# A Theory of Solvability for Lossless Power Flow Equations -- Part I: Fixed-Point Power Flow

## TL;DR

This paper introduces a new fixed-point formulation of lossless power flow equations, providing explicit approximations and convergence guarantees, advancing understanding of power flow solvability in both meshed and radial networks.

## Contribution

It presents a novel fixed-point power flow model parameterized by graph-theoretic matrices, with proven convergence and solvability conditions for lossless power networks.

## Key findings

- Iterates rapidly converge to high-voltage solutions
- Explicit approximation accurately predicts power flow near base case
- Derived conditions guarantee solution existence and uniqueness

## Abstract

This two-part paper details a theory of solvability for the power flow equations in lossless power networks. In Part I, we derive a new formulation of the lossless power flow equations, which we term the fixed-point power flow. The model is stated for both meshed and radial networks, and is parameterized by several graph-theoretic matrices -- the power network stiffness matrices -- which quantify the internal coupling strength of the network. The model leads immediately to an explicit approximation of the high-voltage power flow solution. For standard test cases, we find that iterates of the fixed-point power flow converge rapidly to the high-voltage power flow solution, with the approximate solution yielding accurate predictions near base case loading. In Part II, we leverage the fixed-point power flow to study power flow solvability, and for radial networks we derive conditions guaranteeing the existence and uniqueness of a high-voltage power flow solution. These conditions (i) imply exponential convergence of the fixed-point power flow iteration, and (ii) properly generalize the textbook two-bus system results.