Convex optimal power flow based on power injection-based equations and its application in bipolar DC distribution network

Optimal power flow (OPF) is a fundamental tool for the operation analysis of bipolar DC distribution network (DCDN). However, existing OPF models based on current injection-based equations face challenges in reflecting the power distribution and exchange of bipolar DCDN directly since its decision variables are voltage and current. This paper addresses this issue by establishing a convex OPF model that can be used for the planning and operation of bipolar DCDN. As a novel OPF model, the power flow characteristics of bipolar DCDN are revealed through power injection rather than current injection, ensuring its applicability and convenience in power optimization problems. Furthermore, the original OPF model based on power injection-based equations is introduced, and for the first time, second-order cone programming (SOCP) is utilized to relax the non-convex terms within it. Notely, McCormick envelopes are used to restrict the feasible region of the convex model, thereby reducing the influence of SOCP relaxation. To enhance the tightness of the feasible region of the convex model, the refined sequence bound tightening algorithm (SBTA) is employed to adjust the boundaries within McCormick envelopes. This refinement aims to enhance the efficacy of the boundaries associated with the second-order cone constraints. The effectiveness of the proposed OPF model of the bipolar DCDN is verified through two typical power optimization problems related to operation and planning, i.e., capacity configuration of distributed generation (DG) and operation optimization of the bipolar DCDN. Compared to employing the SOCP relaxation method alone, the relaxation gap is reduced by nearly 1000 times.