An Exact Spectral Refinement Method for Nonconvex Branch-Flow Feasibility in Active Distribution Networks

High penetration of distributed photovoltaics (PV) makes hosting-capacity assessment and active distribution network operation challenging, primarily due to the need to accurately restore nonconvex branch-flow equalities rather than relying on relaxations that may produce physically inconsistent solutions. This paper develops an ADMM-coordinated framework with an exact spectral refinement for QP1QC subproblems, which converts the semidefinite characterization into a tractable one-dimensional refinement over a generalized-eigenvalue-defined interval and enables reliable primal recovery of the original equality constraints. Numerical tests on modified IEEE 33-, 792-, and 1137-bus feeders show that the proposed method substantially improves equality restoration: the normalized mismatch of nonconvex equalities is reduced from 82–108% under SOCP/SDP relaxations to 0.004% on the 33-bus system, and from 94–98% to 0.67% on the 792-bus system; on the 1137-bus system, the mismatch remains 6.4%, still far below the relaxation baselines. Compared with an SDP-based hidden-convex benchmark, the proposed approach preserves essentially the same optimization outcomes while achieving 7–16× lower runtime and converging in 8–13 ADMM iterations.