Least Squares Estimation Based SDP Cuts for SOCP Relaxation of AC OPF

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.