Certificate-based Synthesis of Coordinated Droop Control for Heterogeneous Radial Distribution Networks

Voltage certificates for droop-controlled radial distribution networks are often constructed from worst-case quantities. In heterogeneous radial networks, this approach can hide where voltage deviations are most likely. Furthermore, such certificates become increasingly conservative as the network sensitivities accumulate. Leveraging the structure of the linearized DistFlow model and slope-restricted droop controllers, we derive tighter deterministic voltage certificates that retain heterogeneous network, disturbance and inverter characteristics of each bus. The certificates reveal the buses and local limitations that dominate certified voltage performance. Worst-case bounds are recovered as a special case. Although tighter, the heterogeneous certificates can still deteriorate downstream due to the network structure. To address this, we leverage our certificates as design variables for coordinated voltage control. Specifically, we develop a virtual droop architecture with coordination and affine feedforward compensation to reshape the effective voltage sensitivity, and formulate a linear program that jointly synthesizes the controller and minimizes its heterogeneous voltage certificates under operational, communication and inverter placement constraints. The resulting controller guarantees the voltage and inverter bounds for all admissible disturbances. Evaluation on two network benchmarks, a five-customer residential feeder and a 26-customer rural network comprising four feeders, demonstrates improved certificates and voltage behaviour under inverter limits.