# Convergence of the Z-Bus Method for Three-Phase Distribution Load-Flow with ZIP Loads

arXiv: https://arxiv.org/abs/1605.08511 · HTML: https://mlsanity.com/paper/1605.08511/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 May 2016.

## Built on by (most cited first, 3 of 3)

- [Enhancing the Spatio-temporal Observability of Grid-Edge Resources in Distribution Grids](https://mlsanity.com/paper/2102.07801/) (2021)
- [Learning AC Power Flow Solutions using a Data-Dependent Variational Quantum Circuit](https://mlsanity.com/paper/2509.03495/) (2025)
- [Scalable Self-Supervised Learning for Multiphase AC-OPF in Distribution Systems with Topology Reconfiguration](https://mlsanity.com/paper/2608.25095/) (2026)

## Abstract (arXiv)

This paper derives a set of sufficient conditions guaranteeing that the load-flow problem in unbalanced three-phase distribution networks with wye and delta ZIP loads has a unique solution over a region that can be explicitly calculated from the network parameters. It is also proved that the well-known Z-Bus iterative method is a contraction over the defined region, and hence converges to the unique solution.