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Modelling the induction, thrust and power of a yaw-misaligned actuator disk

Published online by Cambridge University Press:  16 March 2023

K.S. Heck ,
H.M. Johlas  and
M.F. Howland Open the ORCID record for M.F. Howland [Opens in a new window]
K.S. Heck
Affiliation:
Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
H.M. Johlas
Affiliation:
Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
M.F. Howland*
Affiliation:
Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
*
†Email address for correspondence: mhowland@mit.edu
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Abstract

Collective wind farm flow control, where wind turbines are operated in an individually suboptimal strategy to benefit the aggregate farm, has demonstrated potential to reduce wake interactions and increase farm energy production. However, existing wake models used for flow control often estimate the thrust and power of yaw-misaligned turbines using simplified empirical expressions that require expensive calibration data and do not extrapolate accurately between turbine models. The thrust, wake velocity deficit, wake deflection and power of a yawed wind turbine depend on its induced velocity. Here, we extend classical one-dimensional momentum theory to model the induction of a yaw-misaligned actuator disk. Analytical expressions for the induction, thrust, initial wake velocities and power are developed as a function of the yaw angle ($\gamma$) and thrust coefficient. The analytical model is validated against large eddy simulations of a yawed actuator disk. Because the induction depends on the yaw and thrust coefficient, the power generated by a yawed actuator disk will always be greater than a $\cos ^3(\gamma )$ model suggests. The power lost due to yaw misalignment depends on the thrust coefficient. An analytical expression for the thrust coefficient that maximizes power, depending on the yaw, is developed and validated. Finally, using the developed induction model as an initial condition for a turbulent far-wake model, we demonstrate how combining wake steering and thrust (induction) control can increase array power, compared to either independent steering or induction control, due to the joint dependence of the induction on the thrust coefficient and yaw angle.

JFM classification

Wakes/Jets: Wakes Flow Control: Control theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 959 , 25 March 2023 , A9
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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References

REFERENCES

Ainslie, J.F. 1988 Calculating the flowfield in the wake of wind turbines. J. Wind Engng Ind. Aerodyn. 27 (1–3), 213–224.CrossRefGoogle Scholar
Annoni, J., Gebraad, P.M., Scholbrock, A.K., Fleming, P.A. & van Wingerden, J.-W. 2016 Analysis of axial-induction-based wind plant control using an engineering and a high-order wind plant model. Wind Energy 19 (6), 1135–1150.CrossRefGoogle Scholar
Barthelmie, R.J., Hansen, K., Frandsen, S.T., Rathmann, O., Schepers, J., Schlez, W., Phillips, J., Rados, K., Zervos, A. & Politis, E. 2009 Modelling and measuring flow and wind turbine wakes in large wind farms offshore. Wind Energy 12 (5), 431–444.CrossRefGoogle Scholar
Bastankhah, M. & Porté-Agel, F. 2014 A new analytical model for wind-turbine wakes. Renew. Energy 70, 116–123.CrossRefGoogle Scholar
Bastankhah, M. & Porté-Agel, F. 2016 Experimental and theoretical study of wind turbine wakes in yawed conditions. J. Fluid Mech. 806, 506–541.CrossRefGoogle Scholar
Bastankhah, M. & Porté-Agel, F. 2019 Wind farm power optimization via yaw angle control: a wind tunnel study. J. Renew. Sustain. Energy 11 (2), 023301.CrossRefGoogle Scholar
Boersma, S., Doekemeijer, B., Gebraad, P.M., Fleming, P.A., Annoni, J., Scholbrock, A.K., Frederik, J. & van Wingerden, J.-W. 2017 A tutorial on control-oriented modeling and control of wind farms. In 2017 American Control Conference (ACC). IEEE.CrossRefGoogle Scholar
Burton, T., Jenkins, N., Sharpe, D. & Bossanyi, E. 2011 Wind Energy Handbook. John Wiley & Sons.CrossRefGoogle Scholar
Calaf, M., Meneveau, C. & Meyers, J. 2010 Large eddy simulation study of fully developed wind-turbine array boundary layers. Phys. Fluids 22 (1), 015110.CrossRefGoogle Scholar
Choukulkar, A., Pichugina, Y., Clack, C.T., Calhoun, R., Banta, R., Brewer, A. & Hardesty, M. 2016 A new formulation for rotor equivalent wind speed for wind resource assessment and wind power forecasting. Wind Energy 19 (8), 1439–1452.CrossRefGoogle Scholar
Dahlberg, J. & Montgomerie, B. 2005 Research program of the Utgrunden demonstration offshore wind farm, final report: Part 2, wake effects and other loads. Rep. FOI, pp. 2–17. Swedish Defence Research Agency.Google Scholar
Fleming, P., Annoni, J., Churchfield, M., Martinez-Tossas, L.A., Gruchalla, K., Lawson, M. & Moriarty, P. 2018 A simulation study demonstrating the importance of large-scale trailing vortices in wake steering. Wind Energy Sci. 3 (1), 243–255.CrossRefGoogle Scholar
Fleming, P., Gebraad, P.M., Lee, S., van Wingerden, J.-W., Johnson, K., Churchfield, M., Michalakes, J., Spalart, P. & Moriarty, P. 2015 Simulation comparison of wake mitigation control strategies for a two-turbine case. Wind Energy 18 (12), 2135–2143.CrossRefGoogle Scholar
Fleming, P., King, J., Dykes, K., Simley, E., Roadman, J., Scholbrock, A., Murphy, P., Lundquist, J.K., Moriarty, P., Fleming, K., et al. 2019 Initial results from a field campaign of wake steering applied at a commercial wind farm – Part 1. Wind Energy Sci. 4 (2), 273–285.CrossRefGoogle Scholar
Frandsen, S., Barthelmie, R., Pryor, S., Rathmann, O., Larsen, S., Højstrup, J. & Thøgersen, M. 2006 Analytical modelling of wind speed deficit in large offshore wind farms. Wind Energy 9 (1–2), 39–53.CrossRefGoogle Scholar
Gebraad, P., Teeuwisse, F., Van Wingerden, J., Fleming, P.A., Ruben, S., Marden, J. & Pao, L. 2016 Wind plant power optimization through yaw control using a parametric model for wake effects – a CFD simulation study. Wind Energy 19 (1), 95–114.CrossRefGoogle Scholar
Ghate, A.S. & Lele, S.K. 2017 Subfilter-scale enrichment of planetary boundary layer large eddy simulation using discrete Fourier–Gabor modes. J. Fluid Mech. 819, 494–539.CrossRefGoogle Scholar
Glauert, H. 1926 A general theory of the autogyro. Tech. Rep. 1111. HM Stationery Office.Google Scholar
Gottlieb, S., Ketcheson, D.I. & Shu, C.-W. 2011 Strong Stability Preserving Runge–Kutta and Multistep Time Discretizations. World Scientific.CrossRefGoogle Scholar
Hansen, M. 2015 Aerodynamics of Wind Turbines. Routledge.CrossRefGoogle Scholar
Howland, M.F., Bossuyt, J., Martínez-Tossas, L.A., Meyers, J. & Meneveau, C. 2016 Wake structure in actuator disk models of wind turbines in yaw under uniform inflow conditions. J. Renew. Sustain. Energy 8 (4), 043301.CrossRefGoogle Scholar
Howland, M.F., Ghate, A.S. & Lele, S.K. 2020 a Influence of the geostrophic wind direction on the atmospheric boundary layer flow. J. Fluid Mech. 883, A39.CrossRefGoogle Scholar
Howland, M.F., Ghate, A.S., Lele, S.K. & Dabiri, J.O. 2020 b Optimal closed-loop wake steering – Part 1: conventionally neutral atmospheric boundary layer conditions. Wind Energy Sci. 5 (4), 1315–1338.CrossRefGoogle Scholar
Howland, M.F., Ghate, A.S., Quesada, J.B., Pena Martínez, J.J., Zhong, W., Larrañaga, F.P., Lele, S.K. & Dabiri, J.O. 2022 a Optimal closed-loop wake steering – Part 2: diurnal cycle atmospheric boundary layer conditions. Wind Energy Sci. 7 (1), 345–365.CrossRefGoogle Scholar
Howland, M.F., González, C.M., Martínez, J.J.P., Quesada, J.B., Larranaga, F.P., Yadav, N.K., Chawla, J.S. & Dabiri, J.O. 2020 c Influence of atmospheric conditions on the power production of utility-scale wind turbines in yaw misalignment. J. Renew. Sustain. Energy 12 (6), 063307.Google Scholar
Howland, M.F., Lele, S.K. & Dabiri, J.O. 2019 Wind farm power optimization through wake steering. Proc. Natl Acad. Sci. USA 116 (29), 14495–14500.CrossRefGoogle ScholarPubMed
Howland, M.F., Quesada, J.B., Martinez, J.J.P., Larrañaga, F.P., Yadav, N., Chawla, J.S., Sivaram, V. & Dabiri, J.O. 2022 b Collective wind farm operation based on a predictive model increases utility-scale energy production. Nat. Energy 7, 818–827.CrossRefGoogle Scholar
Hur, C., Berdowski, T., Simao Ferreira, C., Boorsma, K. & Schepers, G. 2019 A review of momentum models for the actuator disk in yaw. AIAA Paper 2019-1799.CrossRefGoogle Scholar
Jiménez, A., Crespo, Á. & Migoya, E. 2010 Application of a LES technique to characterize the wake deflection of a wind turbine in yaw. Wind Energy 13 (6), 559–572.CrossRefGoogle Scholar
Jonkman, J., Butterfield, S., Musial, W. & Scott, G. 2009 Definition of a 5-MW reference wind turbine for offshore system development. Tech. Rep. NREL/TP-500-38060. National Renewable Energy Laboratory.Google Scholar
Jonkman, J.M. & Buhl, M.L. 2005 FAST user's guide. Tech. Rep. NREL/EL-500-38230. National Renewable Energy Laboratory.Google Scholar
Kheirabadi, A.C. & Nagamune, R. 2019 A quantitative review of wind farm control with the objective of wind farm power maximization. J. Wind Engng Ind. Aerodyn. 192, 45–73.CrossRefGoogle Scholar
Krogstad, P.-r. & Adaramola, M.S. 2012 Performance and near wake measurements of a model horizontal axis wind turbine. Wind Energy 15 (5), 743–756.CrossRefGoogle Scholar
Liew, J.Y., Urbán, A.M. & Andersen, S.J. 2020 Analytical model for the power–yaw sensitivity of wind turbines operating in full wake. Wind Energy Sci. 5 (1), 427–437.CrossRefGoogle Scholar
Martínez-Tossas, L.A., Churchfield, M.J. & Leonardi, S. 2015 Large eddy simulations of the flow past wind turbines: actuator line and disk modeling. Wind Energy 18 (6), 1047–1060.CrossRefGoogle Scholar
Martínez-Tossas, L.A., King, J., Quon, E., Bay, C.J., Mudafort, R., Hamilton, N., Howland, M.F. & Fleming, P.A. 2021 The curled wake model: a three-dimensional and extremely fast steady-state wake solver for wind plant flows. Wind Energy Sci. 6 (2), 555–570.CrossRefGoogle Scholar
Micallef, D. & Sant, T. 2016 A review of wind turbine yaw aerodynamics. In Wind Turbines – Design, Control and Applications (ed. A.G. Aissaoui & A. Tahour), chap. 2, pp. 27–54. InTechOpen.CrossRefGoogle Scholar
Milne-Thomson, L.M. 1973 Theoretical Aerodynamics. Courier Corporation.Google Scholar
Munters, W. & Meyers, J. 2017 An optimal control framework for dynamic induction control of wind farms and their interaction with the atmospheric boundary layer. Phil. Trans. R. Soc. Lond. A 375 (2091), 20160100.Google ScholarPubMed
Munters, W. & Meyers, J. 2018 Dynamic strategies for yaw and induction control of wind farms based on large-eddy simulation and optimization. Energies 11 (1), 177.Google Scholar
Nagarajan, S., Lele, S.K. & Ferziger, J.H. 2003 A robust high-order compact method for large eddy simulation. J. Comput. Phys. 191 (2), 392–419.Google Scholar
Nicoud, F., Toda, H.B., Cabrit, O., Bose, S. & Lee, J. 2011 Using singular values to build a subgrid-scale model for large eddy simulations. Phys. Fluids 23 (8), 085106.CrossRefGoogle Scholar
Nordström, J., Nordin, N. & Henningson, D. 1999 The fringe region technique and the Fourier method used in the direct numerical simulation of spatially evolving viscous flows. SIAM J. Sci. Comput. 20 (4), 1365–1393.CrossRefGoogle Scholar
Pao, L.Y. & Johnson, K.E. 2009 A tutorial on the dynamics and control of wind turbines and wind farms. In 2009 American Control Conference, pp. 2076–2089. IEEE.CrossRefGoogle Scholar
Schreiber, J., Nanos, E., Campagnolo, F. & Bottasso, C.L. 2017 Verification and calibration of a reduced order wind farm model by wind tunnel experiments. J. Phys.: Conf. Ser. 854, 012041.Google Scholar
Shapiro, C.R., Gayme, D.F. & Meneveau, C. 2018 Modelling yawed wind turbine wakes: a lifting line approach. J. Fluid Mech. 841, R1.CrossRefGoogle Scholar
Shapiro, C.R., Gayme, D.F. & Meneveau, C. 2019 a Filtered actuator disks: theory and application to wind turbine models in large eddy simulation. Wind Energy 22 (10), 1414–1420.CrossRefGoogle Scholar
Shapiro, C.R., Starke, G.M., Meneveau, C. & Gayme, D.F. 2019 b A wake modeling paradigm for wind farm design and control. Energies 12 (15), 2956.CrossRefGoogle Scholar
Sørensen, J.N. 2011 Aerodynamic aspects of wind energy conversion. Annu. Rev. Fluid Mech. 43 (1), 427–448.CrossRefGoogle Scholar
Speakman, G.A., Abkar, M., Martínez-Tossas, L.A. & Bastankhah, M. 2021 Wake steering of multirotor wind turbines. Wind Energy 24 (11), 1294–1314.CrossRefGoogle Scholar
Stevens, R.J., Gayme, D.F. & Meneveau, C. 2015 Coupled wake boundary layer model of wind-farms. J. Renew. Sustain. Energy 7 (2), 023115.CrossRefGoogle Scholar
Wilson, R.E. & Lissaman, P. 1974 Applied Aerodynamics of Wind Power Machines. Oregon State University.Google Scholar
Zong, H. & Porté-Agel, F. 2021 Experimental investigation and analytical modelling of active yaw control for wind farm power optimization. Renew. Energy 170, 1228–1244.CrossRefGoogle Scholar