# High-voltage solution in radial power networks:Existence, properties and equivalent algorithms

## TL;DR

This paper establishes theoretical conditions for the existence and uniqueness of high-voltage solutions in radial power networks and demonstrates that common algorithms reliably find this solution when it exists.

## Contribution

It provides necessary and sufficient conditions for solution existence and proves the convergence of multiple algorithms to the high-voltage solution in radial networks.

## Key findings

- Existence and uniqueness conditions for power flow solutions.
- Algorithms reliably find the high-voltage solution if it exists.
- The high-voltage solution's properties include monotonicity and continuity with respect to injections.

## Abstract

The AC power flow equations describe the steady-state behavior of the power grid. While many algorithms have been developed to compute solutions to the power flow equations, few theoretical results are available characterizing when such solutions exist, or when these algorithms can be guaranteed to converge. In this paper, we derive necessary and sufficient conditions for the existence and uniqueness of a power flow solution in balanced radial distribution networks with homogeneous (uniform $R/X$ ratio) distribution lines. We study three distinct solution methods: fixed point iterations}, convex relaxations, and energy functions we show that the three algorithms successfully find a solution if and only if a solution exists. Moreover, all three algorithms always find the unique high-voltage solution to the power flow equations, the existence of which we formally establish.}%, which can be formally defined as the high-voltage solution -- any other solution of the power flow equations has smaller or equal voltage magnitudes at every bus. At this solution, we prove that (i) voltage magnitudes are increasing functions of the reactive power injections, (ii) the solution is a continuous function of the injections, and (iii) the solution is the last one to vanish as the system is loaded past the feasibility boundary.