Uniqueness of Power Flow Solutions Using Graph-Theoretic Notions

Haixiang Zhang, Sang Woo Park, Javad Lavaei, Ross Baldick

Research output: Contribution to journalArticlepeer-review


This article extends the uniqueness theory in (Park et al., 2021) and establishes general necessary and sufficient conditions for the uniqueness of P-Θ power flow solutions in an AC power system using some properties of the monotone regime and the power network topology. We show that the necessary and sufficient conditions can lead to tighter sufficient conditions for the uniqueness in several special cases. Our results are based on the existing notion of maximal girth and our new notion of maximal eye. Moreover, we develop a series-parallel reduction method and search-based algorithms for computing the maximal eye and the maximal girth, which are necessary for the uniqueness analysis. Reduction to a single line using the proposed reduction method is guaranteed for 2-vertex-connected series-parallel graphs. The relations between the parameters of the network before and after reduction are obtained. It is verified on real-world networks that the computation of the maximal eye can be reduced to the analysis of a much smaller power network, while the maximal girth is computed during the reduction process.

Original languageEnglish (US)
Pages (from-to)100-112
Number of pages13
JournalIEEE Transactions on Control of Network Systems
Issue number1
StatePublished - Mar 1 2022
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Control and Systems Engineering
  • Signal Processing
  • Computer Networks and Communications
  • Control and Optimization


  • Graph theory
  • monotone operators
  • power flow analysis
  • power systems


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