Solvability Regions of Affinely Parameterized Quadratic Equations

Quadratic systems of equations appear in several applications. The results in this letter are motivated by quadratic systems of equations that describe equilibrium behavior of physical infrastructure networks like power and gas grids. The quadratic systems in infrastructure networks are parameterized-the parameters can represent uncertainty or controllable decision variables. It is then of interest to understand conditions on the parameters under which the quadratic system is guaranteed to have a solution within a specified set. Given nominal values of the parameters at which the quadratic system has a solution, we develop a general framework to construct regions around the nominal parameter value such that the system is guaranteed to have a solution within a given distance of the nominal solution. The regions are described by explicit norm-like constraints on the parameters. We compare the results to previous approaches in the context of power systems.