New theoretical results on stability regions and bifurcations of nonlinear dynamical systems and their applications to electric power systems analysis

The concept of stability regions (or domains of attraction) of nonlinear dynamical systems and bifurcation theory are of fundamental importance to the modern theory of stability for many disciplines in engineering and applied mathematics. Working on the concept of stability regions, the Lyapunov approach has been found to give rather conservative estimations of stability regions of many systems. New results on the theory of stability regions and bifurcations are given in this thesis. The concept of practical stability regions is introduced and a comprehensive theory is developed for this concept. With the new concept of practical stability regions, one can greatly overcome the problem of conservative estimations of stability regions using the Lyapunov function approach. The topological and dynamical properties of stability regions of a general class of differential-algebraic equations are presented. It is shown that the stability region of an asymptotically stable equilibrium point can be characterized via the stability region of an associated singularly perturbed system. The structure and the nature of impasse points that can be present via the stability boundary are identified. Estimation of the stability region using energy functions is also included. A constructive methodology to optimally estimate stability regions of large-scale interconnected nonlinear systems is proposed. One appealing feature of the constructive methodology is that it significantly reduces the undesirable conservativeness of the Lyapunov function approach in estimating the stability regions of interconnected nonlinear systems. This constructive methodology is iterative in character and yields a sequence of estimated stability regions which is shown to be a strictly monotonic increasing sequence and yet each of these regions lies inside the entire stability region. The robustness of saddle-node bifurcation under the addition of unmodeled dynamics is investigated. The unmodeled dynamics can be fast and/or slow dynamics. It is shown that, under a fairly general condition, the saddle-node bifurcation persists for general nonlinear systems under unmodeled dynamics of their vector fields. Furthermore, it is shown that the system behaviors after the saddle-node bifurcation of the underlying vector field and that of the vector field with unmodeled dynamics are close to each other in state space. Some particular results obtained in the thesis are applied to the problem of transient stability of structure preserving models and to the analysis of voltage collapse in electric power systems.