# Probabilistic $N$-$k$ Failure-Identification for Power Systems

## TL;DR

This paper introduces a probabilistic approach to the $N$-$k$ failure-identification problem in power systems, formulating it as a complex optimization problem and proposing algorithms to find near-optimal solutions, validated on various test systems.

## Contribution

It formulates a probabilistic $N$-$k$ failure-identification problem as a bilevel mixed-integer nonlinear program and develops algorithms to solve it efficiently.

## Key findings

- Algorithms effectively solve small to large test instances.
- Proposed methods achieve solutions close to the optimal.
- Numerical results demonstrate the approach's scalability and accuracy.

## Abstract

This paper considers a probabilistic generalization of the $N$-$k$ failure-identification problem in power transmission networks, where the probability of failure of each component in the network is known a priori and the goal of the problem is to find a set of $k$ components that maximizes disruption to the system loads weighted by the probability of simultaneous failure of the $k$ components. The resulting problem is formulated as a bilevel mixed-integer nonlinear program. Convex relaxations, linear approximations, and heuristics are developed to obtain feasible solutions that are close to the optimum. A general cutting-plane algorithm is proposed to solve the convex relaxation and linear approximations of the $N$-$k$ problem. Extensive numerical results corroborate the effectiveness of the proposed algorithms on small-, medium-, and large-scale test instances, the test instances include the IEEE 14-bus system, the IEEE single-area and three-area RTS96 systems, the IEEE 118-bus system, the WECC 240-bus test system, the 1354-bus PEGASE system, and the 2383-bus Polish winter-peak test system.