Bogdanov-Takens bifurcation points and Sil´nikov homoclinicity in a simple power-system model of voltage collapse

The bifurcation structure of a simple power-system model is investigated, with respect to changes to both the real and reactive loads. Numerical methods for this bifurcation analysis are presented and discussed. The model is shown to have a Bogdanov-Takens bifurcation point and hence homoclinic orbits; these orbits can be of Sil´nikov type with many coexisting periodic solutions. We may use the bifurcation calculations to divide the two-parameter plane into a number of regions, for which there are qualitatively different dynamics. We classify and further investigate the dynamical behavior in each of these regions, using a Monte Carlo method to investigate basins of attraction of various stable states. We then show how this classification can be used to denote each regions as either safe or unsafe with respect to the likelihood of voltage collapse