Voltage instability: Phenomena, countermeasures, and analysis methods

Proceedings of the IEEE

Volumn 88, Issue 2, 2000, Pages 208-227

  Van Cutsem, Thierry ^a  

^a UNIVERSITY OF LIÈGE   (Belgium)

Author keywords

[No Author keywords available]

Indexed keywords

BIFURCATION (MATHEMATICS); ELECTRIC LOADS; ELECTRIC POWER TRANSMISSION NETWORKS; ENERGY POLICY; NONLINEAR SYSTEMS; NUMERICAL ANALYSIS; OUTAGES; VOLTAGE CONTROL;

ELECTRIC LOAD DYNAMICS; POWER SYSTEM SECURITY; VOLTAGE INSTABILITY;

ELECTRIC UTILITIES;

EID: 0033900542     PISSN: 00189219     EISSN: None     Source Type: Journal    
DOI: 10.1109/5.823999     Document Type: Article

References (132)

1. B. M. Weedy and B. R. Cox, "Voltage stability of radial power links," Proc. his:. Elect. Eng., vol. 1 15, pp. 528-536, 1968.
2. J. Zaborszky and J. W. Rittenhouse, Electric Power Transmission. Troy, NY: Rensselaer, 1969.
3. T. Nagao, "Voltage collapse at load ends of power systems," Elect- Eng. Japan, vol. 95. pp. 62-70, 1975.
4. B. M. Weedy, Electric Power Systems. 3rd ed. New York: Wiley, 1979."
5. [51 W. R. Lachs, "Voltage collapse in EHV power systems," in Proc. IEEE PES Winter Meeting, Paper A 78 057-22, 1978.
6. W. R. Lachs, "System reactive power limitations," in Proc. IEEE PES Winter Meeting, Paper A 79 015-9, 1979.
7. C. Barbier and J.-P. Barret, "An analysis of phenomena of voltage collapse on a transmission system," RevoieGénérale de l'Electricité (Special Issue), pp. 3-21. 1980.
8. Y. Tarnura, H. Mory, and S. Iwamoto, "Relationship between voltage instability and multiple load flow solutions," IEEE Trans. Power App. Syst., vol. 102, pp. 1115-1125, 1983.
9. P. Borremans, A. J. Caivaer. J.-P. De Reuck, J. Goossens, E. Van Geerf, J. Van Hecke, and A. Van Ranst, "Voltage stability-Fundamental aspects and comparison of practical criteria," in CIGRE Proc., Paper 38-11, 1984.
10. T. J. E. Miller, Ed., Reactive Power Control in Electric Power Systems. New York: Wiley, 1982.
11. A. J. Caivaer, "Voltage stability and collapses: A simple theory based on real and reactive currents," Revue Générale de l'Electricité, vol. 8, pp. 1-17, 1986.
12. P. Kessel and H. Glavltsch, "Estimating the voltage stability of a power system," IEEE Trans. Power Delivery, vol. 1, 1986.
13. [13J H. K. Clark, "Voltage control and reactive supply problems," IEEE Tutorial on Reactive Power: Basics, Problems and Solutions, pp. 17-27, 1987.
14. Planning against voltage collapse," Electro, vol. Ill, pp. 55-75, 1987.
15. A. E. Hammad and M. Z. El-Zadek, "Prevention of transient voltage instabilities due to induction motor loads by static var compensators," IEEE Trans.PowerSv.it.. vol. 4, pp. 1182-1190, 1989.
16. C. Concordia, "Voltage instability," Int. J. Elect. Power Energy SW., vol. 13, pp. 14-20, 1991.
17. Bulk Power System Voltage Phenomena 1 - Voltage Stability and Security, L. H. Fink, Ed., Potosi, MO, 1988.
18. Bulk Power Svstem Voltage Phenomena II - Voltage Stability and Security, L. H. Fink, Ed., Deep Creek Lake, MD, 1991.
19. Bulk Power Svstem Voltage Phenomena III - Voltage Stability and Security, L. H. Fink, Ed., Davos, Switzerland, 1994.
20. Bulk Power Systems Dynamics and Control IV - Restructuring, L. H. Fink and C. Vournas, Eds., Santorini. Greece, 1998.
21. Modelling of voltage collapse including dynamic phenomena," CIGRE Publication, CIGRE Task Force 38-02-10, 1993.
22. "Indices predicting voltage collapse including dynamic phenomena," CIGRE Publication, CIGRE Task Force 38-02-11, 1994.
23. "Criteria and countermeasures for voltage collapse," CIGRE Publication, CIGRE Task Force 38-02-12, 1994.
24. [241 "Protection against voltage collapse," CIGRE Publication, CIGRE Working Group 34-08, 1998.
25. "Voltage stability of power systems: concepts, analytical tools, and industry experience," IEEE Working Group on Voltage Stability, IEEE Special Publication 90TH0358-2-PWR, 1990.
26. "Suggested techniques for voltage stability analysis," IEEE Working Group on Voltage Stability, IEEE Publication 93TH0620-5PWR, 1993.
27. "Voltage collapse mitigation," IEEE Working Group K12, IEEE Power System Relaying Committee, 1996.
28. P. Kundur, Power S'.'Mm Stability and Contra'. New York: McGraw-Hill, 1994.'
29. C. W. Taylor, Power System Voltage Stability. New York: McGraw-Hill, 1994.
30. T. Van Cutsern and C. VoBrnas, Voltage Stability of Electric Power Systems. Norweil, MA: Kiuwer, 1998.
31. V. Ajjarapu and B. Lee, "Bibliography on voltage stability," IEEE Trans. Power Syst., vol. 13, pp. 115-125, 1998.
32. P. W. Sauer and M. A. Pai, Power Systems Dynamics and Stability. Englewood Cliffs, NJ: Prentice-Hall, 1997.
33. L. O. Chua, C. A. Dcsoer. andE. S. Kuh, Linear and Nonlinear Circuits. New York: McGraw-Hill, 1987.
34. M. S. Calovic, "Modeling and analysis of under-lead tap-changing transformer control systems," IEEE Trans. Power App.Syst.,vol. 103, pp. 1909-1915, 1984.
35. P. W. Sauer and M. A. Pas, "A comparison of discrete vs. continuous dynamic models of tap-changing-under-load transformers," in Bulk Power System Voltage Phenomena III - Voltage Stability ar.d Security, Davos, Switzerland, 1994.
36. "Lone Term Dynamics - Phase if. Final Report," CIGRE Task Force 38-02-08, CIGRE Publication, 1995.
37. "Recommended models for overexcitation limiting devices," IEEE Trans. Energy Com-ersion, vol. 10, pp. 706-713, 1995.
38. H. Ohtsuki, A. Yokoyama, and Y. Sekine. "Reverse action of on-load tap changer in association with voltage collapse," IEEE Trans. Power SyM., vol. 6, pp. 300-306, 1991.
39. T. K. Vu and C. C. Liu, "Analysis of tap-changer dynamics and construction of voltage stability regions," in Proc. IEEE ISCAS, vol. 3, 1988, pp. 1615-1618.
40. K. T. Vu and C.-C. Liu, "Shrinking stability regions and voltage collapse in power systems," IEEE Trans. Circuits Svst. 1, vol. 39, pp. 271-289, 1992.
41. R. J, Thomas and A. Tiranuchit, "Dynamic voltage instability," in 26th CDC Proc., Los Angeles, CA, 1987, pp. 53-58.
42. Y. Sekine and H. Ohtsuki, "Cascaded voltage collapse," IEEE Trans. Power Syst., vol. 5, pp. 250-256, 1990.
43. C. D. Voumas and G. A. Manos, "Modeling of stalling motors during voltage stability studies," IEEE Trans, Power Svst., vol. 13, pp. 775-781, 1998.
44. B. R. Williams, W. R. Schmus, and D. C. Dawson, "Trans-mission vottage recovery delayed by stalled air conditioner compressors," IEEE Trans. Power Syst., vol. 7, pp. 1173-1181, 1992.
45. J. Deuse, J. Dubois, R. Fanna, and I. Hanza, "Undervoltage load shedding scheme," IEEE Trans. Power Syst., vol. 12. pp. 1446-1454, 1997.
46. K.-M. Graf, "Dynamic simulation of voltage collapse processes in EHV systems," in Bulk Power System Voltage Phenomena 1 - Voltage Stability ans Security, K.-M. Fink, Ed., Potosi, MO, 1988, pp. 6.45-6.54.
47. D. Karlsson, "Voltage stability simulations using detailed models based on field measurements," Ph.D. dissertation, Chalmers University of Technology, Sweden, 1992.
48. L. M. Hajagos and B. Danai, "Laboratory measurements and models of modern loads and their effect on voltage stability studies," IEEE Trans. Power Syst., vol. 1998, pp. 584-590, 1997.
49. D. J. Hill, "Nonlinear dynamic load models with recovery for voltage stability studies," IEEE Trans. Power Syst.. vol. 8, pp. 166-176, 1993
50. D. Karisson and D. J. Hill, "Modelling and identification of nonlinear dynamic loads in power systems," IEEE Trans. Power Syst., vol. 9, pp. 157-166, 1994.
51. W. Xu and Y. Mansour, "Voltage stability analysis using generic dynamic load models," IEEE Trans. Power Syst., vol. 9, pp. 479-493, 1994.
52. W. Xu, E. Vaahedi, Y. Mansour, and J. Tamby, "Voltage stability load parameter determination from field tests on B.C. Hydro's system," IEEE Trans. Power Syxl., vol. 12, pp. 1290-1297, 1997.
53. T. Van Cutsern and C. D. Vournas, "Voltage stability analysis in transient and mid-term time scales," IEEE Trans. Power Syst., vol. 1 l,pp. 146-154, 1996.
54. S. Bernard, G. Trudel, and G. Scott, "A 735-kv shunt reactors automatic switching system for Hydro-Quebec network," IEEE Trans. Power Syst.. vol. 11, pp. 2024-2030, 1996.
55. R. A. Schlueter, "A voltage stability security assessment method," IEEE Tram. Power Syst., vol. 13, pp. 1423-1438, 1998.
56. (1998) Voltage stability criteria, undervoltage load shedding strategy and reactive power reserver monitoring methodology. Reactive Power Reserve Work Group. [Online"]. Available: http://www.wscc.com/criteria.htm
57. J.-P. Paul, J.-Y. Leost, and J.-M. Tesseron, "Survey of the secondary voltage control in France: Present realization and investigations," IEEE Trans. Power SVSL, vol. 2, pp. 505-511, 1987.
58. J.-P. Paul, C. Corroyer, P. JeanneS, J.-M. Tesseron, F. Maury, and A. Torra, "Improvements in the organi/ation of secondary voltage control in France," in CIGRE P roc.. Paper 38/39-03, 1990.
59. [591 H. Vu, P. Pruvot, C. Launay, and Y. Harrnand, "An improved voltage control on large-scale power system," IEEE Trans. Power Syst., vol. Il, pp. 1295-I303, 1996.
60. V. Archi'diacono, S. Corsi, A. Natale, C. Raffaelli, and V. Menditto, "New developments in the application of enel transmission system voltage and reactive power automatic control," in CIGRE Pmc., Paper 38/39-06, 1990.
61. C. W. Taylor, "Concepts of undervoltage load shedding for voltage stability," IEEE Trans. Power Delivery, vol. 7, pp. 480-488, 1992.
62. S. A. Nircnberg and D. A. Mclnnis, "Fast acting loud shedding," IEEE Trans. Power Syst., vol. 7, pp. 873-877, 1992.
63. C. W. Taylor, G. L. Comegys, F. R. Nassief, D. VI. Eiwood, and P. Kundur, "Simulation and implementation of undervoltage load shedding for pacific northwest voltage stability," Pmc. CIGRE '94, 1994.
64. T. Van Cutsem. Y. Jacquemart, J.-N. Marquet, and P. Pruvot, "Extensions and applications of a mid-term voltage stability analysis method," in Bulk Power System Phenomena U-Voltage Stability and Security, Davos, Switzerland, 1994, pp. 251-270.
65. T. J. Overbye, "A power flow solvability measure for unsolvable cases," IEEE Trans. Power Syst., vol. 9, pp. 1359-1365, 1994.
66. T. J. Overbye, "Computation of a practical method to restore power flow solvability," IEEE Trans. Power Syst., vol. 10, pp. 280-287, 1995.
67. [67J S. Arnborg, G. Andersson, D. J. Hill, and I. A. Hiskens, "On undervoltage load shedding in power systems," Int. J. Elect. Power Energy Syst., vol. 19, pp. 141-149, 1997.
68. D. S. Popovic, V. A. Levi, and Z. A. Gorecan, "Co-ordination of emergency secondary-voltage control and load shedding to prevent voltage instability," P roc. I nut. Elect. Eng. Generation, Transmission, Distrih., vol. 144, pp. 293-300, 1997.
69. C. Moors and T. Van Cutsem, "Determination of optimal load shedding against voltage instability," in P roc. 13th PSCC, Trondheim, 1999.
70. L. A. S. Pilotto, M. Szechtman, and A. E. Hammad, "Transient ac voltage related phenomena for HVDC schemes connected to weak ac systems," IEEE Trans. Power Delivery, vol. 7, pp. 1396-1404, 1992.
71. A. J. Calvaer and E. V. Geert, "Quasi steady-state synchronous machine linearization around an operating point and applications," IEEE Trans. Power App. Syst., vol. 103, pp. 1466-1472, 1984.
72. T. Van Cutsem, "A method to compute reactive power margins with respect to voltage collapse," IEEE Trans. Power Syst., vol. 6, pp. 145-156, Feb 1991.
73. S. Iwamoto and Y. Tamura, "A load flow calculation method for ill-conditioned power systems," IEEE Trans. Power App. Syst., vol. 100, pp. 1736-1743, 1981.
74. N. D. Reppen, R. R. Austria, J. A. Uhrin, M. C. Patel, and A. Galatic, "Performance of methods for ranking and evaluadon of voltage collapse contingencies applied to a large-scale network," in IEEEINTUA Athens Power Tech. Pmc., 1993, pp. 337-343.
75. B. J. Stott, "Power system dynamic response calculation," Proc. IEEE, vol. 67, pp. 219-241", 1979.
76. A. Kurita, H. Okubo, K. Oki, S. Agematsu, D. B. Klapper, N. W. Miller, W. W. Price, J. J. Sanchez-Gasca, K. A. Wirgau, and T. D. Younkins, "Multiple time-scale power system dynamic simulation," IEEE Trans. Power Syst., vol. 8, pp. 216-223, 1993.
77. J. J. Sanchez-Gasca. R. D'Aquila, J. J. Paserba, W. W. Price, D. B. Klapper, and I. Hu, "Extended-term dynamic simulation using variable time step integration," Commit. Apolicat. Power, vol. 5, pp. 23-28, 1993.
78. J. Deuse and M. Slubbe, "Dynamic simuiation of voltage collapses," IEEE Trans. Power Syst., vol. 8, pp. 894-904, 1993.
79. G. K. Morison, B. Gao, and P. Kundur, "Voltage stability analysis using static and dynamic approaches," IEEE Trans. Power Sy.it., vol. 8, pp. 1159-1J71, 1993.
80. R- J- Koessler and J. W. F'eltes, "Time-domain simulation investigates voltage collapse," Comput. Amplicai. Power, vol. 5, pp. 18-22, 1993.
81. C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations. Enslewood Cliffs, NJ: Prentice-Hall, 1971.
82. Y. Sekine, K. Takahashi, Y. Ichida, Y. Ohura, and N. Tsuchimori, "Method of analysis and assessment of power system voltage phenomena, and improvements including control strategies for greater voltage stability margins," in CIGRE Proc., Paper 38-206, 1982.
83. T. Van Cutsem, Y. Jacquemart, J.-N. iMarquet, and P. Pruvot, "A comprehensive analysis of mid-term voltage stability," IEEE Tram,. Power Syst., vol 10, pp. 1173-1182, 1995.
84. T. Van Cutsem and R. Mailhot, "Validation of a fast voltage stability analysis method on the hydro-québec system," IEEE Trans. 'Power Syst., vol. 12, pp. 282-292, 1997.
85. H. G. Kwatny, A. K. Pasrija, and L. H. Bahar, "Static bifurcations in electric power networks: Loss of steady-state stability and voltage collapse," IEEE Trans. Circuits Syst. I vol. 33, pp. 981-991, 1986.
86. 1. Dobson and H. D. Chiang, "Towards a theory of voltage collapse in electric power systems." Syst. Control Lett., vol. 13, pp. 253-262, 1989.
87. P. W. Sauer and M. A. Pai, "Power system steady-state stability and the load flow Jacobian," IEEE Traits. Power Svst., vol. 5, pp. 1374-1383, 1990.
88. V. Venkatasubramanian, H. Schattier, and J. Zaborszky, "A taxonomy of the dynamics of the large power system with emphasis on its voltage stability," in Bulk Power System Voilage Phenomena 11-Voltage Stability and Security, Deep Creek Lake, MD, pp. 9-44,
89. C. A. Canizares, F. L. Alvarado, C. L. DeMarco, l. Dobson, and W. F. Long, "Point of collapse methods applied to ac/dc power systems," IEEE Trans. Power Svst., vol. 7, pp, 673-683, 1992.
90. V. Ajjarapu and B. Lee, "Bifurcation theory and its application to nonlinear dynamic phenomena in an electric power system," IEEE Trans. Power Syst., vol. 7, pp. 424-431, 1992.
91. M. A. Pai, P. W. Sauer, and B. C. Lesieutre, "Static and dynamic nonlinear loads and structural stability in power systems," Proc. IEEE (Soecial Issue on Nonlinear Phenomena in Power Svsterns), vol. 83, pp. 1562-1572, 1995.
92. V. A. Venikov, V. A. Stroev, V. I. Idelchick, and V. I. Tarasov, "Estimation of electrical power system steady-state stability," IEEE Trans. Power App. Svst.. vol. 94, pp. 1034-1040, 1975.
93. A. Tiranuchit and R. J. Thomas, "A posturing strategy against voltage instabilities in electric power systems," IEEE Trans. Power Syst.. vol. 3, pp. 87-93, 1988.
94. F. L. Aivarado and T. H. Jung, "Direct detection of voltage collapse conditions," in Bulk Power System Voltage Phenomenon I-Voltage Stability and Security, Potosi, MO, pp. 5.23-5.38.
95. R. A. Schlueter, A. G. Costi, J. E. Sekerke, and H. L. Forgey, "Voltage stability and security assessment,", EPR! Tech. Rep. EL-5967, 1988.
96. P.-A. Löf, T. Smed, G. Andersson, and D. J. Hill, "Fast calculation of a voltage stability index," IEEE Trans. Power Syst., vol. 7, pp. 54-64. 1992.
97. A. Berizzi, P. Bresesti, P. Marannino, G. P. Granelli, and M. Montagna, "System-area operating margin assessment and security enhancement.against voltage collapse," IEEE Trans. Power Syst., vol. 11, pp. 1451-146211996.
98. B. Gao. G. K. Morison, and P. Kundur, "Voltage stability evaluation using modal analysis," IEEE Trans. Power Sxt., vol. 7, pp. 1529-1542, 1992.
99. N. Flatabø, R. Ognedal, and T. Carlsen, "Voltage stability condition in a power system calculated by sensitivity methods," IEEE Trans. Power Syst., vol. 5, pp. 1286-1293, 1990.
100. J. Carpentier, R. Girard, and E. Scano, "Voltage collapse proximity indicators computed front an optimal power flow." in Proc. 8th PSCC, Helsinki, 1984, pp. 671-678.
101. M. Bcgovic and A. Phadke, "Control of voltage stability using sensitivity analysis," IEEE Trans. Power Syst., vol. 7, pp. 114-123,1992.
102. T. Van Cutsem, "An approach to corrective control of voltage instability using simulation and sensitivity," IEEE Trans. Power Syst.. vol. 10, pp. 616-622, 1995.
103. I. Dobson and L. Lu. "Immediate change in stability and voltage collapse when generator reactive power limits are encountered," Bulk Power System Voltage Phenomena 11-Voilage Stability ans Security, pp. 65-73.
104. K. Iba, H. Suzuki, M. Egawa, and T. Watanabe, "Calculation of critical loading condition with nose curve using homoiopy method," IEEE Trans. Power Syst.. vol. 6, pp. 584-593, 1991.
105. H. D. Chiang, W. Ma, R. J. Thomas, and J. S. Thorp, "A tool for analyzing voltage collapse in electric power systems," in Proc. 10ft; Power System Computation Conf, Graz, Austria, 1991. pp. 1210-1217.
106. V. Ajjarapu and C. Christy, "The continuation power flow: a tool for steady state voltage stability analysis," IEEE Trans. Power Syst., vof. 7, pp. 416-423, 1992.
107. C. A. Cañizares and F. L. Alvarado, "Point of collapse and continuation methods for large ac/dc systems," IEEE Trans. Power Svst., vol. 8, pp. 1-8, 1993.
108. H. D. Chiang, A. J. Flueck, K. S. Shah, and N. Balu, "Cpflow: A practical tool for tracing power system steady-state stationary behavior due to load and generation variations," IEEE Trans. Power Syst., vol. 10, pp. 623-630, 1995.
109. Y. Mansour, C. D. James, and D. N. Fettet, "Voltage Stability and Security-B.C. Hydro's Operating Practice," In Bulk Power System Voltage Phenomena II-Voltage Stability and Security, Potosi. MO, pp. 2.9-2.25.
110. T. Van Cutsem, "Network-optimization based reactive power margin calculation," in Proceedings IFAC Symposium on Power Systems Modelling and Control Applications, Paper No 7.1.1, Brussels, Belgium, 1988.
111. O. O. Obadina and G. L Berg, "Determination of voltage stability limit in multimachine power systems," IEEE Trans. Power Syst., vol. 3, pp. 1545-1554, 1988.
112. C. J. Parker, I. F. Morrison, and D. Sutanto, "Application of an optimisation method for determining the reactive margin from voltage collapse in reactive power planning," IEEE Trans. Power Syst., vol. 11, pp. 1473-1481, 1996.
113. G. D. Irisarri, X. Wang, J. Tong, and S. Mokhtari, "Maximum Soadability of power systems using interior point non-linear optimisation method," IEEE Trans. Power Syst., vol. 12, pp. 162-172. 1997.
114. R. R. Austria, N. D. Reppen, J. A. Uhrin, M. C. Patel, and A. Galatic, "Applications of the optimal power flow to analysis of voltage collapse limited power transfer," in Bulk Power System Voltage Phenomena 11-Voltage Stability and Security, Deep Creek Lake, MD, pp. 311-319.
115. B. Gao, G. K. Morison, and P. Kundur, "Towards the development of a systematic approach for voltage stability assessment of laree-scale power systems," IEEE.'1'rans. Power Syst., vol. 11, pp. 1314-1324, 1996.
116. T. Van Cutsem, C. Moisse, and R. Mailhot, "Determination of secure operating limits with respect to voltage collapse," IEEE Trans. Power Syst., vol. 14, pp. 327-335, 1999.
117. G. C. Ejebe, G. D. Irisarri, S. Mokhtari, O. Obadina, P. Ristanovic, and J. Tong, "Methods for contingency screening and ranking for voltage stability analysis of power systems," in Proc. PICA Conf., 1995, pp. 249-255.
118. H. D. Chiang, C. S. Wang, and A. J. Flueck, "Look-ahead voltage and load margin contingency selection functions for large-scale power systems," IEEE Trans. Power Svst., vol. 12, pp. 173-180, 1997.
119. T. Van Cutsem, "A comprehensive voltage security assessment," presented at the IEEE 1998 Winter Power Meeting,
120. E. Vaahedi, C. Fuchs, W. Xu, Y. Mansour, H. Hamadanizadeh, and G. K. Morison, "Voltage stability contingency screening and ranking," IEEE Tram. Power Syst., vol. 14, pp. 256-265, 1999.
121. R. J. Marceau, M. Sirandi, S. Soumare, X. D. Do, F. Gatiana, and R. Mailhot, "A review of signal energy analysis for the rapid determination of dynamic security limits," Canadian J. Elect. Comput. Eng., vol. 21, pp. 125-132, 1996.
122. 1. Dobson, "Observations on the geometry of saddle node bifurcation and voltage collapse in electric power systems," IEEE Trans. Circuits Syst. l, vol. 39, no. 3, pp. 240-243, 1992.
123. S. Greene, I. Dobson, and F. L. Alvarado, "Sensitivity of the loading margin to voltage collapse with respect 10 arbitrary parameters,'' IEEE Trans. Power Syst., vol. 12. pp. 262-272, 1997.
124. F. L. Alvarado, I. Dobson. and Y. Hu, "Computation of closest bifurcations in power systems," IEEE Trans. Power Syst., vol. 9, pp. 918-928. 1994.
125. C. Cafiizares. "Calculating optimal system parameters to maximize the distance to saddle-node bifurcations," IEEE Trans. Circuits Syst. I, vol. 45, pp. 225-237, 1998.
126. Z. Feng, V. Aijarapu, and D. J. Maratukulam, "A practical minimum load shedding strategy to mitigate voltage collapse," IEEE Trans. Pon-er Syst., vol. 13, pp. 1285-1291. 1998.
127. E. De Tuglie. M. La Scala, and P. Scarpellini, "Real-time preventive actions for the enhancement of voltage-degraded trajectories,""IEEE Power Syst., vol. 14, pp. 561-568, 1999.
128. S. Granville, J. C. O. Mello, and A. C. G. Melo, "Application of interior point methods to power flow unsolvabiiity," IEEE Trails. Power Syst., vol. 11, pp. 1096-1104, 1996.
129. X. Wans, O. S. Ejebc, J. Tong, and J. G. Waight, "Preventive/corrective control for voltage stability using direct interior point method," IEEE Trans. Power Syst., vol. 13, pn. 878-883, 1998.
130. T. Van Cutsem, L. Wchenkel, M. Pavella, B. Heilbronn, and M. Goubin, "Decision tree approaches to voltage security assessment," Proc. Inst. Elect. Eng. C, vol. 140, pp. 189-198, 1993.
131. L. Wehenkei, T. Van Cutsem, M. Pavella, B. Heilbronn, and P. Pruvot, "Machine learning, neural networks and statistical pattern recognition for voltage security : A comparative study," Eng. Imell Syst., vol. 2. pp. 233-245, 1994.
132. L. A. Wehcnkel, Automatic Learning Techniques In Povt'er Systems. Norwell, MA: Kluwer, 1997.