  Interconnection studies for distributed energy resources (DERs) can currently
take months since they entail simulating a large number of power flow
scenarios. If DERs are to be actively controlled, probabilistic hosting
capacity analysis (PHCA) studies become more time-consuming since they require
solving multiple optimal power flow (OPF) tasks. PHCA is expedited here by
leveraging the powerful tool of multiparametric programming (MPP). Using an
approximate grid model, optimal DER setpoints are decided by a quadratic
program, which depends on analysis and uncertain parameters in a possibly
nonlinear fashion. By reformulating this program, feasible and infeasible OPF
instances alike are handled in a unified way to uniquely reveal the location,
frequency, and severity of feeder constraint violations. The effect of voltage
regulators is also captured by novel approximate models. Upon properly
extending MPP to PHCA, we were able to find the exact minimizers for 518,400
OPF instances on the IEEE 123-bus feeder by solving only 6,905 of them, and
86,400 instances on a 1,160-bus feeder by solving only 2,111 instances. This
accelerated PHCA by a factor of 10. Thus, a utility can promptly infer grid
statistics using real-world data without a probabilistic characterization of
uncertain parameters.
