Power Flow Analysis in Unbalanced Distribution Networks with Single-Phase and Two-Phase Laterals Using a Fixed-Point Formulation Implemented in MATLAB

Abstract

This paper proposes a fixed-point methodology for power flow analysis in unbalanced three-phase distributionnetworks featuring mixed single-phase, two-phase, and three-phase laterals—a common configuration in rural and low-voltage distribution systems. The method addresses the convergence challenges posed by low X/R ratios, radial topologies,and inherent phase unbalance that hinder classical power flow solvers. The formulation employs a per-unit impedance model,constructs a detailed three-phase nodal admittance matrix, and incorporates explicit mathematical models for wye and deltaload connections. A fixed-point iterative scheme updates nodal voltages using a partitioned network representation. Thealgorithm was implemented in MATLAB® and validated against the industry-standard software DIgSILENT PowerFactory®using three test systems of 4, 8, 25 and 124 nodes. Results demonstrate exceptional accuracy, with voltage magnitudedeviations below 2 × 10−3 p.u., phase angle differences under 0.05◦, and relative power balance errors consistently below0.05%. The method exhibits robust and rapid convergence, reaching a tolerance of ε = 1 × 10−10 in only 4 iterations acrossall test cases, with near-linear computational scalability. This was observed with standard load systems, but to test themethodology’s performance with heavy load systems, the active power consumed by the loads was increased tenfold. Weobserved that execution times and the number of iterations increased, and also that, as the system load increased, the methodbecame less contractive. This combination of high numerical precision, deterministic convergence, and low computationalcost establishes the proposed fixed-point method as an effective and reliable tool for the analysis and planning of modern,unbalanced distribution networks.