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Summary 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 CN2CN2 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.

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