Nonlinear analysis of limit cycles in power system models

The focus of the dissertation is on the study and analysis of limit cycles and Hopf bifurcations in power system models. Both analytical and computational results are pursued for different power system models.; Analysis of Hopf bifurcations is extended to a four equation SMIB model with excitation control. Under the assumptions of a fast high gain exciter, singular perturbation theory is applied to reduce the original four equation system onto a two-equation slow model. Thus Hopf bifurcation coefficient a can be calculated analytically. The sign of the coefficient a gives the supercritical or subcritical nature of the Hopf bifurcations, thus predict the existence of stable or unstable limit cycle. Formulas are derived for estimating the size of the limit cycles analytically. Hopf bifurcation coefficient a along the Hopf bifurcation locus of the SMIB model is computed numerically. We observe that the Hopf bifurcations are mostly subcritical. When the exciter control is a fast high gain control and when the Thevenin equivalent transmission line impedance is high, the Hopf bifurcation can become supercritical leading to birth of stable limit cycles locus on the SMIB system which verified the analytical results.; Computational results for several power system examples are used to study the relevance of ULC's in power system analysis. The size of ULC's indicates the size of the region of attraction. The behavior of ULC's changes with system parameters and generator loadings. Therefore, ULC computation gives us insight into dynamic security, that is, on how large the region of attraction for the operating point is. In such cases, the relative amplitudes of various state variables on the ULC can be used to identify the critical portion of the power system.; A heuristic algorithm for locating ULC is studied. The algorithm is based on reverse time integration with a quadratic stable manifold approximation. The method is investigated on a detailed SMIB system, 9 bus test system and two area system. This method can make insightful analysis of the transient behavior of power systems when the system matrix has some modes with small positive damping.