A phase transition model for cascading network failure

We consider a special structure of dynamic system model that admits a very tractable inclusion of element failure phenomena, for which a global system Lyapunov function can be constructed. This class includes Hamiltonian systems as a special case, with a wide class of R-L-C circuits and mechanical spring-mass-damper systems in which branch failures are induced by exceeding thresholds of inductor current or spring force magnitude. Using a detailed R-L-C circuit as our illustrative example, this article describes how geometric features of the global Lyapunov function constructed, along with partial trajectory information from time domain simulations, can be used to more efficiently predict which branches are subject to failure in a specific disturbance scenario. The underlying concepts are closely related to techniques of merging families of Lyapunov functions in hybrid system analysis. It is hoped that these techniques will add to the set of tools available for predicting and preventing cascading failure in large scale networks.