This paper reports on the latest developments in taxonomy theory originally developed for dynamic systems represented by ordinary differential equations constrained by algebraic equations (DAEs). It gives some new results on the behavior of a second order state constrained system. It shows that the interplay between an unstable system and the hard limit can result generically in stable non-smooth limit cycles. These results are of particular importance for Hopf bifurcations which occur near the state limit since they point towards the possibility of stable operation even after a subcritical Hopf bifurcation has occured.

This paper provides an overview of the taxonomy theory which has been proposed as a fundamental platform for solving practical stability related problems in large constrained nonlinear systems such as the electric power system. The theory reveals a two-level intertwined cellular nature of the constrained system dynamics which serves as a unifying structure, a taxonomy, for analyzing nonlinear phenomena in large system models. These broadly divide into the state space aspects (related to dynamic stability issues among others) and the parameter space aspects (connected with bifurcation phenomena among others). In the state-space formulation, the boundary of the region of attraction for the operating point is shown (under certain Morse-Smale like assumptions) to be composed of stable manifolds of certain anchors and portions of the singularity surface. Such boundary characterization provides the foundation for rigorous Lyapunov theoretic transient stability methods. In the parameter space analysis, the feasibility region which is bounded by the feasibility boundary provides a safe opening region for guaranteeing local stability at the equilibrium under slow parametric variations. The feasibility boundary where the operating point undergoes loss of local stability is characterized in the form of three principal bifurcations including a new bifurcation called the singularity induced bifurcation.

This paper presents a time-delay formulation of the large power system dynamics which includes modeling the fast distributed parameter transmission system dynamics from within the familiar phasor domain of the equipment dynamics. The notion of fast time-varying phasors proposed recently for the balanced power system leads to explicit solutions in the form of time-delayed algebraic equations for the transmission system dynamics which is represented by a large interconnected set of partial-differential wave equations, under certain assumptions.>