5 .>° 9 9 Analysis, Monitoring and Control of Voltage Stability ln Electric Power Systems by Miroslav M. Begovic Dissertation submitted to the Faculty ot' the Virginia Polytechnic institute and State University ot' in partial fultillment of the requirements for the degree PhD in Electrical Engineering in The Bradley Department of Electrical Engineering APPROVED: Arun G. Phadke, Chairman ti Jaime De La Rec Lopez Saifur Rahman / V fx Kwa Sur Tam John W. Laym July, 1989 Blacksburg, Virginia Analysis, Monitoring and Control of Voltage Stability ln Electric Power Systems by Miroslav M. Begovic Arun G. Phadke, Chairman The Bradley Department of Electrical Engineering (ABSTRACT) The work presented in this text concentrates on three aspects of voltage stability studies: analysis and determination of suitable proximity indicators, design of an effective real-time monitoring system and determination of appropriate emer- gency control techniques. A simulation model of voltage collapse was built as analytical tool on'39-bus, lO·generator power system model . Voltage collapse was modeled as a saddle node bifurcation of the system dynamic model reached by increasing the system loading. Suitable indicators for real-time monitoring were found to be the minimum singular value of power flow Jacobian matrix and generated reactive powers. A study of possibilities for reducing the number of measurements of voltage phasors needed for voltage stability monitoring was also made. The idea of load bus coherency with respect to voltage dynamics was in- troduced. An algorithm was presented which determines the coherent clusters of load buses in a power system based on an arbitrary criterion function, and the analysis completed with two proposed coherency criteria. Very good agreement was obtained by simulation between the results based on accurate and approxi- mate measurements of the state vector. An algorithm was presented for identifi- l cation of critical sets of loads in a voltage unstable power system, defined as a subset of loads whose changes have the most pronounced effect on the changes of minimum singular value of load fiow Jacobian or generated reactive powers. Effects of load shedding of critical loads were investigated by simulation and fa- vorable results obtained. An investigation was also done by sensitivity analysis of proximity indicators of the effects that locations and amounts of static var compensation have on the stability margin of the system. Static compensation was found to be of limited help when voltage instabilities due to heavy system loading occur in power systems. The feasibility of implementation of the analyses and algorithms presented in this text relies on development of a feasible inte- grated monitoring and control hardware. The phasor measurcment system which was designed at Virginia Polytechnic institute and State University represents an excellent candidate for implementation of real·time monitoring and control pro- cedures. Dedication Dcdication iv Acknowledgements Writing this section for a dissertation is one of the more pleasant duties. As with any large study of this nature, it was necessary for me to obtain the help and support of many persons. lt is difficult to meaningfully thank all the people who have made this long sought dream a reality. I could not possibly do justicc to all who have hclped in large and small ways; whether they know it or not they have my gratitude. v Acknowlcdgements vi Acknowlcdgcments Table of Contents Chapter I. Introduction .................................................... 1 Chapter 2. A Survey of Research On Voltage Instability ............................ 8 2.1 Introduction ......................................................... 8 2.2 Voltage Collapse: Observations ........................................... 9 2.3 Power System Modeling and Voltage Instability .............................. 16 2.3.1 Modeling of loads ................................................. 18 2.3.2 Generator modeling ................................................ 21 2.3.3 Modeling of ULTC transformers ...................................... 26 2.3 Voltage stability studies ................................................ 29 2.4 Conclusion ......................................................... 34 Chapter 3. Voltage Stability Analysis: Simulation Approach ........................ 37 3.1 Introduction ........................................................ 37 3.2 Model definition ..................................................... 39 3.3 Simulation algorithm .................................................. 43 3.4 Power system at static bifurcation ......................................... 56 vii Table of Contents 3.5 Steady state analysis .................................................. 70 3.6 Dynamic simulation analysis ............................................ 78 3.7 Conclusions ....................................................... 115 Chapter 4. Voltage Stability Monitoring: Phasor Measurements .................... 117 4.1 lntroduction ....................................................... 117 4.2 Clustering of load buscs in power systems .................................. 124 4.2.1 Coherency Criterion 1: Sensitivity Analysis .............................. 132 4.2.2 Cohercncy Criterion 2: Dynamic Analysis ............................... 135 4.2.3 Guidelines for Application .......................................... 138 4.3 Simulation Results .................................................. 142 4.4 Conclusions ....................................................... 163 Chapter 5. Voltage Stability Emergency Control ................................ 166 5.1 Introduction ....................................................... 166 5.2 Sensitivity Analysis of Proximity lndicators ................................. 170 5.2.1 Sensitivity analysis of minimum singular values of Jacobian .................. 171 5.2.2 Sensitivity analysis of generated rcactivc powers ........................... 181 5.2.3 Selective load shcdding ............................................ 187 5.3 Simulations and Comments ............................................ 190 5.4 Conclusions ....................................................... 195 Chapter 6. Voltage Stability and Static Compensation ............................ 215 6.1 Introduction ....................................................... 215 6.2 Sensitivity Analysis of Proximity lndicators ................................. 219 6.2.1 Sensitivity analysis of minimum singular value of Jacobian .................. 227 6.2.2 Sensitivity analysis of generated reactive powers ........................... 229 Table of Contents viii 6.3 Simulations and Comments ............................................ 232 6.4 Conclusions ....................................................... 237 Chapter 7. Conclusions .................................................. 253 Bibliography .......................................................... 260 Appendix A. Parameters of Power System Model ............................... 276 Vita ................................................................ 282 Table of Contents ix List of Illustrations Figure 1. Voltage profile in major 500 kV substations of TEPCO system before the collapse. .................................. 14 Figure 2. Phasor diagram of the generator model .................. 42 Figure 3. Simulation algorithm ............................... 55 +.5x’ Figure 4. Phase portrait of: x" + sin x - sin .412 = 0 . ........ 62 Figure 5. Phase portrait of: x" +.5x’ + sin x - sin 1.56 = 0 . ........ 63 Figure 6. Voltage collapse of a two generator system. ............... 66 Figure 7. Phase angle changes before voltage collapse. .............. 67 Figure 8. Power system model used for simulations. ................ 71 Figure 9. Critical load factor estimation ........................ 74 Figure 10. Phasor diagram of the system in normal operating regime. .. 79 Figure ll. Phasor diagram of the system close to steady state instability. 80 Figure 12. Phasor diagram of the system near voltage stability boundary. 81 Figure 13. Phase angle gradients near stability boundary. ............ 82 Figure 14. Voltage gradients near stability boundary. ............... 83 Figure 15. Voltage profile and gradients in normal and voltage unstable case. 84 Figure 16. Load factor patterns for step change simulations. .......... 96 Figure 17. Voltage profile corresponding to Figure 16 (1). ............ 97 l List or Illustrations x Figure 18. Voltage profile corresponding to Figure 16 (2). ............ 98 Figure 19. Voltage profile corresponding to Figure 16 (3). ............ 99 Figure 20. Minimum eigenvalues of Jacobian for Figures 17,18,19. .... 100 Figure 21. Voltage response for various step changes of load factor. .... 101 Figure 22. System loading pattern for ramp loading. ............... 102 Figure 23. Directions of approach to stability boundary. ............ 103 Figure 24. Voltage profiles at stability boundary (scenarios i, ii, iii). .... 104 Figure 25. Voltage collapse at 10 most critical buses (scenario i). ...... 105 Figure 26. Phase angle trajectories corresponding to collapse in Figure 25. 106 Figure 27. Minimum singular values of Jacobian for collapse in Figure 25. 107 Figure 28. Reactive power generation corresponding to collapse in Figure 25. ............................................. 108 Figure 29. Reactive power generation when generation limits are observed. 109 Figure 30. Voltage collapse due to step change of k applied