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This paper uses convex inner approximations (CIA) of the AC power flow totackle the optimization problem of quantifying a three-phase distributionfeeder's capacity to host distributed energy resources (DERs). This is oftenconnoted hosting capacity (HC), but herein we consider separative bounds foreach node on positive and negative DER injections, which ensures thatinjections within these nodal limits satisfy feeder voltage and current limitsand across nodes sum up to the feeder HC. The methodology decomposes athree-phase feeder into separate phases and applies CIA-based techniques toeach phase. An analysis is developed to determine the technical condition underwhich this per-phase approach can still guarantee three-phase constraints. Newapproaches are then presented that modify the per-phase optimization problemsto overcome conservativeness inherent to CIA methods and increase HC, includingselectively modifying the per-phase impedances and iteratively relaxingper-phase voltage bounds. Discussion is included on trade-offs and feasibility.To validate the methodology simulation-based analysis is conducted with theIEEE 37-node test feeder and a real 534-node unbalanced radial distributionfeeder.

Decomposed Phase Analysis using Convex Inner Approximations: a Methodology for DER Hosting Capacity in Distribution Systems