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

TL;DR

This paper introduces a scalable lower bound method for the worst-case relay attack problem on transmission grids by relaxing the DC optimal power flow to a network flow model, enabling efficient and tight bounds especially in uncongested networks.

Contribution

The authors propose a novel lower bound approach by relaxing Ohm's law constraints, providing theoretical guarantees for uncongested networks and demonstrating computational scalability on large power system networks.

Findings

The network flow relaxation yields the same injections as DCOPF in uncongested networks.

Duals in the network flow relaxation are bounded by 1, allowing exact solutions.

The method is approximately 150 times faster than traditional dual-bound guessing approaches.

Abstract

We consider a bilevel attacker-defender problem to find the worst-case attack on the relays that control the transmission grid. The attacker maximizes load shed by infiltrating a number of relays and rendering the components connected to them inoperable. The defender responds by minimizing the 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 are well-studied, there remains a need for exact and scalable methods. Methods based on using duality on the inner problem rely on the bounds of the dual variables of the defender problem in order to reformulate the bilevel problem as a mixed integer linear problem. Valid dual bounds tend to be large, resulting in weak linear programming relaxations and making the problem difficult to solve at scale. Often smaller heuristic bounds are used, resulting in a lower bound. In this work we also consider 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. We present theoretical results showing that, for uncongested networks, approximating DCOPF with network flow yields the same set of injections, which suggests that this restriction likely gives a high-quality lower bound in the uncongested case. Furthermore, we show that in the network flow relaxation of the defender problem, the duals are bounded by 1, so we can solve our restriction exactly. Last, we see empirically that this formulation scales well computationally. Through experiments on 16 networks with up to 6468 buses, we find that this bound is almost always as tight as we can get from guessing the dual bounds, even for congested networks. In addition, calculating the bound is approximately 150 times faster than achieving the same bound with the reformulation guessing the dual bounds.