The number of (equilibrium) steady-state solutions of models of power systems

We use powerful analytical tools and modern techniques from algebraic geometry to infer the number of (equilibrium) steady state solutions of models of power systems. The theorems developed also infer the upper bound on the number of solutions of the full-fledged (equilibrium) steady-state equations for various levels of detailed models of power systems. Sufficient conditions are provided which determine the precise number of complex solutions to the load flow. The sufficient conditions are cast in terms of properties of the admittance matrix of the power grid. Consequently, these sufficient conditions are placed on the topology (configuration) of the given power network. When the sufficient conditions are not satisfied, the determined precise number becomes an upper bound on the number of solutions. >

[1]  A. Wright.  Finding all solutions to a system of polynomial equations , 1985 .

[2]  Carlos J. Tavora,et al.  Equilibrium Analysis of Power Systems , 1972 .

[3]  F.M.A. Salam,et al.  Determining the solutions of the load flow of power systems: Theoretical results and computer implementation , 1990, 29th IEEE Conference on Decision and Control.

[4]  J. Yorke,et al.  Numerical solution of a class of deficient polynomial systems , 1987 .

[5]  Christopher I. Byrnes,et al.  The singularity theory of the load flow equations for a 3-node electrical power system☆ , 1983 .

[6]  F. Salam,et al.  Asymptotic stability and estimating the region of attraction for the swing equatons , 1986 .

[7]  W. Zulehner.  A simple homotopy method for determining all isolated solutions to polynomial systems , 1988 .

[8]  S. Sastry,et al.  Analysis of power-flow equation , 1981 .

[9]  Ahmed H. El-Abiad,et al.  Transient Stability Regions of Multimachine Power Systems , 1966 .

[10]  Alexander P. Morgan.  A homotopy for solving polynomial systems , 1986 .

[11]  J. Baillieul,et al.  Geometric critical point analysis of lossless power system models , 1982 .

[12]  Lionel M. Ni,et al.  PowerCube-a software package for solving load flow problems , 1989, Proceedings of the 32nd Midwest Symposium on Circuits and Systems,.

[13]  L. Ni,et al.  Parallel processing for the load flow of power systems: the approach and applications , 1989, Proceedings of the 28th IEEE Conference on Decision and Control,.

[15]  Chaotic dynamics even in the highly damped swing equations of power systems , 1988, Proceedings of the 27th IEEE Conference on Decision and Control.

[16]  Fathi M. A. Salam,et al.  Parallel processing for the steady state solutions of large-scale non-linear models of power systems , 1989, IEEE International Symposium on Circuits and Systems,.

[17]  A. Morgan,et al.  Errata: Computing all solutions to polynomial systems using homotopy continuation , 1987 .

[18]  Christopher I. Byrnes,et al.  The load flow equations for a 3-node electrical power system☆ , 1983 .

[19]  Ieee Report.  Computer representation of excitation systems , 1968 .

[20]  F. S. Prabhakara,et al.  A simplified determination of transient stability regions for Lyapunov methods , 1975, IEEE Transactions on Power Apparatus and Systems.

[21]  F.M.A. Salam.  Current issues in the stability and control of interconnected power systems , 1985 .

[22]  James A. Yorke,et al.  The random product homotopy and deficient polynomial systems , 1987 .

[23]  J. Yorke,et al.  A homotopy method for locating all zeros of a system of polynomials , 1979 .

[24]  Arthur R. Bergen,et al.  Power Systems Analysis , 1986 .