Least Square Estimation-Based SDP Cuts for SOCP Relaxation of AC OPF

TL;DR

This paper introduces a novel method using least square estimation to generate SDP cuts that strengthen SOCP relaxations in AC OPF problems, improving solution accuracy and scalability.

Contribution

It proposes a new LSE-based approach to generate SDP cuts that enhance SOCP relaxation for AC OPF, balancing computational efficiency and solution tightness.

Findings

SDP cuts significantly reduce the feasible region outside AC OPF

The method improves the gap between SOCP and SDP relaxations

Case studies show effectiveness on systems with up to hundreds of buses

Abstract

This paper presents a method that generates affine inequalities to strengthen the second-order conic programming (SOCP) relaxation of an alternating current optimal power flow (AC OPF) problem. The affine inequalities serve as cuts to get rid of points outside of the feasible region of AC OPF with semi-definite programming (SDP) relaxation. Hence, the affine inequalities are names as SDP cuts. While AC OPF with SDP relaxation has a high computational complexity, AC OPF with SOCP has a much lower computational complexity. Recent research has found that the feasible region of SDP relaxation is contained inside the feasible region of the SOCP relaxation. Therefore, the integration of SDP cuts into SOCP relaxation provides better scalability compared to the SDP relaxation and a tighter gap compared to the SOCP relaxation. The SDP cuts are generated by solving least square estimation (LSE) problems at cycle basis and further exploring the geometric characteristic of LSE. General feasibility cuts generating method is also employed for analysis. We found that the SDP cuts generated by LSE method are indeed feasibility cuts. The SDP cuts effectively reduce the search space. Case studies of systems with several buses to hundreds of buses have demonstrated the method is very effective in reducing the gaps.