Title: Solvability regions of affinely parameterized quadratic equations,
Abstract: Quadratic systems of equations appear in several applications. The results in
this paper are motivated by quadratic systems of equations that describe
equilibrium behavior of physical infrastructure networks like the power and gas
grids. The quadratic systems in infrastructure networks are parameterized- the
parameters can represent uncertainty (estimation error in resistance/inductance
of a power transmission line, for example)or controllable decision variables
(power outputs of generators,for example). 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 (for example, bounds on voltages and
flows in a power grid). Given nominal values of the parameters at which the
quadratic system has a solution and the Jacobian of the quadratic system at the
solution i snon-singular, we develop a general framework to construct convex
regions around the nominal value such that the system is guaranteed to have a
solution within a given distance of the nominal solution. We show that several
results from recen tliterature can be recovered as special cases of our
framework,and demonstrate our approach on several benchmark power systems.