Accurate and Efficient Derivative-Free Three-Phase Power Flow Method for Unbalanced Distribution Networks

Oscar Danilo Montoya, Juan S. Giraldo, Luis Fernando Grisales, Harold R. Chamorro, Lazaro Alvarado

Research output: Contribution to journalArticleAcademicpeer-review

18 Citations (Scopus)
82 Downloads (Pure)


The power flow problem in three-phase unbalanced distribution networks is addressed in this research using a derivative-free numerical method based on the upper-triangular matrix. The upper-triangular matrix is obtained from the topological connection among nodes of the network (i.e., through a graph-based method). The main advantage of the proposed three-phase power flow method is the possibility of working with single-, two-, and three-phase loads, including Δ- and Y-connections. The Banach fixed-point theorem for loads with Y-connection helps ensure the convergence of the upper-triangular power flow method based an impedance-like equivalent matrix. Numerical results in three-phase systems with 8, 25, and 37 nodes demonstrate the effectiveness and computational efficiency of the proposed three-phase power flow formulation compared to the classical three-phase backward/forward method and the implementation of the power flow problem in the DigSILENT software. Comparisons with the backward/forward method demonstrate that the proposed approach is 47.01%, 47.98%, and 36.96% faster in terms of processing times by employing the same number of iterations as when evaluated in the 8-, 25-, and 37-bus systems, respectively. An application of the Chu-Beasley genetic algorithm using a leader–follower optimization approach is applied to the phase-balancing problem utilizing the proposed power flow in the follower stage. Numerical results present optimal solutions with processing times lower than 5 s, which confirms its applicability in large-scale optimization problems employing embedding master–slave optimization structures
Original languageEnglish
Article number61
Issue number6
Publication statusPublished - 27 May 2021


  • Banach fixed-point theorem
  • three-phase power flow
  • upper triangular representation
  • recursive formulation
  • Genetic algorithm
  • phase balancing


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