# Load-Flow in Multiphase Distribution Networks: Existence, Uniqueness, Non-Singularity and Linear Models

arXiv: https://arxiv.org/abs/1702.03310 · HTML: https://mlsanity.com/paper/1702.03310/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Feb 2017.

## Built on by (most cited first, 5 of 5)

- [Risk-Aware Learning for Scalable Voltage Optimization in Distribution Grids](https://mlsanity.com/paper/2110.01490/) (2021)
- [Enhancing the Spatio-temporal Observability of Grid-Edge Resources in Distribution Grids](https://mlsanity.com/paper/2102.07801/) (2021)
- [Data-driven Modeling for Distribution Grids Under Partial Observability](https://mlsanity.com/paper/2108.08350/) (2021)
- [BOOST-RPF: Boosted Sequential Trees for Radial Power Flow](https://mlsanity.com/paper/2603.21977/) (2026)
- [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 considers unbalanced multiphase distribution systems with generic topology and different load models, and extends the Z-bus iterative load-flow algorithm based on a fixed-point interpretation of the AC load-flow equations. Explicit conditions for existence and uniqueness of load-flow solutions are presented. These conditions also guarantee convergence of the load-flow algorithm to the unique solution. The proposed methodology is applicable to generic systems featuring (i) wye connections; (ii) ungrounded delta connections; (iii) a combination of wye-connected and delta-connected sources/loads; and, (iv) a combination of line-to-line and line-to-grounded-neutral devices at the secondary of distribution transformers. Further, a sufficient condition for the non-singularity of the load-flow Jacobian is proposed. Finally, linear load-flow models are derived, and their approximation accuracy is analyzed. Theoretical results are corroborated through experiments on IEEE test feeders.