Analysis of Semidefinite Programming Relaxation of Optimal Power Flow for Cyclic Networks

The optimal power flow (OPF) problem determines an optimal operating point of the power network that minimizes a certain objective function subject to physical and network constraints. There has been a great deal of attention in convex relaxation of the OPF problem in recent years, and it has been shown that a semidefinite programming (SDP) relaxation can solve different classes of the nonconvex OPF problem to global or near-global optimality. Although it is known that the SDP relaxation is exact for radial networks under various conditions, solving the OPF problem for cyclic networks needs further research. In this paper, we propose sufficient conditions under which the SDP relaxation is exact for special but important cyclic networks. More precisely, when the objective function is a linear function of active powers, we show that the SDP relaxation is exact for odd cycles under certain conditions. Also, the exactness of the SDP relaxation for simple cycles of size 3 and 4 is proved under different technical conditions. The existence of rank-1 or -2 SDP solutions for weakly-cyclic networks is also proved in both lossy and lossless cases. In addition, when the objective function is an increasing function of reactive powers, we prove that the SDP relaxation is exact under certain conditions. This result justifies why the sum of reactive powers acts as a low-rank-promoting term for OPF. The findings of this paper provide intuition into the behavior of SDPs for different building blocks of cyclic networks.