  Swing equations are an integral part of a large class of power system
dynamical models used in rotor angle stability assessment. Despite intensive
studies, some fundamental properties of lossy swing equations are still not
fully understood. In this paper, we develop a sufficient condition for
certifying the stability of equilibrium points (EPs) of these equations, and
illustrate the effects of damping, inertia, and network topology on the
stability properties of such EPs. The proposed certificate is suitable for
real-time monitoring and fast stability assessment, as it is purely algebraic
and can be evaluated in a parallel manner. Moreover, we provide a novel
approach to quantitatively measure the degree of stability in power grids using
the proposed certificate. Extensive computational experiments are conducted,
demonstrating the practicality and effectiveness of the proposal.
