Applying the Crow Search Algorithm for the Optimal Integration of PV Generation Units in DC Networks

This paper presents an efficient master–slave methodology to solve the problem of integrating photovoltaic (PV) generators into DC grids for a planning period of 20 years. The problem is mathematically formulated as Mixed-Integer Nonlinear Programming (MINLP) with the objective of minimizing the tot...

Full description

Autores:
Grisales-Noreña, Luis Fernando
Cortés-Caicedo, Brandon
Alcalá, Gerardo
Montoya, Oscar Danilo
Tipo de recurso:
Fecha de publicación:
2023
Institución:
Universidad Tecnológica de Bolívar
Repositorio:
Repositorio Institucional UTB
Idioma:
eng
OAI Identifier:
oai:repositorio.utb.edu.co:20.500.12585/12282
Acceso en línea:
https://hdl.handle.net/20.500.12585/12282
https://doi.org/10.3390/math11020387
Palabra clave:
Placement;
Active Distribution Network;
Voltage Stability
LEMB
Rights
openAccess
License
http://creativecommons.org/licenses/by-nc-nd/4.0/
id UTB2_5789a9e5487878c5236948445d2e300d
oai_identifier_str oai:repositorio.utb.edu.co:20.500.12585/12282
network_acronym_str UTB2
network_name_str Repositorio Institucional UTB
repository_id_str
dc.title.spa.fl_str_mv Applying the Crow Search Algorithm for the Optimal Integration of PV Generation Units in DC Networks
title Applying the Crow Search Algorithm for the Optimal Integration of PV Generation Units in DC Networks
spellingShingle Applying the Crow Search Algorithm for the Optimal Integration of PV Generation Units in DC Networks
Placement;
Active Distribution Network;
Voltage Stability
LEMB
title_short Applying the Crow Search Algorithm for the Optimal Integration of PV Generation Units in DC Networks
title_full Applying the Crow Search Algorithm for the Optimal Integration of PV Generation Units in DC Networks
title_fullStr Applying the Crow Search Algorithm for the Optimal Integration of PV Generation Units in DC Networks
title_full_unstemmed Applying the Crow Search Algorithm for the Optimal Integration of PV Generation Units in DC Networks
title_sort Applying the Crow Search Algorithm for the Optimal Integration of PV Generation Units in DC Networks
dc.creator.fl_str_mv Grisales-Noreña, Luis Fernando
Cortés-Caicedo, Brandon
Alcalá, Gerardo
Montoya, Oscar Danilo
dc.contributor.author.none.fl_str_mv Grisales-Noreña, Luis Fernando
Cortés-Caicedo, Brandon
Alcalá, Gerardo
Montoya, Oscar Danilo
dc.subject.keywords.spa.fl_str_mv Placement;
Active Distribution Network;
Voltage Stability
topic Placement;
Active Distribution Network;
Voltage Stability
LEMB
dc.subject.armarc.none.fl_str_mv LEMB
description This paper presents an efficient master–slave methodology to solve the problem of integrating photovoltaic (PV) generators into DC grids for a planning period of 20 years. The problem is mathematically formulated as Mixed-Integer Nonlinear Programming (MINLP) with the objective of minimizing the total annual operating cost. The main stage, consisting of a discrete-continuous version of the Crow search algorithm (DCCSA), is in charge of determining the installation positions of the PV generators and their corresponding power ratings. On the other hand, at the slave level, the successive approximation power flow method is used to determine the objective function value. Numerical results on 33- and 69-bus test systems demonstrate the applicability, efficiency and robustness of the developed approach with respect to different methodologies previously discussed in the scientific literature, such as the vortex search algorithm, the generalized normal distribution optimizer and the particle swarm optimization algorithm. Numerical tests are performed in the MATLAB programming environment using proprietary scripts. © 2023 by the authors.
publishDate 2023
dc.date.accessioned.none.fl_str_mv 2023-07-21T15:45:30Z
dc.date.available.none.fl_str_mv 2023-07-21T15:45:30Z
dc.date.issued.none.fl_str_mv 2023
dc.date.submitted.none.fl_str_mv 2023
dc.type.coarversion.fl_str_mv http://purl.org/coar/version/c_b1a7d7d4d402bcce
dc.type.coar.fl_str_mv http://purl.org/coar/resource_type/c_2df8fbb1
dc.type.driver.spa.fl_str_mv info:eu-repo/semantics/article
dc.type.hasversion.spa.fl_str_mv info:eu-repo/semantics/draft
dc.type.spa.spa.fl_str_mv http://purl.org/coar/resource_type/c_6501
status_str draft
dc.identifier.citation.spa.fl_str_mv Grisales-Noreña, L.F.; Cortés-Caicedo, B.; Alcalá, G.; Montoya, O.D. Applying the Crow Search Algorithm for the Optimal Integration of PV Generation Units in DC Networks. Mathematics 2023, 11, 387. https://doi.org/10.3390/math11020387
dc.identifier.uri.none.fl_str_mv https://hdl.handle.net/20.500.12585/12282
dc.identifier.doi.none.fl_str_mv https://doi.org/10.3390/math11020387
dc.identifier.instname.spa.fl_str_mv Universidad Tecnológica de Bolívar
dc.identifier.reponame.spa.fl_str_mv Repositorio Universidad Tecnológica de Bolívar
identifier_str_mv Grisales-Noreña, L.F.; Cortés-Caicedo, B.; Alcalá, G.; Montoya, O.D. Applying the Crow Search Algorithm for the Optimal Integration of PV Generation Units in DC Networks. Mathematics 2023, 11, 387. https://doi.org/10.3390/math11020387
Universidad Tecnológica de Bolívar
Repositorio Universidad Tecnológica de Bolívar
url https://hdl.handle.net/20.500.12585/12282
https://doi.org/10.3390/math11020387
dc.language.iso.spa.fl_str_mv eng
language eng
dc.rights.coar.fl_str_mv http://purl.org/coar/access_right/c_abf2
dc.rights.uri.*.fl_str_mv http://creativecommons.org/licenses/by-nc-nd/4.0/
dc.rights.accessrights.spa.fl_str_mv info:eu-repo/semantics/openAccess
dc.rights.cc.*.fl_str_mv Attribution-NonCommercial-NoDerivatives 4.0 Internacional
rights_invalid_str_mv http://creativecommons.org/licenses/by-nc-nd/4.0/
Attribution-NonCommercial-NoDerivatives 4.0 Internacional
http://purl.org/coar/access_right/c_abf2
eu_rights_str_mv openAccess
dc.format.extent.none.fl_str_mv 18 páginas
dc.format.mimetype.spa.fl_str_mv application/pdf
dc.publisher.place.spa.fl_str_mv Cartagena de Indias
dc.source.spa.fl_str_mv Mathematics 2023, 11, 387
institution Universidad Tecnológica de Bolívar
bitstream.url.fl_str_mv https://repositorio.utb.edu.co/bitstream/20.500.12585/12282/1/mathematics-11-00387.pdf
https://repositorio.utb.edu.co/bitstream/20.500.12585/12282/2/license_rdf
https://repositorio.utb.edu.co/bitstream/20.500.12585/12282/3/license.txt
https://repositorio.utb.edu.co/bitstream/20.500.12585/12282/5/mathematics-11-00387.pdf.txt
https://repositorio.utb.edu.co/bitstream/20.500.12585/12282/4/mathematics-11-00387.pdf.jpg
bitstream.checksum.fl_str_mv 9357fa910c3ebe7b0c693e21bdf2363b
4460e5956bc1d1639be9ae6146a50347
e20ad307a1c5f3f25af9304a7a7c86b6
5754e54e5453f7fbe4a232fbf6579265
41f765df6bd4f3f4a67cb87e4d507c6e
bitstream.checksumAlgorithm.fl_str_mv MD5
MD5
MD5
MD5
MD5
repository.name.fl_str_mv Repositorio Institucional UTB
repository.mail.fl_str_mv repositorioutb@utb.edu.co
_version_ 1814021793857404928
spelling Grisales-Noreña, Luis Fernando7c27cda4-5fe4-4686-8f72-b0442c58a5d1Cortés-Caicedo, Brandon0b676225-338d-48dc-8f2a-694085d9bb42Alcalá, Gerardo04e0a0ab-2e1e-47e9-aa21-c0cb13133579Montoya, Oscar Danilo9fa8a75a-58fa-436d-a6e2-d80f718a4ea82023-07-21T15:45:30Z2023-07-21T15:45:30Z20232023Grisales-Noreña, L.F.; Cortés-Caicedo, B.; Alcalá, G.; Montoya, O.D. Applying the Crow Search Algorithm for the Optimal Integration of PV Generation Units in DC Networks. Mathematics 2023, 11, 387. https://doi.org/10.3390/math11020387https://hdl.handle.net/20.500.12585/12282https://doi.org/10.3390/math11020387Universidad Tecnológica de BolívarRepositorio Universidad Tecnológica de BolívarThis paper presents an efficient master–slave methodology to solve the problem of integrating photovoltaic (PV) generators into DC grids for a planning period of 20 years. The problem is mathematically formulated as Mixed-Integer Nonlinear Programming (MINLP) with the objective of minimizing the total annual operating cost. The main stage, consisting of a discrete-continuous version of the Crow search algorithm (DCCSA), is in charge of determining the installation positions of the PV generators and their corresponding power ratings. On the other hand, at the slave level, the successive approximation power flow method is used to determine the objective function value. Numerical results on 33- and 69-bus test systems demonstrate the applicability, efficiency and robustness of the developed approach with respect to different methodologies previously discussed in the scientific literature, such as the vortex search algorithm, the generalized normal distribution optimizer and the particle swarm optimization algorithm. Numerical tests are performed in the MATLAB programming environment using proprietary scripts. © 2023 by the authors.18 páginasapplication/pdfenghttp://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/openAccessAttribution-NonCommercial-NoDerivatives 4.0 Internacionalhttp://purl.org/coar/access_right/c_abf2Mathematics 2023, 11, 387Applying the Crow Search Algorithm for the Optimal Integration of PV Generation Units in DC Networksinfo:eu-repo/semantics/articleinfo:eu-repo/semantics/drafthttp://purl.org/coar/resource_type/c_6501http://purl.org/coar/version/c_b1a7d7d4d402bccehttp://purl.org/coar/resource_type/c_2df8fbb1Placement;Active Distribution Network;Voltage StabilityLEMBCartagena de IndiasValencia, A., Hincapie, R.A., Gallego, R.A. Optimal location, selection, and operation of battery energy storage systems and renewable distributed generation in medium–low voltage distribution networks (2021) Journal of Energy Storage, 34, art. no. 102158. Cited 60 times. http://www.journals.elsevier.com/journal-of-energy-storage/ doi: 10.1016/j.est.2020.102158Abdelgawad, H., Sood, V.K. A comprehensive review on microgrid architectures for distributed generation (2019) 2019 IEEE Electrical Power and Energy Conference, EPEC 2019, art. no. 9074800. Cited 13 times. http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=9061108 ISBN: 978-172813406-2 doi: 10.1109/EPEC47565.2019.9074800Li, J., Liu, F., Wang, Z., Low, S.H., Mei, S. Optimal Power Flow in Stand-Alone DC Microgrids (2018) IEEE Transactions on Power Systems, 33 (5), art. no. 8279503, pp. 5496-5506. Cited 113 times. doi: 10.1109/TPWRS.2018.2801280Garces, A. On the convergence of Newton's method in power flow studies for dc microgrids (2018) IEEE Transactions on Power Systems, 33 (5), art. no. 8327530, pp. 5770-5777. Cited 119 times. doi: 10.1109/TPWRS.2018.2820430Alassi, A., Bañales, S., Ellabban, O., Adam, G., MacIver, C. HVDC Transmission: Technology Review, Market Trends and Future Outlook (Open Access) (2019) Renewable and Sustainable Energy Reviews, 112, pp. 530-554. Cited 186 times. https://www.journals.elsevier.com/renewable-and-sustainable-energy-reviews doi: 10.1016/j.rser.2019.04.062Saeed, M.H., Fangzong, W., Kalwar, B.A., Iqbal, S. A Review on Microgrids' Challenges Perspectives (2021) IEEE Access, 9, pp. 166502-166517. Cited 82 times. http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=6287639 doi: 10.1109/ACCESS.2021.3135083López González, D.M., Garcia Rendon, J. Opportunities and challenges of mainstreaming distributed energy resources towards the transition to more efficient and resilient energy markets (Open Access) (2022) Renewable and Sustainable Energy Reviews, 157, art. no. 112018. Cited 18 times. https://www.journals.elsevier.com/renewable-and-sustainable-energy-reviews doi: 10.1016/j.rser.2021.112018Kadir, A.F.A., Khatib, T., Elmenreich, W. Integrating photovoltaic systems in power system: Power quality impacts and optimal planning challenges (2014) International Journal of Photoenergy, 2014, art. no. 321826. Cited 47 times. http://www.hindawi.com/journals/ijp/contents/ doi: 10.1155/2014/321826Lamb, W.F., Grubb, M., Diluiso, F., Minx, J.C. Countries with sustained greenhouse gas emissions reductions: an analysis of trends and progress by sector (2022) Climate Policy, 22 (1), pp. 1-17. Cited 32 times. http://www.tandfonline.com/loi/tcpo20 doi: 10.1080/14693062.2021.1990831Moreno, C., Milanes, C.B., Arguello, W., Fontalvo, A., Alvarez, R.N. Challenges and perspectives of the use of photovoltaic solar energy in Colombia (Open Access) (2022) International Journal of Electrical and Computer Engineering, 12 (5), pp. 4521-4528. Cited 11 times. https://ijece.iaescore.com/index.php/IJECE/article/view/26056/15877 doi: 10.11591/ijece.v12i5.pp4521-4528López, A.R., Krumm, A., Schattenhofer, L., Burandt, T., Montoya, F.C., Oberländer, N., Oei, P.-Y. Solar PV generation in Colombia - A qualitative and quantitative approach to analyze the potential of solar energy market (2020) Renewable Energy, 148, pp. 1266-1279. Cited 46 times. http://www.journals.elsevier.com/renewable-and-sustainable-energy-reviews/ doi: 10.1016/j.renene.2019.10.066Gao, S., Jia, H., Marnay, C. Techno-Economic Evaluation of Mixed AC and DC Power Distribution Network for Integrating Large-Scale Photovoltaic Power Generation (2019) IEEE Access, 7, art. no. 8782088, pp. 105019-105029. Cited 14 times. http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=6287639 doi: 10.1109/ACCESS.2019.2931985Radosavljevic, J., Arsic, N., Milovanovic, M., Ktena, A. Optimal Placement and Sizing of Renewable Distributed Generation Using Hybrid Metaheuristic Algorithm (Open Access) (2020) Journal of Modern Power Systems and Clean Energy, 8 (3), art. no. 9086996, pp. 499-510. Cited 49 times. https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=8685265 doi: 10.35833/MPCE.2019.000259Cardona Isaza, J.A. (2022) Ubicación y Dimensionamiento óptimo de Generadores Distribuidos en Redes DC Universidad Tecnológica de Pereira, Pereira, ColombiaEl-Ela, A.A.A., El-Seheimy, R.A., Shaheen, A.M., Wahbi, W.A., Mouwafi, M.T. PV and battery energy storage integration in distribution networks using equilibrium algorithm (Open Access) (2021) Journal of Energy Storage, 42, art. no. 103041. Cited 40 times. http://www.journals.elsevier.com/journal-of-energy-storage/ doi: 10.1016/j.est.2021.103041Gil-González, W., Molina-Cabrera, A., Montoya, O.D., Grisales-Noreña, L.F. An mi-sdp model for optimal location and sizing of distributed generators in dc grids that guarantees the global optimum (Open Access) (2020) Applied Sciences (Switzerland), 10 (21), art. no. 7681, pp. 1-19. Cited 7 times. https://www.mdpi.com/2076-3417/10/21/7681/pdf doi: 10.3390/app10217681Basto-Gil, J.D., Maldonado-Cardenas, A.D., Montoya, O.D. Optimal Selection and Integration of Batteries and Renewable Generators in DC Distribution Systems through a Mixed-Integer Convex Formulation (Open Access) (2022) Electronics (Switzerland), 11 (19), art. no. 3139. Cited 3 times. www.mdpi.com/journal/electronics doi: 10.3390/electronics11193139Molina-Martin, F., Montoya, O.D., Grisales-Noreña, L.F., Hernández, J.C. A mixed-integer conic formulation for optimal placement and dimensioning of DGs in DC distribution networks (2021) Electronics (Switzerland), 10 (2), art. no. 176, pp. 1-15. Cited 9 times. https://www.mdpi.com/2079-9292/10/2/176/pdf doi: 10.3390/electronics10020176Cortés-Caicedo, B., Molina-Martin, F., Grisales-Noreña, L.F., Montoya, O.D., Hernández, J.C. Optimal Design of PV Systems in Electrical Distribution Networks by Minimizing the Annual Equivalent Operative Costs through the Discrete-Continuous Vortex Search Algorithm (Open Access) (2022) Sensors, 22 (3), art. no. 851. Cited 18 times. https://www.mdpi.com/1424-8220/22/3/851/pdf doi: 10.3390/s22030851Montoya, O.D., Gil-González, W., Grisales-Noreña, L.F. Solar Photovoltaic Integration in Monopolar DC Networks via the GNDO Algorithm (2022) Algorithms, 15 (8), art. no. 277. Cited 7 times. www.mdpi.com/journal/algorithms/ doi: 10.3390/a15080277Yang, Z., Xie, K., Yu, J., Zhong, H., Zhang, N., Xia, Q.X. A General Formulation of Linear Power Flow Models: Basic Theory and Error Analysis (2019) IEEE Transactions on Power Systems, 34 (2), art. no. 8468081, pp. 1315-1324. Cited 108 times. https://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=59 doi: 10.1109/TPWRS.2018.2871182Askarzadeh, A. A novel metaheuristic method for solving constrained engineering optimization problems: Crow search algorithm (Open Access) (2016) Computers and Structures, 169, pp. 1-12. Cited 1409 times. http://www.journals.elsevier.com/computers-and-structures/ doi: 10.1016/j.compstruc.2016.03.001Jain, M., Rani, A., Singh, V. An improved Crow Search Algorithm for high-dimensional problems (Open Access) (2017) Journal of Intelligent and Fuzzy Systems, 33 (6), pp. 3597-3614. Cited 42 times. http://iospress.metapress.com/content/300180/ doi: 10.3233/JIFS-17275Hussien, A.G., Amin, M., Wang, M., Liang, G., Alsanad, A., Gumaei, A., Chen, H. Crow search algorithm: Theory, recent advances, and applications (Open Access) (2020) IEEE Access, 8, pp. 173548-173565. Cited 95 times. http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=6287639 doi: 10.1109/ACCESS.2020.3024108Montoya, O.D., Garrido, V.M., Gil-Gonzalez, W., Grisales-Norena, L.F. Power Flow Analysis in DC Grids: Two Alternative Numerical Methods (2019) IEEE Transactions on Circuits and Systems II: Express Briefs, 66 (11), art. no. 8606244, pp. 1865-1869. Cited 59 times. http://www.ieee-cas.org doi: 10.1109/TCSII.2019.2891640Sahoo, R.R., Ray, M. PSO based test case generation for critical path using improved combined fitness function (2020) Journal of King Saud University - Computer and Information Sciences, 32 (4), pp. 479-490. Cited 29 times. www.journals.elsevier.com/journal-of-king-saud-university-computer-and-information-sciences/ doi: 10.1016/j.jksuci.2019.09.010Zhang, X., Beram, S.M., Haq, M.A., Wawale, S.G., Buttar, A.M. Research on algorithms for control design of human–machine interface system using ML (Open Access) (2022) International Journal of System Assurance Engineering and Management, 13, pp. 462-469. Cited 4 times. http://www.springer.com/engineering/production+eng/journal/13198 doi: 10.1007/s13198-021-01469-1Harman, M., Jia, Y., Zhang, Y. Achievements, open problems and challenges for search based software testing (2015) 2015 IEEE 8th International Conference on Software Testing, Verification and Validation, ICST 2015 - Proceedings, art. no. 7102580. Cited 149 times. ISBN: 978-147997125-1 doi: 10.1109/ICST.2015.7102580Kaur, S., Kumbhar, G., Sharma, J. A MINLP technique for optimal placement of multiple DG units in distribution systems (2014) International Journal of Electrical Power and Energy Systems, 63, pp. 609-617. Cited 193 times. doi: 10.1016/j.ijepes.2014.06.023Baran, M.E., Wu, F.F. Network reconfiguration in distribution systems for loss reduction and load balancing (Open Access) (1989) IEEE Power Engineering Review, 9 (4), pp. 101-102. Cited 493 times. doi: 10.1109/MPER.1989.4310642Sahoo, N.C., Prasad, K. A fuzzy genetic approach for network reconfiguration to enhance voltage stability in radial distribution systems (Open Access) (2006) Energy Conversion and Management, 47 (18-19), pp. 3288-3306. Cited 164 times. doi: 10.1016/j.enconman.2006.01.004Baran, M.E., Wu, F.F. Optimal capacitor placement on radial distribution systems (Open Access) (1989) IEEE Transactions on Power Delivery, 4 (1), pp. 725-734. Cited 1568 times. doi: 10.1109/61.19265Monteiro, V., Monteiro, L.F.C., Lo Franco, F., Mandrioli, R., Ricco, M., Grandi, G., Afonso, J.L. The role of front-end ac/dc converters in hybrid ac/dc smart homes: Analysis and experimental validation (Open Access) (2021) Electronics (Switzerland), 10 (21), art. no. 2601. Cited 9 times. https://www.mdpi.com/2079-9292/10/21/2601/pdf doi: 10.3390/electronics10212601Castiblanco-Pérez, C.M., Toro-Rodríguez, D.E., Montoya, O.D., Giral-Ramírez, D.A. Optimal placement and sizing of d-statcom in radial and meshed distribution networks using a discrete-continuous version of the genetic algorithm (Open Access) (2021) Electronics (Switzerland), 10 (12), art. no. 1452. Cited 26 times. https://www.mdpi.com/2079-9292/10/12/1452/pdf doi: 10.3390/electronics10121452Wang, P., Wang, W., Xu, D. Optimal sizing of distributed generations in DC microgrids with comprehensive consideration of system operation modes and operation targets (Open Access) (2018) IEEE Access, 6, pp. 31129-31140. Cited 52 times. http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=6287639 doi: 10.1109/ACCESS.2018.2842119Montoya, O.D., Gil-González, W., Grisales-Norena, L., Orozco-Henao, C., Serra, F. Economic sispatch of BESS and renewable generators in DC microgrids using voltage-dependent load models (Open Access) (2019) Energies, 12 (23), art. no. 4494. Cited 32 times. https://www.mdpi.com/1996-1073/12/23 doi: 10.3390/en12234494Grisales-Noreña, L.F., Montoya, O.D., Ramos-Paja, C.A. An energy management system for optimal operation of BSS in DC distributed generation environments based on a parallel PSO algorithm (Open Access) (2020) Journal of Energy Storage, 29, art. no. 101488. Cited 58 times. http://www.journals.elsevier.com/journal-of-energy-storage/ doi: 10.1016/j.est.2020.101488Montoya, O.D., Grisales-Noreña, L.F., Perea-Moreno, A.-J. Optimal investments in PV sources for grid-connected distribution networks: An application of the discrete–continuous genetic algorithm (Open Access) (2021) Sustainability (Switzerland), 13 (24), art. no. 13633. Cited 16 times. https://www.mdpi.com/2071-1050/13/24/13633/pdf doi: 10.3390/su132413633Grisales-Noreña, L.F., Montoya, O.D., Marín-García, E.-J., Ramos-Paja, C.A., Perea-Moreno, A.-J. Integration of PV Distributed Generators into Electrical Networks for Investment and Energy Purchase Costs Reduction by Using a Discrete–Continuous Parallel PSO (Open Access) (2022) Energies, 15 (20), art. no. 7465. Cited 3 times. http://www.mdpi.com/journal/energies/ doi: 10.3390/en15207465Beasley, J.E., Chu, P.C. A genetic algorithm for the set covering problem (Open Access) (1996) European Journal of Operational Research, 94 (2), pp. 392-404. Cited 541 times. doi: 10.1016/0377-2217(95)00159-Xhttp://purl.org/coar/resource_type/c_6501ORIGINALmathematics-11-00387.pdfmathematics-11-00387.pdfapplication/pdf421364https://repositorio.utb.edu.co/bitstream/20.500.12585/12282/1/mathematics-11-00387.pdf9357fa910c3ebe7b0c693e21bdf2363bMD51CC-LICENSElicense_rdflicense_rdfapplication/rdf+xml; charset=utf-8805https://repositorio.utb.edu.co/bitstream/20.500.12585/12282/2/license_rdf4460e5956bc1d1639be9ae6146a50347MD52LICENSElicense.txtlicense.txttext/plain; charset=utf-83182https://repositorio.utb.edu.co/bitstream/20.500.12585/12282/3/license.txte20ad307a1c5f3f25af9304a7a7c86b6MD53TEXTmathematics-11-00387.pdf.txtmathematics-11-00387.pdf.txtExtracted texttext/plain60429https://repositorio.utb.edu.co/bitstream/20.500.12585/12282/5/mathematics-11-00387.pdf.txt5754e54e5453f7fbe4a232fbf6579265MD55THUMBNAILmathematics-11-00387.pdf.jpgmathematics-11-00387.pdf.jpgGenerated Thumbnailimage/jpeg7747https://repositorio.utb.edu.co/bitstream/20.500.12585/12282/4/mathematics-11-00387.pdf.jpg41f765df6bd4f3f4a67cb87e4d507c6eMD5420.500.12585/12282oai:repositorio.utb.edu.co:20.500.12585/122822023-07-23 00:17:33.098Repositorio Institucional UTBrepositorioutb@utb.edu.coQXV0b3Jpem8gKGF1dG9yaXphbW9zKSBhIGxhIEJpYmxpb3RlY2EgZGUgbGEgSW5zdGl0dWNpw7NuIHBhcmEgcXVlIGluY2x1eWEgdW5hIGNvcGlhLCBpbmRleGUgeSBkaXZ1bGd1ZSBlbiBlbCBSZXBvc2l0b3JpbyBJbnN0aXR1Y2lvbmFsLCBsYSBvYnJhIG1lbmNpb25hZGEgY29uIGVsIGZpbiBkZSBmYWNpbGl0YXIgbG9zIHByb2Nlc29zIGRlIHZpc2liaWxpZGFkIGUgaW1wYWN0byBkZSBsYSBtaXNtYSwgY29uZm9ybWUgYSBsb3MgZGVyZWNob3MgcGF0cmltb25pYWxlcyBxdWUgbWUobm9zKSBjb3JyZXNwb25kZShuKSB5IHF1ZSBpbmNsdXllbjogbGEgcmVwcm9kdWNjacOzbiwgY29tdW5pY2FjacOzbiBww7pibGljYSwgZGlzdHJpYnVjacOzbiBhbCBww7pibGljbywgdHJhbnNmb3JtYWNpw7NuLCBkZSBjb25mb3JtaWRhZCBjb24gbGEgbm9ybWF0aXZpZGFkIHZpZ2VudGUgc29icmUgZGVyZWNob3MgZGUgYXV0b3IgeSBkZXJlY2hvcyBjb25leG9zIHJlZmVyaWRvcyBlbiBhcnQuIDIsIDEyLCAzMCAobW9kaWZpY2FkbyBwb3IgZWwgYXJ0IDUgZGUgbGEgbGV5IDE1MjAvMjAxMiksIHkgNzIgZGUgbGEgbGV5IDIzIGRlIGRlIDE5ODIsIExleSA0NCBkZSAxOTkzLCBhcnQuIDQgeSAxMSBEZWNpc2nDs24gQW5kaW5hIDM1MSBkZSAxOTkzIGFydC4gMTEsIERlY3JldG8gNDYwIGRlIDE5OTUsIENpcmN1bGFyIE5vIDA2LzIwMDIgZGUgbGEgRGlyZWNjacOzbiBOYWNpb25hbCBkZSBEZXJlY2hvcyBkZSBhdXRvciwgYXJ0LiAxNSBMZXkgMTUyMCBkZSAyMDEyLCBsYSBMZXkgMTkxNSBkZSAyMDE4IHkgZGVtw6FzIG5vcm1hcyBzb2JyZSBsYSBtYXRlcmlhLgoKQWwgcmVzcGVjdG8gY29tbyBBdXRvcihlcykgbWFuaWZlc3RhbW9zIGNvbm9jZXIgcXVlOgoKLSBMYSBhdXRvcml6YWNpw7NuIGVzIGRlIGNhcsOhY3RlciBubyBleGNsdXNpdmEgeSBsaW1pdGFkYSwgZXN0byBpbXBsaWNhIHF1ZSBsYSBsaWNlbmNpYSB0aWVuZSB1bmEgdmlnZW5jaWEsIHF1ZSBubyBlcyBwZXJwZXR1YSB5IHF1ZSBlbCBhdXRvciBwdWVkZSBwdWJsaWNhciBvIGRpZnVuZGlyIHN1IG9icmEgZW4gY3VhbHF1aWVyIG90cm8gbWVkaW8sIGFzw60gY29tbyBsbGV2YXIgYSBjYWJvIGN1YWxxdWllciB0aXBvIGRlIGFjY2nDs24gc29icmUgZWwgZG9jdW1lbnRvLgoKLSBMYSBhdXRvcml6YWNpw7NuIHRlbmRyw6EgdW5hIHZpZ2VuY2lhIGRlIGNpbmNvIGHDsW9zIGEgcGFydGlyIGRlbCBtb21lbnRvIGRlIGxhIGluY2x1c2nDs24gZGUgbGEgb2JyYSBlbiBlbCByZXBvc2l0b3JpbywgcHJvcnJvZ2FibGUgaW5kZWZpbmlkYW1lbnRlIHBvciBlbCB0aWVtcG8gZGUgZHVyYWNpw7NuIGRlIGxvcyBkZXJlY2hvcyBwYXRyaW1vbmlhbGVzIGRlbCBhdXRvciB5IHBvZHLDoSBkYXJzZSBwb3IgdGVybWluYWRhIHVuYSB2ZXogZWwgYXV0b3IgbG8gbWFuaWZpZXN0ZSBwb3IgZXNjcml0byBhIGxhIGluc3RpdHVjacOzbiwgY29uIGxhIHNhbHZlZGFkIGRlIHF1ZSBsYSBvYnJhIGVzIGRpZnVuZGlkYSBnbG9iYWxtZW50ZSB5IGNvc2VjaGFkYSBwb3IgZGlmZXJlbnRlcyBidXNjYWRvcmVzIHkvbyByZXBvc2l0b3Jpb3MgZW4gSW50ZXJuZXQgbG8gcXVlIG5vIGdhcmFudGl6YSBxdWUgbGEgb2JyYSBwdWVkYSBzZXIgcmV0aXJhZGEgZGUgbWFuZXJhIGlubWVkaWF0YSBkZSBvdHJvcyBzaXN0ZW1hcyBkZSBpbmZvcm1hY2nDs24gZW4gbG9zIHF1ZSBzZSBoYXlhIGluZGV4YWRvLCBkaWZlcmVudGVzIGFsIHJlcG9zaXRvcmlvIGluc3RpdHVjaW9uYWwgZGUgbGEgSW5zdGl0dWNpw7NuLCBkZSBtYW5lcmEgcXVlIGVsIGF1dG9yKHJlcykgdGVuZHLDoW4gcXVlIHNvbGljaXRhciBsYSByZXRpcmFkYSBkZSBzdSBvYnJhIGRpcmVjdGFtZW50ZSBhIG90cm9zIHNpc3RlbWFzIGRlIGluZm9ybWFjacOzbiBkaXN0aW50b3MgYWwgZGUgbGEgSW5zdGl0dWNpw7NuIHNpIGRlc2VhIHF1ZSBzdSBvYnJhIHNlYSByZXRpcmFkYSBkZSBpbm1lZGlhdG8uCgotIExhIGF1dG9yaXphY2nDs24gZGUgcHVibGljYWNpw7NuIGNvbXByZW5kZSBlbCBmb3JtYXRvIG9yaWdpbmFsIGRlIGxhIG9icmEgeSB0b2RvcyBsb3MgZGVtw6FzIHF1ZSBzZSByZXF1aWVyYSBwYXJhIHN1IHB1YmxpY2FjacOzbiBlbiBlbCByZXBvc2l0b3Jpby4gSWd1YWxtZW50ZSwgbGEgYXV0b3JpemFjacOzbiBwZXJtaXRlIGEgbGEgaW5zdGl0dWNpw7NuIGVsIGNhbWJpbyBkZSBzb3BvcnRlIGRlIGxhIG9icmEgY29uIGZpbmVzIGRlIHByZXNlcnZhY2nDs24gKGltcHJlc28sIGVsZWN0csOzbmljbywgZGlnaXRhbCwgSW50ZXJuZXQsIGludHJhbmV0LCBvIGN1YWxxdWllciBvdHJvIGZvcm1hdG8gY29ub2NpZG8gbyBwb3IgY29ub2NlcikuCgotIExhIGF1dG9yaXphY2nDs24gZXMgZ3JhdHVpdGEgeSBzZSByZW51bmNpYSBhIHJlY2liaXIgY3VhbHF1aWVyIHJlbXVuZXJhY2nDs24gcG9yIGxvcyB1c29zIGRlIGxhIG9icmEsIGRlIGFjdWVyZG8gY29uIGxhIGxpY2VuY2lhIGVzdGFibGVjaWRhIGVuIGVzdGEgYXV0b3JpemFjacOzbi4KCi0gQWwgZmlybWFyIGVzdGEgYXV0b3JpemFjacOzbiwgc2UgbWFuaWZpZXN0YSBxdWUgbGEgb2JyYSBlcyBvcmlnaW5hbCB5IG5vIGV4aXN0ZSBlbiBlbGxhIG5pbmd1bmEgdmlvbGFjacOzbiBhIGxvcyBkZXJlY2hvcyBkZSBhdXRvciBkZSB0ZXJjZXJvcy4gRW4gY2FzbyBkZSBxdWUgZWwgdHJhYmFqbyBoYXlhIHNpZG8gZmluYW5jaWFkbyBwb3IgdGVyY2Vyb3MgZWwgbyBsb3MgYXV0b3JlcyBhc3VtZW4gbGEgcmVzcG9uc2FiaWxpZGFkIGRlbCBjdW1wbGltaWVudG8gZGUgbG9zIGFjdWVyZG9zIGVzdGFibGVjaWRvcyBzb2JyZSBsb3MgZGVyZWNob3MgcGF0cmltb25pYWxlcyBkZSBsYSBvYnJhIGNvbiBkaWNobyB0ZXJjZXJvLgoKLSBGcmVudGUgYSBjdWFscXVpZXIgcmVjbGFtYWNpw7NuIHBvciB0ZXJjZXJvcywgZWwgbyBsb3MgYXV0b3JlcyBzZXLDoW4gcmVzcG9uc2FibGVzLCBlbiBuaW5nw7puIGNhc28gbGEgcmVzcG9uc2FiaWxpZGFkIHNlcsOhIGFzdW1pZGEgcG9yIGxhIGluc3RpdHVjacOzbi4KCi0gQ29uIGxhIGF1dG9yaXphY2nDs24sIGxhIGluc3RpdHVjacOzbiBwdWVkZSBkaWZ1bmRpciBsYSBvYnJhIGVuIMOtbmRpY2VzLCBidXNjYWRvcmVzIHkgb3Ryb3Mgc2lzdGVtYXMgZGUgaW5mb3JtYWNpw7NuIHF1ZSBmYXZvcmV6Y2FuIHN1IHZpc2liaWxpZGFkCgo=

Publicaciones similares