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

TL;DR

This paper introduces a novel cycle-based mathematical formulation for the DC Optimal Transmission Switching problem, proves its NP-hardness, and develops strong valid inequalities to improve solution methods, with promising computational results.

Contribution

It establishes the NP-hardness of DC-OTS, proposes a cycle-based formulation inspired by Kirchhoff's Law, and derives valid inequalities for enhanced solving techniques.

Findings

NP-hardness of DC-OTS proved

Cycle-based formulation developed

Strong valid inequalities for solution improvement

Abstract

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.