{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:27:15Z","timestamp":1760149635818,"version":"build-2065373602"},"reference-count":42,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,9,2]],"date-time":"2023-09-02T00:00:00Z","timestamp":1693612800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ibero-American Science and Technology for Development Program (CYTED)","award":["723RT0150","6-23-7","6-23-5"],"award-info":[{"award-number":["723RT0150","6-23-7","6-23-5"]}]},{"name":"Desarrollo de una metodolog\u00eda para la compensaci\u00f3n \u00f3ptima de  potencia reactiva en sistemas el\u00e9ctricos de distribuci\u00f3n from at Universidad Tecnol\u00f3gica de Pereira","award":["723RT0150","6-23-7","6-23-5"],"award-info":[{"award-number":["723RT0150","6-23-7","6-23-5"]}]},{"name":"An\u00e1lisis de estrategias de control Jer\u00e1rquico para la operaci\u00f3n estable de microrredes ac y dc considerando energ\u00edas renovables from at Universidad Tecnol\u00f3gica de Pereira","award":["723RT0150","6-23-7","6-23-5"],"award-info":[{"award-number":["723RT0150","6-23-7","6-23-5"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>This research addresses the efficient integration and sizing of flexible alternating current transmission systems (FACTS) in electrical distribution networks via a convex optimization approach. The exact mixed-integer nonlinear programming (MINLP) model associated with FACTS siting and sizing aims for the minimization of the expected annual operating costs of the network (i.e., energy losses and FACTS purchasing costs). The constraints of this problem include power equilibrium equalities, voltage regulation bounds, and device capacities, among others. Due to the power equilibrium constraints per node and period, the MINLP model is a non-convex optimization problem. To transform the exact MINLP model into a mixed-integer convex one, the approximation of the product between two variables in the complex domain is relaxed through its hyperbolic equivalent, which generates a set of convex cones. The main advantage of the proposed mixed-integer convex model is that it ensures the global optimum of the problem, even when considering objective multiplexes. Numerical simulations in the IEEE 33-, 69-, and 85-bus grids demonstrate the effectiveness and robustness of FACTS integration via the proposed convex approach in comparison with the exact solution of the MINLP model in the GAMS software as well as with combinatorial optimization algorithms (i.e., the black widow optimizer and the vortex search algorithm). All simulations were carried out in MATLAB with Yalmip optimization and the Gurobi and Mosek solvers. The simulation results show that, for a fixed operation of the FACTS devices (i.e., a VAR compensator) during the day, the annual operating costs are reduced by 12.63%, 13.97%, and 26.53% for the IEEE 33-, 69-, and 85-bus test systems, respectively, while for the operation variable, the reductions are by 14.24%, 15.79%, and 30.31%, respectively.<\/jats:p>","DOI":"10.3390\/a16090420","type":"journal-article","created":{"date-parts":[[2023,9,4]],"date-time":"2023-09-04T02:43:20Z","timestamp":1693795400000},"page":"420","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Optimal Integration of Flexible Alternating Current Transmission Systems in Electrical Distribution Grids Using a Mixed-Integer Convex Model"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7609-1197","authenticated-orcid":false,"given":"Walter","family":"Gil-Gonz\u00e1lez","sequence":"first","affiliation":[{"name":"Department of Electrical Engineering, Universidad Tecnol\u00f3gica de Pereira, Pereira 660003, Colombia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6051-4925","authenticated-orcid":false,"given":"Oscar Danilo","family":"Montoya","sequence":"additional","affiliation":[{"name":"Grupo de Compatibilidad e Interferencia Electromagn\u00e9tica (GCEM), Facultad de Ingenier\u00eda, Universidad Distrital Francisco Jos\u00e9 de Caldas, Bogot\u00e1 110231, Colombia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0985-1472","authenticated-orcid":false,"given":"C\u00e9sar Leonardo","family":"Trujillo-Rodr\u00edguez","sequence":"additional","affiliation":[{"name":"Laboratorio de Investigaci\u00f3n en Fuentes Alternativas de Energ\u00eda (LIFAE), Facultad de Ingenier\u00eda, Universidad Distrital Francisco Jos\u00e9 de Caldas, Bogot\u00e1 110231, Colombia"}]}],"member":"1968","published-online":{"date-parts":[[2023,9,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Wagle, R., Sharma, P., Sharma, C., and Amin, M. (2023). Optimal power flow based coordinated reactive and active power control to mitigate voltage violations in smart inverter enriched distribution network. Int. J. Green Energy, 1\u201317.","DOI":"10.1080\/15435075.2023.2196324"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1080\/22348972.2019.1704353","article-title":"A Study on Improvement of Power Quality by STATCOMs for Low Voltage System","volume":"9","author":"Kawasaki","year":"2019","journal-title":"J. Int. Counc. Electr. Eng."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Zamani, A., Sidhu, T., and Yazdani, A. (2010, January 25\u201329). A strategy for protection coordination in radial distribution networks with distributed generators. Proceedings of the IEEE PES General Meeting, Minneapolis, MN, USA.","DOI":"10.1109\/PES.2010.5589655"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Montoya, O.D., Serra, F.M., Angelo, C.H.D., Chamorro, H.R., and Alvarado-Barrios, L. (2021). Heuristic Methodology for Planning AC Rural Medium-Voltage Distribution Grids. Energies, 14.","DOI":"10.3390\/en14165141"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1016\/j.ijepes.2015.02.026","article-title":"Optimal distribution systems reconfiguration for radial and meshed grids","volume":"72","author":"Hijazi","year":"2015","journal-title":"Int. J. Electr. Power Energy Syst."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1132","DOI":"10.1109\/TPWRD.2015.2462723","article-title":"Coordinated Control of Distributed Energy-Storage Systems for Voltage Regulation in Distribution Networks","volume":"31","author":"Wang","year":"2016","journal-title":"IEEE Trans. Power Deliv."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"6903","DOI":"10.1038\/s41598-023-34057-3","article-title":"Hybrid GWO-PSO based optimal placement and sizing of multiple PV-DG units for power loss reduction and voltage profile improvement","volume":"13","author":"Alyu","year":"2023","journal-title":"Sci. Rep."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"102158","DOI":"10.1016\/j.est.2020.102158","article-title":"Optimal location, selection, and operation of battery energy storage systems and renewable distributed generation in medium\u2013low voltage distribution networks","volume":"34","author":"Valencia","year":"2021","journal-title":"J. Energy Storage"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Nanibabu, S., Shakila, B., and Prakash, M. (2021, January 4\u20136). Reactive Power Compensation using Shunt Compensation Technique in the Smart Distribution Grid. Proceedings of the 2021 6th International Conference on Computing, Communication and Security (ICCCS), Las Vegas, NV, USA.","DOI":"10.1109\/ICCCS51487.2021.9776327"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Santamaria-Henao, N., Montoya, O.D., and Trujillo-Rodr\u00edguez, C.L. (2023). Optimal Siting and Sizing of FACTS in Distribution Networks Using the Black Widow Algorithm. Algorithms, 16.","DOI":"10.3390\/a16050225"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Shaheen, A.M., El-Sehiemy, R.A., Ginidi, A., Elsayed, A.M., and Al-Gahtani, S.F. (2023). Optimal Allocation of PV-STATCOM Devices in Distribution Systems for Energy Losses Minimization and Voltage Profile Improvement via Hunter-Prey-Based Algorithm. Energies, 16.","DOI":"10.3390\/en16062790"},{"key":"ref_12","unstructured":"Lakshmi, S., and Ganguly, S. (2017). Sustainable Energy Technology and Policies, Springer."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"809","DOI":"10.1016\/j.aej.2014.09.012","article-title":"Optimal capacitor placement and sizing in radial electric power systems","volume":"53","author":"Elsheikh","year":"2014","journal-title":"Alex. Eng. J."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Marouani, I., Guesmi, T., Alshammari, B.M., Alqunun, K., Alshammari, A.S., Albadran, S., Abdallah, H.H., and Rahmani, S. (2023). Optimized FACTS Devices for Power System Enhancement: Applications and Solving Methods. Sustainability, 15.","DOI":"10.20944\/preprints202304.1123.v1"},{"key":"ref_15","unstructured":"Eid, A., Kamel, S., and Abdel-Mawgoud, H. (2023). Advanced Control & Optimization Paradigms for Energy System Operation and Management, River Publishers."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"502","DOI":"10.1016\/j.rser.2017.09.062","article-title":"Review of FACTS technologies and applications for power quality in smart grids with renewable energy systems","volume":"82","author":"Gandoman","year":"2018","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"481","DOI":"10.1109\/TPWRD.2011.2176559","article-title":"Multiobjective Optimal Location of FACTS Shunt-Series Controllers for Power System Operation Planning","volume":"27","author":"Ara","year":"2012","journal-title":"IEEE Trans. Power Deliv."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Gil-Gonz\u00e1lez, W. (2023). Optimal Placement and Sizing of D-STATCOMs in Electrical Distribution Networks Using a Stochastic Mixed-Integer Convex Model. Electronics, 12.","DOI":"10.3390\/electronics12071565"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"105549","DOI":"10.1016\/j.cor.2021.105549","article-title":"Exact solution approaches for integer linear generalized maximum multiplicative programs through the lens of multi-objective optimization","volume":"137","author":"Saghand","year":"2022","journal-title":"Comput. Oper. Res."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"44","DOI":"10.20998\/2074-272X.2022.1.06","article-title":"Power quality improvement in distribution system based on dynamic voltage restorer using PI tuned fuzzy logic controller","volume":"1","author":"Reddy","year":"2022","journal-title":"Electr. Eng. Electromech."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"6753","DOI":"10.1007\/s00521-020-05453-x","article-title":"Optimal placement and sizing of FACTS devices for optimal power flow in a wind power integrated electrical network","volume":"33","author":"Biswas","year":"2020","journal-title":"Neural Comput. Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"123959","DOI":"10.1016\/j.jhydrol.2019.123959","article-title":"A novel Master\u2013Slave optimization algorithm for generating an optimal release policy in case of reservoir operation","volume":"577","author":"Turgut","year":"2019","journal-title":"J. Hydrol."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1103","DOI":"10.1287\/ijoc.2020.0993","article-title":"Mixed-Integer Convex Nonlinear Optimization with Gradient-Boosted Trees Embedded","volume":"33","author":"Mistry","year":"2021","journal-title":"INFORMS J. Comput."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"149479","DOI":"10.1109\/ACCESS.2019.2935161","article-title":"First in Canada, Night and Day Field Demonstration of a New Photovoltaic Solar-Based Flexible AC Transmission System (FACTS) Device PV-STATCOM for Stabilizing Critical Induction Motor","volume":"7","author":"Varma","year":"2019","journal-title":"IEEE Access"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1016\/j.swevo.2017.12.008","article-title":"Optimal placement of TCSC and SVC for reactive power planning using Whale optimization algorithm","volume":"40","author":"Raj","year":"2018","journal-title":"Swarm Evol. Comput."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Montoya, O.D., Gil-Gonz\u00e1lez, W., and Hern\u00e1ndez, J.C. (2021). Efficient Operative Cost Reduction in Distribution Grids Considering the Optimal Placement and Sizing of D-STATCOMs Using a Discrete-Continuous VSA. Appl. Sci., 11.","DOI":"10.3390\/app11052175"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"688","DOI":"10.1016\/j.rser.2017.04.035","article-title":"Optimal placement and sizing of distribution static compensator (D-STATCOM) in electric distribution networks: A review","volume":"77","author":"Sirjani","year":"2017","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"783","DOI":"10.1016\/j.asej.2016.05.009","article-title":"Optimal placement of D-STATCOM using sensitivity approaches in mesh distribution system with time variant load models under load growth","volume":"9","author":"Gupta","year":"2018","journal-title":"Ain Shams Eng. J."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"de Koster, O.A.C., and Dom\u00ednguez-Navarro, J.A. (2020). Multi-Objective Tabu Search for the Location and Sizing of Multiple Types of FACTS and DG in Electrical Networks. Energies, 13.","DOI":"10.3390\/en13112722"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"14951","DOI":"10.3182\/20080706-5-KR-1001.02530","article-title":"A Tabu Search Based Method for Optimal Allocation of D-FACTS in Distribution Systems","volume":"41","author":"Mori","year":"2008","journal-title":"IFAC Proc. Vol."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"168","DOI":"10.1016\/j.ifacol.2022.04.028","article-title":"Optimal Sizing and Allocation of DG and FACTS Device in the Distribution System using Fractional L\u00e9vy Flight Bat Algorithm","volume":"55","author":"Reddy","year":"2022","journal-title":"IFAC-PapersOnLine"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1016\/j.ijepes.2012.12.008","article-title":"Optimal location of FACTS devices using Improved Particle Swarm Optimization","volume":"49","author":"Ravi","year":"2013","journal-title":"Int. J. Electr. Power Energy Syst."},{"key":"ref_33","unstructured":"Pappachan, S.N. (2023). Development of optimal placement and sizing of FACTS devices in power system integrated with wind power using modified krill herd algorithm. COMPEL Int. J. Comput. Math. Electr. Electron. Eng."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"100145","DOI":"10.1016\/j.rico.2022.100145","article-title":"Optimal placement and sizing of FACTS devices for optimal power flow using metaheuristic optimizers","volume":"8","author":"Sulaiman","year":"2022","journal-title":"Results Control Optim."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Saravan, M., Slochanal, S., Venkatesh, P., and Abraham, P. (December, January 29). Application of PSO technique for optimal location of FACTS devices considering system loadability and cost of installation. Proceedings of the 2005 International Power Engineering Conference, Singapore.","DOI":"10.1109\/IPEC.2005.207001"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"108346","DOI":"10.1016\/j.epsr.2022.108346","article-title":"Minimization of the distribution operating costs with D-STATCOMS: A mixed-integer conic model","volume":"212","author":"Montoya","year":"2022","journal-title":"Electr. Power Syst. Res."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1016\/j.ejor.2020.01.034","article-title":"A survey on conic relaxations of optimal power flow problem","volume":"287","author":"Zohrizadeh","year":"2020","journal-title":"Eur. J. Oper. Res."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"572","DOI":"10.1109\/TPWRS.2013.2282086","article-title":"Geometry of Power Flows and Optimization in Distribution Networks","volume":"29","author":"Lavaei","year":"2014","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Montoya, O.D., Gil-Gonz\u00e1lez, W., and Hern\u00e1ndez, J.C. (2023). Efficient Integration of Fixed-Step Capacitor Banks and D-STATCOMs in Radial and Meshed Distribution Networks Considering Daily Operation Curves. Energies, 16.","DOI":"10.3390\/en16083532"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"106710","DOI":"10.1016\/j.compeleceng.2020.106710","article-title":"Dynamic active and reactive power compensation in distribution networks with batteries: A day-ahead economic dispatch approach","volume":"85","author":"Montoya","year":"2020","journal-title":"Comput. Electr. Eng."},{"key":"ref_41","unstructured":"Lofberg, J. (2004, January 2\u20134). YALMIP: A toolbox for modeling and optimization in MATLAB. Proceedings of the 2004 IEEE International Conference on Robotics and Automation (IEEE Cat. No. 04CH37508), Taipei, Taiwan."},{"key":"ref_42","unstructured":"Gurobi Optimization, LLC (2022). Gurobi Optimizer Reference Manual, Gurobi Optimization, LLC."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/16\/9\/420\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:45:18Z","timestamp":1760129118000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/16\/9\/420"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,9,2]]},"references-count":42,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2023,9]]}},"alternative-id":["a16090420"],"URL":"https:\/\/doi.org\/10.3390\/a16090420","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2023,9,2]]}}}