Articles | Volume 11, issue 4
https://doi.org/10.5194/wes-11-1487-2026
© 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-11-1487-2026
© Author(s) 2026. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Efficient derivative computation for unsteady fatigue-constrained nonlinear aero-structural wind turbine blade optimization
Brigham Young University, Provo, UT, 84602, USA
Andrew Ning
Brigham Young University, Provo, UT, 84602, USA
National Laboratory of the Rockies, Golden, CO, 80401, USA
Related authors
No articles found.
James Cutler, Christopher Bay, and Andrew Ning
Wind Energ. Sci., 11, 37–49, https://doi.org/10.5194/wes-11-37-2026, https://doi.org/10.5194/wes-11-37-2026, 2026
Short summary
Short summary
Tilting wind turbines change the airflow behind them, which can lower energy production in wind farms. This research tested both a traditional physics-based model and a deep learning method to predict these effects. While the traditional model improved with added optimization, it struggled with complex wake patterns. The deep learning approach was faster and more accurate, showing potential for better wind farm design and control with reduced computational cost.
Jared J. Thomas, Nicholas F. Baker, Paul Malisani, Erik Quaeghebeur, Sebastian Sanchez Perez-Moreno, John Jasa, Christopher Bay, Federico Tilli, David Bieniek, Nick Robinson, Andrew P. J. Stanley, Wesley Holt, and Andrew Ning
Wind Energ. Sci., 8, 865–891, https://doi.org/10.5194/wes-8-865-2023, https://doi.org/10.5194/wes-8-865-2023, 2023
Short summary
Short summary
This work compares eight optimization algorithms (including gradient-based, gradient-free, and hybrid) on a wind farm optimization problem with 4 discrete regions, concave boundaries, and 81 wind turbines. Algorithms were each run by researchers experienced with that algorithm. Optimized layouts were unique but with similar annual energy production. Common characteristics included tightly-spaced turbines on the outer perimeter and turbines loosely spaced and roughly on a grid in the interior.
Marco Mangano, Sicheng He, Yingqian Liao, Denis-Gabriel Caprace, Andrew Ning, and Joaquim R. R. A. Martins
Wind Energ. Sci. Discuss., https://doi.org/10.5194/wes-2023-10, https://doi.org/10.5194/wes-2023-10, 2023
Revised manuscript not accepted
Short summary
Short summary
High-fidelity MDO enables more effective system design than conventional approaches. MDO can shorten the wind turbine design cycle and reduce the cost of energy. We present a first-of-its-kind high-fidelity aerostructural optimization study of a turbine rotor using a coupled CFD-CSM solver. We simultaneously improve the rotor aerodynamic efficiency and reduce the mass of a rotor of a 10 MW wind turbine using 100+ design variables. We discuss the results with unprecedented detail.
Jared J. Thomas, Christopher J. Bay, Andrew P. J. Stanley, and Andrew Ning
Wind Energ. Sci. Discuss., https://doi.org/10.5194/wes-2022-4, https://doi.org/10.5194/wes-2022-4, 2022
Revised manuscript not accepted
Short summary
Short summary
We wanted to determine if and how optimization algorithms may be exploiting inaccuracies in the simple models used for wind farm layout optimization. Comparing optimization results from a simple model to large-eddy simulations showed that even a simple model provides enough information for optimizers to find good layouts. However, varying the number of wind directions in the optimization showed that the wind resource discretization can negatively impact the optimization results.
Andrew P. J. Stanley, Jennifer King, Christopher Bay, and Andrew Ning
Wind Energ. Sci., 7, 433–454, https://doi.org/10.5194/wes-7-433-2022, https://doi.org/10.5194/wes-7-433-2022, 2022
Short summary
Short summary
In this paper, we present a computationally inexpensive model to calculate wind turbine blade fatigue caused by waking and partial waking. The model accounts for steady state on the blade, as well as wind turbulence. The model is fast enough to be used in wind farm layout optimization, which has not been possible with more expensive fatigue models in the past. The methods introduced in this paper will allow for farms with increased energy production that maintain turbine structural reliability.
Cited articles
Akima, H.: A New Method of Interpolation and Smooth Curve Fitting Based on Local Procedures, J. Assoc. Comput. Mach., 17, 589–602, https://doi.org/10.1145/321607.321609, 1970. a
Allen, J., Young, E., Bortolotti, P., King, R., and Barter, G.: Blade planform design optimization to enhance turbine wake control, Journal of Wind Energy, https://doi.org/10.1002/we.2699, 2020. a
American Society for Testing and Materials: Standard Practices for Cycle Counting in Fatigue Analysis, Tech. rep., https://doi.org/10.1520/E1049-85R17, 2017. a
Batay, S., Kamalov, B., Zhangaskanov, D., Zhao, Y., Wie, D., Zhou, T., and Su, X.: Adjoint-Based High Fidelity Concurrent Aerodynamic Design Optimization of Wind Turbine, Fluids, https://doi.org/10.3390/fluids8030085, 2023. a
Bezanson, J., Edelman, A., Karpinski, S., and Shah, V. B.: Julia: A fresh approach to numerical computing, SIAM Rev., 59, 65–98, https://doi.org/10.1137/141000671, 2017. a
Bir, G. S.: Computerized Method for Preliminary Structural Design of Composite Wind Turbine Blades, J. Sol. Energ.-T. Asme., 123, 372–381, https://doi.org/10.1115/1.1413217, 2001. a
Bortolotti, P., Bottasso, C. L., and Croce, A.: Combined preliminary–detailed design of wind turbines, Wind Energ. Sci., 1, 71–88, https://doi.org/10.5194/wes-1-71-2016, 2016. a
Bortolotti, P., Botasso, C. L., Croce, A., and Sartori, L.: Integration of multiple passive load mitigation technologies by automated design optimization – The case study of a medium-size onshore wind turbine, Wind Energy, 22, 65–79, https://doi.org/10.1002/we.2270, 2018. a, b
Bottasso, C., Bortolotti, P., Croce, A., and Gualdoni, F.: Integrated aero-structural optimization of wind turbines, Multibody Syst. Dyn., 38, 317–344, https://doi.org/10.1007/s11044-015-9488-1, 2016. a, b, c
Brent, R. P.: Algorithms for Minimization Without Derivatives, Dover Publications, ISBN 0-486-41998-3, 2013. a
Burton, T., Jenkins, N., Sharpe, D., and Bossanyi, E.: Wind Energy Handbook, John Wiley & Sons, Ltd, 2 edn., https://doi.org/10.1002/9781119992714, 2011. a, b
Cardoza, A.: byuflowlab/Cardoza2026_Efficient_aeroelastic_wind_ gradients: Paper_Release (Paper_Release), Zenodo [code], https://doi.org/10.5281/zenodo.19673037, 2026. a
Chew, K.-H., Tai, K., Ng, E., and Muskulus, M.: Analytical gradient-based optimization of offshore wind turbine substructures under fatigue and extreme loads, Mar. Struct., 47, 23–41, 2016. a
Curtis, A. R., Powell, M. J. D., and Reid, J. K.: On the estimation of sparse Jacobian matrices, IMA J. Appl. Math., 13, 117–119, https://doi.org/10.1093/imamat/13.1.117, 1974. a
Dhert, T., Ashuri, T., and Martins, J. R. R. A.: Aerodynamic shape optimization of wind turbine blades using a Reynolds-averaged Navier-Stokes model and an adjoint method, Wind Energy, 20, 909–926, https://doi.org/10.1002/we.2070, 2017. a, b
Døssing, M., Madsen, H. A., and Bak, C.: Aerodynamic optimization of wind turbine rotors using a blade element momentum method with corrections for wake rotation and expansion, Wind Energy, 15, 563–574, https://doi.org/10.1002/we.487, 2012. a
Fingersh, L., Hand, M., and Laxson, A.: Wind Turbine Design Cost and Scaling Model, Tech. rep., National Renewable Energy Laboratory, https://doi.org/10.2172/897434, 2006. a
Frontin, C., Bortolotti, P., Branlard, E., Martinez, L., Vijayakumar, G., and Barter, G.: Design space exploration for novel reduced-vortex turbine rotors using free wake vortex methods, Rotor Design and Manufacturing, NAWEA WindTech 2023, https://docs.nrel.gov/docs/fy24osti/87915.pdf (last access: 19 November 2025), 2023. a, b
Gill, P. E., Murray, W., and Saunders, M. A.: SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization, SIAM Rev., 47, 99–131, https://doi.org/10.1137/S0036144504446096, 2005. a
Harrison, R. and Jenkins, G.: Cost modelling of horizontal axis wind turbines, Tech. Rep. ETSU-W-34/00170/REP, University of Sunderland, https://www.osti.gov/etdeweb/biblio/7202468 (last access: 19 November 2025), 1994. a
Hermansen, S. M., Macquart, T., and Lund, E.: Gradient-based structural optimization of a wind turbine blade root section including high-cycle fatigue constraints, Eng. Optimiz., https://doi.org/10.1080/0305215X.2024.2428678, 2025. a, b
Ingersoll, B.: Efficient Incorporation of Fatigue Damage Constraints in Wind Turbine Blade Optimization, Wind Energy, 23, 1063–1076, https://doi.org/10.1002/we.2473, 2020. a, b
Jonkman, J. M., Hayman, G. J., Jonkman, B. J., Damiani, R. R., and Murray, R. E.: AeroDyn v15 User's Guide and Theory Manual, National Laboratory of the Rockies, National Laboratory of the Rockies, Golden, CO, draft edn., https://www.nlr.gov/docs/libraries/wind-docs/aerodyn-manual.pdf?sfvrsn=51d961db_1 (last access: 19 November 2025), 2017. a
Kenway, G. and Martins, J. R. R. A.: Aerostructural shape optimization of wind turbine blades considering site-specific winds, AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, 12, https://doi.org/10.2514/6.2008-6025, 2008. a
Kollár, L. P. and Springer, G. S.: Mechanics of Composite Structures, Cambridge University Press, https://doi.org/10.1017/CBO9780511547140, 2009. a
Kwon, H. I., Yi, S., Choi, S., and Kim, K.: Design of efficient propellers using variable-fidelity aerodynamic analysis and multilevel optimization, J. Propul. Power, 31, 1057–1072, https://doi.org/10.2514/1.B35097, 2015. a
Madsen, M. H. Aa., Zahle, F., Sørensen, N. N., and Martins, J. R. R. A.: Multipoint high-fidelity CFD-based aerodynamic shape optimization of a 10 MW wind turbine, Wind Energ. Sci., 4, 163–192, https://doi.org/10.5194/wes-4-163-2019, 2019. a
Madsen, M. H. Aa., Zahle, F., Horcas, S. G., Barlas, T. K., and Sørensen, N. N.: CFD-based curved tip shape design for wind turbine blades, Wind Energ. Sci., 7, 1471–1501, https://doi.org/10.5194/wes-7-1471-2022, 2022. a
Malcolm, D. J. and Hansen, A. C.: WindPACT Turbine Rotor Design Study: June 2000–June 2002 (Revised), Tech. rep., National Renewable Energy Laboratory, Golden, CO, https://doi.org/10.2172/15000964, 2006. a
Mangano, M., He, S., Liao, Y., Caprace, D.-G., Ning, A., and Martins, J. R. R. A.: Wind Turbine Rotor Design Using High-Fidelity Aerostructural Optimization, AIAA J., 63, 3493–3513, https://doi.org/10.2514/1.J064556, 2025. a
Maples, B., Hand, M., and Musial, W.: Comparative Assessment of Direct Drive High Temperature Superconducting Generators in Multi-Megawatt Class Wind Turbines, Tech. Rep. NREL/TP-5000-49086, National Renewable Energy Laboratory, https://docs.nrel.gov/docs/fy11osti/49086.pdf (last access: 19 November 2025), 2010. a
Martins, J. R. R. A. and Ning, A.: Engineering Design Optimization, Cambridge University Press, https://doi.org/10.1017/9781108980647, 2021. a
McDonnell, T. and Ning, A.: Geometrically exact beam theory for gradient-based optimization, Comput. Struct., 298, 107–373, https://doi.org/10.1016/j.compstruc.2024.107373, 2024. a
McWilliam, M. K., Barlas, T. K., Madsen, H. A., and Zahle, F.: Aero-elastic wind turbine design with active flaps for AEP maximization, Wind Energ. Sci., 3, 231–241, https://doi.org/10.5194/wes-3-231-2018, 2018a. a
McWilliam, M. K., Zahle, F., Dicholkar, A., Verelst, D., and Kim, T.: Optimal Aero-Elastic Design of a Rotor with Bend-Twist Coupling, J. Phys. Conf. Ser., 1037, 042009, https://doi.org/10.1088/1742-6596/1037/4/042009, 2018b. a
Mester, R., Landeros, A., Rackauckas, C., and Lange, K.: Differential methods for assessing sensitivity in biological models, PLoS Comput. Biol., 18, e1009-598, https://doi.org/10.1371/journal.pcbi.1009598, 2022. a
Naumann, U.: The art of differentiating computer programs: an introduction to algorithmic differentiation, SIAM, https://doi.org/10.1137/1.9781611972078, 2011. a, b
Ning, A.: Using Blade Element Momentum Methods with Gradient-Based Design Optimization, Struct. Multidiscip. O., 64, 994–1014, https://doi.org/10.1007/s00158-021-02883-6, 2021. a, b
Ning, A. and McDonnell, T.: Automating Steady and Unsteady Adjoints: Efficiently Utilizing Implicit and Algorithmic Differentiation, arXiv [preprint], https://doi.org/10.48550/arXiv.2306.15243, 2023. a, b
Ning, A. and Petch, D.: Integrated design of downwind land-based wind turbines using analytic gradients, Wind Energy, 19, 2137–2152, https://doi.org/10.1002/we.1972, 2016. a, b, c
Pavese, C., Botasso, C. L., Zahle, F., and Kim, T.: Aeroelastic multidisciplinary design optimization of a swept wind turbine blade, Wind Energy, 20, 1941–1953, https://doi.org/10.1002/we.2131, 2017. a, b
Poon, N. M. K. and Martins, J. R. R. A.: An adaptive approach to constraint aggregation using adjoint sensitivity analysis, Struct. Multidiscip. O., 34, 61–73, https://doi.org/10.1007/s00158-006-0061-7, 2006. a
Resor, B. R.: Definition of a 5MW/61.5m Wind Turbine Blade Reference Model, Tech. rep., Sandia National Laboratories, Albuquerque, New Mexico, https://doi.org/10.2172/1095962, 2013. a, b
Revels, J., Lubin, M., and Papamarkou, T.: Forward-Mode Automatic Differentiation in Julia, arXiv [preprint], https://doi.org/10.48550/arXiv.1607.07892, 2016. a
Valotta Rodrigues, R., Pedersen, M. M., Schøler, J. P., Quick, J., and Réthoré, P.-E.: Speeding up large-wind-farm layout optimization using gradients, parallelization, and a heuristic algorithm for the initial layout, Wind Energ. Sci., 9, 321–341, https://doi.org/10.5194/wes-9-321-2024, 2024. a
Santa, F. D.: Automatic differentiation-based multi-start for gradient-based optimization methods, Mathematics, 12, https://doi.org/10.3390/math12081201, 2024. a
Serafeim, G., Manolas, D., Riziotis, V., and Chaviaropoulos, P.: Multidisciplinary aeroelastic optimization of a 10MW-scale wind turbine rotor targeting to reduced LCoE, J. Phys. Conf. Ser., 2265, 04051, https://doi.org/10.1088/1742-6596/2265/4/042051, 2022. a
Varela, B. T. and Ning, A.: Sparsity for Gradient-based Optimization of Wind Farm Layouts, AIAA SCITECH 2023 Forum, https://doi.org/10.2514/6.2023-1543, 2023. a
Yao, S., Chetan, M., Griffith, D. T., Mendoza, A. S. E., Selig, M. S., Martin, D., Kianbakht, S., Johnson, K., and Loth, E.: Aero-structural design and optimization of 50 MW wind turbine with over 250-m blades, Wind Engineering, 46, 273–295, https://doi.org/10.1177/0309524X211027355, 2022. a
Zahle, F., Tibaldi, C., Pavese, C., McWilliam, M. K., Blasques, J. P. A. A., and Hansen, M. H.: Design of an Aeroelastically Tailored 10 MW Wind Turbine Rotor, J. Phys. Conf. Ser., 753, 062008, https://doi.org/10.1088/1742-6596/753/6/062008, 2016. a, b, c
Short summary
New software calculates wind turbine blade design improvements 10 times faster than traditional methods while maintaining accuracy. By combining four advanced mathematical techniques, researchers optimized a blade design to reduce energy costs by 12.78 %, making fatigue-aware design practical for engineering applications.
New software calculates wind turbine blade design improvements 10 times faster than traditional...
Altmetrics
Final-revised paper
Preprint