A Method for Quickly Bounding the Optimal Objective Value of an OPF Problem using a Semidefinite Relaxation and a Local Solution

TL;DR

This paper introduces a method that leverages local solutions to quickly compute bounds on the optimal objective value of the OPF problem using semidefinite relaxation, balancing speed and tightness.

Contribution

It proposes a novel approach that combines local solutions with SDP relaxation to accelerate objective value bounding in OPF problems.

Findings

The method is faster than traditional SDP relaxation.

It provides weaker bounds compared to SDP but faster than other relaxations.

Numerical results demonstrate the speed-tolerance trade-off.

Abstract

Optimal power flow (OPF) is an important problem in the operation of electric power systems. Due to the OPF problem's non-convexity, there may exist multiple local optima. Certifiably obtaining the global solution is important for certain applications of OPF problems. Many global optimization techniques compute an optimality gap that compares the achievable objective value corresponding to the feasible point from a local solution algorithm with the objective value bound from a convex relaxation technique. Rather than the traditional practice of completely separating the local solution and convex relaxation computations, this paper proposes a method that exploits information from a local solution to speed the computation of an objective value bound using a semidefinite programming (SDP) relaxation. The improvement in computational tractability comes with the trade-off of reduced tightness for the resulting objective value bound. Numerical experiments illustrate this trade-off, with the proposed method being faster but weaker than the SDP relaxation and slower but tighter than second-order cone programming (SOCP) and quadratic convex (QC) relaxations for many large test cases.