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

Andrey Bernstein, Emiliano Dall-Anese, Cong Wang, Jean-Yves Boudec, Changhong Zhao

Research output: Contribution to journalArticlepeer-review

110 Scopus Citations


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.
Original languageAmerican English
Pages (from-to)5832-5843
Number of pages12
JournalIEEE Transactions on Power Systems
Issue number6
StatePublished - 2018

NREL Publication Number

  • NREL/JA-5D00-67920


  • Delta
  • distribution network
  • linear model
  • load flow
  • Wye


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