New Formulation and Strong MISOCP Relaxations for AC Optimal Transmission Switching Problem

As the modern transmission control and relay technologies evolve, transmission line switching has become an important option in power system operators' toolkits to reduce operational cost and improve system reliability. Most recent research has relied on the DC approximation of the power flow model in the optimal transmission switching problem. However, it is known that DC approximation may lead to inaccurate flow solutions and also overlook stability issues. In this paper, we focus on the optimal transmission switching problem with the full AC power flow model, abbreviated as AC OTS. We propose a new exact formulation for AC OTS and its mixed-integer second-order conic programming (MISOCP) relaxation. We improve this relaxation via several types of strong valid inequalities inspired by the recent development for the closely related AC Optimal Power Flow (AC OPF) problem. We also propose a practical algorithm to obtain high quality feasible solutions for the AC OTS problem. Extensive computational experiments show that the proposed formulation and algorithms efficiently solve IEEE standard and congested instances and lead to significant cost benefits with provably tight bounds

A Cycle-Based Formulation and Valid Inequalities for DC Power Transmission Problems with Switching

It is well-known that optimizing network topology by switching on and off transmission lines improves the efficiency of power delivery in electrical networks. In fact, the USA Energy Policy Act of 2005 (Section 1223) states that the U.S. should "encourage, as appropriate, the deployment of advanced transmission technologies" including "optimized transmission line configurations". As such, many authors have studied the problem of determining an optimal set of transmission lines to switch off to minimize the cost of meeting a given power demand under the direct current (DC) model of power flow. This problem is known in the literature as the Direct-Current Optimal Transmission Switching Problem (DC-OTS). Most research on DC-OTS has focused on heuristic algorithms for generating quality solutions or on the application of DC-OTS to crucial operational and strategic problems such as contingency correction, real-time dispatch, and transmission expansion. The mathematical theory of the DC-OTS problem is less well-developed. In this work, we formally establish that DC-OTS is NP-Hard, even if the power network is a series-parallel graph with at most one load/demand pair. Inspired by Kirchoff's Voltage Law, we give a cycle-based formulation for DC-OTS, and we use the new formulation to build a cycle-induced relaxation. We characterize the convex hull of the cycle-induced relaxation, and the characterization provides strong valid inequalities that can be used in a cutting-plane approach to solve the DC-OTS. We give details of a practical implementation, and we show promising computational results on standard benchmark instances

Algorithms for DC Programming via Polyhedral Approximations of Convex Functions

There is an existing exact algorithm that solves DC programming problems if one component of the DC function is polyhedral convex (Loehne, Wagner, 2017). Motivated by this, first, we consider two cutting-plane algorithms for generating an $\epsilon$-polyhedral underestimator of a convex function g. The algorithms start with a polyhedral underestimator of g and the epigraph of the current underestimator is intersected with either a single halfspace (Algorithm 1) or with possibly multiple halfspaces (Algorithm 2) in each iteration to obtain a better approximation. We prove the correctness and finiteness of both algorithms, establish the convergence rate of Algorithm 1, and show that after obtaining an $\epsilon$-polyhedral underestimator of the first component of a DC function, the algorithm from (Loehne, Wagner, 2017) can be applied to compute an $\epsilon$ solution of the DC programming problem without further computational effort. We then propose an algorithm (Algorithm 3) for solving DC programming problems by iteratively generating a (not necessarily $\epsilon$-) polyhedral underestimator of g. We prove that Algorithm 3 stops after finitely many iterations and it returns an $\epsilon$-solution to the DC programming problem. Moreover, the sequence $\{x_k\}_{k\geq 0} outputted by Algorithm 3 converges to a global minimizer of the DC problem when$\epsilon$ is set to zero. Computational results based on some test instances from the literature are provided

Contingency-Constrained Unit Commitment With Intervening Time for System Adjustments

The N-1-1 contingency criterion considers the con- secutive loss of two components in a power system, with intervening time for system adjustments. In this paper, we consider the problem of optimizing generation unit commitment (UC) while ensuring N-1-1 security. Due to the coupling of time periods associated with consecutive component losses, the resulting problem is a very large-scale mixed-integer linear optimization model. For efficient solution, we introduce a novel branch-and-cut algorithm using a temporally decomposed bilevel separation oracle. The model and algorithm are assessed using multiple IEEE test systems, and a comprehensive analysis is performed to compare system performances across different contingency criteria. Computational results demonstrate the value of considering intervening time for system adjustments in terms of total cost and system robustness.Comment: 8 pages, 5 figure