Admissibility of Uncertain Injections in Quadratic Algebraic Systems

We study the admissibility problem in multivariate algebraic systems, such as ac electrical networks, where the power injection is quadratic in the state. The goal of such systems is to ensure that the state stays in some security set (e.g., magnitudes of nodal voltages and branch currents are within safety bounds). A common practice is to implicitly control the state by controlling the injection; a difficulty is that the number of states that correspond to a given injection can be zero or many. Further, the injection is subject to some uncertainty. The admissibility problem is whether it is possible to ensure that the state stays in the security set, given that the only available information is some uncertainty set that constrains the injection. We extend the recently proposed V-control theory, design a solution framework to test if a given uncertainty set is admissible, and develop a concrete method for ac electrical networks.