## Solution Techniques for Large-Scale Optimization Problems on the Transmission Grid

##### Abstract

In this thesis, we are interested in solution techniques and primal heuristics for several large-scale optimization problems on the transmission grid. While some of these problems have been studied for a long time, none of the techniques proposed previously allowed them to be solved exactly on large-scale networks, rendering them of little use in practice. We will present methodology which yields high quality solutions on large networks.
In Chapter 2, we consider the DC optimal transmission switching (DCOTS) problem. In this problem, we simultaneously optimize the grid topology (i.e., choose which lines are on and off) and the generator dispatch, using a DC optimal power flow (DCOPF) model. It is well-known that transmission switching is an affordable way to reduce congestion in the grid, reducing generation costs. However, DCOTS has so far been a prohibitively difficult model to solve, particularly given that dispatch problems are solved every 5 to 10 minutes for most independent system operators. We present a data-driven approach which assumes that DCOTS has been solved to optimality offline for a variety of demand profiles. We then use the k-nearest neighbors (KNN) method as a primal heuristic, directly mapping from demand profiles to topologies. This scales well since the computational time is dominated by the time to solve k linear programs. We find that we can generate high-quality primal solutions within the time constraints imposed by real-time operations. In addition, we find that defining the feature space for KNN differently can also yield equally good results: In particular, using dual information from the DCOPF problem can be effective.
In Chapter 3, we propose a scalable lower bound for a worst-case attack on transmission grid relays. This is a bilevel interdiction problem in which an attacker first targets relays within an attack budget, compromising all components controlled by the relays he chooses, and aiming to maximize load shed. Then, a defender redispatches the generators, solving a DCOPF model and minimizing the load shed. Since the inner problem is convex, it is possible to dualize it, resulting in a mixed-integer single-level reformulation. However, the difficulty arises in linearizing this reformulation. Without bounds on the dual variables of DCOPF, this is not possible to do exactly. Prior literature has used heuristic bounds on the duals. However, in addition to providing a lower bound only, this comes at a computational cost: The more conservatively the bounds are chosen, the larger the big-M values are in the resulting mixed-integer programming (MIP) formulation. This worsens the continuous relaxation and makes it increasingly difficult for even commercial solvers. Instead, we propose using a different lower bound: We relax DCOPF to capacitated network flow, dropping the constraints corresponding to Ohm's law. We show that, on uncongested networks, the injections we get from solving this relaxation are a good approximation of those from solving DCOPF. We can again dualize this problem, creating a single-level formulation. The duals of the relaxed defender problem are bounded in absolute value by 1, meaning we can linearize the single-level formulation and solve it exactly. Furthermore this MIP scales extremely well when solving with a commercial solver.
Last, in Chapter 4, we present methodology to solve a trilevel interdiction problem where the inner bilevel problem is the worst-case relay attack problem from Chapter 3. In this problem, we optimize the design of the Supervisory Control and Data Acquisition (SCADA) network controlling the transmission grid in order to minimize the impact of a worst-case cyberattack. Specifically, we decide where the cyber networks in the SCADA system should be subdivided, with communication limited between these subdivisions, a technique called network segmentation. The resulting problem is a trilevel interdiction model with pure integer first and second player problems and a convex third-player problem. We show that it can be solved for large-scale power networks using a covering decomposition approach in which we iteratively fix a network design and generate a worst-case attack. We then find a new design that makes all the generated attacks infeasible. When there is no such design, then the design corresponding to the least-damaging attack generated so far is optimal. We show empirically that this method is scalable for large power networks and moderate network designer and attacker budgets.