IIIII|I'IIIIII I I I I I III! This is to certify that the dissertation entitled VOLTAGE STABILITY AND SECURITY ASSESSMENT FOR POWER SYSTEMS presented by Antonios G. Costi has been accepted towards fulfillment of the requirements for Ph.D. degree in Electrical Engineering : ‘l U, I Major professor Date 1/ I 9/ 87 MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 MSU LlBRARlES .—;—. RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. VOLTAGE STABILITY AND SECURITY ASSESSMENT FOR POWER SYSTEMS By Antonios G. Costi A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering and Systems Science 1987 Copyright by ANTONIOS G. COSTI 1987 ABSTRACT VOLTAGE STABILITY AND SECURITY ASSESSMENT FOR POWER SYSTEMS BY Antonios G. Costi Voltage collapse and abnormal high and low voltages have been ob- served with even greater frequency and severity on large interconnected systems. Recurrent problems exist in Europe, Japan, Southern California, Florida, Pennsylvania Jersey Maryland, New York Power Pool, and Ontario Hydro. These voltage problems are associated with the insufficient local reactive reserves, the transfer of power across long geographical dis- tances, and the ever increasing loading of the long transmission lines of these interconnected systems. For the study of the voltage problems, a sensitivity model was developed based on the linearized decoupled loadflow model. Definitions of P0 and PV stability and controllability that express a healthy cause- effect relationship between the states AV and A06, and the controls AE and disturbances AQL, were developed. Then sufficient conditions on the sensitivity matrices that guarantee PV and P0 stability and controlla- bility were derived. More stringent conditions on the sensitivity matrices were developed not only to guarantee P0 and PV controllability but also assure that load disturbances AQL cause small voltage deviations at the PQ buses, small fl «.9 Antonios G. Costi reactive losses in the system, AE that and that there are always voltage changes can control both the voltage changes at the PQ buses, and the reactive generation of the system. A method for determining the voltage control areas (VCA) and the weak transmission boundaries was developed. Computational results reveal that voltage collapse problems in a VCA are associated with the inability of transfering large amounts of reactive power across the weak trans- mission boundary, and that excessive reactive capacitive support in a VCA causes reactive flows from the VCA to the rest of the system worsening a voltage problem rather than improving it. Real and reactive load in- crease at a bus in a VCA was found to affect the voltages at all buses in the VCA. The VCAS, that are experiencing or are having voltage collapse problems, can be identified using the strong PV and P0 controllability conditions. The degree of violation of these conditions at buses in a VCA can be used to determine the degree of severity of voltage problems in the VCA. Dedicated to my father, George Costi and my brother Panayiotis who are both missing since the 1974 Turkish invasion of Cyprus. ii '5. r” h".. ' a I." . u I 0' . I 9 | i . ." p. . n... \ \ . . . ACKNOWLEDGMENTS Where we see smoke there is fire, and where we see success, there is a whole bunch of people contributing. I would like to thank everyone of them. I would like to thank my family for getting me started right and providing me with good values in life; my grandmother Maria, and my aunt Katina, for convincing my parents to let me go to high school; my high school math teachers, Andreas Kakoulis, Andreas Hatzithas and Costas Mavros that Opened my eyes to see the beauty of mathematics; my wife Coula, without her patience, understanding and love, nothing would have become reality; and Father John Poulos, who with love guided my family all these six difficult but beautiful years at MSU. I would especially like to thank my advisor, Dr. Robert A. Schlueter for his true friendship, constant encouragement, and professional criti- cisms made during the last three years of my Ph.D. studies at MSU. For their interest and advise, I also thank the other members of my committee: Drs. Robert Barr, Hassan Khalil and Habib Salehi. The preparation of the manuscript could have not been done without the excellent job and the extreme patience of my typist, Linda Sonier. TABLE OF CONTENTS Page LIST OF TABLES .......................... vi LIST OF FIGURES .......................... xiv CHAPTER 1. INTRODUCTION ....................... 1 1.1 Voltage Collapse .................. 1 1.2 Voltage Collapse Mechanism ............. 2 1.3 Research Contributions of this Thesis ........ 4 2. LITERATURE STUDY AND STUDY OBJECTIVES .......... 8 2.1 Introduction .................... 8 2.2 Objectives ..................... 24 3. MODEL DEVELOPMENT .................... 28 3.1 Introduction .................... 28 3.2 Model Development .................. 29 3.2.1 Loadflow Equations .............. 33 3.2.2 Polar Form of Loadflow Equations ....... 35 3.2.3 Hybrid Form of Loadflow Equations ...... 35 3.2.4 Evaluation of the Partial Derivatives . . . . 36 3.2.5 Building the Jacobian Submatrices ...... 38 3.2.6 Decoupled-Model Development ......... 45 3.2.7 Sensitivity-Model Development ........ 46 3.3 Network Topology and Properties of the Sensitivity Matrices ...................... 49 iv 4. VOLTAGE STABILITY AND CONTROLLABILITY .......... 52 4.1 Introduction .................... 52 4.2 PO Stability and Controllability .......... 53 4.3 PV Stability and Controllability .......... 63 4.4 Stability and Controllability Constraints ...... 71 4.5 Discussion on Quantitative Controllability ..... 82 5. VOLTAGE CONTROL AREA AND WEAK TRANSMISSION BOUNDARY DETERMINATION ...................... 88 5.1 Introduction .................... 88 5.2 Determination of Voltage Control Areas and Weak Transmission Boundaries ............ 100 6. COMPUTATIONAL RESULTS ON STABILITY AND CONTROLLABILITY . . 120 6.1 Introduction .................... 120 6.2 Computational Results Based on Sensitivity Analysis ...................... 127 6.2.1 Experiment I ................. 129 6.2.2 Experiment 11 ................ 150 6.2.3 Experiment III ................ 194 6.2.4 Experiment IV ................ 222 6.2.5 Experiment V ................. 225 7. CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH . . . 239 7.1 Review ....................... 239 7.2 Recommendations for Future Research ......... 244 APPENDIX A ............................ 248 APPENDIX B ............................ 253 APPENDIX C ............................ 256 BIBLIOGRAPHY ........................... 259 v Table .1(a) .1(b) .1(c) .1(d) .1(e) .1(f) .1(9) LIST OF TABLES Loadflow solution for the base case (Case 1) ........ 93 Loadflow solution when 50 MVAR load is connected at Bus 11 . 94 Loadflow solution when 100 MVAR load is connected at Bus 11 . 95 Loadflow solution when 200 MVAR load is connected at Bus 11 . 96 Loadflow solution when 400 MVAR load is connected at Bus 11 . 97 Loadflow solution when 600 MVAR load is connected at Bus 11 . 98 Loadflow solution when 700 MVAR load is connected at Bus 11 . 99 The lines of the system ranked based on the number of clusters when they appear as weak boundary lines for the first time ......................... 118 Voltage profiles - voltage magnitudes and voltage angles for - the 6 cases ...................... 124 Reactive losses on the weak transmission boundary lines and reactive flows from VCA 1 to the rest of the system ..... 125 The sensitivity matrix SQ Q for the base case (Case 1) . . . 135 G L The sensitivity matrix SQGQL for the case when the reactive load at Bus 11 is 400 MVAR ................. 136 The sensitivity matrix SQGQL for the case when the reactive load at Bus 11 is 600 MVAR ................. 137 The sensitivity matrix SQ for the base case (Case 1) . . 139 LV The sensitivity matrix SQ v'1 for the case when the L reactive load at Bus ll is 400 MVAR ............. 140 vi Table 6.8 6.15(a) 6.15(b) 6.15(c) 6.15(d) 6.15(a) 6.15(f) 6.15(g) 6.15(b) The sensitivity matrix SQ V'1 for the case when the L reactive load at Bus ll is 600 MVAR ............ 141 The sensitivity matrix SVE for the base case (Case 1) . . . 143 The sensitivity matrix SVE for the case when the reactive load at Bus ll is 400 MVAR ................. 144 The sensitivity matrix SVE for the case when the reactive load at Bus ll is 600 MVAR ................. 145 The sensitivity matrix The sensitivity matrix S for the base case (Case 1) . . . 147 QGE S for the case when the reactive QGE load at Bus ll is 400 MVAR ................. 148 The sensitivity matrix S E for the case when the reactive QG load at Bus ll is 600 MVAR ................. 149 Loadflow solution for Case 5 ................ 152 Loadflow solution when 1.00 pu. capacitive susceptance is connected at Bus 11 ................... 152 Loadflow solution when 3.00 pu. capacitive susceptance is connected at Bus 11 ................... 153 Loadflow solution when 5.00 pu capacitive susceptance is connected at Bus 11 ................... 153 Loadflow solution when 7.00 pu capacitive susceptance is connected at Bus 11 ................... 154 Loadflow solution when 10.00 pu capacitive susceptance is connected at Bus 11 ................... 154 Loadflow solution when 10.50 pu capacitive susceptance is connected at Bus 11 ................... 155 Loadflow solution when 10.97 pu capacitive susceptance is connected at Bus 11 ................... 155 vii Table Page 6.16 Reactive losses on the weak boundary and reactive flows in VCA 1 for different values of capacitive susceptance placed at Bus 11 ...................... 157 6.17 The sensitivity matrix sQ V'1 for the case where 0.00 L pu. capacitive susceptance is placed at Bus 11 ....... 159 1 for the case where 1.00 6.18 The sensitivity matrix SQLV' pu. capacitive susceptance is placed at Bus 11 ....... 160 6.19 The sensitivity matrix SOLV-l for the case where 3.00 pu. capacitive susceptance is placed at Bus 11 ....... 161 6.20 The sensitivity matrix SQ v'1 for the case where 5.00 L pu. capacitive susceptance is placed at Bus 11 ....... 162 6.21 The sensitivity matrix SQ v-1 for the case where 7.00 L pu. capacitive susceptance is placed at Bus 11 ....... 163 1 6.22 The sensitivity matrix SQLV' for the case where 10.00 pu. capacitive susceptance is placed at Bus 11 ....... 164 6.23 The sensitivity matrix SQLV'1 for the case where 10.50 pu. capacitive susceptance is placed at Bus 11 ....... 165 1 for the case where 10.97 6.24 The sensitivity matrix SQLV' pu. capacitive susceptance is placed at Bus 11 ....... 166 6.25 The sensitivity matrix SVE for the case where 0.00 pu. capacitive susceptance is placed at Bus 11 ....... 168 6.26 The sensitivity matrix SVE for the case where 1.00 pu. capacitive susceptance is placed at Bus 11 ....... 169 6.27 The sensitivity matrix SVE for the case where 3.00 pu. capacitive susceptance is placed at Bus 11 . ...... 170 6.28 The sensitivity matrix SVE for the case where 5.00 pu. capacitive susceptance is placed at Bus 11 ....... 171 viii Table 6.29 The pu. The pu. The pu. The pu. The pu. The pu. The pu. The pu. The pu. The pu. The pu. The pu. The pu. sensitivity matrix SVE for the case where 7.00 capacitive susceptance is placed sensitivity matrix SVE for the case where 10.00 capacitive susceptance is placed sensitivity matrix SVE for the case where 10.50 capacitive susceptance is placed sensitivity matrix SVE for the case where 10.97 capacitive susceptance is placed sensitivity matrix SQGQL for the capacitive susceptance is placed sensitivity matrix SQGQL for the capacitive susceptance is placed sensitivity matrix SQGQL for the capacitive susceptance is placed sensitivity matrix SQGQL for the capacitive susceptance is placed sensitivity matrix SQGQL for the capacitive susceptance is placed sensitivity matrix SQGQL for the capacitive susceptance is placed sensitivity matrix SQGQL for the capacitive susceptance is placed sensitivity matrix SQGQL for the capacitive susceptance is placed sensitivity matrix SQGE for the capacitive susceptance is placed ix Page at Bus 11 ....... 172 at Bus 11 ....... 173 at Bus 11 ....... 174 at Bus 11 ....... 175 case where 0.00 at Bus 11 ....... 176 case where 1.00 at Bus 11 ....... 177 case where 3.00 at Bus 11 ....... 178 case where 5.00 at Bus 11 ....... 179 case where 7.00 at Bus 11 ....... 180 case where 10.00 at Bus 11 ....... 181 case where 10.50 at Bus 11 ....... 182 case where 10.97 at Bus 11 ....... 183 case where 0.00 at Bus 11 ....... 186 Table Page 6.42 The sensitivity matrix SQGE for the case where 1.00 pu. capacitive susceptance is placed at Bus 11 ....... 187 6.43 The sensitivity matrix SQGE for the case where 3.00 pu. capacitive susceptance is placed at Bus 11 ....... 188 6.44 The sensitivity matrix SQGE for the case where 5.00 pu. capacitive susceptance is placed at Bus 11 ....... 189 6.45 The sensitivity matrix SQGE for the case where 7.00 pu. capacitive susceptance is placed at Bus 11 ....... 190 6.46 The sensitivity matrix SQGE for the case where 10.00 pu. capacitive susceptance is placed at Bus 11 ....... 191 6.47 The sensitivity matrix SQGE for the case where 10.50 pu. capacitive susceptance is placed at Bus 11 ....... 192 6.48 The sensitivity matrix SQGE for the case where 10.97 pu. capacitive susceptance is placed at Bus 11 ....... 193 6.49 Loadflow solution when 930 MVAR load and 14.30 pu. capacitive susceptance are connected at Bus 11 ....... 198 6.50 Loadflow solution when 1410 MVAR load and 19.10 pu. capacitive susceptance are connected at Bus 11 ....... 199 6.51 The sensitivity matrix S0 0 for the case where 0 MVAR G L load and 5.00 pu. capacitive susceptance are connected at Bus 11 ......................... 200 6.52 The sensitivity matrix SQ Q for the case where 600 MVAR G L load and 11.00 pu. capacitive susceptance are connected at Bus 11 ......................... 201 6.53 The sensitivity matrix SQ Q for the case where 930 MVAR G L load and 14.30 pu. capacitive susceptance are connected at Bus 11 ......................... 202 . n .5 I .- 0" .- n- . v- o no u.- - o- v .- vi-h ‘ o! 0 n n, e '4 a ‘1 ll" Table 6.54 6.55 6.56 6.57 6.58 6.60 6.61 6.62 6.63 The sensitivity matrix SVE for the case where 0 MVAR load and 5.00 pu. capacitive susceptance are connected at Bus 11 ........................ The sensitivity matrix SVE for the case where 600 MVAR load and 11.00 pu. capacitive susceptance are connected at Bus 11 ........................ The sensitivity matrix SVE for the case where 930 MVAR load and 14.30 pu. capacitive susceptance are connected at Bus 11 ........................ The sensitivity matrix SQ V'1 for the case where 0 MVAR L load and 5.00 pu. capacitive susceptance are connected at Bus 11 ........................ The sensitivity matrix SQ V'1 for the case where 600 MVAR L load and 11.00 pu. capacitive susceptance are connected at Bus 11 ........................ The sensitivity matrix SQ v'1 for the case where 930 MVAR L load and 14.30 pu. capacitive susceptance are connected at Bus 11 ........................ The sensitivity matrix SQ E for the case where 0 MVAR G load and 5.00 pu. capacitive susceptance are connected at Bus 11 ........................ The sensitivity matrix SQ E for the case where 600 MVAR G load and 11.00 pu. capacitive susceptance are connected at Bus 11 ........................ The sensitivity matrix SQ E for the case where 930 MVAR G load and 14.30 pu. capacitive susceptance are connected at Bus 11 ........................ The sensitivity matrix SQ v'1 for the case where 1410 L MVAR load and 19.10 pu. capacitive susceptance are connected at Bus 11 ................... xi Page . 203 . 204 Table Page 6.64 The sensitivity matrix SVE for the case where 1410 MVAR load and 19.10 pu. capacitive susceptance are connected at Bus 11 ......................... 213 6.65 The sensitivity matrix SQ Q for the case where 1410 G L MVAR load and 19.10 pu. capacitive susceptance are connected at Bus 11 .................... 214 6.66 The sensitivity matrix SQ E for the case where 1410 G MVAR load and 19.10 pu. capacitive susceptance are connected at Bus 11 .................... 215 6.67 Loadflow solution when 1500 MVAR load and 20.00 pu. capacitive susceptance are connected at Bus 11 ....... 217 6.68 The sensitivity matrix sQ V'1 for the case where 1500 L MVAR load and 20.00 pu. capacitive susceptance are connected at Bus 11 .................... 218 6.69 The sensitivity matrix SVE for the case where 1500 MVAR load and 20.00 pu capacitive susceptance are connected at Bus 11 .................... 219 (5.70 The sensitivity matrix SQ Q for the case where 1500 G L MVAR load and 20.00 pu. capacitive susceptance are connected at Bus 11 .................... 220 6.71 The sensitivity matrix SQ E for the case where 1500 MVAR G load and 20.00 pu. capacitive susceptance are connected at Bus 11 ......................... 221 6.72 Loadflow solution when 200 MVAR load and 5.00 pu. capacitive susceptance are connected at Bus 11 ....... 223 6.73 Loadflow solution when 300 MVAR load and 5.00 pu. capacitive susceptance are connected at Bus 11 ....... 224 6.74 Loadflow solution when 100 MW load and 5.00 pu. capacitive susceptance are connected at Bus 11 ....... 226 6.75 Loadflow solution when 200 MW load and 5.00 pu. capacitive susceptance are connected at Bus 11 ....... 227 xii Table Page 6.76 Loadflow solution when 250 MW load and 5.00 pu. capacitive susceptance are connected at Bus 11 ....... 228 6.77 Loadflow solution when 270 MW load and 5.00 pu. capacitive susceptance are connected at Bus 11 ....... 229 6.78 Loadflow solution when 280 MW load and 5.00 pu. capacitive susceptance are connected at Bus 11 ....... 230 6.79 Loadflow solution when the charging of the lines is reduced to half of the original one, and reactive loads are modeled as reactive impedances to ground (Case 7) . . . 232 6.80 The sensitivity matrix SQLV'1 for Case 7 ......... 235 6.8]. The sensitivity matrix SVE for Case 7 ........... 236 6.82 The sensitivity matrix SQGQL for Case 7 .......... 237 6.83 The sensitivity matrix SQGE for Case 7 ........... 238 xiii Figure 2.1 4.1 5.1 5.2(a)-(b) 5'-2(c)-(d) 5-2(e)-(f) 5-2(9)-(h) 5 - 2(i)-(J') 5-2(k)-(l) 5.2(m) (5 .1 5 .2(a) 6 .2(b) 5.2M 6.2(d) LIST OF FIGURES Simple two-bus system ................. Equivalent n-Model for a Transmission Line ...... The 30 Bus New England System ............. Clustering schemes with (a) 2, (b) 3 clusters for the 30 Bus New England System ........... Clustering schemes with (c) 5, (d) 6 clusters for the 30 Bus New England System ........... Clustering schemes with (e) 8, (f) 9 clusters for the 30 Bus New England System ........... Clustering schemes with (g) 10, (h) 12 clusters for the 30 Bus New England System ........... Clustering schemes with (i) 15, (j) 17 clusters for the 30 Bus New England System ........... Clustering schemes with (k) 20, (l) 23 clusters for the 30 Bus New England System ........... Clustering scheme with 24 clusters for the 30 Bus New England System ............... The 30 Bus New England System and the voltage control area VCA 1 .................. Voltage magnitude profile for the base case (Case 1) OOOOOOOOOOOOOOOOOOOOOO Voltage magnitude profile for the case when the reactive load at Bus 11 is 400 MVAR ........ Voltage magnitude profile for the case when the reactive load at Bus 11 is 600 MVAR ........ Voltage magnitude profile for the case when the reactive load at Bus 11 is 900 MVAR ........ xiv Page 19 62 92 111 112 113 114 115 116 117 121 130 131 132 133 Chapter 1 Introduction 1;; Voltage Collapse Voltage collapse and abnormal high and low voltages have been ob- served with even greater frequency and severity on large interconnected systems. Recurrent problems exist in Europe, Japan, Southern California, 15lorida, Pennsylvania Jersey Maryland, New York Power Pool, and Ontario FU/dro. These voltage problems are associated with the increased loading of tJie transmission lines, the insufficient local reactive supply and the sJiipping of power across long geographical distances. Voltage problems are considered the principal threat to stability, security and reliability to the French and Belgian systems as well as to ITuany'Systems in the United States. PJM and NYPP in the East Coast import power through long transmission lines and they operate very close to tlhe voltage stability boundary. The December 19, 1978 blackout in France find the August 4, 1982 blackout in Belgium are attributed to voltage (lollapse. The transmission group in a recent EPRI sponsored meeting to establish future directions of operation, indicated the My o_f the lFtauses Egg solutions 9f these voltage problems as the number one priority "bor research. Conventional loadflow and eigenvalue programs are capable of simu- ~lation and analysis of voltage security and stability problems, respec- ‘tively. These programs are useful tools for determining the existence of voltage problems and establishing security constraints for particular voltage stability or security problems on a particular utility for 1 “on... p in. u 40'. CI" ., "Q l‘ 0- “its specific operating conditions. The fundamental cause for voltage col- lapse and abnormal high or low voltages is a loss of voltage controlla- bnity or stability at PO buses or loss of controllability or stability in reactive power supplied by the PV buses. Although loadflow or eigen- value methods can detect the consequences and find a solution for a particular problem for a given operating condition, through iterative sinmlation and eigenvalue analysis, they cannot diagnose the fundamental cause of the problem. Recent research on voltage security, stability and controllability recognized the local nature of voltage control and the difficulty in shipping reactive power over long geographical distances. lejg Voltage Collapse Mechanism [13]. A system that serves a heavily loaded industrial area, and does not have enough local reactive reserves is usually a system where voltage Collapse threatens to occur. If the system relies heavily on the re- iicrtive power from remote generation sources that are connected to the 55.)!stem through long transmission lines, then when the operating con- <1‘itions are normal, the demanded reactive power will be met. However, when the loading of the system is suddenly increased, or a long transmis- ‘3"ion line that transfers a good portion of power is tripped off, or a ‘I (:55 of local generation occurs, then the long transmission lines cannot DWovide all the reactive power needed. The local generators will try to “"set the reactive mismatch. If there are enough local reactive reserves, the system will settle to another operating point and operation will Continue safely. If on the other hand, there are not enough local re- active reserves then the local generators' MW and MVAR loadings will DI. ‘erplw . .oq ”out" '1 v. " In 1 I’.' n iv. a O I. I I 5 fire u... 3 increase. The generator transformer taps will be raised to their maximum settings, and then one after another the generators will reach full excitation. The generators can be at full excitation only for a limited time period for the safety of the machines. 50 after that time period, the MVAR outputs of the generators are reduced and the voltage at their terminals drop. As a consequence, the voltages at the load buses will drop sharply below 90%. Load tap changing transformers connecting the high voltage network tt) the distribution network will operate in order to restore the voltage 311d load at the distribution level. Since the voltages on the primary sdides of the load tap changers are smaller, the currents in the high ‘vc31tage network are substantially increased. As a consequence of that, there are more losses on transmission lines that reduce the voltages at ‘tlie'load buses even further. The voltages and the system collapse. Therefore, more emphasis should be given to the local reactive reserves. These reserves must be estimated based on the worst single and double contingencies that the system might experience under the worst Elllowable operating conditions. Synchronous condensers operated at low Y‘eactive output can be used in conjunction with (switchable) shunt ca- t>acitors and reactors to increase reactive reserves. Permanently con- nected or switchable capacitors or inductors cannot solve the problem of ‘U'oltage collapse alone. These condensers must be scattered throughout ‘tzhe system to (a) provide reactive power right where it is needed and (b) ‘1:o avoid isolation of these units in case of line outages. TE 1‘5“ 4 1:3 Research Contributions 9: thi; Ihggig Our discussion above has indicated the severity of the problem and then a practical description of how voltage collapse occurs and a general philosophy on solving it. In practical engineering terms, the voltage collapse problem has been an enigma because several important questions are at present unanswered. These questions, the contributions of this Thesis and very recent pertinent literature are now discussed. The first question is whether the voltage collapse is properly unadeled and analyzed as a dynamic problem that can be handled using a tncansient stability model, or as a static problem that can be handled uSing a loadflow model. Kwatny [1] has shown that the voltage collapse problem is due to a static bifurcation of the transient stability model that results in rnLtltiple loadflow'solutions and the lack of a stable equilibrium point. lfrnas, voltage collapse is most properly treated as a dynamic problem and addressed using a transient stability model. Kwatny defined a concept (:ailled strong causality that amounts to being able to solve the loadflow Eitquations for the voltages at all PQ buses and angles at all but genera- tor buses for all points in some neighborhood of an operating point. 53‘trong causality is assured if the Jacobian of the network is nonsingular El‘t the operating point. The second question is what buses and subregion of the network will be affected by voltage collapse. This is difficult to determine because the loadflow will have trouble converging and it will be sensitive to Slight operating condition changes as the system approaches voltage itollapse. One has had difficulty knowing whether the buses and subregion that appear to be vulnerable based on a sequence of loadflow solutions 5 that approach voltage collapse, will actually be the buses that experience voltage collapse. Two methods for determining the buses that belong to voltage control areas are given in Chapter 5. Both methods have been adapted from the literature on determining coherent groups [8, 9] that should be aggre- gated to form a dynamic equivalent for transient stability studies. The coherent groups identified based on the real power/voltage angle dynamics attempt to determine coherent groups based on a two time scale property. 11ne boundaries of these coherent groups has been Shown to be weak and be ‘tJ1e points where loss of synchronism will occur for loss of transient stability due to faults or loss of generation contingencies [8]. The \r<)ltage control areas identified by the same algorithm applied to the r‘eeactive power/voltage Jacobian can be inferred to detect the groups of tJLJSES with weak boundaries in terms of transfering reactive power. Results in Chapter 6 show that all of the buses in one of these ‘v'CJltage control areas experience voltage collapse for a number of diffe- Y‘ent types of contingencies and operating condition changes that effect ‘tZIIat.voltage control area. The buses experiencing voltage collapse are a lways those identified as belonging to a voltage control area. The Voltage control area, that is detected based on a secure base case con- (1'1 tion, correctly identifies the buses that experience voltage collapse ‘FOr the various contingency and operating condition changes. The third question is what causes voltage collapse to occur. Given a. certain operating point for a particular system, voltage collapse is Shown to occur because real and reactive transfers across the weak boundary of a voltage control area increase and cause singularity of the Jacobian which implies loss of voltage stability. These increased real 6 and reactive transfers across the weak boundary may be due to: (a) loss of generation contingencies when there is insufficient real and reactive reserves within the voltage control area, (b) line outages that affect the flows on the weak transmission boundary of the voltage control area, (c) tap changers that increase the loads seen by the system as the taps are set to higher values in an effort to support the voltage in the voltage control area, (d) capacitors that reverse the reactive flows across the weak transmission boundaries, as they are used to support voltages in the voltage control area. The argument that the increased real and reactive flows on branches in the voltage control area boundary cause singularity of the Jacobian and thus, loss of strong causality is contained in Chapter 5. The next question is whether there are tests and measures that indicate the vulnerability or proximity of the system to voltage collapse. A definition of PO stability and controllability and a definition of PV stability and controllability are given in Chapter 4 of this Thesis. A set of conditions on the sensitivity matrices SQLV, SVE’ SQGQL’ SQGE, and a set of constraints that assure that these matrices satisfy the conditions that guarantee voltage controllability are derived. Strong PQ controllability and strong PV controllability are then defined in terms of a set of more stringent constraints on these four sensitivity ma- trices. A system with a healthy voltage profile for which strong PO and PV controllability constraints on these four sensitivity matrices are satisfied should theoretically have no voltage problems and be voltage secure. This is confirmed by computational results in Chapter 6. A system for which the PQ stability constraints are violated, should experience voltage collapse which is confirmed by computational results. A system which is voltage insecure but is voltage stable will satisfy the PQ controllability constraints but will not satisfy the strong PO and PV controllability constraints. The proximity to voltage collapse is then measured by the proximity to violating the controlla- bility constraints and the proximity to voltage security is measured by the proximity to satisfying strong P0 and PV controllability constraints. This is also verified by computational results in Chapter 6. The last question is whether these constraints indicate operating and control changes that may alleviate the insecure voltage and voltage collapse situations. The satisfaction of strong PO and PV controllability constraints assure voltage security and the satisfaction of PO and PV controllability assures prevention of voltage collapse. Moreover, in Chapter 4, these constraints are written in terms of reactive injections, line charging, reactive load, reactive flows across voltage control area boundaries, reactive losses, and angle differences across branches. Thus, these constraints can be analyzed based on changes in these quantities to assess both causes and cures for voltage collapse. This investigation of causes and cures has not been pursued extensively in this Thesis and is a subject for future research. Chapter 2 Literature Study and Study Objectives 2:1 Introduction The literature on voltage stability is categorized according to the approach of analysis employed. There are basically four different approaches. (1) The Sensitivity Analysis Approach (2) The Linearized Dynamic Approach (3) The Multiple Solution Approach; and (4) The Nonlinear Dynamic Approach. We are going to present the present literature on voltage stability according to these analysis approaches. Several papers by Lachs [13,14,30,31] present the voltage collapse problem very descriptively and highlight the causes as being the inability of the system to meet the reactive demand locally. Lachs also supports the idea that utilities that import power from remote locations must have synchronous condensers scattered throughout the system and operated at low reactive output, ready to meet the reactive demand in case a reactive load increase or a loss of supply form a remote area occurs. We proceed with the literature study by presenting the first analy- sis approach. I; The Sensitivity Analysis Approach In the Sensitivity Approach, the voltage stability is treated as a steady state phenomenon. Loss of voltage stability occurs when the system is unable to meet sudden increase in reactive demand. Chapter 2 Literature Study and Study Objectives 2;; Introduction The literature on voltage stability is categorized according to the approach of analysis employed. There are basically four different approaches. ( ) The Sensitivity Analysis Approach (2) The Linearized Dynamic Approach ( ) The Multiple Solution Approach; and ( ) The Nonlinear Dynamic Approach. We are going to present the present literature on voltage stability according to these analysis approaches. Several papers by Lachs [13,14,30,31] present the voltage collapse problem very descriptively and highlight the causes as being the inability of the system to meet the reactive demand locally. Lachs also supports the idea that utilities that import power from remote locations must have synchronous condensers scattered throughout the system and operated at low reactive output, ready to meet the reactive demand in case a reactive load increase or a loss of supply form a remote area occurs. We proceed with the literature study by presenting the first analy- sis approach. I; Ihg Sensitivity Analysis Approach In the Sensitivity Approach, the voltage stability is treated as a steady state phenomenon. Loss of voltage stability occurs when the system is unable to meet sudden increase in reactive demand. Steady-state sensitivity analysis is used to determine voltage stability. Venikov [11] suggests a criterion for voltage stability using a simple dE two bus system. The voltage Vr at the load bus is stable if -—§ > 0, dV where ES is the voltage at the generating bus. Thus, if the volzage at the load bus is stable then any positive (negative) change of voltage dES must cause a positive (negative) change of voltage at the load bus. Borremans et. al., [7] suggest four voltage stability criteria that can be applied to a multibus system. V _1 1 Ei > .___——————— (2.1) 2 cos (5&9) \where E1 is the open circuit voltage at bus i, Vi’ is the voltage at bus i with the load connected, the R/X ratio of all the lines is assumed to be the same and E ==tan'1(X/R), and the loads are assumed to have the same resistive to reactive proportion and 0 = tan-1(XL/RL). AQ dQ . (2) AOL. = 36L] > O (2.2) G 63 PL-PLo In the simple two bus system, for every realistic operating point, vol- tage stability is assumed when at a given real output power, PLo’ a small increase in reactive demand dQL is met by a finite increase of reactive power generation dQG' In the multibus case, the constraint (2.2) is also assumed to hold when a reactive power variation AOL is imposed on a given load bus i or a group of load buses in a given area and the variation AQG is chosen to 10 represent the system's total reactive power generated by all the genera- tors, the synchronous condensers and the shunt VAR sources in response to AOL. The vulnerability of a system to voltage stability is claimed to increase as the ratio AQL/AQG decreases. These authors appear to imply that the system is less vulnerable if the reactive losses are reduced 00 when ZOL": 1, which indicates that the reactive losses dominate the G 60 charging and switchable capacitance. If'EfiL>’1 the shunt capacitance and G line charging of long and underground lines exceeds the reactive losses in the system. If an additional long or underground line is switched in, the system may be more vulnerable to voltage problems rather than less vulnerable as would be suggested by the decrease AQLflIQG that would occur if such a long or underground line was switched in. Thus, the claim that decreasing the ratio AOL/ADG makes the system less vulnerable to voltage stability is not always true. ( ) dVi/vi 3 —_ dQLj/QLj For a certain operating point, the voltage stability is assured when a Small increase in reactive demand dQL causes a decrease in voltages at the buses of the system; and (4) P .i P x and 0L.: (2.4) L Lma QLmax Starting from a stable operating point and increasing the load, the system remains stable as long as the maximum loading is not exceeded. The two sensitivity conditions (2) and (3) can be tested by (a) computing the sensitivity matrices 11 30 S00 = 30—6 (2.5) G L L and -1 . 8.1. SQLV [NJ (2.6) for a certain operating condition, as derived in Section 3, or computing loadflow solutions where the load at each bus i is changed by QLi and the associated changes AQGj’ 1 g j g_M and .Avj, 1 5 j g N are noted. The research performed in Chapter 4 defines voltage stability and controllability at both PV and PO buses and shows that the conditions (2) and (3) are related to sufficient conditions that preserve PV and PO controllability and stability. Carpentier [15] analytically developed a set of constraints for the elementary case of a two bus system, namely (a) (b) a constraint on the reactive load rn N C A D 0 II b xlo A N \J v that can be served by a voltage controlled bus, and a constraint on the reactive supply E02 Q : QC 3 TX—- (2.8) that could be provided by a voltage controlled bus, where E0 is the {MI—i144“ 12 voltage at the voltage controlled bus and X is the reactance of the feeding line. The reactive losses on this elementary system are shown to satisfy q = Q - D = 20C [1-”U1- %— J ' D (2'9) c which increases rapidly with load 0 and reaches the value qc = DC at D = Dc’ The voltage V1 at the load bus satisfies the equation V E_1=—;—[1+ 1-3—J (2.10) O which is one at D = o and drops rapidly toward 1/2 as D approaches Dc' These equations were developed based on the simple two bus system but their significance is tremendous because they can be used to predict voltage collapse in a voltage control area, if that voltage control area appears as a single bus and the rest of the system as another bus. Increasing load in the voltage control area can cause increased losses across the weak transmission boundary and eventually voltage collapse in the voltage control area. Carpentier [15] also derives a voltage stability indicator (2.11) As D increases toward Dc’ we saw earlier that the voltage V1 at the load 13 bus collapses. From the above expression, 2 approaches infinity as 0 increases and the voltage V1 collapses. This result was generalized in [15] for the mhltibus system where voltage collapse occurs if and only if 39— or 99.. approaches infinity, where 30 EUR - M Q = Z QGj (2.12) j=1 is the total reactive power supplied by all the reactive sources in the system, 0 is the total demand throughout the system and DR is the total demand in some region R. A necessary and sufficient condition for PV controllability that is derived is in Chapter 4 of this Thesis, is related to the requirements that 19, and 39— be bounded. It will be shown that the sensitivity an 30R 30 SQ Q = {569.} (2.13) G L L evaluated at an operating point must be nonnegative with no zero columns. matrix, Moreover, it will be shown that gg— must be close to one for a healthy R system. The above is an extension of the requirement (2) in [7] that AQLi AGE; > O (2.14) to the stronger requirement that 0 S ———— < k (2.15) dQLj 14 The results in this Theses even further strengthen this requirement to dQLj i dQ . T361:1 (2-15) 1 Lj "(‘42 If the PQ buses j for which the above sum is much greater than one are identified, then the buses of the system that are overloaded can be determined, and action can be taken. Glavitsch [6] developed a voltage stability indicator 2 F.. E. 160 L = max 1 - V. < 1 (2.17) JeoL J where (IL is the set of load buses, (16 is the set of generator (voltage controlled) buses, Vj is the voltage at a load bus, and Fji are elements of the matrix (2.18) where YLL and YLG are submatrices of the Y matrix in the voltage vs current relation < ll .< (2.19) "'1 ll .< This voltage stability indicator predicts voltage collapse as L ap- proaches 1, which implies that one or more rows of F are orthogonal to E, that is X F.. E. = O (2.20) 15 for some jeaL. The calculation of the indicator L requires the calcula- tion of the matrix F which is dependent on loading and existence of a loadflow solution. Therefore, L cannot be computed if the loadflow does not converge. Galiana [10] proposed a method for the analysis of voltage collapse which does not rely on loadflow or optimal loadflow simulations. Galiana wants to avoid loadflow simulations because when the voltage collapse region is approached, the loadflow algorithm converges slowly or not at all, and it is very difficult to find the step size to be used for the next iteration. The proposed method is based on the feasibility region of loadflow maps and the feasibility margin. The feasibility region is the set of generalized bus injections (P, Q, or V2 at each bus) for which a loadflow solution exists. The feasibility margin is a scalar between 0 and 1 which measures the proximity of a bus injection vector from the boundary of the feasibility region. Voltage collapse is avoided as long as the bus injection vector is such that the feasibility margin is not close to 1. II; Linearized Dynamic Approach Abe [2] developed stability conditions for a multisource power system model from a steady state analysis in which a time lag of the load admittance change was assumed for a small step change in source voltage or shunt capacitance. In [2] and [3] the stability conditions dV. EE_ > O i=1,...,M and j=1,...,N (2.21) dVi d—bT > 0 1,,j = 1,...,N (2.22) J were established and then evaluated using 3% = [SQLv-IJMYE = SYE (2'23) and 31—: = SQLV'1 0 (2.24) where SQLV and SVE are sensitivity matrices and 0 = diag {v12,v22,...,vN2} (2.25) It is established that a sufficient condition for voltage stability is that SVE has nonnegative elements and SQLV be an M-matrix. Under light load c0nditions, it is shown that SVE will have almost all nonnegative elements if SQLv is an M-matrix and thus, all that is required for voltage stability is that SQ V be an M-matrix. Abe [2] extended theseLresults to investigate voltage stability of the dynamics of tap changing transformers. Abe [2] derived an equivalent first order delay model for the load addmittance YL = GL + jBL as seen by the primaries of all tap changing transformers, namely dGL 1 F = - T (GL - fG(V)) (2°26) 17 dBL 1 at— = - T (BL - fB(V)) (2.27) where T is a time constant and fG(V) and fB(V) are steady state voltage characteristics for GL and BL respectively. The above differential equations are functions of the voltage V at the load buses“ . Abe [2] linearized this equation around an equilibrium point and obtained x __1_ 1.33% 4.322 [x B T 8V BB T 3V G B l L L d - EH?" ' + x _1_3_f_ea_v_ -_1_ 1-3161 x G TavaL T 8V aGL G in.) 1- '] L- .1 4.51" T 3V 8R + AR (2.28) 13:61" T 8V 3R where XB = BLi(t) - BL1(«>) (2.29) XG = GLi(t) - GLi(*,V*, P*) must satisfy the load flow equations F(6.¢.v,P) = 0 (2.34) where 0 - the vector of voltage angles at the internal generator buses 8 - the vector of voltage angles at the PO and PV buses V - the voltages at the PQ buses P - the vector of parameters such as mechanical power inputs, real and reactive load at load buses, tap settings, capa- citor settings, etc. A bifurcation point is an equilibrium (9*,¢>*,V*,P*) in every neighborhood of which there are equilibria (81*, QL = fQL(5a69E9V) and E - is the vector V - is the vector 5 - is the vector 8 - is the vector We define developed based on the load flow equations E,V) E,V) (3.1) E,V) E,V) - is the vector of the real power flow equations at all PV-buses. - is the vector of the real power flow equations at all PQ-buses where the voltage V is not controlled. - is the vector of the reactive power flow equations at all PV-buses. - is the vector of the reactive power flow equations at all PQ-buses where the voltage V is not controlled. of voltage magnitudes at the PV buses. of voltage magnitudes at the PQ buses. of phase angles at the PV buses. of phase angles at the PQ buses. 30 )- - P " P 1 PG 6 fPG (6,6,E,V) P e f (6,6,E,V) Y = L , x = and F(X) = PL (3.2) 0L V fQL (6.6.E.V) I . - . 1 Then Y = F(X) (3 3) Using a Taylor series expansion of F(X0+AX) about a particular operating condition, X0, and neglecting higher order partial derivatives of F(X) we obtain F(xo +AX) = F(XO) + 0(x0)ax (3.4) or Y = F(Xo +AX) - F(X ) = J(X0)AX (3.5) where _ 8F x 4(xo) - ax (3 6) x=xO So A1 Bl CI' 01' A 0 c ' 0 ' mo) = 2 2 2. 2. (3.7) A3 B3 c3 D3 1.A4 B4 4' 94' where Q) —h 0.) F.” (T) 31 0 ll Q) 09 -h a)... ‘— L2 Q) .q,’ .—___=s| h _ __._‘ (3.8) All the above partial derivatives are evaluated at the operating point X 0. Then the following expression can be written .A PG- ”A1 B1 A PL = A2 32 A 06 A3 B3 A QL A4 B4 L . . The above system can APG A1 B1 APL A2 B2 AQG A3 B3 AQL A4 B4 where AE/E = diag (E1,E2,... AV/V = diag (V1,V2,... Cl' 01'. F A6 C2' D2' A6 03' 03' AE C4' 04' AV 1 L C1 D1 A6 C3 03 A E/E .1 L -1 ’EN) AE -1 ,VM) AV (3.10) (3.11) are the normalized voltages at the PV buses and PO buses respectively. and C1' diag(E Di 1,E2,... diag(V1,V2,... do I - l,2,3,4. (3.12) l,2,3,4. 33 3.2.1 Loadflow Equations We now present two forms of the loadflow equations - the Polar form and the Hybrid form - that we use to derive the expressions for the Jacobian submatrices A1, A2,...,D4. Consider a system with n buses Let Vi = Vizei , i = 1,2,...,n, (3.13) be the voltage at each bus i. Let vLij = Gij - jBij , ifj (3.14) be the admittance of the line from bus i to bus j, where Gij and Bij are nonnegative quantities. Then the admittance matrix is 7 = [Vij] (3.15) where Yij = -YLij = Yij Y1. = - Gij + 3313 , ifj (3.16) and Y n v .. = Z .. = 11 j=o LTJ .ifi n . n ji‘i .ifi 34 It must be mentioned that for lines with X/R ratios greater than 3 the angle _1 Bi. _1 xi. Yij = tan ———l = tan -——1- (3.18) -Gij -Rij is 90° 5 ya. 3 110°. (3.19) We also mention that the bus i to ground shunt admittance Yio = Gio + JBio =G. +j(B. - B1. 10 1c (3'20) L) Where G1.0 is very small and it is usually neglected; B is due to the ic charging of the lines connected to bus i as well as the capacitor banks placed at bus i; and BiL is due to the load or reactor banks connected to bus i. Let G.. = - ( z 6.. + G. ) (3.21) 11 jfi Tj TO and B.. = - z 8.. + B 11 jfi 13 TO Then vii ‘ Yii Yii ‘ ' Gii + JBii (3°22) 35 3.2.2 Polar form 9: Loadflow Equations * From the vector power equation, S = Diag(V) I and the vector equa- tion I = V V we have 3* = 0iag(v*)I = 0iag(v*) V V (3.23) Therefore, -* .. P - jQ = Diag(V ) Y v (3.24) and n ‘ * Y T P. -JQ. = Z V .. 1 l j=1 l Ij j " -j(a - 6 - Y ) = Z V.Y..V.e i j ij (3.25) 3:1 1 TJ 3 Then n n Qi = jil viYijVj Sin(01 - OJ - Yij) (3.27) are the loadflow equations in Polar form. 3.2.3 Hybrid Form hf Loadflow Equations In the equation 0 n * Pl -301 = jil V1 vlj Vj (3.28) 36 above, we can substitute Y. be 431.11 j 331j and Vk by Vk (0k. Then P - Q = 2 v v e‘3(ai ' 9') (-G + 'B ) i Ji J.=1i.i 3 iJ'JiJ' n = Z V [cos(9 - 8 ) - jSTn(8 - 8.)] (-G + jB. j=1 1 J 13 Then n P1 = jEI V1Vj[-G1j cos(91 - 81) + B11 Sin(01 - 01)] n . Qi = 3:1 V1Vj[-G1j Sin(91 - 83) - B11 cos(81 - 61)]. are the loadflow equations in Hybrid form. 3.2.4 Evaluation hf hhg Partial Derivatives The loadflow equations in Polar form are given by n P1 - g V1Y1jvj cos(61- 6 1J) j-I n Qi = .2 V1Y1jVj sin(9103 - Y1J) J-l Then 3P. n .——l = - 2 Y. Y. .Y. sin(9. - Y..) = 361j= i ii i 13 j'fi (3.30) (3.31) (3.32) (3.33) Q) Q) 07 T! ll 37 2 . _ - jzl V1Y11V1 Sin(61- 91- Y11) + V1 Y11 s1n(-Y11) - 2 2 - 01- V1 Y11sin(y11) - Q1 - V1 811 (3.34) V1Y11V1 sin(81 - 81- Y1. for i P j (3.35) n jzl V1Y11V1 cos(81 - 9j - 1.11) - jfi " 2 jfl V1Y11V1 c0s(81- 91 - Y11) - V1 Y11 cos(Y11) = 2 - 2 P1 - V1 Y11 cos(Y11) - P1 + V1 G11. (3.36) -V1Y11Vj cos(81- 81 - Y11) for T¢j (3.37) n jil Y11V1 cos(81 - 91 - v11) + 2Y11V1 cos(Y11) = jfi n jgl Yijvj COS(91 - ej - Yij) + Yiivi COS(Yii) " 2 jEI V1Y11V1 cos(91 - 81 - Y11) + V1 Y11 cos(Y11) = 2 _ 2 P1 + V1 Y11 COS(Y11) - P1 - V1 G11. (3.38) ViYij COS(91 ' Qj - ij and V. 5—— = V Y. V. cos(01 - 81 - Y11) for i f j. (3.39) n -——— = E Y..V. Sin(81 -81 - Y11) + 2V1Y11 $1n(-Y11) = f n = jgl Y11V1 Sin(81 - 81 - Y11) - V1Y11 STn(Y11) 301 " . 2 . 1 V1 BVT - jgl V1Y11V1 SIn(O1 - 01 - Y11) - V1 Y11 Sln( 11) - l = 0 - v 2v sin(Y ) = Q - v 28 (3 40) i i ii ii 1 i ii' ' aQi . -—— - V1Y1J Sin(81 - 91 - Y11) and 3V. J 301. V —' = V.Y..V. sin(61- 63° - Y1J)f0rlfj. (3.41) i 3 1 1 1J J VJ 3.2.5 Building hhg Jacobian Submatrices In the study to follow, we are going to consider a power system that consists of N PV-buses (buses 1 through N) M PQ-buses (buses N+1 through N+M) where N and M are arbitrary. However in order to make the development easier to follow, we are going to use specific values for N and M so that the Jacobian submatrices A1, A2 through D4 as well as the sensitivity matrices fit the size of a page. In so doing, we will be careful so that our results and conclusions are not based on the simplicity of the 39 small system. Whenever the size and complexity of the system are impor- tant the more general system will be considered. N and M will be taken as 2 and 3 respectively. We also define Q.. V.Y..V. Sin(61 - 81 - Y 11 1 11 J ..) (3.42) 13 ,,) (3.43) P.. V.Y..V. cos(6. - 81 - Y1J 1.] 113,] 1 This simplifies the expressions for the Jacobian submatrices. The Jaco- bian submatrices are now presented. . q P - 8P 8P 1 1 2 361 56; '01 ' V1 B11 Q12 A1 = = (3.44) 332 332 Q -0 -v 28 a6 36 21 2 2 22 1 2 I. .. b J (- 5 " 3P 3P I 361 362 31 32 A2= 86— 36— ‘ Q41 042 (3~45) 1 2 _ _ Q Q 861 362 51 52 . .- i- .1 861 361 351 O) O H O) 0'; N 0) -O to Q) N 12 P+V G22 (3.46) (3.47) (3.48) 40 P1 1 V1 G11 1 P2 + V 2 2 G22 (3.46) (3.47) (3.48) 301 aa 302 36 BQl 39 802 89 41 Q35 Q45 '05 ' V5 855 -P 'P14 -P 'P24 (3.49) (3.50) 303 39 304 39 BQS 39 P + V -P p + 34 V 42 G 303 30 304 89 305 395 44 'P35 45 P + v 5 5 G55 PI ’ v1 G11 P21 P12 P2 ' v2 G22 (3.51) (3.52) Finally, 43 P P 31 32 P P 4l 42 P51 P52 Q1 ' V1 B11 Q21 031 Q32 Q41 042 Q51 Q52 Q2 ' V2 B22 (3.53) (3.54) (3.55) 1 I 02 = L I I I 03 = 001 302 3V 1:5 3 4 4 av4 3v5 4 3V4 3v5 1:6 3:4 4 av4 3v5 4 av4 av5 P4 ' V4 G44 1:5 .33 4 3v4 av5 4 av4 av5 44 ‘ T P13 P14 P23 P24 J .. 1 P35 P45 2 5 ‘ V5 G55 J (3.56) (3.57) 45 013 Q14 015 I = (3.58) Q23 Q24 Q25 1 L .I 3 av 4 av 5 av 3 4 5 4 ' 3 ‘_ 4 av 5 av 3V3 4 5 3 av 4 av 5 av 3 4 5 1 0 -v% Q Q 3 3 33 34 35 = Q Q-VZB Q I as” 43 4 4 44 45 ° Q Q Q - v 28 I 53 54 5 5 55 I 3.2.6 Decoupled-Model Development Going back to the linearized model, and using the fact that real power injections and voltage angles at the buses are decoupled from the reactive power injections and voltages at the buses, we can approximate the linearized model by the decoupled model 46 . 1 l APG ) A1 B1 0 0) f A6 All A B o 0 A8 L = 2 2 (3.60) A06 0 0 c3 D3 AE L AQL o 0 c4 04 AV 0?" AP A B ' A6 5 = 1 1 (3.61) APL A2 32 A9 AQ c D AE G = 3 3 (3 62) A01. , C4 D4 AV Our interest will be focused on the second part. This is because it is known that voltage collapse and abnormal high or low voltages are related to inability of the system to provide reactive power to an area in which the reactive power demand is increased. 3.2.7 Sensitivity-Model Development If we assume that all the PV buses are regulated then the voltage magnidutes E are fixed, implying that AE = 0. Then (3.62) implies that AQG = D AV 3 (3.63) AQL D4AV or equivalently 47 AV = D4-IAQL (3.64) This is the Decoupled Model I. In general we will allow changes of voltage magnitudes at the PV buses. The PV bus connected to the largest unit in the system is proba- bly the only bus where we don't want to change the voltage magnitude; but for the rest of the PV buses we change the voltage magnitudes to improve the stability of the system. So the Decoupled Model 11 can be derived from A0 - C AE +[)AV G 3 3 (3.65) AQL C4 AE + D AV 4 as follows. Solving the second equation for AV and substituting it into the first equation, we obtain -1 -1 AV = -D c AE + 0 A0 A0 = (c - D D 'lc )AE + D D 'lAQ e 3 3 4 4 3 4 L We define the sensitivity matrices as sQLV = 04 (3 67) s = -D '16 (3 68) vs 4 4 ° 5 = c - D D '16 (3 69) QGE 3 3 4 4 ° 48 -1 SQGQL D3‘34 (3.70) Then the Decoupled Models are: Model 1 AV = s ’lAQ QLV L (3.71) = - A AQG SQGQL QL Model __I_ AV= s AE+S 'le VE QLV L (3.72) A0 = S AE - S AQ G 065 QGQL L 'Tl1ee vectors AV and AQG are considered as the states of the system; the ‘Veacztor AE is considered as the control and the vector AQL as the di sturbance. The changes AQL are random and operators at the control centers can have only estimates of reactive load changes in the system. That's why 't"€3 changes AQL are considered to be disturbances. The voltage changes 1355 are called controls because we can adjust them to (a) achieve desired V01 tage changes at the PQ buses and (b) obtain reactive generation Changes for better stability of the system. Model 11 is linear with respect to.AE and AOL. So superposition can be applied. In other words, we can set AE to zero and determine the 1 Changes AV and A061 for a change AOL and then set AQL to zero and 2 datermine the changes AV and AQG2 for a change AE. 49 Then 1 2 AV =AV +AV (3.73) A06 = ”61 + AQGZ are the changes due to simultaneous changes AE and AQL in the system. This property is used in defining the stability and controllability of the system. 3.3 Network Topology and Properties of the Sensitivity Matrices (a) Load Centers Consider the system With N PV buses and M PQ buses introduced earlier. If the PV buses and the lines connecting them to other buses are removed, then the PQ buses that remain form isolated groups. Each of these groups is called a Load Center. Assume that there are S Load Centers. Let N. be the set of PO J buses that belong to the jth Load Center, 1 g,j g_S. We also assume that the PQ buses are numbered so that buses in the same Load Center have consecutive numbers, and for i < j, buses in the ith Load th Center have numbers smaller than buses in the j Load Center. Let M. be the set of PV buses that are directly connected to J the jth Load Center, 1 g_j g_S. We notice that the off diagonal elements of the SQ V = D4 matrix are of the form L where i and j are PQ buses. If there is a physical line between the PQ buses i and j then this entry is in general nonzero; otherwise (b) 50 Qij is zero. With the numbering of the PQ buses just introduced the matrix D4 is a block diagonal matrix and each diagonal block corresponds to a Load Center in the system. Also because the PQ buses in a Load Center are either connected directly or through some other PQ buses, the diagonal blocks of D4 are irreducible [Appendix A]. These are two nice mathematical properties that the matrix 04 has. Voltage Support Centers If, now, the PQ buses and the lines connecting them to other buses are removed, then the PV buses of the system form isolated groups. Each of these groups is called a Voltage Support Center. Assume that there are t such Voltage Support Centers. Then we define the set K. , 1.: j‘: t, as the set of PV buses belonging J th to the j Voltage Support Center. Let Lj be the set of P0 buses directly connected to the jth Voltage Support Center, 1 _<_ j: t. Voltage Control Areas These are groups of PV buses and P0 buses that are stiffly interconnected. By stiffly interconnected we mean that flows of real and reactive power over the lines connecting these buses cause no problem. The cutset of branches that isolate each voltage con- trol area are weak in that relatively small reactive flows across this boundary can cause significant voltage changes in the voltage control area. when that tage ting 51 Let Jp be the set of P0 buses in the pth th VCA and let Ip be the set of PV buses in the p h VCA. Let 5p be the set of PO buses outside the pt VCA directly connected to it and let Ip be the set of PV buses outside the pth VCA directly connected to it. Clearly the connections of buses in IpUJp are stiff connections and the connections of buses from IpUJp to buses in IpUdp are weak. Voltage Control Areas are probably the most important groups because a voltage collapse appears it always extends to all buses in an area are stiffly interconnected and relatively isolated from other vol- control areas. This is very related to the inability of the genera- units in a voltage control area to meet the reactive load demand. In the next chapter we develop the stability and controllability definitions and theorems that guarantee stability and controllability. Chapter 4 Voltage Stability and Controllability 4.1. Introduction In this chapter we state the Stability and Controllability defini- tions and discuss them using Model 11 developed in Chapter 3. Then that sufficient conditions on the sensitivity matrices SQ and S V VE assure PQ Stability and Controllability are obtained. L These sufficient conditons are further explored to obtain easily testable conditions that require less computational effort. Also conditions on the sensitivity matrices SQGQL and SQGE that assure PV Stability and Controllability are developed. In Section 4.4 we develop Security Constraints for the system. These are derived from the conditions on the sensitivity matrices of the system that guarantee Stability and/or Controllability. We derive con- straints for both buses and Voltage Control Areas. Then in Section 4.5, a set of Strong Controllability constraints is developed. All these constraints will be tested on the 30 Bus New England System. This system is small enough so that the sensitivity matrices of Model II can be computed and checked if they have the appropriate proper- ties. So the Stability and Controllability of the system as well as Strong Controllability will be tested in two different ways and the results will be compared [Chapter 6]. On .a large system we check only the constraints because the sensi- tivity requires excessive amounts of memory storage as well as compu- tational time. 52 53 4;; 29 Stability and Controllability Definition A system is called PQ stable if (i) when AE=0, any nonzero nonnegative disturbance AQL* causes the PQ state AV to become nonnegative; and (ii) when AQL=0, any nonzero nonnegative control AE causes the PQ state AV to become nonnegative. Note that the above definition could be stated using "positive" where we presently have "negative". The above definition states that the system is PQ stable if when the voltage at the PV buses is held fixed then a decrease in load causes some of the voltages at the PQ buses to increase, and when the load is fixed a voltage increase at the PV buses causes some of the voltages at the PQ buses to increase. Logically, this is what one expects from a well behaving power system. Definition A system is called PQ controllable if the system is PQ stable and (i) when AE=0, for each j there is a nonzero nonnegative disturbance AQL that causes AVj to become positive; and (ii) when AQL=0, for each j there is a nonzero nonnegative control AE that causes AVj to become positive. This definition could also be stated with "positive" and "negative" interchanged everywhere. * QLi is a power injection and as such is a negative number for P0 buses. So, AQLi > 0 means that the load decreased at the ith PQ bus. 54 So in addition to the PQ stability requirements, we require that. the voltage at any PQ bus can be increased (decreased) by changing the load at some PQ buses or by increasing (decreasing) the voltages at some PV buses. The manner in which this voltage control can be achieved will be obvious after the proof of a theorem on sufficient conditions for P0 controllability. Some mathematical tools that are useful for the theoretical develop- ment to follow are presented in Appendix A. Theorem I A sufficient condition for P0 stability is that SQ V is an M-matrix L and SVE lS nonnegative. Proof '1 is a nonnegative If SQLV 15 an M-matrix then by definition SQ V L matrix. Then if AE=0, _ -1 AV - SQLV AQL (4 1) and for any nonzero nonnegative disturbance AQL, AV is nonnegative. Also, if AQL=0 then AV = SVEAE. (4.2) Since SVE is nonnegative by hypothesis, any nonzero nonnegative control AE, makes AV nonnegative. Therefore, the system is PQ-stable. 55 Theorem g A sufficient condition for PQ controllability is that SQ V is an M- L matrix and SVE is nonnegative with no zero rows. Proof -1 The matrix SQ V is a nonnegative matrix, by the definition of an L M-matrix. Then if AE=O AV = SQLV AQL (4.3) and for any nonzero nonnegative disturbance AQL, AV is nonnegative. Also, since SQ v'1 is nonsingular, all of its rows and columns are ' L nonzero. Then for each i there is j so that [SQ V-IJij:> 0. L Furthermore, if AQLj > 0 then AVi = [0,...,1,...0]AV = ith place _ -1 ' [09 0919 so] SQLV AQL =[s "1144 = 3 [s '1] AQ QLV i L k=1 QLV ik Lk -1 Z [SQLV JijAQLj > 0 (4.4) where -1 . .th -1 [SQLV ]i 15 the 1 row of SQLV and -1 . ..th -1 [SQLV Jij 15 the lJ entry of SQLV . 56 Also, if AQL = 0 then AV = SVEAE. (4.5) and for any AE nonzero nonnegative AV is nonnegative because SVE is nonnegative by assumption. If now 1 5 i g_M then since SVE has no zero row there is a column j such that [SVEJij > 0. Choosing AE to be nonnegative with AEj> 0 we have AVi = [O,...,1,...,O] SVEAE = ith place = [SVEJiAE = M (<51 [5V5]ik AEk 3[SVE]1.J.AEJ. > 0. (4.6) This completes the proof. From the above derivation it is clear that if we want to increase the voltage at the PQ bus i, we can reduce the load connected at PO buses j for which [SQLV-IJij are positive. It is also clear that we can increase the voltage at a PD bus i by increasing the voltage at PV buses j for which [SVEJij are positive. Later on, we are going to see that there are some limitations in the way we increase the voltage at PV buses in order to achieve voltage incre- ments at P0 buses. These limitations are dictated by the sign of the column sums of the SQGE matrix. This will be clear after the definitions of PV stability and controllability and necessary conditions in the accompaning theorems are presented. 57 We now consider the elements of the matrices D3 and C4. These ele- ments are of the form Qij = vivijvj Sln (ei - ej - Yij) (4 7) where the indices correspond to a PV bus and a PD bus. If there is no physical connection between the buses i and j, then Qij = 0 because Yij = D. For the cases where these is a physical connection between the buses i and j and the X/R ratio of the line is greater than 3 (which is usually the case) the angle‘Yi. is such that 3 90° 5 v1.3. 5 110° (4.8) Thenif 0 0 -180 + yij 5 0i - ej : 180 - v13 (4.9) the elements 01.3. and jS are nonpositive. Theorem 3 If the matrix SQ V is an M-matrix and L O 0 -180 + vii 5 6i - ej : 180 - Yij (4.10) for all PV buses i and P0 buses j that are directly connected, then SVE is a nonnegative matrix. Proof The assumption that 0 O -180 + yij 5 61 - 83 5 180 - yij (4.11) 58 implies that Therefore, the matrix C4 is a nonpositive matrix. . . . . -1 _ -1 V is an M-matrix implies that D4 - SQLV L -1 The assumption that SQ Therefore, SVE = -D4 C4 is a nonnegative is a nonnegative matrix. matrix. Theorem‘fl If the matrix SQ V is an M-matrix and L o o -180 + Yij 5 9i - 83 g 180 - ij (4.13) ft)r'all PV buses i and P0 buses j that are directly connected, then the s.Ystem is PQ stable. pY‘oof Then From Theorem 3, the sensitivity matrix SVE is nonnegative. SSince SQ V is an M-matrix and SVE is nonnegative, Theorem 1 implies that L the system is PQ stable. Theorem‘g If each diagonal block of SQ V is an M-matrix, each Load Center of L the system is directly connected to at least one PV bus, and 0 0 for all PV buses i and PD buses j for which a line ij exists, then SVE is nonnegative and has no zero rows. 59 Proof The inequality -180° +vi. < 61. - ej < 180° - v (4.15) J 1'3 implies that Qij‘< 0 for all PV buses i and PD buses j for which a line ij exists. Then the matrix C4 is a nonzero nonpositive matrix. Also, since each diagonal block of SQLV is an irreducible M-matrix, then each 1 I diagonal block of SQ V- is positive. L Thus VE = '04 C4 = -SQLV C4 (4016) ‘i s nonnegative. Assume now that SVE has a zero row. Without loss of generality, we can assume that the first row of SVE is zero. Let S1 be the first diagonal block of s '1 of dimension n x n . Then the first row of s is the QLV 1 1 VE negative of the product of the first row of $1 and the submatrix M1 of C4 consisting of the first n1 rows of C4. 1080, 60 Q1,N+1 Q2,N+1 °°° QN,N+1 Q1,N+2 Q2,N+2 1 °'° QN,N+2 M1 = . . . (4.17) Q Q ... Q _ 1,N+n1 2,N+n1 N,N+n1 d Then since the first row of S1 is positive and M1 is nonpositive, the first row of SVE is zero if an only if M1 is identically zero. This then implies that the PQ buses of the first Load Center are not connected tti any of the PV buses. This is a contradiction. Therefore, SVE has no Zero rows. The above theorem uses a stronger condition on SQLV than that of Theorem 3 in order to prove that SVIE is nonnegative and has no zero rows. However, in practice, if the sparse matrix SQ V is an M-matrix, then -1 L SE) V is a full nonnegative matrix and thus, the matrix SVE can very L well be nonnegative and has no zero rows without any additional require- ment on SQLV. \ Theorem 6 If each diagonal block of SQ V is an M-matrix, each Load Center of the L ) system is directly connected to at least one PV bus, and / o 0 -180 + Y” <91. - 93. < 180 - v1.3. (4.18) for all PV buses i and P0 buses j for which a line ij exists, then the system is PQ controllable. 61 Proof From Theorem 5 the sensitivity matrix SVE is nonnegative and has no zero rows. Then since the diagonal blocks of SQLV are irreducible M- matrices, from Theorem 2 the system is PQ controllable. The proof is thus completed. Since we require that SQLV is an M-matrix or it has irreducible M- diagonal blocks, it is necessary that the off-diagonal elements of this matrix are also negative. These off-diagonal elements are of the form 0.. = V.Y..V. sin (9 ij i ij j i ' ej - Y ij) (4.19) Therefore, 01.3. and jS must be negative for all PQ buses i and j that are di rectly connected. Then 0 o -180 + Yij < 61 - Bj < 180 - Yij (4.20) ft3r all PQ buses i and j that are directly connected. As we are going to show shortly when 61 - Bj gets large in absolute value then the losses on the line from bus i to bus j is also large for fixed voltage magnitudes. So a constraint as the one above is good to have for all buses i and j that are directly connected. Theorem‘z Consider the line connecting the two buses i and j as shown in Figure 4.1. Then the reactive power flow from the bus i to the bus j is given by (4.21) 62 and the reactive power losses on the line are given by 2 2 QLOSS = [Vi + Vj - 2ViVj cos (9i - Oj)] Bij‘ (4.22) Proof The power 5 flowing from i to j is given by * * 'k S = ViI = V1(V1 - Vj) VLij = _ 2 - - * - * _ ’(Vi ' ij)YLij ' K V V1- 3. 3123— _I_> iVV' YLii Yio on Figure 4.1 Equilavent n-Model for a Transmission Line - 2 'k 'k 'k _ = -V 2V cos y + jV 2Y sin Y + i ij ij i ij ij The imaginary part of S is the reactive power flow from i to j. i.e., 2 . . _ 2 s .)= lJ (4.24) 63 In a similar fashion, one can prove that the reactive power flow from j to i is s ..=V.B..+Q.. (4.25) = ..+ .....= ..+ $013 SQJl 1 l3 J 13 TJ 2) QLoss (v1. J. ..) + B.. + V V Y [Sln(91 - OJ - YU 13 i j ij + ..)] = sin (93 - 61 - v13 (Vi + Vj Bij + viVjYij [-2 cos(61 - 93) sin Yij] = [V.2 + V.2 - 2ViV. cos(6 1 J - 8.)]8.. (4.26) J i J 10 From this expression it is clear that as the angle difference gets trigger, the subtracted quantity 2V1Vj cos(91 - Dj) gets small; thus, the losses are larger. 4.3 3V Stability and Controllability Definition A system is called PV stable if (i) When AE = 0, any nonzero nonpositive disturbance AQL causes the PV state AQG to become nonnegative; and (ii) when AQL = 0, there is a nonzero nonnegative control AE that causes the PV state AQG to be such that AQGi >.0 (4.27) 64 Given that the voltages at the PV buses are held fixed, then for any load increase, the voltages at the PQ buses will decrease; therefore, the flow of power from the PV buses to the PQ buses will increase. For the system to be PV stable, we want to see none of the generators put less reactive power into the system than before the load increased. In case the load is fixed and we want to increase the total reactive power flow into the system, we must be able to do so by increasing the voltages at some of the PV buses. For the system to be PV controllable, however, the reactive power injection at the PV buses should increase when either: (a) the voltage at the PV buses is held fixed and the load is in- creased, or (ID) the load is held fixed and the voltage at any of the PV buses is increased Definition A system is called PV controllable if (a) when AE = 0, any nonzero nonpositive disturbance AQL causes the PV state AQG to become nonzero nonnegative; and (b) when AQL = 0, any nonzero nonnegative control.AE causes the PV state AQG to become such that AQGi' > 0 (4.28) Theorem‘g A system is PV stable if and only if (i) the matrix S is nonnegative and QGQL 65 (ii) the matrix SQ E is such that G E]" _>_ 0 (4.29) for at least one j , 1 5 j g N. Proof Assume that AE = 0 and AQL‘; 0. Then by the definition of PV stability A0 = -S AQ > 0 4.30 h Ftir 1 g_j §_M we choose AQL such that the jt element is -ej, where e. is J Positive, and all other elements are zero, then A0 = e.[S ]. > 0 (4.31) G J 060L .1 - Where . .th . [SQGQLJj 15 the 3 column of the matrix SQGQL. Then [SQ Q ]j is nonnegative for any j , j = 1,2,...,N. Therefore, G L SQ Q is a nonnegative matrix. If we now assume that AQL = D, then there G L exists AE 3, 0 such that N 2 AQ . > 0 (4.32) i=1 9‘ — by the definition of PV stability. Then since AQG = SQ E AE, G "(‘42 A0 . = 1 6‘ i=1 J l Interchanging the order of summation we obtain AE > O (4.34) N E [SQGEJij i-— IIMZ j 1 i 1 Because AEj's are nonnegative and at least one is positive foratleastonej , 1§j_<_ N. Conversely, assume that (i) and (ii) are satisfied. Then when AE = D and ”Li“ because SQ Q is nonnegative. When AQL = 0, then since there is at G L least one j such that [S 1 0 (4.37) "NZ 1.: i QGE ‘3 we choose AE such that all the components are zero except for the AEj that we choose positive. Then AE is nonzero nonnegative and N 121 [SQGE]ik AEk = [SQGEJijAEj 3_0 (4.38) Therefore, when AQL is zero there is a nonzero nonnegative AE such that 67 A061 >0 (4.39) IIMZ i 1 Thus, the system is PV stable. Theorem 2 A sVstem is PV controllable if and only if (i) the matrix SQ Q is a nonnegative with no zero columns, and G L (ii) the matrix SQ E is such that G E]1j > O a j=190~°9N (4.40) Proof Assume that AE = 0 and AQL §_0. Then by the definition of PV- controllability AQG = -sQGQL 40L ; 0 (4.41) h For 113 j 5.“ we choose AQL such that the jt element is -ej, where ej is positive, and all other elements are zero, then AQG = ej[SQGQL]j 3_0 . (4.42) where h . .t . [SQGQLJJ is the 3 column of the matrix SQGQL. Then [SQ Q ]j is nonzero nonnegative and since j is arbitrary, the matrix G L S is nonnegative and has no zero columns. QGQL Now assume that AQL = 0 and AE 73 D, then 68 AQG = so E AE (4.43) G and by the definition of PV controllability N AQ . > 0. (4.44) i=1 6‘ Then N N [ 1 z X S .. AE. > 0 (4.45) i=1 j=1 QGE ‘3 3 If we interchange the order of summation, we obtain 53" AE. > 0 (4.46) E1.. > 0 (4.47) for all j=1,2,...,N Conversely, assume that (i) and (ii) are satisfied. Assume 14E = 0 and AQL §_0. Since SQ Q is nonnegative and has no zero G L columns and AQLj<0 for at least one j A0G = -SQGQL A0L = N - 121 [SQGQLJj A0Lj 3,0 (4.48) 69 where . .th [S ]. is the 3 column of S . QGQL J QGQL Finally, if AQL = 0 and AE :0 then N -E [SQ EJij > O for j=1,...,N (4.49) i-1 G implies that N N .. A. . jfl 1:1 [SQGEJTJ EJ > 0 , (4 50) and N N N [ ] ( ZAQ.= 2 2 S .. AE.>0 4.51) i=1 3‘ i=1 j=1 00E ‘J J and the converse has been proved. Theorem 19 If each diagonal block of SQ V is an M-matrix, each Load Center of L the system is directly connected to at least one PV bus, and -180° *“Yij‘< Bi - ej <180o - Yij (4.52) for all PQ buses i and PV buses j for which a line ij exists, then SQGQL is nonnegative with no zero columns. £399: The inequality -180° + Vii < 0, - ej '<1800 - Yij (4.53) In £7“. ,1 70 implies that jS‘< D for all PV buses j and PD buses i for which a line ij exists. Then the matrix 03 is a nonzero nonpositive matrix. T= -(0 T)“1 0 (4.54) SQGQL 3 and the matrices 03T and D4T have the same properties as the matrices in the proof of Theorem 5. Therefore, we can conclude that SQGQLT is nonnegative and has no zero rows. This then implies the SQGQL is nonnegative and has no zero columns. Theorem 9, then, can be restated as: Theorem 11 If each diagonal block of SQ V is an M-matrix, each Load Center of L the system is directly connected to at least one PV bus, 0 0 -180 + Yij‘< 91 - Bj‘< 180 - Yij (4.55) for all PQ buses i and PV buses j for which a line ij exists, and the matrix SQ E is such that G E].. > 0 for j=1,...,N (4.56) then the system is PV controllable. Remarks 1. There are three assumptions made in Theorem 11 to assure PV controllability. The first two are the same as those made in Theorem 6 to assure PQ controllability. Therefore, the assumptions in Theorem 11 assure both P0 and PV controllability. 71 2. The diagonal blocks of SQLV are irreducible. Should they also be strictly diagonally dominant, they are irreducibly diagonally domi— nant. Also, the off diagonal entries of SQLV are nonpositive should (4.55) hold. Then, by Theorems A7 and A2 the diagonal locks of SQLV are M-matrices. We also notice that the diagonal blocks of SQLV are diagonally dominant if and only if the matrix SQLV is diagonally dominant. Using the above two remarks, Theorem 11 can be restated as a new Theorem for P0 and PV controllabilty. Theorem 12 ' If each Load Center of the system is directly connected to at least one PV bus, the angle differences 6i - Bj satisfy 0 O for all PQ buses i and PV buses j for which a line ij exists, the matrix SQ V is strictly diagonally dominant, and the matrix SQ E is such that L G E].. > 0, j=1,2,...,N (4.58) then the system is PD and PV controllable. 4:4 Stability and Controllability Constraints For PQ stability, it is required that the matrix D4 is an M-matrix, and for P0 controllability, it is required that the diagonal blocks of D4 that correspond to each Load Center are irreducible M-matrices. These requirements give rise to a set of security constraints. Theorem A2 of I!"- 72 Appendix A states that a matrix is an M-matrix if all the leading princi— pal minor determinants of the matrix are positive. The principal minor determinants of a matrix being positive imply that the diagonal entries are positive too. So we can obtain the constraints 2 for all i = N + 1,...,N + M. Replacing B.. by - X B.. + B. and solving ll jfi ij 10 for Bio’ we have that Qi B. < 2 B.. +—-— (4.60) 10 jfi 13 vi for all i = N + 1,...,N + M. We can write the susceptance Bio as the sum of several components B. =8. +B. -B. (4.61) 10 1b iC iL where Bi - is the susceptance of capacitor or reactor banks b Bi - is the changing capacitance of the lines connected to bus i c Bi - is the shunt susceptance due to inductive load connected at L the bus i Then Qi B. < B. -B. + Z B..+-—§ (4.62) ‘b 1L ‘c jfi ‘J v]. for all i = N + 1, N + 2,..., N + M. This set of constraints puts an upper bound on the shunt susceptance 73 that can be connected at a PD bus of the system assuming that the load and the voltage at that bus are fixed. (a) (b) (c) (d) (2) Note, that the shunt susceptance that can be connected at a bus increases with the shunt susceptance due to inductive load, BiL; decreases with the charging susceptance of all the lines connected to the bus. Thus, if several short lines with Bij > Bij , where c B.. is the charging susceptance of the line ij, are out of service 13 the: the shunt capacitance that could be connected at bus i would be reduced if SQLV is to be an M-matrix and PV and PD controllability is to be assured; decreases with reactive load increases (more negative reactive load injections 0i); decreases with voltage magnitude decreases. Thus, one can connect more capacitors when the system has no low voltage problems than when the system is experiencing voltage collapse. Furthermore, one produces more reactive support V1.28ib by inserting capacitors when the system is not experiencing low voltage problems. buses that have large line charging susceptance Bic can accept smaller shunt capacitive support. A second set of constraints 0 0 -180 + Yij < Bi - ej‘< 180 + Yij (4.63) that must be satisfied for all PQ buses i and P0 or PV buses j that are directly connected by a line, is obtained by requiring that all the off diagonal elements of S be nonnegative and the matrices S and S 0L" "5 QGQL have nonnegative elements. It will be shown in Chapter 6 that these angle constraints will be 74 vkflated for branches in the boundary of a voltage control area as vol- tages collapse. A voltage collapse will be shown to occur as the magni- tude of the real power transfer into a voltage control area is increased due to the fact that the angle differences on branches connecting PQ bus in the voltage control area to PD and PV buses in other voltage control (areas violate the constraints. Then off diagonal elements of SQLV become positive causing SQLV to no longer be an M-matrix, and elements in SVE and SQGQL become negative. In Chapter 6, we will also show that voltage collapse occurs for ‘incrreased reactive power transfer across the boundary of a voltage con- tiw>l area due to either increased load or loss of generation in the chltage control area. These operating condition changes reduce the chltage at all buses in a voltage control area and would not necessarily camise increases in the angle differences across branches in the boundary 01’ the voltage control area that experiences voltage collapse. However, if there is a net real power transfer in the voltage control area that experiences voltage collapse, then the angles across branches in the boundary of the voltage control area increase and the voltages at buses in the voltage control area decrease in order to maintain the real power transfer. Thus, the angles differences for branches in the boundary of the voltage control area can actually violate (4.63) for the voltage Collapse caused by the operating condition changes considered above. If the constraints in (4.62) are satisfied for every PQ bus and if the angle constraints (4.63) are satisfied for every branch connecting a PD bus to another PQ bus or a PV bus, one could still conceivably ex- perience voltage problems. An additional set of constraints that are 75 stronger than those in (4.62) must be imposed if SQ V is to be an M- nwtrix. This set of constraints is based on the factLthat the matrix SQLV is an M-matrix if and only if the off diagonal elements of SQLV are nonpositive and all of its eigenvalues are in the right half of the complex plane (see Theorem A2 in Appendix A). Theorem A7 in Appendix A states that if an nxn complex matrix A is strictly diagonally dominant and the diagonal entries are positive real numbers, then the eigenvalues (rf A are in the right half of the complex plane. If in addition the off (Tiagonal elements of A are negative real numbers, then A is an M-matrix. We can apply this result on D4 = SQLV. Since it was required that the diagonal entries of D4 are positive, we now require that D4 is Sirrictly diagonally dominant. This provides us with another set of constraints, namely N+M Q. - Vi Bii > 2 Qi j=N+1 jfi 3' (4.64) for all i = N+1, N+2,..., N+M. Since Qij are nonpositive, based on the above discussion, we can rewrite (4.64) as 2 N+M Q. - vi Bii + . N 1013 > o (4.65) J= + in for all i = N+1, N+2,...,N+M. It is clear that if the constraints in (4.65) are satisfied, then those in (4.59) are satisfied too, provided of course, that all the Qij are nonpositive. 76 Substituting Bii by B.. = - Z 8.. + B. + B. - B. (4.66) ii jfi 13 1b 1 1L in (4.65) the following constraint is obtained 2( ) 2 N+M N+M Q. - V. B. + B — B. + v. 2 B + 2 Q > O l 1 lb lc 'IL 1 3:1 1:] j=N+1 lJ in 121 or Q + V 2(B. - B - B. ) + V 2 g B + l i iL 1C ‘6 i j=1 ij in N+M 2 + 2 (v1 81' + 01.)> 0 ' (4.67) j=N+1 J J in for all i = N+1, N+2,..., N+M. If each of the expressions on the left hand side of (4.67) can be in- creased above a threshold, the more controllable will be the voltages in the network. Since the minimum value of (4.67) is a lower bound on eigenvalues of D4, which is the matrix relating the reactive power load injections at the PQ buses to the voltages at the PQ buses for a de- coupled loadflow, and since the minimum eigenvalue of D4 must be greater than a certain threshold to assure convergence of a decouple loadflow program, one could impose a constraint N 2 2 (Bi - B. - Bi )'*‘h = B..-+ Q. + v. 'I 'l L 1C b jllJ 77 N+M 2 + 2 (Vi Bi' + Qi')> k (4.68) j=N+1 J J .lfi for all i=N+1, N+2,..., N+M, where k is a positive constant. This constraint can be rewritten as 2 Q. + v. (8. - 8. - B. ) + v. z 8.. + l l iL ib iC i jeI 13 P + V 2 g X B . + Z (V 23 + Q ) + i m=1 jeIm ij jeJ l ij ij mfp p K 2 + 2 2 (vi 31' + 01') > k (4.69) m=1 jedm 3 J m¢p for i = N+1, N+2,..., N+M, where; ‘ K - is the number of voltage control areas in the system, p - is the voltage control area containing the bus i, th Im - is the set of PV buses in the m voltage control area, and Jm - is the set of PD buses in the mth voltage control area. This constraint can also be written for a voltage control area by summing over all buses in an area to obtain 2 2 2 Q. + Z V. (Bi - B. - B. ) + z Z V. B.. + . i . l l l . . l ij 6 1 Jp ier L c b 1€Jp 361p K + 2 v1.2( 2 z 81..) + z z (6.281.401) + i€J m=1 jelm J isd jeJ J J 78 K + z z 2: (v.28..+0..) >k (4.70) iEJ m=1 jEJm ‘ ‘3 ‘3' p mi‘P Although the constraints (4.69) may be very effective to predict voltage problems at PQ buses that are radially connected over long lines, it may not be as effective as the voltage control area constraint (4.70) in predicting voltage problems at all buses in a voltage control area. The bus constraint values can vary widely for buses in a voltage control area that is experiencing voltage problems but the voltage control area con- straint should better predict the weak coupling between voltage control areas. Voltage collapse will be shown to occur as the weak boundaries of voltage control areas are stressed. The effects of each term on the margin of satisfaction of the con- straint (4.70) are given as follows: (1) 1:0 01 P Since Qi becomes more negative as the constant power reactive load increases, increasing reactive load in a voltage control area will tend to decrease the margin or ultimately cause violation of (4.70) (2) 2 V1.31. 'EJ L 1 9 Since V1281 is positive and represents impedance reactive load and L Qi is negative and represents constant power reactive load, the percentage of the total reactive load modeled as constant power or constant impedence has great effect on the constraint (4.70) (or (4.69)). This percentage should be known at every bus of the system for better and more accurate prediction of voltage problems in a (4) (5) 79 voltage control area. The effects of reactive load modeling will be demonstrated in Chapter 6. iEJp ic 1'15 This expression represents the reactive support provided by line charging and shunt capacitor and reactor banks in a voltage control area. Line charging and switchable shunt capacitance reduces the constraint satisfaction margin. The deleterious effects of signifi- cant line charging and over use of shunt capacitor banks in a vol- tage control area is investigated in Chapter 6. 2 >3 v.28. 'EJ '61 i ij 1 p 3 p The expression represents the reactive supply capacity of branches connecting PV buses to P0 buses in the same voltage control area. As the reactive load at P0 buses in a voltage control area in- creases, the PV buses in the same voltage control area become PQ buses as their reactive outputs reach their upper limits. As these PV buses become PQ buses, the number of terms in the expression decreases significantly and dissappears entirely if there are no PV buses left in the voltage control area. This expression should generally have a large value indicating that there is significant reactive transmission capacity for shipping reactive generation to the reactive load within a voltage control area. 2 Z 2 (V. B.. + Q..) iEJ jeJ ‘ ‘3 ‘3 9 iii” This expression represents losses on the branches connecting PQ (6) (7) on: (1) (2) (3) (4) 80 buses in a voltage control area. 2 Z (V. ier jeJm This expression represents flows from P0 buses in a voltage control area p to PD buses in a voltage control area m. If the voltage control area p is experiencing voltage problems it will often be negative and thus reduce the margin of satisfaction or cause vio- lation of (4.70). 2 2 v.2 ier jeIm This expression represents the reactive power capacity of branches BU- . mfp connecting PQ buses in a voltage control area p to PV buses in a voltage control area m. Since these branches are in the weak trans- mission boundary of the voltage control area p the expression is not generally large even though it is positive . The set of constraints (4.69), (4.70) can be used to place limits the maximum constant power reactive load at a bus or in a voltage control area, the maximum fixed capacitance at a bus or in a voltage control area for power factor correction or due to line charging of long or underground lines, the maximum reactive flow from P0 buses in other voltage control areas to PD buses in a voltage control area, the minimum number of PV buses and the corresponding reactive power transmission capacity of the branches connecting the PV buses to PO buses in a voltage control area. This constraint is a system's 81 plannning type constraint rather than an operation planning constraint as the above three constraints are; it can be used to assure that there is sufficient redundant reactive supply and sufficient reactive transmission capacity. The final constraints that can be imposed are constraints on the column sums of SQGE N N _1 1:1 [SQGEJij ‘ i=1 ([CBJij ’ [D304 C4Jij) > 0 0r 2 N N -1 .- . ..+ ..- .. 0J EJ 8JJ 1:1 0J1 iil [0304 c4]13-> 0 or ' 0 E 2(8 8 B ) E 2(N+M B ) .+ . . - . — . + . 2 .. + J J 3L 3C 3b 3 i=~+1 Jl ifj N 2 N -1 + 121 (Ej 831 + jS) - iEI [0304 c4]ij > 0 (4.71) for all j = 1,2,...,N. This bus constraint is similar in form to the PQ bus constraint (4.69) but with the additional term i "NZ 1 [D3D4'1C4Jij (4.72) i If the system looses PQ controllabiity and D4 is moving toward singu- larity, 04'1 will become large, the expression (4.72) will become more 82 negative for all j = 1,2,...,N, and if (4.71) is violated for some j , 1 g.j 5.N loss of PV controllability will result. PV controllability can also be lost if at least one column sum of C3 (the first four terms in (4.71)) become negative. It will be shown that if a large switchable shunt capacitor is installed at a PV bus in the system, the PV controlla- bility will either be lost or weaken as can be easily seen by (4.71). The constraints (4.70), (4.71) can be used to place constraints on: (1) the flow from PV buses in other voltage control areas to a bus or buses in a voltage control area. (2) the switchable shunt capacitance at PV buses or in a voltage control area. (3) the tap changing transformer ratio that as the tap setting increases, effectively decreases the branch susceptance and places capacitance at the bus closest to the load'and reactance at the bus closest to the source. A PV constraint on the voltage control area can be obtained by adding up the individual constraints for buses in the voltage control area . 4.5 Discussion on Quantitative Controllability The definitions and sufficient conditions for PV and P0 stability and controllability presented in the preceeding sections are qualitative. In practice one has to deal with the quantitative side of these issues. For PQ controllability, we stated that if the voltage at the PV buses is held fixed (AE=0) and the reactive load is increased at at least one PQ bus (AOL-7- 0) then the voltage at the PQ buses is decreased (MiG) (this was one part of the PQ-controllability definition). From this part we 83 see that not only are we interested in how the voltage will change as the load increases, but also how much the voltage decreases at each bus as a result of a certain load increase. If a small load change causes large voltage decreases, then the system is still PQ-controllable but this controllability is weak. If on the other hand, it causes moderate vol- tage decreases, the controllability is strong. The second part of PQ-controllability states that when the load stays fixed (AQL=D), any nonzero nonnegative voltage changes at PV buses should cause voltage increase at each PQ bus. The results in Chapter 6 show that there can be a situation where some of the elements of SVE become negative. Therefore, voltage increases at some PV buses might cause voltage decreases at PO buses which totally violates the cause effect relationship upon which voltage control (by the use of exciters, tap changers and switchable capacitors) is based. Perfect voltage control would be achieved if every PQ bus was directly connected to every PV bus, there were no connections between PQ buses and the system was at flat start condition. If there were N PV buses, then every element of SVE would be equal to 1/N, indicating that every PV bus would have an equal percentage control on the voltage at PO buses. In a practical power system, the voltage at a PD bus will be affected heavily on the voltage changes at a few PV buses, and very lightly on the voltage changes at the rest of them; there will also be PV buses that in no way can affect the voltage at any PQ bus, by con- trolling their voltages. These buses correspond to zero columns in the SVE matrix. Results in Chapter 6 indicate that there are no voltage problems when all row sums of SVE satisfy 84 N 1 - e :_ Z [S 3:1 VEJij g_1 + E (4.73) for e: of the order of less than .10. The results also show that as voltage problems begin to occur this constraint is violated by having row sums either significantly increased or decreased. Elements in SVE go negative and row sums significantly reduce or even become negative as voltage collapse is imminent. As long as elements in SVE are nonzero nonnegative the system would be PQ controllable. However, in a quantita- tive sense this is not enough; the system will be called weakly PQ controllable if it is controllable and (4.73) is not satisfied. It will be called strongly PQ controllable if it is controllable and (4.73) is satisfied. The operating probiem associated with weak PQ controllability can be illustrated by an example. A very small reactive load change AQLi causes a large 2% voltage decrease at a P0 bus i. In the effort to compensate for this voltage change at the PQ bus i, the voltages at the PV buses that have the largest values in the SVE row corresponding to the PQ bus i, must change by 10%. This example clearly indicates that for the small disturbance AQLi the controls AEj's are unacceptable even though the system is PQ controllable; the system is weakly PQ controlla- ble. Thus, one desires to maintain network operating conditions so that the strong controllability condition (4.73) is satisfied for all PQ buses. A system is considered to be PV controllable if any load increase (AQL < 0) when AE=0 causes the reactive generation to increase (AQG: 0) and if when the load is fixed any voltage increase AE at the PV buses 85 causes the total reactive generation to increase, i.e., "(‘42 A061 > O (4.74) i 1 If for the case when the voltages at the PV buses are fixed (AE=0), then the generation is such that M A061. + 2: AQLj (4.75) N AQloss = El j i 1 the reactive losses in the system don't change significantly then the PV controllability is strong; but if'AQloss is large, then the PV controlla- bility is weak. Since N M AQloss = 1:1 AQGi + jil AQLJ' = N M [ ] M = - z 2 S 40 + 2 40 ~ = i-1 j=1 QGQL ‘3 L3 j=1 L3 M N = z - z [s 1.. + 1 q (4.76) J=1[ l=1 QGQL ‘3 ] L3 Therefore, [101055 is small and the system is strongly PV controllable if and only if 1 - e g. z [s 1.. 531 + e (4.77) i=1 QGQL ‘3 for all j=1,2,...,N, where E is a positive number of order .10 to .20. 86 If the system is PV controllable and (4.77) is violated for at least one j, 1 g j 3.”: then the system is weakly PV controllable. If when the load is fixed, the voltages at the PV buses increases by AE then for PV controllability we require that (4.74) is satisfied. This is equivalent to the requirement that the column sums of SQ E G E). > 0 (4.78) for all j=1,2,...,N. If one requires a significant voltage increase at the PV buses of the system to achieve very moderate increase in total reactive generation then the PV controllability is also weak. Therefore, one should require . > k (4.79) when k is a positive number, for all j, in order to have strongly PV controllable system. One extra factor of interest is the size of the individual terms in the expression of (4.79). If the size of the individual terms is very large for a certain j, then the reactive generation AQGi = [SQGEJijAEj (4.80) of each Generator i, will be exceedingly large causing the reactive output of the generator to saturate in either the positive or the nega— tive direction. This reduces the importance of the PV bus j in control- ling the reactive generation of the system. 87 If any of the column sums of the SQ E matrix goes negative, PV G controllability is lost; the system is PV stable unless all column sums th become negative. In case the j column sum of SQ E is negative, then a G j that was found necessary to raise the vol- tages at some PQ buses will cause the total reactive generation of the positive voltage change AE system to decrease which is undesirable. Thus, AEj cannot be used to control reactive generation increases. 5.1 Chapter 5 Voltage Control Area and Weak Transmission Boundary Determination Introduction The power system operation planners have found the voltage collapse problem particularly difficult because: (1) one could not establish the causes of voltage collapse. The lack of causality [1], which is reflected in singularity of the Jacobian and existence of multiple loadflow solutions, is established in [1,19,24] as the cause of voltage collapse; one could not find tests that would adequately predict the occurrence of voltage collapse and adequately describe all of the symptoms and their degree of severity. The definitions of PV and P0 controllability and the conditions on the sensitivity matrices SQLV, SQGQL, SVE and SQGE provide tests for voltage collapse. Tests for strong controllability on the same matrices indicate when the system would begin to experience voltage problems. Violations of the strong controllability conditions indicate the existence of voltage problems; the degree of violation of the strong controllability conditions determines the severity of the voltage problems. The violation of strong controllability and controllability clearly indicate the symptoms of having operating conditions (load level and distribution network configuration, generation dispatch, bus voltage support, etc.) that cause voltage and voltage collapse problems. All of the test conditions associated with strong controllability are new. The controllability test conditions on SVE’ SQGQL, and SQLV 88 89 are not new but the test condition on SQGE is new. The violation of strong controllability tests indicate symptoms in terms of cause effect relationship that suggest why voltage problems exist. The violation of controllability tests also indicate symptoms that are so distructive to normal cause effect relationships in a power system that the imminence of voltage collapse is obvious. (3) one could not determine operating/security constraints on a bus or subregion (voltage control area) that is vulnerable to voltage collapse. The security/operating constraints developed in Chapter 4 indicate operating condition changes that could alleviate voltage problems at the bus or in a subregion. The margin of satisfaction of such constraints provides a measure on the magnitudes of the operating condition modifications required to obtain a particular margin of satisfaction of the constraints. The determination of the buses that belong to a voltage control area and the branches of the boundary of each voltage control area are abso- lutely essential to proper analysis of the above results since it will be shown that: (1) the singularity of the Jacobian is associated with the existence of weak transmission boundaries between groups of buses and their degree of vulnerability. These weak transmission boundaries cause the Jacobian to become approximately block diagonal if buses in the same voltage control area are numbered sequentially. The elements of the Jacobian outside the diagonal blocks, that are normally negative, will increase toward zero and become positive as the severity of the voltage problems increases. This makes the Jacobian lose its positive definiteness and move toward singularity. The 90 clusters of buses in each diagonal block of the Jacobian indicate the buses in each voltage control area. The nonzero elements out- side the diagonal blocks of the Jacobian indicate the weak transmis- sion boundaries that interconnect voltage control areas. (2) the strong controllability conditions (4.73), (4.77) place con- straints at P0 and PV buses. The controllability constraints (4.67), (4.68) place constraints at P0 buses. The controllability constraint (4.71) places constraints at PV buses and the constraint (4.63) places constraints on angle difference across branches. These constraints are only violated for PV and P0 buses belonging to a voltage control area that experiences problems. Voltage control area constraints (4.70) for P0 buses and (4.71) for PV buses are better measures of the proximity of voltage collapse problems in a voltage control area than the bus constraints (4.59), (4.67), and (4.68) and branch constraints (4.63). The proper analysis of the existence, location, causes and cures of voltage collapse or voltage problems is based on these constraints. Thus, apriori determination of voltage control areas and weak transmission boundaries is essen- tial to the analysis of voltage problems. The objective of this section is to develop methods for determining the voltage control areas and the branches that interconnect buses in different voltage control areas. Voltage control areas (VCA's) were defined in Chapter 3 to be groups of PV and P0 buses that are stiffly interconnected and in which voltage magnitudes and voltage angles vary in a similar fashion as a result of operating condition changes. One method of identifying the voltage control areas depends on determining the groups of PV and P0 buses that respond similarly to an increase in load 91 and loss of generation contingencies [8], based on a "coherency measure". A second method for identifying the voltage control areas and determining weak transmission boundaries is developed in this Thesis. Stress on weak transmission boundaries due to real and reactive flows is shown to cause the loss of strong controllability or controllability that indicate voltage problems or voltage collapse in an area. Before a technical discussion of (a) the methods for identifying voltage control areas and (b) how stress on their weak boundaries cause voltage collapse, an example of voltage collapse is presented. This example is presented at this point to not only indicate the importance of identifying the group of buses that will experience voltage collapse as the operating conditions change, but also indicate the kinds of problems that are observed in the loadflow solutions as the voltage collapse is approached. The example is given on a 30 Bus New England test system. The system has 10 PV buses and 20 P0 buses that are interconnected by 37 branches as shown in Figure 5.1. The base case loadflow data in common format, which is a loadflow data exchange format, is listed in Appendix C. The Bus 30, which is the bus connected to the largest unit in the system, is the swing bus. The voltage collapse problem occurs due to an increase in reactive load at Bus 11. The experimental results in Table 5.1 show that as the load at the PQ Bus 11 increases from 0 MVARS to about 900 MVAR, the region consisting of the Buses (4, 5, 6, 7, 8, 10, 11, 12, 13, 14) experiences problems. The reactive outputs of Generators 6 and 10 increase as the load at Bus 11 increases to about 400 MVAR. The reactive flows on the lines (3, 4) (9, 30) and (14, 15) connecting this group of buses to the rest of the system are not affected by this load increase at Bus 11. 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Then: (a) the voltages at the Buses 6 and 10 are no longer controlled and the buses become PQ buses; (b) the voltages at all the bus of this group drop significanly below acceptable levels, with the voltages at other buses unaffected, or slightly affected; (c) other PV buses of the system start supporting the above group of buses for further increases in reactive demand in the area; and (d) the flows on the lines (3,4), (9,30) and (14,15) that connect this group to the rest of the system increase as well as the losses on these lines. These results show that not only the voltage at Bus 11 where the reactive load is increased experiences problems but also the buses that are stiffly connected to it--the buses that are in the same voltage control area with it. Therefore, it is very important to know the VCAs of the system and the weak transmission boundaries in order to be able to better control the system. The rest of this chapter is devoted to the determination of VCAs and the determination and ranking of weak transmission boundaries. 5.2 Determination of Voltage Control Areas and Weak Transmission Boundaries In [8], the weak transmission boundaries between stiffly inter- connected groups of PV and PO buses were determined by converting all the PV buses to P0 buses so that the PV buses can experience voltage swings that will reflect the stiffness of the transmission grid and not the 101 action of the voltage controls. The weak transmission boundary for real and reactive power disturbances encircle PV and P0 buses that are all interconnected and lie in a small geographical area. The algorithm in [8] uses a measure ieI svnm) = {z (vkm - vln) )2} (5.1) where k and l are buses of the system, I is a set of 100 MW and/or 100 MVAR disturbances, and Vk(i) is the voltage deviation from the base case, at the bus k, due to a disturbance ie I. The procedure for identifying the weak transmission boundaries requires a multiple contingency loadflow output corresponding to the set of contingencies I. Then the following steps are followed: (1) Compute the measures Sv(k,l) for all the bus pairs (k,l). (2) Rank the measures Sv(k,l) from smallest to largest and form a ranking table. (3) Form groups by a set of grouping rules and a group formation table. (4) Rank the boundaries from the weakest to the strongest based on the reverse order in the group formation. This algorithm [8] for identifying voltage control areas is related to a method that identifies coherent generator groups that are aggregated to form dynamic equivalents for transient stability studies. The co- herent groups are based on voltage angle changes produced by a linearized classical transient stability model. The boundaries of these coherent groups are furthermore shown to be weak and vulnerable to a loss of synchronism due to both fault and loss of generation contingencies [35]. The buses in a voltage control area identified using the above algorithm 102 were likewise shown [8] to be vulnerable to voltage collapse if real or reactive flows across the boundary of this voltage control area are increased. The above algorithm is a complicated procedure. It requires a loadflow for a change in load or generation at every bus in the network. The number of pairs (k,l) is exceedingly large even for a small size system. The ordering of the measures in step (2) and of the boundaries in step (4) are very slow due to the large number of cases involved. In this Thesis, we employ a much simpler clustering algorithm that uses a modified incidence matrix of the system. The algorithm is based on a method by Zaborsky [9], that divides the system into a number of clusters with relatively slow intercluster dynamics and relatively fast intracluster dynamics for the individual clusters. This algorithm was also initially developed for producing dynamic equivalents based on the linearized dynamics of the lossless machine angle model of a power system 2 = Ax (5.2) where A is an nxn real diagonally dominant matrix with all diagonal terms negative and all the off diagonal terms nonnegative. Then Zaborsky [9] shows that a separation of the modal frequencies above and below a pre- selected arbitrary value V62 can be effected by direct classification of the terms of A (without computing the eigenvalues). Each modal frequency belowifi? is associated with a square diagonal block Akk, k=1,...,N, of A where _ kl is a certain partition of the matrix A and where renumbering of rows and columns possibly took place before partitioning. Then since the off diagonal entries ai., ifj, represent physical network connections between J 103 the buses i and j, the diagonal blocks Akk, k=1,...,N can be used to determine the clusters of buses of the system where each cluster has one and only one slow eigenvalue. kk The algorithm for identifying the square diagonal blocks A for a preselected arbitrary value a produces a modified incidence matrix C where (1) cij = 0 if aij = 0 (2) for each i, the largest set Bi such that Z | ai.|§_a (5.4) jeB. J 1 is determined and Cij is set to zero for all 3:81 (3) Cij # Cji then cij = Cji = 1 (4) C11 = 1 for all i. Then similar permutations are applied on the rows and columns of A and C until C becomes a block diagonal with all diagonal blocks having all 1's. The partitioning of A is then done based on the blocks of C. The motivating point in this algorithm is that entries aij of A for which jcBi and iij indicate a pair of buses that belong to two different clusters. If jeBi and iij then Iaij|< a,; this shows that if buses i and j are in different clusters, then the term aij is small. The context in which this algorithm is used in this Thesis is to separate the buses of the system so that buses in the same cluster are stiffly intercon- nected and clusters are interconnected by weak transmission lines, which taken together, are the weak transmission boundaries. Although the linearized reactive voltage dynamics do not satisfy a differential equation of the form (5.2), the linearized reactive voltage dynamics satisfy a differential equation Ax = A Ax (5.5) where The 104 x = (E E E v )T 1, 2,..., N’ V V N+1’ N+2""’ N+M A is an (N+M) x (N+M) real diagonally dominant matrix with all diagonal terms negative and all off diagonal terms posi- tive A = A1 - K J v"1 (5.6) A1 = diag {- -%I’ - -%E,..., - -%E, o, o,..., oi} (5.7) K is a diagonal matrix v = diag {E1, E2,..., EN, vN+1, vN+2,..., vN+M}. (5.8) J C3 D3 C4 D4 linearized dynamic model can be obtained by linearizing the flux decay model of voltage at every generator bus [.=; E - E - l Tdo- F i i .=1 1 J 13 l J 13 x . - x' . N ‘ -_d_1__d1[2 E.E.Y..sin(5. -6.-Y..)+ J N+M + . . .. ' . - . - .. . j=l§+1 E1VJY1JSln( <51. BJ YIJ)] (5 9) and a singularly perturbed model at each load bus 105 N ° - ;_1_ . _ _ l J-l N+M + . . .. ° . - . - .. , j=§+1 V1VJY1JS‘"(91 GJ v13) (5 10) The formation of voltage control areas based on J or A would be quite similar since the Zaborsky [9] algorithm determines branches that belong to boundaries of voltage control areas based on the magnitudes of the off-diagonal elements of these matrices and since the elements of J are just scaled by the appropriate elements of the diagonal matrices K and V'l. Singularity of A would cause loss of causality [1] based on a transient stability model that includes flux decay. J can be obtained even when only a loadflow solution is available to investigate voltage collapse. Therefore, the matrix J is used to determine voltage control areas in this Thesis. Singularity of the A matrix may be associated with loss of causality for a transient stability model where flux decay effects are included in the generator models. Singularity of the subblock D4 of J, associated with rows and columns corresponding to P0 buses can lead to loss of causality based on a constant voltage behind transient reactance genera- tor model as shown in [1]. An effort will now be made to argue that weakening of the boundary of a voltage control area is associated with singularity of J (or A) and loss of causality. Each diagonal submatrix Jkk of the matrix J is associated with one eigenvalue with magnitude less thancx. As voltage problems become more severe, we have observed that the negative off- diagonal elements of J associated with elements in the boundary of the 106 voltage control area experiencing voltage collapse 0.. = V.V.Y..sin(61 - 6 u 1.113 ) (5.11) J 'Yij increase and approach zero. Three factors can make the absolute magnitude of Qij small. These are: (i) the magnitude of the admittance Yi' J is small, and the angle Yi' is J much greater than 90°; (ii) the magnitudes of the voltages Vi and/or Vj are small; and (iii) the voltage angle difference 9i - Si is large. The computational results in Chapter 6 show that voltage at buses in the voltage control area decreases as reactive flow across the boundary of the voltage control area increases and the angle differences ul- timately increase in order to maintain real power transfer. Thus, Qij for branches in the voltage control area boundary reduces as the reactive power transfers across the boundary increase. Also increasing real power transfers across the voltage control area boundary results in angle differences on branches to increase. Thus, Qij decreases for increased real and/or reactive power transfer across the boundary of the voltage control area. As the voltage control area boundary is stressed and Qij for the boundary branches decrease, the value ofcx required to detect the voltage control area boundary is decreased. From [9], an eigenvalue with magnitude of less than a corresponds to each diagonal subblock of J associated with buses in a voltage control area. Thus, as the real and reactive transfer across the boundary of the voltage control area in- creases, the eigenvalues of J decrease bringing the system closer to loss of causality and voltage collapse. Although there is a lower bound on the eigenvalue based on the diagonal dominance in a subblock [9], the Si ((5 «We t F‘U 107 upper bound on the eigenvalue will decrease with decreases in Qij lead- ing to small eigenvalues of J, lack of loadflow solutions, and the pos- sibility of singularity of J and loss of causality. The algorithm used to detect voltage control areas is now described more precisely. Algorithm Given the matrix J and a real number a, 0 5 a: 1, we produce the modified incidence matrix C = (Cij) as follows. 5.2221 (i) Find the maximum element of J = (dij) mJ = max{|dij|} (5.12) (ii) Compute a = mJa (5.13) £223 (i) Set Cij = 0 if dij = 0; however, dij = 0 if the buses i and j are not directly connected (ii) For each row i of the J matrix order the nonzero elements as |dij1 ‘3 Id”.2 5 ...: dijk) (5.14) Then find the largest index s such that ldijll +. dijzl + ... + 'dijs‘< a (5.15) and set Cijl = Cij2= ... = Cijs = 0 (5.16) 108 Repeat for all i, 1 :,i : N+M (iv) Set C11 = 1 for all i, I :,i 5 N+M (5.17) and if Cij f Cji’ then set Cij = Cji = 1 (5.18) The resulting matirx C is the modified incidence matrix of the system. A 1 at the ijth position indicates that the buses i and j are stiffly interconnected; a 0 at the ijth position indicates that either there is no connection between the buses i and j or the connection between the two buses is weak. 2:22; The buses of the system are clustered into equivalence groups based on the modified incidence matrix C. From the Theorems 82 and B3 in Appendix B, C must be raised to 2S where s is the smallest integer such that N+M-1 5 25. This is done by multiplying c by itself to obtain c2; . . 2 2 . 4, , . . 25"1 25'1 multiplying C by C to obtain C ,..., multiplying C by C to s obtain C 5 C2 . Clearly only 5 matrix multiplications are needed. The 5 resulting matrix is an equivalence relation matrix. The equivalence groups are identified by the rows of the CS matrix. For example, the equivalence group containing the bus i, is simply the set of j's for which (cs)ij = 1. Two rows of C5 are identical if and only if the corresponding buses are in the same equivalence group. Remarks (1) The algorithm in [9] assumes that the Matrix J is such that mJ = max{|dij|} = 1 (5.19) 109 and 0.: a 3 1. In the algorithm presented above, mJ is not assumed to be 1. Then instead of normalizing J by dividing all of its elements by mJ we preferred to multiply a by mJ. Therefore, in Step 1 we computed mJ = max{|dij|} and &l= mdcx and used it in Step 2 to find the modified incidence Matrix C. (2) The program for finding the clusters using the procedure in Step 3 requires storage of full matrices of dimensions (N+M) x (N+M). For a large system this will be extremely difficult if not even impos- sible. Thus, a second program, that uses only the indices (i,j) of the nonzero entries of C, was created. The second program can handle systems with hundreds of buses and thousands of branches with no storage difficulties. The main idea for the development of the second procedure is that the buses in a cluster can be considered as stiffly connected to a certain bus in the cluster--the one with the smallest number, for example. Then if j is stiffly connected to bus i and k is stiffly connected to bus i, j and k are stiffly connected. Therefore, starting from the pairs (i,j) for which Cij = 1, we modify them as follows: (i) if the indices i and j of a pair (i,j) were satisfying i> j then the pair was written as (j,i), (ii) if there were two pairs (i,j) and (j,k) then the second pair was changing to (i,k). After performing all necessary such modifications on the pairs (i,j), we identify a cluster by the bus numbers that appear as second indices of all the pairs that have the same first index. 110 The value of a determines how stiff a line should be to be con- sidered a stiff connection. As a increases, more lines are considered weak connections and they don't show up in the modified incidence matrix, and possibly more clusters are formed. If the clustering is applied for several values of a, then a set of clusterings is obtained. The Figures 5.2(a) - (m) show a set of cluster- ings with 2, 3, 5, 6, 8, 9, 11, 13, 15, 17, 18, 19, 20, 22, 23 and 24 clusters each for the 30 bus New England system. This set of clusterings can be used to rank the lines of the system. Table 5.2 shows the ranking obtained based on the clusterings of Figures 5.2(a) - (m). The rank is the number of clusters in the system when a line appears as a weak boundary for the first time in the above set of clusterings. Table 5.2 was found based on the clusterings and the number of clusters was found for different values of a. However, starting from Table 5.2, one can reconstruct any clustering with one of the numbers of clusters shown in the Rank column of the table. For example, if we want to reconstruct the case where there are 5 clusters in the system, then we remove all the lines with rank less than 5 and we group the buses of the system based on connectivity. This then produces the 5 cluster case. The results obtained by the clustering algorithm using the modified incidence matrix [9] are very similar to those found in [8] using co- herency measures. Also, the experimental results presented earlier showed that the voltages magnitudes and angles at the buses (4, 5, 6, 7, 8, 10, 11, 13, 14) change in a similar fashion due to operating condition changes. The clustering procedure came to confirm these results by showing that the group of those buses remains intact as the value of'a and the number of clusters increase. 111 EmumAm campmcu 3mz mam om mgu Low .N 3v A? a. ~— Hm n #1.. 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In. - - - - 2 - - - - 2 _ - a - . _ - z a - v~ A m — o~ . om ~ m~ m v mm a... - 3 O S o~ 117 24 3 IS " b ll 20" 13 ' Figure 5.2(m) Clustering schemes with 24 clusters for the 30 Bus New England System 118 NO Line Rank 1 1 - 2 2 2 8 - 9 2 3 26 - 29 3 4 26 - 28 3 5 11 - 12 5 6 12-13 5 7 3 - 4 s 8 14 - 15 5 9 16 - 19 6 10 30 - 9 8 11 25 - 26 8 12 17 - 27 8 13 21 - 22 9 14 23 - 24 9 15 1 - 3O 10 16 2 - 3 12 17 28 - 29 12 18 19 - 20 15 19 21 - 24 15 20 26 - 27 15 21 8 - 5 17 22 7 - 6 17 23 2 - 25 17 24 22 - 23 20 25 3 - 18 20 26 4 - 5 20 27 4 - 14 20 28 15 - 16 23 29 6 - 11 23 30 14 13 23 31 17 - 18 24 Table 5.2 The lines of the system ranked based on the number of clusters when they appear as weak boundary lines for the first time 119 In Chapter 6, we are going to perform several experiments causing voltage problems in the voltage control area consisting of the buses (4, 5, 6, 7, 8, 10, 11, 13), by; (i) outaging the Generators 6 and 10, (ii) removing the line (9, 30) that connects the swing bus (Bus 30) to the VCA, after real generation was redispatched to some PV buses in the system and the swing bus, (iii) adding capacitance in the VCA, (iv) increasing reactive load and placing extra capacitance in the VCA, (v) increasing the real load in the VCA. All these operation changes have one thing in common; they try to in- crease the flows over the weak transmission boundary of that VCA as well as the losses on the boundary and thus create voltage problems. Chapter 6 Computational Results on Stability and Controllability 6.1 Introduction In this chapter we present the results of the simulations performed in the study of PV and P0 controllability and stability of the 30 Bus New England System. As was shown in Chapter 5, the group of buses (4, 5, 6, 7, 8, 10, 11, 13, 14) is a stiffly interconnected voltage control area. This VCA, which is denoted as VCA 1 in Figure 6.1, is connected to the rest of the system via the three weak transmission lines (9, 30), (3, 4) and (14, 15) and to the load Bus 12 via the two transformers (12, 11) and (12, 13). This voltage control area consists of two generator buses generating 1223.12 MW and 225.56 MVAR of power and seven load buses serving load of 1255.85 MN and 444 MVAR. It is then clear that power flows from the rest of the system to VCA 1 through the weak transmission boundary. In these calculations, we didn't consider the line charging of the lines within VCA 1 which would decrease the MVAR flow into the voltage control area, nor did we consider the load at Bus 12 -served solely by flows across the weak transmission boundary lines (9, 30), (3, 4) and (14, 15). Increasing real and reactive power flows across the weak transmis- sion boundary lines (9, 30), (3, 4) and (14, 15) causes voltage and voltage collapse problems. This is because the elements of the Jacobian associated with the branches (9, 30), (3, 4) and (14, 15) reduce signifi- cantly, as the power flowing across these branches increases, causing the Jacobian to move closer to singularity. 120 26 121 28 VCA 1 23 ‘— ll- 1 ? 515:9? M I 20 5 L , Figure 6.1 The 30 Bus New England System and the voltage control area VCA 1 122 If any of these three lines is outaged, then the power flowing through the remaining two lines would increase causing voltage and voltage collapse problems in VCA 1. Many of the power utilities on the East Coast fit the above des- cribed situation. These utilities rely on real and reactive power trans- fers across the weak transmission boundary lines, which connect them to Midwestern utilities with cheap coal and nuclear generation and to Canada with cheap hydro generation. These East Coast utilities have higher cost oil fired generation and wish to purchase cheaper power from the Mid- western and Canadian utilities rather than generating it. These East Coast utilities experience low voltage and voltage collapse problems as they attempt to reduce internal real and reactive generation and import power across long transmission lines. To raise the voltage levels, these East Coast utilities have installed switchable shunt capacitors. When- ever the reactive load served by these utilities increases, they switch more switchable shunt capacitors in to compensate for these increases. It was realized that: (1) as the real load increases in a voltage control area and it is served by transfers across the weak transmission boundary, (2) as the reactive load increases in a voltage control area and shunt capacitors are used to compensate for it, and (3) as the reactive losses on the weak transmission boundary of a voltage control area increase, the system approaches voltage collapse as indicated by the constraints (4.63), (4.70) and (4.71). The above scenario was basically the motivation for the simulations performed. At first we removed Generator 6 from VCA 1 and thus the 123 generation in VCA 1 is reduced. Therefore, more power flows through the weak transmission boundary. Then, the generator at Bus 10 is removed and the flows on the weak transmission boundaries are further increased as well as the reactive losses. The real power that the Generators 6 and 10 were producing is now provided by the swing bus, Bus 30, and almost all of this flow enters VCA 1 through the line (9, 30). An attempt to outage the line (9, 30) gave a nonconvergent loadflow case, which indicates a voltage collapse problem. To relieve line (9, 30), the real power pro- duced by the Generators 2, 25 and 29 is increased by 200 MW and thus, the flows of real power into VCA 1 are distributed to all three lines of the weak transmission boundary. Then the line (9, 30) was outaged. The loadflow program had no convergence problems. However, the voltages in VCA 1 obtained are significantly reduced to about 0.8 pu, which is not an acceptable voltage level for proper operation of the system. To improve the voltages in VCA 1, a capacitor bank is placed at Bus 11, which is at a rather central location in VCA 1. Raising the amount of MVAR capaci- tance at Bus 11, the voltages throughout VCA 1 improved. Table 6.1 shows the voltages and voltage angles throughout the system for the series of actions taken as explained above, i.e., (1) for the base case, (2) after the Generator 6 is removed, (3) after the Generator 10 is removed, (4) after the generation at the Buses 2, 25 and 29 is increased by 200 MW, (5) after the line (9, 30) is outaged, and (6) after a capacitor bank of 500 MVAR is placed at Bus 11. mmmmu o mg. so. . mm—mcm mama—o> we. mocappcmos ammu_o> . mmppwoga mmmupo> ..o a...» 124 .....- ...... .....- ...... ..H..- ...... .....- ...... .....- ...... .....- ...... . ..uz... .. ....~i ...... .....- ....H. .. .. ...... .....- ...... ....- ...... ....i ...... ~ ...~.. .~ .....- ...... ..H..- ...... ....- ...... .....- ...... ...~.- ...... ....i ...... . 3.....u.. .~ .....- ...... .~ ..- .... . .....- ...... .....- ..u... .....- ...... ....i ~..... . 3........ .~ ..n..- ...... .. ..- ....H. .....- ...... .....- ...... .....- ...... ....- ...... . «...... .~ .. ..- ...... .....- ...... .....- ...... .....- ...... .....- ...... an..- ...... ~ .12.... a. .....- ...... .....- ...... .....- ...... ...~.- ...... .....- ...... ....u ~..... . ....... c. .....- ...... .....- ...... .....- ...... .....- ...... .....- ...... ....- ...... u ...... .u .....- ...... u. ..- .... . .....- ...... .....- ...... .....- ...... ....- ...... ~ ...... - .....- ...... .....- ...... .....- ...... ....~- ...... .....- ...... ....i ...... . .=..... .~ .....- ...... .....- ...... .....- ...... .....- ...... .....- ...... ....i ...... u «9.2... .~ .....- ...... ..H..- ....H. .....- ...... .....- ...... .....- .... . ...»- ...... u .m... .. .....- ...... .....- .... . .....- ...... .....- ...... ~....- ...... ....i ...... . .=... .. .....- ...... .. ..- .... . .....- ...... .....- .-... .....- ~..... ....- ...... . .z... .. .....- ...... .....- ...... ..H..- ...... .....- ...... .....- ...... ....i ...... . ca....:.. .. ...... ...... .....- ..~. . .. ..- ...... .....- ...... .....- ...... ....i ...... . 3.... .. .....- ...... .....- .... . .....- ...... .....- ...... .....- ...... ....- ...... . .29.... .. .....- ...... .....- .... . .....- ...... .....- ...... .....- ...... ~...- ...... . .=... .. .....- ...... .....- ...... .....- ...... .....- ...... .....- ...... ....- ...... . .... ~. .....- ...... .. ..- ..~.. ....~. ...... .....- ...... .....- ...... ~...- ...... . cauu.:... .. .....- ...... ..H..- ...... ....~- ...... .....- ...... .....- .... . ....i ...... u .... .. .....- ...... .....- ...... .....- ...... .....- ...... .....- ...... .....- ...... . a... . .....- ...... .....- ....” .....- ...... .....- ...... ...-- .... . .....- ...... . a:.. . .....- ...... .....- ...... .....- ...... .....- ~o.... .....- ...... .....- ...... . ...:c... . .....- ...... .....- ...... .....- ...... .....- ...... .....- ...... ....i ...... u .22.... . .....- ...... .....- ..~. . .....- ...... .....- ...... .....- ...... ....i ...... . ...... a ...~.. ~..... .....- ...... .....- ...... .....- ...... .....- ...... .....- ...... . ...:.~. . ...uoi ...... o. ..- ...... .....- ...... .....- ...... .....- ...... ....i uan... . ..un. . .....- ...... ..H..- .... . .....- ...... .....- ...... .....- ...... ....- ...... u «...... u .....- ...... no ..- .... . .....- ...... .....- use... .....- ...... no..- ...... . .=..... . . m e n N . 2:. ...; be... .cz 3.5 .3452 m3. 125 Case Line Q-Loss SQ(I,J) 4 3 0.0252 -0.8137 1 8 9 0.0147 -0.6296 14 15 0.0023 -0.0863 4 3 0.1271 -0.7955 2 8 9 0.4545 -0.2625 14 15 0.0038 -0.0663 4 3 0.4356 -1.5156 3 8 9 2.0309 0.4009 14 15 0.0752 -0.8882 4 3 0.6724 -0.9724 4 8 9 0.9640 -0.4408 14 15 0.1485 -0.6369 4 3 2.2718 -1.3092 5 8 9 0.0009 -0.1246 14 15 0.6546 -2.1710 4 3 1.6189 1.9260 6 8 9 0.0015 -0.2100 14 15 1.4893 1.4893 Table 6.2 Reactive losses on the weak transmission boundary lines and reactive flows from VCA 1 to the rest of the system 126 Table 6.2 shows the reactive flows through the transmission boundary of VCA 1 as well as the reactive losses on these lines for the above six cases. Note that as the Generators 6 and then 10 are taken out and more real power from the swing bus flows across the boundary, the total reac- tive power flowing in VCA 1 decreases and then increases and the total reactive losses on the three weak transmission boundary lines increase (Cases 2 and 3). When the lost generation is distributed to the swing bus, and the Generators 2, 25 and 29 (Case 4), the total reactive losses across the boundary lines decrease and the total reactive power imported increases, despite the fact that the same real and reactive load is served by flows across the weak transmission boundary in both cases. When the line (9, 30) is outaged (Case 5), then the lost generation (that was distributed to the swing bus and the Buses 2, 25, and 29) flows into VCA 1 through the lines (4, 3) and (14, 15). The line (8, 9) connects the VCA 1 to the isolated Bus 9 that has no load or generation; it has only line charging. The total reactive losses across the weak boundary and the total reactive flows into VCA 1 increase sharply, indicating the extreme stress in the network; this is justified by the low voltages in VCA 1 as ob- served in Table 6.1. The use of a 500 MVAR shunt capacitor (Case 6) reverses the reactive flow across the branches (4, 3) and (14, 15) but the reactive losses on the boundary are further increased. The voltage profile given in Table 6.1 does not indicate vulnerability to voltage collapse. However, the large losses on the weak boundary indicate that the problem is not totally solved. The Cases 1, 5 and 6 will be used as base cases for the simulations to be performed in the next section. It will be shown that the system in 127 Case 6 is vulnerable to voltage collapse, based on several operating condition changes. 6.2 Computational Results Based on Sensitivity Analysis In the introductions of Chapter 5 and 6, we presented two realistic situations that can very easily cause voltage problems and voltage col- lapse. Ne discussed the results in terms of voltage magnitudes and reactive losses on the weak transmission boundary of a VCA. Nothing was said about the sensitivity matrices and stability and controllability conditions that must be satisfied. In this section we are going to show how voltage problems and voltage collapse can be detected and diagnosed in terms of the buses effected and the degree of vulnerability through the use of sensitivity matrices and by testing controllability and strong controllability conditions. The violation of strong controllability conditions warn of voltage and voltage collapse problems long before they become serious problems for the system. The series of experiments per- formed consist of: (1) simulations of reactive load increases at Bus 11, beginning from the base case (Case 1). The load is increased to beyond the point where the PV buses 6 and 10 of VCA 1 turn into PQ buses. The purpose of this experiment is to show that voltage collapse can occur if suf- ficient reactive reserves don't exist in any VCA to meet reactive load increases in that VCA, causing reactive power import across weak transmission boundaries. (2) simulations that show the effects of capacitive MVAR increases at Bus 11 on the voltage profile of the system in Case 5. This experi- ment does not represent a scenario of power system operation but (4) (5) 128 rather a documentation of the harmful effects of excessive reactive flows due to line charging, capacitors used for power factor correc- tion, and switchable shunt capacitors used for voltage support. simulations that show the effects of constant reactive power load increases and compensating capacitve MVAR increases at Bus 11. The base case for this experiment is Case 6 rather than Case 1, and thus, Generator 6 and 10 and line (9, 30) have been removed, leaving the system insecure. Analysis of the sensitivity matrices is used to indicate the severity of the vulnerability of the base case (Case 6) as well as how the deterioration of the voltage security occurs as load and compensating capacitive reactive supply increases. The system would not have experienced voltage collapse if synchronous condensers Were used rather than capacitors to compensate for con- stant power reactive load changes. simulations that show the effects of solely increasing constant power reactive load at Bus 11, on the system in Case 6. The results obtained suggest that use of capacitive reactive support helps maintain a healthier voltage profile and prevents voltage collapse from occurring compared to no reactive voltage support at all. However, if synchronous condensers were used rather than capacitors to compensate for the load increase, voltage collapse would not have occurred. The results of Experiment IV and III will be compared to establish these results. simulations that show the effects of real power load increase at Bus 11, on the system in Case 6. The effects of real power load in- crease on the voltage profile, the voltage angle changes at P0 buses in VCA 1, and the sensitivity matrices are compared to those in 129 Experiment IV, for reactive load increase. 6.2.1 Experiment I The purpose of this section is to show; (1) that the voltage control area that experiences voltage or voltage problems can be predicted based on a base case loadflow, where the system is strong PQ controllable and PV controllable. The voltage control area VCA I, predicted based on base case (Case 1), ex- periences severe voltage problems as the constant power reactive load at Bus 11 increases; (2) that increasing the reactive load in VCA 1 causes generator Bus 6 and 10 in VCA 1 to turn from PV to P0 buses as these generators pick up the reactive load increase at Bus 11 in VCA 1 and finally reach their reactive generation limits. Severe voltage problems begin to occur after Buses 6 and 10 become PQ buses and the reactive power required to meet the reactive load increase at Bus 11 in VCA 1, must flow over the weak transmission boundary of VCA 1. (3) how the sensitivity matrices (a) change as Buses 6 and 10 become PQ buses and (b) indicate that the vulnerability to voltage problems increase. The sensitivity matrices SQLV, SVE’ and SQGQL are shown to cleary indicate the vulnerability of the buses in VCA 1 to vol- tage problems. A voltage profile for the base case (Case 1) and the cases where there is reactive load of 400, 600, and 900 MVAR are shown in Figures 6.2(a), (b), (c), and (d) respectively. We notice that if reactive loads at Bus 11 do not exceed the capabilities of the local Generating Units 6 and 10 (Figures 6,2(a) and (b)) the voltages in VCA 1 are slightly 130 2 19 (.0 1 05 25 1.05 1.00 1.04 1.05 1.01 - 8 _L 7 1.01 11_ 10 VCA 1 Figure 6.2(a) Voltage magnitude profile for the base case (Case 1) 131 Figure 6.2(b) Voltage magnitude profile for the case when the reactive load at Bus 11 is 400 MVAR 132 £1.05 29" " 2 ...; O 0 fi 01 O ... O U! N G.) 105 25 1.05 1.03 245 1.03 1.04 Figure 6.2(c) Voltage magnitude profile for the case when the reactive load at Bus 11 is 600 MVAR 133 1 05 26 1.04 1.02 1.04 Figure 6.2(d) Voltage magnitude profile for the case when the reactive load at Bus 11 is 900 MVAR 134 affected. But as the PV Buses 6 and 10 turn into PQ buses and the VCA 1 relies heavily on the weak transmission boundary, the voltage in VCA 1 decrease sharply below acceptable levels (Figure 6.2(c) and (d)). The voltages elsewhere in the system are not affected or they are slightly affected showing the local nature of voltage and voltage collapse prob- lems. Note also, that VCA 1 that experiences voltage problems as reac- tive load is increased, is accurately predicted from base case (Case 1), where no voltage problems exist and the system is strong PQ controllable. The SQGQL matrix has a row for each PV bus and a column for every PQ bus in the system as shown in Table 6.3, which corresponds to the base case (Case 1). The magnitude of elements in any column indicate the relative importance of each PV bus in picking up load increase at a P0 bus. Note the Generators (PV buses) 6 and 10 will pick up almost all of the load at P0 Buses 4, 5, 7, 8, 11, 13, and 14 in VCA 1 as well at Bus 12 and Bus 9. The rows sums of SQGQL indicate the relative importance of PV buses in providing reactive support to P0 buses. Note that PV Buses 6 and 10 have by far the largest row sums for the base case (Case 1). SQGQL is given in Table 6.4 and 6.5 for reactive load levels of 400 and 600 MVAR connected at Bus 11, respectively. Note that when the load increased to 400 MVAR, the elements of the SQGQL matrix, the row sums of SQGQL and the columns sums of SQGQL almost did not change at all. How- ever, when the reactive load at Bus 11 was further increased to 600 MVAR and Buses 6 and 10 became PQ buses, SQGQL changed radically. When the PV Buses 6 and 10 become PQ buses, the rows of SQGQL corresponding to the PV Buses 6 and 10 disappear and columns corresponding to the PQ buses 6 and 10 appear in SQGQL. The elements in SQGQL as shown in Table 6.5, are very different and indicate that the PQ buses 4, 5, 6, 7, 8, 10, 11, 4.. A. mmmuv m... «we. mg. Low o om xwgume xap>wuwmeom ugh m.o up... .>au.~huu.nu. x~8h¢t U»... as» a. 8.. but. .8. 885.60 h... 0:» a. ...... x~¢pcx .00.. 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This causes the significant voltage drop as shown in Figure 6.2 (d) when load at Bus 11 is increased to 900 MVAR. The elements of SQGQL for the elements in Rows 2, 19, 22 associated with the P0 buses within the boundary of VCA 1 further increase as the load at Bus 11 increased to 900 MVAR, and the column sum for the PQ Buses 4, 5, 6, 7, 8, 10, 11, 12, 13, and 14 increase to above 2.3. This indicates that for any load increase in VCA 1 at Bus i, the percentage of the reactive losses required to provide this load increase equals the column sum for this PQ Bus i minus one. Thus, the reactive losses become so large compared to the load change that the reactive reserves at PV buses in the adjoining voltage control areas would be quickly exhausted, causing these PV buses to become PQ buses further aggravating the voltage collapse problem. The row sums for SQGQL at Buses 2, 19, and 22 in Table 6.5 increase significatnly indicating the stress on the system. The matrix SQ v'1 is given in Tables 6.6, 6.7 and 6.8 for base case (Case 1) and whenLthe load at Bus 11 is 400 and 600 MVAR, respectively. Note that when the load is 600 MVAR, there are two additional rows and columns of SQLV'1 since Buses 6 and 10 have changed from PV buses to P0 buses. We note that as the load increases from 0 to 400 MVAR, there are no significant changes in the elements of SQLV-l’ but as the load at Bus 11 increases to 600 MVAR, the row and column elements associated with P0 buses 4, 5, 7, 8, 11, 12, 13, and 14 within the boundary of VCA I, increase significantly (by a factor of 200 in some cases). The sensi- tivity of all PQ bus voltages in VCA 1 to any reactive load change thus 139 m... as» to» .. ...... ...... ...... ....H. >4. ...... . ...... ...... ...... ...... .... . ...... A. mmmuv m... m x.g..E xum>.u.mcmm ugh ...... -..... ...... ...... 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The matrix SVE has a row for each PQ bus and a column for each PV bus as shown in Table 6.9, which corresponds to the base case (Case 1). The elements in any row of SVE indicate the relative importance of each PV bus voltage setpoint in determining the voltage at the PQ bus asso- ciated with that row. The row sums fo SVE are close to one for a system that is strong PQ controllable and has no voltage problems. Also note that the bus voltages at the PQ Buses 4, 5, 7, 8, 11, 12, 13, and 14 within the boundary of VGA 1 depend heavily on the bus voltages at Generators 6 and 10 based on the relative magnitude of elements in the rows of SVE for these buses. The elements and row sums of SVE changed very little when the load at Bus 11 increased to 400 MVAR as can be seen in Table 6.10, because the Buses 6 and 10 were still PV buses. The elements, and row sums of SVE changed radically when the reactive load at Bus 11 increased to 600 MVAR since Buses 6 and 10 become PQ buses as shown in Table 6.11. Comparison of the matrices SVE in Tables 6.9 and 6.l0 which correspond to the base case and the case when the reactive load at Bus 11 is 400 MVAR, shows only insignificant changes. However, the comparison of the SVE matrices for the case when the reactive load at Bus 11 is 400 MVAR (Table 6.10), and for the case when the reactive load at Bus 11 increased to 600 MVAR (Table 6.11) shows that the columns associated with PV Buses 6 and 10 disappear and rows associated with the new PQ buses appear. The elements in rows corresponding to Buses 4, 5, 6, 7, 8, 10, 11, 12, 13, and 14 within the boundary of VCA 1 now have large values for PV Buses 2, 19 and 22. ' Thus, the whole voltage control area's voltage depends on PV buses outside the voltage control area. 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As load at Bus 11 increased to 900 MVAR, the row sums of SVE increased to above 1.9 indicating the stress on the weak boundary of VCA 1 due to the large reactive transfer from the PV Buses 2, 19, and 22 to Bus 11 across the boundary. The increased stress on the boundary is reflected in the increased row sums of SVE for buses in VCA 1 and for elements in the columns corresponding to PV buses 2, 19, and 22 associated with the PQ buses within the boundary of VCA 1. This is a further evidence of the voltage problems and the ultimate voltage collapse. The SQGE matrix is given in Tables 6.12, 6.13 and 6.14 for base case (Case 1) and for the cases when the reactive load at Bus 11 is equal to 400 and 600 MVAR, respectively. The off diagonal elements are negative and the diagonal elements are positive in all three tables indicating that if voltage is increased at a PV Bus i, the reactive generation at the PV Bus 1 increases but the reactive generation at all other PV buses decrease. The off diagonal elements in Column i of SQGE that have large absolute magnitudes indicate the PV buses that back off reactive genera- tion when the voltage setpoint at Bus i increases. Hopefully, there will be several PV buses that will significantly reduce generation when the setpoint voltage and reactive generation at Bus i increases, if there is good voltage redundant and robust control of how reactive power is sup- plied to the network. Furthermore, if the column sums are positive, the reactive generation increase at Bus i exceeds the total reactive genera- tion decrease at other PV buses indicating good local control of how reactive power is supplied. The columns sums of SQ E as shown in Tables G 147 00N0.0I m~ zu~=x a. m>-¢¢U8 hnot uzp 08:. 88.4.0 u>~h¢°u8 0 us. Hanan 00on.nl 0N nNN0.NI nN ~.00.~i nN 00.~.0 NN N0.n..l 0N ~000.0I 0~ non0.NI 0~ 00N0.0I 0 n.0~.~i N N 08:. Ilaaou U... 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N0~...I 0-0.NI 00-.~I nN00.Ni 0N 0000.0! .00~.~I 00n~.~n 0~00.0-I 0000..0~ 0000.. ~00~.N~| 0N00.0n 0N.0.NI ~..0.0| NN 000... 0.00.. .00... .00... n~00..~ 0000.0~t 0000.. 000... .00... .N . nn00.N| 0000.Ni 0~...0I 0~n~.N~I N000.-| 00-.0- ~000.n| 800.... n~n>~¢8¢ >--00bn u..-.. ~nN0.~i Nn...~l n~ zu~zx a. u>~b09ua ~0.t map at.» 88.~.u u0-¢uu8 0 U80 unfit» n~0n.nl 0N -N0.Na 0N 00~0.~I 0N .0.~.0 NN 0~0n.0n 0N N00...I 0~ ~0N0.n| 0~ 0000.00 0 n~n~.~| N w m<>x 0.0 m_ - was an uuo— w>~uummg may ems: mmmu use go~ m cm xpguus ~u~>pupmcmm mg» m~.m «pamh 02.. 88.... U... U8» 0-0..I 0~00.0i n.0n.Nl n0N0..~I .-..NI N000.nl ...... 00N0.0I 0.00..- N x~¢~0¢ no.” Nth 8089.00 n~n>~¢8¢ >--0¢~n u0¢~090 149 m<>z COD mw fig mam um two— fl>puumws use ems: wmmu mnu Law NN a... a we: 00\0N\00 00\0N\.0 .mpga .u~¢a ~n0n.0~ 0~N~.n~| 0~.0.~I 00N~.Nu 00.0.. 0000.Ni ~00~.nl 0N N~00.0I N-0.N~I 0000.00 nNnn.~n 0000.N| 00.0.. 0N00.Ni ~nno.N0u n~ zu-zx u. u.-¢.u: hm.l mzh 08.0 It.~.u u>~hcow8 ~ 000n.~i mug ~..~.NI 0000.Ni 0000.0N~I 800.... n~n>~¢l¢ >--0¢bn wad—a.) £000.00 0~m>~08¢ >--00~n v.0»... ml. UBUSb 000~.n| 0N 0.0N.nl 0N 00~0.NI 0N n~00.0l NN N000..| .N .000.NI 0~ N~00..I N m xptuae xgvsvupmcam ugh ¢~.m apnah 02.0 8:.~.u u... nah 0000.N| 0-0.Na 00~0.0i 00N0.n~o 00.0.-I n00»..0~ n000.-l 0~ n0N0.nI ~0~0.0~I 0~0n.nc 000~.0n .00... N0nn.-l 00MN.~0~ x~0~¢l a... map 150 6.12 and 6.13 are all negative except for PV Bus 22. Thus, the system is PV stable and remains PV stable as long as the load at Bus 11 is not greater than 400 MVAR. All column sums of SQGE for the case of 600 MVAR reactive load at Bus 11 are negative as shown in Table 6.13. This result is due to the fact that the 30 Bus New England System has only high voltage transmission with significant line charging. The significant reactive support due to line charging doens't allow for local control of reactive support which is desired for strong PV controllability. 6.2.2 Experiment 11 The purpose of this experiment is to show (a) both the beneficial and harmful effects of switchable shunt capacitors and (b) that the sensitivity matrices can detect vulnerability to low voltage and high voltage problems. The experiment is performed for the base case as Case 5 where Generators 6 and 10 are removed and the real generation is redispatched to the swing bus, Bus 30, and the Generators 2, 25, and 29. The line connecting Buses 9 and 30 is also removed. The voltages at the buses in VCA 1 are reduced to 0.9 pu. Then a switchable shunt capacitor is connected at Bus 11 and its reactive susceptance in pu. is increased from 0 to 10.97 after which the loadflow failed to convergence. As the shunt susceptance increases from zero to 10.97 the effects on: (1) the voltages in VCA 1 and in the rest of the system, (2) the reactive generation of the PV buses 2, 19, 20, 22, 23, 25, 29 and the swing bus, Bus 30, (3) the reactive losses on the weak transmission boundary lines (4, 3) and (14, 15) of VCA 1 and the reactive flows across these lines and into VCA 1, 12 (4) (5) (6) (7) (8) (9) are 151 the form and elements of the sensitivity matrix SQLV'1 relating voltages at P0 buses in VCA 1 to reactive load injections at PO buses in the same voltage control area, the form and row sums of SVE corresponding to PO buses in VCA 1, P0 stability and controllability, the form and column sums of SQGQL corresponding to the PQ buses in VCA 1 that reflect increases in reactive losses in the system, the form and column sums of SQGE PV stability and controllability, studied in that order. The beneficial and deleterious effects of capacitors become evident in part from this experiment. Furthermore, it is shown that the sensitivity matrices and their appropriate row and column sums can detect and measure relative vulnerability to both low voltage and high voltage problems. (1) (2) The results of the experiment are now discussed. The voltages at buses in VCA 1 increase monotonically as shown in Tables 6.15(a)-(h) as the capacitive susceptance at Bus 11 in- creases. The voltages increase from the unacceptable low values in base case (Case 5) due to the real and reactive transfer caused by the loss of the Generators 6 and 10 and the outage of the line connecting Buses 9 and 30. The voltages reach unacceptable high values as the capacitive susceptance approaches 10.97 pu. The Generator Buses 2, 19, 25 change from PV buses to PO buses as the system changes from Case 1 to Case 5 because they provide the reactive generation required to support the lost reactive generation at Buses 6 and 10 in base case (Case 5). As the capacitive suscep- tance at Bus 11 increases, some of the reactive generation support 9917992 97691117? 99617919 999996! 99126196 909 9676 29119N9 1 1 9191296 2 2 9126996 I I I276 6 6 927N276 9 9 9I1976 6 6 9996flfl6 7 7 97163996 I I IIHU 9 9 9999 19 19 9921 11 11 969019999 12 12 9 '1 1I 13 9 999 16 16 99919I6 19 19 I 769 16 16 979291199 17 17 97'91 19 19 99991 19 19 99'91 29 29 999N296 21 21 II6LPH6 22 22 979276 2I 2 999276 26 26 9792176 29 29 9196Hfl6 26 26 9192176 27 27 992291199 29 29 91229119. 29 29 912276 99 I9 7'27 Table 6.15(a) “NCOCNONNONNCCOOOOCO......OCN. 006-0-waOH~H0~OOOOOOOOOOOOOOOIO~ O I I I I I I I I 0 I - v o O I I 9 t 6 6 I 9 I 9 I o - I 0 I9 172HI 9977 -19.62 9.99 9.99 9796 -II.9I 9.99 9.99 9179 '66.26 922.99 2.69 9699 '99.92 999 II 96.99 9216 -66.76 9.99 9.99 9299 “66.99 9.29 9.99 9997 '67.99 2II II 96.99 9996 '69.66 922 II 76.99 9162 °69.66 9.99 9.99 9137 -62.99 9.99 9.99 9279 '62 99 9.99 9.99 9167 -62 19 9.99 99.99 9I99 '61.96 9.99 9.99 9699 -99.72 9.99 9.99 9292 '92.66 I29 II 199.99 9997 -6I.29 I29.69 I2.I9 9667 -66.II 9.99 9.99 9669 '67.19 199.99 II.II 9679 -6I.66 9.99 9.99 9919 ~69.96 699.99 191.99 9961 '69.79 276.99 119.99 9919 '61 I2 .99 9.99 9669 -61.92 267.99 96.69 9919 -6I.1I III.69 '92.29 9299 °II.66 226.99 9.99 9992 -II.9I 1I9.99 17.99 9799 '6I.26 291.99 79.99 9997 'I1.99 296.99 27.69 9919 '29.I7 29I.99 9.99 IIII °1I.99 1196.99 299.99 152 O . ... C O 9 9 I I I O I I I I l U. U. a 2' .N O COUCCOOCOOCOCOOICCO V......................... V V. 0"” ~90... 9 .0 C .. C O 9"“ 9"“ h a: O :6. M. ....O “Ow-......CCOOOOCOCC .N......‘IOUOCCOODOCOOCCCCOCO II Loadflow solution for Case 5 9917992 97691117? 99917919 9999969 99’26’96 999 9976 9911 1 1 9161296 2 2 9196996 I I 93276 6 6 9279276 9 9 9I1976 6 6 I 7 7 9716l999 I I 9990 9 9 9999 19 19 9921 11 11 969819.99 12 12 9791 1I 19 99999 16 16 99919I6 19 19 99769 16 16 970991199 17 17 97'": 19 19 99991 19 19 99931 29 29 99992 21 21 9961996 22 22 979276 29 2I 9 9276 26 26 9792176 29 29 26 26 9192176 27 27 992291199 29 29 912291199 29 12276 99 I9 9179279 .999 ”flu-I‘VHI’HFCF................9‘9‘ I i I . C C I C O l I I O I O I 9 O I I O I I O D O I I 9 II 172'! 9199 19.29 9 99 9.99 IIII °II.99 9.99 9.99 969 ~66.II I22.99 2.69 99I9 “97.19 999.99 96.99 9779 '61.96 9.99 9.99 9772 '61.7I 9.29 9.99 9669 .66.29 2II.99 96.99 I66I -66.77 922.99 76.99 9726 '66.79 9.99 9.99 9991 '99.16 9.99 9.99 9969 °99.99 9.99 9.99 I7I9 '99.26 9.99 99.99 9991 °99.I2 9.99 9.99 9969 °96.29 9.99 9.99 9 '99.62 I2I.99 199.99 9969 '66.61 I29.69 I2.II 9969 -69.99 9.99 9.99 9799 '69.II 199.99 II.II 9919 '61.99 9.99 9.99 9919 -6I.21 699.99 199.99 91I9 -6I.96 276.99 119.99 9919 -I9.69 9.99 9.99 9669 099.69 267.99 96.69 9969 .66.I9 I99.69 '92.29 9616 '92.62 226.99 9.99 9226 'I6.67 199.99 17.99 9966 -61.79 291.99 79.99 22 °I9.69 296.99 27.69 .993: '27.97 299.99 9.99 II '19.99 1199.99 299.99 0 0 C. C . 39999999999999!!! “DO: 39???? III- Q C . ... I I I I :0...‘........... I ~ I l 9 9 0 9 O I 9 O 9 I C O 9 cm; c—N .0 U C. 2 ..N 3000 ::::::::=::::::::::::::::::: 9 9' unuwuwwwuuno-wuuwo—ur—rumpus-unwant- ...-......OOOOOOOB0.00.0000... ....................D°........ I I 9 I I O 9 I I I I I I I 0 I 6 O I 0 - o 0 I I I I I I 9 .u...”.fi‘t‘..~........00......“O . 9 I O I I I 9 I j l I I I I I I I I 9 ~ I - I I I I I I I .0....II-.00—OO~OOCCOOOOOOOOOOOOI—. . C I I O I O l D 9 9 O I O O C I I I O I O 9 Q j I I I C O 8 C . N“ U. ... ......O........... 9 9 9 I 9 9 9 C I U 9 I U C C U 9672; 96126196 9.99 9.9999 9. 9.99 9.9999 9. 9.99 9.9999 9. 9.99 9.9999 9. 9.99 9.9999 9. 9.99 9.9999 '9. 9.99 9.9999 9. 9.99 9.9999 9. 9 99 9.9999 9, 9.99 9.9999 9. 9.99 9.9999 9. 9.99 9.9999 9. 9.99 9.9999 9. 9.99 9.9999 9. 9.99 9.9999 9. 9.99 9.9999 9. 9.99 9.9999 9. 9.99 9.9999 9. 99 I 9999 9 99 9.9999 9. 99 9.9999 9. 99 9.9999 9. 99 9.9999 9. 99 9.9999 9. 99 9.9999 '9. .99 9.9999 9. 9.99 9.9999 9. .II I 9999 9. 99 9.9999 '9. 99 9.9999 9. 9972' 99126196 9.99 9.9999 9.9999 9.99 9.999 9.9999 9.99 9.9999 9.9999 9.99 9.9999 9.9999 9.99 9.9999 9.9999 9.99 9.999 '9.9669 9.99 9.9999 9.9999 9.99 9.9999 9.9999 9.99 9.9999 9.9999 9.99 9.9999 9.9999 9.99 9.9999 1.9999 9.99 9.9999 9.9999 9.99 9.9999 9.9999 9.99 9.9999 9.9999 9.99 9.9999 9.9999 9.99 9.9999 9.9999 9.99 9.9999 9.9999 9.99 9.9999 9.9999 99 9.9999 9.9999 99 9.999 9.9999 9 9.9999 9.9999 99 9.9999 9.9999 99 9.9999 9.9999 99 9.9999 9.9999 .99 9.9999 '9.6729 9.99 9.9999 9.9999 9.99 9.9999 9.9999 9 99 9.9999 9.9999 99 9.9999 '9.2699 9.99 9.9999 9 Table 6.16(b) Loadflow solution when 1.00 pu capacitive susceptance is connected at Bus 11 153 9917692 676611177 66617616 6666666 6676- 66’26166 66l26/66 60: 9676 7611696 66 17lfl6 1 1 6161666 6 1.6262 -16.66 6.66 6.66 6.66 6.66 166.66 6.6666 6.66 6.66 6 6666 6 6666 2 2 6 66996 2 1.6271 -62.67 6.66 6.66 666.66 166 66 166 66 1.6666 166.66 6.66 6 6666 6 6666 6 6 61276_ 6 6.6662 -66.66 622.66 2.66 6.66 6.66 66.66 6.6666 6.66 6.66 6 6666 6 6666 6 6 17w216 6 6.66 -66.26 666.66 66.66 9.69 6.96 166.66 6.6666 9.99 6.66 6 6666 6 6666 6 6 011676 6 6.6666 -66 61 6.66 6.66 6.66 6.66 166.66 6.6666 6.66 6.66 6 6666 6 6666 6 6 uncann6 6 9.9679 ~66 17 9.29 9.99 6.66 6.66 166.66 6.6666 6.66 6.66 6 6666 -6 6 7 7 H‘l6l996 6 9.9969 -66.26 266.66 66.66 9.99 6.66 166.66 6.6666 6.96 6.66 6 6666 6 6666 6 6 IIHU 6 6.6676 .66.66 922.66 76.66 6.66 9.69 66.66 6.6696 6.69 6.66 9 6666 6 6666 6 6 6660 6 6.6662 ~66.66 6.66 6.66 6.66 6.99 166.66 6.6666 6.99 9.69 6 6666 6 6666 16 16 66x1 6 6.6617 -66.66 6.66 6.66 6.66 6.66 166.66 6.6666 6.66 6.66 6 6666 6 6666 11 11 666616666 6 9.9699 -66.76 6.66 6.66 6.66 6.66 196 66 6.6666 6.66 6.66 6 6666 6 6666 12 12 9791 6 6.6676 -66.16 6.66 66.66 6.66 6.66 166 66 6.6666 6.66 6.66 6 6666 6 6666 16 16 66666 9 6.6766 -66. 6.66 6.66 6.66 6.66 166.66 6.6666 6.66 6.66 6 6666 6 6666 16 16 66616I6 6 6.6776 -66.66 6.66 6.66 6.66 6.66 66.66 6.6666 6.66 6.66 6 6666 6 6666 16 16 6 760 6 6.6666 -66.26 626.66 196.66 6.66 6.66 66.66 6.6666 6.66 6.66 6 6666 6 6666 16 16 670661166 6 1.6161 066.66 626 66 62.66 6.66 6.66 166.66 6.6666 6.66 6.66 6 6666 6 6666 17 17 979" 6 1.6166 -66.26 6.66 6.66 6.66 6.66 166.66 6.6666 6.66 6.66 6 6666 6 6666 16 16 66cn1 6 1.6666 -66.66 166.66 66.66 6.66 6.66 166.66 6.6666 6.66 6.66 6 6666 6 6666 16 16 66661 2 1.6616 -66.66 6.66 6.66 626.16 166.76 166.66 1.6616 666.66 -666 66 6 6666 6 6666 26 26 666H666 2 6.6616 -61.26 666.66 166.66 666.66 116.66 166.66 6.6616 266.66 -266.66 6 6666 6 6666 21 21 99619I6 6 1.9269 -62.66 276.66 116.66 6.66 6.66 66 66 6.6666 .66 6.99 6 6666 6 6666 22 22 979276 2 1.6616 -67.66 6.66 6.66 666.66 216.77 66.66 1.6616 666.66 ~266 66 6 6666 6 6666 26 26 999276 2 1.6 -67.76 267.66 66.66 666. 6 66.66 99.99 1.6666 666.66 -266 66 6 6666 6 6666 26 26 9792176 6 1.9261 -66.66 666.66 -62.26 6.99 6.66 66.66 6.6666 6.96 6.66 6 6666 6.6666 26 2 6166HI6 2 1.6 6 -61.66 226.66 6.66 766.66 221.26 166.66 1.6 6 666.66 -226.66 6 6666 -6 6726 26 26 9192176 6 1.6666 -66.16 166.66 17.66 6.66 6.66 66.66 6.6666 .66 6.66 6 6666 6 6666 27 27 692991199 6 1.6167 069.96 261.66 76.66 6 66 6.66 166.66 6.6666 6.66 6.66 6 6666 6 6666 26 26 616661166 6 1.6667 -26.17 266.66 27.66 6.66 6.66 166.66 6.6666 6.66 6.66 6 6666 6.6666 26 26 612276 2 1.6616 -26 66 266.66 6.66 1626.66 -16.66 166. 6 1.6 2 .6 -266. 6 6 6666 -6.2666 6 1.6666 -16 66 1166 66 296.66 1662.66 266.67 166.66 6.6666 6.66 6.66 6 6666 6.6666 II II 9179276 0999 Table 6.15(c) Loadflow solution when 3.00 pu capacitive susceptance 15 connected at Bus 11 9917999 976911177 6.617919 999996! .699: ..133166 99126196 999 9676 99119H9 II 172fl9 1 1 9161996 I 1.9I26 '19 96 9.99 9.99 9.99 9.99 199.99 I. 99 9.99 9.9999 I 9999 2 2 9166996 2 1.9176 '11.79 9.99 9 99 699.99 199.99 199.99 1.9699 199 99 9.99 I 9999 9 9999 I I 93276 9 1.9192 662.26 I22.99 2.69 9.99 9.99 199.99 9.9999 9.99 9.99 9 9999 9 9999 6 6 927N276 9 1.9112 '92.69 999.99 96.99 9.99 9.99 199.99 9.9999 9.99 9.99 9 9999 I 9999 9 I 931976 9 1.9699 '99.96 9.99 9.99 9.99 9.99 199.99 9.9999 9.99 9.99 9.9999 9 9999 6 6 9996H66 9 1.9996 '96.99 9.29 9.69 9.99 9.99 199.99 9.9999 9.99 9.99 9 9999 -9 9669 7 7 97166996 9 1.9669 -57.99 299.99 96.99 9.99 9.99 199.99 9.9999 9.99 9.99 9.9999 9 9999 I 9 99"” 9 1.9692 '99.19 922.99 76.99 9.99 9.99 199.99 9.9999 9.99 9.99 I 9999 9 9999 9 9 9960 9 1.9999 -9I.21 9.99 9.99 9.99 9.99 199.99 9.9999 9.99 9.99 9.9999 9 9999 19 19 9911 9 1.9699 696.19 9.99 9.99 9.99 9.99 199.99 9.9999 9.99 9.99 9 9999 9 9999 11 11 969N19999 9 1.9779 -96.97 9 99 9.99 9.99 9.99 199.99 9.9999 9.99 9.99 9 9999 9 9999 12 12 9791 9 1.9976 -96.69 9.99 99.99 9.99 9.99 199.99 9.9999 9.99 9.99 I 9999 I 9999 19 1I 99999 9 1.9616 -§;.1; 9.99 9.99 9.99 9.99 199.99 9.9999 9.99 9.99 I 9999 I 9999 16 16 99919fl6 9 1.9696 “92.92 9.99 9.99 9.99 9.99 199.99 9.9999 9.99 9.99 I 9999 9 9999 19 19 99769 9 1.9292 -67.19 I29 99 199.99 9.99 9.99 199.99 9.9999 9.99 9.99 I 9999 9 9999 16 16 970991199 9 1.9I2I ~69.99 69 I2.I9 9.99 9.99 199.99 9.9999 9 II 9.99 9.9999 9 9999 17 17 979"! 9 1.9299 .62.96 9.99 9.99 9.99 9.99 199.99 9.9999 9 99 9.99 9.9999 9 9999 19 19 992N1 9 1.9299 -62.76 199.99 II.II 9.99 9.99 199.99 9.9999 9.99 9.99 9 9999 9 9999 19 19 99P91 2 1.9919 -II.II 9.99 9.99 629.19 59.77 199.99 1.9919 999.99 'III.II I 9999 I 9999 29 29 0999296 2 9.9919 -69.29 699.99 199.99 999.99 119.96 199.99 6.9919 299.99 0299.99 9.9999 9 9999 21 21 9161996 9 1.9929 .61.99 276.99 119.99 9.99 I 99 199.99 9.9999 9.99 9.99 9.9999 9 9999 2 22 979276 2 1.9919 .II.69 9.99 9.99 699.99 192 99 199.99 1.9919 699.99 '299.99 9.9999 9 9999 29 29 999276 2 1.9669 °II.99 267 99 96.69 9 .69 2 .97 199.99 1.9669 I99.99 '299.99 9.9999 9 9999 26 26 9792176 9 1.9996 .6I.II III 69 692.29 9.99 9.99 199.99 9.9999 9.99 9.99 I 9999 9.9999 29 29 OIHH6 2 1.9 099.96 226 99 9.99 799.99 76.77 199.99 1.9999 I99.99 '229.99 9 9999 -I.6729 26 26 9192176 9 1.9692 '96 69 199 99 17.99 9.99 9.99 199.99 9.9999 9.99 9.99 9 9999 9.9999 27 27 99299119. 9 1.9279 '19.21 291 99 79.99 9.99 9.99 199.99 9.9999 9.99 9.99 9 9999 9.9999 29 29 91299119. 9 1. 2 '29.61 296. 9 27.69 9.99 9.99 199.99 9.9999 9.99 9.99 I 9999 9.9999 29 29 912276 2 1.9919 '26.99 299.99 9.99 1926.99 '91.97 199.99 1.9 19 2 9.9 .299.99 I 9999 '9.2699 ’3: I9 9179976 I 1.9999 '19.99 1196.99 299.99 1696.79 216.I9 199.99 9.9999 9.99 9.99 9 9 9.9999 Table 6.15(d) Loadflow solution when 5.00 pu capacitive susceptance is connected at Bus 11 VIII... .7..111fY .II...3. .....Il «mm In: .... Ifltt.l. x a «Lnnrna z z mu». 3 s HJHYI o s unvntta s s u LOYIA o s -n- 7 7 "1&AIIII a a ouwo . 9 osuu I. to 03:! u :1 common 12 12 are: :3 13 oonno :. to oasxonn I. t! .3710 no no aruvsxtol 17 17 are»: x. 1. one»! 1. x9 ..PSI 20 z. osoncoa z: 21 osn.vua z: z: I7..7. z: z: I... 2. 2. 0791.11 z: 29 noun 2. z. uxoeuta 27 21 userstto. z. 2. cacrsxtou :9 :9 011.1. 3. s. oxtulta Table 6.15(a) “~OOCNCNNONNOCOOOCCCOC...-CON. 3. 172”. L .36. -|.. . .... .... L..... -31..2 .... I... L...2 -.1..3 322... 2... L 1..1 ..1..3 ...... ..... L.1.17 '5..33 .... .... L 1... -..... ..2. .... L 1..2 '..... 233... ..... 13.. -...22 .22... 7.... L.1.I ....2. .... .... L.1..1 '.3..3 .... .... L.1... -.3... .... .... .1.91 -.3... .... ..... .1... 0.2..7 .... .... L.117. -....2 .... .... L....2 ... 32.... 1.3... L...1. -.2... 32.... 32.3. L....2 0.1.77 .... .... 1....2 0.2.21 1..... 3.... L...1. '3..2. .... .... I...1. ...... ...... 1.3... L...2. ...... 27.... 11.... L...1. '36... .... .... L.... 036.2. 2.7... ..... L....3 0.2.7. 3..... -.2.2. L..... -3.... 22.... ... L..3.3 '33..2 13.... 17 .. L...11 '3.... 2.1... 7 ... L.... '27... 2.... 27... L...1. ‘2...2 2.3.. .... L.. '1.... 11..... 2..... Loadflow solution when 7.00 pu 154 .... .... ...... 17.... .... .... ...n .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .2..1. ..1.1. ...... 11.... .... .... ...... ...2. ...... '23..2 .... .... 73.... ..3..1 .... .... .... .... .... .... 1.2.z.. ...... 1....7. 1....1 connected at Bus 11 ......2 ....11177 I..1Y.1. 'I..IAI ..l2.l.. .0. .II. “.0DENFCOCVOUOKNFOCCVOuOuflw ”Nu—unnuwwwu Table 6.15(f) ...V'U5MN-COCVOUOMufl . O O O . UN...”CNN-NNCCCCOCCOCCCOOOCON. 3. 172K. 1....7 '1 .17 ... .... 1....2 '32.31 .... .... 1.1... -.2.3. 322... 2... 1.2237 -....7 ...... ..... 1.2.7. -.3 .. .... .... 1.3111 '93 .3 ..2. .... 1.3.1. °.. 6. 233... ..... 1.2... -.. .. .22... 7.... 1.3.7. -5. .. .... .... 1.3372 '52 6. .... .... 1.36.3 -.2 .6 .... .... 1.33.. '.2 .3 .... .3... 1.3... -.2 .. .... .... 1.2.13 °....2 .... .... 1.11.2 °....7 32.... 1.3... 1...12 -.3.1. 32.... 32.3. 1..... '62.12 .... .... 1...?1 -.2. 1..... 3.... 1...1. '3...2 .... .... ....1. °3...3 ...... 1..... 1...7. -....2 27.... 11.... 1...1. -3...1 .... .... 1..... '3...1 2.7... ..... 1..7.7 -.3..1 3..... -.2.2. 1..... '3.... 22.... .... 1...1. '3...1 13.... 17... 1...1 O...11 2.1... 7.... 1...1. “2.... 2..... 27... 1...1. '2.... 2.3... .... 1..3.. '1.... 11..... 2..... .... .... ...... ..1. .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .2..1. ‘2..... ...... 11..3. .... .... ...... ...... ...... ...... .... .... 73.... '2.3..I .... .... .... .... .... .... 1.2.... .. O. . 1712.12 1. ..3 LII... ...... I... L...II 1....I 1.I.II II... I I... ..II LII... I.IIII I... II... ...... I.II LII... ..IIII I... LII... I.IIII I... LII... I .II. I... LII... ...... I.II LII... I.IIII I... II II I.III. I... LII... ...... I... LII... I.IIII I.II LII... I.IIII .... L.I.II ...... I... LI..II I.IIII I.II LII... I.III. I... I ... ...... I... L...I. 1...1I .II... II... I...1. 2....I LII... I.IIII I... LI.... I.ISI. .II... I ... 1.... 3....I LII... ...... I.II LII... 1.... .II... LII... .....I I... L.I.II ...... I... L...II I.II. I... LII... 1.I.1. ...... L I. I. I... capac1t1ve LII... ...... .... LII... 1....I III... L.I.I. I .... ..II I .II ...... I.II LII... I I... I... L...II ...III I... LI..I. ...... .... LII... ...... I... LII... ...... I... .I II I.III. I... .I... ...... I... II... I.III. I... LII... I.IIII I... . II.II ...... I... LII... ...... I... II... ...... I... L.I.II ...... I.II LII... ...III I... L.I.II 1...}. 3II.II LII... I...|I 29.... II... ...... I... LII... I.ISII .II... II... 1....I ...... L.I..I ...... ..II LII... l..... 3..... II... ...... I... II... ...... ..II L.I.II I.IIII I... LII... 1...1I ...... LI..II I.I I.II ..72| .Illlll. .... I...» 0.0... o... 0.0.00 0.0.0. I... ...... 0.00.. I... 0.0... 0.0... I... 0.0... 0.0... o... 0.0... oo.o¢o. a... 0.0... 0.0... a... 0.0... 0.00.. a... ...-o. ...... o... 0.0000 0.00.. a... 0.0... 7....0 0.00 a...” 0.0... a... 0.0... 0.00.. a... 0.000. ...... 0... 0.0... 0.0... a... 0.0... ...... a... 3.0... 0.0... o... 0.00.. 0.0... -s..... 0.0... ...... -zs..oo 0.0... 0.00.. I... 0.0... 0.0... -....oo 0.0... 0.0... -zs.... 0.0.0. 0.0... I... 0.0... 0.0... -z:..oo ...... -0.o72. .... ...... ...... I... 0.0... 0.0... a... 0.0... 0.0... -.....o ...... -o.ao .... 0.0... .. susceptance is ..72' ..12.l.. .... ...... .... ...... .... ...... .... ...... .... ...... .... ...... .... ...... .... ...... .... ...... .... ...... .... ...... .... ...... .... ..... .... ...... .... ...... .... ...... .... ...... .... ...... °3..... ...... '2..... ...... .... ...... °2.. .. ...... '2..... ...... ... ...... '22.... ...... .... ...... .... ...... .... ...... '23.... ...... .... .. 0§999§9999999999 Loadflow solution when 10.00 pu capacitive susceptance is connected at Bus 11 155 ..17ICI .7II1117Y .I....l. I.II... ..72' I.IIIII. III2./I. "*3": mm . .... .. .z' "m . - . I... .II I... I... II... I.IIII I... I I. I. ...... 2 2 IIII'PI 2 L.I.I3 “32.3. I... 3... ...... I... II... 1....I 1II.II I II I. II ...... 3 3 .127. I .11.. ..2.1. 322... 2 .I I... I... LII I. I.IIII I... I... I.IIII ...... I . IUYNIYI I L.2..2 -.I... .II... II... I... I... I I I.IIII I... I... I.IIII I.IIII . . IJ107I I .33.. -.2..1 I... I... I... I... LII II I.IIII I... I... I.IIII I.IIII . . If.‘ I L.3.13 ’.3.I. ..2. I... I... I... LII II I.IIII I... I... I.IIII -I.I..I 7 7 I t I L.3.1I -...1I 233... II... I... I... LII II I.IIII I... I... I.IIII I.IIII I I II". I L.33II -...3I .22... 7.... I... I... LII I. I.IIII I... I II I.IIII I.IIII . . .... I .3.I1 '...32 I... I... I... I... LII II I.IIII I... I II I.III. I.IIII 1. 1. I.II I L 37.. '.2.I7 I... I... I... I... I II I.IIII I... I II I.IIII I. III 11 11 II.IICIOI I L .II. 0.2..2 I... I... I... I... LII II I.IIII I... I... I.IIII 1...... 12 12 I7?! I L 372. '.2.1I I... .I... I... I... LII II I.IIII I... I... I.IIII I.IIII 13 13 .II”. I L 3... -.1... I... I... I... I... II II I.IIII I... I... I.IIII I.IIII 1. 1. II.1GII I L 272. -...2. I... I... I... I... II I. I.IIII I... I... I.IIII I.IIII 1. 1. I 7.0 I L 13.2 -...31 32.... 1.3... I... I. LII II I.IIII I... I... I.IIII I.IIII 1. 1. 0707311.. I L I.I1 -.2..2 32.... 32.3. I... I... LII... I.IIII I... I... I.IIII I.IIII 17 17 I7?! I L I.1. 0.1..3 I... I... I... I... dII.II I.IIII I... I... I.IIII I.III. 1. 1. IICI! I L 1II7 0.2... 13.... 3.... I... I... LII... I.IIII I... I... I.IIII I.IIII 1. 1. ISIS! 2 I31. °3I.3I I... I... .2..1I -2.I.I. LII... 1...1I .II... -3II.II I.III. I.IIII 2. 2. I.II. 2 H ..1I '3..71 .II... 1.3... .I.... 11.. II. I I...1I 2.I.II '2.I.II I.IIII I.IIII 21 21 I3ILIHI I L 2. .....2 27.... 113... I... I... II... I.IIII I. ..II I.IIII I.IIII 22 22 I7I27I 2 L I.1. '3..21 I... I... ...... -.I.71 II.II 1...1I .II... '2.I.II I.IIII I.IIII 23 2 .IIET. 2 L I '3...1 2.7... II... ...... °122..2 LII... 1.... ...... °2.I.II I.IIII I.IIII 2. 2. .70217I I L II73 -.2.7. .II... ..2.2I I... I... II... I.IIII I... I... I.IIII I.IIII 2. 2. I10.“ 2 7. '3...I 22.... I... 73.... ’22.... LII... 1.I.II .II... '22.... I.IIII -I..72I 2. 2. I1.217. I ...I2 '3..33 13.... 17... I... I... LII... I.IIII I... I... I.IIII I.IIII 27 27 I.I'.11.I I L.I7I. -3...7 2.1... 7.... I... I. I LII... I.IIII I... I... I.IIII I.IIII 2. 2. .1273110l I L I... '2...3 .II 27... I... I... LII... I.IIII I... I... I.IIII I.IIII 2. 2. .1227. 2 L I.1. '2..I. 2.3... I... 1.2.... '11.. LII I 1...1I 2.I.II '2.I II I. I .I.2..I ’3: 3. I17I27I 3 L “1.... 11 .II 2.I.II 171..13 17.... lII I. I... I II I. I.IIII Table 6.15(g) Loadflow solution when 10.50 pu capacitive susceptance is connected at Bus 11 I.1.... .7II1L177 III1731. "U‘III .IYII .IIIIIII IIIIIII. .0. III. P.110I. 3. 172B. 1 I It'll I M I... -1..31 I... I... I... I... LII... I.II. I... I... ...III I.IIII 2 2 I1IIPP. 2 h I... '32.3. .II ...I ...... I II LII... 1... I 1II.II I.II I.IIII I.IIII 3 3 27. I L 13.3 -.1..3 322... 2... I... I... II... I.IIII I... I... I.IIII I.IIII . . .27N27I I L 2... °.I..I .II... ...II I... . C. II... I.IIII I... I... I.IIII I.IIII . . .31... . I -.2.32 I... I... I... I... LII II I.IIII I... I... I.IIII I.IIII . . GIN“. I 3... -.2... ..2. I... I... I... LII II I.IIII I... I... I.III. -I....I 7 7 I71IIII. I 3I.2 “.3... 233... I...I I... I... LII I. ..IIII I... I... I.IIII ...... I I .IfiU I L 3.17 '.3..3 .22... 7.... I... I... LII I. ..IIII I... I... I.IIII I.IIII . I I... I L.3.13 °.3... I... I... I... I... LII I. I.IIII I... I... I.IIII I.IIII 1. LI I.I1 I L..23I -.1... I... I... I... I... LII I. I.IIII I... I... I.IIII I.IIII 11 1 IION1CIII I L... “.1... I... I... I... I... I I I.IIII I... I... I.IIII 1I..7II 12 L2 .7?! I L..17. -.1... I... II... I... I... LII II I.III. I... I... I.IIII I.IIII 13 3 I I”. I .3.I1 -.1.1I I... I... I... I... LII II I.IIII I... I... I.IIII I.IIII 1. . II.1.II I .3.71 -...II I... I... I... I... I I I III. I... I... ...III ..IIII 1. . I...” I L.1.1I -....7 32.... 1.3... I... I... II I. I.IIII I... I... I.IIII I.IIII 1. 6 I7U'.11II I L.1.I1 ~.2..3 ..I 32.3. I... I... LII II I.III. I... I... I.IIII I.IIII 17 7 .7Pfl1 I .1I.2 ..1..I I... I... I... I... LII II I.IIII I... I... I.IIII I.IIII II I IICII I L.11.. -.2.1. 1.:.II 3.... I... I... LII II I.IIII I... I... I.IIII I.IIII 1. 0 ..PS! 2 L.I.12 ‘3I.I2 .II I... .2..1I -3II.II LII... 1...1I 3II.I. -3II.II I.IIII I.IIII 2. VI ..."... 2 H...1I '3...2 .II... 1.3... .I.... 1...I. LII... I...1I 23.... '2.I II I.IIII I.IIII 21 V1 I3ILPN. I L.I.7. ~.I.3. 27.... 11.... I... I... .I... I.IIII I I.IIII I.IIII 22 42 .7.27I 2 L.I.1I ‘33... I... I... ...... '1I7..1 II.II 1...1I .II... '2.I.II I.IIII I.IIII 23 i3 IIETI 2 .I... '3..1. 2.7... I...I ...... '1.I.1. LII... 1....I ...... '2.I.II I.IIII I.IIII 2. V. I7.E17I I .I... -.2..I .II... 0.2.2. I... I... II... I.IIII .II I... I.II. I.IIII 2. d. 1 2 L.II23 '3...1 22.... I... 73.... '22.... LII... 1....I .II... '22.... I.IIII -I..72I 2. J. I1.217. I L.I77. '3..22 13.... 17... I... I... LII... I.IIII I... I... I.IIII I.IIII 27 I7 .....110l I L.II17 '3I.77 2.1... 7.... I... I... LII... I.IIII ... I.II I.IIII I.IIII 2. 2. I1.'.11I. I L..... ’2I.3. 2....I 27... I... I... LII... I.IIII I.I I.II I.IIII ... 2. 1227. 2 L.I.1I '2...1 2.3... I... 1.2.... '1.7.7I lII.II 1...1I 23.... '2.I.II ...... ‘I.2..I .3: 3. I1 I27. 3 L.I3II '1.... 11.. I. 23. I. 1727.11 1.2.32 LII... ... I... I... I. I. Table 6.15(b) Loadflow solution when 10.97 pu capacitive susceptance is connected at Bus 11 156 is provided by the capacitive susceptance and the Buses 2, 19, and 25 become PV buses. When Generators 2, 19, and 25 become PV buses, the need to rely on reactive generation from Generators 20, 22, 23 and 29 will be shown based on analysis on S and S to be re- duced. As the capacitive susceptance further increases and ap- proaches 10.97 pu., the reactive output of the generators at the Buses 2, 19, and 25 reaches their lower limit Q and the PV buses min 2, 19 and 25 again become PQ buses. Further reduction in reactive generation at generators outside VCA 1 must come from Generators 20, 22, 23 and 29. The above observations can be made from Tables 6.15(a)-(h) as well as the analysis of SVE and SQGQL that follow. The reactive flows in VCA 1 across the weak transmission boundary lines (4, 3) and (14, 15) for the different values of capacitance at Bus 11 are shown in Table 6.16. It is clear that reactive power flows into the VCA 1 originally, but as the capacitance further increases, reactive power flows from VCA 1 back into the system (negative flows). The reactive losses on the weak transmission lines (4, 3) and (14, 15) follow a different pattern as can be seen in Table 6.16. They decrease monotonically up to a certain point and then they start increasing. The minimum appears to be at a point between 5.00 and 6.00 pu. of capacitive susceptance at Bus 11. We also observe that the voltages in VCA 1 are normal for this amount of capacitive susceptance. This observation is very impor- '1 and the structure tant because as we will see, the entries of SQ V L and row sums of SVE as well as the structure and column sums of SQ Q will confirm that the system is most secure when the shunt G L susceptance at Bus 11 is between 5.00 and 6.00 pu. The analysis of 157 Capacitor Reactive Losses Reactive Flows on Boundary in VCAl 0 2.2727 1.3130 0.6553 2.1749 100 2.0311 0.8556 0.5097 1.4751 200 1.8555 0.3211 0.4184 0.7435 300 1.7459 -0.3356 0.3759 0.0856 400 1.6676 -1.0804 0.3572 -0.6546 500 1.6189 -1.9260 0.3662 -1.4893 600 1.6089 -2.8846 0.4088 -2.4335 700 1.6473 -3.9843 0.4883 -3.5146 800 1.7488 -5.2461 0.6033 -4.6311 900 1.9168 -6.6808 0.7690 -5.9203 1000 2.1485 -8.3454 1.0108 -7.5243 1050 2.2864 -9.3181 101944 -806149 1080 2.3829 -9.9671 1.3299 -9.3760 1095 2.4368 -10.3071 1.4026 -9.7738 1096 2.4392 -10.3298 10‘073 -907980 1097 2.4448 -10.3548 1.4116 -9.8202 Table 6.16 Reactive losses on the weak boundary and reactive flows in VCA 1 for different values of capacitive susceptance placed at Bus 11 (4) 158 these sensitivity matrices also indicates that the system's vul- nerability to voltage problems increases as the shunt capacitive susceptance at Bus 11 approaches 10.97 or as it approaches zero, moving away from a nominal level between 5.00 and 6.00 pu. where the system is most secure. The sensitivity matrices do not indicate whether the voltage problems are due to low voltages or high vol- tages. The matrices 04"1 ( = SQLV-l) corresponding to 0.00 1.00, 3.00, 5.00, 7.00, 10.00, 10.50, and 10.97 pu. capacitive susceptance at Bus 11, are shown in Tables 6.17-6.24 respectively. It can be verified that the elements of these matrices are always nonnegative. -1 LV lie within the boundary of VGA 1, indicate how sensitive their The entries of SQ associated with the Buses 4-8, and 10-14, that voltages are to load changes at buses in VGA 1. The magnitude of 1 decrease and then increase the elements in this submatrix of SQLV' as the capacitance at Bus 11 increases. This result implies that the vulnerability of the system to voltage problems first decrease and then increase as the capacitive susceptance increases. Note that rows and columns for Buses 2, 19, and 25 are present when the capacitive susceptance is small and when it is near 10.97 pu but they are not present when the capacitive susceptance is near 5.00- 6.00 pu. Note also, that when the capacitive susceptance is near 5.00-6.00 pu, the voltages are near normal, the reactive losses on ‘1 in the branches (4, 3) and (14, 15) are minimum, and elements SQ V L subblock associated with buses within the boundary of VCA 1 are minimum. 159 1. 9.02 III2.II. II!!! .017... .YIIILXYY II.17.1. IIOOIAI TI! I. 1...... H.7I1! . .273333217.‘1133222271.7. .117.7‘3.3'.23‘33. ..33327' VIo01.000.11.1111111111311°. 1°.......III............... I0.......I.....o....'..... 2’02723‘323 .3.7‘...zz‘z’, 111...!‘31‘. ""‘77777‘723 .......................... .......................... .......................... .......................... .73.3'.32.7...1I3I““11. I73 72.13.221.3723.3.4. .. 3.7 31.112...1...11111111111111 oooooooooooooooooooooooooo A.02.2...‘.7.3‘3..,,,zz,1.“ .I371.21.7.’...’.,’,.’.., .00111......I..'......'°.'. z.o0.000....0.............. oo........................ 3°..1“37z3..z..2.'....‘7” 33.3. 3211.7..11111111...‘ 7ooz31...11..111111111111'. 2°OOOOIOIO......o.....ooo'. o0...I.....o.............. .7117.....1...211333221231 213.2.00211112222222222222 .o1oooooooooo...°°°°'°0.... 2°0......o.........o....... .......................... 7 130.1..32.:‘7".3,s..“3“. 02..7.322...3......... 7 I nu.ooo1ooo1...111111111111z 30°°°°°°°°....°°°.o....... ..7.3.3z1...7‘5‘321 1111111111 311.23....133..7.77.3IUUI“ .21 3.....1..72 ....”7 . .1... 1...11113‘, “““u ,“zII. IIIIII.IIIIIIIII II II. IIIIIIIIIIIIIIIIIIIIIIIII C .................'........ 311.23....133.””.17.3I u.” .21 3....31..7 .3..”7 . .1I1 1...11113.53.2““ 3 ”I- IIIIIIIIIIIIIIIII1 II .J ...... I I I I I I I I I I I I I I I I I I I II.II...II.IIIIIIIIIIIIIII .3.3.3...3333,,”””. .II. II.......................... IIIIIIIIIIIIIIIIIIIIIIIII ’.‘.,1‘.3".1‘s‘23”’7’ . 320.2’3.‘ .."“733332.’ 11.1.1III11113. ......33 1.I.......I...I.I..I...I. IIIIIIIIIIIIIIIIIII 0 I C I I I I I....I......IIIIIII.....II 212 I1.. I1.1 I.7 1.I 13. 11. 2.3 .1. .3. .2. I..7 I..2 I.37 .337 I... I.2I I.12 I.11 I22. oooooooooooooooooooooooooo 33.“2I113.3.2..7“111“... . 717 7.5‘.’..,," ... ’71 I. ...0.09.01.12112112221111 I 1.......I...I...II..II.IIII .......................... 31......3.’.71.32777.3.237 31......II..I111111111.I 3 .I......ODII111111111111. . 1...........o.............. ”fl7..MW.I7...321I.I7...321 IIIIIIIIIIIIIIIIIIIIIIIIII ................. ....... O I I I O O O C O C I I I O O C I O . ........'................. .21 1.. .II 1.. ... I.1 I33 172 12. II2 11. 1.. 21. 22. 221 21. 23. 23. 237 23. 23. 22. 2.. 1.3 1.. ”.....111713 .31.““”.z.... 20.2....ZI7 71., “231.. .101I3...111133“3““‘ ..113 ......o.......... ....... IIIIIIIIIIIIIIIIIIIIIIIIII I....1.3.1II13.7..3.”.712“ .2..2....3I. .II 12. .1.1I1...1111”.33“u”“.“‘211 I.II.....I..I...II.I...I.I IIIIIIIIIIIIIIIIIIIIIIII uuuuuuusuuu.suun..uxuuuuu .1.1.1I..u111”“.3.””.”3.211 I......................... 7.3.321I.I7...321 3133..--.. 1. 1. m “721! 1. ~22... “20.71.. 7.2 III1III IIIIIUILI. II I. 1. 1 for the case where 0.00 pu V L capacitive susceptance is placed at Bus 11 ty matrix SQ ivi Table 6.17 The sensit 160 H I... 3.72: III2.II. .311... 371.11177 IIILYSI3 PROGRAM TN! 0. 1NVEISE "17.11 ......................... 3.332.333.1333 3111111111133 17 .II. I.1. .1.1 ..7. ..77 I... ..3. 12. 13. I71 II2 IIIIIIIIIIIIIIIIIIIIIIIII 3733333233333331333333 3: 31.7333327333333333‘33 I... 31333333113311111111111111 13333333333333333.33333333 IIIIIIIIIIIIIIIIIIIIIIIII 3373333331233333777“33373 23212333333333333‘3 3332: 13333333333333333333333333 23333333333333333333333333 ......................... ooooooooooooooooooooooo 33.333333333333333333333Q 32333337 3337 221333222337 M ......................... 33 3.733323313137333113337 8326032773.77I77IIIII7I.I .00111......0......II...II 20......0.....I....I.III.I ......................... ...2..7.2..72...111I...2. 32323 62333333333333.3333. 73311333113333333111.33333 2333333330.333333333333333. 333333333333333333333.333 37333333739.733333333333351 2133:33113111111:22111222 3313333333333333333333.333 2333333333.333333333333333 ......................... a.133337333333332333123133 1.23 38.2133723303333333731 3330.3133133311111111111123 733333333333333333333333333 ......................... .1.7...321..I7...321 .L-‘qs111111111 33333333171u7 7 3337733““ 31.717..13. 3 3 1113““. 3133313311313... 3333 111 333.333.333.333 333333333 I I I O IIIII O IIIIIIIIIIIII O 3 3333333333 33333333333333. “33733““33..231“..3313333 13717 133“13 773137333 713331331131 .3 .33u333z11 33333333333333 333 333333 IIIIIIIIIIIIIIIIII I I I I I I 3 33333333333333.3333333333 3373333333 3373”333733333 13717331333133 3373373.. 3.1333133113133333 3333311 nnnnnnnnnnnnnnnnnnnnnnnnn ..7...“.I.3I7.““3.733I.3 1.717. 1...133 .7337... .13331II11313.....“...3211 33333333333333331 3333333 ooooooooooooooooooooooooo 373233373337 33333333133”, 21371733333. .3u3‘3‘3z‘3 2 31333133113133. 3.“.‘33111 13.333333333333333 3333333 ......................... 33‘ 337 33.312332233337333, 31,7 17 33137333333333333. 21.331331131333333 333.3111 133333333333333333333.3333 ......................... 3731323...I.....377333233. 213737.333Ifi333311113333z 313.31.311.13.........3111 1333333333333033333333333. ......................... ‘1 ......................... .1I7232313237..7333I31331 .1...7.I.3...7..777....77 33333333333111111111111333 13...I......3..I..33333333 ......................... fl 733332133 37 333321 11133111 11213.71..7112“.3333 232. 12333322337333 .3533 37... 1333313313331111111 111113 IIIIIIIIIIIIIIIIIIIIIIIII 32.37.3.133..317 22177377 22.33.22.373.33.II.3337. 22333133133311111111111122 3333333333333333333333333 ooooooooooooooooooooooooo 371.33113233333333.333337 3137133333333u3.”37“77333 31333133113133 33 33 33111 33333.3333333333333333333 IIIIIIIIIIIIIIIIIIII O 3 I I I 3333333333333333333333333 733 137 3333213337 333331 .5222asz111111111 7.. H.7I13 13 H.322... IIIIIOIY1VI 7'2 RXIIIUH 213333.102 I. I. 13 I...7. L SQ v'1 for the case where 1.00 pu capacitive susceptance is placed at Bus 11 ty matrix ivi Table 6.18 The sensit 161 0017002 57001111Y 08017515 000600" g‘tg: 00/20/0‘ '50; 17 7H! 09 1002052 HAY!!! 50 20 27 20 29 21 10 17 10 15 50 0 0757 0.0002 0.0017 0 0009 0.0015 0 0009 0 0090 0 0027 0 0017 0 0020 20 0 0002 0 0115 0.0010 0.0027 0.0009 0 0002 0 0007 0 0000 0 0009 o 0005 27 0 0019 0 0019 0.0155 0 0079 0.0029 0 0010 0 0057 0 0007 0 0055 0 0057 20 0 0009 0 0027 0.0070 0 0115 0.0015 0 0009 0 0029 0 0059 0 0010 0.0019 29 0 0010 0 0009 0 0050 0.0015 0 0090 0 0050 0 0091 0 0090 0 0059 0 0050 21 0 0009 0 0002 0.0010 0 0009 0 0050 0 0009 0 0025 0 0027 0 0055 0 0055 10 0 0099 0 0007 0.0050 0 0050 0 0091 0 0025 0 0195 0 0009 0 0099 0 0059 17 0 0029 0 0009 0.0000 0 0055 0 0090 0 0020 0 0009 0 0105 0 0059 0 0050 10 0 0019 0 0009 0.0055 0 0010 0 0059 0 0055 0 0099 0 0059 0 0009 0 0000 15 0.0020 0 0005 0.0050 0 0020 0 0050 0 0055 0 0000 0.0059 0 0009 0 0150 19 0.0050 0 0005 0 0095 0 0022 0 0055 0 0052 0 0002 0 0007 0 0005 0 0120 15 0 0055 0 0000 0 0095 0 0025 0 0059 0 0052 0 0000 0 0009 0 0009 0 0120 12 0 0057 0 0000 0 0097 0 0029 0 0050 0 0059 0 0091 0 0075 0 0000 0 0152 11 0 0050 0 0000 0 0090 0 0029 0 0059 0 0052 0 0009 0 0071 0 0009 0 012 10 0 0059 0 0000 0 0095 0.0025 0 0059 0 0052 0 0000 0 0070 0 0009 0 0127 9 0 0059 0 0000 0 0097 0.0029 0 0050 0 0055 0 0095 0 0075 0 0009 0 0129 0 0 0059 0 0000 0 0097 0 0029 0 0059 0 0055 0 0095 0 0075 0 0009 0 0120 7 0 0059 0 0000 0 0097 0 0029 0 0059 0 0055 0 0095 0 0075 0 0009 0 0120 0 0 0050 0 0000 0 0090 0 0029 0 0055 0 0052 0 0091 0 0071 0 0005 0 0129 5 0 0050 0 0000 0 0090 0 0029 0 0056 0 0052 0 0091 0 0071 0 0005 0 0125 9 0 0050 0 0000 0 0099 0.0025 0 0099 0 0050 0 0009 0 0000 0 0050 0 0111 5 0 0000 0 0005 0 0090 0.0021 0 0059 0 0020 0 0099 0 0002 0 0090 0 0000 2 0 0100 0 0002 0 0010 0 0009 0 0015 0 0009 0 0092 0 0020 0.0010 0.0027 1 0 0500 0 0002 0 0017 0 0009 0 0015 0 0009 0 0091 0 0027 0 0017 0 0020 19 15 12 11 10 9 0 7 0 5 50 0 0090 0 0090 0 0051 0 0099 0.0090 0 0052 0.0052 0 0052 0.0051 0 0051 20 0 0005 0 0005 0 0005 0 0005 0.0005 0 0005 0.0005 0 0005 0.0005 0 0005 27 0 0092 0 0095 0 0095 0 0099 0.0095 0 0095 0.0095 0 0095 0.0099 0 0099 20 0 0021 0 0022 0 0025 0 0022 0.0022 0 0025 0.0025 0 0025 0.0022 0 0022 29 0 0052 0 0052 0 0059 0.0052 0.0052 0 0052 0.0052 0 0052 0.0051 0 0051 21 0.0051 0 0051 0 0055 0.0051 0.0051 0 0051 0.0051 0 0051 0.0051 0 0051 10 0 0000 0 0009 0 0009 0 0000 0.0005 0 0090 0.0090 0 0090 0.0000 0 0000 17 0 0005 0 0007 0 0070 0.0000 0.0000 0 0070 0.0070 0 0070 0.0009 0 0009 10 0 0001 0 0002 0 0009 0.0001 0.0002 0 0002 0.0002 0 0002 0.0001 0 0000 15 0 0120 0 0125 0 0150 0.0129 0.0129 0 0122 0.0122 0 0122 0.0121 0 0119 19 0 0275 0 0209 0 0270 0.0209 0.0207 0 0259 0.0259 0 0200 0.0250 0.0255 15 0 0271 0 0559 0 0550 0 0559 0.0599 0 0510 0.0510 0 0512 0.0510 0.0501 12 0 0200 0 0559 0.0025 0 0575 0.0507 0 0595 0.0595 0 0595 0.0595 0 0552 11 0 0200 0 0555 0 0575 0 505 0.050 0 0590 0.0590 0 0550 0.0590 0.05 10 0 0200 0 0595 0 0507 0 0559 0.0575 0 0529 0.0529 0 0551 0.0529 0 0519 9 0.0209 0 0515 0.0597 0 0551 0.0555 0 0055 0.0901 0 0950 0.0507 0 0509 0 0 0209 0 0515 0 0597 0 0551 0.0555 0 0900 0.0900 0 0950 0.0507 0 0509 7 0 0209 0 0510 0 0590 0 0555 0.0559 0 0950 0.0950 0 0957 0.0509 0 0502 0 0 0200 0 0512 0.0595 0 0550 0.0551 0 0509 0.0509 0 0507 0.0500 0 0570 5 0 0250 0 0505 0.0559 0 0557 0.0520 0 0501 0.0501 0 0500 0.0570 0 0570 9 0 0250 0 0290 0.0205 0 0200 0.0259 0 0277 0.0277 0 0270 0.0271 0 0275 5 0 0105 0 0110 0.0110 0 0115 0.0115 0 0121 0.0121 0 0121 0.0119 0 0120 2 0 0090 0 0099 0.0052 0 0051 0.0050 0 0059 0.0059 0 0059 0.0055 0 0055 1 0 0095 0 0090 0.0051 0 0050 0.0099 0 0052 0.0052 0 0052 0.0051 0 0052 9 5 2 1 50 0.0052 0.0005 0.0095 0.0999 20 0.0005 0.0005 0.0002 0.0002 27 0.0092 0.0090 0.0010 0.0019 20 0.0022 0.0020 0.0009 0.0009 29 0.0090 0.0059 0.0010 0.0010 21 0 002 0.0020 0.0009 0.0009 10 0.0007 0.0099 0.0095 0.0099 17 0 0000 0 0002 0.0029 0.0029 10 0.0057 0.0090 0.0010 0.0019 15 0.0110 0.0001 0.0020 0.0020 19 0.0229 0.0100 0.0099 0.0099 15 0.0299 0.0119 0.0052 0.0055 12 0.0200 0.0121 0.0050 0.0050 11 0.0202 0.0119 0.0055 0.0055 10 0.0255 0.0110 0.0055 0.0059 9 0.0201 0.0120 0.0050 0.0059 0 0.0251 0.0120 0.0050 0.0059 7 0.0200 0.0120 0.0050 0.0059 0 0.0275 0.0125 0.0057 0.0057 5 0.0275 0.0129 0.0057 0.0057 9 0.0277 0.0125 0.0057 0.0057 5 0.0120 0.0195 0.0007 0.0000 2 0.0055 0.0005 0.0099 0.0100 1 0.0052 0.0005 0.0090 0. 90 m "7.1! 15 m. ””7192 VII II.INUI 2102000102 Table 6.19 0' 09 15 0.5955 The sensitivity matrix SQ v'1 for the case where 3.00 pu L capacitive susceptance is placed at Bus 11 162 0017000 010011170 00017010 000000" 0070* 00120100 000: 17 tut 00 1000000 H01!!! 00 20 27 20 20 21 10 17 10 10 00 0 0700 0 0002 0.0010 0 0000 0.0010 0 0000 0.0001 0.0020 0.0010 0.0027 20 0.0002 0.0110 0 0010 0 0027 0.0000 0.0002 0.0007 0 0000 0 0000 0.0000 27 0.0010 0.0010 0.0100 0.0070 0.0020 0 0010 0.0000 0.0000 0 0000 0.0007 20 0.0010 0.0027 0.0070 0.0111 0 0010 0 0000 0.0.20 0 0030 0 0017 0.0010 20 0.0010 0.0000 0.0020 0.0010 0.0000 0 0000 0.0001 0.0000 0 0000 0.0000 21 0 0010 0.0002 0 0010 0 0000 0 0020 0 0000 0.0020 0 0027 0 0000 0.0000 10 0.0000 0. 007 0.0007 0 0020 0 0001 0 0020 0.0100 0.0007 0 0000 0.0000 17 0.0000 0.0000 0.0007 0 0000 0 0000 0 0027 0.0007 0 0102 0 0000 0.0007 10 0.0010 0.0000 0 0000 0 0010 0.0000 0 0000 0.0000 0 0000 0 0000 0.0000 10 0.0020 0.0000 0 0000 0 0020 0.0007 0.0000 0.0000 0 0000 0 0007 0.0100 10 0.0000 0.0000 0 0002 0 0022 0 0002 0.0001 0.0070 0 0000 0 0001 0.0122 10 0.0000 0.0000 0 0000 0 0020 0.0002 0 0002 0.0000 0 0007 0.0001 0.0121 12 0.0000 0.0000 0 0000 0 0020 0 0000 0 0000 0.0007 0 0070 0.0000 0.0120 11 0.0000 0.0000 0 0000 0 0020 0 0002 0 0002 0.0000 0 0000 0.0001 0.0120 10 0.0000 0.0000 0 0000 0 0020 0 0002 0 0002 0.0000 0 0007 0.0001 0.0120 0 0.0000 0.0000 0 0000 0 0020 0 0002 0 0002 0.0000 0 0070 0.0002 0.0110 0 0.0000 0.0000 0 0000 0 0020 0 0:0: 0 0002 0. 000 0 0070 0.0002 0.0110 7 0.0000 0.0000 0 0000 0 0020 0 0002 0.0002 0.0000 0 0070 0 0002 0 0110 0 0.0007 0.0000 0 0000 0 0020 0 0002 0 0001 0.0007 0 0000 0.0001 0.0110 0 0.0000 0.0000 0 0000 0 0020 0 0091 0.0001 0.0007 0 0000 0.0000 0.0110 0 0.0000 0.0000 0 0000 0 0022 0 0000 0 0020 0.0000 0 0000 0.0000 0.0107 0 0.0000 0.0000 0 0000 0 0021 0 0000 0 0020 0.0001 0 0001 0.0000 0.0000 2 0.0102 0 0002 0 0010 0 0000 0 0010 0 0000 0.0002 0 0020 0.0010 0.0027 1 0.0000 0.0002 0 0010 0 0000 0 0010 0 0000 0.0001 0 0027 0.0010 0.0027 10 10 1 11 10 0 0 7 0 0 00 0.0000 0.0000 0.0000 0 0000 0.0000 0 0002 0.0002 0.0002 0.0001 0.0001 20 0.0000 0.0000 0.0000 0 0000 0.0000 0 0000 0.0000 0.0000 0.0000 0.0000 .7 0.0001 0.0002 0.0000 0 0002 0.0002 0 0000 0.0000 0.0000 0.0000 0.0000 0 0.0021 0.0021 0.0022 0 0022 0.0022 0 0022 0.0022 0.0022 0.0022 0.0022 .0 0.0000 0 0001 0.0002 0 0000 0.0000 0 0000 0.0000 0.0000 0.0000 0.0000 .1 0.0000 0 0001 0.0002 0 0000 0.0000 0 0000 0.0000 0.0000 0.0000 0.0000 LI 0.0077 0.0001 0.0000 0 0000 0 0002 0 0000 0.0000 0.0000 0.0000 0.0000 17 0.0000 0.0000 0.0000 0 0000 0.0000 0 0007 0 0007 0 0007 0.0000 0.0000 10 0.0000 0.0000 0.0002 0 0000 0.0000 0 0000 0.0000 0.0000 0.0000 0.0000 10 0.0120 0.0110 0.0122 0 0117 0.0110 0 0000 0.0110 0.0110 0.0110 0.0110 L0 0.0202 0.0207 0 0200 0 0202 0.0200 0 200 0.0200 0.0200 0.0200 0.0200 L0 0.0200 0.0010 0.0022 0 0002 0.0010 0 0201 0.0201 0.0200 0 0201 0.0270 2 0.0200 0.0022 0.0001 0 0000 0.0020 0 0007 0.0000 0.0000 0.0000 0.0200 L1 0.0200 0.0000 0.0000 0 0000 0.0020 0 0012 0.0012 0.0010 0.0010 0.0000 L0 0.0200 0.0011 0.0020 0 0020 0.0000 0 0207 0.0207 0.0200 0.0207 0.0200 0 0.0201 0 0200 0.0010 0 0010 0.0000 0 0700 0.0000 0.0007 0.0000 0.0002 0 0.0201 0.0200 0.0010 0 0010 0.0000 0 0000 0.0000 0.0000 0.0000 0.0002 7 0 0202 0.0200 0.0012 0 0010 0.0001 0 0007 0.0007 0.0000 0.0007 0.0001 0 0.0200 0.0200 0.0000 0 0010 0.0200 0 0000 0.0000 0.0000 0.0000 0.0002 0 0.0200 0.0270 0.0000 0 0000 0.0200 0 0000 0.0000 0.0000 0.0002 0.0000 0 0.0210 0.0220 0.0202 0 0200 0.0200 0 0202 0.0200 0.0202 0.0200 0.0200 0 0.0100 0.0100 0.0112 0 0110 0.0100 0 0110 0.0110 0.0110 0.0110 0.0110 2 0.0000 0.0000 0 0001 0 0000 0.0000 0 0002 0.0000 0.0002 0.0002 0.0002 1 0.0000 0.0000 0.0000 0 0000 0.0000 0.0002 0.0002 0.0002 0.0001 0.0001 2 1 00 0.0002 0.0000 0.0007 0.0001 20 0.0000 0.0000 0.0002 0.0002 27 0.0001 0.0000 0.0010 0.0010 20 0 0021 0.0020 0.0000 0 0010 20 0.0007 0.0000 0 0010 0.0010 21 0.0020 0.0020 0.0010 0 0010 10 0.0000 0.0001 0.0000 0.0000 17 0.0000 0.0001 0.0020 0.0020 10 0.0000 0.0000 0.0010 0.0010 10 0.0100 0.0000 0.0020 0.0020 10 0.0210 0.0102 0.0000 0.0000 10 0.0220 0 0100 0.0001 0.0002 12 0.0200 0.0110 0.0000 0.0000 11 0.0200 0.0110 0.0000 0.0000 10 0.0200 0.0111 0.0002 0.0000 0 0.0200 0.0020 0.0000 0.0000 0 0.0200 0.0120 0.0000 0.0000 7 0.0200 0.0120 0.0000 0.0000 0 0.0200 0.0117 0.0000 0.0000 ‘ 0 0.0201 0.0110 0.0000 0.0007 0 0.0200 0.0110 0.0000 0.0007 0 0.0110 0.0100 0.0000 0.0000 2 0.0002 0.0000 0.0000 0.0100 1 0.0002 0.0000 0.0000 0.0000 Tl! HIT!!! 18 Iflllllflo IOIIEIIYIVI I‘lllflfl IUHIII 0' ITEIAYIIIS IIICIII 0N! It'll"! EIGEIVALUI 0' It 18 0.0060 Table 6.20 The sensitivity matrix SQ v'1 for the case where 5.00 pu L capacitive susceptance is placed at Bus 11 163 0017000 37601117' 60617313 PIOOIAI 0672: 00126106 P000 17 700 06 1000030 "0701! 30 20 27 26 26 21 0 17 16 13 30 0 0633 0.0000 0 0000 0 0000 0.0000 0 0000 0 0000 0.0000 0 0000 0 0000 20 0 0000 0.0112 0 0010 0 0026 0 0003 0 0002 0 0006 0.0007 0 0006 0 0006 27 0 0000 0.0010 0 0163 0 0073 0 0026 0 0016 0.0067 0.0030 0 0030 0 0031 26 0 0000 0.0027 0 0073 0 0100 0 0013 0 0000 0 0026 0.0030 0 0016 0 0016 26 0 0000 0.0003 0 0026 0 0016 0.0001 0 0020 0 0033 0.0060 0 0033 0 0030 21 0 0000 0.0002 0 0016 0 0000 0 0020 0 0001 0.0020 0.0026 0 0032 0 0031 10 0 0000 0.0006 0 0067 0 0023 0.0033 0 0020 0.0113 0.0072 0 0030 0 0063 17 0 0000 0 0000 0 0060 0 0031 0 0060 0 0026 0.0072 0.0001 0 0066 0 0060 16 0 0000 0 0006 0 0030 0 0016 0 0033 0 0033 0.0030 0.0067 0 0063 0 0030 13 0 0000 0 0006 0 0032 0 0017 0 0031 0 0031 0 0066 0.0060 0 0060 0 0136 16 0 0000 0 0006 0 0032 0 0017 0 0063 0 0026 0 0036 0.0060 0 0030 0 0102 13 0 0000 0 0006 0 0033 0 0017 0 0062 0 0026 0.0030 0 0030 0 0030 0 0101 12 0 0000 0 0006 0 0036 0 0010 0 0063 0 0026 0.0061 0 0032 0 0031 0 0103 11 0 0000 0 0006 0 0033 0 0017 0 0062 0 0026 0.0060 0 0030 0 0060 0 0000 10 0 0000 0 0006 0 0033 0 0017 0 0062 0 0026 0 0030 0 0030 0 0060 0 0100 0 0 0000 0 0006 0 0036 0 0010 0 0062 0 0023 0 0062 0 0031 0 0060 0 0007 0 0 0000 0 0006 0 0036 0 0010 0 0062 0 0023 0 0062 0 0031 0 0060 0 0007 7 0 0000 0 0006 0 0036 0 0010 0 0062 0 0023 0 0062 0 0031 0 0060 0 0000 6 0 0000 0 0006 0 0033 0 0017 0 0061 0 0023 0 0061 0 0031 0 0060 0 0006 3 0 0000 0 0006 0 0033 0 0017 0 0061 0 0023 0 0061 0 0030 0 0060 0 0003 6 0 0000 0 0006 0 0032 0 0017 0 0030 0 0023 0 0060 0 0060 0 0063 0 0007 3 0 0000 0 0003 0 0027 0 0016 0 0023 0 0016 0 0061 0 0061 0 0027 0 0060 1 0 0600 0 0000 0 0000 0 0000 0 0000 0 0000 0 0000 0 0000 0 0000 0 0000 16 13 12 11 10 0 0 7 6 3 30 0.0000 0 0000 0 0000 0 0000 0.0000 0.0000 0 0000 0 0000 0 0000 0 0000 20 0.0006 0 0006 0 0006 0 0006 0.0006 0.0006 0 0006 0 0006 0 0006 0 0006 27 0.0031 0 0032 0 0033 0 0032 0.0032 0.0032 0 0032 0 00 0 0032 0 0032 26 0 0016 0 0016 0 0017 0 0016 0 0016 0.0017 0 0017 0 0017 0 0016 0 0016 26 0 0062 0 0061 0 0062 0 0061 0.0061 0.0060 0 0060 0 0060 0 0060 0 0060 21 0.0023 0 0023 0 0026 0 0023 0 0023 0.0026 0 0026 0 0026 0 0026 0 0026 10 0 0033 0 0037 0 0030 0 0030 0 0050 0.0060 0 060 0 0060 0 0030 0 0030 17 0 0060 0 0060 0 0030 0 0060 0 0060 0.0060 0 0060 0 0060 0 0060 0 0060 16 0 0060 0 0060 0 0030 0 0060 0 0060 0.0067 0 0067 0 0067 0 0067 0 0067 13 0 0101 0 0000 0 0101 0 0007 0 0000 0.0003 0 0003 0 0003 0 0006 0 0003 16 0 0200 0 0206 0 0207 0 0100 0 0201 0.0103 0 0103 0 0103 0 0102‘ 0 0100 13 0 0206 0.0263 0 0263 0 0267 0 0233 0.0220 0 0220 0 0230 0 0220 0 0223 12 0 0200 0.0263 0 0662 0 0273 0 0260 0.0260 0 0260 0 0260 0 0260 0 0262 11 0 0200 0.0260 0 0273 0 0202 0 0263 0.0233 0 0233 0 0233 0 0233 0 266 10 0 0202 0.0236 0 0260 0 0263 0 0276 0.0261 0 0261 0 0262 0 0262 0 0233 0 0 106 0.0231 0 0230 0 0233 0 0263 0.0600 0 0330 0 0316 0 0270 0 0277 0 0 0106 0.0231 0 0230 0 0233 0 0263 0.0330 0 0330 0 0316 0 0270 0 0277 7 0 106 0.0232 0 0231 0 0236 0 0266 0.0316 0 0316 0 0320 0 0201 0 0276 6 0 0106 0.0230 0 0230 0 0236 0 0263 0.0270 0 0270 0 0200 0 0201 0 0260 3 0 0101 0.0226 0 0262 0.0267 0 0233 0.0276 0 0276 0 0273 0 0260 0 0273 6 0 0172 0.0106 0 0103 0 0102 0 0100 0.0202 0 0202 0 0201 0 0100 0 0200 3 0 0066 0.0070 0 0073 0.0072 0 0071 0.0073 0 0073 0 0073 0 0076 0 0073 1 0 0000 0.0000 0 0000 0 0000 0 0000 0.0000 0 0000 0.0000 0 0000 0 0000 1 30 0.0000 0.0000 0.0606 20 0.0 0.0003 0.0000 27 0.0031 0.0027 0.0000 26 0.0016 0.0016 0.0000 26 0.0037 0.0023 0.0000 21 0 0022 0.0016 0.0000 10 0.0030 0.006 0.0000 17 0.0067 0.0061 0.0000 16 0.0066 0.0027 0.0000 13 0.0006 0.0061 0.0000 16 0.0172 0.0067 0.0000 13 0.0106 0.0072 0.0000 12 0.0106 0.0073 0.0000 11 0.0103 0.0076 0.0000 10 0.0100 0.0073 0.0000 0 0.0206 0.0070 0.0000 0 0.0206 0.0070 0.0000 7 0.0206 0.0070 0.0000 6 0.0200 0.0070 0.0000 3 0.0201 0.0077 0.0000 6 0.0203 0.0077 0.0000 3 0.0076 0.0002 0.0000 1 0.0000 0.0000 0.0300 III "0703! 13 0002000. 00000007102 I6l1lfll IUHICI 0' 1700671003 0000320 70! "Illflll 2100l00102 0' 06 13 0.0033 Tabie 6.21 The sensitivity matrix sQ v'1 for the case where 7.00 pu L capacitive susceptance is placed at Bus 11 164 0017602 37601117Y 6N617313 000000! 0672: 00/26l06 0062 17 1"! 06 1'02032 H0101! 30 20 27 26 26 21 10 17 6 13 30 0 0636 0.0000 0.0000 0 0000 0.0000 0 0000 0.0000 0.0000 0 0000 0 0000 20 0 0000 0 0112 0.0017 0 0026 0.0003 0 0002 0.0006 0.0007 0 0006 0 0006 27 0 0000 0 0010 0 0130 0 0073 0.0023 0 0013 0.0063 0.0037 0.0020 0 0030 26 0 0000 0 0026 0 0071 0 0107 0.0013 0 0000 0.0023 0.0030 0.0013 0 0016 26 0.0000 0 0003 0 0023 0 0013 0 0007 0 0027 0.0032 0.0030 0.0031 0 0060 21 0 0000 0 0002 0 0016 0 0000 0 0027 0 0070 0.0020 0.0023 0.0031 0 0030 10 0 0000 0 0006 0 0066 0 0026 0 0032 0 0020 0.0100 0.0060 0 0030 0 0063 17 0 0000 0 0007 0 0037 0 0030 0 0030 0 0026 0.0060 0.0007 0 0063 0 0066 16 0 0000 0 0006 0 0030 0 0016 0 0031 0 0032 0 0030 0.0063 0 0060 0 0036 13 0 0000 0 0006 0 0031 0 0016 0 0060 0 0030 0.0066 0.0066 0 0037 0 0126 16 0 0000 0 0006 0 0031 0.0016 0.0060 0 0023 0.0033 0.0067 0 0067 0 0006 13 0 0000 0 0006 0 0031 0.0016 0.0060 0 0023 0.0033 0.0067 0 0067 0 0003 12 0 0000 0 0006 0 0032 0 0017 0.0061 0 0023 0.0037 0.0060 0 0060 0 0006 11 0 0000 0 0006 0.0031 0 0017 0.0060 0 0026 0.0036 0.0060 0 0067 0 0001 10 0 0000 0 0006 0 0031 0 0017 0.0060 0 0023 0.0036 0.0067 0 0067 0 0002 0 0 0000 0 0006 0 0032 0 0017 0 3060 0 0026 0.0030 0.0060 0 0066 0 0000 0 0 0000 0 0006 0 0032 0 0017 0 0060 0 0026 0.0030 0.0060 0 0066 0 0000 7 0 0000 0 0006 0.0032 0 0017 0.0063 0 0026 0.0037 0.0060 0 0066 0.0000 6 0 0000 0 0006 0.0031 0 0017 0.0000 0 0026 0.0037 0.0060 0 0066 0 0000 3 0 0000 0 0006 0 0031 0 0017 0.0030 0 0026 0.0037 0.0067 0 0063 0.0000 6 0 0000 0 0006 0 0030 0 0016 0 0036 0 0022 0.0036 0.0066 0 0063 0 0001 3 0 0000 0 0003 0 0026 0 0016 0 0023 0 0016 0.0030 0.0060 0 0027 0.0030 1 0 0600 0 0000 0.0000 0 0000 0 0000 0 0000 0.0000 0.0000 - 0 0000 0 0000 16 13 12 11 10 0 0 7 6 3 30 0.0000 0 0000 0 0000 0 0000 0.0000 0 0000 0 0000 0 0000 0.0000 0.0000 20 0.0006 0 0006 0 0006 0 0006 0.0006 0 0006 0 0006 0 0006 0.0006 0.0006 27 0.0030 0 0030 0 0031 0 0030 0.0030 0 0031 0 0031 0 0031 0.0030 0.0030 26 0.0016 0 0016 0 0016 0 0016 0.0016 0 0016 0 0016 0 0016 0.0016 0.0016 26 0.0060 0 0030 0 0060 0 0030 0.0030 0 00 0 0030 0 0030 0.0030 0.0030 21 0.0026 0 0026 0 0026 0 0026 0.0026 0 0023 0 0023 0 0023 0.0023 0.0023 10 0.0032 0.0036 0 0033 0 0033 0.0036 0 0036 0 0036 0 0036 0.0033 0.0033 17 0.0066 0 0066 0 0067 0.0066 0.0066 0 0067 0 0067 0.0067 0.0066 0.0066 16 0.0066 0 0066 0 0067 0 0063 0.0066 0 0063 0 0063 0.0063 0.0063 0.0066 13 0. 003 0 0001 0 0002 0.0000 0.0000 0 0000 0 0000 0.0000 0.0007 0.0006 16 0.0103 0 0170 0 0100 0 0176 0.0176 0 0160 0 0160 0.0160 0.0160 0.0167 13 0.0170 0 0223 0 0223 0.0212 0.0210 0 0107 0 0107 0.0100 0.0100 0.0103 12 0.0101 0 0223 0 0350 0.0230 0.0227 0 0211 0 0211 0.0212 0.0212 0.02 11 0.0173 0 0212 0 0231 0.0230 0.0223 0 0216 0.0216 0.0217 0.0217 0.0211 10 0.0177 0 0210 0 0227 0.0223 0.0236 0 0207 0.0207 0.0200 0.0200 0.0202 0 0.0171 0 0100 0 0212 0.0217 0.0200 0.0600 0.0276 0.0262 0.0233 0.0236 0 0.0171 0 0100 0 0212 0.0217 0.0200 0.0276 0.0276 0.0262 0.0233 0.0236 7 0.0171 0 0100 0 0213 0.0210 0.0200 0.0261 0.0261 0.0272 0.0237 0.0233 6 0.0170 0 0100 0 0213 0.0210 0.0200 0.0236 0.0236 0.0236 0.0237 0.0220 3 0.0160 0 0106 0 0207 0.0211 0.0202 0.0233 0.0233 0.0232 0.0220 0.0233 6 0.0133 0 0162 0 0160 0 0160 0.0163 0.0176 0.0176 0.0173 0.0173 0.0173 3 0.0062 0 0063 0 0060 0 0067 0.0066 0.0070 0.0070 0.0070 0.0060 0.0060 1 0.0000 0 0000 0 0000 0 0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3 1 30 0 0000 0.0000 0 0603 20 0 0006 0.0003 0 0000 27 0 0020 0.0026 0 0000 26 0 0013 0.0016 0 0000 26 0 0036 0.0023 0 0000 21 0 0022 0.0016 0 0000 10 0 0033 0.0030 0 0000 17 0 0063 0.0060 0 0000 16 0 0062 0 0027 0 0000 13 0 0000 0.0060 0 0000 16 0 0133 0.0066 0 0000 13 0 0163 0.0067 0 0000 12 0 0170 0.0070 0 0000 11 0 0160 0.0060 0 0000 10 0 0166 0.0060 0 0000 0 0 0177 0.0072 0 0000 0 0 0177 0.0072 0 0000 7 0 0177 0.0072 0 0000 6 0 0173 0.0071 0.0000 3 0 0176 0.0071 0.0000 6 0 0170 0.0072 0 0000 3 0 0071 0.0007 0 0000 1 0 0000 0.0000 0.0307 702 “1012 13 “0. ”07102 ”1” ma 0' 112.071” am m I101” amuse: 00 06 13 0.0032 Table 6.22 The sensitivity matrix sQ v'1 for the case where 10.00 pu L capacitive susceptance is placed at Bus 11 165 10 P602 001 26/ 06 0670' 0011630 3760311" 66617313 m 1“ I 1W2I30 ”701! ......................... 006313 6.3333313666332326 113730 23733311111111.5111 71013100110011111111111111 1303330300033333333333333. 213303013633 33 32032333273336 3366 31313133113311111111 10033333333333333333333333 ttttttttttttttttttttttttt 030 006 031 026 033 033 003 030 030 061 063 063 063 067 066 066 066 066 066 066 066 063 060 030 033 ......................... uuuuuuuuuuuuuuuuuuuuuu 3313333323323333333632313 633333333333333333333333 23333333333333.33333333333 3 33 3 ooooooooooooooooooooooooo 71333013300310.3337111333313 263333233633333333333332: 31311133333333.33333333113 33.333.333333333333333333333 ......................... 3107012730363130333221033 2332233333733333333333112 71011100110333113111111111 ......................... 133233“233311223333222333 313331 22.112222222222223.) 331333.3333333333333333333 2333333333333333.333333333 ......................... 62n00326302036310003 360“ 03 3.3360231773330003 021 30311233113111111111111237 30003333003330333003003033 763‘R0JM36WR1W007636321 ------------------------- ......................... 1336333333313n367 720337631 613.33 33.. 3133313311 122 3233111 333333033333333333J330303. 13”6””|6.”033113I3.110 1373 36" 0"“.“10311u1uun3”“3 unu1n1 3333333333333333.33333333 uuuuuuuuuuuuuuuuuuuuuuuuu sun-un«uu.uuiuuuss«znunuu 3 3130”10011“12”333”"2 266111 133333.3333333333333333333 ......................... 603262633632220u336267702 620337633u63011 333303733 1133313311 122333333322111 13333003333333333333333333 3333333 3333333333333333333 733333370330333333333n333 620u37630363331300300 733 2133 133113123633233222111 ......................... 133312623636233733623333“ 323337 63.363333333337373 3133313311312303223322.6211... 13333333333333333333333333 ......................... ------------------------- 103 016 071 037 006 060 062 006 006 000 136 163 166 160 166 166 166 166 166 163 166 130 0100 1111 "”3 373363313037 336331 Mann 7033033u373623766631330u3 2322366 270677773333331 1 1731133311.111111111111337 3333333333333333333333333. cccccccccccccccccccccccc 33 3 36333337333311361 3173333337777773 233113 31131111111111113 oooooooooooooooooooooooo 3333333 2376013333““33““30037133 720337630“ 3 0060 733 6100010011 12 3“,.2222 2111 333333333333 3 3333333333 0073”"“fl06za730 711330323 720“ 3 6 332211 30 13311 12232233333 111 337 3 1373363313037336331 333 33“ ”3111111111 m M1018 13 mo. “67100 in I303” 210006“! N 06 13 0.6311 Table 6.23 The sensitivity matrix SQ v'1 for the case where 10.50 pu L capacitive susceptance is placed at Bus 11 166 10 P600 0670‘ 00126106 0017600 376011177 60617313 PIOQIAI 7H! 06 1000030 06701! 360321333362603 023333333333333 M0 M3 H1 H7 092 7 7.01010 1000000000000300000333.0303 .......................... 06072306336 030“103666222730 72206336060 0307063300737 310101000110111111111111111 1030000000000030000.3000300 .......................... 7733303336701332233333133, 60323620666633043333330.3666 00°0000010003000003.3003... 10000000300003000003.0303... 0.063663236333333 333 3 3 3 1000000000333300300330.3333 20000000003033030.033333033 000033.0033033330333303... ........................ 0.000030033.330033003333333 33033601330276030333122066 33361766323130063777777067 320113000110111111111111123 20000030000003.03030 3000303 .......................... 3306322013160032133366633 6637633200070300000000013 610111000103031111111111111 2000003003000000000.3000003 -------------------------- .......................... nnnnnnnnnnnnnnnnnnnnnnnnnn 0003733“3 1 0.3063030003037320 63360036360300110 111113 7 000112000111112222222222237 313300000003300000003000300 nnnnnnnnnnnnnnnnnnnnnnnnnn 76676271017300600110736361 021030.3333006011036733300 0 7 1010100011112233333333221 1 00000000000000000000000000 ......................... a 00000000000000000000000000 7667 2710173036732203 0210“03333000011077 3““ 1010100011112233333n332. 0030003003 0330330000000 nnnnnnnnnnnnnnnnnnnnnn 30203 7667“27101730367“120366361 0310 03333060011 7 30.3030 010131030111132333 0. 333211 330000033300.03300003333030. 330330330003.30033000333000. 013”...- 133303033033333030030.0333 3. uuuuuuuuuuuuuuuuuuuuuuuuu 232321“””u“30 7310090267 03 uuuuuuuuuuuuuuuuuuuuuuuuu 3636337170333 0210. r303333337 31010 103311113I3663333333 10.3.0.000300300333333.3333. ooooooooooooooooooooo 2" "0 0100 733301u32 32 021060 33 07 31013130011112 ooooooooooooooooooooooooo 3370303033663332211107333 6 0200633662037777777733333 7 61010100011112222322222211 1 1330330003033333.030333... 3. ......................... 377037 6366200717733370300 3 21070036100766766666666212 31000100011111111111111111 1 133000030303000033.0330333 3. ......................... 363”062“62030 7310011130 716 017 003 00000 010011110 00 00030003333311111111111133 1330000300030003.30300030.3. 0. 03003033'000000000033000.0. 3. uunuumn..zmsmaun 1 "037 33633 1 3223300672327 736 3133137 ”33 37660302 33 0000 2 0112000113111111111111 33 0000000033.0.0.0.0.0003033.0.3.03.3. ”0 7131373 03.3“310uu 2 3673 30 000 3201u1 00113111 123133 33000300000030303000 .......................... U" 08“ I!“ 0207 0260 0216 0226 «an». uuunuu mm..su.n «nun .x.-. :...-: mnuuuz. nzzxt 3033330300333 33333 33333 066731 0 3 303 033 2220 021030“3”3“30 ”310“““mflu033 310131330111 12” 333333 3 211 030300330003303030003.3.0.0.0.0. oooooooooooooooooooo 330 “77673“ 360.631 ounusucusz. an unnusswm mmmm 1 1 1 1111 33.0.0.0.3.0.03000030300300.030030 ..... 6.60-urn 066-‘0 ”0783610076 6363210007636321 22 232 ll ill fl'lflfl“!13l.¢flfl.lllfllflwl “.IUIHIIIflIIfll010'0613 0.6727 Table 6.24 The sensitivity matrix SQ v'1 for the case where 10;97 pu L capacitive susceptance is placed at Bus 11 167 (5) Tables 6.25-6.32 show the matrices SVE corresponding to capacitive (6) (7) susceptance values of 0, 1.00, 3.00, 5.00, 7.00, 10.00, 10.50, and 10.97, respectively. The elements of these matrices are nonnegative. The row sums corresponding to buses in VCA 1 are greater than 1.2 for all cases. Note that the row sums of SVE decrease as the capacitive susceptance increases from zero and then increase as the capacitive susceptance approaches 10.97 pu. which indicates that the row sums of SVE measure relative vulnerability to voltage collapse due to low or high voltage problems. For the case where there is no capacitance connected at Bus 11 (Case 5), only the generating 20, 22, 23, and 29 are PV bus; the generating buses 2, 19 and 25 are PQ buses. As we add capacitance at Bus 11 the Buses 2, 19 and 25 turn into PV buses and columns corresponding to these buses appear in SVE' These columns of SVE are large for rows corresponding to P0 buses 4-8, and 10-14 within the boundary of VCA 1 indicating that these buses have more control over the vol- tages at the PQ buses in VCA 1. Note that as the capacitive suscep- tance increases further the columns associated with PV buses 2, 19, and 25 disappear again because these buses become PQ buses. The sensitivity matrix SVE indicates that Generators 20, 22, 23, and 29 determine the voltage at buses within VCA 1 boundary when Generator Buses 2, 19, and 25 are PQ buses. The results on SQLV'1 and SVE for the different values of capacitive susceptance at Bus 11 discussed in (4) and (5), indicate that the system is always PQ controllable. Tables 6.33-6.40 show the matrices SQGQL for the capacitive suscep- tance values 0.00, 1.00, 3.00, 5.00, 7.00, 10.00, 10.50, and 10.97 1653 umumpa m_ mucmaamumzm w>Pppumauu an oo.o mews: mmau on“ Low m> 0N uo_awm=mm use m~.m o_gap \ m>~p¢°u88€8 .OKUNZOI mu x~¢hb~aun¢hm thpaQ» vw—~.. con~.c noon.o anon.e no~c.o v9~c.. -N09.0 ¢N00.0 0va.o ”awe.- noNc.o one... oo—v.o snav.e gnaw.- nnnn.. no~n.a °~Nn.o nan—.9 9095.9 Dean.- oce~.. och—.0 hoc~.o noce.o voo~.o NN new-.0 awn-.9 smnN.e v—en.e nan.. o-n.o hvnn.o annn.a usnn.o cwwn.o on~n.e o~vn.° saun.a onen.o vaoN.. ~vo~.. noc~.¢ ~nv~.9 n~oo.o o-c~.. NonN.o son—.0 OQNOU‘CMN-COQROBCVDN-O HH—Hflflflfldd Xuxbcl w>m Nth EH1»! I1.1?— MH mam an coua_a m? mucmuaoumsm m>wu_umamu an oo.~ mews: mmmu as» com m>m xwgume »u_>_u_mcmm och o~.o opac» 99—99998898 .999N898 nu x9999! 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The elements in columns associated with buses within the boundary of VCA 1 have large elements in the rows associated with Buses 2, 19, and 25 since when these buses are PV buses, they are responsible for adjusting for the reactive load and losses in VGA 1. When the Generator Buses 2, 19, and 25 are PQ buses, which occurs when the capacitive susceptance is near zero and when it approaches 10.97 pu, the Generator Buses 20, 22, 23, and 29 share the reactive load and losses in VCA 1. The column sums of SQGQL associated with P0 buses lying within the VCA 1 boundary are always above 1.2 and decrease as capacitive susceptance increases above zero but then increase as capacitive susceptance increases toward 10.97 pu. Thus, the column sums of SQGQL measure vulnerability to voltage collapse due to both low and high voltage problems. Since the column sums of SQGQL are above 1.2, the system is not strong PV controllable. The row sums of SQGQL indicate which buses are most responsible for providing reac- tive support. Large values for Buses 20, 22, 23, and 29 of these row sums suggest the system is insecure. These row sums are quite large when Buses 2, 19, and 25 are PQ buses but they are even larger for Buses 2 and 25 when they are PV buses, because the system relies so heavily on their support when the voltages in the system are healthy. The large row sums of SQGQL indicate that although the voltage profile is reasonably healthy, the heavy dependence on just two PV buses ( 2 and 25) will make the system 185 susceptible to severe voltage problems if these buses become PQ buses. (8) The matrices SQGE and their column sums for the different capacitance values considered are shown in Tables 6.41-6.48. The column sums of SQGE are originally positive indicating that even though the voltages in VCA 1 are extremely low, the system is PV controllable. This of course was not a healthy situation, but as capacitors are added, the column sums of SQGE decrease and some turn negative indicating that the system is only PV stable. As more capacitance is added, the column sums further decrease and they eventually become negative indicating that PV stability is lost too. The results show that PV stability and controllability are affected more than PQ stability and controllability due to increases in capacitance in the system. This experiment doesn't represent a scenario of power system operation, but rather a documentation of the beneficial and harmful effects of excessive reactive flows due to line charging, capacitors used for power factor correction and switchable shunt capacitors. This experiment showed that a system can be both PV and P0 controllable even for the case of a very undesirable operating point. The capacitive susceptance cannot only improve the voltage profile, but move the system closer to being strongly PQ controllable but further from being PV controllable. Furthermore, excessive line charging, and power factor correction capacitance can cause voltage collapse as shown when the capacitive susceptance increases toward 10.97 pu. Long and underground lines and power factor correction capacitors are switched out for light load conditions to avoid the harmful effects of excessive shunt 186 o H. mam an umuupq m. mucouqmumam m>wuwuuamu an oo.o mean: ammo one co; m cm x_cuws Aa_>_uwm=mm mg» .e.o o_nmh N..h..cs Ot.N. Nto h.¢N. h. th.t Na.h¢ON=tOt N80 N60. no .t:. tttaou Nth ...... 0N ...... .N .9.N.0 NN s99... .N .t:. ttaaou N... Nth 0N Nets ..\.N\0. .Nh¢fl £08058. ...»..t‘ >h.4..¢h. 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In this experiment, we increase the reactive load at Bus 11 and add the same amount of capacitance to compensate for the load increase; keeping the capacitance always 5.00 pu larger than the reactive load in per unit on a 100 MVAR base. This experiment was performed in order to show that one cannot compensate for reactive load increase by connecting shunt capacitive susceptance in the same manner as one would do by using synchronous condensers. If synchronous condensers were connected to Bus 11 and were used to compensate for reactive load in- crease, the voltage profile and the vulnerability of the system to vol- tage problems (as measured by the sensitivity matrices) would not change at all. This would be confirmed based on the security constraint (4.73) derived in Chapter 4. However, if shunt capacitive susceptance is used to compensate for the reactive load increase, the magnitude of the reac- tive load that can be accepted before voltage collapse occurs, is dra- matically increased over the case where no capacitive susceptance is used (Experiment IV). The use of compensating shunt capacitive support for load increase helps maintain a healthy voltage profile and thus helps prevent voltage collapse up to some point. If too much shunt capacitive susceptance is used in compensating for reactive load increase, con- straint (4.73) indicates the capacitance actually helps aggravate voltage 195 collapse in a manner just like reactive load increase does in Experiment I. The deleterious effects of capacitance in this experiment is very different than in Experiment 11 where it causes high voltages. In this experiment, it causes voltage drops by weakening the voltage control area network structure once its beneficial effects of providing local reactive and voltage support are exhausted. A second purpose of this experiment is to document the existence and characteristics of the static bifurcation point that exists when the load and compensating capacitive susceptance are increased. The existence of the static bifurcation point is now addressed. It was found that as the reactive load is increased to 930 MVAR and the capacitance increased to 14.30 pu, the loadflow did not converge but again converged after the reactive load increased above 1410 MVAR and the capacitance increased above 19.10 pu. For reactive load between 930 MVAR and 1410 MVAR to loadflow program experiences problems due to the pre- sence of a bifurcation point. In fact, this can be seen by the loadflow solution tables, Tables 6.49 and 6.50 that show the voltage profiles when reactive load at Bus 11 is 930 and 1410 MVAR, respectively. As the load increases from 0 to 930 MVAR, the voltages increase in VCA 1 and as the load increase beyond 1410 MVAR, the voltages appear very low, about 0.9 pu and increase very slowly. Note, that the voltages at buses within VCA 1 boundary are above 1.15 when the load is 930 MVAR but are below .90 when load is above 1410 MVAR. Therefore, when the reactive load is between 930 MVAR and 1410 MVAR the system experiences voltage collapse. The following is a situation that can very well appear in real life. Assume that the load suddenly increases in an area and in response the operators connect more capacitance to alleviate problems caused by 196 the load increase. Such an action might cause the operating point of the system to move into the second region of operation in which much lower voltages could be obtained, destroying the cause-effect relation of capacitance versus voltage magnitudes. If the lower voltages obtained are considered as due to excessive capacitance used, then the operators might reduce the capacitance hoping to increase voltages. In which case, the operating point of the system is shifted much closer to the bifurca- tion point. Therefore, voltage collapse could be caused by mishandling the use of capacitor banks. . . . . -1 The sen51tiv1ty matrices SQGQL, SVE’ SQLV and SQGE appear in Tables 6.51-6.53, 6.54-6.56, 6.57-6.59, and 6.60-6.62 for reactive load and shunt capacitance values of (O, 5.0) (600, 11.0) and (930, 14.30), respectively. Note, that the column sums of SQGQL for buses within the VCA 1 boundary increase from about 1.25 to 1.93 as the reactive load and compensating shunt capacitive susceptance increase. This increase in the column sums of SQGQL for buses within the VCA 1 boundary reflects the increasing reactive losses that are incurred in attempting to supply reactive load in VCA 1 from PV buses outside VCA 1 through a continually weaker transmission boundary. These increasing column sums of SQGQL for buses within the VCA 1 boundary is a clear indication of proximity to voltage collapse. The row sums of SVE for buses within the VCA 1 boundary increase from 1.3 to 2.3 as reactive load and compensating shunt capacitive sus- ceptance increase. The deviation of these rows sums from 1.0 indicate the weakness of voltage control. Thus, the deviation of the row sums of SVE from 1.0 is also a proximity indicator for voltage collapse problems. 197 The elements of SQLV'1 for row/columns entries associated with P0 buses within VCA 1 increase as reactive load and compensating shunt capacitive susceptance increase. The system is PQ controllable based on the fact that SQLV'1 and SVE is nonnegative and that SVE has no zero rows but the system is certainly not strong PQ controllable. The column sums of SQGE for Buses 19, 22, 23, and 29 decrease monotonically to more negative values as reactive load and compensating shunt capacitive susceptance increases. The column sum for Bus 25 de- creases as reactive load and capacitance increases to (600 MVAR, 11.00 pu) but then increases as reactive load and shunt capacitance further increase to (930 MVAR, 14.30 pu), due to the fact that Bus 2 becomes a PV bus and begins to support load in VCA 1. Since the column sums are negative for all three reacitve load and shunt capacitive susceptance levels, the system is not PV stable. The very severe problems that are associated with reactive load and shunt capacitive susceptance values above (1410 MVAR, 19.10 pu) are now discussed. The system is not P0 or PV stable and is so vulnerable to voltage collapse that even obtaining a loadflow solution is surprising. Tables 6.63-6.66 show the sensitivity matrices for reactive load and shunt capacitive susceptance of (1410 MVAR, 19.10 pu). Matrices SQ V'l, SVE and SQGQL have both positive and negative entries so the system is not PQ stable or PV stable. The SQLv-l elements that are negative are associated with P0 buses that lie inside as well as outside VCA 1. All rows of SVE associated with buses within the VCA 1 boundary are negative. Elements in rows associated with Buses 1, 2, 3, 15, 18, 25, 26, 27, 28, and 30 are also negative. The row sums of Buses 1-15, and 18 are all negative indicating that any rise in voltage setpoints at PV Buses 19, . '8 l..l‘ an. .In'. 198 .. mam um umuumccou mt. wucmuamumzm m>wumuwacu 3Q Om.¢~ fit. Owo— m<>£ 0mm :NEZ coqu—o. 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NO¢5¢.> 206 Y"! 06 INVEISI H0701! ...-......OOCOCCOQOOOCC. .- N N O 30 20 27 20 20 21 10 17 10 10 00 0 0700 0 0002 0 0010 0.0000 0 0010 0.0000 0.0001 0.0020 0.0010 0. 20 0 0002 0 0110 0.0010 0 0021 0 0000 0.0002 0.0007 0.0000 0 0000 0. 27 0 0010 0.0010 0 0100 0 0070 0 0020 0.0010 0.0000 0.0000 0.0030 0. 20 0 0010 0 0027 0.0070 0 0111 0 0010 0.0000 0.0020 0.0000 0.0017 0. 2 0 0010 0 0000 0.0020 0.0010 0 0000 0.0000 0.0001 0:0000 0.0000 0. 21 0 0010 0 0002 0.0010 0 0000 0 0020 0.0000 0.0020 0.0027 0.0000 0. 10 0 0000 0.0007 0 0007 0 0020 0 0001 0.0020 0.0100 0.0007 0.0000 0. 17 0 00:0 0 0000 0 0007 0 0000 0 0000 0.0027 0.0007 0.0102 0.0000 0. 10 0 0010 0 0000 0.0000 0 0010 0 0000 0.0000 0.0000 0.0001 0.0000 0 10 0 0020 0 0000 0.0000 0 0020 0 0007 0.0000 0.0000 0.0000 0.0007 0. 10 0 0000 0 0000 0.0002 0 0022 0 0002 0.0001 0.0070 0.0000 0.0001 0. 10 0 0000 0 0000 0.0000 0 0020 0 0002 0.0002 0.0000 0.0001 0.0001 0. 12 0 0030 0 0006 0.0003 0 0020 0 00 0.0033 0.0007 0.0070 0.0003 0. 11 0 0000 0 0000 0.0000 0 0020 0 0002 0.0002 0.0000 0.0000 0.0001 0. 10 0 0000 0 0000 0.0000 0 0020 0 0002 0.0002 0.0000 0.0007 0.0001 0. 0 0 0000 0 0000 0 0000 0 0020 0 0‘02 0.0002 0.0000 0.0070 0.0002 0. 0 0 0000 0 0000 0.0000 0 0020 0 0002 0.0002 0.0000 0.0070 0.0002 0. 7 0 0000 0 0000 0.0000 0 0020 0 0002 0.0002 0.0000 0.0070 0.0002 0. 0 0 0037 0 0000 0.0063 0 0023 0 0032 0.0031 0.0007 0.0000 0.0001 0. 0 0 0000 0 0000 0.0000 0 0020 0 0001 0.0001 0.0007 0.0000 0.0000 0. 0 0 0000 0 0000 0.0000 0 0022 0 0000 0.0020 0.0000 0.0000 0.0000 0. 3 0 0000 0 0003 0.0000 0 0021 0 0030 0.0020 0.0001 0.0001 0.0000 0. z 0 0102 0 0002 0.0010 0 0000 0 0010 0.0000 0.0002 0.0020 0.0010 0. 1 0 0306 0 0002 0.0010 0 0000 0 0013 0.0000 0.0001 0.0027 0.0010 0. 10 10 12 11 10 0 0 7 0 00 0 0000 0 0000 0 0000 0 0000 0.0000 0.0002 0.0002 0.0002 0.0051 20 0.0003 0 0003 0 0003 0 0003 0.0003 0.0003 0.0003 0.0003 0.0003 27 0.0001 0 0002 0 0000 0 0002 0.0002 0.0003 0.0003 0.0003 0.0003 20 0.0021 0 0021 0 0022 0 0022 0.0022 0.0022 0.0022 0.0022 0.0022 20 0 0030 0 0031 0 0032 0 0030 0.0030 0.0030 0.0030 0.0030 0.0030 21 0 0000 0.0001 0 0002 0 0000 0.0000 0.0000 0.0000 0.0000 0.0030 10 0.0077 0 0001 0 0003 0 0003 0.0002 0.0000 0.0000 0.0000 0.0000 17 0 0003 0 0003 0 0000 0 0000 0.0003 0.0007 0.0007 0.0007 0.0000 10 0 0030 0 0000 0 0002 0 0039 0.0030 0.0030 0.0030 0.0030 0.0030 10 0 0120 0 0110 0 0122 0 0117 0.0110 0.0110 0.0110 0.0110 0.0110 10 0.0202 0 0207 0.0200 0 0202 0.0100 0.0200 0.0200 0.0200 0.0200 13 0.0200 0.0310 0 0322 0 0302 0.0310 0.0201 0.0201 0.0203 0.0201 12 0 0233 0 0322 0.0301 0 0333 0.0320 0.0307 0.0300 0.0300 0.0300 11 0.0200 0 0303 0.0333 0 0300 0.0320 0.0312 0.0312 0.0310 0.0313 10 0.0260 0 0311 0 0320 0 0323 0.0336 0.0297 0.0207 0.0200 0.0207 0 0.0261 0 0203 0.0310 0 0313 0.0300 0.0730 0.0000 0.0307 0.0303 3 0.0201 0 0203 0 0310 0 0313 0.0300 0.0000 0.0000 0.0300 0.0303 7 0 0202 0 0200 0.0012 0.0010 0.0001 0.0007 0.0007 0.0000 0.0007 0 0 0230 0 0203 0 0300 0.0310 0.0200 0.0303 0.0303 0.0303 0.0303 0 0 0200 0 0270 0 0000 0.0000 0.0200 0.0000 0.0000 0.0000 0.0331 6 0 0213 0 0220 0 0202 0 0230 0.0230 0.0232 0.0233 0.0232 0.0200 3 0 0100 0 0100 0 0112 0 0110 0.0100 0.0113 0.0113 0.0113 0.0113 2 0 0000 0.0000 0 0001 0 0000 0.0000 0 0002 0.0000 0.000: 0.0002 1 0 0063 0.0000 0 0030 0.0000 0.0000 0.0032 0.0032 0.0032 0.0031 0 3 2 1 so 0 0002 0.0000 0 0007 0.0001 20 0 0003 0 0003 0 0002 0.0002 27 0 0001 0.0000 0 0010 0.0010 :0 0 0021 0 0020 0 0000 0.0010 :0 0 0001 0.0000 0 0010 0.0010 21 0 0020 0 0020 0 0010 0.0010 10 0 0000 0 0001 0 0000 0.0000 17 0 0000 0 0001 0 0020 0.0020 10 0 0000 0 0000 0 0010 0.0010 10 0 0100 0 0000 0 0020 0.0020 10 0 0213 0.0102 0 0000 0.0000 13 0 0229 0.0100 0 0031 0.0032 12 0 0200 0.0110 0.0000 0.0000 11 0 0200 0.0110 0.0000 0.0000 10 0 0233 0.0111 0.0032 0.0030 0 0 0230 0.0120 0.0030 0.0030 0 0 236 0.0120 0.0030 0.0030 7 0 0233 0.0120 0.0030 0.0030 0 0.0200 0.0117 0.0000 0.0000 3 0.0231 0.0110 0.0033 0.0037 0 0 0233 0.0110 0.0033 0.0037 3 0 0113 0.0100 0.0060 0.0060 2 0 0032 0.0000 0.0000 0.0100 1 0 0032 0.0003 0.0000 0.0000 7"! 30701! 13 IOIIIIO. I.Illfl07IVI Illllfll IUIIII 0' 170l0710l3 0006300 VII I3IIIII II.II001UI 0' 00 13 0.3300 Tab1e 5.57 The sensitivity matrix 50 '1 for the case where o MVAR load V L and 5.00 pu capacitive susceptance are connected at Bus 11 .Ffi‘v-D-‘t‘NNNN 207 1H! 00 1'00032 IQYIIX 30 20 27 20 20 21 10 17 10 10 30 0 0703 0.0003 0.0023 0 0012 0.0021 0.0013 0.0030 0.0033 0.0020 0.0030 20 0 0003 0 0113 0 0010 0 0027 0 0000 0.0002 0.0000 0.0000 0.0003 0.0000 27 0 0020 0.0010 0.0130 0 0070 0 0033 0.0020 0.0002 0.0070 0.0030 0.0000 20 0.0012 0.0027 0.0077 0 0111 0 0017 0.0010 0.0032 0.0030 0.0020 0.0023 20 0 0022 0.0000 0.0033 0 0017 0.0000 0.0032 0.0000 0.0001 0.0002 0.0000 21 0.0013 0.0003 0.0020 0 0010 0 0032 0.0000 0.0020 0. 031 0.0030 0.0000 10 0 0030 0.0000 0.0002 0.0033 0.0000 0.0020 0.0107 0.0000 0.0037 0.0070 17 0 0037 0.0000 0.0071 0 0037 0.0031 0.0031 0.0003 0.0100 0.0000 0.0070 10 0 0020 0.0003 0.0030 0 0020 0 0002 0.0030 0.0037 0.0000 0.0073 0.0070 13 0 0062 0.0000 0.0007 0 0020 0 0007 0.0001 0.0070 0.0071 0.0070 0.0100 10 0 0073 0.0000 0.0000 0 0031 0.0073 0.0000 0.0113 0.0002 0.0000 0.0100 13 0 0003 0.0000 0.0000 0 0030 0 0070 0.0003 0.0123 0.0101 0.0002 0.0170 12 0 0000 0.0000 0.0070 0 0030 0 0002 0.0030 0.0133 0.0107 0.0007 0.0100 11 0 0000 0.0000 0.0070 0 0030 0 0002 0.0030 0.0133 0.0100 0.00 0.0100 10 0 0000 0.0000 0.0000 0 0033 0 0030 0.0000 0.0120 0.0100 0.0000 0.0102 0 0 0000 0.0000 0.0000 0 0030 0 0070 0.0000 0.0132 0.0103 0.0003 0.0170 0 0 0000 0.0000 0.0000 0 0030 0.0070 0.0000 0.0132 0.0103 0.0003 0.0170 7 0 0000 0.0000 0.0000 0 0030 0.0000 0.0000 0.0133 0.0103 0.0000 0.0170 0 0 0000 0.0000 0.0000 0.0033 0.0070 0.0000 0.0131 0.0100 0.0003 0.0170 3 0 0000 0.0000 0.0007 0.0033 0.0070 0.0007 0.0120 0.0102 0.0001 0.0170 0 0 0002 0.0000 0.0001 0 0032 0.0000 0.0002 0.0110 0.0003 0.0001 0.0133 3 0 0000 0.0000 0.0000 0 0023 0.0000 0.0027 0.0100 0.0070 0.0032 0.0032 2 0 0100 0.0003 0.0022 0.0012 0 0020 0.0012 0.0000 0.0030 0.0020 0.0030 1 0 0010 0.0003 0.0022 0.0012 0.0020 0.0012 0.0000 0.0030 0.0020 0.0030 10 13 12 11 10 0 0 7 0 3 30 0.0007 0 0073 0.0000 0 0000 0.0070 0 000 0.0000 0.0000 0.0070 0.0070 20 0 0007 0 0000 0 0000 0 0000 0.0000 0 0000 0.0000 0.0000 0.0000 0.0000 27 0 0030 0 0003 0 0007 0.0007 0 0003 0 0000 0.0003 0.0000 0. 003 0.0000 20 0 0030 0 0033 0 0033 I 0030 0 0030 0 0030 0.0030 0.0030 0.0033 0.0033 20 0.0071 0 0070 0 0000 0 0070 0 0070 0 0077 0.0077 0.0077 0.0070 0.0073 21 0.0003 0 0000 0 0000 0 0000 0 0007 0 0000 0.0000 0 0007 0.0000 0.0003 10 0.0111 0 0122 0 0130 0 0130 0 0120 0 0120 0.0120 0 0120 0.0127 0.0120 17 0 0000 0 0000 0 0100 0 0103 0 0101 0 0101 0.0101 0 0101 0.0100 0.0000 10 0 0000 0 0000 0 0000 0 0003 0 0001 0 0000 0.0000 0 0000 0.0000 0.0030 13 0 0100 0 0170 0 0100 0 0102 0 0170 0 0170 0.0170 0.0173 0.0170 0.0171 10 0 0330 0 0333 0 0371 0 300 0 0300 0 0300 0.0300 0.0330 0.0300 0.0302 13 0 0330 0 0007 0 0002 0 0030 0 0000 0 0017 0.0017 0.0010 0.0017 0.0000 12 0.0370 0 0003 0 0000 0.0003 0 0070 0 0030 0.0030 0 0037 0.0030 0.0002 11 0.0300 0 0032 0 0000 0.0310 0 0001 0 0000 0.0000 0.0007 0.0000 0.0032 10 0 0303 0 0000 0 0070 0 0001 0 0002 0 0001 0.0001 0.0003 0.0002 0.0020 0 0 0330 0 0021 0 0030 0 0000 0 0003 0 0322 0.0331 0 0310 0.0070 0.0071 0 0 0330 0 0021 0 0030 0 0000 0 0003 0 0331 0.0331 0.0310 0.0070 0.0071 7 0 0333 0 0023 0 0000 0 0070 0 0000 0 0313 0.0313 0.0331 0.0001 0.0071 0 0 0332 0 0020 0 0030 0 0007 0 0000 0 0070 0.0070 0.0070 0.0070 0.0002 3 0 0303 0 0000 0 0000 0 0033 0 0031 0 0000 0.0000 0.0000 0.0002 0.0003 0 0 0300 0 0330 0 0330 0 0301 0 0300 0 0300 0.0300 0.0301 0.0337 0.0330 3 0 0100 0 0100 0.0171 0 0171 0 0100 0 0170 0.0170 0.0170 0.0100 0.0107 2 0 0007 0.0070 0 0070 0 0070 0 0077 0 0070 0.0070 0.0070 0.0073 0.0070 1 0 0007 0.0070 0 0070 0 0070 0 0077 0 0070 0.0070 0.0070 0.0070 0.0077 0 3 2 1 30 0.0070 0.0070 0 0103 0.0010 20 0.0007 0.0000 0 0003 0.0003 27 0.0030 0.0003 0 0023 0.0020 20 0.0030 0.0023 0 0012 0.0012 20 0.0007 0.0000 0 0021 0.0022 21 0.0001 0.0027 0 0013 0.0013 10 0.0117 0.0100 0 0031 0 0033 17 0.0001 0.0070 0 0033 0.0037 10 0.0070 0.0032 0 0023 0.0020 13 0.0131 0.0033 0 0000 0.0001 10 0.0300 0.0107 0.0070 0.0073 13 0.0337 0.0100 0 0070 0.0001 12 0.0301 0.0170 0 0000 0.0007 11 0.0302 0 0170 0 0030 0.0007 10 0.0330 0 0170 0 0001 0.0000 0 0.0300 0.0170 0 0000 0.0007 0 0.0303 0.0170 0 0000 0.0007 7 0.0303 0.0170 0 0000 0.0007 0 0.0300 0.0170 0 0003 0.0030 3 0.0337 0.0172 0 0032 0.0000 0 0.0330 0.0101 0 0077 0.0000 3 0.0130 0.01 0 0070 0.0070 2 0.0073 0.0073 0.0102 0.0100 1 0.0073 0.0073 0102 0.0300 "02 M701! 13 0000000. “71'! 7”! Illllll 21.2.0000! 0' I0 10 0.3107 Table 6.58 The sensitivity matrix SQ v'1 for the case where 600 MVAR load L and 11.00 pu capacitive susceptance are connected at Bus 11 208 tn! I. 1...... «I'll! go 2. 27 26 3‘ 21 I. l7 [6 3. . ..,. . .... . .... ...... ...... . .... . .... ...... ...... 2. . oooo . .112 ....1. . 0.2. ...... . ...2 . ...7 ...... ...... 27 . .... . ..1. . .1.. ....7. ...... . ..1. ...... ...... ...... z. .‘o... . ..27 ....7. . .1.. ....1. . .... ....27 .....s . ..1. 2. . .... . .... . 0030 0.001. 0.0093 0 ...1 .....1 ...... ...... 21 ...... . ...2 . ..1. ....1. .....1 . ...2 ....2. ....2. .... 1. .7.... . ...7 . .... . ..2. .....1 . ..2. ...12. .....1 ...... 17 . o... . .... . .... . .... ...... . ..2. .....1 ...... ...... 1. ...... . .... . .... ....1. ...... ...... ..... ...... ....7. 1. ...... .....s . ...2 . ..22 ...... ...... ...... ...... ....7: 1. ...... . ...7 . .... . ..2. ...... .....2 ...... .....1 . ...1 1: ...... ...... . .... . .031 ....7s ...... ...1.. ...... ...... 12 ...... ...... o ...2 . ...: ....7. ...... ...112 ...... ...... 11 ...... ...... . .... . ..ss ....7. ...... ...11. .....s ...... 1. ...... . .... .....1 . ...2 ....77 .....7 ...1.. ...... .....1 . ...... . .... . .... . ..12 ....7. ...... ...1.. .....2 ...... . ...... ...... . .... . 0.12 ..no7. ...... ...1.. .....2 ...... 7 ...... ...... . .... . ..12 ....7. ...... . .11. .....2 ...... . ...... ...... . ... . ...2 ....7. ...... ...1.. .....1 . .... s . .... ...... . .... .....1 ....7. . .... . .1.7 ...... .....7 . . .... . ...7 . ...2 . ..27 ...... . .... ...... ....7. ....7. 1 ...... . ...s . ...s . ..1. ...... . ..2. ....7. ...... ...... 1 ...... . .... . .... . .... ...... . .... ...... ...... ...... 1. 13 12 11 1. . . 7 . 1. ...... . .... . .... ...... . .... . .... ...... ...... ...... 2. ...... . ...7 . .... . .... . ...7 . ...7 . ...7 .....7 .....7 27 .....2 . ..s7 . .... ...... . .... . .... .... ...... ...... 2. . ..27 . ..2. .....1 ....11 . .... . .... ...... ...... ...... 2. ...... . ..73 . ..77 . ..77 . ..7. . ..7. ....71 ....7. ....7: 21 .....1 . .... . ...7 .....7 . .... . .... ...... ...... ...... 1. .....2 . .1.. . .1.. . .11. . 1.. . .1.. ...1.. ...1.. ...1.. 17 ....79 . ...7 o ...2 ...... . .... . .... ...... ...... ...... 1. ....7. . .... . ...1 ...... . .... . .... ...... ...... ...... 1. ...1.1 . .17. . .1.2 ...1.1 . .177 . .171 ...171 ...172 . .171 1. ....2. . ...7 . .... ...1.2 . .3.. . .1. . .1. .....1 ...3. 1s ...... . .... . .... ...... . ...1 . .... . ....1. ....1. 12 ...... . .... . ..2. ...... . ..71 . .... . ...... ...... 11 ...1.. . ...7 ...... . .3.. . ..77 . s. . .... ...... ...... 1. ...137 .....2 ....71 . ..7. . ..71 . ...2 . ...: ...... ...... o ...1.. ....12 . ...7 . .... . .... . .7.. . ... ...... ...... . ...... ....12 .....7 . .... . .... . .... . .... ...... ...... 7 ...1.. ....1. ...... . ...1 . .... . .... . .... ...... ...... . ...... ....12 ...... . .... . s. . .... . .... .....1 ...... s ...... ...... ...... . .... . ..23 . . . .... ...... .....s . ...2.. . .32. ...s.. . .ss. . .337 . .3.. . ...2 ...... .....1 s . .112 . .12. . .11: . .13. . .131 . .132 . .132 ...1.. ...132 1 ...... . .... . .... . .... . .... . .... . .... ...... ...... . s 1 1. ...... . .... ...... 2. ...... ...... . .... 27 ...... ...... ...... 2. ....2. . ..1. ...... 2‘ I 1 II62 I . II.“ I. IIII 21 ...... ....2. ...... 1. ...... ....7. ...... 17 o...77 ...... ...... 16 I.II?! I.II” I.IIII 1s ...1.. ...... ...... 1. ...2.s ...11. . .... 1s ...12. ...12. ...... 12 ...1.7 ...1.. ...... 11 ...1.2 ...1.. ...... 1. ...11. ...11. ...... 9 ...1.. ...137 . .... . ...... ...137 ...... 7 ...... ...137 ...... . ...1.. ...1.. ...... s ...:s. ...1.. ...... . ...111 ...122 ...... 1 ...11. ...1.. . .... 1 ...... ...... ...1.. VI! natnxx 1s IIIIIII. IIIIIIIYIVI “I!" nun I? I'll!!!“ m m ”It- "MM“ ' u I! I.II” Tabie 6.59 The sensitivity matrix SQ V L and 14.30 pu capacitive susceptance are connected at Bus 11 .II ...-....GCCOCCCCCOCC... .. ... ........... ...-......OCCCCCCICCCCC . . . . C U . I U . . C O O I O I l O Q I I I for the case where 930 MVAR load Hfi mzm um umuumccou mam mucmuamumzm u m>wuwumamu 2a oo.m new umop m<>z o wcmzz mmmu use new m cm xwgace »u_>wuwmcmm ugh om.o o_amp ~0N9.n- an 80—... "—0 Marga: has 9: 959 £900 m>~h¢ouz 9 5.9 33.: 209 ~N U099 99\o~\99 “mud: Itcooxm m~m>a¢8¢ rand-999m wathao, 9995.9N 999n.nn1 9ann.~1 nacn.~u 9009.9 unON.nI 9~ soon.n—1 N09c.9n 099—.91 90n~.~1 9099.0 v~0~.0~1 9N 00~9.~1 nunn.nu ~0ns.cn~ ~099.—-1 9999.9 vh~h.01 n~ N0~v.~- n900.91 h~n9.-~1 mean at. 9009.0 N00s.9—1 NN 9999.9 0999.9 0999.9 9090.9 n—9o.9~ ~090.-I 0N nncv.n1 ne9~.-1 9995.9- 99-.c—- 9090.0s1 ~000.00n 0n 9~ nN nN - 9N 0a chhdl meow map - 0°99 99x9~\99 “up90 8996089 m~n>a¢8¢ >9—a-9‘pfl methao> ~00».N1 9N N0N0.n1 9N 990s.~1 9N 90-.9- NN c~9n.01 9N ~0~0.n1 Du ea «tan 8:3400 Uccm Nth 210 HH mam an umuumccou mam mucmpamumzm w m>wuwumamu an oo.HH new umop m<>z com mgmsz mmmu mgu Low m cm xmgums auw>pu_mcmm mg» 00.0 wpam» snn~.n~1 mu suns: a0 u>~b¢oux ~90: map 0:09 8:0000 u>~b¢0uz 9 09¢ Manx» s-0.01 9N unn~.n—1 9N nun-.51 9N 0~99.51 NN 9~0n.91 9N v0~v.-1 0n 0— 0100 880000 wccm nth ~N U099 99\9~\09 “Upca t<¢°0¢9 m~n>a¢8¢ rung-90hw 0099009 -99.9N mes—.0ul 9c09.—1 c~n~.~- 9999.9 0»~9.nn 9N N0cn.0—I 00cn.~n 090m..- 9999.9u1 9999.9 n900.n~1 9N c~09.~- nnn0.01 9959.nn~ ~999.~N~- 0999.9 0999.9—1 nN Nnn9.~- ~9N0.0~I 9099.~N~I ~n9v.nv~ 9909.9 -9c.0-- NN 9909.9 9999.9 0099.0 99.—.9 n~09.9s N090.-1 9N 9~90.n- m9~9.v—1 ~0—9.0—1 9090.9u- 9990.9h- 99—0.n9~ 9— 9N nN n~ NN 9N 0— x~¢~¢8 000m 08p ~N u0¢9 99\0N\99 ”0—90 td¢8¢ >h~a~9¢~n uO<~00> 211 ”H mam um umpumccou use mucmuamumzm a m>wuwuaaau 3a om.¢~ uca uao_ ¢<>z 0mm meme; ammo as“ to. m cm chuas »~_>Pu0m=mm ugh Nc.a m_aac 909-.9No 99 80988 90 u>-90u9 5908 085 9809 880000 99999098 5 U99 09085 9999.9- 9N 9999.91 9N 9995.91 99 N999.—-1 99 9599.91 9N 99—0.5—1 9- 900-.991 u N 9809 880000 9009 98h NN 0099 99\9~\99 .9590 8990099 nunha989 >h~d~9959 ”095009 59—9.9N 0—~0.~—1 ~909.—1 ~N5n.~1 9909.9 9999.91 9uvn.91 ON 99n~.nn1 nN99.09 5999.~- 0~nn.~1 9999.9 95Nm.91 ~99N.991 9N 9995.~1 9-5.~1 ~999.nn~ 59~5.—~u1 9999.9 99N9.9—1 5999.51 99 9999.91 9509.91 9959.--1 5909.99— 9999.9 —N~—.9~1 9959.~—1 NN 9999.9 9999.9 9099.9 9009.9 n—9m.95 N999.~51 9990.9 -95.n1 9.99.91 9999.0—1 —0-.9~- 9999.951 -99.n9~ 999~.9~1 9999.91 9099.n9- 9959.51 -09.-1 9999.9 9——9.9—1 99n~.~0 9~ 9N nN Nu 99 9— N x~9598 0099 0:5 - 9099 99\9~\99 .9590 8990099 n—nhd989 >5~0~995w u095000 212 722 26 1292222 II'213 32 22 27 26 22 26 21 12 17 16 22 I 2711 2.2226 2.2222 2.2217 2 2262 O2.2226 O2.2216 O2.2216 O2.2217 O2 2229 22 I 2226 2.2116 2.2212 2.2222 2 2212 O2.2221 O2.2221 I 2221 2.2222 O2.2221 27 I 2222 2.2219 2.2129 2.2277 2.2226 O2.2221 O2.2221 I 2212 2.2222 O2.2221 26 I 2212 2.22 2.2 2.212 2.2269 O2.2222 O2.2222 I 2226 2.2216 O2.2226 22 I 2271 2.2212 2.2222 2.2269 2.2122 O2.2219 O2.2211 O2 2229 O2.2222 O2.2222 26 -2 2226 O2.2221 O2.2221 O2.2222 O2.2219 I 2272 2.2212 O2 2212 2.2226 2.2226 21 O2 2216 O2.2221 I 2222 O2.2222 O2.2212 I 2212 2.2272 O2 2227 2.2222 2.2216 1. -2 2016 I 2221 I 2212 2.2226 O2.2212 O2 2212 O2.2227 2.222 O2.2216 17 O2 2212 2226 I 2222 2.2212 O2.2222 2.2226 2.2222 I 2221 2.22 2.2222 16 O2 2221 O2 2221 O2 2221 O2.2226 O2.2222 2.222 2.2216 O2 2216 I 2222 2.2222 12 O2 2222 O2 2229 O2 2229 O2.2226 O2.2 O2.221 O2.III7 O2 I O2 2261 O2.2216 16 O2 2226 O2 2226 O2 2126 O2.211I O2.2172 O2.2122 OI.II62 O2 2226 O2 2122 O2 2122 12 O2 2222 O2 2227 O2 2122 O2.2122 O2 2227 O2.2166 O2.2227 O2 2221 O2 2222 O2 2172 12 O2 2222 O2 2262 O2.2216 O2.2177 O2 2276 O2 2171 O2.2122 O2.2227 O2.22 O2 2222 11 O2 2262 O2 2266 O2 2222 O2.2126 O2 2226 O2.2179 O2.2127 O2 2262 O2.2267 O2 2212 12 O2 2211 52 2262 O2.2222 O2.2162 O2 2762 O2.2162 O2.2297 O2 2211 O2.2262 O2 2192 9 O2 2296 O2 2229 O2.2197 O2.2161 O2 2269 O2.2122 O2.2292 O2.2297 O2.2226 O2 2122 I O2 2292 O2 2229 O2 2197 O2.2161 O2 2269 O2.2122 O2.2292 O2.22 O2.2226 O2 2122 7 -2 2292 O2 2229 O2.2192 O2.2162 O2 2221 O2.2129 O2.229 O2.2299 O2.22 O2 2129 6 O2 2292 OI 2229 OI 2196 O2.2161 OI 2262 O2.2122 O2.2296 O O2.2226 OI 2127 2 OI 2222 OI 2227 OI 2126 O2.2122 O2 2226 O2.2122 '2.2292 O2 2221 O2 2222 '2 2172 6 O2 2129 O2 2222 O2.2122 O2 2126 O2.2162 O2.2127 O2 2 O2 2192 O2 21 O2 2126 2 O2 2211 O2 2226 O2.2226 O2 2212 O2 2212 O2.2229 O2 2222 O2 2222 O2 2222 O2 22 2 I 2122 2 2222 2.2222 2 2219 I 2272 O2.2227 O2.2216 O2.2216 O2 2212 O2.2222 1 I 2622 I 2226 2.2222 2 2212 2 2272 O2.2222 O2.2212 O2.2212 O2 2217 O2.2229 12 16 12 12 11 12 9 2 7 6 22 O2.2272 O2.2126 O2.2221 O2.2292 O2.2221 O2.2276 O2.2222 O2.2222 O2.2262 O2.2229 22 O2.2222 O2.2222 O2.2226 O2.2229 O2.2261 '2.2227 O2.2222 O2.2222 O2.2222 O2.2222 27 O2 2 O2 2129 O2 2172 O2.2226 O2.2216 O2.2196 O2.2122 O2.2122 O2.2127 26 O2 2222 O2 2126 O2 2162 O2 2166 O2.2172 O2.2 O2 2169 O2 2169 O2 2 22 O2 2122 22 O2 2 O2 2161 O2 2222 O2 2222 O2.2266 O2.2262 O2 2227 O2 2227 O2 22 O2 2222 26 O2 2212 O2 2122 O2 2161 O2 2162 O2 2172 O2.2127 O2 2121 O2 2121 O2 2122 O2 2121 21 O2 2227 O2 2 O2 2226 O2 2292 O2 2122 O2.2292 O2 2292 O2 2292 O2 2292 O2 2292 12 O2.2262 O2.2199 O2.2272 O2.2216 O2.2222 O2.2222 O2.2222 O2.2222 O2.2222 O2.2226 17 O2.2261 O2.2126 O2.2212 O2.2267 O2.2227 O2.2222 O2.2222 O2.2222 O2.2222 O2.2226 16 O2.2216 O2.2119 O2.2162 O2.2196 O2.2226 O2.2126 O2.2179 O2.2179 O2.2122 O2.2179 12 2.2222 O2 2222 O2.2216 O2.2262 O2.2226 O2.2222 O2.2262 O2.2262 O2.2262 O2 2262 16 O2 2222 O2 2 O2 2666 O2.2722 O2.2212 O2.2761 O2.2726 O2 2726 O2 2729 O2 2722 12 O2 2221 O2 O2 O2 2972 O2.1229 O2.2922 O2.2922 O2 2922 O2 2922 O2 2921 12 O2 2277 O2 2727 O2 2972 O2 2222 O2.1166 O2.1272 O2.126 O2 126 O2 1271 O2 1261 11 O2 2296 O2 2226 O2 1226 O2 1169 O2.1127 O2.1112 O2.122 O2 1229 O2 1 O2 1 12 O2 2222 O2 2767 O2 2922 O2.1272 O2.1127 O2.2922 O2.1211 O2.1212 O2 1217 O2 1227 9 O2 2222 O2 2761 O2 2962 O2 1222 O2.1122 O2.1222 O2.2261 O2. O2 2921 2 O2 2222 O2 2762 O2.2962 O2.1222 O2.1122 O2 1222 O2.2222 22 O2.2262 O2 21 7 O2 2222 O2 2762 O2.2922 O2.1222 O2.1127 O2 1222 O2 2267 O2 O O2.2926 6 O2 2221 O2 2726 O2.2929 O2.1262 O2.129I O2 1216 O2 2916 O2.2916 O2 2919 O2.2922 2 O2 2222 O2 2726 O2 2922 O2.1221 O2 1226 O2 2979 O2 2271 . . O2.2221 6 O2 2262 O2 2212 O2 269 O2.2222 2227 O2 2729 O2 2722 O2.2722 O2.2712 O2.2712 2 O2.2112 O2.2272 O2.2271 O2.2629 O2.2666 O2.2627 O2.2222 O2.2 2 O2.2226 O2.2222 2 O2.2222 O2.2222 O2.2276 O2.2222 O2.2221 O2.2222 O2.2226 O2.2226 O2.2226 O2.2226 1 O2.2277 O2.2129 O2.2222 O2.2292 O2.2229 O2.2222 O2.2262 O2.2262 O2.2227 O2.2267 6 2 1 22 -2 2266 O2.2162 O2 2212 2 2297 2.262 22 O2 2222 O2.2222 O2 2222 2 2222 2.2226 27 O2 2177 O2.2122 O2.2222 I 2222 2.2222 26 O2 2162 O2.2292 O2.2212 I 2 2219 22 O2 2216 O2.21 O2.2212 I 2276 I 2272 26 OI 2166 O2 2122 O2.2229 O2 2227 O2 2227 21 O2 2226 O2 2 -2.2222 O2 2216 O2 2216 1 O2 2272 O2 2127 O2.2222 O2 2216 O2 2216 17 O2.2212 O2.2169 O2.2222 O2.2219 O2.2212 16 O2.2171 O2.2122 O2.2266 O2.2222 O2.2222 12 O2.2222 O2.2227 O2 2116 O2.2226 O2 2226 16 O2 2696 O2.2211 O2.2279 O2.2212 '2 2212 12 O2 2297 O2.26 O .2222 O2.2292 O2.2227 12 O2 1222 O2.2226 O2.2662 O2.2262 O2 2 2 11 O2 1269 O2. O2.2662 O2.2222 O2 22 12 O2 2972 O2.2762 O2.2621 O2.2222 O2 2216 9 O2 2272 O2.2719 O2.2299 O2.2222 O2 2.0) 2 O2 2277 O2.2719 O2.2299 O2.2222 O2 2222 7 O2 2226 O2.2722 O2.2622 O2.2211 O2 2222 6 O2 2222 O2.2719 O2.2299 O2.2222 O2 2222 2 O2 2227 O2.2622 O2.2272 O2.2 O2 2226 6 O2 26 2 O2.2626 O2.2222 O2.2197 O2 2192 3 I: 127? '3'3111 1'13} 1'31: ’1 1133 1 O2.2222 O2.2172 O2.2211 2 72222 222 262 2'227192 ILIIIIVI 12 722 I2721l. TI! III2? 2' IIICI 12 O2.1127 I222... IUIIII 2' 172l2722l2 2222222 VII Illllll 2222I22122 2' 26 12 O1.2626 Table 6.63 The sensitivity matrix SQ v'1 for the case where 1410 MVAR Toad L and 19.10 on capacitive susceptance are connected at Bus 11 “a mam an umuumccou mgm mocmuamumam a>_a.umaau 3a ofi.m~ new umop m<>z oflefi «can: «we. we“ go. m>m xrcaus »p_>_u.m=mm ash em.m «.3.» 213 99 U099 99\9~\99 "U590 9~9~.—1 9— 80—88 90 59908 USP .9—9598 U85 89 959U8UdU U>~590U9 95 U99 U9U85 ~99~.91 9—99.9 9959.91 9N~—.91 9999.9 959—.91 5 9909.91 9999.9 5959.91 9~N-.91 9999.9 999—.91 H ~099.91 9—99.91 99-.91 9955.91 9999.9 9999.91 9 9999.-1 5999.91 09—9.91 9999.91 9999.9 9959.91 9 5999.N1 9999.91 ~999.91 9999.91 9999.9 9-99.91 9 9995.91 9~99.91 9N59.91 59n5.91 9999.9 ~999.91 9 9595.91 9999.91 9559.91 99N5.91 9999.9 9999.—1 5 9995.N1 5~99.91 9959.91 9955.91 9999.9 u~99.-1 9 9995.~1 ”~99.91 9959.01 5905.91 9909.9 9N99.-1 9 99N9.N1 9959.91 9999.91 9595.91 9999.9 95~0.-1 9— 99-.91 9999.91 9999.01 9-9.91 9999.9 9—9~.—1 - 9099.~1 N999.91 9-—9.91 ~555.91 9990.9 9999.91 - N999.~1 9999.91 9599.91 9999.91 9999.0 ~9N9.91 9— 9999.—1 9959.91 u999.91 9599.91 9999.9 9.99.91 9- 9599.91 999-.91 9999.91 9999.91 9999.9 9950.91 9- a9~9.9 99N9.91 9-59.9 9995.9 9999.9 9—9~.9 9- 999—.9 9999. 9N-9.9 ~9—9.u 9099.9 59N0.9 5- -9~.91 99—9.9 9990.91 9999 91 9999.9 ~959.91 9— 9959.9 99~9.91 9999.9 9999.0 9999.9 9959.9 5N 9-9.9 95~9.91 5999.9 9599.9 9999.9 95-.0 9N 9-99.91 959—.9 9990.91 9999.91 9999.9 ~9—~.91 9N ~999.9 9—99.9 99—9.91 5NN9.91 9999.9 5-99.01 9N 9999.9 9999.9 9~00.91 9999.91 9999.9 9999.91 59 9999.9 9959.9 9999.91 9999.91 9999H9 9599H91 9N 995N.91 9999.9 9~59.91 999—.91 9999 9 999- 91 99 99 99 99 99 9- x~9598 U99 U85 8990099 9~9>0989 >5—d—9959 U09500> Hg mam um cmaumccou mum mucmuamumam S m>_u_umamu an ofi.mH ucm vac, m<>z oH¢H mgmgz ammo «:9 so; g cm x_ggos »u_>_p_m=mm cg? me.o mpamp .59U995UU99U9 X99598 U9999 U85 90 899 5999 989 883909 5999 U85 99 99U999 X99598 49999 U85 99 9899 889999 989 9839 899 U85 214 9959.91 99 89988 99 59998 U85 .999598 U85 89 958U8U9U U9959°U8 95 U99 U9U85 9909.91 9999.91 9559.91 9995.91 9999.91 9999.91 9999.9 9990.0 9999.01 9599.91 5959.01 99 9999.91 9959.91 9090.91 9999.01 9999.91 9999.01 99 9990.51 5999.01 9999.01 9959.91 9999.91 9999.01 99 0090.9 9999.9 9090.0 9099.9 0999.9 0909.9 99 9999.091 9999.91 9959.91 9999.91 9999.9 9909.91 99 9 9 9 9 , 9 9999.91 9959.91 9999.91 9999.91 9059.91 9999.91 9999.91 9099.91 9995.91 9959.91 9999.91 9009.01 9999.91 9999.91 9999.01 9959.91 9999.91 9559.01 9599.91 5999.01 99 9999.01 9999.91 9999.01 9999.91 9999.91 9999.01 9599.01 9999.91 9999.91 9999.01 99 9559.91 9099.01 9959.91 9959.01 9995.01 9955.91 9995.91 9999.01 9999.91 9990.01 99 9999.9 9090.9 9909.9 9000.9 0009.0 9090.9 0000.9 9009.0 9000.9 9000.0 99 9999.91 9999.91 9—99.91 9999.91 9959.01 9959.91 9990.91 9999.01 9999.01 9950.01 9— 9 5 9 9 99 99 99 99 99 99 9999.9 9909.9 9999.01 9999.9 9909.9 9999.91 5999.9 5999.9 9999.9 9999.01 9999.91 9599.9 9990.0 0090.91 5990.01 9999.9 9999.0 9999.0 9999.9 9990.9 99 9959.9 9999.9 9990.01 9990.0 9909.0 9590.91 0990.91 9909.01 9900.01 9550.91 99 9999.9 9999.9 9990.01 9999.0 9090.0 9590.01 9990.91 5909.01 9900.91 9599.01 99 9909.9 9999.9 0000.0 0009.0 0000.0 0000.0 0000.0 0000.0 0900.9 0000H0 09 9999.9 9590.9 9950.01 9950.9 5999.0 9999.01 9990.91 9900.01 9590.01 9999 01 99 99 59 99 99 99 99 99 59 99 09 X99598 99699 U85 99 U099 99x99x90 ”U999 8999999 99959989 559999959 U09599> 215 HH mam an umuumccou mgm mucmuamumam a m>wuwumnmu an OH.¢H ucu too. ¢<>z ofiefi mgmg: mung mzu Lo; m cm x_gyms xuv>wp_m=mm mgh mo.o «Pack «N He‘s nN uo~h¢ouz knot Us» «tan xtadau u>-¢9uz a u¢¢ mama» nan.eN oe-n.e 09.5.9 0999.. anon.— ON ooav.e ONnn.nn~ ccc~.o—~I nee... ono~.nl nN neNu.NN 0N oncn.o— nN unnc.~n NN vsen.°| 0N 09"".uc o— mtam Itadcu uaom Nth 8¢¢O°¢& m~n>a¢8< >h~d~¢ nhn~.. 99.0.. anne.~ 0N —N0~.e-| can... cans.nl nN qu. nm— 0909.9 no.5.nu NN 099‘.- n~e¢.°~ None.—~I 0N nv—~.nn 0.90.05: noon.~N~ .— NN 0N o— x~8b¢t moon uxh ttcoeus n~m>ab~d~¢_u_umnmu an oo.o~ new use. z<>z com“ cog; cowuzpom zepwuuod mm m mpnmp ...: . ...... ...... .... .... ...... .....u a...~N .....su .....N ...v.- .....u ...... n ..u:-—. .n .n . ...N..: ...... .....Nu ....nN ...... ...... ...nn ...cN.~ .... ...».N ~..sNu ...... N ...N.. .N .N . ...... ...... .... .... ...... ...... .... .... ...sN .....N va..nu .Nc..u . sea-nsua. .N .N . ...... ...... .... .... ...... ...... .... .... ...ns ...—.N ...-ca Nsoo.. . zed—mm... 5N NN . ...... ...... .... .... ...... ...... .... .... ...su ...... a...nu nn~e.— . chau._. .N .N . .Nsv..u ...... ....NN: ...... ..n..~ ...... ...... ....nn .... ...cNN sm.Nna m~vo.~ N «tzap—.~.¢pn and...» TI! IQ I'VEISI RIVIi‘ so ' so 0.0720 0.0006 20 0.0000 0 0115 21 0.0010 0.0019 20 0.0025 0.0031 25 0.0001 0.0013 20 -0.oozo 0.0000 21 -o 9012 0.0000 10 -o 0000 0.0002 17 -0.ooov 0.0005 10 -o.ooz0 0.0000 1s -o.0o00 -0.0007 10 -0.0170 -0.0022 13 -0.0200 -0.0011 12 -0.0200 -0.0037 11 -0.0290 -0.0050 10 -0.0270 -0.0039 0 -0.0250 o0.0033 0 -0.0250 -0.0033 7 -0.0250 -0.0033 0 -0.0250 -0.0033 5 -0.0200 -0.0031 0 -0.01so -0.0021 3 0.0005 -0.0001 2 0.0110 0.0000 1 0.0502 0.0006 1s 0 :0 -0 0002 -0 0137 20 -. 0000 .. 0021 27 -0 0020 -0 0100 20 -0 0027 -0 0000 25 -0 0052 -0 01:7 20 -0 0000 -0 0000 21 -0.0003 -0.0050 10 -0.0050 -0.0100 17 -0 0030 -0 0120 10 -0 0005 -0 0100 15 0 0010 -0 0103 10 -0 0100 -0 0300 13 -0 0271 -0 0 0 12 -0 0322 -0 0000 11 -0 0130 -0 070: 10 -0 0300 -0 0035 0 -0 0101 —0 0 0 -0 0301 —0 0020 7 -. 0303 -0 0032 0 -0 0300 -0 0 s -0 0200 -0 0:90 0 -0 0200 -0 0011 3 -0 0093 -0 0230 2 -I 0068 -O 0172 1 -0 0000 -0 0101 5 0 10 -0.0211 -0.0101 20 -0.0020 -0.0010 27 -0.01s1 -0.0103 20 -0.o121 -0 0002 25 -0.0100 -0 0123 20 -0.0123 -0 0007 21 -0.0073 -0 0052 10 -o.ozso -0 01s: 17 -0.o101 -0 0120 10 -0.0100 o0.010s 15 -0.0217 -0.0200 10 -O.DSIQ -I.O¢30 1s -0 0703 -0.0300 12 -0.0071 -0. 11 -0 0003 -0.0710 10 -0 0020 -0.0050 0 -0 0735 -0.0001 0 -0 0735 -0.0007 1 -0 0703 -0. 2 0 -0 0101 -0.0000 3 -I 0691 -I.IS7Q 0 -. 0:07 -0 031 3 -0 0:00 -0 0200 2 -0.0231 -0.01s¢ I -I.0217 -0.0l¢! VIII! All 327 Ilfll'l'l IIIIIIVS II TIC IIVIII. TI! IIISY I! IIICI 13 VII Ifllfllll [III-I‘LII I' I‘ 18 ......C... I I u 0 1 D I O I O -....37 218 -..1.IQ Table 6.68 The sensitivity matrix SQ v'1 for the case where 1500 MVAR load L and 20.00 pu capacitive susceptance are connected at Bus 11 f “9 mam an cmuumccou mew mucmpqmumam m>_u.uaaau 3a oo.o~ new use. m<>z com“ «can; ammo we” to. u>m xwapae »o_>_u_m=om one $0.0 mpg.» 219 9N 9099 99\9N\99 "unca 99h9.91 9n 80—38 59 99898 Nth .xuflhtt wt» 8n 9»:&!an U>uh<¢w8 99 ~89 unmzh u~9~.91 NN99.9 9999.91 9999.91 9999.9 99N~.91 a 9N9~.91 9999.9 9999.91 9N99.91 9999.9 N9N~.91 N «999.91 9—N9.91 9999.91 Nun—.91 9999.9 ~—9-.91 9 9999.~1 ~99n.91 999N.91 n999.91 9999.9 9999.91 9 999N.N1 9999.91 9999.91 9999.91 9999.9 99‘9.91 9 N999.N1 9NN9.91 9999.91 99—9.91 9999.9 9999.91 9 9N99.N1 99s9.91 9999.91 99~9.91 9999.9 9999.91 9 9999.N1 9NN9.91 9999.91 99~9.91 9999.9 ~N99.91 9 9999.N1 ~N~9.91 9999.91 9999.91 9999.9 NN99.91 9 9~n9.N1 9999.91 nu~9.91 9999.91 9999.9 9N99.91 9a ~999.N1 9N99.91 n—99.91 9999.91 9999.9 9959.91 ‘— 9599.N1 99N9.91 «999.91 9999.91 9999.9 99N9.91 N— N99—.N1 9999.91 9959.91 NN99.91 9999.9 9999.91 9— 9999.~1 9—Nn.91 ~99N.91 NN99.91 9999.9 9999.91 9— 999—.91 9599.91 99~9.91 99—9.91 9999.9 9NN9.91 9— 9599.9 9599.91 9N99.9 99N~.9 9999.9 999—.9 9— 999n.9 9N99.9 99N9.9 9~99.9 9999.9 NN99.9 N— ~N99.91 9999.9 99-9.91 99N9.91 9999.9 N999.91 9n ~99N.9 9999.91 ~—99.9 9999.9 9999.9 N999.9 ~N 9n99.9 9999.91 ~9—N.9 N99~.9 9999.9 959—.9 9N 9999.91 999—.9 9999.91 9a99.91 9999.9 ~999.91 9N 99a9.9 NN99.9 «999.91 9999.91 9999.9 99—9.91 9N 99N9.9 99NN.9 9999.9 99~9.9 9999.9 99N9.9 NN 9959.9 99s9.9 N999.91 9~99.91 9999.9 9N99.91 9N s9s—.91 N999.9 9999.91 9999.91 9999.9 959—.91 99 ON 9N NN 9N 9n xnxhtt u>9 nth t<¢09¢9 9~9>9b~a~9¢hm 909999) Hg mam an umuumccou mew mucmuamumam S ms.».umamu an oo.o~ ecu two, m<>z coma aces: ammo as“ co9 c om chaas xyw>wupm=um asp ofi.m o_na» .>du>«90999u9 x«ub¢t u>99¢ 93h 99 398 9994 989 883990 ~99. map 99 999999 x«sh«h¢Gu8 9N met 9898b 999«.91 999«.91 9999.91 9999.«1 9N99.N1 «999.N1 9999.9 9999.9 «9N9.91 99NN.91 9999.91 9N N9N9.91 N999.91 9«99.91 9999.91 959N.91 9999.91 9N 9999.91 9«99.91 9999.91 999«.91 9999.91 9999.91 NN 9999.9 9999.9 9999.9 9999.9 9999.9 9999.9 9N 9«9N.91 9NN«.91 N99«.91 9N9«.91 N999.91 99N9.91 9« « N 9 9 9 999«.N1 NN9N.N1 9N9«.N1 999«.N1 999N.N1 99N9.N1 99«9.N1 9999.N1 N999.«1 9«9«.91 99N9.91 N9N9.91 N9N9.91 N9N9.91 9999.91 9999.91 9599.91 N9«9.91 N99N.91 ««99.91 9N 9N99.91 9999.91 9999.91 9999.91 NN99.91 «999.91 9N«9.91 9999.91 N99N.91 9N«9.91 9N 9999.91 9N99.91 9~N9.91 9999.91 9999.91 9999.91 9999.91 9«99.91 «999.91 99«9.91 NN 9999.9 9999.9 9999.9 9999.9 9999.9 9999.9 9999.9 9999.9 9999.9 9999.9 9N 9999.91 9N«9.91 9999.91 9999.91 N«99.91 N~N9.91 9999.91 9999.91 N9N9.91 9NN9.91 9« 9 N 9 9 9« «« N« 9« 9« 9— NN~9.9 999«.9 9«99.91 9999.9 ««99.9 NN«9.91 9999.9 N««9.9 9999.9 9N9«.91 5999.91 9999.9 9N99.9 9999.91 9999.91 9«9«.9 99N9.9 999N.9 9999.9 9N99H9 9N h«99.9 NNN9.9 99«9.91 9«99.9 99«N.9 N«99.91 9999.91 9999.9 N«99.91 9999.91 9N N9N«.9 9«99.9 N9N9.91 9N~9.9 999«.9 9999.91 9999.91 99«9.9 9«99.91 9999.91 NN 9999.9 9999.9 9999.9 9999.9 9999.9 9999.9 9999.9 9999.9 9999.9 9999.9 9N «9N«.9 9599.9 9999.91 9999.9 999«.9 9999.91 99«9.91 N9N9.9 9N99.91 99N« 91 9« 9« N« 9« «N 9N 9N 9N 9N 9N 99 x«¢b¢8 90099 uzh NN 9099 99\9N\99 .9999 $990999 9«9>a<2¢ >h~9«9 221 99 mam um umaumccou mgm mucmaqmumzm m>_u_umqmu an oo.o~ ucm a uno_ m<>z oomH mgmgz ammo mg» .O$ m cm x_.~ma »u_>_~_m=mm «‘9 -.m m_amp 9999.91 99 209:: 99 99999992 9998 9:9 9:59 823990 u>999auz 9 999 99939 N999.9N 9N 9999.99 9N 9999.9N NN 9999.91 9N N999.99 99 99 9899 889990 9099 0:9 9N 9099 99\9N\99 .9999 3990999 99999989 999999999 909999) 9N99.99 9999.9 999N.9 9999.9 9999.9 9N 9999.9 9999.999 N99N.9991 9999.9 9999.91 9N 999N.9 9999.9991 9999.N99 9999.9 9999.91 NN 9999.9 9999.9 9999.9 9999.99 N999.991 9N 9999.9 9N99.91 9N99.91 9999.991 9N99.999 99 9N 9N NN 9N 99 x9999: 9999 9:9 9N 9999 99\9N\99 .9999 8999999 99999989 999999999 9999999 222 6.2.4 Experiment I! After the generation at Buses 6 and 10 was removed, after the gene- ration at the Buses 2, 25, and 29 was increased by 200 MW and the line (9, 30) was outaged, the voltages in VCA 1 became extremely low, of the order of 0.8 pu. To improve the voltages (but not necessarily the PV and P0 stability or controllability of the system ) a 5.00 pu capacitor was connected at Bus 11 (Case 6). Then the question is whether the system is vulnerable to further reactive load increases in VGA 1. The results show that the reactive load need not increase significantly before we ex- perience voltage problems. Table 6.72 shows the voltage profile for a load of 200 MVAR at Bus 11. The voltages in VCA 1 drop below 1.0 pu and at some buses they drop down to 0.95 pu. Table 6.73 shows that for 300 MVAR reactive load at Bus 11, the voltages drop below 0.9 pu. For reactive load beyond 346 MVAR, the loadflow program has convergence problems. Comparison of the results from Experiments III and IV shows that: (a) a reactive load increase of 930 MVAR could be sustained if a com- pensating capacitor of 14.30 pu is connected in parallel but only a 346 MVAR load increase could be sustained if a 5.0 pu capacitor is connected in parallel at Bus 11. Thus, use of capacitors prevents loadflow convergence problems up to some point. (b) compensating reactive load increases by synchronous condensers rather than capacitors, causes no worsening of voltage problems or vulnerability to voltage collapse. (c) increasing capacitance has beneficial effects in maintaining voltage profile within a voltage control area, and it prevents reactive trans- fer across the weak transmission boundary up to some point. 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The very rapid increase in vulnerability to uncompensated reactive load increase occurs because the reactive power to supply this load increase occurs over the weak boundary of the voltage control area caus- ing it to further weaken. Our analysis in Chapter 5 indicates that his causes singularity of the Jacobian and ultimate voltage collapse due to loss of strong causality. 6.2.5 Experiment 1 The purpose of Experiment V is to show that voltage collapse will occur for a real power load increase at Bus 11 that is picked up by the swing bus, Bus 30, causing an increasing real power transfer across the Branches (4, 3) and (14, 15) of the weak transmission boundary of VCA 1. We succeeded in increasing real load to only about 280 MW before we experienced loadflow convergence problems. The voltage profiles provided in Table 6.74-6.78 corresponding to real load of 100, 200, 250, 270, and 280 MW at Bus 11, show that the voltages in VCA 1 decrease down to unacceptable levels and the voltage angles decrease (increase in magnitude). 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Real load, real generation as well as con- trolled voltages at the PV buses were left unchanged. This case is considered as Case 7. Table 6.79 shows that the voltage magnitudes are above 1.0 pu. and voltage angle differences are very small. From the same table, it can be noted that only the reactive generation at Bus 2 is close to Q The max‘ rest of the PV buses have large reactive reserves. The sensitivity matrix SQLV'1 has very small off-diagonal elements as can be seen in Table 6.80, indicating that a reactive load increase at a P0 bus will affect almost exclusively the voltage at that same bus. The matrix SVE is nonnegative and has no zero rows as can be seen in Table 6.81. Also, all the row sums of SVE are within .10 away from one. Therefore, the system is PQ strongly controllable. The matrix SQGQL as shown in Table 6.82 is nonnegative and has no zero rows. All its column sums are within less than .10 from one showing that the losses in the system are very small. Finally, the column sums of SQGE shown in Table 6.83 are all positive. Thus, the system is PV strongly controllable. These last results show that if the charging of the lines is signi- ficantly large, the system might have controllability problems that make it more vulnerable to voltage problems. Also, the modelling of the reactive loads in the system is a fundamental issue. The reactive loads should be modelled as partly reactive load and partly reactive impedance to ground, i.e. .— 1 234 - 2 QLi ’ kli 011 + k2: YQi Vi where 01' 1 is reactive load injection at Bus i YQi is reactive admittance to ground at Bus i kli’ k2i are positive numbers such that kli + k21 = 1. 235 ”unusuuus u--uu "mm. mmuuumuuu «ammummu ............... .................... ’33.".3Cu ”Rut” "133. 7 «an». uuum «in no. .I .. . . . ................... -.....e; .. “a... ---...m um mmmmmmm ................ 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Chapter 7 Conclusions and Recommendations for Future Research 7.1 Review System planners and operators realized that high or low voltage and voltage collapse problems threatening a voltage control area are related to insufficient reactive reserves in the area and misuse of switchable shunt capacitors to support voltages. Experience shows that the weak transmission boundary is incapable of large real and reactive transfers from one voltage control area to another. The voltage collapse problem was shown [1] to be associated with singularity of the Jacobian matrix. The Jacobian matrix is moving toward singularity as the negative off-diagonal elements Qij corresponding to lines on the weak transmission boundary get larger and eventually become positive. Identifying the buses that belong to a voltage control area and the branches that belong to the weak boundaries connecting voltage control areas is, thus, essential. A "coherency" based method for determining voltage control areas was developed in [8]. A voltage coherency measure was used to establish the groups of buses that have similar voltage changes for the set of 100 MVAR load increases at all PQ buses and 100 MVAR generation losses at all PV buses. This is similar to a method developed in [34], to determine coherent groups to be aggregated to form dynamic equivalents for tran- sient stabiltiy studies. A second method for identifying the voltage control areas and the branches in the weak transmission boundaries was developed in this Thesis. The algorithm is based on a similar one proposed by Zaborski [9] 239 240 for identifying coherent groups of generators that are aggregated to obtain dynamic equivalents for transient stability studies. This method determines a modified incidence matrix that is used to identify the groups of buses that are stiffly interconnected and, thus, identify the voltage control areas. The branches connecting buses in different voltage control areas are the weak transmission boundaries. This method requires significantly less computational effort than the coherency method because: (1) only a base case loadflow solution is required rather than a multiple contingency loadflow corresponding to the set of 100 MVAR reactive load increases at all PQ buses and 100 MVAR loss of genera- tion at all PV buses. (2) coherency measures need not be computed and ranked for all bus combinations (i,j) in the system. The number of combinations can be extremely large even for a small system, and (3) the determination of the branches that belong to the weak transmis- sion boundaries require little computation and storage; the deter- mination of voltage control areas requires modest computations. The simulations performed showed that for load increases in a vol- tage control area the generators in that area are the only ones that respond. This clearly shows the isolation of that voltage control area from the rest of the system. As the reactive load increases beyond the capabilities of the local generators and the corresponding buses become PQ buses, then generators outside the voltage control area start pro- viding the extra reactive power needed in the voltage control area. The reactive transfers across the voltage control area boundary stresses the boundary. The losses on the weak transmission boundary increase and the 241 voltages inside the voltage control area drop. Switchable shunt capaci- tors connected at some buses in the voltage control area succeed in bringing the voltages back to acceptable levels, but as it was shown this causes the system to become extremely vulnerable to load increase in the voltage control area as well as to line outages, especially those on the weak transmission boundary. An argument was made that real and/or reactive load increases weaken the voltage control area boundary and lead to singularity of the loadflow Jacobian, causing loss of causality and voltage collapse. Retention of strong causality which is a system condition that is derived based on a transient stability model, prevents voltage collapse but does not prevent severe bus voltage problems, and does not assure loadflow convergence. Causality can be tested by determining whether the Jacobian is singular at any particular operating point. Controllability as defined in Chapter 4 has not been shown to assure causality, although this may be possible. PV controllability describes the proper direction of the cause effect relationship between the changes in reactive power generated at every PV bus and both the reactive power injection changes at PO buses and the voltage setpoint changes at PV buses. PQ controlla- bility describes the proper direction of cause effect relationship be- tween voltage changes at PO buses and both the reactive power injection changes at P0 buses and the voltage setpoint changes at PV buses. Con- trollability can be tested by computing the sensitivity matrixes SQLV-l’ SVE’ SQGQL and SQGE and testing the following conditions and properties: (a) SQ V is an M matrix L (b) SVE is nonnegative and has no zero rows 242 (c) SQGQL is nonnegative and has no zero columns (d) the column sums of SQGE are all positive. Stability is also defined in Chapter 4 that shows the proper direction of cause effect between voltage changes at the PQ buses and reactive genera- tion changes at PV buses, and reactive load injection changes at P0 buses and voltage setpoint changes at PV buses, but with weaker requirements than PQ controllability. For stability the four sensitivity matries must satisfy the following weaker conditions: (a) SQLV is an M matrix (b) SVE is nonnegative (c) SQ Q is nonnegative G L (d) at least one of the column sums of SQGE is positive. Strong controllability establishes both magnitude and direction of the cause-effect relationship between the voltage changes at PO buses and reactive generation changes at PV buses, and reactive load injections at P0 buses and voltage setpoint changes at PV buses. Strong controlla- bility assures a tight or stiff control of voltage at P0 buses and reactive power generated at PV buses. Strong PQ controllability assures that the voltages at P0 buses are not very sensitive to reactive load disturbances. Strong PV controllability assures that the total reactive generation increase is almost the same as the reactive load increase and that the total reactive generation increases significantly if the voltage setpoint at any PV bus is raised. The system is strongly con- trollable if it is controllable and the following constraints are satis- fied: 243 (a) SQ V'1 has relatively smaller off diagonal entries in each column L than the diagonal entry, N (b)1-e_<_ X [S j= -1 for all i = 1,2,...,M and e of the order of 10%, VEJij < 1 + e (7.1) N (c) 1 -e: < 21 [S Jlj < 1+5: (7.2) i= QGQLU for all j = 1,2,...,M and c of the order 10 to 20%, . > k (7.3) for j = 1,2,...,N and k positive. The derivation of the sensitivity matrices, the definitions of sta- bility and controllability, and the derivations of conditions that assure stability, controllability and strong controllability are given in Chap- ters 3 and 4. A computer program was developed to compute the sensitivity matrices -1 , S and SQGE as well as row sums of SVE’ the column SQLV’ SoLv VE’ SQGQL sums of SQ Q and SQ E and check for the appropriate properties of these G L G sensitivity matrices. The computer program was applied to the 30 bus New England System to test its capabilities, to determine voltage control areas and the branches that belong to the weak transmission boundary. Five experiments were conducted to show that: (1) voltage collapse can occur if sufficient reactive reserves don't exist in a VCA, to meet the reactive load increases in that VCA, causing reactive power import across the weak transmission boundary, (2) voltage collapse can occur if excessive shunt capacitance exists in 244 a VCA due to line charging, power factor correction capacitors, or switchable shunt capacitors used for voltage control, (3) capacitive compensation for constant reactive load increase will aggravate voltage collapse. Synchronous condensers used to compensate for reactive load increases will not cause any change in security of the network, (4) real and reactive transfers across the weak transmission boundary of a voltage control area cause voltage and voltage collapse problems. These results suggest that satisfaction of P0 and PV controllability indicates voltage security and violation of PO controllability indicates imminent voltage collapse. Proximity to voltage collapse or voltage security is measured by the row sums of SVE and column sums of SQGQL and their relative closeness to conditions associated with voltage collapse and voltage security, respectively. 7.2 Recommendations for Future Research Future research on both PV and P0 stability and controllability could: (1) develop a sensitivity model that does not use the decoupled loadflow as a starting point. For example, if we assume APL=0 andibd=0, we can use the three equations; 0 = BZAB + CZAE + 02AM B A6 + C AE + 0 AV (7.4) 3 3 A06 3 AQL B A6 + C4AE + D4AV 4 to develop a sensitivity model similar to Model 11. This can be 245 achieved by solving the first equation for 0 to obtain 1 1 A6 = -32 CZAE - 32 024v (7.5) and then substitute it in the last two equation to obtain A0 = C 'AE + D 'AV G 3 3 (7_6) AQL = C4'AE + D4'AV where c ' = c - B '1c 3 3 3 2 2 c ' = c - B B '1c 4 4 4 2 2 (7 7) . _ -1 D3 ‘ D3 ' B3B2 D2 D ' = D - B B '10 4 4 4 2 2 The equations (7.6) are identical to the ones in (3.62). Therefore, a sensitivity model _ -1 av - sVE AE + SQLV AQL A0 = 5 AE - 5 A0 (7.8) G QGE QGQL L A9 = 56E AE + 560L AQL can be obtained, where the last equation was obtained by substi- tuting the first equation of (7.8) in (7.5). The principal advantage of this model is that it could be used to develop the relationship between strong causality, controllability, and strong controllability since the model used to analyze strong (3) 246 causality and controllability and strong controllability are iden- tical. Strong causality by its title only infers that there are cause-effect relationships that hold when strong causality holds but does not describe them. Controllability describes the direction of cause-effect relationships and shows that they apply to each bus and that they are different at PV and PO buses. Strong controllability describes quality and the direction the control of reactive power at PV buses and voltages due to the voltage setpoints AE and the dis- turbance AQL so that the controlled variables can be robustly and precisely controlled regardless of disturbances or operating con- dition changes. The derivation of the theoretical relationships between causality controllability and strong controllability may provide additional information and justification of these three conditions. investigate whether causality, controllability and strong controlla- bility tests can be applied at an equilibrium point (loadflow so- lution) or for any point along a transient trajectory resulting due to a fault, loss of generation, etc. disturbance. These causality, controllability, and strong controllability tests can be applied to at time steps in a transient stability simulation to investigate whether: (a) there are voltage collape problems that can only be investi- gated based on a transient stability simulation, (b) a modified loadflow formulation could be developed to test for voltage collapse due to loss of generation or fault contin- gencies. determine a pattern recognition based screening tool that could 247 screen line outage and loss of generation contingencies that could cause voltage collapse. The methodology would use A.C. distribution factors and would be based on a particular base case solution. (4) determine causes and cures for voltage collapse. The results in Chapter 5 show that weakening the boundary of a voltage control area causes loss of causality and thus voltage collapse. The determination of causes that weaken the boundaries of voltage con- trol areas can be undertaken based on analysis of the loadflow Jacobian. The identification of changes in real and reactive unit commitment, network configuration, voltage profile, real and reac- tive power dispatch that help relieve or alleviate voltage problems could be investigated. Changes in primary and secondary voltage controls could be investigated. Finally, discrete supplementary control (switchable series capacitors, reactors, line switching, etc.) that could relieve or alleviate voltage problems could also be investigated. APPENDIX Appendix A Definition A; (a) An n x m real matrix A = (aij) is called positive (nonnegative) and is denoted by A > 0 (A 3 0) if aij > O (aij 3 O) for all i,j, 1_<_i_<_nand15j<_m. (b) If -A > 0 (-A 3_0) then A is called negative (nonpositive) and is denoted byA< 0 (A30). (c) If A Z. 0 (A §_0) and there are i,j such that aij # 0 then A is called nonzero nonnegative (nonzero nonpositive) and is denoted by A > 0 (A < O). + + (d) If A = (aij) and B = (bij) are n x m real matrices and aij > bij (aij.i bij)’ then A is said to be greater than (greater or equal to) B and is denoted by A > B (A.: B). This can also be read as B is less than (less than or equal to) A and written as B3< A (3.3 A). Definition‘Ag Let Zn be the set of all n x n real matrices A = (aij) for which aijio for all i =j, 1:i,j in. Definition A; Let At: Zn' Then A is said to be an M-matrix if A is nonsingular 1 and A- > 0. Defnition 53 Let A be an n x n complex matrix. Then A is convergent if the sequence of matrices A, A2, A3,... converges to the null matrix 0, and is divergent otherwise. 248 249 Theorem A; If A is an n x n complex matrix, then A is convergent if and only if p(A) < 1, where p(A) = max {Ikil : l 5_i 5 n } (A1) Theorem Ag Let A 5 Zn' Then the following six statements are equivalent: (1) The matrix A is an M-matrix (2) There exists x > 0 such that Ax> 0 (3) There exists y > 0 such that Aty2> O (4) All the leading principal minor determinants of A are positive det I I > o k = 1,2,...,n (A2) (5) All the eigenvalues of A have positive real parts. (6) The diagonal entries of A are positive real numbers, and if we let D be the diagonal matrix whose entries are - -1 . dii - aii , 1 g_i g n (A3) then the matrix B=I-DA is nonnegative and convergent. Theorem A; Let A,B 6 Zn’ A 5_B and A be an M-matrix. Then 1 1 (a) A and B are nonsingular and A' .3 B' .1 0; and (b) q(B) z q(A) where q(A) = min {Re(AA)} (A4) 250 Definition 5; An n x n matrix A is called reducible if there is a permutation matrix P such that PAP = (A5) where A11 is an r x r matrix, 1.: r<: n. A matrix is irreducible if no such a permutation matrix exists. Theorem 53 An nxn matrix A = (aij) is irreducible if for all i,j , 1 §_i,j §_n, either aij f 0 or there are k,l,...,m such that aik akl"'amj f 0. Theorem Ag 1 Let A 6 Zn be an M-matrix. Then A' >’ 0 if and only if A is irre- ducible. Definition Ag i 1,2,...,n (A5) A matrix is called strictly diagonally dominant if for at least one i the inequality in (A6) is a strict inequality. An irreducible matrix is called irreducibly diagonally dominant if it is strictly diagonally dominant. 251 Theorem Ag Let A = (aij) be an irreducible n x n complex matrix, and x1,x2,..., xn be any n positive real numbers. If n 2 |a i=1 13' "i E a (A7) for all 1 g i g n, with strict inequality for at least one i, then p(A)_>_ on. TheoremlAZ Let A = (aij) be an n x n strictly or irreducibly diagonally domi- nant complex matrix. Then, the matrix A is nonsingular. If all the diagonal entries of A are in addition positive real numbers, then the eigenvalues A1 of A satisfy Re(>\i) >0 , 1 [A ..n |/\ 3 (48) Theorem A§ Let A = (aij) be an n x n irreducible matrix which is diagonally dominant. If a Z 0, 1 g,i g.n and a.. S 0 for all i f j. then A=0 is ii ij an eigenvalue of A if an only if n X a.. = 0 for all 1 < i < n (A9) i=1 ‘3 ' ‘ Theorem‘Ag Let A = (aij) 5 Zn have aii > 0 for all i. Then (a) if A is irreducibly diagonally dominant, A"1 > 0. (b) if A is diagonally dominant and equalities in (A6) hold for all i, 252 then A is singular, and (c) if the inequality in (A6) doesn't hold for some i then A is not an M- matrix. Theorem A10 Let A 6 Zn be an irreducibly diagonally dominant M-matrix, and A.1 = B = (bij)' Then bii-i bij > 0 i,j=1,2,...,n (A10) Theorem A11 Let A = (aij) be an arbitrary n x n real matrix and n A1 = .§ laijl , 1_: l §.n (A11) j-l in Then min {afi -A1. : liiin}: Rerlfor all k where A1, A2,...,k are the eigenvalues of the matrix A. n Appendix B Definition Bl Consider a network with n buses. We define the incidence matrix B = (Bij) of the network as follows: 1 if bus i is directly connected to bus j ij 0 otherwise for all i,j , 1 g i,j g n. The ”bus i is directly connected to bus i" is considered true for all i. The matrix B is a binary matrix. In other words, it consists of 0's and 1's. We define some binary operations on binary matrices that will be used in some Theorems later. Definition B2 ij) be two nxm binary matrices. Then the binary addition of A and B is the binary matrix A+B where Let A = (Aij) and B = (B (A+B)ij = Aij + Bij (Bl) Definitionlgg Let A = (Aij) be an nxm binary matrix and B = (Bkl) be an mxs binary matrix. Then the binary multiplication of A and B is the nxs binary matrix A-B where (A-B)ij = Ail'Bij + Ai2°32j + ... + A B (82) mi In both definitions, the operations "+" and u." are binary operations as . 0 1m known from Boolean algebra. 253 254 Theorem Bl Let B = (Bij) be the incidence matrix for a given network with n buses. Then (Bz)ij = (B-B) = 1 if and only if the buses i and j are i3 either directly connected or they are connected through one other bus. Proof Assume that i< j. Then (8 ij = BilBlj + ... + BiiBij + + Biijj + ... + Bianj (83) 2 _ . . - = ' (B )ij - 1 if and only if a term BikBkj - 1 for some k. If k 1 or k = 3, then Bii = 1 and Bij = 1, or Bij = 1 and Bjj = 1. In either case, Bij = 1 which implies that the Buses i and j are directly connected. If k f i and k f j, then Bik = 1 and Bkj = 1 implying that the Bus i is directly connected to Bus k and Bus k is directly connected to Bus j. Therefore, the two buses i and j are connected to each other through one other bus. The intervening Bus k might not be unique. This completes the proof. Theorem pg Let B = (Bij) be the incidence matrix of a given network with n buses. Then (Bm)ij = 1 if and only if the Buses i and j are connected to each other through as many as m - 1 other buses. If m 3,n-1, then (Bm)ij = 1, if and only if there is a path from the Bus i to the Bus j in the network. 4 9 w I DEA?!) 255 Theorem B; 1 . n- . . . . The matrix B 15 an equ1valence relation matrix. Epppf We need to show that the three properties - reflexive, symmetric and transitive - hold. (i) Reflexive (Bn'l)ii = 1 because the Bus i is trivially connected to itself. (ii) Symmetric (Bn'l)ij = 1 implies that (Bn'l)ji = 1. This is true 'because (Bn'1)ij = 1 implies that the Bus i is connected to Bus j through a path. Therefore, the Bus j is connected to Bus i through a path and hence, n-l _ (B )ji -1- (iii) Transitive n-1 = n-l = n-l = . . If (B )ij 1 and (B )jk 1, then (B )ik 1. This 15 true because if there is a path from i to j and a path from j to k, then there is a path from i to k. The critical thing in the proof of this theorem is that the paths have at most n-2 intervening buses. "'1 can be used to cluster the buses of the Therefore, the matrix B network into equivalence groups. The equivalence groups are disjoint, and for any two buses in a group there is a path connecting them, and for any two buses in two different groups there is not path connecting them. APPENDIX C Common Format 1" Base Case Loadflow Data 256 .= I00. N O O O. O. 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Fltlflfls HaslUKSFflJOI “” 257 Nfl'fiOhdozuflnon s ..uNnvnoh...uNnQn ~.o - —----:-:a—NNNNN~~~NNNNI’|”Gaga-nun nNhnans..~~.0-nh.NQOONOnhOnOOOOflhCOhOO .ununouono-ans .N n0.l\=f..h:..h.”- N :3.“ -.Ncnuh :- annual-unann- I-N NNNNNNA unnNNnnnthChhvvvno~~~QCOOON~OHQOO~~Ouv h --------—--—-~~~ NN N VIE tilts FflllON 258 at .0 v — ch: 5. Qty a v o..- On nu BIBLIOGRAPHY L 1] [ 2] [ 3] [ 4] [ 5] [ 6] [ 7] L 8] [ 9] [10] [11] Bibliography H.G. Kwatny, A.K. Pasrija, L.Y. Bahar, "Loss of Steady State Sta- bility and Voltage Collapse in Electric Power Systems," Pro- ceedings of the 24th Conference on Decision and Control, Ft. Lauderdale, FL, Dec. 1985. S. Abe, Y. Fukunaga, A. Isono, and B. Kondo, "Power Systems Voltage Stability,“ IEEE Trans. on Power Apparatus and Systems, Vol. PAS 101, No. 10. pp. 3830-3840, Oct. 1982. S. Abe, A. Isono, "Determination of Power System Voltage Stability, Parts I and 11," IEEE of Japan Vol. 968, No. 4, pp. 171-186, 1976. C.C. Liu and F.F. 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