Fair Allocation of Operating Envelopes for Distribution Networks Considering Voltage Unbalance

Operating envelopes (OEs) are increasingly used to allocate limits to distributed energy resources (DERs) while maintaining secure distribution network operation. In unbalanced low-voltage feeders, OE calculation based only on voltage magnitude and thermal constraints can yield overly optimistic limits because power quality constraints such as voltage unbalance are neglected. This paper proposes a three-phase unbalanced AC optimal power flow framework for computing coupled P--Q OEs with explicit voltage unbalance factor (VUF) constraints. In addition, two fairness mechanisms for allocating the available P--Q flexibility across multiple PV units are embedded and compared: (i) network-weighted proportional fairness and (ii) lexicographic max--min fairness. Case studies on unbalanced test feeders illustrate how VUF constraints reshape the P--Q feasible region and the impact of power quality-constrained operation. The comparison highlights the trade-off between the efficiency, equity, and practicality of fairness allocation methods.