Rank-One Solutions for SDP Relaxation of QCQPs in Power Systems

It has been shown that a large number of computationally difficult problems can be equivalently reformulated into quadratically constrained quadratic programs (QCQPs) in the literature of power systems. Due to the NP-hardness of general QCQPs, main effort of this stream of problems has been put into deriving near-optimal solutions with low computational complexity. Recently, semidefinite programming (SDP) relaxation has been recognized as a promising technique to solve QCQPs from various applications such as the alternating current (ac) optimal power flow (OPF) problem. However, this technique has not been guaranteed to achieve a rank-one solution, which is a necessary condition to recover a feasible solution of the original QCQPs. In this paper, instead of investigating the conditions under which a rank-one solution exists, we propose a general solution framework to derive near-optimal but rank-one solutions for the SDP relaxation of QCQPs. In the proposed algorithm, all the parameters are provided in a systematic manner. In order to demonstrate the effectiveness of our method, the proposed algorithm is applied to solve the ac-OPF and state estimation problems in various settings. Extensive numerical results show that our method succeeds in obtaining rank-one solutions in all our case studies and only small optimality gaps are induced by our approach.