  Estimating the stability boundary is a fundamental and challenging problem in
transient stability studies. It is known that a proper level set of a Lyapunov
function or an energy function can provide an inner approximation of the
stability boundary, and the estimation can be expanded by trajectory reversing
methods. In this paper, we streamline the theoretical foundation of the
expansion methodology, and generalize it by relaxing the request that the
initial guess should be a subset of the stability region. We investigate
topological characteristics of the expanded boundary, showing how an initial
guess can approach the exact stability boundary locally or globally. We apply
the theory to transient stability assessment, and propose expansion algorithms
to improve the well-known Potential Energy Boundary Surface (PEBS) and Boundary
of stability region based Controlling Unstable equilibrium point (BCU) methods.
Case studies on the IEEE 39-bus system well verify our results and demonstrate
that estimations of the stability boundary and the critical clearing time can
be significantly improved with modest computational cost.
