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We consider a bilevel attacker-defender problem to find the worst-case attackon the relays that control the transmission grid. The attacker maximizes loadshed by infiltrating a number of relays and rendering the components connectedto them inoperable. The defender responds by minimizing the load shed,re-dispatching using a DC optimal power flow (DCOPF) problem on the remainingnetwork. Though worst-case interdiction problems on the transmission grid arewell-studied, there remains a need for exact and scalable methods. Methodsbased on using duality on the inner problem rely on the bounds of the dualvariables of the defender problem in order to reformulate the bilevel problemas a mixed integer linear problem. Valid dual bounds tend to be large,resulting in weak linear programming relaxations and making the problemdifficult to solve at scale. Often smaller heuristic bounds are used, resultingin a lower bound. In this work we also consider a lower bound, where instead ofbounding the dual variables, we drop the constraints corresponding to Ohm'slaw, relaxing DCOPF to capacitated network flow. We present theoretical resultsshowing that, for uncongested networks, approximating DCOPF with network flowyields the same set of injections, which suggests that this restriction likelygives a high-quality lower bound in the uncongested case. Furthermore, we showthat in the network flow relaxation of the defender problem, the duals arebounded by 1, so we can solve our restriction exactly. Last, we see empiricallythat this formulation scales well computationally. Through experiments on 16networks with up to 6468 buses, we find that this bound is almost always astight as we can get from guessing the dual bounds, even for congested networks.In addition, calculating the bound is approximately 150 times faster thanachieving the same bound with the reformulation guessing the dual bounds.

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