PV1ESI_} RETURNING MATERIALS: P1ace in book drop to meanings remove this checkout from 4—1:...- your record. FINES win be charged if book is returned after the date stamped below. VOLTAGE STABILITY AND SECURITY FOR ELECTRIC POWER SYSTEMS By Louis Pariai Shu A THESIS Submitted to Michigan State University in partia] fulfiilment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of E1ectrica1 Engineering 1984 ABSTRACT VOLTAGE STABILITY AND SECURITY FOR ELECTRIC POWER SYSTEMS By Louis Parlai Shu This thesis presents a new approach to voltage stability prob- lems in electric power systems. The study develops (a) the theory that explains the different types and causes of voltage stability and (b) the computer based methods for detecting when a power system will experience voltage stability problems. The rapid growth of the interconnections between different electric utilities in recent years has brought on very serious and frequent voltage stability problems. Voltages can collapse in cer- tain regions and/or large sustained voltage oscillations have been experienced. The lack of theoretical explanation of these stability problems is due to their being a rather recent phenomenon and that they are described by very large scale models that involve thousands of nonlinear equations. The voltage stability problem is shown here as a lack of suf- ficient reactive power support within specific stiffly interconnected groups of buses. Attempting to supply reactive power across the weak transmission boundaries causes voltage collapse. Weak boundaries Louis Parlai Shu are theoretically shown to be a cause of a loss of voltage control- lability and a loss of reactive load observability that ultimately can lead to voltage stability or voltage collapse problems. Voltage problems can result from insufficient transmission capacity, heavy current loads on transmission lines in these weak boundaries. Shunt capacitance for long transmission lines or for voltage control was shown to be another cause for this loss of voltage controllability, observability, and stability. Methods for determining and ranking weak boundaries were tested on a 30 bus New England system. The buses in the stiffly connected groups were shown to act as an equivalent bus for loadflow simula- tions of multiple line outage or loss of generation contingencies. The buses in weak boundaries were also shown to experience large voltage variations for these contingencies. Increasing reactive flows across weak transmission boundaries or providing capacitive reactive power for a stiffly connected group requiring reactive support were both shown to further weaken the weak transmission boundary and lead to voltage collapse. These com- putational results thus confirm the theoretical results on the causes of voltage stability and security problems on power systems. To my mother Jane and my wife Arlene ii 0-. n. - ; 1" m ACKNOWLEDGMENTS I wish to gratefully acknowledge my indebtness to my major advisor, Professor R.A. Schlueter, for his invaluable assistance in the planning and preparation of this dissertation. I can never repay the tremendous amount of time he devoted to my research, espe- cially on those weekends and holidays. Also, I thank the other members of my committee, Dr. M.A. Shanblatt, Dr. H.K. Khalil, and Dr. David Yen, for their suggestions to improve the quality of this thesis. Special acknowledgment is made of the contribution of Dr. G.L. Park and Dr. R.0. Barr, for reviewing my original research proposal. I thank the system engineers of Consumers Power Company. In particular, I wish to thank Mr. P.A. Rusche, P.E., Executive Engineer, for constant reviewing of my research, Mr. R.H. Roades for coaching me on the data preparation, simulation, and analysis, and Mr. J.E. Sekerke and Mr. C.A. Bunnell for 'their help in programming. Mrs. Carol A. Cole improved the overall style and presentation of the manuscript. She also conscientiously typed the manuscript and its revisions in minimum time. I wish to express my deep appreciation to my wife, Arlene w. Chun, for her constant encouragement and helpful suggestions in the preparation of this thesis. Without her encouragement and consent, iv this thesis would not have been completed simultaneously with my third masters degree. My mother fostered my ambition to become a fighter pilot and love for my engineering profession. v F A d A.- F I . TABLE OF CONTENTS LIST OF TABLES ....................... LIST OF FIGURES ....................... CHAPTER 1. INTRODUCTION ..................... 1.1. The Objective of This Thesis .......... 1.2. The Natural Structure of Voltage Problems in Large Scale Power Systems ........ , . . . 1.3. State of the Art ................ 1.4. An Overview of the Thesis ........... 2. 'METHODS FOR THE DETERMINATION OF WEAK TRANSMISSION BOUNDARIES ...................... 2.1. Introduction .................. 2.2. The Linearized Loadflow Models and Stiffly Connected Buses ................ 2.3. Detection of SSC Group in P - e Loadflow Model . 2.4. Detection of SSC Group in Q - V Loadflow Model . 2.5. Detection of SSC Group in A.C. Loadflow Model 2.6. The Limited Sources and Weak Boundaries DETERMINATION OF WEAK TRANSMISSION BOUNDARIES FOR THE 30 BUS NEW ENGLAND SYSTEM .............. 3.1. Introduction .................. 3.2. Weak Phase Transmission Boundaries ....... 3.3. Weak Voltage Transmission Boundaries . . . '.' . 3.4. Weak Current Transmission Boundaries ...... V 10 14 14 16 20 26 32 4O 44 44 53 64 71 vi CHAPTER 4. CONDITIONS FOR VOLTAGE CONTROLLABILITY, OBSERVABILITY, AND STABILITY .................... 76 4.1. Introduction .................. 76 4.2. The Model Development ............. 77 4.3. The Controllability and Observability Equations and the Sensitivity Matrices Under Light Load Conditions ................... 91 4.4. The Theorems for Local and Global Voltage Stability ................... 107 4.5. Theorems Integrating the Weak Boundaries and Sensitivity Analysis .............. 121 5. VOLTAGE STABILITY AND SECURITY EVALUATION ON THE 30 BUS NEW ENGLAND SYSTEM ................ 140 5.1. Introduction .................. 140 5.2. The Systematic Impacts of the Weak Boundary to the Voltage Stability ............. 142 5.3. Local and Global Effects of Capacitance on Voltage Stability ............... 146 5.4. The Effects of Shunt Capacitor on the Control- lability and Observability of the Steady State Voltage Problem ................ 157 6. CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH . 181 6.1. Review ..................... 181 6.2. Recommendations for Future Research ...... 185 APPENDIX 1. BASE CASE LOADFLOW DATA IN COMMON FORMAT . . . . 187 BIBLIOGRAPHY ........................ 189 TABLE \l 01 01 «F o o o o 01 U1 U" 01 01 (J1 01 (11 01 U1 U1 U1 0 O I O 5.11a. .8b. .8c. . Eigenvalues and eigenvectors of SQ .9a. .9b. .9c. .9d. 5.10a. 5.10b. 5.10c. 5.10d. LIST OF TABLES Summary of results for Case 1 ............ Summary of results for Case 2 ............ New groups obtained from the weak boundary identifica- tions based on voltage measure ............ Summary of results for Case 3 ............ Summary of results for Case 4 ............ Summary of results for Case 5 ............ Summary of results for case 6 ............ . Sensitivity matrix SQ E for Case 3 .......... G Eigenvalues and eigenvectors of SQ V for Case 3 L Sensitivity matrix S for Case 3 .......... QLV V for Case 3 L Sensitivity matrix S E for Case 4 .......... QG Eigenvalues and eigenvectors of SQ E for Case 4 G Sensitivity matrix SQ V for Case 4 .......... L Eigenvalues and eigenvectors of SQ V for Case 4 L Sensitivity matrix SQ E for Case 5 .......... G Eigenvalues and eigenvectors of SQ E for Case 5 G Sensitivity matrix S for Case 5 .......... QLV Eigenvalues and eigenvectors of SQ V for Case 5 L Sensitivity matrix SQ E for Case 6 .......... G . vii 145 148 149 150 151 152 154 158 158 160 161 163 163 164 165 168 168 170 171 173 I"; hook- TABLE 5.11b. Eigenvalues and eigenvectors of SQ 5.11c. Sensitivity matrix SQ V for Case 6 .......... L 5.11d viii . Eigenvalues and eigenvectors of SQ G L E for Case 6 V for Case 6 173 175 176 'A an ad. . ‘q 1 II ‘A ‘ n '1 "in 1.) ‘- V". a“ FIGURE .1. 3. 3. 1 3.10. 1. 2. LIST OF FIGURES A hybrid electromechanical analog model ....... The original 30 bus New England system ........ The weak boundaries for a 3-group partition based on real power disturbance and phase rms coherency measure The weak boundaries for a 4-group partition based on real power disturbance and phase rms coherency measure The weak boundaries for a 6-group partition based on real power disturbance and phase rms coherency measure The weak boundaries for a 3-group partition based on complex power disturbance and phase rms coherency measure The weak boundaries for a 4-group partition based on real power disturbance and phase rms coherency measure The weak boundaries for a 5-group partition based on real power disturbance and phase rms coherency measure The weak boundaries for a 3-group partition based on real power disturbance and voltage rms coherency measure The weak boundaries for a 4-group partition based on real power disturbance and voltage rms coherency measure The weak boundaries for a 6-group partition based on real power disturbance and voltage rms coherency measure ix 48 55 56 57 61 62 63 65 66 67 l 0...! V 'hv I. .1 li. . 0 h '3‘. FIGURE 3.11. 3.12. 3.13. 3.14. 4.1. 4.2. 4.3. 5.1. The weak boundaries for a 5-group partition based on reactive power disturbance and voltage rms coherency measure ....................... 70 The weak boundaries for a 6-group partition based on reactive power disturbance and voltage rms coherency measure ....................... 72 The weak boundaries for a 5-group partition based on reactive power disturbance and voltage rms coherency measure when all generator buses are regulated . . . . 73 The weak boundaries for a 3-group partition based on the current coherency measure for both real and com- plex power disturbances ............... 74 The Jacobian matrix of five-bus system in polar form . 96 The partitioned Jacobian matrix of a three-SSC-group network ....................... 123 The equivalent sensitivity matrix SQLV ........ 137 The four group partition based on reactive power dis- turbance and voltage nms coherency measure after gen- erators 6 and 10 are removed ............. 144 CHAPTER 1 INTRODUCTION 1.1. The Objective of This Thesis The purpose of this chapter is to give an orientation of the voltage problems and define the objectives of the thesis. There are two kinds of voltage problems: (a) voltage collapse, and (b) volt- age oscillations. The consequences of these voltage problems are the following: the voltage collapse will cause the blackouts and economic loss to the utilities and customers. The voltage oscilla- tions will cause the maintenance problem and eventually lead to the stability problems of the whole power system. Voltage collapse and voltage oscillations can also cause the inability to transfer real power from the location of generation to the location of the cus- tomers. Today those transfers are widespread practice and are essen- tial to the economic and reliable operation of power systems. Back in the 1950's and 1960's utilities began to interconnect their networks. Initially, they were reluctant to admit that they had voltage problems. As they relied on those interconnections more, these voltage problems became more frequent and severe. It is only recently that utilities have been willing to admit the severity of the voltage problems. V on tlb ‘- PM " \ l".t 0'. ‘1 o w .55“ c.‘ ‘ . F4, “ h 4' I l I f l 7. C" A - c ‘ '. , a The following areas have experienced voltage problems and have had to limit transfers of power: Florida, Pennsylvania/Maryland/New Jersey power pool“, Ontario, and California. Some common characteristics of the power systems in the above geographical areas are: (1) they all heavily import power from other utilities; and (2) the generation stations which supply the power to those areas are located at far away remote sites. ‘Tfieev~e are many other utilities that have recently reported problems ‘311<1 have indiCated the severity of the problems. Although it is u"derstood how severe the problems are, very little research has b€3E371 performed to date, and the causes and solutions are not well "'ltiearstood. The study of the causes and solution of these voltage p'"C>t>lems has been established as the number one priority for research as established by the transmission planning group within a recent EPRI sponsored meeting organized to establish priorities for research 'i" ‘power system operation and planning. One may ask why these inter- <3S>Flnections cause problems and what are the causes. This thesis i r‘dicates that: (1) the weak boundary; (2) the insufficient local reactive supply to maintain voltage; (3) the installation of capacitors for reactive supply rather than generators; and (4) the capacitance associated with long transmission lines in weak transmission boundaries 'Ii :4 bl . uvuld .F’ n.- l' v i.- T In}, nu .‘ I I i 0.. ‘. .A. I“ W. , i l ‘0. 'Q. :..l~'.l I I 1 "a 'Q a . .4 ‘ V v 7 v a... , . ~ u 5 ~ '. A I I 'J (I, I- a ,_1 are some of the causes of the voltage stability problems. This thesis will establish theoretically as well as experimentally that the above four network conditions will cause voltage problems. The objectives of this research are defined as follows: (1) to define the structural causes for weak transmission boundaries; (2) to develop methods for identifying locations of weak transnfission boundaries; (3) to categorize the causes of voltage stability problems as due to: (a) inadequate resources (b) loss of voltage controllability (c) loss of reactive Observability (4) to determine the necessary conditions that insure voltage :stability-will not occur for the above categories; and (5) to determine conditions that will cause loss of voltage :stability for the above categories. 3L.2. ;Ihe Natural Structure of Voltage Problems in _Large Scale Power Systems .A typical interconnected power network can be visualized by a hybrid electromechanical analog model as shown in Figure 1.1. This model highlights the structure of several stiffly interconnected groups, The nodes with support of capacitor banks represent the sources of reactive power. The big and small masses represent the r"hacthE loads of different sizes. The heavy duty springs represent uv \ v- w c I o. Q t I“ s .I .0. Us '- ”a." I V _ 9-5." -. ' Pf 0.; In. ll cl: U' III I O J“ ' ‘~ \fl 1‘ u. I" '9 In. - ' I“ '- ~ p A \ i "- Q I. a h . a ,3. - . u . v‘ :a‘ I~.| - 4 n .t . -6 . ' wt s. . U ‘- .- " ‘.F‘. 1': n I. l I . u v A C. 4‘ u o I o. a .q! 'I‘L '- the stiff transmission lines with large power carrying capability. The light springs are those weak tie-lines interconnecting different utilities. The light springs that surround each stiffly connected group can also be regarded as a weak boundary for that particular group of buses. Imagine that one of the masses was suddenly hit by a hammer, which is an analog to a sudden impulse of load to this group. The sti ffly connected group of buses will oscillate together as one bus, and this oscillation is analogous to the os'cillation of the phase angles or voltage magnitudes of the voltages at buses. Another situation that might occur is when someone connects an additional mass at a load bus (the mass pointed to by the arrow in Figure 1.1). Suppose the originally designed supports (reactive sources) are not strong enough to sustain the additional load. Then the whole group will sag. The analog in a power system is the decrease of voltage magnitudes at each bus of this group. This electromechanical model ‘is used to indicate and provide insight into the voltage problems. The electromechanical analog is not a one-to-one mapping of our theory but will be referred to in order to help provide understanding. Figure 1.1 has shown the structural nature of a typical inter- connected network. Based on these concepts this thesis will develop a 5915 of reasonable assumptions, models, theorems, and operating C°"5tV'a'lnts for each bus as well as for each strongly interconnected group of buses in a power system transmission network related to voItage stability and security. The stiffly interconnected groups and their boundaries. Figure 1.1. A hybrid electromechanical analog model. '.‘.. ,‘ 3:”t: 'ivd "' O . ‘.‘-1‘ ‘u ‘ . a vi a ‘c‘ ‘V i 1.3. State of the Art The stability of power systems has generally been associated with real power such as steady state stability for real power flows across transmission elements, transient stability for large contin- gencies such as faults, and eigenvalue determined steady state (dynamic) stability [4]. The relationship between steady state stability, eigenvalue based steady state stability, and steady state security has been developed based on'the work of Venikov [4] on eigenvalue based steady state stability, recent research on power system dynamic equivalents [2,3], and the recent EPRI 1999-1 project entitled "Methods of Analysis of Generators Governor Response and System Security" [1]. A similar set of analyses on stability and security methods does not at present exist for voltage stability and security but is the subject of this thesis. The analysis and compu- tational methods for assessing steady state disturbance and eigen- value based Lhag stability and security are now discussed. The relatively poorly developed state of the art of present analysis and computational methods for steady state disturbance and eigenvalue based voltage stability and security will then be discussed. Venikov [4] developed a method for analyzing voltage stability by determining eigenvalues of the linearized transient stability model u. ‘ 4.‘ O. ..D 5 .- '2'. A ‘ V “'n h '3 P "in. ‘ - I- ~ 0 ‘. (I l ‘ l. h i a ML" 4 "a ‘A W ‘A ., .:\ S . [LII '9 f '42" r9 O‘ILPM‘ =L + at [[11 9 u.» -U'1 1141) 4_PLJ I - n x n identity matrix T - n x n Jacobian of the network reduced to internal generator buses H H H . 1 2 n U diag - —7r-, - -?r', ...., - " s " 2 " s f nominal system frequency 5 It was shown that the determinant of the matrix A has the same sign as the determinant of the Jacobian of the deter- minant of 1' since n H. det 1(sk) = E ajA-det 5 = “f *1 j=1 5 j=1 5 i=1 A where )‘i are eigenvalues of matrix A. A The changes in sign of T(Sk) for operation condition Sk is used to check the eigenvalue based steady state stability for I. A A 0 “am I u v '. I. I” r v w I 0 II IA h.- 0 \ l t O‘C‘ I o All g‘ I' ’\ O ‘5' ' n... (s \ .0. u. . TV I;: A’. T; "v. is :‘:‘F" F a“ ,‘I 1 i. y .“9- \ , - u.“ ”'2 (1‘ ‘ I ' ":“o s.‘\ s \9 ’ ' ‘ a \ v'. ,. a.” '33. It, ~'~ k ‘ -; a. A .FA‘ ' ‘a V. o L;. ._l l. f' I V f 'a I .“ 0"! I ~ I . -‘ \§. different operating conditions Sk' If the small positive eigenvalues of I go negative a loss of eigenvalue based phase stability results. The small eigenvalues of 1’ correspond to the small imaginary eigen- values of A (from this investigator's work [2,3] on the theoretical basis for dynamic equivalents based on the model (i)), which always describe the oscillations of stiffly connected generator groups against one another. Thus, the loss of eigenvalue based steady state phase stability occurs across these weak transmission bounda- ries when for some reason the small positive eigenvalues of 1' go to zero and then negative, causing the associated eigenvalues of A with small imaginary parts to go to zero and then become positive and real. Positive real eigenvalues for A will obviously indicate an eigenvalue based loss of, steady state stability. The recent work under RP 1999, "Methods of Analysis of Gen- erators Governor Response and System Security," showed that a loss of stability can occur across these weak boundaries between stiffly Connected generator groups for loss of generation contingencies. The lost generation for a loss of generation contingency is picked up by every generator in proportion to its inertia and thus these he“ Power flows focus back to the generator bus where the loss of generation contingency occurred. These real power flows all cross the weak boundary of this stiffly connected generator group that exPerlences the loss of generation contingency causing the already Weak boundary to experience thermal insecurity or loss of steady State stability problems. Thus these weak boundaries connecting stiffly connected generator groups have been shown through this analysis to be associated with both eigenvalue based and loss of generator disturbance based steady state stability problems. This RP 1999 project also developed algorithms and computer programs for: (1) identifying stiffly connected generator groups and the associated weak transmission boundaries; (2) determining and ranking (according to vulnerability to stability or security problems) the network elements that belong to the weak boundary surrounding each stiffly connected generator group; (3) a security assessment methodology that includes: (a) a contingency measure that can accurately detect single, double, or triple loss of generation or line outage contingencies that experience thennal or steady state stability limit violations on one or more network elements; (b) a network element measure that can detect whether a network element experiences one or more thermal overload or steady state stability limit violations over the set of single, double, and triple contingencies evaluated; and (c) a system security index that measures all the rela- tive thermal or stability limit violations over each contingency and network element combination times the probability of occur- rence of the contingency and times the relative weighting of ‘the network element's importance to security for the base case Operati ng conditions. U- 1:- r ”U A b oi... ‘N n v .0. u do i | 1 u ‘5 A- Y F ' O I .. U .1.. .\1 Q d I. R U 0 u :I R i In, t .s a V A n i. 1-. u o u n In ' . I. I I .\v II n V I I. ai- O h \‘I \l. .1. I. , .. 10 The existing steady state voltage stability methods [5-8] address the eigenvalue based steady state stability problem. Sen- sitivity matrices relating operating or dependent variables such as load bus voltage to control variables such as terminal voltage E on generators, turns ratios on off nominal tap changing transformer, and susceptance of capacitors have been developed [58]. The papers [5,6] showed that the sensitivity matrices, which are products of an inverse matrix times another matrix, will have positive elements as long as the matrix being inverted is approximately an M matrix and thus has strictly positive eigenvalues. The sensitivity matrices will then also have strictly positive entries. The paper [8] did not attempt to analyze the structure of the sensitivity matrix relat- 'i ng operating and control variables as in [5,6]. The loss of sta- bility was inferred if a significant percentage of the elements in this sensitivity matrix change sign as the operating conditions Change. Such changes in sign obviously reflect changes in the sign of eigenvalues of the matrix that is inverted. Thus both methods assess eigenvalue based steady state voltage stability. 1 - 4- Merview of the Thesis In order to achieve the objectives defined in section 1.1, two parallei approaches will be used to explain the causes of the v("II-39¢? Problems. Since the nature of the structure dictates the behaVIOY‘ and response of the network, both approaches concentrate on the Structural causes of these voltage problems. The first approach ‘sabgnl. _.: p. h 11 is the identification of weak boundaries. The second approach is the sensitivity analysis. Chapter 2 will develop the weak boundary identification pro- cedure based on a linearized loadflow model. In a power network each source bus has its own upper and lower limits for reactive power generation. 0n the other hand, the reactive power at each ‘load bus is specified. If the reactive requirement at a source bus exceeds the limit the reactive power injection will be fixed at that 'limit. The source bus is then considered as a load bus, because it can no ‘longer match the demands of the load reflected to it or adjust for disturbances. If one increases the reactive power requirement at load buses in a group until it exceeds the capability of all the source buses in_a group, all source buses in that group will be con- Verted to load buses. This will cause that group of buses to lose the voltage control. This problem is due to the fact that the ‘3 ncreased reactive load is only reflected to reactive sources in its 0"" group, and because the weak boundary isolates one group from the Other groups. If reactive load continues to increase and there are "0 SOurces in that group, sources in other groups will attempt to Supply the reactive power but in so doing cause the voltage to col- 1 3P59- Once that specific group of buses loses voltage control, its helghbori ng groups will not be able to use that part of the network 110 transfer real power. Voltage collapse in the group can also Cause V0] tage problems for the whole system. 12 Chapter 2 defines what is meant by a stiffly connected group in terms of voltage phase and/or voltage magnitude. A method is then proposed for determining the stiffly connected groups and is then shown theoretically to detect such groups. Finally, the above discussion of how these weak transmission boundaries can cause volt- age col 'lapse due to insufficient reactive reserve within each group is discussed. Chapter 3 then presents computational results on the New England system that determines the stiffly connected groups and indicates how the weak boundaries surrounding these groups are affected by load level, line outages, capacitors, and local reactive reserve. In the second approach, the sensitivity analysis, the same 1 ineari zed loadflow model will be used but for a different purpose. I n order to determine the controllability. of voltage magnitudes at the load buses and the Observability of changes in reactive power 1 oad at the source buses, two algebraic equations will be developed ‘3 n the sensitivity analysis. These two algebraic equations will be Called the controllability equation and the Observability equation, resilectively. Based on these two equations it can be shown that the 1 ack of controllability will induce stability problems and the lack or Ob$eY‘vability will induce stability problems. Chapter 4 will discuss the theoretical part of the sensitivity approach, which includes the mathematical model, a set of theorems QQrived based on these two equations. These results indicate that a 1°55 0f voltage controllability or Observability can occur due to 13 weak boundaries, thus combining the results of the weak boundary and sensitivity approaches. It is also shown that the capacitive reactive sources at any bus or stiffly interconnected group are constrai ned by the stiffness of the elements connecting the bus or connecting this group to the rest of the system, respectively. Chapter 5 will discuss the experimental results of the sensitivity In both Chapter 3 and Chapter 5 the 30 bus New England analysis. system will be used. Chapter 6 provides an overview of the theorems and results of the thesis and their application to the power industry. A briei= discussion of future research is also included in Chapter 6. l on. ;,f so I“ l.- u...’ . SR ‘~ a ~I~r u ‘ Eff. 'v1' 0 ‘ b 'o . u“\;H V‘ I o.- n:';o1 u": v ‘,. u. “a " :n. CHAPTER 2 METHODS FOR THE DETERMINATION OF WEAK TRANSMISSION BOUNDARIES 2.1. Introduction The linearized load flow model developed in this chapter will be uti‘l ized in this research to study interrelationship of sources and loads. This load flow model will be used to first define what i s meant by a stiffly connected group and a weak transmission bound- ary. The basic concept of a stiffly interconnected group is similar i n nature to the concept based on the electromechanical analog dePicted in Figure 1.1. As pointed out in the previous chapter, these weak boundaries decouple the stiffl y~ connected groups and pre- Vent the real power from being transferred from one group to another. A‘Ithough the existence of weak transmission boundaries is well under— stood, there is no method for determining the location of these weak boundaries. Under normal operating conditions an experienced opera- tor may be able to tell where these weak boundaries are in his area. Howaver, for the larger regional interconnected networks that are more C°mpl i cated and for the case where abnormal operating conditions Caused by multiple line outage or loss of generation contingencies. Operator eXperience may not be sufficient to locate these weak boundaries. Systems planners also need methods to determine the 14 15 location of weak transmission boundaries and their relative level of insecurity to ensure the security and reliability of the system. Therefore the concept of strict synchronizing coherency for both voltage and phase, and a set of measures will be developed for locat- ing and ranking of the insecurity of weak boundaries. The measures will then be proven to detect strict synchronizing and thus the location of weak boundaries. Finally, the decoupling of stiffly connected groups of buses that characterize the weak transmission boundaries will be justified as the cause of voltage problems, because (a) the voltage changes at buses due to all line outage or loss of reactive source contingencies are used in the algorithms developed to detect the location and rank the insecurity of weak boundaries; ( b) the weak boundaries decouple the buses in different stiffly ”i nterconnected groups and thus prevent the requirement for reactive Sapply to cross such boundaries and the reactive supply to cross the weak boundaries; and (c) the local reactive reserve within each Stiffly interconnected group may not be sufficient to handle the i"eactive load change which cannot be met by sources in other groups due to the weak boundaries. The sensitivity analysis in Section 4 will develop the neces- Sary C°nditions for loss of voltage stability due to loss of control- 1abfllt)’ and loss of Observability. The existence of weak transmis- sim‘ bWildaries will then be shown to cause the loss of controllability ahd Observability induced stability problems. 16 2.2. The Linearized Loadflow Models and Stiffly Connected Buses The following three linearized loadflow models will be dis- cussed in this chapter: 1) A.C. loadflow model: "AP [aP/ae aP/av [ae‘ .. (2.1) [A0] _aQ/ae aQ/avj AVJ 2) Real power and phase angle (P - e) loadflow model: [AP] = [aP/ae][ae] (2.2) 3) Reactive power and voltage magnitude (Q - V) loadflow model: [40] = [aQ/aVJEAVJ (2.3) Where P is the vector of real power injections or residuals at the buses Q is the vector of reactive power injections or residuals at the buses 9 is the vector of voltage phase angles at the buses V 1's the vector of voltage magnitude at the buses I n genera] , at any bus 1 we have n P. = - - i=1 Where 17 fl i=1 n i=1.j#1 n aQi/aei =2 viijij cos (9]. - ej - (1.3.) (2.3) 3:1,.13‘1 n 3P1./avi =Zijij cos (61. - ej - yij) + ViYfi cos (41.1.) J=1 (2.10) n 3‘1 (2.12) 18 P and Q are the real and reactive power injections, respectively i,j are the index of buses Y.. is the magnitude of the admittance between buses i 13 . and J yij is the phase angle of the admittance between buses i and j Since it is assumed that there is no power dissipation in the network the Jacobian matrices are singular, therefore the inverse matrix of each of the Jacobians does not exist unless we take one of the buses as reference. Without loss of generality let the buses in the network be indexed from O to N, with bus 0 as the reference bus, and define the reduced Jacobian matrix as FAB. PaP/aé aP/aV. [25‘ ~ [AO‘ _aO/aa aO/aV] _AV] q aPS/aeS aPS/aec .aPc/aes aPc/aec d where J6 is the phase Jacobian matrix P is the vector of real power injections or residuals at the buses except the reference bus Q is the vector of reactive power injections or residuals at the buses except the reference bus . 7.3? n. pa. , i l .b._ 1'- ‘. -As ,. I) I U..“.. .n. b . 1". I O .4: I i I. n’. I A s: ' s 19 5 is the vector of differences of voltage phase angle at the buses with respect to the phase angle of the reference bus 9 is the normalized vector of voltage magnitudes at the buses with respect to the voltage magnitude of the reference bus Note that from now on we will drop the symbol “”" for the referenced power model and the term Jacobian matrix will replace the term reduced Jacobian matrix in this thesis. Definition: Strict Synchronizing Coherent Group (SSC) A group of buses are called strict synchronizing coherent if the voltage angles and relative voltage magnitudes of each pair of buses in the group respond identically for any disturbance. This definition of strict synchronizing coherency can be applied to the nonlinear model and to all three of the linearized loadflow models. It will now be shown that a sufficient condition for SSC to hold is that there be n - 1 elements with infinite admit- tance that form a tree and connect all n buses in the SSC group. It is intuitively clear that this sufficient condition will cause SSC to hold in the nonlinear model since all n buses are shorted together and thus form a single equivalent bus. This sufficient condition for strict synchronizing coherency is now proven to hold in each of the linearized loadflow models and is then proven to be detected by an appropriate coherency measure. The proofs are developed for the P- a, Q - V, and AC loadflow models, respectively.. It can be proved that the n- 1 admittances connecting the n buses in a group to form a tree will cause phase, voltage, and both voltage and phase angles for buses in the same SSC group to respond identically for appropriate 20 disturbance. Then a phase coherency measure and a composite voltage/ phase coherence measure can be established to detect this SSC property. 2.3. Detection of SSC Group in P - e Loadflow Model The P - e loadflow model is now broken into a study group and a test group and the group is then assumed to be connected by n - 1 infinite admittance elements forming a tree. This group of n buses is then proven to be an SSC group. Now let S be the index set of the study group, and S ={ 1, 2, ...., m } C be the index set of the coherency test group, and C ={ m + 1, m + 2, ...., m + n } and the real power/phase angle Jacobian becomes .- - I- - u- 1 APs aPs/aeS aPS/aec ass 3 (2.14) LAPC. LaPc/aeS aPc/aec_ (Paco where , - aPs/aeS aPS/aec J = (2.15) _aPc/aes aPC/aecJ is the phase Jacobian matrix. The sufficient condition for SSC requires that there are at least n - 1 interconnections such that Y.. + a that connect all n buses in the test group. Therefore sub- 13 matrix [aPc/aec] has the following properties: 21 1) It is a symmetric matrix. 2) It has at least 2(n - 1) infinitely large off-diagonal elements. 3) All the n diagonal elements are also infinitely large and thus can be expressed as [aPc/aeCJ'l = l/uEH] for u > o and u + o (2.16) where the Hij is the element of H at the ith row and jth column. Then H = 0 if no connection between buses i and j ij = “eij if it is not an infinitely stiff connection = Eij if it is an infinitely stiff connection where EIj'S are non-zero real numbers in the same order of magnitude. Property 2.1: If there is a set of n - 1 infinite admittances connecting all n buses forming a tree in the n bus test group, then (1) [aPc/aech + [0] (2.17) (ii) aec + 0 for any disturbance AP and this test group is a strict synchronizing coherent group. Proof (i): Since SSC holds in the P - 6 model, it can be shown that this condition causes 22 1 {I/uIHJI'l [aPc/aeCJ' (2.18) u{[H]}'1 where u > O and u + 0 implies [aPc/aecl‘1 + [0] Proof (ii): Given that [BPS/36$] and [aPC/aec] are nonsingular square matrices, and the corresponding partitioned Jacobian inverse is K11 K12 1‘1 = (2.19) K 21 “22‘ L such that K 11 {[aPS/aes] - [aPS/aec][aPC/aec]'1[aPc/aes]l"; K12 = -K11[3Ps/aec][aPc/aec]'1 (2.20) K21 = -[3Pc/aec]'1[aPc/865]K11 K22 = [aPc/aecj'l - [aPc/aec]'1[aPc/aes]K12 Now if [aPc/aGCJ'l + 0, then 23 K12 * [0] l‘21 * [0] K22 ‘* [0] A9 5 Solving for that A6c A6 AP K11 0 AP ll Ca ll LABC. LAPC_ L 0 0. LAPC. (2.21) in (2.14) and substituting (2.20), it is clear (2.22) Therefore aec-+ O for any APs and APc and thus the test group is SSC. Having shown that the SSC sufficient condition holds in the linearized P - e loadflow model, a measure 2 1/2 Ce(k,1) = E{[A6k - aei] } APs E{AP} = E = 0 APC t _ EIAPAP } - Re is proposed and is then shown to detect the SSC property. Before proceeding, the coherency measure is written as (2.23) (2.24) 24 _ t 1 Ce(k,1) - {eklseekl} (2.25) where the jth element of the vector ek1 is defined as 1 if j = k 0 otherwise with k,1 = 1, 2, ...., m + n 59 = EiAeABt} (2.27) Property42.2: The coherency measure satisfies C9(k,l) ;:s for any small 6 > 0 if bus k,1 belong to the coherent group i.e., k,1 = m + 1, m + 2, ...., N Proof: Since , . K11 K12 J‘l _K21 K22 which is partitioned into the study and SSC group respectively, then - - )- - I- -t K11 K12 APs K11 K12 t _ t t 1629 - [aPS aPc] (2.28) [K21 K22] _Apc_ _K21 K22] 25 but from property 2.2 we have K12 + [0] K21 + [0] (2.29) K22 + [0] so that t t ‘ KHAPSAPSK11 0 . aeaet-+ (2.30) . 0 0 In order to detect the SSC group we can artificially create a set of loss-of-generation contingencies such that E{APAPt} has all elements on the principle diagonal nonzero. Therefore. the coherency measure between each pair of buses becomes _ t 1 ={ e:1E{AeA6t}ek1}I 1/2 ’ t t O t Hence Ce(k’1) + 0 if both buses k and I belong to the SSC coherent group (k,1 = m + 1, m + 2, ...., N). 26 2.4. Detection of SSC Group in 97- V Loadflow Model The Q - V loadflow model is now broken into a study group and a test group, and the group is then assumed to be connected by n - 1 infinite admittance elements fanning a tree. This group of n buses is then proven to be an SSC group. Now let - . F q r - AQs aQS/aVS aQS/aVc AVS = (2.32) LAOS. Lan/avS an/Bvc‘ Lavc‘ Define aQS/avS sQS/BVc JV 3 (2.33) _an/avs aQC/avc- If buses i and j belong to the SSC group, then and the submatrix [an/avc] has the following properties: 1) It is an asymmetric* matrix. 2) It has at least 2(n - 1) infinitely large-off diagonal elements. 3) All the n diagonal elements are also infinitely large. Let *It can be structurally symmetric with all symmetric elements. 27 [ace/ave] = 17111111 (2.35) u>Oandp+O Hij be the element of k at the ith row and jth column Hij = 0 if no connection between buses i and j = "eij if it is not infinitely stiff connection Eij if it is an infinitely stiff connection where eij's are non-zero real numbers in the same order of magnitude. Property 2.3: If there is an n - l admittance connecting all n buses ferming a tree in the n bus test group, then -1 (1) [soc/ave] + [0] (ii) AVc + [O] (2.36) for any disturbance AQ and this test group is a strict synchronizing coherent group. Proof (i): Since SSC holds in the Q - V model, it can be shown that [soc/ave)“1 {l/uEHJI'l (2.37) 11111114 28 where u > O and u + 0 implies [30 /av 1'1 + [0] c c _ Proof (ii): Assume that [aQS/BVS] and [an/aVc] are nonsingular square matrices, and the corresponding partitioned Jacobian inverse is K11 K12 1'1 (2.38) _ 21 K22 such that 1 K11 — {[aQS/avsj — [aQS/avc][an/avc]'1[an/avs]l- K12 = ‘KIIERQS/PVCJEPQc/avc1-l (2.39) _ -1 K21 - ’[an/avc] [BQc/SVSJKll K22 = [soc/aver1 - [an/aYCJ'IIaQC/avslk12 Note that if [an/avcj’1 + 0, then K12'+ [0] K21 +,[o] K22+ [0] 29 AV 5 Solving for in (2.32) and substituting (2.39), we get AVc [Avs AQS [kll 0 ADS _ -1 _ - JV - AV A0 0 0 AQ .- c—n L CJ b J E. c-L Therefore AVc + [O] for any AQS and ADC and thus the test group is SSC. Having shown that the SSC sufficient condition holds in the linearized Q - V loadflow model, a measure - 2 Cv(k,1) - E [AVk/Vk - AVl/Vl] p I! AQS EIAQ} = E = 0 AQ - C‘ EIAQAQt} = RV is proposed and is then shown to detect the SSC property. Define a diagonal matrix r . As O [A] = ..0 AC1 30 .th where the 1 element on the principle diagonal of [A] is l/Vi with i = 1, 2, ...., m + n, ek1 as defined in (2.26), and define the ' EIAYAYt} (A II (2.40) E{[A][AVAVt][A]} where AVk = AVk/Vk, then the coherency measure has the following form: C (k 1) = {et 5 e 1* (2 41) v ’ k1 V k1 ° Property 2.4: The coherency measure satisfies Cv(k,1) ; e for any small a > 0 if bus k,1 belong to the coherent group, k,1 = m + 1, m + 2, ...., m + n. Proof: Since K11 K121 -1 JV - , k K L 21 22_ which is partitioned into the study and SSC group, respectively, then D < D < II D < D < l K11 K12 F _K21 “22‘ K11 [A] h K1 _K21 “22‘ A05 AQc - [AJIAVAVtJIA] q 2 - -K11 K12Wt [40: 40:] _ [K21 K22, [1‘qu 7‘11 K12- [40: A02] ch‘ 3‘21 K22 31 [1] But from property 2.3, matrices K12, K21, and K22 satisfy K12 * [0] K21'+ [0] so that T t t - KIIAQSAQSKII AVAV 1+ [A] L O ' t t AsKllAQsAQsKllxs AYAYt'+ 0 L. [A] (2.42) (2.43) QR D u 4 or: "r .“ f If! ' (I) 32 In order to detect the SSC group in the Q - V model, we can artificially createa set of loss-of—generation (reactive power) contingencies such that E{AQAQt} has all elements on the principle diagonal non-zero. Therefore, the coherency between each pair of buses in the system becomes _ t 1 Cv(k,1) - {eklsvekl} _ t - -t § ASKIIAQS AQ :Kllx S 0 5 i t > T ekIE‘ ’ekl . t 0 0) I 1 {A SSK11E{AQ AQ :Kllks}} O A t 1 ) l 0 0) J Hence Cv(k,1) + 0 if both buses k and I belong to the SSC group. 2.5. Detection of SSC Group in A.C. Loadflow Model The A.C. loadflow model is now broken into a study group and a 'test group and the group is then assumed to be connected by n - 1 irrfinite admittance elements forming a tree. This group of n buses is then proven to be an SSC group. has the following properties: 1) It is an asymmetric matrix. r . APS (BPS/ass aPS/avS aPS/aec AQs 305/365 aQs/avS AQS/aeC APC BPC/BOS BPc/BVS ape/39c -AQQJ baQC/aes aQC/avS aoc/aeC aPs/aes ap /avS E aPS/ae aPS/avc I 305/395 30 /avS i aQS/ae aQS/avc J9: ------------------- I -------------------- ape/39$ 3P /3Vs E BPC/BO BPc/BVC _ BQc/BOS BQ /3VS E BQc/BB BQc/BVC )- J11 ‘112-1 €21 J22‘ and the submatrix aPc/aec ape/avc J22 ’ Lan/aec age/avg- (2.45) (2.46) 2) It has at least 4(n - 1) infintely large off-diagonal elements. *It can be structurally symmetric with all symmetric elements. 34 3) All the Zn diagonal elements are also infinitely large and thus can be expressed as J22 = 1/u[H] (2.47) for u > O and u + 0 where h J.’(‘.h H.. be the element of k at the it row and column lJ I ll ij 0 if no connection between buses i and j “eij if it is not an infinitely stiff connection Eij if it is an infinitely stiff connection where Sij's are non-zero real numbers in the same order of magnitudes. In order to simplify the expressions in the following proofs, let AW t t tt=tttt [AWSAWc] [APSAQSAPCAQCJ t t tt=tttt [AXSAXc] [ABSAVSAOCAV ] AX C Property 2.5: If there is an n - 1 infinite admittance connect- ing all n buses forming a tree in the n bus test group, then (i) 022 + [O] (2.48) (ii) AXC + O for any disturbances AP and A0 ABC i.e. -+O AVc 35 Proof (i): Since SSC holds in the A.C. model, it can be shown that 15; {I/uEHJI‘l (2.49) u{[H]}"1 but u > O and u + 0 implies 022 + [0] Proof (ii): Assume that 011 and 022 are nonsingular square matrices, and let the corresponding partitioned Jacobian inverse be , . K11 K12 J.1 3 (2.50) _ K21 K22 such that -1 -1 K11 = [J11'J12J22J21] K = -K a 0‘1 12 11 12 22 (2.51) _ -1 K21 ' 'J22J21K11 _ -1 -1 K22 ‘ J22"]22J21K12 36 Note that if 15% + 0, then K12'+ [0] K21 + [O] K22+ [03 AXS Solving for in (2.45) and substituting (2.51) gives AXc r T " ‘ r “r - AXs AXs K11 O AWS =J-1 = LAXC‘ ”AXC'J .. O 0.. _ch_) Therefore AXC + [0] for any AWS and ANC and thus the test group is SSC. As a result, the SSC sufficient condition has been shown to hold ir1 the linearized A.C. loadflow model. A measure [9] 1 C(k,1) = E{(Aek - A01)2 + (AVk/Vk - AVl/V1)2} (2.52) = {at [s ]e + at [s ] 1é k1 6 k1 k1 v ek1 AP E = 0 40 A E [4PtAQt] = R 37 is proposed and then shown to detect the SSC property. Let 59’ Sv’ AS, and Ac be defined as in the previous sections of this chapter. Additionally, let , 1 for j = k or j = k + m {ek1)j = l-l for j = 1 or j = 1 + m (2.53) ‘ 0 otherwise 5 = EIXAxAxti} (2 54) with '. ] AS O 1: L.0 AC- '1 ol XS 3 (2.55) 1° *s. '1 ol Ac: 1° *c- where 38 IS is 2m x 2m matrix 1c is 2n x 2n matrix 1 is 2(m + n) x 2(m + n) matrix Then from (2.53), (2.54), and (2.55) I _ 2 C(k,1) - E{[Aek - A81] + [AVk/Vk - AVl/V1]} = {at E{A6Aet}e + et E{AVAV }e % k1 k1 k1 k1} ”A6; § = 't t -t t 't - } (eklEi Avs [ABSAVSABCAVC])ek1 (2.56) Aec .PVC = {EEIAEIAXAthekI )5 Property 2.6: If k,1 belong to SSC then C(k,1):é e for any small e > 0 Proof: Since K11 K12 J'l - , _K21 K22_ 39 which is partitioned into the study and SSC groups, respectively, then I F q I- q t K11 Klzl Aws K11 K12 AxAxt = [AWEAWE] (2.57) _K21 K22_ _9”c, _K21 K22 . But from property 2.5 we get K12 * [0] K22 * [0] so that r - t t Kllawsawskll 0 AXAXt'+ (2.58) In order to detect the SSC group in the A.C. model, again we canartificially'create a set of loss-of-generation (real and reac- tive power) contingencies such that EIAWAWt} has all elements on the Principle diagonal non-zero. Therefore, when measuring the coherency between each pair of buses in the system we get 4O _ - 1 _ -t t - C(k,1) - {ek1E(AAXAX Held} —- G I = 0-] AS O AXs AS i - -t t t - - ‘eklEl [AXS AXC] *ekl] .0 Ac‘ [Axe .0 Ac‘) ll 0'! .K AWS AWth 0. pi 0“ I s 11 11 s _ -t )e +1 {eklEY k1} (:0 )‘c. L 0 0. L0 20‘ V i SK AWS Awthi 0) 11 11 S 1 - ‘t )" ‘ (ekiE‘s eKI) 1 0 0.1 XSK11E{{AwSAw§K111 S 0 § - -t 4 )- ‘ (9k) ekl ) L O 01 since _ l -t ’- C(k,1) +~ ‘ek1w< ekl} ( 0 . 0J Therefore C(k,1) + 0 if both buses k and 1 belong to the SSC group. 2.6. The Limited Sources and Weak Boundaries Physically for each source bus or generator bus there is a set of lower and upper limits of reactive power injection, i.e., o" 41 Qmin,i fi-Qi f-Qmax,i at source or generator bus i. Whenever Qi hits one of the limits, the reactive power injection will stay there. As far as we are concerned in the loadflow solutions, this particular source bus now is regarded as a load bus, and it can no longer take action in responding to any reactive power disturbances to match demands beyond its capability. If a stiffly connected group always has the reactive power demands exceeding the total capability of the group, eventually all its source buses will be converted into load buses. under this situation, due to the fact that the reactive power cannot be transmitted over long distance (across weak bounda- ries). this group will lose the voltage stability. If each stiffly connected group is regarded as a single bus, this group with heavy reactive load will be regarded as a load bus to those interconnected networks. One may ask if this is the case then can a set of guidelines or operational constraints be developed for each of these stiffly connected groups based on our understanding about the behavior of each individual bus.‘ The answer is yes. If the constraints and rules for voltage stability can be developed at each bus then these constraints can be generalized to the groups. We will show this in Chapter 4 using the sensitivity analysis. The weak transmission boundaries for a power system are the branches that connect the buses in different SSC groups. It is clear that these boundaries are referred to as weak because none of the branches that connect the buses in these groups has infinite admittance where within each SSC group of buses there is an infinite .riao~ n a I I. .y no - uvrd . that get IPOI'LA o I. vv "A. b ...3 no. r 1.. l. I: , 1‘ " pnA.1 - .. _ I "1..” c n“, “P.“ u ‘- 42 admittance path connecting all buses. In practical networks the SSC groups will not be connected by infinite admittance branches but rather the collection of buses within each such group are more stiffly interconnected than the buses between such groups. Operating condi- tions such as loadflow can possibly modify the SSC groups in a net- work. Moreover, the SSC groups defined based on phase, voltage, and both voltage and phase (current) may be different. The next chapter will investigate the SSC groups based on the phase, voltage, and current coherency measures. The weak transmission boundaries can cause the security and stability problems for power systems. It was shown that the phase coherency measure (regulated loss of generated contingencies) based on inertial loadflow identified the weak transmission boundaries that decoupled the dynamics of the classical transient stability model in [1]. This decoupling could cause the phase oscillations between regions in power systems that have recently been observed. Furthermore, the weak boundaries identified by the phase coherency measure have been shown to identify the branches that are severely affected by any line outage or loss of generation contingency in a 49 bus test system. Thus the branches in the weak boundaries are the insecure elements in the network and once identified indicate the contingencies that could most severely affect system security by affecting particular branchescn~groups of branches in these weak boundaries. 43 The above study [1] was performed for thermal and steady state phase stability problems. The weak boundaries will be shown in Chapter 4 to prevent changes in reactive load from being reflected to reactive sources in other SSC groups. Thus, sources in one SSC group will not provide reactive power for reactive load requirements in another SSC group- Furthermore, the weak boundaries will be shown in Chapter 4 to require voltage support for load buses be provided by the voltages established at reactive source buses in an SSC group. The voltage at load buses in one SSC group of load and generator buses will therefore not be significantly affected by the voltage magnitudes established at source buses in other SSC groups. It is thus clear that voltage control is local within each SSC group. Therefore, if reactive load increases or decreases, the source buses in an SSC group must compensate. If the reactive load continues to change such that all source buses in an SSC group reach the upper or lower limit, these source buses in the SSC can no longer provide voltage control. Additional changes in reactive load will also-cause large voltage changes within the SSC group as the network attempts to provide reactive power through the weak boundaries. It is thus clear that there must be sufficient positive and negative reactive reserve in each SSC group to preserve voltage stability problem and is investigated in the computational results on the 30 bus New England system given in Chapter 5. CHAPTER 3 DETERMINATION OF WEAK TRANSMISSION BOUNDARIES FOR THE 30 BUS NEW ENGLAND SYSTEM 3.1. Introduction The purpose of this chapter is to determine the weak trans- mission boundaries for phase, voltage magnitude, and current on a 30 bus New England system. The weak transmission boundary for phase is based on a phase coherency used to determine the coherent groups to be aggregated to produce dynamic equivalents [3] for transient stability studies. This same phase rms coherency measure evaluated for all inertial loadflow simulated loss of generation contingencies was shown [1] (a) to detect the weak transmission boundaries to cause the phase oscillation problem based on an analysis of the classical transient stability model, and (b) to determine the weak transmission boundaries and the associated branches that experience thermal or phase stability problems for line outage or inertial loadflow simulated loss of generation contingencies. The weak phase transmission boundary based on the rms phase coherency measure is evaluated in this chapter based on load contin- gencies at all buses in the network rather than just loss of genera- tion contingencies. The loss of load disturbances at all buses should be more robust and should establish the weak transmission boundaries 44 45 between groups of load and generation buses and not just between groups of generator buses. The weak transmission based on this phase coherency measure should establish the boundaries and associated branches that are vulnerable to phase stability problems. The weak phase transmission boundaries are determined for three cases: active power load disturbances at every bus, reactive power load disturbances at every bus, and complex power load disturbances at every bus. It is found that the weak phase transmission boundaries for active and complex power disturbances are quite similar and reflect the electrical distance from the swing bus. This occurred because the mismatch due to the loss of load is eliminated by a similar loss of generation at the swing bus causing a power flow to the swing bus from the disturbed bus. The weak phase boundaries for loss of reactive disturbance had no pattern and reflect the weak coupling of phase and reactive power. A voltage coherency measure is used to determine the weak volt- age transmission boundaries for loss of load at all buses. The weak voltage transmission boundaries reflect the boundaries between groups of PO and PV buses where voltage security and stability problems should occur. This is confirmed by the results in Chapter 5 based on a set of multiple loss of generation and line outage contingencies. The weak voltage tranmission boundaries between groups of PV and P0 buses are determined by converting all PV buses to PO buses so that the PV buses can experience voltage swings that will reflect the stiffness of the transmission grid and not the action of the voltage controls. The weak voltage transmission boundary for active power 46 and reactive power disturbances encircle PO and PV buses that are all interconnected and lie in a small geographical area. The weak boundaries for reactive power disturbances and real power distur- bances are quite similar showing that weak voltage boundaries are sensitive to both active and reactive power flows. The principal difference between the weak voltage transmission boundaries for active and reactive power disturbances is that there appears to be additional separation of groups of buses near the swing bus for real power disturbances, and the groups of buses for reactive power dis- turbances appear to be based more on local transmission network characteristics. A current coherency measure is used to determine the weak current transmission boundaries for loss of load disturbances at all buses. The weak current transmission boundaries represent the bounda- ries between groups of PV and P0 buses where thermal security problems should occur. All PV buses are converted to PO buses so that the PV buses can experience both phase and voltage swings that are determined by the network and not the voltage controls. The weak current bounda- ries determined for active and complex power disturbances are iden- tical. Moreover, the weak current boundaries are very similar to the weak phase transmission boundaries, reflecting that the phase stability and thermal overload problems are related and are likely to occur on the same transmission boundaries. This was tacitly assumed in the recent EPRI study 47 All computer simulations in this chapter as well as in Chap- ter 5 use the 30 bus New England system. This system has 10 generator buses, 20 load buses, and 37 branches as shown by Figure 3.1. The solved loadflow data or the base case data is listed in Appendix 1, using common format for loadflow data exchange. The phase, voltage, and current coherency measures defined in Chapter 2 were derived based on a linearized loadflow model and a probablistic disturbance model. Although the coherency measures defined in this manner were shown to detect the SSC groups, it is not convenient for computing the nns measure. It can be shown that the phase and current coherency measures can be evaluated using measures 1 S6 = i Z (A6k(i) - A51(i))2} (3.1) 181 . . 2 % sv = { Z(:V—§:-1—)— - AVE/3:”) } (3.2) 181 . . 2 A S ={ZI {(A8k(i) - A61(i))2 + (A—WG-El - fill—:11) }} (3.3) is where A5k(i) is the phase deviation and AVk(i) is the voltage devia- tion at bus k for contingency i. The set of contingencies{I}must be selected so that it models the statistical description of the disturbance. U) 48 \0 Figure 3.1. if 29 o -——-- .. 25 “T 25 2h 27 —-—— 16 J 21 15___J.. 22 lb 23 - l9 —— 1.2 MA NW NV\ 11 20 -—-—- 13 10 The original 30 bus New England system. 49 It can be shown in a manner similar to that in [2] that if the set of contingencies ieI are a set of load disturbances at each of the N buses in the network, the statistics of the disturbance will be an identity matrix and the coherency measure is strictly based on the relative stiffness of the network model connecting buses k and 1. The phase, voltage, and current coherency measures are evaluated for a set of N 100 MW loss of active power disturbances at each of the N buses. The phase, voltage, and current coherency measures are also evaluated for a set of N 100 MVAR loss of reactive power dis- turbances at all N network buses as well as a set of N 100 MW and 100 MVAR loss of complex load disturbances at all N buses. The weak transmission boundaries are determined for each coherency measure for all three sets of active, reactive, and complex power disturbances. An improved current rms coherency measure is now proposed. Let AVk(i)€jAek(i) AIk(i) = IZkllchkl where IZkll and Ykl are the magnitude and phase of the impedance of the branch connecting buses k and 1. The natural logarithm is a monotonically increasing function of the argument, and thus will preserve the relative magnitude of AIk(i) and AI1(i). Therefore define AIk1(i) 50 AIk1(i) = 1n AIk(l) - 1n AI1(i) AVk(i) = 1n -tj(A0k(i) - Ykl) lzkil AV1(l) ' 1" +j(A91(l) ‘ Ylk) (21k! Wk“) . . . =11) AVl i + J(A8k(l) - Ael(1)) The improved rms current coherency measure is defined as {ZIAIEIII icI 1 2 2 . 2“” mm} + (e (1) - e (1)) (3.4) icI .ZVITTY k 1 The rms current coherency measure can be shown to detect stiffly interconnected groups of buses and is zero if the AVk(i) = AV1(i) and Aek(i) = A81(l) which indicates buses k and 1 are coherent. The identification of weak transmission boundaries is based on a grouping method that utilizes the coherency measures evaluated for all bus pairs k and 1. The grouping method is based on the commuta- tive rule [22]. This rule for forming a group requires that a group be formed if and only if all the generators are coherent with respect to each other; that is, if the Group CI is a group containing 51 buses A and B, then bus C is added to this group if and only if gen- erator C is coherent with A, and C is coherent with B. This method has been used [2,3] for clustering generators in coherent groups for producing dynamic equivalents of the system for transient stability studies. The values of the coherency measures are ranked from the smallest to the largest forming a ranking table; then the groups are formed based on the following algorithm: (1) Fonnthe first group (a pair) from the smallest coherency measure at rank 1, r = 1. (2) Decide which of the following possibilities apply to buses k, 1 at the rank r = r + 1. (3) If r = N x ifl—%—ll , stop. N = number of buses. (a) If neither k nor 1 has been previously identified as belonging to a group, then this pair becomes a new group. (b) If bus k(1) belongs to a group but bus 1(k) does not, then (i) If 1(k) has been previously recognized as coherent with all members of the group to which k(1) belongs except for k(1), then add 1(k) to the group con- taining k(1). (ii) If 1(k) has not been found previously to be coherent with all other members of the group to which k(1) belongs, then recognize that k and 1 are coherent but do not add 1(k) to the coherent group containing k(1). Return to (2). 52 (c) If buses k and 1 belong to different groups, then (i) If all possible bus pairs which can be selected from the members of the two groups except k and 1 have been previously recognized as being coherent, then merge the two groups to form a single group containing all members of the separate group. (ii) If at least one pair of buses which can be selected from the two groups other than k and 1 has not yet been recognized as a coherent pair, then recognize the pair k and 1 as coherent but do not merge the groups. Return to (2). The algorithm continues the procedure to the bottom of the ranking table, and when every bus pair is checked, it terminates. As one proceeds down the ranking table, individual buses are included in groups and later groups are merged to form larger groups. As groups are merged, the boundaries between groups should be continuously weaker since a coherency measure between bus pairs indicates stiff connection of the buses, and the coherency measures are ranked from the smallest to the largest in this ranking table. Thus, the bounda- ries may be ranked from the weakest to strongest based on the reverse order of the group formation; that is, the last two groups to be lumped into a single group have the weakest boundary between them, and the second weakest belongs to the second-to-the-last group aggre- gated, etc. 53 The commutative grouping rules assure that all the buses in a group formed haVe stiff connection by requiring a bus to join the group if and only if it is coherent with all the buses in the group, and groups are merged if and only if all the buses in group one are coherent with every bus in group two. This ensures at least n - 1 stiff connections exist between buses in an n bus group. Now to sum up the procedure for identifying vulnerable bounda- ries, the following steps are given: (1) Compute the coherency measure C(k,1) for all bus pairs. (2) Rank the coherency measures from smallest to largest and form a ranking table. (3) Form groups by the commutative grouping rules and set a group formation table. (4) Rank the boundaries from the weakest to the strongest based on the reverse order in the group formation table. A Fortran computer program has been developed to implement this procedure. It requires files of the base cases loadflow angles and loadflow simulations of all loss of load contingencies to compute the coherency measure. The output gives the groups of buses in reverse order to formation indicating the increasingly stronger boundaries as existing groups are broken up to form larger numbers of groups. 3.2. Weak Phase Transmission Boundaries The weak phase transmission boundaries are identified based on the rms coherency measure (3.1) evaluated for loss of load distur- bances at every bus. As indicated above, the rms coherency measure 54 has been utilized to (1) determine groups to be aggregated to produce dynamic equivalents; (2) to identify the weak transmission boundaries that cause decoupling leading to phase oscillation problems based on classical transient stability model; and (3) to identify weak bounda- ries that experience thermal overload and steady state stability problems for line outage and loss of generation contingencies. The rms phase coherency measure has previously been simulated for strictly loss of generation contingencies using an inertial loadflow simula- tion. The results in this section are based on loss of load contin- gencies and a conventional loadflow simulation which reduces the generation of the swing bus to compensate for the loss of load. The inertial loadflow would reduce the power at every generator bus in proportion to the ratio of its inertia to the total inertia of all generators in the system. .Cg§g_l: Real power loss of load disturbances The phase coherency measure is computed for the set of 100 MW loss of load disturbances and the grouping method is utilized to identify the stiffly connected groups and associated weak transmission boundaries. Figures 3.2, 3.3, and 3.4 show the weak boundaries for a 3, 4, and 6 group partition of the network. The three-group parti- tion shows the three weakest transmission boundaries. The largest group contains the swing bus (bus 30), most of the load buses in the network, as well as generator buses 2 and 6. All of the remaining buses are grouped based on their electrical distance to the swing bus. The large group III containing the swing bus is split in the \ \ ’ Q \\N) ‘25 \ \Q\ I ‘ --"-b-----~ \ \\ 2 5 --dl ~ ’ ~--‘---‘-.P—--~-- “ \ \\ I 26 ~‘~‘\ \ \ \ Figure 3.2. The weak boundaries for a 3-group partition based on real power disturbance and phase rms coherency measure. 56 "6 ‘\ (£9.1____ ‘ \sqL—u-4 ,. . '¢“ ":1: :33 \‘ 28.. 7.1—— ‘ --‘ I Z s ‘---)--.h- I, 25 "qb- --‘---_~ -‘p- ------‘ '\ I ’I 1"""'" 27 """0 O——— 7..— Iif lei—LL) \ L» l JI {i I z: 4:- t; ‘A \ 1 I ~ 4 ‘~‘-_-___-- “aid. Figure 3.3. The weak boundaries for a 4-group partition based on real power disturbance and phase rms coherency measure. \ § ‘ 57 27 ~- ~~~_‘1’-¢—-¢ ’- -)-I- -- C - The weak boundaries for a 6-group partition based on Figure 3.4. real power disturbance and phase rms coherency measure. 58 four-group partition separating generator buses 2 and 6 from a group of buses containing the swing bus. The buses joining generators 2 and 6 in this group are again the buses in group III furthest elec- trically from the swing bus. Both groups II and IV in four-group partition are further split in the six-group partition again based on electrical distance to the swing bus. The groupings of buses based on electrical distance to the swing bus appear to be based on the method of simulating the loss of 100 MW load contingencies. The loss of 100 MW in active power is matched by a decrease in 100 MW of generation at the swing bus causing an active power flow to the swing bus from the particular disturbed bus. This method of simulating loss of load contingencies identifies the sequence of continually weaker transmission boundaries between the swing bus and the rest of the system. An alternate method of simulating the loss of load contingencies is an inertial loadflow where the 100 MW of generation required to match the 100 MW of load is distributed to all generators based on their inertia. This procedure used in [1] would identify the weak transmission boundaries for the inadvertant flows caused by the decrease in frequency after loss of load contingencies. The weak boundaries based on the use of a swing bus to match loss of genera- tion would be based on an operating procedure which utilizes a single generator in the utility to perform regulation or alternatively is an equivalent external system representation. 59 A set of weak phase boundaries could also be identified by simulating loss of load contingencies using distribution factors to allocate the 100 MW decrease in generation among the generators in a utility. The comparison of results of determining the stiffly intercon- nected groups for this same system based on an inertial loadflow simulation of loss of generation contingencies and based on this set of loss of load simulated contingencies indicates the weak transmis- sion boundaries are similar. Thus it appears that the weak transmis- sion boundaries depend more on the network rather than the type of contingency (loss of load or loss of generation) and the type of simu- lation (inertial distribution or swing bus). The weakest boundaries are those observed in the three-group partition and the boundaries formed by splitting these three groups into four and six groups are successively stronger. It was observed based on a study of 49 bus system in [1] that the branches in the weaker phase transmission boundaries experience thermal overloads ,more often and more severely than the branches in stronger phase boundaries. Moreover, it was observed that all thermal overloads for a set of multiple line outage and all loss of generation contingencies occurred on branches in these weak boundaries. The branches in the weak phase transmission boundaries should be vulnerable to steady state stability problems rather than the thermal overload problems. The weak current transmission boundaries based on the current coherency measure (3.4) should determine the branches that 60 experience thermal overloads. However, the results obtained for the weak current transmission boundaries in section 3.4 indicate they are nearly identical to the phase transmission boundaries, thus explaining why thermal overloads would occur on the weak phase trans- mission boundaries. Cg§g_g: Reactive loss of load disturbances The weak phase transmission boundaries were determined based on a set of 100 MVAR loss of load disturbances at every network bus. The results from this case do not have any value for applications. Moreover, the grouping and weak boundary identification are not creditable. The groups are formed based on the weak coupling of reactive power disturbances to the phase deviation. Since the results have no value for applications and show no viable weakness of the transmission grid, the numerical results will not be shown here. l§g§g_§: Complex loss of load disturbances The weak phase transmission boundaries were determined based on a set of 100 MW and 100 MVAR loss of load disturbances at every net- work bus. The results of three-, four-, and five-group partition are shown in Figures 3,5, 3.6, and 3.7, respectively. The coherent groups in each case are very similar to the groups obtained from case 1 with only real power disturbances. The reason for this simi- larity is due to the fact that phase response is not sensitive to the reactive disturbances. These results also show slight differences on the groups obtained for the real and complex disturbances. 61 a"”—-H\‘~ I \ :29 \\ \" \. a”’ I \ I? 25 —- “:2.‘~ \\ \ -Q \ I "‘:‘ \ -:db---¢D‘ ‘1‘ ‘~ 21.; )\ ’-- _’-‘-‘—--- ~ \ ’\\ /’1.-—-—- 27 :0 I ‘ 1 1‘ 1’ l l \16 I ’ \\, l 4" ’I 9 f at” \ L—J \ x 1’ 21 \ ~ 15 “l": 3‘“"""' I! l 171‘ (’22 _- I 4 ———— 14 --~flfll 5 l J— I/’ I ’ 4) ' ’45! ’1] S-—---' ’}" I I ,r, / 1 I; I ,’ 23-—-- l9 6 * [’I’ I I \ ’1’ :’ ‘ 8——-———- 1’1’ '1’... 2’ \ r I l I" ,..v’ \ ’{fflz l " :: I l . I ”f“‘ I \ __ l .fi” IVVN :4 l \~_d’ I l-__L l‘l‘zo . , l 13-—-———n4 ’ i 1.0 I I \ A) I \ "/ x I \\ I \ z \ 1’ .‘\‘_4’ Figure 3.5. The weak boundaries for a 3-group partition based on complex power disturbance and phase rms coherency measure. 62 F‘ \ 3\ D - _-‘P~ 2 27 1.7 ‘h---- ’ --.- ‘- -- -- ---- Q The weak boundaries for a 4-group partition based on Figure 3.6. rea] power disturbance and phase rms coherency measure. 63 I ~ 1” ::T"~\~ C29 —” ‘\‘\ \\ 0’ ' “\ u-----~ O \ ' on“ Q ’-.P“ ‘2‘ ’0 Q r— 2 . ‘\ / 1*: '. \‘ \\ I ‘\ ‘ ‘~ ‘ \ I 2 s s--- ~----_--‘ \‘\ I -‘D-¢-------:-- ~ 2“, ‘5‘ “‘\‘ ' ’ ‘3. 27 ‘4 \ a"I’/” " ‘43 ‘\16 ’ O’d’ I ‘ l '“s \ Q‘s . '.' ’ ’ I g '{”” ‘fl “ I; c ”’ " ’ I : “30 —"""‘ I 18 “—E 17 """—| ‘ ’4' ‘ ” ‘ .H I; ~~~ 1" 230 'I ! ." I. ’ ls ‘ I 0.-.“: ‘ i W. ' I 3“?"— ,:' {22_—.— u ' I ‘\! ' l .q I o c z. ——- - nil—~10 u ; v... ' : I z’ a : \ g “‘ r: 'g 1” 0' ‘ ‘\ ' “‘ . K ” ’ ‘, ‘ 91 ’ x“ \‘ \\ ” ‘ “ ‘\ _d—— I \“9 \ ; 6 1’ O. ‘ ‘ I ‘ ' I I ‘ " \‘ ~..a o , \ \ \‘ "r‘ I . ‘ ‘~~ “\ [é ’ ---‘ {19""'“_—" \‘ ‘ " p’Tz' 3 I ‘ \ \ / ’1 \‘ ' g \‘ \ ’ l . AAA . \\§’ 1 I’I’ /VV\ ‘t ' \\\‘7— ’l I | NV\ \‘ 0 \\‘\~"’ 1’ 11——-— :‘20-——- ' \ 13.______.J \ I \ i' s‘ a \\ m- , ~ I \ I \v «g‘ i, "~. I’ “~- an” Figure 3.7. The weak boundaries for a 5-group partition based on reaT power disturbance and phase rms coherency measure. 64 3.3. Weak Voltage Transmission Boundaries Weak voltage transmission boundaries are known to be much more local than the phase transmission boundaries identified in the pre- vious section. Previous efforts have focused on identification of local voltage control areas for the purpose of providing proper volt- age control. Recent papers have documented efforts to identify the voltage control areas based on the magnitude of elements in sensi- tivity matrices [15]. No effort has been made to determine and rank the weak transmission boundaries for the purpose of voltage security assessment. The identification and ranking voltage of weak trans- mission boundaries would indicate the branches and boundaries where voltage security problems exist and where line outage, loss of genera- tion, or loss of switchable capacitors, or reactors could lead to voltage collapse, low voltage profiles, or voltage oscillations. §a§g_lz Real power loss of load disturbance The weak transmission boundaries based on the voltage coherency measure evaluated for the set of 100 MW real power loss of load con- tingencies are shown in Figure 3.8, 3.9, and 3.10 for the 3, 4, and 6 group partition of the system. The local nature of the stiffly interconnected groups or voltage control areas is clear in each case. The transmission boundary and the associated branches are the loca- tion where large voltage deviations occur for loss of load or genera- tion contingencies as shown in Chapter 5. Moreover, line outages of branches in these boundaries will be shown to severely aggravate the Voltage security or stability problems. 65 I \ I, Q \ \ / 29,—“ | l’ \ 4"” -‘ ” \ I \s‘ ' 2 ‘s‘ \---‘- 25 ‘ ~\ I 21‘ q -- \‘ ‘Od ‘ "/‘§:—n——::} : 27'L"" x . 16 I I l / 9 x J z I '30 -——--I ’ 18-—[ 17 l ‘ ' t3 -——-——- 15--—- 22 l ‘ " ‘--‘. -‘fl------- -_‘-‘ _______ “ ‘ ’a-v—----"'------"~‘ \ l .1 ____ lh—I—--—m- ‘ l 3' ' I, h I ” ’ $ ‘ U I“, [I V \ ' ‘ ‘ ’ ,I : “\ / l 23"— ‘ \ 9___,‘_‘ \ 6 { \\ 3 \“J.’—" \\ \\ ’ ’8’ ‘~ ‘5 l9--——" ’ t \\ \ ’1 ‘s 7—3 | n } ~v~ ,/ \\7______ MA W " , / s ‘ 11--—-—-L :{20 d—— I, s I \ 13——7 \ l \\ 10- I ’ \ (g I -’ Figure 3.8. The weak boundaries for a 3-group partition based on real power disturbance and voltage rms coherency measure. 66 ’z’-\\ l’zo Q \ I ’ \ I’ \ ‘ ‘-~ Q 28' Kc :71— , ‘2 “ \ " I \ \ ’ ...-.:">\ \ \ ’ ,/ 2b. \ I’f—d"), \ \ \s 2" ,’ // 18‘- 1.7-— \ ’ I I I \ I I I 21 ‘ ’ I 3—---«-—--- 15 ' --1-.------_----:-‘ 'l ' ,"7+ :— lhT—E“, ' ‘, S—'""—_ ”"” ‘ : \‘\ I" ” I" 2'2 . ‘ I ‘ \ I I - ' \c: \.\ , _ I <~~dP " O ’\ ‘ I -v \:\ } ‘ 12 }‘~ // \ L AAA I I \~- 1- ’-C--- I Figure 3.9. The weak boundaries for a 4-group partition based on real power disturbance and voltage rms coherency measure. 67 ’---¢b- ’ --- -“-- ~ 5 _— ‘ Figure 3.10. measure. ‘ 15 \ ‘s _l lhT—M I \ fi‘ "’-& ‘~-’----- - 22 21 The weak boundaries for a 6-group partition based on real power disturbance and voltage rms coherency 68 The three-group partition shown in Figure 3.8 has both reactive sources (generators) and loads in each localarea. The weak boundary, which divides the system, is clearly defined by the lines connecting the following pairs of buses: (1,2), (3,4), (8,9), and (14,15). This is the weakest boundary of the system. The reactive power supply in each of these groups must be sufficient to meet the local loads. The four-group partition of the system, shown in Figure 3.9, also has local control areas composed of both sources and loads. The weak boundary in this case is defined by the following lines: (1,2), (3,4), (8,9), (14,15), (2,25), and (17,27). The difference between the three-group and four-group boundaries is that the latter contains additional but less vulnerable elements (2,25) and (17,27). For a very large system this procedure for identifying and ranking the boundaries and branches in terms of voltage security provides the system planner or the operator a very useful on-line tool to predict the worst contingencies that would affect the weakest trans- mission boundaries. Corrective actions could then be determined to relieve this vulnerability. The weak boundaries could also be dis- played graphically on the control panel to the system operator. The operator would thus have a better picture to select the correct zones to perform the contingency analysis. Another interesting case is shown in Figure 3.10, where the system is divided into six groups. From the base case data it is found that bus 4 is carrying 500.0 MW and 184.0 MVAR load, and bus 14 is an intermediate bus which has no load and no generation. These 69 two buses are at the center of the system and electrically close to the swing bus making them a sort of buffer zone between different control areas. The results for loss-of-reactive-load contingencies are similar to those for the loss-of-real-load contingencies. But the difference of these results is that they can identify the local voltage control areas more precisely and always maintain sources and load buses in each control area. This six-group partition based on real power disturbance did not have a source in the bus 4 and bus 14 group. The interesting result of the identification of weak trans- mission boundaries for real power loss of contingencies is that the weak voltage boundaries identified for reactive power disturbances in Case 2 are also vulnerable based on real power disturbances. .Qa§g_2: Reactive loss of load disturbances The weak voltage transmission boundaries are identified based on the voltage rms coherency measure evaluated for the set of loss of 100 MVAR reactive load disturbances at each bus. Since reactive power is much more strongly related to voltage than is real power, the weak voltage transmission boundaries should be identified based on the set of reactive power loss of local disturbances rather than the loss of real power load disturbances. The five and six group partition of the network using the reactive power disturbances, shown in Figures 3.11 and 3.12 respectively, are similar to the four-group partition in Figure 3.9 for real power disturbance. Group IV in Figure 3.9 is split into three groups in both the five and six group partition shown in Figures 3.11 and 3.12 for reactive poWer‘ 70 u “ ‘ ~------ 15 ——.---~-—" ~- Figure 3.11. The weak boundaries for a 5-group partition based on reactive power disturbance and voltage rms coherency measure. 71 disturbance. Group II in Figure 3.10 is ungrouped in Figures 3.11 and 3.12, which suggests it is a group separate from the other groups in these 5 and 6 group partitions. Finally, group III in the five- group partition in Figure 3.11 is broken into two groups each with a source in the six—group partition in Figure 3.12. Note that all groups in Figures 3.11 and 3.12 have a source bus and surrounding load buses except for group II. Group II is a buffer zone between groups I and II, and group IV and the ungrouped buses. Figure 3.13 shows the partition of the network based on the voltage coherency measure evaluated for reactive disturbance when the generator bus is regulated. Note that all the generators and closely related load buses form a single group because their voltage controls hold their terminal voltages constant. These groups really reflect the action of voltage controls rather than the voltage control areas and weak voltage transmission boundaries in the network. This is confirmed by the results in Chapter 5 that indicate the weak voltage transmis- sion boundaries of Figure 3.11 are indeed the location where voltage problems occur for loss of generation and line outage contingencies. 3.4. Weak Current Transmission Boundaries Weak current transmission boundaries should indicate the bounda- ries and branches where thermal overload problems should occur. (The three-group partition of the transmission grid based on the current coherency measure evaluated for real power and for complex power disturbances are identical and shown in Figure 3.14. This three-group partition based on the current coherency measure is 72 Figure 3.12. The weak boundaries for a 6-group partition based on reactive power disturbance and voltage rms coherency measure. 73 27 \0 ’18 ‘7; __ JLT” _ _. 3 —--1—-— I I, Figure 3.13. :—I /’-‘~ 7’0 n‘--‘-’----4 w \ 'EZ-W " "" \ \ \ I I .3. j - ---"""‘"EL\ “‘ I ?l’ l5 ‘5 \M 23‘” The weak boundaries for a 5-group partition based on reactive power disturbance and voltage rms coherency measure when all generator buses are regulated 74 — -—- \ ‘ .__.....32.— —--":':..—'2-—--‘-‘::~§ a”’ 21+ 5’ \ i ’\ .l d fl ~~~~---‘ 27 — 4 I ‘ / I V . 16 I 1"- :\\ /’ ’ ’3°——\ I! ’/ 7 r \ l 1 l l l I '22 i I I l l l I I l l0 ltd ’ ‘_....- / I 4KEF'F=_Z____ ’1' I I V (f) .I‘ ’ \ \ L1 / l . \ I I’ ’ ‘ ~§~° v” I] Figure 3.14. The weak boundaries for a 3-group partition based on the current coherency measure for both real and complex power disturbances. 75 similar to the six-group partition (but not the three-group partition) based on the phase coherency measure shown in Figure 3.4 that group III in Figure 3.14 is broken into four groups. This suggests that the thermal overload security problems may be more severe on bounda- ries closer to the swing bus and that phase stability problems may be more severe on the weak boundaries of the subgroups that compose group III in Figure 3.14. However, the results suggest that thermal overload problems will likely occur on the weak phase boundaries. This severity of the violations and the number of contingencies causing thermal problems on a boundary are not likely to be proportional to the ranking of the weak phase boundaries. CHAPTER 4 CONDITIONS FOR VOLTAGE CONTROLLABILITY, OBSERVABILITY, AND STABILITY 4.1. Introduction The objective of this chapter is to lay the groundwork for the study of the structural causes of voltage stability problems. This is accomplished by first developing sensitivity analysis based on the linearized loadflow equations in section 4.2. A set of sensitivity matrices is defined which relates the voltage magnitudes at load buses and reactive power injections at the generator buses to the voltage at generator buses and the reactive and real power at load buses. Two algebraic equations are developed by this sensitivity analysis in section 4.2. The first equation relates the voltage at P0 buses to the voltage at PV (reactive source) buses and the reac- tive load at P0 buses. Since the voltage at PV buses acts as a control and the load at P0 buses acts at a disturbance to the voltage at P0 buses, the equation is called the controllability equation. The other equation will reflect the reactive demands of the system at the source buses. Since under normal operations the reactive demands from P0 can be observed at or reflected to the source buses by this equation, it is called the Observability equation. 76 77 The voltage stability of an interconnected power system can be affected by the lack of resources and the weakened boundary between two regions. An example mathematical model of five buses system will be introduced and the corresponding sensitivity matrices will be determined in section 4.3 under light-load conditions. In section 4.4 a set of theorems for voltage stability will be derived based on the controllability and Observability equations. Finally, in section 4.5 we will integrate the two parallel approaches about voltage stability problems, which will post the very useful local and global operational constraints for the interconnected power systems. 4.2. The Model Development It is intended to establish the relationship among the various controlled and observed variables of the system in this section. This leads to a set of sensitivity matrices which can be derived from the following loadflow equations: fP (a,e,E,V) = 0 G fP (6,0,E,V) = 0 L f (6,6,E,V) = 0 06 f (a,e,E,V) = 0 QL where V 6 6 78 is the vector of the real power flow equations at all generator (PV) buses is the vector of reactive power flow equations at all generator (PV) buses is the vector of the real power flow equations at load (PQ) buses where voltage V is not controlled is the vector of reactive power flow equations at all load (PQ) buses where voltage V is not controlled is the vector of voltage magnitude at the generator (PV) buses is the vector of voltage magnitude at the load (PQ) buses is the vector of phase angle at the generator (PV) buses is the vector of phase angle at the load (PQ) buses Then the Jacobian matrix which relates the change of input vector E. - E. V. - V. = t . A._1____12 A._1___;uz AX [A5,Ae,AE,AVJ , aEi = E, , AVi = Vi to the change of output function r 1 afp (6,6,E,V) G af(a,e,E,V) = afp (5,9,E,V) (4.1) L Af (5,9,E,V) QG AfQ (a,e,E,V) L is A1 B1 C1 D1 J = A2 82 C2 02 (4 2) A3 B3 C3 D3 A4 B4 C4 D4 .1 79 so that the following expression can be written F ' ’ ‘ ' " APG A1 B1 C1 D1 A6 AP A B C D A0 L = 2 2 2 2 (4.3) If the phase angle changes at generator buses are assumed to be neglectable in the calculation of voltage magnitudes, then A6 can be set to zero to solve for A6 in (4.3) using APL = Ber + CZAE + DZAV (4.4) to obtain _ -1 A6 - 82 (APL - CZAE - DZAV) (4.5) Substituting for A6 in (4.3) when A5 = 0, the following expressions for AQG and AQL are obtained AQé = B3A0 + C3AE + D3AV 1 -1 1 = (D3 - 3382 DZ)AV + (C3 - B382 C2)AE + 8382 APL (4-5) AQL = B4A6 + C4AE + D4AV = (D - B 3‘10 )AV + (c - B B‘lc )AE + B B-IAP (4 7) 4 4 2 2 4 4 2 2 4 2 L ° 80 Solving for AV in (4.7) the following expression is obtained AV =' [D4 ‘ 8482102]-1[C4 ‘ B4Bfi21AE + [04 - B48 2 02] 1[A0L - B 482 1AP L] Now define the following sensitivity matrices: -1 _ -1 -1 SVE ' ‘ [D4 ‘ B482 Dz] [C4 ' B482 C2] (4.8) SOL": -4[o -3465102] (4.9) Then the above expression for AV becomes _ -1 -1 AV - [SVE]AE + [SQLv] [AQL - B4B2 APL] (4.10) where SVE relates the voltage vector V at the PQ (reactive load) buses to the voltage vector E at PV (reactive generation) generator buses and SQLV relates the vector of reactive power disturbances at the load buses QL to the change in voltage V. Note that the voltages at load buses are controlled by the voltages at generator buses. In order to have the proper control- lability in the system, the matrix SVE must have all positive ele- ments. This implies that changes in any elements of E will cause changes in like sign in all elements of V. Similarly, the sensitivity matrix Sélv must have non-negative elements and be nonsingular so L 81 that changes in any element of reactive injections QL will cause changes of like sign in all elements of V. Now substituting AV in the expression in (4.6), the following expression is obtained _ -1 -1 A06 ‘ [(53 ‘ B382 C2) ‘ (D3 ' B3B2 D2W4 ‘ B482 Dz) -1 (C4 ’ B482 C2)]AE -1 -1 -1 -1 + (03 ' 8382 D2)(D4 ‘ B482 Dz) (AQL ' B432 APL) + B B'lAP 3 2 L Now define another set of sensitivity matrices s = (c B B'lC ) - (D - B 6‘10 )(0 - B 3'10 ).1 QGE 3 ' 3 2 2 3 3 2 2 4 4 2 2 (c - B B‘lc ) (4 11) 4 4 2 2 ' -1 _ -1 -1 SQGQL ‘ ' (DB ' 8382 D2)(D4 ‘ B482 D2) (4.12) Then the expression for AQG becomes -1 1 AQG = [SQGEJAE - [SQGQL](AQL - B4B2 APL) + 838% APL (4.13) 82 If the system is assumed to be operating normally, then a positive reactive power injection AQL at any load buses will induce a negative power injection AQG at all generator buses. As a result, the matrix SQGQL must have all positive elements under normal conditions. The reactive injections at generators should increase with increase in the magnitudes of the generator voltages E, so that the matrix SQ E G normally must have nonsingular with non-negative elements. Based on the controllability and observability equations a set of theorems regarding necessary conditions for voltage stability will be derived. Changes in voltage AE at PV (source) buses and changes in reactive injection QL at P0 (load) buses are considered the control and disturbance input. The state of the network is considered to be the voltage V at P0 (load) buses. The network governed by (4.10) is said to be stable if a vector of solely positive (negative) incremental changes AE on one or more elements of E will cause only positive (negative) increments on elements of V, and if a vector of solely positive (negative) increments AQL on one or more elements of QL will cause only positive (negative) increments in V. If there exists a vector of solely positive (negative) incremental changes on elements of E which produces no change on any of the elements of V, the network is said to be structurally uncontrollable. Likewise, if some vector of solely positive (negative) incremental change on elements of QL produces no change on any of the elements of V, the network is said to be uncontrollable. The network is said to be structurally 83 unstable if there exists a vector of solely positive incremental changes AE that will produce a vector AV that has one or more nega- tive incremental changes. Similarly, a network is said to be unstable if there exists some vector of positive (negative) incremental changes AQL that will produce a vector AV with one or more negative elements. Thus from equation (4.10), the network is stable and controllable if (a) SVE has no negative elements and no zero rows, and (b) the square matrix SQLV is positive definite with no negative elements. The network is uncontrollable if either SVE has no negative elements but has one or more rows, or if SQ V is positive semi-definite. A system is unstable if SVE has one tr more negative elements, or if SQLV is indefinite (or negative definite) or has negative elements. A power system network that is structurally uncontrollable will have multiple solutions, and a structurally unstable network will have no solution and experience voltage collapse or voltage rise. The reactive injection AQG at PV buses governed by (4.13) is considered an output of the network because it is the reactive power reflected to or requested of PV buses. If one or more PV buses cannot serve the requested reactive injection due to constraints 0-1010 Gi,min Gi Gi,max then the reactive injection at that bus is set to Q . or Q Gi,m1n Gi,max and the bus converts from a PV bus to P0 bus. In this case the system loses some degree of Voltage controllability due to the lack 84 of reactive source at that bus. If all PV buses in a local area are converted to P0 buses, then a voltage collapse will occur as dis- cussed in Chapter 2. 0G is considered an output and (4.13) is an output equation for the network. Either one of the following two conditions will cause the output stability of the network: 1) If some vector of solely positive (negative) incremental changes AQG in 0G do not cause a vector of solely positive (negative) changes AEG in EG, or 2) If some vector of solely negative (positive) incremental changes AQL in QL do not cause a vector of solely positive (negative) change in QG(AQG), then the network loses observability. It is said to be structurally output unstable. If the square sensitivity matrix [SQGEJ'1 is positive definite with no negative elements, and sensitivity matrix SQGQL has no negative elements or zero columns, then the network is said to be observable and output stable. If the matrix SQGE is semi-definite or SQGQL has no negative elements but one or more zero columns, then the network is said to be unobservable. A network is output unstable if either SQGE is indefinite or has negative elements, or if SQGQL has one or more negative elements. Now let us summarize the above concepts in more rigorous mathematical format. The following definitions and facts are needed for our theorems [13]- 85 Definition: A real matrix A is called positive if all of its elements have positive value, A > 0. A.is non-negative if all of its elements are non-negative, A :_0. Definition: A real n x n matrix A = (a ij) with Qij 5_0 for all i f j is an M-matrix, if A is nonsingular, and A'1 :_0. fag£_1: If A = (aij) is a real symmetric and nonsingular n x n irreducible matrix, where aij §_0 for all i f j, then A'1 3_0 if and only if A is positive definite. EEEEL§3 The Jacobian matrix of the linearized loadflow equa- tion for any non-islanded power system network is irreducible. From the discussion of the controllability and observability equations the following theorems are established. Theorem 1: A necessary condition for the network to be con- trollable and stable is that SQ V be an M-matrix.' L Theorem 2: A necessary condition for the network to be observ- able and output stable is that SQ E be an M-matrix. G Theorem 3: A necessary condition for the network to be con- trollable and stable is that SVE be a positive matrix. Theorem 4: A necessary condition for the network to be observ- able is that SQ be a positive matrix. GQL The necessary conditions that will cause S and S to be QLV QGE M matrices and S and S to be positive will be investigated for VE QGQL light load conditions in Section 4.4. The conditions that assure the above properties in these sensitivity matrices will indicate the 86 causes for loss of observability and controllability induced sta- bility problems. Facts 1 and 2 indicate that a necessary and sufficient condi- tion for the matrices SQLV and SQGE to be M matrices is that these matrices be positive definite if the network is not islanded or has nonsymmetric transmission elements. Therefore one can investigate the satisfaction or the degree of satisfaction of the M matrix con- dition by computing the eigenvalues of these matrices and noting the magnitude of the small positive eigenvalues if a loss of controlla- bility or observability is possible and the negative eigenvalues if a loss of controllability or observability induced voltage stability has occurred. It is emphasized here that the sensitivity matrix SVE is the matrix which reflects the voltage controllability of the system and SQGQL represents the reactive power observability of the system. They will be discussed in detail later. In the analysis that follows, different forms of loadflow equations and Jacobians are required for the analysis. The polar form is given first and then the hybrid form. Relationships between the elements in the bus admittance matrix are also given which will be useful in this analysis. 87 lj = Yij(COS Yij + 3 Sin Ylj) = -Gij + JBij Slj = ViVjYij (4.14) Oij=8i ' 6:} Then the power injections can be written in the polar form as n 3:1 n 3:1 _ 2 aq./ae. = P - vge. (4.17) 1 l 1 1 11 _ 2 ViBPi/aVi - Pl + ViGll (4.19) VjBPi/BVJ = Sij COS (Oij - Yij) (4.20) where 88 _ 2 viaqi/avi - Q1 - V1811 (4.21) th V1. is the (i,j) element of bus admittance matrix J ViO is the shunt capacitance at bus i qLij is the admittance of line that connects busi and bus j ..< Lij = Gij - JBij Wlth Gij 3_0 and Bij 3_0 The elements of the bus admittance matrix satisfy where ij = 'YLij ‘ 'Gij + JBij ‘ C J n n 11 = :E:: Lij + Yi0 = ' 12:: 13 + YlO i=lajfl J=1,j#l ReIYij} = Yij cos (Yij) = 'Gij Im{Y1j} = Yij Sln (Yij) = Bij Re{Yi0} ‘ GiO 3_0 H 3 r‘fi -_. .o xwguas cc_noomq mg. ...¢ m.:m.. QVQN ¢> Iva Am? INVOvc—mmwmm ANQLINVOvcmevm A~¢LI~V®V:—m~¢m m QQUN Q) IQQ AMQQIMQOvaUMVmI ANQLINVSVmOUN¢WI Advgl~63vaU—¢ml .qm.-.mo.=.mqmm mam. m>.;:.m.~m.-~move.m~mm ..m.-_mo.=.m.mm m.¢ma-v.o.mou.mm- MMAN m. -m. .Nm.-~ma.m66~mm- ..m.-_.a.moa.mm- ooooooooooooooooooooooo.coon-000moooooooooooooooococo000......cooomcoco-o...00.000.000.00...coo-cocoo- coo-coco...-oooooooooooooocooooooooo ..N -.No.=.m.~m Am. -mwo.=.4m~mm NNAN N. -Nc ..~.-_~o.=_m.wm m..~.-e~o.moo.~m- .m~2-m~o.mcum~m- NNAN N. -N. ..N.-.Novmoo_~m- .¢_2-..o.=.me.m .m..-m.o.=.mm. u.~..-~.avc.m~.m ..a _> -.c m....-¢.a.mou¢.m- .m..-m_o.moum.m- .m..-~.o.moow.m- ..o .> -.a OOOOOOIOOOOOIOCOOOOOO0.00.0.0...“OOOOOOOOOOOO...OOOOOOOOOWOOOOOOOOCOO0.0...O0.000000000000000000000... N «vow q. .4. Am..-m¢o.moum.m m.~.2-~.®.mou~4m ..c.-..a.moo_em «4.. e. -.c- .m¢.-meo.=.mmem .~.2-~.3.=.m~.m ..e.-..3.=_m_.m .qm.-¢ma.mouemm mac. n. .ma m.~n.-~mo.mou~mm ..m.-.ma.mou.mm ..m -qmovc.menm n... m. -mc- .Nm.-~mo.=_m~mm ..m.-_ma.=_m_mm ..NL-.~®.mooe~m .nwa-m~o.moum~m mmm N. -Nc- ..N.-.Na.=_m.~m ..N.-¢~e.=.m.~m Amwanm~¢.=.mm~m N -u~ N, +N. ..N.-.No.mou.~m .N..-~_o.mou~.m ..c .> ..a 4-.- N m m > c ...L-..m.moae.m ...L-m_e.mcun.m N ....-¢.o.=.m¢.m .m..-m.o.=_mm.m .N..-~.3V=_m~. .. 97 2 P1 ‘ V1611 ‘512 C05 (912 ‘ Y12)1 A3 = 2 (‘521 C05 (621 ‘ Y21) P2 ‘ V2G22 _ ‘2911 G12 1 = .li‘l I ‘321 'ZGZJ' 3752 d 1 ‘531 C05 (631 ‘ Y31) '532 C05 (632 ‘ 1(32)‘1 A4 = L'541 °°S (e41 ‘ 141) '542 C05 (942 ‘ Y42)_ lG31 G321 L641 G42 1 1513 51" (913 ‘ Y13) S14 51" (914 ' 1’14)1 B1 - ‘ L523 51” (923 ' 723) S24 51” (924 ' Y24)) ' l “313 '314 (‘323 '324) 98 P 2 . .1 '03 ‘ V3333 S34 5‘" (e34 ' Y34) . 2 S43 5‘" (943 ' 143) ‘04 ' V4344 . . 2333' B34 1263 . '343 2341' ) _ ‘523 C05 (923 723) '324 C05 (924 l24)J ' G13 G14 G23 G24 L. .l -P -V2G -S cos(6 - )- 3 3 33 34 34 Y34 L‘543 C°S (943 ' 743) p4 ‘ v4944 _ ‘ 293.) G34 .lf3 1 G43 ‘ 2643'. #4 P1 + vle11 $12 cos (912 le) c1 = 2 L $21 cos (621 - y21) p2 + V2622 - 26 + 26 -G .1 13 10 12 = 37‘1 L “821 2623' + 2G20 - J'fZ S31 C05 (631 ' Y31) S32 C05 (932 - Y32) c2 = L541 cos (641 - 141) S42 cos (e42 - y42) J "631 ‘932 1 L‘G41 “642} ' 2 . 1 Q1 " V1811 S12 51” (612 ' Y12N c3 = 5 $1n(9 -Y) Q-VZB 121 21 21 2 2 22 _ [:532 - 23 -B T 11 10 12 _ J'fl L ’821 2323' ' 23203 J'i‘Z 100 S41 Si" (941 ‘ Y41) II T ..B' 31 'B 32 ‘841 ‘842 L . 513 C05 (913 ‘ Y13) S23 C05 (923 ‘ Y23) “G13 'G14 ‘623 “524 L. J 2 3633 P3 + V 543 C05 (943 ' Y43) L F 2E3F3j + 2930 G34 J’i‘3 L #4 43 2% + 2G40 S32 5‘” (932 ‘ 732) S42 5‘" (642 ‘ Y42) S34 COS (934 ‘ 734) P4 + V4644 .J "1 I r S13 51" (e13 ' Y13) S23 5‘" (923 ' 123) -B 13 '314 “323 ‘324 2 Q3 ' V3333 S43 51" (e43 ' 143) L :E:P3j ' 2830 '334 43 IE:?41 ' 284o Jf3 -B if4 S34 5‘” (934 ' Y34) 2 Q4 ' V4344 4 From equations (4.8), (4.9), (4.11), and (4.12) the sensitivity matrices can be expressed as S SQGQL = MQGQL SQLV QGE = -1 VE = M QLV MVE -1 -1 S - S S QGE QGQL QLV VE (4.33) (4.34) (4.35) where SQLV, M SQLV 102 , M , M are given below: VE QGQL QGE ‘ [D4 B432 D2J r 2333' T 2330 ”B34 3243 '343 2.34.1"2 ‘ 2533' G34 1% 32643J #4 f . 2333' "B34 ‘1 (- . 2633 + 2G3o “634 #3 “G43 2643' + 2640 _ 37‘4 J :22P3j ‘ 2830 B34 #3 VE- 103 Jf4 J?3 '643 2643' + 2G40 1744 J —1 '331 '332 i _ B41 '842J '2931' G34 '2331‘ ‘334 "1 jf3 J Jf3 G43 ' 2643' 1 '343 2343' L 3244 . L 1'24 1‘ FM + 3 G31 G41 QGQL 104 #3 #3 ‘943 2643' '343 124 j G32. G42) 1 [D3 - 8382 02] -B '313 14 ‘323 '324‘ '834 1:5:P41 jf4 J G13 G14 _ ( G23 G24 . _ .L 2633. + 2630 #3 '543 LL313 B14 = ( _323 B24) F - G13 G14 L + 1 G23 G24 . - J , 2633. + 2630 #3 _ 'G43 '34 2843' 1714 1 _ 7 2313' ' 2810 ‘312 #1 'le 2823' ' 2820 if? L .- - 1 ' 1 G13 G14 2833' "B34 ( jfB G23 G24 k ‘843 284.1 Ji‘4 - . L d _ ‘G31 ‘G32 ‘ L 'G41 "G42 . 2313' ‘ 2810 “312 = 57‘1 “321 2323' ' 2320 if? L - l ' 3 G13 G14 2333 '834 + ‘ 123 GI23 G24 L "843 2343' 'G31 G32) L641 G42. 107 At this point all important equations that are needed for sen- sitivity analysis in sections 4.4 and 4.5 have been derived. Thus the sufficient conditions for voltage stability are derived in the next two sections. 4.4. The Theorems for Local and Global Voltage Stability The system that will be studied here is assumed to be operated under light load conditions, as it is defined in the previous section. The local control at load buses will be discussed first. Setting APL = 0 in (4.8), the following expression relating the voltage V at P0 bus to the control voltage E at PV buses and the reactive load disturbance AQL at PO buses is obtained V = [SVEJAE + [SQLleQL It is shown in section 4.2 that for proper control of AV the sensitivity matrices SVE must have all positive elements and SQLV must be positive definite. One way to show the positive definiteness is using the properties of M-matrix, which will be shown later. The positiveness of both SVE and SQGE and the positive definite- ness of SQGE and SQLv are the basis for proper voltage control. If these properties are violated the loss of voltage control would finnediately result. Thus, the large capacitances of long trans- Inission lines or underground transmission lines will have been shown to cause voltage control problems for light load conditions. These PY‘Oblems are known to exist and are often solved by removing these 108 long tranmission lines or underground transmission under light load conditions. 1 LVMVE must be positive for the PV buses to properly support voltage at P0 All elements of the sensitivity matrix SVE = $6 in (4.10) buses. If SVE loses its positiveness, indicating a loss of control- lability, then there are some PV buses that have no effect or have reversed effect on some PQ buses. It will cause a loss of voltage stability. This loss of controllability and stability would be seen if raising E at PV buses has no effect at all on raising V at par- ticular PQ buses and if increasing E at some PV buses actually would cause decreases in V at some PQ buses, respectively. Theorem 5: Under light load condition, a sufficient condition for SVE to be positive is that SQL of all lines are equal. Proof: - Recall that _ -1 "vs ‘ ‘ [C4 ' B432 C2J B31 B32 , B41 B42 _ GBj '634 Vizzst “B34 L ‘1 + 123 { j¢3 > ‘643 2643' 1 ‘343 2343 , jf4 jf4 V be an M-matrix and the R/X ratios “1_ [HOW]: .".S- I 111 109 G31 G32 G41 G42 , B31 B32 G31 G32 2 + R/X I ( ) r... I B41 B42 G41 G42 l B31 B32 = (1 + (R/X)2) B41 B42 Since all Bij's are positive, matrix MVE is always positive. Since . . _-1 . SQLV lS assumed to be an M matrix, then SVE - SQLVMVE TS always positive. Next, a sufficient condition for SQ Q to be positive, which G L in part assures observability, is considered. Recall that if APL = 0, equation (4.13) becomes A06 1 [SQGEJAE ‘ [SQGQLJAQL _ -1 SQGQL - MQGQL SQLV (4.34) = M - S S S (4.35) QGE QGE QGQL QLV VE 110 Theorem 6: Under light load condition, a sufficient condition that S be positive is that QGQL 1) SQLv be an M-matrix, 2) the shunt conductances GiO are negligible at all buses, and 3) R/X ratios of all lines are equal. Proof: Since _ -1 B13 B23T . B14 324. '1 L (- .. ‘ 613 614* 12:3331 ‘334 ‘1 + ( Jf3 , G23 624. . _ ‘343 2343; l Jf4 ' ll 26:”. + 2630 G34 jf3 , L ‘643 2643' + 2(340. if4 B13 B23 G13 G14 “=' + (WM .314 B24 G23 G24 ._,._.1 111 B13 323-L B13 314) = + (M)2 L814 824_ L323 B24_ B13 314) -- (1 + (we) 8 L323 24‘ then MQ is positive. Assuming that S V is an M matrix, then GQL Q L _ -1 . . . . SQGQL - MQGQL SQLV 15 a pOSitive matrix. Remarks: so 0 must be positive if a positive reactive load G L injection QL is to produce a negative reactive generator injection at PV buses. If SQ Q has negative elements a continual voltage rise G L or voltage collapse would occur based on the change in AOL because the reactive sources would not compensate for changes in AOL but actually reduce support. Theorem 7: Under light load condition, a sufficient condition for S to be an M-matrix is QLV 1) if D4 is an M-matrix, and 2) R/X ratios of all lines are equal. Proof: _ -1 2333 ' ZB3o '334 - if3 '343 2343' ' 2B40 #4 1:1"..fi D pr. ”in"; + (NM The above matrix L 112 113 Z 83.1 E 238332] -834 2 . . 1+ 2 J“ = [1+ (3) 104 X -B43 28 843' [1+(: BXOZJJ m a A H + A ><|Jo V v 3'24 If 630=G40=0 for light load conditions 83 and B4> O, and D4 is assumed to be an M matrix, then SQLV must be an M matrix. From fact 2, since SQLV is irreducible, then it is also positive definite. Remarks: SQLV can be positive definite, but could also be semi-definite, indefinite, or even negative definite, since it depends on the shunt Bio's in D4. B1.0 can represent either shunt capacitances of transmission lines or switchable shunt capacitors banks at each bus. Under light load conditions switchable shunt capacitors banks would never be in the network. D4 could be indefinite or negative definite if the shunt capacitance (830,840) of a long or an under- ground transmission system was sufficient to overcome the positive definiteness of 04 with these shunt capacitances eliminated. The shunt capacitance of the transmission line not only makes D4 less positive definite but possibly indefinite or negative definite. The shunt capacitance can also possibly make SQLV indefinite or negative deffinite. If SQLV should ever become negative definite, the effects of"load QL injection on voltage would be reversed from normal condi- tions. More important, the positiveness of the matrices SVE’ SQGQL’ E’ can only be assured if SQ V and the positive definiteness of SQ G L is Positive definite. 114 Theorem 8: Under light load condition, a sufficient condition for S is positive definite is that QGE 1) S be an M-matrix (D be an M-matrix), QLV - 4 2) R/X ratios of all lines are equal, and 3) Matrix C3 D3 pi c4 D4 is positive definite. Proof: We know that SQGE = MQGE - SQGQL SQLv SVE‘ Since 81.0 s and 610's are assumed to be negligible, SQ V can be expressed as from L Theorem 7. 2331 834 g jf3 sQLv ‘343 2345 jf4 L _ ‘ - 2933' '934 + (M) “‘3 ‘G43 2543 m P L ‘ d 2331 B34 _ jf3 + (R/X)2 3’3 2331 ‘334 = (1 + (R/X)2) 3’3 '343 2343' J J'f4 b The matrix product SQGQL SQLV SVE becomes _ -1 SQGQL SQLV SVE ' (-MQGQL)SQLv(-MVE) L B13 B14 =L(1+ (Fa/oz) , B23 B24 (1 + (R/X)2)'1( J*3 i L #4 B31 B32 L (1 + (Rm?) , B41 B42 Fifi—573.7- ‘31-. _ i (1 I _ 116 1313 B14 = (1 + (Rmz) .323 B241 L L:§:}3 B - ‘ 1 P B B ‘ 3j ' 34 ' 31 32 ( ‘j#3 > ,_ ‘343 23411 L -341 3421 m The matrix MQ E can now be expressed as G M -1 QGE [C3 ‘ B332 52] L2 ‘ Blj ' 2310 812 = #1 _ '321 2323' ' 2320‘ #2 L613 G14 LLZB33 B34 '1 + p 123 } G'23 G24 11 ‘343 2343' 1 - ' 17‘4 4 G31 G32 G41 G42 F-ii—fl? 1 117 LZBIJ' B12 .1111 - '321 2321, .1392 LG'13 G14 L233” '334 ‘1 + { Ji‘3 ’ G23 G241 ,_ “B43 2343;, #4 r331 G32. .641 G421 LZBIJ ‘312 .-. #1 L 'le 2321. 31‘? LB _ _ B -‘ 1 13 B14 :33 “B34 ‘ + (R/X)2 ‘ J#3 p _323 3241.. '343 2343'“ #4 B31 B32- B41 B421 The matrix SQ S 065 G = M E now becomes 118 2311' ‘312 = 3’1 L ‘le 23211 #2 B13 B14 + (R/x)2 _323 824 . (B31 B32 . B41 B42. 2331 B34 ( jf3 ._ '343 2341', 1144 L331 B32 . B41 B42 . - S S S QGE QGQL oLv VE 2331 Jf3 . L ‘343 2341. L ”B '334 m 13 314' 23 324. It: I STA— 1| m l...— 119 and D4 are both positive definite. The inverse matrix has the form (C3 - o3oglc4) and since the inverse matrix is positive definite if the original matrix is positive definite, the submatrix [C3- D30;1C4] is positive definite. 120 Remarks: If SQGE is positive definite then increasing voltage at PV buses requires a large positive reactive injection. However, if SQGE is semi-definite, then raising E at some PV buses has no effect on raising QG at these buses. If SQGE is indefinite, then raising E at PV buses could indeed require reduced positive reactive injection which is contrary to how a power system is supposed to support voltage. The above results indicate that if SQLV is negative definite, then a controllability induced loss of stability can occur directly or because SVE has negative elements and a loss of observability induced loss of stability can occur because SQGQL has negative ele- ments or SQGE is negative definite. The results for light load con- ditions indicate that for SQLV to be positive definite D4 must be gsitgve definite, and for SQGE to be positive definite both D4 and 3 4 must be positive definite. These matrices will be negative gfingte if the capacitive admittance at generator or load bus exceeds half the sum of all branch admittance connected to that bus. This explains the Operation practice of switching out certain long trans- mission lines with large shunt capacitive admittance under light load conditions. Although an analysis is not performed for normal or heavy load conditions, it is hypothesized that the admittance of switchable shunt capacitance at any bus should also be less than half the admit- tance of all branch elements connected to that bus. This hypothesis will be studied experimentally in Chapter 5. 121 The above analysis is quite important because it has been known that capacitance from long transmission lines can cause voltage problems under light load conditions but the above analysis is the first that analytically derives a limit on the capacitive admittance at a bus to assure voltage controllability and observability induced stability. Deriving a similar limit for normal and heavy load con- ditions is a subject for future research. 4.5. Theorems Integrating the Weak Boundaries and Sensitivity AnalySis In this section, two parallel approaches of the identifica- tion of weak boundary and sensitivity analysis are integrated, and summarize the corresponding mathematical results as theorems. In order to show that the SSC groups preserve their coherency structure in all sensitivity matrices, the following model of a three SSC group network is introduced. _ t oPG [APGIAPGZAPG3] APL - [APLIAPLZAPL3]t AQG = [AQelAqezAQoalt AQL ‘ [AQL1AQLzAQLa]t _ t A6 - [A61A62A63] 122 _ t A6 - [A61A62A63] oE = [AE AE AE 1t 1 2 3 - t AV - [Av11v21v3] (4.36) The partitioned Jacobian matrix for this three-SSC-group system can be expressed as in Figure 4.2. Property 1: All submatrices in Figure 4.2 are square matrices. Definition: (Dominant diagonal block) If a square matrix is partioned such that the absolute value of any nonzero element in its diagonal blocks is strictly greater than the absolute value of any element off the diagonal blocks, then those diagonal blocks are called dominant diagonal blocks of the matrix. Property 2: Matrices 82, B4, C3, 02’ and D4 have dominant diagonal blocks corresponding to the three SSC groups. Property 3: Let matrices F ‘ F ' A11 A12 B11 B12 Amxk g and kan = A A B B _ 21 224 _ 21 22J If the submatrices A11, A22, 311, and 322 are dominant diagonal blocks in A and B respectively, then the product matrix C = AB also has the corresponding diagonal blocks dominant. 123 3p(53 3"(:3 [”63] (:3 3963 apes] 861 362 363 . 361 392 393 3E1 3E2 3E3 3V1 3V2 3V3 A B C D [31961] 31361 aPGl 31361 3P61 3PGl §[3p61] 3P61 3pc31 aPGl 3pc;1 3P61 351 862 353 5 391 362 863 5 3E1 3E2 3E3 5 av1 3v.2 3V3 aPG2 [31362] zaps2 5 31362 31962 3P62 31362 [”62] 31962 5 3PGZ aPGz 31362 351 352 353 g 391 332 363 5 3E1 aEzJ 3E3 5 W1 3V2 3V3 5 3P63 aPG3 aPGB 5 8P 5 3PG3 aPG3 8PG3 ale 3PL1 3PL1 Erspr] 3pL1 ap11 5 3P11 3p11 3PL1 [5PL1] ale ale 351 362 363 ELael 362 363 5 361 352 3E3 val 3V2 3V3 apLz 3PL2 3pL2 § aP12 [pré] apL2 § apLz 3P12 3P12 3p12 I3p12] 3p12 : 351 362 353 5 331 362 393 5 3E1 3E2 3E3 3v1 Lav2 3V3 l 3PL3 3P13 3PL3 § § apL3 3PL3 3p13 3p13 3pL3 [3PL3]1 9V1 BVZ L3V3 OOOOOCOOCOOOOCOCOOOOOO a061 a061 a061 av1 3V2 3V3 ”L3 3PL3 [spa] 361 352 353 361 362 393 . 351 3E2 3E3 (8061] 3°61 3061 3061 3061 306113061] 3061 3°61 861 352 363 a 1 352 3E3 391 382 363 3°62 '3062] 3°32 3°52 [3Qc2] ach 361 362 363 351 3:2 353 3°63 a063 [?QGB 3°53 3063 3063 361 362 363 . 861 392 363 aQLI 3°L1 aQL1§E0L1]'aQL1 301.1 361 362 363 'L?61 392 863 IEQLz] 2)012 362 363 901.3 IBQL3] 391 382 393 Figure 4.2. The partitioned Jacobian matrix of a three-SSC group network. 1 3062 3062 3062 3V1 3V2 0V3 3°52 3°62 3°32 391 362 363 3°63 3°63 a033 3°63 3°63 3063 3E1 352 L3E‘; 3V1 3V2 3V3 3°11 301.1 30L1 §I30L1 3011 30L1 3E1 352 3E3 :LPVI 3V2 3V3 30L2 [“12] 3012 av1 "“' "“' 30L2 30L2 3°12 3E1 352 3E3 80L2 30L2 3012 301 362 363 Q, Q) 0’ r5: (9‘53 on *J n) a013 3013 301.3 8E1 352 353 30L3 3013 3013 301 352 363 301.3 a013 a013] av1 av2 3V3J‘ 124 Property 4: The inverse of a matrix with dominant diagonal blocks has the same diagonal blocks dominant. - It is essential to prove the following property, and then use the mathematical induction to complete the proof of Property 4. If matrix “11’“ H12 H = . (4.37) where (1) H11 and H22 are nonsingular square matrices, and (2) Hlllu and H22/u are dominant diagonal blocks in H, when u + 0, then the inverse of H, which can be written as K11 K12 H‘1 = (4.38) with the same partitioning as H, also have dominant diagonal blocks. Proof: By the inverse matrix lemma the following results are obtained: . _ -1 -1 (1) K11 ‘ [”11/“ ' ”12(“H22)“21] when u + 0, so that K11 3 (U)[H11]-1 (ii) when u + 0 k12 3 (iii) when u + 0 K21 - (iV) when u + 0 K22 - Ill Summarizing 21 ‘ Ill 22 125 K12 = ’ [K11H12(H22/U)-1] 2 -1 -1 '(“) [”11H12H22] _ -1 K21 ’ ‘ [(“22/“) ”21K11] ~ 2 -1 -1 ‘ ‘(“) “22“21H11] -1 -1 K22 ‘ “”22 ‘ H22H21K12 ~ -1 -1 2 -1 -1 ‘ uH22 ’ “”22H21 '(“) [“11H12”22] (111112214 (i) to (iv), the following relationships hold: (4.39) - 2 -1 -1 ‘ ‘(“) [“22H21H11] (111142211 126 Therefore H'1 has diagonal dominant blocks as declared. Finally, the proof of Property 4 can be accomplished by mathe- matical induction as shown in the following example: Let ”11“L1 H12 H13 H 3 H21 ”22/“ H23 H l. 31 H32 “33/31 where u + 0, and I K11 K12 5 K13 -1 __ I H ‘ 521---’f22-=.-52§- I _K31 K32 5 K31 The matrix can be partitioned into four blocks. Then by (4.39) it can be asserted that submatrix K33 is a dominant diagonal block. Similarly, by symmetry, if H is partitioned such that rk i K K ) -ll- ' “13-----13 -1 _ ' H ‘(K21 : K22 K23 K31 5 K32 K33 then the submatrix K11 is a dominant diagonal block. Also, when u + 0 ‘-1 K11 K12 LHII/u H12 Ill H H /u 127 then by (4.39) K22 is a dominant diagonal block. Therefore Prop- erty 4 has been proven with three diagonal blocks and the generali- zation into r finite dominant diagonal blocks can be easily obtained by the reader. Applying Property 1 to Property 4 under light load conditions, the following sensitivity matrices _ -1 2333' '334 = (1+ (R/x12) ”‘3 '343 2341' L 124 and S = M - S S S QGE QGE QGQL QLV VE ‘ F 2313' '312 - 121 ‘321 2321 122 L. .- B13 B14 L LLBZ3 B24 , 128 '2331‘ ‘834 ‘1 B31 832 Jf3 ( ) ‘343 LEE:?41 B41 B42 324 , 1 = c - o o‘lc 3 3 4 4 will both have dominant diagonal blocks. In the following two theorems a detailed sensitivity model will be used, where one of the SSC group in the system has no source bus as shown in (4.40). " -" " * * z z * * : - r . PGl A1 811 5 B12 5 C1 D11 5 D12 021 * * E g * * g PL1 A21 B211 5 8212 5 C21 0211 5 0212 QL1 .C...C.CCCC..0..E...;....§.......C.....C...§...;C.. PLz A22 B221 5 B222 5 c22 D221 5 D222 Q12 ‘ 000*. 0000000 *0 oooogoooooooogoooo; ....... ; 00.03 ..... 000 061 A3 B31 5 B32 5 C3 D31 5 D32 V61 4 2 5 5 . 4 5 QL1 . A41 8411 5 B412 5 C41 D411 5 D412 V11 ..‘C............§.0.;0..Og..........C......§...;.... QL2 A42 8421 E B422 5 C42 D421 E D422 V12 .- ..l - il l- u (4.40) This system has two SSC groups. The first group consists of both source and load buses which are denoted as G1 and L1, respectively. The second SSC group has only load buses L2. The original Jacobian 129 matrix for the sensitivity analysis is partitioned as shown in (4.40). Note that each SSC group will have the corresponding dominant diago- nal blocks as shown above, where the submatrices with "*" denote the dominant blocks. Those dominant blocks are larger than the other block by an order of l/u, where 0 < u < 1. For example, 0411:=%'D411’ where D411 has the same order as D412 does. The leading subscript of each submatrix in (4.40) is identical to the subscript of the original submatrix of (4.2). The second or third subscript indicates the partitioned blocks of each original submatrix. Based on the above model (4.40) the following theorems can be proven, respectively. Theorem 9: If there is no source in an SSC group, then SVE has zero rows and loss of controllability occurs. Proof: It is known that from the original sensitivity model _ -1 SVE ‘ SQLV MVE sQLV = [04 - 34agloz] 2 (1 + (R/X)2)D4 MVE = -[c4 - B4B§1c2] s (1 + (R/X)2)(-C4) -1- s 04 (-c4) Ill VE Now apply these results to the new model - - 40 -c - f 4 -1T * 411 -1 T 41 D411 D412 ‘C41 A u D412 ";“ SVE = = D 0* -C D 2.52.2. -C 421 422 L 42 421 u l 42 L- d d h- L— J Let 130 11 d21 d22 L. .L Similar to (4.39) as u + 0, then 11 12 21 22 and VE Therefore II! III "I ll! (u)[D411]‘1 '(“)2[°411°412°4223 ‘(“) ZED4220421D 411] (41104221‘1 - -c d 41 11 ( u ) + d12L'C42) L 1 ‘“41 d21 L L + d22(‘°42) L. L0411L C41) ' L“ )ZED411D4129422](C4 2) '(“)[D422”421° 4111‘ C41) + (“)(0412)(‘C42) (4.41) 131 SVE + as u + 0 Thus the rows corresponding to the SSC groups without source bus will go to zero, as the stiffness of the SSC group with the source buses goes to infinity. A loss of controllability occurs for the SSC group without a reactive source bus. Theorem 10: If there is no source in an SSC group, then SQGQL has zero columns and loss of observability occurs. Proof: It is known that s = $1 QGQL MQoQL SVQL MQGQL = -[03 - B335 021 (1 +(R/x)2)(-D3) _ -1 2 50301. =(- -03)(% ) Applying these results to the new model DGDL = -[D* 31 D32] LD411 _D421 132 U 412 * D422_ u 412 -o 31 422 1, where Ld11 d12 -1 D; 3 d21 d22 .- ..L matrix D4 has dominant diagonal blocks given in (4.41). Matrix SQ Q is G L D D - g .31 SQGQL ‘ L‘ n d11 ' D32d21 3 d12 ' D32d22] - [L'D D31)[D411] + L D 32)L “2)[D412D421D411] L D31)L “)[D411D412D412] + L D32)L“)D412] Therefore 0] + [("' D31)(D411) as 11 + 0 . SDGDL 133 Thus the columns corresponding to the SSC groups without source buses will go to zero as the stiffness of the SSC group with source buses goes to infinity. A loss of observability thus occurs for the SSC group without a reactive source. From the above results, the effects of the weak boundaries corresponding to the SSC groups (or the stiffly interconnected groups) can be determined. If the dependent and independent variables of the linear loadflow model are rearranged, then a partitioned Jacobian matrix and a set of partitioned sensitivity matrices can be obtained with dominant diagonal blocks corresponding to the SSC groups. By the definition of SSC group, each SSC group can be modeled as a single bus with respect to the whole system. Then, the constraints derived for each bus in section 4.4 must be satisfied by the SSC group for the voltage stability of the whole system. In other words, the SSC group can be modeled as an equivalent bus to obtain a new Jacobian matrix and carry out the sensitivity analysis. The following is immediately valid from the new Jacobian matrix with the equivalent bus representing each SSC group. Theorem 11: A sufficient condition for the equivalent sensi- tivity matrix SQLV to be an M matrix is that the sum of B1.0 over all buses in the SSC group should not exceed the sum of Bij over all transmission lines of the weak boundary surrounding the SSC group. Proof: Let AQL = SQ VAVL where L 134 t 40L = [401 402 403 404 405 406 407] t AVL = [AV1 Av2 AV3 AV4 AVS AV6 AV7] They are the deviations of reactive power injections and voltages at all load buses, respectively. Without loss of generality, let buses {1, 2, 3, 4} belong to Group 1 and buses {5, 6, 7} belong to Group 2. Then the control- lability equation can be transformed in the linear space as the following: L1 0 o o 50 o o ' o 1 o 0 go o o o o 1 o (0 o o u = 1 1 1 1 go o o 6-47—4-51??? o o o 0 go 1 0 Lo 0 o 011 1 1j and T T '1 UAQL = UCSQLVJU (U ) AVL = SQV(UT)-1AVL A0L - S AV 135 where 30 V 3 UESQ VJUT (4.42) L L 39L 9 (UT)-1AVL '1 o o -1510 o o. o 1 o -1 101 o o o o 1 -1 10 o o (UT)-1 = o o o 1 10 o o au.an.an.uapgguuéuurI o o o o 50 1 -1 Lo 0 o o :0 o 1‘ Therefore PAVI - AV4- sz - AV4 AV3 - AV4 AVL = AV4 AVS - AV7 AV6 - AV7 1v7 and 136 - A A0 = UAQ = L L :E:DQ° 1 The matrix U in (4.42) sums up rows 1, 2, 3, 4, and rows 5, 6, 7 of SQ V’ then put the result in row 4 and row 7, respectively. L T in (4.42) sums up the columns 1, 2, 3, 4 and columns 5, The matrix U 6, 7 of the preprocessed matrix [USQLV]’ The result of the equiva- lent sensitivity matrix SQLv given in Figure 4.3 carries the informa- tion about the global constraints on area-shunt-capacitance on columns and rows 4 and 7. All admittances of the lines that interconnect groups are eliminated from the (4,4) and (7,7) elements during this trans- formation. The only terms left on the equivalent diagonal elements are capacitances of that group, and the negative of the sum of total admittances of its boundary. Therefore, if the total shunt capaci- tance of that group exceeds the total admittances of its boundary, then the corresponding diagonal element will no longer dominate that row and column. Furthermore, the equivalent sensitivity matrix SQLV is irreducible and symmetric. Since the original SQ V is positive L 137 0mm mmw ..v "'3 “3 U1 {A} U3 U3 U1 U5 “3 U5 0.00000000000...00.0.0000000000000.0...... 000.0. “Hop H"? ..w gm e e 0mm mmm .5 N 8w 3m 5 N Sm am A U) n F‘ "1 00.000000000000000ooooooooooooooooooooooooooooooo mx Cums >u_>pp_mcmm a=m_m>_:cm mg» mi .-.N N mmm mmm ma .mé 85a: mu... 3 ..w 5w m R Now How Nmm Hmm 000.00.00.00 mmm Hmm Nam flaw 138 definite, then gQLV must be diagonally dominant to be an M-matrix. Therefore the sum of the area shunt capacitance must not exceed the total admittances of the weak boundary of that area for EQLV to be an M matrix and for the system to remain controllable, observable, and stable from Theorems 5-8. Now it becomes very clear that the voltage stability problems gare not simply caused by the lack of reactive sources in an SSC group, but also are caused by the wrong type of sources (capacitor banks). The installation of the capacitor banks as reactive sources is much cheaper than the generation of synchronous condensers. There- fore when the reactive supply or reserve is not sufficient to meet the demand, it is preferred to install more capacitor banks in that area without knowing the side effects to the whole system. In sum- marizing all theoretical results up to this point, it is understood that (1) from the proof and identification of weak boundaries of Chapter 2, there are weak boundaries to prevent the transmission of reactive supply over long distances; and (2) from sensitivity analysis of this chapter, the weaker the boundary around an SSC group the less the amount of area reactive supply can rely on the capacitor banks. Now one may ask why a utility does not eliminate the weak boundaries of the system by requiring all utilities to be uniformly stiffly interconnected, such that there will be no weak boundaries. At 139 this point it might appear to be a good idea, but any contingency and even system operations may break the uniformity of the intercon- nections. The cost-benefit considerations and the impossibility of obtaining rights of way for such redundant transmission makes such an idea impossible for the EHV transmission grid, although it can be approximated on distribution networks. The following property will show that such a uniform connectivity will cause the singularity for the suggested system. By pivotal condensation and mathematical induction the following property can be proven. Property: Let c f 0 and d be constants and A be an n x n matrix with aii = c and aij = d if i f j. Then det {A} = (c - d)".1 (c + (n - 1)d) From the above property, A will be singular if and only if c = d or d = -(1/(n - 1))c. Physically, this implies that an isolated, uni- formly interconnected non-SSC group can lose controllability and observability under light conditions, due to its structural uniformity. In conclusion, one has to live with the existing weak bounda- ries in the large scale power networks. The voltage problems can be solved by understanding the impact and interference of weak bounda- ries to the system controllability and observability. The next chapter will establish experimentally the theoretical results devel- oped in this chapter. CHAPTER 5 VOLTAGE STABILITY AND SECURITY EVALUATION ON THE 30 BUS NEW ENGLAND SYSTEM 5.1. Introduction A methodology for determining and ranking weak transmission boundaries for phase, voltage, or current variations was developed in Chapter 2. The methodology was applied to a 30 bus New England test system in Chapter 3. Voltage controllability, observability, and voltage stability were defined in Chapter 4. A necessary condi- tion for controllability and stability was shown to be that SQLV be an M matrix and that SVE be positive. A necessary condition for observability and stability was shown to be that SQGE be an M matrix and that SQGQL be a positive matrix. It was shown that a sufficient condition for SQGQL to have zero columns and SVE to have zero rows and thus violate the necessary conditions for observability and con- trollability is that there exist no reactive sources in a stiffly interconnected group of buses. It is thus clear that having deter- mined the stiffly interconnected groups for any operating condition, sufficient reactive power reserve must be maintained in each group to accommodate load change and loss of reactive source contingencies. A further sufficient condition for voltage stability under light load conditions is that the shunt capacitive admittance at all buses in a stiffly interconnected group must not exceed the admittance of the 140 141 lines in the weak boundary connecting the group to other stiffly interconnected groups. This condition may well justify the utility's practice of switching out some of the long transmission lines with large shunt capacitance under light load conditions. Line outage and loss of generation contingencies are simulated for the same 30 bus New England system model used in Chapter 2 to test the methodology for determining and ranking weak transmission boundaries. The results of these simulations using a Philadelphia Electric (PE) Load Flow are presented in Section 5.2. The bus voltages in each stiffly interconnected group respond similarly for each con- tingency. Line outages of elements in the weak voltage boundaries are shown to cause large voltage drops at buses in load groups and large voltage increases at buses in supply groups. Loss of generators in a load group with insufficient reactive supply is shown to cause the-same large voltage drops in load groups and large voltage increases in the supply groups. These results confinn that sufficient reactive reserve must be maintained in each group for proper voltage control and that the weak transmission boundaries decouple the voltage con- trols (reactive sources) in one group from supplying load in another group (loss of voltage controllability). The results confirm that the need for reactive support will be observed and thus occur locally unless insufficient reactive resources exist in a stiffly intercon- nected group. The reactive requirements can be observed across weak boundaries if there is insufficient resources in a group and thus cause the large voltage variations across the weak boundaries. 142 The study, performed in subsections 5.3 and 5.4, is intended to show that adding shunt capacitance in a stiffly interconnected group can cause loss of controllability, observability, and stability for non light load condition, confirming the analysis performed in Chapter 4 for light load conditions. An experiment is performed where a stiffly interconnected group with a weak transmission boundary is forced to be reactive resource deficient due to loss of generation contingencies. Capacitive reactive support and load is simultaneously increased at a bus in this group. The results indicate that the voltages at buses drop with increased capacitive admittance until the loadflow no longer converges. Moreover, eigenvalues of the Jacobian.rapidly decrease to zero from large positive values as the capacitive admittance is increased. The elements in the eigenvectors associated with these rapidly decreasing eigenvalues experience sign changes and magnitude changes for those elements in the group experi- encing the voltage collapse. Thus the addition of capacitances for voltage support can actually weaken the boundary around the group, causing a voltage collapse and a loss of voltage controllability and observability induced stability. 5.2. The Systematic Impacts of the Weak Boundary to the Voltage Stability The original New England System is very stable because each group in the system has sufficient reactive sources. In order to observe the voltage problems, two strategies are applied in this 143 section to simulate contingencies which will cause the voltage sta- bility problems: (1) loss of generation contingencies that cause one group to have insufficient reactive supply, which then causes complex power to flow over the weak transmission boundary; and (2) line outage contingencies that remove elements in weak transmission boundaries, which further weaken these boundaries. Several line outage and loss of generation contingencies were simu- lated. The effects of line outages of elements in weak boundaries always had much greater security problems than outages of branches within stiffly interconnected bus groups. Case 1 is a multiple loss of generation and line outage contin- gency that both affect stiffly interconnected group IV in Figure 5.1. The loss of generation leaves the group with insufficient reactive supply and the line outage weakens the transmission boundary for that group. Note that the results for this case, presented in Table 5.1, show low voltage violations at several buses in this group. Moreover, high voltage violations are observed at generator buses in groups I and II. Case 2 is a multiple loss of generation contingency of genera- tors 6 and 10 in stiffly interconnected group IV in Figure 5.1. Note that in this case all voltage in this stiffly interconnected group have low voltage violations and the generator buses in a neighboring group have high voltage violations. 144 "¢fl ‘\ ’I’ 29 “ av /’ ‘. / <2 ___....., IA” 25“ I 4”; I I , ’l I, 20 I ”£:--~\ I I ’ r’ I] 1 —- 27'- ’l ’ I 4' 4' 4r \ \‘ I” 21‘ s. - ~u [MI \ -,_ , "--_ 1:“, “~_°_::" --~-‘T‘ 22 \ "’ -------------- I I. ‘ ,fi, 1h A s l A) \ I” " i - , I I \ v ’ \ 1’ S i' i! \ a I, IV I I 23 " :9 6 ' ’ \ ' ’ H ' l ’ j \ l I \ 8 - I ’19 ' 4’ \ I l , \s 12- - :l (E I \ , ' nmvx \ I rvvx II {7“""‘ ,Nnvx |‘ (I ‘x 11 I‘20-——- I \. I \ ’ ‘x 13 1' \ 1’ ‘Tiy 10- . - 45’ ‘5 1.05 p.u.) Low: V4 0.937 V5 = 0.944 V6 = 0.946 V7 = 0.938 v8 - 0.939 High: v19 = 1.051 v22 = 1.051 v25 = 1.058 v29 = 1.051 146 These results clearly show that buses in a stiffly connected group behave as a single bus for line outage and loss of generation contingencies. Furthermore, the large voltage variations across elements in the weak boundaries indicate that both a loss of voltage control and a loss of observability of the need for reactive support occur in these weak boundaries. 5.3. Local and Global Effects of Capacitance on voltage Stability The theoretical results in Chapter 4 indicate that under light load conditions a sufficient condition for voltage controllability requires that the magnitude of the shunt capacitance at any bus in a stiffly connected group be less than the admittance of all branches connected to that bus. Likewise, a sufficient condition for voltage controllability for light load conditions is that the sum of the admittance of all shunt capacitance in a stiffly connected group must be less than the sum of the admittance that connects this group to other groups and thus belongs to the weak boundary for that group. These results do not necessarily apply for non light load conditions. A hypothesis, that replacing generation reactive support by capacitive reactive support can cause loss of voltage controllability and observability and lead to a voltage collapse, is tested. A special test case was developed to provide a test for the hypothesis. Gen- erators 6 and 10 are removed from stiffly interconnected group IV in Figure 5.1. Generators 2, 25, and 29 make up for the lost real power generation evenly. Case 1 results in Table 5.1 suggest that these 147 generators also attempt to provide reactive support over the weak transmission boundary. Line (30,9) is removed to prevent the swing bus from directly supplying reactive support for group IV in Fig- ure 5.1. The stiffly connected groups for this system were then recomputed and are given in Table 5.3 and Figure 5.1. The system at this point does not have sufficient reactive supply and the load- flow did not converge. A 600 MVAR shunt capacitor is placed at bus 11 in group 4 of Table 5.3 that should experience voltage problems. The capacitor is placed at bus 11 because it is located electrically between the buses where generators 6 and 10 were lost. The summary of the abnormal voltages from the PE loadflow is shown in Table 5.4. There is no low voltage reported and the high voltages are right on their maximum desired limits, indicating that this case is normal. Case 4 is shown in Table 5.5, where only a 500 MVAR shunt capacitor is inserted at bus 16. One low voltage is reported at bus 8, which indicates that group 4 has difficulty in importing reactive power across its weak boundary. At this point, the voltage problem still appears to be a local problem. The results for Case 5 are shown in Table 5.6, where a smaller 400 MVAR capacitor is inserted. The voltage problem is spread over all the buses in groupIV. It is clear from Cases 4-6 that the sen- sitivity to the capacitive reactive support appears to be large. Four cases were simulated with 700 MVAR, 800 MVAR, 900 MVAR, and 1000 MVAR capacitors, respectively. All of these cases have no low 148 Table 5.2. Summary of results for case 2. System configuration Removed generator(s): 6, 10 Removed line(s): none Chaages of load and shunt capacitance Load: none Total: N/A Shunt capacitor(s): none Abnormal Voltages (V < 0.95 or V > 1.05 p.u.) Low: V4 = 0.917 V5 = 0.909 V6 = 0.908 V8 = 0.904 V10 = 0.914 V11 = 0.911 V13 = 0.917 V14 = 0.928 High: V19 = 1.051 V22 1.051 V25 = 1.058 V7 V12 v29 = 0.902 0.897 1.051 149 Table 5.3. New groups obtained from the weak boundary identifica- tions based on voltage measure. Group 1 Generator(s): 2, 25, 29, 30 Load: 1, 3, 16, 27, 28 Group 2 Generator(s): 19, 20, 22, 23 Load: 21 Group 3 Generator(s): none . Load: 15, 16, 17, 18, 24 Group 4 Generator(s): none Load: 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 150 Table 5.4. Summary of results for case 3. System configuration Removed generator(s): 6, 10 Removed line(s): (30,9) Changes of load and shunt capacitance Load: none Total: N/A Shunt capacitor(s): 600 MVAR at bus 11 Abnormal voltages (V < 0.95 or V > 1.05 p.u.) Low: none High: v = 1.051 v22 = 1.051 v 19 25 = 1.058 V29 = 1.051 151 Table 5.5. Summary of results for case 4. System configuration Removed generator(s): 6, 10 Removed line(s): (30,9) Changes of load and shunt capacitance Load: none Total: N/A Shunt capacitance: 500 MVAR at bus 11 Abnormal voltages (V < 0.95 or V > 1.05 p.u.) Low: V = 0.948 8 High: V1 9 = 1.051 V22 = 1.051 V25 = 1.058 V = 1.051 29 152 Table 5.6. Summary of results for case 5. System configuration Removed generator(s): 6, 10 Removed line(s): (30,9) Chagges of load and shunt capacitance Load: none Total: N/A Shunt capacitance: 400 MVAR at bus 11 Abnormal Voltages (V < 0.95 or V > 1.05 p.u.) Low: V4 = 0.921 V5 = V8 = 0.899 V9 = 0.905 V10 = V12 = 0.924 V13 = 0.938 V14 = High: V19 = 1.051 V22 = 1.051 V25 0.915 V6 = 0.918 V7 = 0.902 0.940 V11 = 0.941 0.938 1.058 V29 = 1.051 153 voltage reported, but all have several generator buses outside group 4 reach their upper voltage limits. Case 4, where the 500 MVAR capacitor is insufficient to meet reactive load in group 4, is studied further to determine if increas- ing reactive power load and increasing capacitive reactive support at bus 11 can cause a weakening of the weak boundary between group 4 and the rest of the system. It was hypothesized that a loss of voltage stability and thus a voltage collapse as predicted in Theorem 11 for light load conditions would occur and this is confirmed in the following results. Case 4 is chosen to initiate the experiment because the low voltage at bus 8 indicates there is both insufficient reactive support within group 4 and a reliance reactive flow that crosses the weak transmission boundary. Case 6 shows the loadflow results when 1250 MVAR of capacitive reactive support and 750 MVAR of reactive load injection are inserted at bus 11. Case 6 is identical to Case 4 in terms of having a net 500 MVAR of reactive support at bus 11. The results in Table 5.7 indicate that buses 4, 5, 6, 7, and 12 now have low voltage problems in addition to the low voltage at bus 8, which was also observed in Case 4. Several cases were run where the capacitive reactive support and reactive load injection at bus 11 were both increased in a manner that maintained a net 500 MVAR of reactive support. The PE loadflow failed to converge for all of these cases. These results from Cases 4, 6, and these additional cases indicate that capacitive reactive support in a stiffly interconnected group that is reactive resource Table 5.7. 154 Summary of results for Case 6. System configuration Removed generator(s): Removed line(s): 6, 10 (30,9) Changes of load and shunt capacitance Load: Total: Shunt capacitance: Abnormal voltages: Low: V4 0.935 V 0.918 8 High: V19 = 1.051 750 MVAR at bus 11 N/A 1250 MVAR (V < 0.95 or V > 1.05 p.u.) V5 = 0.933 V6 = 0.937 V7 = 0.921 V12 = 0.944 V22 = 1.051 V25 = 1.058 V29 = 1.051 155 deficient can cause a loss of controllability and observability across the weak transmission boundary that leads to voltage stability and voltage collapse. A stiffly interconnected group that had suf- ficient reactive support from 600 (Case 3) to 1000 MVAR of capaci- tance reactive power support did not have stability problems. Thus, it appears that capacitive support may only cause voltage stability problems when the group it has been inserted in still has insuffi- cient reactive support and is also relying on weak transmission boundaries for support. Case 6 shows that the voltage problem at bus 8 in Case 4 spreads over the entire group but the voltage at bus 11 can still be maintained within the desired range. It also indicates that the shunt capacitor solves the voltage problem at bus 11 locally, but the neighboring buses start to experience the low voltage problem. In Chapters 2 and 3 the theory and the measures of weak bound- ary for a power system were developed, tested, and confirmed. In this chapter the effects of the weak boundary to the voltage sta- bility are checked experimentally. Except for the base case, all the simulations are carried out under heavy load conditions. Using the same New England 30 bus model, a series of contingencies are simulated in section 5.2. It confirmed that the weak boundary will prevent the reactive power from being transferred from one SSC group to the other. Therefore the voltage problem can only be solved in the local area. 156 In this section and section 5.4 the effects of shunt capacitor under heavy load condition are investigated. In general, there are three sources of reactive power which can be used to support the voltage in the local area: (1) generators, (2) synchronous conden- sers, and (3) shunt capacitors. The costs of these different kinds of equipment are quite different and the shunt capacitor is the cheapest one among them. Since the side effects of the shunt capaci- tor were not well understood before this investigation, the system planner usually preferred to use the shunt capacitor as the supple- mental device for reactive power supply for a local area. It is a widespread practice in the industry to install shunt capacitors as reactive supply to meet reactive load and support voltage in each local area. The results obtained in this section can be further explained by the eigenvalue analysis of the sensitivity matrices in section 5.4. It will be shown that a large positive eigenvalue is reduced to nearly zero as the reactive load injection and capacitive supply are increased. The results obtained in this section are very important in today's power system planning and operations, because the system planners can easily concentrate on the fact that the capacitors can always keep the local bus within the desired voltage limit as it did in the above case at bus 11. When the system has a voltage collapse, it is really difficult to convince oneself that the capacitance at all-looks-normal bus 11 instead of some other factor caused the system collapse. Without the sensitivity analysis, eigenvalue 157 analysis combined with the identification of the weak boundary, the understanding of voltage problems for a large scale system is dif- ficult. The sensitivity/eigenvalue analysis of Cases 3-6 is analyzed in the next subsection. 5.4. The Effects of Shunt Capacitor on the Controllability and Observability of the Steady State voltage Problem The sensitivity matrices SQGE and SQLV are computed for Cases 3-6 and are displayed in Tables 5.8-5.11. The matrices are diagonally dominant and all the large elements lie in diagonally dominant blocks because the generator buses and load buses were reordered according to the buses in the four stiffly interconnected groups in Table 5.3. Thus, the ordering of buses in SQGE is 2, 25, 29, 30, 19, 20, 22, 23. The SQGE matrix has two diagonal sub-blocks of (2, 25, 29, 30) and (19, 20, 22, 23) which are the only two stiffly interconnected groups with PV (generator buses). The ordering of buses in SQLV is (1, 3, 26, 27, 28), (21), (15, 16, 17, 18, 24), and (4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14), where the brackets represent the PQ (load) buses«in the four stiffly interconnected groups in Table 5.3. The eigenvalues and eigenvectors for the SQ E and SQ V matrices L are also displayed in Tables 5.8-5.11 for Cases 396. In Case 3, with 600 MVAR capacitance, the SQLV eigenvalue 6 in Table 5.8 is the smallest of the eigenvalues and has value 237. The largest elements of an eigenvector indicate the buses at which the eigenvalue has the most effect. 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In Case 4 with 500 MVAR capacitance, the eigenvalue 6 has value of only 2.20 and the eigen- vector 6 no longer has a large element value (.88) at only bus 4 at group 4, but now has moderately large values 11-40 at all buses in group 4 except bus 11. In Case 5 with 400 MVAR of capacitance, the eigenvalue 6 drops further and the magnitude of the elements for the eigenvector increases at all buses in group 4 except at bus 4, which originally had the large dominant value in these eigenvectors. The value of the eigenvector element for bus 11 is still small. In Case 6, with the 1250 MVAR capacitance and 750 reactive load, the eigenvalue decreases to 0.13 but the eigenvector for eigenvalue 6 is similar to the values in Case 5. It is evident that reducing the capacitive reactive support in Cases 3-5 causes reliance on the weak transmission boundary and thus causes a weakening of this boundary as evidenced by: (I) A large reduction in the magnitude of eigenvalue 6. (2) A significant increase in the magnitude of the elements of the eigenvector for eigenvalue 6 at all buses in group 4 except buses 4 and 11. Bus 11 with the capacitance is no longer part of the group since its voltage is maintained by the capacitance. The increase in eigenvalue elements at all other buses in group 4 except bus 11 indicates eigenvalue 6 becomes a group against group eigen- value rather than a local bus 4 eigenvalue. The evolution of group eigenvalues reflects the fact that the reactive flows so weaken the boundary of the group that it effectively acts as an equivalent bus. 179 (3) The sign of elements in eigenvector 6 in group 4 changes sign, indicating that eigenvalue 6 is destabilizing the network as the boundary is weakened. In Cases 3-5 the addition of capacitance and reactive load injection confirms that adding capacitance in the group and retaining the reliance on reactive support across the boundary reduces eigen- value from approximately 2.2 to 0.13 from results in Cases 4 and 6. Moreover, adding this capacitance and load simultaneously also increases the magnitude of elements in group 4. Both of these results indicate the addition of capacitance in a group with a weak boundary significantly weakens the boundary and causes the group to act as a single equivalent bus. Further additions of capacitance in this group would appear to cause the eigenvalue 6 to 90 negative and the elements of eigenvector 6 in group IV to further increase. These results were not obtained because the PE loadflow did not converge for the cases run with additional capacitance and load at bus 11. The SQGE eigenvalues 4, 5, and 8 experience major changes as capacitance is added in Cases 3-6 in Tables 5.3-5.8. The eigen- value 4 decreases initially as the capacitance is decreased from 600 MVAR to 500 MVAR in Cases 3 and 4. However, all three eigen- values increase rapidly for Cases 5 and 6, where the boundary is weakened due to additional flows across the boundary or additional capacitive reactive support, respectively. Eigenvector 4 has large values at generators 25 and 19. Eigenvector 5 has a large element at 19, and eigenvector 8 has a large element at 22, all of which 180 experience high voltages in Cases 5 and 6. Thus, the effect of capacitance at bus 11 in group 4 has an effect on the SQGE even when there are no generators in group 4 because the weakened bounda- ries raise voltages at generators in other groups. This raise in voltage at generator buses in other groups is effected by increases in the eigenvalues that have eigenvectors with large components at these generator buses. A loss of observability and stability is thus evidenced by the increases in eigenvalues in 5Q E and the increases in elements of G the associated eigenvectors at generator buses that experience high voltage problem. 50 CHAPTER 6 CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH 6.1. ‘Bgvigw The existence and location of weak transmission boundaries is well known to utility system planners and Operators based on their years of experience with a particular system. Formal methods for detennining the location of weak transmission boundaries, ranking the relative vulnerability of the boundary, and determining the transmission elements that belong to the weak transmission boundaries did not exist. A method for determining and ranking weak phase transmission boundaries was recently developed in [I]. These weak phase trans- mission boundaries were shown to cause phase oscillations and thermal and steady state stability problems for either inertial loadflow simulated loss of generation contingencies or line outage contingen- cies. A computer package was developed that allowed ranking of the network branches in terms of their impact on either thermal security or based on steady state stability. Contingencies that most severely affect each of the most vulnerable network branches are also deter- mined as part of this package [1]. This thesis extends these previous results by defining weak transmission boundaries for voltage and for current variations. The 181 182 groups of buses within a stiffly interconnected group surrounded by a weak voltage transmission boundary will be a voltage control area. The weak voltage and weak current transmission boundaries should be those boundaries across which large voltage variations occur and large current changes that lead to thermal overload occur. The weak phase transmission boundaries defined in [1] should determine the network branches and boundaries where steady state stability problems should occur. A phase, voltage, and current coherency measure is proposed and theoretically shown to detect the weak phase, voltage, and current transmission boundaries. A method for determining and ranking weak phase, voltage, and current transmission boundaries was developed and was applied to the 30 bus New England System model. Weak phase, voltage, and current transmission boundaries were determined and ranked. The current and phase boundaries are similar but the ranking of the boundaries is different. The weak voltage boundaries are quite different from either the phase or current boundaries and separate buses into local voltage control areas as would be expected from engineering judgment. Voltage controllability, observability, and stability are defined in section 4. Necessary conditions on the properties of sensitivity matrices SVE’ SQLV, SQGE, and SQGQL were found that assure voltage controllability, observability, and stability. Finally, it was shown that lack of PV (reactive sources) buses in any stiffly interconnected group is a.sufficient condition for a loss of voltage 183 controllability or observability. A sufficient condition for loss of stability for light load conditions is that the shunt capacitive ..admittance at all buses in a stiffly interconnected group exceed the admittance of all branches in the weak transmission boundary that surrounds the group. These results suggest that sufficient reactive reserve must be maintained in each control area since relying on reactive support across weak transmission boundaries can cause a loss of voltage controllability and observability that may lead to voltage collapse. Moreover, the percentage of reactive support and reactive reserve in any control area made up of switchable shunt capacitance should be limited depending on the weakness or possible contingency induced weakness of the voltage transmission boundary. Thus the more expensive synchronous condensers may have to be used rather than switchable capacitors as is the present practice when insufficient generation reactive support is available in a stiffly connected group. Long transmission lines with large shunt capacitance may also need to be switched out under light load conditions to avoid having total shunt capacitance in a stiffly connected group exceed the admittance of the weak boundaries for that group. A further sufficient condition requires the network to be nonuniform since if all buses in a group are connected together by nearly identical branch elements, a loss of stability can occur. This could occur on a power system distribution network where all buses are often inter- connected by nearly identical transmission elements. 184 The simulation of multiple loss of generation and line outage contingencies confirms that buses in a stiffly connected group behave similarly and that large voltage variations occur across weak bounda- ries. These results confirm that weak transmission boundaries cause voltage security problems due to loss of voltage controllability and observability. A loss of voltage stability is then shown to occur for a network where a group has weak transmission boundaries. The greater the reliance on the boundary for reactive support the greater and more widespread the low voltage problem is within the group. This reduction in the magnitude of eigenvalues, the increase in mag- nitude of elements of the associated eigenvectors within the group experiencing low voltage problems, and the sign change of elements of eigenvectors for buses in the group all indicate that requiring additional reactive power flow across weak boundaries will lead to loss of voltage stability and thus voltage collapse. Addition of capacitance in the stiffly connected group in a like manner caused dramatic reduction of positive eigenvalues, an increase in the mag- nitude of elements for buses in the group, and finally a decrease of voltage at buses in the group. The loadflow would not converge if too much capacitive admittance was inserted. The results clearly confirm that reactive flows across weak transmission boundaries and capacitive support within the stiffly interconnected group can both cause loss of voltage controllability and loss of observability, voltage security problems, and ultimately voltage collapse. 185 6.2. Recommendations for Future Research Future research on voltage stability and security problems could: (1) Determine whether the constraints on the shunt capacitive admittance in stiffly interconnected group developed for light load conditions can be applied or modified for non light load conditions. (2) Define phase controllability, observability, and sta- bility in a similar manner as performed for voltage and detennine necessary conditions for phase controllability, observability, and stability. (3) Determine sufficient conditions that assure phase and voltage controllability, observability, and stability. (4) Relate how phase and voltage stability affect the asymp- totic stability of the non classical transient stability model. (5) Determine a fast computational method for determining and ranking the voltage security of elements based on a voltage network element security measure. The method for determining and ranking the contingencies that most severely affect the most insecure ele- ments would also be desired. This security assessment methodology could be based on the work performed for phase stability or thennal overload security problems [1]. (6) Determine a pattern recognition procedure that could identify voltage security and stability problems based on steady state estimation data or for a simulated contingency. The problems would be identified without the checking of voltage constraints, 186 reactive reserve constraints, or capacitive support constraints in each voltage control area [14]- (7) Development of robust operating constraints that can assure voltage security and stability in each stiffly interconnected group. APPENDIX 1 BASE CASE LOADFLOW DATA IN COMMON FORMAT APPENDIX 1 BASE CASE LOADFLOW DATA IN COMMON FORMAT DAY 317 OF 1980 TEST 30 ITEMS 100.. 1980 ODOOOODOOOOOOOCOOOOOOOOOOOOOOD 990090009006000OOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOO . u u o o I o o I a v I c v o o o 6000000009006000OOQOCOOOOOOOOQ 6 0° 0°. 0 Q BOOOCOOOOOOOOOOOOOGOOO60000000 I I O I I I I C 000000OOOOOOOQO.-Q°°Q°O°°O°°°° N “‘0 90“ '1‘ IN N NC nnN N N l OI III I I .0 00 00° C o .°°°°°°°°°°°°°°°°°°C°°°°°°°°°° . C . . Q I O I I I O I I I I I I C O 0 C I . . C I I I . . 000000000000000000399900000090 no 0°“ 9 n u.” N? nnN c n 9° 9° 9°. . . . 9‘ —. Qfifl d C 0 ‘7“ “n I‘m, n O ' °°°°°°°°°°°°°°0°°°096060090000 C C . I I O I I O C I I 0 C C C O O C . C O C . I O . I .a‘oflOCd—OOOOCOGHOOO°°°°H°~°°°°°~ 0°00°°°°°0°°°°00°°°0900669900- °°O°°°°°°°°°°°°°G°°°°°°°°°°°6° C I I I C D I I C I I I O I I I I O O O I I O 0 I I O I O O OOOOGCOOOOOGCOOOOOOOOO00000009 OOOOOOCOOOOOOOQOOOC0569600099. ~~~~~fl~~~~H~fi—~fldnfifl~flflfl~~~~fl~ ° 00. O N 00000000000000H0°0°°0°00~0000°O I O U C I C C I I Q I 660009000000°0Ifl°°°0°0°°fi°°°°°fi .mn N0 nNa n h N“ 00 eon 0 n h ‘ 00000 0 00 000 000 000 oeuooococoaoooenoeeneoNooooeoo 0 CI 0.. I I I O I I I I I I I I O. I I I I I I I I I I 00hhh000~0n000nm000NO0N0000n09 v—mNnQ an on 00" 0 h Int-00 Nu — n I a a 00000 0 00 000 000 000 ooooonooooooooeoeomvooONeoecan oevoonvotvoecoco—encoooooumbeeas NneocoN Is on M N a N Nmnwp N—oNNo—nr‘l N ON N If! If! n cos-um fl - oeunnoeooONnNa—onomNmoeeaneNno vqnanavonNnNocheocoohncnooeco I III... 0 I I I I I I I I I I I I I O I O I I I I I I Ocnon00000mn~on¢flh0hh00'50—‘000‘00 IlllllnulllntIIIIAIIIIII—IIlll II t I I 0000-00ln00—0M000h#HONNON0n—IQRO0 saownnoeonnoon—onsoneuocomNo—c vvnmmmnoncnN~m0nu¢cnnnnoo~~~nv flan—I‘fl—Iulfla-Ir‘c-I—Io-IH—OQI-IfiOHfi—O—I—n—Odd—dud 0NN°0NN°°°°°NNN°00000NOM0000°N ~~~~~~~Nnnnccvvcnoooooosnshuns mum—«A—NnnnvvvccmoooooohnnnNun 37 ITEMS ~N~1qmomwco~r4m¢u~or~~¢c~n .—-————.———.——.——.r;r.r; OOOOOOOODOOOOOOOOODOOO OOOOOOOQOQQOOOOOOOQOOO OOOOOOOOOQOOOOOOOGGODO DOOOOOQOOOOOOOOOOOOOQO GOOOOOOOOOOQOOOOOOOOOQ QGQOQOGOOGOOQOOOOOOOOD I .QOOQCCOGOOOOOOOOOOOOC I I I I I I o o n o I o I I I 6’ an ~00 OOCOOOOOOOOGGOOOOGQOOO 0000000000090000000—1—00 °°°°°°°°°°°°°°°°°°°O°° OOOOOOOOOOOOOOOOOOOOOO 0000000000000000000 0 000NON000NNQQ¢00°QQ 0 oceanoaoa‘ro—mnsoomw Q onvmnoNncnnNnevmonm h ON—INU‘NONN‘F‘NNOF‘NNOOQOO I I I I I I I 0 I I I 0 I I I I I I I I I I 000000—000000000—000000 0000000000000000000000 0000000000000000000000 ace—unvms—noennoNoonv-fiuono HnflmNthmNN~nN~¢m¢¢nn¢ ¢N=~n¢o~~~mwuo~~Noe—¢e 0000000000600000000000 I O I I I I O I O O I I I I I 0000000000000000000000 0000000000000000000000 OOOOOOOQOOOOOOQOOOOOOO moonNnmeconnmNno°¢¢mo~r n~~~n¢m~—oe—~ooo~oco—o OOOOOOOOODOOOOOGOOOOOO OOOOOOOOOOOOGOOOOOOQOO I O O O I I I O I O O I I I I O I I I I I I 0000000000000000000000 WOOOOOOOOOOGQQOOOOOO—IHO Oa-Ia-o—I—I—Ia-OI-I—I—IHI-I—O—I-Id—IHHH—i—o— 11/14/50 CONSUMERS PONER C0. m d 3 .‘H—cmn—n—ndanNlflnnnl-‘v‘f‘rcng O 2 z 2 O a O OZ< O u J¢<<