The application of numerical analysis method on Hopf bifurcation in power system

Abstract

The problem of determining the Hopf bifurcation point in an autonomous nonlinear system with bifurcation parameters is converted into solve a set of nonlinear algebraic equations , which can be constructed by a semi-analytical method and solved by an iterative numerical algorithm as proposed, according to the necessary and sufficient conditions for existing of Hopf bifurcation. First of all, the eigen-polynomial of Jacobian matrix of the system can be decomposed into product of two parts under the condition for existing of Hopf bifurcation. And then, the coefficients of the eigen-polynomial can be decided though the semi-analytical method by using the relations of the coefficients and the elements of Jacobian matrix, and deducing the conditions which should be decided by the coefficients at Hopf bifurcation point. The nonlinear equation set is constructed through the above condition and the one that Hopf bifurcation point must be the balance point of the system. The nonlinear equation set is constructed through the above condition and the one that Hopf bifurcation point must be the balance point of the system. And last, the nonlinear equation set is solved by using Newton2Raphson iterative method. The proposed method can determine both Hopf bifurcation point and the pure imaginary eigenvalue pair of the Jacobin matrix of the system at the same time, thus avoiding the shortcoming for repeatedly solving the eigenvalue problems for every change of the chosen parameters as in some other existing methods. The proposed method is feasible and effective which is illustrated in General Electric Power Systems. © Springer-Verlag 2012.