A Closed-Form Topological Supervisory Filter for Real-Time Voltage Collapse Prevention and Dynamic Capacity Routing in Radial Distribution Grids

Conventional continuation power flow and Newton-Raphson solvers scale cubically with network dimension and exhibit severe Jacobian ill-conditioning near saddle-node bifurcation boundaries. In distribution networks hosting high penetrations of volatile distributed energy resources and battery energy storage systems (BESS), millisecond-scale latency is required to trigger corrective load-shedding and rerouting. In this paper, we adapt the combinatorial mechanics of the Topological Discrete Universe (TDU v4.9) framework into a deterministic, closed-form macro-controller. The effective branch saturation u = S/S_max is mapped onto a bounded topological stress functional Sigma(u) possessing an analytical bifurcation limit at u_c = 1 / (1 + sqrt(delta_EMT)) ~ 0.7359. Evaluated on the benchmark IEEE 33-bus radial system, the proposed filter executes in 2.5 µs per decision cycle—representing an over 180x speedup compared to standard iterative solvers—and provides an early trigger threshold u_trigger ~ 0.6034 that precedes physical voltage collapse without matrix inversion.