A Phase Transition Model for

As the scale of engineered systems such as electric power grids, communication networks, and the Internet expands and as society’s dependence upon reliable operation of these networks increases, it is vital that system engineers seek a better understanding of how small-scale failures of individual elements may propagate to produce global, system-wide failures. Within this general class of problems, importance sampling methods have improved the efficiency of search techniques for identifying rare failures in probabilistic models for communication networks [1], [2]; similar progress has been made in characterizing the role of graph interconnection structure in propagation of failures in electric power networks [3], [4]. The work here is motivated largely by the electric power systems application. Efforts to date on these problems for power networks have focused primarily on Markov chain-like representations, in which the state of the system is represented as discrete and the engineering constraints impacting failure transitions are those of the steady state. Our goal here will be somewhat different in flavor. In a deterministic model, we seek to capture the role of transient dynamic response following a specified initiating disturbance and examine subsequent (“cascading”) element failures that are induced when operating thresholds for individual elements are exceeded along the state trajectory. In the formulation to be presented here, the potentially probabilistic nature of initiating events is not treated. To allow tractable analysis of continuous dynamics in state-space models of high dimension, we will restrict attention to a specialized class of system representation, one in which energy storage associated with individual elements is