{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,24]],"date-time":"2026-06-24T21:39:03Z","timestamp":1782337143657,"version":"3.54.5"},"reference-count":45,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2022,6,27]],"date-time":"2022-06-27T00:00:00Z","timestamp":1656288000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Center for Research and Scientific Development of Universidad Distrital Francisco Jos\u00e9 de Caldas","award":["1643-12-2020"],"award-info":[{"award-number":["1643-12-2020"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computers"],"abstract":"<jats:p>The problem regarding the optimal location and sizing of fixed-step capacitor banks in distribution networks with radial configuration is studied in this research by applying a two-stage optimization approach. The first stage consists of determining the nodes where the capacitor banks will be placed. In this stage, the exact mixed-integer nonlinear programming (MINLP) model that represents the studied problem is transformed into a mixed-integer quadratic convex (MIQC) model. The solution of the MIQC model ensures that the global optimum is reached given the convexity of the solution space for each combination of nodes where the capacitor banks will be installed. With the solution of the MIQC, the suitable nodes for the installation of the fixed-step capacitors are fixed, and their sizes are recursively evaluated in a power flow methodology that allows for determining the optimal sizes. In the second stage, the successive approximation power flow method is applied to determine the optimal sizes assigned to these compensation devices. Numerical results in three test feeders with 33, 69, and 85 buses demonstrate the effectiveness of the proposed two-stage solution method for two operation scenarios: (i) operation of the distribution system under peak load conditions throughout the year, and (ii) operation considering daily demand variations and renewable generation penetration. Comparative results with the GAMS software confirm the excellent results reached using the proposed optimization approach. All the simulations were carried out in the MATLAB programming environment, version 2021b, as well as using the Gurobi solver in the convex programming tool known as CVX.<\/jats:p>","DOI":"10.3390\/computers11070105","type":"journal-article","created":{"date-parts":[[2022,6,27]],"date-time":"2022-06-27T22:31:14Z","timestamp":1656369074000},"page":"105","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Selection and Location of Fixed-Step Capacitor Banks in Distribution Grids for Minimization of Annual Operating Costs: A Two-Stage Approach"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6051-4925","authenticated-orcid":false,"given":"Oscar Danilo","family":"Montoya","sequence":"first","affiliation":[{"name":"Grupo de Compatibilidad e Interferencia Electrom\u00e1gnetica, Facultad de Ingenier\u00eda, Universidad Distrital Francisco Jos\u00e9 de Caldas, Bogot\u00e1 110231, Colombia"},{"name":"Laboratorio Inteligente de Energ\u00eda, Facultad de Ingenier\u00eda, Universidad Tecnol\u00f3gica de Bol\u00edvar, Cartagena 131001, Colombia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2372-8056","authenticated-orcid":false,"given":"Edwin","family":"Rivas-Trujillo","sequence":"additional","affiliation":[{"name":"Grupo de Compatibilidad e Interferencia Electrom\u00e1gnetica, Facultad de Ingenier\u00eda, Universidad Distrital Francisco Jos\u00e9 de Caldas, Bogot\u00e1 110231, Colombia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9983-4555","authenticated-orcid":false,"given":"Diego Armando","family":"Giral-Ram\u00edrez","sequence":"additional","affiliation":[{"name":"Facultad Tecnol\u00f3gica, Universidad Distrital Francisco Jos\u00e9 de Caldas, Bogot\u00e1 110231, Colombia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Picard, J.L., Aguado, I., Cobos, N.G., Fuster-Roig, V., and Quijano-L\u00f3pez, A. 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