ACOPF for three-phase four-conductor distribution systems: semidefinite programming based relaxation with variable reduction and feasible solution recovery

Emerging distribution systems with a proliferation of distributed energy resources are facing with new challenges, such as voltage collapse and power flow congestion in unsymmetrical network configurations. As a fundamental tool that could help quantify these new challenges and further mitigate their impacts on the secure and economic operation of distribution systems, effective AC optimal power flow (ACOPF) models and solution approaches are in urgent need. This study focuses on ACOPF of three-phase four-conductor configured distribution systems, in which neutral conductors and ground resistances are modelled explicitly to reflect practical situation. In addition, by leveraging the Kirchhoff's current law (KCL) theorem and the effect of zero injections, voltage variables of neutrals and zero-injection phases can be effectively eliminated. The ACOPF problem is formulated as a convex semidefinite programming (SDP) relaxation model in complex domain. In recognising possible solution inexactness of SDP relaxation model, a Karush-Kuhn-Tucker condition based process is further proposed to effectively recover feasible solutions to the original ACOPF problem by calculating a set of computational-inexpensive non-linear equations. Numerical studies on a modified IEEE 123-bus system show the effectiveness of the proposed SDP relaxation model with variable reductions and the feasible solution recovery process for three-phase four-conductor configured distribution systems.