This paper is concerned with solving the nonconvex problem of optimal power flow (OPF) via a convex relaxation based on semidefinite programming (SDP). We have recently shown that the SDP relaxation has a rank-1 solution from which the global solution of OPF can be found, provided the power network has no cycle. The present paper aims to provide a better understating of the SDP relaxation for cyclic networks. To this end, an upper bound is derived on rank of the minimum-rank solution of the SDP relaxation, which depends only on the topology of the power network. This bound is expected to be very small in practice due to the mostly planar structure of real-world networks. A heuristic method is then proposed to enforce the low-rank solution of the SDP relaxation to become rank-1. To elucidate the efficacy of this technique, it is proved that this method works for weakly-cyclic networks with cycles of size 3. Although this paper mainly focuses on OPF, the results developed here can be applied to several OPF-based emerging optimizations for future electrical grids.
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