{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T07:33:18Z","timestamp":1777447998172,"version":"3.51.4"},"reference-count":30,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2021,5,27]],"date-time":"2021-05-27T00:00:00Z","timestamp":1622073600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>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 \u0394- 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\u2013follower 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\u2013slave optimization structures.<\/jats:p>","DOI":"10.3390\/computation9060061","type":"journal-article","created":{"date-parts":[[2021,5,27]],"date-time":"2021-05-27T04:30:46Z","timestamp":1622089846000},"page":"61","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":29,"title":["Accurate and Efficient Derivative-Free Three-Phase Power Flow Method for Unbalanced Distribution Networks"],"prefix":"10.3390","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6051-4925","authenticated-orcid":false,"given":"Oscar Danilo","family":"Montoya","sequence":"first","affiliation":[{"name":"Facultad de Ingenier\u00eda, Universidad Distrital Francisco Jos\u00e9 de Caldas, Bogot\u00e1 D.C. 11021, Colombia"},{"name":"Laboratorio Inteligente de Energ\u00eda, Universidad Tecnol\u00f3gica de Bol\u00edvar, Cartagena 131001, Colombia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2154-1618","authenticated-orcid":false,"given":"Juan S.","family":"Giraldo","sequence":"additional","affiliation":[{"name":"Electrical Engineering, Mathematics and Computer Science\u2014MOR Group, University of Twente, 7522 NB Enschede, The Netherlands"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1409-9756","authenticated-orcid":false,"given":"Luis Fernando","family":"Grisales-Nore\u00f1a","sequence":"additional","affiliation":[{"name":"Grupo GIIEN, Facultad de Ingenier\u00eda, Instituci\u00f3n Universitaria Pascual Bravo, Campus Robledo, Medell\u00edn 050036, Colombia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1746-5560","authenticated-orcid":false,"given":"Harold R.","family":"Chamorro","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering at KTH, Royal Institute of Technology, SE-100 44 Stockholm, Sweden"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6030-9582","authenticated-orcid":false,"given":"Lazaro","family":"Alvarado-Barrios","sequence":"additional","affiliation":[{"name":"Department of Engineering, Universidad Loyola Andaluc\u00eda, 41704 Sevilla, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,5,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Montoya, O.D., Gil-Gonz\u00e1lez, W., and Hern\u00e1ndez, J.C. 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