Iterative convex relaxation of unbalanced power distribution system integrated multi-energy systems

This study describes a multi-period, multi-energy operation of a three-phase unbalanced Electric Distribution System (EDS) that coordinates with a District Heating System (DHS) and a Natural Gas (NG) distribution system. The electrical subsystem problem is formulated as a bi-level programming problem, with levels 1 and 2 solving the subsystem’s linearized and relaxed nonlinear versions, respectively. Second-Order Cone Programming (SOCP) and polyhedral relaxations are availed to circumvent the non-convexities involved in the subsystems. The SOCP relaxation for an unbalanced power distribution network is inexact; also, the subsystems applying polyhedral relaxation may not generate a meaningful solution as the relaxations are not tight. Therefore, a solution recovery algorithm is developed for each subsystem to recover a feasible solution from their relaxed counterparts. The successive bound tightening algorithm employing a solution recovery procedure is proposed for each subsystem, improving solution quality and strengthening the relaxations with the desired computational efficiency. The proposed solution strategy to optimize the Multi-Energy System (MES) operation cost is verified on the three-phase IEEE-13 and IEEE-123 bus systems, each coordinating with a 30-node DHS and a 6-bus NG network. The results analyses demonstrate that the proposed solution strategy efficiently achieves an optimal solution, reducing maximum relaxation error below 0.1% for each subsystem. The proposed strategy delivered a 14.58% reduction in real power losses and a 12.73% decrease in phase voltage unbalance rate for the EDS in MES. Furthermore, a 1.96% decrease in operational cost demonstrates the techno-economic benefits of the proposed strategy.