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

Abstract

We consider a bilevel attacker-defender problem to find the worst-case attack on the relays that control transmission grid components. The attacker infiltrates some number of relays and renders all of 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 (MILP). 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. Through our empirical experiments, we find that this bound is almost always as tight as we can get from guessing the dual bounds, even for more congested networks. We demonstrate our results in a case study on ten networks with up to 6468 buses.

Comparing Machine Learning and Optimization Approaches for the N − k Interdiction Problem Considering Load Variability

“…Due to the challenges inherent to this formulation, many solution approaches simplify the problem to yield a structure suited for single-level reformulation. The network flow model used in [11], for example, linearly relaxes the power flow equations in the lower-level problem. Conversely, the two new approaches presented in the next section of this paper train NNs with sampled solutions to the nonlinear lower-level problem (6c)-(6d) to preserve some of the nonlinear behavior and allow for fast online detection of multiple dangerous contingencies across a range of operation.…”

“…This section presents three approaches for solving the N − k interdiction problem ( 6): a state-of-the-art bilevel programming approach that uses a network flow relaxation of the power flow equations [11], a new approach that uses a NN reformulated as a MILP, and a new multi-step NN regression approach.…”

“…As a benchmark, we consider the recently proposed approach in [11] that solves the N − k interdiction problem using a network flow relaxation of the power flow equations [17] in the lower-level problem. While the upper-level problem is similar to that of model ( 6), the lower-level problem is much simpler in that it (i) is linear, (ii) cannot remove other network components, and (iii) focuses only on minimizing the load shed.…”

“…Most formulations therefore rely on linearizing the lower-level problem, which can lead to "false positives" or "false negatives" when identifying dangerous contingencies [3], [8]- [11]. In essence, solving this problem involves a tradeoff between solution correctness and computational tractability.…”

“…We also consider a range of load variation, i.e., the factor by which loads may randomly vary from nominal values during training and run-time testing, in both of these approaches to make the predictions flexible to different load patterns and/or inclusion of distributed solar. We compare the performance of these new ML approaches to the state-of-the-art linearized bilevel optimization approach in [11]. The results demonstrate that the ML methods outperform this linearized approach at finding the most dangerous contingencies on average over several test cases and contingency sizes.…”