A SOCP Relaxation for Cycle Constraints in the Optimal Power Flow Problem

This article presented a convex relaxation approach for the optimal power flow problem. The proposed approach leveraged the second-order cone programming (SOCP) relaxation to tackle the non-convexity within the feasible region of the power flow problem. Recovering an optimal solution that is feasible for the original non-convex problem is challenging for networks with cycles. The main challenge is the lack of convex constraints to present the voltage angles within a cycle. This article aims to fill this gap by presenting a convex constraint enforcing the sum of voltage angles over a cycle to be zero. To this end, the higher-order moment relaxation matrix associated with each maximal clique of the network is formed. The elements of this matrix are utilized to form a convex constraint enforcing the voltage angle summation over each cycle. To keep the computation burden of leveraging the higher-order moment relaxation low, a set of second-order cone constraints are applied to relate the elements of the higher-order moment relaxation matrix. The case study presented the merit of this work by comparing the solution procured by the introduced approach with other relaxation schemes.