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It is well-known that optimizing network topology by switching on and offtransmission lines improves the efficiency of power delivery in electricalnetworks. In fact, the USA Energy Policy Act of 2005 (Section 1223) states thatthe U.S. should "encourage, as appropriate, the deployment of advancedtransmission technologies" including "optimized transmission lineconfigurations". As such, many authors have studied the problem of determiningan optimal set of transmission lines to switch off to minimize the cost ofmeeting a given power demand under the direct current (DC) model of power flow.This problem is known in the literature as the Direct-Current OptimalTransmission Switching Problem (DC-OTS). Most research on DC-OTS has focused onheuristic algorithms for generating quality solutions or on the application ofDC-OTS to crucial operational and strategic problems such as contingencycorrection, real-time dispatch, and transmission expansion. The mathematicaltheory of the DC-OTS problem is less well-developed. In this work, we formallyestablish that DC-OTS is NP-Hard, even if the power network is aseries-parallel graph with at most one load/demand pair. Inspired by Kirchoff'sVoltage Law, we give a cycle-based formulation for DC-OTS, and we use the newformulation to build a cycle-induced relaxation. We characterize the convexhull of the cycle-induced relaxation, and the characterization provides strongvalid inequalities that can be used in a cutting-plane approach to solve theDC-OTS. We give details of a practical implementation, and we show promisingcomputational results on standard benchmark instances.

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