  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.
