{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:57:23Z","timestamp":1760241443391,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2018,2,25]],"date-time":"2018-02-25T00:00:00Z","timestamp":1519516800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Energies"],"abstract":"<jats:p>This article addresses the problem of optimizing electrical power generation using kite power systems (KPSs). KPSs are airborne wind energy systems that aim to harvest the power of strong and steady high-altitude winds. With the aim of maximizing the total energy produced in a given time interval, we numerically solve an optimal control problem and thereby obtain trajectories and controls for kites. Efficiently solving these optimal control problems is crucial when the results are used in real-time control schemes, such as model predictive control. For this highly nonlinear problem, we derive continuous-time models\u2014in 2D and 3D\u2014and implement an adaptive time-mesh refinement algorithm. By solving the optimal control problem with such an adaptive refinement strategy, we generate a block-structured adapted mesh which gives results as accurate as those computed using fine mesh, yet with much less computing effort and high savings in memory and computing time.<\/jats:p>","DOI":"10.3390\/en11030475","type":"journal-article","created":{"date-parts":[[2018,2,27]],"date-time":"2018-02-27T03:36:12Z","timestamp":1519702572000},"page":"475","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Optimal Control Algorithms with Adaptive Time-Mesh Refinement for Kite Power Systems"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3606-1695","authenticated-orcid":false,"given":"Lu\u00eds","family":"Paiva","sequence":"first","affiliation":[{"name":"SYSTEC\u2013ISR, Faculdade de Engenharia, Universidade do Porto, 4200-465 Porto, Portugal"},{"name":"Instituto Superior de Engenharia do Porto, Polit\u00e9cnico do Porto, 4249-015 Porto, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3516-5094","authenticated-orcid":false,"given":"Fernando","family":"Fontes","sequence":"additional","affiliation":[{"name":"SYSTEC\u2013ISR, Faculdade de Engenharia, Universidade do Porto, 4200-465 Porto, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2018,2,25]]},"reference":[{"key":"ref_1","unstructured":"(2017, October 01). Global Wind Report 2016 | GWEC. Available online: http:\/\/gwec.net\/publications\/global-wind-report-2\/global-wind-report-2016\/."},{"key":"ref_2","unstructured":"(2017, October 01). Ampyx Power: Airborne Wind Energy. Available online: http:\/\/www.ampyxpower.com."},{"key":"ref_3","unstructured":"(2017, October 01). KiteGen. Available online: http:\/\/kitegen.com."},{"key":"ref_4","unstructured":"Lind, D.V.M. (2015, January 15\u201316). Developing a 600 kW Airborne Wind Turbine. Proceedings of the 2015 Airborne Wind Energy Conference, Delft, The Netherlands."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Ahrens, U., Diehl, M., and Schmehl, R. (2013). High Altitude Wind Energy from a Hybrid Lighter-than-Air Platform Using the Magnus Effect. Airborne Wind Energy, Springer. Green Energy and Technology.","DOI":"10.1007\/978-3-642-39965-7"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Ahrens, U., Diehl, M., and Schmehl, R. (2013). Airborne Wind Energy, Springer. Green Energy and Technology.","DOI":"10.1007\/978-3-642-39965-7"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Fagiano, L., and Milanese, M. (2012, January 27\u201329). Airborne Wind Energy: An overview. Proceedings of the 2012 American Control Conference (ACC), Montreal, QC, Canada.","DOI":"10.1109\/ACC.2012.6314801"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1461","DOI":"10.1016\/j.rser.2015.07.053","article-title":"Airborne Wind Energy Systems: A review of the technologies","volume":"51","author":"Cherubini","year":"2015","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"106","DOI":"10.2514\/3.48021","article-title":"Crosswind kite power","volume":"4","author":"Loyd","year":"1980","journal-title":"J. Energy"},{"key":"ref_10","unstructured":"Paiva, L.T., and Fontes, F.A.C.C. (2015). Mesh-Refinement Strategies for Fast Optimal Control and Model Predictive Control of Kite Power Systems. Book of Abstracts of the 2015 International Airborne Wind Energy Conference, Delft University of Technology."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"302","DOI":"10.1016\/j.egypro.2017.10.254","article-title":"Optimal control of kite power systems: Mesh\u2013refinement strategies","volume":"136","author":"Paiva","year":"2017","journal-title":"Energy Procedia"},{"key":"ref_12","unstructured":"Findeisen, R., and Allg\u00f6wer, F. (2002, January 19\u201321). An Introduction to Nonlinear Model Predictive. Proceedings of the 21st Benelux Meeting on Systems and Control, Veldhoven, The Netherlands."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1016\/S0959-1524(01)00023-3","article-title":"Real-time optimization and nonlinear model predictive control of processes governed by differential-algebraic equations","volume":"12","author":"Diehl","year":"2002","journal-title":"J. Process Control"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1016\/S0167-6911(00)00084-0","article-title":"A general framework to design stabilizing nonlinear model predictive controllers","volume":"42","author":"Fontes","year":"2001","journal-title":"Syst. Control Lett."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Fontes, F.A.C.C., Magni, L., and Gyurkovics, \u00c9. (2007). Sampled-data model predictive control for nonlinear time-varying systems: Stability and robustness. Assessment and Future Directions of Nonlinear Model Predictive Control, Springer.","DOI":"10.1007\/978-3-540-72699-9_9"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"4553","DOI":"10.3934\/dcds.2015.35.4553","article-title":"Adaptive time-mesh refinement in optimal control problems with state constraints","volume":"35","author":"Paiva","year":"2015","journal-title":"Discret. Contin. Dyn. Syst."},{"key":"ref_17","unstructured":"Paiva, L.T. (2014). Numerical Methods in Optimal Control and Model Predictive Control. [Ph.D. Thesis, Universidade do Porto]."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1109\/TCST.2009.2017933","article-title":"High Altitude Wind Energy Generation Using Controlled Power Kites","volume":"18","author":"Canale","year":"2010","journal-title":"IEEE Trans. Control Syst. Technol."},{"key":"ref_19","unstructured":"Diehl, M. (2001). Real-Time Optimization for Large Scale Nonlinear Processes. [Ph.D. Thesis, University Heidelberg]."},{"key":"ref_20","unstructured":"Vinter, R.B. (2000). Optimal Control, Springer."},{"key":"ref_21","unstructured":"Betts, J.T. (2001). Practical Methods for Optimal Control Using Nonlinear Programming, SIAM."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Gerdts, M. (2011). Optimal Control of ODEs and DAEs, De Gruyter.","DOI":"10.1515\/9783110249996"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1016\/j.arcontrol.2011.10.009","article-title":"Regularity of minimizers for higher order variational problems in one independent variable","volume":"35","author":"Gavriel","year":"2011","journal-title":"Ann. Rev. Control"},{"key":"ref_24","unstructured":"Paiva, L.T. (2013). Optimal Control in Constrained and Hybrid Nonlinear System: Solvers and Interfaces, Faculdade de Engenharia, Universidade do Porto. Technical Report."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1002\/(SICI)1099-1514(199801\/02)19:1<1::AID-OCA616>3.0.CO;2-Q","article-title":"Mesh refinement in direct transcription methods for optimal control","volume":"19","author":"Betts","year":"1998","journal-title":"Optim. Control Appl. Methods"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"398","DOI":"10.1002\/oca.2114","article-title":"A ph mesh refinement method for optimal control","volume":"36","author":"Patterson","year":"2014","journal-title":"Optim. Control Appl. Methods"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"271","DOI":"10.2514\/1.45852","article-title":"Density Functions for Mesh Refinement in Numerical Optimal Control","volume":"34","author":"Zhao","year":"2011","journal-title":"J. Guid. Control Dyn."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1007\/s10957-015-0704-1","article-title":"Normality and Nondegeneracy for Optimal Control Problems with State Constraints","volume":"166","author":"Fontes","year":"2015","journal-title":"J. Optim. Theory Appl."},{"key":"ref_29","unstructured":"Falugi, P., Kerrigan, E., and Van Wyk, E. (2010). Imperial College London Optimal Control Software. User Guide (ICLOCS), Department of Electrical Engineering, Imperial College London."},{"key":"ref_30","unstructured":"Fontes, F.A.C.C., and Paiva, L.T. (2017). Guaranteed Collision Avoidance in Multi\u2013Kite Power Systems. Book of Abstracts of the 2017 International Airborne Wind Energy Conference, Albert-Ludwigs University Freiburg."}],"container-title":["Energies"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1996-1073\/11\/3\/475\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T14:56:17Z","timestamp":1760194577000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1996-1073\/11\/3\/475"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,2,25]]},"references-count":30,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2018,3]]}},"alternative-id":["en11030475"],"URL":"https:\/\/doi.org\/10.3390\/en11030475","relation":{},"ISSN":["1996-1073"],"issn-type":[{"type":"electronic","value":"1996-1073"}],"subject":[],"published":{"date-parts":[[2018,2,25]]}}}