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