A non-convex alternating direction method of multipliers heuristic for optimal power flow

Abstract

Abstract-The optimal power flow (OPF) problem is fundamental to power system planing and operation. It is a nonconvex optimization problem and the semidefinite programing (SDP) relaxation has been proposed recently. However, the SDP relaxation may give an infeasible solution to the original OPF problem. In this paper, we apply the alternating direction method of multiplier method to recover a feasible solution when the solution of the SDP relaxation is infeasible to the OPF problem. Specifically, the proposed procedure iterates between a convex optimization problem, and a non-convex optimization with the rank constraint. By exploiting the special structure of the rank constraint, we obtain a closed form solution of the non-convex optimization based on the singular value decomposition. As a result, we obtain a computationally tractable heuristic for the OPF problem. Although the convergence of the algorithm is not theoretically guaranteed, our simulations show that a feasible solution can be recovered using our method. NOTATIONi is the imaginary unit. W * is the Hermitian of W , Tr (W ) is a trace of W , and W F = Tr (W W * ) is the Frobenius norm of W . The generalized inequality, W 0, means W is a positive semidefinite matrix.The projection operator Π S (W ) = argmin Z∈S W − Z 2 F , is the projection of W onto the set S.

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