Quadratic Convex Relaxation based centralized OPF for Multi-Phase Active Distribution Networks

Semi-Definite Programming (SDP) and Second Order Cone Programming (SOCP) relaxations are state-of-the-art lift-and-project based techniques for solving the non-convex AC optimal power flow problem in multi-phase active distribution networks. A novel centralized Quadratic Convex (QC) relaxation for such networks, which is a significant departure from the liftand-project based approaches, is developed in this paper. The proposed scheme encloses the non-convex region, associated with the non-linear terms of power flow equations represented in the polar form, by appropriate convex envelopes. The envelopes for the trigonometric terms are based upon the first order Taylor series approximation and cosecant/secant functions, whereas bilinear terms are enclosed by McCormick envelopes. Furthermore, appropriate bounds on the lifted variables are also introduced and the trilinear function is enclosed by the recursive application of the McCormick envelope. The application of the proposed scheme on several test cases reveals that it outperforms SOCP in all the cases and is close to the SDP relaxation. Furthermore, it has also shown to have high computational efficiency as compared to the SDP approach due to the existence of mature solving technology for quadratically constrained problems.