Abstract
A detailed example of a power system model with load dynamics is studied by investigating qualitative changes or bifurcations in its behaviour as a reactive power demand at one load bus is increased. In addition to the saddle-node bifurcation often associated with voltage collapse, we find other bifurcation phenomena which include Hopf bifurcation, cyclic fold bifurcation, period doubling bifurcation, and the emergence of chaos. The presence of these dynamic bifurcations motivates a re-examination of the role of saddle-node bifurcations in the voltage collapse phenomenon. In fact, simulation results suggest that voltage collapse may take place before the reactive power demand is increased to the system steady-state operating limit where a saddle-node bifurcation is detected. We also consider the role that the algebraic constraints imposed by some load models may play in the global analysis of the attractors of the system. Implications for power system operations are drawn.
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References
Abed E H 1985 Singularly perturbed Hopf bifurcation.IEEE Trans. Circuits Syst. CAS-32: 1270–1280
Abed E H, Alexander J C, Wang H, Hamdan A M A, Lee H C 1992 Dynamic bifurcations in a power system model exhibiting voltage collapse. University of Maryland, Technical Report no.TR 92–96
Abed E H, Varaiya P P 1984 Nonlinear oscillations in power systems.Int. J. Electr. Power Energy Syst. 6: 37–43
Alexander J C 1986 Oscillatory solutions of a model system of nonlinear swing equations.Int. J. Electr. Power Energy Syst. 8: 130–136
Arapostathis A, Sastry S S, Varaiya P P 1982 Global analysis of swing dynamics.IEEE Trans. Circuits Syst. CAS-29: 673–679
Chen R L, Varaiya P P 1988 Degenerate Hopf bifurcation in power systems.IEEE Trans. Circuits Syst. CAS-35: 818–824
Chiang H D 1989 Study of the existence of energy functions for power systems with losses.IEEE Trans. Circuits Syst. CAS-36: 1423–1429
Chiang H D, Hirsch M, Wu F F 1988 Stability regions of nonlinear autonomous dynamical systems.IEEE Trans. Autom. Control 33: 16–27
Chiang H D, Liu C W, Varaiya P P, Wu F F, Lauby M G 1992 Chaos in a simple power system model.IEEE Winter Power Meeting Paper No. 92WM 151-1PWRS
DeMarco C L, Bergen A R 1987 A security measure for random load disturbances in nonlinear power system models.IEEE Trans. Circuits Syst. CAS-34: 1546–1557
Dobson I, Chiang H D 1989 Towards a theory of voltage collapse in electric power systems.Syst. Control Lett. 13: 253–262
Dobson I, Delchamps D F 1989 Basin boundaries in the pendulum with nonperiodic forcing.Proceedings of the 23rd Conference of Information Sciences and Systems, Baltimore
Fekih-Ahmed L, Chiang H D 1992 Analysis of voltage collapse in structure preserving models of the power system.Proceedings of the International Symposium on Circuits and Systems, San Diego, pp. 2525–2528
Grebogi C, Ott E, Yorke J A 1983 Crises, sudden changes in attractors, and transient chaos.Physica D7: 181–200
Guckenheimer J 1983 Persistent properties of bifurcations.Physica D7: 105–110
Guckenheimer J, Holmes P J 1986Nonlinear oscillations, dynamical systems, and bifurcations of vector fields.Applied Mathematical Sciences, vol. 42 (Berlin: Springer-Verlag)
Kopell N, Washburn R B 1982 Chaotic motions in the two-degree-of-freedom swing equations.IEEE Trans. Circuits Syst. CAS-29: 738–746
Kwatny H G, Pasrija A K, Bahar L Y 1986 Static bifurcations in electric power systems: Loss of steady state and voltage collapse.IEEE Trans. Circuits Syst. CAS-33: 981–991
Milnor J 1963Morse theory (Princeton,NJ: University Press)
Narasimhamurti N 1984 On the existence of energy functions for power systems with transmission losses.IEEE Trans. Circuits Syst. CAS-31: 199–203
Rajagopalan C, Sauer P W, Pai M A 1989 Analysis of voltage control systems exhibiting voltage collapse.Proceedings of the 28th Conference on Decision and Control (New York:IEEE Press) pp. 332–335
Salam F M A, Marsden J, Varaiya P P 1984 Arnold diffusion in swing equations of a power system.IEEE Trans. Circuits Syst. 30: 199–206
Sastry S S, Desoer C A 1981 Jump behaviour of circuits and systems.IEEE Trans. Circuits Syst. 28: 1109–1124
Takens F 1976Constrained equations: a study of implicit differential equations and their discontinuous solutions.Lecture Notes in Mathematics, 525 (Berlin: Springer-Verlag)
Thom R 1975Stability and morphogenesis (Transl. by D H Fowler) (Paris: Addison-Wesley)
Tsolas N A, Arapostathis A, Varaiya P P 1985 A structure preserving energy function for power system transient stability analysis.IEEE Trans. Circuit Syst. CAS-32: 1041–1049
Varaiya P P, Wu F F, Chiang H D 1990 Bifurcation and chaos in power systems: A survey.ERL/UCB Memo no. M90/98, University of California, Berkeley
Varghese M, Chiang H D, Thorp J S 1993 Computer algorithms in power systems: From constructive methods to truncated fractals.IEEE Trans. Educ. 36: 36–41
Venkatasubramanian V, Schattler H, Zaborsky J 1992 A stability theory for differential algebraic systems such as the power system.Proceedings of the International Symposium of Circuits and Systems. (New York:IEEE Press) pp. 2517–2520
Wang H, Abed E H 1992 Bifurcation control of chaotic dynamical systems.Proceedings of the Nonlinear Control Systems Design Symposium (Paris:IFAC Press) pp. 57–62
Zaborsky J, Huang G, Zheng B, Leung T C 1988 On the phase portraits of a class of large nonlinear dynamical systems such as the power system.IEEE Trans. Autom. Control 33: 4–15
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Tan, C.W., Varghese, M., Varaiya, P. et al. Bifurcation and chaos in power systems. Sadhana 18, 761–786 (1993). https://doi.org/10.1007/BF03024224
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DOI: https://doi.org/10.1007/BF03024224