Analysis of Failures in Power Grids

This paper focuses on line failures in the transmission system of power grids. Recent large-scale power outages demonstrated the limitations of percolation- and epidemic-based tools in modeling failures and cascades in power grids. Hence, we study failures and cascades by using computational tools and a linearized power-flow model. We first obtain results regarding the Moore-Penrose pseudoinverse of the power grid admittance matrix. Based on these results, we analytically study the impact of a single-line failure on the flows on other lines and introduce metrics to evaluate the robustness of grids to failures. We also illustrate via simulation the impact of the distance and resistance distance on the flow increase following a failure, and discuss the difference from the epidemic models. We use the pseudoinverse of admittance matrix to develop an efficient algorithm to identify the cascading failure evolution, which can be a building block for cascade mitigation. Finally, we show that finding the lines whose removal results in the minimum yield (the fraction of demand satisfied after the cascade) is NP-Hard and present a simple heuristic for finding such a set. Overall, the results demonstrate that using the resistance distance and the pseudoinverse of the admittance matrix provides important insights and can support the development of algorithms for designing robust power grids and controlling the evolution of a cascade upon failures.