Near-optimal loop opening with a rotary power flow controller: Global-optimality conditions, certified domains, and practical-band fragmentation

Opening an RPFC-supported loop without a fresh voltage disturbance hinges on how the controller withdraws to zero injection. Field practice uses a two-step rule—suppress the injection magnitude, then reset the residual angle—yet no path-level theory explains when this rule is optimal or how it fails. This paper casts the exit stage as a continuous path problem on the control plane with voltage excursion as cost. An exact identity equates the two-step excess above the endpoint lower bound to the sum of two endpoint-envelope violations; when both vanish, the path is globally optimal. A counterexample rules out any unconditional theorem; implicit-function arguments then yield checkable pointwise and domain certificates. On the IEEE 69-bus feeder, a 625-point operating rectangle is certified with zero excess and a monitored, nonsingular load-flow Jacobian; certification persists out to ± 30% load, carries over to a meshed variant, a non-ideal excitation-branch model, and a 118-bus feeder, and survives 500 paired Monte Carlo draws on two topologies with heterogeneous load and photovoltaic uncertainty. On the reduced plane, 78.1% of feasible points are safely near-optimal at the 0.01 kV threshold, with the worst path-induced gap at 0.0063 kV. Execution stress contracts the practical band from 1.389 to 0.906 kV and fragments it into up to four islands; sensitivity sweeps expose a sharp bias-driven collapse between 0.75 ∘ and 1.25 ∘ , whereas ± 20% load shifts barely matter. External reactive support of up to 1 Mvar widens the band edges yet heals none of the bypass-limited gaps, confirming where the remediation budget belongs.