Rank-1 positive semidefinite matrix-based nonlinear programming formulation for AC OPF

Abstract

Semidefinite programming (SDP) relaxation offers a tight relaxation to nonconvex alternating current optimal power flow (AC OPF) problems. When the solution obtained from SDP relaxation of AC OPF is a rank-1 positive semidefinite (PSD) matrix, this solution is exact to the original problem. Research efforts have been devoted to find a rank-1 PSD matrix. In this paper, a nonlinear programming formulation with the PSD matrix as the decision variable is proposed. The rank-1 PSD matrix constraint is equivalent to all 2×2 minors of the PSD matrix being zero. The main challenge of the proposed formulation is the large number of the quadratic equality constraints. For a system of N buses, there are (Formula presented.) minor related constraints (For a 10-node system, this number is 2025). Graph decomposition–based approach is then implemented in this research to decompose a power grid into radial lines and three-node cycles. Enforcing the related submatrices PSD and rank-1 guarantees a full PSD rank-1 matrix. Case study results demonstrate that the proposed formulation can provide similar quality results with the original AC OPF formulation.