Exploiting Symmetry in the Power Flow Equations Using Monodromy

TL;DR

This paper introduces a novel approach using monodromy to solve power flow equations, demonstrating its effectiveness in decomposing solution varieties, revealing symmetries, and outperforming other homotopy methods in finding all solutions.

Contribution

It presents a new monodromy-based method for solving power flow equations, with theoretical proofs and numerical comparisons showing its advantages over existing techniques.

Findings

Monodromy decomposes the solution variety into trivial and nontrivial parts.

The nontrivial subvariety is proven to be irreducible.

Monodromy outperforms polyhedral and total degree homotopy methods in certain cases.

Abstract

We propose solving the power flow equations using monodromy. We prove the variety under consideration decomposes into trivial and nontrivial subvarieties and that the nontrivial subvariety is irreducible. We also show various symmetries in the solutions. We finish by giving numerical results comparing monodromy against polyhedral and total degree homotopy methods and giving an example of a network where we can find all solutions to the power flow equation using monodromy where other homotopy techniques fail. This work gives hope that finding all solutions to the power flow equations for networks of realistic size is possible.