A differential analysis of the power flow equations

TL;DR

This paper introduces a new convex domain-based method for efficiently solving AC power flow equations, providing guarantees on solution existence, uniqueness, and convergence, grounded in monotone operator theory.

Contribution

It develops a family of convex domains characterized by Linear Matrix Inequalities that ensure at most one solution and enable efficient solution or non-existence certification.

Findings

Efficient solution when the solution lies within the convex domain

Ability to certify non-existence of solutions

Validation on IEEE test networks confirms practical relevance

Abstract

The AC power flow equations are fundamental in all aspects of power systems planning and operations. They are routinely solved using Newton-Raphson like methods. However, there is little theoretical understanding of when these algorithms are guaranteed to find a solution of the power flow equations or how long they may take to converge. Further, it is known that in general these equations have multiple solutions and can exhibit chaotic behavior. In this paper, we show that the power flow equations can be solved efficiently provided that the solution lies in a certain set. We introduce a family of convex domains, characterized by Linear Matrix Inequalities, in the space of voltages such that there is at most one power flow solution in each of these domains. Further, if a solution exists in one of these domains, it can be found efficiently, and if one does not exist, a certificate of non-existence can also be obtained efficiently. The approach is based on the theory of monotone operators and related algorithms for solving variational inequalities involving monotone operators. We validate our approach on IEEE test networks and show that practical power flow solutions lie within an appropriately chosen convex domain.