A Scalable Lower Bound for the Worst-Case Relay Attack Problem on the Transmission Grid

We consider a bilevel attacker-defender problem to find the worst-case attack on the relays that control transmission grid components. The attacker infiltrates relays and renders the components connected to them inoperable, with the goal of maximizing load shed. The defender responds by minimizing the resulting load shed, re-dispatching using a DC optimal power flow (DCOPF) problem on the remaining network. Though worst-case interdiction problems on the transmission grid have been studied for years, none of the methods in the literature are exact because they rely on assumptions on the bounds of the dual variables of the defender problem in order to reformulate the bilevel problem as a mixed integer linear problem. The result is a lower bound, and additionally, the more conservatively the dual variables are bounded, the weaker the linear programming relaxations are and hence the more difficult it is to solve the problem at scale. In this work we also present a lower bound, where instead of bounding the dual variables, we drop the constraints corresponding to Ohm's law, relaxing DCOPF to capacitated network flow. This is a restriction of the original bilevel problem. We present theoretical results showing that, for uncongested networks, approximating DCOPF with network flow yields the same set of injections, and thus the same load shed, which suggests that this restriction likely gives a high-quality lower bound in the uncongested case. Furthermore, we show that in this formulation, the duals are bounded by 1, so we can solve our restriction exactly. Last, because the big-M values in the linearization are small and network flow has a well-known structure, we see empirically that this formulation scales well computationally with increased network size. Last, we find that this bound is almost always as tight as we can get from guessing the dual bounds, even for more congested networks.