# A Theory of Solvability for Lossless Power Flow Equations -- Part II: Conditions for Radial Networks

## TL;DR

This paper develops a theoretical framework for understanding when lossless power flow equations are solvable in radial networks, providing conditions that ensure unique high-voltage solutions and insights into grid stability.

## Contribution

It introduces parametric conditions for power flow solvability in radial networks, extending previous results and linking network structure to solution existence and uniqueness.

## Key findings

- Derived conditions guarantee solution existence and uniqueness.
- Conditions imply convergence of fixed-point iteration.
- Generalizes classical two-bus system results.

## Abstract

This two-part paper details a theory of solvability for the power flow equations in lossless power networks. In Part I, we derived a new formulation of the lossless power flow equations, which we term the fixed-point power flow. The model is parameterized by several graph-theoretic matrices -- the power network stiffness matrices -- which quantify the internal coupling strength of the network. In Part II, we leverage the fixed-point power flow to study power flow solvability. For radial networks, we derive parametric conditions which guarantee the existence and uniqueness of a high-voltage power flow solution, and construct examples for which the conditions are also necessary. Our conditions (i) imply convergence of the fixed-point power flow iteration, (ii) unify and extend recent results on solvability of decoupled power flow, (iii) directly generalize the textbook two-bus system results, and (iv) provide new insights into how the structure and parameters of the grid influence power flow solvability.