PowerModels.jl: An Open-Source Framework for Exploring Power Flow Formulations

In recent years, the power system research community has seen an explosion of novel methods for formulating and solving power network optimization problems. These emerging methods range from new power flow approximations, which go beyond the traditional DC power flow by capturing reactive power, to convex relaxations, which provide solution quality and runtime performance guarantees. Unfortunately, the sophistication of these emerging methods often presents a significant barrier to evaluating them on a wide variety of power system optimization applications. To address this issue, this work proposes PowerModels, an open-source platform for comparing power flow formulations. From its inception, PowerModels was designed to streamline the process of evaluating different power flow formulations on shared optimization problem specifications. This work provides a brief introduction to the design of PowerModels, validates its implementation, and demonstrates its effectiveness with a proof-of-concept study analyzing five different formulations of the Optimal Power Flow problem

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

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.Ph.D

Optimization of Critical Infrastructure with Fluids

Many of the world's most critical infrastructure systems control the motion of fluids. Despite their importance, the design, operation, and restoration of these infrastructures are sometimes carried out suboptimally. One reason for this is the intractability of optimization problems involving fluids, which are often constrained by partial differential equations or nonconvex physics. To address these challenges, this dissertation focuses on developing new mathematical programming and algorithmic techniques for optimization problems involving difficult nonlinear constraints that model a fluid's behavior. These new contributions bring many important problems within the realm of tractability. The first focus of this dissertation is on surface water systems. Specifically, we introduce the Optimal Flood Mitigation Problem, which optimizes the positioning of structural measures to protect critical assets with respect to a predefined flood scenario. Two solution approaches are then developed. The first leverages mathematical programming but does not tractably scale to realistic scenarios. The second uses a physics-inspired metaheuristic, which is found to compute good quality solutions for realistic scenarios. The second focus is on potable water distribution systems. Two foundational problems are considered. The first is the optimal water network design problem, for which we derive a novel convex reformulation, then develop an algorithm found to be more effective than the current state of the art on select instances. The second is the optimal pump scheduling (or Optimal Water Flow) problem, for which we develop a mathematical programming relaxation and various algorithmic techniques to improve convergence. The final focus is on natural gas pipeline systems. Two novel problems are considered. The first is the Maximal Load Delivery (MLD) problem for gas pipelines, which aims at finding a feasible steady-state operating point that maximizes load delivery for a severely damaged gas network. The second is the joint gas-power MLD problem, which couples damaged gas and power networks at gas-fired generators. In both problems, convex relaxations of nonconvex dynamical constraints are developed to increase tractability.PHDIndustrial & Operations EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/169849/1/tasseff_1.pd