A sparse convex AC OPF solver and convex iteration implementation based on 3-node cycles

Abstract

In this paper, we exploit sparsity technique and strengthen second-order cone programming (SOCP) relaxation of alternating current optimal power flow (AC OPF) through enforcing submatrices corresponding to maximal cliques and minimal chordless cycles in the cycle basis positive semi-definite (PSD). The proposed method adds virtual lines in minimal chordless cycles to decompose each of them into 3-node cycles. By enforcing the submatrices related to 3-node cycles PSD, the resulting convex relaxation has a tight gap. For majority of the test instances, the resulting gap is as tight as that of a semi-definite programming (SDP) relaxation, yet the computing efficiency is much higher. Further, convex iteration is implemented on the proposed solver to achieve exactness by enforcing all submatrices corresponding to lines and virtual lines rank-1. Test results from small size to large size with thousands of buses demonstrate the capability of the proposed solver and the feasibility of the convex iteration implementation.