  This paper focuses on cascading line failures in the transmission system of
the power grid. Recent large-scale power outages demonstrated the limitations
of percolation- and epid- emic-based tools in modeling cascades. Hence, we
study cascades by using computational tools and a linearized power flow model.
We first obtain results regarding the Moore-Penrose pseudo-inverse of the power
grid admittance matrix. Based on these results, we study the impact of a single
line failure on the flows on other lines. 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 then study the
cascade properties, considering metrics such as the distance between failures
and the fraction of demand (load) satisfied after the cascade (yield). We use
the pseudo-inverse 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 set of lines whose
removal has the most significant impact (under various metrics) is NP-Hard and
introduce a simple heuristic for the minimum yield problem. Overall, the
results demonstrate that using the resistance distance and the pseudo-inverse
of admittance matrix provides important insights and can support the
development of efficient algorithms.
