A Bound Strengthening Method for Optimal Transmission Switching in Power Systems

TL;DR

This paper introduces a simple bound strengthening method for the optimal transmission switching problem in power systems, significantly improving solver speed for large-scale real-world systems by preprocessing convex relaxations.

Contribution

It presents a novel, NP-hardness proof for bound strengthening and offers an effective preprocessing technique that enhances mixed-integer solver performance.

Findings

Speedup in solver runtime for medium- and large-scale systems

NP-hardness of finding the strongest variable bounds

Effective bound strengthening as a preprocessing step

Abstract

This paper studies the optimal transmission switching (OTS) problem for power systems, where certain lines are fixed (uncontrollable) and the remaining ones are controllable via on/off switches. The goal is to identify a topology of the power grid that minimizes the cost of the system operation while satisfying the physical and operational constraints. Most of the existing methods for the problem are based on first converting the OTS into a mixed-integer linear program (MILP) or mixed-integer quadratic program (MIQP), and then iteratively solving a series of its convex relaxations. The performance of these methods depends heavily on the strength of the MILP or MIQP formulations. In this paper, it is shown that finding the strongest variable upper and lower bounds to be used in an MILP or MIQP formulation of the OTS based on the big-$M$ or McCormick inequalities is NP-hard. Furthermore, it is proven that unless P=NP, there is no constant-factor approximation algorithm for constructing these variable bounds. Despite the inherent difficulty of obtaining the strongest bounds in general, a simple bound strengthening method is presented to strengthen the convex relaxation of the problem when there exists a connected spanning subnetwork of the system with fixed lines. The proposed method can be treated as a preprocessing step that is independent of the solver to be later used for numerical calculations and can be carried out offline before initiating the solver. A remarkable speedup in the runtime of the mixed-integer solvers is obtained using the proposed bound strengthening method for medium- and large-scale real-world systems.