An Exact Determinantal Calculus for Reliability and Reconfiguration of Radially Operated Distribution Networks

A distribution feeder is built meshed and operated radially, so at any instant it occupies one of a combinatorial family of admissible configurations. We show that this family, weighted in the natural maximum-entropy way, is a determinantal point process whose kernel is the transfer-current matrix of the network, and we read that kernel in the operator's language: the probability that a line section is energised equals its own self transfer-current factor, Foster's sum rule is the trace identity, and the covariance of two switching states is minus the square of their transfer current. Independent faults leave the feeder exactly within this family for any number of faults, whereas no restoration mechanism ignorant of the section resistances can return it there; among those that can, one is canonical, being the unique mechanism that reverses the fault, and its weight is the section's transfer-current factor in the post-fault network with everything still energised shorted. We show, and report, that these weights are a structural diagnostic and not a repair priority. The maintained feeder is solved in closed form, the reported reliability indices prove robust to the dispatch policy, the same calculus governs a fleet of distributed generators with a different kernel, and an exact transport equation prices what a reinforcement programme costs the feeder's ability to reconfigure. The central spanning-tree and sector identities are verified against exhaustive enumeration; the dynamical and sensitivity statements are checked by exact master-equation computations and finite differences.