Controlling the Electrical State via Uncertain Power Injections in Three-Phase Distribution Networks

We consider the problem of controlling the electrical state of a three-phase distribution network by enforcing the explicit power injection. More precisely, we assume that the power injection is constrained to reside in some uncertainty set, and the problem is to ensure that the electrical state remains in a set that satisfies feasibility constraints. To formalize this, we say that a set S of power injections is a "domain of V-control" if any continuous trajectory of the electrical state that starts in the set V must stay in V as long as the corresponding trajectory of the power injection stays in S. First, we show that the existence and uniqueness of load-flow solution is not enough to guarantee V-control, and give sufficient additional conditions for V-controllability to hold. Incidentally, the derivation of our conditions establishes that local uniqueness implies nonsingularity of the load-flow Jacobian (the converse is well known by the inverse function theorem). Then, we give a concrete algorithm to determine whether a given set of power injections is a domain of V-control for some feasible and nonsingular set V. The algorithm is evaluated on IEEE test feeders. Our method can be used to perform admissibility tests in the control of distribution networks by power injections.