Strengthening the Conic Quadratic Relaxation of the Optimal Transmission Switching Problem

The goal of the Optimal Transmission Switching (OTS) problem is to identify a topology of the power grid that minimises the total energy production costs, while satisfying the operational and physical constraints. The problem is formulated as a non-convex mixed-integer nonlinear program, which poses extraordinary computational challenges. A common approach is to solve the mixed-integer second-order conic programming (MISOCP) relaxation of the OTS problem instead of the original non-convex problem. In this talk, we introduce new valid inequalities, called disjunctive cycle-bound inequalities, to strengthen an existing MISOCP relaxation of the OTS problem. We propose additional valid inequalities based on integer programming theory to mitigate the extra computational burden imposed by the big-M formulation of the disjunctive cycle-bound inequalities. We prove that some of these valid inequalities are facet-defining and also provide an efficient separation algorithm. Computational experiments conducted on benchmark instances from PGLib demonstrate the effectiveness of the proposed approach.