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DATE DLMATE DUE' ‘,,_,,\.\\\‘ b“;\\ 0* - DATE DUE VOLTAGE COLLAPSE BIFURCATION OF A POWER SYSTEM TRANSIENT STABILITY MODEL By I-Pung Hu A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering 1990 r..-_--._3_i_ ABSTRACT VOLTAGE COLLAPSE BIFURCATION OF A POWER SYSTEM TRANSIENT STABILITY MODEL By I-Pung Hu A complete power system model, which is composed of an algebraic load flow model and dynamic generator/exciter model, is developed. This complete power system model is for- mulated to point out the similarities and differences between the load flow models and the complete power system model that includes electrical generator/exciter and generator me- chanical dynamics. Comparison of the load flow and the complete power system model simulation results indicate a converged load flow simulation may not imply voltage stabil- ity and will not accurately assess the proximity to voltage instability in the complete pow- er system model. The effects of line drop compensation, excitation system control, machine saturation, and field current limits must be modelled precisely if accurate assess- ments of proximity to voltage collapse are to be obtained. These effects can be accurately modelled in a transient stability simulation but are not accurately modelled in current load flow models. A modified load flow model and simulation method is proposed that in- cludes the effects of line drop compensation, excitation system control, machine satura- tion, and field current limits. Voltage instabilities are classified into two categories. Load flow voltage instability is caused by supply and demand problem. Dynamic voltage instability is caused by the insta- bility of the flux decay and exciter dynamics. Four voltage bifurcation tests, algebraic, al- gebraic/dynamic, dynamic/algebraic, and flux decay bifurcation tests are developed in this thesis. The algebraic bifurcation test can identify the supply and demand problem in the distribution system. algebraic/dynamic and dynamic/algebraic bifurcation tests can detect the instability of the generator dynamics. These tests are applied to analyze a two bus sys- tem and a twelve bus system. The results indicate the dynamic generator/exciter portion of the complete model becomes voltage unstable before the algebraic load flow portion of the complete model violates the widely used load flow based tests for voltage instability. Thus, new limits for stable operation must be established based on the instability of the dynamic portion of the complete power system model. Simulations show that generators in a coherent group of buses will lose their flux decay voltage stability when the reactive generation reserves in that coherent group of buses are exhausted. ACKNOWLEDGEMENTS Special thanks go to my thesis advisor, Dr. Robert A. Schlueter.This dissertation would never have been done without his excellent directions. Most importantly, the many quali- ties he demonstrated as a distinguished scholar and professional will always be my guid- ance as I pursue my career. I would also like to thank my committee members, Dr. DemosGelopulos, Dr. Hassan Kha- lil, Dr. Fathi Salam, and Dr. Norman Hills, who provided professional opinions for me af- ter reading my dissertation. Finally, the patience and support of my family and my fiancee, which is greatly appreciat- ed, helped me through this time of great challenges. This thesis is supported by EPRI (Electric Power Research Institute) in project RP-3040 and by Detroit Edison Company. TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES LIST OF SYMBOLS CHAPTER 1 INTRODUCTION 1.1 Voltage Collapse 1.2 The Purpose of This Thesis 1.3 Load Flow Collapse 1.4 Review of Load Flow Voltage Collapse Methods ..... 1.5 Review of Dynamic Voltage Collapse Methods ........ 1.6 Thesis Contribution CHAPTER 2 POWER SYSTEM MODELING 2.1 Introduction 2.2 Single Bus System Model 2.2.1 Mechanical Dynamics ...................................... 2.2.2 Flux Decay Dynamics ...................................... 2.2.3 Excitation System Dynamics .......................... 2.2.4 Algebraic Equation Model ................................ xiv CHAPTERS CHAPIER4 2.3 General Power System Model ...................................... 2.4 Comparison of Load Flow and General Power System Steady State Simulations TESTS FOR VOLTAGE COLLAPSE ................................ 3.1 Introduction 3.2 Classification of Types of Voltage Instability .............. 3.2.1 Load Flow Voltage Instability .......................... 3.2.2 System Voltage Instability ............................. 3.3 Voltage Instability Tests 3.3.1 Model Linearization 3.3.2 System Bifurcation Test ................................... 3.3.3 Algebraic Bifurcation Test ................................ 3.3.4 Algebraic/Dynamic System Bifurcation Test 3.3.5 Dynamic/Algebraic System Bifurcation Test 3.3.6 Flux Decay Bifurcation Test ............................. 3.4 Relationship to Literature SIMULATION RESYULTS ON VOLTAGE INSTABILITY OF POWER SYSTEM MODELS ...................................... 4.1 Simulation Results of A TWO Bus Power System Model 4.1.1 Mathematical Models ...................................... 4.1.2 Two Bus System Simulation With Excitation Control Included 4.2 Load Flow and Algebraic Bifm'cations .......................... 4.2.1 Introduction 4.2.2 Load Flow Voltage Instability Simulations ..... 28 36 36 36 37 38 4o 41 42 43 46 48 51 57 62 67 68 78 94 94 4.2.3 Algebraic Voltage Instability Simulations ..... 4.3 Dynamic/Algebraic Voltage Instability Simulations 4.4 Algebraic/Dynamic Voltage Instability Simulations CHAPTER 5 REVIEW AND TOPICS FOR FUTURE REASEARCH 5.1 Review 5.2 Topics for Future Research APPENDIX A MODEL LINEARIZATION A.1 Mechanical Dynamics A.2 Saturation Function SD A.3 Flux Decay Dynamics A.4 Excitation System Dynamics ...................................... A.4.l Line Drop Compensation ................................ A.4.2 Linearization of Excitation System Dynamics .. A.5 Power Flow A.5.1 Real Power Linearization for Generator and Terminal Buses A.5.2 Reactive Power Linearization for Generator Buses A.5.3 Reactive Power Linearization for Terminal Buses A.5.4 Real Power Linearization for The Network ..... A.5.5 Reactive Power Linearization for The Network APPENDIX B SENSITIVITY MATRD( DEVELOPMENT AND MATHEMATICAL BACKGROUND ................................ 111 132 150 150 155 157 157 157 159 161 161 163 163 170 B.1 Sensitivity Matrix Development ....... B.2 Condition Number ......................... B.3 Nonsingularity of Matrix A APPENDIX C SIMULATION RESULTS C.l Load Flow Voltage Instability ............. C.2 Algebraic Voltage Instability ............. C.3 Dynamic/Algebraic Voltage Instability C.4 Algebraic/Dynamic Voltage Instability LIST OF REFERENCES ......................... ......................... ....................... ....................... 177 178 185 185 191 239 283 Table 4.1 Table 4.2 Table 4.3 Table 4.4 Table 4.5 Table 4.6 Table 4.7 Table 4.8a Table 4.8b Table 4.9 Table 4.10 Table 4.11a Table 4.11b Table 4.12 Table 4.13 LIST OF TABLE Base case data for the two bus system simulation ............. Voltage instability tests for Q=0.07 Voltage instability tests for Q=0.27 Voltage instability tests for Q=0.32 Voltage instability tests for Q=0.37 Voltage instability tests for Q=0.76 Voltage instability tests for Q=0.8326 .................................. Summary of voltage instability test for high voltage solution Summary of voltage instability test for low voltage solution Load flow simulation for the increase of line reactances at L2 and L3 Changes of the ratio of the determinant of the algebraic test matrix and the jacobian matrix Generator data for the twelve bus system ............................ Exciter data for the twelve bus system .................................. The equilibrium point of 10 War load at bus TERM3 .......... The eigenvalues of the dynamic/algebraic bifurcation test 84 85 87 88 89 91 93 93 97 . 115 Table 4.14 Table 4.15 Table 4.16 Table 4.17 Table 4.18 Table 4.19 Table 4.20 Table 4.21 Table 4.22 Table 4.23 Table 4.24 Table 4.25 Table 4.26 matrix and the flux decay bifurcation test matrix for 10 MVar load at bus TERM3 115 The equilibrium point of 50 MVar load at bus TERM3 ........... 117 The eigenvalues of the dynamic/a1 gebraic bifurcation test rmtrix and the flux decay bifurcation test matrix for 50 MVar load at bus TERM3 117 The equilibrium point of 100 MVar load at bus TERM3 ........ 119 The eigenvalues of the dynamic/algebraic bifurcation test matrix and the flux decay bifurcation test matrix for 100 MVar load at bus TERM3 119 The equilibrium point of 110 MVar load at bus TERM3 ........ 120 The eigenvalues of the dynamic/algebraic bifurcation test matrix and the flux decay bifurcation test matrix for 110 MVar load at bus TERM3 120 The equilibrium point of 120 MVar load at bus TERM3 ........ 121 The eigenvalues of the dynamic/algebraic bifurcation test matrix and the flux decay bifurcation test matrix for 120 MVar load at bus TERM3 121 The equilibrium point of 135 MVar load at bus TERM3 ........ 122 The eigenvalues of the dynamic/algebraic bifurcation test matrix and the flux decay bifurcation test matrix for 135 MVar load at bus TERM3 (without exciter in) Ratio of the condition number (Algebraic) Ratio of the condition number (algebraic/dynamic) ................. 139 The eigenvalues of the algebraic/dynamic bifurcation test matrix and the flux decay bifurcation test matrix for 100 MVar load at bus LOAD2 139 Table 4.27 Table 4.28 Table 4.29 Table 4.30 Table 4.31 Table 4.32 Table 4.33 Table C.1.l Table C.1.2 Table C.1.3 Table C. 1.4 Table C.1.5 Table C.2.1 Table C.2.2 The eigenvalues of the algebraic/dynamic bifurcation test matrix and the flux decay bifurcation test matrix for 120 MVar load at bus LOAD2 The eigenvalues of the algebraic/dynamic bifurcation test matrix and the flux decay bifurcation test matrix for 140 MVar load at bus LOAD2 The eigenvalues of the algebraic/dynamic bifurcation test matrix and the flux decay bifurcation test matrix for 160 MVar load at bus LOAD2 The eigenvalues of the algebraic/dynamic bifurcation test matrix and the flux decay bifurcation test matrix for 170 MVar load at bus LOAD2 The eigenvalues of the algebraic/dynamic bifurcation test matrix and the flux decay bifurcation test matrix for 170 MVar load at bus LOAD2 (satrn‘ation) The eigenvalues of the algebraic/dynamic bifurcation test matrix and the flux decay bifurcation test matrix for 150 MVar load at bus LOAD2 (saturation) Eigenvalues for different load levels Load flow simulation for the regular line reactances Load flow simulation for two times line reactances Load flow simulation for three times line reactances Load flow simulation for four times line reactances Load flow simulation for three times line reactances Equilibrium point (algebraic 40 MVar) Equilibrium point (algebraic 45 MVar) cccccccccccccc ooooooooooooooooo .............. oooooooooooooo ccccccccccccccccccccccccccccccccccc 141 141 142 142 148 148 149 Table C.2.3 Table C.2.4 Table C.2.5 Table C.2.6 Table C.2.7 Table C.2.8 Table C.2.9 Table C.2.10 Table C.2.11 Table C.2.12 Table C.3.1 Table C.3.2 Table C.3.3 Table C.3.4 Table C.3.5 Table C.3.6 Table C.3.7 Table C.3.8 Table C.3.9 Table C.3.10 Table C.3.ll Table C.3.12 Table C.3.13 Table C.4.1 Table C.4.2 Table C.4.3 Table C.4.4 Equilibrium point (algebraic 50 MVar) ................................... 194 Equilibrium point (algebraic 55 MVar) ................................... 194 Equilibrium point (algebraic 58 MVar) ................................... 195 Equilibrium point (algebraic 60 MVar) ................................... 195 Output for algebraic 40 MVar 196 Output for algebraic 45 MVar 198 Output for algebraic 50 MVar 200 Output for algebraic 55 MVar 202 Output for algebraic 58 MVar 204 Output for algebraic 60 MVar 206 Equilibrium point (dynamic/algebraic 10 MVar) .................... 209 Equilibrium point (dynamic/algebraic 50 MVar) .................... 209 Equilibrium point (dynamic/algebraic 100 MVar) .................... 210 Equilibrium point (dynamic/algebraic 110 MVar) .................... 210 Equilibrium point (dynamic/algebraic 120 MVar) .................... 211 Equilibrium point (dynamic/algebraic 135 MVar) .................... 211 Output for dynamic/algebraic 10 MVar ................................... 212 Output for dynamic/algebraic 50 MVar ................................... 216 Output for dynamic/algebraic 100 MVar ................................ 220 Output for dynamic/algebraic 110 MVar ................................ 224 Output for dynamic/algebraic 120 MVar ................................ 228 Output for dynamic/algebraic 135 MVar (with exciter) ........ 232 Output for dynamic/algebraic 135 MVar (without exciter) ..... 236 Equilibrium point (algebraic/dynamic 100 MVar) .................... 240 Equilibrium point (algebraic/dynamic 120 MVar) .................... 240 Equilibrium point (algebraic/dynamic 140 MVar) .................... 241 Equilibrium point (algebraic/dynamic 160 MVar) .................... 241 xii Table C.4.5 Table C.4.6 Table C.4.7 Table C.4.8 Table C.4.9 Table C.4.10 Table C.4.11 Table C.4.12 Equilibrium point (algebraic/dynamic 170 MVar) .................... 242 Output for algebraic/dynamic 100 MVar ................................ 243 Output for algebraic/dynamic 120 MVar ................................ 249 Output for algebraic/dynamic 140 MVar ................................ 255 Output for algebraic/dynamic 160 MVar ................................ 261 Output for algebraic/dynamic 170 MVar ................................ 267 Output for algebraic/dynamic 170 MVar(without exciter in) .. 273 Output for algebraic/dynamic 150 MVar(without exciter in) .. 278 Figure 1.1 Figure 2.1 Figure 2.2 Figure 2.3 Figure 2.4 Figure 2.5 Figure 2.6 Figure 3.1 Figure 4.1 Figure 4.2 Figure 4.3 Figure 4.4 Figure 4.5 LIST OF FIGURE Voltage Collapse Phenomenon Air Gap Saturation Excitation System Model Freld Current Limit Controller Twelve bus test system model Comparaison of the solutions of load flow model and general power system model (K A=50 ) Comparaison of the solutions of load flow model and general power system model (K A=200 ) Algorithm for identifying the linearly dependent rows in the matrix [Cl D102] TWO Bus Power System Model Load flow and equilibrium manifolds for two stable equilibrium points Load flow and equilibrium manifolds for loss of causality at low voltage solution Load flow and equilibrium manifolds with one stable and one unstable equilibrium points Load flow and equilibrium manifolds with one intersection xiv 31 32 49 67 71 71 73 73 Figure 4.6 Figure 4.7 Figure 4.8 Figure 4.9 Figure 4.10 Figure 4.11 Figure 4.12 Figure 4.13 Figure 4.14 Figure 4.15 Figure 4.16 Figure 4.17 Figure 4.18 Figure 4.19 Figure 4.20 Figure 4.21 Figure 4.22 Figure 4.23 Figure 4.24 Figure 4.25 Figure 4.26 Figure 4.27a Load flow and equilibrium manifolds with no intersection ..... Load flow and equilibrium manifolds for different field voltages Load flow and equilibrium manifolds for different reactive loads Load flow and equilibrium manifolds for different real power loads Algorithm of a computer program that includes the exciter effects in a two bus system mode Procedure of two bus simulation Phase portrait for Q=0.07 Phase portrait for Q=0.27 Phase portrait for Q=0.32 Phase portrait for Q=0.37 Phase portrait for Q=0.76 Phase portrait for Q=0.8326 Phase portrait for Q=0.85 A twelve bus test system Load bus voltages at different line reactances Line losses for diflerent line reactances Angle differences of HST2/LOAD2 and HST3/LOAD2 at different line reactances ooooooooooooooooooooooo oooooooooooooooooooooooooooooooo Algebraic bifurcation simulation (field voltage) ....................... Algebraic bifurcation simulation (internal bus voltage) Algebraic bifurcation simulation (field current) ....................... Algebraic bifmcation simulation (terminal bus voltage) Algebraic bifurcation simulation (Q-V curve) XV ooooooooooooooooooooooo 74 76 76 79 81 84 85 87 88 89 91 92 95 99 99 101 105 106 _ r..- .———-—-————_._ Figure 4.27b Figure 4.28 Figure 4.29 Figure 4.30 Figure 4.31 Figure 4.32a Figure 4.32b Figure 4.33a Figure 4.33b Figure 4.33c Figure 4.33d Figure 4.34 Figure 4.35 Figure 4.36 Figure 4.37 Figure 4.38 Figure 4.39 Figure 4.40 Figure 4.41 Q—V curve (2 time shunt capacitance, 1.5 times series reactance) Time simulation of E'q at 135 MVar (without exciter in) ..... Time simulation of [id at 135 MVar (without exciter in) ........ Trme simulation of VT at 135 MVar (without exciter in) ........ Q-V curve at bus TERM3 The reverse action of the reactive power generation ................. The reverse action of the reactive load/generation ................. Simplified exciter/flux decay model Simplified exciter/flux decay model (SVE) .................................. Root Locus for a generator/exciter model (positive gain) ........... Root Locus for a generator/exciter model (negative gain) .......... Q-V curve for algebraic/dynamic simulation .......................... Generator capability curve Time simulation of the internal bus voltage (170 MVar) ........ Time simulation of the field current (170 MVar) .................... Time simulation of the terminal bus voltage (170 MVar) ..... Time simulation of the internal bus voltage (150 MVar) ........ Time simulation of the field current (150 MVar) .................... Time simulation of the terminal bus voltage (150 MVar) ..... 109 125 125 126 127 129 129 130 130 131 131 LIST OF SYMBOLS number of generators with no field current upper limit violation. Internal, terminal, or high side transformer buses connected to these generators are called controllable. number of generators with field current limits violations. Internal, terminal, or high side transformer buses that are connected to these generators are called uncontrol- lable. number of load buses. i=1,...,l j=1,...,m k=1,...,n controllable internal bus voltage uncontrollable internal bus voltage conuollable terminal bus voltage uncontrollable terminal bus voltage controllable high side transformer bus voltage uncontrollable high side transformer bus voltage ¢ . Tr" Tj: load bus voltage controllable internal bus angle uncontrollable internal bus angle controllable terminal bus angle uncontrollable terminal bus angle controllable high side transformer bus angle uncontrollable high side transformer bus angle load bus angle controllable internal bus frequency uncontrollable internal bus frequency controllable internal bus field voltage uncontrollable internal bus field voltage, a constant mechanical power supplied by the steam turbine at controllable generator mechanical power supplied by the steam turbine at uncontrollable generator real power generated at controllable generator real power generated at uncontrollable generator real power received at controllable terminal bus real power received at unconnellable terminal bus xviii d . QHE' real power load at controllable high side transformer bus real power load at uncontrollable high side transformer bus real power load at load bus reactive power generated at controllable generator reactive power generated at uncontrollable generator reactive power received at controllable terminal bus reactive power received at uncontrollable terminal bus reactive power load at controllable high side transformer bus reactive power load at uncontrollable high side transformer bus reactive power load at load bus measured voltage stabilizer output voltage amplifier output voltage VHF-i: reference voltage ' o 1 d0!” ' o 1 mi. generator direct axis transient open circuit time constant at controllable internal bus generator direct axis transient open circuit time constant at uncontrollable internal bus generator per unit inertia constant at controllable internal bus generator per unit inertia constant at uncontrollable internal bus generator load damping coefficient at controllable internal bus generator load damping coefficient at unconu'ollable internal bus regulator input filter time constant regulator stabilizing circuit time constant exciter time constant amplifier input filter time constant line drop compensation reactance regulator stabilizing circuit gain exciter self excitation at full load field voltage amplifier circuit gain rotating exciter saturation at ceiling voltage generator saturation function applied to direct axis reactance at controllable internal bus generator saturation function applied to direct axis reactance at uncontrollable in- ternal bus transient direct axis reactance at controllable internal bus transient direct axis reactance at controllable internal bus steady state direct axis reactance at controllable internal bus steady state direct axis reactance at uncontrollable internal bus xqi: steady state quadrature axis reactance at controllable internal bus 14].: steady state quadrature axis reactance at uncontrollable internal bus CHAPTER 1 INTRODUCTION 1.1 Voltage Collapse Voltage collapse of the interconnected power system has been observed fiequently and is of concern to utilities around the world. Voltage collapse has been associated with the transfer of power over long distances and the trend of siting generators far from load cen- ters due to environmental and political concerns. Voltage collapse is a slow continuous decline of voltage over a 10 to 20 minute interval and followed by a rapid decline of voltage (Fig. 1.1). In some cases, this voltage decline can occur over an interval as short as one minute and yet in other rare cases the voltage de- cline can occur over a several hour period. Although much research has been undertaken to establish the precise model and causes of voltage collapse, there still isn't a complete understanding or agreement on the exact mod- el to be used to simulate voltage collapse. This thesis develops a precise model and inves- tigates the contribution of each element in this precise model to voltage collapse. After the precise model is developed and understood, the causes of voltage collapse, the measures or conditions for assessing the proximity to voltage collapse are investigated. The operat- ing and planning criteria that will ensure adequate safety margins against voltage collapse, and the network enhancement su'ategy that would most effectively and economically in- crease margins for voltage collapse problems are then topics for future investigation. Voltage Figure 1.1 Voltage collapse phenomenon 1.2 The Purpose or This Thesis The purpose of this thesis is to (a) greatly extend the understanding of the modeling of generator dynamics needed to accurately assess proximity to voltage collapse; (b) show that the causes of voltage collapse not only occur in the algebraic equa- tions of the model but also in the differential equations of the model; (c) determine tests that indicate the proximity to voltage collapse; (d) extend the understanding of the various types of voltage collapse and how each is caused to occur. 1-3 Load Flow Collapse Tr_———_———'———__——" 3 The Electric Power Research Institute (EPRI) funded research at Michigan State Universi- ty [1,2] focuses on the causes of voltage collapse in a load flow model that does not in- clude generator dynamics. The EPRI funded study of voltage collapse is thus restricted to only the algebraic equations (load flow) model of a power system. This EPRI research has shown that voltage collapse problems are associated with the following three factors (a) the increased loading of the transmission lines; (b) shunt capacitive reactive power withdrawal and increased line losses (12X) with voltage drop; (0) inadequate local and neighboring reactive generation support. The research pointed out that voltage collapse is associated with reactive power supply de- mand problems in voltage control areas. These voltage control areas are groups of buses that are coherent in both transient and steady state response in both voltage magnitude and phase to all disturbances that occur outside the voltage control area. This type of coheren- cy is due to the weakness of the transmission network outset that isolates the coherent gTOuP of buses called a voltage control area. An algorithm was developed that could deter- mine these voltage control areas for very large data bases. Voltage collapse can occur if (a) there are no reactive generation reserves on synchronous generation or on static var compensators, or if under load tap changers are at the limits on tap changing action, or if there is no capacitive reactive reserve on mechanically switched capacitors. If a voltage control area has no supply of reactive power fi'om any of the above sources, maybe vulnerable to severe voltage decline. The voltage decline can be large because the reactive power voltage jacobian elements associated with branches in the voltage control area boundary are small compared with branches connecting buses within the voltage control 1F— 4 areas. (The algorithm for determining voltage control areas identifies and eliminates the branches with the smallest reactive power jacobian elements connected to each bus. The groups of buses that are isolated by the elimina- tion of these branches with the smallest reactive power voltage jacobian ele- ments are the voltage control areas) The voltage decline needed to import reactive power across the branches in the voltage control area boundary with small jacobian element values must necessarily be large; (b) The weak voltage control area boundary branches clog up and can’t effec- tively transfer reactive power between voltage control areas. These boundary branches not only clog up but become “drains”, where both ends of the branch send reactive power into the branch to meet the large 12X losses on the branch. Branches become drains with large SIL loading levels as bus voltage drops on both ends of the branch or as real power transfer over the branch increases. A boundary branch that acts as a drain does not effectively ship reactive power from one voltage control area with reactive reserves to a voltage control area that has no reactive reserves and needs reactive power. A branch that acts as a drain sucks the reactive power needed to meet it’s 12X reactive losses from both buses it connects in the two voltage control areas. The voltage control area with the higher voltage level will provide more of the reactive power needed to meet 12X losses on the branch; (c) Voltage control areas that do not have reactive power reserves and have boundary branches with drain problems have difficulty in meeting increased reactive demand. If the voltage control area is at EHV (Extra High Voltage) voltage level and is connected by long transmission lines or underground lines, the shunt capacitive support may be large. If voltages drop in a voltage control area with (1) no reactive reserves (2) a weak boundary with a significant number of branches with drain problems (3) significant shunt capacitive reactive power support, the large shunt capacitive reactive power withdrawal at buses in the voltage control area with voltage drop may not be met by the limited amount of reac- tive power that can be drawn in across the voltage control area boundary. This results in a lack of a stable load flow solution and thus voltage collapse. As a summary, the increased loading of transmission lines clog up the ability of these lines to supply reactive power to the load area, line losses increase the reactive power load in the load area, and capacitor reactive power withdrawal decreases the local reactive power supply in load areas. If there is no other way to bring in reactive power support, the volt- age will keep decreasing until a point where the load flow solution associated with the buses in the load area can not be solved. This kind of voltage instability is caused by reac— tive power supply and demand problems and has nothing to do with the flux decay and ex- Citcr dynamics of the generator. This kind of voltage instability is called load flow voltage ininability. It will be shown in this thesis that load flow voltage instability could be one of the causes of system voltage instability. 1.4 Review of Load Flow Voltage Collapse Methods Load flow voltage collapse methods, developed by using load flow model and solution al- gorithms, have been widely used in the utilities for voltage collapse analysis. It is assumed that a lack of convergence to a load flow solution is related to voltage collapse. The singu- x511'in of a jacobian matrix or part of a jacobian matrix indicates that the load flow solution mgorithm may not converge to a solution and that the solution (if computed) may be ‘a -a‘ ‘F_———— 6 point of bifurcation (a point where two or more solutions merge). Tamura[12,13] showed that existence of closely related multiple load flow solutions were likely to appear under heavy load condition. A pair of closely located load flow solutions seems to be related to the voltage collapse. The closely located load flow solutions are caused by the singularity of the Jacobian matrix. The voltage instability proximity indica- tor developed in [12,13] is basically another method to detect the singularity of the jacobi- an matrix. Sensitivity analysis based voltage collapse tests place conditions on the relationship be- tween changes in states or outputs and changes in inputs. Bonemans [28] provided condi- tions of sensitivity matrices S 9091. and SE}, for voltage stability. Carpentier [8] provided a condition for SQGQL' Schlueter [3] defined PO and PV controllability and developed sen- sitivity matrix tests on S Qa E, SVE, S QaQL’ and S3”, that assure PQ and PV controllability. It is shown that all lmown tests for voltage collapse can be derived based on assuming PQ and PV controllability hold. It Will be shown in this thesis that the load flow model based voltage collapse tests are in- COI'rect, especially when the system is stressed. The load flow voltage collapse methods only investigate the supply and demand problems of the transmission and distribution sys- tem. The instability of the generator dynamics can not be analyzed by load flow voltage COllapsc methods. It is shown in this thesis that the instability of generator dynamics will cause one type of voltage instability of the power system and that load flow methods in- VCStigate the other type of voltage instability. 15 Review of Dynamic Voltage Collapse Methods Dynamic voltage collapse methods take the dynamics of the generator into account. Only mechanical dynamics of the generator have been included in the dynamic model used in L A [7, 27]. The importance of flux decay and excitation system dynamics to voltage collapse have been pointed out by Schlueter [1 ,2,3,4]. Venikov [7] recognized that a sign change of the determinant of the jacobian of equilibri- um equations of a transient power system stability model may indicate voltage instability of the power system. Kwatny [27] showed that the static bifurcations of the equilibrium (load flow) equations were associated with either voltage collapse or steady state angle stability. Kwatny [27] separated divergence instability (singularity of the jacobian of the equilibrium equations of both the differential and algebraic equations) from loss of causal- ity (singularity of the jacobian of just the algebraic equations). The implications of diver- gence instability and loss of causality with regard to voltage instability will be studied more completely in this thesis. The differential equations used in this model represented only the mechanical dynamics of the generator. The algebraic equations represent the real and reactive power balance equations at load buses. Schlueter [1,2,3,4] extended Kwat- ny’s [27] work by developing conditions for static voltage collapse that included the flux decay dynamics and excitation system dynamics of the generators. Loss of causality in this extended model will be associated with load flow voltage collapse and divergence sta- bility will be associated with dynamic voltage instability. Schlueter [1,2,3,4] defined pa- rameters 7.. that change slowly over time and can cause the equilibrium point to move to a mint of bifurcation. Schlueter [1,2,3,4] also showed that PV and PQ controllability could asSure. that the transient stability model of [27, 7] could not experience divergence insta- bility or loss of causality. Sauer and Pai [19] presented the relationship between a power SYstem dynamic model and standard load flow model. Only generator mechanical dynam- ics Were included in his model. This thesis shows that instability of the flux decay and excitation system dynamics may mslllt in voltage collapse. The causes of flux decay and excitation system instability are identified. TF—_——_—_ 8 1.6 Thesis Contribution This thesis will investigate system voltage instability which occurs in a power system model that includes both generator and exciter dynamics and the algebraic equations asso- ciated with the real and reactive power balance equations at buses in the transmission net- work. A necessary condition for system voltage instability is that the jacobian of the equations that describe the equilibrium point of this set of differential and algebraic equa- tions (general power system model) be singular. A necessary condition for load flow volt- age instability is that the jacobian of the real and reactive power balance equations of a load flow model be singular at some equilibrium point. Since the real and reactive power balance equations in a load flow and in the mid term transient stability model are different, the necessary conditions for load flow voltage collapse may not indicate a system voltage instability and vice versa. The primary contribution of this thesis is the development of a power system model that includes the following three factors: (1) air gap saturation in the synchronous machine model, (2) line drop compensation in the excitation system model, (3) field current limitation and the field current limit controller in the excitation system, and Shows their influence on proximity to voltage collapse. All of these factors are associ- ated with how voltage control is accomplished in the set of differential and algebraic equa- ti9118 of the system voltage instability model. These factors are greatly simplified in a load flow model and can cause large error in predicting proximity to voltage collapse. It will be shown that the load flow equilibrium point and the equilibrium point of the set of differential and algebraic equations (general power system model) that describe system I'VE—— voltage instability model diverge as the equilibrium points of the two models approach voltage instability. These three factors (a) are the factors that cause the differences in the equilibrium point of the load flow and the equilibrium point of the set of differential and algebraic equa- tions; (b) cause the tests for system voltage instability to give different results from the test for load flow voltage instability even when the same equilibrium point is used in both models. The line drop compensation decides the bus or fictitious point in the network where volt- age is going to be controlled by the generator exciter. In most cases, the generator terminal voltage and current are measured because these variables are easier to measure. Line drop compensators utilize the measured terminal voltage and current and a model of the net- work connected to the generator to calculate the voltage and current at some other point (fictitious or real) where the exciter attempts to hold voltage to some reference value. The actual point in the network where the exciter’s line drop compensator selects to control VOItagc, significantly changes the amount of the local reactive power demand supplied by this generator and can significantly affect the voltages observed in the network after a con- tingttncy or a change in operating conditions. The air gap saturation of the generator is the magnetic saturation of iron in the rotor and in the Stator. Before saturation, the field current will generate enough flux to induce suffi- °i°nt stator voltage to control the generator terminal bus voltage. When air gap saturation happens, the ability of the exciter to increase the induced stator voltage will be reduced Wen though field current is increased. Field current limits are thermal limits of generator rotor and prevent overheating of the Smerator rotor. The field current limit is given as a curve that plots the level of field cur- 10 rent versus the time duration of that level of field current. If the field current limit is ex- ceeded, there will be a camel to disable the exciter in the generator and reduce the field current down to continuous rating levels that can be sustained indefinitely. Air gap satura- tion effects certainly contribute to a machine reaching its field current limit. The second contribution of this thesis is the classification of the various types of system voltage instability and the development of proximity test for each type of system voltage instability. The necessary condition for system voltage instability is based on singularity of the jacobian of the set of differential and algebraic equations that describe the equilibri- um point for this general power system model. The necessary conditions for load flow voltage instability are based on the singularity of the jacobian of the algebraic equations of the load flow model. The jacobian for testing for system voltage instability is evaluated at the equilibrium point for the set of differential and algebraic equations. The jacobian for testing for load flow voltage instability is evaluated at the equilibrium point of the load flow algebraic equations. It should be noted that the focus of this thesis is to study only those bifurcations and sin gu- lal'ilzies that can occur when the state of the dynamic and algebraic model is at the equilib- rium point. Furthermore, the focus is toward describing the necessary conditions for different types of bifurcations and singularities that cause voltage collapse at the equilibri- “m points rather than describing the bifurcation itself or the dynamical behavior near or after the bifurcation occurs. The test conditions for system voltage instability and load flow instability do not necessar- ily indicate that a bifurcation will occur at that equilibrium point because the test condi- fiofis are necessary and not sufficient conditions. Furthermore, the test conditions do not i“dictate that a voltage collapse bifurcation occurs even if a bifurcation (change in the number of solutions) occurred at a specific equilibrium point 100-0) as Operating condi- tion I. is varied over D. 6.1,] and passes through 3.0. Several different types of bifurcation 11 could occln' (saddle node bifurcation, steady state angle stability bifurcation) and thus the necessary conditions for bifurcation do not indicate that a voltage collapse bifurcation has occurred. A system algebraic voltage instability bifurcation can occur due to a bifurcation of the algebraic equations when coupling of the algebraic and differential equations is ig— nored or there are linearly dependent rows in the linearized real and reactive power bal- ance equations of load buses. This system algebraic voltage instability is related to load flow voltage instability but is based on the equilibrium point of the set of algebraic and differential equations whereas load flow voltage instability is based on the load flow set of algebraic equations and their equilibrium point. A system dynamic bifurcation occurs when rows of the jacobian of the system voltage instability model associated with genera- tor flux decay and exciter dynamics are linearly dependent with the rows associated with the real and reactive power balance equations at generator terminal, high side transformer, or load buses. The third contribution is to develop and test proximity measures for dynamic system volt- age instability and algebraic system voltage instability. There are four test conditions de- veloped in this thesis. The algebraic bifurcation test can be used to test for algebraic (load flow) system voltage instability. Testing singularity of the submatrix of the system voltage stability model jacobian associated with the real and reactive power balance of generator terminahhigh side transformer, and load buses (the algebraic bifurcation test jacobian) es- tablishes whether loss of causality occurs and whether reactive demand supply problems exist. If the reactive demand supply. problems occur at load and high buses, the test is sim- ilar to a load flow biftu'cation test and would satisfy the system voltage collapse bifurca- tion test. If the row dependence of the algebraic bifurcation test includes rows associated with generator terminal buses, the algebraic bifurcation test indicates possible singularity of the power system model but does not assure that the system bifurcation test is satisfied. The algebraic/dynamic test is equivalent to the system jacobian matrix test since the alge- braic/dynamic test matrix is singular if and only if the system voltage stability model jaco- 12 bian is singular. The dynamic/algebraic test shows themtability of the dynamic states of the power system with the assumption that no algebraic bifurcation has occurred. The flux decay bifurcation test can be used to explain the cause of system dynamic voltage instabil— ity since it indicates whether system voltage instability is related to instability of the flux decay dynamics. This thesis has shown that the tests for voltage instability based exclusively on the load flow are invalid because the load flow does not accurately model the exciter and genera- tor. The model which is developed in this thesis is necessary not only to compute the true equilibrium point of the system but also to develop tests for system voltage instability. The primary contribution of this thesis is the development of a power system model that permits simulation of voltage instability. The development of a complete dynamic system analysis of the bifurcations and singularities of this model that are related to voltage col- lapse is impossible. A test condition for system bifurcation (change in the number of solu- tions at equilibria) is developed and an effort is made toward a classification of the types of bifurcations and singularities (loss of causality) of the model. It should be noted that the development of a complete dynamical system analysis of the power system model devel- oped in this thesis is not easy since the theory for describing bifurcations and singularities of constrained differential equations is not complete. In Chapter 2, a precise power system model is developed. The inability of getting correct steady state solution of the load flow model is also shown in Chapter 2. The tests for volt- age collapse and the classification of voltage collapse are provided in Chapter 3. In Chap- ter 4, a two bus system and a twelve bus system are tested using the theory developed in Chapter 3. In Chapter 5, conclusions and topics for future research are given. CHAPTER 2 POWER SYSTEM MODELING 2.1 Introduction A general power system model, which includes mechanical dynamics, flux decay dynam- ics, and excitation system dynamics of a generator and real and reactive power balance equations for each network bus, is developed in this chapter. This model can be used to test for algebraic voltage instability and system voltage instability. There are two different kinds of nonlinear equations in this general power system model. One is a set of nonlinear differential equations which represents the dynamics of the generator. Another is a set of nonlinear algebraic equations which represents the real and reactive power balance equa- tion for each bus in the network. Differential Equation Model x(r) = foo).y(t),r(t)) Mr) e [14.1,] Algebraic Equation Model 0 = g(x(t).y(t).7~(t)) where x (t) : state vector of the generator dynamics, 14 y (t) 2 state vector of bus voltage and angle of terminal buses, high side trans- former buses, and load buses, and IL (I) : state vector of the slow varying operating parameter. 1(1) is the set of operating parameters that change over time. It (I) can be used to repre- sent real and reactive power load, generation dispatch, under load tap changers, and switchable shunt capacitors. As 1(t) varies over [1“.ka , there is assumed to be at least one equilibrium point for each 7», (x0 (A) , y0 (l) ) . A necessary condition for the system to experience system voltage instability is that the jacobian 3; i J _ 8x 8y 13 38 8x Ty be singular at some 10 and (x0 (2.0) , y0 (9.0)) . Section 2.2 discusses the derivation of each element in the general power system model. 'I‘he air gap saturation, line drop compensation, field current limit, and excitation system control are discussed in detailed. Section 2.3 presents a general power system model which has I controllable generator buses, m uncontrollable generator buses, and n load buses. We also show that the conventional voltage instability test, which is implemented using a load flow power system model simulation, has problems getting correct solutions when the system is suessed. Simulation results which show the inability of load flow simulation to get the correct solution are provided in Section 2.4. 22 Single Bus System Model The differential equation model includes the mechanical dynamics, flux decay dynamics, 15 and excitation system dynamics of the generator. The air gap saturation, field current lim- it, and line drop compensation should be included in this model. 2.2.1 Mechanical Dynamics ' MS+DS = PM—P‘(E, 8, V, e) (2.1) where M: generator per unit inertia constant D: generator load damping coefficient 5: internal bus angle with respect to a synchronous rotating reference line P M: input mechanical power P‘: output electrical power The nonlinear differential equation (2.1) is called the swing equation because it is the same equation which describes the “swinging” of a pendulum in a uniform gravitational field. It describes the “swings" in the power angle 5 during a transient. If it is assumed that there is no angle stability problem, PM = P‘(E,5,V,e), the nonlinear differential equation (2.1) becomes a nonlinear algebraic equation. It is as- sumed there is no angle stability problem in the analysis of flux decay bifurcation test in Chapter 3 and the two bus system simulation in Chapter 4, . 2 .22 Flux Decay Dynamics The flux decay equation for each generator is 16 Q = Efd- (Eq+ g—E (Ex-1'0) q xdli"q + (xd—x'd) Vcos (8 — 0) xld xld = E”- E : internal generator voltage proportional to field flux linkage behind steady state direct axis reactance E' : internal generator voltage proportional to field flux linkage behind tran- sient direct axis reactance E fd: generator field voltage I d: projection of terminal current on direct axis V: generator terminal voltage 1: d: transient direct axis reactance x d: steady state direct axis reactance QGE: generator reactive power generation 1'40: generator direct axis transient open circuit time constant Notice that QGE is the reactive power out of the generator. If x' d = xq, QGE is the reactive power out of the internal bus. It is necessary to include the air gap saturation for studying voltage instability. In Fig 2.1, E! is the voltage behind Potier leakage reactance. Before the air gap saturates, E P < A , the increase of Ifd can effectively increase EP. If Ep 2 A, I" can no longer control 15‘p effec- tively due to the air gap saturation. The ability of the exciter to control the induced voltage has subsequently been reduced. Ifd Figure 2.1 Air gap saturation The model that includes air gap saturation is 1' doE'q = Efd— (1 +SD(EP))Eq . [dud—rd) . Quad-fa) where B(E -—A) 50(3) = __P_. p E, de V‘d = ”-1qup = vd- x P q: (Ea-Vent! V‘q = Vq+Idxp = V — q x d: I 2 2 15'p = V'd + V‘ 4 V4, Vq: Projections of terminal voltage on generator d and q axes E p: Voltage behind the potier leakage reactance 18 The parameters A and B of the generator saturation function S D are evaluated from the following equations B(1.0-A)2 B(l.2—A)2 SOLO = T and 5012 = T The 361.0 and 8612 are provided by the specification of the generator. 2.2.3 Excitation System Dynamics In the modeling of excitation system control, we include the transfer function of the line drop compensator, amplifier, exciter, stabilizer, and the measurement device (Figure 2.2). The line drop compensator is selected in the design of the excitation system. The excita- tion system feedback voltage is determined from measuring the generator terminal voltage V and current I and computing Vc = V+ 1’ch using a fictitious reactance xc. By varying xc, Vc can be the voltage close to the internal bus, terminal bus, or a fictitious bus out somewhere in the network. The generator’s excitation system will react to a disturbance very differently for different line drop compensation values xc. VREF Amplifier f K A/(1+sT A) 7 V3 V1 Line Drop Compensation Figrn'e 2.2 Excitation system model 19 Field current limit is a curve that relates the field current value to the duration of the inter- val during which the field current exceeds that value. If the field current exceeds the field current limit, the generator rotor will over heat and a field current limit control automati- cally disables the excitation system control and brings the field current down to a level (continuous rating) that can be sustained indefinitely without overheating the generator ro- tor. For example, in Figure 2.3, if the field current is at Ifdz, the time period that allows field current to remain at that value is t2. After that period of time (t2), the field current will be brought down to its continuous rating to prevent any damage of the exciter. If the field current is lower than its continuous rating (Ifdo), it can remain at that value for infi- nite time. The action of the field current control will be approximated by setting Ka to zero and setting Efd to the value appropriate to produce the continuous rating of field current. This model of the field current limiter is not available in all transient stability programs and this approximate model is quite adequate for assessing retention or loss of voltage sta- bility. t(time) l t: t‘ It‘d) 1de 1de 1de Ifd(field current) continuous rating Figure 2.3 Field current limit controller 20 If we write a set of state equations to represent excitation system dynamics we get the fol- lowing state equations. V1: — V1+IV+jch| "R V3 = I: 6413+]? _ KFEfd(SE:Efd) +KE)) F E E VR = 7:1;(KA(VREF_ V1 ' V3) ' V12) 2,4 = 1:in — (SE (Efd) + K5) Efd) where V1 : measured voltage V3: stabilizer output voltage VR: amplifier output voltage E fd: exciter field voltage from exciter 1R: regulator input filter time constant 1: F: regulator stabilizing circuit time constant 15: exciter time constant 1: A: amplifier input filter time constant xc: line drop compensation reactance K F: regulator stabilizing circuit gain K E: exciter self-excitation at full load field voltage K A: amplifier circuit gain S E: rotating exciter sann'ation at ceiling voltage 21 VREF: reference voltage 2.2.4 Algebraic Equation Model The algebraic equation model is the real and reactive power balance equation which repre- sent the power flow at each bus in the network. If the air gap saturation is considered at the generator internal bus and terminal bus, the algebraic equation model is B(Ep-A)2 D = —E— q—xp+x 4" 1+SQ P a" ' — 2 pe=_i’s‘“_,(i_°_’+1(_1_-,i)sin(2(s—e)) Id: 2 ‘4: x4, E'quin(8-6) (1 +sD) = + xld+SDxP KT 1:,”qu _ (1+SD))sin(2(5-9)) 2 chxq+1thDrcp I'd-i-SDXP e _ 2E'quin(6—0) —E'chos(8—9) -V2sin2(8-0) + I x d: xq,(E‘q—Vsin(5-0) )2 .2 x d: (1+SD) _ . - _ _. _ _2-2 _ _WOEqumw 9) Echos(8 9) Vsrn (8 0))+ 22 (xdanqxpsp) (1 +SD)2 (E'q— Vsin(8— e) )2 (x'a,+s,,1:p)2 (xd+quD) E' Vsin (5-9) V2 1 1 _ q _ ___ ' _ a— I.“ + 2 (qu 164.)” (2(6 9)) = E'qVSin (5-9) (1+SD) + Jt'a,+SDxp K3 ‘4”qu _(HSD) sin(2(5—6)) 2 xdxq+quDxp x'd+SDxp E'V 5—9 QG=_«E<___)_V2( xds cos(8—9)2+sin(8—6)2 x'ds xqs = E'chos(6-6) (1+SD) _ 1'4+prp V2(cos(5-9)2(1+SD) + sin(8—6)2(xd+quD)) x'd+ SD):p xdxq+quDxp n i=1 3 Qa - 9?: = 2 VuVLjYLjSi“ (9 ’ 9L1 ’ 7L1) ,-= 1 P‘: generator real power generation at internal bus Q‘: generator reactive power generation at internal bus PG: real power injection at generator terminal bus QG: reactive power injection at generator terminal bus Pg: realpower load at high side transformer bus Qfiz reactive power load at high side transformer bus 23 generator saturation function applied to direct axis reactance generator saturation function applied to quadrature axis reactance potier reactance internal bus voltage internal bus angle terminal bus voltage terminal bus angle high side transformer bus voltage high side transformer bus angle load bus voltage load bus angle 2.3 General Power System Model The variable and parameters that are used in this section are defined in LIST OF SYM- BOLS. . 1 e . l e 8,. = (0‘. (2.3) 6]- = to} (2.4) . -xdflE' q‘+ (xdfl. —x 'm) VT‘cos (5.— —6T‘.) ) J E": (1+Si)+E,- q (%—1_Td0i)(( x dsi X dsr' D fd = 1+sDi( (Idi+snixpz)5'qr + (xdi’x'ai)Vrr°°s(5r-'eri) ) +93 140: fat + SDr'xpi 1'4: + SDixpi 1aor _ Q? (1411' ‘ 1'41) J _ (Trio—7X5 m— (”Squr'T (2.5) E' - = _1’ (( xdsiE'qi (’44 ‘x 4:1) Vrj°°s (51" a”) J (1 + SD.) + E 4) q] ‘40} I‘m 1: dsj I f I = 1 + SD’ ( > as 0 , 4 ”5'“ r -0.005 . 1 . 1 1 . . 50 60 70 80 90 100 110 120 130 REACTIVE LOAD AT BUS LOAD2 110-3 (9 BUS GROUP 2 LF.) vs. (N.LD.C. “:50 1:5.) 16 r r 1 fi r 77 r 14 ~ LOAD2; 12 ~ I. 1 m 1.011131 U 10 b 1'". -1 a 8 I- I . II a a 6» 4 z '0') 4 1- . > 2 P . '1 0 .1 -2 r 4 a a A_ r r 50 60 70 80 90 100 110 120 130 REACTIVE LOAD AT BUS LOAD2 Figure 2.5 Comparaison of the solutions of load flow model and general power system model (K A=50 ) 32 (9 BUS GROUP 2 LF.) vs. (N.L.D.C. KAIZOO T.S.) T 0.02 HSTS 0.015 ’ .° C H I I 1 1 P I l VOLTAGE DIFFERENCE ”.ms 1 1 L A A 1 A 50 60 70 80 90 100 110 120 130 REACTIVE LOAD AT BUS LOAD2 1110'3 (9 BUS GROUP 2 LP.) vs. (N.L.D.C. KA=200 T.S.) 18 1 r . , . , 16- , « LOAD2 ;' l4 - I. l I. I-' l.’ 12 ~ 'LO VOLTAGE DIFFERENCE oo 50 60 70 80 90 100 l 10 120 130 REACTIVE LOAD AT BUS LOAD2 Figure 2.6 Comparaison of the solutions of load flow model and general power system model (K A=200 ) 33 bus or high side transformer bus voltage as with load flow but regulates some fictitious point that can be anywhere between the internal generator bus and some point out in the network. Finally, the reactive power limit in a load flow model is an approximation to the reactive power produced by the generator after the field current limit of the generator is hit and the field current limit controller has brought the field current down to its continuous rating level. Thus, the general power system model would allow a much higher field cur- rent level than indicated by the continuous field current level for durations indicated by a field current capability curve. The computational results presented compare the equilibri- um point of the general power system model before field current limits are hit (the exciter is not disabled by the field current limit controller and field cunent is not reduced to con- tinuous rating level) with that obtained fi'om a load flow that contains a reactive generation limit that is related to continuous rating limit. This result will show a divergence between the general power system model equilibrium point and the load flow power system model equilibrium point even before field current limit is hit in the general power system model; that is when the system has not yet progressed to the point where generators have lost con- trol of voltage. A second comparison of the equilibrium points of the load flow with con- tinuous rating reactive limits with the equilibrium of the general power system model could be conducted when the field current limit is hit and the field current limiter has dis- connected the exciter and reduced field cm'rent down to continuous rating levels with the equilibrium. The comparison of the load flow and general power system model equilibria should be quite close if the continuous rating of field current is at a level where saturation effects are negligible or have been accurately modeled and the generator is operating close to the point at which the correspondence between the field cmrent and reactive power lim- it is computed. It shows in our simulations that the solution of the general power system model diverged at 125 MVar which is much less than the reactive load level of 182 MVar at which the load flow diverges. The assumption that the load flow simulation is conservative, which has 34 been widely applied, is not true. The general power system model is more vulnerable to voltage collapse and lack of a converged solution than the load flow model. Figure 2.5 and Figure 2.6 show that the divergence of solutions of HST3, LOAD2, and LOAD3 are much larger than the other buses. This is because the reactive reserve at bus GEN3 has been exhausted in the heavy load condition and LOAD2 and LOAD3 are close to GEN 3. The solutions in Figure 2.6 for an exciter amplifier gain of K A=200 are far closer than the solutions in Figure 2.5 for K A=50 in light load conditions. The reason that there is better agreement between the equilibria of the load flow and general power system model at large exciter gain is that the load flow model assumes the exciter gain is infinite so that the control of terminal voltage has no error. The general power system model has a finite exci- tation system closed loop gain KEX and an error in regulating voltage proportional to the inverse of KEX (K'IEX). Since KEX is proportional to the exciter amplifier gain K A and in— versely related to air gap saturation, the agreement between the models increases with K A and decreases as the air gap saturation and reactive generation increase. The reasons for the divergence of the load flow model fi'om the general power system model are summarized in the following: (a) the load flow simulation uses reactive power generation to regulate bus volt- age and the general power system model uses field current to regulate bus voltage, (b) the general power system model takes into account the air gap saturation, which affects the closed loop gain of the exciter control loop, but the load flow model does not, (c) the load flow model assumes KA is infinite but K A is a finite value in the gen- eral power system model; The exciter loop gain KEx is proportional to the (d) 35 exciter amplifier gain and inversely proportional to air gap saturation. The er- ror in regulating voltage is proportional to K'IEX. generators in the load flow model regulate either their high side transformer bus voltage or terminal bus voltage but the general power system model which takes line drop compensation into account allows any point between the generator internal bus and some fictitious point out in the network to be regulated. CHAPTER 3 TESTS FOR VOLTAGE COLLAPSE 3.1 Introduction Voltage instability problems are classified into two difl‘erent categories in this thesis. These two kinds of voltage instability are results of very different types and locations of su'ess. Load flow voltage instability is caused by the inability of the transmission system to supply the reactive load when there is no reactive power supply at that load voltage con- trol area. System voltage instability is caused by either the instability of generator dynam- ics or the links between dynamic states and algebraic states in a stressed network. Four voltage instability tests are discussed in this chapter. 3.2 Class'fication of Types of Voltage Instability In a power system, there are strong connections and weak connections in the transmission system. We define the strength of the branch in terms of reactive power transferring capa- bility of the branch. A strong connectiOn has no problem in transferring both real and reac- tive power. A weak connection has difficulty in transferring reactive power in the sense that if the operating condition of the system is changed the weak transmission line gets clogged up and can not transfer reactive power without significant losses and voltage dif- ference across the branch. It can be shown that the strength of a connection is dependent 37 on the reactance of the transmission line and the operating condition. A heavily loaded transmission line with large line reactance usually has difficulty in transferring reactive power. If we put any kind of disturbance in the system, the buses connected with strong connections will respond to the disturbance coherently in both voltage magnitude and phase. If we colleCt those buses with strong connections, we can form a coherent group of buses and we call this coherent group of buses a Voltage Conu'ol Area(VCA). (Schlueter/ Costi[9]) 3 2.1 Load F low Voltage Instability Load flow voltage instability problems are reactive supply and demand problems. They are caused by the weak boundary connections among voltage control areas and capacitive reactive withdrawal with voltage drop inside the voltage control areas. A voltage decrease in a voltage control area will increase line reactive losses and weaken the strength of boundary connections to the neighboring areas. Because of the decrease of voltage, the re- active power supplied by line charging and shunt capacitors in that area will be decreased. Thus the voltage decrease at an area will not only decrease the shunt capacitive reactive power supply in that area but also reduce the capability of boundary transmission lines to transfer reactive power from other areas. It will come to a point where the load flow equa- tions associated with the buses in that area can not be solved and the jacobian matrix be— comes singular at that point. We call this kind of voltage instability load flow instability. In this thesis, we show that (1) weak boundary transmission lines may get clogged up and become a reac- tive drain that sucks motive power from both buses they are connected to; (2) the voltage instability caused by supply and demand problems can be ana- lyzed by load flow simulation and the singularity of the load flow jacobian matrix indicates this kind of problem; 38 (3) load flow voltage instability, a particular type of system voltage instability, can not occur when there is plenty of reactive supply close to the point of stress. In contrast, system dynamic voltage instability may occur in an area even when there is plenty of reactive power supply close to it. It will be shown in the Voltage Instability Tests chapter that load flow voltage instability (supply and demand problems) satisfy the necessary condition for bifurcation of the con- strained difl‘erential equation if the jacobian matrix associated with the real and reactive power equations at high side transformer buses or load buses have linearly dependent rows. This would confirm the validity of the methods that apply load flow jacobian analy- sis to investigate voltage instability problems. It should be noted that load flow instability, that includes the real and reactive power balance equations at generator terminal buses, may not imply system voltage instability. More importantly, system voltage instability will be shown to occur without load flow voltage instability. This indicates that the load flow jacobian analysis is not conservative enough to predict all the possible types of voltage in- stability. The load fiow instability test can be used to test for system voltage instability that result from reactive supply/demand problems at high side transformer and load buses. 3.22 System Voltage Instability System voltage instability occurs due to two possible reasons. One reason is the reactive demand supply imbalance and the other is the instability of generator dynamics. There are four factors, air gap saturation, field cmrent limit controller, line drop compensator, and excitation system control which may cause the instability of generator dynamics. These four factors have been modelled in Chapter 2. Air gap saturation reduces the ability of a generator to create reactive power and control voltage. As air gap saturation increases, the increase in field current needed to control a specific terminal voltage change increases. If the field current limit is hit, the excitation 39 system is disconnected and field current is reduced to it’s continuous rating level. The bus voltage controlled by the exciter is no longer controlled It will be shown that increased air gap saturation and increased reactive generation in a stressed network will move the ei- genvalues of dynamic states toward the right half plane and system voltage instability may occur. If the fictitious bus voltage controlled by the line drop compensator is located far out in the network, field current will be closer to saturation and the generator will be re- quired to produce more reactive generation. It will be shown that the flux decay and excit- er eigenvalues are generally complex if the field current limit is not hit and the excitation system is still in control of terminal bus voltage. It will be shown that, in some cases, these complex eigenvalues of the flux decay and exciter or the generator mechanical dynamics move to the right half plane and a Hopf bifurcation may occur. The loss of stability associ- ated with Hopf bifurcation may cause the field current limit to be hit before the reactive loads on the machine would have caused the field current limit to be hit. Thus, the genera- tor excitation system may be disabled and the field current reduces to continuous rating if either the field current limit is hit or if the flux decay and exciter or generator mechanical dynamics experience a Hopf bifurcation. Both air gap saturation and increased reactive generation will be shown to move the flux decay eigenvalues close to the right half plane. If the field current limit is hit, the exciter is disconnected, and field current is reduced to it’s continuous rating. It will be shown that the flux decay eigenvalues will generally be real rather than complex as they were before the field current is hit and will be far closer to the fa) axis. After the field current limit is hit, the flux decay eigenvalue may be positive or become positive with increased reactive load. The generator’s response after the flux de- cay eigenvalue becomes positive will be shown to result in monotone decreasing field cur- rent, induced voltage, and reactive power output. All of these would explain why voltage collapse is observed to be a continuous decline of voltage. This kind of voltage instability can be detected by using a dynamic/algebraic test to be defined. It will be also shown that this dynamic voltage collapse may occur even if the field current limits aren’t reached and the exciter is not disabled. The other type of system voltage instability is caused by the stress on the transmission and distribution system and is tested using the algebraic/dynamic test to be defined. This type of voltage instability will be shown to not only capture load flow instability at high side transformer and load buses that do not interact with generator flux decay dynamics as well as the voltage instability caused by interaction between the generator and the network at generator terminal, high side transformer, and load buses. The system voltage instability caused by interaction of flux decay dynamics with the network at terminal, high side trans- former, and load buses can be tested by the algebraic/dynamic test since the algebraic/dy- namic test can test for every type of voltage instability. 3.3 Voltage Instability Tests It should be noted that the focus of this thesis is to study only those bifurcations and singu- larities that occur when the state of the algebraic and dynamic models are at the equilibri- um point. Furthermore, the focus is toward describing the necessary conditions for different types of bifurcations and singularities rather than describing the bifurcation or describing the dynamical behavior before or after the bifurcation occurs. This thesis pro- vides different types of tests for voltage instability. The algebraic bifurcation test will be used for testing for load flow voltage instability. The algebraic/dynamic and dynamic/al- gebraic bifurcation tests are tests for dynamic voltage instability. The algebraic, algebraic/ dynamic, and dynamic/algebraic tests for voltage instability are not tests for the same type of voltage instability. The types of voltage instability tested for in each of these three tests will be described. A flux decay bifurcation test is a test for instability of the flux decay dy- namics under the assumption that there are no angle stability problem. This flux decay bi- furcation test shows theoretically that the sensitivity matrix 5905, S VB and the air gap saturation, excitation system control, and reactive power generation decide the stability of 41 generator flux decay dynamics. 3 3 .1 Model Linearization x(t) =f(x(t).y(t).?~(t)) (3.1) 0 = 8(X(t).y(t).l(t)) (3.2) where x(t): state vector of the generator dynamics y(t): state vector of bus voltage and angle of terminal buses, high side trans- former buses, and load buses l. (t) : state vector of the slow varying operating parameter, 1 (t) e [2.0, lb] df 4f J = a: a; = A (x0! y0’ Av0) B (x0, y0’ 1'0) (3.3) E _d_g_ C (x09 yO’ 1'0) D (x0: y0’ A'0) dx dy where matrix A, B, C, and D are a function of initial values of x0, yo, and 9.0. We can write a set of linearized equations A: (t) = AAx (t) +BAy (t) (3.4) 0 = an (t) +DAy (t) (3.5) A y contains the changes of bus voltage and angle of terminal buses, high side transformer buses, and load buses. We can further divide A y into Ay1 and Ay2 where Ay1 is the changes of bus voltage and angle of terminal buses, and A y2 is the changes of bus voltage and angle of high side transformer buses and load buses. Equation (3.4) and (3.5) become Arm = AAx(t) + [3132:][318] (3.4.1) 2 42 0 = C1Ax(t)+ DID? ”1“) (3.5.1) o D3D4 Ay2(t) where .3182] = B 01] - c c, ___ 0 _C2 D1 D2 = D D304 The C2 matrix is zero because there is no direct connection from internal buses to high side transformer buses or load buses. 3 3 .2 System Bifurcation Test It should be pointed out that for each I. (t) value there are several solutions (equilibrium points) (x3 (0 0’: (0) of equations 3.1 and 3.2. Some of these equilibria are stable and oth- ers are not stable. It is assumed that the system is operating at a stable equilibrium at t=0 and that the bus voltages are near 1.0 p.u. and angle difl‘erences are less than 45° in this so- lution. As i. (t) slowly varies for (>0, the equilibrium point on}, ‘0 .yf; “’) changes until at some point (3. = l. (o) e [lash], the jacobian matrix I becomes singular. The condition det{J(x3, y3,5.)} = 0 is a necessary condition for static bifurcation of the general power system model. Singu- larity of I does not imply a bifurcation (change in the number of solutions at [x3, y3‘, at] as 2. (t) passes through 3.). If a system bifurcation occm's, it may not be a bifurcation that 43 causes voltage instability but could result in angle instability or some other type of bifur- cation. If the bifurcation occurs that results in voltage instability and it may result in one of at least three types of system voltage instability. It should be noted that the development of a complete dynamical system analysis of the power system model developed in this thesis is not easy. The theory for describing bifur- cation and singularities of constrained differential equation system models is not complete and the degree of complexity is too high to be handled by today’s state of the art computer system. For example, it takes six nonlinear differential equations and four nonlinear a1 ge- braic equations to describe a generator and its terminal and high side transformer buses. A simplified two bus system with one generator bus and one terminal bus will be investigat- ed in this thesis. It will be shown that we have to make crude assumptions to make this two bus system model valid and this model can only be used to investigate a limited num- ber of causes of voltage instability. 33.3 Algebraic Bifurcation Test Algebraic bifurcation is a change in the number of solutions in the algebraic equation 0 = g(x(t).y(t).1(t)) (3.7) as a function 0f y (I (t) . l11(1) ) in a neighborhood around a point (555»). A necessary con- dition for algebraic bifurcation is that gyms.) =0 is singular at (i, 51.2.) . The test condition for algebraic bifm-cation is a test condition for loss of causality [27] of the transient stability model. Loss of causality indicates that the transient stability simulation packages that iteratively updated x using the differential equations and update y using the algebraic equations may not obtain unique solutions and will generally terminate due to numerical failure as singularity of D is approached. Alge- braic bifurcation does not guarantee that a system bifurcation will occur as will be dis- cussed in more detail shortly. However, algebraic bifurcation indicates a point where the transient stability simulation will fail and where strange (chaotic) behavior may occur[21] under stressed operating conditions. The algebraic bifurcation test is similar to the widely used load flow jacobian test. If det (DIX, ya 10) = 0 then algebraic bifurcation occurs. At the bifurcation point (x0, yo, 10), two closely located load flow solutions merge into one and matrix D becomes singular at this bifurcation point. The singularity of matrix D indicates that there is lack of a solution to satisfy the supply and demand problem. It is usually caused by the weak boundaries of voltage con- trol areas that reduce the ability to import reactive power, the shunt capacitive reactive supply withdrawal with voltage decline, and the lack of sufficient reactive supply in criti- cal voltage control areas. The difference between algebraic bifurcation test and load flow jacobian test is that the steady state solutions we use to determine the singularity of matrix D are difierent In an algebraic bifurcation test, we use the solutions of 0 =f(X(t).)’(t).?~(t)) (3.6) 0 = g(x(t).y(t).l(t)) (3.7) to test the singularity of matrix D. In widely used load flow jacobian test, the solution of 0 = §(xo. (yw1(t)).l(t)) (3.8) is used, where E includes the real power balance equations at terminal high side trans- 45 former, and load buses; retains reactive power balance equations at high side transformer and load buses, but eliminates reactive power balance equation at generator terminal buses that are included in g. The y(t) vector is thus divided into a vector y1(t) of the angle at all buses and the voltage at all high side transformer and load buses. yo represents the genera- tor voltage setpoint at terminal buses. The test for load flow bifurcation is then based on the singularity or nonsingularity of 3?. By] We have shown that the solutions of (3.6) and (3.7) and the solutions of (3.8) tend to di- verge at heavy load condition but before the field current limits are hit. The results of the algebraic bifurcation test and the load flow jacobian test will also diverge as we move the system close to load flow voltage instability. Note that if algebraic bifurcation test shows that the matrix D is singular, the system jacobian matrix J will not be singular unless D - CA'IB is singular as will be discussed in the next subsection. If the singularity of matrix D is caused by the linear dependency of two or more rows in matrix [D3 D J in equation (3.5.1), this implies that there are linearly dependent rows in the system jacobian matrix J because the submatrix 02 is a zero matrix. Linearly depen- dent rows of [D3 D 4] thus implies the system jacobian matrix I becomes a singular matrix and that a system bifurcation may occur. This indicates that the linear dependency of the rows of the real and reactive power jacobian associated with high side transformer and load buses is one of the ways to cause system bifurcation. It shows that the methods which use the algebraic bifurcation test to investigate voltage instability problems are valid if the voltage instability is caused by linearly dependent rows of [D3 D 4] associated with high side transformer buses or load buses. The matrix [D1 D2] hastheform "3101.310; 3 D1 = 39—1-37?» = [Dinb’l'fn QT QT D23 D24 sop—v, "" "" aPT 3P7. — 15 6 D2: ml; 0 8VI! 0 = Dn,n 0mm Drlun 0mm $31: 0 $91 0 0:5" 0",," Di?" 0mm V H H where , , , , , , , and are diagonal matrices represented 30—, 3'7; 3?); av, 30,, av” 89,, 37,, by the notation D2. n. Rows of [D1 Dz] can not be dependent unless the ith row of [D1 D2] is linearly depen- dent with the i+nth row. If [Dr DJ has linearly dependent rows or if one or more rows of [D1 Dz] are linearly dependent with one or more rows of [D3 D 4] , then an algebraic bi- furcation (loss of causality) may exist that will not necessarily cause system bifurcation which requires J to be singular. An algebraic bifurcation (loss of causality) that does not cause a static system bifurcation can result in chaos [21] and possibly other unacceptable behavior which may or may not be associated with voltage collapse. A loss of causality that does not cause system bifurcation is studied for a two bus example system in the next chapter. A general investigation of loss of causality (algebraic bifurcation) and it’s impacts on the behavior of the system behavior is beyond the scope of this thesis. 33.4 Algebraic/Dynamic System Bifurcation Test If man-ix A is nonsingular, the system jacobian J is nonsingular(singular) if and only if 47 M1=D - CA‘IB is nonsingular(singular)[24]. Since it is shown that matrix A is always nonsingular in Appendix B.3, the algebraic/dy- namic system bifurcation test is always valid for testing the singularity of the system jaco- bian matrix J. We have indicated that the (loss of causality) algebraic bifurcation test on D can be used to test the singularity of the system jacobian matrix if [D3 D ‘ll have linearly dependent rows of D. If these linearly dependent rows of D belong to [Dr D2] or both [Dr D2] and [D3 D 4] , there is a loss of causality (possibly chaotic behavior) but no steady state system bifurcation. Likewise, there are system bifurcations of the algebraic equations (and obviously also the set of differential and algebraic equations) that can’t be detected by the algebraic bifurcation test. Two such cases are (a) linearly dependent rows in matrix [C1 D1 Dz] , or (b) linearly dependent rows in matrix [D3 1);] and [C1 D1 D2]. A method that will identify the linearly dependent rows in the matrix [C1 D1 D2] is pre- sented here. The matrix [C1 D1 D2] can be represented by P —l 13 14 15 16 on." DilnDrltznomtn DmannDnn 0mmDmnomm [C1 0102] = = 2 22 B 24 25 26 on." DmlannOnAn DflfllDfleflDflrfl OAMDflrflonrm k1] 3‘2 3P BP 11 T 12 T where Du”. = as r Dun = a—Et—qr nal matrices that belong to D1 and D2 have been previously defined in section 3.3.4. 'Ihere BQT BQT r)21 = , and D22 = and the other diago- n- a 3’6 "- n 3?: are n generator buses and m high side transformer buses and load buses in this example. If 48 there are linearly dependent rows in the matrix [C1 D1 D2] , it has to be one row in the matrix K1 and another row in the matrix K2. It is only possible that the ith row of K1 is lin- early dependent on the ith row of K2. The ith row in the matrix chan never be linearly de- pendent on the jth row of the matrix K2 if i is not equal to j. The algorithm for identifying the linearly dependent row in the matrix [C1 D1 02:] is given in Figure 3.1. The computa- tion effort is less than or equal to 6n (scalar division) + 7n (scalar comparison) + n (scalar addition) where n is the number of generator buses. 33 5 Dynamic/Algebraic System Bifurcation Test If matrix D is nonsingular, the system jacobian matrix J is nonsingular(singular) if and only if M2 = A -BD"1C is nonsingular(singular). If D is nonsingular, both M1 and M2 can be used to test for system bifurcation (singularity of J). Both M1 and M2 can be used to test for singularity of J when D and A are nonsingu- lar because the singularity of J depends on the row dependence of rows of [C D] and [A B] rather that row dependence in [A B] or [C D] alone. M2 represents the system ma- trix of the nonlinear constrained difi'erential equation modeled linearized at an equilibrium point x0, yo, 10 and thus defines the eigenvalues of the equivalent unconstrained dynami- cal system. If the real parts of the eigenvalues of M2 are all negative, the equivalent un- constrained dynamical system is locally stable in the neighborhood of the equilibrium point. A dynamic/algebraic system bifurcation test can be used to test the stability of the dynamic states if the matrix D is nonsingular. If there are complex eigenvalues with zero 49 i no lineme dependent row Nonzero elements of the ith row of the nonzero elements of the ith row of the matrix K2 save the result in the matrix Y matrix K1 are divided by the corresponding ® Figure 3.1 Algorithm for identifying the linearly dependent rows in the matrix [c112l Dz] 50 real parts, the system may experience Hopf bifurcation and yet J and M2 will be nonsingu- lar. Thus, singularity of J and M2 will not indicate all possible bifurcations but only static bifurcations. A fundamental assumption, which is confirmed by our computational results, is that voltage collapse bifurcations are static bifurcation that can be tested for by singular- ity of J, M2, or M1. Examples of singularities, that are not detected by singularity of J, M1, and M2 and are not necessarily associated with voltage collapse, are (a) Hopf bifurcations which occur when eigenvalues of M2 are complex with zerorealpartsand (b) algebraic bifurcation (loss of causality) det(D) = detqbl DZD = 0 D3D4 where rows of [Dr DJ are dependent on rows of [D3 D 4] . (Note that row dependence in [03 D 4] are detected in J and M1 and is associated with load flow bifurcation which are reactive demand/supply related voltage collapse bifurcations) It should be noted that de- spite the above theoretical results and the computational results in the next chapter that confirms this theory, there is no doubt that our understanding of bifurcations of this system is not complete. This M2 test for system static bifurcation is a test for bifurcation of a type that are not re- lated to algebraic bifurcation or the bifurcations that occur in equation (3.8) are related to load flow bifurcation. The singularity of M2 as 2. (t) varies indicates that J is singular and that a bifurcation may have occurred in the dynamical system where the algebraic con- straints have been eliminated. Whether a bifurcation occurs or not and whether the bifur- cation is related to voltage instability or not, the system will become unstable if the real part of the eigenvalue that became zero becomes positive as 3. (t) continues to vary. This instability could be related to voltage collapse even if no bifurcation occurs when M2 and 51 J were singular. Zaborsky[21] discusses this type of dynamic/algebraic bifurcation on a two bus system and provides some preliminary results on it’s dynamical system behaviors. 3 .3 .6 F lux Decay Bifurcation Test In flux decay system bifurcation test, we assume that there is no angle instability. 110) =f1(x,(t).x2(t).y(t).>~(t)) x2 (I) = f2(x1(t),12(t),)’(t),l(t)) O = g(x1(t)sx2(t)ry(t)sx(t)) ran an ail 3x75172279 _ a2 afz afz 33153655 3g gag :371-312 5; _l where x(t) in equation (3.1) is divided into x1(t) and x2(t). x1(t) represents the flux decay states of the generator. x2(t) represents the states of mechanical and exciter dynamics. The jacobian of the above equations is _. T1 T2 T3 T4 r 52 3f2 5; T = 1 3 ag 97a af2 afz 797255 as as a; 32 1 If there are linearly dependent rows where one or more occur in [T1 T2] and the remaining rows occur in [T3 T4] , the dynamic/algebraic test of this new jacobian will be singular if J is singular and [T1 T3 is linearly independent and [T3 T4] is linearly independent Singu- larity or existence of positive eigenvalues of M3 = T1 - 1213314 = T indicates that the voltage instability may be related to dynamic voltage instability associat- ed with flux decay dynamics. Matrix T represents the matrix associated with flux decay dynamics when the excitation system dynamics, generator mechanical dynamics, and al- gebraic equations have been eliminated at the equilibrium point. Matrix T is used as a test for the flux decay dynamics in a manner similar to the AESOPS algorithms tests for the stability of mechanical system eigenvalue. The results indicate M3 does not change signif- icantly as the excitation system of every generator remains in control of terminal bus volt- age. However, if the field ctnrent limit is hit on one generator, then M3 varies significantly. Note that singularity of M2 may not be an appropriate test for dynamic instability of the flux decay dynamics when D is singular. However M3 may be a valid test for flux decay bifurcation when D is singular since D is a submatrix of T3 which mat be nonsingular when D is singular. The flux decay bifurcation test can point out how the air gap satura- tion, line drop compensation, field current limit, sensitivity matrices, and reactive genera- tion will influence the stability of flux decay dynamics. 53 Computational results in the next chapter indicate that when a generator has not hit field current limit the eigenvalues associated with both the mechanical dynamics or the genera- tor flux decay dynamics and the exciter dynamics are generally complex. Although real parts of these eigenvalues will be shown to approach zero and possibly result in Hopf bi- furcation that can cause field current limit violation as reactive load network and network stress, it does not appear that it will cause voltage instability. The disablement of the exci- tation system and the reduction of the field current to continuous rating levels caused by the oscillation generally seems to extinguish the Hopf bifurcation. The resultant system af- ter the excitation system is disabled has real eigenvalues rather than complex eigenvalues in the examples discussed in the next chapter. If the real eigenvalues associated with the generator flux decay dynamics are negative the system is stable and 1 remains nonsingular. If the eigenvalues associated with the flux decay dynamics are positive, the system experi- ences a dynamic voltage collapse where the induced flux, field current, and reactive power out of the generator approach zero forcing the voltage collapse in the system. When the ei- genvalues associated with this flux decay dynamics become positive, the determinant of J changes sign. A specific flux decay bifurcation test is desired which would indicate when the eigenvalues associated with the flux decay dynamics become real rather than being complex and indicate whether the eigenvalue is positive or negative. It should be noted that it is possible to effectively disable the excitation system if the field current is above continuous rating and the air gap saturation has reduced the excitation loop gain to small values. In this case, the flux decay eigenvalue will be shown to ap- proach the right half plane if SE}, and S vs have negative elements. In this case, the al ge- braic voltage collapse test was violated indicating a reactive demand supply problem has occurred that could have brought on dynamic collapse where generator field current flux are unstable and approach zero. An analysis of flux decay bifurcation in undertaken. To perform this analysis, it will be 54 necessary to assume that steady state angle instability has not occurred so that the required sensitivity matrices are well defined. Since voltage collapse is assumed to be a static bifur- cation, the derivatives 3 and (1') can be set to zero and the swing equations are deleted and replaced by an algebraic real power balance equation of the generator. The reactive power balance equation of the generator internal bus is also added because it will be used for sen- sitivity matrix analysis in this section. I" = f1(5, 6, E, V) Q‘ = f2(5. 9.13. V) The sensitivity matrix model has the form (Appendix II) AQ‘ = SQOEAE + SQGQLA Q1. (3.9) AV = SEVAQL + SVEAE (3.10) We also set the derivative of exciter states belonging to x2 in the general power system model to be zero and we obtain another set of algebraic equations. We can solve for the field voltage in terms of the terminal bus voltage. -K .K . E = diag( A' R' )V fl SEr' (Em) + K51 T "KMKRt )A VT (3.11) SE! (5)308.) + KE; + SE! (E13098 dei AEfd = diag[ where 8,4 is the vector of generator field voltage and V, is the vector of terminal bus volt- age. The only linearized dynamic equations left are for flux decay equations (3.12) A5" = $015 —AE' +Q°(x"-x")AE — (Id-I'dmgj fd" 2 4 do Eqr’O 8‘in 55 Substituting (3.9), (3.10), and (3.11) into (3.12) we obtain AB": = (diag(1;))(diag(— l + Q“) (31'2" di) ) 401 (1:0 _KAiKRr‘Vr‘O 551' + K5: + SEiEfdiO + diag( )WSVE x .—x' . I -diag( 42.2 d’)SQGE )AE'qHMQIJ: TAE+nAQL (3.13) in where _W selects the terminal bus voltage vector out of the vector 1’ that includes voltage at terminal, high side transformer, and load buses. Matrix T is a diagonally dominant matrix and all the eigenvalues of matrix T has nega- tive real parts in the normal operating condition. As the system becomes stressed, the reac- tive power generation Qfo of generators start to increase in a speed faster than E310 This will reduce the negative diagonal dominance of matrix T and move T toward singularity. When the generator experiences excitation system saturation, the term 551' (Efdio) "’ KEr' + $151 (EfdiO) Efdio may increase and force the diagonal elements of matrix T to become less negative. If there is a terminal bus voltage deviation such that S vs elements and row sums increase, the diagonal dominance of matrix T is even further reduced. The loss of diagonal domi- nance of T indicates the matrix is approaching singularity since T is a diagonally dominant M matrix. The sensitivity matrix S Q 05 has the property that the negative off diagonal ele- ments become more negative and the positive diagonal elements become less positive as the system is stressed. This would also contribute to the loss of diagonal dominance and singularity of matrix T . The air gap saturation will help to move the field current to its up- 56 per limit. Air gap saturation can significantly reduce x'ds and x33. Thus, both the diagonal and off diagonal elements of the third term of equation 3.13 associated with or diagonally dominant matrix SQGE will move toward zero which has the effect of moving the eigen- values associated with equation 3.13 toward the right half plane. If the field current limit is hit, the excitation system is disabled and the second term -KAiKRiVi0 SEi (Exam) '1' K51"' 5121' ( (Efdr'o) EfdiO) becomes zero. This means that a large negative diagonal element of matrix T will be re- moved. The eigenvalues of matrix T will be moved to a point where either it is very close to the origin or even becomes positive. When the excitation system is disabled, the field current is reduced to continuous rating which reduces air gap saturation. The reduction of field current and air gap satmation has the effect of causing the eigenvalues associated with the third term in equation 3.13 to be more negative. PQ controllability requires S VI; to be nonnegative. Strong PQ controllability requires row sums of SVE to be near 1. PV controllability requires that S 90 5 have positive diagonal el- ements, be an M matrix, and be a diagonally dominant. Loss of PQ controllability occurs when elements of row sums of SW? approach infinite and then instantly go negative. Note that if elements of S vs are negative, it is virtually certain that the eigenvalue of matrix T will be positive since the diagonal matrix (3.14) d‘ ( ‘KAiKtho ) rag 551' (51410) + K514” 35: ( (51410) 51410) is so large. Loss of PV controllability results in loss of the M matrix and a diagonal dom- inance property of S Qa 5° When the exciter is disabled due to field current limitation and the second term in equation 3.13 is missing, the loss of PV controllability can cause the ei- genvalues of T to become positive, since the term 57 . l . xd' 'x'd' drag (a; )drag( IE2. ‘)SQ¢;E qr is not small compared to the first term. If S965 is a diagonally dominant, the above term has positive eigenvalues, but if S Qa E is no longer a diagonally dominant or loses its M matrix property, then the eigenvalues of the term will be negative and the eigenvalues of T can become positive. Loss of PV controllability can also contribute to eigenvalues of T becoming positive when the excitation system is not disabled due to field current limitation, but its efi'ect should be small compared to loss of PQ controllability. This simplified power system model shows clearly how a dynamic voltage instability would happen and indicates the contribution of each element to voltage instability. 3.4 Relationship to Literature There is a large amount of literature on the voltage stability problem. Most of this literature deals with the existence of load flow solutions in the steady state or static condition of the power system. Research efforts have resulted in several difi‘erent methodologies for the coordination and utilization of the reactive power and voltage control resources of the system. It has been known that there will be two static load flow solutions which eventually merge into one in a two bus system when a bus with an increasing real or reactive power load is fed through a transmission line from a fixed magnitude voltage source. With increasing load beyond the critical load value, there exists no load flow solution. In mathematical terms, this is a static or saddle node bifurcation. A similar event occurs on a large power system if real or reactive load (1‘. (t) ) is increased at a critical bus. Before 7": (t) is increased, there are many solutions under normal operating conditions. One or more of the 58 numerous load flow solutions will converge together and merge into one and the jacobian matrix becomes singular, which indicates a bifurcation and voltage collapse if the bifurcation is associated with voltage collapse. Many research eflorts of voltage stability are devoted to this static bifurcation event. Tamura [12,13] has confirmed that closely related multiple load flow solutions that merge are likely to appear under heavy load conditions. He showed that a pair of load flow solutions located close to each other is related to voltage instability. A voltage instability proximity index which measures the closeness of a pair of load flow solutions and proximity of the jacobian matrix to singularity are developed in his research. A method for computing these multiple load flow solutions are provided in his recent research. Much of the voltage collapse literature [28,29,8,12,13] uses a conventional load flow model to analyze voltage instability. Generator dynamics are not included in their research. This kind of voltage stability problem has been classified as a demand/supply problem. In this thesis, we use algebraic bifurcation test to investigate the supply demand problems of the power system. The difl’erence between an algebraic bifurcation test and the widely used load flow jacobian test is that in an algebraic bifurcation test we use the equilibrium point from a general power system model which includes the mechanical dynamics, flux decay dynamics, and excitation system dynamics and the test for voltage collapse is a test for row dependence of D = [DI Dz]. D3 D 4 Load flow bifurcation utilizes the equilibrium point of just the real power balance equation at terminal, high side transformer, and load buses and the reactive power balance equations at high side transformer and load buses. The load flow bifurcation test of the singularity of [D3 D 4] are based on the load flow equilibrium points. It has been shown in this thesis that the conventional load flow model is not suitable for voltage instability 59 analysis under heavy load conditions and that generator dynamics are strongly related to the voltage stability. Venikov [7] recognized the significance of a degeneracy in the real power angle jacobian matrix with respect to the steady state stability of a transient power system stability model. He observed that in load flow calculations performed by Newton’s method one can estimate the steady state stability of the operating condition in question. Under certain conditions, a change in the sign of the determinant of the jacobian matrix during a continuous variation of parameters coincides with the movement of the operating condition from a voltage stable to a voltage unstable condition. Kwatny[27] attempted to relate static voltage instability and voltage collapse to bifurcation phenomena. Only generator mechanical dynamics and real and reactive power balance load flow equations were included in this model. It was shown that static bifurcations of the load flow equations were associated with either divergence type instability or loss of causality. The bifurcation tests were shown to be tests for steady state angle stability or voltage collapse in a load flow model. Schlueter[l,2,3,4] pointed out that the dynamics of flux decay and excitation system has tremendous impact on the voltage instability of a power system. These generator dynamics are included in his model for theoretical analysis. He also identified the component of A which is the operating parameters which may drive the power system to bifurcation. 1 may be used to represent changes in real or reactive power load, real or reactive power transfer, changes in real power generation on generators due to inertial, governor, automatic generation control, or economic dispatch response to loss of generation contingencies, or actions of under load tap changers or switchable shunt capacitors in the distribution system. A definition of PQ controllability is hypothesized on the proper response of voltage at load buses and voltage set point changes at generator buses. The PQ controllability definition results in an array of tests for voltage collapse that 60 encompass all the tests that developed for voltage collapse. The PQ controllability tests on sensitivity matrix SQLV and S vs assure that the load flow and transient mid term stability model will not experience bifurcation. Sauer and Pai[19] presented the relationship between a detailed power system dynamic model and a standard load flow model. The field current limit controller and line drop compensator were not included in their model and the flux decay and excitation system dynamics were ignored in the analysis.The linearized dynamic model is examined to show how the load flow jacobian appears in the system dynamic state jacobian for evaluating steady state stability in two special cases where the load flow jacobian is the jacobian of the constrained differential equation model. The singularity of the load flow jacobian was shown to imply singularity of the system dynamic state jacobian for a special case where reactance and resistances of the generator model are ignored, exciter dynamics and flux decay dynamics are ignored and field current limits and line drop compensation are ignored. If all of these factors are taken into account, this thesis shows that the tests for system bifurcation are very difi'erent fiom those for load flow bifurcation. The difierences between load flow and system bifurcation not only occur due to these model difi'erences and their resulting differences in equilibrium points but also because eliminating the modeling differences eliminates the property of dynamic/algebraic bifurcation that can’t occur in a load flow or in Pai and Sauer’s simplified dynamic model [19]. There are papers which discuss the stability of the voltage controllers, such as under load tap changer and switchable shunt capacitors in the distribution system. Abe[14], Liu and Wu[22], and Illic[l6,17] have developed test conditions for stability of the control of under load tap changers that are similar to the test conditions developed in this thesis for flux decay bifurcation tests of the general power system model. There are papers which apply sensitivity analysis to determine voltage instability. The sensitivity analysis develops the relations of the changes of states and inputs. Borremans 61 [28] suggested two conditions SQGQL>O and S‘1V>O L for voltage stability. SQan. > O is the requirement that the generator correctly responds to a reactive load increase and say > 0 is the requirement that an increase in reactive demand will cause a decrease in voltages. Carpentier[8] also derived a voltage stability indicator and pointed 'out that This is a stronger condition than Borremans’, and can be used as an indicator to determine an overloaded system. Glavitsch[29] provided another sensitivity test, S V5, to investigate the relationship of load bus voltage and generator bus voltage. Schlueter[3] has shown that all the sensitivity tests are a subset of the tests that can be derived based on assuming PQ and PV controllability hold. The sensitivity matrices used in the flux decay bifurcation tests in this thesis have the same definition as Schlueter [3] using a load flow model but use the constrained dynamical model equilibrium rather than a load flow equilibrium to evaluate the sensitivity matrices. CHAPTER 4 SIMULATION RESULTS ON VOLTAGE INSTABILITY OF POWER SYSTEM MODELS A two bus and a twelve bus power system models are simulated in this chapter. The sim- plicity of the two bus system model allows us to investigate the voltage instability prob- lems by using a two dimensional phase portrait. The simulation results of a two bus system model confirm the validity of our voltage instability tests. Since the air gap satura- tion, field current limit, line drop compensation, exciter system control, and the distribu- tion system are eliminated in the two bus system model, the voltage instability caused by those factors can not be seen in a two bus system model. The twelve bus system model includes the air gap saturation, field current limit, line drop compensation, exciter system control, and the network. Because of the complexity of the twelve bus system, it is impossible to investigate the system voltage instability by using a phase portrait. The voltage instability tests are applied to determine the system voltage sta- bility and the causes of voltage instability. A Q—V curve for bus LOAD2 is also provided for each simulation case to monitor the efl‘ects of the change of the network operating con- dition to the system voltage instability. All the data shown in this chapter are in per unit base. It will be shown in section 4.1 that the only system bifurcation which can occur in a two 62 63 bus system where exciters are disabled is when the high and low voltage solutions merge together. The eigenvalue of the flux decay dynamics are negative real values which ap- proach zero as the solutions merge. It indicates that the merging point is a bifurcation point. A simple exciter control is included and the effects of this exciter control on the tra- jectory of the system is discussed. An algebraic bifurcation due to linearly dependence in [D1 Dig] can occur which does not cause system bifurcation. This singularity is associated with possible chaotic behavior. The two bus system model is shown to have a stable low voltage solution at small reactive load levels and an unstable low voltage solution at hi gh- er reactive load levels. The boundary where the low voltage solution is no longer stable is where the chaotic behavior can occur. In section 4.2, algebraic bifurcation and load flow bifurcation of a 12 bus model are stud- ied. Two cases are studied. In the first case, line reactances of the two lines that connect a load bus to the rest of the system are increased. This bus is a voltage control area. The weakening of the boundary of the voltage control area is theoretically sufficient to cause a voltage collapse bifurcation. The collapse must occm' because a theoretical upper bound on the small eigenvalue associated with a voltage control area is the maximum of the sum of the magnitude of boundary jacobian elements connected to a bus. Since there is only one bus and the boundary jacobian elements approach zero, the upper bound on the small eigenvalue associated with the voltage control area approaches zero. Since an eigenvalue approaches zero, the necessary conditions for a bifurcation must occur. The computational results confirm that collapse occurs but also indicates that the algebraic bifurcation is due to a reactive demand supply problem associated with shipping reactive power into this voltage control area. The shunt capacitive reactive supply at the bus is shown to be re- duced due to the large bus voltage drop needed to ship the reactive and real power to the bus over the weakened voltage control area boundary. The branches in the voltage control area boundary are shown to experience geometrically increased reactive losses as the line reactances of the voltage control area boundary increases. The ability of the boundary 64 branches to ship reactive power declines as the branches become drains and suck reactive power from both buses they are connected to. Thus, the reactive demand supply problem develop at a bus despite the fact that there is ample reactive generation reserves on all of the generators in the system. The second case confirms that an algebraic bifurcation at a load bus will cause a system bifurcation in the general power system model. The results also indicate that as algebraic bifurcation is approached, the transient stability program will not converge to a solution. This is similar to a Newton Raphson load flow where voltage collapse occurs when an ei— genvalue of the load flow jacobian is zero but may not converge when an eigenvalue of the jacobian is less than one. Since the EPRI Transient Mid Term Stability Program utilizes a Newton Raphson algorithm to solve the algebraic equations, the program puts out a diag- nostic that a solution can not be found before voltage collapse bifurcation occurs. The second case shows theoretically and computationally that the bifurcation test results are confirmed by a Q—V curve. Q—V curves are widely used in the utility industry to assess proximity to voltage collapse. The Q-V curve is a stress test where reactive load is added at a bus until the system can no longer supply that load. It has been shown that Q—V curve is lower bound on the small eigenvalue of the reactive power voltage jacobian matrix for the reactive deficient voltage control area where bus at which the Q—V curve is calculated resides. The reactive power voltage jacobian SQLV is a submatrix of D and singularity of this submatrix does not necessarily indicate singularity of J. However, singularity of S QLV has been shown to be associated with violation of virtually every known test for voltage collapse. Since the Q—V curve is a lower bound on the small eigenvalue of SQLV associated with a voltage control area, it is theoretically related to load flow bifurcation, transient mid term stability model bifurcation tests, load flow sensitivity voltage collapse tests, and lin- earized dynamic voltage collapse test. The Q-V curve never reaches the knee at which voltage collapse bifurcation should occur on the Q-V curve. These results indicate the sys- 65 tem under study comes close but never achieves a voltage collapse bifurcation as reactive load is increased at the bus which is an isolated voltage control area from the first case. This confirms the results of the algebraic bifurcation and system bifurcation tests where the system bifurcation test matrix and algebraic bifurcation test matrix approach singulari- ty but are never singular for increasing reactive load at this same bus. The test results indicate that the 12 bus test system is not vulnerable to algebraic bifurca- tion and reactive demand and supply problem. The system was modified to make it far more susceptible to reactive demand supply problem by increasing series reactances and shunt susceptances of lines. However, the EPRI Transient Mid Term Stability Program would not converge to an equilibrium when the 12 bus system model was modified to make it more susceptible to reactive demand supply voltage collapse problems. The New- ton Raphson algorithm in the Transient Mid Term Stability Program attempts to solve the real and reactive power balance equations together with the difi‘erential equations. Since the transient stability model has no generator terminal PV buses, as the load flow model has, the Newton Raphson algorithm in the transient stability program does not converge but the load flow Newton Raphson program converges. A program for solving for the equilibrium of the general stability model is needed if proximity to voltage collapse is to be accurately assessed. DynamiCIalgebraic bifurcation problems are analyzed in Section 4.3 and 4.4. In Section 4.3, a generator and its terminal bus are isolated fiom the rest of the system by increasing the transformer reactance that connects the generator to the rest of the twelve bus system. The purpose of this particular case is to establish whether the results obtained for the two bus system composed of the generator internal and terminal bus in Section 4.1 are valid in a general system. The conclusion is that the results on the two bus system can be used to describe the dynamic/algebraic bifurcation that can occur due to the bifurcation of the load flow, flux decay manifold, and control manifold of that generator. The generator dynamics 66 are stable with no oscillations for increase in reactive load at the generator terminal bus if the field current limit has not been reached and the exciter has not been disabled. Howev- er, if the continuous rating field current limit is exceeded and the excitation system is not disabled, the excitation system dynamics approach a dynamic bifurcation. Static relation- ship between reactive generation and reactive load and reactive load and voltage indicate the element SE}, and S Q GQL associated with the generator terminal bus are negative. A root locus analysis indicates that the system could experience a dynamic/algebraic bifur- cation although none occurred as reactive load was increased before the transient stability program would no longer converge to an equilibrium. The system experienced a dynamic/ algebraic voltage collapse bifurcation when the excitation system was disabled at the reac- tive load level where the field current was at the continuous rating limit. A time simulation indicates that there is a rapid decline of terminal voltage, field current, and reactive power output. The flux decay bifurcation test, and the dynamic algebraic bifurcation test confirm the eigenvalue analysis and transient stability simulation results that there is a dynamic voltage collapse problem. The two bus voltage control area case of a generator internal and terminal bus is extended to a three bus voltage control area composed of a generator, terminal, and load bus in Sec- tion 4.4. The test for voltage collapse is performed by increasing reactive load at the load bus rather than at the generator terminal bus as in the case studied in Section 4.3. The re- sults indicate an oscillation develops before the field current limit is reached. A eigenval- ueleigenvector analysis as reactive load is increased indicates the oscillation occurs at the generator flux decay and exciter dynamics. A root locus is performed to explain why the oscillation develops. The system never approaches an algebraic system bifurcation based on a Q—V curve and on the algebraic bifurcation test. The system never approaches a dy- namic] algebraic bifm'cation as reactive load and the field current limit is reached. After the continuous field current limit is hit and the excitation system is disabled, the system BXperiences a dynamic/algebraic bifurcation. The transient stability program fails to be hh 67 able to simulate this unstable system response. 4.1 Simulation Results of A No Bus Power System Model The two bus power system model is composed of a generator internal bus and terminal bus. The analysis used is taken from [21] but the computational results and the relation- ships to the theory in Chapter 3 are new. This two bus model is shown in Figure 4.1. 5'48 is the internal bus voltage magnitude and phase angle. The PL and QL are real and reactive power loads at the terminal bus and are assumed to be constant. The generator is assumed to be a round rotor machine, so x'd = xq. This model is valid for simulating large power system if (a) large reactances of the transformers isolate every generator from the rest of the system, and (b) the rest of the system can be modeled as an equivalent bus with constant real and reactive power loads. Allthedatashowninthissectionareinperunitbase. 325' x'd 529 Pc= O__.| g r—‘PDQL E f d Figm'e 4.1 Two Bus Power System Model 4.1.1 Mathematical Models The mathematical model without exciter dynamics of a two bus power system can be stat- edasfollows . -x x -x' 1' 4015' = —,—"E'+ " , dEcosS+Efd (4.1) x d x d E'E ' PL = ,m5 (4.2) x d E'EcosS-E2 QL = 1 (4'3) x d where 8 = 8—0 Since we assume that there is no angle instability problem, 8 can be eliminated and leave a second order nonlinear system. From (4.2) . PLx'd ““5 ' W From (4.3) x' +32 c038 = gigs—— (4.4) and (PLx'd) 2+ (QLx'd-t- £2) 2 w328+sin25 = l = 2 (5'5) =>g(E,E') = (E'E)2- (PLx'dV- (QLx'd+E2)2 = o (4.5) Plug equation (4.4) into equation (4.1) we get 69 . -. _ 'xd . xd—x'd 2 xd-x'd "aoE — EB +WE +TQL+Efd (4.6) Equation (4.6) is a constrained differential equation. It is constrained by equation(4.5). The states F. and E’ have to always stay in a manifold defined by equation (4.5). We call this manifold the load flow manifold. If we set E" = 0, equation (4.6) becomes f(E, E') = —E'2+ +f—ff’E E2+ (xd-x'd)QL+Efd = o (4.7) It turns out that the equilibrium point of the flux decay equation (4.6) has to always stay in a manifold defined by equation (4.7). We call this manifold the equilibrium manifold or the flux decay manifold. The intersections of the load flow manifold (4.5) and flux decay manifold (4.7) are the equilibrium points of the system. In Figure 4.2, the dotted line is the load flow manifold and the solid line is the flux decay manifold. There are two intersec- tions that stand for two steady state solutions or equilibrium points. One is at (Eh, E'h) = (0.99, 1.01) and we call it high voltage solution. This solution is usually within the acceptable voltage range. Another one is at (E ,, E‘ ,) = (0.22, 0.4129) and we call it low voltage solution. (Eu, E‘ u) = (0.26, 0.404) is the point that has the mini- mum E' of a load flow manifold. Since the flux decay manifold is derived by setting E." = 0, E" > 0 for the area below flux decay manifold and E“ < 0 for the area above flux decay manifold. Since the trajectory is constrained within the load flow manifold by (4.5), it follows that ag . EEdE-t-g— gal? = 0 as as» 272 _E’-2 (48) “HE“ E: E5 ' 65‘ At the minimum point of E' of the load flow manifold (5,, E‘ u) 70 dB“ _ a? _ o =E'fi = 2(x'dQL+E3) (4.9) If E>Eu, by equations (4.5) and (4.8) dB. d—E >0, and if E E“. It indicates that both (Eh, E'h) and (E,, 5",) are stable equilibrium points in this case. It is known that the equilibrium point after sudden rise in load for the recent Tokyo voltage collapse was a sta- ble low voltage solution. Thus, voltage collapse, that occurs after a disturbance or contin- gency, may be due to the system being dislodged from a stable high voltage solution and converges to a low voltage stable solution. A system that experiences load flow demand/ supply problems could experience a voltage decline and ultimately converges to a low voltage solution. It is also possible that the low voltage steady state solution intersects with the lowest point 71 1.2 U I I I I ‘ I l 1 _ Eh=0.9950. Eh(Prime)=l.Oll4 ' El=0.2200, E1(Prime)=0.4129 Eu=0.2600, Eu(Prime)=0.4037 EPrime 0.3 1 . . . 1 P=0.33,Q=0.07 ,Efd=1 .055 ,Xd=0.66.Xd(Prirne)=0.2 I l 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 E Figure 4.2 Load flow and equilibrium manifolds for two stable equilibrium points 1.1 P=0.33,Q=0.32.Efd=l.22,Xd=0.66,Xd(Prime)=0.2 1 .2 I l r r T r I t w Eh=0.9959. Eh(Prirne)=l.0614 ." 1-1 ’ I=0.3000, El(Pr'ime)=0.5585 '1 Bu=0.3000, Eu(Prime)=0.5585 III 0.4 1 1 1 1 1 1 _L 1 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 B Figme 4.3 Load flow and equilibrium manifolds for loss of causality at low voltage solution 72 of the load flow manifold if reactive power QL increases to 0.32. In Figure 4.3, ( E ,, E',) = (Eu, 5"“) . The system loses its causality (det(D)=0) at the unstable equilibri- um point (E u, E ' u) . The system would remain at the unstable equilibrium point (E u, E ' u) unless some disturbance occurs. The disturbance would cause the trajectory to either approach (E h, 5",.) or to experience dynamic collapse as E declines and E’increas- es. If there were external dynamics in the system 4.1-4.3 , the dynamical system theory may not be mature enough to define what will happen at this point. It has been shown that chaotic behavior occurs in the simulation [21] if there are external dynamics. It will be shown in the following section that as the system moves to algebraic bifurcation, the sta- bility of the system at low voltage equilibrium point becomes unpredictable because the simulation does not converge and dynamical system theory can not currently predict what will happen. Notice that the high voltage solution (E h, E ' h) where the power system is usually operated at is still a stable equilibrium point. The chaotic behavior or loss of volt- age stability would occur only if the system is operated at the low voltage solution. Note that the reactive load increase causes the stable low voltage solution to experience loss of causality and the possible loss of voltage stability or chaotic behavior. Figure 4.4 is the case where E, > Eu for an additional increase in QL to 0.47. There are two intersections of load flow and equilibrium manifolds. The high voltage solution (E h, E") is stable. The low voltage solution (E p E',) is unstable. The equilibrium points are closer if we compare with the case in Figure 4.2 and Figure 4.3. It will be shown in Figure 4.5 and Figure 4.6 that these two equilibrium points will merge together and disap- pear as QL continuous to increase. Figme 4.5 is the case where there is only one intersection of the load flow manifold and the equilibrium manifold at Q=0.8326. This is the point that violates the conditions of dynamic/algebraic and algebraic] dynamic tests and the system jacobian matrix becomes singular. It will be shown in the following section that two equilibrium points merge to- 73 1 2 P=O.33.Q=0.47,Efd=l.315.Xd=0.66.Xd(Prime)=0.2 3, 1211:09950. 131101111».me 1., _ El=0.3750, El(Prirne)=0.6500 Eu=o.3425. Eu(Prime)=0.6463 EP' 0.5 1 1 1 1 1 1 1 1 . 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 l 1.1 E Figure 4.4 Load flow and equilibrium manifolds with one stable and one unstable equilibrium points P=0.33,Q=0.8326.Efd=l.5,Xd=0.66.Xd(Prime)=0.2 2 I I I I I I I I T Bt=0.7650. Et(Prime)=0.9863 1.3 Eu=0.4250, Eu(Prirne)=0.8314 1.64, 1.4*- | | \ Q \ EPrime Figure 4.5 Load flow and equilibrium manifolds with one intersection EPrime EPrirne 74 P=0.33,Q=0.85,Efd=1.5 ,Xd=0.66,Xd(Primc)=0.2 2 I I T I I I Y I Eu=0.4250, Eu(Prime)=0.8395 Figure 4.6 Load flow and equilibrium manifolds with no intersection Equilibrium Manifolds for different 13de 1.2 I V I Solid Line: Equilibrium Manifold(Efd=0.8) 1.1 _ Dashed Line: Equilibrium Manifold(Efd=l.05) , Dotted Line: Load Flow Manifold 3 --:>' lanaicpcczxmaaaixdrnmepaz . 0.3 L . . . . . _. . . 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 E Figure 4.7 Load flow and equilibrium manifolds for different field voltages 75 gether and the system becomes unstable with the voltage E and E ' decaying toward zero. This type of loss of voltage stability is observed in the slow continuous decline in voltage. It will be clear in the 12 bus system example that when a generator hits field current limit and the excitation system is disabled, a dynamic voltage instability results in continuous decline in induced voltage field current and terminal bus voltage that are caused by the loss of stability of generator dynamics. Figure 4.6 is the case where QL=0.85 and there exists no solution when field current limits are hit and the excitation system is disabled. The case where the exciter is not disabled is discussed in the next section. It will be shown that if the exciter is not disabled, this case would never happen. An increase of field current when there is a real or reactive load in- crease will move the high and low voltage solutions far away from each other to preserve the stability of the system voltage. The efi‘ects of changes of field current, real power load, and reactive power load will be now discussed in Figure 4.7, 4.8, and 4.9. Position and shape of these two manifolds change with parameter values like Em, PL, and QL. Figure 4.7 shows the changes in the flux decay manifold for difl‘erent Em values. Since load flow manifold is not a function of Em, there is no change in both position and shape of the load flow manifold. In Figure 4.7, the solid line represents the equilibrium manifold with Efd = 0.8 , and the dotted line represents the load flow manifold. The inter- sections of these two manifolds are at (0.7, 0.71) and (0.29, 0.405). Notice that the high voltage solution is not within the acceptable range in a normal system. If we increase the Efd fi'om 0.8 to 1.05, the equilibrium manifold can be move up and represented by a dashed line. The new intersections of the load flow manifold and equilibrium manifold be— come (0.98, 1.05) and (0.215, 0.42). The high voltage solution is now in the acceptable range. By increasing field voltage (field current), the system voltage E can be increased. This indicates the important role that adjustment of Em has in maintaining the system volt- age at the desired value. 76 Equilibrium and Load Flow Manifolds for different Qs 1.2 Solid line: Equilibrium Manifold(Q=0.07) 1.1 _ dotted line: Equilibrium Manifold(Q=O.30) . dashed line: Load flow Manifold(Q=0.07) ’ dashdot line: Load flow Manifold(Q=0.30) Efd=l.05. P=0.33, Xd=0.66, Xd(Prime)=O.2 pa I EPrimc 0.6 0.7 0.8 0.9 E Figure 4.8 Load flow and equilibrium manifolds for different reactive loads Load Flow Manifolds for different Ps 1.6 Solid line: Load Flow Manifold(P=0.33) 1‘ Dashed line: Load Flow Manifold(P=0.83) 1.4 H, Dotted line: Equilibrium Manifold(P=0.33) 4 1. Efd=l.05. Q=0.07, Xd=0.66. Xd(Prime)=0.2 1.2- EPrime 0.5 0.6 0.7 0.8 0.9 l 1.1 E Figure 4.9 Load flow and equilibrium manifolds for different real power loads 77 In Figure 4.8, solid line and dotted line represent the equilibrium manifold and load flow manifold at reactive power load at 0.07 respectively. The high and low voltage steady state solutions are (0.95, 0.95) and (0.215, 0.42) respectively. Since both the load flow manifold and equilibrium manifold are functions of the reactive power load, the position and shape of both of the manifolds will change if there is a change in the reactive power load. If the reactive power load is increased from 0.07 to 0.3, the new equilibrium manifold and load flow manifolds are represented by dashed line and dashdotted line. The new high and low voltage steady state solutions become (0.76, 0.85) and (0.37, 0.58). This is the case where there is no excitation system control. It indicates that if the system loses its excitation sys- tem control, a reactive power load increase will cause a system voltage decrease. In this special case, we can also see that the increase of reactive power load will move the system from two stable equilibrium points to one stable equilibrium point (E h, E'h) and one un- stable equilibrium point (E ,, E' ,) . The equilibrium points become closer to each other as we increase the reactive power load. It will be shown that as we increase the reactive pow— er load even further, the two equilibrium points will merge together and become an unsta- ble equilibrium point. This is the point where the algebraic/dynamic test condition is violated and bifurcation may occur at this point. Figure 4.9 shows the efi‘ects of changes of the real power load. Since the real power load is a function of the load flow manifold but not a function of the equilibrium manifold, only the position and shape of the load flow manifold will be changed for a real power load change. The dotted line represents the equilibrium manifold. The solid line represents the load flow manifold where the real power load is at 0.33. The high and low voltage solu- tions are (0.95, 1.0) and (0.215, 0.43). Equation (4.5) g(E,E’) = (E'E)2- (PLx'd)2— (QLx',,+E2)2 = o (4.5) 78 shows that real power load is neither a coeflicient of E nor E’. It indicates that the change of real power load will influence the solutions of lower voltage and has less impact on high voltage solutions. The load flow manifolds for difi‘erent real power loads will con- verge as E and E’ increase. This phenomenon can be seen in the Figure 4.9. The dashed line is the load flow manifold when the real power load is at 0.83. The new high and load voltage steady state solutions become (0.88, 0.9) and (0.53, 0.62). The sys— tem is moving from two stable equilibrium points, (0.95, 1.0) and (0.215, 0.43), to one sta- ble equilibrium point (0.88, 0.9) and one unstable equilibrium point (0.53, 0.62). If there is no excitation system control and real power load is further increased, those two equilibri- um points will merge together just like the case of increasing reactive power load and the algebraic/dynamic test conditions will be violated. 4.1 2 Two Bus System Simulation With Excitation System Control included The mathematical model developed in section 4.1.1 does not include the excitation system control. The field voltage (Em) is an assigned value for each simulation in section 4.1.1. In a real power system, field voltage plays an important role in maintaining terminal bus volt- age at a preset value. In this section, the exciter system control is included. The exciter is assumed to have infinite gain so that as long as the field current is below its upper limit the terminal bus voltage can be held at its preset value. It is also assumed that the exciter can respond to disturbances infinitely fast. If the field current hits its upper limit, a field current limiter will disable the excitation system control and field voltage is assume to remain at its upper limit. A computer program is developed for the two bus system with the exciter control simula- tion. The algorithm for this computer program is in Figure 4.10. The program reads in the initial values of the field voltage (Em), the desire terminal bus voltage (El-gt"). and the re- active power load condition. In the first iteration, the program solves equations (4.5) and 79 ® Read in Bide Eros and QL ‘ Solve equations (4.7) and (4.5) for E and E‘ I Erd=Eid+7L / ' Output / Figure 4.10 Algorithm of a computer program that includes the exciter effects in a two bus system model 80 (4.7) for terminal bus voltage (E) and internal bus voltage (E '). E is then compared with the Eref. If the absolute value of the difference of E and Emf is less than the tolerance 8, the program stops at this iteration and provide an output for the specified reactive power load condition. If the difl‘erence between E and Eref is larger than 8, the program checks if the field current is on its upper limit or not. If the field current is on its upper limit, it indi- cates that the generator loses its capability of controlling voltage. The program provides the output E and E ' and stops. If the field current is still within its acceptable operating range, the field voltage will be increased (or decreased) by a small value 7». The increase or decrease of field voltage is dependent on whether the terminal bus voltage is lower or higher than the desired terminal bus voltage. After the field voltage is adjusted, the pro- gram starts its second iteration to solve the equations (4.7) and (4.5). The program will stop when either the difference of terminal bus voltage and desired terminal bus voltage is within the tolerance or the field current hits its upper limit. The procedure of this simulation is shown in Figure 4.11. The purpose of this simulation is to (a) confirm the validity of voltage instability tests by using phase portrait analy- sis method, (b) show the voltage instability occurs in the flux decay dynamics much earlier than the load flow jacobian matrix becomes singular in the case that the sys- tem is operated at the neighborhood of the high voltage solution, and (c) investigate the voltage instability in the neighborhood of low voltage solu- tion. The line drop compensation, air gap saturation, and precise exciter control are not includ- ed in this simulation and will be included in the twelve bus system simulation. Table 4.1 81 l increaseQL I simulation program diverges Yes (no solution) save results voltage instability tests phase portrait Figure 4.11 Procedure of two bus simulation 82 shows the base case data for this two bus system simulation. All of these data are in per unit base. Table 4.1 Base case data for the two bus system simulation P Q En Emma. Era xr x'r 0.33 0.07 1.055 1.5 1.0 0.66 0.2 The equations are restated here to formulate the voltage instability matrices. . -x x -x' 1." doE' = —,—"E'+ " , dEcosS-t-Efd (4.1) x d x d E'E ’ pL = f“ (4.2) x d E'Ecosfi - E2 QL = 1 (4'3) x d The jacobian matrix of these nonlinear equations is _xd (Id-I'd) 00380 (I'd-1d) 5.08il'1801 x d x d x d J ___ E,,sin50 E'osinfi0 E'oEocosfi0 rd 1'4 I'd Eocos 80 E '0 c0880 - 2Eo -E'0Eo sin 80 1'4 I'd x'd - where [273] 83 x d x',, -Eosin501 C = ”4 Eocosfi0 _ 1.4 J _ E'asinb0 E'oEocosfio- D = x'd Jr'rl _ x',, x',, _ The algebraic bifurcation test matrix is D. The dynamic/algebraic bifurcation test matrix is A = A - BD'IC The algebraic/dynamic bifurcation test matrix is f) = D - CA'IE Since in this two bus system model we assumed there is no angle stability problem and the excitation system control can respond to the disturbance infinitely fast, the dynamic/alge- braic bifurcation test is the same as the flux decay bifurcation test. Figure 4.12 is the phase portrait for the base case data. Since E,=0.22 is less than Eu=0.26, both high and low voltage steady state solutions are stable. It is confirmed by the voltage instability tests in Table 4.2. The eigenvalues of dynamic/algebraic bifm'cation test matrix A are negative for both high and low voltage solutions. The condition number of the alge- braic bifurcation test matrix D for low voltage solution indicates that if the system is oper- ated at a point near the low voltage solution, the system is very close to a loss of causality P=0.33,Q=0.07.Efd=l.055,Xd=0.66,Xd(Prime)=0.2 1.2 EPrime Eh=0.9950, Eh(Primc)=l .0114 El=0.2200, El(Prime)=0.4129 Eu=0.2600, Eu(Prime)=0.4037 j j 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 l 0.1 1.1 E Figure 4.12 Phase portrait for Q=0.07 Table 4.2 Voltage instability tests for Q=0.07 E E' l/cond(D) l/cond(b) eig(A) High voltage 0.995 1.0114 0.8732 0.2683 -0.9364 Low voltage 0.22 0.4037 0.077 0.1736 -9.1857 1.2 85 P=0.33,Q=0.27,Efd=1.185,Xd=0.66,Xd(Prime)=0.2 1.1 e EP' Eh=0.9950, Eh(Prime)==l.0514 El=0.2850, E1(Prime)=0.5285 Eu=0.2925, Eu(Prime)=0.5278 0'31 0:2 0:3 0:4 0:5 0:6 0:7 0T8 0:9 1.1 E Figure 4.13 Phase portrait for Q=0.27 Table 4.3 Voltage instability tests for Q=0.27 E E' 1/cond(D) 1/cond(D) eig(A) High voltage 0.995 1.0514 0.844 0.2225 -0.8379 Low voltage 0.285 0.5285 0.0285 0.3161 -48.2318 86 problem. The condition number of the algebraic/dynamic bifurcation test matrix D indi- cates the system is far away from the system bifurcation point for both the high and low voltage solutions. In Figure 4.13, the reactive power load is increased to 0.27. Since the field voltage Efd is within its acceptable limit, the high voltage solution of the terminal bus is close to its pre- set value l.0. The high and low voltage solutions are both stable. The condition number of the algebraic bifurcation test matrix D in Table 4.3 for the low voltage solution indicates that the increase of reactive power load may cause loss of causality problems in the alge- braic equations if the system is operated at the low voltage solution. Comparing Figure 4.12 and 4.13, the high and low voltage solutions move closer to each other for an increase in reactive power load even when there is still plenty of field current reserve. The field cur- rent reserve can only guarantee two steady state solutions but can not guarantee both of these two steady state solutions to be stable. Figure 4.14 and Table 4.4 are for the case where the reactive power load is further in- creased to 0.32. The low voltage solution is “almost” at the minimum of the load flow manifold. Because of the round ofi‘ error of the computer, it is impossible to find the exact point. Table 4.4 shows that the system is very close to loss of causality in low voltage so- lution (l/cond(D)=0.014). The eigenvalue of A for low voltage solution starts to move to negative infinity if E, is less than but very close to E“. If E, is larger than but very close to Eu, we will see a very large positive eigenvalue of A . It indicates that at the neighborhood of the unstable equilibrium point of 4.1-4.3 (E a, E' u) = (E,, E‘ ,) , the eigenvalue of A can be either close to positive infinity or negative infinity. The loss of causality of the alge- braic equation causes this kind of chaotic phenomena. The eigenvalue of A for high volt- age solution is negative in Table 4.4 but is moving toward the right half plane for increase in QL if we compare the eigenvalue of Ii in Table 4.3. If the reactive power load is increased to be at 0.37, the low voltage solution becomes an 1.2 P=0.33,Q=0.32.Efd=l.22.Xd=0.66.Xd(Prime)=0.2 1.11- EPrirne Eh=0.9959, Eh(Prime)=l.0614 El=0.3(X)0. El(Prime)=0.5585 Eu=0.3000, Eu(Prime)=0.5585 0.4 ‘ ‘ ‘ ‘ 4 ‘ L 0.1 0.2 0.3 0.5 0.7 0.8 0.9 l 1.1 Figure 4.14 Phase portrait for Q=0.32 Table 4.4 Voltage instability tests for Q=0.32 E E' l/cond(D) 1/oond(D) eig(A) High voltage 0.995 1.0614 0.8329 0.2115 -0.81 19 Low voltage 0.3 0.5585 0.014 0.3556 -1 10.1627 1.2 1.1 I EPr-irne P=0.33,Q=0.37,Efd= l .25,Xd=0.66,Xd(Prime)=0.2 Eh=0.9950, Eh(Prime)=l.07 ll El=0.3300. E1(Prime)=0.589l Eu=0.3125, Eu(Prime)=0.5885 Tr 1 0"(11.1 0:2 0:3 0:4 0.5 0:6 0:7 0:3 0:9 1.1 E Figure 4.15 Phase portrait for Q=0.37 Table 4.5 Voltage instability tests for Q=0.37 E E' l/cond(D) 1/cond(D) eig(A) High voltage 0.995 1.07 1 1 0.8214 0.2 0.7861 Low voltage 0.33 0.5891 0.0696 0.3915 22.3132 1.8 P=0.33,Q=0.76,Efd=1 .5,Xd=0.66,Xd(Prime)=0.2 1.64. EPrime 1.4» '1. Eh=0.9750, Eh(Prime)=l.1329 El=0.5550. El(Prime)=0.8374 Eu=0.4075, Eu(Prime)=0.7972 1 Figure 4.16 Phase portrait for Q=0.76 Table 4.6 Voltage instability tests for Q=0.76 E E’ 1/cond(D) 1/cond(D) eig(A) High voltage 0.975 1.1329 0.7196 0.1171 -0.5318 Low voltage 0.555 0.8374 0.4855 0.2544 1.6653 90 unstable equilibrium point because of E, > E u. The instability of the low voltage equilibri- um point (E ,, E',) = (0.33, 0.589) is also shown in Table 4.5 because the eigenvalue of A becomes positive. The trajectory close to this unstable equilibrium point will move away from it. The magnitude of the eigenvalue of A for the high voltage solution is also decreased if it is compared with the eigenvalues in the Table 4.3 and 4.4. Although the ex- citer can still hold the terminal bus voltage at its preset value as we increase the reactive power load, the level of the stability of the dynamic states is decreased because the eigen- value of A is moving toward the right half plane. When Q is at 0.76, Efd=Efdm in Figure 4.16. The exciter control is disabled and the ter- minal bus voltage can not be held at its desired value. The indicators of system bifurcation l/cond(D) for both high and low voltage solutions decrease. It indicates that the system jacobian becomes more singular and the system is moving to a system bifurcation point. Since the equilibrium points are far away from the minimum point of the load flow mani- fold, the indicator of algebraic bifurcation l/cond(D) is large. In Figure 4.17, there is only one intersection of the load flow and equilibrium manifolds at Q=0.8326. This point is the system bifurcation point. The high and low voltage solutions merge together. Our system bifurcation test indicator 1/cond(D) shows the algebraic/dy- namic test matrix is extremely close to a singular matrix. Because of the limitation of the precision of computer, the eigenvalue of A is still negative but very close to zero. This is the point of dynamic voltage instability since the equilibrium point is unstable since all points on g (E, E') above (E,,, E',) would converge to (E,,, E”) . The trajectory would result in E and E' approaching zero that results in voltage collapse due to the dynamic sta- bility problem. Note that the dynamic voltage collapse occurred after the field current limit is hit which disabled the excitation system. The dynamic instability that caused dynamic voltage instability occurred for very small reactive load increase above that where the field current limit is hit which disabled the excitation system. The dynamic instability that 91 P=0.33,Q=O.8326,Efd= l .5,Xd=0.66,Xd(Prime)=0.2 1.8: 1.6-‘2 1.4 - EP' Em0.7650, Et(Prime)=0.9863 Eu=0.4250, Eu(Prime)=0.8314 Figure 4.17 Phase portrait for Q=0.8326 Table 4.7 Voltage instability tests for Q=0.8326 E E' l/cond(D) l/cond(b) eig(A) High voltage 0.765 0.9863 0.6722 0.000472 -0.0023 Low voltage 0.765 0.9863 0.6722 0.000472 -0.0023 EPr-ime P=0.33,Q=0.85,Efd=1.5,Xd=0.66,Xd(Prime)=O.2 j I I I T I Eu=0.4250, Eu(Prime)=0.8395 I Figure 4.18 Phase portrait for Q=0.85 93 Table 4.8a Summary of voltage instability test for high voltage solution Q E E' 1/cond(D) det(D) l/cond(b) det(D) eig(A) Em 0.07 0.995 1.01 14 0.8732 24.51 0.2683 6.9552 —0.9364 1.055 0.17 0.995 1.0313 0.8624 24.489 0.2452 6.5929 -0.8884 1.12 0.27 0.995 1.0514 0.844 24.444 0.2235 6.206 -0.8379 1.185 0.32 0.995 1.0614 0.8329 24.41 0.21 15 6.01 -0.81 18 1.22 0.37 0.995 1.0711 0.8214 24.381 0.201 5.808 -0.7861 1.25 0.76 0.975 1.1329 0.7196 22.467 0.1171 3.62 —0.5318 1.5 0.81 0.875 1.0628 0.7079 15.874 0.0732 1.6138 -0.3355 1.5 0.8326 0.765 0.9863 0.6722 10.145 0.00047 0.00698 -0.0023 1.5 Table 4.8b Summary of voltage instability test for low voltage solution Q E E 1/cond(D) det(D) 1/cond(D) det(D) eig(A) 1:.rd 0.07 0.22 0.4037 0.077 02126 0.1736 -0.6754 -9.l857 1.055 0.17 0.24 0.4704 0.0941 -0.229 0.2358 -0.7762 -11.2086 1.12 0.27 0.285 0.5285 0.0285 -0.0611 0.3161 -0.888 48.2318 1.185 0.32 0.3 0.5585 0.014 -0.029 0.3556 -0.98 -110.1627 1.22 0.37 0.33 0.5891 0.0696 0.154 0.3915 -1.042 22.3132 1.25 0.76 0.555 0.8374 0.4855 3.036 0.2544 -1.5325 1.6653 1.5 0.81 0.65 0.9049 0.5973 5.689 0.1144 -1.051 0.6096 1.5 0.8326 0.765 0.9863 0.6722 10.145 0.00047 0.111698 -0.0023 1.5 94 caused dynamic voltage instability occurred for very small reactive load increase above that where the field current limit is reached. If the reactive power load is increased to be at 0.85, there is no intersection of the load flow and equilibrium manifolds. This is shown in Figure 4.18. Table 4.83 and 4.8b are summaries of voltage instability tests for high and low voltage so- lutions respectively. Note that the signs of the determinant of D are always negative for low voltage solutions and positive for high voltage solutions. The sign of D and the sign of the eigenvalue of A are always in agreement indicating the determinant of D is a good test for dynamic voltage stability. This would be an important indicator for determining where the operating point of the system is, especially when the system is close to system bifurcation. 4.2 Load Flow and Algebraic Bifurcations 42.1 Introduction A twelve bus system (Figure 4.19) with generator dynamics included is the sample system of this section. The air gap saturation, excitation system control, field current limit, and line drop compensation are precisely modeled. It is shown in this section that voltage in- stability may occur due to load flow or algebraic bifurcation. 4 2.2 Load Flow Voltage Instability Simulations This section demonstrates that (a) the load flow voltage instability may occur at a voltage control area of the transmission network which is weakly connected to the rest of the system, 95 GEN2 L7 HSTZ —+ HSTl L6 GENl | I 3 1.3 L9 81 i I TERM2 TERM] fl _. L2 L1 LOAD2 LOAD3 L3 L4 L5 TERM3 GEN3 Figure 4.19 A twelve bus test system (b) (C) (d) 96 reactive generation reserve can be available on generators at the outside of the voltage control areas but the reactive power can not be transferred through the weak connected transmission lines to the area that needs the re- active power support, the load flow voltage instability is a supply and demand problem, and the transient stability program is not robust enough to simulate a load flow voltage collapse problem. In the simulation, the reactances of the transmission lines L2 and L3 are increased by fac- tors of 2.0, 3.0, 4.0, and 4.1. The simulation was performed using the load flow program because (a) (b) it is a reactive supply and demand problem and the load flow program is much more robust than the transient stability pro- gram in solving a stressed network since the load flow assumes the generator terminal bus voltages are specified and need not be solved for but the genera- tor terminal bus voltage must be solved for in the transient stability model. Solving for the algebraic equations for bus voltages at generator terminal buses as well as the high side transformer and load buses and the angles at terminal, high side transformer and load buses in the transient stability model is like solving a load flow without generator PV buses, which is very difficult to solve. Complete load flow simulation results are given in Appendix C. It should be noted that in- creasing reactances on L2 and L3 has the efi‘ect of causing bus LOAD2 to become a volt- age control area. It has been shown that the maximum of the small eigenvalue associated with a voltage control area is the maximum of the sum of the voltage control area bound- 97 Table 4.9 Load flow simulation for the increase of line reactances at L2 and L3 Reactance l 2 3 4 4.1 (times) Voltage“ 0.9576 0.9239 0.8740 0.7946 0.7106 (LOAD2) Angle" -9.1 «14.5 -20.4 -27.4 -3l.8 (LOAD2) Voltage 0.9962 0.9910 0.9864 0.9804 0.9719 (HSTZ) Angle -1.0 0.3 1.0 1.7 1.8 (HSTZ) Voltage 0.9870 0.9833 0.9773 0.9603 0.9206 (HST3) Angle -7.2 -9.1 -10.3 -11.5 -11.6 (HST3) Losses“ -l6.8 -8.09 0.6 11.38 22.83 (Ll) Losses -l4.38 -9. 15 -0.69 12.96 23.67 (L3) Angle Dif. 8.1 14.8 21.4 29.1 33.6 (LZ) Angle Dif. 1.9 5.5 10.1 15.9 20.2 (L3) *Angle in degree, voltage in p.u., and losses ill MVar. 98 ary branches connected to any bus. Increasing the reactances of L2 and L3 have the efl'ect of reducing the maximum eigenvalue of this LOAD2 bus voltage control area to zero, as- suring that a voltage collapse bifurcation will ultimately occur. These results will establish that load flow bifurcation is a reactive demand supply problem. Table 4.9 shows (a) the changes of voltages, (b) angle differences between buses HST2 and LOAD2 and buses HST3 and LOAD2, and (c) the line losses of the transmission lines which are connected to bus LOAD2 as reactances L2 and L3 are increased. In Figure 4.20, load bus voltages are plotted as a function of the magnitude of the line re- actances. Figure 4.21 shows dramatic increase in line losses for the increase of the line re- actance. Figure 4.22 shows the angle difl'erences of HST2 and HST3 versus the increase in line reactances. In Figure 4.20, the voltage of bus LOAD2 decreases as we increase the reactances. When the reactances are foru' times greater than the base case line reactances, the voltage of bus LOAD2 has reached an unacceptable value of 0.7946. The voltages of buses LOAD] and LOAD3 remain high before the reactances are increased by a factor of 4.1. The PV buses TERMl and TERM2 have plenty of generation reserve for each of these five cases. The reactive generation reserves at buses TERMl and TERM2 can not be transferred to LOAD2 area because of the weak connections. It indicates that the load flow voltage insta- bility is a reactive supply and demand problem at the LOAD2 bus. As the reactances are increased to two times the base case value, the voltage at bus LOAD2 decreases to 0.9239. This voltage decline results in a reactive power supply with- Voltage Line Losses 0.95 0.9 0.85 0.8 0.75 0.7 25 20 15 99 Load Bus Voltages Dashed line: Bus voltage at LOADl 1 Solid line: Bus voltage at LOAD2 _ Dorted line: Bus voltage at LOAD3 l 1.5 2 2.5 3 3.5 4 4.5 Reactance(times) Figure 4.20 Load bus voltages at different line reactances Line Losses at L2 and L3 - Dashed line: Line losses(L3) ‘ Solid line: Line losses(LZ) 1.5 2 2.5 3 3.5 4 4.5 Reactance(tirnes) Figure 4.21 Line losses for different line reactances 100 drawal associated with line charging (Figure 4.21) since the losses are negative when the reactances of L2 and L3 are small and positive as 12 and L3 reactances increase. The neg- ative line losses on L2 and L3 indicate the shunt reactive power provided by the line charging of long transmission lines are greater than the 12X reactive power losses on the lines when the line reactances are small. As the series reactances increase, the 12X losses increase dramatically and the reactive power supplied by line charging on these long trans- mission lines decrease due to voltage drop at LOAD2. Thus, the total reactive losses in Figure 4.21 increase with line reactances. Note that the line losses increase geometrically when line reactance is increased by a factor greater than 4. This geometric increase in re- active losses occurs when the line becomes a drain with SIL greater than 1 and sucks large amounts of reactive power from both buses it is connected to. The development of severe drain problems on voltage conuol area boundary branches can be shown to occur when branch angle differences exceed a threshold. Note that the angle diflerences in Figure 4.21 increase and suddenly grow geometrically as drains developed on the boundary branches. Notice that the voltage at buses LOADl and LOAD3 do not respond to the decrease of voltage at bus LOAD2 because the bus LOAD2 is isolated by weak connection and there is plenty of reactive power reserve close to buses LOADl and LOAD3 to hold their volt- ages high. As the reactances become three times larger than the base case values, the increase of an- gle differences (Figme 4.22), the decrease of voltage at bus LOAD2, and the increase of line losses at L2 and L3 further weaken the ability of bus LOAD2 to obtain reactive power supply. The voltage at bus LOAD2 decreases to 0.874. The voltage of bus LOAD2 is fur- ther decreased to 0.7946 when the reactances of L2 and L3 are increased to four times of their regular values. This voltage is well below the acceptable range. The line losses are about fifteen times larger than the line losses at smaller reactances (3 times the normal val- ues). Angle Difference 35 30 101 Angel Differences of HST2/LOAD2 and HST3/load2 I T 7 T ' Dashed line: Angle difference(HST3/LOAD2) Solid line: Angle difference(HST2/LOAD2) p- >- l.5 2 2.5 3 3.5 4 Figure 4.22 Angle differences of HST2/LOAD2 and HST3/LOAD2 at difierent line reactances 4.5 102 As the reactances of L2 and L3 are increased to be 4.1 times of their base case values, the voltage at bus LOAD2 decreases sharply. The line losses and angle differences are in- creased tremendously. This indicates this voltage problem is truly a reactive demand and supply problem at bus LOAD2. These results confirm that the algebraic bifurcations are reactive demand supply problems. Large shunt capacitive reactive supply withdrawal at buses in the voltage control area and geometrically increasing reactive losses on voltage control area boundary branches bring on the reactive demand supply problem. Note that the results also confirm that voltage collapse can be forced to occur when the voltage con- trol area boundary is weakened causing the upper bound on the small eigenvalues associ- ated with that voltage control area to approach zero. 4.2.3 Algebraic Voltage Instability Simulations In Section 4.2.2, It was suggested and analytically justified that if the transmission system is stressed by increasing the reactances of L2 and L3, the transient stability program has difficulty in obtaining converged equilibrium solutions. These equilibrium solutions are obtained by simulating the transient stability model based on the converged load flow so- lution. The lack of convergence of the equilibrium program (steady state solution of the transient stability program) is caused by large real and reactive power load on the trans- mission system and the increase of the reactances at L2 and L3. In this section, the reac- tances of L2 and L3 are increased but the real power loads at buses LOADl, LOAD2, and LOAD3 are decreased by a total of 150 MW. The decrease of the real power load can in- crease the strength of the network and help the equilibrium program to obtain converged results. In the simulation, the reactive power load at bus LOAD2 is increased from 40 MVar to 61 MVar. If the reactive power load is equal or larger than 62 MVar, an error message says “error in solving the network discontinuity” would be produced by the equilibrium pro— 103 gram. The ill-conditioned network matrix prevents the equilibrium program from obtain- ing a converged solution. The eigenvalues, eigenvectors, condition number, and determinant of the system jacobian matrix, algebraic/dynamic test matrix, and flux decay test matrix are computed for each converged steady state solution. The complete simula- tion results are provided in Appendix C. Table 4.10 Changes of the ratio of the determinant of the algebraic test matrix and the jacobian matrix Load level(MVar) 40 45 50 55 58 60 det(Dload)/det(D40) 1 0.7486 0.5129 0.2816 0.1490 0.0669 det(Jload)/det(l4o) 1 0.8381 0.6892 0.5180 0.4201 0.3608 Table 4.10 shows the changes of the ratio of the determinant of the algebraic bifurcation test matrix and the jacobian matrix as the reactive power load at bus LOAD2 is increased from 40 MVar to 60 MVar. The ratio drops dramatically in the algebraic bifmcation test matrix. It indicates the transmission network is under stressed and has dificulty in trans- ferring reactive power to bus LOAD2. The ratio for the system jacobian matrix is also de- creased from 1.0 to 0.3608. It indicates that the system jacobian matrix is becoming singular and is caused by the singularity of the algebraic bifmcation test matrix because the algebraic bifurcation test is approaching singularity faster. This confirms that a reac- tive supply and demand program in serving reactive load at load bus is one of the reasons that will cause system voltage instability. 104 All of the eigenvalues of the jacobian matrix (eigjj) and algebraic bifurcation matrix (eigd) in Appendix C decrease as we increase the reactive power load. This is another method of showing both the system jacobian matrix and the algebraic bifurcation test matrix are ap- proaching singularity. We have shown that the simulation algorithm of the equilibrium program is less robust than the Newton-Raphson algorithm of the load flow program in the last section. The nec- essary condition for a N ewton-Raphson method to obtain a converged solution is that '1. >1 ‘l min where [H m," is the minimum absolute eigenvalue of the matrix. If any eigenvalue of the load flow jacobian matrix has an absolute value less than one may cause the divergence of the Newton-Raphson algorithm. In the simulation results (Appendix C), the smallest ei- genvalue of the system jacobian matrix decreases to 0.0511 and the smallest eigenvalue of the algebraic bifm'cation test matrix decreases to 0.1268. These eigenvalues are small enough to violate the necessary condition for Newton Raphson method to guarantee a con- verged solution. Thus, the equilibrium program that uses 8 Newton Raphson algorithm to solve the algebraic equations no longer converges when the reactive load at LOAD2 ex- ceeds 62 MVar. A more robust algorithm might allow the load flow and equilibrium pro- grams to converge to solutions at points right up to the point where bifurcation occurs 2. .n = 0. Since the equilibrium and load flow program utilize a Newton Raphson algo- M8 rithm, the solutions very close to bifurcation can not be computed. Tables in Appendix C.2 shows the equilibrium point for each load level. All of the genera- tors still have plenty of reactive generation reserve since there is no field current limit vio- lation. This result implies that the algebraic bifurcation in the network does not cause significant reactive generation response from the generators due to the weakness of lines L2 and L3. Field voltage(p.u.) Flux linkage(p.u.) 105 Algebraic bifurcation simulation(5 time, 55 MVar) 1.7 I I I I I 1.6 ~ ...... ------------------------------------------------------- a 1.5 - W 1.4 - 4 Solid line: GEN3 1.3 ‘ Dashed line: GENZ ~ Doued line: GENl 1.2 ~ - 1.1 " 3; ‘ -4 1 1 4 L 1 l 0 5 10 15 20 25 30 time(second) Figure 4.23 Algebraic bifurcation simulation (field voltage) Algebraic bifurcation simulation(5 time, 55 MVar) 1.06 . . . . * 1.04 '- /\ 1.02 - 1 l L , ‘ 0.98 ~ ‘ Solid line: GEN3 0-96 ' Dashed line: GENZ . 0.94 _ Dotted line: GEN l a 0.92 - ‘ 0.9 - ,. ......................... ‘ 0.88 1 J ‘ ‘ ‘ o 5 10 15 20 25 30 time(second) Figure 4.24 Algebraic bifurcation simulation (internal bus voltage) Field current(p.u.) 1&5 Algebraic bifurcation simulation(5 time, 55 MVar) 1.45 A "3v". ' ‘ j .......... fl. _______ ' .......... 1.4 - ' 1.35 - 1.3 r ’/\/\ 1.25 P 1.2 ~ 1 15 _ Solid line: GEN3 Dashed line: GENZ 1.1 ~ Dotted line: GENl 1.05 - 1r ................................................................................................................ .1 0.95 ‘ ‘ ‘ ' L 0 5 10 15 20 25 time(second) Figure 4.25 Algebraic bifurcation simulation (field current) Algebraic bifurcation simulation(5 time, 55 MVar) 1.004 fl . . . ‘ 1.“)2 .- .._;r\.o “‘\ 2 1 .:' Q 0998 - " g i g 0.996’ I: a g Solid line: GEN3 E 0.9941- ; Dashed line: GEN2 ,_ ; Dotted line: GENl 0.992h .g' 0.99' 0.988 . ‘ ‘ ‘ ‘ o 5 10 15 20 25 time(second) Figure 4.26 Algebraic bifurcation simulation (terminal bus voltage) 107 Time simulations for field voltage, internal bus voltage, field current, and terminal bus voltage are provided in Figure 4.23, 4.24, 4.25, and 4.26 for the reactive power load at 60 MVar. In Figure 4.23, the field voltages of all three generators are increased to pick up the reactive power load increase at the bus LOAD2. All the field voltage converged to a steady state solution and bus GEN3 and GEN 2 pick up most of the reactive power load in- crease. No field voltage upper limit is hit in this case. Figure 4.24 shows that internal bus voltages of all the generators reach a stable steady state solution. No field current limit (2.2 p.u.) violation occurs as shown in Figure 4.25. In Figure 4.26, all the terminal bus voltages are decreased by the disturbance at the time when the disturbance is introduced. The terminal bus voltages are restored at about 13 seconds. Small oscillations occur in the voltage at the terminal buses. The magnitude of these small oscillations are decreasing and will be damp out if the time simulation is continued. The time simulation results confirm there is no dynamic voltage collapse when the reactive load at LOAD2 is increased from 40 to 60 MVars. These results agree with the results in Table 4.10. Thus, although the sys— tem is approaching algebraic bifurcation and a system bifurcation, no voltage collapse bi- ftncation has occmred. A lower bound on the minimum eigenvalue associated with a voltage control area is kmSmM 2 {SW} .1 i=1 " If the reactive power angle coupling is ignored A n+m A, 5 mini {Qi - VgBii + 2 ViVjBijSin (9, " 9,- - 75)} {0' = 1). 0:0} 5 min, {2Q,} since there are no PV buses in the network algebraic equations. Thus, if the reactive load QLi (Qi=Qoi-Qu) at a bus in increased as a stress test for a voltage control area, the lower 108 bound on the minimum eigenvalue decreases. The Q-V curve; which is produced by mak- ing a bus in a reactive deficient region a PV bus, reducing the setpoint voltage, and plot- ting the reactive power generated at the bus, is a plot of the stress test of adding reactive load at the test bus. The parabolic shape of the Q-V curve is shown in Figure 4.28. The knee of the Q-V curve (at which 3%, = 0) occurs when no more reactive load can be add- ed since the series 12X losses with voltage drop and the shunt capacitive reactive supply withdrawal with voltage drop overcome the ability to obtain reactive power from the sources of reactive power (synchronous generators). It has thus been shown that the Q-V curve in the general power system model, is a test to determine how small the minimum eigenvalue of a submatrix of [D3 D 4] associated with a voltage control area can be. It is clear that the fact that the minimum of a lower bound on a minimum eigenvalue estimate of a submatrix (SQLV) of [D3 D 4] doesn’t necessarily indicate a point of singularity of [D3 D 4] and J. However, the singularity of S QtV has been theoretically shown to relate to every known test [1, 12, 13, 28, 29, 8, 27, 14, 22, 16, 17] for voltage collapse. Showing the Q-V curve is related to a lower bound on the small eigenvalue of S Q." associated with a voltage control area links the Q-V curve to all other tests for voltage collapse including the algebraic bifurcation test [27]. The Q—V curve is a standard tool used by industry for as- sessin g proximity to voltage collapse in a load flow. It has been shown to relate the sin gu- larity and possible bifurcation of the reactive power balance equations in the general power system model as well as every known test for voltage collapse. The bifurcation tests results in Table 4.10 indicate that the system is approaching an alge- braic bifurcation but the load flow and equilibrium programs do not converge and the EPRI load flow will not solve. The Q-V curve is computed using the equilibrium program and load flow program when reactive load is increased at bus LOAD2. Figure 4.27 a shows the Q—V curve for both load flow and equilibrium program simulation. Since there is no PV bus in the equilibrium program computation, the Q—V curve for equi- 109 5 times reactance Q-V curve at LOAD2 0.9 I l I I 0.88 0.86 A . 0.84 =1 ii (182 3 ‘§ «as E '8 0.78 .3 0.76 .. 4 Solid line: General model 0'7 ' Dashed line: Load flow model ‘ 0.72 - : 0.7 ‘ 1 ‘ ‘ 40 45 50 55 60 65 Reactive power load(MVar) Figure 4.27a Algebraic bifurcation simulation (Q-V curve) Q-V curve(charging 0.3, 1.5 times) 1 I I I 0.95 - 0.9 1- Voltage(p.u.) c 8 0.8 - 0.75 - _L L 0.7 1 L . 45 46 47 48 49 50 Reactive Power Load(MVar) Figure 4.27b Q-V curve (2 times shunt capacitance, 1.5 times series reactance) 51 110 librium simulation is performed by increasing the reactive power load at bus LOAD2 and computing it’s bus voltage. The Q-V curve for load flow simulation is performed by (a) setting bus LOAD2 as a fictitious PV bus with infinite positive and negative reactive generation, (b) changing the voltage set point of the fictitious PV bus, and (c) computing the reactive generation at that fictitious PV bus. The negative reactive generation is the reactive power which can be supplied by the sys— tem to this fictitious PV bus. The knee of the Q-V curve (g2 = 0) is the point where the maximum reactive power can be supplied by the system to the fictitious PV bus. The Q-V curve for the load flow simulation in Figure 4.27a shows that at 65 MVar the low voltage at bus LOAD2 does not cause a change sign of the Q-V curve. It indicates that the low voltage at bus LOAD2 does not result in large line losses and large capacitive withdrawal of the shunt capacitance. Since the low voltage occurs before the maximum reactive pow- er supply of the system (the knee of the Q-V curve), the system will not experience a volt- age collapse at 65 MVars based on either the load flow or general power system model. This confirms the results of Table 4.10 where bifurcation has not occurred at 60 MVars. The slope of the Q-V curve and the bifurcation test results in Table 4.10 indicate the sys- tem is close to satisfying the condition for bifurcation. To establish that the test system is not very vulnerable to load flow bifmcation, the test results in Table 4.10 and the Q—V curve are sumcient. However, if the line reactances are multiplied by 1.5 and the shunt susceptances are multiplied by 2, the system should be more vulnerable to voltage col- lapse. This is confirmed by the Q-V curve for this case given in Figure 4.27b. The knee of the Q—V curve occurs at 0.81 and with a minimum of 51 MVars rather than 65 MVars. Thus, adding shunt reactive supply by increasing line charging did not overcome the ef- fects of increased 12X losses on branches and increased shunt capacitive reactive with- 111 drawal. An extensive set of tests were performed to determine if the system could be modified so that the knee of the Q-V curve would occur at voltage above Vmin=.88 and the reactive margin QM," = minv (Q (Vk) ) would decrease. Although increasing the series reactance and shunt susceptances of the transmission lines caused vmin to increase and Qmin to de- crease for load flow computed Q—V curves, the equilibrium program could not even com- pute one point on some of these curves. An improved equilibrium program is needed to investigate algebraic bifurcation on a general power system model with reactive demand supply problems. Without this improved equilibrium point program, our research had to be content with studying an example system that did not have severe reactive demand sup- ply problems. 4.3 Dynamic/Algebraic Voltage Instability Simulations Because of the large reactance of the transformer compared with the transmission lines, generator terminal buses in a power system can generally be assumed to be isolated from the rest of the system. A system based on this assumption is developed. If there is an in- crease of reactive power load close to the terminal bus, it will be shown that the generator dynamics will become unstable and cause dynamic voltage instability after the field cur- rent limit is reached and the field current limiter disables the exciter control. The simula- tion shows a rapid decline of the terminal bus voltage immediately occurs after the generator loses the capability of controlling voltage. The air gap saturation and the satura- tion of the exciter system are shown to contribute to the field current reaching it’s upper limit. Complete computer simulation results are in Appendix C. The twelve bus system (Figtne 4.19) is the test system in this section. The generator and exciter data is given in Table 4.11a and 4.11b. The data is taken from [23]. The reactance of L5 is increased by ten times to ensure the buses GEN3 and TERM3 are a voltage con- Table 4.11a Generator data for the twelve bus system GENl GEN2 GEN3 Rated MVA 250 192 125 Rated KV 18 18 15.5 Rated PF 0.85 0.85 0.85 SCR 1.05 0.64 0.9 1'40 9.2 5.9 8.97 r', 0.195 0.232 0.174 X, 0.995 1.651 1.22 x, 0.568 1.59 1.16 r, 0.16 0.102 0.078 561.0 0.0769 0.105 0.1026 561.2 0282 0.477 0.432 D 2.0 2.0 2.0 A 0.8178 0.85 0.8397 B 2.3167 4.67 3.9928 w, 1603 634 596 Table 4.11b Exciter data for the twelve bus system GENl GEN2 GEN 3 t, 0.06 0.06 0.06 KR 1.0 1.0 1.0 t A 0.02 0.2 0.2 K, 25 25 25 VRW 5.99 1.0 1.0 VRMIII '5.99 -l.0 -1.0 1:3 0.1 0.5685 0.6758 K E -0.02 -0.0505 -0.0601 830.75,“, 0.127 0.0778 0.0924 $51.0” 0.3 0.303 0.3604 AEX 0.0096 0.0013 0.0016 35,, 1.1461 1.3733 1.6349 Efdm 3.10257 3.79307 2.3393 t, 0.48 0.35 0.35 K, 0.0317 0.091 0.108 114 trol area that is isolated from the rest of the system. The base case data for the simulation of the twelve bus system is in Tables of Appendix C.3. For each simulation, the reactive power load at the bus TERM3 is increased. The equilibrium point of the general power system model is computed. The system jacobian matrix, dynamic/algebraic, and algebraic/ dynamic bifurcation test matrices, algebraic bifurcation test matrix, and flux decay bifur- cation test matrix and the corresponding eigenvalues and eigenvectors are calculated. The stability of the dynamic states is then being tested. If there is an eigenvalue which has pos- itive real part, a time simulation is performed. It will be shown that reverse actions, such as a decrease of the terminal bus voltage results a decrease of the reactive power genera- tion at a generator bus, may occur in some cases. This case is a generalization of the dynamic voltage instability observed in the two bus system when the high side transformer and load bus voltage solution of the load flow and equilibrium manifolds converged. The generalization allows for the connection of the gen- erator internal and terminal buses to a power system. The bifurcation occurs due to bifur- cation of the generator flux decay and the real and reactive power balance equations at terminal bus since the transformer isolates the generator from the rest of the system and the load at TERM3 is increased. The reactive power load at the bus TERM3 is 5 MVar in the base case. It is increased to 10 MVar in the first simulation. Table 4.12 shows the equilibrium point of this simulation. The first column and the second column are the number of the snapshot of the transient stability simulation and the time duration at which this snapshot is taken. “11” means the eleventh snapshot and 49.9995 means the eleventh snapshot is taking at 49.9995 seconds and the system solutions have converged to a steady state solution. The third, fourth, and fifth columns are the steady state solutions for the buses GENl, GEN2, and GEN3 respec- tively. There are eleven columns for the A-C bus voltage magnitude and A-C bus voltage angle in the last two rows. The first two columns represent the number of snapshot and 115 Table 4.12 The equilibrium point of 10 MVar load at bus TERM3 GENERATOR ANGLE IN DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 10.4736 45.1464 19.3101 GENERATOR FIELD VOLTAGE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0706 1.5243 1.2870 GEN. FLUX LINKAGE (Q-AXIS) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.9861 0.8227 0.9992 GEN. ELECTRICAL POWER - (MW) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 82.5440 123.0011 31.9911 GENERATOR EXCITER SATURATION -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.0431 0.0146 0.0388 GENERATOR MEGAVAR OUTPUT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -11.2795 -1.7768 13.7459 GENERATOR FIELD CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 ~49.9995 0.9983 1.3775 1.1563 GENERATOR TERMINAL CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.8331 1.2301 0.3482 GEN. TERM. CURR. ANGLE DEGREES-NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 7.6718 2.6678 -17.5473 A-C BUS VOLTAGE MAGNITUDE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0000 1.0000 0.9998 1.0078 1.0041 0.9955 0.9924 0.9850 0.9899 A-C BUS VOLTAGE ANGLE DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -0.1094 1.8402 5.7049 -2.8605 -2.5509 -4.9643 -5.0428 -6.1330 -5.3102 Table 4.13 The eigenvalues of the dynamic/algebraic bifmcation test matrix and the flux decay bifurcation test matrix for 10 MVar load at bus TERM3 eigda - vtt - -26.4462 +52.18041 1.0000 0.0442 0.0069 -26.4462 -52.1804i -0.6955 1.0000 1.0000 —3.8967 +14.9033i -0.3321 0.6017 -0.4808 -3.8967 -14.90331 -3.9132 +14.99011 -3.9132 -14.99011 eigtt - -15.1523 -l6.4814 1.0e+03 . -16.5558 -0.3525 + 1.73381 -0.1470 -0.3525 - 1.73381 -l.1285 -0.1340 + 1.06601 —0.5760 -0.1340 - 1.06601 -0.0342 + 0.97361 -0.0342 - 0.97361 -0.0835 + 0.07051 -0.0835 - 0.07051 -o.0487 + 0.59291 -0.0487 - 0.59291 —0.0108 + 0.59141 -0.0108 - 0.59141 116 time duration. The third to the fifth column represent the voltage magnitudes and voltage angles of buses TERMl to TERM3. The next three columns are for the buses HSTl to HST3. The last three columns are for buses LOAD] to LOAD3. All the angles are in de- grees. The real power is in megawatts. The reactive power is in megavars. The voltage magnitude and current are in per unit. Table 4.12 indicates that the system is operated in a normal condition. No saturation or upper limit violation occurs. The high voltages at buses HSTl and HST2 are caused by large line charging of long transmission lines. In Table 4.13, the “eigda” represents the eigenvalues of the dynamic/algebraic bifurcation test matrix and the “eigtt” represents the eigenvalues of the flux decay bifurcation test ma- trix. The negative real parts of the eigenvalues indicates that the system is dynamically stable in this simulation. It will be shown in the simulations that the condition number re- mains small and will not change dramatically as we change the reactive power load at bus TERM3. It indicates that there is no supply and demand problems in this series of simula- tion. It is due to the isolation of the generator from the rest of the system and the increase of the reactive power load at bus TERM3 is picked up by the bus GEN3. A complete sim— ulation result which includes condition number of the algebraic bifurcation test matrix and eigenvalues and eigenvectors for the dynamic/algebraic bifurcation test matrix and flux decay bifurcation test matrix are in Tables of Appendix C.3. Table 4.14 is the equilibrium point of the system when the reactive power load at bus TERM3 is increased to 50 MVar. The field voltage of the bus GEN3 is increased from 1.287 at 10 MVar to 1.6919 and there are small changes in the field voltages of GENl and GEN2. It is due to the large reactance of the transformer that isolates the buses GEN3 and TERM3 from the rest of the system. The bus GEN3 has to pick up almost all the reactive power load increase at the bus TERM3. The flux linkage (internal bus voltage 8' q) of the bus GEN 3 is also increased to support the reactive power generation. The voltage profile of the system is normal and there is no saturation or field current limit violation. The real 117 Table 4.14 The equilibrium point of 50 MVar load at bus TERM3 GENERATOR ANGLE IN DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 6.6330 41.2437 12.3914 GENERATOR FIELD VOLTAGE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0719 1.5263 1.6919 GEN. FLUX LINKAGE (O-AXIS) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.9864 0.8236 1.0554 GEN. ELECTRICAL POWER - (MM) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 82.5348 123.0010 32.0007 GENERATOR EXCITER SATURATION -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.0431 0.0147 0.0984 GENERATOR MEGAVAR OUTPUT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -10.9684 -1.4629 53.2513 GENERATOR FIELD CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.9994 1.3791 1.4841 GENERATOR TERMINAL CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.8326 1.2301 0.6236 GEN. TERM. CURR. ANGLE DEGREES-NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 3.6286 -1.3086 -57.0667 A-C BUS VOLTAGE MAGNITUDE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0000 1.0000 0.9962 1.0076 1.0039 0.9947 0.9922 0.9845 0.9894 A-C BUS VOLTAGE ANGLE DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -3.9413 -1.9900 1.9319 -6.6926 -6.3820 -8.7883 -8.8752 -9.9605 -9.1373 Table 4.15 The eigenvalues of the dynamic/algebraic bifurcation test matrix and the flux decay bifurcation test matrix for 50 MVar load at bus TERM3 eigda - vtt = -26.4473 +52.18061 -0.6029 0.0333 -0.6322 -26.4473 -52.18061 1.0000 1.0000 -0.2536 -3.9097 +14.89961 -0.4735 0.0371 1.0000 -3.9097 -14.89961 -3.9365 +14.98971 -3.9365 -14.98971 eigtt - -16.4719 -16.5512 -171.1429 -15.1482 -852.9462 -0.3374 + 1.73601 -114.6348 -0.3374 - 1.73601 -0.1189 + 1.06891 -0.1189 - 1.06891 -0.0402 + 1.00361 -0.0402 - 1.00361 -0.0835 + 0.06731 -0.0835 - 0.06731 -0.0646 + 0.63311 -0.0646 - 0.63311 -0.0257 + 0.60151 -0.0257 - 0.60151 118 part of the eigenvalues of the dynamic/algebraic bifurcation test matrix and the flux decay bifurcation test matrix are all negative. Notice that the eigenvalues of the flux decay bifur- cation test matrix have smaller magnitudes compared with the data in Table 4.13. It indi- cates that the flux decay eigenvalues are moving toward the right half plane as we increase the reactive power load at the bus TERM3. Table 4.16 shows the equilibrium point of the system as the reactive power load at bus TERM3 is increased to 100 MVar. The field voltage at the bus GEN3 has been increased to 2.1685. The voltage at the terminal bus can not be held close to its preset value and drops to 0.9765 because of the air gap saturation and the excitation system saturation. The eigen- values in Table 4.17 indicates that the dynamic states are still stable but the flux decay ei- genvalues are closer to the origin compared with the data in Table 4.15. The result of the increase of the reactive power load at the bus TERM3 to be 110 MVar is shown in Table 4.18 and 4.19. The field voltage and reactive power generation of the bus GEN3 are further increased. The voltage magnitude at bus TERM3 is decreased to suck more reactive power from the bus GEN3 to supply its reactive power load increase. The low voltage at bus TERM3 is also caused by the large air gap saturation and exciter satura- tion. The negative real part of the dynamic/algebraic and flux decay eigenvalues in Table 4.19 indicate that the system is stable. It can be seen in Table 4.20 that the field current is very close to it’s upper limit as we in- crease the reactive power load to 120 MVar. The terminal bus voltage has dropped to 95 percent of its base case value. It will be shown in the following cases that the terminal bus voltage will decrease tremendously as we increase the reactive power load. It is because the generator can no longer control its terminal bus voltage. Although the upper limit of the field current is not exceeded. it is heavily saturated. The negative real eigenvalues of the dynamic/algebraic test matrix in Table 4.21 indicates that the system is stable. 119 Table 4.16 The equilibrium point of 100 MVar load at bus TERM3 GENERATOR ANGLE IN DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -22.0389 12.2170 -18.1092 GENERATOR FIELD VOLTAGE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0789 1.5374 2.1685 GEN. FLUX LINKAGE (O-AXIS) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.9877 0.8289 1.1037 GEN. ELECTRICAL POWER - (MN) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 82.6329 123.0139 31.9007 GENERATOR EXCITER SATURATION -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.0433 0.0149 0.3166 GENERATOR MEGAVAR OUTPUT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -9.2917 0.2288 100.6401 GENERATOR FIELD CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0056 1.3881 1.8765 GENERATOR TERMINAL CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.8316 1.2301 1.0810 GEN. TERM. CURR. ANGLE DEGREES-NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -26.1684 -30.7394 -98.8884 A-C BUS VOLTAGE MAGNITUDE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0000 1.0000 0.9765 1.0066 1.0029 0.9908 0.9912 0.9818 0.9868 A-C BUS VOLTAGE ANGLE DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -32.5838 ~30.6328 -26.4567 -35.3410 -35.0296 -37.4042 -37.5278 -38.5924 -37.7648 Table 4.17 The eigenvalues of the dynamic/algebraic bifurcation test matrix and the flux decay bifurcation test matrix for 100 MVar load at bus TERM3 eigda - vtt - -26.4522 +52.18191 -0.9917 0.0350 -0.2730 -26.4522 -52.18191 1.0000 1.0000 -0.6262 -3.9206 +14.91771 -0.0815 0.0113 1.0000 -3.9206 -14.91771 -4.0319 +14.97501 -4.0319 -14.97501 eigtt - -16.4550 -16.5434 -152.8097 -15.1372 -815.9868 -0.3148 + 1.74241 -42.9767 -0.3148 - 1.74241 -0.0353 + 1.10191 -0.0353 - 1.10191 -0.1296 + 1.03231 -0.1296 - 1.03231 -0.0835 + 0.06491 -0.0835 - 0.06491 -0.0664 + 0.66481 -0.0664 - 0.66481 -0.0265 + 0.61281 -0.0265 - 0.61281 120 Table 4.18 The equilibrium point of 110 MVar load at bus TERM3 GENERATOR ANGLE IN DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -41.6573 -7.5377 -37.8437 GENERATOR FIELD VOLTAGE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0818 1.5420 2.2561 GEN. FLUX LINKAGE (Q-AXIS) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.9882 0.8310 1.1094 GEN. ELECTRICAL POWER - (MN) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 82.6391 123.0167 31.8963 GENERATOR EXCITER SATURATION -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.0433 0.0149 0.3948 GENERATOR MEGAVAR OUTPUT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -8.6070 0.9225 109.6247 GENERATOR FIELD CURRENT ~NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0081 1.3917 1.9561 GENERATOR TERMINAL CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.8309 1.2302 1.1786 GEN. TERM. CURR. ANGLE DEGREES-NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -46.2405 -50.6628 -119.7442 A-C BUS VOLTAGE MAGNITUDE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0000 1.0000 0.9686 1.0062 1.0024 0.9892 0.9908 0.9807 0.9858 A-C BUS VOLTAGE ANGLE DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -52.1862 -50.2332 -45.9373 -54.9448 -54.6319 -56.9922 -57.1326 -58.1876 -57.3587 Table 4.19 The eigenvalues of the dynamic/algebraic bifurcation test matrix and the flux decay bifurcation test matrix for 110 MVar load at bus TERM3 eigda - vtt - -26.4541 +52.18251 1.0000 0.0359 -0.2668 -26.4541 -52.18251 -0.9899 1.0000 -0.6454 -3.9191 +14.91971 0.0652 0.0097 1.0000 ~3.9191 -14.91971 -4.0663 +14.97421 -4.0663 -14.97421 eigtt - -16.4498 -16.5413 -151.3925 -15.1337 -803.2924 -0.3087 + 1.74471 -36.6830 -0.3087 - 1.74471 -0.0293 + 1.12271 -0.0293 - 1.12271 -0.1404 + 1.02851 -0.1404 - 1.02851 -0.0835 + 0.06461 -0.0835 - 0.06461 -0.0662 + 0.66911 -0.0662 - 0.66911 -0.0257 + 0.61501 -0.0257 - 0.61501 121 Table 4.20 The equilibrium point of 120 MVar load at bus TERM3 GENERATOR ANGLE IN DEGREES ~NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -81.3496 -47.4235 -77.6249 GENERATOR FIELD VOLTAGE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0857 1.5481 2.3393 GEN. FLUX LINKAGE (Q-AXIS) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.9889 0.8337 1.1130 GEN. ELECTRICAL POWER - (MN) ~NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 82.7351 123.0285 31.7987 GENERATOR EXCITER SATURATION -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.0434 0.0150 0.4876 GENERATOR MEGAVAR OUTPUT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -7.6940 1.8459 118.2906 GENERATOR FIELD CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0115 1.3967 2.0380 GENERATOR TERMINAL CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 ~49.9995 0.8310 1.2304 1.2783 GEN. TERM. CURR. ANGLE DEGREES-NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -86.5553 -90.7772 -160.4985 A-C BUS VOLTAGE MAGNITUDE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0000 1.0000 0.9580 1.0057 1.0019 0.9872 0.9902 0.9793 0.9844 A-C BUS VOLTAGE ANGLE DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -91.8678 -89.9176 -85.4970 -94.6310 -94.3190 -96.6641 -96.8218 -97.8672 -97.0354 Table 4.21 The eigenvalues of the dynamic/algebraic bifurcation test matrix and the flux decay bifurcation test matrix for 120 MVar load at bus TERM3 eigda - vtt 2 -26.4567 +52.18341 1.0000 0.0370 -0.2646 -26.4567 —52.18341 -0.9762 1.0000 -0.6620 -3.9179 +14.92061 0.0539 0.0086 1.0000 -3.9179 -14.92061 -4.1042 +14.97381 -4.1042 -14.97381 eigtt - -16.4432 -16.5388 -149.7845 -15.1296 -786.6969 -0.3014 + 1.74771 -31.9402 -0.3014 - 1.74771 -0.0251 + 1.14561 -0.0251 - 1.14561 -0.1518 + 1.02591 -0.1518 - 1.02591 -0.0835 + 0.06421 -0.0835 - 0.06421 -0.0659 + 0.67261 -0.0659 - 0.67261 -0.0245 + 0.61751 -0.0245 - 0.61751 122 Table 4.22The equilibrium point of 135 MVar load at bus TERM3 GENERATOR ANGLE IN DEGREES ~NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -378.5388 -346.0558 -372.2955 GENERATOR FIELD VOLTAGE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.1185 1.6006 2.3393 GEN. FLUX LINKAGE (Q-AXIS) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.9949 0.8562 1.0562 ' GEN. ELECTRICAL POWER - (MN) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 82.7741 123.0988 31.8060 GENERATOR EXCITER SATURATION -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 ~49.9995 0.0442 0.0159 0.4876 GENERATOR MEGAVAR OUTPUT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.2228 9.9667 123.5647 GENERATOR FIELD CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0401 1.4404 2.2099 GENERATOR TERMINAL CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.8278 1.2344 1.4658 GEN. TERM. CURR. ANGLE DEGREES-NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -29.0325 -31.5311 -96.8258 A-C BUS VOLTAGE MAGNITUDE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.9999 1.0001 0.8692 1.0013 0.9973 0.9694 0.9855 0.9671 0.9726 A-C BUS VOLTAGE ANGLE DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -28.8783 -26.9001 -20.9484 -31.6552 -31.3244 -33.5055 -33.8568 -34.7921 -33.9472 Table 4.23 The eigenvalues of the dynamic/algebraic bifurcation test matrix and the flux decay bifurcation test matrix for 135 MVar load at bus TERM3 (wrthout excrter 1n) eigda - vtt - -26.4792 +52.19111 1.0000 0.0469 -0.2255 -26.4792 -52.19111 -0.9066 1.0000 -0.6684 -3.9046 +14.92301 -0.0013 -0.0004 1.0000 -3.9046 -14.92301 -16.4891 -15.1344 ett - -0.3244 + 1.71961 -0.3244 - 1.71961 -137.9121 0 0 -0.1269 + 1.10851 0 -659.2447 0 -0.1269 - 1.10851 - 0 0 0.2214 -0.0838 + 0.68051 -0.0838 - 0.68051 -0.0199 + 0.67101 -0.0199 - 0.67101 -0.0836 + 0.06291 -0.0836 - 0.06291 0.2956 123 Table 4.24 shows the changes of the ratio of the condition number of the jacobian matrix and the algebraic bifurcation test matrix before the exciter on GEN 3 is disabled. There are less than 15 percent change for the algebraic bifurcation test matrix and 10 percent change for the jacobian matrix as reactive load at TERM3 is increased. It indicates that no reactive power supply and demand problem occurs. This is confirmed by plotting the Q-V curve for the reactive load increase at TERM3 in Figure 4.31. Table 4.24 Ratio of the condition number (Algebraic) Load cond(D1°ad)/cond(Dlo) cond(Jload)/cond(J 10) 10 1.0 1.0 50 0.9365 1.0376 100 0.8758 1.0671 110 0.8661 1.0714 120 0.8574 1.0763 130 0.8642 1.0848 135 0.8886 1.0935 None of the real part of the eigenvalues calculated in this simulation are zero or approach- ing zero, which means there is no oscillation problem as reactive load increases. As the reactive power load at bus TERM3 is increased to 135 MVar, the field current up- per limit is hit. Since the simulation program does not have the capability to disable the exciter on line, the program converged to a steady state solution. Notice that this equilibri- um point is the steady state solution with the exciter. Eigenvalues of the dynamic/algebraic test matrix and flux decay test matrix are computed with K A3=0 to represent the loss of the excitation problem. Note that the equilibrium point is computed with AN exciter imple- mented but the dynamic algebraic test is computed assuming there is no exciter because it is assumed that the state does not change when the machine is operating at the field current 124 limit before and after the exciter is disabled. The results are shown in Table 4.23. It shows that there is a positive real eigenvalue in both of the dynamic/algebraic and the flux decay test matrices. It indicates that the system becomes unstable. The eigenvector in Appendix C.3 corresponding to this unstable eigenvalue shows that the flux decay state E ' q of GEN 3 has the major contribution in this unstable mode. The time simulation of the system with- out the exciter at GEN3 is calculated for a step reactive load increase of 135 MVars at bus TERM3. Figure 4.28 shows that the internal bus voltage of GEN 3 continuously decreases after the disturbance. The field current in Figure 4.29 increases dramatically in the first four seconds and is evidence of the increase in reactive power on GEN3. At 9.8 second, a sharp decrease of the field current occurs and the simulation program stops due to numeri- cal problems. The terminal bus voltage at bus TERM3 in Figure 4.30 is decreased slowly to 0.2 and then experiences a dip to 0.145 at 9.8 seconds in the simulation. The time simu- lation confirms the voltage instability does occur. This dynamic voltage instability occurs due to action of the uncontrolled generator flux decay dynamics which force the collapse to occur by reducing induced voltage behind transient reactance, the field current, and the reactive power generated by the machine. If the exciter is not disabled after the field current upper limit is hit and the reactive power load at bus TERM3 is increased, we may still obtain a converged steady state solution but with a reverse action of generator control. Figure 4.32a describes the actions of generator control in the simulation and is a Q-V curve at the generator terminal bus. Note that the Q- V curve indicated by circles in Figure 4.32a is nearly identical to the Q—V curve of Figure 4.31. The circle represents the cases before the reverse action occurs. A decrease in the terminal bus voltage will cause an increase in the reactive power generation to support the reactive load increase. This control action tries to stabilize the system and maintain the ter- minal bus voltage close to it’s preset value. The x-mark represents the cases after the re- verse action occurs. A decrease in the terminal bus voltage will cause a decrease in the reactive generation. Thus, the diagonal element of SE}, are associated with the terminal Flux linltage(p.u.) Field current(p.u.) 125 Reactive load increase at TERM3(no exciter) 1.1 I r I 1 1 y r ' 1 1. '4;;;;;;;:;;; ____________________________ . 0.9 ~ . 0.8 - . 0.7 r- . 0.6 "' ,4 Solid line: GEN3 Dashed line: GEN2 0-5 * Dorted line: GENl 1 0.4 - . 0.3 ‘ ‘ ' 1 1 1 . 1 . 1 2 3 4 5 6 7 8 9 10 ll time(second) Figure 4.28 Time simulation of E ' 4 at 135 MVar (without exciter in) Reactive load increase at TERM3(no exciter) 2.8 1 . . Y 1 . , 1 1 Solid line: GEN3 2.6 - Dashed line: GEN2 . Dotted line: GENI 2.4 L- 4 2.2 ~ . 2 b - 1.8 - 1‘ . 1.6 - - 1.4 :---- . . 1.2 " ............................................................................... A -1 1 p- ........ -4 0.8 ‘ ‘ 1 l 1 1 2 3 4 5 6 7 8 9 10 ll time(second) Figure 4.29 Trme simulation of I id at 135 MVar (without exciter in) Terminal bus voltage(p.u.) 126 Reactive load increase at TERM3(no exciter) 1.2 T fl I T T 7 F T T Solid line: GEN3 0.2 . Dashed line: GEN2 Dotted line: GENl C L. b. Figure 4.30 Time simulation of VT at 135 MVar (without exciter in) 11 Terminal bus voltage(p.u.) 127 Q-V curve(dynamic/algebraic) 0.98 - 0.96 l 0.94 - O.92 - 0.9 r 0.88 ~ l 0.86 0 20 4O 60 80 100 Reactive load(MVar) Figru'e 4.31 Q-V curve at bus TERM3 120 140 128 bus of GEN3 must be negative at points marked by X’s where reverse action occurs. Fig- ure 4.32b describes the actions of the reactive generation for a reactive load increase. The dashed line shows the reverse action, an increase in the reactive power load will decrease the reactive power generation. In this case, the diagonal element of S QGQL associated with the terminal bus of GEN3 is negative. The fact that S Qan. and SE}, have negative ele- ments indicate that the closed loop system loses PQ and PV controllability at the points by X’s in Figure 4.32a. The points below the knee of the Q—V curve (marked by X’s) in Fig- ure 4.32a are also understood to be unstable from the load flow based Q-V curve methods for assessing proximity to voltage collapse. Although Figure 4.32a indicates that a bifur- cation point has been reached for system operating at points below the knee of the curve. The dynamic] algebraic and algebraic dynamic tests at points below the knee of the Q-V curve indicate the system is still stable but is approaching instability as reactive load in- creases toward the knee. This kind of voltage instability is caused by a change sign of SVE and can be explained by root locus. Figure 4.33a is a simplified exciter/flux decay model for a generator. The termi- nal bus voltage VT which is controlled by the exciter and the network is fed back to the ex- citer. Since the network efiects can be estimated by the sensitivity matrix SVB: Figure 4.33a can be linearized and represented by Figure 4.33b. It is shown in [1] that SVE is pos- itive in the normal operating conditions. Before the reverse action occurs, SVE is positive and the gain is positive. The root locus is shown in Figure 4.33c. The shape of the root lo- cus will change depending on the location of poles and zeros. For a non self excited exci- tation system, the flux decay pole is generally real and is the closest pole to the j (0 axis. If the reverse action occurs, the sign change of SVE will cause a sign change of the gain. For the negative gain, since the number of poles minus the number of zeros is two, the asymp- tote is on the real axis which means the root locus has to stay on the real axis for negative gain. Since the flux decay pole is always the closest pole to the j (0 axis, it indicates that Terminal bus voltage Reactive power generation(MVar) 129 The normal and reverse actions of the reactive generation 0.9 ~ 0.8 I 0.7 ~ I 0.6 0.5 *- O.4 - V 1' 7 U I I I T o o o circle: normal action X-mark: reverse action 1 0.3 0 20 4O 60 80 100 120 Reactive generation Figure 4.32a The reverse action of the reactive power generation The reverse action of reacrive generation/load 140 140 120 - N O r I I T T T I I I ‘§ ‘ ‘Q Q § 5 § ‘\ ‘ Solid line: Stable case Dashed line: Unstable case L 1 1 l 1 1 1 l 20 40 60 80 100 120 140 160 Reactive power load(MVar) Figure 4.32b The reverse action of the reactive load/generation 180 130 VREF Amplifier V Bf E f + KA/(l+sTA) R [1/(KE+er) d l' G(s) ‘1 f 7 Exciter V 3 sKF/(l+sTF) Stabilizer V1 Measurement V f l' 1/(1+sTR) : Network Figure 4.33a Simplified exciter/flux decay model AVREF Amplifier + + AVR I AEfd ___-‘5 Eq g - KA/(l'l-STA) 1/(KB+STE)I ‘ 6(8) _— Exciter A V3 7 :st/(HSTF) : Stabilizer Measurement AVl AV l"——'| I l1/(1+sTR)I 1 SVE |-——— Figure 4.33b Simplified exciter/flux decay model (SW) 131 F: Flux decay pole E: Exciter pole S: Stabilizer pole M: Measurement pole G: Amplier gain pole wrulllllu Figure 4.33c Root Locus for A Generator/exciter model(positive gain) Figure 4.33d Root Locus for A Generator/exciter model(negative gain) 132 the dynamic instability is caused by the instability of flux decay equations if the reverse action occurred. The gain of the exciter loop becomes more negative and approaches mi- nus infinity as the system approaches the knee of the Q—V curve from below this knee. The decrease in negative gain causes the flux decay pole to move to the right half plane in Fig- ure 4.33d. This explains the loss of dynamic algebraic stability as the knee of the Q—V cm've is approached from below the knee. 4.4 Algebraic/Dynamic Voltage Instability Simulations The results in Section 4.3 indicate that the system bifurcation observed in a two bus sys- tem composed of a generator and it’s terminal can occur in an N bus system if the genera- tor internal and terminal bus of a particular generator are a voltage control area with a weak boundary and the reactive load at the terminal bus is increased. The system experi- ences a dynamic/algebraic bifurcation after the field current limit is hit and the exciter is disabled. The dynamic/algebraic bifurcation causes the field current and voltage behind transient reactance to approach zero that forces the system into voltage collapse. This re- sult is identical to that on the two bus system. Furthermore, the system has a stable equi- librium point at low voltage when the exciter is not disabled just as in the two bus system model given in [21]. However, if reactive load is increased, the system could experience sustained oscillations. This simple extension of the two bus system result to the more general 12 bus model is now followed by a slightly more complex case. Lines L2 and L4 reactances are increased so that buses GEN3, TERM3, and LOAD2 are a voltage control area. The reactive load at LOAD2 is increased. In this case, it is hoped that an algebraic bifurcation (reactive de- mand supply problem) will occur as the network attempts to serve the reactive load at LOAD2. This algebraic bifurcation will hopefully lead to a dynamic/algebraic bifurcation and a dynamic voltage collapse at GEN3. The system has been shown to be not very vul- 133 nerable to reactive demand supply problems from results in Section 4.2. Thus, the results in this section do not indicate that a reactive demand supply problem (algebraic bifurca- tion) develops but a dynamic/algebraic bifurcation occurs after the field current limit is reached and the exciter is disabled. This section is titled Algebraic/Dynamic Voltage Insta- bility Simulation because of the assumption that a system can experience dynamic/alge- braic voltage instability and algebraic voltage instability simultaneously in an actual power system when the reactive demand supply problem leads to a dynamic voltage insta- bility problem. It is believed that this is a very common type of voltage instability on actu- al power systems that are vulnerable to reactive demand supply problems. The algebraic bifurcation test, system bifurcation test, and Q-V curve indicates the system never experiences a reactive demand supply problem (load flow bifurcation) before the field current limit on GEN 3 is reached. However, it will be shown that an oscillation prob- lem may develop as the reactive power load is increased and the field current of GEN3 is below but approaches it’s upper limit. The real part of the eigenvalues of the dynamic/al- gebraic bifurcation test matrix approaches zero as LOAD2 reactive load increases. If the exciter is disabled at the point where the field current is very close to it’s upper limit, ei- genvalues of dynamic /a1gebraic test is positive. This indicates a dynamic voltage instabil- ity has occurred which results in field current and voltage behind transient reactance approaching zero. An attempt was made to simulate the voltage collapse problem using the EPRI Extended Transient Mid Term Stability Program. The simulation program pro- vides the “error in solving differential equations” error message. It indicates that the cur- rent simulation program is not suitable for the voltage instability simulation. A new program that includes the air gap saturation, field current limit, line drop compensation with much more robust algorithm needs to be developed to investigate the voltage insta- bility of a power system. Figure 4.34 is the Q-V curve at bus LOAD2 for the general model and load flow model. Load bus voltage(p.u.) 134 Regular 12 L4 Q-V curve at LOAD2 0.95 0.9 0.85 I j 180 0.8 - . Solid line: General model 0,75 _ Dashed line: Load flow model . 0.7 1 1 1 1 1 60 80 100 120 140 160 Reactive power load(MVar) Figure 4.34 Q-V curve for algebraic/dynamic simulation 135 The solid line shows that the field current limit is hit before the Q-V curve reaches the knee for the Q-V curve produced using the general power system model. The result indi- cates that the system did not reach the knee of the Q—V curve before the field current limit is reached at reactive load of 170 MVars. The equilibrium program did not converge at re- active load above 170 MVars. The load flow model (Dashed line) diverges earlier than the general model in this case because the reactive generation limit defined in the load flow model is conservative as shown in Figure 4.35. Since the increase of the reactive power load has to be picked up by TERM3 in the load flow model, the small reactive reserve at generator connected to TERM3 is quickly exhausted and causes large voltage decrease (Dashed line). The reactive generation limit of the load flow model is decided by the capability curve. The capability curve of a synchronous machine shows the limits placed on the electrical watts and VARs (a) by the permissible temperature rise of the windage, and (b) by the me- chanical system connected to the shaft, and (c) assuming operation at rated terminal volt- age. Figure 4.35 shows a typical capability curve for a generator, plotted on the S plane, where P is the vertical axis and Q is the horizontal axis. Operation within the boundaries of the curve is safe from the standpoints of heating and stability. The maximum reactive generation is decided by the level of real power generation. Since the real power genera- tion is fixed in the load flow model, the maximum reactive power generation could be de- cided by the capability curve at that specific real power generation level. The actual load flow reactive generation limit is specified as some approximation to the field current limit in the transient stability model. The field current limit which is the rotor heat limit, is specified at the intersection of the stator heat limit and rotor heat limit. This intersection point is point B in Figure 4.35 . This point B is specified in terms of the stator heat limit IS] and the power factor of the generator. The stator current phasor can be calcu- lated hour this information, and the voltage at the internal bus can be calculated. Since the 136 *Paxis 3V 6" 18 '4 Rated P of prime mover +0 axrs Motor ( Generator 0‘ Stator heat limit Rated IS I of sync. machine Rotor heat limit r—géil—J (Am V, . V,.) Figure 4.35 Generator capability curve 137 magnitude of the generator internal voltage is proportional to the field current, the maxi- mum field current can be calculated. It should be noted that this is the continuous rating level and that the field current can exceed this continuous rating level for short period as long as the field current capability curve shown in Figure 2.3 is not exceeded. Generally, the reactive generation limit in a load flow is taken as points A, B, or C in Figure 4.35 de- pending on the application and operating condition. If point A, is chosen, the generator can produce its maximum real power generation without exceeding the stator heat limit or rotor heat limit. However, the generator could produce far more reactive generation if the generator operated at a real power generation level far below the maximum real power generation level. Point A is a conservative choice for reactive generation limit. If the reac- tive generation limit was chosen to be at point B, the continuous rating rotor field current limit, the stator heat limit would be exceeded if the generator produced real power above that at point B, PB = S x PfB . If point C is chosen, the stator heat limit and rotor heat limit would be exceeded if real power is generated and the stator heat limit would be ex- ceeded for real power generation levels above PB- Note that although the transient stability model can exceed the rotor heat limit for short periods without having to modify the exci- tation control, the load flow must utilize the rotor heat limit since a load flow is a model of the steady state operating condition of the generator. If the rotor heat limit were ever ex- ceeded in the steady state operating condition, physical damage to the generator would re- sult. The load flow reactive generation limit on a generator is often chosen at point A since the rotor heat limit and stator heat limit will never be exceeded. The reactive generation limit was always chosen as point A in this case. Sometimes, point B can be used to specify the rotor heating limit as long as the generator is operating far below PB. Point C might be used for a synchronous condenser. It should be noted that the transient stability model can produce more reactive power in steady state based on the rotor heat limit in Figure 4.35 than the reactive power limit based on point A in Figure 4.35 . Furthermore, the transient stability model can produce more reactive power than the rotor heat limit as long as the 138 field cmrent level and duration not above the field current capability curve of Figure 2.3. Thus, the load flow would appear to be conservative. The load flow’s conservatism in pre- dictingproximity to voltage collapse depends on (a) how effective the generator excitation control is in controlling the voltage at some point in the network called the control point. (b) how the load flow approximates the actual control point determined from the load drop compensator. The voltage control point is decided by the setting of line drop compensation in the power system. The voltage control point can be at a point between internal bus and terminal bus, terminal bus and high side transformer bus, or out in the network. The voltage control point out in the network is generally less vulnerable as long as the control is maintained but the control will be lost more quickly. Conversely, the voltage control point close to the generator internal bus is more vulnerable (less effective in controlling voltage) but the control is lost more slowly. There are generally two possible control points in the load flow model, which are the generator terminal bus and high side transformer bus. If the ac- tual voltage control point is chosen to the generator internal bus than is modeled in the load flow, the load flow modeling would make the system appear less vulnerable to volt- age problems as long as the voltage control is active since network voltage is more effec- tively controlled than in the transient stability model. However, the generator would generally lose its reactive generation reserve in the load flow due to the more efi‘ective control of network than in the transient stability model. Losing the control of voltage due to exhaustion of reactive generation reserves in the load flow and not in the ttransient sta- bility model makes the load flow far more vulnerable to loss of voltage stability than the transient stability model. The reactive power load at bus LOAD2 is increased from 100 MVar to 170 MVar. Table 4.25 shows the change of the ratio of condition numbers. The changes of the condition 139 Table 4.25 Ratio of the condition number (a1 gebraic/dynamic) load (MVar) cond(D]oad)/cond(D100) cond(Jload)/cond(J 100) 100 1.0 1.0 120 0.9727 1.0271 140 0.9513 1.0650 160 0.9385 1.1000 170 0.9481 1.1194 Table 4.26 The eigenvalues of the algebraic/dynamic bifurcation test matrix and the flux decay bifurcation test matrix for 100 MVar load at bus LOAD2 eigda = vtt = 1.0000 0.0009 -0.5571 —0.7446 1.0000 -0.1589 0.6679 0.0009 1.0000 ~26.4918 +52.21031 -26.4918 -52.21031 -3.8952 +14.90511 -3.8952 -14.90511 -3.9486 +14.98621 -3.9486 -14.98621 eigtt - -15.2041 -16.4660 1.06+04 . -16.5489 -0.3457 + 1.69671 -0.0146 -0.3457 - 1.69671 -2.3122 -o.0397 + 1.11761 ‘ -0.0094 -0.0397 - 1.11761 -0.0841 + 0.02001 -0.0841 — 0.02001 “‘1" -0.0599 + 0.90931 -0.0598 - 0.90931 -0.0716 + 0.61731 -0.0716 - 0.61731 -0.0479 + 0.57701 -0.0479 - 0.57701 140 eigenvalues and eigenvectors of the dynamic/algebraic bifurcation test matrix and flux de- cay bifurcation test matrix at 100 MVar reactive power load. The real parts of the eigen- values are negative which indicate the system is dynamically stable. Table 4.27 shows the eigenvalues of the dynamic/algebraic model for the load at 120 MVar. All the eigenvalues have negative real parts. A pair of complex eigenvalues (with arrow sign in Table 4.27) will be shown to approach the 1‘00 axis as the reactive load is fur- ther increased. Another pair of complex eigenvalues (with dashed arrow sign in Table 4.27) will be shown to become real eigenvalues and stay negative as reactive load is in- creased toward 170 MVars. These results are similar to those in Section 4.3 and is ex- plained in Section 4.3. Table 4.28 shows the eigenvalues of the dynamic/algebraic model for the reactive power load at 140 MVar. No positive real part of the eigenvalue indicates the system is dynami- cally stable. Comparing with Table 4.27, the complex eigenvalues with arrow sign in Ta- ble 4.28 are closer to 1'0) axis than the ones in Table 4.27 and the complex eigenvalues with dashed arrow sign in Table 4.27 become real eigenvalues in Table 4.28. Table 4.29 shows the dynamic/algebraic model eigenvalues for 160 MVar. The real parts of the complex eigenvalues with arrow sign changed from 0.0196 in Table 4.28 to 0.007 8 in Table 4.29 which indicates this pair of complex eigenvalues are moving further toward ja) axis. One of the eigenvalues with dashed arrow sign becomes more negative and the other one moves toward the origin. All the eigenvalues have negative real parts and thus the system is stable. Notice that the flux decay eigenvalues are moving toward the right half plane as the reactive power load is increased. It indicates that the instability of flux de- cay dynamics may occur. Table 4.30 shows the dynamic model eigenvalues at 170 MVar which is the last converged case of this simulation. All the real parts of the eigenvalues stay negative. Figure 4.36, 141 Table 4.27 The eigenvalues of the algebraic/dynamic bifurcation test matrix and the flux decay bifurcation test matrix for 120 MVar load at bus LOAD2 eigda = vtt = 1.0000 0.0031 -0.3339 -0.5743 1.0000 -0.2918 0.2355 0.0020 1.0000 -26.4999 +52.21341 -26.4999 -52.21341 -3.8978 +14.90561 -3.8978 -14.90561 -3.9807 +14.98621 -3.9807 —14.98621 eigtt = -15.1974 -16.4610 l.0e+03 . -16.5452 -0.3373 + 1.70191 -o.1335 —0.3373 - 1.70191 -6.7572 -0.0305 + 1.12691 -0.0630 -0.0305 - 1.12691 "’ -0.0841 + 0.01401 -0.0841 - 0.01401 “III“ -0.0662 + 0.91871 -0.0662 — 0.91871 -0.0727 + 0.63241 -0.0727 - 0.63241 «0.0512 + 0.58271 -0.0512 - 0.58271 Table 4.28 The eigenvalues of the algebraic/dynamic bifurcation test matrix and the flux decay bifurcation test matrix for 140 MVar load at bus LOAD2 eigda = -26.5105 +52.21761 -26.5105 -52.21761 -3.8992 +14.90571 -3.8992 -14.90571 -4.0274 +14.98771 vtt 8 1.0000 0.0062 -0.2789 -0.5335 1.0000 -0.3489 0.1256 0.0027 1.0000 ~4.0274 -14.9877i eigtt - -15.1892 -16.4542 1.0e+03 * -16.5408 -0.3255 + 1.70861 -0.1274 -0.3255 - 1.70861 -3.5600 -0.0196 + 1.14141 ‘_ -0.0436 -0.0196 - 1.14141 -0.0925 m. -0.0758 -0.0550 + 0.58821 -0.0550 - 0.58821 -0.0743 + 0.64541 -0.0743 - 0.64541 -0.0770 + 0.92711 ~0.0770 - 0.92711 142 Table 4.29 The eigenvalues of the algebraic/dynamic bifurcation test matrix and the flux decay bifurcation test matrix for 160 MVar load at bus LOAD2 eigda = ~26.5260 +52.22351 -26.5260 ~52.22351 -3.9004 +14.90371 -3.9004 -l4.90371 -4.0921 +14.99121 Vtt = 1.0000 0.0107 -0.2629 -0.5245 1.0000 ~0.3896 0.0815 0.0034 1.0000 -4.0921 -14.99121 eigtt = —16.4447 —15.1758 1.0e+03 * -16.S354 -0.3094 + 1.71911 -o.1213 —0.3094 - 1.71911 -2.1531 -0.0078 + 1.16381 -0.0316 -0.0078 - 1.16381 " -0.0668 -0.1015 will" -0.0593 + 0.59431 -0.0593 - 0.59431 —0.0760 + 0.65571 -0.0760 - 0.65571 -0.0932 + 0.93641 -0.0932 - 0.93641 Table 4.30 The eigenvalues of the algebraic/dynamic bifurcation test matrix and the flux decay bifurcation test matrix for 170 MVar load at bus LOAD2 eigda - vtt = -26.5444 +52.23051 -26.5444 -52.23051 -3.9033 +14.89891 -3.9033 -14.89891 -4.1104 +14.98641 1.0000 0.0156 -0.2700 -0.5389 1.0000 -0.4010 0.0836 0.0051 1.0000 -4.1104 ~14.98641 eigtt = -16.4349 -15.1595 1.0e+03 * -16.5306 -0.2952 + 1.73201 -0.1158 -0.2952 - 1.73201 -1.5233 -0.0016 + 1.18901 ‘_ -0.0302 -0.0016 - 1.18901 -0.0636 ‘I -0.1048 -0.0630 + 0.60041 -0.0630 - 0.60041 -0.0770 + 0.65521 -0.0770 - 0.65521 -0.1122 + 0.94301 -0.1122 - 0.94301 143 4.37, and 4.38 show the time simulation of the internal bus voltage, field current, and ter- minal bus voltage. These figures state that the system is stable at this load level. Figure 4.39, 4.40, and 4.41 are the time simulation of the internal bus voltage, field cur- rent, and terminal bus voltage when the exciter is disabled but the reactive load is reduced to 150 MVars. These figures show that there is no system instability occursat this load lev- el which is consistent with the eigenvalue results in Table 4.32. The dynamic/algebraic matrix eigenvalues with the solid arrow for the reactive load from 100 MVar to 170 MVar are collected in Table 4.33. It shows that this pair of eigenvalues approaches the jtu axis. It indicates that an oscillation problem may occur if the reactive load at LOAD2 is further increased. The eigenvectors in Appendix C.4 associated with this pair of eigenvalues show that this pair of eigenvalues are heavily related to the inter- nal bus angle of GEN2 and the field voltage of all three generators especially GEN2. This indicates that the oscillation may develop before the field current limit is reached if the re- active power load is further increased. Since the oscillation problem is closely related to the internal bus angle and field voltage, the oscillation could cause field current limit vio- lation, and excitation system disablement. If we use the same equilibrium point at 170 MVar and assume the exciter is disabled, the eigenvalues of the flux decay bifurcation test and the reduced dynamic/algebraic bifurcation test matrices, shown in Table 4.31, have one positive real eigenvalue which indicate the system is unstable. A time simulation was attempted. Unfortunately the simulation program did not provide any results if the exciter was disabled. It is caused by the inability of the simulation program to solve the difl'eren- tial equations close to the bifurcation point. The results indicate that the excitation system would be disabled either due to field current limit violation due either to the average level of the field current or the development of the oscillation. Once the excitation system is disabeld, the system is unstable. Although the Extended Transient Mid Term Stability Program wap not robust enough to simulate the Flux linkage(p.u.) 144 Regular system with exciter(l70MVar) 1.25 . . 1 T , 1.2 ~ J 1.15 .- d 1‘1 5 Solid line: GEN3 " Dashed line: GEN2 1'05 P Dotted line: GENI ‘ 1 b "X “-1 0.95 - ........... ~ 0.9 ~ J 0.85 1 1 A 1 1 0 5 10 15 20 25 30 time(second) Figure 4.36 Time simulation of the internal bus voltage (170 MVar) 145 Regular system with exciter(170MVar) Field current(p.u.) 2.4 T Y 2.2 ~ 2 .. 1-3 ' Solid line: GEN3 Dashed line: GEN2 1-6 " boned line: GEN] 1.4 - 1.2 P \\\\\ a'v."- """""""" - ................................ 1 .- 0.8 ‘ ‘ 0 10 15 20 time(second) Figure 4.37 Time simulation of the field current (170 MVar) Regular system with exciter(l70MVar) 1.05 Y I I 1 r 1 7" 2 0.95 - 3' g 1,} Solid line: GEN3 ’53 Dashed line: GEN2 § 0-9 ' Doned line: GENI 7.: E o... . 12 0.8 ~ 0.75 . . 1 * 0 10 15 20 time(second) Figure 4.38 Time simulation of the terminal bus voltage (170 MVar) Flux linkage(p.u.) Field current(p.u.) 146 Regular system without exciter( l SOMVar) 1.2 I T I 7 1.15 - Solid line: GEN3 4 Dashed line: GEN2 1,1 » Dotted line: GEN l _ 1.05 ~ .. 1 '- .1 0.95 ~ - 0.9 e """"""""""""""" : 0.85 ‘ L m 1 0 10 15 20 25 30 time(second) Figure 4.39 Time simulation of the internal bus voltage (150 MVar) Regular system without exciter( ISOMVar) 1.9- l.8~ 1.7L- l.6 - l.5~ 1.4- 1.3 '- 1.2- 1.1--' V Y I 1 Solid line: GEN3 Dashed line: GEN2 Dotted line: GEN 1 . ‘- ‘08 . ‘-- --- .......................... time(second) Figure 4.40 Time simulation of the field current (150 MVar) 147 Regular system without exciter( l SOMVar) 1.05 T I T I 1 _ {1 .u T ,,,,,,, -- -- """"" ‘ -------------------------------------------- A E Solid line: GEN3 g Dashed line: GEN2 g 0.95. boned line: GEN] . "6 > a .8 g 0.9 . « E I- 085r ‘ ~ 0.8 1 1 1 1 1 o 5 10 15 20 25 30 time(second) Figure 441 Time simulation of the terminal bus voltage (150 MVar) 148 Table 4.31 The eigenvalues of the algebraic/dynamic bifurcation test matrix and the flux decay bifurcation test matrix for 170 MVar load at bus LOAD2 (saturation) eigda = ' vtt = -26.5444 +52.23051 1.0000 0.0155 -0.2002 -26.5444 -52.23051 -0.4906 1.0000 -0.4246 -3.8928 +14.9079i -0.0017 -0.0002 1.0000 -3.8928 —14.90791 -16.4681 -15.1809 ett = -0.3277 + 1.69901 -0.3277 - 1.69901 1.0e+03 * -0.0478 + 1.11541 -0.0478 - 1.11541 -0.1140 -0.0683 + 0.69821 —l.5191 -0.0683 - 0.69821 0.0002 -0.0630 + 0.61241 -0.0630 - 0.61241 -0.1048 -0.0636 0.2004 ‘— Table 4.32 The eigenvalues of the algebraic/dynamic bifurcation test matrix and the flux decay bifurcation test matrix for 150 MVar load at bus LOAD2 (saturation) eigda - vtt - -26.4666 +52.26841 1.0000 0.0025 -0.0674 -26.4666 -52.26841 -0.1803 1.0000 -0.1320 -3.8998 +14.9699i -0.0006 0.0000 1.0000 -3.8998 -14.96991 -16.5234 -15.6538 etc - -0.3235 + 1.39871 -0.3235 - 1.39871 1.0e+03 ‘ -0.2121 + 1.08171 -0.2121 - 1.08171 —0.0935 0 0 -0.0688 0 -2.4353 0 -0.0932 + 0.79241 0 0 -0.0001 -0.0932 - 0.79241 -0.0632 + 0.63681 -0.0632 - 0.63681 -0.0340 + 0.60391 *0.0340 0.60391 149 number for both system jacobian and algebraic bifurcation test matrices are less that 11 percent which indicate both matrices are not approaching singularity. Table 4.26 shows the system, it is anticipated that the field current, internal generator and terminal voltage would approach zero as in Section 4.3. A more robust algorithm is needed for the Extended Transient Mid Term Stability Pro- gram if it is to be able to simulate the actual voltage collapse problems and calculate con- tigency equilibria for voltage collapse problems. Such an algorithm may exist in version 2.0 of the Extended Transient Mid Term Stability Program, that was not available at the time this research was conducted. Table 4.33 Eigenvalues for different load levels Load (MVar) Eigenvalues 100 — 0.0397 1 1.1176i 120 - 0.0305 :l: 1.1269i 140 — 0.0196 :l: 1.1414i 160 - 0.0078 :l: 1.1638i 170 -0.0016:|:1.1890i CHAPTER 5 REVIEW AND TOPICS FOR FUTURE RESEARCH 5.1 Review A precisely modelled power system which includes the generator dynamics and real and reactive power balance equations was developed in this thesis. It has been shown that the voltage instability problems may occur in the dynamics of the generator before the jacobi- an matrix of the real and reactive power balance equations become singular. It indicates that the load flow based voltage instability methods can not predict the voltage instability problems which occur in the generator dynamics. Furthermore, the thesis also showed that the equilibrium point calculated by using the general power system model was different from the one calculated by using load flow model in the case that both models were using the same system configuration and initial value. The difference of the equilibrium point of the general power system model and the load flow model increases as the system was stressed. The reasons for the divergence of the load flow model from the general power system model are as follows: (a) the load flow simulation uses reactive power generation to regulate bus volt- age and the general power system model uses the field current to regulate bus voltage, (b) the general power system model takes into account the air gap saturation 150 151 which affects the closed loop gain of the exciter control loop, but the load flow model does not, (c) the load flow model assumes KA is infinite but KA is a finite value in the general power system model; the exciter loop gain KEX is proportional to the exciter amplifier gain and inversely proportional to air gap saturation, (d) generators in the load flow model regulate either their high side transformer bus voltage or terminal bus voltage but the general power system model which takes line drop compensation into account allows any point between the generator internal bus and some fictitious point out in the network to be regulated. Voltage instability problems were classified into two different categories in this thesis. These two kinds of voltage instability are results of very different types and locations of stress. Load flow voltage instability is caused by the inability of the transmission system to supply the reactive load when there is no reactive power supply at that load voltage con- trol area. System voltage instability is caused by either the instability of the generator dy- namics or the coupling between the generator dynamic states and algebraic states in a stressed network. This thesis provided different types of tests for different voltage instability. The m bifurcation test was used for testing for load flow voltage instability. The test condition for algebraic biftn'cation test is a test for loss of causality[27]. Transient stability simulation packages that iteratively update dynamic states using the differential equations and update states of the network using the algebraic equations may not obtain unique solutions and will generally terminate due to numerical failure as singularity of matrix D is approached. It has been shown that the algebraic bifurcation(loss of causality) does not necessarily cause a system bifurcation, but can result in a system bifurcation. An algebraic bifurcation that does not cause a static system bifurcation can result in chaos[21] and possibly other 152 unacceptable behavior which may or may not be associated with voltage collapse. The algebraigldmamic system bifurcation test was shown to be always valid for testing the singularity of the system jacobian matrix J. The algebraic/dynamic system bifurcation test can detect the system bifurcations of the algebraic equations like (a) linearly dependent rows in matrix [C1 D1 D7], or (b) linearly dependent rows in matrix [D3 D4] and [C1 D1 D2]. which can not be detected by the algebraic bifurcation test. The algebraic/dynamic test can also be used to test for voltage instability of the generator dynamics. The dmamiclalgebraic system bifurcation test represents the system matrix of the nonlin- ear constrained differential equation model linearized at an equilibrium point and thus de- fine the eigenvalues of the equivalent unconstrained dynamical system. The dynamic/ algebraic system bifurcation test can be used to test the stability of the dynamic states if the matrix D is nonsingular. The singularity of this test indicates that the system jacobian matrix I is singular and that a bifurcation may have occurred in the dynamical system where the algebraic constraints have unique solutions and thus can be eliminated. If the al- gebraic equations can have bifurcation (D is regular ), the dynamic algebraic test is not defined. The flux decay bifurcation test can indicate whether the voltage instability is related to dy- namic voltage instability associated with flux decay dynamics. The derivation of the flux decay bifurcation test pointed out how the air gap saturation, line drop compensation, field current limit, sensitivity matrices, and reactive power generation would influence the sta- bility of flux decay dynamics. A computer program was developed to compute the system jacobian and eigenvalues and eigenvectors of the dynamic states. The computer program was applied to a two bus sys- 153 tem and a twelve bus system to determine the stability of the system. The results were compared with time simulations to confirm their validity. A series of experiments were designed and conducted as follows: Experiment 1: A two bus system with flux decay dynamics and real and reactive power balance equations was investigated. The only system bifur- cation which could occur in this two bus system where exciters were disabled was when the high and low voltage solutions merge. The system bifurcation resulted in a loss of stability where both generator internal and terminal voltage dropped to extremely small value. An algebraic bifurcation due to linear dependence in the lin- earized jacobian of real and reactive power balance equations also occurs and does not cause system bifurcation, but is an unstable equilibrium of the transient stability model. This unstable equilibri- um point is associated loss of voltage stability or possible chaotic behavior. Experiment 2: Algebraic bifurcation of the general power system model and load flow bifurcation of a twelve bus system model were studied. A load bus was isolated fiom the rest of the system by increasing the reac- tance of lines that connect the load bus to the rest of the system. In the load flow bifurcation case, it was shown that weak boundaries developed and became drains to block the reactive power supply to the load bus; Thus a reactive power demand supply type of voltage instability problem developed at a bus despite the fact that there is ample reactive generation reserves on all the generators in the sys- tem. Algebraic bifurcation of the equivalent general power system model was studied. The algebraic bifurcation test results were shown theoretically and computationally to be confirmed by a Q-V Experiment 3: Experiment 4: 154 curve. The test results showed that the twelve bus system was not vulnerable to algebraic bifurcation and reactive demand and supply problem. The system was modified to be more susceptible to reac- tive demand and supply problem. However, the EPRI Transient Mid Term Stability Program would not converge to an equilibrium point when the system was susceptible to reactive demand supply problems. A generator and its terminal bus were isolated from the rest of the system by increasing the transformers reactance that connected the generator to the rest of the twelve bus system. The purpose of this particular case was to establish whether the results obtained for the two bus system composed of generator internal bus and terminal bus were valid in a general system. The conclusion is that the re- sults on the two bus system could be used to describe the dynamic/ algebraic bifurcation that can occur due to the bifurcation of load flow manifold, flux decay manifold, and control manifold of the generator. The system experienced a dynamic/algebraic voltage collapse bifurcation when the excitation system was disabled at the reactive load level where the field current was at the continuous rat- ing limit. A time simulation indicated that there was a rapid decline of terminal bus voltage, field cmrent, and reactive power output. The flux decay bifurcation test and the dynamic/algebraic bifurca- tion test confirmed the eigenvalue analysis and transient stability simulation results that indicated there was a dynamic voltage prob- lem. The two bus voltage control area case of a generator internal and terminal buses was extended to a three bus voltage control area 155 composed of a generator, terminal, and load bus in this experiment. The test for voltage collapse was performed by increasing reactive load at the load bus rather than at the generator terminal bus as in the previous experiment. The results indicated an oscillation devel- oped before the field current limit was reached. The system never approached an algebraic system bifurcation based on a Q—V curve and based on the algebraic bifurcation test. The system never ap- proached dynamic/algebraic bifurcation as reactive load and the field current limit was reached. After the continuous field current limit was hit and the excitation system was disabled, the system ex- perienced a dynamic/algebraic bifurcation. The Transient Stability Program failed to simulate this unstable response. 5.2 Topic for Future Research An equilibrium program which can compute the equilibrium point of a precisely modeled power system needs to be developed. This thesis uses the steady state solution of a pre- cisely modeled transient midterm stability program to compute the equilibrium point. This method is very time consuming and does not guarantee obtaining converged solutions. Since the correct equilibrium point is the basis of the voltage instability analysis and the computation of the equilibrium points have to be repeated many times for different operat- ing conditions, a fast and precise equilibrium program which can handle large scale sys- tem databases is a must. Voltage instability usually occurs at the equilibrium point which is close to bifurcation point. The transient mid term transient stability program also had difficulty performing time simulations when the system was close to loss of voltage stability. A robust numeri- cal algorithm for simulating transient mid term trajectories needs to be developed. If this 156 algorithm were available, the trajectories of the dynamic states could then be simulated and the system behavior close to voltage instability could be more fully explored. A theory that describes the bifurcations and singularities of constrained differential equa- tions is not complete. The analysis of voltage stability at the bifurcation point for a large system would be a very interesting topic if that theory was available. Since dynamic voltage instability is caused by instability the flux decay and excitation control system , new excitation system controls are needed. Nonlinear or adaptive control theory could be applied to develop a control that can prevent a dynamic/algebraic voltage instability and algebraic voltage instability. A secondary voltage control that slowly adjusts voltage setpoints could also be developed. The secondary voltage control should help prevent both algebraic and dynamic algebraic voltage collapse by preventing field current limit violations on appropriate generators. APPENDICES APPENDIX A MODEL LINEARIZATION The general power system model developed in chapter 2 is linearized in this appendix. The subscripts i and j are removed in the cases where there is no difference of the linear- ization for both controllable buses and uncontrollable buses. The definition of variables and parameters are defined in the LIST OF SYMBOLS. A.l Mechanical Dynamics . 1 Arm = (Ky-0,167,111?) l . 1 A5, = Ami A8]. = A01]. A.2 Saturation Function SD 2 i -V 2 E» = ((9—27) + (V4 “3" "‘11 I q: xd: Nlh‘ 157 158 = ((VT51D(8 - 9T) — VTSin (8- 6T) xp (Xd+quD) )2 + xaxwxqsox. l (E’ —V cos(5—0 ))x (1+8) 2 2 (VTcos(5—0T)+ q T , T p D ) ) xd+SDxp Let v (E' )x V'do = VdO- dep and V'qo = Vq0+ :10. 0 xqsO xdsO l V 2 E' -v x 2 2 p xqso q xdr-O then .. AV . AE'q AEP = {—l—X {—T)+B(—— , “TJ-i-CMHDAO +EASD) EpO Vro qu where ~ x’ -x xd+Sdoxp B = V‘qu‘qopr +SD) x'd+Sdoxp C: V' V 5 -0 (14(1)) do TOCOS(° 7‘0) xfiq+xqsdoxp xd+xp ) V‘quTosm (80- 0 T0) (FT-___;fi'sdoxp x ( -xq +xp) D: V‘ V 80 -0 (d4 )+ do "CO“ 70) xdxq-lvxq Sdoxp V. V - (5 0 )(——— “’7’ ) srn - 4° To 0 To x’d+Sdoxp 159 E _ o{-Vmsin (50-910)xdqup (xq-xp))+ - d (xdxq 1" qudoxp) 2 V' ((qu‘ VTocos (SO—GT0) )xp (x'd-xp)) 4° (1',,+s,,(,xp)2 23 Ep -A AEp A50: ( P02 ) Epo 23 E -A .. AV A3,! = ( P30 )( (— T)+B(— q)+CA8+DA6T +EASD) Ep0 V70 E90 Solvefor ASD 213 E -A AV AB“ 115,, = 3 ( P0 ) . [A[ —T)+D(— qTJ+CA8+DAOJ Epo - 23 (Epo -A) E V10 5' q0 AVT AE' =A(——— )+B(— , :)+CA8+DAOT VT0 E A.3 Flux Decay Dynamics The fius decay equations for controllable internal buses are . 1+S. —x.+S. .5”. AE' . = ( Dr)( ( dr. Dioxpt) 980 )AEqi + [ 1 )AEfdi'1' "dor x 41' "' SDiOxpi “”4033in (1 4’ Sm) Vrro (Id: ’ 1'41) C05 (570 ’ 91:0) ‘40: (for: + SDiOxpi) 5'qu Ti (1 + Sm) V110 (Id: ‘ 1'41) Si“ (510 " 9m) A 5 + ‘40: (I'd: + SDioxpi) E'qro . i ' 0T1. TdOi (x 41' '1' SDioxpi) 5 gm 160 (‘ (141+ SDi0xpi) E'qio + V170 (xdi ‘ 1'41) cos (5:0 " eno) + 1'4: ‘1‘ 5010th (1 "’ SDiO) ("1,113.in (1'41 'xdi” (3541+ 501015192 (1 + 5010) xpi (’Vrzo (xdi 'x'di) “’5 (810 ' 9710)) 1 (I'd; 1' 5010x1592 Since the field voltages are assumed to be held constant for uncontrollable internal buses, the linearization of flus decay equations for uncontrollable buses become . 1+S- -x-+S. .E'. A qu = ( DJ)( ( d1. 010x101) 410)A qu + ‘40; x dj + SDioxpi Tao,- (1'4; 7‘ 51210pr) 3410 T} (1 + 50;) V170 (de ’x'dj) sin (5,0 " 0Tjo) I.10} (x'dj '*' 5010110) E 410 A8j+ (1 + 50].) VTjO (xdj -x'dj) 8111(8j0 - 91-1-0) , - A0 140;“ 41+ 5070’») E410 + Ti [- (de "' SDjoxpj) E'qjo 4' Vrjo (xdj 'x'dj) °°s (8,0 " 91,0) + 1'0" 5010‘» (I'd; + 5019292 (1 '1' Sojo) xpj ("VTjO (xdj “x'dp cos (81.0 - 9170)) J ASDj (1549+ SDijpj) 2 1.1015. qu A.4 Excitation System Dynamics 161 A.4 .1 Line Drop Compensation If the line drop compensator is considered, the fictitious bus voltage will be Vc = |VTA(97- 5) “77%| where [T = Iad+Iaq E'q — VTcos (5 - ST) + VTsin (5- 97) 1x ds xqs VT4(0T— 5) +jITxc = VTcos (5-0T) -jVTsin (5—0T) + [ (1 + SD) (E'q- VTcos (5- 0T)) (xd-l-quD) (VTsin (5- 0T») J . . + x 1(x d-t-SDxp) xdxq+xpquD " (1+SD) (E'q-VTcos(5-01.))xc)2 + V. = ((VTCOS (5 - 9T) + (for 1' sDxp) 1 2 (xd +quD) (VTsin (5 — GT) )xc xdxq-l-xpquD 2 - VTsin (5— 07.)) ) A.4 .2 Linearization of Excitation System Dynamics Let Vc = JV?+V§ The linearized difi‘erential equation for the voltage V1 out of measurement device will be AV; = — {—AV + V ———+ V + (11,) 1 55",, 9 KW 3V, 6 V1.0 a 8 _3_ Evans + 36—1ch 0, + asDVcASD ) 162 where 8V 1+S V'IV DO 51:?6 = 00 ”[x' +S ) q d 00",» (1+ 300) cos (5O- 910) XL.) + c _ 1 _ _ BVT — CO (910 (cos (50 em) 1'4"“ SDO’xp V (x d+quDO) Sin (80-0T0)xc_sin(6 _e ) 20 14x4 +xpquDO 0 T0 +(1+SDo)VTosin(80- 9T0)xc) + 1.4+ 5001;) 5’ ((xd d+x quo)VTo°°S(5o —Tc00)x 0 —V cos(5 -0 )j) 1 c0 x'd+300xp =(V10 [V10 sin (50 -910)- 351 .3 020 =v;3[t>.. V (V10 sin (50 - 0T0) qudxc (xq —xp) ) ) ’° (xix. Mason 2 xaacq '1’qu 500 + Vro cos (5O — 0T0) ) ) ((- (xd'1'quDo) ) VTOCOS (50 - 91.0)1‘. (E'q- VTcos (5- 91))(1'4‘xp11c)+ (n’,,,+sl,,,x‘,)2 The linearization of the rest of excitation system states is K -AV. AVsi= —1- -AV.+ F' R‘- t 3‘ ‘t n Bi Kr; (5'51“ (Erato) Erato + 581' (51410) 1’ KEi) AEfdi‘ ) ‘5: 163 _ KAi (‘ Ayn" AVBi) ' AVRi AV“ - "Ai AE _ AVR,- (351' (514:0) Efdi0+SEi(Efdi0) +KEi)AEfdi fdi " ‘Ei A.5 Power Flow Real and reactive power balance equations for generator, terminal, high side transformer, and load buses are linearized. A 5 .1 Real Power Linearization for Generator and Terminal Buses , _ BP‘ AF, 31" AVT BP‘ 310‘ 3]" where a!" E'qOVTOSin (50- 670) (1+ S00) E'q x'd+SDoxp ape quVToSin (50-910) (1+SDO) + BVT x'd + 5001p 2 xd+quDo _ 1+SDO . _ To (xdxq + qu 001p x'd + S Doxp sm (2 (80 970)) 35 - x'd+SDoxp 2 xd+quDo 1+SDO ) - . cos(2(8 -9 )) To (xdxq+quDoxp x 4+SDOXP 0 To 164 EFT- x'd-i-SDoxP 2 xd+quDO 1+SDO ) — , cos 2(8 -6 )) To (xdxq+quDO.xp xd+SDoxp ( 0 To ape = quVTosin (80-970) (I'd-xp) + 53‘; (x'¢1'*'SDO’xp)2 v§03m(2(so—em)) ( xdxquq-xp) x'd—xp J 2 (xdxq + quDoxp) 2 (I'd + SDOXp) 2 AP;— AP; = —VTVHYmsin (e,— e” - 1m) A6T+ (VTVHYTHcos (91 - 9H - 71H) + 2VTYmcosym) A VT + VTVHYTHCOS (or ' 9H ’ 7m) A V}: A 5 2 Reactive Power Linearization for Geneer Buses e _ 3Q‘ A5. BQ‘ AVT 3Q‘ 3Q‘ 3Q‘ AQ — fi;(E—,qf)+m(m)+a-s A8+5§;AOT+BS_DASD aQe _ (ZEmeSin (80" 010) " E'quToCOS (80- 010)) (1+ SDO) 5E; — x'd+SDOIp + 2(xgq+quPSDo)qu(qu-Vmfin (50-910)) (1 +300)2 (1', + S Doxp) 2 (xd + S Doxq) = (1 +500) (ZEmesin (so-em) -2v%osin2(60-em)) _ x d+SDoxp 3451 165 x'd+ Snnxn 2 (xdxq +qupspo) (1+ 300) 2 (—quvmsin (80 — 910) + 11%,,st (t50 — 97.0)) (x'd “900150)2 (181+ SDqu) aQe _ (1 + S00) (23‘ quTocos (5O - 9T0) - V§osin (2 (50 - em) )) + 35 x'd+ Sooxp (1 + 500) (E'qurosin (50’ 910)) + x'd-t-SDoxp (xdxq + qupSDO) (1 + 5110)2 (—2E'quTocos (5O — 9T0) + V31) sin (2 (50 -— 9T0) ) ) (’54 + SDOxp) 2 (14 + Sooxq) BQ‘ _ (1 + $00) (—2E'q0VTocos (aso - em) + V%osin (2 (50 — 6,0) )) EFT x'd + S Doxp (1 + Soc) (E'quToSin (50 ’ 910)) + x'd + S Doxp (x'd + smxp) (1 + $00) 2 (215* qumcos (250 — em) - V%osin (2 (80 - em) )) (I'd + 5001p) 2 (x4 ‘*' Sooxq) Let K = -E'qOVTocos (60 - em) - V§,,sin2 (80 - on) + 2153mm0 sin (60 - em) , . 2 . 2 12 K = ‘ZE'qurosm(50'9m)+VTo-“n (50-970)+qu SD (X.d+SDoxp)2 Q.) (2 (1 + $00) (xdxq + qupspo) + (1+ S00) prxq) K' _ (x'd + SDOIp) 2 (‘4 + 50014) A.5.3 AQ; = aQ;~ _ 3?;- aQ;_ 3V, ‘ aQ;_ 35 _ 3Q;_ 9.6; _ aQ;_ as;- 166 (1 + 500) 2 (2):" (xd+ SDoxq) +xq (x'd + spoxpn ((xdxq +qupsDo) )K‘ (x'd + SDGXp) 3 (Id + SDqu) 2 Reactive Power Linearization for Terminal Buses 8Q; AE'q 3Q; AVT 3Q; 8Q; 8Q; E'qOVTocOS (80" 9T0) (1+ SDO) x'd+Sooxp E'qOVTocos (50 - em) (1+ 500) - 211%,,cos2 (as0 - 9T0) (1+ 500) _ x'd+SDOxp 2"?“st (50 - 010) (xd + SDO-xq) (xdxa + x4501») x'd+SDOxp —(1+S ) 1 +5 0" 2 , _ __fl_ 4 D " VTosm (2 (50 9T0) ) (rd-+5005 +xdxq+xquxP) x'd-I-Sooxp (1+S ) x +5 0:: 2 . _ Do _ d D (1 Vmsm (2 (80 670)) (rd-+8001], xdxq-I-quDxP) E'quTo sin (50 - 01.0) (x'd -—xp) (x'd+SDoxp)2 2 (cos2(80-9m) (x'd-xp) + sin2(80-GTo)qud(xq—xd) ) To (x'd + S 002p) 2 (xdxq + quDxp) 2 167 AQ;- AQ; = VTVHYmcos (9T- 9,, - 7m) ABT + —VTV”YT”cos (9T - 6H — 7“,) A6” + . (VTVHYTHsin (91,— 9,, - 7m) - 2V§YTHs VTVHYTH Si“ (91 ' 9H ’ 7m) A VI! A 5 .4 Real Power Linearization for The Network 1 —A P f1,- = [2 VHiOVHsOYHiHs-cos (emo " 911:0 ’ 7mm) + s=l i= 1 VmoVTioYHm'COS (91m ’ eno " 7mm) 2 VmoVkoYHikCOS (Gm-0 - 9m " YHtk) )A VHF... k = l l 2 VHiOVHsOYHiHsCOS (emu " 911:0 " 7115113) A Vin)“ = 1 "I (Z VHioVHjOYHiijs (911:0 " eHjO " 7mm) A VHj)+ . = 1 n . (‘2 VmOVwYmkcos (OHiO " 9w "' 7H“) A Vi)..- = 1 z (" 2 Vmovmoyumssm (911m ' 91m “ Yams) ’ s = l (s at i) m 2 VHiOVHjOYHiHj Si“ (emu " 6Hjo ' 7mm) ' i = 1 168 VHiOVkOYHikSin (91110 ‘ 91:0 " 71m) ' n z VHiOVkOYHikSin (91110 " 91:0 "’ 71111:) )A9H1+ k = 1 l 2 VHiOVHSOYHiHsSin (63m - 91150 " YHiHs) A 6H5)+ = 1(s¢ i) m (2 VHiOVHjOYHiHj Sin (61150 - eyjo "' 71.15;”) Aeflj)+ i=1 n 2 VHiOVkOYHik Si“ (91110 "’ eko " 71111) A 9k) + = 1 VHiOVTiOYHiTiCOS (91110 ’ 91:0 ‘ 711m) A VTi + VHiOVTiOYHiTiSin (91110 ' em " Yum) A 0no 1 ‘AP :1; = (2 VHjonoYHjmcos (eHjO " emu ' 7mm) + .= 1 m 2 VHjoVHsoYHszCOS (OHjO ’ 91m ' 7mm) + s=1 VHJ'OVTjOYHfl'jCOS (91m ’ 9m " 7mm) + n 2 VujoVkoYijCOs (91m ‘ 9m " Yujk) )A VHJ‘" k = 1 I 2 "movmoYum:cos (91m "' emu "Yujml A Vm + i=1 0' 2 VujovmoYHjusms (emu ' 911:0 ’ 7mm) A VH3 + s = l k = l 169 1 (‘ 2 VHjOVHiOYHjHISin (931-0 - 9”“, — 7mm) - i= 1 M 2 VHjOVHSOYHsz'Sin (eyjo " 91150 " 71.11115) ' s = 1(sat I) VHjoVTjoYHm' Sin (911,11 ‘ eTjo " 711m) ' =1 1 ZVHjOVHiOYHjHiSin(eujo” 0H“) 7313i)A9H,-+ i=1 2 VHiOVHwYuiui Si“ (91le ’ 91130 ‘ 7mm) A 6H: + s = 1 (s at i) n 2 VHIOVOkYijSi“ (91m ‘ 91:0 " Yujk) A 91; + k = 1 VHjOVTjOYl-Ifl'j “’5 (OHjO 9no 711111) A VTj+ VHjoVTjoYHm 3i“ (91:10 ’ 9m " 711m) A 9n l -APg = LEvafl,oYm,ws (13 91110-71111) + j= l n . . Z Vmechos (9"0 - 0,0 - 7”) )A Vk+ 3:1 I 2 VkoVHionmcos (6"0 OHIO ’71:}!1') AVm+ i— - 1 m 21 VmVHjoYmJ-cos (91:0 0”].0 ~7mj) AVHJ. + i = . n X Vstothos ((9,:o 3:1 1 ( 2 VkOVHiOYkHiSin (910 i=1 "3 Z VkoVHjonHJ-sin (91:0 i=1 n 2 VkoVsoYhsin (9m s=1(lc¢s) l 2 VkOVHIOYkHISin (910 i=1 2 VmVHJOijsin ((1,:0 j— = 1 2 VkoVsonsin (6w 3 = 1(kats) 170 -OSo-yb)AVS+ 91110 “71111) 6Hjo "'71:!!1') - 050—7“) )A9k+ 91110 ' 71111) A 9m + 91H 0 7111;) AeHj + 630—79136 A 5 5 Reactive Power Linearization for The Network I —A QHi= (z VHiOVHJOYHiHsSin (611“)- 011,0 _ 71.1"“) + 3:1 2 VmoVHjoYmHj Sin (Gum-6 i=1 HjO -7HiHj) + VHiOVTiOYHiTj Sin (euro " 9m " 711111) + n 2 VmthoYmkSin (91m ' k=l =1 91:0 ' 71111:) )A Vm+ l ('2 Vl-liOVIlsOYHiHs'sin (01150 - 93,0 - Yang) A VHSJ+ 171 m ( 2 VHiOVHIOYHiHj Si“ (911:0 " eHjO "' 7mm) A V111)" i = 1 fl 2 VHiOVkOYHikSin (emu ‘ eko ’ 71111:) A Vic]+ = 1 1 L 2 Vmovusoymmcos (euro ’ 911:0 " 7mm) + =l(s¢i) m 2 VHiOVHjOYHiHJ'COS (91m " 91110 " 7mm) + ,- = 1 VHiOVkOYHik 005 (euro ‘ 01:0 " 71m) "' 2 VHiOVkOYHikCOS (euro ’ e1:0 ‘ 71111:) )A 9m ‘ k = 1 1 L 2, VHiOVHsOYHiHsCOS (91110 " 911:0 " 7mm) A91“) _ = I (sat i) "I ( VHiOVHjOYHiHj °°s (euro ' eHjO " 7mm) A 9w) " j = 1 n L: VmoVkoYkaOS (9mg - 910 - 71111) A 9k) + = 1 VHiOVTiOYHiTi Sin (91110 ’ 9110 ' 711m) A VTi ' V1110 VnoYHmCOS (91110 " 9110 ' 7am) A 9110 l d _ o "A Qflj - (‘2 VHjOVHiOYHjHism (GHjO "' 91110 ’ 7mm) + .= 1 m 2 VHjOVHs-Oylljflsm (91110 - 93,0 - 73,-”) + s = l VHJ'O VTJ‘O Yum Sin (Oujo " eTjO " Yum) + 172 n 2 VHjOVkOYij Sin (eHjO ’ 91:0 ' 71111:) )A VHI+ k = l l 2 VHjoVHioYHjmsm (91th ’ 91m " 7mm) A V111 + i=1 2 VHjOVHsOYHsz Sin (91m " 911:0 " 7mm) A VHS + 3:1 I! k=1 1 (Z VHjonoYHijOS (9310 " 9mg - 711,111) + . = 1 m 2 VHjoVHsoYHszCOS (ewe " 91m " 7mm) + s=l(s¢0 VHjO VTjO Yum C05 (eHjO " eTjO ' 71111,“) + II 2 VHjO VwYijcos (911,0 ' 910 " 711,11) )A 9111‘ k = 1 I 2 VHionoYHjmws (eujo " 9am " 7mm) A 9m " i=1 "3 2 VHjOVHJOYuiujcos (91m ' 911:0 " 7mm) A 9H: " s=l(s$0 i=1 VHjOVTjOYHJTj 3i“ (91m " 9110 ' 711m) A VT} — VujoVTjoYum “’3 (91110 "’ 9m " 711m) A 913' l “A Q: = (‘2 VkonoYm: Si“ (910 ' 91110 ‘ 7m) + .= 1 173 Ill ZVkOVHjOijsinwko ’91110“ 7111,) + -=1 5:1 I 2 VkonoYuh s1n (9110 6H1 .0 '71:!!1‘) AVm+ i- — l 2 VkoVHJ-oijsin (9110 61110-711111“) AVHj+ j- — l 2 VstoYucos (910‘ ego-7,“) AVS+ 3:1 1 (‘2, VkoVH1onH1C°S (9 6mg 71”,.) + 1 m 2 VkOVHjOYkHjcos (910 -,-"9H(, 71",) + i= 1 n s= 1(kats) l 2 VkononmCOs (9110 91110 71111) A611: i- — l m i= 1 n 2 VkoVsoncos (9,0 - 9 1o - 7,9119, 3 =1(k¢s) APPENDIX B SENSITIVITY MATRIX DEVELOPMENT AND MATHEMATICAL BACKGROUND 8.1 Sensitivity Matrix Development Sensitivity matrices were used to show how the reactive power generation, air gap satura- tion, field current limit, and excitation system control will influence the flux decay stabili- ty of a power system in chapter 3. The development of sensitivity matrices is based on the real and reactive power balance equations PC = g1(8, 9,15, V) PL = g2 (8, 0,5, V) QG = g3 (5,9,E,V) Q, = 84(5191E1V) where E, 8: voltage magnitude and phase angle of generator internal bus, 174 The jacobian matrix for the real and reactive power balance equations is and V,0 81(5161E1V)3 82(8191E1V)3 83(696’E1V): 84(6361E9V): arc aPG 3‘8 56 an an 38 1')? 3Q0 3Q; 58 33 3Q], 3Q], 178 56 voltage magnitude and phase angle of terminal, high side 175 transformer, and load buses, the vector of the real power balance equations at generator buses, the vector of the real power balance equations at terminal, high side transformer, and load buses, the vector of the reactive power balance equations at gener- ator buses, and the vector of the reactive power balance equations at termi- nal, high side transformer, and load buses. aPG aPG fi 37 BPL at», 37:“- 317 an 3Q; 375’— 57 aQL aQL 272311 APG APL AQG AQL A131 A2 32 A3 33 A4 84 C101 C2 Dz C3 03 C4 D4 A8 A0 A8 AV A1 B1 A2 32 A3 33 A4 B4 C1 D1 C2 Dz C3 Ds C4 D4 (13.1) 176 Set APG = 6 APL Solve for [2‘1] =12: 2:112: 3:] [2’3 and substitute back to equation(B.l) to obtain a reduced model AQG = .7 AE AQL AV where J: 63123 = C303 .. A333 A131 C101 C4D4 C404 A434 A232 C2122 The sensitivity model has the form _ 1 AV — SELVAQL +SVEAE WMR 177 —1 AB D s =D-AB 11 1 1[11,2,2 [1] -1 AB Q SVE = _SQV[C4-[A4B4] l 1 [ D -1 A B D1 5 =-D-AB 11 1.1,, [ [1 3] 1,1, [0]] —1 _ A 3 C1 ‘9ch - [C3’ [A3 33;] A; B; [CJ]_SQGQLSQLVSVE The sensitivity matrix S 911 E has the following properties (a) positive diagonal elements and negative off-diagonal elements, and (b) the magnitude of the negative ofi-diagonal elements would increase and the positive diagonal elements would decrease as the network is getting stress. It is shown in chapter 3 that these properties explain that one of the reason of flux decay dynamics being instability is caused by the stressed distribution system. Schlueter and Costi [1,9] discussed the derivation and characteristics of sensitivity mani- ces in much more detailed. 8.2 Condition Number Theorem 4.8 [26]: For any invertible n x n matrix A and any matrix norm, the condition number of A indi- cates the relative distance of A from the nearest noninvertible n x n matrix, Specially, 178 ——1— = min M B is not invertible cond (A) "All B.3 Nonsingularity of Matrix A This section shows that the submatrix A of the jacobian matrix of a general power system model is always nonsingular. A general power system model is Differential Equation Model 1&0) = f(x(t) .y(1),).(1)) Ht) 6 [1,11,] Algebraic Equation Model 0 = g(x(t),y(t).>~(t)) where x (t) state vector of the generator dynamics, y (I) state vector of bus voltage and angle of terminal buses, high side trans- former buses, and load buses, and l (t) state vector of the slow varying operating parameter. Thejacobianmatrixis afaf J: 3323} ___ [A B] @533 CD array The sub matrix can be represented by = AMP A1 0 A5 179 where AMP: linearized mechanical and flux decay dynamics, AE: linearized exciter dynamics, and A1: interface of AMP and AE. We first show matrix AB is nonsingular. AE can be represented by 222 0 0 0 0 224 225 226 A E = 227228 229 0 0 0 z302311 where 222 = diagfl?) Ri lxl . -1 224 = dragfi—J Pi lxl Z = diag K“ 25 2171151 1x1 226 = diag (— Km (355 (chdr'o) chdio 4' 551' (chdio) + K51) ) ‘Fi‘er lxl -K , ‘ lxl -K , 228 = diag (7f) 1 IX! . 229 = diag ($1111); 1 X 180 Z30 = diag(T—lE—‘) ‘ lxl Z3l = diag (_ (SE! (chdlo) chdio + S51. (ECfdiO) + K51.) ] lxl 2151' Since Z22 is a diagonal matrix and has no zero diagonal element, detZ22 at 0. 224 225 Z26 detAE = detZ22 x det 228 229 0 0 230 Z31 The term 3151' (chdio) 5111110 + SE: (chdio) 4‘ K121> 0 03-3-1) (a) if K590, equation 3.3.1 is true, and (b) if KEi<0, equation (b.3. l) is still true because in the normal condition as long as the generator is supplying reactive power (field voltage is large), the mag- nitudes of the first two positive elements will dominate the whole term. In the heavy load condition, the field voltage and the saturation function are larger than the normal condition, (3.3.1) is true. Since (b.3.l) is always true, Z31 is nonsingular. the determinant of AE can be represented in the following form Z Z z detAE = detZzzxdet (Z31) xdet(|:z?A 25]- [313“?3] [0 23(3) = 28229 l 1111th x 11¢th x det( 224 225 ‘ 2212312311 ) :- z28 z29 181 Z deth x detZ31 x det[ 2’4 O] 228 229 Since 7/14 and Z29 are nonsingular, detAE = detZ22 x detZ31 x detZ24 x cletZ29 at O the matrix AB is nonsingular. We now show the matrix AM}: is nonsingular. The matrix AMI: can be represented by 2,022 0230 A _ 21100 000 MP- 02,20 0 o 0 00213 0Z140 L00 0 218021, where . _Di 21 = “filial , X Z _ diag (’Etovriows (510 ’ 9110) 2 - M 1x111 - Viioms (2 (510 ‘ 9m) ) (1'11: ‘ Jr41') Mix'dixqi )lxl )lxl . “EtoVrto Si" (510 ‘ 91'10) Z3 = drag Mix'd- I . _Dj "IX"! Z d' -Ej0VTjoc°s (5,0 " erjo) 7 = mg M . . - ix 4: 182 V§iocos (2 (510- e no) ) (x'di " x41) Mix'djxqi Z __ d' —Ej0VTjOSin (51-0-9110) 3 - ‘08 M 1 1111,- 11111111 Zn = lel 212 = Imxm VT“) Sin (8‘0‘ 9T1'0) (141-1'41) = diag 213 . 51021101" 111' -V sin (8— : diag Tjo ’0 9rjo) (xdj -x'd,-) ( (11.7“... ( EjO‘tdOj‘x dj ' 1.... 2110;" J} 219 = 8diag( Since the matrices 21 and 26 are nonsingular, J mxm )lxl 1,... r" - _ _ O 0 0 0 211 0 O 0 0 O z 1 det[z1 Zo]xdet - O 12 [2; 01] (02110211 _0 0_ {p 1 ‘21121122 9 “21121 23 9 1 1 det|:zol 20] xdet 0 .2122; 27 9 “21223 28 6 213 0 214 0 K- O 213 0 219 183 Since mauices 214 and 219 are nonsingular diagonal matrices, !- — ( 1 ’erzilzz 0 “21121123 9 l 1 det 9 -ZuZ; Zr 9 “2122; 28 = 213 9 Z14 9 (L o 213 0 2,9 _ j l l darn 9},“ ‘21121 22 9 _ ‘21121123214213 0 0 219 0 —zuz;‘z1 0 41121121215211 1 1 z “[2” 0] 1111 -z11r1 (21412151211) 0 11 0 4112? 0 which implies 184 —VTicost]‘.K1 S 0 (B32) Normally, x'd « xd or xq, K2 < O which implies 11%,.11111211‘11'2 s 0 (13.3.3) But (B32) and (3.3.3) can not be equal at the same time, 22 + 23211213 < 9 for all n which implies det (z2 + Z3232”) at 0 The same procedure can be applied to the matrix Z7 + 2823218 and shows I is matrix is nonsingular. The matrix AMP is nonsingular. APPENDIX C SIMULATION RESULTS C.l Load flow voltage instability Table C. 1.1 to Table C.1.5 are the outputs from EPRI load flow program. The reactances of L2 and L3 are increased by 2, 3, 4, 4.1 times. Refer to the discussion in section 4.2.2. 185 186 mm onm new: H o.on~ \m.mH “<9 o.v1 >2 n.mH 11111111 1 11 111 11 o.o o.o o.o o.o «H.v~ o.~> coco.H m.nH manna v>.oH oo.o m.vH o.noH on~ New: H o1on~ \o.oH «<9 a.. >2 o.oH av 111111111111111111111111111111111 o.o o.o 0.0 0.0 an..H o.noH oooo.H o.oH «sans mm «~.. oo.o >.n1 o.mu on~ Ham: H o.on~ \o.aH may o.o >2 o.oH 1111111 1111111111 c.o o.o o.o o.o >.n1 o.mo coco.H o1oH Hxxuu M 2.21 2.0 p.21 ed? as 21m: H m>.o~1 m..H n.eH1 ..mm1 onm Hem: H o.u1 >2 v..- 1mw 111111111111 11111111 1111111111 1 111111111 1111111111111 o.o o.o o.on o.oa o.o o.o mmpo.o on~ na2 n.o- mm 111111111111111111111111111111111111 1111111 11111 11111 o.o o.o 0.0m o.m~H o.o o.o onno.o can «neon .n.oH1 Hm.o o.oH1 o.e>1 ona ~am= H mm oo.aH1 0H.o ..v~1 o.v~1 onw Hem: H H..1 >2 >.o- :1 11111111111111111111111111111111111111111111111111111 o.o o.o o.mn o.ooH o.o o.o omoo.o can Hneoa n 2.” 85 =6? 0.2.1 92 222. H .mw >~..H1 FH.o o.o1 o.on om~ no1 >2 o.>- nu 11111111111111111111111111111111111111111111111111111 o.o o.o o.o o.o o.o o.o o>om.° onw nan: m 1.3 86 n.~ 99:1 ex: 223 H .m 2.31 3d m.~1 new 88 ~33 H «n.oH1 Hm.o m.o m.o> om~ Hox H.a- w 11111111111 111 1111111 1 11111111111111111111 111 1111111 1 o.o o.o o.o o.o o.o O... $36 SN «5: mm v~.. oo.o m.> o.mo1 o.cH Hague H m>.o~1 m..H q.~H1 o.oo omw noa o.on~ “w 111111111111 111111111111111111111111111111 1 1111111 111 0.0 o.o o.o o.o o.o o.o .noo.H can Hum: 11 4>o ¢<>x :2 a<>z 3: mzx 3: ¢<>z :2 ¢<>x 3: uqoze >2 H4aao¢ 1L pom mummoH quH mzoga mzHH oh oH azazm oHau¢uu 11 H . Heo¢m¢ mmquHm «on zaemun anus amuse oH 187 Table C. 1.2 Load flow simulation for two times line reactances mmtu muzh mom HdOmmd no 02m mm.n 00.0 0.0n 0.~P 0mm man: a n0.o~ 00.0 0.- 0.no~ 0n~ «km: a o~.v 00.0 N.0I n.m0 0mm aka: a vo.vfia 00.0 0.onl n.0«n 0n~ mam: a oh.m~u 50.~ v.n~l 9.051 can "km: a mm.mn 0v.0 n1om1 m.mmo 0mm nhm: a 00.01 on.“ h.mH1 h.~>o 0m~ NPMI n on.0| 00.0 0.51 0.501 0mm ukm: a m0.0~1 00.0 0.0N1 v.~nl 0m~ dam: H mm.n 00.0 ~.h~| 0.~hn m.n~ HZ¢MH H vo.vul 00.0 0.H n.m~ 0n~ no0.~ n.m~1 o.mh 0m~ n0<04 a m0.0~| 00.0 m.h m.- 0m~ ~0o «(>2 3: ¢<>I 3: mid: mam hum mummoa uzwa mxoqh mzHA OH OH XunllluoIIUIIIInIIIIII2 3: Hzaxm F.0I 0000.~ «.0 0000.“ 0.0 0000." 0.0a: -h0.0 n.val on~o.0 n.nl v~00.0 H.00 nnoo.0 n.0 0H00.0 01nl v~00.n mqoz< .D.m >2 n.mH n.n~ nzsuh >2 0.0a 0.0m «tank >2 0.0a 0.0a atauh >2 v.n- onN n02 n.-~ onw motda >2 b.n~w 0n~ andda >2 H.o- 0n~ new: >2 o.h- onu «hm: >2 n.0n~ 0mm "hm: >2 A‘DHU‘ mm= 3: «(>2 3! 0(04 zouh~ku~hu ¢Ou Zahmrm hmuh mmmam an 188 Table C.1.3 Load flow simulation for three times line reactances 00(0 0Hza «Oh h«0«m« no 02m 00.0 00.0 0.0V 0.~0 00m 000: H 0H.0H 00.0 «.00 0.00H 00w ~00: H mn.v 00.0 0.0 0.00 00m H00: H 00.vH1 00.0 0.0H1 0.HH1 .0mm 000: H 0n.n~1 001N m.~H1 H.001 0m~ H00: H 00.01 00.0 0.001 0.001 00w 000: H 00.0 00.H n.0H1 0.001 00w ~00: H 00.01 00.0 0.v1 0.001 00m ~00: H 00.0H1 00.0 0.001 H.01 00m H00: H 00.0 00.0 0.001 0.~01 0.0H 02«m& H 00.vH1 v0.0 0.0 0.HH 00m 00(04 H 00.01 00.0 0.vn H.00 00m ~0(Oq H 0H.0H 00.0 0.nH1 0.00H1 0.0H ~2«mh H 00.0 00.H 0.0H 0.00 00w ~0(OA H 00.01 00.0 0.~1 0.00 00w H0(Oq H 00.0 00.0 0.0 0.001 0.0H H2«mh H 00.nm1 00.~ 0.HH1 0.00 00m 00(04 H 00.0H1 00.0 v.0H H.0 00w HQ(OA H H>o «(>2 :2 «(>2 32 02(2 000 000 000004 mzHA 03040 02H; OH OH 2111111111111111111111(H(0 mzHA1 ttttttttttttttttttttt 21111111111 MNH0HU0H 000H\0~\v H 0 (m«( 20 H H .02 00(0 . mzoN smegma zoqh 2030“ 0.000 \0.0H 0.0 0.0 0.0nw \0.0H 0.0 0.0 0.00m \0.0H 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 «(>2 :2 920:0 0(04 «0300 A(m« «:UHH «(>2 0(04 0.0 0.0 0.0NH 0.00H 32 .H «0.0v 0.N0 «~.00 0.00H 0.0 0.00 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 «(>2 :2 onH(«mzmo (0(0 00011 0.01 0000.H 0.0 0000.H 0.0 0000.H 0.0H1 0000.0 v.0N1 0000.0 0.01 0000.0 n.0H1 0000.0 0.H «000.0 0.~1 H000.0 HHUZ( 10.0 >2 0.0H 0.0H M2009 >2 0.0H 0.0H «2209 >2 0.0H 0.0H H2209 >2 0.~NN 00w 00(04 >2 0.H0~ 00m «0(Qa >2 0.0NN on“ H0404 >2 0.0NN 000 000: >2 0.0«0 00w ~80: >2 0.0mm 00m H90: >2 H(DHU( 00(01m2(8 .oz maomo on><2mzmo m>HauHaoeHHHm «oh :u9m2m puma mamas oH 189 Table C.1.4 Load flow simulation for four times line reactances 00(0 0H=h «Oh B«Omm« no 020 o>.. oo.o m..m o.~> on~ new: H o.on~ Hm.mH >2 v.mH 1111111111111111111111111111111 11 11 o.o o.o o.o o.o =m..m o.~> HHoo.o 0.0H «tame oo.>H oo.o m.an o.noH on~ Hem: H o.on~ \o.oH mca >.> >2 o.oH 1 1 1111111111 o.o o.o o.o o.o 2o.on o.noH oooo.H o.oH «same ov.¢ oo.o H.HH >.o¢ on~ Ham: H o.onm \o.oH m2 o.oH 111111111111111111111111111 o.o o.o o.o o.o H.HH >.oo oooo.H o.oH Hague «n.2H1 ~o.o m.mH1 ~.m1 on~ namx H on.o~1 oH.n n.VH1 o.vo1 can Hem: H >.HH1 >2 o.oH~ 11111111111111111111111111111111111111111111111111111 o.o o.o o.on o.om o.o o.o nHmm.o on~ no.H v.vH1 o.am1 on~ ~em= H v.>~1 >2 o.~oH 111111111111111111111111111111111 11 o.o o.o o.on o.m~H o.o o.o eco>.o ann ~o.~1 >2 n.n- 11111 111111111111111111111111111111111111111111111111 o.o o.o o.mn o.ooH o.o o.° oHpa.o onu Ha.. oo.o >.mv1 o.~>1 m.mH mamas H vn.vH1 ~o.o ~.H ~.m on~ no<0H H oa.~H mm.o v.2. n.0o on~ ~o2 o.o- 11111 11111 111 11 o.o o.o o.o o.o o.o o.o nooa.o onm mam: oo.>H oo.o H.HN1 o.moH1 o.mH Hzmma H on.HH o~.H o.m~ a.ow omw mecca H wn.o1 ~m.o o.n1 «.NoH on~ HocoH H >.H >2 n.n- 11111111111111-11111111111111111111 o.o o.o o.o o.o o.o o.o coma.o onw ~am= cv.v oo.o 0.91 >.oo1 o.oH Hzame H an.o~1 oH.m a.m1 >.~o omm na2 o.o- 11111111111111111111111111111111111111111111111111111 o.o o.o o.o o.o 0.0 0.: ovao.o onw Ham: H>o ¢<>z :: 2<>x 3: mz2 3: ¢<>2 xx 22>: 3: mHoz< >2 H<=eo< you mummoH mzHH monu mzHH oH azazm oHauHpo «on 209020 9009 00000 0H 19!) Table C.1.5 Load flow simulation for three times line reactances 00(0 0H20 «00 b«000« 00 020 .H.m oo.o >..m m.H> on~ new: H o.on~ \n.mH >29 «.01 >2 o..H 1111111111111111111111111 1111 0.0 o.o o.o o.o :m.vn o.~> >~ma.o «.mH «tame ov.oH oo.o m.mn o.noH on~ «an: H o.on~ \o.aH «<9 o.> >2 o.oH 11 o.o 0.0 0.0 o.o 2o.ooH o.noH coco.H o.oH «same «o.c oo.o o..~ m.>u onw Hem: H o.om~ \o.oH >49 o.o >2 o.oH 11111111111111111 111111111111111111111111111111111111 o.o o.o o.o o.o o..~ m.>a oooo.H o.oH quma om.nH1 oo.o n.m1 «.21 can new: H >¢.>H1 om.n >.v~1 n.mo1 omm Hem: H >.HH1 >2 n.HH~ 11111 111111111111 11 1111111111111111 1 11111 11111 111111 1 o.o o.o o.on o.oa o.o o.o ~H~o.o can no2 ..noH 11 1111 111111111111 11 o.o o.o o.om o.m~H o.o o.o ooH>.o on“ ~oo.n1 va.o o.~1 n.HoH1 on~ New: H Hm.oH1 Ho.o v.~n1 n.H onm Hem: H >.~1 >2 m.H- 1-1111111111111111 111111 111111 o.o o.o o.mn o.ooH o.o o.o ~noo.° onm Ho1 m.mH «rams H mn.nH1 oo.o H.o1 m.v 0mm mecca H po.n~ No.H c.5m m.oo onN Nocoq H o.HH1 >2 >.HH~ 111111111111111111111111111 11111111111111111111111111 o.o o.o o.o o.o o.o o.o oo~>.o onu new: ov.oH oo.o ~.mn1 o.noH1 o.oH ~zmme H Ho.- oH.~ m.mn n.oo onm ~oo.n1 .o.o v.n1 ~.~oH onm HocoH H o.H >2 m.n- 11111111111111111111111111 111111111111111111111111111 o.o o.o o.o o.o o.o o.o aHpm.o onn ~am= «o.v oo.o o.aH1 m.>c1 o.oH Hzamp H H2.>H1 on.n m.o >.oo can mo2 o.>- 111111 11 11 1 o.o o.o o.o o.o c.o o.o nooa.o on~ Ham: H>o «(>2 32 22>: 3: mz: :2 ¢<>z 22 22>: 2: uHoz¢ >2 Hcaao< pom mummoH mzHH mzogu mzHH oh oH Hzazm o um¢u1mzHaoHau¢um um9HOH0(00 hz0H02(«b 000(000 00(hao> 000 209000 h00h 00000 0H 191 C.2 Algebraic Voltage Instability Table C.2.l to C.2.6 are converged equilibrium poimts of the simulation in section 4.2.3. Table C.2.7 to C.2.12 are the output from the eigenvalue program. detjj: determinant of the system jacobian matrix detd: determinant of the algebraic bifurcation test matrix detda: determinant of the dynamic/algebraic bifurcation matrix detad: determinant of the algebraic/dynamic bifurcation matrix condjj: condition number of the system jacobian matrix condd: condition number of the algebraic bifurcation test matrix condda: condition number of the dynamic/algebraic bifurcation test matrix condad: condition number of the algebraic/dynamic bifurcation test matrix vtt: eigenvector of the flux decay matrix eigtt: eigenvalue of the flux decay matrix eigij: eigenvalue of the system jacobian matrix vjj: eigenvector of the system jacobian matrix eigd: eigenvalue of the algebraic bifurcation test matrix vd: eigenvector of the algebraic bifurcation test matrix eigda: eigenvalue of the dynamic/algebraic bifurcation test matrix vda: eigenvector of the dynamic/algebraic bifurcation test matrix 192 eigad: eigenvalue of the al gebraic/dynamic bifurcation test matrix vad: eigenvector of the algebraic/dynamic bifurcation test matrix Only those variables that are necessary for the discussion will be shown in the tables. 193 Table C.2.1 Equilibrium point (algebraic 4O MVar) 1 9 BUS TEST SYSTEM(NO LINE DROP COMPENSATION) PAGE NUMBER 7 Dl-Z.O, DZ-Z.0, 03-2.0, KA-SO, Rc-0.0, XC-0.0 DATE 4/26/90 LOADZ-do, Load flow test(roactances are 5 times bigot) TIME 23.52.10 GENERATOR ANGLE IN DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 480.1023 521.9312 478.4626 GENERATOR FIELD VOLTAGE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 ~49.9995 1.0488 1.5377 1.3202 GEN. FLUX LINKAGE (Q-AXIS) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.9932 0.8697 1.0167 GEN. ELECTRICAL POWER - (MN) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 32.5898 112.8315 22.1943 GENERATOR EXCITER SATURATION -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.0426 0.0149 0.0418 GENERATOR MEGAVAR OUTPUT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -8.1783 7.1539 18.5161 GENERATOR FIELD CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.9757 1.3729 1.1799 GENERATOR TERMINAL CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.3350 1.1315 0.2906 GEN. TERM. CURR. ANGLE DEGREES-NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 130.0460 119.7484 69.7633 A-C BUS VOLTAGE MAGNITUDE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0001 1.0001 1.0005 1.0048 0.9972 0.9898 0.9895 0.8869 0.9844 A-C BUS VOLTAGE ANGLE DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 ~49.9995 115.9183 123.3732 109.3142 114.8294 119.3184 108.5745 116.2736 98.6373 109.4202 Table C.2.2 Equilibrium point (algebraic 45 MVar) 1 9 BUS TEST SYSTEM(NO LINE DROP COMPENSATION) PAGE NUMBER 7 Dl-2.0, D2-2.0, D3-2.0, KA-SO, RC-0.0, XC-0.0 DATE 4/26/90 LOADZ-dS, Load flow test(reactancoe are 5 times bigot) TIME 23.26. S GENERATOR ANGLE IN DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 ~49.9995 271.2339 312.5252 269.0554 GENERATOR FIELD VOLTAGE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0555 1.5583 1.3726 GEN. FLUX LINKAGE (O-AXIS) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.9944 0.8774 1.0238 GEN. ELECTRICAL POWER - (MN) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 32.8513 112.9162 21.9617 GENERATOR EXCITER SATURATION -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.0427 0.0152 0.0469 GENERATOR MEGAVAR OUTPUT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME N0 11 -49.9995 -6.4683 10.5992 23.7105 GENERATOR FIELD CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 ~49.9995 0.9821 1.3930 1.2236 GENERATOR TERMINAL CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 ~49.9995 0.3344 1.1349 0.3242 GEN. TERM. CURR. ANGLE DEGREES-NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -81.8151 -90.8223 -146.6501 A-C BUS VOLTAGE MAGNITUDE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0000 1.0000 1.0001 1.0038 0.9954 0.9865 0.9880 0.8603 0.9818 A-C BUS VOLTAGE ANGLE DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -92.9684 -85.4633 -99.6677 -94.0672 -89.5291 -100.4023 -92.5988 -110.7306 -99.5299 194 Table C.2.3 Equilibrium point (algebraic 50 MVar) 1 9 BUS TEST SYSTEM(NO LINE DROP COMPENSATION) PAGE NUMBER 7 Dl-2.0, D2-2.0, D3-2.0, KA-SO, RC-0.0, xc-o.o DATE 4/25/90 LOAD2-50, Load flow test(reactances are 5 times biqer) TIME 8. 3.57 GENERATOR ANGLE IN DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -119.9527 4.0186 44.7801 1.4807 GENERATOR FIELD VOLTAGE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -119.9527 1.0631 1.5818 1.4313 GEN. FLUX LINKAGE (Q-AXIS) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -119.9527 0.9956 0.8857 1.0315 GEN. ELECTRICAL POHER - (MN) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -119.9527 32.8573 112.9999 22.0002 GENERATOR EXCITER SATURATION -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -119.9527 0.0429 0.0156 0.0536 GENERATOR MEGAVAR OUTPUT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -119.9527 -4.6168 14.4625 29.4330 GENERATOR FIELD CURRENT ~NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -119.9527 0.9891 1.4154 1.2719 GENERATOR TERMINAL CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -119.9527 0.3318 1.1392 0.3675 GEN. TERM. CURR. ANGLE DEGREES-NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -119.9527 7.8302 0.1097 -60.1378 A-C BUS VOLTAGE MAGNITUDE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -119.9527 1.0000 1.0000 1.0000 1.0029 0.9935 0.9831 0.9864 0.8312 0.9792 A-C BUS VOLTAGE ANGLE DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -119.9527 -0.1681 7.4031 -6.9146 -1.2681 3.3266 -7.6531 0.2286 -18.4519 -6.7644 Table C.2.4 Equilibrium point (algebraic 55 MVar) 1 9 BUS TEST SYSTEM(NO LINE DROP COMPENSATION) PAGE NUMBER 7 D1-2.0, D2-2.0, D3-2.0, KA-SO, RC-0.0, XC-0.0 DATE 4/25/90 LOADZ-SS, Load flow tost(reactancea are 5 times bigot) TIME 11. 9. 6 GENERATOR ANGLE IN DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -345.4698 -305.3192 -348.5387 GENERATOR FIELD VOLTAGE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0718 1.6086 1.4973 GEN. FLUX LINKAGE (Q-AXIS) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.9971 0.8949 1.0400 GEN. ELECTRICAL POWER - (MN) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 33.0882 113.1236 21.8248 GENERATOR EXCITER SATURATION -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.0431 0.0161 0.0623 GENERATOR MEGAVAR OUTPUT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -2.3933 19.0317 35.9617 GENERATOR FIELD CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.9974 1.4419 1.3262 GENERATOR TERMINAL CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.3320 1.1456 0.4192 GEN. TERM. CURR. ANGLE DEGREES-NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 14.4675 8.4112 -55.6365 A-C BUS VOLTAGE MAGNITUDE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.9999 0.9999 0.9996 1.0017 0.9912 0.9790 0.9845 0.7980 0.9759 A-C BUS VOLTAGE ANGLE DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 ~49.9995 10.3333 17.9748 3.4834 9.2241 13.8841 2.7473 10.7532 -8.6329 3.6667 195 Table C.2.5 Equilibrium point (algebraic 58 MVar) 1 9 BUS TEST SYSTEM(NO LINE DROP COMPENSATION) PAGE NUMBER 7 D1-2.0, 02-2.0, D3-2.0, KA-SO, RC-0.0, XC-0.0 DATE 4/25/90 LOADZ-SB, Load flow tost(roactances are 5 times bigor) TIME 11.38.37 GENERATOR ANGLE IN DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -614.1379 -574.3732 -617.4922 GENERATOR FIELD VOLTAGE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0787 1.6278 1.5435 GEN. FLUX LINKAGE (Q-AXIS) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.9981 0.9009 1.0458 GEN. ELECTRICAL POWER - (MW) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 33.1591 113.2174 21.7917 GENERATOR EXCITER SATURATION -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.0433 0.0164 0.0694 GENERATOR MEGAVAR OUTPUT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -0.8631 22.2199 40.4071 GENERATOR FIELD CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0031 1.4603 1.3631 GENERATOR TERMINAL CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.3319 1.1506 0.4564 GEN. TERM. CURR. ANGLE DEGREES-NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 103.1594 98.2279 32.4044 A-C BUS VOLTAGE MAGNITUDE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.9999 0.9999 0.9993 1.0009 0.9897 0.9763 0.9833 0.7755 0.9737 A-C BUS VOLTAGE ANGLE DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -258.3309 -250.6364 -265.2353 -259.4433 -254.7369 -265.9724 -257.8909 82.2170 -265.0356 Table C.2.6 Equilibrium point (algebraic 60 MVar) 1 9 BUS TEST SYSTEM(NO LINE DROP COMPENSATION) PAGE NUMBER 7 Dl-2.0, D2-2.0, D3-2.0, KA-SO, RC-0.0, xc-o.o DATE 4/25/90 LOAD2-60, Load now tost(roactancos are 5 times bigot) TIME 10.35.28 GENERATOR ANGLE IN DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -829.4944 -789.9480 -832.7736 GENERATOR FIELD VOLTAGE -NUMEER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 ~49.9995 1.0829 1.6420 1.5771 GEN. FLUX LINKAGE (Q-AXIS) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.9988 0.9054 1.0499 GEN. ELECTRICAL POWER - (MN) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 32.8113 113.2591 22.1776 GENERATOR EXCITER SATURATION -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.0434 0.0167 0.0750 GENERATOR MEGAVAR OUTPUT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.1422 24.5224 43.5964 GENERATOR FIELD CURRENT -NUMEER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0068 1.4733 1.3899 GENERATOR TERMINAL CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.3281 1.1539 0.4853 GEN. TERM. CURR. ANGLE DEGREES-NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -113.8827 -118.1580 175.5071 A-C BUS VOLTAGE MAGNITUDE ~NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0000 1.0001 0.9995 1.0005 0.9888 0.9747 0.9825 0.7598 0.9725 A-C BUS VOLTAGE ANGLE DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -113.6344 -105.8894 -120.4918 -114.7355 -109.9946 -121.2430 -113.1653 -133.3874 -120.3143 eigjj - -26.2673 -26.2673 49.1836 46.1034 39.8293 -12.5990 -12.5990 -10.3545 -10.3545 25.4839 23.8411 21.2110 -15.7823 -16.6154 -16.5977 -4.9845 -4.9845 -4.1796 ~4.1796 -3.9324 -3.9324 11.5052 10.6917 +52.36411 -52.36411 +25.4489i -25.4489i +20.66541 -20.6654i +14.11621 -14.11621 +15.06591 -15.06591 +14.99011 -14.99011 8.1602 + 1.18391 8.1602 - 1.18391 1.6114 -0.5621 -0.5621 -0.2152 -0.2152 0.7097 -0.0925 -0.0925 0.3103 0.0545 -0.0635 -0.0635 -0.0025 -0.0025 eigd - 49.1836 46.1038 39.8328 -12.4443 -12.4443 -10.1708 -10.1708 + + 1.56091 1.56091 1.21451 1.21451 0.90921 0.90921 0.57361 0.57361 0.60561 0.60561 +25.66101 -25.66101 +20.65571 -20.65571 -5.1331 +14.18941 -5.1331 -14.18941 25.4946 23.8789 21.2135 8.1761 + 1.23091 8.1761 - 1.23091 1.7777 11.4941 11.0223 0.9511 eigad - 1.0o+02 * -0.2224 + 6.12021 -0.2224 - 6.12021 -2.4305 + 3.90951 -2.4305 - 3.90951 2.1554 + 3.15531 2.1554 - 3.15531 0.3776 + 0.55851 0.3776 - 0.55851 196 Table C.2.7 Output for algebraic 40 MVar 0.4905 0.4502 0.3510 .3313 .0003 .0449 .1206 .0676 + 0.00581 .0676 - 0.00581 .1296 0000000 detjj - -6.8110e+30 detd - 1.6769e+21 detda - -4.0616e+09 detad - 1.1185e+29 condjj - 1.6309e+06 condd - 66.7979 condda - 7.7767e+05 condad - 2.9371o+06 vtt - 1.0000 0.0495 -0.0046 -0.7183 1.0000 1.0000 -0.3687 0.3889 -0.5528 eight - 1.0e+03 * -0.1555 -1.1772 -0.4878 197 Table C.2.7 (cont’d) 81911 ‘- -26.2712 -26.2712 48.9269 45.8832 39.7197 -12.5574 -12.5574 25.4149 23.8081 20.9556 -10.2499 -10.2499 -15.7839 -16.6157 ~16.5981 -4.1703 -4.1703 -3.9369 -3.9369 -4.8887 -4.8887 11.3732 10.2916 +52.36591 -52.36591 +25.33251 -25.33251 +20.55091 -20.55091 +15.09711 -15.09711 +14.98931 -14.98931 +13.95391 -13.95391 8.1280 + 1.17381 8.1280 - 1.17381 1.4979 -0.5685 -0.5685 -0.2225 -0.2225 0.7014 -0.0988 -0.0988 0.3112 0.0538 -0.0643 -0.0643 -0.0042 -0.0042 eigd - 48.9268 45.8835 + + 1.55541 1.55541 1.21511 1.21511 0.91581 0.91581 0.58051 0.58051 0.60811 0.60811 -12.4006 +25.54871 -12.4006 -25.54871 39.7233 -10.0677 +20.53691 -10.0677 -20.53691 -5.0251 +14.05731 -5.0251 -14.05731 25.4254 23.8462 20.9590 1.7386 8.1361 + 1.22171 8.1361 - 1.22171 11.4645 10.5266 0.8296 eigad - 1.0o+02 -0.2195 -0.2195 -2.1702 ‘8' + 4. 5.56451 5.56451 3.57881 -2.1702 - 3.57881 1.8949 + 2.79971 198 Table C.2.8 Output for algebraic 45 MVar 1.8949 2.79971 0.3774 + 0.56021 0.3774 - 0.56021 0.4878 0.4485 0.3498 .3302 .0003 .0414 .1167 .0668 + 0.00541 0.0668 - 0.00541 0.1284 OOOOO detjj = -5.7086e+30 detd - 1.2554e+21 detda I -4.5473e+09 detad - 5.0494e+28 condjj - 1.6453e+06 condd - 77.7386 condda - 7.8439e+05 condad - 2.3740e+06 vtt - 1.0000 0.0540 -0.0256 -0.6724 1.0000 1.0000 -0.4136 0.2844 -0.6042 eigtt = 1.0o+03 * -0.1518 -1.0892 -0.3871 199 Table C.2.8 (cont’d) eigjj - -26.2756 -26.2756 48.6824 45.6562 39.6142 -12.5173 ~12.5173 25.3522 23.7724 ~10.1363 -10.1363 20.7064 -15.7855 -16.6162 -16.5986 -4.1544 -4.1544 -3.9425 -3.9425 -4.7854 -4.7854 11.2511 9.8822 +52.36801 -52.36801 +25.21511 -25.21511 +20.43121 -20.43121 +15.12441 -15.12441 +14.98871 -14.98871 +13.78131 -13.78131 8.0822 + 1.16441 8.0822 - 1.16441 -0.5770 + 1.54671 -0.5770 1.3701 -0.2325 -0.2325 0.6834 -0.1077 -0.1077 0.3129 0.0532 -0.0649 -0.0649 -0.0062 -0.0062 eigd = 48.6823 45.6565 39.6177 4. 1.54671 1.21621 1.21621 0.92531 0.92531 0.58771 0.58771 0.61051 0.61051 -12.3580 +25.43531 -12.3580 -9.9559 -9.9559 -4.9025 -4.9025 25.3624 23.8107 20.7105 -25.43531 +20.41281 -20.41281 +13.91101 -13.91101 8.0811 + 1.21141 8.0811 - 1.21141 11.3876 1.7030 10.0815 0.6667 eigad - 1.0o+02 -0.2232 -0.2232 -1.9220 -1.9220 t + + 5.12161 5.12161 3.28991 3.28991 1.6527 + 2.47961 1.6527 - 2.47961 0.3778 + 0.56361 200 Table C.2.9 Output for algebraic 50 MVar 201 Table C.2.9 (cont’d) 0.3778 - 0.56361 0.4851 0.4468 0.3487 0.3291 0.0003 0.1278 0.0370 0.0660 + 0.00491 0.0660 - 0.00491 0.1125 detjj = —4.6493e+30 detd = 8.6008e+20 detda - -5.4056e+09 detad a 2.2694e+28 condjj = 1.6577e+06 condd - 98.1072 condda - 7.9004e+05 condad = 1.9277e+06 vtt - 1.0000 0.0615 -0.0553 -0.5948 1.0000 1.0000 -0.4797 0.2343 -0.5970 eigtt - 1.0o+03 * -0.1471 -1.0789 -0.3113 eigij - -26.2808 -26.2808 48.4166 45.3777 39.4845 -12.4688 -12.4688 25.2769 23.7165 -10.0005 -10.0005 20.4420 -15.7873 -16.6169 -16.5992 -4.1316 -4.1316 -3.9496 -3.9496 -4.6618 -4.6618 11.1177 9.4755 8.0079 + 8.0079 - -0.5884 -0.5884 1.2337 -o.2493 -0.2493 0.6399 -0.1212 -0.1212 0.3179 0.0522 -0.0657 -0.0657 -0.0084 -0.0084 eigd = 48.4165 45.3779 39.4880 -12.3069 +25.30251 -12.3069 -25.30251 -9.8223 -9.8223 -4.7522 -4.7522 25.2868 23.7555 20.4469 11.2804 7.9968 + 7.9968 - 0.4393 9.6582 1.6724 eigad - 1.0o+02 -0.2367 -0.2367 -1.6787 -1.6787 1.4255 + +52.37041 -52.3704i +25.07791 -25.07791 +20.27821 -20.27821 +15.14761 ~15.14761 +14.98801 -14.98801 +13.57961 -13.57961 1.14491 1.14491 + 1.53121 - 1.53121 + 1.21781 - 1.21781 + 0.93801 - 0.93801 + 0.59541 - 0.59541 + 0.61331 - 0.61331 +20.25521 -20.25521 +13.73101 -13.73101 1.18821 1.18821 1* + 4.76641 - 4.76641 + 3.01711 - 3.01711 2.17471 202 Table C.2.10 Output for algebraic 55 MVar 1.4255 - 2.17471 0.3784 + 0.56881 0.3784 - 0.56881 0.4821 0.4447 0.3473 0.3277 0.0002 0.0316 0.1272 0.0650 + 0.00401 0.0650 - 0.00401 0.1082 detjj - -3.5278e+30 detd - 4.7215e+20 detda = -7.4718e+09 detad - 9.5557e+27 condjj s 1.6780e+06 condd - 150.5746 condda - 7.9943e+05 condad - 1.5734o+06 vtt - 1.0000 0.0733 ~0.0932 -0.4550 1.0000 1.0000 -0.5731 0.2158 -0.5289 eigtt - 1.0o+03 * -0.1411 -1.2327 -0.2568 203 Table C.2.10 (cont’d) eigjj - ~26.2849 -26.2849 48.2570 45.1980 39.4003 -12.4391 -12.4391 25.2325 23.6762 -9.9074 -9.9074 20.2775 -15.7886 -16.6174 -16.5996 -4.1159 -4.1159 -3.9550 -3.9550 -4.5740 -4.5740 11.0307 9.2492 7.9455 + 7.9455 - -0.5966 -0.5966 1.1557 -0.2653 -0.2653 -0.1328 -0.1328 0.5897 0.3246 0.0515 -0.0662 -0.0662 -0.0098 -0.0098 eigd - -12.2753 +25.21901 -12.2753 -25.21901 +52.37231 -52.37231 +24.99161 -24.99161 +20.17141 -20.17141 +15.15881 -15.15881 +14.98761 -14.98761 +13.44441 -13.44441 1.12351 1.12351 + 1.51541 - 1.51541 1.21851 1.21851 0.94841 0.94841 l +l + + 0.60061 - 0.60061 + 0.61501 - 0.61501 48.2570 45.1983 39.4038 -9.7308 +20.1454i -9.7308 -20.14541 -4.6458 +13.60601 -4.6458 -13.60601 25.2423 23.7155 20.2827 0.2617 1.6573 11.2040 7.9291 + 1.16301 7.9291 - 1.16301 9.4251 eigad - 1.0o+02 * -0.2506 + 4.57971 -0.2506 - 4.57971 -1.5289 + 2.84751 -1.5289 - 2.84751 1.2909 + 1.98591 1.2909 - 1.98591 0.3793 + 0.57401 204 Table C.2.ll Output for algebraic 58 MVar 0.3793 — 0.57401 0.4803 0.4434 0.3465 0.3267 0.0002 0.0276 0.1269 0.0644 + 0.00321 0.0644 - 0.00321 0.1056 detjj a -2.8610e+30 detd - 2.4987e+20 detda - -1.1450e+10 detad = 5.3121e+27 condjj - 1.6936e+06 condd - 254.1842 condda = 8.0650o+05 condad - 1.3904o+06 vtt - 0.0840 1.0000 -0.1177 1.0000 -0.3165 1.0000 0.2176 -0.6403 ~0.4592 eigtt - 1.0o+03 * -1.6489 -0.1360 -0.2318 205 Table C.2.ll (cont’s) eigjj - -26.2875 -26.2875 48.1655 45.0979 39.3536 -12.4227 -12.4227 25.2130 23.6544 +52.37351 -52.37351 +24.93881 -24.93881 -9.8454 +20.10841 -9.8454 -20.10841 20.1756 -15.7892 -16.6178 -16.5999 -4.1059 -4.1059 -3.9591 -3.9591 -4.5119 -4.5119 10.9728 9.1219 + + + 15. 15. 14. 14. 13. 13. 16471 16471 98751 98751 35531 35531 7.8968 + 1.10461 7.8968 - 1.10461 -0.6017 -0.6017 1.1104 -0.2791 -0.2791 -0.1423 -0.1423 0.5411 0.3323 0.0511 -0.0664 -0.0664 -0.0108 -0.0108 oigd - 1.50081 1.50081 1.21871 1.21871 0.95721 0.95721 0.60411 0.60411 0.61591 0.61591 -12.2575 +25.16791 -12.2575 48.1654 45.0982 39.3571 -9.6699 -9.6699 -4.5715 -4.5715 25.2226 23.6938 20.1811 0.1268 1.6488 11.1513 + + -25.16791 20.08041 20.08041 13.52201 13.52201 7.8780 + 1.14101 7.8780 - 1.14101 9.2936 eigad - 1.0o+02 * -0.2622 + 4.47381 -0.2622 - 4.47381 -1.4301 + 2.73341 -1.4301 - 2.73341 1.2042 + 1.86041 1.2042 - 1.86041 0.3803 + 0.57901 0.3803 - 0.57901 206 Table C.2. 12 Output for algebraic 60 MVar 0.4793 0.4427 0.3460 0.3261 0.0002 0.0247 0.1267 0.0640 + 0.00251 0.0640 - 0.00251 0.1040 detjj = -2.4574e+30 detd = 1.1225e+20 detda = -2.1893e+10 detad - 3.5268e+27 condjj = 1.6994e+06 condd - 526.8583 condda - 8.0890e+05 condad = 1.2736e+06 Vtt '- 0.0937 1.0000 -0.1321 1.0000 -0.1928 1.0000 0.2258 -0.6827 -0.4044 eigtt. - 1.0o+03 * -2.8355 -0.1323 -0.2186 207 Table C.2.12 (cont’d) 208 C.3 Dynamic/Algebraic Voltage Instability Table C.3.1 to Table C.3.6 are the converged equilibrium point of the simulation in section 4.3. Table C.3.7 to C.3.12 are the output from the eigenvalue program. Table C.3.13 is the case where field current is saturated. 209 Table C.3.1 Equilibrium point (dynamic/algebraic 10 MVar) 1 9 BUS TEST SYSTEM(NO LINE DROP COMPENSATION) PAGE NUMBER 7 Dl-2.0, D2-2.0, D3-2.0, KA-25, RC-0.0, xc-o.o DATE 5/ 2/90 LOADZ-SO, REAL POWER LOAD - regular, no load at LOADI and LOAD3 TIME 22.30.13 GENERATOR ANGLE IN DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 ~49.9995 10.4736 45.1464 19.3101 GENERATOR FIELD VOLTAGE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0706 1.5243 1.2870 GEN. FLUX LINKAGE (Q-AXIS) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.9861 0.8227 0.9992 GEN. ELECTRICAL POWER - (MW) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 82.5440 123.0011 31.9911 GENERATOR EXCITER SATURATION -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.0431 0.0146 0.0388 GENERATOR MEGAVAR OUTPUT ~NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -11.2795 -1.7768 13.7459 GENERATOR FIELD CURRENT ~NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.9983 1.3775 1.1563 GENERATOR TERMINAL CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.8331 1.2301 0.3482 GEN. TERM. CURR. ANGLE DEGREES-NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 7.6718 2.6678 -17.5473 A-C BUS VOLTAGE MAGNITUDE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0000 1.0000 0.9998 1.0078 1.0041 0.9955 0.9924 0.9850 0.9899 A-C BUS VOLTAGE ANGLE DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -0.1094 1.8402 5.7049 -2.8605 -2.5509 -4.9643 -5.0428 -6.1330 -5.3102 Table C.3.2 Equilibrium point (dynamic/algebraic 50 MVar) 1 9 BUS TEST SYSTEM(NO LINE DROP COMPENSATION) PAGE NUMBER 7 D1-2.0, D2-2.0, D3-2.0, KA-ZS, RC-0.0, XC-0.0 DATE 5/ 3/90 torm3-50, REAL POHER LOAD - rogular TIME 7.45.22 GENERATOR ANGLE IN DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 6.6330 41.2437 12.3914 GENERATOR FIELD VOLTAGE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0719 1.5263 1.6919 GEN. FLUX LINKAGE (O-AXIS) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.9864 0.8236 1.0554 GEN. ELECTRICAL POWER - (MW) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 82.5348 123.0010 32.0007 GENERATOR EXCITER SATURATION -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.0431 0.0147 0.0984 GENERATOR MEGAVAR OUTPUT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -10.9684 -1.4629 53.2513 GENERATOR FIELD CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.9994 1.3791 1.4841 GENERATOR TERMINAL CURRENT ~NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.8326 1.2301 0.6236 GEN. TERM. CURR. ANGLE DEGREES-NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 3.6286 -1.3086 -57.0667 A-C BUS VOLTAGE MAGNITUDE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0000 1.0000 0.9962 1.0076 1.0039 0.9947 0.9922 0.9845 0.9894 A-C BUS VOLTAGE ANGLE DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -3.9413 -1.9900 1.9319 -6.6926 -6.3820 -8.7883 -8.8752 -9.9605 -9.1373 210 Table C.3.3 Equilibrium point (dynamic/a1 gebraic 100 MVar) 1 9 BUS TEST SYSTEM(NO LINE DROP COMPENSATION) PAGE NUMBER 7 Dl-2.0, D2-2.0, D3-2.0, KA-25, Rc-0.0, XC-0.0 DATE 5/ 3/90 torm3-100, REAL POHER LOAD - rogular TIME 8.14.13 GENERATOR ANGLE IN DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -22.0389 12.2170 -18.1092 GENERATOR FIELD VOLTAGE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0789 1.5374 2.1685 GEN. FLUX LINKAGE (Q-AXIS) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.9877 0.8289 1.1037 GEN. ELECTRICAL POWER - (MW) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 82.6329 123.0139 31.9007 GENERATOR EXCITER SATURATION -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.0433 0.0149 0.3166 GENERATOR MEGAVAR OUTPUT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -9.2917 0.2288 100.6401 GENERATOR FIELD CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0056 1.3881 1.8765 GENERATOR TERMINAL CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.8316 1.2301 1.0810 GEN. TERM. CURR. ANGLE DEGREES-NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 ~49.9995 -26.1684 -30.7394 -98.8884 A-C BUS VOLTAGE MAGNITUDE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0000 1.0000 0.9765 1.0066 1.0029 0.9908 0.9912 0.9818 0.9868 A-C BUS VOLTAGE ANGLE DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -32.5838 -30.6328 -26.4567 -35.3410 -35.0296 ~37.4042 -37.5278 -38.5924 -37.7648 Table C.3.4 Equilibrium point (dynamic/algebraic 110 MVar) l 9 BUS TEST SYSTEM(NO LINE DROP COMPENSATION) PAGE NUMBER 7 D1-2.0, D2-2.0, D3-2.0, KA-25, RC-0.0, XC-0.0 DATE 5/ 4/90 torm3-110, REAL POWER LOAD - regular TIME 9.39.11 GENERATOR ANGLE IN DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -41.6573 -7.5377 -37.8437 GENERATOR FIELD VOLTAGE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0818 1.5420 2.2561 GEN. FLUX LINKAGE (Q-AXIS) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.9882 0.8310 1.1094 GEN. ELECTRICAL POWER - (MW) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 82.6391 123.0167 31.8963 GENERATOR EXCITER SATURATION -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.0433 0.0149 0.3948 GENERATOR MEGAVAR OUTPUT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -8.6070 0.9225 109.6247 GENERATOR FIELD CURRENT ~NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0081 1.3917 1.9561 GENERATOR TERMINAL CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.8309 1.2302 1.1786 GEN. TERM. CURR. ANGLE DEGREES-NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -46.2405 -50.6628 -119.7442 A-C BUS VOLTAGE MAGNITUDE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0000 1.0000 0.9686 1.0062 1.0024 0.9892 0.9908 0.9807 0.9858 A-C BUS VOLTAGE ANGLE DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -52.1862 -50.2332 -45.9373 -54.9448 -54.6319 -56.9922 -57.1326 -58.1876 -57.3587 211 Table C.3.5 Equilibrium point (dynamic/a1 gebraic 120 MVar) 1 9 BUS TEST SYSTEM(NO LINE DROP COMPENSATION) PAGE NUMBER 7 D1-2.0, DZ-2.0, D3-2.0, KA-25, RC-0.0, XC-0.0 DATE 5/ 3/90 term3-120, REAL POWER LOAD - regular TIME 8.41.23 GENERATOR ANGLE IN DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -81.3496 -47.4235 -77.6249 GENERATOR FIELD VOLTAGE ~NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0857 1.5481 2.3393 GEN. FLUX LINKAGE (Q-AXIS) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.9889 0.8337 1.1130 GEN. ELECTRICAL POWER - (MW) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 82.7351 123.0285 31.7987 GENERATOR EXCITER SATURATION -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.0434 0.0150 0.4876 GENERATOR MEGAVAR OUTPUT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -7.6940 1.8459 118.2906 GENERATOR FIELD CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0115 1.3967 2.0380 GENERATOR TERMINAL CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.8310 1.2304 1.2783 GEN. TERM. CURR. ANGLE DEGREES-NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -86.5553 -90.7772 -160.4985 A-C BUS VOLTAGE MAGNITUDE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0000 1.0000 0.9580 1.0057 1.0019 0.9872 0.9902 0.9793 0.9844 A-C BUS VOLTAGE ANGLE DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -91.8678 -89.9176 -85.4970 -94.6310 -94.3190 -96.6641 -96.8218 ~97.8672 -97.0354 Table C.3.6 Equilibrium point (dynamic/algebraic 135 MVar) 1 9 BUS TEST SYSTEM(NO LINE DROP COMPENSATION) PAGE NUMBER 7 D1-2.0, DZ-2.0, D3-2.0, KA-25, RC-0.0, XC-0.0 DATE 5/ 4/90 torm3-135, REAL POWER LOAD - regular TIME 9.59.34 GENERATOR ANGLE IN DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -378.5388 -346.0558 -372.2955 GENERATOR FIELD VOLTAGE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.1185 1.6006 2.3393 GEN. FLUX LINKAGE (O-AXIS) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.9949 0.8562 1.0562 GEN. ELECTRICAL POWER - (MW) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 82.7741 123.0988 31.8060 GENERATOR EXCITER SATURATION -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.0442 0.0159 0.4876 GENERATOR MEGAVAR OUTPUT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.2228 9.9667 123.5647 GENERATOR FIELD CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0401 1.4404 2.2099 GENERATOR TERMINAL CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.8278 1.2344 1.4658 GEN. TERM. CURR. ANGLE DEGREES-NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 ~29.0325 -31.5311 ~96.8258 A-C BUS VOLTAGE MAGNITUDE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.9999 1.0001 0.8692 1.0013 0.9973 0.9694 0.9855 0.9671 0.9726 A-C BUS VOLTAGE ANGLE DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49. 9995 -28. 8783 ~26. 9001 -20. 9484 -31. 6552 -31. 3244 -33. 5055 -33. 8568 -34. 7921 -33. 9472 eigjj - -26.2805 +52.36981 -26.2805 -52.36981 50.9883 47.7837 40.6818 -12.9438 +26.14311 -12.9438 -26.14311 -11.2838 +21.82191 -11.2838 -21.82191 25.1042 22.0818 20.3194 -15.8215 -16.5987 -16.6157 -5.6622 +14.96751 -5.6622 -14.96751 -3.9241 +15.00831 -3.9241 -15.00831 -4.0810 +14.93151 -4.0810 -14.93151 12.2866 + 0.92571 12.2866 - 0.92571 8.0250 + 1.11061 8.0250 - 1.11061 1.9009 -0.5395 + 1.55761 -o.5395 - 1.55761 -0.1966 + 1.23371 -0.1966 - 1.23371 0.9130 -0.1214 + 0.91711 -0.1214 - 0.91711 0.2535 0.0692 Table C.3.7 Output for dynamic/algebraic 10 MVar 0.0153 + 0.58151 0.0153 - 0.58151 -0.0418 + 0.58571 -0.0418 - 0.58571 eigda - -26.4462 +52.18041 -26.4462 -52.18041 -3.8967 +14.90331 -3.8967 -14.90331 -3.9132 +14.99011 -3.9132 -14.99011 -15.1523 —16.4814 -16.5558 —0.3525 + 1.73381 -0.3525 - 1.73381 -0.1340 + 1.06601 -0.1340 - 1.06601 —0.0342 + 0.97361 -0.0342 - 0.97361 -0.0835 + 0.07051 -0.0835 - 0.07051 -0.0487 + 0.59291 -0.0487 - 0.59291 -0.0108 + 0.59141 -0.0108 - 0.59141 vda - Columns 1 through 4 0.0000 - 0.00001 0.0000 0.0000 + 0.00001 0.0000 0.0000 + 0.00001 0.0000 0.0000 + 0.00001 0.0000 Ill-4- 212 0.00001 0.0000 0.00001 0.0000 0.00001 0.0000 0.00001 0.0000 |+I+ eigd - 50.9884 47.7842 40.6840 -12.8059 -12.8059 -11.121 -11.121 -5.7241 -5.7241 25.1119 22.0913 20.3424 12.4410 12.4410 8 8 + + +26.30811 -26.30811 +21.84681 -21.84681 14.91011 14.91011 1.05341 1.05341 8.0781 + 1.16571 8.0781 - 1.16571 1.3527 1.9337 eigad = 1.0e+03 * -0.0535 + 3.89611 -0.0535 - 3.89611 -0.2283 + 0.51191 -0.2283 - 0.51191 0.2267 + 0.44871 0.2267 - 0.44871 0.0388 + 0.05571 0.0388 - 0.05571 0.0511 0.0452 0.0357 0.0321 0.0000 0.0146 0.0058 0.0070 + 0.00041 0.0070 - 0.00041 0.0125 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 0.00001 0.00001 0.00001 213 Table C.3.7 (cont’d) 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 -0.0002 + 0.00031 -0.0002 - 0.00031 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 -0.0013 + 0.00081 -0.0013 - 0.00081 0.0000 + 0.00001 0.0000 - 0.00001 -0.0005 + 0.00031 -0.0005 - 0.00031 0.0001 + 0.00001 0.0001 - 0.00001 0.0000 + 0.00021 0.0000 - 0.00021 0.0000 + 0.00001 0.0000 - 0.00001 -0.0001 + 0.00071 -0.0001 - 0.00071 0.0000 + 0.00001 0.0000 - 0.00001 -0.0001 + 0.00061 -0.0001 - 0.00061 -0.0048 - 0.01051 -0.0048 + 0.01051 0.0000 - 0.00021 0.0000 + 0.00021 0.0000 - 0.00001 0.0000 + 0.00001 -0.0021 - 0.03061 -0.0021 + 0.03061 0.0000 - 0.00001 0.0000 + 0.00001 -0.0023 - 0.02491 -0.0023 + 0.02491 1.0000 + 0.00001 1.0000 - 0.00001 0.0054 — 0.00001 0.0054 + 0.00001 0.0000 + 0.00031 0.0000 - 0.00031 1.0000 + 0.00001 1.0000 — 0.00001 0.0000 + 0.00031 0.0000 0.00031 0.8175 - 0.01771 0.8175 + 0.01771 -0.0764 - 0.15371 -0.0764 + 0.15371 -0.0008 - 0.00351 -0.0008 + 0.00351 0.0000 - 0.00001 0.0000 + 0.00001 -0.0288 - 0.11061 -0.0288 + 0.11061 0.0000 - 0.00001 0.0000 + 0.00001 -0.0214 - 0.07561 —0.0214 + 0.07561 Columns 5 through 8 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 0.0000 0.0000 + 0.00001 0.0000 - 0.00001 -0.0001 0.0000 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 0.0000 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 0.0000 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 0.0000 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 0.0000 0.0000 - 0.00001 0.0000 + 0.00001 0.0049 -0.0003 0.0011 - 0.00061 0.0011 + 0.00061 0.0004 0.0013 -0.0006 + 0.00031 -0.0006 0.0000 + 0.00001 0.0000 - 0.0001 - 0.00031 0.0001 + -0.0001 + 0.00031 -0.0001 0.0000 — 0.00001 0.0000 + 0.0017 + 0.02461 0.0017 - -0.0021 - 0.03031 -0.0021 0.0000 + 0.00001 0.0000 - -0.8089 - 0.00051 -0.8089 1.0000 + 0.00001 1.0000 - 0.0000 - 0.00001 0.0000 + 0.0230 + 0.08891 0.0230 - -0.0240 - 0.09251 -0.0240 - 0.00031 0.0001 0.0004 0.00001 0.0454 -0.0025 0.00031 0.0222 0.0566 - 0.00031 0.0190 0.0352 0.00001 -0.0524 0.0030 0.02461 -0.0144 -0.0336 + 0.03031 -0.0123 -0.0209 0.00001 1.0000 -0.0623 + 0.00051 0.3852 1.0000 0.00001 0.3307 0.6233 0.00001 -0.6839 0.0390 0.08891 -0.0448 -0.1069 + 0.09251 -0.0323 -0.0560 Columns 9 through 12 0.0000 0.0060 + 0.00091 0.0060 - 0.00091 -0.0656 + 0.01761 0.0000 -0.0115 - 0.00611 -0.0115 + 0.00611 0.2602 + 0.02051 0.0000 -0.0039 + 0.00301 -0.0039 - 0.00301 -0.0883 - 0.04021 0.0000 -0.0002 - 0.00341 -0.0002 + 0.00341 0.0238 + 0.05851 0.0000 -0.0021 + 0.00711 -0.0021 - 0.00711 -0.0113 - 0.24271 0.0000 0.0021 + 0.00181 0.0021 - 0.00181 -0.0269 + 0.08621 0.0000 -0.0194 - 0.05441 -0.0194 + 0.05441 0.0353 + 0.02971 -0.0009 -0.0312 + 0.00731 -0.0312 - 0.00731 -0.0100 - 0.17861 0.0006 -0.0135 + 0.00661 -0.0135 - 0.00661 -0.0718 + 0.00941 -0.0003 -0.0265 - 0.04151 -0.0265 + 0.04151 0.0160 + 0.00571 -0.0367 -0.0278 - 0.01101 -0.0278 + 0.01101 -0.0215 - 0.09841 0.0565 -0.0229 - 0.00861 -0.0229 + 0.00861 -0.0528 - 0.00151 0.0003 0.0264 + 0.03981 0.0264 - 0.03981 -0.0155 - 0.00551 0.0217 0.0272 + 0.00921 0.0272 - 0.00921 0.0235 + 0.09261 -0.0334 0.0224 + 0.00711 0.0224 - 0.00711 0.0501 - 0.00021 -0.0068 0.0177 + 0.17341 0.0177 - 0.17341 -0.0547 - 0.02081 -0.6487 0.1156 + 0.15181 0.1156 - 0.15181 -0.0640 + 0.60601 1.0000 0.0969 + 0.12331 0.0969 - 0.12331 0.3005 + 0.11131 0.0043 1.0000 + 0.00001 1.0000 - 0.00001 -0.3384 + 0.38741 0.0690 0.1269 - 0.14161 0.1269 + 0.14161 1.0000 + 0.00001 -0.0895 0.0859 - 0.09941 0.0859 + 0.09941 0.1069 - 0.42891 Columns 13 through 16 -0.0656 - 0.01761 -0.0358 - 0.04081 -0.0358 + 0.04081 -0.0835 + 0.07051 0.2602 - 0.02051 -0.0099 + 0.17681 -0.0099 - 0.17681 -0.0761 + 0.06691 -0.0883 + 0.04021 0.0830 - 0.06731 0.0830 + 0.06731 -0.0815 + 0.07031 0.0238 - 0.05851 -0.0405 + 0.03821 -0.0405 - 0.03821 1.0000 - 0.00001 -0.0113 + 0.24271 0.1818 + 0.00381 0.1818 - 0.00381 0.9269 - 0.01891 -0.0269 - 0.08621 -0.0721 - 0.08271 -0.0721 + 0.08271 0.9849 - 0.00991 0.0353 - 0.02971 0.0078 + 0.05321 0.0078 - 0.05321 0.0003 - 0.00131 -0.0100 + 0.17861 0.0694 - 0.08121 0.0694 + 0.08121 0.0121 + 0.00511 -0.0718 - 0.00941 -0.0386 - 0.12541 -0.0386 + 0.12541 -0.0012 + 0.00021 214 Table C.3.7 (cont’d) 0.0160 - 0.00571 0.0090 + 0.01361 0.0090 - 0.01361 0.0006 - 0.00071 -0.0215 + 0.09841 0.0419 - 0.05931 0.0419 + 0.05931 -0.0007 + 0.00041 -0.0528 + 0.00151 -0.0281 - 0.10281 -0.0281 + 0.10281 0.0001 — 0.00011 -0.0155 + 0.00551 -0.0086 - 0.01311 -0.0086 + 0.01311 -0.0007 + 0.00071 0.0235 - 0.09261 -0.0379 + 0.05721 -0.0379 - 0.05721 0.0007 - 0.00041 0.0501 + 0.00021 0.0295 + 0.09631 0.0295 - 0.09631 -0.0001 + 0.00011 —0.0547 + 0.02081 -0.0363 - 0.04841 -0.0363 + 0.04841 0.0122 + 0.00221 -0.0640 - 0.60601 -0.3494 + 0.28021 -0.3494 - 0.28021 0.0033 - 0.00261 0.3005 - 0.11131 -0.0125 + 0.65791 -0.0125 - 0.65791 -0.0002 + 0.00041 -0.3384 - 0.38741 -0.5452 + 0.09511 -0.5452 - 0.09511 0.2750 + 0.00501 1.0000 - 0.00001 0.5146 + 0.62451 0.5146 - 0.62451 -0.0770 - 0.01511 0.1069 + 0.42891 1.0000 - 0.00001 1.0000 + 0.00001 0.0067 - 0.00121 Columns 17 through 20 -0.0835 - 0.07051 0.0104 - 0.18951 0.0104 + 0.18951 0.0524 - 0.03261 -0.0761 - 0.06691 0.1035 + 0.13401 0.1035 - 0.13401 -0.1307 + 0.18531 -0.0815 - 0.07031 -0.1859 + 0.38901 -0.1859 - 0.38901 -0.0492 - 0.12701 1.0000 + 0.00001 -0.3188 + 0.00871 -0.3188 - 0.00871 -0.0535 - 0.08951 0.9269 + 0.01891 0.2103 - 0.19181 0.2103 + 0.19181 0.3092 + 0.22671 0.9849 + 0.00991 0.6773 + 0.25791 0.6773 - 0.25791 -0.2161 + 0.07931 0.0003 + 0.00131 0.0273 + 0.02261 0.0273 - 0.02261 -0.0042 + 0.01691 0.0121 - 0.00511 -0.1317 + 0.09191 -0.1317 - 0.09191 -0.1388 - 0.20711 -0.0012 - 0.00021 -0.0585 - 0.12061 -0.0585 + 0.12061 0.0799 + 0.03681 0.0006 + 0.00071 0.0041 + 0.00511 0.0041 - 0.00511 -0.0003 + 0.00271 —0.0007 - 0.00041 -0.0345 - 0.00301 -0.0345 + 0.00301 -0.0113 - 0.05461 0.0001 + 0.00011 -0.0073 — 0.06741 -0.0073 + 0.06741 0.0384 + 0.01581 -0.0007 — 0.00071 -0.0039 - 0.00501 -0.0039 + 0.00501 0.0004 - 0.00261 0.0007 + 0.00041 0.0326 + 0.00231 0.0326 - 0.00231 0.0115 + 0.05121 -0.0001 - 0.00011 0.0080 + 0.06351 0.0080 - 0.06351 -0.0364 - 0.01431 0.0122 - 0.00221 -0.0236 - 0.01091 -0.0236 + 0.01091 -0.0036 - 0.01111 0.0033 + 0.00261 0.1984 + 0.04921 0.1984 - 0.04921 0.0182 + 0.33621 -0.0002 - 0.00041 -0.0223 + 0.40071 -0.0223 - 0.40071 -0.2137 - 0.13111 0.2750 - 0.00501 -0.3053 + 0.15031 -0.3053 - 0.15031 -0.1326 - 0.06061 -0.0770 + 0.01511 0.1186 - 0.59401 0.1186 + 0.59401 1.0000 + 0.00001 0.0067 + 0.00121 1.0000 + 0.00001 1.0000 - 0.00001 -0.3513 + 0.51911 Column 21 0.0524 + 0.03261 -0.1307 - 0.18531 -0.0492 + 0.12701 -0.0535 + 0.08951 0.3092 - 0.22671 -0.2161 - 0.07931 -0.0042 - 0.01691 -0.1388 + 0.20711 0.0799 - 0.03681 -0.0003 - 0.00271 -0.0113 + 0.05461 0.0384 - 0.01581 0.0004 + 0.00261 0.0115 - 0.05121 -0.0364 + 0.01431 -0.0036 + 0.01111 0.0182 - 0.33621 -0.2137 + 0.13111 -0.1326 + 0.06061 1.0000 - 0.00001 -0.3513 - 0.51911 detjj - ~1.6291e+31 detd - 3.9878e+21 detda - -4.0852e+09 detad - 2.2963e+31 condjj - 1.4060e+06 condd = 60.7094 condda - 6.7747e+05 condad - 1.3290e+08 vtt - 1.0000 0.0442 0.0069 -0.6955 1.0000 1.0000 -0.3321 0.6017 -0.4808 eigtt - 1.0o+03 * -0.1470 -1.1285 -0.5760 215 Table C.3.7 (cont’d) eigjj - -26.2812 ~26.2812 50.9551 47.7583 40.6669 -12.934 -12.934 -11.321 -11.321 25.0956 22.0582 20.4673 -15.8216 -16.596 -16.613 -5.6908 -5.6908 -3.9536 -3.9536 -4.0934 -4.0934 12.2942 12.2942 3 3 4 4 2 1 216 Table 03.8 Output for dynamic/algebraic 50 MVar +52.37011 -52.37011 +26.10201 -26.10201 +21.66351 -21.66351 +14.93541 -14.93541 +15.00791 -15.00791 +14.94071 -14.94071 4. 0.94431 0.94431 8.0423 + 1.05811 8.0423 - 1.05811 1.9557 -0.5412 -0.5412 0.9595 -0.2020 -0.2020 0.2503 0.0659 + + -0.0450 + -0.0450 - 0.62861 0.0250 + 0.59121 0.0250 - 0.59121 -0.1225 + 0.93001 —0.1225 - 0.93001 eigda - 1.56171 1.56171 1.23831 1.23831 0.62861 -26.4473 +52.18061 ~26.447 3 -3.9097 +14.89961 -3.9097 -14.89961 -3.9365 +14.98971 -14.98971 -3.9365 -16.4719 -16.551 -15.148 -0.3374 -0.3374 -0.1189 -0.1189 -0.0402 -0.0402 -0.0835 -0.0835 -0.0646 -0.0646 -0.0257 -0.0257 vda - 2 2 l+l+l+l+|+l+ 1.73601 1.73601 1.06891 1.06891 1.00361 1.00361 0.06731 0.06731 0.63311 0.63311 0.60151 0.60151 -52.18061 Columns 1 through 4 0.0000 0.0000 0.0000 0.0000 0.0000 I +-+-+| 0.00001 0.00001 0.00001 0.00001 0.00001 0.0000 0.0000 0.0000 0.0000 0.0000 +lll+ 0.00001 0.00001 0.00001 0.00001 0.00001 0.0000 0.0000 0.0000 0.0000 0.0000 eigd - 50.9553 47.7590 40.6690 -12.7930 +26.26971 -12.7930 -26.26971 -11.1508 +21.68571 -21.68571 +14.88221 -l4.88221 -11.150 -5.7533 -5.7533 25.1034 22.0691 20.4957 12.4480 12.4480 8 + 1.06871 - 1.06871 8.0977 + 1.12121 8.0977 - 1.12121 1.3188 2.0642 eigad - 1.0e+03 * -0.0529 + 2.74391 -0.0529 - 2.74391 -0.0701 + 0.42581 -0.0701 - 0.42581 0.0663 + 0.22361 0.0663 - 0.22361 0.0412 + 0.06121 0.0412 - 0.06121 0.0511 0.0451 0.0357 0.0317 0.0000 0.0146 0.0058 0.0070 + 0.00041 0.0070 - 0.00041 0.0125 0.00001 0.0000 - 0.00001 0.00001 0.0000 + 0.00001 0.00001 0.0000 - 0.00001 0.00001 0.0000 + 0.00001 0.00001 0.0000 - 0.00001 217 Table C.3.8 (cont’d) 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 -0.0002 + 0.00031 -0.0002 - 0.00031 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 -0.0013 + 0.00081 -0.0013 - 0.00081 0.0000 + 0.00001 0.0000 - 0.00001 -0.0003 + 0.00041 -0.0003 - 0.00041 0.0001 + 0.00001 0.0001 - 0.00001 0.0000 + 0.00021 0.0000 - 0.00021 0.0000 + 0.00001 0.0000 - 0.00001 -0.0001 + 0.00081 -0.0001 - 0.00081 0.0000 + 0.00001 0.0000 - 0.00001 0.0001 + 0.00061 0.0001 - 0.00061 -0.0048 - 0.01051 -0.0048 + 0.01051 -0.0001 - 0.00021 -0.0001 + 0.00021 0.0000 - 0.00001 0.0000 + 0.00001 -0.0021 - 0.03061 -0.0021 + 0.03061 0.0000 - 0.00001 0.0000 + 0.00001 -0.0103 - 0.02121 -0.0103 + 0.02121 1.0000 - 0.00001 1.0000 + 0.00001 0.0055 - 0.00081 0.0055 + 0.00081 0.0000 + 0.00031 0.0000 - 0.00031 1.0000 + 0.00001 1.0000 - 0.00001 0.0000 + 0.00031 0.0000 - 0.00031 0.7114 - 0.29261 0.7114 + 0.29261 -0.0764 - 0.15371 -0.0764 + 0.15371 -0.0013 - 0.00341 -0.0013 + 0.00341 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 Columns 5 through 8 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 0.0007 - 0.00091 0.0007 + 0.00091 -0.0006 + 0.00031 -0.0006 0.0000 + 0.00001 0.0000 - -0.0001 - 0.00031 -0.0001 -0.0001 + 0.00041 -0.0001 0.0000 - 0.00001 0.0000 + 0.0096 + 0.02061 0.0096 - -0.0020 - 0.03041 -0.0020 0.0006 + 0.00091 0.0006 - -0.6976 + 0.26621 -0.6976 1.0000 + 0.00001 1.0000 - 0.0005 - 0.00051 0.0005 + 0.0493 + 0.06901 0.0493 - -0.0237 - 0.09261 -0.0237 Columns 9 through 12 0.0000 0.0060 + 0.00111 0.0060 - 0.00111 -0.0288 - 0.11051 -0.0288 + 0.11051 -0.0442 - 0.05931 -0.0442 + 0.05931 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0003 0.0000 0.0013 -0.0010 - 0.00031 0.0004 0.0006 0.00001 -0.0029 -0.0002 + 0.00031 0.0566 -0.0430 - 0.00041 0.0431 0.0566 0.00001 0.02061 0.0034 0.0002 -0.0336 0.0255 + 0.03041 -0.0256 -0.0335 0.00091 -0.0716 -0.0047 - 0.26621 1.0000 -0.7610 0.00001 0.00051 0.06901 + 0.09261 0.7600 1.0000 0.0449 0.0029 -0.1069 0.0810 -0.0687 -0.0899 -0.0622 + 0.01581 -0.0120 - 0.00641 -0.0120 + 0.00641 0.2487 + 0.03771 - 0.00291 -0.0873 - 0.05381 0.00351 0.0210 + 0.05591 - 0.00741 0.0093 - 0.23371 0.00151 -0.0407 + 0.08621 + 0.05451 0.0429 + 0.03321 - 0.00801 -0.0099 - 0.16901 - 0.00731 -0.0828 + 0.00111 + 0.04111 0.0188 + 0.00591 + 0.01001 -0.0227 - 0.09771 + 0.00711 -0.0697 - 0.01311 0.0264 - 0.03941 -0.0182 - 0.00571 0.0279 - 0.00821 0.0246 + 0.09191 -0.0001 0.0000 -0.0033 + 0.00291 -0.0033 0.0000 0.0000 - 0.00351 0.0000 + 0.0000 -0.0022 + 0.00741 -0.0022 0.0000 0.0019 + 0.00151 0.0019 - 0.0049 -0.0188 - 0.05451 -0.0188 0.0004 -0.0315 + 0.00801 -0.0315 0.0001 -0.0135 + 0.00731 -0.0135 0.0454 -0.0266 - 0.04111 -0.0266 0.0223 -0.0286 - 0.01001 -0.0286 0.0198 -0.0253 - 0.00711 -0.0253 -0.0524 0.0264 + 0.03941 -0.0144 0.0279 + 0.00821 -0.0128 0.0245 + 0.00561 0.3870 0.1244 + 0.14911 0.1244 - 0.14911 0.0245 - 0.00561 0.0663 + 0.01051 1.0000 0.0193 + 0.17361 0.0193 - 0.17361 -0.0650 - 0.02181 -0.0554 + 0.60761 0.3425 0.1197 + 0.11671 0.1197 - 0.11671 0.3888 + 0.18231 1.0000 + 0.00001 1.0000 - 0.00001 0.1240 - 0.14871 0.1240 + 0.14871 1.0000 + 0.00001 0.0837 - 0.11361 0.0837 + 0.11361 0.2413 - 0.54311 13 through 16 -0.6841 -0.0450 -0.0337 Columns -0.0622 -0.0828 - 0.00111 -0.0420 - 0.11481 -0.0420 + 0.11481 - 0.01581 -0.0377 - 0.03791 -0.3819 + 0.46121 -0.0377 + 0.03791 -0.0835 + 0.06731 0.2487 - 0.03771 0.0093 + 0.18291 0.0093 - 0.18291 -0.0762 + 0.06401 -0.0873 + 0.05381 0.0690 - 0.07931 0.0690 + 0.07931 -0.0818 + 0.06721 0.0210 - 0.05591 -0.0362 + 0.03901 -0.0362 - 0.03901 1.0000 + 0.00001 0.0093 + 0.23371 0.1816 - 0.01661 0.1816 + 0.01661 0.9275 - 0.01811 -0.0407 - 0.08621 -0.0817 - 0.06541 -0.0817 + 0.06541 0.9873 - 0.00911 0.0429 - 0.03321 0.0081 + 0.05741 0.0081 - 0.05741 0.0004 - 0.00121 -0.0099 + 0.16901 0.0774 - 0.08511 0.0774 + 0.08511 0.0123 + 0.00491 -0.0006 + 0.00011 0.0188 - 0.00591 0.0099 + 0.01541 0.0099 - 0.01541 0.0006 - 0.00061 218 Table C.3.8 (cont’d) -0.0227 + 0.09771 0.0452 - 0.06411 0.0452 + 0.06411 -0.0007 + 0.00041 -0.0697 + 0.01311 -0.0299 - 0.10581 -0.0299 + 0.10581 0.0001 - 0.00011 -0.0182 + 0.00571 -0.0096 - 0.01491 -0.0096 + 0.01491 -0.0007 + 0.00061 0.0246 - 0.09191 -0.0409 + 0.06191 -0.0409 - 0.06191 0.0007 - 0.00041 0.0663 - 0.01051 0.0309 + 0.09901 0.0309 - 0.09901 -0.0001 + 0.00011 -0.0650 + 0.02181 -0.0388 - 0.05551 -0.0388 + 0.05551 0.0122 + 0.00211 -0.0554 - 0.60761 -0.3809 + 0.29951 -0.3809 - 0.29951 0.0033 - 0.00251 0.3888 - 0.18231 0.0385 + 0.67831 0.0385 - 0.67831 0.0001 + 0.00031 -0.3819 - 0.46121 -0.5992 + 0.09401 -0.5992 - 0.09401 0.2744 + 0.00481 1.0000 - 0.00001 0.5372 + 0.65761 0.5372 - 0.65761 -0.0779 - 0.01441 0.2413 + 0.54311 1.0000 - 0.00001 1.0000 + 0.00001 0.0065 - 0.00061 Columns 17 through 20 -0.0835 - 0.06731 -0.0025 - 0.30551 -0.0025 + 0.30551 0.0187 - 0.04411 -0.0762 - 0.06401 -0.0130 + 0.18161 -0.0130 - 0.18161 -0.1192 + 0.17771 -0.0818 - 0.06721 -0.0646 + 0.63311 -0.0646 - 0.63311 0.0280 - 0.08161 1.0000 - 0.00001 -0.4771 + 0.05261 -0.4771 - 0.05261 -0.0718 - 0.03421 0.9275 + 0.01811 0.2860 - 0.00871 0.2860 + 0.00871 0.2864 + 0.21051 0.9873 + 0.00911 1.0000 + 0.00001 1.0000 - 0.00001 -0.1334 - 0.05231 0.0004 + 0.00121 0.0340 + 0.02941 0.0340 - 0.02941 0.0018 + 0.01371 0.0123 - 0.00491 -0.0710 - 0.01681 -0.0710 + 0.01681 -0.1269 - 0.20651 -0.0006 - 0.00011 -0.0599 - 0.06701 -0.0599 + 0.06701 0.0372 + 0.04291 0.0006 + 0.00061 0.0063 + 0.00861 0.0063 - 0.00861 0.0009 + 0.00211 -0.0007 - 0.00041 -0.0104 - 0.02421 -0.0104 + 0.02421 —0.0129 - 0.05501 0.0001 + 0.00011 -0.0166 - 0.05891 -0.0166 + 0.05891 0.0223 + 0.01741 -0.0007 - 0.00061 -0.0059 - 0.00841 -0.0059 + 0.00841 -0.0008 - 0.00211 0.0007 + 0.00041 0.0102 + 0.02261 0.0102 - 0.02261 0.0131 + 0.05151 -0.0001 - 0.00011 0.0162 + 0.05531 0.0162 - 0.05531 -0.0211 - 0.01611 0.0122 - 0.00211 -0.0352 - 0.02051 -0.0352 + 0.02051 -0.0072 - 0.00701 0.0033 + 0.00251 0.0362 + 0.15071 0.0362 - 0.15071 0.0269 + 0.34191 0.0001 - 0.00031 0.0771 + 0.35181 0.0771 - 0.35181 -0.1333 - 0.11221 0.2744 - 0.00481 -0.4755 + 0.20581 -0.4755 - 0.20581 -0.1217 + 0.00631 -0.0779 + 0.01441 0.4100 - 0.12841 0.4100 + 0.12841 1.0000 + 0.00001 0.0065 + 0.00061 0.8294 - 0.13741 0.8294 + 0.13741 -0.3293 + 0.26071 Column 21 0.0187 + 0.04411 -0.1192 - 0.17771 0.0280 + 0.08161 -0.0718 + 0.03421 0.2864 - 0.21051 -0.1334 + 0.05231 0.0018 - 0.01371 -0.1269 + 0.20651 0.0372 - 0.04291 0.0009 - 0.00211 -0.0129 + 0.05501 0.0223 - 0.01741 -0.0008 + 0.00211 0.0131 - 0.05151 -0.0211 + 0.01611 -0.0072 + 0.00701 0.0269 - 0.34191 -0.1333 + 0.11221 -0.1217 - 0.00631 1.0000 - 0.00001 -0.3293 - 0.26071 detjj - -2.0475e+31 detd - 4.1246e+21 detda - -4.9640e+09 219 Table C.3.8 (cont’d) detad c 1.6198e+30 condjj - 1.4589e+06 condd - 56.8584 condda - 7.0325e+05 condad - 6.8366e+07 vtt - -0.6029 0.0333 -0.6322 1.0000 1.0000 -0.2536 -0.4735 0.0371 1.0000 eigtt - -171.1429 -852.9462 -114.6348 Table C.3.9 Output for dynamic/algebraic 100 MVar eiqjj - -26.2854 +52.37211 -26.2854 -52.37211 50.7730 47.6031 40.5914 -12.8643 +25.96491 -12.8643 -25.96491 -11.1901 +21.07341 -11.1901 -21.07341 25.0576 21.9705 20.2247 -15.8216 -16.6106 -16.5913 -5.6948 +14.78691 -5.6948 -14.78691 -4.1666 +14.99441 -4.1666 -14.99441 -4.0085 +14.98881 -4.0085 -14.98881 12.2683 + 0.96041 12.2683 - 0.96041 8.0234 + 0.96851 8.0234 - 0.96851 2.0079 —0.5451 + 1.56701 -0.5451 - 1.56701 0.9982 -0.2075 + 1.24491 -0.2075 - 1.24491 -0.1431 + 0.96321 -0.1431 - 0.96321 0.2478 0.0634 -0.0341 + 0.66271 -0.0341 - 0.66271 0.0228 + 0.60121 0.0228 - 0.60121 eigda - -26.4522 +52.18191 -26.4522 -52.18191 -3.9206 -3.9206 -4.0319 -4.0319 -16.4550 -16.5434 -15.1372 -0.3148 -0.3148 -0.0353 -0.0353 -0.1296 -0.1296 -0.0835 -0.0835 -0.0664 -0.0664 -0.0265 -0.0265 vda - +14.91771 -14.91771 +14.97501 -14.97501 1.74241 1.74241 1.10191 1.10191 1.03231 1.03231 0.06491 0.06491 0.66481 0.66481 0.61281 0.61281 I+l+l+l+l+l+ Columns 1 through 4 0.0000 0.0000 0.0000 0.0000 0.0000 |+++l 0.00001 0.00001 0.00001 0.00001 0.00001 0.0000 0.0000 0.0000 0.0000 0.0000 +lll+ 0.00001 0.00001 0.00001 0.00001 0.00001 220 0.0000 0.0000 0.0000 0.0000 0.0000 eigd - 50.7732 47.6038 40.5936 -12.719 -12.719 -11.013 -11.013 -5.7709 -5.7709 25.0651 21.9803 20.2582 1.2702 2.1924 12.4206 12.4206 1 1 7 7 + + +26.14111 -26.14111 +21.08741 -21.08741 14.74821 14.74821 1.08001 1.08001 8.0821 + 1.04621 8.0821 - 1.04621 eigad - 1.0o+03 -0.0519 -0.0519 -0.0438 I + 4. 1.35441 1.35441 0.44691 -0.0438 - 0.44691 0.0029 + 0.08531 0.0029 - 0.08531 0.0784 + 0.06351 0.0784 - 0.06351 0.0509 0.0449 0.0356 0.0304 0.0000 0.0124 0.0058 0.0070 + 0.00031 0.0070 - 0.00031 0.0145 0.00001 0.0000 - 0.00001 0.00001 0.0000 + 0.00001 0.00001 0.0000 - 0.00001 0.00001 0.0000 + 0.00001 0.00001 0.0000 - 0.00001 0.0000 - 0.00001 0.0000 + -0.0002 + 0.00031 -0.0002 0.0000 + 0.00001 0.0000 - + 0.00001 0.0000 - + 0.00001 0.0001 - + 0.00001 0.0000 - + 0.00001 0.0000 - 0.0000 0.0001 0.0000 0.0000 1.0000 0.0000 + + 0.0000 0.0000 -0.0048 - 0.01051 -0.0048 0.0000 - 0.00001 0.0000 - 0.00001 1.0000 + 0.00001 0.0000 + 0.00031 0.0000 + 0.00031 0.0000 -0.0763 - 0.15371 -0.0763 0.0000 - 0.00001 0.0000 + 0.0000 - 0.00001 0.0000 + Columns 5 through 8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 - 0.00061 + + + + + 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 + + 4. -0.0005 + 0.00031 -0.0005 -0.0001 + 0.00011 -0.0001 -0.0002 + 0.00011 -0.0002 -0.0002 + 0.00061 -0.0002 0.0000 - 0.00011 0.0000 + 0.0109 + 0.00511 0.0109 - -0.0018 - 0.03051 -0.0018 0.0026 + 0.00131 0.0026 - -0.1929 + 0.34351 -0.1929 - 0.34351 1.0000 -0.9490 1.0000 - 0.00001 1.0000 + 0.0004 - 0.00181 0.0004 + 0.0433 + 0.01111 0.0433 - -0.0230 - 0.09311 -0.0230 Columns 9 through 12 221 Table C.3.9 (cont’d) 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 - 0.00031 0.0000 + 0.00001 0.0000 - 0.00001 0.00001 -0.0013 + 0.00081 -0.0013 - 0.00081 0.00001 0.0000 + 0.00031 0.0000 - 0.00031 0.00001 0.0000 + 0.00021 0.0000 - 0.00021 0.00001 0.0000 + 0.00071 0.0000 - 0.00071 0.00001 0.0002 + 0.00041 0.0002 - 0.00041 + 0.01051 -0.0001 - 0.00021 -0.0001 + 0.00021 0.00001 -0.0021 - 0.03051 -0.0021 + 0.03051 0.00001 -0.0119 - 0.00541 -0.0119 + 0.00541 0.00001 0.0042 - 0.00131 0.0042 + 0.00131 0.00031 1.0000 - 0.00001 1.0000 + 0.00001 0.00031 0.1954 - 0.37971 0.1954 + 0.37971 + 0.15371 -0.0014 - 0.00251 -0.0014 + 0.00251 0.00001 -0.0288 - 0.11041 -0.0288 + 0.11041 -0.0400 - 0.00981 -0.0400 + 0.00981 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00061 - 0.00031 - 0.00011 - 0.00011 - 0.00061 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0004 0.0000 0.0013 -0.0012 0.0005 0.0006 -0.0036 0.0000 0.0566 0.0570 0.0571 -0.0537 0.00011 0.0042 0.0000 0.00511 -0.0337 0.0318 + 0.03051 -0.0342 -0.0340 0.00131 -0.0888 0.0002 0.00001 0.9977 1.0000 0.00181 0.0558 0.01111 -0.1070 0.1010 + 0.09311 -0.0916 -0.0913 -0.0001 0.0000 0.0060 + 0.00141 0.0060 - 0.00141 -0.0540 + 0.01291 -0.0001 -0.0124 - 0.00671 -0.0124 + 0.00671 0.1636 + 0.05061 -0.0030 + 0.00241 -0.0030 - 0.00241 -0.0269 - 0.06631 0.0002 - 0.00351 0.0002 + 0.0000 0.0000 0.0000 0.0000 0.0049 0.0004 0.0001 0.0455 0.0226 0.0220 -0.0525 0.0266 -0.0146 0.0292 -0.0143 0.0288 -0.6850 1.0000 + 0.00001 1.0000 - 0.00001 0.00351 0.0133 + 0.04861 - 0.00761 0.0411 - 0.14981 0 + .00141 -0.0593 + 0.02631 0.05441 0.0605 + 0.04441 - 0.00901 -0.0306 - 0.14831 - 0.00821 -0.0937 - 0.05011 0.04061 0.0270 + 0.00471 0.00861 -0.0301 - 0.09541 0.00541 -0.0955 - 0.06891 + + + 0.0266 - 0.03891 -0.0260 - 0.00441 0.0292 - 0.00671 0.0316 + 0.08921 -0.0025 + 0.00761 -0.0025 0.0017 + 0.00141 0.0017 - -0.0179 - 0.05441 -0.0179 -0.0319 + 0.00901 -0.0319 -0.0141 + 0.00821 -0.0141 -0.0267 - 0.04061 -0.0267 -0.0300 - 0.00861 -0.0300 -0.0300 - 0.00541 -0.0300 + 0.03891 + 0.00671 + 0.00391 0.0288 - 0.00391 0.0908 + 0.06281 1.0000 0.0225 + 0.17421 0.0225 - 0.17421 -0.1000 - 0.02051 0.3924 0.1381 + 0.14661 0.1381 - 0.14661 -0.0071 + 0.62641 0.3770 0.1648 + 0.10071 0.1648 - 0.10071 0.5863 + 0.47951 -0.4969 + 0.68011 -0.0457 0.1212 - 0.15981 0.1212 + 0.15981 1.0000 - 0.00001 -0.0377 0.0866 - 0.13931 0.0866 + 0.13931 0.7969 - 0.57531 Columns 13 through 16 -0.0540 - 0.01291 -0.0246 - 0.05641 -0.0246 + 0.05641 -0.0835 + 0.06491 0.1636 - 0.05061 -0.0485 + 0.29331 -0.0485 - 0.29331 -0.0763 + 0.06171 -0.0269 + 0.06631 0.0838 - 0.14581 0.0838 + 0.14581 -0.0820 + 0.06491 0.0133 - 0.04861 -0.0509 + 0.03021 -0.0509 - 0.03021 1.0000 + 0.00001 0.0411 + 0.14981 0.2856 + 0.01111 0.2856 - 0.01111 0.9276 - 0.01741 -0.0593 - 0.02631 -0.1491 - 0.06251 -0.1491 + 0.06251 0.9885 - 0.00861 0.0605 - 0.04441 -0.0158 + 0.05701 -0.0158 - 0.05701 0.0004 - 0.00121 -0.0306 + 0.14831 0.1486 - 0.01061 0.1486 + 0.01061 0.0124 + 0.00461 -0.0937 + 0.05011 -0.0440 - 0.11511 -0.0440 + 0.11511 -0.0001 + 0.00001 0.0270 - 0.00471 0.0004 + 0.02041 0.0004 - 0.02041 0.0006 - 0.00061 -0.0301 + 0.09541 0.0892 - 0.03351 0.0892 + 0.03351 -0.0007 + 0.00041 222 Table C.3.9 (cont’d) -0.0955 + 0.06891 -0.0257 - 0.11411 -0.0257 + 0.11411 0.0000 - 0.00011 -0.0260 + 0.00441 -0.0004 - 0.01971 -0.0004 + 0.01971 -0.0007 + 0.00061 0.0316 - 0.08921 -0.0835 + 0.03441 -0.0835 - 0.03441 0.0007 - 0.00041 0.0908 - 0.06281 0.0258 + 0.10701 0.0258 - 0.10701 -0.0001 + 0.00011 -0.1000 + 0.02051 -0.0026 - 0.07021 -0.0026 + 0.07021 0.0125 + 0.00201 -0.0071 - 0.62641 -0.5738 + 0.02931 -0.5738 - 0.02931 0.0032 - 0.00241 0.5863 - 0.47951 0.1343 + 0.69761 0.1343 - 0.69761 0.0011 + 0.00031 -0.4969 - 0.68011 -0.5963 - 0.21151 -0.5963 + 0.21151 0.2738 + 0.00461 1.0000 + 0.00001 0.1495 + 0.96231 0.1495 - 0.96231 -0.0781 - 0.01351 0.7969 + 0.57531 1.0000 + 0.00001 1.0000 - 0.00001 0.0067 - 0.00021 Columns 17 through 20 -0.0835 - 0.06491 0.0416 - 0.31961 0.0416 + 0.31961 0.0085 - 0.05661 -0.0763 - 0.06171 -0.1275 + 0.16001 -0.1275 - 0.16001 -0.1135 + 0.17951 -0.0820 - 0.06491 -0.0664 + 0.66481 -0.0664 - 0.66481 0.0474 - 0.04741 1.0000 - 0.00001 -0.4822 - 0.01451 -0.4822 + 0.01451 -0.0916 - 0.01781 0.9276 + 0.01741 0.2572 + 0.16611 0.2572 - 0.16611 0.2844 + 0.19761 0.9885 + 0.00861 1.0000 - 0.00001 1.0000 + 0.00001 -0.0738 - 0.08061 0.0004 + 0.00121 0.0227 + 0.03231 0.0227 - 0.03231 0.0047 + 0.01411 0.0124 - 0.00461 0.0234 - 0.08981 0.0234 + 0.08981 -0.1212 - 0.20161 -0.0001 - 0.00001 -0.0251 - 0.03941 -0.0251 + 0.03941 0.0205 + 0.03291 0.0006 + 0.00061 0.0046 + 0.01041 0.0046 - 0.01041 0.0015 + 0.00221 -0.0007 - 0.00041 0.0233 - 0.03091 0.0233 + 0.03091 -0.0134 - 0.05591 0.0000 + 0.00011 -0.0056 - 0.04441 -0.0056 + 0.04441 0.0162 + 0.01381 -0.0007 - 0.00061 -0.0043 - 0.01011 -0.0043 + 0.01011 -0.0014 - 0.00221 0.0007 + 0.00041 -0.0215 + 0.02961 -0.0215 - 0.02961 0.0135 + 0.05241 -0.0001 - 0.00011 0.0050 + 0.04171 0.0050 - 0.04171 -0.0150 - 0.01301 0.0125 - 0.00201 -0.0311 - 0.03011 -0.0311 + 0.03011 -0.0098 - 0.00631 0.0032 + 0.00241 -0.1684 + 0.15451 -0.1684 - 0.15451 0.0280 + 0.34841 0.0011 - 0.00031 0.0996 + 0.26411 0.0996 - 0.26411 -0.1335 - 0.06331 0.2738 - 0.00461 -0.5212 + 0.09711 -0.5212 - 0.09711 -0.1351 + 0.03491 -0.0781 + 0.01351 0.4363 + 0.41681 0.4363 - 0.41681 1.0000 + 0.00001 0.0067 + 0.00021 0.5844 + 0.00861 0.5844 - 0.00861 -0.2542 + 0.17511 Column 21 0.0085 + 0.05661 —0.1135 - 0.17951 0.0474 + 0.04741 -0.0916 + 0.01781 0.2844 - 0.19761 -0.0738 + 0.08061 0.0047 - 0.01411 -0.1212 + 0.20161 0.0205 - 0.03291 0.0015 - 0.00221 -0.0134 + 0.05591 0.0162 - 0.01381 -0.0014 + 0.00221 0.0135 - 0.05241 —0.0150 + 0.01301 -0.0098 + 0.00631 0.0280 - 0.34841 -0.1335 + 0.06331 -0.1351 - 0.03491 1.0000 - 0.00001 -0.2542 - 0.17511 detjj - -2.3623e+31 detd - 3.7768e+21 detda - -6.2547e+09 detad - 223 Table C.3.9 (cont’d) 9.6026e+28 condjj 8 1.5003e+06 condd - 53.1719 condda n 7.2324e+05 condad - 1.7124e+07 vtt - -0.9917 0.0350 -0.2730 1.0000 1.0000 -0.6262 -0.0815 0.0113 1.0000 eigtt - -152.8097 -815.9868 -42.9767 81911 - -26.2872 -26.2872 50.6961 47.5341 40.5593 -12.8396 -12.8396 -11.1220 -11.1220 25.0429 21.9409 20.0631 ~15.8216 -16.6102 -16.5899 -5.6908 -5.6908 -4.2012 -4.2012 -4.0142 -4.0142 12.2529 12.2529 8.0054 + 8.0054 - 2.0161 -0.5464 -0.5464 1.0013 -0.2087 -0.2087 -0.1508 -0.1508 0.2476 0.0631 -0.0321 -0.0321 0.0216 + 0.0216 - eigda - Table C.3.10 Output for dynamic/algebraic 110 MVar +52.37291 -52.37291 +25.92081 -25.92081 +20.86261 -20.86261 +14.72731 -14.72731 +15.01271 -15.01271 +14.98341 -14.98341 + 0.95921 - 0.95921 0.94461 0.94461 1.56821 1.56821 1.24671 1.24671 0.97241 0.97241 + 0.66751 - 0.66751 0.60301 0.60301 -26.4541 +52.18251 -26.4541 -52.18251 -3.9191 -3.9191 -4.0663 -4.0663 -16.4498 -16.5413 -15.1337 -0.3087 -0.3087 -0.0293 -0.0293 -0.1404 -0.1404 -0.0835 -0.0835 -0.0662 -0.0662 -0.0257 -0.0257 vds - +14.91971 -14.91971 +14.97421 -14.97421 1.74471 1.74471 1.12271 1.12271 1.02851 1.02851 0.06461 0.06461 0.66911 0.66911 0.61501 0.61501 O+I+I+I+I+l+ Columns 1 through 4 0.0000 0.0000 0.0000 0.0000 0.0000 n + +-+I 0.00001 0.00001 0.00001 0.00001 0.00001 0.0000 0.0000 0.0000 0.0000 0.0000 +lll+ 224 0.00001 0.0000 0.00001 0.0000 0.00001 0.0000 0.00001 0.0000 0.00001 0.0000 eigd - 50.6964 47.5348 40.5615 -12.6936 -12.6936 -10.9456 -10.9456 -5.7720 -5.7720 25.0503 21.9500 20.0974 1.2561 2.2121 12.4047 12.4047 +26.09941 -26.09941 +20.87441 -20.87441 +14.69531 -14.69531 4. 1.07801 1.07801 8.0648 + 1.02611 8.0648 - 1.02611 eigad - 1.0o+03 -0.0517 -0.0517 -0.0426 -0.0426 * + + 1.17971 1.17971 0.44711 0.44711 0.0828 + 0.05521 0.0828 - 0.05521 -0.0027 + 0.07741 -0.0027 - 0.07741 0.0508 0.0448 0.0355 0.0300 0.0000 0.0124 0.0058 0.0069 + 0.00021 0.0069 - 0.00021 0.0145 0.00001 0.00001 0.00001 0.00001 0.00001 0 0 0 0 0 .0000 .0000 .0000 .0000 .0000 |+I+I 0.00001 0.00001 0.00001 0.00001 0.00001 0.0000 - 0.00001 0.0000 + -0. 0002 + 0. 00031 -0. 0002 0.0000 + 0.00001 0.0000 0.0000 + 0.00001 0.0000 - 0.0001 + 0.00001 0.0001 - 0.0000 + 0.00001 0.0000 - 0.0000 + 0.00001 0.0000 -0. 0048 - 0. 01051 -0. 0048 0.0000 - 0.00001 0.0000 + 0.0000 - 0.00001 0.0000 + 1.0000 - 0.00001 1.0000 + 0.0000 + 0.00031 0.0000 - 0.0000 + 0.00041 0.0000 - -0.0763 - 0.15371 -0.0763 0.0000 - 0.00001 0.0000 + 0.0000 - 0.00001 0.0000 + Columns 5 through 8 225 Table C.3.10 (cont’d) 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 0.00031 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 - 0.0000 + 0.00001 0.0000 - 0.0000 - 0.00001 0.0000 + 0.0000 + 0.00001 0.0000 - 0.0000 - 0.00001 0.0000 + 0.0000 + 0.00001 0.0000 - 0.0000 + 0.00001 0.0000 - 0.0000 - 0.00051 0.0000 + -0.0005 + 0.00031 -0.0005 -0.0001 + 0.00011 -0.0001 -0.0002 + 0.00011 —0.0002 -0.0002 + 0.00061 -0.0002 0.0000 - 0.00011 0.0000 + 0.0093 + 0.00361 0.0093 - -0.0017 - 0.03051 -0.0017 0.0029 + 0.00111 0.0029 - -0.1408 + 0.29361 -0.l408 1.0000 - 0.00001 1.0000 + 0.0003 - 0.00201 0.0003 + 0.0363 + 0.00671 0.0363 - -0.0228 - 0.09321 -0.0228 Columns 9 through 12 0.0000 0.0060 + 0.00151 0. -0.0001 -0.0125 - 0.00671 0.00001 -0.0013 + 0.00081 -0.0013 - 0.00081 0.00001 0.0000 + 0.00021 0.0000 - 0.00021 0.00001 0.0000 + 0.00021 0.0000 - 0.00021 0.00001 0.0000 + 0.00071 0.0000 - 0.00071 0.00001 0.0002 + 0.00041 0.0002 - 0.00041 + 0.01051 -0.0001 - 0.00021 -0.0001 + 0.00021 0.00001 -0.0021 - 0.03051 -0.0021 + 0.03051 0.00001 -0.0103 - 0.00391 -0.0103 + 0.00391 0.00001 0.0041 - 0.00111 0.0041 + 0.00111 0.00031 1.0000 + 0.00001 1.0000 - 0.00001 0.00041 0.1426 - 0.33021 0.1426 + 0.33021 + 0.15371 -0.0013 - 0.00251 -0.0013 + 0.00251 0.00001 -0.0288 - 0.11041 -0.0288 + 0.11041 0.00001 -0.0342 - 0.00611 -0.0342 + 0.00611 0.00001 0.0000 0.0000 0.00001 0.0000 0.0001 0.00001 0.0000 0.0000 0.00001 0.0000 0.0000 0.00001 0.0000 0.0000 0.00001 0.0000 0.0000 0.00001 -0.0004 0.0000 0.00051 0.0012 0.0013 - 0.00031 0.0005 -0.0006 - 0.00011 -0.0036 -0.0001 - 0.00011 0.0528 0.0566 - 0.00061 0.0573 -0.0571 0.00011 0.0042 0.0001 0.00361 -0.0314 -0.0335 + 0.03051 -0.0344 0.0341 0.00111 -0.0879 -0.0018 0.29361 0.9332 1.0000 0.00001 1.0000 -0.9975 0.00201 0.0553 0.0011 0.00671 -0.0999 -0.1065 + 0.09321 -0.0922 0.0914 0060 - 0.00151 -0.0519 - 0.01521 -0.0125 + 0.00671 0.1119 + 0.11301 0.0000 -0.0029 + 0.00221 -0.0029 - 0.00221 0.0118 - 0.06051 0.0000 0.0002 - 0.00351 0.0002 + 0.00351 -0.0123 + 0.04651 0.0000 -0.0025 + 0.00761 -0.0025 - 0.00761 0.0980 - 0.10221 0.0000 0.0015 + 0.00141 0.0015 - 0.00141 -0.0541 - 0.00911 0.0049 -0.0177 — 0.05441 -0.0177 + 0.05441 0.0294 + 0.07271 0.0004 -0.0320 + 0.00931 -0.0320 - 0.00931 0.0471 - 0.14461 0.0001 -0.0145 + 0.00841 -0.0145 - 0.00841 -0.0468 - 0.09301 0.0455 -0.0268 - 0.04051 -0.0268 + 0.04051 0.0222 + 0.01931 0.0227 -0.0304 - 0.00821 -0.0304 + 0.00821 0.0227 - 0.09861 0.0228 -0.0316 - 0.00511 -0.0316 + 0.00511 -0.0400 - 0.11501 -0.0525 0.0266 + 0.03881 0.0266 - 0.03881 -0.0215 — 0.01851 -0.0147 0.0295 + 0.00641 0.0295 - 0.00641 -0.0182 + 0.09391 -0.0149 0.0302 + 0.00361 0.0302 - 0.00361 0.0393 + 0.10691 1.0000 0.0234 + 0.17451 0.0234 - 0.17451 -0.0803 - 0.07571 0.3940 0.1419 + 0.14611 0.1419 - 0.14611 -0.3317 + 0.54281 0.3893 0.1791 + 0.09561 0.1791 - 0.09561 0.2500 + 0.75871 -0.6854 1.0000 + 0.00001 1.0000 - 0.00001 -0.8280 + 0.33591 -0.0459 0.1206 - 0.16281 0.1206 + 0.16281 0.8532 + 0.51511 -0.0391 0.0887 - 0.14731 0.0887 + 0.14731 1.0000 - 0.00001 Columns 13 through 16 -0.0519 + 0.01521 -0.0177 - 0.05381 -0.0177 + 0.05381 -0.0835 + 0.06461 0.1119 - 0.11301 -0.0766 + 0.28371 -0.0766 - 0.28371 -0.0764 + 0.06141 0.0118 + 0.06051 0.0948 - 0.14551 0.0948 + 0.14551 -0.0820 + 0.06461 -0.0123 - 0.04651 -0.0490 + 0.02391 -0.0490 - 0.02391 1.0000 - 0.00001 0.0980 + 0.10221 0.2807 + 0.03621 0.2807 - 0.03621 0.9276 - 0.01731 -0.0541 + 0.00911 -0.1513 - 0.07151 -0.1513 + 0.07151 0.9888 - 0.00851 0.0294 - 0.07271 -0.0168 + 0.05191 -0.0168 - 0.05191 0.0004 - 0.00121 0.0471 + 0.14461 0.1383 + 0.01251 0.1383 - 0.01251 0.0124 + 0.00461 -0.0468 + 0.09301 -0.0418 - 0.11431 -0.0418 + 0.11431 0.0000 + 0.00001 0.0222 - 0.01931 -0.0009 + 0.01881 -0.0009 - 0.01881 0.0006 - 0.00061 226 Table C.3.10 (cont’d) 0.0227 + 0.09861 0.0856 - 0.02081 0.0856 + 0.02081 -0.0007 + 0.00041 -0.0400 + 0.11501 '0.0250 - 0.11461 -0.0250 + 0.11461 0.0000 - 0.00011 -0.0215 + 0.01851 0.0009 - 0.01811 0.0009 + 0.01811 -0.0007 + 0.00061 -0.0182 - 0.09391 -0.0805 + 0.02231 -0.0805 - 0.02231 0.0007 - 0.00041 0.0393 - 0.10691 0.0247 + 0.10751 0.0247 - 0.10751 -0.0001 + 0.00011 -0.0803 + 0.07571 0.0017 - 0.06431 0.0017 + 0.06431 0.0125 + 0.00201 -0.3317 - 0.54281 -0.5277 - 0.03701 -0.5277 + 0.03701 0.0032 - 0.00241 0.2500 - 0.75871 0.1750 + 0.69501 0.1750 - 0.69501 0.0015 + 0.00031 -0.8280 - 0.33591 -0.5362 - 0.22671 -0.5362 + 0.22671 0.2736 + 0.00461 0.8532 - 0.51511 0.0390 + 0.89811 0.0390 - 0.89811 -0.0780 - 0.01331 1.0000 + 0.00001 1.0000 - 0.00001 1.0000 + 0.00001 0.0069 - 0.00011 Columns 17 through 20 -0.0835 - 0.06461 0.0471 - 0.32081 0.0471 + 0.32081 0.0076 - 0.05861 -0.0764 - 0.06141 -0.l416 + 0.15541 -0.1416 - 0.15541 -0.ll32 + 0.18061 -0.0820 - 0.06461 -0.0662 + 0.66911 -0.0662 - 0.66911 0.0492 - 0.04291 1.0000 + 0.00001 -0.4816 - 0.02271 -0.4816 + 0.02271 -0.0946 - 0.01631 0.9276 + 0.01731 0.2507 + 0.18681 0.2507 - 0.18681 0.2855 + 0.19601 0.9888 + 0.00851 1.0000 - 0.00001 1.0000 + 0.00001 -0.0663 - 0.08271 0.0004 + 0.00121 0.0210 + 0.03251 0.0210 - 0.03251 0.0051 + 0.01431 0.0124 - 0.00461 0.0369 - 0.09661 0.0369 + 0.09661 -0.1202 - 0.20031 0.0000 - 0.00001 -0.0199 - 0.03651 -0.0199 + 0.03651 0.0187 + 0.03111 0.0006 + 0.00061 0.0044 + 0.01061 0.0044 - 0.01061 0.0016 + 0.00231 -0.0007 - 0.00041 0.0281 - 0.03111 0.0281 + 0.03111 -0.0134 - 0.05611 0.0000 + 0.00011 -0.0039 - 0.04321 -0.0039 + 0.04321 0.0156 + 0.01311 -0.0007 - 0.00061 -0.0041 - 0.01031 -0.0041 + 0.01031 -0.0015 - 0.00221 0.0007 + 0.00041 -0.0259 + 0.02991 -0.0259 - 0.02991 0.0135 + 0.05261 -0.0001 - 0.00011 0.0032 + 0.04041 0.0032 - 0.04041 -0.0143 - 0.01241 0.0125 - 0.00201 -0.0305 - 0.03121 -0.0305 + 0.03121 -0.0102 - 0.00631 0.0032 + 0.00241 -0.1963 + 0.15071 -0.1963 - 0.15071 0.0278 + 0.34971 0.0015 - 0.00031 0.1112 + 0.26071 0.1112 - 0.26071 -0.l386 - 0.05281 0.2736 - 0.00461 -0.5249 + 0.08141 -0.5249 - 0.08141 -0.l381 + 0.03821 -0.0780 + 0.01331 0.4276 + 0.48851 0.4276 - 0.48851 1.0000 + 0.00001 0.0069 + 0.00011 0.5601 + 0.03291 0.5601 - 0.03291 -0.2420 + 0.16631 Column 21 0.0076 + 0.05861 -0.1132 - 0.18061 0.0492 + 0.04291 -0.0946 + 0.01631 0.2855 — 0.19601 -0.0663 + 0.08271 0.0051 - 0.01431 -0.1202 + 0.20031 0.0187 - 0.03111 0.0016 - 0.00231 -0.0134 + 0.05611 0.0156 - 0.01311 -0.0015 + 0.00221 0.0135 - 0.05261 -0.0143 + 0.01241 -0.0102 + 0.00631 0.0278 - 0.34971 -0.1386 + 0.05281 -0.1381 - 0.03821 1.0000 - 0.00001 -0.2420 - 0.16631 detjj - -2.3599e+31 detd - 3.5874e+21 detda - -6.5782e+09 detad - 227 Table C.3.10 (cont’d) 5.6399e+28 condjj - 1.5064e+06 condd - 52.5824 condda = 7.2616e+05 condad 8 l.3045e+07 vtt - 1.0000 0.0359 -0.2668 -0.9899 1.0000 -0.6454 0.0652 0.0097 1.0000 eigtt - -151.3925 -803.2924 -36.6830 eiqjj - -26.289 -26.289 50.6033 47.4511 40.5218 -12.808 -12.808 -ll.024 -ll.024 25.0234 21.9046 19.8396 -15.821 -16.609 -16.588 -5.6860 ~5.6860 -4.2433 -4.2433 -4.0178 -4.0178 12.2312 12.2312 5 5 8 8 9 9 5 7 2 228 Table C3. 11 Output for dynamic/algebraic 120 MVar +52.37401 -52.37401 +25.86721 -25.86721 +20.58661 -20.58661 +14.64321 -l4.64321 +15.03291 -15.03291 +14.98021 -14.98021 4. 0.95581 0.95581 7.9789 + 0.91531 7.9789 - 0.91531 2.0248 -O.5480 -0.5480 1.0022 -0.2103 -0.2103 -0.1604 -0.1604 0.2476 0.0626 -0.0304 + -0.0304 - 0.67161 0.0202 + 0.60501 0.0202 - 0.60501 eigda - 1.56951 1.56951 1.24881 1.24881 0.98301 0.98301 0.67161 -26.4567 +52.18341 -26.456 7 -3.9179 +14.92061 -3.9179 -l4.92061 -4.1042 +14.97381 -14.97381 -4.1042 -16.443 -16.538 -15.129 -0.3014 -0.3014 -0.0251 -0.0251 -0.1518 -0.1518 -0.0835 -0.0835 -0.0659 -0.0659 -0.0245 -0.0245 vda - 2 8 6 l+l+l+l+l+l+ 1.74771 1.74771 1.14561 1.14561 1.02591 1.02591 0.06421 0.06421 0.67261 0.67261 0.61751 0.61751 -52.18341 Columns 1 through 4 0.0000 0.0000 0.0000 0.0000 0.0000 l +-+-+I 0.00001 0.00001 0.00001 0.00001 0.00001 0.0000 0.0000 0.0000 0.0000 0.0000 +lll+ 0.00001 0.00001 0.00001 0.00001 0.00001 0.0000 0.0000 0.0000 0.0000 0.0000 +1 +1 + eigd - 50.6035 47.4517 40.5241 -12.6620 -12.6620 -10.8491 -10.8491 -5.7734 -5.7734 25.0307 21.9129 19.8742 1.2392 2.2303 12.3822 12.3822 +14.62151 -14.62151 + 1.07361 1.07361 8.0390 + 1.00141 8.0390 - 1.00141 eigad - 1.0o+03 -0.0514 -0.0514 -0.0417 It 4. + 1.02591 1.02591 0.44651 -0.0417 - 0.44651 0.0854 + 0.04751 0.0854 - 0.04751 -0.0062 + 0.07061 -0.0062 - 0.07061 0.0508 0.0447 0.0355 0.0294 0.0000 0.0124 0.0058 0.0069 + 0.00021 0.0069 - 0.00021 0.0145 0.00001 0.00001 0.00001 0.00001 0.00001 0.0000 0.0000 0.0000 0.0000 0.0000 I+I+I +26.04861 -26.04861 +20.59611 -20.59611 0.00001 0.00001 0.00001 0.00001 0.00001 229 Table C.3.ll (cont’d) 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 -0.0002 + 0.00031 -0.0002 - 0.00031 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 -0.0013 + 0.00081 -0.0013 - 0.00081 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 + 0.00021 0.0000 - 0.00021 0.0001 + 0.00001 0.0001 - 0.00001 0.0000 + 0.00021 0.0000 - 0.00021 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 + 0.00071 0.0000 - 0.00071 0.0000 + 0.00001 0.0000 - 0.00001 0.0002 + 0.00041 0.0002 - 0.00041 -0.0048 - 0.01051 -0.0048 + 0.01051 -0.0001 - 0.00021 -0.0001 + 0.00021 0.0000 - 0.00001 0.0000 + 0.00001 -0.0021 - 0.03051 -0.0021 + 0.03051 0.0000 - 0.00001 0.0000 + 0.00001 -0.0091 - 0.00301 -0.0091 + 0.00301 1.0000 - 0.00001 1.0000 + 0.00001 0.0041 - 0.00111 0.0041 + 0.00111 0.0000 + 0.00031 0.0000 - 0.00031 1.0000 + 0.00001 1.0000 - 0.00001 0.0000 + 0.00041 0.0000 - 0.00041 0.1095 - 0.29141 0.1095 + 0.29141 -0.0763 - 0.15371 -0.0763 + 0.15371 -0.0013 - 0.00241 -0.0013 + 0.00241 0.0000 - 0.00001 0.0000 + 0.00001 -0.0288 - 0.11041 -0.0288 + 0.11041 0.0000 - 0.00001 0.0000 + 0.00001 -0.0298 - 0.00401 -0.0298 + 0.00401 Columns 5 through 8 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 0.0000 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 0.0001 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 0.0000 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 0.0000 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 0.0000 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 0.0000 0.0000 + 0.00001 0.0000 - 0.00001 -0.0004 0.0000 0 0000 - 0.00041 0.0000 + 0.00041 0.0011 0.0013 - .0005 + 0.00031 -0.0005 -0.0001 + 0.00011 -0.0001 -0.0002 + 0.00011 -0.0002 -0.0002 + 0.00061 -0.0002 0.0000 - 0.00011 0.0000 + 0.0079 + 0.00261 0.0079 - -0.0016 - 0.03061 -0.0016 0.0031 + 0.00101 0.0031 - -0.1078 + 0.25301 -0.1078 1.0000 - 0.00001 1.0000 + 0.0001 - 0.00211 0.0001 + 0.0309 + 0.00421 0.0309 - -0.0226 - 0.09341 -0.0226 - 0.00031 0.0005 -0.0005 - 0.00011 -0.0035 -0.0001 - 0.00011 0.0486 0.0566 - 0.00061 0.0575 -0.0538 0.00011 0.0041 0.0002 0.00261 -0.0289 -0.0335 + 0.03061 -0.0346 0.0322 0.00101 -0.0868 -0.0036 - 0.25301 0.8578 1.0000 0.00001 1.0000 -0.9370 0.00211 0.0546 0.0023 0.00421 -0.0919 -0.1065 + 0.09341 -0.0927 0.0863 Columns 9 through 12 0.0000 0.0061 + 0.00161 0.0061 - 0.00161 -0.0505 - 0.00911 -0.0001 -0.0125 - 0.00671 -0.0125 + 0.00671 0.1147 + 0.08661 0.0000 -0.0028 + 0.00191 -0.0028 - 0.00191 0.0080 - 0.05001 0.0000 0.0000 0.0000 0.0003 - 0.00351 0.0003 + -0.0025 + 0.00761 -0.0025 0.0013 + 0.00141 0.0013 - 0.00351 -0.0070 + 0.04421 - 0.00761 0.0734 - 0.10171 0.00141 -0.0438 - 0.00601 0.0049 -0.0174 - 0.05431 -0.0174 + 0.05431 0.0354 + 0.07201 0.0004 -0.0321 + 0.00961 -0.0321 - 0.00961 0.0309 - 0.14501 0.0001 -0.0150 + 0.00861 -0.0150 - 0.00861 -0.0456 - 0.08911 0.0456 -0.0269 - 0.04031 -0.0269 + 0.04031 0.0250 + 0.01841 0.0228 -0.0309 - 0.00781 -0.0309 + 0.00781 0.0134 - 0.09861 0.0238 -0.0336 - 0.00471 -0.0336 + 0.00471 -0.0423 - 0.11661 -0.0525 0.0267 + 0.03861 0.0267 - 0.03861 -0.0241 - 0.01761 —0.0148 0.0300 + 0.00601 0.0300 - 0.00601 -0.0094 + 0.09371 -0.0156 0.0321 + 0.00321 0.0321 - 0.00321 0.0410 + 0.10821 1.0000 0.0246 + 0.17481 0.0246 - 0:17481 -0.0904 - 0.07431 0.3958 0.1465 + 0.14561 0.1465 - 0.14561 -0.2808 + 0.56231 0.4052 0.1971 + 0.08981 0.1971 - 0.08981 0.3066 + 0.77421 -0.6858 1.0000 + 0.00001 1.0000 - 0.00001 -0.8351 + 0.40821 -0.0461 0.1201 - 0.16651 0.1201 + 0.16651 0.8638 + 0.43031 -0.0409 0.0920 - 0.15761 0.0920 + 0.15761 1.0000 - 0.00001 Columns 13 through 16 -0.0505 + 0.00911 -0.0107 - 0.05041 -0.0107 + 0.05041 -0.0835 + 0.06421 0.1147 - 0.08661 -0.1037 + 0.26941 -0.1037 - 0.26941 -0.0764 + 0.06101 0.0080 + 0.05001 0.1050 - 0.14261 0.1050 + 0.14261 -0.0821 + 0.06421 -0.0070 - 0.04421 -0.0466 + 0.01741 -0.0466 - 0.01741 1.0000 - 0.00001 0.0734 + 0.10171 0.2716 + 0.06091 0.2716 - 0.06091 0.9276 - 0.01721 -0.0438 + 0.00601 -0.1508 - 0.08001 -0.1508 + 0.08001 0.9891 - 0.00841 0.0354 - 0.07201 -0.0172 + 0.04661 -0.0172 - 0.04661 0.0004 - 0.00121 0.0309 + 0.14501 0.1256 + 0.03491 0.1256 - 0.03491 0.0124 + 0.00451 -0.0456 + 0.08911 -0.0393 - 0.11321 -0.0393 + 0.11321 0.0000 + 0.00001 0.0250 - 0.01841 -0.0020 + 0.01711 -0.0020 - 0.01711 0.0006 - 0.00061 0.0134 + 0.09861 0.0805 - 0.00821 0.0805 + 0.00821 -0.0007 + 0.00041 230 Table C.3.ll (cont’d) -0.0423 + 0.11661 -0.0244 - 0.11521 -0.0244 + 0.11521 0.0000 - 0.00011 -0.0241 + 0.01761 0.0020 - 0.01651 0.0020 + 0.01651 -0.0007 + 0.00061 -0.0094 - 0.09371 -0.0761 + 0.01011 -0.0761 - 0.01011 0.0007 - 0.00041 0.0410 - 0.10821 0.0237 + 0.10801 0.0237 - 0.10801 -0.0001 + 0.00011 -0.0904 + 0.07431 0.0053 - 0.05811 0.0053 + 0.05811 0.0127 + 0.00201 -0.2808 - 0.56231 -0.4738 - 0.09981 -0.4738 + 0.09981 0.0031 - 0.00241 0.3066 - 0.77421 0.2210 + 0.69331 0.2210 - 0.69331 0.0019 + 0.00031 -0.8351 - 0.40821 -0.4757 - 0.23511 -0.4757 + 0.23511 0.2734 + 0.00461 0.8638 - 0.43031 -0.0688 + 0.82091 -0.0688 - 0.82091 -0.0777 - 0.01311 1.0000 + 0.00001 1.0000 + 0.00001 1.0000 - 0.00001 0.0072 - 0.00011 Columns 17 through 20 -0.0835 - 0.06421 0.0518 - 0.32141 0.0518 + 0.32141 0.0068 - 0.06061 -0.0764 - 0.06101 -0.1538 + 0.15071 -0.1538 - 0.15071 -0.1132 + 0.18211 -0.0821 - 0.06421 -0.0659 + 0.67261 -0.0659 - 0.67261 0.0508 - 0.03891 0.0000 0.00001 -0.0149 - 0.03401 -0.0149 + 0.03401 0.0170 + 0.02941 0.0006 0.00061 0.0041 + 0.01081 0.0041 - 0.01081 0.0017 + 0.00231 -0.0007 - 0.00041 0.0321 - 0.03121 0.0321 + 0.03121 -0.0134 - 0.05641 0.0000 + 0.00011 -0.0022 - 0.04231 -0.0022 + 0.04231 0.0150 + 0.01251 -0.0007 - 0.00061 -0.0038 - 0.01051 -0.0038 + 0.01051 -0.0016 - 0.00221 0.0007 + 0.00041 -0.0297 + 0.03001 -0.0297 - 0.03001 0.0135 + 0.05281 -0.0001 - 0.00011 0.0013 + 0.03951 0.0013 - 0.03951 -0.0136 - 0.01191 0.0127 - 0.00201 -0.0299 - 0.03221 -0.0299 + 0.03221 -0.0106 - 0.00641 0.0031 + 0.00241 -0.2198 + 0.14691 —0.2198 - 0.14691 0.0275 + 0.35111 0.0019 - 0.00031 0.1249 + 0.26201 0.1249 - 0.26201 -0.1443 - 0.04201 0.2734 - 0.00461 -0.5272 + 0.06671 -0.5272 - 0.06671 -0.l415 + 0.04121 -0.0777 + 0.01311 0.4185 + 0.54861 0.4185 - 0.54861 1.0000 - 0.00001 0.0072 + 0.00011 0.5413 + 0.05741 0.5413 - 0.05741 -0.2298 + 0.15841 1.0000 + 0.00001 -0.4808 - 0.02991 -0.4808 + 0.02991 -0.0975 - 0.01491 0.9276 + 0.01721 0.2441 + 0.20471 0.2441 - 0.20471 0.2871 + 0.19481 0.9891 + 0.00841 1.0000 + 0.00001 1.0000 - 0.00001 -0.0596 - 0.08461 0.0004 + 0.00121 0.0194 + 0.03261 0.0194 - 0.03261 0.0055 + 0.01461 0.0124 - 0.00451 0.0482 - 0.10191 0.0482 + 0.10191 -0.ll91 - 0.19881 + Column 21 0.0068 + 0.06061 -0.1132 - 0.18211 0.0508 + 0.03891 -0.0975 + 0.01491 0.2871 - 0.19481 -0.0596 + 0.08461 0.0055 - 0.01461 -0.ll91 + 0.19881 0.0170 - 0.02941 0.0017 - 0.00231 -0.0134 + 0.05641 0.0150 - 0.01251 -0.0016 + 0.00221 0.0135 - 0.05281 -0.0136 + 0.01191 -0.0106 + 0.00641 0.0275 - 0.35111 -0.1443 + 0.04201 -0.1415 - 0.04121 1.0000 + 0.00001 -0.2298 - 0.15841 detjj - -2.3191e+31 detd - 3.3387e+21 detda - -6.9461e+09 detad - 3.30129+28 231 Table C.3.ll (cont’d) condjj - 1.5133e+06 condd = 52.0531 condda a 7.2942e+05 condad - 9.9080e+06 Vtt 3 1.0000 0.0370 -0.2646 ~0.9762 1.0000 -0.6620 0.0539 0.0086 1.0000 eigtt - -149.7845 -786.6969 -31.9402 232 Table C3. 12 Output for dynamic/algebraic 135 MVar (with exciter) -4.0725 -14.91771 -16.5262 -16.3923 -15.0917 -0.2489 + 1.79741 -0.2489 - 1.79741 -0.0627 + 1.26921 -0.0627 - 1.26921 -0.2073 + 1.02841 -0.2073 - 1.02841 -0.0835 + 0.06281 -0.0835 - 0.06281 -0.0673 + 0.65971 -0.0673 - 0.65971 -0.0173 + 0.63021 -0.0173 - 0.63021 vda - Columns 1 through 4 0.0000 - 0.00001 0.0000 0.0000 + 0.00001 0.0000 0.0000 + 0.00001 0.0000 0.0000 + 0.00001 0.0000 0.0000 - 0.00001 0.0000 +lll+ 0.00001 0.0000 0.00001 0.0000 0.00001 0.0000 0.00001 0.0000 0.00001 0.0000 0.8529 + 0.38761 0.8529 - 0.38761 -0.0792 + 0.63331 -0.0792 - 0.63331 0.5003 0.4420 0.3508 0.2733 0.0003 0.0726 0.1229 0.0628 0.1413 0.0576 0.00001 0.0000 0.00001 0.0000 0.00001 0.0000 0.00001 0.0000 0.00001 0.0000 l+l+l e1 d - eigjj - 9 -26.3100 +52.38371 22:33:; -26.3100 -52.3837i -12.4724 +25.73721 49.8143 -12.4724 -25.73721 46.7068 40.1889 40.1863 - -12.6240 +25.54311 -3;3§§§ iiS;23281 -12.6240 -25.54311 -5,7933 +13_37351 24.8738 -5.7988 -13.87351 -10.0765 +18.6717i 24.8799 -10.0765 -18.6717i 21.5507 21.6456 17.6370 —16.6080 1.1086 -15.8202 2,1907 -16.5817 ' -5.7156 +13.80811 1311283 1 82333:: -5.7156 -13.80811 7.7099 + 0.88991 —4.2625 +15.10421 7.7099 - 0.33991 -4.2625 -15.10421 -4.0018 +14.9819i -4.0018 -14.98191 17.6120 12.0060 + 0.85531 12.0060 - 0.85531 7.6532 + 0.79231 7.6532 - 0.79231 2.0066 -0.5653 + 1.57181 -0.5653 - 1.57181 0.9580 -0.2299 + 1.25771 -o.2299 - 1.25771 -0.2269 + 1.04601 -0.2269 - 1.04601 0.2525 0.0611 -0.0376 + 0.66071 -0.0376 - 0.66071 0.0137 + 0.61651 0.0137 - 0.61651 eigad - eigda - 1.0o+02 * -26.4792 +52.19111 -26.4792 -52.19111 -0.4740 + 5.56501 -3.9388 +14.91161 -0.4740 - 5.56501 -3.9388 -14.91161 -0.4325 + 4.27671 -4.0725 +14.91771 -0.4325 - 4.27671 0.00001 0.00001 0.00001 0.00001 0.00001 0.0000 - 0.00001 0.0000 + -0.0002 + 0.00031 -0.0002 0.00001 0.0000 - 0. 0. 0. 0. 0. 0000 0000 0001 0000 0000 ++ + + + 0.00001 0.00001 0.00001 0.00001 0.0000 0.0001 0.0000 0.0000 -0.0048 - 0.01051 -0.0048 0.0000 - 0.00001 0.0000 + 0.0000 - 0.00001 0.0000 + 1.0000 + 0.00001 1.0000 - 0.0000 + 0.00041 0.0000 - 0.0000 + 0.00051 0.0000 - -0.0763 - 0.15381 -0.0763 0.0000 - 0.00001 0.0000 + 0.0000 - 0.00001 0.0000 + Columns 5 through 8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 + + + Table C.3.12 (cont’d) 233 0.00001 0.0000 - 0.00001 0.0000 + - 0.00031 0.0000 + 0.00001 0.0000 -0.0012 + 0.00071 -0.0012 0.0002 + 0.00041 0.0002 - 0.0001 + 0.00021 0.0001 - 0.0000 + 0.00071 0.0000 - 0.0005 + 0.00051 0.0005 - + 0.01051 -0.0001 - 0.00021 -0.0001 + 0.00021 0.00001 -0.0022 - 0.03051 -0.0022 0.00001 -0.0171 - 0.00121 -0.0171 0.00001 0.0043 - 0.00231 0.0043 + 0.00041 1.0000 - 0.00001 1.0000 + 0.00051 0.0639 - 0.55231 0.0639 + + 0.15381 -0.0021 - 0.00241 -0.0021 + 0.00241 0.00001 ~0.0289 - 0.11041 -0.0289 + 0.11041 -0.0534 + 0.00611 -0.0534 - 0.00611 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 + 0.00001 -0.0002 - 0.00051 -0.0006 + 0.00031 -0.0001 + 0.00021 -0.0003 + 0.00021 -0.0003 + 0.00091 -0.0002 -0.0006 -0.0001 -0.0003 -0.0003 0.0001 - 0.00021 0.0001 + 0.0118 + 0.00091 0.0118 - -0.0016 - 0.03071 -0.0016 0.0038 + 0.00161 0.0038 - -0.0607 + 0.38091 -0.0607 1.0000 + 0.00001 1.0000 - 0.0004 - 0.00271 0.0004 + 0.0437 - 0.00471 0.0437 + -0.0226 - 0.09381 -0.0226 + 0.09381 0.0667 -0.0930 Columns 9 through 12 + 0.00051 - 0.00031 - 0.00021 - 0.00021 - 0.00091 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0. 0. 0. 0. 0. 0. 0000 0000 0000 0000 0000 0000 -0.0001 -0.0003 0.00001 - 0.00001 - 0.00071 0.00041 0.00021 0.00071 0.00051 + 0.03051 + 0.00121 0.00231 0.00001 0.55231 0.0013 0.0006 -0.0004 0.0006 -0.0005 -0.0034 0.0566 0.0290 -0.0416 0.0575 0.00021 0.0006 0.0039 0.00091 -0.0335 -0.0173 + 0.03071 0.0249 -0.0348 0.00161 -0.0123 -0.0818 - 0.38091 1.0000 0.5111 0.00001 -0.7236 1.0000 0.00271 0.0077 0.0518 0.00471 -0.1066 -0.0549 0.0000 0.0057 + 0.00271 0.0057 - 0.00271 0.0335 + 0.01041 -0.0001 -0.0126 - 0.00631 -0.0126 + 0.00631 -0.0685 - 0.03461 0.0000 -0.0018 - 0.00121 -0.0018 + 0.00121 -0.0139 + 0.00061 0.0010 - 0.00331 0.0010 + 0.0000 0.0000 0.0000 0.0049 0.0004 0.0003 0.0459 0.0238 0.0337 -0.0528 0.0278 -0.0154 0.0345 -0.0221 0.0528 -0.0025 -0.0005 -0.0151 -0.0336 -0.0219 -0.0281 -0.0357 -0.0554 0.00741 0.00111 0.05261 0.01151 0.00721 0.03931 0.00631 0.00691 + + + + + + + 0.03751 0.00411 0.00431 -0.0025 -0.0005 -0.0151 -0.0336 -0.0219 -0.0281 -0.0357 -0.0554 0.0278 - 0.03751 0.0172 + 0.0345 - 0.00411 0.0247 - 0.0528 - 0.00431 0.0328 - 0.00331 0.0069 - 0.02671 -0.0245 + 0.05521 0.0010 + 0.01091 -0.0056 - 0.07911 + + + + 0.00741 0.00111 0.05261 0.01151 0.00721 0.03931 0.00631 0.00691 -0.0361 -0.0190 -0.0177 -0.0287 -0.0375 + + + + 0.09761 0.07751 0.03201 0.06731 0.11911 0.03071 0.06481 0.11221 1.0000 0.0341 + 0.17971 0.0341 - 0.17971 0.0527 + 0.12691 0.4123 0.1829 + 0.15591 0.1829 - 0.15591 0.3224 - 0.33481 0.5742 0.3350 + 0.14961 0.3350 - 0.14961 0.2787 - 0.77701 -0.6895 1.0000 + 0.00001 1.0000 - 0.00001 1.0000 + 0.00001 -0.0482 0.1288 - 0.19471 0.1288 + 0.19471 -0.4753 - 0.43461 -0.0581 0.1559 - 0.25591 0.1559 + 0.25591 -0.7219 - 0.56151 Columns 13 through 16 0.0335 - 0.01041 0.0211 - 0.02671 0.0211 + 0.02671 -0.0835 + 0.06281 -0.0685 + 0.03461 -0.1993 + 0.13361 -0.1993 - 0.13361 -0.0764 + 0.05961 -0.0139 - 0.00061 0.1286 - 0.07841 0.1286 + 0.07841 -0.0824 + 0.06311 0.0069 + 0.02671 -0.0290 - 0.01471 -0.0290 + 0.01471 1.0000 + 0.00001 -0.0245 - 0.05521 0.1624 + 0.16111 0.1624 - 0.16111 0.9273 - 0.01661 0.0010 - 0.01091 -0.0975 - 0.10541 -0.0975 + 0.10541 0.9930 - 0.00831 -0.0090 + 0.02241 -0.0090 - 0.02241 0.0005 - 0.00111 0.0356 + 0.11411 0.0356 - 0.11411 0.0120 + 0.00411 -0.0205 - 0.11111 -0.0205 + 0.11111 0.0004 - 0.00001 -0.0031 + 0.00861 -0.0031 - 0.00861 0.0006 - 0.00061 0.0347 + 0.04351 0.0347 - 0.04351 -0.0007 + 0.00041 -0.0056 -0.0361 -0.0190 -0.0177 -0.0287 l+ll+ 0.07911 0.09761 0.07751 0.03201 0.06731 234 Table C.3.12 (cont’d) -0.0375 - 0.11911 -0.0197 - 0.11931 -0.0197 + 0.11931 0.0001 - 0.00011 0.0172 - 0.03071 0.0030 - 0.00831 0.0030 + 0.00831 -0.0007 + 0.00061 0.0247 + 0.06481 -0.0342 - 0.04031 -0.0342 + 0.04031 0.0006 - 0.00041 0.0328 + 0.11221 0.0194 + 0.11221 0.0194 - 0.11221 -0.0001 + 0.00011 0.0527 - 0.12691 0.0085 - 0.02861 0.0085 + 0.02861 0.0137 + 0.00191 0.3224 + 0.33481 -0.1158 - 0.30571 -0.1158 + 0.30571 0.0027 - 0.00231 0.2787 + 0.77701 0.1834 + 0.69501 0.1834 - 0.69501 0.0030 + 0.00041 1.0000 - 0.00001 -0.2l72 - 0.16371 -0.2172 + 0.16371 0.2706 + 0.00451 -0.4753 + 0.43461 -0.4745 + 0.28001 -0.4745 - 0.28001 -0.0748 - 0.01161 -0.7219 + 0.56151 1.0000 + 0.00001 1.0000 - 0.00001 0.0112 - 0.00011 Columns 17 through 20 -0.0835 - 0.06281 0.0483 - 0.31981 0.0483 + 0.31981 0.0063 - 0.06241 -0.0764 - 0.05961 -0.1461 + 0.16051 -0.1461 - 0.16051 -0.1195 + 0.19211 -0.0824 - 0.06311 —0.0673 + 0.65971 -0.0673 - 0.65971 0.0577 - 0.04381 1.0000 - 0.00001 -0.4871 - 0.02351 -0.4871 + 0.02351 -0.0986 - 0.01271 0.9273 + 0.01661 0.2631 0.19461 0.2631 - 0.19461 0.2994 + 0.19791 0.9930 + 0.00831 1.0000 0.00001 1.0000 + 0.00001 -0.0669 - 0.09341 0.0005 + 0.00111 0.0187 0.03221 0.0187 0.03221 0.0063 + 0.01521 0.0120 — 0.00411 0.0092 0.09881 0.0092 0.09881 -0.1120 - 0.19061 0.0004 + 0.00001 0.0105 0.03111 0.0105 0.03111 0.0110 + 0.02721 0.0006 + 0.00061 0.0039 0.01031 0.0039 0.01031 0.0018 + 0.00251 -0.0007 - 0.00041 0.0218 - 0.03351 0.0218 + 0.03351 -0.0132 - 0.05771 0.0001 + 0.00011 0.0078 - 0.05021 0.0078 + 0.05021 0.0125 + 0.01241 -0.0007 - 0.00061 -0.0036 - 0.01001 -0.0036 + 0.01001 -0.0017 - 0.00241 0.0006 + 0.00041 -0.0200 + 0.03201 -0.0200 - 0.03201 0.0134 + 0.05411 -0.0001 - 0.00011 -0.0082 + 0.04681 -0.0082 - 0.04681 -0.0113 - 0.01171 0.0137 - 0.00191 -0.0311 - 0.03021 -0.0311 + 0.03021 -0.0119 - 0.00651 0.0027 + 0.00231 -0.1602 + 0.17301 -0.1602 - 0.17301 0.0268 + 0.35831 0.0030 - 0.00041 0.0874 + 0.33851 0.0874 - 0.33851 -0.1240 - 0.04961 0.2706 - 0.00451 -0.5111 + 0.06661 -0.5111 - 0.06661 -0.1478 + 0.04661 -0.0748 + 0.01161 0.4840 + 0.39951 0.4840 - 0.39951 1.0000 - 0.00001 0.0112 + 0.00011 0.6347 + 0.19971 0.6347 - 0.19971 -0.2134 + 0.12321 +ll+|+ + + Column 21 0.0063 + 0.06241 -0.1195 - 0.19211 0.0577 + 0.04381 -0.0986 + 0.01271 0.2994 - 0.19791 -0.0669 + 0.09341 0.0063 - 0.01521 -0.1120 + 0.19061 0.0110 - 0.02721 0.0018 - 0.00251 -0.0132 + 0.05771 0.0125 - 0.01241 -0.0017 + 0.00241 0.0134 - 0.05411 -0.0113 + 0.01171 -0.0119 + 0.00651 0.0268 - 0.35831 -0.1240 + 0.04961 -0.1478 - 0.04661 1.0000 + 0.00001 -0.2134 - 0.12321 detjj - -1.4207e+31 detd - 1.5970e+21 detda - -8.8961e+09 detad - 5.4416e+27 235 Table C.3.12 (cont’d) condjj = 1.5374e+06 condd - 53.9462 condda = 7.4085e+05 condad = 2.9271e+06 vtt a -0.9950 0.0478 -0.3074 1.0000 1.0000 -0.6522 -0.0870 0.0167 1.0000 eigtt 8 -140.1835 -668.6855 -41.0645 236 Table C.3.13 Output for dynamic/algebraic 135 MVar (without exciter) 7.6339 + 0.77341 7.6339 - 0.77341 1.7916 -0.5340 + 1.56031 -0.5340 - 1.56031 -0.2013 + 1.23501 -0.2013 - 1.23501 -0.0865 + 0.71801 -0.0865 - 0.71801 -0.0001 + 0.62991 -0.0001 - 0.62991 0.4741 + 0.14521 0.4741 - 0.14521 ‘ 61 d - elgjj = 9 -26.3100 +52.38371 2223333 -26.3100 -52-3837i -12.4724 +25.73721 49.8143 -12.4724 -25.73721 46.7068 40 1889 40.1863 - ' -12.6242 +25.54261 -3'3§§§ :13'23231 -12.6242 -25-54261 -S:7988 +13:87351 24.8737 -5.7988 -13.87351 -10.1048 +18.64801 24 3799 -10.1048 —18.64801 21°5507 21.6455 17°6370 ~16.6012 1 1086 -15.8222 2.1907 -4.0397 +14.99351 ' -4.0397 -14.99351 ii'iigi : 8'33331 -5.8019 +13.88561 7 5099 + 0 59991 -5.8019 -13.88561 7°7099 - 0 88991 17.6090 ° ' 12.0060 + 0.85551 12.0060 - 0.85551 0.0596 0.2686 eigad - eigda ' 1.06+02 * -26.4792 +52.19111 -26.4792 -52.19111 ‘g-gggg * g-gligi -3.9046 +14.92301 ‘ - ‘ - 1 1 -3.9046 ‘14.9230i "0.3552 + 4.38301 _16.4891 -0.3552 - 4.38301 —15.1344 1-§4%5 -0.3244 + 1.71961 3'4351 -0.3244 — 1.71961 0'4296 -0.1269 + 1.10851 0°3503 3:12;: : 3:12:21 + _000838 _ o_58051 , -0.0391 - 0.10611 -0.0199 + 0.67101 '0-0355 -0.0199 - 0.67101 g-gggi -0.0836 + 0.06291 - -0.0836 - 0.06291 0-0520 0.2956 °-°997 0.0726 0.1411 vda - Columns 1 through 4 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 -0.0002 + 0.00031 -0.0002 - 0.00031 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 -0.0013 + 0.00071 -0.0013 - 0.00071 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 0.0001 + 0.00001 0.0001 - 0.00001 0.0000 + 0.00021 0.0000 - 0.00021 0.0000 + 0.00001 0.0000 - 0.00001 -0.0001 + 0.00071 -0.0001 - 0.00071 -0.0048 - 0.01051 -0.0048 + 0.01051 0.0000 - 0.00021 0.0000 + 0.00021 0.0000 - 0.00001 0.0000 + 0.00001 -0.0021 - 0.03051 -0.0021 + 0.03051 1.0000 - 0.00001 1.0000 + 0.0000 + 0.00041 0.0000 - -0.0763 - 0.15381 -0.0763 0.0000 - 0.00001 0.0000 + Columns 5 through 8 237 Table C.3.13 (cont’d) 0.00001 0.0040 - 0.00011 0.0040 + 0.00011 0.00041 1.0000 - 0.00001 1.0000 + 0.00001 + 0.15381 -0.0006 - 0.00261 -0.0006 + 0.00261 0.00001 -0.0287 - 0.11041 -0.0287 + 0.11041 0.0000 0.0000 0.0061 + 0.00101 0.0061 - 0.00101 0.0000 -0.0001 -0.0126 - 0.00651 -0.0126 + 0.00651 0.0000 0.0000 -0.0028 + 0. 0.0000 0.0000 -0.0001 - 0. 0.0000 0.0000 -0.0023 + 0. 00311 -0.0028 - 0.00311 00351 -0.0001 + 0.00351 00771 -0.0023 - 0.00771 0.0000 0.0000 0.0020 + 0.00121 0.0020 - 0.00121 -0.0002 0.0049 -0.0186 - 0.05471 -0.0186 + 0.05471 0.0013 0.0004 -0.0300 + 0. 00891 -0.0300 - 0.00891 0.0000 -0.0001 -0.0059 + 0.01111 -0.0059 - 0.01111 -0.0020 0.0457 -0.0263 - 0.04081 -0.0263 + 0.04081 0.0566 0.0223 -0.0280 - 0. 0.0023 -0.0527 0.0261 + 0. 00771 -0.0280 + 0.00771 03911 0.0261 - 0.03911 -0.0336 -0.0144 0.0272 + 0.00601 0.0272 - 0.00601 -0.0491 1.0000 0.0265 + 0. 17201 0.0265 - 0.17201 1.0000 0.3864 0.1303 + 0.13281 0.1303 - 0.13281 0.0309 -0.6875 1.0000 - 0. 00001 1.0000 + 0.00001 -0.1069 -0.0450 0.1098 - 0.15211 0.1098 + 0.15211 Columns 9 through 12 -0.0560 + 0.01561 -0.0560 0.2274 - 0.00451 0.2274 + -0.0793 - 0.01361 -0.0793 0.0196 + 0.04831 0.0196 - -0.0272 - 0.20201 -0.0272 -0.0041 + 0.07201 -0.0041 0.0333 + 0.04151 0.0333 - -0.0223 - 0.17151 -0.0223 -0.0318 + 0.01111 -0.0318 0.0173 + 0.01031 0.0173 - -0.0245 - 0.10091 -0.0245 -0.0167 - 0.00991 -0.0167 0.0264 + 0.09481 0.0264 - -0.0611 - 0.03681 -0.0611 -0.0552 + 0.63021 -0.0552 -0.4787 + 0.35101 -0.4787 1.0000 + 0.00001 1.0000 - Columns 13 through 16 -0.0233 - 0.07111 -0.0233 -0.0880 + 0.20691 -0.0880 0.1036 - 0.02521 0.1036 + -0.1069 + 0.03161 -0.1069 0.3042 + 0.14011 0.3042 - -0.0330 - 0.15531 -0.0330 0.0160 + 0.02031 0.0160 - -0.0996 - 0.16881 -0.0996 —0.0137 + 0.00191 -0.0137 0.0042 + 0.00261 0.0042 - -0.0149 - 0.06101 -0.0149 -0.0040 - 0.00261 -0.0040 0.0151 + 0.05711 0.0151 - -0.0211 - 0.00381 -0.0211 0.0282 + 0.38151 0.0282 - -0.1879 + 0.14401 -0.1879 1.0000 + 0.00001 1.0000 - Column 17 -0.0020 -0.0880 0.1236 -0.0067 -0.2976 0.4180 0.1203 0.3874 -0.5088 -0.0045 -0.0263 - 0.01561 0.1022 - 0.23261 0.1022 + 0.23261 0.00451 -0.2205 - 0.11901 -0.2205 + 0.11901 + 0.01361 -0.0838 + 0.68051 -0.0838 - 0.68051 0.04831 -0.3549 - 0.10651 -0.3549 + 0.10651 + 0.20201 -0.1330 + 0.34031 ~0.1330 - 0.34031 - 0.07201 1.0000 + 0.00001 1.0000 - 0.00001 0.04151 -0.0132 - 0.00211 -0.0132 + 0.00211 + 0.17151 0.1840 - 0.09681 0.1840 + 0.09681 - 0.01111 0.0337 + 0.04491 0.0337 - 0.04491 0.01031 -0.0030 + 0.00551 -0.0030 - 0.00551 + 0.10091 0.0654 + 0.00831 0.0654 - 0.00831 + 0.00991 0.0030 - 0.00521 0.0030 + 0.00521 0.09481 -0.0620 - 0.00671 -0.0620 + 0.00671 + 0.03681 0.0021 - 0.02381 0.0021 + 0.02381 - 0.63021 -0.3696 - 0.11601 -0.3696 + 0.11601 - 0.35101 -0.2106 - 0.18671 -0.2106 + 0.18671 0.00001 -0.2227 + 0.97301 -0.2227 - 0.97301 + 0.07111 -0.0836 + 0.06291 -0.0836 - 0.06291 - 0.20691 -0.0767 + 0.05991 -0.0767 - 0.05991 0.02521 -0.0821 + 0.06301 -0.0821 - 0.06301 - 0.03161 1.0000 - 0.00001 1.0000 + 0.00001 0.14011 0.9299 - 0.01621 0.9299 + 0.01621 + 0.15531 0.9897 - 0.00881 0.9897 + 0.00881 0.02031 -0.0002 - 0.00121 -0.0002 + 0.00121 + 0.16881 0.0092 + 0.00361 0.0092 - 0.00361 - 0.00191 0.0034 + 0.00051 0.0034 - 0.00051 0.00261 0.0006 - 0.00061 0.0006 + 0.00061 + 0.06101 -0.0007 + 0.00041 -0.0007 - 0.00041 + 0.00261 -0.0007 + 0.00061 -0.0007 - 0.00061 0.05711 0.0006 - 0.00041 0.0006 + 0.00041 + 0.00381 0.0137 + 0.00191 0.0137 - 0.00191 0.38151 0.0027 - 0.00231 0.0027 + 0.00231 - 0.14401 0.2704 + 0.00401 0.2704 - 0.00401 0.00001 -0.0742 - 0.01251 -0.0742 + 0.01251 0.0041 0.0244 0.0438 0.1850 0.4949 1.0000 detjj = 8.4838e+26 detd = 1.5970e+21 detda - 5.3124e+05 detad - -3.7038e+25 condjj - 1.5371e+06 condd = 53.9462 condda - 7.4053e+05 condad - 2.9308e+06 vtt - 1.0000 0.0469 -0.2255 -0.9066 1.0000 -0.6684 -0.0013 -0.0004 1.0000 ett - -137.9121 0 0 0 -659.2447 0 0 0 0.2214 238 Table C.3.13 (cont’d) CA Tat 4.4 C8: 239 C.4 Algebraic/Dynamic Voltage Instability Table C.4.l to Table C.4.5 are the converged equilibrium point of the simulation in section 4.4. Table C.4.6 to C4. 10 are the output from the eigenvalue program. Table C.4.11 is the case where field current is saturated. 240 Table C.4.l Equilibrium point(algebraicldynamic 100 MVar) 1 9 BUS TEST SYSTEM(NO LINE DROP COMPENSATION) PAGE NUMBER 7 Dl-2.0, 02-2.0, 03-2.0, KA-ZS, RC-0.0, xc-0.0 DATE 5/ 6/90 load2-100, REAL POHER LOAD - regular, real system TIME 21.58. 9 GENERATOR ANGLE IN DEGREES ~NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995-1177.6327-1158.5438-1178.7778 GENERATOR FIELD VOLTAGE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.1505 1.2844 1.7498 GEN. FLUX LINKAGE (Q—AXIS) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0009 0.9183 1.0624 GEN. ELECTRICAL POWER - (MW) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 81.9070 73.2209 32.2941 GENERATOR EXCITER SATURATION -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.0451 0.0114 0.1130 GENERATOR MEGAVAR OUTPUT ~NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 9.3730 3.3089 59.8387 GENERATOR FIELD CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0722 1.1746 1.5360 GENERATOR TERMINAL CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.8221 0.7323 0.6751 GEN. TERM. CURR. ANGLE DEGREES-NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -114.2417 ~109.8750 -171.9371 A-C BUS VOLTAGE MAGNITUDE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0000 1.0001 0.9953 0.9982 1.0000 0.9783 0.9903 0.9443 0.9664 A-C BUS VOLTAGE ANGLE DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME N0 11 -49.9995 -107.6946 -107.2852 -108.9968 -109.0723 -108.5942 -109.5441 -109.9591 -109.9120 -109.9659 Table C.4.2 Equilibrium point(algebraic/dynamic 120 MVar) 1 9 BUS TEST SYSTEM¢NO LINE DROP COMPENSATION) PAGE NUMBER 7 D1-2.0, 02-2.0, D3-2.0, KA-ZS, Rc-0.0, xc-0.0 DATE 5/ 5/90 load2-120, REAL POWER LOAD - regular, real system TIME 13.37.32 GENERATOR ANGLE IN DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995-2051.0349-2032.4624-2052.9587 GENERATOR FIELD VOLTAGE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.1617 1.3131 1.9371 GEN. FLUX LINKAGE (Q-AXIS) ~NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0028 0.9259 1.0831 GEN. ELECTRICAL POWER - (MM) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 81.9194 73.3913 32.5427 GENERATOR EXCITER SATURATION -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.0454 0.0118 0.1781 GENERATOR MEGAVAR OUTPUT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 ~49.9995 13.5463 8.2481 78.8409 GENERATOR FIELD CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 ~49.9995 1.0871 1.2076 1.6889 GENERATOR TERMINAL CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.8247 0.7353 0.8473 GEN. TERM. CURR. ANGLE DEGREES-NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 89.5024 92.9486 27.6158 A-C BUS VOLTAGE MAGNITUDE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49. 9995 0.9999 1. 0001 0.9900 0. 9974 0.9989 0.9675 0. 9894 0.9246 0.9645 A-C BUS VOLTAGE ANGLE DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -261.0426 -260.6111 -262.3363 -262.4216 -261.9244 -262.8969 -263.2994 -263.1671 -263.2791 241 Table C.4.3 Equilibrium point(algebraic/dynamic 140 MVar) 1 9 BUS TEST SYSTEM(NO LINE DROP COMPENSATION) PAGE NUMBER 7 Dl-2.0, D2-2.0, D3-2.0, KA-ZS, RC-0.0, xc-0.o DATE 5/ 5/90 load2-140, REAL PONER LOAD - regular, raal system TIME 13.55.36 GENERATOR ANGLE IN DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 ~49.9995-3264.8188-3247.0376-3267.9502 GENERATOR FIELD VOLTAGE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.1762 1.3488 2.1176 GEN. FLUX LINKAGE (O-AXIS) ~NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0054 0.9352 1.0997 GEN. ELECTRICAL POWER - (MN) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 83.1457 73.6318 31.6923 GENERATOR EXCITER SATURATION -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.0458 0.0122 0.2787 GENERATOR MEGAVAR OUTPUT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 ~49.9995 19.3959 14.6628 97.6530 GENERATOR FIELD CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.1088 1.2496 1.8409 GENERATOR TERMINAL CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.8414 0.7416 1.0261 GEN. TERM. CURR. ANGLE DEGREES-NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -48.2228 -45.8850 -112.5751 A-C BUS VOLTAGE MAGNITUDE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 ~49.9995 0.9999 1.0001 0.9801 0.9964 0.9976 0.9521 0.9881 0.8999 0.9616 A-C BUS VOLTAGE ANGLE DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49. 9995 -394. 8943 ~394. 4821 ~396. 4099 -396. 2954 -395. 8015 -396. 9703 -397. 1780 -397. 1258 -397.1205 Table C.4.4 Equilibrium point(a1gebraic/dynamic 160 MVar) 1 9 BUS TEST SYSTEM(NO LINE DROP COMPENSATION) PAGE NUMBER 7 Dl-2.0, 02-2.0, D3-2.0, NA-ZS, RC-0.0, XC-0.0 DATE 5/ 5/90 load2-160, REAL POWER LOAD - regular, rsal system TIME 14.10.46 GENERATOR ANGLE IN DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995-5102.7803-5085.8179-5106.4595 GENERATOR FIELD VOLTAGE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.1967 1.3968 2.2925 GEN. FLUX LINKAGE (Q-AXIS) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0090 0.9468 1.1112 GEN. ELECTRICAL POWER - (MN) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 83.7227 73.9501 31.6281 GENERATOR EXCITER SATURATION -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.0464 0.0128 0.4329 GENERATOR MEGAVAR OUTPUT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 27.3257 23.1931 116.2336 GENERATOR FIELD CURRENT -NUMEER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.1369 1.3042 1.9989 GENERATOR TERMINAL CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.8553 0.7534 1.2193 GEN. TERM. CURR. ANGLE DEGREES-NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -91.4546 -90.3235 -155.6560 A-C BUS VOLTAGE MAGNITUDE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.9999 1.0001 0.9645 0.9950 0.9958 0.9307 0.9865 0.8684 0.9578 A-C BUS VOLTAGE ANGLE DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 -792.8209 -792.3969 -794.3982 -794.2339 -793.7243 -794.9797 -795.1110 -795.0235 -794.9921 242 Table C.4.5 Equilibrium point(algebraic/dynamie 170 MVar) 1 9 BUS TEST SYSTEM(NO LINE DROP COMPENSATION) PAGE NUMBER 7 01-2.0, D2-2.0, D3-2.0, KA-ZS, RC-0.0, XC-0.0 DATE 5/ 5/90 load2-170, REAL POWER LOAD - regular, real system TIME 14.47. 8 GENERATOR ANGLE IN DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995-6788.8491-6772.3994-6791.4521 GENERATOR FIELD VOLTAGE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.2204 1.4455 2.3393 GEN. FLUX LINKAGE (Q-AXIS) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.0130 0.9581 1.1031 GEN. ELECTRICAL POWER - (MN) -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 82.4783 74.1971 33.2760 GENERATOR EXCITER SATURATION -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.0472 0.0135 0.4876 GENERATOR MEGAVAR OUTPUT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 34.8703 31.2340 123.3748 GENERATOR FIELD CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 1.1618 1.3542 2.0799 GENERATOR TERMINAL CURRENT -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.8550 0.7685 1.3178 GEN. TERM. CURR. ANGLE DEGREES-NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 17.2926 17.8892 -43.1107 A-C BUS VOLTAGE MAGNITUDE -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995 0.9998 1.0002 0.9426 0.9935 0.9942 0.9061 0.9849 0.8378 0.9540 A-C BUS VOLTAGE ANGLE DEGREES -NUMBER BUS/GEN NAME NO NUMBER BUS/GEN NAME NO 11 -49.9995-1038.6335-1038.1355-1039.8147-1040.0276-1039.4691-1040.4576-1040.8798-1040.4736-1040.6694 243 Table C.4.6 Output for algebraic/dynamic 100 MVar e1 d - eigjj - 9 -26.3306 +52.39381 2213333 -26.3306 -52.39381 39.3335 48.8605 -12.7885 +25.87911 33.2332 -12.7885 -25.87911 -12.9499 +25.70271 -1812132 138332381 -12.9499 -25.70271 -5.6040 +14.61421 -10.6054 +20.54821 -5_5040 —14.61421 -10.6054 -20.54821 25.0320 25.0774 23 6173 23.5798 21:7496 21.7340 1.4243 -15.8133 1.9648 -16.5808 11.6231 + 0.80971 -16.6084 11.6231 - 0.80971 -5.4912 +14.65261 3.2102 + 0.45331 —5.4912 -14.65261 0,2102 - 0.45331 -4.1864 +14.93861 -4.1864 -14.93861 -3.9697 +14.98651 -3.9697 -14.98651 11.5361 + 0.69271 11.5361 - 0.69271 8.1440 + 0.19591 8.1440 - 0.19591 2.0359 -0.5034 + 1.57631 -0.5034 - 1.57631 -0.1350 + 1.19381 -0.1350 - 1.19381 0.8303 -0.1017 + 0.89991 -0.1017 - 0.89991 0.3216 0.0368 -0.0570 + 0.61651 -0.0570 - 0.61651 -0.0430 + 0.57211 -0.0430 - 0.57211 eigda - -26.4918 +52.21031 -26.4918 -52.21031 -3.8952 +14.90511 -3.8952 -14.90511 -3.9486 +14.98621 -3.9486 -14.98621 —15.2041 -16.4660 -16.5489 -0.3457 + 1.69671 -0.3457 - 1.69671 -0.0397 + 1.11761 -0.0397 - 1.11761 -0.0841 + 0.02001 -0.0841 — 0.02001 -0.0598 + 0.90931 -0.0598 - 0.90931 —0.0716 + 0.61731 —0.0716 - 0.61731 —0.0479 + 0.57701 -0.0479 - 0.57701 vda - Columns 1 through 4 (1.0000 - 0.00001 0.0000 + 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 (3.0000 + 0.00001 0.0000 - 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 2A4 Table C.4.6 (cont’d) 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 -0.0002 + 0.00031 -0.0002 - 0.00031 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 -0.0012 + 0.00071 -0.0012 - 0.00071 0.0000 + 0.00001 0.0000 0.00001 -0.0001 + 0.00031 -0.0001 - 0.00031 0.0001 + 0.00001 0.0001 0.00001 0.0000 + 0.00021 0.0000 - 0.00021 0.0000 + 0.00001 0.0000 - 0.0000 + 0.00001 0.0000 0.00001 0.0001 + 0.00041 0.0001 - 0.00041 -0.0048 - 0.01051 -0.004 + 0.01051 0.0000 - 0.00021 0.0000 + 0.00021 - 0.00001 -0.0001 + 0.00071 -0.0001 8 0.0000 - 0.00001 0.0000 + 0.00001 -0.0021 - 0.03051 -0.0021 + 0.03051 + + 0.00071 0.0000 - 0.00001 0.0000 0.00001 -0.0090 - 0.01251 -0.0090 + 0.01251 1.0000 - 0.00001 1.0000 0.00001 0.0046 - 0.00061 0.0046 + 0.00061 0.0000 + 0.00021 0.0000 0.00021 1.0000 + 0.00001 1.0000 - 0.00001 0.0000 + 0.00021 0.0000 - 0.00021 0.4246 - 0.26741 0.4246 + 0.26741 -0.0762 - 0.15381 -0.0762 + 0.15381 -0.0011 - 0.00281 -0.0011 + 0.00281 0.0000 - 0.00001 0.0000 + 0.00001 -0.0289 - 0.11051 -0.0289 + 0.11051 0.0000 - 0.00001 0.0000 + 0.00001 -0.0350 - 0.03331 -0.0350 + 0.03331 Columns 5 through 8 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 0.0000 0.0000 + 0.00001 0.0000 - 0.00001 -0.0001 0.0000 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 0.0000 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 0.0000 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 0.0000 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 0.0000 0.0000 + 0.00001 0.0000 - 0.00001 0.0048 -0.0003 0.0003 - 0.00041 0.0003 + 0.00041 0.0003 0.0012 -0.0006 + 0.00031 -0.0006 - 0.00031 0.0001 0.0003 0.0000 + 0.00001 0.0000 - 0.00001 0.0456 -0.0026 -0.0001 - 0.00011 -0.0001 + 0.00011 0.0167 0.0565 -0.0001 + 0.00041 -0.0001 - 0.00041 0.0156 0.0299 0.0000 - 0.00011 0.0000 + 0.00011 -0.0525 0.0031 0.0068 + 0.00951 0.0068 - 0.00951 -0.0108 -0.0336 -0.0020 - 0.03041 -0.0020 + 0.03041 -0.0101 -0.0178 0.0013 + 0.00071 0.0013 - 0.00071 1.0000 -0.0643 -0.3265 + 0.20061 -0.3265 - 0.20061 0.2906 1.0000 1.0000 - 0.00001 1.0000 + 0.00001 0.2701 0.5277 0.0003 - 0.00091 0.0003 + 0.00091 -0.6862 0.0406 0.0315 + 0.03001 0.0315 - 0.03001 -0.0336 -0.1068 -0.0237 - 0.09271 -0.0237 + 0.09271 -0.0265 -0.0478 Columns 9 through 12 0.0000 0.0055 - 0.00041 0.0055 + 0.00041 -0.0353 + 0.00951 0.0000 -0.0106 - 0.00311 -0.0106 + 0.00311 0.1203 + 0.00711 0.0000 -0.0036 + 0.00341 -0.0036 - 0.00341 -0.0309 - 0.01911 0.0000 -0.0008 - 0.00311 -0.0008 + 0.00311 0.0096 + 0.03121 0.0000 -0.0005 + 0.00631 -0.0005 - 0.00631 0.0025 - 0.10771 0.0000 0.0023 + 0.00161 0.0023 - 0.00161 -0.0161 + 0.02821 -0.0001 -0.0198 - 0.05501 -0.0198 + 0.05501 0.0376 + 0.03821 -0.0005 -0.0242 + 0.00901 -0.0242 - 0.00901 -0.0393 - 0.15011 0.0006 -0.0109 + 0.00551 -0.0109 - 0.00551 -0.0456 - 0.02141 -0.0005 -0.0257 - 0.04131 -0.0257 + 0.04131 0.0189 + 0.00661 -0.0239 -0.0231 - 0.00501 -0.0231 + 0.00501 -0.0305 - 0.09661 0.0567 -0.0190 - 0.00601 -0.0190 + 0.00601 -0.0392 - 0.03041 0.0006 0.0255 + 0.03961 0.0255 - 0.03961 -0.0181 - 0.00641 0.0141 0.0224 + 0.00361 0.0224 - 0.00361 0.0322 + 0.09041 -0.0336 0.0185 + 0.00481 0.0185 - 0.00481 0.0378 + 0.02761 -0.0123 0.0286 + 0.16971 0.0286 - 0.16971 -0.0729 - 0.02391 -0.4222 0.1094 + 0.10301 0.1094 - 0.10301 -0.0221 + 0.63541 1.0000 0.0888 + 0.08861 0.0888 - 0.08861 0.1922 + 0.24451 0.0077 1.0000 - 0.00001 1.0000 + 0.00001 -0.4365 + 0.42081 0.0449 0.0804 - 0.12981 0.0804 + 0.12981 1.0000 - 0.00001 -0.0900 0.0655 - 0.08641 0.0655 + 0.08641 0.3393 - 0.23161 Columns 13 through 16 -0.0353 - 0.00951 -0.0841 + 0.02001 -0.0841 - 0.02001 -0.0059 - 0.02161 0.1203 - 0.00711 -0.0799 + 0.01951 -0.0799 - 0.01951 -0.1066 + 0.08581 -0.0309 + 0.01911 -0.0833 + 0.02001 ~0.0833 - 0.02001 0.1130 - 0.03991 0.0096 - 0.03121 1.0000 + 0.00001 1.0000 - 0.00001 -0.0232 + 0.00811 0.0025 + 0.10771 0.9518 - 0.00631 0.9518 + 0.00631 0.1016 + 0.11051 -0.0161 - 0.02821 0.9915 - 0.00271 0.9915 + 0.00271 -0.0518 - 0.12091 0.0376 - 0.03821 0.0006 - 0.00031 0.0006 + 0.00031 -0.0048 + 0.03191 -0.0393 + 0.15011 0.0086 + 0.00131 0.0086 - 0.00131 0.0722 + 0.00761 -0.0456 + 0.02141 -0.0008 + 0.00011 -0.0008 - 0.00011 -0.0311 - 0.12001 0.0189 - 0.00661 0.0006 - 0.00021 0.0006 + 0.00021 0.0016 + 0.00841 -0.0305 + 0.09661 -0.0002 + 0.00001 -0.0002 - 0.00001 0.0486 - 0.00341 245 Table C.4.6 (oont’d) -0.0392 + 0.03041 0.0000 - 0.00001 0.0000 + 0.00001 -0.0228 - 0.09881 -0.0181 + 0.00641 -0.0007 + 0.00021 -0.0007 - 0.00021 -0.0015 - 0.00811 0.0322 - 0.09041 0.0002 - 0.00001 0.0002 + 0.00001 -0.0458 + 0.00451 0.0378 - 0.02761 0.0000 + 0.00001 0.0000 - 0.00001 0.0235 + 0.09271 -0.0729 + 0.02391 0.0144 + 0.00061 0.0144 - 0.00061 -0.0123 - 0.03001 -0.0221 — 0.63541 0.0014 - 0.00021 0.0014 + 0.00021 -0.2892 - 0.06321 0.1922 - 0.24451 0.0000 + 0.00001 0.0000 - 0.00001 0.0375 + 0.61451 -0.4365 - 0.42081 0.2629 + 0.00141 0.2629 - 0.00141 -0.2976 - 0.05221 1.0000 + 0.00001 -0.0286 - 0.00251 -0.0286 + 0.00251 -0.0857 + 0.56501 0.3393 + 0.23161 0.0007 - 0.00001 0.0007 + 0.00001 1.0000 - 0.00001 Columns 17 through 20 -0.0059 + 0.02161 0.2355 - 0.17011 0.2355 + 0.17011 0.0402 - 0.07921 -0.1066 - 0.08581 -0.1710 + 0.09761 -0.1710 - 0.09761 “0.2319 + 0.22091 0.1130 + 0.03991 -0.5062 + 0.28651 -0.5062 - 0.28651 0.0785 - 0.05531 -0.0232 - 0.00811 -0.3155 - 0.34491 -0.3155 + 0.34491 -0.1421 - 0.05781 0.1016 - 0.11051 0.1877 + 0.25531 0.1877 - 0.25531 0.4133 + 0.36751 -0.0518 + 0.12091 0.5518 + 0.75591 0.5518 - 0.75591 -0.1063 - 0.12731 —0.0048 - 0.03191 0.0091 + 0.04891 0.0091 - 0.04891 0.0044 + 0.01431 0.0722 - 0.00761 -0.0182 - 0.06211 -0.0182 + 0.06211 -0.1290 - 0.21531 -0.0311 + 0.12001 -0.0601 - 0.15641 -0.0601 + 0.15641 0.0257 + 0.06021 0.0016 - 0.00841 -0.0015 + 0.00951 -0.0015 - 0.00951 0.0005 + 0.00281 0.0486 + 0.00341 0.0127 - 0.02011 0.0127 + 0.02011 -0.0056 - 0.05531 -0.0228 + 0.09881 -0.0067 - 0.07101 -0.0067 + 0.07101 0.0121 + 0.02351 -0.0015 + 0.00811 0.0016 - 0.00911 0.0016 + 0.00911 -0.0004 - 0.00271 -0.0458 - 0.00451 -0.0116 + 0.01921 -0.0116 - 0.01921 0.0063 + 0.05211 0.0235 - 0.09271 0.0069 + 0.06691 0.0069 - 0.06691 -0.0115 - 0.02211 -0.0123 + 0.03001 -0.0146 - 0.03671 -0.0146 + 0.03671 -0.0084 - 0.00921 -0.2892 + 0.06321 -0.0964 + 0.10141 -0.0964 - 0.10141 -0.0268 + 0.32801 0.0375 - 0.61451 0.0295 + 0.41721 0.0295 - 0.41721 -0.0720 - 0.13941 -0.2976 + 0.05221 -0.4437 - 0.16611 -0.4437 + 0.16611 -0.1514 - 0.00761 -0.0857 - 0.56501 0.3163 + 0.23851 0.3163 - 0.23851 1.0000 - 0.00001 1.0000 + 0.00001 1.0000 - 0.00001 1.0000 + 0.00001 -0.3739 + 0.14111 Column 21 0.0402 + 0.07921 -0.2319 - 0.22091 0.0785 + 0.05531 -0.1421 + 0.05781 0.4133 - 0.36751 -0.1063 + 0.12731 0.0044 - 0.01431 -0.1290 + 0.21531 0.0257 - 0.06021 0.0005 - 0.00281 -0.0056 + 0.05531 0.0121 - 0.02351 -0.0004 + 0.00271 0.0063 - 0.05211 -0.0115 + 0.02211 -0.0084 + 0.00921 -0.0268 - 0.32801 —0.0720 + 0.13941 -0.1514 + 0.00761 1.0000 + 0.00001 -0.3739 - 0.14111 eigad - 1.0o+03 * -0.0510 + 1.92641 -0.0510 — 1.92641 -0.0647 + 0.39781 -0.0647 — 0.39781 0.0626 + 0.19191 0.0626 - 0.19191 0.0408 + 0.06271 0.0408 - 0.06271 0.0488 0.0439 0.0327 0.0349 0.0000 0.0122 246 Table C.4.6 (cont’d) 0.0054 0.0068 + 0.00041 0.0068 - 0.00041 0.0134 vad - Columns 1 through 4 0.0001 + 0.00061 0.0001 - 0.00061 0.0085 + 0.04111 0.0085 - 0.04111 0.0324 + 0.00801 0.0324 - 0.00801 —0.0071 - 0.00131 -0.0071 + 0.00131 0.0000 + 0.00001 0.0000 - 0.00001 0.0018 + 0.00261 0.0018 - 0.00261 0.0696 - 0.00371 0.0696 + 0.00371 1.0000 - 0.00001 1.0000 + 0.00001 1.0000 + 0.00001 1.0000 - 0.00001 -0.1105 + 0.07211 -0.1105 - 0.07211 0.0002 - 0.00001 0.0002 + 0.00001 0.0708 - 0.02551 0.0708 + 0.02551 -0.0001 - 0.00121 -0.0001 + 0.00121 -0.0125 - 0.07871 -0.0125 + 0.07871 -0.0005 - 0.01801 -0.0005 + 0.01801 0.0103 + 0.00761 0.0103 - 0.00761 0.0001 - 0.00001 0.0001 + 0.00001 -0.0035 - 0.00591 -0.0035 + 0.00591 0.0002 + 0.00741 0.0002 - 0.00741 -0.0035 + 0.02181 -0.0035 - 0.02181 -0.0001 + 0.00291 -0.0001 - 0.00291 0.0000 - 0.00011 0.0000 + 0.00011 0.0000 + 0.00021 0.0000 - 0.00021 0.0028 + 0.01501 0.0028 - 0.01501 -0.0001 + 0.00011 -0.0001 - 0.00011 0.0005 + 0.00621 0.0005 - 0.00621 0.0000 + 0.00141 0.0000 - 0.00141 -0.0007 - 0.00051 -0.0007 + 0.00051 0.0000 + 0.00001 0.0000 — 0.00001 -0.0002 + 0.00071 -0.0002 - 0.00071 -0.0001 - 0.00101 -0.0001 + 0.00101 0.0002 - 0.00291 0.0002 + 0.00291 0.0000 - 0.00071 0.0000 + 0.00071 0.0001 + 0.00011 0.0001 - 0.00011 0.0000 - 0.00001 0.0000 + 0.00001 -0.0009 - 0.00351 -0.0009 + 0.00351 Columns 5 through 8 -0.0199 + 0.08401 -0.0199 - 0.08401 -0.0896 + 0.23761 -0.0896 - 0.23761 0.0201 - 0.04271 0.0201 + 0.04271 0.1540 - 0.04211 0.1540 + 0.04211 -0.0269 + 0.00651 -0.0269 - 0.00651 -0.0727 - 0.10421 -0.0727 + 0.10421 1.0000 + 0.00001 1.0000 - 0.00001 1.0000 + 0.00001 1.0000 - 0.00001 -0.4968 - 0.35351 -0.4968 + 0.35351 -0.3501 - 0.63641 -0.3501 + 0.63641 0.1415 + 0.29211 0.1415 - 0.29211 -0.2884 + 0.41651 -0.2884 - 0.41651 0.0492 - 0.14891 0.0492 + 0.14891 0.2087 - 0.35871 0.2087 + 0.35871 -0.0756 + 0.06941 -0.0756 - 0.06941 -0.3085 + 0.05201 -0.3085 - 0.05201 0.0575 - 0.01251 0.0575 + 0.01251 0.1127 + 0.19221 0.1127 - 0.19221 0.0035 + 0.01051 0.0035 - 0.01051 0.0023 + 0.00341 0.0023 - 0.00341 0.0001 - 0.00021 0.0001 + 0.00021 0.0000 - 0.00011 0.0000 + 0.00011 -0.0237 + 0.02281 -0.0237 - 0.02281 -0.0191 - 0.00071 -0.0191 + 0.00071 -0.0034 + 0.01261 -0.0034 - 0.01261 -0.0064 + 0.03541 -0.0064 - 0.03541 0.0051 - 0.00621 0.0051 + 0.00621 0.0206 - 0.01441 0.0206 + 0.01441 -0.0061 - 0.00051 -0.0061 + 0.00051 -0.0124 - 0.01671 -0.0124 + 0.01671 -0.0010 - 0.00101 -0.0010 + 0.00101 -0.0044 + 0.00301 -0.0044 - 0.00301 -0.0012 + 0.00081 -0.0012 - 0.00081 -0.0016 - 0.00061 -0.0016 + 0.00061 0.0045 - 0.00661 0.0045 + 0.00661 0.0052 - 0.00681 0.0052 + 0.00681 Columns 9 through 12 0.2346 0.2420 -0.9939 0.4225 0.2158 0.0291 -0.8998 0.2693 0.1966 0.0698 -0.9850 0.2055 -0.4241 -0.3856 0.9156 -0.4203 -0.4290 -0.0904 1.0000 -0.3784 -0.3646 -0.1006 0.9222 -0.2207 -0.2415 -0.3l44 0.9762 -0.3819 -0.2012 -0.0284 0.9499 -0.3545 -0.1951 -0.0102 0.9668 -0.1785 -0.0007 -0.0011 0.0019 0.0000 0.0000 0.0000 0.0001 0.0000 -0.0028 0.0063 0.0154 -0.0045 0.5748 -0.1471 0.0686 1.0000 1.0000 -0.5768 -0.0084 -0.6227 0.7328 1.0000 0.0130 -0.0699 -0.7482 0.4470 -0.0573 -0.1028 -0.4081 -0.2608 -0.0037 0.2160 -0.3182 -0.3309 -0.0354 -0.2460 Columns 13 through 16 1.0000 0.0171 0.2148 1.0000 - 0.00001 0.9879 0.1488 -0.0757 -0.3162 - 0.17151 0.9986 —0.2145 -0.0784 -0.5512 + 0.14101 0.9989 0.0126 0.1563 0.6012 - 0.02301 0.9926 0.0671 -0.0255 -0.1875 - 0.09471 p.11. hly . n a.» (I r\ .n. . AL .I. a.» #0 . rl . u a Q 9 .4. u . AU AIL M.» OU 80 010 Q “IV. “1‘ I “HM O '6“ $1.: 8U 0V. “V I. . 07 . 4U A . Fb fl 1 Q o C00 CCU< 64. C000 C000 5.3 L & 247 Table C.4.6 (cont’d) 0.9985 -0.0634 -0.0462 -0.3261 + 0.09651 0.9952 0.0115 0.0541 0.1404 - 0.06501 0.9965 0.0193 -0.0740 -0.2788 + 0.03131 0.9986 0.0571 -0.0290 -0.0015 + 0.05661 -0.0002 0.0003 0.0004 0.0001 - 0.00001 0.0000 0.0000 0.0000 0.0000 - 0.00001 0.0000 -0.0032 0.0052 -0.0011 + 0.00011 0.0000 0.3245 0.4209 0.0834 - 0.02231 -0.0001 0.5382 0.4575 0.0379 - 0.02791 0.0001 -0.3295 0.6234 0.0065 - 0.01891 -0.0001 1.0000 0.6012 0.0870 - 0.03541 0.0002 -0.1934 1.0000 0.0314 - 0.03551 0.0001 -0.5994 0.9307 0.0468 - 0.03481 Columns 17 through 18 1.0000 + 0.00001 -0.0982 -0.3162 + 0.17151 0.1642 -0.5512 - 0.14101 -0.0917 0.6012 + 0.02301 -0.0306 -0.1875 + 0.09471 0.0347 -0.3261 - 0.09651 -0.0184 0.1404 + 0.06501 0.0185 -0.2788 - 0.03131 -0.1057 -0.0015 - 0.05661 0.1111 0.0001 + 0.00001 -0.0004 0.0000 + 0.00001 0.0000 —0.0011 - 0.00011 0.0006 0.0834 + 0.02231 -0.3908 0.0379 + 0.02791 0.1051 0.0065 + 0.01891 0.0873 0.0870 + 0.03541 -0.2529 0.0314 + 0.03551 1.0000 0.0468 + 0.03481 -0.8546 detjj - -7.7919e+30 detd - 3.19812+21 detda - -2.4364e+09 detad - 2.5928e+29 condjj - 2.2617e+06 condd - 64.9835 condda - 1.0843e+06 condad - 5.3892e+07 vtt I 1.0000 0.0009 -0.5571 -0.7446 1.0000 -0.1589 0.6679 0.0009 1.0000 eigtt - 1.0e+04 * -0.0146 -2.3122 -0.0094 248 Table C.4.6 (cont’d) eigjj = -26.3382 -26.3382 48.5461 44.8936 39.7860 -12.8916 -12.8916 -10.4749 -10.4749 24.9942 23.4238 21.5370 -15.8138 -16.5808 -16.6075 -5.4317 -5.4317 -3.9957 -3.9957 -4.2132 -4.2132 11.3305 11.3305 8.4729 7.7830 1.9690 -0.5044 -0.5044 -0.1370 -0.1370 0.8575 ~0.1080 -0.1080 0.3181 0.0356 -0.0549 -0.0549 -0.0459 -0.0459 eigda - -26.4999 -26.4999 -3.8978 -3.8978 -3.9807 -3.9807 —15.1974 -16.4610 -16.5452 -0.3373 -0.3373 -0.0305 -0.0305 -0.0841 -0.0841 -0.0662 -0.0662 -0.0727 -0.0727 —0.0512 -0.0512 vda - Columns 0.0000 - 0.0000 + 0.0000 + 0.0000 + 0.0000 - 249 Table C.4.7 Output for algebraic/dynamic 120 MVar +52.39741 ~52.39741 +25.57131 -25.57131 +20.31351 -20.31351 +14.48651 -14.48651 +14.98101 -14.98101 +14.98061 -14.98061 + 0.74271 - 0.74271 + 1.57731 - 1.57731 + 1.19511 — 1.19511 + 0.90881 - 0.90881 + 0.63231 - 0.63231 + 0.57781 - 0.57781 +52.21341 -52.21341 +14.90561 -14.90561 +14.98621 -14.98621 1.70191 1.70191 1.12691 1.12691 0.01401 0.01401 0.91871 0.91871 0.63241 0.63241 0.58271 0.58271 1 +1 + l4-l-+l +1 + 1 through 4 0.00001 0.00001 0.00001 0.00001 0.00001 0.0000 0.0000 0.0000 0.0000 0.0000 +lll+ 0.00001 0.00001 0.00001 0.00001 0.00001 0.0000 0.0000 0.0000 0.0000 0.0000 eigd - -12.7286 +25.75311 -12.7286 -25.75311 48.5460 44.8949 39.7902 -10.2868 +20.31891 -20.31891 -10.2868 -5.5572 +14.47881 -14.47881 -5.5572 24.9982 23.4628 21.5542 1.4136 1.9238 11.4166 + 0.84031 11.4166 - 0.84031 8.1916 + 0.23131 8.1916 - 0.23131 0.00001 0.00001 0.00001 0.00001 0.00001 0.0000 0.0000 0.0000 0.0000 0.0000 l+l+l 0.00001 0.00001 0.00001 0.00001 0.00001 0.0000 - 0.00001 0.0000 + 250 Table C.4.7 (cont’d) 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 -0.0002 + 0.00031 -0.0002 - 0.00031 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 -0.0012 + 0.00071 -0.0012 - 0.00071 0.0000 + 0.00001 0.0000 - 0.00001 -0.0001 + 0.00031 -0.0001 - 0.00031 0.0001 + 0.00001 0.0001 - 0.00001 0.0000 + 0.00021 0.0000 - 0.00021 0.0000 + 0.00001 0.0000 - 0.00001 -0.0001 + 0.00071 -0.0001 - 0.00071 0.0000 + 0.00001 0.0000 - 0.00001 0.0001 + 0.00041 0.0001 - 0.00041 -0.0048 - 0.01051 -0.0048 + 0.01051 0.0000 - 0.00021 0.0000 + 0.00021 0.0000 - 0.00001 0.0000 + 0.00001 -0.0021 - 0.03051 —0.0021 + 0.03051 0.0000 - 0.00001 0.0000 + 0.00001 -0.0095 - 0.00891 -0.0095 + 0.00891 1.0000 - 0.00001 1.0000 + 0.00001 0.0045 - 0.00081 0.0045 + 0.00081 0.0000 + 0.00031 0.0000 - 0.00031 1.0000 - 0.00001 1.0000 + 0.00001 0.0000 + 0.00021 0.0000 - 0.00021 0.3066 - 0.29121 0.3066 + 0.29121 -0.0762 - 0.15381 -0.0762 0.0000 - 0.00001 0.0000 + + 0.15381 -0.0011 - 0.00271 -0.0011 + 0.00271 0.00001 -0.0289 - 0.11051 -0.0289 + 0.11051 0.0000 - 0.00001 0.0000 + 0.00001 -0.0344 - 0.02191 -0.0344 + 0.02191 Columns 5 through 8 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 0.0000 0.0000 + 0.00001 0.0000 - 0.00001 -0.0001 0.0000 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 0.0000 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 0.0000 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 0.0000 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 0.0000 0.0000 + 0.00001 0.0000 - 0.00001 0.0048 -0.0003 0.0001 - 0.00041 0.0001 + 0.00041 0.0003 0.0012 -0.0006 + 0.00031 -0.0006 - 0.00031 0.0001 0.0003 0.0000 + 0.00011 0.0000 - 0.00011 0.0456 -0.0028 -0.0001 - 0.00001 -0.0001 + 0.00001 0.0169 0.0566 -0.0001 + 0.00041 -0.0001 - 0.00041 0.0163 0.0331 0.0000 - 0.00011 0.0000 + 0.0071 + 0.00661 0.0071 - -0.0019 - 0.03041 -0.0019 0.0017 + 0.00081 0.0017 - -0.2343 + 0.21621 -0.2343 1.0000 + 0.00001 1.0000 - 0.0003 - 0.00121 0.0003 + 0.0305 + 0.01941 0.0305 - -0.0235 - 0.09281 -0.0235 0.00011 -0.0526 0.0033 0.00661 -0.0109 -0.0336 + 0.03041 -0.0106 -0.0198 0.00081 1.0000 -0.0687 - 0.21621 0.2948 1.0000 0.00001 0.2823 0.5821 0.00121 -0.6873 0.0435 0.01941 -0.0341 -0.1069 + 0.09281 -0.0278 -0.0529 Columns 9 through 12 0.0000 0.0054 - 0.00031 0.0054 + 0.00031 -0.0340 + 0.00941 0.0000 -0.0105 - 0.00321 -0.0105 + 0.00321 0.1142 + 0.00681 0.0000 -0.0035 + 0.00341 -0.0035 - 0.00341 -0.0285 - 0.01911 0.0000 -0.0008 - 0.00301 -0.0008 + 0.00301 0.0092 + 0.02991 0.0000 -0.0006 + 0.00631 -0.0006 - 0.00631 0.0033 - 0.10141 0.0000 0.0023 + 0.00161 0.0023 - 0.00161 -0.0163 + 0.02571 0.0000 -0.0194 - 0.05471 -0.0194 + 0.05471 0.0405 + 0.03981 -0.0005 -0.0243 + 0.00941 -0.0243 - 0.00941 -0.0401 - 0.14711 0.0006 -0.0111 + 0.00581 -0.0111 - 0.00581 -0.0465 - 0.02551 -0.0005 -0.0259 - 0.04111 -0.0259 + 0.04111 0.0204 + 0.00661 -0.0260 -0.0237 - 0.00461 -0.0237 + 0.00461 -0.0316 - 0.09671 0.0568 -0.0204 - 0.00571 -0.0204 + 0.00571 -0.0424 - 0.03641 0.0006 0.0256 + 0.03941 0.0256 - 0.03941 -0.0196 - 0.00641 0.0154 0.0230 + 0.00321 0.0230 - 0.00321 0.0332 + 0.09041 -0.0337 0.0198 + 0.00451 0.0198 - 0.00451 0.0408 + 0.03321 -0.0118 0.0310 + 0.17021 0.0310 - 0.17021 -0.0798 - 0.02361 -0.4606 0.1149 + 0.10311 0.1149 - 0.10311 -0.0157 + 0.64061 1.0000 0.1010 + 0.08661 0.1010 - 0.08661 0.2217 + 0.27591 0.0074 1.0000 - 0.00001 1.0000 + 0.00001 -0.4595 + 0.45631 0.0490 0.0801 - 0.13451 0.0801 + 0.13451 1.0000 - 0.00001 -0.0905 0.0672 - 0.09361 0.0672 + 0.09361 0.3955 - 0.23521 Columns 13 through 16 -0.0340 - 0.00941 -0.0841 + 0.01401 -0.0841 - 0.01401 -0.0044 - 0.02101 0.1142 - 0.00681 -0.0801 + 0.01371 -0.0801 - 0.01371 -0.1125 + 0.08831 -0.0285 + 0.01911 -0.0834 + 0.01411 -0.0834 - 0.01411 0.1139 - 0.04421 0.0092 - 0.02991 1.0000 + 0.00001 1.0000 - 0.00001 -0.0224 + 0.00641 0.0033 + 0.10141 0.9529 - 0.00441 0.9529 + 0.00441 0.1043 + 0.11491 -0.0163 - 0.02571 0.9925 - 0.00191 0.9925 + 0.00191 -0.0568 - 0.11981 0.0405 - 0.03981 0.0006 - 0.00021 0.0006 + 0.00021 -0.0060 + 0.03141 -0.0401 + 0.14711 0.0082 + 0.00091 0.0082 - 0.00091 0.0756 + 0.01611 —0.0465 + 0.02551 -0.0007 + 0.00011 -0.0007 - 0.00011 -0.0310 - 0.11671 0.0204 - 0.00661 0.0006 - 0.00011 0.0006 + 0.00011 0.0012 + 0.00851 -0.0316 + 0.09671 -0.0002 + 0.00001 -0.0002 - 0.00001 0.0512 + 0.00081 251 Table C.4.7 (cont’d) -0.0424 + 0.03641 0.0000 - 0.00001 0.0000 + 0.00001 -0.0232 - 0.10011 -0.0196 + 0.00641 -0.0007 + 0.00011 -0.0007 - 0.00011 -0.0011 - 0.00821 0.0332 - 0.09041 0.0002 - 0.00001 0.0002 + 0.00001 -0.0485 + 0.00061 0.0408 - 0.03321 0.0000 + 0.00001 0.0000 - 0.00001 0.0236 + 0.09381 -0.0798 + 0.02361 0.0148 + 0.00041 0.0148 - 0.00041 -0.0108 - 0.03101 -0.0157 - 0.64061 0.0013 - 0.00011 0.0013 + 0.00011 -0.2973 - 0.09251 0.2217 - 0.27591 0.0000 + 0.00001 0.0000 - 0.00001 0.0834 + 0.62081 -0.4595 - 0.45631 0.2628 + 0.00101 0.2628 - 0.00101 -0.2943 - 0.06821 1.0000 + 0.00001 -0.0271 - 0.00171 -0.0271 + 0.00171 -0.1373 + 0.57871 0.3955 + 0.23521 0.0005 + 0.00001 0.0005 - 0.00001 1.0000 + 0.00001 Columns 17 through 20 -0.0044 + 0.02101 0.2427 - 0.13001 0.2427 + 0.13001 0.0488 - 0.09351 -0.1125 - 0.08831 -0.1306 + 0.02201 -0.l306 - 0.02201 -0.2388 + 0.23021 0.1139 + 0.04421 -0.5473 + 0.26081 -0.5473 - 0.26081 0.0594 - 0.02891 -0.0224 - 0.00641 -0.2464 - 0.35561 -0.2464 + 0.35561 -0.1665 - 0.06901 0.1043 - 0.11491 0.0578 + 0.19991 0.0578 - 0.19991 0.4277 + 0.37221 -0.0568 + 0.11981 0.5053 + 0.80751 0.5053 - 0.80751 —0.0580 - 0.09681 -0.0060 - 0.03141 0.0044 + 0.04491 0.0044 - 0.04491 0.0059 + 0.01641 0.0756 - 0.01611 0.0265 - 0.00821 0.0265 + 0.00821 -0.1274 - 0.21091 -0.0310 + 0.11671 -0.0517 - 0.15381 -0.0517 + 0.15381 0.0179 + 0.04981 0.0012 - 0.00851 -0.0024 + 0.00871 -0.0024 - 0.00871 0.0006 + 0.00331 0.0512 - 0.00081 0.0172 - 0.00251 0.0172 + 0.00251 -0.0055 - 0.05601 -0.0232 + 0.10011 -0.0077 - 0.07271 -0.0077 + 0.07271 0.0105 + 0.01991 -0.0011 + 0.00821 0.0025 - 0.00841 0.0025 + 0.00841 -0.0005 - 0.00321 -0.0485 - 0.00061 -0.0162 + 0.00271 -0.0162 - 0.00271 0.0062 + 0.05271 0.0236 - 0.09381 0.0075 + 0.06841 0.0075 - 0.06841 -0.0098 - 0.01871 -0.0108 + 0.03101 -0.0096 - 0.03621 -0.0096 + 0.03621 -0.0101 - 0.01071 -0.2973 + 0.09251 -0.1022 - 0.00541 -0.1022 + 0.00541 -0.0275 + 0.33131 0.0834 - 0.62081 0.0790 + 0.42731 0.0790 - 0.42731 -0.0766 - 0.11201 -0.2943 + 0.06821 -0.3884 - 0.20211 -0.3884 + 0.20211 -0.1765 - 0.00741 -0.1373 - 0.57871 0.0162 + 0.28251 0.0162 - 0.28251 1.0000 + 0.00001 1.0000 - 0.00001 1.0000 + 0.00001 1.0000 - 0.00001 -0.3129 + 0.12021 Column 21 0.0488 + 0.09351 -0.2388 - 0.23021 0.0594 + 0.02891 -0.1665 + 0.06901 0.4277 - 0.37221 -0.0580 + 0.09681 0.0059 - 0.01641 -0.1274 + 0.21091 0.0179 - 0.04981 0.0006 - 0.00331 -0.0055 + 0.05601 0.0105 - 0.01991 -0.0005 + 0.00321 0.0062 - 0.05271 -0.0098 + 0.01871 -0.0101 + 0.01071 -0.0275 - 0.33131 -0.0766 + 0.11201 -0.1765 + 0.00741 1.0000 - 0.00001 -0.3129 - 0.12021 eigad - 1.0e+03 * -0.0497 + 1.00031 -0.0497 - 1.00031 -0.0516 + 0.40591 -0.0516 - 0.40591 0.0433 0.12071 0.0433 0.12071 0.0461 0.07431 0.0461 0.07431 0.0485 0.0431 0.0322 0.0348 0.0000 0.0122 l+l+ 0.0053 0.0068 + 0.00031 0.0068 - 0.00031 0.0130 vad - Columns 1 through 4 0.0009 + 0.00491 0.0009 - 0.0317 + 0.01541 0.0317 - 0.0000 + 0.00001 0.0000 - 0.2943 - 0.03041 0.2943 + 1.0000 + 0.00001 1.0000 - 0.0018 - 0.00041 0.0018 + -0.0014 - 0.00941 -0.0014 -0.0016 - 0.03451 -0.0016 0.0003 - 0.00011 0.0003 + 0.0009 + 0.01631 0.0009 - -0.0003 + 0.00561 -0.0003 0.0002 + 0.00161 0.0002 - -0.0001 + 0.00081 -0.0001 0.0001 + 0.00261 0.0001 - 0.0000 + 0.00001 0.0000 - -0.0002 - 0.00231 -0.0002 0.0000 - 0.00121 0.0000 + -0.0001 - 0.00041 -0.0001 Columns 5 through 8 -0.0292 + 0.13461 -0.0292 0.0514 - 0.06541 0.0514 + -0.0547 - 0.00321 ~0.0547 1.0000 + 0.00001 1.0000 - -0.5237 - 0.43591 -0.5237 0.0603 + 0.40431 0.0603 - 0.0815 - 0.22911 0.0815 + -0.1380 + 0.11221 -0.1380 0.1095 + 0.01071 0.1095 - 0.0024 + 0.00661 0.0024 - 0.0002 - 0.00051 0.0002 + -0.0394 + 0.02101 -0.0394 -0.0038 + 0.01981 -0.0038 0.0084 - 0.01091 0.0084 + -0.0112 - 0.00421 -0.0112 -0.0013 + 0.00031 -0.0013 -0.0018 + 0.00081 -0.0018 0.0074 - 0.00821 0.0074 + Columns 9 through 12 252 Table C.4.7 (cont’d) 0.00491 0.01541 0.00001 0.03041 0.00001 0.00041 0.0069 + 0.04081 0.0069 - -0.0039 - 0.00121 -0.0039 0.0007 + 0.00151 0.0007 - 1.0000 + 0.00001 1.0000 - -0.0683 + 0.03631 -0.0683 0.0392 - 0.01041 0.0392 + 0.04081 + 0.00121 0.00151 0.00001 - 0.03631 0.01041 + 0.00941 -0.0097 - 0.07781 -0.0097 + 0.07781 + 0.03451 0.0063 + 0.00471 0.0063 - 0.00471 0.00011 -0.0014 - 0.00321 -0.0014 + 0.00321 0.01631 -0.0028 + 0.02241 -0.0028 - 0.02241 - 0.00561 -0.0001 - 0.00021 -0.0001 + 0.00021 0.00161 0.0019 + 0.01401 0.0019 - - 0.00081 0.0003 + 0.00611 0.0003 0.00261 0.00001 -0.0004 - 0.00031 -0.0004 -0.0003 + 0.00041 -0.0003 + 0.00231 0.0001 - 0.00301 0.0001 0.00121 0.0001 + 0.00011 0.0001 * 0.01401 - 0.00611 + 0.00031 - 0.00041 + 0.00301 0.00011 + 0.00041 -0.0007 - 0.00331 -0.0007 + 0.00331 - 0.13461 -0.0775 + 0.19871 -0.0775 - 0.19871 0.06541 0.1161 - 0.04851 0.1161 + 0.04851 + 0.00321 -0.0664 - 0.06381 -0.0664 + 0.06381 0.00001 1.0000 + 0.00001 1.0000 - 0.00001 + 0.43591 -0.3984 - 0.55641 -0.3984 + 0.55641 0.40431 -0.1710 + 0.41001 -0.1710 - 0.41001 0.22911 0.1774 - 0.30781 0.1774 + 0.30781 - 0.11221 0.01071 0.00661 0.00051 0.1149 + 0.12231 0.1149 - 0.0026 + 0.00411 0.0026 - 0.0001 - 0.00031 0.0001 + -0.2474 + 0.07311 -0.2474 - 0.07311 0.12231 0.00411 0.00031 - 0.02101 -0.0327 + 0.00511 -0.0327 - 0.00511 - 0.01981 -0.0075 + 0.03001 -0.0075 - 0.03001 0.01091 0.0166 - 0.01281 0.0166 + 0.01281 + 0.00421 -0.0116 - 0.01371 -0.0116 + 0.01371 - 0.00031 -0.0032 + 0.00171 -0.0032 - 0.00171 - 0.00081 -0.0017 + 0.00001 -0.0017 - 0.00001 0.00821 0.0078 - 0.00721 0.0078 + 0.00721 0.1906 0.2824 0.9827 0.3679 0.2027 0.0675 0.9099 0.2285 0.1776 0.0923 1.0000 0.1636 -0.3385 -0.4352 -0.8788 -0.3940 -0.3344 -0.1854 -0.1903 -0.l816 —0.0004 -0.1537 -0.1362 -0.3549 -0.0689 -0.0341 -0.0012 -0.9823 -0.9396 -0.9483 -0.9603 -0.9854 -0.0018 -0.3606 -0.3230 ~0.1803 -0.3237 -0.3099 -0.1404 0.0002 0.0000 -0.0001 -0.0002 -0.0001 -0.0055 0.0091 0.5611 -0.0792 -0.0207 1.0000 1.0000 -0.4967 -0.0153 -0.6178 0.5887 1.0000 -0.0265 -0.0916 0.3758 0.0443 ~0.1093 -0.2670 0.0194 0.2136 -0.3504 0.0333 -0.2355 -0.7514 -0.3470 -0.2715 Columns 1.0000 0.9880 0.9989 0.9989 0.9927 -0.0263 13 through 16 -0.0067 0.0297 0.1796 1.0000 + 0.00001 0.1200 -0.0661 ~0.3000 ~ 0.13901 0.1925 -0.0651 -0.5674 + 0.11491 0.0171 0.1333 0.6044 - 0.01871 0.0581 -0.0220 -0.1774 - 0.07741 253 Table C.4.7 (cont’d) 0.9987 -0.0553 -0.0378 -0.3350 + 0.07871 0.9953 0.0082 0.0469 0.1478 - 0.05311 0.9966 0.0310 -0.0696 -0.2812 + 0.02531 0.9988 0.0455 -0.0256 -0.0045 + 0.04591 -0.0002 0.0003 0.0004 0.0001 - 0.00001 0.0000 0.0000 0.0000 0.0000 - 0.00001 0.0000 -0.0051 0.0083 -0.0018 + 0.00011 0.0000 0.3593 0.3840 0.0861 - 0.01751 -0.0001 0.5149 0.4281 0.0418 - 0.02231 0.0002 -0.3233 0.6029 0.0100 - 0.01561 -0.0001 1.0000 0.5516 0.0916 - 0.02791 0.0002 -0.2820 1.0000 0.0407 - 0.03061 0.0001 -0.4760 0.8592 0.0515 - 0.02741 Columns 17 through 18 1.0000 - 0.00001 -0.1027 -0.3000 + 0.13901 0.1876 -0.5674 - 0.11491 -0.1264 0.6044 + 0.01871 -0.0333 -0.1774 + 0.07741 0.0454 -0.3350 - 0.07871 -0.0275 0.1478 + 0.05311 0.0197 -0.2812 - 0.02531 -0.1071 -0.0045 - 0.04591 0.1238 0.0001 + 0.00001 -0.0004 0.0000 + 0.00001 0.0000 -0.0018 - 0.00011 0.0000 0.0861 + 0.01751 -0.3763 0.0418 + 0.02231 0.1533 0.0100 + 0.01561 0.0219 0.0916 + 0.02791 -0.1620 0.0407 + 0.03061 1.0000 0.0515 + 0.02741 -0.9890 detjj - —7.1532e+30 detd - 2.7013e+21 detda - -2.6481e+09 detad - 3.4097e+28 condjj - 2.3231e+06 condd - 63.2110 condda - 1.114le+06 condad - 1.5128e+07 vtt - 1.0000 0.0031 -0.3339 -0.5743 1.0000 -0.2918 0.2355 0.0020 1.0000 eigtt - 1.0e+03 * -0.1335 -6.7572 -0.0630 254 Table C.4.7 (cont’d) eigjj - ~26.3481 -26.3481 48.2581 43.6372 39.6573 —12.8140 -12.8140 24.8905 23.1327 21.2563 -10.2704 -10.2704 -15.8155 -16.5809 -16.6067 255 Table C.4.8 Output for algebraic/dynamic 140 MVar +52.40221 -52.40221 +25.40191 -25.40191 +19.95171 -19.95171 -5.3659 -5.3659 -4.0238 -4.0238 -4.2413 -4.2413 11.0753 11.0753 8.6791 7.5169 1.8834 -0.5052 -0.5052 0.8780 -0.1406 -0.1406 0.3163 0.0342 -0.1181 -0.1181 -0.0535 -0.0535 -0.0490 -0.0490 eigda - +14.24411 -14.24411 +14.97251 -14.97251 +15.04471 -15.04471 + 0.76591 - 0.76591 + 1.57781 - 1.57781 + 1.19691 - 1.19691 + 0.91801 - 0.91801 + 0.64611 - 0.64611 + 0.58331 - 0.58331 -26.5105 +52.21761 -26.5105 -3.8992 -3.8992 -4.0274 -4.0274 -15.1892 -16.4542 -16.5408 —0.3255 -0.3255 -0.0196 -0.0196 -0.0925 -0.0758 -0.0550 -0.0550 -0.0743 -0.0743 -0.0770 -0.0770 vda - + + 0.64541 + -52.2176 +14.90571 -14.90571 +14.98771 -14.98771 1.70861 1.70861 1.14141 1.14141 I+l+ 0.58821 0.58821 0.64541 0.92711 0.92711 1 Columns 1 through 4 0.0000 ~ 0.0000 + 0.0000 + 0.0000 + 0.0000 - 0.0000 - 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 eigd - -12.6492 +25.59031 -25.59031 -12.6492 48.2580 43.6386 39.6612 -10.0834 +19.95011 -10.0834 -l9.95011 -5.4933 -5.4933 24.8936 23.1728 21.2755 1.3797 1.8751 11.1611 11.1611 7.7588 8.5551 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 +14.28381 -14.28381 + 0.84621 - 0.84621 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 256 Table C.4.8 (cont’d) -0.0002 + 0.00031 -0.0002 - 0.00031 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 -0.0011 + 0.00071 -0.0011 - 0.00071 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 + 0.00021 0.0000 - 0.00021 0.0001 + 0.00001 0.0001 - 0.00001 0.0000 + 0.00021 0.0000 - 0.00021 0.0000 + 0.00001 0.0000 - 0.00001 -0.0001 + 0.00071 -0.0001 - 0.00071 0.0000 + 0.00001 0.0000 - 0.00001 0.0001 + 0.00031 0.0001 - 0.00031 -0.0048 - 0.01051 -0.0048 + 0.01051 0.0000 - 0.00021 0.0000 + 0.00021 0.0000 - 0.00001 0.0000 + 0.00001 -0.0021 - 0.03051 -0.0021 + 0.03051 0.0000 - 0.00001 0.0000 + 0.00001 -0.0087 - 0.00591 -0.0087 + 0.00591 1.0000 - 0.00001 1.0000 + 0.00001 0.0044 - 0.00081 0.0044 + 0.00081 0.0000 + 0.00031 0.0000 - 0.00031 1.0000 - 0.00001 1.0000 + 0 00001 0.0000 + 0.00031 0.0000 - 0.00031 0.2067 - 0.27181 0.2067 + 0.27181 -0.0761 - 0.15381 -0.0761 0.0000 - 0.00001 0.0000 + + 0.15381 -0.0011 - 0.00271 -0.0011 + 0.00271 0.00001 -0.0289 - 0.11051 -0.0289 + 0.11051 0.0000 - 0.00001 0.0000 + 0.00001 -0.0302 - 0.01321 —0.0302 + 0.01321 Columns 5 through 8 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 0.0000 0.0000 + 0.00001 0.0000 - 0.00001 -0.0001 0.0000 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 0.0000 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 0.0000 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 0.0000 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 0.0000 0.0000 + 0.00001 0.0000 - 0.00001 0.0048 -0.0003 0.0001 - 0.00031 0.0001 + 0.00031 0.0003 0.0012 -0.0005 + 0.00031 ~0.0005 0.0000 + 0.00011 0.0000 - -0.0001 + 0.00001 -0.0001 -0.0001 + 0.00051 -0.0001 0.0000 - 0.00011 0.0000 + 0.0064 + 0.00431 0.0064 - -0.0018 - 0.03051 -0.0018 0.0022 + 0.00081 0.0022 - -0.1568 + 0.19841 -0.1568 1.0000 + 0.00001 1.0000 - 0.0002 - 0.00151 0.0002 + 0.0263 + 0.01131 0.0263 - -0.0232 - 0.09301 -0.0232 - 0.00031 0.0001 0.0004 0.00011 0.0458 -0.0031 - 0.00001 0.0173 0.0566 - 0.00051 0.0174 0.0374 0.00011 -0.0527 0.0036 0.00431 -0.0111 -0.0337 + 0.03051 ~0.0113 -0.0224 0.00081 1.0000 -0.0749 - 0.19841 0.3002 1.0000 0.00001 0.2992 0.6543 0.00151 -0.6886 0.0474 0.01131 -0.0348 -0.1069 + 0.09301 -0.0297 -0.0599 Columns 9 through 12 0.0000 0.0054 - 0.00021 0.0054 + 0.00021 -0.0326 + 0.00951 0.0000 -0.0105 - 0.00341 -0.0105 + 0.00341 0.1068 + 0.00511 0.0000 -0.0034 + 0.00351 -0.0034 - 0.00351 -0.0249 - 0.01781 0.0000 -0.0007 - 0.00301 -0.0007 + 0.00301 0.0088 + 0.02841 0.0000 -0.0008 + 0.00631 -0.0008 - 0.00631 0.0028 - 0.09361 0.0000 0.0023 + 0.00161 0.0023 - 0.00161 -0.0152 + 0.02201 0.0000 —0.0190 - 0.05441 -0.0190 + 0.05441 0.0442 + 0.04221 -0.0006 -0.0245 + 0.00991 -0.0245 - 0.00991 -0.0413 - 0.14351 0.0006 -0.0115 + 0.00611 -0.0115 - 0.00611 -0.0464 - 0.03141 -0.0005 -0.0261 - 0.04081 -0.0261 + 0.04081 0.0226 + 0.00671 -0.0289 -0.0246 - 0.00401 -0.0246 + 0.00401 -0.0331 - 0.09711 0.0571 0.0005 0.0171 -0.0224 - 0.00531 —0.0224 + 0.00531 -0.0454 - 0.04501 0.0258 + 0.03911 0.0258 - 0.03911 -0.0217 - 0.00651 0.0238 + 0.00261 0.0238 - 0.00261 0.0347 + 0.09071 -0.0340 0.0216 + 0.00411 0.0216 - 0.00411 0.0436 + 0.04121 -0.0111 0.0342 + 0.17091 0.0342 - 0.17091 -0.0896 - 0.02401 -0.5107 0.1224 + 0.10321 0.1224 - 0.10321 -0.0080 + 0.64891 1.0000 0.0070 0.0543 0.1178 + 0.08321 0.1178 - 0.08321 0.2604 + 0.32021 1.0000 - 0.00001 1.0000 + 0.00001 -0.4959 + 0.50371 0.0798 - 0.14091 0.0798 + 0.14091 1.0000 + 0.00001 -0.0911 0.0700 - 0.10341 0.0700 + 0.10341 0.4689 - 0.22551 Columns 13 through 16 -0.0326 - 0.00951 -0.0925 -0.0758 0.0583 - 0.10401 0.1068 - 0.00511 -0.0885 -0.0721 -0.2496 + 0.24031 -0.0249 + 0.01781 -0.0920 -0.0753 0.0425 - 0.01411 0.0088 0.0028 - 0.02841 1.0000 1.0000 -0.1845 - 0.08181 + 0.09361 0.9569 0.9517 0.4444 + 0.38271 -0.0152 - 0.02201 0.9951 0.9929 -0.0305 - 0.06931 0.0442 - 0.04221 0.0008 0.0005 0.0065 + 0.01831 -0.0413 + 0.14351 0.0072 0.0082 -0.1258 - 0.20721 -0.0464 + 0.03141 -0.0007 -0.0007 0.0146 + 0.04271 0.0226 - 0.00671 0.0007 0.0005 0.0006 + 0.00381 -0.0331 + 0.09711 -0.0002 -0.0002 -0.0054 - 0.05661 -0.0454 + 0.04501 0.0000 0.0000 0.0104 + 0.01751 257 Table C.4.8 (cont’d) -0.0217 + 0.00651 -0.0008 -0.0007 -0.0004 - 0.00371 0.0347 - 0.09071 0.0002 0.0002 0.0061 + 0.05331 0.0436 - 0.04121 0.0000 0.0000 -0.0096 - 0.01651 -0.0896 + 0.02401 0.0153 0.0158 -0.0114 - 0.01251 -0.0080 - 0.64891 0.0012 0.0011 -0.0282 + 0.33441 0.2604 - 0.32021 0.0000 0.0001 -0.0918 - 0.08991 -0.4959 - 0.50371 0.2661 0.2673 -0.1987 - 0.01361 1.0000 - 0.00001 -0.0245 -0.0264 1.0000 + 0.00001 0.4689 + 0.22551 0.0003 0.0003 -0.2746 + 0.12021 Columns 17 through 20 0.0583 + 0.10401 0.2469 - 0.10221 0.2469 + 0.10221 -0.0012 - 0.02011 -0.2496 - 0.24031 -0.0826 - 0.01481 -0.0826 + 0.01481 -0.1216 + 0.08851 0.0425 + 0.01411 -0.5955 + 0.22991 -0.5955 - 0.22991 0.1134 - 0.04761 -0.1845 + 0.08181 -0.1998 - 0.35951 -0.1998 + 0.35951 -0.0214 + 0.00311 0.4444 - 0.38271 -0.0081 + 0.12891 -0.0081 - 0.12891 0.1056 + 0.12241 -0.0305 + 0.06931 0.4564 + 0.87021 0.4564 - 0.87021 -0.0611 - 0.11721 0.0065 - 0.01831 0.0024 + 0.04171 0.0024 - 0.04171 -0.0073 + 0.03011 -0.1258 + 0.20721 0.0390 + 0.03851 0.0390 - 0.03851 0.0769 + 0.02981 0.0146 - 0.04271 -0.0432 - 0.15071 -0.0432 + 0.15071 -0.0304 — 0.11391 0.0006 - 0.00381 -0.0030 + 0.00821 -0.0030 - 0.00821 0.0005 + 0.00851 -0.0054 + 0.05661 0.0151 + 0.01071 0.0151 - 0.01071 0.0528 + 0.00811 0.0104 - 0.01751 -0.0088 - 0.07421 -0.0088 + 0.07421 -0.0235 - 0.10161 -0.0004 + 0.00371 0.0031 - 0.00781 0.0031 + 0.00781 -0.0004 - 0.00821 0.0061 - 0.05331 -0.0145 - 0.00981 -0.0145 + 0.00981 -0.0502 - 0.00621 -0.0096 + 0.01651 0.0079 0.06971 0.0079 - 0.06971 0.0232 + 0.09521 + -0.0114 + 0.01251 -0.0063 - 0.03571 -0.0063 + 0.03571 -0.0085 - 0.03131 -0.0282 - 0.33441 -0.0748 - 0.07961 -0.0748 + 0.07961 -0.2932 - 0.13791 -0.0918 + 0.08991 0.1473 + 0.43611 0.1473 - 0.43611 0.1455 + 0.62651 -0.1987 + 0.01361 -0.3490 - 0.22281 -0.3490 + 0.22281 -0.2815 - 0.08791 1.0000 - 0.00001 -0.1930 + 0.22451 -0.1930 - 0.22451 -0.2174 + 0.57311 -0.2746 - 0.12021 1.0000 + 0.00001 1.0000 - 0.00001 1.0000 - 0.00001 Column 21 -0.0012 + 0.02011 -0.1216 ~ 0.08851 0.1134 + 0.04761 -0.0214 - 0.00311 0.1056 - 0.12241 -0.0611 + 0.11721 -0.0073 - 0.03011 0.0769 - 0.02981 -0.0304 + 0.11391 0.0005 - 0.00851 0.0528 - 0.00811 -0.0235 + 0.10161 -0.0004 + 0.00821 -0.0502 + 0.00621 0.0232 - 0.09521 -0.0085 + 0.03131 -0.2932 + 0.13791 0.1455 - 0.62651 -0.2815 + 0.08791 -0.2174 - 0.57311 1.0000 + 0.00001 eigad - 1.0e+02 * -0.4749 + 6.94291 -0.4749 - 6.94291 -0.4694 + 4.12191 -0.4694 - 4.12191 0.7469 + 0.67411 0.7469 - 0.67411 0.0822 + 0.84941 0.0822 - 0.84941 0.4806 0.4193 0.3465 0.3149 0.0002 0.1208 0.0503 258 Table C.4.8 (cont’d) 0.0669 + 0.00141 0.0669 - 0.00141 0.1262 vad 8 Columns 1 through 4 0.0047 + 0.01911 0.0047 - 0.01911 0.0063 + 0.04031 0.0063 - 0.04031 0.0311 + 0.02211 0.0311 - 0.02211 -0.0026 - 0.00091 -0.0026 + 0.00091 0.0001 + 0.00011 0.0001 - 0.00011 0.0004 + 0.00091 0.0004 - 0.00091 0.7964 - 0.12061 0.7964 + 0.12061 1.0000 - 0.00001 1.0000 + 0.00001 1.0000 - 0.00001 1.0000 + 0.00001 -0.0484 + 0.02391 -0.0484 - 0.02391 0.0064 - 0.00191 0.0064 + 0.00191 0.0233 - 0.00541 0.0233 + 0.00541 -0.0080 - 0.03621 -0.0080 -0.0028 - 0.04951 -0.0028 0.0006 - 0.00041 0.0006 + 0.0025 + 0.03051 0.0025 - -0.0005 + 0.00791 -0.0005 0.0014 + 0.00611 0.0014 - 0.0001 + 0.00301 0.0001 - 0.0002 + 0.00371 0.0002 - -0.0002 + 0.00011 -0.0002 -0.0006 - 0.00421 -0.0006 0.0000 - 0.00171 0.0000 + -0.0004 - 0.00151 -0.0004 Columns 5 through 8 -0.1245 + 0.14571 -0.1245 0.0819 - 0.00801 0.0819 + -0.0208 - 0.04521 -0.0208 1.0000 + 0.00001 1.0000 - -0.3280 - 0.41361 -0.3280 -0.1150 + 0.25011 -0.1150 0.2286 - 0.21641 0.2286 + -0.1855 + 0.01391 -0.1855 0.0409 + 0.08021 0.0409 - 0.0041 + 0.00371 0.0041 - 0.0000 - 0.00061 0.0000 + -0.0345 + 0.01461 -0.0345 -0.0171 + 0.02511 -0.0171 0.0158 - 0.00631 0.0158 + -0.0042 - 0.01271 -0.0042 -0.0033 - 0.00071 -0.0033 -0.0015 + 0.00051 -0.0015 0.0097 - 0.00771 0.0097 + Columns 9 through 12 + 0.03621 -0.0085 - 0.07671 -0.0085 + 0.07671 + 0.04951 0.0049 + 0.00321 0.0049 - 0.00321 0.00041 -0.0007 - 0.00181 -0.0007 + 0.00181 0.03051 -0.0025 + 0.02251 -0.0025 - 0.02251 - 0.00791 -0.0001 - 0.00021 -0.0001 + 0.00021 0.00611 0.0015 + 0.01331 0.0015 - 0.01331 0.00301 0.0002 + 0.00611 0.0002 ~ 0.00611 0.00371 -0.0003 - 0.00021 -0.0003 + 0.00021 - 0.00011 -0.0004 + 0.00031 -0.0004 - 0.00031 + 0.00421 0.0001 - 0.00301 0.0001 + 0.00301 0.00171 0.0000 + 0.00011 0.0000 - 0.00011 + 0.00151 -0.0006 - 0.00321 -0.0006 + 0.00321 - 0.14571 0.0206 + 0.19801 0.0206 - 0.19801 0.00801 0.0801 - 0.14111 0.0801 + 0.14111 + 0.04521 -0.l319 + 0.01641 -0.1319 - 0.01641 0.00001 1.0000 + 0.00001 1.0000 - 0.00001 + 0.41361 -0.6277 — 0.60301 -0.6277 + 0.60301 - 0.25011 0.0165 + 0.70021 0.0165 - 0.70021 0.21641 0.0346 - 0.34881 0.0346 + 0.34881 - 0.01391 -0.2056 + 0.24511 -0.2056 - 0.24511 0.08021 0.2629 + 0.00001 0.2629 - 0.00001 0.00371 0.0005 + 0.00461 0.0005 - 0.00461 0.00061 0.0005 - 0.00071 0.0005 + 0.00071 - 0.01461 -0.0808 + 0.00961 -0.0808 - 0.00961 - 0.02511 0.0036 + 0.02601 0.0036 - 0.02601 0.00631 0.0084 - 0.02041 0.0084 + 0.02041 + 0.01271 -0.0252 - 0.00801 -0.0252 + 0.00801 + 0.00071 0.0002 + 0.00201 0.0002 - 0.00201 - 0.00051 -0.0032 + 0.00051 -0.0032 - 0.00051 0.00771 0.0134 - 0.00971 0.0134 + 0.00971 0.1357 0.3301 0.3042 0.9407 0.1842 0.1148 0.1808 0.9033 0.1569 0.1200 0.1143 1.0000 -0.2348 -0.3481 ~0.3046 -0.1158 -0.1737 -0.1675 -0.0001 1.0000 -0.4844 -0.2258 ~0.1801 -0.4002 -0.1205 -0.0636 -0.0012 -0.2915 -0.2592 -0.1318 -0.2560 -0.2585 -0.0953 -0.8016 -0.9305 -0.9458 -0.8831 -0.9517 -0.9922 0.0003 -0.0016 0.0000 -0.0002 -0.0002 -0.0003 -0.0100 0.0116 -0.0089 -0.0417 0.5430 0.0007 1.0000 0.0270 -0.4252 -0.6110 -0.0367 0.4500 1.0000 -0.1241 -0.0455 -0.7522 0.3047 -0.1180 0.0274 -0.2886 -0.2700 0.2127 0.0364 -0.2242 -0.3762 -0.2198 0.0365 Columns 13 through 16 1.0000 0.0556 0.1315 1.0000 - 0.00001 0.9881 0.0629 -0.0578 -0.2796 - 0.06291 0.9994 -0.1466 -0.0417 -0.5886 + 0.05241 0.9990 0.0268 0.1009 0.6088 - 0.00851 0.9928 0.0409 -0.0196 -0.1640 - 0.03541 0.9990 -0.0407 -0.0225 -0.3457 + 0.03581 259 Table C.4.8 (cont’d) 0.9953 0.0024 0.0351 0.1576 - 0.02431 0.9968 0.0554 —0.0597 -0.2838 + 0.01131 0.9990 0.0178 -0.0228 -0.0076 + 0.02081 -0.0002 0.0004 0.0003 0.0001 - 0.00001 0.0000 0.0001 0.0001 0.0000 - 0.00001 0.0000 -0.0072 0.0126 -0.0027 + 0.00011 0.0000 0.4355 0.3412 0.0903 - 0.00781 -0.0001 0.4659 0.3936 0.0486 — 0.01021 0.0002 —0.2986 0.5803 0.0170 - 0.00761 -0.0001 1.0000 0.4941 0.0992 - 0.01251 0.0003 -0.4777 1.0000 0.0591 - 0.01561 0.0001 -0.2004 0.7774 0.0590 - 0.01251 Columns 17 through 18 1.0000 + 0.00001 0.0839 -0.2796 + 0.06291 -0.1895 -0.5886 - 0.05241 0.1638 0.6088 + 0.00851 0.0275 -0.1640 + 0.03541 -0.0547 -0.3457 - 0.03581 0.0374 0.1576 + 0.02431 -0.0177 -0.2838 - 0.01131 0.0820 -0.0076 - 0.02081 -0.1223 0.0001 + 0.00001 0.0002 0.0000 + 0.00001 -0.0001 -0.0027 - 0.00011 0.0027 0.0903 + 0.00781 0.2396 0.0486 + 0.01021 -0.2451 0.0170 + 0.00761 0.0959 0.0992 + 0.01251 -0.1003 0.0591 + 0.01561 -0.7730 0.0590 + 0.01251 1.0000 detjj - -6.0033e+30 detd - 2.103le+21 detda - -2.8546e+09 detad - 8.0480e+27 condjj - 2.4087e+06 condd - 61.8176 condda - 1.1557e+06 condad - 7.6703e+06 vtt - 1.0000 0.0062 -0.2789 260 Table C.4.8 (cont’d) -0.5335 1.0000 -0.3489 0.1256 0.0027 1.0000 eigtt - 1.0e+03 * -0.1274 -3.5600 -0.0436 81931 " -26.362 -26.362 47.9863 41.9016 39.4275 -12.718 -12.718 24.7753 5 5 8 8 261 Table C.4.9 Output for algebraic/dynamic 160 MVar +52.4091 -52.4091 +25.l943 -25.1943 -9.9658 +19.42801 -9.9658 -19.42801 22.7029 20.8268 -15.816 -16.581 -16.606 -5.2985 -5.2985 -4.2669 -4.2669 -4.0509 -4.0509 10.7543 10.7543 8.8583 7.2567 1.7696 -0.5073 -0.5073 0.8926 -0.1464 -0.1464 0.3148 0.0329 -0.1347 -0.1347 -0.0524 -0.0524 -0.0527 -0.0527 eigda - 8 2 4 +13.91751 -13.91751 +15.11411 -15.11411 +14.96801 -14.96801 + 0.72911 0.72911 1.57781 1.57781 1.19951 1.19951 0.93011 0.93011 0.58961 0.58961 0.65701 0.65701 1 1 1 1 -26.5260 +52.22351 -26.526 0 -52.2235 -3.9004 +14.90371 -3.9004 ~14.90371 -4.0921 +14.99121 -4.0921 -14.99121 -16.444 -15.l75 -16.535 -0.3094 -0.3094 -0.0078 -0.0078 -0.0668 -0.1015 -0.0593 -0.0593 -0.0760 -0.0760 -0.0932 -0.0932 vda - 7 8 4 1.71911 1.71911 1.16381 1.16381 I+l+ 0.59431 0.59431 0.65571 0.65571 0.93641 0.93641 1 Columns 1 through 4 0.0000 0.0000 0.0000 0.0000 0.0000 1 +-+ +1 0.00001 0.00001 0.00001 0.00001 0.00001 0.0000 0.0000 0.0000 0.0000 0.0000 +lll+ 0.00001 0.00001 0.00001 0.00001 0.00001 0.0000 0.0000 0.0000 0.0000 0.0000 +1 +1 + eigd - -12.5516 +25.39081 -25.39081 -12.5516 47.9862 41.9035 39.4309 -9.7834 -9.7834 -5.4046 -5.4046 24.7776 22.7421 20.8486 1.3117 1.8161 10.8424 10.8424 7.4401 8.7768 0.00001 0.0000 0.00001 0.0000 0.00001 0.0000 0.00001 0.0000 0.00001 0.0000 +19.4176i -19.41761 +14.01091 -14.01091 + 0.79741 - 0.79741 I+I+l 0.00001 0.00001 0.00001 0.00001 0.00001 262 Table C.4.9 (cont’d) 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 ~0.0002 + 0.00031 -0.0002 - 0.00031 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 -0.0011 + 0.00071 -0.0011 - 0.00071 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 + 0.00021 0.0000 - 0.00021 0.0001 + 0.00001 0.0001 - 0.00001 0.0000 + 0.00021 0.0000 - 0.00021 0.0000 + 0.00001 0.0000 - 0.00001 -0.0001 + 0.00071 —0.0001 - 0.00071 0.0000 + 0.00001 0.0000 - 0.00001 0.0001 + 0.00031 0.0001 - 0.00031 -0.0048 - 0.01051 -0.0048 + 0.01051 0.0000 - 0.00021 0.0000 + 0.00021 0.0000 - 0.00001 0.0000 + 0.00001 -0.0021 - 0.03051 -0.0021 + 0.03051 0.0000 - 0.00001 0.0000 + 0.00001 -0.0075 - 0.00401 -0.0075 + 0.00401 1.0000 - 0.00001 1.0000 + 0.00001 0.0044 - 0.00081 0.0044 + 0.00081 0.0000 + 0.00031 0.0000 - 0.00031 1.0000 + 0.00001 1.0000 - 0.00001 0.0000 + 0.00031 0.0000 0.00031 0.1416 - 0.23931 0.1416 + 0.23931 -0.0761 - 0.15391 -0.0761 + 0.15391 -0.0011 - 0.00271 -0.0011 + 0.00271 0.0000 - 0.00001 0.0000 + 0.00001 -0.0289 - 0.11051 -0.0289 + 0.11051 0.0000 — 0.00001 0.0000 + 0.00001 -0.0256 - 0.00811 -0.0256 + 0.00811 Columns 5 through 8 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 0.0000 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 -0.0001 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 0.0000 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 0.0000 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 0.0000 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 0.0000 0.0000 + 0.00001 0.0000 - 0.00001 -0.0004 0.0048 0.0000 - 0.00021 0.0000 + 0.00021 0.0011 0.0003 -0.0005 + 0.00031 -0.0005 0.0000 + 0.00011 0.0000 - -0.0001 + 0.00011 -0.0001 -0.0001 + 0.00051 -0.0001 0.0000 - 0.00011 0.0000 + 0.0054 + 0.00281 0.0054 - -0.0017 - 0.03051 -0.0017 0.0026 + 0.00071 0.0026 - -0.1057 + 0.16881 -0.1057 1.0000 + 0.00001 1.0000 - 0.0001 - 0.00181 0.0001 + 0.0216 + 0.00651 0.0216 - -0.0227 - 0.09321 -0.0227 - 0.00031 0.0004 0.0001 0.00011 -0.0035 0.0459 - 0.00011 0.0566 0.0178 - 0.00051 0.0434 0.0191 0.00011 0.0040 -0.0529 0.00281 -0.0337 -0.0115 + 0.03051 -0.0261 -0.0124 0.00071 -0.0838 1.0000 - 0.16881 1.0000 0.3094 0.00001 0.7568 0.3260 0.00181 0.0532 -0.6907 0.00651 -0.1070 -0.0359 + 0.09321 -0.0699 -0.0327 Columns 9 through 12 0.0000 0.0054 - 0.00011 0.0054 + 0.00011 -0.0312 + 0.00971 0.0000 -0.0103 - 0.00351 -0.0103 + 0.00351 0.0975 + 0.00181 0.0000 -0.0034 + 0.00341 -0.0034 - 0.00341 -0.0198 - 0.01561 0.0000 -0.0006 - 0.00301 -0.0006 + 0.00301 0.0085 + 0.02681 0.0000 -0.0009 + 0.00621 -0.0009 - 0.00621 0.0009 - 0.08381 0.0000 0.0023 + 0.00161 0.0023 - 0.00161 -0.0133 + 0.01711 0.0000 -0.0184 - 0.05401 -0.0184 + 0.05401 0.0490 + 0.04581 -0.0007 -0.0248 + 0.01051 -0.0248 - 0.01051 -0.0435 - 0.13931 0.0006 -0.0122 + 0.00641 -0.0122 - 0.00641 -0.0451 — 0.03931 -0.0004 -0.0264 - 0.04051 -0.0264 + 0.04051 0.0258 + 0.00701 -0.0325 -0.0260 - 0.00331 -0.0260 + 0.00331 -0.0350 - 0.09791 0.0574 -0.0254 - 0.00501 -0.0254 + 0.00501 -0.0480 - 0.05731 0.0005 0.0261 + 0.03881 0.0261 - 0.03881 -0.0247 - 0.00671 0.0193 0.0251 + 0.00191 0.0251 - 0.00191 0.0366 + 0.09131 -0.0343 0.0243 + 0.00381 0.0243 - 0.00381 0.0457 + 0.05271 -0.0098 0.0388 + 0.17191 0.0388 - 0.17191 -0.1044 - 0.02511 -0.5755 0.1331 + 0.10401 0.1331 - 0.10401 0.0009 + 0.66161 1.0000 0.1434 + 0.07791 0.1434 - 0.07791 0.3161 + 0.38801 0.0062 1.0000 + 0.00001 1.0000 - 0.00001 -0.5527 + 0.56931 0.0613 0.0802 - 0.15021 0.0802 + 0.15021 1.0000 + 0.00001 -0.0919 0.0755 - 0.11801 0.0755 + 0.11801 0.5648 - 0.19581 Columns 13 through 16 -0.0312 - 0.00971 -0.0668 -0.1015 0.0682 - 0.11351 0.0975 - 0.00181 -0.0635 -0.0976 -0.2643 + 0.25231 -0.0198 + 0.01561 -0.0663 -0.1012 0.0290 - 0.00471 0.0085 - 0.02681 1.0000 1.0000 -0.2005 - 0.09471 0.0009 + 0.08381 0.9507 0.9615 0.4643 + 0.39841 -0.0133 - 0.01711 0.9927 0.9973 -0.0127 - 0.04751 0.0490 - 0.04581 0.0004 0.0010 0.0068 + 0.01991 -0.0435 + 0.13931 0.0080 0.0060 -0.1238 - 0.20351 -0.0451 + 0.03931 -0.0006 -0.0008 0.0132 + 0.03861 0.0258 - 0.00701 0.0004 0.0007 0.0005 + 0.00421 -0.0350 + 0.09791 -0.0002 -0.0002 -0.0053 - 0.05731 263 Table C.4.9 (cont’d) -0.0480 + 0.05731 0.0000 0.0000 0.0110 + 0.01621 -0.0247 + 0.00671 -0.0006 -0.0009 -0.0003 - 0.00411 0.0366 - 0.09131 0.0002 0.0002 0.0060 + 0.05401 0.0457 - 0.05271 0.0000 0.0000 -0.0100 - 0.01541 -0.1044 + 0.02511 0.0170 0.0159 -0.0126 - 0.01441 0.0009 - 0.66161 0.0008 0.0011 -0.0284 + 0.33791 0.3161 - 0.38801 0.0000 0.0000 -0.1154 - 0.06991 -0.5527 - 0.56931 0.2697 0.2672 -0.2180 - 0.02211 1.0000 - 0.00001 -0.0252 -0.0216 1.0000 - 0.00001 0.5648 + 0.19581 0.0001 0.0000 -0.2543 + 0.12761 Columns 17 through 20 0.0682 + 0.11351 0.0245 - 0.25751 0.0245 + 0.25751 0.0028 - 0.01831 -0.2643 - 0.25231 -0.0420 + 0.02401 -0.0420 - 0.02401 -0.1340 + 0.08531 0.0290 + 0.00471 -0.0760 + 0.65571 -0.0760 - 0.65571 0.1137 - 0.04991 -0.2005 + 0.09471 -0.3918 + 0.00801 -0.3918 - 0.00801 -0.0197 - 0.00101 0.4643 - 0.39841 0.0435 + 0.05911 0.0435 - 0.05911 0.1043 + 0.13271 -0.0127 + 0.04751 1.0000 - 0.00001 1.0000 + 0.00001 -0.0647 — 0.11501 0.0068 - 0.01991 0.0361 + 0.01391 0.0361 - 0.01391 -0.0084 + 0.02741 -0.1238 + 0.20351 0.0794 - 0.00491 0.0794 + 0.00491 0.0744 + 0.04911 0.0132 - 0.03861 -0.1472 - 0.02511 -0.1472 + 0.02511 -0.0288 - 0.11151 0.0005 - 0.00421 0.0057 + 0.00611 0.0057 - 0.00611 -0.0004 + 0.00811 -0.0053 + 0.05731 0.0219 - 0.00201 0.0219 + 0.00201 0.0524 + 0.01901 0.0110 - 0.01621 -0.0720 - 0.02071 -0.0720 + 0.02071 -0.0235 - 0.10361 -0.0003 + 0.00411 -0.0053 - 0.00601 -0.0053 + 0.00601 0.0004 - 0.00781 0.0060 - 0.05401 -0.0207 + 0.00231 -0.0207 - 0.00231 -0.0501 - 0.01661 -0.0100 + 0.01541 0.0671 + 0.02041 0.0671 - 0.02041 0.0224 + 0.09691 -0.0126 + 0.01441 -0.0336 - 0.00971 -0.0336 + 0.00971 -0.0054 - 0.03031 -0.0284 - 0.33791 -0.1291 - 0.01351 -0.1291 + 0.01351 -0.2706 - 0.19991 -0.1154 + 0.06991 0.4945 - 0.04291 0.4945 + 0.04291 0.2293 + 0.63291 -0.2180 + 0.02211 -0.3340 + 0.19611 -0.3340 - 0.19611 -0.2548 - 0.10671 1.0000 + 0.00001 -0.0011 + 0.34661 -0.0011 - 0.34661 -0.3273 + 0.53771 -0.2543 - 0.12761 0.3937 - 0.90201 0.3937 + 0.90201 1.0000 - 0.00001 Column 21 0.0028 + 0.01831 -0.1340 - 0.08531 0.1137 + 0.04991 -0.0197 + 0.00101 0.1043 - 0.13271 -0.0647 + 0.11501 -0.0084 - 0.02741 0.0744 - 0.04911 -0.0288 + 0.11151 -0.0004 - 0.00811 0.0524 - 0.01901 —0.0235 + 0.10361 0.0004 + 0.00781 -0.0501 + 0.01661 0.0224 - 0.09691 -0.0054 + 0.03031 -0.2706 + 0.19991 0.2293 - 0.63291 -0.2548 + 0.10671 -0.3273 - 0.53771 1.0000 + 0.00001 eigad - 1.0e+02 * -0.4274 + 5.15591 -0.4274 - 5.15591 -0.4694 + 4.10961 -0.4694 - 4.10961 0.8182 + 0.49551 0.8182 - 0.49551 -0.0305 + 0.69591 -0.0305 - 0.69591 0.4767 0.4029 0.3443 0.3041 0.0002 0.0470 -0.0050 - 0.06151 -0.0050 + 0.06151 —0.0086 - 0.07671 -0.0086 + 0.07671 0.0043 - 0.02391 0.0043 + 0.0003 - 0.00061 0.0003 + -0.0023 + 0.02821 -0.0023 -0.0011 + 0.00371 -0.0011 0.0008 + 0.01041 0.0008 - -0.0002 + 0.00491 -0.0002 -0.0003 + 0.00181 -0.0003 -0.0003 + 0.00011 -0.0003 0.0002 - 0.00081 0.0002 + 264 Table C.4.9 (cont’d) 0.0689 0.1178 0.0627 0.1237 vad - Columns 1 through 4 0.0038 + 0.03251 0.0038 - 0.03251 0.0063 + 0.04031 0.0063 - 0.0086 + 0.01271 0.0086 - 0.01271 -0.0025 - 0.00101 -0.0025 0.0001 + 0.00031 0.0001 - 0.00031 0.0002 + 0.00051 0.0002 - 1.0000 + 0.00001 1.0000 - 0.00001 1.0000 - 0.00001 1.0000 + 0.3530 + 0.07201 0.3530 - 0.07201 -0.0499 + 0.02411 -0.0499 0.0095 - 0.00161 0.0095 + 0.00161 0.0149 - 0.00351 0.0149 + 0.04031 + 0.00101 0.00051 0.00001 - 0.02411 0.00351 0.02391 0.0049 + 0.00331 0.0049 - 0.00331 0.00061 -0.0005 - 0.00111 —0.0005 + 0.00111 0.02821 -0.0025 + 0.02251 -0.0025 - 0.02251 0.00371 -0.0002 - 0.00041 -0.0002 + 0.00041 .01041 0.0015 + 0.01311 0.0015 - 0.01311 0.00491 0.0002 + 0.00611 0.0002 - 0.00611 0.00181 -0.0003 - 0.00021 -0.0003 + 0.00021 0.00011 -0.0004 + 0.00021 -0.0004 - 0.00021 .00081 0.0001 + 0.00011 0.0001 - 0.00011 0 0.0000 - 0.00381 0.0000 + 0.00381 0.0001 - 0.00301 0.0001 + 0.00301 0 + -0.0003 - 0.00251 -0.0003 Columns 5 through 8 -0.1653 + 0.12641 -0.1653 0.0696 + 0.01381 0.0696 - -0.0006 - 0.03971 -0.0006 1.0000 + 0.00001 1.0000 - -0.2498 - 0.34281 -0.2498 -0.1246 + 0.17101 -0.1246 0.2815 - 0.17931 0.2815 + -0.1650 - 0.01721 -0.1650 0.0074 + 0.06861 0.0074 - 0.0045 + 0.00271 0.0045 - -0.0001 - 0.00081 -0.0001 -0.0389 + 0.01591 -0.0389 -0.0250 + 0.02421 -0.0250 0.0165 - 0.00311 0.0165 + 0.0009 - 0.01351 0.0009 + -0.0030 - 0.00211 -0.0030 -0.0015 + 0.00081 -0.0015 0.0120 - 0.00751 0.0120 + I+OOI I+OO+OI+O+OI Columns 9 through 12 0.0714 0.3903 0.2296 -0.8755 0.1604 0.1761 0.1242 -0.8885 0.1357 0.1572 0.0577 -0.9973 0.12641 -0.2034 - 0.11451 .01381 0.0840 + 0.20681 0.0840 - 0.20681 0.03971 0.0983 - 0.17951 0.0983 + 0.17951 .00001 -0.7101 + 0.69871 -0.7101 - 0.69871 0.34281 1.0000 - 0.00001 1.0000 + 0.00001 0.17101 -0.6803 - 0.65691 -0.6803 + 0.65691 .17931 0.2961 + 0.28511 0.2961 - 0.28511 0.01721 -0.0885 - 0.42411 -0.0885 + 0.42411 .06861 -0.2570 + 0.32101 -0.2570 - 0.32101 .00271 -0.0025 - 0.00281 -0.0025 + 0.00281 0.00081 -0.0001 + 0.00141 -0.0001 - 0.00141 0.01591 0.1004 - 0.09521 0.1004 + 0.09521 0.02421 -0.0251 - 0.01391 -0.0251 + 0.01391 .00311 0.0139 + 0.02381 0.0139 — 0.02381 .01351 0.0373 - 0.01941 0.0373 + 0.01941 0.00211 -0.0031 - 0.00051 -0.0031 + 0.00051 0.00081 0.0035 - 0.00311 0.0035 + 0.00311 0.00751 -0.0077 + 0.02401 -0.0077 - 0.02401 -0.1163 -0.5337 -0.2924 -0.2764 -0.0346 -0.1518 -0.1524 0.3310 1.0000 -0.1814 -0.3090 -0.2386 -0.4537 -0.1889 -0.1031 -0.2114 0.6914 -0.1847 0.8503 -0.0719 0.9508 -0.1769 0.7869 -0.1977 0.9323 -0.0437 1.0000 0.0002 -0.0012 0.0005 0.0013 0.0000 —0.0004 -0.0003 0.0006 -0.0168 0.0123 -0.0111 0.0632 0.5213 0.1023 1.0000 -0.0711 1.0000 -0.3815 -0.6012 0.0544 0.0732 -0.7510 0.2405 -0.1309 -0.0061 -0.2371 -0.2655 0.2170 -0.0553 -0.1814 -0.4132 -0.1933 -0.0494 Columns 13 through 16 1.0000 0.9882 0.9997 0.9990 0.9928 0.0939 1.0000 0.1019 -0.0498 -0.3898 -0.0318 -0.0303 -0.5035 -0.0608 0.0751 0.5961 0.0453 -0.0178 -0.2244 0.0112 0.00251 -0.0006 - 0.00311 -0.0006 + 0.00311 -0.2034 + 0.11451 265 Table C.4.9 (cont’d) 0.9993 -0.0150 -0.2814 -0.0174 0.9954 0.0257 0.1171 -0.0060 0.9970 -0.0555 -0.2628 0.0954 0.9992 -0.0200 0.0343 -0.0379 -0.0002 0.0003 0.0001 0.0005 0.0000 0.0001 0.0000 0.0001 0.0000 0.0178 -0.0038 -0.0076 0.0000 0.2936 0.0767 0.5721 -0.0001 0.3542 0.0326 0.3868 0.0002 0.5544 0.0057 -0.2225 -0.0001 0.4295 0.0784 1.0000 0.0003 1.0000 0.0416 -0.8107 0.0002 0.6835 0.0376 0.3108 Columns 17 through 18 1.0000 0.0522 -0.1155 -0.1889 —0.7347 0.2106 0.6332 0.0150 -0.0676 -0.0649 -0.4393 0.0464 0.2244 -0.0150 -0.3113 0.0466 ~0.0564 -0.1222 0.0001 0.0000 0.0000 -0.0001 -0.0044 0.0091 0.1081 0.0338 0.0748 -0.4045 0.0385 0.2461 0.1290 -0.5052 0.1136 -0.4809 0.0871 1.0000 detjj = -4.5853e+30 detd - 1.4648e+21 detda ' ~3.1304e+09 dethd - 2.0335e+27 condjj - 2.4879e+06 condd - 60.9866 condda - 1.1942e+06 condad - 4.3957e+06 vtt - 1.0000 0.0107 -0.2629 -0.5245 1.0000 -0.3896 0.0815 0.0034 1.0000 eigtt = 1.0e+03 * -0.1213 -2.1531 —0.0316 266 Table C.4.9 (cont’d) 267 Table C.4.10 Output for algebraic/dynamic 170 MVar e1 d - eigjj - 9 -26.3797 +52.41731 -iijfiggi 132:38321 -26.3797 -52.41731 47 7325 47.7826 40°2851 40.2813 38 8534 38.8520 -9°4218 +18 78361 -12.6302 +24.99801 -9'4213 -13'73351 - .6302 -24 99801 ' ' 12 . -5.3122 +13.70081 24.6874 ~5.3122 -13.70081 22.2377 24.6893 20.1966 22 2731 -9.5934 +18.80251 20'2200 -9.5934 -18.80251 1 5223 -15.8158 1'7551 -16.6073 7°1557 ~16.5818 10 5407 + 0 66071 -4.2540 +15.14641 10'5407 - 0'66071 -4.2540 -15.14641 3,§2so ' -4.0594 +14.97001 -4.0594 -14.97001 -5.2398 +13.57301 -5.2398 —13.57301 10.4425 + 0.58441 10.4425 - 0.58441 9.0127 7.0083 1.6632 -0.5111 + 1.57611 -0.5111 - 1.57611 0.8882 -0.1538 + 1.20091 -0.1538 - 1.20091 -0.1563 + 0.94061 -0.1563 - 0.94061 0.3149 0.0323 -0.0535 + 0.65661 -0.0535 - 0.65661 -0.0552 + 0.59601 -0.0552 - 0.59601 eigda - -26.5444 +52.23051 -26.5444 -52.23051 -3.9033 +14.89891 -3.9033 -14.89891 -4.1104 +14.98641 -4.1104 -14.98641 -16.4349 -15.1595 -16.5306 -0.2952 + 1.73201 -0.2952 — 1.73201 -0.0016 + 1.18901 -0.0016 - 1.18901 -0.0636 -0.1048 -0.0630 + 0.60041 -0.0630 - 0.60041 -0.0770 + 0.65521 -0.0770 - 0.65521 —0.1122 + 0.94301 -0.1122 - 0.94301 vda - Columns 1 through 4 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 268 Table C.4.10 (cont’d) 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 -0.0002 + 0.00031 -0.0002 - 0.00031 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 -0.0011 + 0.00071 -0.0011 - 0.00071 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 + 0.00021 0.0000 - 0.00021 0.0001 + 0.00001 0.0001 - 0.00001 0.0000 + 0.00021 0.0000 - 0.00021 0.0000 + 0.00001 0.0000 - 0.00001 -0.0001 + 0.00081 -0.0001 - 0.00081 0.0000 + 0.00001 0.0000 - 0.00001 0.0001 + 0.00041 0.0001 - 0.00041 -0.0048 - 0.01051 -0.0048 + 0.01051 -0.0001 - 0.00021 -0.0001 + 0.00021 0.0000 - 0.00001 0.0000 + 0.00001 -0.0021 - 0.03061 -0.0021 + 0.03061 0.0000 - 0.00001 0.0000 + 0.00001 -0.0082 - 0.00411 -0.0082 + 0.00411 1.0000 - 0.00001 1.0000 + 0.00001 0.0047 - 0.00091 0.0047 + 0.00091 0.0000 + 0.00031 0.0000 - 0.00031 1.0000 - 0.00001 1.0000 + 0.00001 0.0000 + 0.00031 0.0000 - 0.00031 0.1422 - 0.25971 0.1422 + 0.25971 -0.0761 - 0.15391 ~0.076l + 0.15391 -0.0012 - 0.00291 -0.0012 + 0.00291 0.0000 - 0.00001 0.0000 + 0.00001 -0.0289 - 0.11051 -0.0289 + 0.11051 0.0000 - 0.00001 0.0000 + 0.00001 -0.0276 - 0.00781 -0.0276 + 0.00781 Columns 5 through 8 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 0.0000 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 -0.0001 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 0.0000 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 0.0000 0.0000 - 0.00001 0.0000 + 0.00001 0.0000 0.0000 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 0.0000 0.0000 + 0.00001 0.0000 — 0.00001 -0.0004 0.0048 0.0000 - 0.00021 0.0000 + 0.00021 0.0011 0.0003 -0.0005 + 0.00031 -0.0005 - 0.00031 0.0005 0.0001 0.0000 + 0.00011 0.0000 - 0.00011 -0.0039 0.0461 -0.0001 + 0.00011 -0.0001 - 0.00011 0.0566 0.0184 -0.0001 + 0.00061 -0.0001 - 0.00061 0.0501 0.0212 0.0000 - 0.00011 0.0000 + 0.00011 0.0045 -0.0531 0.0055 + 0.00261 0.0055 — 0.00261 -0.0337 -0.0119 -0.0016 - 0.03061 -0.0016 + 0.03061 -0.0302 -0.0138 0.0028 + 0.00071 0.0028 - 0.00071 -0.0932 1.0000 -0.1002 + 0.17231 -0.1002 - 0.17231 1.0000 0.3199 1.0000 - 0.00001 1.0000 + 0.00001 0.8703 0.3612 0.0001 - 0.00191 0.0001 + 0.00191 0.0593 -0.6932 0.0218 + 0.00581 0.0218 - 0.00581 -0.1071 -0.0372 -0.0226 - 0.09331 -0.0226 + 0.09331 -0.0807 -0.0364 Columns 9 through 12 0.0000 0.0053 + 0.00021 0.0053 - 0.00021 -0.0305 + 0.01021 0.0000 -0.0101 - 0.00351 -0.0101 + 0.00351 0.0887 - 0.00301 0.0000 -0.0032 + 0.00301 -0.0032 - 0.00301 -0.0139 - 0.01241 0.0000 -0.0004 - 0.00301 -0.0004 + 0.00301 0.0086 + 0.02571 0.0000 -0.0010 + 0.00601 -0.0010 - 0.00601 -0.0026 - 0.07461 0.0000 0.0020 + 0.00151 0.0020 - 0.00151 -0.0104 + 0.01171 0.0000 -0.0178 - 0.05341 -0.0178 + 0.05341 0.0532 + 0.05031 -0.0007 -0.0252 + 0.01121 -0.0252 - 0.01121 -0.0465 - 0.13571 0.0006 -0.0134 + 0.00651 -0.0134 - 0.00651 -0.0418 - 0.04831 -0.0003 -0.0267 - 0.04021 -0.0267 + 0.04021 0.0292 + 0.00791 -0.0352 -0.0274 - 0.00271 -0.0274 + 0.00271 -0.0368 - 0.09911 0.0575 -0.0290 - 0.00511 -0.0290 + 0.00511 -0.0475 - 0.07131 0.0004 0.0263 + 0.03851 0.0263 - 0.03851 -0.0280 - 0.00761 0.0208 0.0264 + 0.00121 0.0264 - 0.00121 0.0383 + 0.09231 -0.0344 0.0277 + 0.00381 0.0277 - 0.00381 0.0454 + 0.06571 -0.0084 0.0437 + 0.17321 0.0437 - 0.17321 -0.1205 - 0.02841 -0.6219 0.1441 + 0.10581 0.1441 - 0.10581 0.0068 + 0.67601 1.0000 0.1685 + 0.08251 0.1685 - 0.08251 0.3259 + 0.48341 0.0053 1.0000 - 0.00001 1.0000 + 0.00001 -0.6252 + 0.62911 0.0662 0.0815 - 0.15961 0.0815 + 0.15961 1.0000 - 0.00001 -0.0922 0.0848 - 0.13501 0.0848 + 0.13501 0.6583 - 0.14141 Columns-13 through 16 -0.0305 - 0.01021 -0.0636 -0.1048 0.0750 - 0.12081 0.0887 + 0.00301 -0.0605 -0.1010 -0.2792 + 0.26401 -0.0139 + 0.01241 -0.0631 -0.1046 0.0246 - 0.00031 0.0086 - 0.02571 1.0000 1.0000 -0.2120 - 0.10271 -0.0026 + 0.07461 0.9517 0.9643 0.4832 + 0.41431 -0.0104 - 0.01171 0.9927 0.9981 -0.0047 - 0.04041 0.0532 - 0.05031 0.0004 0.0011 0.0069 + 0.02071 -0.0465 + 0.13571 0.0074 0.0052 -0.1215 - 0.20021 -0.0418 + 0.04831 -0.0006 -0.0008 0.0132 + 0.03861 0.0292 - 0.00791 0.0004 0.0008 0.0004 + 0.00451 -0.0368 + 0.09911 -0.0001 -0.0002 -0.0052 - 0.05791 269 Table C.4.10 (cont’d) -0.0475 + 0.07131 0.0000 0.0000 0.0119 + 0.01631 -0.0280 + 0.00761 -0.0006 -0.0009 -0.0003 - 0.00431 0.0383 - 0.09231 0.0001 0.0002 0.0059 + 0.05461 0.0454 - 0.06571 0.0000 0.0000 -0.0107 - 0.01551 -0.1205 + 0.02841 0.0177 0.0165 -0.0137 - 0.01551 0.0068 - 0.67601 0.0007 0.0010 -0.0281 + 0.34131 0.3259 - 0.48341 0.0001 0.0000 -0.1271 - 0.06431 -0.6252 - 0.62911 0.2650 0.2620 -0.2281 - 0.02701 1.0000 + 0.00001 -0.0232 -0.0193 1.0000 + 0.00001 0.6583 + 0.14141 0.0002 0.0000 -0.2537 + 0.13761 Columns 17 through 20 0.0750 + 0.12081 0.0223 - 0.25031 0.0223 + 0.25031 0.0067 - 0.01561 -0.2792 - 0.26401 -0.0328 + 0.00951 -0.0328 - 0.00951 -0.1436 + 0.07451 0.0246 + 0.00031 -0.0770 + 0.65521 -0.0770 - 0.65521 0.1123 - 0.04691 -0.2120 + 0.10271 -0.3807 + 0.01071 -0.3807 - 0.01071 -0.0171 - 0.00501 0.4832 - 0.41431 0.0200 + 0.04781 0.0200 - 0.04781 0.0958 + 0.14091 -0.0047 + 0.04041 1.0000 - 0.00001 1.0000 + 0.00001 -0.0631 - 0.11151 0.0069 - 0.02071 0.0343 + 0.01201 0.0343 - 0.01201 -0.0081 + 0.02371 -0.1215 + 0.20021 0.0794 + 0.00041 0.0794 - 0.00041 0.0641 + 0.06751 0.0132 - 0.03861 -0.1410 - 0.02191 -0.1410 + 0.02191 -0.0256 - 0.0004 - 0.00451 0.0054 + 0.00561 0.0054 - 0.00561 -0.0010 + 0 -0.0052 + 0.05791 0.0212 - 0.00011 0.0212 + 0.00011 0.0469 + 0.03031 0.0119 - 0.01631 -0.0694 - 0.01831 -0.0694 + 0.01831 -0.0225 - -0.0003 + 0.00431 -0.0051 - 0.00551 -0.0051 + 0.00551 0.0010 - . 0.0059 - 0.05461 -0.0201 + 0.00051 -0.0201 - 0.00051 -0.0452 - 0.02741 -0.0107 + 0.01551 0.0646 + 0.01841 0.0646 - 0.01841 0.0213 + 0.09871 -0.0137 + 0.01551 -0.0326 - 0.00801 -0.0326 + 0.00801 -0.0031 - 0.02741 -0.0281 - 0.34131 -0.1228 - 0.02321 -0.1228 + 0.02321 -0.2182 - 0.25391 -0.1271 + 0.06431 0.4832 - 0.07711 0.4832 + 0.07711 0.2477 + 0.63731 -0.2281 + 0.02701 -0.3096 + 0.18801 -0.3096 - 0.18801 -0.2180 - 0.11011 1.0000 - 0.00001 -0.0302 + 0.33271 -0.0302 - 0.33271 -0.4264 + 0.45161 -0.2537 - 0.13761 0.3597 - 0.87071 0.3597 + 0.87071 1.0000 + 0.00001 Column 21 0.0067 + 0.01561 -0.1436 - 0.07451 0.1123 + 0.04691 -0.0171 + 0.00501 0.0958 - 0.14091 -0.0631 + 0.11151 -0.0081 - 0.02371 0.0641 - 0.06751 -0.0256 + 0.11121 -0.0010 - 0.00721 0.0469 - 0.03031 -0.0225 + 0.10541 0.0010 + 0.00691 -0.0452 + 0.02741 0.0213 - 0.09871 -0.0031 + 0.02741 -0.2182 + 0.25391 0.2477 - 0.63731 -0.2180 + 0.11011 -0.4264 - 0.45161 1.0000 - 0.00001 eigad - 1.0e+02 * -0.3418 + 4.36391 -0.3418 - 4.36391 -0.5307 + 3.80431 -0.5307 - 3.80431 0.8175 + 0.42941 0.8175 - 0.42941 -0.0495 + 0.64121 -0.0495 - 0.64121 0.4738 0.3852 0.3417 0.2942 0.0002 0.0438 270 Table C.4.10 (cont’d) 0.0696 0.1143 0.0598 0.1229 vad - Columns 1 through 4 0.0045 + 0.03831 0.0045 - .03831 0.0080 + 0.04311 0.0080 - 0.04311 0.0023 + 0.00291 0.0023 - .00291 -0.0054 - 0.00421 -0.0054 + 0.00421 0.0001 + 0.00041 0.0001 - .00041 0.0003 + 0.00051 0.0003 - 0.00051 1.0000 + 0.00001 1.0000 - .00001 1.0000 - 0.00001 1.0000 + 0.00001 0.0814 + 0.00081 0.0814 - .00081 -0.1300 + 0.02951 -0.1300 - 0.02951 0.0115 - 0.00181 0.0115 + -0.0055 - 0.07251 -0.0055 0.0018 - 0.00661 0.0018 + -0.0001 - 0.00081 -0.0001 -0.0019 + 0.02481 -0.0019 -0.0001 + 0.00111 -0.0001 0.0009 + 0.01231 0.0009 - -0.0001 + 0.00581 -0.0001 -0.0001 + 0.00051 -0.0001 -0.0003 + 0.00011 -0.0003 0.0000 - 0.00331 0.0000 + 0.0000 - 0.00021 0.0000 + -0.0004 - 0.00291 -0.0004 .00181 0.0147 - 0.00421 0.0147 + 0.00421 0.07251 -0.0113 - 0.08211 -0.0113 + 0.08211 .00661 0.0070 + 0.01021 0.0070 - 0.01021 0.00081 -0.0007 - 0.00101 -0.0007 + 0.00101 0.02481 -0.0030 + 0.02171 -0.0030 - 0.02171 0.00111 -0.0001 - 0.00161 -0.0001 + 0.00161 .01231 0.0020 + 0.01401 0.0020 - 0.01401 0.00581 0.0004 + 0.00651 0.0004 - 0.00651 0.00051 -0.0005 - 0.00071 -0.0005 + 0.00071 0.00011 -0.0004 + 0.00021 -0.0004 - 0.00021 .00331 0.0001 - 0.00281 0.0001 + 0.00281 .00021 0.0001 + 0.00031 0.0001 - 0.00031 0.00291 -0.0007 - 0.00331 -0.0007 + 0.00331 +OOI 1 ID! I +O+OOOOOO Columns 5 through 8 -0.1817 + 0.12061 -0.1817 0.0650 + 0.02141 0.0650 - 0.0052 - 0.03671 0.0052 + 1.0000 - 0.00001 1.0000 + -0.2165 - 0.32001 -0.2165 -0.1308 + 0.15111 -0.1308 0.3028 - 0.16771 0.3028 + -0.1572 - 0.02681 -0.1572 -0.0026 + 0.06391 -0.0026 0.0047 + 0.00241 0.0047 - 0.12061 0.2245 - 0.07251 0.2245 + 0.07251 .02141 -0.2190 - 0.08991 -0.2190 + 0.08991 .03671 0.0663 + 0.21091 0.0663 - 0.21091 .00001 0.0163 - 0.94341 0.0163 + 0.94341 0.32001 -0.7161 + 0.68671 -0.7161 - 0.68671 0.15111 1.0000 - 0.00001 1.0000 + 0.00001 .16771 -0.4135 + 0.01591 -0.4135 - 0.01591 0.02681 0.3863 + 0.24421 0.3863 - 0.24421 0.06391 -0.0516 - 0.44511 -0.0516 + 0.44511 .00241 0.0034 + 0.00031 0.0034 - 0.00031 -0.0001 - 0.00111 -0.0001 0.00111 -0.0013 - 0.00151 -0.0013 + 0.00151 -0.0414 + 0.01581 -0.0414 0.01581 -0.0121 + 0.15611 -0.0121 - 0.15611 -0.0284 + 0.02401 -0.0284 - 0.02401 0.0269 - 0.00851 0.0269 + 0.00851 0.0168 - 0.00201 0.0168 + 0.00201 -0.0275 - 0.00671 -0.0275 + 0.00671 0.0030 - 0.01321 0.0030 + 0.01321 -0.0151 + 0.04341 -0.0151 - 0.04341 -0.0029 - 0.00261 -0.0029 + 0.00261 0.0026 - 0.00231 0.0026 + 0.00231 -0.0015 + 0.00091 -0.0015 - 0.00091 -0.0009 + 0.00461 -0.0009 - 0.00461 0.0129 - 0.00731 0.0129 + 0.00731 -0.0113 - 0.02591 -0.0113 + 0.02591 I+OI+OI+OOOI Columns 9 through 12 0.0267 0.4460 0.1591 -0.8247 0.1433 0.2380 0.0711 -0.8855 0.1173 0.1971 0.0050 -0.9813 -0.0356 -0.5623 ~0.1374 0.6047 -0.2536 -0.3793 -0.1160 0.7878 -0.2553 -0.3035 -0.0099 0.9683 0.0210 -0.4949 -0.1031 0.7121 -0.1363 -0.2605 -0.1411 0.9242 -0.1381 -0.1460 0.0030 1.0000 0.0004 -0.0011 0.0007 0.0011 0.0001 -0.0007 -0.0004 0.0008 -0.0206 0.0074 -0.0108 0.0729 ‘ 0.5036 0.2350 1.0000 -0.0926 1.0000 -0.3880 -0.5867 0.0620 0.2451 1.0000 -0.2792 0.0968 -0.7483 0.1895 -0.1498 0.0090 -0.2002 -0.2525 0.2345 -0.0669 -0.1506 -0.4624 -0.1497 -0.0634 Columns 13 through 16 1.0000 0.0840 1.0000 0.1269 0.9883 -0.0383 -0.4421 -0.0836 0.9998 -0.0437 -0.4693 -0.0040 0.9990 0.0685 0.5911 0.0572 0.9929 -0.0124 -0.2527 -0.0070 271 Table C.4.10 (cont’d) 0.9994 -0.0243 -0.2496 -0.0026 0.9954 0.0262 0.0985 -0.0099 0.9970 -0.0614 —0.2520 0.1158 0.9993 —0.0192 0.0582 -0.0706 -0.0002 0.0002 0.0001 0.0006 0.0000 0.0002 0.0000 0.0002 0.0001 0.0188 -0.0041 -0.0049 0.0000 0.2555 0.0697 0.6519 -0.0001 0.3216 0.0232 0.3530 0.0002 0.5320 -0.0012 -0.1347 -0.0001 0.3777 0.0666 1.0000 0.0003 1.0000 0.0285 -0.9601 0.0002 0.6023 0.0274 0.6374 Columns 17 through 18 1.0000 0.0332 ~0.0070 -0.1815 —0.8430 0.2334 0.6500 0.0073 -0.0004 -0.0673 -0.5035 0.0443 0.2709 -0.0125 -0.3271 0.0274 -0.0846 -0.1254 0.0001 -0.0001 0.0000 -0.0003 -0.0055 0.0132 0.1117 -0.0732 0.0831 -0.4905 0.0441 0.3337 0.1358 -0.7182 0.1375 -0.3446 0.0908 1.0000 detjj - -3.2336e+30 detd a 9.6001e+20 detda - -3.3683e+09 detad - 7.7356e+26 condjj - 2.5317e+06 condd - 61.6129 condda - 1.2151e+06 condad - 2.9276e+06 vtt - 1.0000 0.0156 -0.2700 -0.5389 1.0000 -0.4010 0.0836 0.0051 1.0000 eigtt - 1.0e+03 * -0.1158 -1.5233 -0.0302 272 Table C.4.10 (cont’d) Table C.4.11 Output for algebraic/dynamic 170 MVar(without exciter in) eigjj - -26.3797 +52.41731 -26.3797 -52.41731 47.7826 40.2813 38.8519 -12.6304 +24.99741 -12.6304 -24.99741 -9.6232 +18.78061 -9.6232 -18.78061 24.6874 22.2368 -16.588 20.1955 -15.817 -4.1193 -4.1193 -5.2809 -5.2809 10.4417 10.4417 9.0174 6.9834 -0.4966 -0.4966 1.4438 -0.1346 -0.1346 -0.0585 -0.0585 -0.0544 -0.0544 1 2 +15.01611 -15.01611 +13.64341 -l3.64341 + 0.58331 0.58331 + 1.56891 1.56891 1.17941 1.17941 0.71731 0.71731 0.60621 - 0.60621 + + + 0.5180 + 0.10801 0.5180 - 0.10801 0.0338 0.1944 eigda - -26.5444 +52.23051 -26.544 4 -52.23051 -3.8928 +14.90791 -3.8928 -14.90791 -16.468 -15.180 -0.3277 -0.3277 -0.0478 -0.0478 -0.0683 -0.0683 -0.0630 -0.0630 -0.1048 -0.0636 0.2004 vda a 1 9 1.69901 1.69901 1.11541 1.11541 0.69821 0.69821 0.61241 0.61241 Columns 1 through 4 0.0000 - 0.00001 0.0000 0.0000 0.0000 0.0000 0.0000 + + + 0.00001 0.0000 0.00001 0.0000 0.00001 0.0000 0.00001 0.0000 0.0000 - 0.00001 0.0000 -0.0002 + 0.00031 -0.0002 0.0000 + 0.00001 0.0000 - 0.0000 + 0.00001 0.0000 - 0.0001 + 0.00001 0.0001 - 0.0000 + 0.00001 0.0000 - -0.0048 - 0.01051 -0.0048 0.0000 - 0.00001 0.0000 + + + + 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 - 0.00031 0.0000 273 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 eigd - -12.4605 +25.20251 -12.4605 47.7826 40.2851 38.8534 -9.4218 -9.4218 -5.3122 -5.3122 24.6893 22.2731 20.2200 1.2223 1.7561 7.1667 10.5407 10.5407 8.9260 + 0.00001 - 0.00001 + 0.00001 - 0.00001 + 0.00001 - 0.00001 -25.2025 +18.78361 -18.78361 +13.70081 -13.70081 + 0.66071 - 0.66071 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 + + + + 0.00001 0.0000 0.00001 -0.0011 + 0.00071 -0.0011 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 + 0.00021 0.0000 - 0.00001 -0.0001 + 0.00071 -0.0001 + 0.01051 0.0000 - 0.00021 0.0000 0.00001 -0.0021 - 0.03051 -0.0021 1 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 - 0.00001 - 0.00071 0.00001 0.00021 - 0.00071 + 0.00021 + 0.03051 274 Table C.4.ll (cont’d) 1.0000 + 0.00001 1.0000 - 0.00001 0.0042 - 0.00011 0.0042 + 0.00011 0.0000 + 0.00031 0.0000 - 0.00031 1.0000 + 0.00001 1.0000 - 0.00001 -0.0761 - 0.15391 -0.0761 + 0.15391 -0.0006 - 0.00271 -0.0006 + 0.00271 0.0000 - 0.00001 0.0000 + 0.00001 -0.0288 - 0.11051 -0.0288 + 0.11051 Columns 5 through 8 0.0000 0.0000 0.0053 - 0.00051 0.0053 + 0.00051 0.0000 -0.0001 -0.0098 - 0.00361 -0.0098 + 0.00361 0.0000 0.0000 -0.0036 + 0.00441 -0.0036 - 0.00441 0.0000 0.0000 -0.0008 - 0.00301 -0.0008 + 0.00301 0.0000 0.0000 -0.0010 + 0.00601 -0.0010 - 0.00601 0.0000 0.0000 0.0029 + 0.00161 0.0029 - 0.00161 -0.0002 0.0048 -0.0194 - 0.05421 -0.0194 + 0.05421 0.0011 0.0003 -0.0236 + 0.00951 -0.0236 - 0.00951 0.0000 -0.0001 -0.0050 + 0.00871 -0.0050 - 0.00871 -0.0022 0.0460 -0.0259 - 0.04091 -0.0259 + 0.04091 0.0566 0.0177 -0.0241 - 0.00381 -0.0241 + 0.00381 0.0026 -0.0530 0.0256 + 0.03921 0.0256 - 0.03921 -0.0336 -0.0114 0.0233 + 0.00251 0.0233 - 0.00251 -0.0538 1.0000 0.0405 + 0.16991 0.0405 - 0.16991 1.0000 0.3070 0.1203 + 0.09941 0.1203 - 0.09941 0.0342 -0.6921 1.0000 + 0.00001 1.0000 - 0.00001 -0.1069 -0.0356 0.0772 - 0.13881 0.0772 + 0.13881 Columns 9 through 12 -0.0307 + 0.00751 -0.0307 0.1203 + 0.00401 0.1203 - -0.0429 - 0.01031 -0.0429 0.0079 + 0.02721 0.0079 - -0.0011 - 0.10781 -0.0011 -0.0075 + 0.03881 -0.0075 0.0305 + 0.03631 0.0305 - -0.0376 - 0.14641 -0.0376 -0.0210 + 0.00691 -0.0210 0.0160 + 0.00761 0.0160 - -0.0300 - 0.09691 -0.0300 -0.0154 - 0.00741 -0.0154 0.0316 + 0.09071 0.0316 - -0.0658 - 0.02441 -0.0658 -0.0194 + 0.63411 -0.0194 -0.4222 + 0.33081 -0.4222 1.0000 + 0.00001 1.0000 - Columns 13 through 16 0.0757 - 0.16541 0.0757 + -0.2599 + 0.30431 -0.2599 -0.0132 + 0.08291 -0.0132 -0.2799 - 0.09481 -0.2799 0.5350 + 0.36931 0.5350 - 0.1361 + 0.00761 0.1361 - 0.0175 + 0.02861 0.0175 - -0.1202 - 0.17931 -0.1202 -0.0217 + 0.00821 -0.0217 0.0020 + 0.00601 0.0020 - -0.0056 - 0.05901 -0.0056 -0.0018 - 0.00591 -0.0018 0.0063 + 0.05561 0.0063 - -0.0233 - 0.01711 -0.0233 -0.0281 + 0.34811 -0.0281 -0.3167 + 0.03401 -0.3167 1.0000 + 0.00001 1.0000 - Column 17 -0.0166 -0.0651 0.0428 -0.0828 -0.3247 0.2137 0.1094 0.2565 -0.5464 -0.0022 -0.0183 I |+O++O| +OOO+I IO 0.00751 -0.1340 + 0.05621 -0.1340 - 0.05621 .00401 -0.1202 + 0.31211 -0.1202 - 0.31211 0.01031 .02721 0. 0.10781 0.03881 .03631 0. 0.14641 0.00691 .00761 0. 0.09691 0.00741 .09071 0. 0.02441 0.63411 0.33081 .00001 1. .16541 0.30431 0.08291 0.09481 .36931 0. .00761 0. .02861 0. 0.17931 0.00821 0.4469 - 0.45061 0.4469 + 0.45061 0983 + 0.18231 0.0983 - 0.18231 0.4594 + 0.12721 0.4594 - 0.12721 -0.7012 - 0.57141 -0.7012 + 0.57141 0180 + 0.01391 0.0180 - 0.01391 -0.0956 - 0.13931 -0.0956 + 0.13931 -0.0271 + 0.01991 -0.0271 - 0.01991 0055 - 0.00041 0.0055 + 0.00041 -0.0085 - 0.06661 -0.0085 + 0.06661 -0.0053 + 0.00021 -0.0053 - 0.00021 0093 + 0.06281 0.0093 - 0.06281 -0.0199 + 0.01271 -0.0199 - 0.01271 -0.0311 + 0.39691 -0.0311 - 0.39691 -0.0470 + 0.24001 -0.0470 - 0.24001 0000 + 0.00001 1.0000 - 0.00001 -0.1048 -0.0636 -0.1011 -0.0605 -0.1046 -0.0631 1.0000 1.0000 9643 0.9517 9981 0.9926 0011 0.0004 0.0052 0.0074 -0.0008 -0.0005 .00601 0.0008 0.0004 0.05901 0.00591 -0.0002 -0.0001 -0.0009 -0.0006 .05561 0.0002 0.0001 0.01711 0.34811 0.03401 0.0165 0.0177 0.0010 0.0007 0.2620 0.2650 0.00001 -0.0193 -0.0233 7.. 0.013.161.1100 000A4013»v O I l I O 8 0080000011 I \ eiga QI.‘ ~16 n. . 4 4 u H Al». A U . . . 1) in n 6 Pa 60 C h 0AVCfiH . V . 0 u . 050 Pu 6U 3..» AU 1; a E .144 0.0019 0.0170 0.0298 0.1217 0.3195 1.0000 eigad - 1.0e+02 * -0.5787 + ~0.5787 - —0.2333 + -0.2333 - 1.1720 0.5331 0.4652 0.3840 0.3406 -0.0397 + —0.0397 - -0.0272 0.1799 0.0002 0.0717 0.1080 0.0578 0.1147 4.15511 4.15511 4.07271 4.07271 0.09471 0.09471 vad - Columns 1 through 4 0.0071 + 0.03951 0.0071 - -0.0030 + 0.00191 -0.0030 0.0000 + 0.00001 0.0000 - 1.0000 + 0.00001 1.0000 - -0.0117 + 0.07401 -0.0117 0.0003 - 0.00011 0.0003 + -0.0103 - 0.07521 -0.0103 0.0084 - 0.00051 0.0084 + 0.0000 + 0.00021 0.0000 - -0.0033 + 0.02361 -0.0033 -0.0009 - 0.00031 -0.0009 0.0017 + 0.01251 0.0017 - 0.0003 + 0.00601 0.0003 - -0.0006 + 0.00001 ~0.0006 -0.0004 + 0.00011 -0.0004 0.0002 - 0.00311 0.0002 + 0.0002 + 0.00001 0.0002 - -0.0006 - 0.00301 -0.0006 Columns 5 through 8 -0.1691 -0.4676 -0.5694 0. 275 Table C.4.11 (cont’d) 0.03951 0.0041 + 0.04121 0.0041 - 0.04121 - 0.00191 0.0015 - 0.00231 0.0015 + 0.00231 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 0.00001 1.0000 + 0.00001 1.0000 - 0.00001 - 0.07401 -0.0185 - 0.05441 -0.0185 + 0.05441 0.00011 0.0003 - 0.00001 0.0003 + 0.00001 + 0.07521 -0.0043 - 0.07791 -0.0043 + 0.07791 0.00051 -0.0019 + 0.00161 -0.0019 - 0.00161 0.00021 0.0000 - 0.00011 0.0000 + 0.00011 - 0.02361 -0.0013 + 0.02321 -0.0013 - 0.02321 + 0.00031 0.0007 - 0.00021 0.0007 + 0.00021 0.01251 0.0007 + 0.01291 0.0007 - 0.01291 0.00601 -0.0002 + 0.00621 -0.0002 - 0.00621 - 0.00001 0.0002 - 0.00011 0.0002 + 0.00011 - 0.00011 -0.0004 + 0.00011 -0.0004 - 0.00011 0.00311 -0.0001 - 0.00311 -0.0001 + 0.00311 0.00001 -0.0001 + 0.00001 -0.0001 - 0.00001 + 0.00301 -0.0004 - 0.00311 -0.0004 + 0.00311 5856 0.0391 -0.1514 -0.1552 0.3322 0.0006 0.0135 0.0684 0.2440 1.0000 1.0000 1.0000 -0.7344 -0.3048 0.3848 0.3641 -0.0043 -0.0353 -0.5210 -0.1438 -0.3762 0.2728 0.6283 0.7716 -0.6520 -0.1205 0.1184 0.0072 0.1542 -0.3630 -0.0367 -0.1035 -0. 1917 0.0067 0.0030 0.0030 -0.0015 -0.0011 0.0006 0.0009 -0.0008 -0.0433 -0.0993 -0.1438 0. 0279 -0.0295 -0.1436 0.2545 0.2850 0.0129 -0.0619 0.8122 -0.4344 0.0051 0.0604 0.3564 -0.0007 0.0711 1.0000 -0.5686 0.2005 -0.0018 -0.0023 -0.2017 -0.2458 0.0132 0.0395 -0.1012 -0.4846 Columns 9 through 12 276 Table C.4.ll (cont’d) 0.0051 -0.1377 + 0.00791 -0.1377 - 0.00791 -0.1132 -0.0612 0.0691 - 0.02771 0.0691 + 0.02771 -0.0588 -0.0944 -0.1689 - 0.28891 -0.1689 + 0.28891 -0.l947 0.0192 -0.1654 + 0.08761 -0.1654 - 0.08761 -0.1294 0.0440 0.0715 - 0.07251 0.0715 + 0.07251 -0.0646 0.1188 -0.3834 - 0.25651 -0.3834 + 0.25651 -0.2127 0.0583 -0.0255 - 0.00371 -0.0255 + 0.00371 -0.0873 -0.0012 -0.2003 - 0.19461 -0.2003 + 0.19461 -0.1526 0.1050 -0.5400 + 0.50911 -0.5400 - 0.50911 -0.3548 0.0010 0.0000 - 0.00011 0.0000 + 0.00011 0.0001 -0.0002 0.0000 + 0.00001 0.0000 - 0.00001 0.0000 -0.0249 1.0000 + 0.00001 1.0000 - 0.00001 1.0000 1.0000 0.0061 + 0.03291 0.0061 - 0.03291 0.0789 -0.5759 0.0145 + 0.03871 0.0145 - 0.03871 0.0888 -0.2920 0.4109 + 0.17601 0.4109 - 0.17601 0.6362 -0.1643 -0.0021 + 0.03161 -0.0021 - 0.03161 0.0781 0.2385 0.1206 + 0.19381 0.1206 - 0.19381 0.4522 -0.1387 0.0497 + 0.22181 0.0497 - 0.22181 0.3293 Columns 13 through 16 0.2274 0.9991 1.0000 0.1573 0.1805 0.9876 -0.5230 0.0608 1.0000 1.0000 -0.3l65 0.3496 -0.0120 0.9981 0.5796 0.0643 -0.0303 0.9921 -0.2894 0.0523 -0.2176 0.9994 -0.1645 0.0860 -0.0073 0.9945 0.0719 0.0327 -0.1766 0.9968 -0.2198 0.0384 -0.7867 0.9990 0.0632 -0.1530 0.0000 -0.0002 0.0001 0.0003 -0.0001 0.0000 0.0000 0.0003 -0.4410 -0.0082 -0.0904 -0.4647 0.0833 -0.0013 0.1077 0.2990 -0.1380 -0.0016 0.0821 0.6510 -0.0202 -0.0068 0.0169 -0.3734 -0.2326 -0.0015 0.1437 1.0000 0.1317 -0.0062 0.1683 0.5480 0.4848 -0.0047 0.0621 -0.6801 Columns 17 through 18 1.0000 -0.0649 0.0677 0.1430 -0.8430 0.0536 0.6633 -0.0307 0.0530 0.0438 -0.5141 0.0095 0.3084 0.0222 -0.3284 -0.0886 -0.1242 0.0305 0.0002 0.0001 -0.0787 0.1873 0.1875 0.1438 0.2626 0.4132 0.2254 detjj - -0.0004 0.0000 -0.1402 -0.4009 0.0488 -0.1015 -0.2914 1.0000 -0.9740 1.8175e+26 detd - 9.6001e+20 detda - 1.8932e+05 detad - 277 Table C.4.11 (cont’d) -4.9559e+24 condjj - 2.5408e+06 condd - 61.6129 condda = 1.2208e+06 condad - 2.9173e+06 vtt - 1.0000 0.0155 -0.2002 -0.4906 1.0000 -0.4246 -0.0017 -0.0002 1.0000 ett - l.0e+03 * -0.1140 -1.5191 0.0002 Table C4. 12 Output for algebraic/dynamic 150 MVar(without exciter in) eigjj - -26.3461 -26.3461 51.3629 46.8155 42.7175 -14.6136 -14.6136 -12.2939 -12.2939 27.4704 25.4227 -7.5743 —7.5743 23.4856 -16.5863 -15.9151 -3.9477 -3.9477 13.4571 13.4571 11.7788 9.8842 5.6831 3.5289 -0.3585 -0.3585 -0.2236 -0.2236 -0.0709 -0.0843 -0.0843 -0.0497 -0.0497 -0.0349 -0.0349 eigda - +52.40581 -52.40581 +27.98291 —27.98291 +22.57711 -22.57711 +16.79801 ~16.79801 +14.94441 -14.94441 + 0.89961 0.89961 1.37491 1.37491 1.10361 1.10361 0.79791 0.79791 0.64021 0.64021 0.59841 0.59841 -26.4666 +52.26841 -26.4666 -52.26841 —3.8998 -3.8998 -16.5234 -15.6538 -0.3235 -0.3235 -0.2121 -0.2121 -0.0688 -0.0932 -0.0932 -0.0632 -0.0632 -0.0340 -0.0340 vda - +14.96991 -14.96991 1.39871 1.39871 1.08171 1.08171 0.79241 0.79241 0.63681 0.63681 0.60391 0.60391 Columns 1 through 4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0002 + + + -0.0048 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 + 0.00031 - 0.01051 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 + + + -0.0002 0.0000 + 0.00001 0.0000 - 0.0000 + 0.00001 0.0000 - 0.0001 + 0.00001 0.0001 - 0.0000 + 0.00001 0.0000 - -0.0048 0.0000 - 0.00001 0.0000 + 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 278 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 + + + - 0.00031 0.0000 0.00001 -0.0012 + 0.00061 -0.0012 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 0.00001 -0.0001 + 0.00051 + 0.01051 0.0000 - 0.00001 0.0000 -0.0021 - 0.03041 0.00001 eigd - -14.4530 +28.14741 -14.4530 -28.14741 -12.1199 +22.56811 -12.1199 -22.56811 51.3628 46.8168 42.7216 -7.4686 +16.75921 —7.4686 -16.75921 27.4743 25.4621 3.7091 23.5039 5.7289 10.0391 11.7581 13.5198 + 0.91731 13.5198 - 0.91731 0.00001 0.0000 - 0.00001 0.00001 0.0000 + 0.00001 0.00001 0.0000 - 0.00001 0.00001 0.0000 + 0.00001 0.00001 0.0000 - 0.00001 0.00001 0.0000 + 0.00001 + 0.00001 0.0000 - 0.00001 - 0.00061 -0.0001 -0.0021 - 0.00051 + 0.00001 + 0.03041 279 Table C.4.12 (cont’d) 1.0000 + 0.00001 1.0000 - 0.00001 0.0011 + 0.00001 0.0011 - 0.00001 0.0000 + 0.00011 0.0000 - 0.00011 1.0000 - 0.00001 1.0000 + 0.00001 -0.0759 - 0.15381 -0.0759 + 0.15381 -0.0001 - 0.00071 —0.0001 + 0.00071 0.0000 - 0.00001 0.0000 + 0.00001 -0.0286 - 0.11011 -0.0286 + 0.11011 Columns 5 through 8 0.0000 0.0001 0.0098 - 0.00571 0.0098 + 0.00571 0.0001 0.0000 -0.0057 - 0.00851 -0.0057 + 0.00851 0.0000 0.0000 -0.0025 + 0.00141 -0.0025 - 0.00141 0.0000 0.0000 -0.0054 - 0.00581 -0.0054 + 0.00581 0.0000 0.0000 -0.0049 + 0.00521 -0.0049 - 0.00521 0.0000 0.0000 0.0014 + 0.00151 0.0014 - 0.00151 -0.0001 0.0046 -0.0239 - 0.06881 -0.0239 + 0.06881 0.0012 0.0001 -0.0141 - 0. 0.0000 0.0000 -0.0027 + 0. 00061 -0.0141 + 0.00061 00201 -0.0027 - 0.00201 -0.0008 0.0440 -0.0184 - 0.03951 -0.0184 + 0.03951 0.0565 0.0062 -0.0088 - 0. 0.0010 -0.0508 0.0181 + 0. 00391 -0.0088 + 0.00391 03811 0.0181 - 0.03811 -0.0335 -0.0039 0.0086 + 0.00331 0.0086 - 0.00331 -0.0207 1.0000 0.0344 + 0. 13991 0.0344 - 0.13991 1.0000 0.1086 0.0376 + 0.04461 0.0376 - 0.04461 0.0131 -0.6673 1.0000 - 0. 00001 1.0000 + 0.00001 -0.1065 -0.0122 0.0431 - 0.05701 0.0431 + 0.05701 Columns 9 through 12 -0.0211 + 0.00961 -0.0211 0.3153 - 0.05921 0.3153 + -0.0225 + 0.00561 -0.0225 0.0123 + 0.01711 0.0123 - -0.1077 - 0.27041 -O.1077 0.0089 + 0.01911 0.0089 - 0.0080 + 0.03521 0.0080 - 0.0044 - 0.18191 0.0044 + -0.0079 + 0.00161 -0.0079 0.0047 + 0.01191 0.0047 - -0.0169 - 0.10401 -0.0169 -0.0045 - 0.01151 -0.0045 0.0194 + 0.09841 0.0194 - -0.0184 - 0.03901 -0.0184 -0.1174 + 0.61501 -0.1174 -0.3674 + 0.01571 -0.3674 1.0000 - 0.00001 1.0000 + Columns 13 through 16 -0.0014 + 0.05571 -0.0632 -0.0190 + 0.01151 -0.0696 —0.0932 - 0.79241 -0.0692 - 0.00991 ~0.0116 ~ 0.02531 1.0000 - 0.00001 0.1753 - -0.0001 + 0.00041 -0.0454 0.0318 + 0.00071 0.0555 - - 0.00961 -0.0028 -0.0014 - 0.05571 0.05921 -0.0161 -0.0190 - 0.01151 - 0.00561 0.0325 -0.0932 + 0.79241 0.01711 0.0410 -0.0692 + 0.00991 + 0.27041 0.2339 -0.0116 + 0.02531 0.01911 -0.4726 1.0000 + 0.00001 0.03521 -0.0674 -0.0001 - 0.00041 0.18191 -0.1980 0.0318 - 0.00071 - 0.00161 1.0000 0.0010 + 0.02521 0.01191 0.0000 0.0003 + 0.00121 + 0.10401 -0.0018 0.0115 + 0.00341 + 0.01151 0.0001 -0.0003 - 0.00121 0.09841 0.0017 -0.0110 - 0.00291 + 0.03901 -0.0016 -0.0026 - 0.00401 - 0.61501 0.0095 -0.0614 - 0.03611 - 0.01571 -0.0261 -0.0484 - 0.00211 0.00001 -0.2649 -0.0643 + 0.14351 + 0.63681 -0.0632 - 0.63681 0.0247 + 0.02521 + 0.03441 -0.0696 - 0.03441 -0.2307 + 0.15921 0.0137 + 0.11411 0.0137 - 0.11411 0.0055 + 0.01811 1.0000 + 0.00001 1.0000 - 0.00001 0.0393 - 0.04311 0.0643 + 0.10291 0.0643 - 0.10291 0.2842 + 0.36611 0.03891 0.1753 + 0.03891 0.0293 - 0.01071 - 0.02051 -0.0454 + 0.02051 0.0011 + 0.01271 0.05611 0.0555 + 0.05611 -0.0855 - 0.23501 0.0010 - 0.02521 -0.0030 + 0.00951 -0.0030 - 0.00951 -0.0092 + 0.00211 0.0003 - 0.00121 -0.0076 - 0.01261 -0.0076 + 0.01261 0.0000 + 0.00101 0.0115 - 0.00341 0.0168 - -0.0003 + 0.00121 -0.0110 + 0.00291 -0.0156 -0.0026 + 0.00401 -0.0614 + 0.03611 -0.1135 -0.0484 + 0.00211 -0.0643 - 0.14351 Column 17 0.0247 - 0.02521 -0.2307 - 0.15921 0.0055 - 0.01811 0.0393 + 0.04311 0.2842 - 0.36611 0.0293 + 0.01071 0.0011 - 0.01271 -0.0855 + 0.23501 -0.0092 - 0.00211 0.0000 - 0.00101 -0.0078 + 0.05721 0.01391 0.0168 + 0.01391 -0.0078 - 0.05721 0.0071 + 0.01231 0.0071 - 0.01231 0.0000 - 0.00101 + 0.01351 -0.0156 - 0.01351 0.0084 + 0.05381 0.0549 + 0.02981 0.0549 - 0.02981 -0.0026 - 0.00381 + 0.06141 -0.1135 - 0.06141 -0.0162 + 0.34331 0.6760 - 0.21971 0.6760 + 0.21971 -0.0514 - 0.01151 0.1964 + 0.29581 0.1964 - 0.29581 1.0000 - 0.00001 0.0000 + 0.00101 0.0084 - 0.05381 -0.0026 + 0.00381 -0.0162 - 0.34331 -0.0514 + 0.01151 1.0000 + 0.00001 eigad = 1.0e+02 -0.5717 -0.5717 -0.4252 -0.4252 6.60501 6.60501 4.22821 4.22821 280 1.3999 0.6388 0.5068 0.4490 -0.0727 + -0.0727 - 0.3764 0.0147 .2142 .0423 .1057 .1361 .0972 .1411 0.12841 0.12841 OOOOOO vad - Columns 1 through 4 0.0229 + 0.01841 0.0229 - 0.1072 + 0.02101 0.1072 - 0.0000 + 0.00001 0.0000 - 0.8852 - 0.17411 0.8852 + 1.0000 - 0.00001 1.0000 + 0.0001 - 0.00011 0.0001 + -0.0130 - 0.04571 -0.0130 -0.0039 - 0.05551 -0.0039 0.0008 - 0.00011 0.0008 + 0.0036 + 0.03351 0.0036 - -0.0007 + 0.00821 -0.0007 0.0020 + 0.00701 0.0020 - 0.0003 + 0.00351 0.0003 - 0.0003 + 0.00381 0.0003 - -0.0002 + 0.00011 -0.0002 -0.0008 - 0.00461 -0.0008 0.0000 - 0.00181 0.0000 + -0.0006 - 0.00171 -0.0006 Columns 5 through 8 -0.1203 -0.3594 -0.2669 0. Table C.4.12 (cont’d) 0.01841 0.0237 + 0.03851 0.0237 - 0.03851 0.02101 0.0014 - 0.00031 0.0014 + 0.00031 0.00001 0.0000 + 0.00001 0.0000 - 0.00001 0.17411 1.0000 + 0.00001 1.0000 - 0.00001 0.00001 0.0107 - 0.00551 0.0107 + 0.00551 0.00011 0.0003 - 0.00011 0.0003 + 0.00011 + 0.04571 -0.0079 - 0.08171 -0.0079 + 0.08171 + 0.05551 0.0020 — 0.00131 0.0020 + 0.00131 0.00011 0.0000 + 0.00001 0.0000 - 0.00001 0.03351 -0.0024 + 0.02381 -0.0024 - 0.02381 - 0.00821 0.0000 + 0.00011 0.0000 - 0.00011 0.00701 0.0012 + 0.01251 0.0012 - 0.01251 0.00351 0.0001 + 0.00591 0.0001 - 0.00591 0.00381 -0.0001 + 0.00011 -0.0001 - 0.00011 - 0.00011 -0.0004 + 0.00011 -0.0004 - 0.00011 + 0.00461 0.0000 - 0.00311 0.0000 + 0.00311 0.00181 0.0000 - 0.00001 0.0000 + 0.00001 +.0.00171 -0.0005 - 0.00301 -0.0005 + 0.00301 4988 0.0036 -0.0703 0.0203 0.1766 0.0005 0.0107 0.1237 0.2187 1.0000 1.0000 0.5347 -0.3274 0.3434 0.1300 -0.0037 -0.0295 -0.2408 -0.7785 -0.5131 -0.3498 0.2483 0.5604 0.4327 -0.6367 -0.1097 0.0954 0.0536 -0.3101 0.0063 -0.0293 0.0079 0.0036 0.0020 -0.0006 0.0003 0.0003 -0.0367 -0.0242 -0.1529 -0. -0.0020 -0.0004 -0.0850 -0.1081 0. -0.0946 0.4244 0.0180 1900 0337 0.0107 -0.0226 1.0000 -0.5059 0.0040 0.0527 0.6754 1.0000 -0.0004 0.0348 -0.6918 0.3307 -0.0018 -0.0052 -0.3365 -0.2443 0.0110 0.0279 -0.2298 -0.3698 Columns 9 through 12 281 Table C.4.12 (cont’d) - 0.01461 0.0837 0.1190 0.00611 -0.0385 0.0679 -0.1095 + 0.01461 -0.1095 0.0594 + 0.00611 0.0594 - -0.0834 - 0.25111 -0.0834 + 0.25111 -0.0423 0.0985 -0.1622 + 0.10341 -0.1622 - 0.10341 -0.0481 0.1328 0.0988 - 0.02561 0.0988 + 0.02561 -0.0363 0.0898 -0.3278 - 0.32851 —0.3278 + 0.32851 0.0494 0.1087 —0.0078 + 0.02681 -0.0078 - 0.02681 -0.0169 0.1033 -0.1466 - 0.20441 -0.1466 + 0.20441 -0.0533 0.0888 -0.6487 + 0.60861 -0.6487 - 0.60861 0.0612 0.1898 0.0000 - 0.00011 0.0000 + 0.00011 0.0008 0.0001 0.0000 + 0.00001 0.0000 - 0.00001 -0.0001 0.0000 1.0000 - 0.00001 1.0000 + 0.00001 -0.0221 1.0000 0.0060 + 0.01381 0.0060 - 0.01381 1.0000 0.1271 0.0132 + 0.02281 0.0132 - 0.02281 -0.5807 0.1387 0.3465 + 0.11381 0.3465 - 0.11381 -0.1321 0.7413 0.0014 + 0.01441 0.0014 - 0.01441 -0.1446 0.1326 0.0937 + 0.11201 0.0937 - 0.11201 0.1934 0.6406 0.0063 + 0.15621 0.0063 - 0.15621 -0.1934 0.5611 Columns 13 through 16 0.2272 0.9374 1.0000 0.1261 0.1514 1.0000 -0.4078 —0.3692 1.0000 0.9405 -0.2802 -0.2107 -0.0216 0.8469 0.5439 0.0407 -0.0508 0.8768 -0.1946 -0.1984 -0.1948 0.8099 -0.1330 -0.0582 —0.0224 0.7822 0.1012 -0.0702 -0.1591 0.7752 -0.1506 -0.0434 -0.7483 0.6601 0.0275 0.0590 0.0000 -0.0001 0.0003 -0.0001 0.0000 0.0000 0.0000 -0.0002 -0.4308 -0.4482 -0.1129 0.2820 0.0669 -0.1554 0.1136 -0.1066 -0.0983 -0.1753 0.1024 -0.5877 -0.0567 -0.5488 0.0011 0.2598 -0.1926 -0.2014 0.1751 -0.7983 0.1602 -0.6647 0.1677 -0.7431 0.3786 -0.5734 0.0447 1.0000 Columns 17 through 18 1.0000 -0.2090 -0.0630 0.2264 -0.4876 -0.1397 0.5959 -0.0901 -0.0143 0.0554 -0.2566 -0.0353 0.2172 0.0106 -0.1708 -0.1073 -0.0464 0.1091 0.0003 -0.0005 0.0000 0.0000 -0.1414 0.0354 0.1581 -0.4904 0.1561 0.0465 0.2394 0.2777 0.1374 detjj - -0.0571 0.0567 -0.5149 1.0000 -0.9298 -6.0095e+29 detd - 1.7207e+23 detda - -3.4925e+06 detad = 282 Table C.4.12 (cont’d) 3.4649e+28 condjj = 1.4727e+05 condd = 17.6983 condda = 1.2449e+05 condad - 9.0097e+04 vtt = 1.0000 0.0025 -0.0674 -0.1803 1.0000 -0.1320 -0.0006 0.0000 1.0000 ett B 1.0e+03 * ~0.0935 0 0 0 -2.4353 0 0 0 -0.0001 LIST OF REFERENCES LIST OF REFERENCES R.A. 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