  Analysis of transient stability of strongly nonlinear post-fault dynamics is
one of the most computationally challenging parts of Dynamic Security
Assessment. This paper proposes a novel approach for assessment of transient
stability of the system. The approach generalizes the idea of energy methods,
and extends the concept of energy function to a more general Lyapunov Functions
Family (LFF) constructed via Semi-Definite-Programming techniques. Unlike the
traditional energy function and its variations, the constructed Lyapunov
functions are proven to be decreasing only in a finite neighborhood of the
equilibrium point. However, we show that they can still certify stability of a
broader set of initial conditions in comparison to the traditional energy
function in the closest-UEP method. Moreover, the certificates of stability can
be constructed via a sequence of convex optimization problems that are
tractable even for large scale systems. We also propose specific algorithms for
adaptation of the Lyapunov functions to specific initial conditions and
demonstrate the effectiveness of the approach on a number of IEEE test cases.
