Power system transient stability: The critical clearing time

Energy and Lyapunov methods, termed direct methods, have been developed for the study of transient stability of power systems. Although they reduce the amount of computations involved, they have been found to be conservative, i.e. the critical clearing time predicted by these methods is found in most cases to be too small compared to the critical clearing time obtained via the conventional integration routines. Here we address the conservativeness problem and seek to remedy the direct methods approach to remove this obstacle. First we restate and prove a theorem upon which the analysis of the Lyapunov methods in the power system context have been based. The theorem originally is due to Willems but for which there has been no proof in the literature. We discuss the existing methods such as the controlling UEP and the PEBS (Potential Energy Boundary Surface), identify the sources of conservativeness in these methods and carefully illustrate these issues using a one machine-?? bus example. A new view that aims at resolving the pessimistic (sometimes over optimistic) results of direct methods is presented, then a new approach which is based on parameterizing a "piece" of the boundary associated with the "Controlling" UEP is introduced. The concept of Controlling UEP appears to offer a feasible approach that can be made analytically sound. Finally, the new approach is applied to the one machine-?? bus for the purpose of an illustration.