Certifying Feasibility of Supervised Negative-Sequence Directional Protection Under Grid-Following and Grid-Forming Control Uncertainty

Preprint notice: This manuscript was submitted to IEEE Transactions on Power Delivery on August 16, 2026, and has not yet been peer reviewed. All network data, scenario matrices, result files, and reproduction scripts are openly archived on Zenodo at https://doi.org/10.5281/zenodo.21964797. Abstract: Negative-sequence directional elements near inverter-based resources are routinely assessed against a single nominal converter model, yet the quantities available to the relay vary materially with grid-following (GFL) sequence-control gains and with the grid-forming (GFM) current-limiting strategy—details that are proprietary and rarely available to the protection engineer. This paper reframes the setting problem as certification under declared model uncertainty. Each observable network state is checked hierarchically for a robust current-supervision interval, negative-sequence polarizing-voltage availability, and separability of forward and reverse apparent-impedance sets in the setting space of a commercial directional element, across an envelope anchored in IEEE Std 2800-2022 and spanning four physically defined current-limiting laws represented exactly in the phasor domain. Each state receives one of three certificates: robustly feasible, robustly infeasible with an identified mechanism, or uncertified. On a fully specified 69-kV benchmark (10080 fault cases, 40-member envelope), 8 of 14 relay–state pairs are certified with a minimum supervision margin of 0.0033 pu and 6 are uncertified with the failing members named explicitly; 32 cases admit no steady-state equilibrium, which a smooth limiter surrogate had silently "solved." A dynamic average-value model confirms the certificates within the relay measurement window—including sustained limit cycles in the no-equilibrium cases—and the full certification transfers to an independent second system with swapped converter placement, where only 4 of 40 envelope members would individually certify all uncertified states. Settings are synthesized only for certified states; all data and scripts are openly archived.