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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.

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As is known, the insertion of new generation technologies, whether it is on transmission or distribution, has several effects on the traditional electric network, from technical changes to its regulation. There is a global interest on finding the best way to take advantage of those emerging technologies and the best way they can interact with the traditional power system. This paper proposes a methodology to exploit the maximum potential of the traditional electric network and the integration of a well-known generation technology such as the wind generation. To do so, we present a Probabilistic AC Optimal Power Flow (POPF) that takes into account load variation, wind's stochastic behavior and variable line's thermal rating which is usually used as a deterministic value in several studies. A validation of two proposed schemes of the Point Estimate Method (PEM) is made, not only for normal distributions but for different kinds of probability distributions, such as Weibull and generalized extreme value. Obtained results are compared with the Monte Carlo (MC) simulation indicating that the proposed method significantly reduces the computational burden while maintaining a high level of accuracy.

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