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

Abstract

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.

An Improved Crow Search Algorithm Applied to the Phase Swapping Problem in Asymmetric Distribution Systems

“…where V g3ϕ is the vector containing all the voltages at the slack node that are known for power flow purposes [21]. V d3ϕ is the vector containing all the unknown variables of interest, i.e., the demand voltages.…”

“…In the case where node m has a constant power load with a Y structure (assuming it is solidly earthed [33]), the demand current can be shown as in Equation ( 9), as reported in [21].…”

“…If node m has a load with a connection ∆, the demand current can be as shown in Equation ( 10), as reported in [21].…”

“…In the specialized literature, the balance phase problem, with the minimizing power losses approach, has been solved using different optimization methods, including the Chu and Beasley genetic algorithms [8,16,[21][22][23][24], particle swarm optimization [9], mixedinteger convex optimization [25], bat optimization algorithm [26], differential evolution algorithm [27], simulated annealing optimizer [28], and vortex search algorithm [15], among others.…”

Improved Genetic Algorithm for Phase-Balancing in Three-Phase Distribution Networks: A Master-Slave Optimization Approach

“…Furthermore, an additional important contribution presented in this study corresponds to the extension of the recently developed successive approximation power flow method [28] to three-phase unbalanced distribution networks. The resulting method could be applied to loads with ∆− and Y−connections including possible meshed grid configurations with the possibility of demonstrating its convergence by applying the Banach fixed-point theorem as recently proposed in [29] for the upper-triangular power flow method.…”