arXiv:2610.27897v1 Announce Type: new Abstract: Accurate representations of the flexibility in distribution grids via tractable algorithms are essential for TSO-DSO coordination. However, flexibility region (FR) estimation based on nonlinear optimal power flow (OPF) generally lacks feasibility guarantees, while conventional convex relaxations may suffer from inexactness. This paper proposes a framework for constructing explicit inner approximations of active-reactive power FRs in radial networks based on the augmented relaxed OPF (AR-OPF). First, less-conservative sufficient conditions for the AR-OPF exactness are derived and extended to generalized objectives. Furthermore, an interior-feasibility analysis is established, guaranteeing that the entire region enclosed by the traced boundary is AC feasible under the proposed sufficient conditions. Building on these theoretical results, two adaptive algorithms are developed to construct explicit inner approximations and numerically convex representations of FRs. Numerical evaluations on different benchmark networks demonstrate consistently small relaxation residuals and high flexibility area retention in most cases, showing that the proposed methods provide a grounded approach to construct operationally admissible FRs for TSO-DSO coordination.