Quantifying the effect of k-line failures in power grids

In this paper, we quantify the effect of k-line failures on flow changes using the DC power flow model. We demonstrate that our approach results in efficient tools that can be used to reduce both the total number of cases needed to be analyzed and the computational complexity in power grid contingency analysis. After providing an analytical update of the pseudo-inverse of the admittance matrix following a k-line failure, we compute the k-line outage distribution matrix. The k-line outage distribution matrix is a generalization to the line outage distribution factor for single line failures. We obtain a matrix equation based on the submatrices of the matrix of equivalent reactance values, relating changes in power flows to the initial flows on the failed lines. We also define and analytically compute the disturbance value of a failure - the weighted sum of squares of the flow changes - and show that it can be computed for any set of failures in O(1) independent of the size of the power grid. Finally, we numerically compute disturbance values for all possible choices of 3-line failures in IEEE 118-bus and 300-bus systems and show that the disturbance values provide a clear separation between failures with higher impact and lower impact.