-~ wwwuw. memes l T' l|‘:lu ll‘E ' . tn" ”ll; r bileHLLv‘ A R"' f"( hU‘lyLl DY"V'C E: l‘!‘ h ‘ TESTliG Ti BUS nu [- REHIEH’ C; RESEARIH TABLE OF CONTENTS Page List of Tables . ...................... v List of Figures . ..................... vii Chapter 1 IMPROVING THE THEORETICAL BASIS FOR THE COHERENCY METHOD OF PRODUCING DYNAMIC EQUIVALENTS . ................ 1 2? A REVIEW OF METHODS OF PRODUCING DYNAMIC EQUIVALENTS . . . . . ............ 8 £3 STRUCTURAL CONDITIONS UNDER WHICH A GROUP OF MACHINES BEHAVES AS A SINGLE MACHINE 46 THE LINEAR MODEL IDENTIFYING THE COHERENCY EQUIVALENT WITH THE MODAL EQUIVALENT 70 A REDUCTION ALGORITHM FOR DETERMINING DYNAMIC EQUIVALENTS . . . . . ........ 135 ‘5' TESTING THE REDUCTION ALGORITHM ON THE 39 BUS NEH ENGLAND SYSTEM ...... . . . . . 173 7' REVIEW, CONTRIBUTIONS, AND TOPICS FOR FUTURE RESEARCH . . . . . . . . . . . ..... . . 243 BI BLIOGRAPHY ........................ 255 (f ‘ ~—— 0 Railing Table C T 'Y' ‘ c. ' Jul Bummer.- E'I; ovalue Data Sene'ators The Matrix -T The Matrix -T, 39 BUS NEH Erg‘. Ccnerency Data Generators FETUS 0f Cone Gifveterators ] Eigenvalue saga 1.8.9.10 Coterency Data . 1,899.10 MUG 0f Ethel 0f GenEratOrs 8 {TEEN/aloe Data Band 9 . . P U?- referent Heasni he”6mm 8 am at *“eratOr 8 ' 6~14 LIST OF TABLES Page Ranking Table of R.M.S. Coherency Measures for ZMIIN Disturbance of all Ten Generators ...... l77 Eigenvalue Data for ZMIIW Disturbance of All Ten Generators . . . . . . . . . . . ....... l78 The Matrix -1_ for the 39 Bus New England System . 180 The Matrix -I_1 for the 6-7 Aggregation of the 39 Bus New England System . . . . . ........ 182 Coherency Data for ZMIIW Disturbance of All Ten Generators . . . . . . . . ............ 186 Ranking of Coherency Measures for ZMIIW Disturbance of Generators l,8,9,lO . . . . . . . . . . . . . . 19l Eigenvalue Data for ZMIIW Disturbance of Generators l,8,9,lO o o o o o o o o o o o ..... o o o o o 193 Coherency Data for ZMIIN Disturbance of Generators 1,899.10 0 o o o o oooooooooooooooo 194 Ranking of Coherency Measures for ZMIIW Disturbance of Generators 8 and 9 . . . . . . . . . . . . . . . 204 Eigenvalue Data for ZMIIW Disturbance of Generators 8 and 9 O I O O O O O ...... O ...... I 0 205 Coherency Measure Data for ZMIIW Disturbance of Generators 8 and 9 . . . . . . . . . ....... 206 Coherency Measure Ranking for ZMIIW Disturbance at Generator 8 . . . . . . . . . . . . . . . . . . 218 Eigenvalue Data for ZMIIH Disturbance of Generator 8 O O O O O O O O O O O O O I O O O O O O O O O O O 219 goherency Data for ZMIIW Disturbance of Generator I O O O O O O O O O O O O O O O O O O O ..... 220 Fart-ing of Care Of Bus] 0 O 0 Eigenvalue 3a .a Canerency Peasa Eere'ator l . ‘T‘efi. ..._es._r_.———— Table 6-15 6-l6 6-17 Page Ranking of Coherency Measures ZMIIW Disturbance of Bus 1 . . ...... . ............. 229 Eigenvalue Data for ZMIIW Disturbance on Generator 1 230 Coherency Measure Data for ZMIIW Disturbance of Generator 1 . . . . . ............... 23] vi a ‘0', I . 1" c, O "D tc'eerent Ge Equivaie't t'plex Ra‘. Generation, Original E. Bus . , Original 3e bl Series C Uriginal Er Two Ge'era‘. it) After A Four SEWEV-a F n Utespling REIOIIVE R3 Coherency ’Sitesation Three Zone Line Diagra: Reiati've Ha. ( {seating D ZR.” 0‘. FR ‘V U Smuhfions e . 'ep DlSturt Figure 2-la Z-Ib 2—lc ZZ-Vlci LIST OF FIGURES Configuration of Coherent Generator Buses in Original Network ................. Coherent Generator Buses are Connected to an Equivalent Bus Through Ideal Transformers with Complex Ratio .................. Branch Between Coherent Buses 2 and 3 is Replaced by Equivalent Shunt Admittance on Buses 2 and 3 Generation, Loads and Shunt Admittances on Original Buses are Transferred to the Equivalent Bus ..... . ................. Original Generator Terminal Buses are Eliminated by Series Combination of Ideal Transformers with Original Branches ................ Two Generator External Group (a) Before and (b) After Aggregation of Generator Terminal Buses Four Generator System Exhibiting Strong Linear .Decoupling .................... Relative Ranking of Structural Conditions for Coherency .................... Four Generator System (a) Before and (b) After Aggregation of Generators 2 and 3 ........ Three Zone Partition of Power System ....... Line Diagram of 39 Bus New England System Relative Magnitude of Coherency Measure vs. Ag regation Level (a) ZMIIN of all Generators (b ZMIIW of Generators l,8,9,lO . . . . . . . . . Simulations of System Response to One Per Unit Step Disturbances on Generators 8 and 9 ..... vii Page 28 50 120 I66 I74 CD. Relative “53 fizcregattcr [kl wu°'-' A‘ \y" 6-06. V Si7.7 ticrs i‘ep Sis‘ r: Uri UH v:- c...‘ ‘ 4.qu e C v-lodu :‘Rflrenao; an ‘- Hub V‘I v. a gi-UIe’;P"S Ste; Cist.r: Hie Sene'at Secerators 2 Ee'eratcrs 2 Figure 6-4 6—5 Ei-G t5-é7 Relative Magnitude of Coherency Measure vs. Aggregation Level (a) ZMIIW of Generators 8,9 (b) ZMIIW of Generator 8 ....... . ..... Simulations of System Response to One Per Unit Step Disturbances on Generators 8 and 9 ..... Simulations of System Response to One Per Unit Step Disturbance on Generator 8 . . . . . . . . . Magnitude of Coherency Measure Threshold vs. Aggregation Level . . . . . . . . . . . ..... Simulations of System Response to a One Per Unit Step Disturbance on Generator l . . . . . . . . . Five Generator System (a) Before Aggregation of Generators 2 and 3 (b) After Aggregation of Generators 2 and 3 . . . . . . . . . . . . . . viii Page 207 249 an fi-r‘.0.'¢-lq . A ' 5. - .. 'ezesic P": t" The StJC'j Of C :r secarity aSSEE :'>~ -:n‘ IA I1! 4‘.‘ ...a:..='.; {Clara 9-- fiteitii'fi require: site's response to a FISI-Ol‘lcally t '3- In the conerercJ 2'5 DESI dlSIUl‘tancg :T' ' c ‘ Met-d. to See 1 is. ‘ Hereby naintair -. 4‘? e' C cg: . “I Such a «were . II ngp IS tT‘Er Few ’ thus l““duci 5er CHAPTER I IMPROVING THE THEORETICAL BASIS FOR THE COHERENCY METHOD OF PRODUCING DYNAMIC EQUIVALENTS I- The Basic Problem The study of power system disturbances for the purpose of plan- ning or security assessment requires the solution of potentially over a thousand coupled differential equations. To obtain the time domain '“353F>c>rnse of these equations, even in linearized form, for as little as one 59<2<3r1c1, is a very expensive computational task. Therefore, there has been a considerable effort over the last ten years to reduce the number of ecluaizions required to perform a satisfactory analysis of a power s‘y's’tem's response to a disturbance. Historically there have been two major approaches to this prob- lem. In the coherency method, the full set of equations is solved for a test disturbance. The accelerations of all the generators are the“ checked, to see if a group of generators accelerate at the same rate . thereby maintaining their initial angle differences with respect to each other. Such a group of generators is called coherent. Each Coherent group is then replaced by one equivalent, or "aggregated" gener-ator, thus reducing the numberof equations by reducing the number “f generators. This reduced order model is then used to analyze the Mer system response for all disturbances that occur in the general a"ea of the test disturbance used to derive the equivalent. ‘ 'l"'fl£.w ‘ i are; If the 3009' 596" §e< as t"e ‘3‘“ 3512' is retaired; it strety finding a l metrical for", State no ir;a:t or :E'ariiitj, an: ‘as .zstarzard 0:9?31‘39‘3 :th "e‘n’S -* e:s_:es and certain C 335% a;:eal beca. she'ent lines and g ear-t lines and gener l’:ie;'ee of detail 1:72:th is not are .L , he may shor Terrie I ....ai foundation I'm.- ' .n.) ill th Using t? {ETIirjin “9 0f the t i better Mr U'IJ 1 Int modal ana‘. h, =5; 'r ‘, *' Witor legs tr 4,2”r T a «a. Pornt of v: n ‘l .Dia‘e . ad ”‘- IEO'S of '1' ifIlncar ' F differ-E The modal analysis method defines the internal system as the area of the power system where the disturbances will occur, and every- thing else as the external system. A detailed model of the internal system is retained; it is the external system that is reduced. This is done by finding a linear model of the external system, transforming it to canonical form, and then eliminating canonical states, or modes, that have no impact on the internal system, using controllability, observability, and fast eigenvalue arguments. That is, one follows the standard operating procedure of linear systems theory. Both methods have advocates; both methods have certain ad- vantages and certain disadvantages. The coherency method has great int“-u‘it‘ive appeal because it yields a reduced order model composed of equi Valent lines and generators. Further, the models for the equi- v . . . . a1 ent lines and generators can be either linear or nonlinear, and of a ny degree of detail desired. As will be shown, this degree of f] - e><‘I bility is not available in the modal analysis method. The major shortcoming of the coherency method is that its th at) Y‘etical foundations are incomplete. This causes no operational D rob“ ems with using the method in its present form, but a more complete u ndQV‘standing of the theory of coherency equivalents would undoubtedly Wead to a better means of implementing the coherency method. The modal analysis method of producing reduced order equivalents has no shortcomings theoretically, but it has some drawbacks from the fu'mtional point of view. First of all, the reduced order model is not f0Y‘mulated in terms of equivalent lines and generators, but in terms of a Set of linear differential equations. This is not a serious problem — ‘ -.."""9"‘§ ”van-w :t':se who ‘5“? 5" fzieutillty “"3 r5 1'3'35‘15 agreed” is 1;; 3‘ tire externe' If one met-h: 1221‘sr did not, t :::”eti*~:ds have he :i'ietl'tis are at: :e’fes of the power If tine last : 3'1 iiit the cotere' 2's Izelly correct. . Jae next section. I F ‘ mgr "a at? ’i {-9 There are $6: tuition: between 1 0".) . The most 5' a- i at News that by t iim' iate statistic FEW» m can be sta .v » -I\ :E’E'lh WV COTErF-nr yr- to those who formulate the model, but it is to the operating personnel 0f the utility who have to use it. The second shortcoming of the modal analysis approach is that it cannot provide a nonlinear reduced order model of the external system. If one method of forming equivalents produced good results and the other did not, the issue would be settled. As it would happen, both methods have been used successfully. This argues strongly that both methods are utilizing the same fundamental set of structural pro— perties of the power system to form equivalents. If the last supposition is correct, then what is required is to S how that the coherency method can be used to produce equivalents that are modally correct, that is, equivalents that will preserve both co- h«event properties and also retain the same modes as the modal analysis method. This is the basic problem that will be addressed in the present AWOVk. The details of the analytical approach to the problem are given 1 h the next section. I I - Prior Work that Helps Show _th_e ”El There are several recent develOpments that point to strong cohnections between the modal and coherency methods of forming equi- va‘lents. The most significant is the work by Schlueter [5, 6, 7, 8] Nh ich shows that by using an r.m.s. coherency measure and the fitDtiropriate statistical disturbance, rules for aggregating coherent 9QIterators can be stated that are also rules for properly eliminating "‘Odes in terms of controllability and observability argunents. 5‘3 gnificantly, for the appropriate statistical disturbance, the inter- QEnerator coherency measures can be shown to depend only on the plant '5 l of the “"9? it” i '_’ . Asecfiid 1":“3rt et71sibta‘3'ied bf Ci: 211::1nditior-s 0n t"? 151:3.se a sceci ‘1 9’3" f'a'i tigi for 6r" "‘5 :17rerei-CY C5“ t9 5 i’jhéi‘m and 5‘39" Translated to t 155:.ations for the 5 iii the power syst'i' Efiittifiio two groups. :1ez“iedgrodd, the at hosted. This se:ar Tit "fen the structura 351‘S‘ied, the cohereoc iitiesaire. This is, The results of lhsiesisoint of the s i’iset of conditions e a". '- fie mdal equivaie 5" 35‘6“ conditions ti £151.15 ihitial tWSt ‘t "33"“ Conditions fi Eli‘s?» are identica' matrix, A, of the linear state model of the power system, i=es+§u. A second important pre-cursor to the present work is the results obtained by Dicaprio and Marconato [l0, ll]. These results state conditions on the structure of the power system at time t = 0 that cause a specified group of generators to remain perfectly coherent for any t 1 0, for any disturbance that occurs outside the group. This coherency can be shown to hold for the nonlinear model, making it a Very powerful and significant result. Translated to the linearized model these conditions decouple the equations for the specified group of generators from those for the rest of the power system, and thereby divide the eigenvalues of the sJ’S‘l:em into two groups, one associated with the equations for the sF>"i‘<:'ified group, the other set with the equations for the remainder of the system. This separation of eigenvalues is strong intuitive proof th at when the structural conditions of Dicaprio and Marconato are Satisfied, the coherency equivalent and the modal analysis equivalent ““6 the same. This is, in fact, shown in Chapter 4. The results of Dicaprio and Marconato have importance also from the standpoint of the strategy of the present research. The fact that QhQ set of conditions exist which identify the coherency equivalent “i th the modal equivalent, leads one to speculate as to whether there are other conditions that produce the same result. In fact there are, an<1 the initial thrust of this research is to find the complete set of Str‘uctural conditions for which the modal equivalent and the coherency eQuivalent are identical. The other Curr ‘ar perturbatio'i r'e‘san, their, Alle. 'i:":l'4 t'lat this K II. iStracta-ral 5‘. The first :"35 riders on the 3355 “Petzyeqaivalerts tc iii Strict 5;” i2} Strict Ge: (3) Strict Str iii Pseudo-Cor i5} bleak Linea These cooditic "i" Wily Satisfie :‘ii'li'rartaoce, becé ig'SjStfiir. COhQVEDC‘y. 253-31 by near BDDmx' W Of these E-JE in. conceptual iz' (“Mm cohc t0 condmm. IT'H' ' .. 1.:tions that cause stiat in the limit a w satisfies the ct The other current work of importance is the application of singular perturbation techniques to power systems due to Kokotovic, Ninkelman, Chow, Allemong, et al. [l2, l3, l5]. It will be shown in Chapter 4 that this work has a natural place in the present research. III. A Structural Outline 9_f_Lh_e_ Present Research The first phase of the analysis develops a set of structural conditions on the power system, which if true, cause the modal and co- herency equivalents to be identical. These conditions are called: (l) Strict Synchronizing Coherency . (2) Strict Geometric Coherency (3) Strict Strong Linear Decoupling (4) Pseudo-Coherency (5) Weak Linear Decoupling These conditions are all hypothetical in the sense they are never exactly satisfied in a real power system. This hardly diminishes their importance, because these five conditions are really archetypes FOr system coherency. Actual coherency in a real power system will be Qaused by near approximations to one of the five conditions or by \Q OInbinations of these conditions. Thus the five archetypes have great value in conceptualizing coherent behavior in a power system. Geometric coherency is simply a renaming of the Dicaprio- Marconato conditions. The name is chosen to reflect the structural ‘ltbnditions that cause the condition. Further, it is shown in Chapter 4’. that in the limit as the parameter p + 0, the singular perturbation model satisfies the condition for strict synchronizing coherency. Thus int, 151‘e1ry deve . c ,e 5":*& current via The sect": tartan: satset 0*. 7*W.fimag :5 detecte: t Ifié'iy 59"“.-4 ‘ 5.5:; ' 75“: lfiortart L«cg... ‘M I 1rd 1 Ala . ,6 FE .Zie and] the theory developed in this present work is general enough to subsume ifll the current viable theories on the formation of dynamic equivalents. The second phase of the analysis demonstrates that the most hmmrtant subset of the five conditions, namely strict synchronizing coherency, strict geometric coherency and strict strong linear decoupling can be detected by the r.m.s. coherency measure, in consort with a properly selected statistical disturbance. These three conditions are the most important because there are strong guarantees that if any one (Jf’ these conditions exist in the linear model then that condition will a1so exist in the nonlinear model. The same cannot be said for pseudo- coherency and weak linear decoupling. Further, by using the right sequence of statistical disturbances i t is possible to distinguish between two .levels of aggregation. A general statistical disturbance of all generators detects syn- chronizing coherency and determines the principle, system-wide coherent 9"‘Oups. This is called the global model or aggregation. A second ES"I'Eiltistical disturbance of only selected generators detects geometric QCherency and strong linear decoupling. This second disturbance can be e‘S'smciated with a local or parochial model. The ability to distinguish 91 Qbal and local levels of aggregation has broad consequences that are 1 argely beyond the reach of this present work, but have great promise fOr future research. The third phase of the analysis is the integration of the theory fir11:0 a formal reduction algorithm. This is done in the latter part of tJ‘Iapter 5.' The results of testing the algorithm on the 39 Bus New E“gland system are excellent and follow the theory very well, indicating that the analysis is sound. Ii 3&3?de PTECU The next Cid?- i5se'cv and mdal me Elé'liii see 5339 5' $1.151} is regretta ':.'15rsta'.d Ctacters gr; 151ail, of both t 252?], trat is wta IV. Background Preparation The next chapter is wholly a review, in some detail, of the coherency and modal methods of producing dynamic equivalents. The reader will see some of the same phrases he has seen here. The redundancy is regrettable, unavoidable, and to some extent beneficial. To understand Chapters 3, 4 and 5 requires a good understanding, in some detail, of both the coherency and modal equivalencing techniques. Hopefully, that is what Chapter 2 provides. L 4‘. ‘ u 1"": ";,"P 1 a . yi'yblur .r ,p ‘ ‘ 7! P ale bbi'sfiruyl :1.‘:es ar eaten?“ II ‘3'. I v; I 4' "T.”F ,.._, large 1 553.5: :C‘e‘il‘ distr' half of the i win ' ”he cecaod of i‘ Ezraril yfrort its 5 g . - 'r ' " :iar‘l cg the risl "a .Ser ii' ith a more s use in the diff 11,5, (3.3 “tit \ v in “he Siv-nl I :ff'n ‘ Crdn is. "in 4:98 . rs th g . a?" .e .1“ Land CHAPTER 2 A REVIEW OF METHODS OF PRODUCING DYNAMIC EQUIVALENTS I. Introduction The construction of dynamic equivalents for power systems provides an excellent example of the difficulties inherent in modeling and analyzing large scale systems. To improve the reliability of the e1 ectric power distribution system, all the generators in the eastern half of the United States are interconnected. Thus if operating problems develop in one area and a particular utility finds ‘3 tself in a position where it cannot produce sufficient power to meet the demand of its users, that utility can "borrow" power temporarily from its neighbors through interconnecting lines. By th us sharing the risk of operating problems the utilities provide the user with a more reliable source of power. Part of the price paid for this improved reliability is an i '1 Crease in the difficulty of analyzing the dynamic behavior of the pt)err system. There are approximately two thousand generators in the eastern half of the United States, all connected in parallel. wk)cleled in the simplest way possible, every generator requires a SeCond order nonlinear differential equation. Even with modern high SDeed computers, the pr05pect of numerically evaluating four thousand coupled differential equations is not very appetizing. ' Ila-N 12' —_——F _r' . :““ec or late: i :115;"*as great 5:: 15.2.5 Lasticcs a :1‘:‘-~"i‘y is EC". Hun-~llo :H.:PEP|: is aficep.p.4 5'56 Eecaise the .3 ‘z a -. wrai a:;rc:n* ‘0 - ny‘cgt tu't h" """”'er of differential equations. Thus for the analysis of large F3‘3‘Mer systems there is really no choice but to use the simplest heIi>r~esentation possible. The simpler representation used here ne91 ects the effects of exciter and turbine governor control, at 1 east in detail. The damping constant D]. serves to represent in a Qeneral way the overall effect of these control systems. In a p<3V~Ier system the various control systems tend to dampen the response 01: the power system without greatly affecting its natural fre- quencies [2]. In Chapter 4, it will be shown that the assumption of u‘ni form damping, i.e. the ratio Di/Mi is the same for all generators, ‘eads to a precise formulation of this empirical idea. The assumption ST? uniform damping will be made shortly when the equations for the Power system are put in state-space form. The second observation worth noting is that the real power equations can be decoupled from the reactive power equations. This i .4 menu me a '-:;:a'tes in tre C m generator M l P \‘V‘ " «V'J 4r 1 __ _ '0 " ‘— -. .c .. . - fl. 4 .v .. - _ a. I ny’L . I - ’n — ‘— CI . - I .— ~- 1. v - | . '0 .93 I: :‘J ‘ .J: -. - I --. ._ .- g I a. V - I I "Nv ‘. a . I .‘b a -a{ ~ — ' J H‘ . ‘A v~ ‘ - . I'E'E 3", . ‘4" real bases 3‘ n 4’; real M. 3; V0”: 5‘ , ‘ V01t5 . (v .of' ,. 9‘ \ <‘>I: 0A ~~ + ‘ § ‘ 29' 'DI '{ "K: \ >5 2‘ - + v ‘ 14 is a comnonly made'approximation, based on the following reasoning. The changes in the complex voltages and power injections at the net- work generator and load buses may be expressed as: '- 7 fl 5'5 5.67 5E 52’ As?- SPL BPL BPL 3PL APL _: _: :. ; A9. — a§_ a_e_ a_E_ av Age 3.6. 19.3: 395 399— A; as 39. a; 5T A—Q—L i}... 3% 3: 3.]... AV a ag 3g a_E_ a! L- —-J L. J Where £_G_, gg real and reactive power injections at internal generator buses - p.u. EL, ELL. real and reactive power residuals at the load buses - p.u. §, §_ voltages and angles at generator internal buses y_, _6_ voltages and angles at load buses. Now the first two equations can be written in the form aB_G_ 33g 33g egg =—--— +——— +—— -—— agg 3,; Ag 39— Ag 8.5. “3+3! A! 33!; a£_L_ 32L 33L ABL‘EA-é*s_e:'A9-*SEA§*EIA1 To first order, the power flows are largely dependent on the voltage ,\ angles at the generator and load buses, not the voltage magnitudes. is tne same as ..... ' at h- r"‘ «,efi tit L 9 3‘: . AA l‘ f‘ P}. :1 uv'r)IE'L. ' ' .‘V‘ .r. , .4 'V‘ - O 0- - -— ,_, J - - -. _ CA '53: J .I A .u. "" .FE 'he b.»-u pm this t 'F. t' I A Q. 3,316 euuctTCf‘s u I a. A. . J: 39'9””?‘5 - I.“ . ' 3'9 ‘11". {Keri-.4 ullrA l.':, 5'5'393: 15 This is the same as saying BEE/81 = BEL/31 = 2. Further, it can be assumed that the generator voltages behind the transient reactances are constant. That is, Ag = 0. These two approximations result in egg 33g ABE‘WA—‘S—Wfl‘i agL_ BEL Aye—WE‘VE These are the decoupled equations for real power flow. With this background information, and assuming uniform damp- 1'I'lg, the equations for the power system can now be put in state- sDance model form as follows. First, a reference frame is chosen for the angles and Speeds of the generators. The reference frame chosen is the generator angle of the Nth machine, ASN. That is, one establishes N-l angle dif- fe rences: A A61: A51. - A6N ’ 1: l,2,...,N-1 and N-l speed differences Ami =Awi - AUJN ’ 1 = 1,2,0009N'10 Next consider the N equations of the form represented by (2.1a), and Subtract equation N from equation i to get: , , APMi Ape, APMN APGN Di oN MI’MN=(‘”T'T)'(TN—'W)'(”§A‘”"Ti°‘”")' Now the left hand side of this equation is simply All} . However, 1 the right-hand side contains all N speeds. Making the assumption ' ; A 4 ”".rfi 9' " "Um “any I ‘3 r":." n" . ' . n- \uol “ reassr"lcn f3f 'L 22".21 by 25-2 0' :4. “.0 H. . ' N: :JIUCL1‘. S 0 'CL (1 f, w L t' a. 4.] ecueh‘l’flr‘ & L; : L {-113 :t 1 ‘4. ._- l (‘9: - T . 10 ”eke tr- ":0 [’3’ l- (1'!) (Ft) —4 I‘)’> I I 1 ( )I) 16 of uniform damping allows the last term on the right hand side to be written 0(Awi - Aw") = 0(Ami), where o ='Di/Mi = DN/MN. Thus under the assumption of uniform damping, the power system can be repre- sented by 2N-2 differential equations, Specifically the N-l de- fining equations of the form 0.0. r,- A31 =Aai i l,2,...,N-l (2.2a) and N-l equations of the form, d A - 1—. - 1—. - - A ' = - '3? Ami - Mi (APMi - APGi) MN (APMN APGN) koi, l l,2,...,N l Di Where o=M—-, i =l,2,...,N. To make the model complete, the power equation must be \nlr~itten in terms of the new variables, i.e. ., PAPGfi FBPG 33E {- A1 —‘ x ‘7.— AQ = 3§_ 39- (2.2c) 32L. 33L APL T -—"" A9: L --.J 8_5_ 3_§_ L. .J L. J where, notationally, 61 = 51 - SN 1 = 1,2, ,N-l 6k=ek-6N k=192900090 and AT _ A A A AT- A A A g " [61962900095N-1]9 9 ’[91,92,...,eq]. The next S‘.E :2: :e accorzll ' — }:—'A+- . A ~ - - -. ‘K :ar. :e reunite 39% .233 . ~ ' L 21‘ ~ ..I\ A 0‘1... - F" ._ n :‘ 3'? . \‘ - ., '1 'Dr‘ - ~.‘J - W .- + & "e Shape 7C 0 I Lt) = Tent. .. . , V‘WWS ngd‘c O _ ’ - I1. ’9) O l O l 1‘2-) 17 The next step is to express agg in terms of A_3_ and All. This can be accomplished by solving the second power equation in (2.2c) for AE and substituting this expression for Ag into the first equation in (2.2c). The result is: we. a$.ut" WL. AE§=TA§+T x All-TALE 3g a_e_ 3g ag This can be rewritten as we we @anfl.x am.afl.“ ALG= “7"“? T "‘7.— A§_+T 7— APL a§_ a_8 3g 3g 3g a_e_ '— or, Xu)=eyu+§uu) ago then follows inmediately with A A = -99... U = -ég‘tl. @. ” Ml r- m r- 1 Q(N-l)x(N-l) Zflfol Q Q A = §_ = ........... 4i T film-”MN-” E ’13—" L-” " J L. J . C |:' . l ' J l - 5 p 9:3 :9'3 f: —. __ _ o ' A A i'hq -‘ K ’:': .~ . I _ -; -‘ u ’1‘ , i315 ls tr '45.". throw. “it that per raw, but this c '1‘ ‘fe linear mp 25., generatOr l8 l. _ 1.. M1 1 MN _ 0 ' M _ . 2 n= ' - .1 -1— MN-Z MN 1 l 0 -———-— -— c" MN-l MN 629. 32.6. 81L. " 82L. . 432 33L. “ T = 7—"? “:- ‘T‘ L="‘"Z‘ x 3g 33 3;; ea 33 as This is the basic linear model of the power system that will £362? used throughout the following development. In Chapter 4, it will be shown that there is a more convenient reference than the Nth gen- eY‘ator, but this change of reference will not alter the basic form of the linear model. With rare exceptions, the first m gen- e‘F‘ators will be considered the internal system, and the last '1 = N-m generators, the external system. That is, the first m gen- ei"ators correspond to the subsystem where the disturbances occur and Whose detailed behavior is to be studied. The last n generators (Itaierespond to the remaining part of the grid to which the internal System is connected. The linear model plays an important part in both coherency and modal analysis, although the role in the two methods is quite different. If the dynamic equivalent is produced using the modal analysis technique, the internal system is represented in detail, With each generator being described by high order (seven or eight) a. fat cart?“ on A ‘ ‘ r” Tigers-r. ge : 2's cement are ZI'Cilll the line :72er swing tag lf‘ii‘ir cauld juf 7.3% system, bu! ':"=512€'lsive. l . .,-e sewer syste 9* in reference The cot-ere "-“~=’ System is be 533?: that main: ’eszonge to a d ' "'J. In the Stl “I43. . . :a- h; f: “Md to on? 19 linear or nonlinear differential equations. The external system is divided into subsections and each subsection is represented by a linear model. Linear system theory, in particular the concepts of fast modes, controllability and observability, is then used to reduce the order of each linearized subsection of the external system. In the coherency method, a linearized model of the entire power system, internal system and external system,is subjected to a disturbance and the coherent generator groups determined. Those generators that are coherent are then replaced by a single "equivalent" machine, and a 1 inear or nonlinear model can be produced. Thus in the coherency apl-‘H‘oach the linearized model is used only to determine which gen- elr‘ators swing together in response to a disturbance. This coherent behavior could just as .well be determined using a nonlinear model 0f the system, but this would, of course,be computationally much m0 Y‘e expensive. The justification for assuming that the linear model 01: the power system captures the coherency behavior of the system is 9‘3 van in reference [2] and is discussed in the next subsection. 1 I I . Reduced Order Eggivalents by the m g Coherency The coherency method of forming a dynamic equivalent of a power system is based on the intuitive idea that a group of gen- eY‘ators that maintain the same relative voltage angles to each other in response to a disturbance act, in principle,like one large gen- eV‘ator. In the strictest sense, two generators, or buses, are de- fined as coherent if the ratio of their complex bus voltages is \ constant over time. In practice this definition of coherency has i been relaxed to only requiring that the voltage angle between two - ‘ 3 r -~'rv2:n cars.» V: .J ;- - r “r: 3 RC? E x: U. L "'1‘ cvervi es- of m: voceqt ref; - .,. d bi Mistxta' cote'est 9‘. We???” ' erator, 1i ie'erator t l5 crucial E'EtOTS lr‘ The static first rep},- buses by a load buses. W3 netwcri W the ne‘ Dmtlers a reduced ne‘ ”fever (34 that ONE (:5 The genera' one 0" a Sr equivalent be used at are Slfifla] "SF-an ELITE herent 96W Characteri: here-”t gene the” t“0 e< bUS SlnCe . Satlsiactm (kWh-arms USUaHy’ bf ”h 20 buses remain constant over time. The principal work in coherency has been done by Podmore and Germond [2, 3, l4]. This section contains a brief overview of the main points of reference [2], as they pertain to the present research. The procedure for forming power system dynamic equivalents as outlined in reference [2] can be summarized into the following steps. (1) A disturbance is applied to the power system and the (2) (3) coherent groups determined. The determination of coherency is done at the terminal buses of the gen- erator. It is assumed that the fictitious internal generator buses are also coherent. This assumption is crucial to the dynamic aggregation of the gen- erators into a single equivalent generator. The static network equations are reduced in order by first replacing all the coherent generator terminal ' buses by a Single equivalent bus, before eliminating load buses. This replacement allows the order of the network representation to be greatly reduced. And the network reduction and dynamic equivalencing problems are decoupled. In addition, the reduced network representation is applicable to whatever generator model, complicated or simple, that one cares to choose. The generators at the coherent buses are replaced by one or a small number of equivalent machines at the equivalent terminal bus. One equivalent machine will be used at the bus if the set of coherent generators are Similar enough in response characteristics. A "small number" of generators will be used if the co- herent generators are of very different response characteristics. For instance, if the set of co- herent generators includes both steam and hydro units, then two equivalent generators will be used at the bus since it has been found empirically that a satisfactory single machine equivalent for a group of generators that include both steam and hydro cannot, usually, be found. The equivalencing procedure assumes that each generator in the group (or subgroup) to be equivalenced can be represented by a block dia- gram of transfer functions of the same form. The equivalent machine is assumed to have a block diagram of the same form and an identification procedure, based 0' parafete the em" r9530"SE The deter '. ...' J ' 29: a. rczel a::':'-:-:e ncclir 'e'e': grc._:s. 9" :‘zre acne-r Syst :1::‘:al euerie xie‘ can be use: I‘E'..':5.'.CES, or :53 w t. charze t «with?! that 4 H I...»* 21 based on least square error is used to identify the parameters that best match the frequency response of the equivalent machine to the cumulative frequency response of the coherent group (or subgroup). [2] The determination of the coherent groups is done using the linearized model of the power system described in Section II of this chapter. In early work on coherency, Podmore and GErmond used a complete nonlinear model of the power system to determine the co- herent groups. However, experience showed that a linearized model of the power system would suffice. The justification, based on this practical experience is the following. The assumption that a linear undel can be used implies that coherency can be detected by small disturbances, or viewed another way, that the length of the disturbance does not change the coherent groups. This assumption is based on the (abservation that if a certain bus is faulted, the coherency behavior is not significantly changed by increasing the fault clearing time. Since the 1mm of the fault essentially determines the size of the d‘isturbance by determining how much energy is put into the accelera- tion of the generator, the linear model will suffice. The second important assumption is that in the linear model, the very simple ‘3<>nstant voltage behind a transient reactance model of the generators can be used. The justification for this is based on the empirical EVidence that the amount of detail in the generating unit model has sOme effect upon the swing curve, particularly the damping, but does ‘NOt radically affect the more basic characteristics such as natural fre- QUencies and mode shapes. Reference [2] illustrates this argument by Showing that good estimates of system modes and mode shapes result from considering the classical constant voltage behind transient reactance ;‘.e":r represert ='.a:":n rereseri fiertfal e:.atio' Zirtise-gcven F'fllre mig‘rt Dace t'e i at: of the me“ mi map of i571, lode elir 1553559255159, The replac .295“ has is dope 3'9 33-19.? fig at :5. sea. COOSIOEr t .’ ‘ P .i Y” ..3‘ = Ylll] :E “L: "l‘hrgn+ ‘ ” grOup 22 model of the generators and ignoring excitation system and turbine- governor representations. This greatly simplifies the differential equation representation by reducing the number of first order dif- ferential equations per machine to two, whereas with the excitation and turbine-governor represented the number of first order equations per machine might very well be seven to ten. Once the coherent groups are determined, step 2 is the re- cmction of the network. This proceeds in two stages. First, each coherent group of buses is replaced by a single equivalent bus. Second, node elimination techniques are used to remove as many load buses as possible. The replacement of a group of coherent buses by a single co- herent bus is done under a power conservation assumption. That is, the power M at each boundary bus of the coherent group is con- served, and the power production of the coherent group is also con- Served. E4 Consider the following algebraic network equations - #- I I : : E : O . : 0 J9 . 391_-----::--‘.'Ta----5,-3ese:12-- --:-:--3'1'n-_-- -‘."1‘- I Iqu Y(nul)l "' Y(n+1) : Y(m+l)(m+l) Y(m+l)n vm+l 2 : E : : : o g : o o o I In a LYnl Ynm : Yn(mfl) Ynn J bvn a Where the first m equations refer to the buses on the boundary of the coherent group and the last n-m equations are for the buses of on .' ‘ :521.:F.S.On .O S as: The tune pa . . .i .F .: .3". EX 93m?“ .‘..E'.",‘.,‘_‘,'.g the u r a- I c "i 'EbuitS I“: P; h S - l 5-: k=‘. 4:: in the bO'UT‘. “I {‘9 Went; N0" the ~‘1C-3d by One 9 23 the coherent group. (Only one coherent group is considered because the extension to several coherent groups is obvious and straightfor- ward.) The current Ib, for a boundary bus is: m n Y Vk + I = X v v b k=1 bk k=m+l bk k the complex power injection at bus b is given by * Sb = Ibe, where * denotes the complex conjugate. Substituting the expression for Ib into the formula for the complex power results in: n ** s ? v*v* v + 2 v v v (2 3 ) = 0 o a b k5] k bk b k=m+1 k bk b The first term is the contribution to the power at bus b by other buses in the boundary, and the second term is the power contribution f.V‘om the generators of the coherent group. Now, the algebraic network equations with the coherent buses ‘“‘3l>laced by one equivalent bus are: q r— . a r- -q I‘l Yll YIZ "° Y1m i Ylt V1 0 . I . : : E . v2 I . ' C = : . I I -32 --mi.-------_-::--Ym-LTr.nz- .Ya I ltd 31:1 Ytn: Yttg .th Repeating the analysis which was done for the unreduced equations leads to '"z‘irst tent In . . f , sis in tne bear. "Wind term I". ::»:-r ccrseria 3'55 the equivaler 3’ a: the EQUlva‘. ii. \ 24 m *1: +*Y*V Sb - kg] kakab Vt bt b . (2.3b) The first term in (2.3b) is the power contributed at b by the other buses in the boundary and is identical to the first term in (2.3b). The second term in (2.3b) is the power of the equivalent bus and for power conservation must be set equal to the second term in (2.3a). That is, v*v* - E v*v* v t bt b k=m+1 k bk b which reduces to: i V" < Y = — Y o 203C) bt k=m+1 vt bk 'Tfan the equivalent impedance th can be determined once the voltage, Vt, at the equivalent bus is selected. This voltage is normally taken 13C) be M: 1 " 1 | = _ e . t n m k k k=m+l 55C) far the equivalent Y-bus elements th, b = l,...,m have been dQtermined. The next step is to make use of the conservation of power of the buses in the coherent group to determine Ytb’ b = l,...,m and Ytt' Let c denote a bus in the coherent group. Then the total Power of the coherent group is: n s - Z 1*v C c=m+l C c Using the unreduced network equations Ic can be written as: b=l c=r+l :;...;‘ II::I."fl . . .1 b=m+l Cb b ° Substituting this expression into the power equation for SC, and interchanging the order of summation results in m n s X y v*v* v § E v*v* v (2 4 ) = + . . a C b=l c=m+l b Cb C b=m+l c=m+l b Cb C Repeating this same line of analysis for the reduced network results in m * * S = X VbYt V *‘k + v v v C b=l t t tt t ° (2'4b) The first term in (2.4a) or (2.4b) is the power flow from the coherent group, or the equivalent, to the boundary. Equating these terms gives m n v*v v ~ ? v*v* v which reduces to: y = 2 y ( C) , b = l,2,...,m. (2.4c) tb c=m+l vat The second term in (2.4a) or (2.4b) is the internal power of the Coherent group, or the power of the equivalent. Equating these terms ‘ eads to: =nan Y _ tt b=m+l c=m+l vt V * ch(vio . (2.4d) Equations (2.3c), (2.4c) and (2.4d) give expressions for the equivalent elements of the reduced bus admittance matrix, under the assumptions j‘IJiEl‘ :cnservati If: det: t;- 3"‘15'g trarsisre s‘e'eit t- each afar difference chincre an: :e'e'tj reductic' 21:27 results jus‘ Isis used to il‘ av.” . ' ‘ Tum V is e ' t z'er'us analysis tetgmp) is c 'st ratio to the S'Jr in Figure 2. ltzitions, the ra i‘,‘ 'i ' ' in Circulating 533;? T ii? ge'erally be Litre rest of the assay have been “Ii ‘te 99-”.eratcr 'Ecial bUSES may 9 “TC" L. netveen bug Cl .m‘ -‘5wlvalent S 26 of power conservation. Note that Ytb f th. In fact, the form of th and Ytb shifting transformer has been introduced into the line from the indicates that a phase equivalent to each boundary bus b. The phase shift is half the angular difference of th and Ytb [2]. Podmore and Germond provide a physical interpretation of the coherency reduction that gives some valuable insight into the mathe- nnticalresultsjust derived. The simple network of Figures 2.la to 2.le is used to illustrate the procedure. §tgp_l. The voltage, Vt’ at the equivalent bus is defined. Although V is essentially arbitrary the definition given in the t previous analysis is usually used. Each terminal bus (of the co- herent group) is connected through an ideal transformer with complex tuirns ratio to the equivalent bus. The turns ratio is directed as Shown in Figure 2.lb and calculated as 3k = Vk/Vt' Under coherent c0nditions, the ratio 5k is a constant for each bus in the group and no circulating power flows through any of the phase shifters. §t§p_g, The generator terminal buses, of the coherent group will generally be connected radially through a step-up transformer ‘tC> the rest of the network. However, in some cases the low voltage b"as may have been eliminated by combining the transformer reactance “With the generator internal reactance. In this circumstance non- ‘Fadial buses may exist in the coherent group and a common branch may Connect them. Thus, any intragroup branch, in this example the branch between buses 2 and 3 in Figure 2.lb, is removed by replacing it by equivalent shunt admittances. Consider the current flow in -.- 731;“). t-Et'n'EE.’ Se:a.se o.‘ '22 ::'.star.t are 72' :*' either l‘. L '5 i“e:t of the U l "l ‘ "3‘5 5'19 retwcrk ‘p l U' . A n ehr .. "t IDUSES are I?" i“ Fl'Sure 2. .2 .FinSfer. This 27 the branch between buses 2 and 3. I23 = (V2 ‘ v3)v23 ° Because of the assumption of coherency in the group, Vz/V3 is a constant and the current, 123, can be written as a linear func- tion of either V2 or V3. That is, V2 I23 ‘ V3(Vg-- l)Y23 or V3 123 = V2“ - V—Z‘)Y23 . The effect of the branch can be replaced by a shunt admittance (l - V3/V2)Y23 at bus 2 and (l - V2/V3)Y23 at bus 3. Figure 2.lc Shows the network after the intragroup branch is removed. The generation, load and shunt admittances on the Step 3. COherent buses are transferred to the equivalent bus and sumned as Shown in Figure 2.ld. The generation and load are not modified by the transfer. The shunt admittance is scaled to account for the off- nominal tap ratio of the ideal transformer. Step 4. The original coherent buses are eliminated by a seeries combination of the original branch and the ideal transformer. lb? several original branches connect to the eliminated bus, (see tflls 2), the ideal transformer is combined with each of them. At this point the network reduction is half complete. To Complete it, load buses are eliminated. Algebraically, the elimina- tion of load buses reduces the order of the bus admittance matrix . The unreduced YBus is a very sparse matrix. Initially, the YBus REV; WG’JTATION CF COHERE 28 REMAINDER OF ORIGINAL NETWORK §23 FIGURE 2-Ia COWIGURATION W COI'ERENT GENERATOR BUSES IN ORIGINAL NETWORK REMAINDER OF ORIGINAL hETWORK Igt.15h%h. "I‘ "I‘ rI" FIGURE 2- lb COI-ERENT GEERATm BUSES ARE CONhECT ED TO AN EOUIWLENT BUS THROUGH IDEAL TRANSFORMERS WITH COMPLEX RATIO REMAIN: W WEN COHERENT SHUNT ADM RE MAI 29 REMANDER OF ORIGINAL NETWORK I, I T TIW' III: II [I023 "I" MI rI“ MANCH BETWEEN COHERENT BUSES 2 AND 3 IS REPLACED BY EMENT SHUNT ADMITTANCE ON BUSES 2 AND 3 4 REMANDER OF ORIGINAL I‘ETWORK 2 a2 ulna! UL, M "I“ % ’1“ "1“. MB GENERATION, LOADS AND SHUNT ADMITTANCES ON (RENAL HJSES ARE TRANSFERRED TO TI'E EOWLENT BUS 3 30 REMAINDER OF ORIGINAL INETWORK " I. a. I "I "I” I FIGURE 2-le ORIGINAL GENERATOR TERMINAL BUSES ARE ELMNAT ED BY SERIES CWBINATION OF IDEAL TRANSFWERS WITH ORIGINAL BRANCI'ES 9"”31‘3r. of load buses f“ 21in YEIS' “New“ 6 m. That is,ten*S ' '-:.':' :.‘ :uo buses that "‘4" :.:-.::cf'rectly to each 0‘; :aa"'r"at‘?on of additio' :E'ancn-zero terms if 2;“,2r:;-crtional to to 23::useIimination IS EFL titans in YBus begir’. Etc) ~:ased reduction to ii'éfie Ioad bus elimina’ a"fifteetvork can be grea‘ 3'37! rIon-essential node tithe nurber of branche ”flies “0 Passes. A fi Ts'ecuction process at k ‘afiiVEI. A second pazl I Nos [2]. . . III“ the coherent «w ..e, the next Step I: '1“ sure-5r ' ESent eaCh cohercl TIN“ . 'n , . °I the identific " 5'14‘3val .. Got QEnerator t. 9“ 3l elimination of load buses further reduces the number of non-zero terms in YBus‘ However, as more load buses are eliminated fill-in can occur. That is,terms have to be added to account for the inter- action of two buses that were both connected to a bus being eliminated, but not directly to each other. Thus a point is reached at which the elimination of additional load buses actually increases the number of non-zero terms in Y Since the computation time is Bus' roughly proportional to the number of non-zero elements in YBus’ load bus elimination is ended at the point where the number of non- zero terms in YBus begins to increase. By first applying the co- herency based reduction to the coherent generator buses, and then doing the load bus elimination, the number of branches in the equi- valent network can be greatly reduced. To guard against fill-in, certain non-essential nodes are selected for retention, to help mini- mize the number of branches in the reduced Y The node reduction Bus' requires two passes. A first pass is made to determine the point in the reduction process at which the minimum number of terms of YBus is achieved. A second pass is then made and terminated at the point of minimum terms [2]. With the coherent groups determined and the network reduction complete, the next step is the modeling of the equivalent generators that represent each coherent group. In reference [2] Podmore and Germond give a detailed des- cription of the identification procedure used to produce the model of the equivalent generator. In the present research, only a very simple classical model of the generator is used. This involves only the rotor dynamics of the generator. The rotor dynamics for the :na'ent generator are e ”ii? interested in NH r '1'.ia‘.ec' s'roaid COr‘S‘uIt The basic differ? fists used througFCUt L. p.u. Speed devi inertia consta' ! mechanical pews 3 electromagnetic darping constar a machine sutscr ..:.se of the coherency :aee speed deviation °I the 9WD resul 32 equivalent generator are easily derived, and are shown below. The reader interested in how more detailed equivalent generators are fermulated should consult reference [2]. The basic differential equation representing the rotor dynamics used throughout this research is: 2H ff(jg-14M -AP - D.Aw. (2.5) 3' dt Mj 63- J J with Aw p.u. speed deviation from synchronous speed H inertia constant (generator + turbine) in MNS/MVA PM mechanical power in p.u. PG electromagnetic power in p.u. D damping constant in p.u. j machine subscript Because of the coherency assumption, all the machines of a group have the same speed deviation. Thus summing over the machine equations (2.5) of the group results in 2d—Aw— 2Hj=gAPMj-2APGj-MZD. dti J J 3'3 Thus for the equivalent machine: (a) The inertia constant is the sum of the inertia constants of the machines of the coherent group. (b) The damping factor is the sum of the damping factors of the machines of coherent group. (c) The mechanical power is the sum of the mechanical powers of the machines of the coherent group. (d) The electrical power is the sum of the electrical powers of the machines of the coherent group. Results ICI 3rd (d :rassffi‘IO-“I already 73 53? the otter tranSfE 'rseietails are not fed z'es.:-se;aent analysis. The method of for: test-died in detail ir. 'st'attris equivalert i ‘zsaretfies referred to a ardent fcmed by the Téezaivalent forced by stadied both of t». 35'v‘a'erts, it will the". 'ézt'rs between the two 3" Le Eguivalents The main work in EElms has been done t; I" ‘3“ a form that will «can of reference {I ’Eiier at I0 wishes to refe In modal analysis "=!sters): l) The stL “Mod which will b Idha . Iarea, or areas, 9 I itb L $de area to be be "Mel 2= 33 Results (c) and (d) are in agreement with the power conserva- tion assumption already made for the coherent bus reduction. The de— tails of the other transfer functions are given in reference [2]. Those details are not included here because they are not needed in the subsequent analysis. The method of forming equivalents by the use of coherency has been studied in detail in this section. The motivation for doing so is that this equivalent is the one most widely accepted and used. It is sometimes referred to as the "averaged equivalent" or the equivalent formed by the "method of averaging". In the next section, the equivalent formed by using modal analysis will be investigated. Having studied both of the primary methods of forming power system equivalents, it will then be possible to begin investigating con- nections between the two equivalents. IV. Forming Equivalents by_Modal Analysis The main work in forming power system equivalents by modal analysis has been done hYUndriIIEIJ. This section summarizes that work in a form that will be useful in the subsequent analysis. The notation of reference [I] is retained for the convenience of the reader who wishes to refer to this work. In modal analysis the power system is divided into three areas (or systems): l) The study area where the disturbance is assumed to occur and which will be studied (and modeled) in detail; 2) The external area, or areas, that part of the power system close enough to the study area to be influenced by and in turn influence the study area, and will be modelled in some degree of detail; 3) The ." 'ca'Iy dis. , "Y‘OSE re... - S! < ,' "'e :cner . . "e THIS I. u A: I -" 91!. j ’- DIS» ‘5‘ :.""' ..... Th3 study I {differEr 34 electrically distant part of the power system which is first identified and then represented by effective inertias and impedances. Both the identification and representation of the "electrically distant" part of the power system is done by experience and engineering judgement. Thus the models of two of the three partitions of the power system distinguished by modal analysis are determined at the outset of the analysis. That is, the study area, or system , is modelled in great detail, and the electrically distant part of the power system is immediately reduced to a very simple equivalent. Thus the thrust of modal analysis is determining the level of detail required in the model of the external area, or system. For the moment it will be assumed that there is only one external system. The generalization to several external systems can easily be made once the basic ideas have been elucidated for a single external system. The key step in the modal analysis approach is to define the buses that connect the external system to the study (internal) system and the electrically distant parts of the power system. Since the models for the study system and the electrically distant part of the power system are well defined at this stage of the analysis, the electrically distant part can with no loss of generality be made part Of the study system, and hereafter will be considered so. The ljggar_differential and algebraic equations necessary to represent the external system can be formulated as: i=ex+eeq new ALT = 111' + 5 A17 (2.6b) m 1 is a vector of sf 3-5-3 of the external 5.‘ r are vectors of the Ca :22:e:aeen the study 55' Equations (2.6) ar severe the generators re'etregecerators are r 3'32573'. voltage behind a 35-599 the generators i1'3’3'7e't reactance is c The internal volt.- . : “:E‘IIIOI'I. referenCe fl“: 11'; small perturbation :‘l D . l W :.. Q. LE cos 5 *EClagSma] genera. . ' rIECIUClr‘g 1 AV - o. ‘ (-E l LV = . 01 (—c be put 1n man-- . g 1 35 where y_ is a vector of state variables sufficient to describe the behavior of the external system, y} is a subvector of y_ and A11: AMT are vectors of the current and voltage changes at the boundary nodes between the study system and the external system. Equations (2.6) are derived in reference [I] both for the case where the generators are represented in detail and for the case where the generators are represented in the classical form as a constant voltage behind a transient reactance. The derivation for the case where the generators are represented as a constant voltage behind a transient reactance is outlined below. The internal voltage of generator i is determined relative to a network reference frame D-Q. That is, < II -E Sln 61 < II I E cos 5i . Taking small perturbations gives _ I ’ __ I AVD AE s1n 5i E (cos 61)A51 i I _ I - AVQ AE cos 5i E (Sln 6i)A6i . i But the classical generator model of constant internal voltage implies that AE' = O, reducing the expressions for AVD. and AV to: 1 Qi AVD (-E cos 6i)A6. . ‘I 'I - ' . AVQi ( E Sln 5i)A5i . This can be put in matrix form for a system of m generators as III “—1 : re'e EC is a (2". X " :':'e:ia;2r.al for eacr The basic algetr ‘ 2 D h HI L (.1. I 3’. [(n (*7.— 'v-a —-4 I '94 "' 4 . "01 -.A at“ h“, ar‘ '1‘: -1 ‘ 1" "tarsal voltage, a! any: and voltage at ”o 2 Do an“ ‘ ~ costar-y “tween "Entire (2.7) into 36 AV = ED A (2.7) where §Q_ is a (2m x m) matrix with n non-zero 2 x l submatrices on the diagonal for each generator being equivalenced. The basic algebraic network equations are AI 9' D' A_V_I = (2.8) A_I_ A B 2311 as 1 _.. I where AIG and AMI are vectors of changes in generator current and internal voltage, and ALT and A\_/_T are vectors of changes in current and voltage at the equivalent terminals, i.e. the terminals on the boundary between the study system and the external system. Substituting (2.7) into (2.8) gives =C_'§_I2A§+_D_'AV _T . (2.9) A16 The electrical power of each generator is given by which leads to the perturbation equation 01+ v01 A101 + 101 Ain I 1:; ”art 1:; This can be put into matrix form for the m generators as APi = VD1 AIDi+ 101 AV re": I and l’,‘ arr "b s'aests. Tne values ’253599’18'323!‘ in: "2's of the rem; L 5' g - SucScltutirg IF- J I E3 ; - 1-1‘ ‘ = AP :5, 1",“ ' H. : ,‘-. «Lu! the S\ T I I : mi 1"; r5. = l «r (EFE T ‘ 37 AP. = 10 A! + [CALM (2.10) where IO and yo are (m x 2m) matrices of (l x 2) diagonal elements. The values of ID , IQ , VD and VQ can be calculated 1 i i ' 1 once the generator internal voltage phasor Ei£§i has been specified in terms of the network D-Q reference frame. Substituting (2.7) and (2.9) into (2.10) yields Af.= IO §Q_A§_+ yo C §Q_éd + yo 0 AILT =.A_P_A§..e13.AIT (2.11) with = I = I AP. (lacrwswz he Now, the system differential equations are: Im§=-ee_s_-A_e . . (2.l2) 61 - w0(Si - SR) 1 = l,2,...,m where Im = diagonal matrix of inertia constants .QE_= diagonal matrix of damping coefficients. Equation (2.ll) is an expression for Ag in terms of A§_ and A! . T Putting (2.ll) into (2.12) results in - I a. “.I I O I 1 I 1 u I I 1 I 1 0 ' — I I l ‘ I ‘ 1 I . I , 1 .- 1 NI: .0...‘..4----I -- . -l I , i’ 'I API -. .. L -m a -4 '.:s:‘:atirg {2.7} into 1;: ll =A'E3 -r ‘T __ ..._ Lasers (2.l3) are of £3325 that are subi I lie eoaations (, e than state-Space ...... "'5‘3r.'::ation y = 89 §. :39 fonn 38 r- . 'N r- | ‘5 r- - r- 0" I A3 lwo -w ASE 0 1 0 I mo “”0 0 I -w _. , . I I o o I o I o I _ I o -:- - ----;---1 -------- l --------- --- + ----;---- AYT (2-138) 5 -T' API -1’ DP as -1' BP g-J 541—. -m— .JL—La L._m—_J Substituting (2.7) into (2.8) and solving for AIT gives AI .1 = e12 Aa. + e'AyJ. (2.13m Equations (2.13) are of exactly the same form as (2.6). It is these equations that are subjected to modal analysis. The equations (2.6) are next transformed to canonical form by well known state-space analysis techniques. That is a matrix: § whose columns are eigenvectors of the matrix A is found. Then the transformation ‘y =~§p is made. This transforms equations (2.6) to the form~ A‘AE + 9' AMT (2.l4a) ALT = uB."P_ + s AMT (2.l4b) SR =_B"'p where, (l) g== §f1A S is the diagonal matrix whose nonzero iagonaT Elements are the eigenvalues of .A. (2) 3? is the rows of §. corresponding to the subvector 1' ofx (3) BI' is the row of §_ corresponding to SR, the reference speed deviation of the external system. (I) a' 1's We" ate that (I) 655 =32 'egitirate assurmle asztzained by EXtTaCtIr ?es.:1ector 1' of ,V_ a‘afion of the referent related because it is fished in combinatior The method used ' ‘2":115 from an explahat 11's n (2.14), to the . Assuming all the Jerts the steady-st e = .’-1 ~55 1; :ttie transient soluti ~ e“‘) 1 HERE“ 5 ing: lg, Idle? tOf R) ther 39 (4) g] is given by e' = 5‘8 Note that (1) assumes no zero or repeated eigenvalues. This is a legitimate assumption for power systems. The algebraic equations are obtained by extracting the rows from .§ that correspond to the subvector yf of y_ and to the state SR which is the speed deviation of the reference machine in the external system. SR must be retained because it is required when the external system equations are solved in combination with those of the study system. The method used for the deletion of unimportant response modes follows from an explanation of the transient and steady-state solu- tions of (2.l4), to the §£gp_input AIT. Assuming all the eigenvalues are distinct and have negative real parts the steady-state solution is = zit-193A! . -ss T and the transient solution is .. ”At '1 | p_(t) - (.1. - e )A QAILT _ At - <_I_ - e- 1255. At The matrix (I_- e-) is diagonal and thus the transient response of each element p_ is independent of all the others. The response of each element of 9, therefore, has a steady-state component and a decaying transient component. These exponentially decaying components are the natural modes of response of the system. The actual response . lad". seesi‘stefi- 1’ 15 U if rejeCtIr-g frcr :s'a'e'tsof p “hid 32f the grounds for a) The real D5 [ESEUVQ N. that W3 me its III-‘31 S distrutance a) The corresz nuf’ltel‘s 1“" be aSS'J'Ed C) The cor-res? shall nuT'iC‘E mcde may be state VECtC The corrESC small HUT-5E the mode re on the equi In terms of lir -..rs can be categor‘ sat-tion. 40 of the system, y, is built up of linear combinations of the natural modes. The order of the electromechanical equivalent can be re- duced by rejecting from (2.l4) those rows and columns corresponding to elements of .p which do not contribute significantly to .y_=_J1. Some of the grounds for deleting specific modes are a) The real part of the eigenvalue is such a large negative number in relation to other eigenvalues that the mode may be assumed to jump instantly to its final steady-state value in response to a step distrubance. b) The corresponding row of A'Ie' contains such small numbers in relation to othEr'?ows that the mode may be assumed not to be excited by the input Ayw. c) The corresponding column of R" contains such small numbers in relation to other columns that the mode may be assumed to contribute nothing to the state vector y: d) The corresponding column of nR" contains such small numbers in relation to other columns that the mode may be assumed to have negligible effect on the equivalent output vector A_¢. In terms of linear system theory nomenclature these four con- ditions can be categorized as follows. Condition a) is a fast mode condition. Condition b) is a controllability condition. That is the natural modes discarded by this condition are discarded because they cannot be controlled by the input A11. Condition c) is an observability argument that says the modes do not observably affect the measured state of the external system. Condition d) is an observability condition in the sense that modes discarded under this condition have no observable effect on the output vector AI . The application of labels such as "observability" and "con- trollability" to the four conditions of mode elimination may seem 7:3? and at best red: retif certain structur. :e'ercy methods yield as for this claim res :‘Tarllity. It is far :as:::cceats arc the f Filly. oerhaps tedir Gite the I'EQJ‘Irg 3166;;at‘20n (2.7) Cd" _( i:-n‘1 -‘1 .14" 731’ ~ 1199M, f‘Ilp-he l “i ”i 90: ‘JL ~ “2 L 41 trivial and at best redundant. However, in Chapter 4 it is shown that if certain structural conditions are satisfied, the modal and coherency methods yield an identical dynamic equivalent. The argu- ments for this claim rest on the concepts of observability and con- trollability. It is for this reason that the relationship between these concepts and the four conditions for mode elimination is so carefully, perhaps tediously, drawn. Once the required selection of modes to be retained has been made equation (2.7) can be rearranged in the form e= F3 7 El (2.lSa) L.E2_J ra“ "a 9 a"? ”a“ E} = 9 e1 9 £1 + e1 AVE (2.l5b) L.£;J (ES. S2 g2_JI_£2_J Lf%é '9. 31 11,21 '31 F11“ E1 = ' (2.l5c) gas 114 flag 32; ‘er where, g_= subvector of .2 to be retained - subvector of _p assumed to jump immediately to v I steady state subvector of _p assumed to be zero I)? Fora ste: IIIJ‘ IQ ' II to aacn(2J5c) yiel Cl -‘ln :1C {/3 l 133-; ‘ '1. , ‘I t] are a 42 1,9? _e_, 91., g, 11]. = submatrices of Ag' and S obtained after reordering rows and columns. For a step input £4 = Q_ and p2 = 9, Hence multiplying out (2.15b) yields -I B] " '93.] 9.] A11" and £11 1T9. AMT- Equation (2.15c) yields X0 =P£Q+I1131 _ -l ‘ 9.9.+ DJI'Q. 94 M1) Since the speed deviation SR is one of the states that must be retained in yo the vector yb becomes The combining (2.l4b) with (2.16) yields ALT = :1 .0519. + In E] + SWAT SR=929+§2A11 where 9H’ g4 are all rows of g, g. except the last, and g2, fie are the last rows of ‘g, g, Then the equations become: ‘I- - “’1 \ : ". 3, Seco '2‘ '5‘ I< IO .. ..—J | 1 I 3.) J .1 If) m 43 .9. =IQ+§A1T (2-173) 13.1. = 1943+ (11.8.1 + DAY; (2.l7b) 5R=923+§2 ALT (2.l7c) Equations (2.17) are the final form of the reduced equivalent of the external system. There are several points worth noting about the model equi- valent. First, the dynamic simulation of the power system requires the simultaneous integration of the differential equations of the study system and the external system. The integration of these equa- tions is straightforward, gnge_the input AII has been determined. The determination of the All requires the combined solution of the algebraic network equation for the study system and the external system. Second, the modal analysis technique determines a linear equivalent for the external system. In the coherency method, by contrast, the equivalent can be either linear or nonlinear. In one respect this is a drawback. However, an advantage is that a reduced order equivalent can be found for a group of generators that are not necessarily coherent. Third, the input to the modal equivalent model, is the voltage differences AI: at the boundary. This is in contrast to the linear model developed in Section 2 of this chapter, where the input is the step input in mechanical powers. The choice of the model in Section 2 results from the intent to relate coherency and model analysis, but from the coherency perspective. -. rgr'1 .._ n... 036;,5': 0- Coo-0'- V r .\ I1. . I In. Al.» F- n \ I» I . .&b O ‘1 1 ‘y I ‘II ‘ I .r C. E n: . . .qI .\. «L . F 2H F4 In .0 b fill. ” u ,h s P r\.w «Cd PM, BU IV. a. .3 G. at a - I. 3. ._.C .P. In. a . v N: p .u . . ”w” H a. I o e a . - . 5 C.» :s -d .\v :5. r o . o .3 .: .- n.» .3 p s a w 2» S .—. . u .p. o n . v s u a . 3.» d a. e u .‘I. ...w u.. u. :- .P- .u. nw a. .o. a... . z». 0 Phi .0. n I . v M u . .II . ti o ,. .I u‘ o C In “ I “.1: c§ I r b 44 There are some real disadvantages to the modal approach that also deserve some attention. First, the mode elimination procedure requires the calculation of eigenvalues and eigenvectors. This is a computationally expensive step. Even if one is willing to accept this expense there is another difficulty. Thatis the fact that it is not practical to compute eigenvalues for a system of more than lOOth order. Using the simple classical model of each generator this means that the external system cannot have over 50 generators. For a large system the approach adopted by Undrill [I] is to break the external system into sections and construct a linear model of each section. Implicit in this approach is the assumption that specific eigenvalues can be associated with specific sections. This automatically intro- duces another approximation. A second difficulty with the modal equivalent is that the reduced order model of the external system is not expressed in terms of equivalent lines and generators but in terms of retained canonical states. This robs the equivalent of much of its physical insight. It is particularly unappealing to power system operators and planners who are used to thinking in terms of lines and generators. This second disadvantage of the modal method is closely re- lated to a third disadvantage which is that the available transient stability programs are all written to accept dynamic equivalents ex- pressed in terms of equivalent lines and generators. To implement modal equivalent procedures involves modifying the existing transient stability programs. The fourth and most telling disadvantage of the modal approach is that rules exist for aggregating on the basis of coherency while ..:s.;.;;,51y preserv‘s‘ w~ I'ra'v I. .o- ' ‘ Tris crazier ’39 3:21? aid then descri '26 :“:"sg power system i's'.3is. Althea?) IE" resisting necessary ...‘ .‘. ,.; ,. _ :..1:.-:'c1"9 pT‘OeECuT‘c 33:1) examining those area: of generators t 45 simultaneously preserving modal properties [5, 6, 7]. V. Summary This chapter has established a linearized model for a power system and then described, in some detail, the two primary methods of forming power system equivalents, namely coherency and modal analysis. Although lengthy, this chapter has provided the basic understanding necessary to uncover the connections between these two equivalencing procedures. Chapter 3 takes the first step in that pro- cess by examining those system structure conditions that result in a group of generators behaving as a single generator. ' I "F‘- F“p‘n a . __, _b yrl ‘ =~ tho tachire .;P.'I.‘Af‘ . ~ I l I “‘3’: If trig .-:(\ ' -‘\t :C.. J ’ 1. H *-~~?erent if e 1': “'8‘; u r _" ‘ o ""WtOrS wil .;:Ere 1‘3: the a h C-ang ar '-Efi- (dad an 9" . "1 \- 'A ‘v . went DOING!“ - ‘r W‘s. C u a HOWEVEr - I ~. .Ilgdefines . c Jitars 6 cc 1 ._ E‘EFatq CHAPTER THREE STRUCTURAL CONDITIONS UNDER WHICH A GROUP OF MACHINES BEHAVES AS A SINGLE MACHINE I. Introduction The concept of forming a reduced order dynamic equivalent using coherency was first introduced by Chang and Adibi [4]. They defined two machines to be coherent, ”to oscillate together" in their terminology, if there exists a constant c.. such that 13 51(t) - 6j(t) z Cij for O < t < to. A group of generators is said to be coherent if each pai[_of generators in the group is coherent. In the subsequent development two gradations 0f coherency are distinguished. If 51(t) - 5j(t) = Cij for 0 < t < t0 then the generators will be said to be strictly coherent. If the angle difference is only approximately true, i.e. if 5i(t) - ath)‘e cij then the generators will be said to be coherent. Chang and Adibi modeled the generators as current sources and derived an equivalent that could not be expressed in terms of equivalent power system components, i.e. equivalent lines and gen- erators. However, some more recent work by Dicaprio and Marconato [10, 11] defines a strdctural condition under which a group of gen- erators accelerate together and remain strictly coherent. Under‘this condition it can be shown that the coherent group behaves as a single generator. The work of Dicaprio and Marconato is described in detail 46 “:5 “art section 1 a 72's at'orship bet a. 2 ‘ , _ "P .Nrd:“. .sv .I;IOCS ar- . 7353901751 1 v “a my 1S Smili I .10 47 in the next section because it is fundamental to an understanding of the relationship between coherency and modal dynamic equivalents. However, the structural condition of Dicaprio and Marconato is only one of three rather hypothetical conditions under which a group of generators behaves as if it were a single generator. Two other structural conditions can be found which lead to this same result. These conditions are explored in two subsequent sections of this chapter. A word here about nomenclature and notation. In this chapter the coherency of a specific group of n generators will be considered. This group of generators will most often be referred to as "the specified group of n generators".] To be perfectly correct, the group of generators should always be referred with this phrase. However, in consideration of the reader's ears, "the specified group of n gen- erators" will frequently be referred to as "the group of n generators" and occasionally as simply "the group". (Somewhere right now Mary McCarthy is smiling.)' Hopefully the meaning will be clear from the context. In terms of notation, the frequently encountered expression K II —I 0 N v ..,m means that i is an index over a §g3_of n elements and k is an index over a different set of m elements. 1In discussing pseudo-coherency in Section IV, it will also be con- venient to call the generators external to the specified group the study group. ' :wir? -‘~ " u‘ - -————'—_——- Ut‘v' Consider 1 E'é'iZCI‘S G3 and G '5: External 535:9" 2,.zze'ui1l be cars t? :it reactance 8n F1P¢rP .. El "mated ‘ 48 II. Strict Geometric Coherency Consider the simple power system model of Figure 3.la. Let generators G3 and G4 and the admittances y1 through y8 be the external system, with buses l and 2 the boundary between the internal and external system. The generators 63 and G4 in the external system will be considered constant voltages. E3 and E4, behind transient reactances, y7 and y8, respectively. In Figure 3.lb the transient reactances y7 and y8 have been eliminated. This can be accomplished in the simple example of Figure 3.1 by a series of star-mesh transformations, or in a more general setting by writing a set of node equations I_= 1y for the buses l,2,3,4,3',4' and then eliminating buses 3' and 4'. Then the equations for buses 1,2,3 and 4, with I1 and I 2 the equivalent current injections at buses l and 2, are r11“ rYll Yl2 Yi3 “141 W], I2 = Y2] Y22 Y23 Y24 V2 I3 Y3l Y32 Y33 Y34 E3 J4J J41 Y42 Y43 Y44s LE4J Dicaprio shows [l0, ll] that if certain structural conditions exist in the power system at time t = 0', then no matter what disturbance occurs in the internal system at time t 3_0, generators G3 and G4 will remain strictly coherent, for all t 3_0. The conditions that must exist in the system of Figure 3.1 at t = 0- are: IE3I .00 e'J(53 64) = |E4' M3 31 Y (3.la) M4 41 .. .,' a] l'1 I" 49 . 0 O -J(6 -6 ) IE I 32 e 3 4 = ——£L- (3.lb) where, M3 and M4 are the inertias of generators 3 and 4 and 63 and 62 the phase angles of internal generator voltages E3 and E4 respectively, at time t = 0'. Now, express the admittances Y.. in polar form as Y.. = . 1‘] . O O 1‘] JYij . . 3(51'54 [Yi.|e . Multiply equation (3.la) by |51|€ and equation 3 . o o 3(62’64) (3.lb) by IEZIe to obtain: . 0 O - 0 0 3(5l'53+Y3l) J(51'54‘3’41) IE llE IIY le IE llE ||Y le 1 3 31 _ l 4 41 - 0 0 . O 0 IEIIEIIY |e3(52'53+Y32) IEIIEIIY [Eng-54w”) 2 3 32 _ 2 4 42 M " M (3.2b) 3 4 Now equation (3.2a) can be rewritten as IE1||E3IIY31I 0 0 . . O O IE1IIE4||Y41| - 0 o 0 O O O 4 Equating the real parts of this expression yields 0 O O 0 M3 M4 O O 0 0 Now IE1||E3||Y31l Cos(a1 - a3 + Y3]) and IE1||E4||Y41| 605(61-64+v4]) are the synchronizing power coefficients between bus 1 and bus 3 and between (0} TWO (31 E AND 1 50 C INTERNAL SYSTEM > I ’2 2 #«Afi Y6 Y4 Y. ’ l Y3 y5 3'- v" b4' Y7 Va 3 4 © (0) @ (b) FIGURE 3'1 TWO GENERATOR EXTERNAL GROUP (0 )BEFORE AND (MAFTERAGGREGATION CF GEhERAI'm TERMINAL BUSES l “ ' o-'. ’1 ,...'E C 're 1r] »C r ‘ ‘ . “ h‘ I a ah} k" : !:P:v g" e: 5.“, J V' "'":‘A.-' .‘1 - a w ea». ‘- 13‘E’6nt for a 1‘» ls easy Ere 3.] to The C35 In”! buses. . o In. .:-.A ..- .CherEnCy be ”‘4- u /< 51 bus 1 and bus 4, respectively, at time t = 0'. The conditions of (3.l) show that for generators 3 and 4 to remain coherent after the dis- turbance occurs, the synchronizing power coefficients between bus l and buses 3 and 4 are proportional to each respective generator”s inertia. The same is true for the synchronizing power coefficients between bus 2 and buses 3 and 4. The result is that when a disturbance occurs in the internal system, the amount of that diSturbance energy delivered by gagh_boundary bus to the generators of the external system is prorated to each generator's inertia such that all the generators of the external group accelerate at the same rate and remain per- fectly coherent for any t_: 0. It is easy enough to generalize from the simple example of Figure 3.l to the case of n generators in the external system and m boundary buses. In the general case Dicaprio's conditions for perfect coherency become E. -j(a?-a§) En e =-M; Ynk (3.3) for: any i = l,2,...,n-l any k = l,2,...,m Dicaprio calls the conditions specified in (3.3) the condi- tions for "theoretical coherency in the large", with "large” meaning that the conditions imply coherency for the nonlinear representation of the system used to derive (3.3), namely the algebraic equations T I .Y. Y __ —k = kk -—kn k (3.4a) 111 Ink inn -§n 0--.'- ):~» A». «A» L u 3» k 5 OS w ‘- 52 plus a second order differential equation of the form PM1.= PG. + M. 25'. i = l,2,...,n (3.4b) 'I 1 l for each generator in the system with 1* a k x 1 vector of the currents injected at the boundary buses I“ a n x l vector of the currents injected at the internal buses of the n generators of the coherent group .yk a k x 1 vector of the voltages at the k boundary buses g“ a n x l vector of the voltages at the internal buses of the n generators of the coherent group each with magnitude Ei’ phase angle 61, i = l,2,...,n. -1kk a k x k matrix of the admittances between the k boundary buses an a k x n matrix of the admittances between the k boundary buses and the n internal generator buses of the coherent group 1“” an n x n matrix of the admittances between the n internal generator buses of the coherent group an = ilk PM, the constant mechanical input power of the generator i of the coherent group. P61 = Re{Ei - 1:} M. the inertia constant of generator i of the coherent group Although Dicaprio calls the coherency that results from the satisfaction of conditions (3.3) "Theoretical Coherency", in the O ftflS work I. w». {ran the when 7..- re strongiy liter Aproof teat . :t“ident for stri: .5 (l) tuses b netting of tte t (2) tases l buses of That is1 " SEEEI y a nOk {3‘ That tn PEfEfren \4‘ That to are con To prOVe “EC 3’9 Strictly coir nime to Write 1: ‘ PH]. ~ p8. " ‘ N i ‘9' 9v. . “ psi ‘8 th W‘e'itor i ana d the n achl'nes fl . 'hif :657i‘3rs . ‘4 1n the Dr 53 remainder of this work it will be referred to as strict geometric coherency (SGC). The reason for the name change is to distinguish the coherency that results from the structural condition of strict geometric coherency from the coherency that results when the generators of a C group are strongly interconnected electrically. A proof that Dicaprio's condition (3.3) is both necessary and sufficient for strict geometric coherency is now given. First, assume (l) buses k = l,2,...,m are the boundary buses con- necting a group of n generators to the remainder of the power system. (2) buses i = l,2,...,n are the internal voltage buses of the specified group of n generators. That is, assume that the terminal buses of the n generators,and all_load buses, have been removed by a node elimination procedure. (3) That the generators of the specified group are referenced to generator n of the group. (4) That the generator internal voltage magnitudes are constant in the specified group. To prove necessity, assume the n generators of the specified group are strictly coherent. Then all the generators of the specified group accelerate at the same rate, so that, using equation (3.4b) it is possible to write _ PMi - PGi " PM - PG Si"'-_TF-_'= an =-—J%T———fl- i = l,2,...,n-l (3.5) i Now PM, - PGi is the difference between the mechanical power input to generator i and the electrical power output of generator i. Thus if the n machines are strictly coherent then the ratio of generator power difference to generator inertia will be the same for all n generators in the group. The CTN-9X :3 7'75: 372.: goes tn... ear-”fie: group the :7:22? the 9.707.: is '72722.’:s from l) t7e {.5 a result a 77:73? the specified in tte power d‘iS 27.7127, 2.265. This c Let PG. + , :: TQM. Then F‘ n pa]. + J16] : “:72 bar ,. 11+ 4%; 13 1‘. L 9‘ 54 The complex power generated by a particular machine in the specified group goes two places, to the other generators in the group and to the m boundary buses. For a disturbance that occurs outside the specified group the amount of power transmitted gmgng_the gen- erators of the group is the same before and after the disturbance. This results from l) the fact that the voltage magnitudes at the internal generator buses are constant, and 2) the assumption that the voltage angles at these internal buses are strictly coherent. As a result any change in the power distributed by a gen- erator of the specified group after a disturbance occurs will be a change in the power distributed by that generator to the m boundary buses. This can be formalized as follows. Let P61 + jQGi be the power injected by generator i of the group. Then P61 + 3031 = P61 + .1061. + P62 + JQGi where PG; + jQG% is the complex power transmitted from generator i to the other generators of the group and PG? + jQG; is the complex power transmitted from generator i to the m boundary buses. For a disturbance at time t = O, the system is in equilibrium at time t = 0'. That is PMi(0-) = PGi(0'). Putting this information into (3.5) gives PGi(0') - PGi(t) = Pen(o') - PGn(t) or, Mi Mn :7 --) - P3 .I 1 :1’2, ’n-“‘ s "5; -—‘\n". 3.5.1.18: group. The Dica:rl0 ;::n:ers of netw: ..:.f:‘.:ns at tire t = I 2": .y. ?_ .7etor i as g A. - X S ”in - .1. - f F .7.» JJUFESS the PM ~...’. ' 1 .ngattine t = ReiEQe l N :0 m Fri 0 55 Ps;(o') - Pe;(t) + PG;(0‘) - PG”(t) Under the asSumption of strict coherency among the n internal gen- erator buses, this reduces to pG¥(0-) - peg(t) = PG;(0') - PG;(t) Mi M (3.6) n for i = l,2,...,n-l, since the result is true at any generator bus in the specified group. The Dicaprio condition (3.3) is a restatement of equations (3.6) in terms of network parameters and the steady-state load flow conditions at time t = 0'. To see this write the dynamic equation for generator i as PMi - PGi PMn - PGn 1n 1 n Mi Mn Now express the PMi in terms of the steady state load flow conditions existing at time t = 0". That is O j69m oe-jak ja? n -j6° - - = 0 l * o 1 * o . PMi - PGi(0 ) Re{EiekX1Ykine + Ei e jZleiEj e J} . o o 3(6 -6 ) m -J(6°- 0) _ o i n * k n . o o - o 0 3(a.-a ) n * -J(6.-5 ) o 1 n 0 j n + E1 e jginiEj e } 77772 a? t '4 .- xv on ’. J —.,.:_: Fr? 1... . . e a . . 0» IR. -K r P4. . . r1 .‘Ia H— \II at .l» «u Fl. n .. . m, a l g :2 o.» —.h .pr F“ .o:..' l . C 3» n3 WI. 9 II. n .pnu. a" .U P .§ Nd\ .2 i. a: .. .- = a .u“ ~§ , - . .2 ._ .o . x .. u‘. n. vanl. ...o-.\ I...» 2.. 56 The first summation represents the power flows between generator i of the specified group and the m boundary buses. The second summa- tion gives the power flows between generator i and the other n-l generators of the group. The superscript o designates the steady state values at time t = 0-. Similar expressions can be written for PGi’ PM", and PG”, namely, 3(6-- v° " k }} (3.7) E .. E, . ei(5275n) E n j=l Mn 3n 1 ji j(a ~5.) JEJ e n J E° , E9 , j(6°—6°) j(5°-a9) ._Q -.;L i n o n J n 1 for all i = l,2,...,n-l. 7.7:? n generators 31 Q- I q .~‘ 2.2:ee'e.trina:e:, le; 'n. z:- ’15.: V. ' 1-? m ~ber's EigE j! .. . 'u ‘1’. .3 o ‘y. 57 Recall that the disturbances occur outside the specified group of n generators and that the n terminal buses of the group have been eliminated, leaving the internal bus voltages of expression (3.7). That is, E9 = E. 1 1 for any generator in the SPECifled QVOUP- Keeping these facts in mind, consider the expressions E * E. * j(6.-6) ._Q -._l , l n = Mn Ykn Mi Yki e k l,2,...,m (3.8a) 1. :1,2,000,n-] E E j(6.-6 ) ._n -._1 * 1 n - = Mn Jn Mi in e j l,2,...,n (3.8b) i = l,2,...,n-l Since the terms Ei’ E E in (3.8) refer to constant internal bus 3’ n voltages within the specified group of n generators it is possible to write: E E j(6 ~6 ) -._n * _._1 1 n = i = l,2,...,n-1 _ J _ _1_ 'l n . = bji(6in) - n an 1 YJ1 e j l,2,...,n (3.9b) i = l,2,...,n-l Substituting the expressions (3.9) into (3.7) yields 0 0 o m 35 36 n _ . nk o 0 nk 6in ’ R9{k;] aki(6in)vk e ‘ aki(6in)vk e } o (3.l0) n -j6. jd. _ 3n _ 0 3n- Re{_Z [bji(5in)e bji(5in)e JEj} J=l for i = l,2,...,n—l. : g : 1‘ -. "r"’ n [H s l" A :7:‘7_7r :3 l . m: uJ-I »U| \ 0 U. R . . - .. :«e. - Q - ’. .d A‘l b “ v I a R § 1. v; rtn L ‘ q 7. n ..i";rh: I a. (.7 _ A lkhin) T L7 m \- ' I 'C'v’Es rap . l;&ess1tj. IE9 prOOf of 58 Under the assumption of strict coherency in the specified group, and equation (3.lO) reduces to H m Jénk(t) _ 36nk(0 ) Gin - Re{k§1 aki(éin)[vk(t)e - Vk(0 )e l} or, co m 5in = Re{ 2 aik(6in)uk(t)} 1 = l,2,...,n-l (3.ll) k=l Because the n generators of the specified group are strictly coherent, Sin = O, i = l,2,...,n-l. Since the voltage differences Uk(t) at the m boundary buses are arbitrary, the satisfaction of (3.ll) requires aik(6in) = O for k = l,2,...,m (3.12) This proves necessity. The proof of sufficiency proceeds in a straightforward way from equation (3.l0). Assuming that the conditions (3.3) are satisfied or equivalently that conditions (3.l2) are satisfied, consider that since the disturbance occurs outside the specified group of n gen- erators, the angles and speeds of this group cannot change in- stantaneously. That is, + a 0 ) = a. 0 ) = 59 (3.l3a) in( i = l,2,...,n-l + (.Sin(O ) II 0') —l 3 A O V N O -::‘7-r-’.-.'.. "3 ~' “ r'S J LA) 0 in A I. 1.“. i.“ ‘ r‘ ' L > x-__ V) L- t.‘ o ... .1353 arid t Be '3. has . ’L'vJ‘I 1n v, E. p .— ~ ~; ‘ k surdance. i7: .. 5‘" aS-IT‘ . “mtlcn . Q‘:‘ 7.\€7r ‘ Per . 3 ‘. l- A U .Era‘ -»Cr5 9X ‘Lern. whic- ‘ s‘:"r “is C I . h 0" 1:10 . .4; 59 Further (3.l3a) implies that 0 ) = O i = l,2,...,n-l (3.l4) Relationships (3.l3) and (3.l4) guarantee that .. + 6in(t) = 61n(0 ) = 0 o o + 5mm = 3(0 ) = o (3.15) + .. 5mm = 517nm ) = 517nm ) for all t > 0, completing the proof of sufficiency. Dicaprio generalizes his results somewhat by eliminating all load buses and the terminal buses of all generators in the power system, both in the specified group and outside. This leaves only m internal generator buses outside the specified group plus the n in- ternal generator buses of the group itself. Under this configuration Dicaprio uses deviations in the admittances to represent the application of a disturbance. This is necessary since the internal bus voltages are by assumption constant for all time. Dicaprio defines a general "disturbance external to the group" of generator i = l,2,...,n and then shows that conditions (3.3), with k representing now the m generators external to the specified group, are necessary and sufficient conditions for the specified group to be strictly coherent in response to a disturbance "external to the group" [ll]. The generalization is Dicaprio's way of trying to identify coherent groups without having to 60 specify the coherent group in advance. It is worth noting that this is exactly the function fulfilled by a coherency measure. The discussion of coherency in Chapter 2 established that if a group of generators were strictly coherent, then it was possible to replace the group by a single equivalent generator and perfectly pre- serve the response of the system to a disturbance outside the group. The equivalent derived by Podmore [2] required phase shifting trans- formers in the equivalent lines connecting the single equivalent buses to the boundary buses. That is Yke f Yek where e represents the equivalent bus and k = l,2.,...,m is one of the boundary buses. Dicaprio shows [ll] that it is possible to replace the co- herent group by a single equivalent machine, that perfectly preserves the dynamic response, but does not require the phase shifter, i.e. Y = Yek’ k = l,2,...,m. The choice of which equivalent to use is ke a matter of taste. Dicaprio's equivalent has some advantage in that it does not require phase shifting transformers. The important point is that if conditions (3.3) are satisfied then from the perspective of the rest of the power system, the strictly coherent group of n generators looks like one single machine. The coherency that results from the satisfaction of con- ditions (3.3) has been termed strict geometric coherency because it results from the structural geometry (or topology) 0f the network and load-flow conditions. This terminology distinguishes strict geometric coherency from another type of coherency, namely the coherency that results from two generators being very tightly interconnected. An- other look at Figure 3.l shows that generators 3 and 4 can be . ." ' v on 'zf;f:.pf j . - v-v . ‘ § F‘. ' z" .r n— I.’ 1- ‘— o v 0 P r .;-6 '7 o “‘5'! . , V’I ' .‘fl. r- ' Q ~ . 1-- .-.,, a "‘II. A. ‘- rv-w - ' is. a . . z :4-.. «'0‘ he " 5.; HS 5 ‘0 I . ~ ‘ 5.. 'l“‘:§:“ ‘1'»: h...‘ . -... P $p ;» _ . I :H I " 7‘45 17“ .E'F-P‘ v I}: (t I . ‘ C . " etr‘n \ L07. a" ': rcn‘ I wait: 211“ IE. ‘t. .- 1‘54: 8 :LR‘: 61 coherent without satisfying the conditions for SGC. Specifically, these two generators can be coherent if the admittance Y5 of Figure 3.lb is very large in relation to the other admittances Yi, Yé. In fact, in the limit as Yé gets infinitely large, generators Y5, Y4 and 3 and 4 become strictly coherent. The reader may find this to be patently obvious, and hardly worthy of consideration. In addition, actual admittances are not inifintely large in actual power systems. However, in that regard, it is worth pointing out that the structural conditions (3.3) for strict geometric coherency will never be satisfied exactly in any real power system either. And, in point of fact, it is far more likely that in a oeal power system generators 3 and 4 are coherent because admittance Yé is very large compared to the other admittances than because conditions (3.3) are exactly, or even approximately, satisfied. For completeness, it is shown in the next section that if n-l lines connecting all_ n generators of a group are made infinitely stiff then the n generators are strictly co- herent. The reader who considers this to be carrying coals to Newcastle can proceed immediately to Section IV, which deals with the less obvious data of pseudo-coherency. III. Strict Synchronizing Coherency Consider again a specified group of n generators connected to the remainder of the power system through a set of m boundary buses, and assume that the disturbances that occur are external to the specified group. Consider the particular equation from (3.7) for gen- erator l in the specified group. Now divide through this equation by 7k Y2], the admittance connecting generator I to generator 2 in the group, .- c r. A. u- , _, . .< 7' "'- " .0. l o. .«u-. l . ,. .o.~ I l p 5 -Q n D ‘ . n - 'u o —— v ‘ ‘ O . ’. n- - \ I ' - p I. .. -' N . L - V _ ' V ,.- . ~ - ‘ :3". .-‘ ‘C .4 .Q U. “':Ir:‘- ‘ u ‘- ‘.\l ,9...) ‘ l“ I - I no . 9‘ 5‘ 62 'k and let |Y21| + m. The left hand side of (3.7) goes to zero and all * terms on the right hand side vanish except those containing Y2]. Thus equation (3.7) reduces to E E j(d.-d.) E 0 = Re{- el—g e 1 J + Since Ei = E3, i = l,2,...,n this expression reduces to . . O O 3(61-62) - 3(61'62) Thus generators l and 2 are strictly coherent. As a next step take the particular equation from (3.7) for generator 2 and repeat the steps 7* above this time letting |Y32| + m. This yields, 0 52 ' 63 52 53 so that generators l, 2 and 3 are now coherent. Proceding in this way, at step i adding generator i+l to the group by letting * i+l,i strictly coherent. [Y | + m makes, after n-l steps the entire group or n generators Next take the particular equation from (3.7) for generator i of the specified group of n generators and let * IYkil + m, k = 1,2,3,...,n, k f i , (3.16) in such a way that 63 * |Y..| WET=Cjk for J.k=1.2....,n j.kfi 1 Then dividing through by (Y: and invoking (3.l6) reduces the equa- , 1 t1on for din to n E9E9 (o.-a.) 0 = Re 2 {—l—J-C. [e 1 J - e i=1 Mi 3* . O O 3(61‘5j) This expression is true for any 0i = l,2,...,n and for agy_steady 6. state load flow conditions Ege 3, j = l,2,...,n. Hence it must be true that i = l,2,...,n Thus the n generators of the specified group are strictly coherent. It is easy to see at this point that choosing any set of n-l lines that connect all_ n machines and letting these lines become infinitely stiff results in the group of n generators being strictly coherent. The practical case, of course, does not allow for infinitely stiff connections. However, from equations (3.7) it is clear that if a group of n machines can be found whose interconnections are very large relative to the interconnections between the group and the rest of the power system, then the analysis carried through above for strict synchronizing coherency is approximately true. It is also clear from (3.7) that the approximation will be better if the number 64 of boundary buses between the group and the rest of the power system is small. All of this may seem almost trivial. Part of the reason for this is that most of the ideas presented here have been in the folk- lore of power systems for a long time, and the ideas seem so obvious that they hardly need justification. Nonetheless, Chapters 4 and 5 will prove this formalization to be a powerful conceptual tool. The coherency due to a group of machines being tightly bound will be called synchronizing coherency, and in the case of n-l in- finitely strong interconnections, strict synchronizing coherency (SSG). Synchronizing coherency is the predominant cause of coherency in real power systems, and the time devoted to it here is probably well spent. The analysis of this section has resulted in a second set of conditions that result in a specified group of n machines being strictly coherent and appearing to the remainder of the power system to be one single machine. In the next section conditions are formulated under which the n machines of the specified group are no longer coherent, but still appear_to the remainder of the power system to be one single generator. IV. Pseudo-Coherency In Section II, strict geometric coherency resulted from net- work structure and loadflow conditions that prorationed the dis- turbance energy in such a way that all the generators of the specified group accelerated at the same rate, causing them to appear to be a single generator. A natural question to ask is whether this 72717215. mid w 2.72 :uts‘.e a TEENS. ca.sir int‘t‘ens b»: 1'29): still ' 3.23339, .2115 "he $761“: 7:: 50.73am) ”‘7 5‘1’3197. i. :5’5'3173-5 in tr 771% as the a: 3%" (3.1; 3““ v - 65 phenomena could work in reverse. That is, suppose a disturbance occurs outside a specified group of n machines, and propagates into the group, causing the generators to accelerate at different rates. Can conditions be found such that the specified group although not coherent, still "appears" to be coherent to the rest of the system? Suppose, now, there are n generators in the stgdy_group, that is the part of the power system external to the specified group, and m boundary buses between the study group and the rg§t_of the power system, i.e. the specified group. Let i be one of the n generators in the stugy_group. Then equation (3.7) can be inter- preted as the acceleration of generator i, . E . irsi-on) Y.. e . j(o -a.) -._l n J an M. 31 1E3 9 '2' EE” 6. = -Re{ { -—-Y 1n i=1 Mn 0 O - 0 0 ~ 0 0 'I + Ki(t) i = l,2,...,n-l (3.18) E° . E‘,’ . inf-5°) o nag-5:) ki e JVk 9 Equation (3.17) is simply a rearrangement of the terms of equation (3.7). 3‘53: sflath' r7; transfers a.“ 3:12.727. is the ezitte irte'ac 7:27.21”) group as 272752.73 of the 5'in f0 the Stuc aeration of g: :r .1 of the st. 6!. ‘. D “V, i.e. 7:1:rs of the 5: 31.7mm. It is .1537. 3‘3 group are r L‘EEXF'FESSlon 0) Mi trict g. M (3.19) iterate, if the NOW consi :01)!” ‘. "”"mg relati 66 The first summation in (3.l7) is the contribution to 6," due to the energy transfers among the n generators of the study group. The second term is the acceleration of generator i of the study group due to the interaction at the boundary between the n generators of the study group and the generators of the specified group. If the generators of the specified group are to appear as a single gen- erator k) the study group, then this second summation, which is the acceleration of generator i of the study group relative to genera- tor n of the study group caused py_the specified group must be zero, i.e. Ki(t) = 0 , i = l,2,...,n-l, (3.l9) fer 911_ t > 0. In other words, the acceleration caused by the gen- erators of the specified group is the same for every machine of the study group. It is assumed that the angles of the n generators of the study group are not coherent. Therefore it is not possible to factor the expression on the right hand side of (3.l8) as was done for the case of strict geometric coherency. It is apparent that the condi— tions of (3.19) cannot be satisfied exactly for any arbitrary dis- turbance, if they can be satisfied at all. Now consider the following. Suppose that at t = 0' the fbllowing relationships exist. E. -j(o§ - 5°) E n-__n_ = - = _;.Y1ke - Mn Ynk for 1 l,2,...,n l, k l,2,...,m (3.20) 1.73.2.7 5.127359 :5 n generat: 3‘32‘335 :8th 27:72.7 tie an; 75:12 of (3.2 I ll”! 3'9 ifiserators The one 3 iflChlnes 1'5 NH, “HRS be 7‘8} “W t0 so: 67 Further suppose that the disturbance that occurs in the area containing the n generators of the study group is not large, so that the angle deviations between the n generators are only a few degrees. Assuming further the angle differences among the n generators and the m boundary buses at steady state are not large, it follows that the con- ditions of (3.20) are approximate1y_satisfied for all t > 0. It is then possible to write - 0_ O . - , o o E * 3(61 5") 3(5n 5k) 3(5n-6 ) m . K1.(t) . Rd E n - ifi Ykie J[vke - vke " 1}} E * {[1 Y k l Mn k 2 0 s i = l,2,...,n-1. Thus if the conditions of (3.20) are satisfied, then the relative acceleration of the generators of the gtpgy_group does not depend on the generators of the specified group. The analysis above indicates that if the specified group of n machines is not strictly coherent, then it cannot under any con- ditions be represented by a single equivalent generator without dis- torting to some extent the response of the remainder of the system. However, the analysis also indicates that for small disturbances yjthip_the study group, the specified group may, under conditions (3.20), be quite adequately represented by a single generator. In fact, it will be shown in Chapter 4, that the conditions (3.20) are sufficient to decouple the linearized equations for the power system, so that fer the linearized model, the n generators of the :2 :e7spective no... 6961‘ ' 3.2153“ x 7"v‘iil'"S ‘ is u d uiu'l ”'3'. ‘ a .. .. I35 '3: “Hire a sis 1:“ e «i Strict 68 specified group behave exactly as a single generator. Since, from the perspective of the remainder of the power system, the behavior of the specified group of generators is identical to that of a group of strictly coherent generators, the conditions (3.20) will be called the conditions for pseudo-coherency. The absence of the adjective strict means that these conditions result in pseudo-coherency only for the linearized model of the power system. V. Some Observations pf the Relative Utility p:_the Three Types pf Coherency This chapter has explored conditions under which a specified group of n generators responds to a disturbance outside the group as if it were a single generator. The first set of conditions (3.3) which lead to strict geometric coherency are purely hypothetical in the sense that they could probably never be satisfied exactly in any real power system. The utility of an approximate satisfaction of these con- ditions can only be answered empirically. It is interesting to note that the most important use of the conditions (3.3) comes from their trivial satisfaction when the Yik are very, very small. That is, conditions (3.3) explain conceptually the well known empirical fact that generators a long electrical distance from a disturbance accelerate together, even if their inertias are widely different. Strict synchronizing coherency is purely hypothetical for a different reason, namely that real power systems do not have infinite admittances. It is, however, by far the most important set of conditions for coherent behavior of group. Its utility when approximately satisfied has been well established by Podmore and u l- . 9" VAz‘qA' Ju'. ' , 0 .AE .FFE -~: 35: CC'ET r . 10F Strl :..’..'.?.t the g; :‘~‘=":13r real ‘. 2-1‘ I‘ L. p .A I"' ‘3 “Ellyn: .2. it“: work l 69 others. Further, as will be shown in Chpater 4, in the linearized model, the synchronizing coherency can lead to a separation of the power system into fast and slow subsystems, through the techniques of singular perturbation theory. The third condition, pseudo-coherency, is by far the weakest, applying only to small disturbances. VI. Implications pi the Coherency Conditions The three types of coherency discussed in this chapter have important implications for understanding the connection between the modal and coherency methods of forming dynamic equivalents. For strict geometric and strict synchronizing coherency, the fact that the specified group of n generators acts like a single generator really means that the internal behavior of the specified group is beyond the control of disturbances outside the specified group. For the case of pseudo-coherency the internal behavior of the specified group is undetectable by the rest of the power system, that is, unobservable. It was shown in Chapter 2, that observability and controllability conditions are used to discard canonical states in forming the dynamic equivalent of the external system. In Chapter 4, it will be shown that for a linearized model of the power system, if the conditions for SGC, $56 or PC are satisfied for a group of n machines then the modal and coherency dynamic equivalents are identical. The results in this chapter make this result seem fairly evident and it takes only a little work to establish it. CHAPTER 4 THE LINEAR MODEL IDENTIFYING THE COHERENCY EQUIVALENT WITH THE MODAL EQUIVALENT I. Introduction In Chapter 3 conditions were found which caused a specified group of n generators to behave, from the point of view of the rest of the power system, like a single generator. This analysis was done using a nonlinear model of the power system. One would expect those conditions to yield the same result if the equations for the power system are linearized about some stable operating point. Section II is devoted to showing that this is in fact the case, for a modified form of the linear equations developed in Chapter 2. Section III then formalizes the concepts about controllability and observability discussed in a qualitative way at the end of Chapter 3. Specifically, it is shown that if the conditions for strict geometric coherency or strict synchronizing coherency or pseudo-coherency are met by a specified group of n generators, then the coherency and modal analysis methods yield the same equivalent for the specified group of generators. Section IV then extends these results by show- ing that the structural conditions necessary to apply the techniques of singular perturbation theory to a specified group of generators are, in the limit as the parameter p +»0, the conditions for strict synchronizing coherency of the specified group. Section V introduces the concept of linear decoupling. 7O 71 An aside is required here about notation and nomenclature. The notation used here follows that of Chapter 3, but perhaps needs some clarification, because it is not presicely the orthodox notation used with power systems; it has, however, some advantages which will become clearer in Chapter 5. In the typical linearized, N generator, power system model, the generators are numbered sequentially and the last, or the N-th, generator is taken as the reference. That nota- tion will be followed here, but with a second notation superimposed over it. The first m generators will be termed the "study group", and the last n = N-m generators will be termed "the Specified group of n generators", as in Chapter 3. In general, the first m gen- erators, or study group, corresponds to the area of the power system where the disturbances occur. It can be identified with the "study sytem " or "internal system" nomenclature typically found in the power system literature. The specified group of n generators is the group of generators whose behavior is being investigated for the pur- pose of finding some property which will allow the group to be re- placed by a reduced order model. In this chapter the analysis is primarily aimed at showing conditions under which the specified group can be represented by a single generator. As in Chapter 3, in order to save the reader's sanity the term "specified group of n gen- erators" will occasionally be shortened to "the specified group" or just "the group". The meaning should be clear from the context. This chapter also introduces the concept of an archetype. An archetype is a hypothetical set of structural conditions on a group of generators. The conditions are hypothetical in the sense that 7.75.. 3'9 DEiE :72, are ass. s:7.::.ral c: 727. is so 5.7 3.3:: .‘or St) urn-64; ' - I I uu'vel 1 72 they are never exactly satisfied in an actual power system, although they are assumed to hold for the archetype. In some cases the structural conditions are achievable but have a probability of occurring that is so small as to be considered zero. Dicaprio's conditions (3.3) for strict geometric coherency are an example of this kind of archetype. In other cases the archetype is achieved by a limiting pro- cess, such as making synchronizing power coefficients infinitely large. In the linear model this limiting process will result in a matrix going to zero or the product of two matrices going to zero. The notation ad0pted to distinguish this case is A_+»Q_ or A§_+ 9, This will avoid having to write limit Ail t-mo k shorthand notation employed to make the monograph smoother for the = Q, It is simply a reader. It has the added advantage of helping remind the reader that a hypothetical configuration is being considered. The reader should not be concerned if the concept of an archetype seems a little vague. That vagueness will dissipate as various archetypes are considered. 11. Decoupling the Linear Model The conditions found in Chapter 3, which cause a specified group of n generators to behave as a single generator were formulated using a second order nonlinear representation for the generators in the system, and the nonlinear, algebraic, power equa- tions for the system. The same result can be shown for a linearized version of the equations, as follows. :7,:°‘7r in ' only. ‘7‘ "' . duvl 2727777 37: :73 rife 6r .1.. ._, 73 The linear model that will be used in the following discussion is a slight modification of the model deveTOped in Chapter 2. That model was a state-space representation of order 2N-2. Half of the equations in that model are simply defining equations that relate the generator angle excursions A6, to the generator speed excursions Awi’ i = l,2,...,N-l. For the present analysis, it is more convenient to rewrite the 2N-2 state equations as N-l second order equations, A§11-1 ‘ ‘51 $571-1 ‘ CAM-1 + E! (47‘) where, B = [E El!) T : l p [APM], APM2,...,APMN: APL1, APL2,...,APLQ] ii - Fl— 1 7 —' M 'M— ‘ 1__ -11“. M2 MN ' 1.__ -1. L MN-1 “NJ egg aP_L ’1 L.‘ ' "77‘ ‘7:- ag 3g I_= N x N-l matrix of synchronizing power coefficients. . .a h 2' ' - ‘1')- F 1 D . . O, ' ."e I" 0'-.. f I al’ p n .§ I/I 74 The modification required concerns the representation of the forcing function. The linear model of Chapter 2 assumes that load buses can be in the following locations: (l) In the part of the power system outside the specified group of n-generators and not contiguous to this group. (2) On the boundary between the group of n generators and the rest of the power system and contiguous to both areas. (3) Inside the specified group of n generators and not contiguous to the power system outside the group. The analysis of Chapter 3 assumes that only buses of categories (l) and (2) remain. That is, that all the load buses and generator terminal buses of the specified group have been eliminated and only the generator internal buses remain. In the following discussion, the standard 2N-2 model will be rearranged into this semi-reduced form to show how satisfaction of the conditions of Chapter 3 de- couples the linearized equations. First, with N = m+n, let, 1) 6n . i = l,2,...,n, be the internal generator angles of the specified group. 2) 6m , k = l,2,...,m, be the internal generator angles of thb m generators of the study group, i.e. the generators outside the Specified group. Partition the matrices of equation (4.l) to match the dimensions of the vectors gm = [6m ’6m 900.,6 gooogsm J l 2 mk m ,...,On.,...,6n ] = [6 ,6 '1 nl n2 1 n-l 22,, That is, the angles 6m of the vector 6m are the internal generator k angles of the first m generators of the power system; the angles ;, Er.” 75 6ni of vector én-l overall power system. Generator N Then the equations (4.l) can be written .. ' f- . H r- '5 Agm ('31-’11 i ('fi 1)12 [ Aém -:- = ......... r---; ..... ---- f§n~1g J-‘fl I)21 i ("E D224 fén-L r- A | 7" m, 5619, + ----4 ....... A I A .52 I'm-92. where _E, in partitioned form, is, I r_l_ l M1 : 1 E I? i 0 I i I r- s ’ I ii iii l—J _ _ I 1.12-3.3 ”ms ' A 9 :flzz : l \— ._J I M l m+1 0 l — I I I L E r- “N L11 : L12 L.‘ ----f ----- L i I; ‘_-21 I 22“ 1m and -I—n-l respectively. A.P_M AP}. is the reference. II I l I I I I I I I I I I I I I J I I I I I I I I I I I I I I I I I I I I I I l I I I I I are identity matrices of dimension m and n-l are generator angles 6m+l""’6N-l of the (4.2a) (4.2b) fflszl 1 'bkul 76 fl] = [Elli fll2] is m x N and consists of the first m rows of [1 M2 = [Q_E 822] is n-l x N and consists of the last n = N-m rows of ._ (E L) = M L + E L is m x Q and consists of the first m (E_L)2 = @22L22 is n-l x and consists of the last n-l = N-m-l rows of __L, If we assume that equations (4.2) are in the semi-reduced form, then the conditions for strict geometric coherency amount to the condition (fi_L)2 = 9: To see why this is so consider the following. Assume that the system is in the semi-reduced form of Chapter 3, i.e. the load buses have been eliminated from the group of n generators, and that there are qm load buses left in the study group, and qb load buses on the boundary between the study group and the specified group of n generators. Consider the power equation at generator i of the specified group of n generators. . o o . o n J(6.-6.-y..) §b 3(o.-e -y. ) = s 1 j ij 5 1 k 1k PG. Re{ i EiEj yij e + _ Ein yik e } j—l k-l where yij = lYijl is the magnitude of the admittance Yéj between buses i and j and v.. is the phase angle of Y§.. The SUper- 13 1 script 5 indicates the network is in the semi-reduced form. The first summation represents the power exchanged between gen- erator i and the other generators of the group. The second summation is the power exchange between generator i and the m boundary buses. Taking the partial derivative of PGi with respect to eh, h = l,2,...,qb results in 77 3P6. 36 " HEVhyhsmwi -6h-Yflp,i=l£”.um.h=lflp.u% which are the synchronizing power coefficients between the boundary buses and generator i. Now take the partial derivative of P61 with respect to 91 where 92 is the voltage angle at one of the load buses not contiguous to the specified group of n generators, then 339, .____ = o i l,2,...,n (4.8) 86 l _ I - l,2,...,qm Let, . l) 9 =[91,6 2,. . .,6 h” meq ] be the vector of voltage angles at the load buses o? boundary between the study group and the specified group. 2) pq= [61,92,...,e£,. ...,6q ]T be the vector of voltage angles at the load buses thernal to the study group. 3) 39m = [PG],PGZ,...,PGk,...,PGm]T be the vector of electrical powers injected by the generators of the study group. 4) pg" = [PG],PGZ,...,PGi,...,PGn]T be the vector of electrical powers injected by the generators of the specified group. T T 5 e = e ' 6 , PG = PG ' PG . )-[1m'-qb] _ L...__,,1 Then, I I _§§: - 311 :-E12 _ 311 : E12 39 277717;," ' '6"7"f>" —2l l -22 - ' —22 {'16 f9 3 (1) T1 (1; ‘r r ~- ‘UJ‘ V»: 78 where E4] is m x qm _P_12 is m x qb 32] is n x qm E22 is n The matrix 22] = Q_ by (4.8). Now assume the conditions (3.3) are satisfied for the qb boundary buses and premultiply egg/39, by Q, to Obtain A A BPG M M P P 9 : = —]l “12 —11 “-12 “ 39 o M o P —- —22 —- —22 A : A A fl11311: l511312 + 9112322 I = ------ r--: -------------- I 9- : fl22322 where 851 is m x m 852 is m x n M22 is (n-l) x m 9_ is (n-l) x n The elements in @22322 are cm the form A t- tn' =__1_l_.__:_l_ . = _ . ' = {flzzflzzhj Mi “)1 1 l,2,...,n 1, 3 1,2,...,qb where the index i runs over generators of the specified group and the index j runs over the boundary buses of the group. But, satisfie: tOPJJSa‘ NON not of the 79 aPGi s . o o t.. 39, E.V.y.. sm(a. - e. - y..) 13 - g - l J 1J l J 13 Assume the conditions (3.3) for strict geometric coherency are satisfied, namely . o o Ei s _J(5i - 6n) _ En s ~—-Y..e - ——-Y . or, Mi 1:] Mn "3 s i n ij Eiyije _ En s JYnj . — M ynje "H n where jY-. - = S = S 13 II 1,2,... ,n and Yij yije l,2,...,qb (.1. II Now, moving all terms to the left-hand side of equation (4.3), .00 3(5n-6.) conjugating, and multiplying by Eje results in the expression . 0 O . 0 0 a.-a.- .. a —5.- . E.E.y§.eJ( ‘ J Y‘J) E E.ys.eJ( " 3 YnJ) 1 J 13 _ N J DJ. = 0 (4,4) ”1 Mn Now note that the elements {@22222}ij are simply the imaginary part of the expressions in equation (4.4). That is, are EGC .r- 'rU‘ S . . 0 E.E.y. 3(5 -5 'y .) E 3(5 6 “Y -) 1 ij 1 J 1J _fl. 5 n 3 ”3 Imag{——'h—i——e Mn EJynJe } s o EiEinj S1n(6i 613 Yij) EnEJ ys. sin(6o - 59 - Y -) 1 Mn nJ n ”J t. . t . _ lJ "J - -———-—-— 1 - 1,2,. 9", =1’2’ ’q M1 Mn b = 0 . Therefore, if the conditions (3.3) for strict geometric coherency hold, then the matrix 822222 = Q, As a consequence, it is now possible to write, for the semi-reduced model, i.e. with the load buses in the specified group or n generators eliminated, 3g ------- 4E ---------------- (4.5) Now, if the disturbances are confined to the study group then equa- tions (4.2b), the equations of the specified group of n generators, are unforced. If it can now be shown that (-N_I)2] = Q. then, under SGC, the equations for the specified group are completely decoupled from the equations for the study group. The condition (gfl_1)21 = Q_ can be shown to result from the condition $22322 = Q_ as follows. Consider the unreduced equations ++ u 81 r- d r- r- w AEQJ egg1 33g1 33g,1 age1 33§_ Ag, 3A6 3A6 8A6 8A6 3A6 A3§2 -i —2 —l —2 —3 age APL1 = 3392 3392 BEE2 3392 3392 ngl §ZS“' 3A6 3A6 8A6 3A6 ' 2 —2 —4 —2 —3 Age, 492 AELB BELl BELl BELJ BELJ BEL- A93 (4 6) ; _J L. .J ' Bag] 2mg2 313g] 31ng 31x33 33L? BEL; aPL2 aggz agge 53:3' SZEE' 3A9, 3492 5353‘ 333 333 3PL3 3P_L3 3_P_L3 L. 313g] aagz 2mg, aAgz 34939 where fig] and Ag] are the internal generator power and angles for the study group PG and age are the internal generator power and angles for the specified group of n generators 3L4 and qu are the load power and angles for the load buses of the study group 2L2 and A92 are the load power and angles for the boundary buses £53 and A93 are the load power and angles for the load buses of the spec1f1ed group of n generators. It is assumed here that the generator terminal buses have all been eliminated and that there are no connections between the generators of the study group and the specified group, and no connections between the load buses of the study group and the load buses of the specified group. It is implicity assumed that no study group internal generator buses are boundary buses. This causes no lack of generality because ifa study :ciszn in As Sen 6) in by the SE 82 if a study group generator were a boundary bus, the corresponding column in (-fi_1_)21 would, assuming conditions (3.3) are true, be zero. As a consequence equations (4.6) can be put in the form: A391 9-11 9 9—1 3 9—1 4 9— Ag] 1% 2 122 9 124 925 may. ABE] = 93] 9. £33 934 9. A94 (4-7) AP—Lz 941 942 943 944 945 A92 Jig-Lad -9- 952 9 954 955, _A93_, To put (4.7) in the semi-reduced form assumed for equations (4.2) requires the elimination of the load buses of the specified group. Setting AELB = Q, solving fOr Aga, and eliminating A93 from the first fbur equations of (4.6) yields ”A591“ Pin % E 913 {44“ FAQ] AEEZ =, 9‘ ..... g??i--%’ ..... €25- Ag? (4.3) ABE-1 i3i 9: é A33 {—34 A91 £352.. -941 942) 943 944.. £924 where i.22 = 122 ' 92595952 924 "' 924 " 9259339434 942 = 942 ' 9459:5952 —44 = 944 " 9459315954 Next eliminate the load buses Alf];1 and ABE; from equations (4.8) by the same procedure tg_ggt; 83 ape J 0 J J A6 -—4 =< -41 :- - —-SH —512 5 -4 APG o J J J A6 a ‘24 t!) “224 [521 “522.) c “'24 (in - is”) 21.512 Aé, = ~ (4.9a) -J (J - J ) Ag ~— ‘521 ‘22 ‘522 .4 ,_ 2.. where is“ "‘ 3111911131 + Z42941) + 944(221-‘131 + 522941) 1312 = i132—12942 ” 5114222942 iL521 = 9—24‘521931 * 122941) (4-9b) 9522 = 324222512 and -1 —Z-n Z42 = 9—33 %34 121 -Z—22 943 944 Now 924 is the matrix 222, i.e. the synchronizing torque co- efficients between the boundary buses and the generators of the specified group of n generators. If the conditions (3.3) fOr strict geometric coherency are satisfied then A ~ E22 9-24 = 9 ’ and A [122 J52] = _o_ from (4.9b). Now '98 , in turn, is the matrix of synchronizing torque coefficients Zl between the generators of the study group and the generators of the =-T specified group for the fully reduced model. That is -J _2] ‘521 well f'cr the ful l3) |-«4 disturb; Faking Cfltditj Sufficj 84 where I_ is the N x N-l matrix of synchronizing torque coefficients for the fully reduced model, with all_load buses eliminated, i.e., uz> 1‘“ = ("11141 + 512121) i n11142 * fl12122 --; --------------- r-; -------------- I fl22121 : M22122 Thus if condition (3.3) is satisfied then fizzéz4 = 9_ which in its turn implies that @2295 = @2212] = Q, Thus it has been 2l shown that the conditions (3.3) fOr strict synchronizing coherency completely decouple the equations for the specified group from the equations for the study group. That is, under the conditions of SGC, (fl l_._)2 = (fl [)2] = 9_ so that equations (4.2) take the form ' . - 313.312 A?" .. £923-- “€21- . ‘3}! ‘4 "’3’ I . 9_ :_§22 A6 9_ cl" A§n 0 (4 10b) x ABM x EH " [M 1]] Egg and Eij-(-nl)1j’ 19.] - 192 The term fié ABM does not appear in equation (4.l0b) because disturbances on ABM are restricted to generators of the study group, making .82 ABM = Q, The initial conditions A§n(0’)==A§n(0') = Q_ combined with conditions (3.3), the conditions fOr strict geometric coherency, are sufficient to yieldlA§n(t) = Q_ fbr all t > 0. Thus equations (4.l0) reva‘ rc‘ ‘2 5"5' 5" C i I . 0F I- I DU ‘9 d «3 4| '1 'b :‘nq a J: K “fir- _l -.:r In ' 'W Lu," 85 reduce to the differential equations of the m generators outside the specified group of n generators, referenced to generator n of that group. Since the differential equations in (4.l0) now only involve the first m generators, or the_§tugy group, and the last generator of the specified group, the network power equations can be reduced by setting APGi = 0, i = l,2,...,n-l, and eliminating Adi, i = l,2,...,n-l. The resulting equations represent a system of m+l generators, i.e. the m generators of the study group referenced to generator n of the specified group. That is, the specified group of n generators acts like a single generator, for disturbances out; sjgg_that group. In (4.2) the system as a whole is referenced to generator n of the specified group, resulting in the inertia of the equivalent machine in equations (4.l0) being the inertia of machine n of the original group. If the inertia is large, then this choice of reference is probably adequate. It nay be advantageous, however, to use a reference that yields an equivalent generator whose inertia is more representative of the group as a whole, such as the sum of the inertias of the group. This is the approach taken in the next section. The analysis so far has established that if the conditions (3.3) fer strict geometric coherency hold, then in the linear model, the specified group of coherent generators act like a single generator. An analogous line of reasoning shows that the same result is true if the conditions (3.20) for pseudo-coherency are satisifed. This time the structural conditions exist between the boundary buses and the gen- erator buses of the gtugy_group. Since the proof is very similar to that just provided for SGC, it will not be given in full detail. Rather acareful ou‘ Eerin the rim func 3:119 pract Cons tire qb amber of iterator Tetrices c T) [To It Then the 86 a careful outline of the steps will be provided. As will be discussed later in the chapter, pseudo-coherency is a conceptual property whose primary function is to lend completeness to the theory. It has very little practical utility, at least in the present research. Consider the matrix __:-.- 2.1.5.422 3—1 l9 i P -21 ' ~22 I T ‘ where E§_= [gem : Egn] as before, but now, the load buses are eliminated from the study group so that r- 6 H ‘qb 9] = '6'- k. —qn .4 where qb is the number of buses in the boundary and qn is the number of load buses in the specified group. As before assume all generator terminal buses are eliminated. The dimensions of the sub- matrices of 339/39_ are then 341 m x qb 342 m x qn l’21 " X qb -E22 n x qn Then the 3_ matrix has the form O- O i k U g. «in, _l I‘M T:F.J ‘ ‘ x : A. 3? ‘ I C Iii-(Em 87 Since, this time, the load buses in the specified group have no direct effect on changes in generation in the study group. Next, change the reference from the last generator of the specified group to the first generator of the study group. This gives E_ the form "-L l. : T M1 M2 I - L l. i 9. "1 M3 E ° I I : ,. ' i4- ]; u“ .2 - m- i _ ......... fl """""""""""" 5‘? """"""""" ' n n ' —2l —22 imam l. E Ml i 1 . M L. : m+n_'l Then, A A I I egg unis). 2,152 89 " 7.0-: -------- fl12 E E22 E24 5 E22 _ I ‘ fllip—n 5 9 ——————————————— T----—-- 34—12511 ” M22321 l fl22322 where 351 is (m-l) x m ‘fl2l 15 n x m N22 is n x n and the Q_ is (m-l) x n. ’12 and ti :zi‘ltians Trat is th gap, and s: tiat t? Q . ‘ Wit as I. J . 14 ‘5 EERerat. Has bee 88 The conditions (3.20) can be manipulated as was done in the case of conditions (3.3) to yield II N 'é t- 't i M_1k_-_fl= O for k i 1 - That is the subscript i runs over generators 2,...,m of the study group, and the subscript k over the qb buses of the boundary. But to « A _ 1k tlk _ {31-1131in ‘ —M1. " T] " 0 so that the conditions (3. 20) yield A fl11311 = 9’ Just as the conditions (3.3) led to A M P —22-22 = 9° Now eliminate the load buses within the study group by setting APE] = Q, in equation (4.7), solving fOr A94 and back substituting to get: r- - r-~ ' ~ 5 #- fl ABE2 in 9 5114 9 ABE] I I A392 9 ..... %§i-€€&---§§§ 9% = ~ : ~ A52 941 942! —44 945 A92 I I APL o J . J J A6 L—ag ;— —52; —54 —55 B —3J ~ 914 is the matrix of synchronizing torque coefficients between the generators of the study group and the boundary buses where the network has been reduced by eliminating the load buses of the study group. "it )5 J, 75‘?) I'E'E Ir... 1c... But J Cf the f 5‘.de g, 89 That is 844 is the matrix B4]. Hence, if conditions (3.20) hold, then ”.944 = 9. Now eliminate 3L2 and fits to obtain A29. (a - 9. > 4 4.5. APG -J (J - g_ ) A6 where 13—51] = 442-11941 9-512 = 114911942 + £12152) 9-521 = 9.2411194] + i252—21941 51522 = 924(111942 + 512952) " i25(1219-42 + 122952) 2 z 3 J '1 —n —12 [44 —45 321 12 J54 95 But 9512 = 142 the submatrix of synchronizing torque coefficients of the fully reduced model, with the reference generator 1 of the study group. That is A I : 1“-11 i 9- 111 : 112 M"- vii" -2l :—22 —2l :-22 I A I 9111111 E fl11112 I = """"" Z """ + """"""""" I fl21111 * fl22121 : (321112 ” E22122) A P. I, E', :1 ll 4 pr. . U, 511' n I: H. l 1. .1 5‘ Olly) FA. ' v. [We "5. . , are 90 and T -12 = '—11" -s ‘ (”in-‘11:“Z +Z J I111 12 11—42 42—52) =9. when fill-1M = 9, Hence the conditions (3.20) are Sufficient to decouple the equations for the study group from the equations for the specified group of m generators. Thus, if the conditions of (3.20) hold, equation (4.2) can be written. .. - A - T - A§m — (- -M__)12Agm °A§m + [M E (fl_L_)13[A_13_M 5 11ng (4.lla) .. = -A . ' + -.’2 u A he, ( ED221341 0A4" ( £159.52,“ + [11 LJZEAEL] (4.111)) Once again the term flz Am does not appear in (4.llb) because the disturbances are confined to generators of the study group. Equation (4.llb) shows that the specified group of n generators is definitely not coherent. But (4.lla) shows that the motion of the specified group is decoupled from that of the study group, so that the specified group appears, to the study, group to be a single generator. Next assume that the conditions for strict synchronizing cOherency are satisfied, and that ({1}); exists. The inverse 0f (41 D22 can be written ( A ).1 1 Cofn Cof12 ... Cof1n -M T = . 22 Det(-l_~i_ D22 C°f21 Cof22 Conn L. Cofn] Cofn2 . . . Cofnn .4 where Cofij is the ijth cofactor of (181)” *“fitely s was of I l ‘x \n-) Ii ‘ b‘ ’ 1 I. -\ I‘ r,‘/ 1 .I ‘D. . 1 9) Assume as was dbne in Chapter 3 that n-l of the intercon- nections that link all n generators of the Specified group are made infinitely stiff. In the linear model, the corresponding n-l elements of (-E D22 become infinitely large. Now (1&1)22 is (n-l) x (n-l) so that Det(-_|E1: _T__)22 is the summation of (n-l)! terms, each term being the product of n-l elements of (-_N_ _T_)22~‘. One of these terms is the product of all n-l elements that are being allowed to become infinitely large. Now each cofactor Cofij is, in turn, the summation of (n-2)! terms each term being the product of (n-Z) elements of (-_N_ _T__)22. Hence no term in Cofij can be the product of more than n-2 of the elements that are becoming infinitely large. As a result Det(-fi _T_)22 dominates every term in the summation of COfij’ i,j = l,2,...,n-l. Thus in the limit all terms of (..fl 1);; tend to zero. Now assume that (:14 D22 is finite and rewrite equation (4.2b) as _’\ _'| .. A .. A - ° ( 1"- l)2215-‘511-1 ‘ (‘11 D22('fl1)21A§m + Aén-i ‘ ”(‘8- Deg/Pm + (E Dggmzam + (E UZAEL] Now letting the n-l elements of (-fi_l)22 become infinitely large "ESults in 9=A§n~l for all t>0 “hi ch says that there is no perturbation of the specified group. This result in turn reduces equation (4.2a) to the form 92 egm = Ml l) 11Aém—O-Aém + [N AP + (E L) 1 _M APL]. 1 Thus once again, assuming zero initial conditions, the specified group of n generators behaves, from the point of view of the remainder of the system, like a single equivalent generator. The analysis to this point can be summarized as the following, Result 1: Given the linearized model of an N-generator power system (4.2) in the reduced or semi-reduced form. if any of the three conditions $66, $56, or PC holds for a specified group of n generators, that group of n generators can be replaced by a single equivalent generator, and the response of the remainder of the system to a disturbance outside the specified group of n gen- erators will be perfectly preserved. If the proper reference frame is chosen, the form of the single equivalent generator is exactly that proposed by Podmore. However, the N-th generator reference frame is not the proper one. The proper reference frame is defined in the next section. Result l is the first important step in reaching the goal Stated at the end of Chapter 3, namely connecting the modal and COherency equivalents via the concepts of controllability and Observability. That goal is realized in the next section. I] 1. Identifying the Coherency Equivalent with the Modal Equivalent This section uses the results of Section II, to establish c0"ditions under which the modal and coherency equivalents are 1dentic'al. As one might suspect the conditions that lead to identical e(luivalents are the three conditions iof strict geometric coherency, Strict synchronizing coherency and pseudo-coherency. There are two l'ntermediate steps that simplify the proof of the main result. The .5 "‘5?" v.1. \ 0" t I 3 «.54 Q ’ ‘ P .... D .. .p. fiA. M r If 9;. " -ln 93 first intermediate step is a lemma that gives expressions for the eigenvalues of the model of equations (4.2) in terms of the damping coefficient 0, and the elements of the matrix -fi 1, The second step is to establish a more general referencing scheme for the model of (4.2). The following lemma provides some useful insight into the way the system structure affects the eigenvalues of the linear model of the power system. Lemma. Let A be the plant matrix in the state s ace r . . __ . e re- sentation of equat1ons (4.2), w1th the frequency expressedpin perp un1t. If xi, Ai’ i = l,2,...,n-l, are the 2N-2 eigenvalues of A and vi = l,2,...,N-l are the N-l eigenvalues of -fi_I, then v02+4yi .AT=—§9- V02+4Yi l 1 2 Proof: Putting the frequency in per unit results in Then Dett )._I_ - A] = Det 1.. 2 '1 ..... L-------- E 1 i (A+o)I Using the identity, I 5-11 ; A12 _1 net ""7"" = Demfin ‘ 512522521H5223} A21 1 A22 results in a: {d 358 re 94 Det{>._I_ - A} = Det{[AI - (-fgfi1)][(x + o)_I_]} =muuux+ml-cfip} the last step indicates that if Yi’ i = l,2,...,N-l are the eigen- values of £1, then the eigenvalues of _A_ are the solutions of the equation A2 + AU - y = 0, or .\ A. = - g-+ l—loz + 4y. 'I 2 2 T $ i=1,2,...,N-1 ”it: '%‘12’°2+4Y1 It is the usual case with power systems that o is small. Thus in most analysis, including Dicaprio [l0, ll] and Kokotovic [l5], 0 is set equal to zero. For the present analysis 0 will be re- tdined, but the results of the lemma make it clear that the eigen- valees of the linearized model are predominately imaginary and de- Pend almost exclusively on the structure of the matrix -f1_T_. The second intermediate step modifies the reference frame 0f ‘the linear model. To this point,the reference for the 2N-2 State equations has been 6", the angle of generator 11 of the SPECified group, which corresponds to generator N in the state Space representation. A reference that is more in line with the aggregation of generators using in coherency equivalents is a “Kbdification of the reference frame that Meisel [9], calls the Uniform Center of Angle (UCA) reference frame. The reference frame Used here will be referred to as the UCAn reference frame. The , ...r‘fl? I ‘n‘.. .y v 0 v er P ,I: .1 b L, l“ P : I...“ DA 1 ' a. ’5" I'itl bl}: I; o {.5 () _.a. (‘4‘? W "Ere 95 subscript n denotes the fact that the summations involved are taken over the n generators of the specified group. n M Z pfi-éi, where the summation runs over the Define 6e = i=1 e n generators of the group and Me = 1:] Mi' Assume the same con- vention as before, namely that there are N = m+n generators in the power system, the first m being the study group and the last n = N-m being the specified group. Then in the notation of the state model the summation above is over generators m+l,...,N. Then, using the fictitious angle 6e ‘as the reference, the system of N generators can be expressed as, A61 = ankoi (4.12a) ‘ _ 1.. _ - l_. _ _ ~ m1. - Mi (APMi APGi) Me (APMe APGe) 0 Am]. (4.12b) r- s f" 1 f" a,“ ABE. BE§_ BE§_ ag "" = ea a§_ "I," (4.l2c) AP]; Jo \_ _J 32-. BEL- 33 35 L.— —._J “hare ~ \ 51 = 6i ' 6e 61 = 6i ' 6e ~ = ~ 1:192...9N-1 w] mi - we ’ D. o = 1 PE' 1 .J ~ ~ ~ ~ T ~ ~ ~ ~ T a = = \N'] [61,62,...,6N-1] '9‘ [61,629..0’60] LG = [PG],PGZ,...,PGN]T 3L_ = [PL1,PL2,...,PLQ]T 1'37? ~'1 _ n and M = 2 M., e i=1 1 96 n E PMi, PGe = PM 9 i=l i "M: 1 PGi' tm put in the form of a state-space model where, X= [2131.113 A = (1 -& 1 B = I9 1 with, M = r1 Ffi' .1. M2 L. 51¢! 1>\l " (”fl I.)22}° Now [1; - (-11 1122]" in + {- (111)221'1 11; - [_I_+ M11 1);)1'11 .. ... .IJ .- “41 1. (T! 104 so that in the limit, when n-l connections are allowed to grow in- finitely large, -1 _ [Al - (ZN. l)22] 'T XE; ' l] T g and Det[A_I_ - (111)] = Det[A_I_ - (+1 DHJDetEAL - (£11221 Thus, once again, the eigenvalues are segregated precisely as they were in the discussion of SGC, and there are no oscillations within the group due to disturbances in the study group. Hence, as before the modes eliminated as uncontrollable are those associated with (~flI)22. Note that plain N. has been used to emphasize the eigen- values are ppt_reference dependent. I If the conditions for pseudo-coherency are satisfied for the group of n generators then once again mode elimination leads to an equivalent identical to the equivalent obtained by the coherency method. Under PC the submatrix (efl_1)12 = 9, so that Dem; - (21)] Detm - (E I)“ - (1111.)12141 - 1-411223'1('fll)21] x veto; - (-m 11221 DEtEAL " ('fl l)]]]Det[)\l "' ('M. l)22]- The eigenvalues A,, A: of the system model (4.12) that rePY‘esent intermachine oscillations within the specified group are again decoupled from the eigenvalues for the study group. In pseudo-coherency, the n generators of the specified group are no longer coherent but still appear to the remainder of the 105 system to behave as a single machine. This implies that transformed to the canonical form, the modes represented by the eigenvalues of P111122 will be eliminated as unobservable. The m eigenvalue; pairs associated with (M_I)]] represent the intermachine oscilla- tions among the m+1 generators of the reduced system which consists of the m generators of the study group and the reference generator which is just the single generator equivalent of the Specified group. The analysis to this point has now formally established the following important relationship. Result 2. Given a linear model of an N generator power system (472 1, if any of the three conditions SGC, SSC, or PC hold for a specified group of n generators, then the equivalents formed by modal techniques and coherency techniques are identical. Further in the UCA reference frame, the equivalent for the specified group is of the Exact form proposed by Podmore. ~ Result 2 provides some very important insight into the re- lationship between modal and coherency techniques by showing three structural conditions under which the two methods produce identical equivalents. Further each condition could be related to a con- ‘trollability or observability condition for mode elimination. Chapter 2 also listed "fast" eigenvalues as a rule for mode elimination. Ft>r the second order generator models.used in this research, fast eigenvalues, in the classical sense of modes that decay rapidly to Zero are not present. This is apparent from the lenma proved earlier 1" this section which showed that all the eigenvalues have the gamg’ small real part, that is -o/2. Fast eigenvalues do occur in this "“JCIel, however, if fast is interpreted as high frequency. The results (VF. references [5, 6, 7] indicate that for a group of n generators 106 n-l eigenvalue pairs can be associated with the intermachine oscillations of the group. These oscillations are of high fre- quency, relative to other eigenvalues in the system. Intuitively the high frequency of these eigenvalues would seem to be due to strong interconnections between machines of the group. But strong relative to what? The next section provides some analysis of the case where fast (high frequency) modes are present. The results of this analysis help quantify the meaning of "strong" interconnections. IV. Decouplinngjgh Frequency Modes The analysis to this point has all been aimed at decoupling the differential equations for the study group from those for the specified group of n generators. A principal tool for decoupling differential equations is singular perturbation theory. This section investigates how singular perturbation theory can be applied to de- coupling the equations of our example system (4.2). Singular perturbation in the usual sense means that a system contains a set of canonical modes or states that are highly damped and decay rapidly to zero in a short boundary layer of time after a disturbance has been applied to the system. The solution technique is to set the derivatives of these fast modes to zero, changing a sub- set of the dynamic equations for the system from differential to algebraic form. These algebraic equations are then solved for the fast modes and back substituted into the remaining equations to eliminate the fast variables. The formalities of the procedure are as follows. 107 Consider the system 51 = 91151 + 9—1242 + 21’51 + 519 . 2.. p_§2 + C X =9— X —22—2 T 1192392 21—1 where u is a sufficiently small scalar. The form of the equations shown here is often referred to as the two-time scale form. It will subsequently be shown that this form can be obtained not only for the usual interpretation of singular perturbation but also for systems with slightly damped, high frequency modes. The vector N, can be identified with the study group of the example system used throughout this chapter, and 52 with the specified group of n generators. In the singular perturbation approach the effect of the fast transients is neglected by setting p = O in the second set of equations. This makes these equations algebraic. Solving them for 1] and back substituting into the first set of equations yields .0 - p ._1 ° 51 ‘ (911 " 5:129- 29211331 I 9-111 + £1”- This provides a reduced order model of the overall system. Notice that the aggregation is not equivalent to replacing the specified group of n machines by one equivalent machine because there is a change in the equations describing the relative motion of the machines of the study group. This analysis is not directly applicable to the power system model, because, as was shown in the lemma of Section III, all the eigenvalues of the model have the same, small real part._ Thus 108 the power system model has no modes that are highly damped and decay rapidly to zero. However, Chow, Allemong and Kokotovic have shown [15] that this same approach can be applied to the case where a system contains a set of lightly damped high frequency states. An outline of the analysis of reference [15] is given here. The nota- tion follows that of the reference for the benefit of those readers who may wish to examine the subject in more detail. Consider a system of first order differential equations of the form 2=A1+§1 (4mm g=px+pg (4mm w1th 1n1t1al cond1t1ons 5(to) = 50, 2(t0) = go, and assume (1) The norms of the matrices A,§,§,Q_are bounded about u = O and the state 2 is 0 even dimension, that is Z 6 Rzm (2) The matrix D is of the form 9: 1‘91 22 -93 “92 where 92, 93 are m x m non-singular matrices and thg matrix 9293 has simple and negative eigenvalues -wi . In reference [15], Chow, et al. first show that the eigen- values of the matrix fi-Q_ are of the form OI :Jwi/U 1 =1,2,...,m where Oi is the i-th diagonal element of the matrix 4.. l I "J 109 31 = 1194 T 95]Qq__)lr] and I_ is such that I-ngalf] = Diag(-w§,-w§,...,-w§). This establishes that as u + O the eigen- values of the system 1111 = 2! + L1 (4.19) approach infinity along asymptotes parallel to the imaginary axis. 'This guarantees that the states associated with the subsystem 1=2£+21 will be of high frequency, with §_§_ playing the role of p, Chow next examines the system w=21+£ where 9 satisfies assumption (2) given above, and shows that if p(t) = EKt) + §(t) with EKt) the slowly varying part of p(t) and ||jl|5_c], H§]|< c2 for some fixed c],c2, then there is a finite T](u) such that the slowly varying part EXt) of Kit) is .4 mo=-§ mo+gmi where --l _ -l D ' 9. 23 V10 .2 _. This result is then applied to equation (4.l8b), with 9.5_ P1aying the role of y_(t), to obtain the slowing varying part Z of Z as -1 IP% =-§ 21+MM- "5': 110 Next the slowly varying part of Z: is separated from Z. by introducing the change of variables e=gt221+e§121+11 (to) and determining §_ such that equations (4.18) become 1= (50 - 11g _G_)z<_+ Be (4.20a) Q + 11 EM] (4.201)) That is, the slowly varying part of Z_has been transferred to equation (4.20a) in the sense that (4.20b) now only involves the fast variable n. To obtain the form (4.20) requires that g_ satisfy the re- lation -D G + (9:1 £+@H%-u§9=0- Invoking the implicit function gives the solution to this equation as £3.=1)_ 9.110%(141 =0-2CA +0(u) where _ _ ,4 50-11-1111 £- The fast varying part of 1 is then separated by introducing the change of variables §_=X-u(g2"+u)=x_-uhe (4.21) and choosing N such that ‘VOI A‘U 111 P1450 - 11298- 182+ HLE) = 9. which by invoking the implicit function theorem is - -2 —2 -2 -2 1205.2 ~32 -13.9. 9.52 +9111) _— —-2 —-2 —-1 41,82 -e2 9.9.2 +g$.)’stem of (4.18) as Ch (- I. I.U ‘111 I. s .v n h..‘ -1- p n!‘ 1“. \ .>1= 19!. 2<_(t0) = 50 (4.23a) Z= @491. (4.23m The oscillatory subsystem - _ -—-l 0 2-9. _Z_(t0)-;o+p_ ex.) (4.2.) is obtained from (4.22b) by neglecting 0(11) terms in Q. It is interesting to note that the final results in equation (4.23) are the same as would be obtained by setting u = O in (4.18b) solving fer Z_ and substituting into (4.18a) a§_jf_the states of _g were classic fast decaying states. This overview of reference [15] highlights the main points ger- mane to the fellowing discussion of high frequency modes in the example Power system. It has been included to provide continuity and to give the reader without immediate access to the reference some feel fer Singular perturbation techniques. The equations (4.23) show that if the conditions for "fast 8igenvalues" exist then the original system can be represented by a re- duced set of dynamic equations for the lower frequency states and a set Of’ algebraic equations. Further, the subsystem (4.24) provides the QYW1amics for the fast eigenvalues, if this information becomes Signif’icant. The ferm of the singular perturbation equivalent (4.23a) is not: the Podmore or "averaged“ equivalent, since A0 = A_- §_Qf]£g However, it will be shown next that when these results are El131’1‘ied to a power system, the term 13“le is _Q_(u2) so that A ‘= l\ and the singular perturbation and averaged equivalents are \ 9 identi cal. 113 To put the example power system equations in the form of (4.18), let 1).] = (_)_- _I_ _)(_1 + 9. Q l] . a; 2 ~ 12 ('MT)” '01 £2 11 ('M. 1)]2 9. l2 - . 42.11 + [M1 :(ML)]] --- (4.25a) _ ' -.. AB; 2.2.] .0. 9 £1 0— 1- Zl , = ~ ‘I' 2 ~ (4.251)) Then the matrices 5,839.11 01‘ 14-18) are f— a = 9 1 13. = g 9. LI'E I)” '01 UZI'E 1)]2 g .- 4 r. —. (4.26) _C_ = Q Q Q = 9. .1. .. _, 2 ~ L.‘ -I_’1_I_)2] Q _J x.“ ('__ _)22 'UUl-J T0 inSure that these matrices satisfy the conditions necessary for ”During the singular perturbation transformations, it is necessary t0 do the following. 114 (l) Insure that u2(-B_I)]2 and (1MHI)21 are finite for u + 0. (2) Find the conditions on (-M_I)22 to insure that ~ 1 (E 922 "' ‘2 Q3 1.1 where 93 is nonsingular so that 9. of (4.26) satisfies the conditions on 42_ in equations (4.18). To this end consider the four machine system of figure 4.1. Let generator 1, be the study group and gen- erators 2,3,4 be the specified group of n generators. Use the UCAn reference frame over the three generators of the specified group, and discard the equation for generator 4 as redundant. The matrix (-M_I)22 can be written r- “ir- a 1 1 1 1 (--- -i - -—- - ——- t t M2 Me Me Me 22 23 (-M T) =- t23 t33 "‘22 -1. 1_._1__ -1. t , Me M3 Me Me 24 34 \. ...JL. _J " t t t T 22. 22 T 23 T 24 t23 t23 T t33 T t34 2 e 2 e + + + + {t32 t22 Mt23 t24)}{f§§__ (t23 t33 t34)} L. 3 e M3 Me .4 The matrix (-fi]:)]2 can be written (“911112 T 4411112 T 9112122] 1 1 1 1 "T "i -—{t t1-[--— -— -—1 t t M1 12 13 Me Me Me 22 23 115 so that, t12 (t22 T t23 T t24) t13 (t23 T t33 T t34)1 [{M ' }{ ’ ] J 1 Me M1 Me (5111)]; - (4.23) Consider first (-BI)]2. Making the substitutions t22 T ‘t12 T t23 T t24 t33 Tt13 T t23 T t34 ’ it is possible to write (-M_I)]2 as t t t t ~ _ 12 12 13 13 Here the symmetry of the matrix I_ has been used plus the fact that the elements of any column or row of 1_ sum to zero. Note that (-M_I)]2 does not depend on the elements of the submatrix 122 of synchronizing power coefficients between the machines of the specified group. The same is true fer (~M:I)2]. Thus these matrices will always be finite, since only elements {12211jT of 122 will be set equal to %- in the subsequent development. Now consider the matrix (-M_I)22. Make the substitution t22 T ”112 't23 ‘t24 t33 T ‘t13 ’123 “‘34 then P: 'i {_t (L_1._)_Ez_3_-f.211_} {13.1.1.3} ( fi ) 12 M8 M2 M2 M2 M2 Me -_1 =- 22 {122,112 {, (1..-1.., 122-132} L.M3 Me T3 Me M3 M3 M3 .4 116 Suppose that only the interconnection between generators 2 l l and 3 is strong and let t23 = -§3 and factor out -§u Then in the u- u limit as u2 + O, the matrix (-M.I)22 approaches '1 -Lr-1. 1.. “2 M2 M2 1_ _1. M M g 3 3.J that is (-E_I)22 + - ié-QB where 93 is singular. The conditions for applying the singular perturbation transformations are that Q. be 3 nonsingular. Now let both the interconnections between generators 2 and 3 and generators 2 and 4 be strong by letting t24 = t23 = l—-. In 2 u this case, factoring out 1?. and letting u2 + 0 cause the matrix ~ 1J (i1 D22 to approach r“_2 1 h- m— 7 1 2 2 “‘2' “ 1. -1. ."3 ”3.. ‘so that 93 is nonsingular. Generalized to an» n dimensional case, it is easy enough to show that the requirement for the non-singularity of Q_ is that a 3 set of n-l interconnections, linking pl] the machines of the specified group, be set equal to 174 This is the same Specifica- tion for stiff interconnections reguired by strict synchronizing co- herency. Now, returning to equations (4.23), and letting n-l inter- connections among the machines of the specified group equal i}. puts the equations (4.23) in the form 117 i o I x o o z —1 _ — — —1 + — — —1 O - ~ 2 ~ -- 52 ('5 I’11 “0-1— 52 1‘ ('3‘- 1)12 9 £2 +11 :(fl 9,] M11 --- (4.27a) 42L. e i, 9 9. 1, e 1 21 . = ~ + (4.27b) 1:1. .Z_2 (‘11. l)2‘| 9. £2 93 111-UL) 12 which is the form necessary to apply the singular perturbation approaCh. The form of Q_ in (4.27b) makes it easy to calculate the eigenvalues of 111-Q PM -T—I 7 1 ' T “T Det{>\_I_- {1‘2} = Det 1 ‘fig3 (4 +0); K. _J Det1UL - 37,132,111. + 01111 Det{[(>\2T+ 1011 - (12— 23H} 11 d Det{v I.- —§-Q_}, 93 nonsingular. t Thus the eigenvalues of Q_ are yi,y: = :_/7;' where Ai’ i = l,2,...,n-l are the eigenvalues of 15-93 = (-E.I)22 which are known to have infinitely large imagina:y parts as u + O. The process of letting u + 0 corresponds to letting n-l interconnections in the specified group become infinitely stiff. That is u'+ O is the "vehicle" by which one "travels" back to the results of Section II, namely that fbr n-l infinitely stiff connections among 118 the members of the specified group of n generators, the specified group behaves precisely as a single generator. That is, the process pf_sending p §p_zero blends the singular perturbation concept pf_ fast eigenvalues into the concept pf_strict synchronizipg_cohereppy. In addition (4.23) shows that the approximation to a single machine is very good even for u small, since it is only necessary to drop terms 9(22) in order to have 50 = A. Thus it is possible to state the following result. , Result 3. If the states of a specified group of n generators can be identified as high frequency, then the specified group of n generators can be replaced by a single generator without affecting the response of the remainder of the system. Further. in the limit as u +-O, the singular perturbation concept of high frequency modes merges with the concept of strict synchronizing coherency. The singular perturbation equivalent is then identical to the modal and coherency equivalent, and the equivalencing has not introduced any error in the response of the study group. At this point four conditions have been examined, strict geometric coherency (SGC), strict synchronizing coherency (SSC), pseudo-coherency (PC) and fast eigenvalues (FE). FE, however can be identified with synchronizing coherency and hereafter the term synchronizing coherency will stand for both these concepts. In the next section, one last method of decoupling the linear model is in- vestigated. At that point the theory of Chapters 3 and 4 will begin to clearly point the way towards a theoretically sound, and com- I putationally viable algorithm for generating reduced order dynamic equivalents. V. Linear Decoupling So far in this chapter, three archetypal conditions have been considered, each of which when satisfied causes a specified group of 119. machines to behave, from the perspective of the rest of the power system, like a single generator. Each of the archetypes, in turn, has been associated with conditions on one of the submatrices of -M_I, Strict geometric coherency causes (-_M__I_)21 = p, Strict synchronizing co- herency causes (fl 1);; + _O_ in the limit as the inter connections be- tween machines are progressively stiffened. Pseudo-coherency has been shown to cause (-M_I)]2 = p, All of these conditions, in turn, de- couple the differential equations for the study group from those for the specified group to allow a reduction in the order of the system of equations that need to be analyzed fer disturbances that occur within the study system. As has been pointed out before, all of the arche- types are hypothetical in the sense that they are never satisfied exactly in any real power system. It is possible to achieve near approximations to these conditions. That is,||(-M_I)22H < e. or “(111)12“ 9.. The analysis for these types of linear decoupling is not pre- sented because these conditions, and the condition of pseudo-coherency, are not tested in the formal algorithm for producing dynamic equivalents presented in the next chapter. The analysis of Chapters 3 and 4 has shown that pseudo-coherency is a concept that can only be strictly 129 true in the linear model. The conditions for pseudo-coherency depend upon the structure of the study group, or internal system, at ’T. This is reflected in the linear model by the fact that time t = 0 the conditions for pseudo-coherency is (-fl_1)]2 = 9, Since the disturbances occur in the study group these conditions may very well be destroyed by the disturbance. As a consequence, the presence of pseudo-coherency in the linear model does not strongly guarantee that the condition will persist in the nonlinear model. The two types of linear decoupling not analyzed here depend,like pseudo-coherency,on the conditions within the study system, through the matrix (-fi_1)]2. Hence, they are discarded along with pseudo-coherency. In contrast, the conditions of strict synchronizing coherency and strict geometric coherency transfer from the linear model quite strongly because they are not directly dependent on structural condi- tions within the study group. This is reflected in the linear model by the fact that conditions for synchronizing and geometric coherency are expressed in terms of the matrices fl_I42 and (fl_1)é2, namely as (n .1),1 = 2 and my}; +9. Strict strong linear decoupling depends on these same two matrices. Thus it is reasonable to suppose that the structural con- ditions fOr strict strong linear decoupling if present in the linear undel will also be present in the nonlinear model. The hypothetical example adds credibility to this argument since SSLD appears in many cases to be a combination of strict synchronizing coherency and strict geometric coherency. 130 VI. Establishing a_Hierarchy gf_Structural Conditions for Coherency ‘ This very lengthy chapter, has traveled over a lot of material and concepts. It is necessary at this point to summarize the results of this chapter and point out how they might be used in producing dynamic equivalents. This chapter has used a basic example system consisting of a study group and a specified group of n generators. The basic plan of attack has been to establish thbse conditions that cause the specified group to be strictly coherent, gr_appga§_to be strictly coherent to the study group. In other words, conditions were sought under which the specified group could be replaced by a single equi- valent machine without changing the response of the study group to disturbances withjn_the study group. Those conditions can be summarized as (l) Synchronizing Coherency (2) Geometric Coherency (3) Pseudo-Coherency (4) Linear Decoupling In the process of establishing these various conditions, their relative merits, in producing dynamic equivalents have been discussed in an informal way. In the preceding section of this chapter, the ranking of these conditions became less informal when pseudo-co- herency and two types of linear decoupling were discarded as condi- tions for determining dynamic equivalents. The process of ranking these structural conditions is com- pletely fOrmalized by figure 4.2. The position in the table indicates 131 the value assigned to the condition in forming dynamic equivalents. As can be seen strict cynchronizing coherency is ranked high- est. It is easily the single most important condition to be tested for in a power system, for the purpose of forming equivalents. Tightly interconnected machines are coherent under a wide variety of disturbances, because the tight interconnections force the machines to remain in synchronism. Geometric coherency and strong linear decoupling are ranked second in importance. As will become evident in the next chapter, these conditions will be of use in forming equivalents where the dis- turbances are assumed to occur only in a particular area of the power system. This is almost self-evident, because there is an immediate and natural identification of the area in the power system where the disturbances occur, with the study group of this chapter. The concepts of pseudo-coherency and weak linear decoupling rank lowest in importance. As discussed in the preceding section, these conditions are not used in the algorithm presented in Chapter 5 for determining coherent groups. The dotted lines in figure 4.2 that connect the two types of linear decoupling to the other three conditions implicitly categorize linear decoupling as a derivative of the other three conditions. This is really an artificial choice. The example used to introduce strong linear decoupling was a combination of both synchronizing coherency and geometric coherency. Probably most cases of linear decoupling can be broken down in this fashion, but there are potentially many cases of strong lineardecoupling that cannot be categorized in this way. 132 Synchronizin Coherency g “ ~~ ‘§ s “‘ s “ ~~ ‘ ~‘ I ' _ _ : Geometric Strong : Coherency -'"" Linear . j _ Decoupling 1 1 ‘ 1 . |Pseudo-Coherency I I 1 4 1 l ' T . ' 3 1 I ' ' | L _ : L ............. 1 Weak Linear c --------------------- t Decoup1ing Figure 4.2. Relative Ranking of Structural Conditions for Coherency 133 Thus one could just as well categorize SLD as an independent condi- tion. It is worth noting at this point that what has been termed synchronizing coherency includes the singular perturbation ideas of high frequency eigenvalues that can be associated with inter-machine oscillations of the group, allowing the group to be represented as a decoupled subsystem. As has been shown in this chapter the singular perturbation approach is, in the limit of infinitely stiff connections, indistinguishable from the previously introduced idea of strict synchronizing coherency. The choice of terminology is arbitrary, and those who prefer to use the terminology of singular perturbation theory to describe this condition should do so. The writer wishes in no way to obscure or detract from the very fine work of Mssrs. Chow, Allemong, Kokotovic, Winkelman, et al. Finally the reader may have noticed that the adjective strict no longer modifies the conditions for declaring a group of machines coherent. The idea of strict coherency has been very useful in establishing some conceptual classes of conditions under which groups 0f generators are either coherent, or appear to some other part of the system to be coherent. In reality strict coherency is never aChieved, only approximate coherency. In Chapter 5, a method is developed for measuring the structural conditions presented in this chapter. Where it is a useful aid to the analysis process, the conceptual idea of strict coherency will be re-introduced. The goal, however, is a practical scheme for measuring coherency conditions. The reader should recognize that the 134 coherency that is actually measured is seldom, if ever, purely synchronizing coherency or geometric coherency although in many cases one or the other predominates. In all cases the coherency is never perfect and one encounters the sticky task of establishing aggrega- tion threshholds. That is, how big can the measure of coherency become, before the group is no longer considered coherent? CHAPTER 5 A REDUCTION ALGORITHM FOR DETERMINING DYNAMIC EQUIVALENTS I. Introduction Chapter 4 established five hypothetical conditions on the structure of a power system namely, (1) Strict Synchronizing Coherency (2) Strict Geometric Coherency (3) Strict Strong Linear Decoupling (4) Pseudo-Coherency (5) Weak Linear Decoupling that allow a specified group of generators to be replaced by a single generator. For brevity these five conditions will be given the collective name "structural conditions for coherency", even though for conditions (4) and (5) the specified group may not be coherent, but only appear to be coherent. Although it is not feasible to satisfy any of these conditions exactly in a real power system, the assumption is that near satisfac- tion will still preserve modal and coherent properties. There is already empirical evidence to indicate that this assumption is true [6, 7]. The conditions for structural coherency can be considered rules for aggregation, and the next task is to find a means of identifying these conditions when they are satisfied or nearly 135 136 satisfied in an actual power system. Chapter 4 showed that all five conditions could be expressed in terms of the submatrices of -fl 1, The most direct approach would be to test the submatrices themselves. While this is the most direct approach, it may not be the best, or the most convenient to implement com- putationally. An alternative is to find some other measure that can also detect the structural conditions for coherency. As it turns out, the r.m.s. coherency measure is one such measure. The first part of this chapter shows that the r.m.s. coherency measure, when used with the proper statistical disturbances, can identify a major subset of the structural conditions discussed in Chapter 4. It is then a straightforward task, one in fact that has already been accomplished, to modify the software developed by Podmore and Germond for the Electric Power Research Institute (EPRI) [2], to use the r.m.s. coherency measure. This modified software is used extensively in the study of the 39 Bus New England System discussed in Chapter 6. II. The R.M.S. Coherency Measure The possibility of using an r.m.s. coherency measure to de- termine coherent groups of generators which could be aggregated into single generators to form reduced order power system models, was initiated by Schlueter [5]. Three subsequent papers [6, 7, 8] strengthened the connection between modal and coherency equivalents. An optimum form of disturbance for determining coherent groups was also established. That is, disturbances were found that fOrm co- herent groups that depend on the dynamic structure of the power system but do not depend on the disturbance used to determine them. 137 In this section the r.m.s. coherency measure is defined, and the re- sults of references [5, 6, 7, 8] pertinent to the present discussion are reviewed. The r.m.s. coherency, th’ between generators k and 2 of a power system is defined as =/L E{ T [A6 (t) - A6 (t)]2dt} °k4 Tn 0 k 2 where g is the expectation operator. The expectation operator appears, because as shown in [7], the optimum disturbance for detect- ing coherent groups that depend on the power system structure is not_ deterministic. In fact, it is shown in [7] that there is no single deterministic disturbance that will adequately detect structural coherency. The next task is to define the form of the probabilistic input u(t). This g(t) will then be used as an "input disturbance" to drive the linear power system model of Chapter 2, for determining coherent groups. > First, decompose u(t) into two functions 94(t) and u2(t), i.e. u(t) = yh(t) + u2(t). The function 34(t) is defined as 21“) = Q_ for t < 0 That is, 24(t) is a vector step function, initiated at time t = 0. In the linear model g(t) represents deviations in mechanical power on the generators and deviations in electric power at load buses. Since g4(t) is a step function, non zero entries in g4(t) will 138 model (1) Loss of generation (2) Loss of load due to load shedding (3) Line switching If 21(t) is to represent the random occurence of such events, then it is necessary to define, r- - E12 and T _ WENT) - _M_]1[_u_1(t) - M2] } - 311 0 II 50 0 322 _J The reader should note that the vector matrix ”I is not the same as the submatrix M] of the (N-l) x N matrix M_ that appears in -_N_1_ L The matrices Eh] and 54] describe the uncertainty in the location and magnitude of generation changes ABM. The matrices £52 and 322 describe the uncertainty in the location and magnitude of power injections on buses due to either loads being shed or lines being switched. The function u4(t) can only model disturbances that resemble step changes. To model a fault, define r 0 t > T1 220422 0991 0 t < 0 K 139 That is u2(t) represents a pulse of duration T], occurring at time t = 0. Recall from Chapter 2 that faults were represented by changing the mechanical power to a generator. Thus 32(t) = ABM 0 so that the last Q elements of u2(t) are zero, where Q is the number of load buses in the power system. If 32(t) is to be probabilistic, then define 5‘22“”: -’T—'21 ”52 and m rd“ ET 4,. fl V 1 JOE L_J m I: m A ¢+ V 1 3 [Le—J 5,4 11 lo :0 I». IO 11 A)” IO The initial conditions are also assumed to be random, €{EKO)} =.Q mm 5101‘} = 1x(o) This assumption reflects the idea that for a given steady-state operating point, the power system is expected, on the average, to be right at the operating point, although instantaneously it may be subject to transient fluctuations. The initial conditions are assumed to be uncorrelated with 34(t) and 32(t), i.e. 1 lo aye) 9.16)} we) 2%)} 1 lo 140 Finally, it is assumed that 34(t) and u2(t) are un- correlated with respect to one another. This assumption is based on the fact, that the model is only used to represent one type of con- tingency at a time. For the linear model of Chapter 2, the r.m.s. coherency measure, ckg can now be written 1—-€{JT [Ad (t) - A6 (t)]2dt } g Tn 0 k 2 Ckz _ r1 T 2 % _ TE-ETTO [(A6k(t) - A6N(t)) - (A6£(t) - A6N(t)] dt} _"1 22 ' ggkz §x(‘) 2kg] where §(t) is the state vector of the linear model, of the power system, and 1 T T _s_x(t) = 7 _x_(t)_(t) at - (5.1) T 0 is a (ZN-2) x (ZN-2) square matrix, with gki a 2N-2 vector de- fined by - r ' — k {g‘k£}j — 1 J - -1 j = 2 for k f N, A f N 0 J 2 . A T 5 T k for k x N, 2 = N k 0 i f T j = for k = N, 2 f N KP j f R For the input function u(t) = 24(t) + 32(t), 5(t) has the form 141 " t gflmto) + (0 gAVB(1_J_1 + _u_2)dv for t < T] _- A t Av A("T1) TT Av e— (0) + e—-B u dv + e e— B u dv for t > T —- —- O —1 0 —- —--2 1 K Substituting this expression for {(t) into (4.1), carrying out the expectation Operation, and utilizing the assumptions that yq(t), 32(t) and [(t) are uncorrelated, leads to the expression rT sxm = 1—5 gflTyx(0)eflTdt (5.2) 0 T1{ [r1 T +l_.0 eA.VB dv][R + R + m mT + m mT + m mT + m mT][ eflyB dVJT }dT Tn ‘0 —- -4 -2 *1 1 —2-2—1—2-2—1 0 1 T Av T Av +.—— e— B dv][m m + R ][ Wgr- dV]T }dT Tn —- - —4—4 —4 0 T T T A(T-T) 1 Ah-T) 1 + J.rT [ {[g‘ 1 [ g—AV_B_ dv][m2m "1+; _RZJ[_e_ 1 J _AvgvaT 1dr T JT] 0 0 T T Av(T-T ) l T + 1.5. I {[g" A [ gflvg dv][_m2_mI][J gflvé vaT}dT T T.l O 0 T T T A(T-T ) 1 + 171—1 {[J QA‘CB. dvltmflglte— ] I 9.53 deJ T-}dT T T1 0 0 The integer n is chosen to be one if a load shedding, line switching, or generator dropping contingency occurs and zero for a fault. These choices guarantee a finite non-zero value of §XTTT for an infinite observation interval. Equation (5.2) gives the form of §x(t) for a very general stochastic disturbance. However (5.2) is not much help analytically 142 because it is not easy to get a closed form expression for §x(t). The major cause of this difficulty is the pulse portion 22(t). If the disturbances are restricted to step functions, i.e. 9(t) = gq(t), then as shown in reference [7] ‘§X(t) can be put in closed form by letting T +1w. That is lm§(fl=§flfl=[fln T-roo 5.[M T1 (5.3) where, _ T T Eu ' [311 T ”111-"111 T L "‘12 [9-11 T m11512 ‘- T T + £822 + 21.22 2122);. J (5.4) Equation (5.3) reveals that, for step disturbances the r.m.s. coherency measure depends only on power system structure through the matrix [! [J'T. However, it is also dependent on the nature of the disturbance through the matrix Eu which contains the statistics of the disturbance. The next section investigates some fundamental properties of the r.m.s. coherency measure for the general case where 5“ depends on the statistics of u(t). In the subsequent section it is shown that the proper selection of u(t) causes '5“ not to be dependent on u(t). That is, the elements of Eu will no longer be values specific to the statistics of u(t). III. General Properties 9: -S-X(°°)' Assume, as in Chapter 4, a power system with N = m+n gen- erators, where the first m generators constitute the study group of interest and the last n generators constitute a group of generators whose coherency is to be investigated. FOr notational convenience let 143 5111 912 (El) 321 £12 (1:11)” (MT) where the partitioning is conformal with the dimensions m and n of the study group and the specified group of n generators. Similarly, partition K11 K42 r|'__7< K21 K22 where the submatrices Adj contain values that depend on the mean and covariance matrices N4 and 34 of the statistical step dis- turbance vector u(t) = AP” . To make K independent of these _u matrices would mean that the EQj submatrices always had the same constant value, no matter what the statistics of 3(t). Assuming that 94] and 322 are nonsingular, the inverse of [[1 can be written as r' ‘1 -1 -1 -1 9 "5111 912 3 [1111" = 1 -1 -1 p L922 51219- _J where _ -1 9 " 5111' E112 $122 221 _ -1 3‘ $122 ' 5121 $111 912 Then the r.m.s. coherency measure .§x(w) can be written 144 _sx (...) = 0.4. ...TTEuIM. 11“ T 9-] '91; 9123 1T111 512“ P 9-‘T '9-‘T921322T - 9‘11229421971 3.] ,4 .521 522.. [IE-T542513 ET _, rT‘S'XN 3‘12-7 LT‘ 21 T‘2'2_J where AX” T QTTATIQTT ' 9.151234912111‘ " 9119-123-152194 + anaizfl'Tfezfl'TaIzafl (5.5.) §x12 T “9.151151451215122 T 94-15121”:T T 3119123452194921322 ‘ 2119125.]5223TT (535‘) 312] T 11235121945419-T T 9229219-15123-T912911 +EE£VEEJEAUMG §x22 T 92-29219-151194921922 T 5122921951512'3."T ' E'Tfizfl'Tngg; 1‘ 2715223." (5.5d) Now consider §X , the submatrix that contains the informa— 22 tion on the coherency between the n generators of the specified group whose coherency is being investigated. In the linear model for 145 the power system, the condition of strict geometric coherency (SGC) corresponds to 92] =.Q. If .92] =.Q, then -1 - T__ -1 -T i P 522?- ‘922522 922 x22‘— so that §X22 depends only on the structure of the specified group of n generators and not on the study group. This is exactly what is implied by SGC. This means that for a disturbance within the study group, that is, outside the specified group, 5&2 = 9_ which in turn causes §_ - 0. Thus the r.m.s. coherency measure will x -— capture the conditTgn of strict geometric coherency. The only defect is that the expression for §X22 is dependent on the nature of the statistics of the disturbance that occurs in the study group. That is, it is possible that terms in §X22 would go to zero not because 92] = 9_ but because the particular statistics of the disturbance cause fifij = Q, What is needed is a type or category of disturbance that yields the same constant 5%. no matter how the details of its J statistics change. This type of disturbance is formulated in the next section. A parallel argument can be carried out on szz for the con- dition of strict synchronizing coherency (SSC), which in the linear model corresponds to 9;; +'9, FOr this case, using the matrix identity (5.9), -1’ 1 -1 _ -1 -1 _- -1 F— T 9.22 T322321‘911 '. S112922921J 212.922 '* 9 ' so that A "'9 22 for a disturbance either in the study group or within the specified group of n generators. 146 Now consider the condition of pseudo-coherency, which corresponds in the linear model to 942 = Q, Examining §X11 in equa- tion (5.5a), which measures the coherency of the generators of the study group, shows that if the conditions for pseudo-coherency are satisfied by the power system structure then -1 _ -1 -1 5119- ‘ £1115115111 which depends only on the structure of the study group. Thus for a disturbance that occurs outside the study group, i.e. in the specified group of n generators, 51] = 9_ which, in turn, causes Txll = 9, so that the r.m.s. coherency measure will detect the condition of pseudo-coherency if it exists in the structure of the power system, but only by a ZHIIW disturbance of the specified group. It is worth noting that pseudo-coherency does not eliminate terms in S This ...x ' is as expected since in the case of pseudo-coherency the speETfied group is not truly coherent, but only appears coherent to the study group. Finally the case of strict strong linear decouplint (SSLD) is detected by the r.m.s. coherency measure. In the linear model the conditions for SSLD imply 95:92] +-g_ which causes -1 -1 S + P P so that S -X depends only on the structure of the specified group 22 and §X + Q, for a disturbance confined tg_the stugngroup, as 22 was the case with strict geometric coherency. The weak linear de- coupling conditions (94295;) +-Q_ and (942922921) +-Q_ are not 147 detected by the r.m.s. coherency measure, for a disturbance confined to the study group. Chapter 4 concluded by ranking the five structural conditions for coherency and intimating that pseudo-coherency and weak linear decoupling were not conditions that were necessarily worth de- tecting. The foregoing analysis indicates that PC and WLD can be left undected by using only 1) disturbances of the whole system to detect synchronizing coherency and 2) disturbances confined to the study system to detect geometric coherency and strong linear decoupling. This is the strategy adopted in the reduction algorithm fOrmulated at the end of this chapter. It is apparent at this point that the r.m.s. coherency measure is capable of detecting all the major, and most of the minor, system structure conditions of Chapter 4, that permit generators to be aggregated while preserving the modal and coherency properties of the power system. The results in this section have been established using a very general stochastic disturbance. As a result §X(m) is dis- turbance dependent. In the next section, a particular stochastic dis- turbance is chosen, which makes §x(m) independent of the disturbance. That is, for the chosen disturbance §x(w) depends only on the structure of the power system as embodied by the matrix -fl_1, At the same time the next section takes the first step towards the formulation of a general algorithm fOr determining modal-coherent dynamic equi- valents fOr a power system. IV. 'The ZMIIW Disturbance In Section II, the general form of §x(w) for step disturbances was found to be _ -l . -T _S_X(w) - [M11 gum T1 where the expression for 1<-11 512 E“ = K K L:21 —22_J as given by (5.3) depends on the statistics of u(t). Suppose that a type of disturbance could be found that yielded the same constant fifij submatrices when applied to different power system models. That is, the specific details of the In] and 5] matrices would be particular to the power system, but the resulting 533 would always be the same constant matrices that did not depend on 34 and a]. For such a disturbance the expressions (5.5) for the sub- matrices of §x(w) would be ideal measures for determining aggregation conditions that depend solely on system structure. There are, in fact, as shown in reference [7] two particular disturbances that will accomplish this very goal. One of these disturbances, called a ZMIID disturbance has 311Tl’ 322T9’ —T111T9’ -T112T9-' fl22T9- and results in This disturbance has the potential liability that it is reference de- pendent [7]. The second disturbance results in 5.. = 45-111 (5.7) 149 where 2 for i = j {511913 T 1 for i f j The disturbance that yields the 51W of (5.7) is a zero mean, independent disturbance over all_the generators of the system with . 2 2 2 2. _ _ = = d1ag{M M MTTTT°TM }a 322 ‘ Q, [11]“ 9. P112 9° R 1,2,0... m+n -41 where the Mi's are thegenerator inertias of the system. For exposi- tory convenience, this disturbance will be called a zero mean, inde- pendent, inertially weighted (ZMIIW) disturbance. It is worth noting at this point that the effect of the stochastic ZMIIW disturbance can be obtained as the summation of n + m deterministic disturbances [7]. That is, let EX (m) be the matrix 1 that results from a disturbance of M? per unit on bus i only. Then _S_ (0°) = X S ,(w). X 1:] X. FOr the ZMIIW disturbance, the resulting matrix EIW can be partitioned, conformal with the dimensions m and n of the study group and specified group, respectively, as F' '1 K K -IN 5111 22" 150 Then substitution of the 51W for the 5,. in equations (5.5) pro- .. 13 13 duces expressions for the submatrices of’ §x(m) that depend exclusively on the structure of the power system, as embodied in the matrix fi_I_TT. Having established that 51W is disturbance independent, the subscript IW is now omitted and KIw will subsequently be referred to as .5. The disturbance independent form of the equations (5.5) can now be used to begin establishing an algorithm for producing dynamic equi- valents. Consider again the expression (5.5d) for §X where the 22 K. are now the K . It was shown earlier that for a general dis- turbance of all generators §X can be small only if PT] 22 is small. Recall that -1 5 T 922 " EL219115112 (5-3) Applying the matrix identity +a- 1 1 -1-1al 41112322 ' 3219-1151121 —21—11 (5'9) -1 -1_ -1 [9-11 ' 312 222 921] ‘ 311 to (5.8) yields -1 _1 _ _ _1 T 922‘1 T 921L911 ' £112512251211 15112922] T 92211 T 5a] (5-10) Then, -T 21522? ”212” + K11522111+ 1531735; (5.11) Equation (5.11) indicates that the fburth term in the expression for szz depends on _922 and 522, and for ||_22H not small, §X 1+ 0 only if 9221+_ O. For the ZMIIW disturbance over all gen- -1 erators ||K H is not small. Hence S ‘+_Q only if + Q, A similar argument holds for §X for 941 +.0. 11 151 The condition g§;-+ Q. is the condition for strict synchroniz- ing coherency, i.e. tight interconnections among a group of generators. Thus a ZMIIW disturbance over all the generators of a system has the effect of identifying groups of generators that are tightly inter- connected because neither strict geometric coherency or strict strong linear decoupling can cause szz + Q_ for such a general ZMIIW dis- turbance. Only 9;; +.9_ will cause §X22 +-Q, Stated another way, a ZMIIW over all the generators of both_the study group and the specified group of n generators would identify the study group and the specified group as two tightly bound subsystems only if the norms of §K and ll S were both small or zero. -x22 Identifying the tightly interconnected groups of generators is a primary step in forming dynamic equivalents since generators that are very tightly interconnected tend to remain coherent in the face of very strong disturbances. filrther, tightly bound groups are more impervious to the location of the disturbance than groups formed by satisfying the other structural conditions fOr coherency. For in- stance, the specified group of n generators in the example system would have a much greater tendency to remain coherent in response to a disturbance within the specified group if the group coherency were due to synchronizing coherency, as opposed to geometric coherency or strong linear decoupling. Thus a ZMIIW disturbance of all generators in the system would be the ideal first step in determining dynamic equivalents. 152 If the first step in forming dynamic equivalents is to find the tightly bound group, the obvious second step is to test for the other structural conditions for coherency, namely geometric coherency and strong linear decoupling. How that step can be accomplished is the subject of the next section. V The Three Part Partition gf_the Example System Consider again the expression (5.5d) for Sx . The first three 22 terms of Sx depend on 95:92], the fourth term on 922' If a dis- turbance couTg be found which eliminated the fourth term in (5.5d) then that disturbance would detect structural conditions where 9;; +-Q_ is ngt_satisfied, but 92292l +-Q_ is. Since strict geometric co- herency implies ng = Q_ and strict strong linear decoupling results from 95:32] +19, a disturbance that eliminates the fOurth term in §X22 could thus be used to detect these two structural conditions. The requisite disturbance is a zero mean, independent, inertially weighted disturbance such that _ . 2 2 2 _ _ _ 3.” - d1ag{M1,M sooost309-O-90}9 322 " 9., m” ‘9.’ £12 ‘9 that is a ZMIIW disturbance over the generators of the studngroup. Such a disturbance gives K” the fOrm r- H -I-mxm 9'1an = (5.12) that is 542 = 52] = 522 = 0. This reduces szz to the fOrm §x 2122 2219 9.441.125; - (513) 153 BefOre showing how this disturbance over the study group can be used to detect strict geometric coherency and strict strong linear decoupling, first consider a partition of the example system into three parts as show below. ’7.“ 4' '1 rx ‘1 A-1 -"—11 A12 fl13—1 3‘2 T' A21 A22 A23 52 ' x n 11. n 4. _—3_J L_31 32 35st L..._J +[N1flL1 The inverse of the matrix h_= @_I_ can be written -1 _ F' “l [AT-1] ‘ E11 -‘—12 £13 I21 E22 123 f f f L_’—-31 —32 -—33J where, _ -1 111 ' !11 f =(1-'111 h'Th Wu H1 -21 —"21—21T—22—23—31-31 41 f =(-V' +hThV )v —31—31—31—33—32-21—21 —11 f = (-v“h +h'Th V'Th )v“ —21 —12—12 —11—13—32—32 —22 f = v'1 —22 -22 f =(-v +11"11 V'Th ) —32 —32 —T32 —33—31-12-—12 _ - -1 :13 “ ('113913T h11A12—23P—23)".:1}1 .. -1 ( 5343122921 V13—13) APM APL 154 i ll < —33 .33 v =111 -h VTTh +11 h'Th 111 11 "11 —11 ~11 —12—21—21 —12—22—23—31—31 —13—31—31 +h h'T VTTh ] —1 3—33-32 -21-21 v = [11 -11 11411 1 —12 -11 43—33—31 = [h - h h'Th 1 £13 —11 —12—22—21 =[11 -11 v'Th +11 h'Th "11 -11 ‘Th 1‘22 —22 —21-12—12 41—11—13—32—32 —23—32—32 -+ h h.1 V- h ] —23—33—31 —1 2—1 2 _ -1 £21 ' [£22 ' £23£33£321 _ -1 1’23 ' [£22 ' £21£11££121 v =m -11v"11 +11 '1 "11 +11 "11 —33 33 —31—13—13 —31-11-12—2323 -—32—22—21 _ -1 £31 ' Lh33 ' £32£22£231 v = [h - h h'Th 1 —32 —33 -31—11-13 21’ 111 3—13 -1 42123323] First consider a general ZMIIW disturbance over all the gen- erators of both the study group and the specified group. turbance, §*(m) can be written 3x11») 11. 11“ 511 11" r" H'r ' £11 £12 £13 £11 £21 £22 £23 £21 f f f K K :31 —32 —33 _1 C31 -3 ~42 £22 2 £13 —23 -33_J For this dis- (5.15) . “affirm... flnwr.v.rm..1it¥.m .. 155 where 2 i = j {K }. - , m = 1,2,3 “A” 1 1111' {l for all i,j, m,n = 1,2,3, m f n. {Kmn}ij Consider §X which measures the coherency between angles of 33 the vector 33. Carrying out the multiplicationir1(5.l3) yields _ 1 -S-x33 ‘ ‘£31£11 T £32£21 T ‘33£31]£31 T [£31K—12 T£32£22T £33 £321£32 (5'16) -1 1 £33 [£33‘ —31 £13 T £33£32£23 T £331£33 It is possible to make the first two terms vanish. Assume that the group of generators represented by 53 satisfy the conditions of strict geometric coherency. Then £31 :3] = :32 = 9, For the third term in (5.16) to be small £33 must be = 532 = 9, which implies small. The matrix £33 can be written _ -1 1 1 '11‘ -1 ‘33" [A33 " A31£13A13T A31A11A12—V23A23T A32A22A21£1A3—13 A32£23Azs] F'- l -1 v.1 “-1 ‘ A33 " [A31 A32] £13 A11A12—23 -1 -1 1 ’A22A21-V-13 £23 T23 _J L. -1 [A33 431329-31 :1 Applying the matrix identity (5.9) yields _-1-1-1 -1-e-,___h11-1 £33 ‘ A33 TA3391‘92 ' 9-3A339-1] Se—“3A33 A33 [AH—(111' 156 Thus the third term in _s_x depends on A313 and 5x + 0 only if '33 T' —'33 _' fl5;‘+ Q, It is obvious that the same analysis holds for any Sx , kk k = 1,2,3, using the other two groups as the "study group". This establishes fOr the three way partition of the model, the same result shown for the two way partition, namely that for a ZMIIW disturbance over all the generators of the system, the principal structural con- dition detected will be the identification of tightly interconnected groups of generators. This repetition of the analysis already done is somewhat redundant, but it provides a very casual inductive proof that the analysis can be extended to an n-way partition with the same result. Next, assume that the ZMIIW disturbance is applied to the gen- erators of group 1, and that the reference is taken over a machine or a group of machines in either group 2 or group 3 so that "T '1 lmxm A — £11: A 911x11 9— 1 1 (.9 g 9,2,."21 where m, n1, 112 are the number of generators in groups 1,2,3, respectively. Then the expression(5.l6) for 'SX reduces to 1' 33 s =fkf‘=2(f f‘) (517d) SImilar express1ons can be written for szz, §x23, and §X32 namely, . s = f k f‘ = 2(f f‘ ) (5 17a) —X22 ~21-11—21 —21-21 ' s =fkf‘=2(f f‘) (51711) -X23 -21—11-31 -21-31 ' _ T _ T Ax '£31£11£21 ' 2(£31£21) “-1791 32 157 Note that for this disturbance,S and §X32 provide the same in- 423 fOrmation as §X and §X . In expanded form _-1-1-1-1 1 §x33 ‘ ("£31A31TA33A32£21A211£11£111 £31A31 T A33A32£21A211 (5'18“ 3 = (-v'1h + h 11111 1h )v'1 T(-1I v1h + h '1hv h )T(518b) -x22 -21-21 —22—22 V31 -31 —11 £11 -21—21 —22—22 £31 —31 ° The expressions for 1:21 and f3] can be written as if ‘1 ‘— v'1-'(h1h v1) r‘h ‘1 —21 -21 —22 —23 —31 -21 = 1 1 1 (5.19) -_-(11_ .11. v ) I n L. 31—1 L. 33 32 -21 31 _J L. 314 Note that the left matrix on the right hand side of equation (5.19) is the inverse of the matrix F A11 A12 (5.20) A32 A33 L. _J In the present analysis the disturbances are confined to group 1. That is group 1 can be identified as the "study group", and groups 2 and 3 are a two-way partition of the "specified group of n generators". Equation (5.19) is then very important because it establishes that jnga_ZMIIW disturbance over the study group_ §x(g0 provides exactly the same information as E _T_. The conceptual power of (5.19) is shown by some examples. Consider first that a ZiIIW dis- turbance of all the generators has shown groups 2 to be a tightly interconnected group. Equation (5.19) indicates that if the generators of groups 2 and 3 are, together, satisfying the conditions of 158 geometric coherency or strong linear decoupling, then a ZMIIW dis- turbance over the generators of the study group will show this, since the matrix (5.20) is the 922 and the matrix [h2] 53]]T the 92] of the analysis done for the two part partition. If the primary in- terest were in disturbances within the study group (group 1), then groups 2 and 3 could be aggregated into a single equivalent generator. Now suppose that the ZMIIW disturbance over all generators does not indicate that either groups 2 or 3 are separate, tightly interconnected groups. The ZMIIW disturbance over the generators of the study group will still indicate whether groups two or three are together, or individually, exhibiting strict geometric coherency (SGC) or strict strong linear decoupling (SSLD). If all generators of the two groups are exhibiting either SGC or SSLD, then equation (5.19) shows directly that these coherency conditions are satisfied. If either of the two groups is exhibiting SGC or SSLD by it- self, this will also be detected. Consider the expression for §X .. 33 The matrix V3} can be written in the fOrm v" = [h'1 + h'1 h - h h 1 h 1h h 1 —31 -33 -—33 —22 -23 —33 —32 —23 —33 This allows Sx . to be written as 33 - -l -l -1 -1h -T -l -1 §x33 '1‘A33T5c1A33 A31T A33 A32V -21— A211£11 £111‘A33T K -c1-A—33 A31 -1 -l T TA33 A32 £21 £1211 Thus any one of the conditions -1 (l) h33 + 0 (strict synchronizing coherency) (2) [A31 E A3] = [Q_i 0] (strict geometric coherency) (3) h3;[h31 E h32] + Q_ (strict strong linear decoupling) will cause §X33 = Q_ or §X 4-9, In this case, h33 1s 1dent1f1ed with 922 and, [A31 h32] with 92], so that groups 1 and 2 are the "study system" and group 3 "the specified group of n generators." One might very well ask how the last result can be obtained when the ZWIIW disturbance is over group one, but both groups one and two together are being identified as the study group. Suppose that the ZAIIW disturbance is done over both groups one and two. §_X 33 will be modified by the addition of the term -1 -1 -1 -1 -1 E’£32 A32 A -1 —311£12 A12 V22 V22"£32 A32 V -1 T A -12 A121 -1 TA 43%1 -33 This term can be rewritten in the form -1-1 -1 [[h33 + K ]h h +h -1-1-T-1-1 h -33 —32 —33 £12 A121£22 £2211A33 T £ a1A33 —32 A31 -1 -1 —33 A31 £12 T + h h -121 This additional term is also zero under precisely the same conditions just analyzed. Hence the 2411W disturbance of group 1 is sufficient to identify group 3 as coherent. Intuitively this seems to say that 2411W disturbance of group 1 disturbs group 2 strongly enough to detect if either the conditions for BC or SLD are satisfied between groups 2 and 3. An example of strong linear decoupling was considered in Chapter 4; another is considered here. Figure 5.1a shows a four gen- erator system. Let generator 1 be the study group, generators 2 and _‘ilnl 160 (11) FIGURE 5-l FOUR GENERATOR SYSTEM (O’BEFORE “(NAPIER AGGREGATIW 07 GEAERATGRS ZMDS 161 3 be group 2 and generator 4 group 3. Let the reference be generator 2. Then A [T ‘1 IT M T = £—- - 1—- t t t ‘1 -- — M1 M2 11 13 14 1__ _ 1__ ‘12 ‘23 ‘24 M M 3 2 ‘13 ‘33 ‘34 l. 1__ L 111 1.12 _J L‘14 ‘34 ‘44J r- . '5 . (3'11- ‘14)1(i1_3_£2_3 £153-21 M] 142 E M1 M2 E M] 142 ------------ r---—------OO-‘------------ I I t t : t t 1 t t 13 12 . 33 23 . 34 24 (r- 1.( -———)1(——-—-—) 3 'M2—' 3 TM3—‘ M2 1 M3 M2 ------------ E-------------‘:------------ (£13.- £121 1 (t§fl.- £231 1 (£35.- E25) (_M4 2 1 A4 M2 1 M4 M2 .4 t + t t _ 1 12 13 14 Let t - —- and *-——— 23 u M2 + M3 M4 Then M3 M2 M3 3 M3 M2 = - l (M2 + M3) _ —t-J._3_. .. —..2..4_ = h u ‘M§M§"' M M3 22 1*- m - 2.4.. = h M3 M2 23 3.4. - 3.2—3. .. - l 1 + t34 = h M4 M2 u M2 M4 32 162 jig—4:. _ 2.4. = h M4 M2 33 Next write Fh h 1‘1 "T133 - 1.123." 22 23 T3“' d 11 1. - .22. [‘22. k-32 33.1 L. d d .4 M2 T M3 A23 ' where d = h h - h h , and limit pd = (_ h + ___9 = d 22 33 32 23 “+0 M2143 33 M2 Then h uh limit —%§-= 1imit 33 = 0 “*0 111->0 -h -uh 1111111 -—§-3- = limit (123 = 0 U20 n+0 1 '11 -1Jh 14— limit -—%g-= limit d£2 = 3"T‘1 11*0 n+0 l1mit %A = limit £3123 = M2 d. M3 = :3 utO n+0 The condition (t12 + t13)/(M2 + M3) = ‘14/M4 can be used to show that M3 M2 M3 M4 M2 M3 Combining the above results gives r" «_1 w- -s r- “1 r—MZ + M3 ‘1 r- fl h22 h23 hzl o o ‘ ( )c o = M + M h21 h33 "23 1 _( 2 M 3) 3 (5.21) 3 _ . B- ..J h- ..4 K C 0 h. K .J E. .J C..J 163 t where C = (-l£---—l§). 4 Figure 5.lb gives a physical interpretation of this result. Letting p + 0 causes t23 + w, making the line between generators 2 and 3 infinitely stiff. This corresponds to aggregating generators 2 and 3 into one generator, called 2-3 in Figure 5.lb,whose inertia is M23 = M2 + M3. The synchronizing torque coefficient becomes t],2_3 = t12 + t13. Hence,in the limit as u + 0 generator 2-3 and generator 4 satisfy the conditions for strict geometric coherency with respect to generator 1. The four generator model of Figure 5.1 is an example of the case where group 2 satisfies strict synchronizing coherency, while group 3 and the single generator equivalent for group 2 satisfy strict geometric coherency with group 1. In the example group 3 contains only one generator, but the generalization is clear. It is an instructive exercise to consider a five generator model where generator 1 is the study group, generators 2 and 3, group 2 and generators 4 and 5, group 3, where generator 2 is the reference and the power system structure satisfies, t14/M4 = t15/M5 = t12 + t13/M2 + M3. The general form of equation (5.2l) can be achieved without doing all the algebra, which involves two 2 x 2 matrix inversions. Acompletely parallel line of analysis can be developed for the conditions of pseudo-coherency and weak linear decoupling of the types (21 D1201 I) + 0 and (I1 D1201 122 + 9.- This -T 22521 analysis is not presented here for two reasons. First, it is com- pletely analogous to what has already been done. Second, the fbrmal algorithm for producing dynamic equivalents which is presented in the 164 next section does not test for these coherency conditions. The rationale for omitting these conditions goes as follows. It was shown in Chapters 3 and 4 that the pseudo-coherency condition could only be exactly satisfied in the linear model and that the transference of pseudo-coherent behavior observed in the linear model to the nonlinear model could not be strongly guaranteed. The weak linear decoupling conditions, like pseudo-coherency depend on the submatrix (31)”. They are therefore, categorized with pseudo- coherency and not detected. 0n the other hand, all the coherency con- ditions that will be tested for in the formal algorithm are strong conditions in the sense that satisfaction of any of the conditions in the linear model strongly guarantees its satisfaction in the non- linear model. The detection of coherency conditions presented in terms of the three-way partitionm4 PFm cod mmapm>cmmwm Empmxw mo magmaxymcwmmeH.Hm mausuwcmms mcouwcmcmmvcm» __m we mucmngzpmwo 3HH2N go$ mama mspo>cmmwm muwunnmnmuvnm:~up wuuumumu¢amuwup wnnumucumnmup uncle wumuwup uncle mum mup mum-e mn— num mup mum po¢.¢ mcoz corpmmmgau< .~.m mpnmh 179 pairs of the form A1,A: = - %-:_jbi. For the linearized model of the New England System used in the present data collection 0 was set at .275. Since - g- is the same for all eigenvalues and very small, all that is required to characterize the eigenvalue pair Ai,A: is bi’ the magnitude of the imaginary part. Therefore, fOr expository convenience, the imaginary part of the eigenvalue may occasionally be loosely referred to as an eigenvalue. At each level of aggregation in Table 6.2 the system model has one less generator and one less pai§_of eigenvalues, until finally at level 8 the system model consists of two generators, the aggregate of generators 1 through 9, and the reference, generator 10. The eigenvalue information shown for each level of aggregation in Table 6.2 is calculated from EH14, the fl_l_ matrix for the level i aggregation. An explanation is given here of how Ehlj is obtained from the matrices fl_ and I_ of the Full New England System. Table 6.3 gives the matrix -1 for the New England system, with all load buses and all generator terminal buses eliminated. Only the internal generator buses remain. The matrix 8_ for the New England System, with generator 10 the reference is 180 nmo.omn mew.mu No_.p wvm. moo. mum. mum. «we. mum. ecm.~ mmn.m mop.” «No.0Pu mmo. 5mm. N—m. mom. won. wmm. woo.m FOP.~ ¢¢m. mum. wow.o~- mmw.m Nvo. «mm.~ Npm. wmm. “mo.— vmm.m moo. nmm. mmw.m mnm.m~u mmm. mvo.~ mvo.p mmn. mom.~ soumzw ucmpmcu 3oz «No. Fpm.~ mum. mum. Npm. mmu. New. vwm.P wwm. mvo.N mmm.ou m~¢.~ mpv.~ omo.-: woe. Pro.p mam. «mm. ~mm. m~m.~ $3 «we. 3“. «a. as; we. :0; 5.2- 82 m8.— msm mm as» to» P- —¢m.m mum. mmm. wmm. mmu. mmm. Nmu. mmo.m pmm.mu w~o.~ xtfi. .2. 9: evo.m evm.v moo.m mumo.~ mmm.p pmm. u~m.~ wem.p who.— mmp.m~u .m.o wpnwh 181 r L M1 L M2 L M3 L ”4 L ”5 L M6 L M7 L M8 L L ”9 where, M1 = .2228 M6 M2 = .1607 M7 M3 = .1899 M8 M4 = .1517 M9 M5 = .1379 M10 Then -fi31 matrices. I 3 ...: O 3“ ._l -—l O O O O O O 10 .1846 .1400 .1289 .1830 2.6525 is, of course, just the multiplication of these two To find the level 1 aggregation where generators 6 and 7 are combined into a single generator, first form ~11 by 1) adding column 7 of I to column 6, and then 2) adding row 7 of this intermediate matrix to row 6. The result is {1, shown in Table 6.4. To form 95 182 me.F www.mu NoP.F mom.~ mum. mum. «we. mum. vvm.— mmm.m No_.p vmo.o~u mnm.~ Npm. own. «om. wmm. woo.m mmv.m mom.~ mum.p wmu.o~u ~<.F mmm.m mmm.p nvm.p ~N¢.N vmo. mum. Npm. n¢.P mmm.o1 NF¢.N wow. mow. Pmm. Fpo.~ mum. mom. mmo.m NF¢.~ mmo.~Pn _Fo.P Nmn. upm.p mmm.w www. com. mmm.p woe. Fpo.p me.opn wmo.m wvm.F Few.m mum. wmm. mem.~ mom. mmn. wmo.m www.ms w~o.~ Emumxm ucmpmcm zmz mam mm mg» *0 :owumcmmgmm< N-o any to» FF. xwgpmz ms» vqo.m ¢¢m.~ moo.m NN¢.~ me. Rpm.~ mwm.~ m~o.~ mmp.mpu .¢.m mrnoh 183 add the inertias of generators 6 and 7 to get the inertia of the equi- valent machine. Put the inertia of the equivalent machine in place of the inertia for generator 6 in the matrix E_ and eliminate the row of E that contains the inertia for generator 7. The result is r~ '1 3|--‘ I l-J ._a 3 3|-‘ 21d [3 —J I 3"“ 3" b 3" —l 3“ 3" 3"" 3"“ ...; ...: ...: I ol ol ol 3|—‘ 3|—' c—J 9 MTOJ The matrix -fiqiq is then just the multiplication of the matrix 131:] above by the matrix -_T_1 in Table 6.4. Note that the 1 and 1] matrices have one less column that row. The 1. matrix is actually square, but the last column is always deleted because it contains redundant infbrmation (found in the last row) about the reference machine. Returning now to Table 6.2, the arrangement of the eigenvalue data shows clearly why the aggregation scheme used by Podmore is often called an "averaged" equivalent. Each entry in row i of the table 184 can berthought of as a weighted average of the entries directly above it in row i-l. Table 6.2 can be interpreted as meaning that at each level of aggregation a pair of eigenvalues has to be eliminated. The new eigenvalues are a "best fit weighting" of the information available fronithe level above. A careful look at Table 6.2 indicates that the weighting can heavily favor particular pairs of eigenvalues. Fbr instance, the eigen- values in level 1 of the table are all very close to the eigenvalues above to the left in level 0. That is, the eigenvalue pair with imaginary part 9.984 is essentially discarded at the first 1evel«of aggregation. Note that this is the highest frequency pair. This means that the eigenvalue pair - %-:_9.984 can be closely identified ‘with the intermachine oscillations between generators 6 and 7. At Other levels, as in going from aggregation level 4 to 5 the weighting is such that no single eigenvalue absolute value is discarded. That is, there is no single eigenvalue pair that can be associated, in this case, with the oscillation between group 1-8 and group 2-3. Note that at level 1, 9.984 is discarded, at level 2, 9.811 is discarded. At level 3 all the eigenvalues are retained except 8.288 and 9.542 which are averaged into 8.373. At level 4, the eigenvalue 8.314 is discarded and at level 5, 8.372. The general pattern is to always discard the eigenvalue pair of highest frequency. Since the ZMIIW disturbance of all ten generators is designed to detect tightly interconnected groups, this pattern makes sense, since what should be discarded are the high fre- quency oscillations within the groups. 185 It can be expected, then, that when an eigenvalue absolute value is essentially discarded in going from level i-l to level i that the resulting model at level i will be almost as good as the model at level i-l. If this is not the case one might expect a noticeable degradation in the model's ability to reproduce the actual system response. Specifically, for the information about the New England system contained in Table 6.2, one would expect the model to be very good through aggregation level 4, i.e. the 1-8 2-3 4-6-7 grouping, but to deteriorate at about level 5. These predictions are generally born out by the data presented in subsequent sections, but befbre examining those results, it is shown that the eigenvalue infbrma- tion of Table 6.3 is contained in the coherency measure matrix §( (co). Table 6.5 shows the r.m.s. coherency measures between the reference, generator 10, and the individual machines at each level of aggregation. (Senerator 10 is a very "large" generator in the sense that its inertia is approximately ten times that of any other gen- erator in the system. It is also the last generator to be aggregated so that r.m.s. coherency measures are available between it and all other equivalent generators at each level of aggregation. A few words of explanation of how to read Table 6.5 is perhaps in order. The top row of the table contains the coherency measures between generator 10 and the other nine generators of the full New England System. That is, the number directly under (1,10) is the coherency measure between generators l and 10, etc. At the first level of aggregation the remaining machines, in order, are l,2,3,4,5,6-7,8,9,10. The entries in row two, from left 186 NMQ'LOKDNQ Fm>m4 Pemoo. m-w-u-a-m-e-m-m-_ ahmeo. emmmo. m-h-o-m-e-m-m-_ emeeo. Beams. “mama. m-h-m-e-m-~-_ cameo. woeeo. meoao. wommo. “-mhe m-m-~-P mmwoo. mamas. mmomo. NNNmo. Reese. ~-a-e m-~ m-” meao. mmmao. eemmo. mm_mo. mmmeo. eomeo. N.a-e m-P memoo. magma. mamoo. mammo. ewomo. o_meo. ameeo. h-o m-p mmmeo. mmoeo. mmemo. mmmao. «ammo. Feomo. eomeo. _Pmeo. “-0 semao.. mmeeo. mmomo. w_smo. ommao. seams. Fmomo. wwmeo. Fameo. aeoz Sta 813313 873.973 8:: 812 813.31: cowummmemm<.wm m~m>m4 _Pm.mm Lopmcmcmw Emumxm comm ccm.mm gopmgmcmw :mmzumm mmgsmmmz mucmgwsou cowummmgmm< meopaemcmo ewe __< co aoeaaesemwo ZHHEN toe aeaa aucataeou .m.o apaae 187 to right, correspond to the coherency measures between generator 10 and each of the generators, in the order given. That is, the first entry on the left is the coherency measure between generator 1 and 10, the second that between 2 and 10, the sixth entry that between the aggregated generator 6-7 and 10, the seventh that between generators 8 and 10, and so on. As another example at level 5 the generators are the aggregate 1-2-3-8, the aggregate 4-6-7, and generators 5 and 9. The four entries in this row, from left to right, correspond to the coherency measures between generators 10 and generators 1-2-3-8, 4-6-7, 5 and 9, in that order. The rule is to aggregate to the lowest machine in the group and to list the coherency measures, left to right, from the lowest numbered machine to the highest. For the ZMIIW disturbance of all 10 generators, the coherency measure infbrmation in Table 6.5 is essentially the same as the eigen- value infbrmation in Table 6.2. For instance, the coherency measures in the level 1 aggregation are all very close to the corresponding values in level 0, with the exception of generator 6-7 which has a larger coherency measure, with generator 10, than that fOr either machine 6 or machine 7 at level 0. The change in the coherency measure is an order of magnitude greater fbr 6-7 than it is fOr any other machine at that level. At lower levels there is an averaging of coherency measures just as there was an averaging of eigenvalues in Table 6.2. It is more difficult to detect because the entries in Table 6.5 are not rank ordered as they were in Table 6.2. The general interpretation of the shifts in coherency measures is the f01lowing. If two generators with nearly the same coherency 188 measure are aggregated the coherency measure for the aggregate is gen- erally larger, but the behavior of the aggregate should follow that of either of the generators aggregated. If the coherency measures of the two generators being aggregated are quite different the coherency measure of the aggregate may be close to one of the two initial coherency measures or it may be somewhere in between. If the coherency measure of the aggregate is close to one of the two initial coherency measures, the aggregate's behavior will probably be close to that of the generator whose coherency measure it favors. If the aggregate's coherency measure is somewhere in between, its behavior may follow that of neither of the generators being aggregated. This is, generally, the same line of reasoning followed in discussing eigenvalue averaging in Table 6.2. Since each row i of Table 6.5 is generated by forming Ihili’ finding its inverse and then generating §xi(«0 = (8415)-]5K8413I-T. the information in this table is, computationally, no easier to obtain than the eigenvalue information in Table 6.2. All that has been done is verify computationally, what has been shown theoretically in Chapters 3 through 5 of this present work and in references [5-8], namely that the coherency information available from the §((w) matrix is the same information available from the calculation of eigen- values. If coherency measure information is to be useful in assessing the loss of system model accuracy at each level of aggregation, that infbrmation must come from the initial coherency calculation in the LINSII program, modified to use the r.m.s. coherency measure. This 189 computer program generates the matrix (ET)-1 of the unaggregated system directly, i.e. without an inversion. The information available from (E T)'1 is the ranking Table 6.1. Note that to the right of this table is the point at which each level of aggregation occurs. Since the numbers in the table are all of roughly the same order of magnitude there is no dramatic jump in the threshold level of the coherency measure required to reach an additional level of aggrega- tion. Intuitively Table 6.1 seems to indicate that the decay in the quality of the system model will be fairly uniform, rather than taking dramatic jumps. Figure 6.2a shows a plot of the magnitude of the coherency measure at each level of aggregation. Note the plateau at levels 4 and 5. This contradicts only slightly the qualitative assessment made above, and seems to indicate that there might be a drop in model validity around aggregation level 4, 5, or 6. Figure 6.2b is a similar plot for the ZMIIW disturbance of generators l, 8, 9 and 10 discussed in the next section. It's plateau is not as distinct. These results indicate that it mgy_be possible to detect the aggregation cut-off level directly from the ranking table. III. The ZMIIW Disturbance gj_the Internal System The internal system of the New England System consists of gen- erators l, 8, 9 and 10 [2]. Table 6.6 is the ranking table fbr a ZMIIW disturbance of these four generators. The coherency measures thresholds for the first two levels of aggregation are both very small in absolute value and of the same order of magnitude. One would expect the system models at these two levels 190 7.0 8 \ / THRESHOLD 8 . \ 20 1.0 00 I 2 3 4 5 6 7 g AGGREGATION) LEVEL (0 4.0 3.5 / 3.0 9. \ \ 71185311013 6 \ 51 \ / 0.5 / AGGREGATION LEVEL (11) FIGURE 6'2 RELATIVE MAGNITUDE 0F COI'EREMIY NEASLRE \S. AGGREGATICN LEVEL (0)2MIIW'G ALL GEAERN'ORS (b) ZMIIW OF GEhERATWS l,8,9,lO 191 Table 6.6. Ranking of Coherency Measures for ZMIIN Disturbance of Generators 1, 8, 9, 10 Ranking Generator Coherency Aggregation Level Pair Measure 1. C(6,7) .0253 1 2. C(4,7) .0607 3. C(4,6) .0861 2 4. C(2,3) .3069 3 5. C(5,6) 1.1886 6. C(5,7) 1.2140 7. C(3,4) 1.2446 8. C(4,5) 1.2748 4 9. C(3,7) 1.3023 10. C(3,6) 1.3264 11. C(2,4) 1.5513 12. C(2,7) 1.6090 13. C(2,6) 1.6332 14. C(3,8) 2.3847 15. C(2,8) 2.4088 5 16. C(1,2)‘ 2.4794 17. C(3,5) 2.4855 18. C(l,3) 2.5064 19. C(4,8) 2.6775 20. C(7,8) 2.7145 21. C(6,8) 2.7302 22. C(2,5) 2.7888 23. C(1,8) 2.8385 6 24. C(1,4) 2.9826 25. C(l,7) 3.0262 26. C(l,6) 3.0446 27. C(5,8) 3.5920 28. C(l,5) 3.9924 7 29. C(6,9) 4.8028 30. C(7,9) 4.8045 31. C(4,9) 4.8090 32. C(5,9) 4.8748 33. C(3,9) 5.3451 34. C(8,9) 5.5017 35. C(2,9) 5.5044 36. C(l,9) 5.8884 8 37. C(2,10) 8.2068 38. C(l,10) 8.3528 39. C(3,10) 8.5122 40. C(8,lO) 8.7396 41. C(4,10) 9.7495 42. CE7,10) 9.8086 43. C 6.10) 9.8333 44. C(5,10) 10.9923 45. C(9,10) 11.8382 192 to be very near that of the full New England System. The coherency measure threshold used to reach level 3 is 3.56 times that at level 2, but still rather small in magnitude, so that the model at level 3 could still be expected to be quite good. The coherency measure threshold fbr the level 4 aggregation is 4.15 times that fbr level 3 and 14.8 times that for level 2. In addition, its relative size is the same as that fbr the coherency measures of the general ZMIIW disturbance of all ten generators. Therefore, a significant decay in the quality of the system model might be expected to begin about level 4. The plot in Figure 6.2b of the coherency thresholds for the levels of aggrega- tion neither confirms not denies this supposition. Since the coherency measure threshold to reach levels 5,6,7 and 8 are all of the same order of magnitude as the threshold for level 4, the system model might be expected to decay rather slowly and uniformly from level 4 onwards. Essentially the same infbrmation can be de- rived from either the eigenvalue information of Table 6.7 or the co- herency infbrmation of Table 6.8. Table 6.7 shows that the imaginary part 9.984 is essentially discarded at level 1, and 9.543 at level 2, and 8.314 at level 3. Note, however, that the imaginary part 9.811 persists to level 4 and is finally averaged into 9.054 at level 5. This is a somewhat different pattern of eigenvalue elimination than was fbund préVIOUSIY where it ‘was generally the highest frequency eigenvalues that were eliminated. first. In this case the modes being eliminated are those not excited by the partial disturbance, which will not necessarily be the high frequency, intenmachine oscillations of tightly interconnected groups. 193 NMQ'LOLD Pm>m4 «mm.m nom.n m¢¢.m oom.v nnmum1¢ mumnmnp emo.m gem.“ sea.m mae.e . e-m-m-a m-m-~ Pam.m ¢m~.m “mm.“ «mm.m ~m¢.e ~1o1m1e mum —Pw.m Nmm.m mum.n mnm.m vom.o mm¢.¢ muonv muw PFm.m mum.m «Pm.w mum.m mmm.m vom.o ooa.¢ “none ppm.a mcm.m mpm.m mmm.w who.n Nmm.m mmm.o vov.e mum ppw.m me.m mam.m nw~.m eno.n Pmm.m om~.m poc.¢ mcoz 111. 11. cowuflMWme<.Hmmemmwm .11 . xwm cow mmapm>cwmwm Ewpmxw Lo magma xgmcwmmefi mo menswecmms corpummcmm< o_ .m .m .F mtoumemcmw mo mucmngaumpc 3~H2N Lam mums m:_m>cmmmm .N.m open» 194 LC NMQ'LD Pm>m4 Rocco. omemo. momma. mnmumue mnmumup Nmmoo. unmoo. vmeo. mauve. nuoumnv mumum ommoo. mmneo. mmmmo. comma. anvo. numumnw mnm Numoo. mpuwo. mmooo. «come. opmmo. mevo. num1v mum mmmoo. Foneo. mecca. ommmo. uppmo. cameo. mmoeo. uncle mmmmo. mmoco. wmnmo. mmmmo. mmomo. pnomo. womvo. FPoqo. mum epmmo. «move. mmomo. m_¢mo. ommoo. opomo. Pmomo. wwmwo. pmmeo. mcoz 3:3 3:8 8:3 8:3 8:3 8:3 873 8:3 8:: comummugmm<.wm m~w>m4 ka.mm Loumcmcmm Emumxm comm ucm op Lepmgmcww :mmzpmm mmcammms Aucmgmcou cowummmgmm< op .m .m .p mcopmcmcmm we mucmngaumpo 3HHrN com mama zucmsmsou .m.m mpnmh 195 At level 5 the imaginary parts 7.546 and 9.054 are more clearly averages of 7.287, 8.254, and 9.811. At level 6, 9.054 is essentially discarded, indicating that it is representative of the oscillations between generator 1 and the aggregate 2-3-8. Table 6.8, the coherency measure data for the ZMIIW disturbance of generators l,8,9,lO yields the same infbrmation as Table 6.7. The coherency measures at level 1 are slightly larger, but essentially the same as the coherency measures for the corresponding machines at level 0. The biggest increase, as might be expected is the coherency measure of the aggregate generator 6-7 that replaces generators 6 and 7 of level 0. The same analysis can be made of levels 2 and 3. At level 4, the coherency measure for the aggregate 4-5-6-7 is very clearly an average of the coherency measures for generator 5 and the aggregate 4-6—7 at level 4. This could mean that the behavior of the aggregate generator 4-5-6-7 will portray the average response of all f0ur generators, rather than favoring either generator 5 or the aggregate 4-6-7. By contrast at level 5 the coherency measure for the aggregate generator 2-3-8 is significantly closer to that of the aggregate 2-3 at level 4 than to the coherency measure of generator 8 at level 4. Thus the behavior of the aggregate 2-3-8 may follow the behavior of generator 2 or generator 3 more closely than that of generator 8. The same analysis applies to generator 1 when it is aggregated at level 6 Figures 6.3a through 6.3f are simulations comparing the response of the full 39 Bus New England System to the system models at the first 6 levels of aggregation dictated by the ZMIIW disturbance over generators 196 Figures 6.3. Simulations of System Response to One Per Unit Step Disturbances on Generators 8 and 9. S S {3 designates the full 39 Bus New England System <> <> <> designates the aggregated model dictated by the ZMIIW disturbance of generators l, 8, 9, 10 Figure 6.3 Aggregation Level System Generators (a) l l 2 3 4 5 6-7 8 9 l0 (b) 2 l 2 3 4-6-7 5 8 9 10 (c) 3 l 2-3 4-6-7 5 8 9 10 (d) 4 1 2-3 4-5-6-7 8 9 l0 (e) 5 l 2-3-8 4-5-6-7 9 l0 (f) 6 l-2-3-8 4-5-6-7 9 lO Generator 10 is the reference 197 :0 > 828%. H at: <> w _ x < < < mozoumm Zn Utuh New- ”mg. 10' T .8 in flow .*~. 1 -um 88*. -.N- .8. no fiww 1 in. flow flaw- fimm- ”mg- 1m“ 1mm 1 no. 9 'N30 “BOP/GO 310MB 1 'N30 (OBDJ‘ABO 310MB 6 'N30 (030l'A30 310MB 4 O O N no 2 on U I: H .1 ._ m l p b ' P PITT—m ONQOONO 8 'N30 (OBOJ'ABO 310NB 198 TIRE IN SECONDS W o v o o v: o v N I c}: v 0 ’N30 (030l’A30 310MB sEcouoa I 'N30 (0301‘A30 310MB (b) @0006? c- 199 :0 .... ...... §< <> v>< wgzfi us: >. n - I'm: um" I 1 (D l V? N 9 ‘N30 (OBUl'ABO 310MB l ‘NBO (0303'A30 310MB 6 'N30 (030l‘n30 310MB 200 ' m \ / 3e. I ' \ ‘ Q .. x I l- ! ‘2 3I - ‘1, ’ l' .’ .l‘ , __ l- . 3/ - I l I. “‘ X m w ‘3 \ \ ‘3 .‘ A \ ‘ f. ' U D l l "4 ' 03 no u A; o a Z 2 O O D U U U U U U . a I. 0! OJ , , z 2 Oil 0-. M II) t t N H O- _ .- - +- M l n I- l l- r- r- . _ L H—Ifi—Ifi—I F‘I'T'T'W—l Nooovwo GPTDVN I ”N00” *0“ 7 VONVDOOOOV fiITMN-i 'TI 8 'N30 (OBOI’ASO 310MB 0 'N30 (O3OJ‘A30 310915 I ‘N30 (0301130 310?“! (d) T °N30 (030l'A30 310NU A30 310MB 0 ‘N30 [0301' 6 'N30 (030)‘A30 310MB 202 ‘. V. 2 CV mozouww zu u 2 fivNI flow: 1 NMI tout 10— van int 'N30 (030J’A30 310MB 1 8 'N30 (030)'A30 310MB 203 l,8,9,lO. The disturbance used in the simulation is one p.u. steps on bgth_generators 8 and 9. In all the simulation curves of this chapter the curve designated by' Cl is the New England System and the curve designated by O the aggregated model. These simulations verify the conclusions reached by analyzing the data of Tables 6.6 through 6.8. The system model starts to deviate from that of the full system at level 4, although the model at level 4 is still very good for 3 seconds. At level 5, the response quality has decayed badly on generator 8 and is not much better on generator l. By analyzing Table 6.8 it was predicted that the response of the aggregate generator would fOllow that of generator 2 or 3 more than that of generator 8. Figure 6.3e seems to verify this. The same result occurs, as hypothesized, fbr the aggregate l-2-3-8 in Figure 6.3f. Note that the main degradation of the model occurs when a generator is aggregated into a group with which it is not very coherent. Note, in contrast, that the response of gen- erator 9, is quite accurate through six levels of aggregation. Tables 6.9, 6.l0, 6.ll and Figures 6.4a, and 6.5a through 6.5f provide the standard data for a ZFHIW disturbance of generators 8 and 9. The same sort of analysis performed above can be performed here, with about the same result. Table 6.9 shows that the progression in the size of the coherency measures is almost linear through the middle aggregation levels. It takes a dramatic jump between levels 6 and 8 as shown by the plot of coherency measure threshold magnitude versus aggregation level shown in Figure 6.4a. This curve has a somewhat different shape than the cor- responding curves for the previous two disturbances. 204 Table 6.9. Ranking of Coherency Measures for ZMIIW Disturbance , of Generators 8 and 9 Ranking Generator Coherency Aggregation Level Pair Measure 1. C(6,7) .0021 l 2. C(4,7) .0050 3. C(4,6) .0072 2 4. C(2,3) .0500 3 5. C(5,6) .0996 6. C(5,7) .1017 7. C(4,5) .1068 4 8. C(3,5) .1346 9. C(l,4) .1630 10. C(l,7) .1655 11. C(l,6) .1666 12. C(2,5) .1833 13. C(3,6) .2299 14. C(3,7) .2320 15. C(l,5) .2338 5 l6. C(3,4) .2369 17. C(2,6) .2797 18. C(2,7) .2818 19. C(2.4) .2868 20. C(l,3) .3229 21. C(l,2) .3693 6 22. C(2,10) .6290 23. C(3,10) .6789 24. C(5,10) .8079 25. C(6,10) .9068 26. C(7,10) .9090 27. C(4,10) .9140 28. C(l,10) .9878 7 29. C(1,8) 2.0472 30. C(4,8) 2.2072 31. C(7,8) 2.2104 32. C(6,8) 2.2118 33. C(5,8) 2.2764 34. C(3,8) 2.3241 35. C(2,8) 2.3549 36. C(8,10) 2.8131 8 37. C(4,9) 4.7526 38. C(7,9) 4.7569 39. C(6,9) 4.7587 40. C(l,9) 4.7694 41. C(5,9) 4.8421 42. C(3,9) 4.9745 43. C(2,9) 5.0197 44. C(8,9) 5.0326 45. C(9,lO) 5.5806 205 (“03¢ka .szww wmmd mmcé mmmé nnmnmnvumumnp emm.m sev.u mneam cum.e mum nuoumuvup Ppm.m emm.m mm~.n «mm.o mm¢.v unoumle mum Ppm.m Num.m. mum.n mum.o eom.m noe.e n-m-e m-~ pr.m mum.w «Pm.m mum.m mnm.o wom.o one.e numuv Fpm.m Nem.m mpm.m mmm.w m~o.n mmm.m ~m~.m eme.¢ mum ewm.m Ppm.m Fem.m m—m.m ~m~.m one.“ pmm.o mm~.o po¢.¢ mcoz 3% Ho. flan. ,xIfl. Mm. 839,563 532m Hm 3.8“. 32.82; Hm 3.83.; as a; a new m mcouacwcmo mo oucmncsumvo 3HH3N so» mum: mapm>cmmwm .oF.m mpnmh 206 NMQ’LDK) Fm>m4 omemo. Pmpmo. mmpmo. muoumuwumumnp mmmoo. cameo. ommmo. mmmmo. mum muonmueup ommmo. mnmvo. mmmoo. commo. Pmneo. sumnmnv mum Nummo. mpneo. mmmoo. vmmmo. opumo. Fuoco. sum-w mum momma. pomvo. mummo. ommmo. np—mo. weave. “move. mimic mmmoo. mmovo. mummo. mmmoo. moom.o Pnomo. eomwo. Fpmeo._ mum epwmo. mmmeo. mmomo. mpemo. ommoo. mpmmo. Fmomo. wmwco. Pameo. mcoz 3:3 8:8 8:u 8:3 8:3 8:: 8:3 ASS 8:: compmmwcmm<.mm.mpm>m4 xwm mm coumcmcmw Emumaw :umu ucm op copmcmcmw cmwzuwa,mmc:mmmr augmemzoo coppmmmcmm< m new w mgopmcmcmm mo mucmngzumwo 3HH2N so» mama mgzmmms zucmgosoo .Fp.m m—amh 207 2.00 #_ I 75 I50 125 / LOO 07 5 050 / 0.25 THRESHOLD 0 I 2 3 4 5 6 7 8 AGGREGAT':0)N LEVEL 0 0.8 0.7 0.6 0.5 / g / E 03 .2 / OJ 0 I 2 3 4 5 6 7 8 AGGREGATKN LEVEL (6) FIGURE 6-4 RMIVE MAGNI'I'UDEG CO'ERENCY MEASIREVS. AGGREGKI’UII LEVEL (OIZMIIW G" GENERATORS 8,9 (”BMW 01" m 8 208 Figures 6.5. Simulations of System Response to One Per Lhit Step Disturbances on Generators 8 and 9 {3 {3 {3 designates the full 39 Bus New England System <> <> <> designates the aggregated model dictated by the ZMIIW disturbance of generators 8 and 9 Figure 6.5 Aggregation Level System Generators (a) l l 2 3 4 5 6-7 8 9 10 (b) 2 1 2 3 4-6—7 5 8 9 10 (c) 3 1 2-3 4-6-7 5 8 9 10 (d) 4 1 2-3 4-5-6-7 8 9 10 (e) 5 1-4-5-6-7 2-3 8 9 10 (f) 6 1-2-3-4-5-6-7 8 9 10 Generator 10 is the reference 209 i <3... (.... .....q. {a 32< (X) i¢NI um—I r. 8 ’N30 (030)‘A30 310NU 1 'N30 (0301'A30 310MB 8 'N30 (0301'A30 310MB 210 new: ”no- mozouum H quh -o- . ..-o < ...o ”.2 UK [NM lovl . l'N' nozouum z_ “:22 - MF’F'l, . l‘\q\‘quflvr w Ll.fl17fv > . IQI { < <. < ..o iQN Ice T raw 0*: NM! Duwm 2H wtuh mnl W P . — L:I‘D§UQEPHIF. m . . . 0 ‘N30 (0301‘A30 310MB 1 'N30 (0303'A30 310MB 8 'N30 (0301‘A30 310MB 211 .../...... E? > F 0 'N30 [0301‘A30 310MB 1 'N30 (0301‘A30 310MB 8 'N30 I030I'A30 310NU 212 ><> > a) < C... 328% E .uz>< > i f - .1 . A”. ‘ 0 'N30 (0301'A30 310NU 1 ’N30 (030)‘A30 310MB 0 “N30 (0303'A30 310MB 213 g 828% :5 mozouum za wt" ”4“. ”ma- ”8.- .mm. ”as- .8" ”an no. 0 'N30 (0301'A30 310MB 1 ‘N30 I030)'A30 310MB 6 ‘N30 (030l'A30 310MB 214 r: wot: tum: Tmfllu 1: En wo¢ 1. ”*1 1mm: in": 10a inn 10* 0 'N30 (0301'A30 310MB 8 ’N30 (030I'A30 310MB 215 This same infbrmation is available from the eigenvalue data of Table 6.10. Note that the imaginary parts 4.589 and 9.358 at level 6 are fairly close to the 4.461 and 9.541 values of level 0. The value 6.493 at level 6 appears to be an equal weighting of the values 6.296 and 6.921 at level 0. The analysis of the data in Table 6.11 is almost a repeat of that done for Table 6.10. Note in particular that when generator 1 is added to 4-5-6-7 at level 5, that the resulting coherency measure for the aggregate 1-4-5-6-7 is much closer to the measure for 4-5-6-7 at 2 level 4 than to the measure for generator 1 at level 4. This seems to indicate that the behavior of the aggregate 1-4-5-6-7 will be closer *‘ to that of 4-5-6-7 than to that of generator 1. The main reason for including the data for the ZMIIW disturbance of generators 8 and 9, can be seen by comparing the eigenvalue imaginary parts retained at levels of 5 and 6 of Tables 6.7 and 6.10. The aggregations in these two tables are identical through level 4. At level 5, however the 8,9 disturbance retains a slightly higher fre- quency eigenvalue, 9.384 versus 9.054 for the l,8,9,lO disturbance. The 9.054 eigenvalue is discarded at level 6 by the l,8,9,lO disturbance when generator 1 was added to the group 2-3-8. This, as stated earlier, identifies the eigenvalue pair - %-:_9.054 with the oscillations be- tween generators l and 8. Now note that for the 8,9 disturbance gen- erator 1 is added to the group 4-5-6-7 at level 5 and the group 2-3 is added to the group l-4-5-6-7 at level 6. That is, generator 1 is not added to a group containing generator 8 and this is reflected by the retention through level 6 of the eigenvalue pair - %-:_j 9.358 216 which represents the oscillations between generators l and 8. The eigenvalue retention is clearly dependent on the geography of the partial ZMIIW disturbance. Thus applying the commutative aggregation rule to the r.m.s. coherency ranking table seems to provide selective eigenvalue retention, based on the geography of the partial ZMIIW disturbance. This is a rather remarkable result. The 8,9 disturbance also touches on another issue, namely the question of disturbing part of a tightly interconnected group. The ZMIIN disturbance of all ten generators showed generators l and 8 to be tightly interconnected. Tables 6.7 and 6.10 both show that the greatest averaging of eigenvalues occurs when one of these generators is aggregated into a group that does not contain the other. liore will be said about this in the next section. Figures 6.5a through 6.5f are the simulation results for one per unit step disturbances on generators 8 and 9 for the first six levels of aggregation as dictated by the ZMIIW disturbance of gen- erators 8 and 9. The simulation results show that the system model response has its first noticeable decay at level 4. The level 5 and level 6 models are marginally better than those at the same level fbr the ZMIIW disturbance of buses l,8,9,lO. But it is clear from Figure 6.5e that the degradation is pronounced once generator 1 is aggregated with 4-5-6-7. This speaks directly to the matter of disturbing only some of the generators in a tightly interconnected group. It is the reason why the reduction algorithm contains a step in which the boundary between the internal system and buffer zone is redefined in order to avoid disturbing only part of a group of generators that are tightly interconnected and cross the boundary between the internal 217 system and the buffer zone. Recall that the ZMIIW disturbance of all ten generators showed generators l and 8 to be tightly interconnected at aggregation level 2. If generators 8 and 9 are considered to be. the internal system then, then group 1-8 crosses the boundary between internal system and the buffer zone. The consequences of forming equi- valents under these conditions are explored in greater detail in the next section. IV. ZMIIW Distrubances of Buses l and Bus 8 ——————_——-— Tables 6.12, 6.13, 6.14, Figure 6.4b and Figures 6.6a through 6.6f provide the standard information fbr a ZMIIW disturbance of gen- erator 8. An examination of Figures 6.6a through 6.6d shows that the quality of the system models response decays rapidly at level 4 when generator 1 is aggregated with generator 9. A look at Table 6.13 shows that the absolute values of the eigenvalue imaginary parts at level 4 are truly averages of those at level 3. This means that there is no eigenvalue pair that can be associated closely with the intermachine oscillation of generators l and 9. This is born out by Table 6.14 which shows that the coherency measures for generators l and 9 are significantly different from each other. All of this analysis seems to indicate that the process of re— defining the boundary between the internal system and buffer zone is a necessary step in the reduction algorithm fOr producing dynamic equivalents. This conclusion is supported by a similar analysis for ZMIIW disturbances of generators 8 and 10, and generator 9. However, there are cases where the aggregation does seem to work, even though a tightly interconnected group has been broken up. 218 Table 6.12. Coherency Measure Ranking for ZMIIW Disturbance at Generator 8 Ranking Generator Coherency Aggregation Level Pair Measure 1. C(6,7) .0012 l 2. C(4,7) .0029 3. C(4,6) .0041 2 4. C(2,3) .0235 3 5. C(l,9) .0268 4 6. C(3,5) .0300 7. C(2,5) .0536 5 8. C(5,6) .0573 9. C(5,7) .0585 10. C(4,5) .0615 11. C(3,6) .0873 12. C(3,7) .0886 13. C(3,4) .0915 14. C(2,6) .1109 15. C(2,7) .1121 16. C(2,4) .1151 6 17. C(4,9) .1351 18. C(7,9) .1380 19. C(6,9) .1393 20. C(l,4) .1620 21. C(l,7) .1649 22. C(l,6) .1661 23. C(5,9) .1966 24. C(l,5) .2235 25. C(3,9) .2267 26. C(2,9) .2502 27. C(l,3) .2535 28. C(l,2) .2771 7 29. C(2,10) .3272 30. C(3,10) .3508 31. C(5,10) .3808 32. C(6,10) .4382 33. C(7,10) .4394 34. C(4,10) .4423 35. C(9,10) .5775 8 36. C(l,10 .6044 37. C(l,8) 2.0392 38. C(8,9) 2.0661 39. C(4,8) 2.2012 40. C(7,8) 2.2042 41. C(6,8) 2.2054 42. C(5,8) 2.2628 43. C(3,8) 2.2928 44. C(2,8) 2.3164 45. C(8,lO) 2.6436 219 NM'U'LOLO mmm.m mvm.n Foo.v nnmamuvumnm mup omm.m mom.n cum.n wmm.v uncle mumnm map omm.m mmo.w nom.n mww.m omm.¢ uncle mum mnp ppw.m Num.m muw.n mum.m com.o ume.¢ “none mum Pym.m mkm.m epm.m mum.n mum.m com.m omv.e mimic ppm.m Nem.m mpm.w wmm.w who.n mmm.m nmm.o em¢.¢ mum ppw.m Fem.m mpm.w um~.m mum.m Pmm.m nmm.o Fm¢.¢ mcoz www.m co $83.33 Hm m 38.. a Mu film chlqlm .5 simlumxmu Hm REE H8535 Hm 8.63232 compmmmgmg m coumcmcmo mo mocmngapmmo 3HH5N so» some mzpm>=mmwm .m~.m mpnmh 220 NMQLOO ~m>m4 oomeo. ovao. mmmpo. numumneumum mup pmneo. Nommo. mwmmo. mmopo. uncle mumnm mn~ cameo. mnemo. momma. memmo. mmopo. mimic muw mup mpmvo. .mmwmo. mnemo. mommo. mNNNo. mmwpo. uncle mum Pomqo. mmwmo. onomo. momma. mmomo. mkomo. mmv—o. uncle mmmvo. nmwwo. cmmmo. mmomo. Poomo. mnowo. nucmo. mmepo. mum mmovo. mmmmo. mmomo. mmmpo. FmOmc. owomo. mmomo. snows. mnvpo. mcoz 8:3 8.3 8.2 8.3 8.3 8.: 8.3 8.3 8.: :owummmcmmmalx»m.mm Loumgmcma Empmxw comm ucm m Lopmcmcmo cmmzpma mwcammmz aucmcmgou cowpmmmcmm< m Loumcmcmo mo mocmngaumwn 3HHsN go$ mung mucogmcou .e_.m mpnm» 221 Figures 6.6. Simulations of System Response to a One Per Uhit Step Disturbance on Generator 8 {3 {3 E3 designates the fu11 39 Buss New England System <> <> a€> designates the aggregated mode] dictated by the ZMIIW disturbance of generator 8 [figure 6.6 Aggregation Level System Generators r (a) 1 1 2 3 4 5 6-7 8 9 10 (b) 2 1 2 3 4-6-7 5 8 9 10 H (c) 3 1 2-3 4-6-7 5 8 9 10 (d) 4 1-9 2-3 4-6-7 5 8 10 (e) 5 1-9 2-3-5 4-6-7 8 10 (f) 6 1-9 2-3-4-5-6-7 8 10 Generator 10 is the reference 222 > > 853 .. E V P F w b .. m V. mozoumm zu u=_h- ‘. wznh.. I mnzooum zu wznh 3 -“ ivwl IN" 'NBO (0301'A30 310MB 1 8 ‘N30 (O3OJ'A30 310NU 8 ’N30 (OBOJ‘ABO 310MB 223 m p ADV V. .4 mozoou; z_ u:_h.. 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A look at Table 6.15 shows that 5 levels of aggregation can be achieved from the first 9 coherency measures in the table, and that 7 levels of aggregation can be achieved before the absolute value of the coherency measure becomes very large. An examination of Table 6.16 shows that there is definitely an averaging of eigenvalues at levels 4, 5, and 6. However, the re- sult at level 6 is imaginary eigenvalue parts 4.558, 8.248 and 9.808 that are very close to the values 4.461, 8.287, and 9.541 of level 0. Note also that a value near 4.5, one near 8.3 and one near 9.8 are pre- served through all levels of aggregation. Thus it appears that these latter three values are the imaginary parts of three eigenvalue pairs that represent the intermachine oscillations between generators l, 8, and 2-3-4-5-6-7-9 that are very close to the oscillations between gen- erators 1,8, and 2 at level 0. This leads one to suspect that con- ditions fer geometric coherency, strong linear decoupling or a combina- tion thereof are satisfied for the aggregate 2-3-4-5-6-7-9, for a dis- turbance at generator 1. An examination of Table 6.17 shows that the coherency measures between generator 1 and generators 2,3,4,6,7 are very close to the same at level 0, with coherency measures between 1 and 9 and l and 5 some- what higher.. At level 6 the coherency measure between generator 1 and the aggregate 2-3-4-5-6-7-9 is close to the coherency measure between 229 Table 6.15. Ranking of Coherency Measures ZMIIW Disturbance of Bus 1 Ranking Generator Coherency Aggregation Level Pair Measure 1. C(6,7) .0021 l 2. C(4,7) .0050 3. C(4,6) .0071 2 4. C(3,5) .0139 3 5. C(2,3) .0377 6. C(2,5) .0516 4 7. C(4,9) .0711 8. C(7,9) .0762 9. C(6,9) .0783 5 10. C(5,6) .0991 ll. C(5,7) .1012 12. C(4,5) .1063 13. C(3,6) .1130 14. C(3,7) .1151 15. C(3,4) .1202 16. C(2,6) .1508 17. C(2,7) .1529 18. C(2,4) .1580 19. C(5,9) .1775 20. C(3,9) .1914 21. C(2,9) .2291 6 22. C(8,9) .2754 23. C(4,8) .3466 24. C(7,8) .3517 25. C(6,8) .3538 26. C(5,8) .4529 27. C(3,8) .4668 28. C(2,8) .5046 7 29. C(2,10) .5813 30. C(3,10) .6191 31. C(5,10) .6330 32. C(6,10) .7321 33. C(7,10) .7342 34. C(4,10) .7393 35. C(9,10) .8105 36. C(8,10) 1.0859 8 37. C(l,8) 1.9096 38. C(l,9) 2.1850 39. C(l,4) 2.2562 40. C(l,7) 2.2613 41. C(l,6) 2.2634 42. C(l,5) 2.3625 43. C(l,3) 2.3764 44. C(l,2) 2.4142 45. 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F "c-d I new: 1min A30 310NU 'N30 (030)' I N30 (0301'A30 310NU 8 240 l and the individual machines of the aggregate at level 0. The co- herency measure between generators l and 8 are almost identical at levels 0 and 6, while the coherency measure between generators 1 and 10 changes only moderately (about 8%) between levels 0 and 6. This seems to verify the supposition that for a disturbance at generator 1 either geometric coherency or strong linear decoupling is at work. This says that the partial disturbance step of the reduc- tion algorithm can pay a handsome return if the structural conditions ”1 of the power system are satisfying geometric coherency or strong h linear decoupling. The other results in this section, however, offer a strong caveat against ignoring the boundaries of tightly inter- i‘ connected groups of generators when doing ZMIIW disturbances over a subset of generators. Despite the very good results for the ZMIIW disturbance of generator 1, the best rule would be to disturb all generators of a tightly bound group when determining a local model for a particular generator within the group. V. Summary gf_Results The results in this Chapter verify quite conclusively that the reduction algorithm for producing dynamic equivalents proposed in Chapter 5.is a reasonable and viable approach to producing dynamic equivalents. The importance of the general ZMIIW disturbance of all generators in determining tightly interconnected groups has been sub- stantiated, and the validity of the partial ZVIIIN disturbance for de- tecting the conditions of geometric coherency and strong linear de- coupling have been verified. Embedded in this verification is the necessity to be cautious in fbrming local equivalents based on the 241 disturbance of only some of the generators within a tightly inter- connected group. The ZMIIW disturbance over a subset of generator coupled to the commutative rule for aggregation of the r.m.s. coherency ranking table also showed the ability to do eigenvalue retention in a "geo- graphic" way. That is, the eigenvalue retention seems to be based on the modes most excited by a disturbance located in a certain area of the power system. Hence, overall, the ZMIIW disturbance of only part of the power system has proved to be a viable step in the overall reduction algorithm. The question of whether the coherency measure thresholds in the ranking table can predict the point of overaggregation is not answered definitively by the data presented in this chapter. Most of the plots of threshold magnitude versus aggregation level swing upward rapidly at_ or after level 6, while the proper aggregation level for valid system response seems to be level 4 or 5. More data collection may help prove or disprove this trend. Although the data presented in this chapter is more than adequate to verify the analysis of Chapters 3, 4 and 5 some important work remains to be done. One interesting and necessary follow-on would be to test the algorithm on a "large" system, one with at least fifty generators. One reason for doing this would be to test the pro- posed method of refining the boundary between the buffer zone and the far system. This has not been possible in a ten generator system where realistically all that is present is the internal system and the buffer zone. Such a study would also determine if a general ZMIIW disturbance of all generators will identify the same principle groups of tightly 242 interconnected groups as the algorithm based on singular perturbation theory proposed by Kokotovic and Ninkelman [12, 13]. It would also be worthwhile to build some nonlinear system models based on the aggregations dictated by the ZMIIW disturbances. The easiest place to do this would probably be on the New England System by implementing the software package developed for EPRI by Podmore and Germond [2]. This would silence some of the criticism that inevitably results from using linear system theory to analyze ‘ fl nonlinear systems. It would help test the supposition put forward in F Chapter 4 that synchronizing coherency, geometric coherency and strong linear decoupling are structural power system conditions whose presence ‘“5 in the linear model strongly guarantees their presence in the nonlinear model. CHAPTER 7 REVIEW, CONTRIBUTIONS, AND TOPICS FOR FUTURE RESEARCH I. Overview gf_1h§§1§_ This research was initiated primarily to establish a stronger theoretical connection between the two traditional methods of producing reduced order dynamic equivalents for power systems, namely coherency equivalents and modal equivalents. These two methods had both been used successfully, and both had their proponents. The fact that both methods could produce good equivalents was strong, intuitive evidence that both methods must be utilizing the same fundamental properties of the power systems's structure to produce dynamic equivalents. The evidence that this was the case was strong [5, 6, 7, 8, 10, 11], but far from complete. The review of the two methods of producing equivalents in Chapter 2 while aimed at delineating the differences between the two methods also pointed up one similarity. In both methods the search for an equivalent began by assuming that the disturbances would occur in a particular area of the power system, called the internal system. Everything else fell into the category of the external system. The perspective, then, for both methods was to look outward from the internal system and form a reduced order model of the external system. One might call this a local or parochial perspective on the dynamic equivalents problem. Through the course of this research, aimed 243~ ‘1' ... 244 primarily at linking the two traditional techniques of forming equi- valents, a broader, more global, perspective on the problem of dynamic equivalents emerged. More will be said about this in later sections of the chapter. Chapter 3 began the hunt for theoretical connections between the modal and coherency techniques, by reviewing one of the strongest clues, namely the work of Dicaprio and Marconato, on what was to be eventually termed, in this present work, "Strict Geometric Coherency" [10, ll]. Dicaprio and Marconato divided their example power system into a study group and a specified, or in their terminology an "evidenced", group. .They then stated structural conditions between the study group and the specified group, which if true at time t = 0', caused g11_the generators of the specified group to accelerate at the same rate in response to any_disturbance within the stugy_group. This meant that, from the viewpoint of the study group, the specified group appeared to be one generator. The striking feature of Dicaprio and Marconato's result is that it holds for the nonlinear model. The rest of Chapter 3 investigated other conditions that, like the Dicaprio-Marconato condition, might cause the specified group to behave, from the perspective of the study group, like a single generator; two were identified. The first was called strict synchronizing coherency and depended upon progressively stiffening, at least, n-l interconnections linking all n genera- tors of the specified group, until these interconnections were infi- nitely strong (zero impedance). The other condition called, pseudo-co- herency, was a mirror image of the Dicaprio-Marconato conditions for strict 245 geometric coherency, in that it also relied upon structural conditions between the study group and the specified group of n generators at time t = 0-. In pseudo-coherency, however, the specified group was not coherent but only appeared to be coherent to the study group. Thus Chapter 3, determined three hypothetical conditions under which the specified group could be replaced by a single machine. The three conditions are called hypothetical in the sense that they could never be perfectly satisfied in a real power system. Strict synchroniz— ing coherency relies upon infinitely stiff (zero impedance) inter- connections between generators. Strict geometric coherency and pseudo- coherency conditions can be realized with real components, but the probability of the conditions being satisfied for a sizeable group of machines is effectively zero. It could be argued that strict synchronizing coherency is hypothetical in-a different sense than the other two conditions be- cause it relies upon non-finite components. This argument has philosophical but not practical merit, because near approximations to strict synchronizing coherency are more common in power systems than are near approximations of the other two conditions. Further, the argument becomes irrelevant by the end of Chapter 4, since at that point it is evident that the real power of these three conditions is in their use as conceptual tools fer understanding the combination of conditions that lead to coherency in a real power system. That is, the three con- ditions, SSC, SGC, and PC can be viewed as archetypes for group co- herency. In a real system a combination of these archetypes may be at work simultaneously to cause group coherency. 246 Chapter 4 re-examined the three conditions for coherency for the linear model. One of the results of Chapter 3 had been that satisfaction of the conditions for strict synchronizing or strict geometric coherency at time t = O', guaranteed coherency of the specified group for all t > O. This could ngt_be shown for pseudo? coherency and in that sense pesudo-coherency was a far weaker condi- tion than the other two. All three conditions did however decouple the liggar_equations, leading to the specified group behaving as a single machine, for dis- turbances withjg_the study group. This decoupling provided the key to showing that if any one of the conditions SSG, SGC or PC were satisfied, the modal and coherency methods produced the sgmg_equivalent for the specified group. This result depends on the decoupling of the linear equations which in turn separates the eigenvalues fer the system model into two sets, one set (of eigenvalues) associated with the equations for the study group (of generators) through the matrix (-fl__T_)H and the other set with the equations for the specified group through the matrix (-fi_1)22. 'The coherency method offinding an equivalent re- placed the specified group by a single machine because it behaved as a single machine from the perspective of the study group, for dis- burbances withjn_the stugy_group. The modal analysis method produced the identical equivalent by using controllability and observability arguments to discard the modes (canonical states) associated with the specified group of 'n generators, through the matrix (:flj[)22. In the case of strict synchronizing and strict geometric coherency, the modes were discarded as uncontrollable. In the case of strict (linear) 247 pseudo-coherency the modes were discarded as unobservable. Thus Chapter 4, established the conditions under which the modal and coherency equi- valents fbr a specified group of generators were identical. This picture was made even more complete by l) incorporating into the three hypothetical conditions for coherency the work of Chow, Kokotovic, Allemong, Winkelman, et al., on singular perturbation equi- valents, and 2) by introducing the idea of linear decoupling. It was shown that the singular perturbation model, which discards high fre- quency modes as unobservable is almost perfectly congruent with the concept of strict synchronizing coherency. In fact, the limiting pro- cess of sending the parameter p to zero in the singular perturbation model was shown to coincide with the process of letting n-l intercon- nections linking all n machines become infinitely stiff, so that, in the limit, strict synchronizing coherency and the two time scale separa- tion of singular perturbation become identical. The concept of linear decoupling was then introduced to account for those cases where the linear equations were essentially decoupled, but the decoupling could not be attributed completely to any one of the three conceptual conditions for coherency, i.e. SSC, SGC or PC. Once again, the concept of linear decoupling was introduced as a conceptual aid by showing how the specified group could be perfectly decoupled by a combination of strict synchronizing coherency and strict geometric coherency. Three types of linear decoupling were identified. Two of these types were classified as "weak", since like pseudo- coherency the presence of the conditions in the linear model did not strongly guarantee the presence of the conditions in the nonlinear model. 248 One type of linear decoupling, however, did carry this guarantee and could be classified with strict synchronizing coherency and strict geometric coherency. This type of linear decoupling was termed strict strong linear decoupling. It is best understood by an example. Consider the five generator example of figure 7.1a and suppose the following structural conditions hold. where the tij's are synchronizing power coefficients and the Mi's are machine inertias. Let generator 1 be the study group, generators 2 and 3 group 2, andgenerators 4 and 5 group 3. The conditions (1) and (2) are not sufficient to cause groups 2 and 3 to behave like one large coherent group. That requires or a tree of stiff interconnections among generators 2,3,4 and 5. Neither of these conditions is implied by conditions (1) and (2). Thus, the generators 2, 3, 4 and 5 are satisfying neither the conditions for strict geometric coherency or strict synchronizing coherency. Now let the interconnection between generators 2 and 3 become infinitely stiff. This causes generators 2 and 3 to act like a single machine of inertia M2 + M3. The synchronizing power coefficients be- tween the aggregate generator 2-3 and generators l, 4, and 5 are as shown in figure 7.1b. FlGLRE 7-l FIVE GENERATOR SYSTEM u) BEFORE AGGREGATION OF GENERATORS 2 AND 3 (”AFT ER AGGREGATE]! OF GEMRATORS 2 AND 3 250 . 1‘12” t Since fiE——;7MS—-= FT_'= FTT" generators 4, 5, and the aggregate 2-3 Satisfy the conditions for strict geometric coherency. for disturbances at generator 1. Thus a combination of strict synchronizing coherency and strict geometric coherency will cause gen- erators 2, 3, 4, and 5 to act as a single generator for disturbances at generator 1. This example also illustrates the great conceptual power of the archetypal conditions, strict synchronizing coherency and strict geometric coherency. I Chapter 4, then, contains two major accomplishments, l) a set of conditions under which the modal and coherency equivalents were identical and 2) a determination of which of these conditions were worthy of being used to form a dynamic equivalent. Those chosen for use in forming the dynamic equivalent were synchronizing coherency, geometric coherency and the strong type of linear decoupling (SLD). Chapter 5 next provided the means of detecting the selected conditions. It was shown that a particular type of disturbance, called a ZMIIW disturbance made the r.m.s. coherenCy measure depend only on the structure of the linear model, i.e. on the -fl_1_ matrix. Further by using different ZMIIW disturbances one could distinguish synchroniz- ing coherency from geometric coherency and strong linear decoupling. Synchronizing coherency was detected by a general ZMIIW disturbance of all generators. The other two types by a ZMIIW distrubance over a specific subset of°generators called the internal system. This internal system can be identified with the study group of Chapters 4 and 5. The distinction between the types of disturbances leads in a very natural way to a distinction between two types of reduced order 251 models of a power system. The general ZMIIW disturbance over all gen- erators detects those groups of generators that are tightly inter- connected. That is, it divides the overall power system into areas, called principal groups, that react in consort to distrubances any- where in the system. Thus, if the main concern is how a disturbance propagates among the principle groups, then the general ZMIIW disturbance can provide the proper model. This model might be thought of as a global model . The ZMIIW disturbance over a specific subset of machines, on the other hand, provides a means of finding what is coherent, looking outward from that subset of machines. It is not hard to see that this is congruent with the traditional perspective on forming equivalents, discussed in Chapter 2. Chapter 5, concludes by incorporating both the general and the specific ZMIIW disturbances into a reduction algorithm for producing dynamic equivalents. The results of testing the algorithm on the 39 Bus New England System were summarized in Chapter 6, and indicated that the algorithm worked very well. 11. Contributions The ideas of pseudo-coherency and linear decoupling are new. In some limited sense strict synchronizing coherency is also new. The knowledge that tightly interconnected generators remain coherent has existed for a long time, but it was never formalized into a theoretical concept requiring n-l infinitely stiff interconnections among n generators to make them strictly coherent. It was this fermulation that led to the result that, in the limit when the parameter p ..0, 252 the singular perturbation concept of two time scale separation is identical to strict synchronizing coherency. As important as the individual ideas are, the significant con- tribution has been the integration of these individual ideas into a general theory that provides a good conceptual understanding of how a power system responds to a disturbance, both at the global level and the local level. The generality of this theory is demonstrated by its ability to encompass all the current methods employed in constructing ._LZY dynamic equivalents, including the singular perturbation approach. This conceptual understanding can be directly applied to the problems of security assessment and planning. The global modeling level is of great interest for on-line system monitoring and control, since it provides some insight into how major portions of the power system interact in response to a disturbance. The r.m.s. coherency measure may be capable of serving as a security measure that would in- dicate when the system is vulnerable to an unstable condition (called a contingency), so that corrective action can be taken. For power system planning, both the global and local modeling aspects can be utilized, since in planning, both global stability and transient stability from the standpoint of a single machine are of interest. The reduction algorithm itself has already been implemented by making appropriate changes to the EPRI software package [2]. This modified software package provides a computationally efficient means of producing dynamic equivalents for systems of agy_size. It's main virtue is that it does not require the calculation of eigenvalues, which is the main drawback to almost any modal analysis scheme for producing equivalents, including the singular perturbation approach. 253 The data collection done on the 39 Bus New England System not only verified the analysis of Chapters 3, 4, and 5, it added some new infbrmation, in particular, the idea of eigenvalue retention tuned to the location of the disturbance. This is in itself a very interesting result. 111. (Topics for Future Research The analysis in Chapter 4, indicates that the singular per- turbation concept coincides with the concept of synchronizing coherency. This means that the principle groups identified by the reduction al- gorithm of Chapter 5 should coincide with the major groups determined by Ninkelman, Kokotovic, et al. in references [12, 13]. This is purely computational work to be done on a system with at least fifty or sixty generators. Two prime candidates are the 64 generator system used by Ninkelman, Kokotovic, et al. [13], and the model of the western grid used by Podmore [2]. Another useful investigation is the comparison of the performance of the coherency equivalent, i.e. the averaged equivalent machine, with the singular perturbation equivalent fbr the same generator aggregation. The conjecture is that the coherency equivalent will per- form better for short intervals and the singular perturbation equivalent will be better for long intervals. Some computational research can also be done to compare Podmore's and Germond's results with the proposed reduction algorithm. Podmore and Germond use a different coherency measure, a different aggregation rule for determining coherent groups, and disturbances that are not ZMIIW. It might be useful to use the model of the western grid provided 254 by reference [2] and perform ZMIIW disturbances from the same locations as the disturbances in Podmore and Germond's work and compare the equivalents. The present work also provides new insight into the problem of on-line identification of the external system. The identification problem requires an assumption about the structure of the external system. That structure can be obtained from a general ZMIIW disturbance of all the generators. It would provide first of all the order of the state model of the external system and second, the appropriate locations at which to make the measurements. That is, measurements would be taken at machines that belong to the principal groups determined by the general ZMIIW disturbance. It may even be possible to do the identification based only on measurements taken within the internal system and at the boundary between the internal and external systems. Th future research possibilities out of the current work seem fairly rich. The ones discussed here by no means exhaust the list, but only serve to indicate some of the more promising avenues of exploration. BIBLIOGRAPHY 10. 11. BIBLIOGRAPHY J.M. Undrill and A.E. Turner, “Power System Equivalents", Final Report on ERC Project RP904, January 1971. R. Podmore and A. Germond, "Development of Dynamic Equivalents for Transient Stability Studies," Final Report on EPRI Research Project 763, April 1977. R. Podmore, "Identification of Coherent Generators for Dynamic Equivalents", IEEE Trans., Vol. PAS-97, July/August 1978, pp. 1334-1354. A. Chang and M. Adibi, "Power System Dynamic Equivalents", IEEE Transactions on Power Apparatus and Systems, Vol. PAS-89, No. 8, November/December, 1970. R.A. Schlueter, H. Akhtar, and H. Modir, "An RMS Coherency Measure: A Basis fer Unification of Coherency and Modal Analysis Model Aggregation Techniques,“ 1978 IEEE PES Summer Power Meeting. R.A. Schlueter, U. Ahn, "Modal Analysis Equivalents Derived Based on the RMS Coherency Measure", 1979 IEEE PES winter Power Meeting, Paper No. A-79-061-3. J. Lawler, R.A. Schlueter, P. Ruesche, D.L. Hackett, "Modal- Coherent Eguivalents Derived from an RMS Coherency Measure", 1979 IEEE ES Summer Power Meeting. J. Lawler, R.A. Schlueter, "An Algorithm fer Computing Modal Coherent Equivalents", submitted for presentation 1981 IEEE PES Winter Power Meeting. J. Meisel, "Reference Frames and Emergency State Control fer Bulk Electric Power Systems", Proceedings of the 1977 Joint Auto- matic Control Conference, Vol. 2, pp. 747-754. ' U. Dicaprio and R. Marconato, "Structural Coherency Conditions in Multimachine Power Systems", VII IFAC World Congress, Helsinki, Finland. U. Dicaprio, "Conditions for Theoretical Coherency in Multi- machine Power Systems", Centro Ricerca di Automatics, ENEL - Milano, Italy. 255‘ 12. 13. 14. 15. 16. 17. 18. 256 B. Avramovic, P.V. Kokotovic, J.R. Ninkelman, J.H. Chow, "Area Decomposition for Electromechanical Models of Power Systems", submitted for presentation at IFAC Symposium on Large Scale Systems: Theory and Applications - Toulouse, France, 1980. J.R. Ninkelman, J.H. Chow, B.C. Bowler, B. Avramovic, P.V. Kokotovic, "An Analysis of Interarea Dynamics of Multi-machine Systems", submitted. A. Germond, R. Podmore, "Dynamic Aggregation of Generating Unit Models", IEEE Trans., Vol. PAS-97, July/August 1978. PP. 1060- 1068. J.H. Chow, J.J. Allemong, and P.V. Kokotovic, "Singular Perturba- tion Analysis of Systems with Sustained High Frequency Oscilla- tions", Automatica, Vol..l4, pp. 271-279, 1978. GENERAL REFERENCES P. Anderson, Analysis 9f_Fau1ted Power Systems, Iowa State University Press, Ames, IA, 1973. 0.1. Elgerd, Electric Energy System Theory; An Introduction, McGraw-Hill, New York, NY, 1971. P. Anderson, A. Fouad, Power Systems Control and Stability, Iowa State University Press, Ames, IA, 1977. 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