Articles | Volume 11, issue 5
https://doi.org/10.5194/wes-11-1751-2026
© Author(s) 2026. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
https://doi.org/10.5194/wes-11-1751-2026
© Author(s) 2026. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Optimal control of crosswind kite systems with an engineering wake model based on vortex loops and dipoles
Jochem De Schutter
CORRESPONDING AUTHOR
Department of Microsystems Engineering, University of Freiburg, Georges-Köhler-Allee 102, 79108 Freiburg, Germany
Antonia Mühleck
Department of Microsystems Engineering, University of Freiburg, Georges-Köhler-Allee 102, 79108 Freiburg, Germany
Rachel Leuthold
Department of Microsystems Engineering, University of Freiburg, Georges-Köhler-Allee 102, 79108 Freiburg, Germany
Moritz Diehl
CORRESPONDING AUTHOR
Department of Microsystems Engineering, University of Freiburg, Georges-Köhler-Allee 102, 79108 Freiburg, Germany
Department of Mathematics, University of Freiburg, Ernst-Zermelo-Straße 1, 79104 Freiburg, Germany
Related authors
Markus Sommerfeld, Martin Dörenkämper, Jochem De Schutter, and Curran Crawford
Wind Energ. Sci., 8, 1153–1178, https://doi.org/10.5194/wes-8-1153-2023, https://doi.org/10.5194/wes-8-1153-2023, 2023
Short summary
Short summary
This study investigates the performance of pumping-mode ground-generation airborne wind energy systems by determining power-optimal flight trajectories based on realistic, k-means clustered, vertical wind velocity profiles. These profiles, derived from mesoscale weather simulations at an offshore and an onshore site in Europe, are incorporated into an optimal control model that maximizes average cycle power by optimizing the kite's trajectory.
Markus Sommerfeld, Martin Dörenkämper, Jochem De Schutter, and Curran Crawford
Wind Energ. Sci., 7, 1847–1868, https://doi.org/10.5194/wes-7-1847-2022, https://doi.org/10.5194/wes-7-1847-2022, 2022
Short summary
Short summary
This research explores the ground-generation airborne wind energy system (AWES) design space and investigates scaling effects by varying design parameters such as aircraft wing size, aerodynamic efficiency and mass. Therefore, representative simulated onshore and offshore wind data are implemented into an AWES trajectory optimization model. We estimate optimal annual energy production and capacity factors as well as a minimal operational lift-to-weight ratio.
Thomas Haas, Jochem De Schutter, Moritz Diehl, and Johan Meyers
Wind Energ. Sci., 7, 1093–1135, https://doi.org/10.5194/wes-7-1093-2022, https://doi.org/10.5194/wes-7-1093-2022, 2022
Short summary
Short summary
In this work, we study parks of large-scale airborne wind energy systems using a virtual flight simulator. The virtual flight simulator combines numerical techniques from flow simulation and kite control. Using advanced control algorithms, the systems can operate efficiently in the park despite turbulent flow conditions. For the three configurations considered in the study, we observe significant wake effects, reducing the power yield of the parks.
Markus Sommerfeld, Martin Dörenkämper, Jochem De Schutter, and Curran Crawford
Wind Energ. Sci., 8, 1153–1178, https://doi.org/10.5194/wes-8-1153-2023, https://doi.org/10.5194/wes-8-1153-2023, 2023
Short summary
Short summary
This study investigates the performance of pumping-mode ground-generation airborne wind energy systems by determining power-optimal flight trajectories based on realistic, k-means clustered, vertical wind velocity profiles. These profiles, derived from mesoscale weather simulations at an offshore and an onshore site in Europe, are incorporated into an optimal control model that maximizes average cycle power by optimizing the kite's trajectory.
Markus Sommerfeld, Martin Dörenkämper, Jochem De Schutter, and Curran Crawford
Wind Energ. Sci., 7, 1847–1868, https://doi.org/10.5194/wes-7-1847-2022, https://doi.org/10.5194/wes-7-1847-2022, 2022
Short summary
Short summary
This research explores the ground-generation airborne wind energy system (AWES) design space and investigates scaling effects by varying design parameters such as aircraft wing size, aerodynamic efficiency and mass. Therefore, representative simulated onshore and offshore wind data are implemented into an AWES trajectory optimization model. We estimate optimal annual energy production and capacity factors as well as a minimal operational lift-to-weight ratio.
Thomas Haas, Jochem De Schutter, Moritz Diehl, and Johan Meyers
Wind Energ. Sci., 7, 1093–1135, https://doi.org/10.5194/wes-7-1093-2022, https://doi.org/10.5194/wes-7-1093-2022, 2022
Short summary
Short summary
In this work, we study parks of large-scale airborne wind energy systems using a virtual flight simulator. The virtual flight simulator combines numerical techniques from flow simulation and kite control. Using advanced control algorithms, the systems can operate efficiently in the park despite turbulent flow conditions. For the three configurations considered in the study, we observe significant wake effects, reducing the power yield of the parks.
Cited articles
Akberali, A., Kheiri, M., and Bourgault, F.: Generalized aerodynamic models for crosswind kite power systems, J. Wind Eng. Ind. Aerod., 215, https://doi.org/10.1016/j.jweia.2021.104664, 2021. a
Andersson, J. A. E., Gillis, J., Horn, G., Rawlings, J. B., and Diehl, M.: CasADi – A software framework for nonlinear optimization and optimal control, Mathematical Programming Computation, 11, 1–36, https://doi.org/10.1007/s12532-018-0139-4, 2019. a
Branlard, E.: Wind Turbine Aerodynamics and Vorticity-Based Methods, Springer-Cham, https://doi.org/10.1007/978-3-319-55164-7, 2017. a
Crismer, J.-B., Haas, T., Duponcheel, M., and Winckelmans, G.: Large eddy simulation of airborne wind energy systems flying in turbulent wind using model predictive control, Wind Energ. Sci. Discuss. [preprint], https://doi.org/10.5194/wes-2025-288, in review, 2026. a
De Lellis, M., Reginatto, R., Saraiva, R., and Trofino, A.: The Betz limit applied to Airborne Wind Energy, Renewable Energy, 127, 32–40, https://doi.org/10.1016/j.renene.2018.04.034, 2018. a
De Schutter, J.: Periodic optimal control methods for nonlinear energy conversion systems, Ph.D. thesis, University of Freiburg, https://doi.org/10.6094/UNIFR/248910, 2024. a
De Schutter, J., Leuthold, R., and Diehl, M.: Optimal Control of a Rigid-Wing Rotary Kite System for Airborne Wind Energy, in: Proceedings of the European Control Conference (ECC), https://doi.org/10.23919/ECC.2018.8550383, 2018. a
De Schutter, J., Harzer, J., and Diehl, M.: Vertical Airborne Wind Energy Farms with High Power Density per Ground Area based on Multi-Aircraft Systems, Eur. J. Control, 74, 100867, https://doi.org/10.1016/j.ejcon.2023.100867, 2023a. a, b
De Schutter, J., Leuthold, R., and Bronnenmeyer, T.: Code release for “Optimal Control of Crosswind Kite Systems with an Engineering Wake Model based on Vortex Loops and Dipoles” (Wind Energy Science, 2026) (v1.0.0-wes-2026), Zenodo [code], https://doi.org/10.5281/zenodo.20208710, 2026. a
Fagiano, L., Quack, M., Bauer, F., Carnel, L., and Oland, E.: Autonomous Airborne Wind Energy Systems: Accomplishments and Challenges, Annual Review of Control, Robotics, and Autonomous Systems, 5, 603–631, https://doi.org/10.1146/annurev-control-042820-124658, 2022. a
Fritz, F.: Airborne Wind Energy, chap. Application of an Automated Kite System for Ship Propulsion and Power Generation, 391–411, Springer-Verlag Berlin Heidelberg, https://doi.org/10.1007/978-3-642-39965-7_20, 2013. a
Gaunaa, M., Forsting, A. M., and Trevisi, F.: An engineering model for the induction of crosswind kite power systems, J. Phys. Conf. Ser., 1618, 032010, https://doi.org/10.1088/1742-6596/1618/3/032010, 2020. a, b, c, d
Gros, S. and Diehl, M.: Modeling of Airborne Wind Energy Systems in Natural Coordinates, in: Airborne Wind Energy, Springer-Verlag Berlin Heidelberg, https://doi.org/10.1007/978-3-642-39965-7_10, 2013. a, b
Gros, S. and Zanon, M.: Numerical Optimal Control with Periodicity Constraints in the Presence of Invariants, IEEE T. Automat. Contr., 63, 2818–2832, https://doi.org/10.1109/TAC.2017.2772039, 2018. a
Gros, S., Zanon, M., and Diehl, M.: A Relaxation Strategy for the Optimization of Airborne Wind Energy Systems, in: Proceedings of the European Control Conference (ECC), 1011–1016, https://doi.org/10.23919/ECC.2013.6669670, 2013. a
Haas, T., De Schutter, J., Diehl, M., and Meyers, J.: Wake characteristics of pumping mode airborne wind energy systems, J. Phys. Conf. Ser., 1256, 012016, https://doi.org/10.1088/1742-6596/1256/1/012016, 2019. a
Haas, T., De Schutter, J., Diehl, M., and Meyers, J.: Large-eddy simulation of airborne wind energy farms, Wind Energ. Sci., 7, 1093–1135, https://doi.org/10.5194/wes-7-1093-2022, 2022. a
Hart, C.: Kites: a Historical Survey, Paul P. Appel Publisher, 1982. a
Heydarnia, O., Wauters, J., Lefebvre, T., and Crevecoeur, G.: Optimal Path Planning of Airborne Wind Energy Systems with a Flexible Tether, Journal of Guidance, Control and Dynamics, 48, 1–8, https://doi.org/10.2514/1.G008967, 2025. a
Horn, G., Gros, S., and Diehl, M.: Numerical Trajectory Optimization for Airborne Wind Energy Systems Described by High Fidelity Aircraft Models, in: Airborne Wind Energy, Springer-Verlag Berlin Heidelberg, https://doi.org/10.1007/978-3-642-39965-7_11, 2013. a
HSL: A collection of Fortran codes for large scale scientific computation, http://www.hsl.rl.ac.uk (last access: 25 November 2025), 2011. a
Joshi, R., Schmehl, R., and Kruijff, M.: Power curve modelling and scaling of fixed-wing ground-generation airborne wind energy systems, Wind Energ. Sci., 9, 2195–2215, https://doi.org/10.5194/wes-9-2195-2024, 2024. a
Kaufman-Martin, S., Naclerio, N., May, P., and Luzzatto-Fegiz, P.: An entrainment-based model for annular wakes, with applications to airborne wind energy, Wind Energy, 25, 419–431, https://doi.org/10.1002/we.2679, 2022. a
Kheiri, M., Nasrabad, V. S., and Bourgault, F.: A new perspective on the aerodynamic performance and power limit of crosswind kite systems, J. Wind Eng. Ind. Aerod., 190, 190–199, https://doi.org/10.1016/j.jweia.2019.04.010, 2019. a, b
Kokkedee, J.: Wind farm wake flow recovery with the use of kites, Master's thesis, Delft University of Technology, 2022. a
Kruijff, M. and Ruiterkamp, R.: A roadmap towards airborne wind energy in the utility sector, in: Airborne Wind Energy: Advances in Technology Development and Research, Springer, Singapore, https://doi.org/10.1007/978-981-10-1947-0_26, 2018. a
Leuthold, R., Gros, S., and Diehl, M.: Induction in Optimal Control of Multiple-Kite Airborne Wind Energy Systems, in: Proceedings of 20th IFAC World Congress, Toulouse, France, https://doi.org/10.1016/j.ifacol.2017.08.026, 2017. a
Leuthold, R., De Schutter, J., Malz, E. C., Licitra, G., Gros, S., and Diehl, M.: Operational Regions of a Multi-Kite AWE System, in: European Control Conference (ECC), https://doi.org/10.23919/ECC.2018.8550199, 2018. a
Leuthold, R., De Schutter, J., Crawford, C., Gros, S., and Diehl, M.: Rigid-Wake Lifting-Line Vortex Modeling in a Single-Kite AWE Optimal Control Problem, in: Book of Abstracts of the Airborne Wind Energy Conference 2024, edited by: Sánchez-Arriaga, G., Thoms, S., and Schmehl, R., Delft University of Technology, https://doi.org/10.4233/uuid:85fd0eb1-83ec-4e34-9ac8-be6b32082a52, 2024. a
Loyd, M.: Crosswind Kite Power, J. Energ., 4, 106–111, https://doi.org/10.2514/3.48021, 1980. a
Malz, E., Verendel, V., and Gros, S.: Computing the Power Profiles for an Airborne Wind Energy System based on Large-Scale Wind Data, Renewable Energy, https://doi.org/10.1016/j.renene.2020.06.056, 2020. a
Malz, E. C., Koenemann, J., Sieberling, S., and Gros, S.: A Reference Model for Airborne Wind Energy Systems for Optimization and Control, Renewable Energy, 140, 1004–1011, https://doi.org/10.1016/j.renene.2019.03.111, 2019. a
Manwell, J. F., McGowan, J. G., and Rogers, A. L.: Wind Energy Explained: Theory, Design and Application, Second Edition, John Wiley & Sons, Ltd, Chichester, UK, https://doi.org/10.1002/9781119994367, 2009. a
Mühleck, A. and De Schutter, J.: Toni2412/AWEWA: Code release for Section 4 “Optimal Control of Crosswind Kite Systems with an Engineering Wake Model based on Vortex Loops and Dipoles” (Wind Energy Science, 2026) (v1.0.0), Zenodo [code], https://doi.org/10.5281/zenodo.20238024, 2026. a
Noga, R., Paulig, X., Schmidt, L., Karg, B., Quack, M., and Soliman, M.: Optimization of 3-D flight trajectory of variable trim kites for airborne wind energy production, Industrial Abstract accepted to the Europan Control Conference (ECC) 2024, arXiv, https://doi.org/10.48550/arXiv.2403.00382, 2024. a
Ploumakis, E. and Bierbooms, W.: Enhanced Kinetic Energy Entrainment in Wind Farm Wakes: Large Eddy Simulation Study of a Wind Turbine Array with Kites, 165–185, Springer Singapore, Singapore, ISBN 978-981-10-1947-0, https://doi.org/10.1007/978-981-10-1947-0_8, 2018. a
Pynaert, N., Haas, T., Wauters, J., Crevecoeur, G., and Degroote, J.: Aero-servo simulations of an airborne wind energy system using geometry-resolved computational fluid dynamics, Wind Energ. Sci., 10, 2663–2684, https://doi.org/10.5194/wes-10-2663-2025, 2025. a
Trevisi, F., Castro-Fernández, I., Pasquinelli, G., Riboldi, C. E. D., and Croce, A.: Flight trajectory optimization of Fly-Gen airborne wind energy systems through a harmonic balance method, Wind Energ. Sci., 7, 2039–2058, https://doi.org/10.5194/wes-7-2039-2022, 2022. a
Trevisi, F., Riboldi, C. E. D., and Croce, A.: Vortex model of the aerodynamic wake of airborne wind energy systems, Wind Energ. Sci., 8, 999–1016, https://doi.org/10.5194/wes-8-999-2023, 2023. a, b, c, d
Tugnoli, D., Montagnani, D., Syal, M., Droandi, G., and Zanotti, A.: Mid-fidelity approach to aerodynamic simulations of unconventional VTOL aircraft configurations, Aerosp. Sci. Technol., 115, 106804, https://doi.org/10.1016/j.ast.2021.106804, 2021. a
Van Niekerk, T.: Enhancing Wind Farm Wake Recovery Through Kite-Induced Vertical Entrainment, Master's thesis, Delft University of Technology, 2025. a
Vermillion, C., Cobb, M., Fagiano, L., Leuthold, R., Diehl, M., Smith, R. S., Wood, T. A., Rapp, S., Schmehl, R., Olinger, D., and Demetriou, M.: Electricity in the air: Insights from two decades of advanced control research and experimental flight testing of airborne wind energy systems, Annu. Rev. Control, 52, 330–357, https://doi.org/10.1016/j.arcontrol.2021.03.002, 2021. a
Wächter, A. and Biegler, L. T.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program., 106, 25–57, https://doi.org/10.1007/s10107-004-0559-y, 2006. a
Zanon, M., Gros, S., Andersson, J., and Diehl, M.: Airborne Wind Energy Based on Dual Airfoils, IEEE T. Contr. Syst. T., 21, 1215–1222, https://doi.org/10.1109/TCST.2013.2257781, 2013. a, b, c
Zanon, M., Gros, S., Meyers, J., and Diehl, M.: Airborne Wind Energy: Airfoil-Airmass Interaction, in: Proceedings of the IFAC World Congress, 5814–5819, https://doi.org/10.3182/20140824-6-ZA-1003.00258, 2014. a, b
Short summary
The performance of high-performance crosswind kite systems is strongly affected by the complex wake structures they generate. We develop an unsteady, vortex-based wake model that can be efficiently integrated into optimal control frameworks for flight trajectory optimization. The model is shown to provide good agreement with higher-fidelity simulations while incurring only moderate additional computational cost, enabling more reliable performance prediction.
The performance of high-performance crosswind kite systems is strongly affected by the complex...
Altmetrics
Final-revised paper
Preprint