Accelerating Distribution Network Reconfiguration via Perspective Conic Contraction: A Convex Hull Approach

Real-time topology optimization is critical for the resilience of active distribution networks under increasing distributed energy resource (DER) volatility. However, achieving global optimality within operational time limits remains a challenge: existing convex relaxation techniques suffer from weak lower bounds in high-switching-density scenarios, leading to excessive branching. This paper establishes that the fundamental bottleneck lies in the static cone structure of standard secondorder cone (SOC) relaxations, which permits non-physical "ghost flows" with a sub-linear decay rate (O(√ ϵ)) in open branches. To resolve this geometric mismatch, we propose a Hybrid Perspective-Disjunctive Formulation that achieves the closure of the convex hull for branch flow models. By integrating polyhedral slicing with conic volumetric contraction, the proposed method enforces a rigorous linear decay (O(ϵ)) of power flows, effectively sealing the relaxation gap. Theoretical analysis confirms the strict dominance of the proposed feasible region over state-of-the-art formulations. Empirically, the method exhibits a computational paradox: the theoretical attainment of the convex hull enables a solution acceleration of over 50% on the IEEE 141-bus system by drastically pruning the search space, despite the increased number of auxiliary variables. Moreover, the formulation's isotropic nature ensures superior robustness against reverse power flows, providing a reliable and rapid operational mechanism for active distribution networks.