{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,21]],"date-time":"2026-02-21T18:36:23Z","timestamp":1771698983937,"version":"3.50.1"},"reference-count":60,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2022,6,18]],"date-time":"2022-06-18T00:00:00Z","timestamp":1655510400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100018928","name":"Instituto Tecnol\u00f3gico Metropolitano","doi-asserted-by":"publisher","award":["P21101"],"award-info":[{"award-number":["P21101"]}],"id":[{"id":"10.13039\/501100018928","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>Several applications require to estimate the power production of photovoltaic (PV) systems under partial shading conditions. For example, dynamic reconfiguration of the array connections is needed to maximize the power production under partial shading conditions, which requires estimating the power generated by the PV array in several possible configurations. Therefore, a fast and accurate modeling technique is needed to perform those calculations in practical times and with low estimation errors. To address those kinds of problems, this paper proposes a modeling approach based on the double-diode model to provide high accuracy at low voltage and low irradiance conditions, which are important for partial-shading analysis. Moreover, the proposed modeling technique is designed to be implemented in parallel processing devices; thus, the calculation time is much shorter in comparison with classical serial solutions. The proposed model is tested in terms of accuracy and speed, obtaining satisfactory results. Finally, the applicability of the parallel model in reconfiguration applications is demonstrated using an application example.<\/jats:p>","DOI":"10.3390\/computation10060100","type":"journal-article","created":{"date-parts":[[2022,6,18]],"date-time":"2022-06-18T09:51:11Z","timestamp":1655545871000},"page":"100","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Model of Series-Parallel Photovoltaic Arrays Using Double-Diode Model and Parallel Computing"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4634-2642","authenticated-orcid":false,"given":"Juan David","family":"Bastidas-Rodr\u00edguez","sequence":"first","affiliation":[{"name":"Facultad de Ingenier\u00eda y Arquitectura, Universidad Nacional de Colombia, Manizales 170003, Colombia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2231-4177","authenticated-orcid":false,"given":"Carlos Andr\u00e9s","family":"Ramos-Paja","sequence":"additional","affiliation":[{"name":"Facultad de Minas, Universidad Nacional de Colombia, Medell\u00edn 050034, Colombia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1498-7323","authenticated-orcid":false,"given":"Sergio Ignacio","family":"Serna-Garc\u00e9s","sequence":"additional","affiliation":[{"name":"Departamento de Electr\u00f3nica y Telecomunicaciones, Instituto Tecnol\u00f3gico Metropolitano, Calle 54A No. 30-01, Medell\u00edn 050013, Colombia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,18]]},"reference":[{"key":"ref_1","unstructured":"REN21 (2020). Renewables 2020 Global Status Report, Renewable Energy Policy Network for the 21st Century. Technical Report."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"188","DOI":"10.1016\/j.enbuild.2017.02.015","article-title":"Recent advancement in BIPV product technologies: A review","volume":"140","author":"Shukla","year":"2017","journal-title":"Energy Build."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"753","DOI":"10.1016\/j.solener.2020.07.077","article-title":"A fast and accurate generalized analytical approach for PV arrays modeling under partial shading conditions","volume":"208","author":"Kermadi","year":"2020","journal-title":"Sol. Energy"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Serna-Garc\u00e9s, S., Bastidas-Rodr\u00edguez, J., and Ramos-Paja, C. (2015). Reconfiguration of Urban Photovoltaic Arrays Using Commercial Devices. Energies, 9.","DOI":"10.3390\/en9010002"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Petrone, G., Ramos-Paja, C.A., and Spagnuolo, G. (2017). Photovoltaic Sources Modeling, John Wiley & Sons, Ltd.","DOI":"10.1002\/9781118755877"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1016\/j.rser.2018.03.011","article-title":"Identification of unknown parameters of solar cell models: A comprehensive overview of available approaches","volume":"90","author":"Abbassi","year":"2018","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3112","DOI":"10.1016\/j.matpr.2020.11.038","article-title":"Exploratory modelling and simulation of InGaN as a basis of comparison between single and double diode models of PV devices","volume":"49","author":"Hassan","year":"2020","journal-title":"Mater. Today Proc."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1109\/ACCESS.2020.3046536","article-title":"A Forensic-Based Investigation Algorithm for Parameter Extraction of Solar Cell Models","volume":"9","author":"Shaheen","year":"2021","journal-title":"IEEE Access"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"13403","DOI":"10.1109\/ACCESS.2021.3052153","article-title":"Gradient-Based Optimizer for Parameter Extraction in Photovoltaic Models","volume":"9","author":"Ismaeel","year":"2021","journal-title":"IEEE Access"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"107218","DOI":"10.1016\/j.asoc.2021.107218","article-title":"Opposition-based JAYA with population reduction for parameter estimation of photovoltaic solar cells and modules","volume":"104","author":"Yang","year":"2021","journal-title":"Appl. Soft Comput."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"112595","DOI":"10.1016\/j.enconman.2020.112595","article-title":"Comprehensive overview of meta-heuristic algorithm applications on PV cell parameter identification","volume":"208","author":"Yang","year":"2020","journal-title":"Energy Convers. Manag."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"111102","DOI":"10.1109\/ACCESS.2020.3000770","article-title":"Coyote Optimization Algorithm for Parameters Estimation of Various Models of Solar Cells and PV Modules","volume":"8","author":"Diab","year":"2020","journal-title":"IEEE Access"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"35979","DOI":"10.1109\/ACCESS.2020.2975078","article-title":"Parameters Identification of Photovoltaic Models Using a Multi-Strategy Success-History-Based Adaptive Differential Evolution","volume":"8","author":"Hao","year":"2020","journal-title":"IEEE Access"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"694","DOI":"10.1016\/j.solener.2020.09.032","article-title":"Solar photovoltaic parameter estimation using an improved equilibrium optimizer","volume":"209","author":"Mohamed","year":"2020","journal-title":"Sol. Energy"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"192","DOI":"10.1016\/j.solener.2019.01.025","article-title":"Hybridizing cuckoo search algorithm with biogeography-based optimization for estimating photovoltaic model parameters","volume":"180","author":"Chen","year":"2019","journal-title":"Sol. Energy"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"180","DOI":"10.1016\/j.solener.2019.01.026","article-title":"Parameter estimation of photovoltaic cells using improved Lozi map based chaotic optimization Algorithm","volume":"180","author":"Pourmousa","year":"2019","journal-title":"Sol. Energy"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Et-Torabi, K., Nassar-Eddine, I., Rmaily, R., Obbadi, A., Errami, Y., Sahnoun, S., El Fajri, A., and Agunaou, M. (2018, January 5\u20138). Photovoltaic parameter extraction methods for single and double diode models under real changing environmental conditions. Proceedings of the 2018 6th International Renewable and Sustainable Energy Conference (IRSEC 2018), Rabat, Morocco.","DOI":"10.1109\/IRSEC.2018.8702901"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"520","DOI":"10.1016\/j.enconman.2015.11.041","article-title":"Parameter identification and sensitivity analysis of solar cell models with cat swarm optimization algorithm","volume":"108","author":"Guo","year":"2016","journal-title":"Energy Convers. Manag."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Tamrakar, V., Gupta, S., and Sawle, Y. (November, January 30). Study of characteristics of single and double diode electrical equivalent circuit models of solar PV module. Proceedings of the 2015 International Conference on Energy Systems and Applications, Pune, India.","DOI":"10.1109\/ICESA.2015.7503362"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Alrahim Shannan, N.M.A., Yahaya, N.Z., and Singh, B. (December, January 29). Single-diode model and two-diode model of PV modules: A comparison. Proceedings of the 2013 IEEE International Conference on Control System, Computing and Engineering, Penang, Malaysia.","DOI":"10.1109\/ICCSCE.2013.6719960"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Elgohary, R., Elela, A.A.A., and Elkholy, A. (2018, January 18\u201320). Electrical Characteristics Modeling for Photovoltaic Modules Based on Single and Two Diode Models. Proceedings of the 2018 Twentieth International Middle East Power Systems Conference (MEPCON), Cairo, Egypt.","DOI":"10.1109\/MEPCON.2018.8635214"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Suthar, M., Singh, G.K., and Saini, R.P. (2013, January 10\u201312). Comparison of mathematical models of photo-voltaic (PV) module and effect of various parameters on its performance. Proceedings of the 2013 International Conference on Energy Efficient Technologies for Sustainability, Nagercoil, India.","DOI":"10.1109\/ICEETS.2013.6533584"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1016\/j.solener.2020.11.036","article-title":"Implicit modelling of series-parallel photovoltaic arrays using double-diode model and its solution","volume":"214","author":"Correa","year":"2021","journal-title":"Sol. Energy"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"103287","DOI":"10.1016\/j.micpro.2020.103287","article-title":"Reduced Partial shading effect in Multiple PV Array configuration model using MPPT based Enhanced Particle Swarm Optimization Technique","volume":"2020","author":"Shankar","year":"2020","journal-title":"Microprocess. Microsyst."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"438","DOI":"10.1016\/j.renene.2017.06.014","article-title":"Seven indicators variations for multiple PV array configurations under partial shading and faulty PV conditions","volume":"113","author":"Dhimish","year":"2017","journal-title":"Renew. Energy"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"104","DOI":"10.1511\/2005.52.3448","article-title":"Why W?","volume":"90","author":"Hayes","year":"2005","journal-title":"Am. Sci. Mag."},{"key":"ref_27","first-page":"413","article-title":"A Robust Method for Solving Transcendental Equations","volume":"9","author":"Chakraborty","year":"2012","journal-title":"IJCSI Int. J. Comput. Sci. Issues"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"8118","DOI":"10.1016\/j.ijhydene.2011.10.095","article-title":"Numerical evaluations of transcendental equations for transient experiments","volume":"37","author":"Li","year":"2012","journal-title":"Int. J. Hydrogen Energy"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"768","DOI":"10.1109\/JPHOTOV.2019.2898208","article-title":"Mismatched Series\u2013Parallel Photovoltaic Generator Modeling: An Implicit Current\u2013Voltage Approach","volume":"9","author":"Correa","year":"2019","journal-title":"IEEE J. Photovolt."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"432","DOI":"10.1016\/j.solener.2017.10.007","article-title":"Photovoltaic system using Lambert W function-based technique","volume":"158","author":"Tripathy","year":"2017","journal-title":"Sol. Energy"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1016\/j.cam.2012.11.021","article-title":"Precise and fast computation of Lambert W-functions without transcendental function evaluations","volume":"244","author":"Fukushima","year":"2013","journal-title":"J. Comput. Appl. Math."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2622","DOI":"10.1016\/j.cpc.2012.07.008","article-title":"Lambert W function for applications in physics","volume":"183","year":"2012","journal-title":"Comput. Phys. Commun."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"2160","DOI":"10.1109\/TSP.2002.801912","article-title":"Numerical evaluation of the Lambert W function and application to generation of generalized Gaussian noise with exponent 1\/2","volume":"50","author":"Monir","year":"2002","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1145\/361952.361970","article-title":"Solution of the transcendental equation we w = x","volume":"16","author":"Fritsch","year":"1973","journal-title":"Commun. ACM"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1016\/j.solener.2015.03.043","article-title":"A new explicit double-diode modeling method based on Lambert W-function for photovoltaic arrays","volume":"116","author":"Lun","year":"2015","journal-title":"Sol. Energy"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"214","DOI":"10.1016\/j.renene.2014.07.059","article-title":"Lambert W function-based technique for tracking the maximum power point of PV modules connected in various configurations","volume":"74","author":"Fathabadi","year":"2015","journal-title":"Renew. Energy"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1109\/TSTE.2015.2389715","article-title":"Partial Shading Analysis of Multistring PV Arrays and Derivation of Simplified MPP Expressions","volume":"6","author":"Psarros","year":"2015","journal-title":"IEEE Trans. Sustain. Energy"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1109\/TSTE.2013.2282168","article-title":"An Explicit PV String Model Based on the Lambert $W$ Function and Simplified MPP Expressions for Operation Under Partial Shading","volume":"5","author":"Batzelis","year":"2014","journal-title":"IEEE Trans. Sustain. Energy"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1816","DOI":"10.1109\/JPHOTOV.2020.3016607","article-title":"Modeling and Prediction of PV Module Performance Under Different Operating Conditions Based on Power-Law I\u2013V Model","volume":"10","author":"Zhang","year":"2020","journal-title":"IEEE J. Photovolt."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"100312","DOI":"10.1016\/j.segan.2020.100312","article-title":"One shape parameter-based explicit model for photovoltaic cell and panel","volume":"21","author":"Oulcaid","year":"2020","journal-title":"Sustain. Energy Grids Netw."},{"key":"ref_41","first-page":"3534","article-title":"A Photovoltaic Model with Reduced Computational Time","volume":"62","author":"Mahmoud","year":"2014","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"562","DOI":"10.1016\/j.asoc.2018.09.003","article-title":"Solving transcendental equation using artificial neural network","volume":"73","author":"Jeswal","year":"2018","journal-title":"Appl. Soft Comput. J."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"232","DOI":"10.1515\/phys-2018-0034","article-title":"Siewert solutions of transcendental equations, generalized Lambert functions and physical applications","volume":"16","author":"Barsan","year":"2018","journal-title":"Open Phys."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1016\/j.matcom.2020.04.019","article-title":"SoC-based embedded real-time simulation of mismatched photovoltaic strings","volume":"184","author":"Guarino","year":"2021","journal-title":"Math. Comput. Simul."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1003","DOI":"10.1016\/j.epsr.2010.12.008","article-title":"Modeling of photovoltaic fields in mismatched conditions for energy yield evaluations","volume":"81","author":"Petrone","year":"2011","journal-title":"Electr. Power Syst. Res."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1016\/j.renene.2013.01.009","article-title":"A technique for mismatched PV array simulation","volume":"55","author":"Spagnuolo","year":"2013","journal-title":"Renew. Energy"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1652","DOI":"10.1016\/j.solmat.2007.05.021","article-title":"Analytical model of mismatched photovoltaic fields by means of Lambert W-function","volume":"91","author":"Petrone","year":"2007","journal-title":"Sol. Energy Mater. Sol. Cells"},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Satpathy, R., and Pamuru, V. (2021). Manufacturing of crystalline silicon solar PV modules. Solar PV Power, Elsevier.","DOI":"10.1016\/B978-0-12-817626-9.00005-8"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"120390","DOI":"10.1016\/j.energy.2021.120390","article-title":"TVACPSO-assisted analysis of the effects of temperature and irradiance on the PV module performances","volume":"227","author":"Gong","year":"2021","journal-title":"Energy"},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Lindfield, G., and Penny, J. (2019). Solution of Non-Linear Equations. Numerical Methods, Elsevier.","DOI":"10.1016\/B978-0-12-812256-3.00012-9"},{"key":"ref_51","unstructured":"Nocedal, J., and Wright, S.J. (2006). Trust-Region Methods. Numerical Optimization, Springer. [2nd ed.]. Chapter 4."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"102842","DOI":"10.1016\/j.trc.2020.102842","article-title":"Synchronous and asynchronous parallel computation for large-scale optimal control of connected vehicles","volume":"121","author":"Li","year":"2020","journal-title":"Transp. Res. Part C Emerg. Technol."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"915","DOI":"10.1109\/JPHOTOV.2014.2307161","article-title":"On the parameter extraction of a five-parameter double-diode model of photovoltaic cells and modules","volume":"4","author":"Hejri","year":"2014","journal-title":"IEEE J. Photovolt."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"119480","DOI":"10.1016\/j.energy.2020.119480","article-title":"Analysis of electrical and thermal characteristics of PV array under mismatching conditions caused by partial shading and short circuit failure of bypass diodes","volume":"218","author":"Lee","year":"2021","journal-title":"Energy"},{"key":"ref_55","doi-asserted-by":"crossref","unstructured":"Accarino, J., Petrone, G., Ramos-Paja, C.A., and Spagnuolo, G. (2013, January 11\u201313). Symbolic algebra for the calculation of the series and parallel resistances in PV module model. Proceedings of the 2013 International Conference on Clean Electrical Power (ICCEP), Alghero, Italy.","DOI":"10.1109\/ICCEP.2013.6586967"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.rser.2017.10.051","article-title":"Parameter estimation for photovoltaic system under normal and partial shading conditions: A survey","volume":"84","author":"Kumari","year":"2018","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_57","doi-asserted-by":"crossref","unstructured":"Bhol, R., Dash, R., Pradhan, A., and Ali, S.M. (2015, January 19\u201320). Environmental effect assessment on performance of solar PV panel. Proceedings of the 2015 International Conference on Circuits, Power and Computing Technologies [ICCPCT-2015], Kumaracoil, India.","DOI":"10.1109\/ICCPCT.2015.7159521"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"112407","DOI":"10.1109\/ACCESS.2020.3000420","article-title":"A Robust Strategy Based on Marine Predators Algorithm for Large Scale Photovoltaic Array Reconfiguration to Mitigate the Partial Shading Effect on the Performance of PV System","volume":"8","author":"Yousri","year":"2020","journal-title":"IEEE Access"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"2406","DOI":"10.1109\/TII.2017.2700327","article-title":"Rapid MPPT for Uniformly and Partial Shaded PV System by Using JayaDE Algorithm in Highly Fluctuating Atmospheric Conditions","volume":"13","author":"Kumar","year":"2017","journal-title":"IEEE Trans. Ind. Inform."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"4319","DOI":"10.1109\/TIE.2009.2024664","article-title":"Electrical PV Array Reconfiguration Strategy for Energy Extraction Improvement in Grid-Connected PV Systems","volume":"56","year":"2009","journal-title":"IEEE Trans. Ind. Electron."}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/10\/6\/100\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:34:48Z","timestamp":1760139288000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/10\/6\/100"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,18]]},"references-count":60,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2022,6]]}},"alternative-id":["computation10060100"],"URL":"https:\/\/doi.org\/10.3390\/computation10060100","relation":{},"ISSN":["2079-3197"],"issn-type":[{"value":"2079-3197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,6,18]]}}}