Solvability regions of affinely parameterized quadratic equations

TL;DR

This paper introduces a framework to identify convex regions of parameters ensuring solutions to affine quadratic systems, with applications to power and gas networks, extending recent results and validated on benchmark systems.

Contribution

It develops a general convex region construction method for quadratic systems with parameters, unifying and extending existing results, and applies it to infrastructure network models.

Findings

Framework guarantees solutions within specified parameter regions.

Several recent results are recovered as special cases.

Validated approach on benchmark power systems.

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.