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Optimal control of energy extraction in wind-farm boundary layers

Published online by Cambridge University Press:  27 February 2015

Jay P. Goit  and
Johan Meyers
Jay P. Goit
Affiliation:
Department of Mechanical Engineering, KU Leuven, Celestijnenlaan 300A, B3001 Leuven, Belgium
Johan Meyers*
Affiliation:
Department of Mechanical Engineering, KU Leuven, Celestijnenlaan 300A, B3001 Leuven, Belgium
*
†Email address for correspondence: johan.meyers@kuleuven.be
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Abstract

In very large wind farms, the vertical interaction with the atmospheric boundary layer plays an important role, i.e. the total energy extraction is governed by the vertical transport of kinetic energy from higher regions in the boundary layer towards the turbine level. In the current study, we investigate optimal control of wind-farm boundary layers, considering the individual wind turbines as flow actuators, whose energy extraction can be dynamically regulated in time so as to optimally influence the flow field and the vertical energy transport. To this end, we use large-eddy simulations of a fully developed pressure-driven wind-farm boundary layer in a receding-horizon optimal control framework. For the optimization of the wind-turbine controls, a conjugate-gradient optimization method is used in combination with adjoint large-eddy simulations for the determination of the gradients of the cost functional. In a first control study, wind-farm energy extraction is optimized in an aligned wind farm. Results are accumulated over one hour of operation. We find that the energy extraction is increased by 16 % compared to the uncontrolled reference. This is directly related to an increase of the vertical fluxes of energy towards the wind turbines, and vertical shear stresses increase considerably. A further analysis, decomposing the total stresses into dispersive and Reynolds stresses, shows that the dispersive stresses increase drastically, and that the Reynolds stresses decrease on average, but increase in the wake region, leading to better wake recovery. We further observe also that turbulent dissipation levels in the boundary layer increase, and overall the outer layer of the boundary layer enters into a transient decelerating regime, while the inner layer and the turbine region attain a new statistically steady equilibrium within approximately one wind-farm through-flow time. Two additional optimal control cases study penalization of turbulent dissipation. For the current wind-farm geometry, it is found that the ratio between wind-farm energy extraction and turbulent boundary-layer dissipation remains roughly around 70 %, but can be slightly increased by a few per cent by penalizing the dissipation in the optimization objective. For a pressure-driven boundary layer in equilibrium, we estimate that such a shift can lead to an increase in wind-farm energy extraction of 6 %.

JFM classification

Boundary Layers: Boundary layer control Turbulent Flows: Turbulence simulation Turbulent Flows: Turbulent boundary layers
Type
Papers
Information
Journal of Fluid Mechanics , Volume 768 , 10 April 2015 , pp. 5 - 50
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© 2015 Cambridge University Press 

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References

Abkar, M. & Porté-Agel, F. 2013 The effect of free-atmosphere stratification on boundary-layer flow and power output from very large wind farms. Energies 6, 2338–2361.Google Scholar
Andersen, S. J., Sørensen, J. N. & Mikkelsen, R. 2013 Simulation of the inherent turbulence and wake interaction inside an infinitely long row of wind turbines. J. Turbul. 14, 4, 1–24.Google Scholar
Badreddine, H., Vandewalle, S. & Meyers, J. 2014 Sequential quadratic programming (SQP) for optimal control in direct numerical simulation of turbulent flow. J. Comput. Phys. 256, 1–16.CrossRefGoogle Scholar
Barthelmie, R. J., Pryor, S. C., Frandsen, S. T., Hansen, K. S., Schepers, J. G., Rados, K., Schlez, W., Neubert, A., Jensen, L. E. & Neckelmann, S. 2010 Quantifying the impact of wind turbine wakes on power output at offshore wind farms. J. Atmos. Ocean. Technol. 27, 1302–1317.CrossRefGoogle Scholar
Bewley, T. R., Moin, P. & Temam, R. 2001 DNS-based predictive control of turbulence: an optimal benchmark for feedback algorithms. J. Fluid Mech. 447, 179–225.Google Scholar
Borzi, A. & Schultz, V. 2012 Computational Optimization of Systems Governed by Partial Differential Equations. SIAM.Google Scholar
Bou-Zeid, E., Meneveau, C. & Parlange, M. B. 2005 A scale-dependent Lagrangian dynamic model for large eddy simulation of complex turbulent flows. Phys. Fluids 17, 025105.Google Scholar
Burton, T., Sharpe, D., Jenkins, N. & Bossanyi, E. 2001 Wind Energy Handbook. Wiley.CrossRefGoogle Scholar
Cal, R. B., Lebron, J., Castillo, L., Kang, H. S. & Meneveau, C. 2010 Experimental study of the horizontally averaged flow structure in a model wind-turbine array boundary layer. J. Renew. Sustain. Energy 2, 013106.Google Scholar
Calaf, M., Meneveau, C. & Meyers, J. 2010 Large eddy simulation study of fully developed wind-turbine array boundary layers. Phys. Fluids 22, 015110.Google Scholar
Calaf, M., Parlange, M. B. & Meneveau, C. 2011 Large eddy simulation study of scalar transport in fully developed wind-turbine array boundary layers. Phys. Fluids 23, 126603.CrossRefGoogle Scholar
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 1988 Spectral Methods in Fluid Dynamics. Springer.Google Scholar
Chamorro, L. P. & Porté-Agel, F. 2011 Turbulent flow inside and above a wind farm: a wind-tunnel study. Energies 4, 1916–1936.CrossRefGoogle Scholar
Chang, Y. & Collis, S. S.1999 Active control of turbulent channel flows based on large eddy simulation. In Proceedings of the 1999 ASME/JSME Joint Fluids Engineering Conference, FEDSM99-6929.Google Scholar
Choi, H., Hinze, M. & Kunisch, K. 1999 Instantaneous control of backward-facing step flows. Appl. Numer. Maths 31, 133–158.Google Scholar
Chowdhury, S., Zhang, J., Messac, A. & Castillo, L. 2012 Unrestricted wind farm layout optimization (UWFLO): investigating key factors influencing the maximum power generation. J. Renew. Energy 38, 16–30.Google Scholar
Collis, S. S., Chang, Y., Kellogg, S. & Prabhu, R. D.2000 Large eddy simulation and turbulence control. In Proceedings of the 2000 AIAA Fluids Conference, AIAA-2000-2564.Google Scholar
Delport, S., Baelmans, M. & Meyers, J. 2009 Constrained optimization of turbulent mixing-layer evolution. J. Turbul. 10, 18,1–26.Google Scholar
Delport, S., Baelmans, M. & Meyers, J. 2011 Maximizing dissipation in a turbulent shear flow by optimal control of its initial state. Phys. Fluids 25, 045105.Google Scholar
Fleming, P., Gebraad, P., van Wingerden, J., Lee, S., Churchfield, M., Scholbrock, A., Michalakes, J., Johnson, K. & Moriarty, P.2013 The SOWFA super-controller: a high-fidelity tool for evaluating wind plant control approaches. In Proceedings of the EWEA 2013, 4–7 February 2013, Vienna, Austria, pp. 1–12. EWEA.Google Scholar
Frandsen, S. 1992 On the wind speed reduction in the center of large clusters of wind turbines. J. Wind Engng Ind. Aerodyn. 39, 251–265.CrossRefGoogle Scholar
Frigo, M. & Johnson, S. G. 2005 The design and implementation of FFTW3. Proc. IEEE 93 (2), 216–231; Special issue on ‘Program Generation, Optimization, and Platform Adaptation’.CrossRefGoogle Scholar
Hamilton, N., Kang, H. S., Meneveau, C. & Cal, R. B. 2012 Statistical analysis of kinetic energy entrainment in a model wind turbine array boundary layer. J. Renew. Sustain. Energy 4, 063105.Google Scholar
Hansen, K. S., Barthelmie, R. J., Jensen, L. E. & Sommer, A. 2011 The impact of turbulence intensity and atmospheric stability on power deficits due to wind trubine wakes at Horns Rev wind farm. Wind Energy 15, 183–196.CrossRefGoogle Scholar
Hansen, A. D., Sørensen, P., Iov, F. & Blaabjerg, F. 2006 Centralised power control of wind farm with doubly fed induction generators. J. Renew. Energy 31, 935–951.Google Scholar
Hinze, M. & Kunisch, K. 2001 Second order methods for optimal control of time-dependent fluid flow. SIAM J. Control Optim. 40, 925–946.CrossRefGoogle Scholar
Ivanell, S., Sørensen, J. N., Mikkelsen, R. & Henningson, D. 2009 Analysis of numerically generated wake structures. Wind Energy 12, 63–80.Google Scholar
Jameson, A. 1988 Aerodynamic design via control theory. J. Sci. Comput. 3, 233–260.Google Scholar
Jensen, N. O.1983 A note on wind generator interaction. Tech. Rep. Risø-M-2411. Risø National Laboratory, Roskilde, Denmark.Google Scholar
Jimenez, A., Crespo, A., Migoya, E. & Garcia, J. 2007 Advances in large-eddy simulation of a wind turbine wake. J. Phys.: Conf. Ser. 75, 012041.Google Scholar
Johnson, K. E. & Thomas, N. 2009 Wind farm control: addressing the aerodynamic interaction among wind turbines. In American Control Conference, 2009. ACC’09, pp. 2104–2109. IEEE.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, Golden, CO.CrossRefGoogle Scholar
Kaminsky, F. C., Kirchhoff, R. H. & Sheu, L.-J. 1987 Optimal spacing of wind turbines in a wind energy power plant. Solar Energy 39, 467–471.CrossRefGoogle Scholar
Kim, J. 2003 Control of turbulent boundary layers. Phys. Fluids 15, 1093–1105.Google Scholar
Kusiak, A. & Song, Z. 2010 Design of wind farm layout for maximum wind energy capture. J. Renew. Energy 35, 685–694.Google Scholar
Larsen, J. W., Nielsen, S. R. K. & Krenk, S. 2007 Dynamic stall model for wind turbine airfoils. J. Fluids Struct. 23, 959–982.Google Scholar
Lebron, J., Castillo, L. & Meneveau, C. 2012 Experimental study of the kinetic energy budget in a wind turbine stream-tube. J. Turbul. 13 (43), 1–22.Google Scholar
Li, Z. J., Navon, I. M., Hussaini, M. Y. & Le Dimet, F. X. 2003 Optimal control of cylinder wakes via suction and blowing. Comput. Fluids 32 (2), 149–171.Google Scholar
Lissaman, P. B. S.1979 Energy effectiveness of arbitrary arrays of wind turbines. In Proceedings of the 17th AIAA Aerospace Sciences Meeting, New Orleans, LA, AIAA-1979-114.Google Scholar
Lu, H. & Porté-Agel, F. 2011 Large-eddy simulation of a very large wind farm in a stable atmospheric boundary layer. Phys. Fluids 23, 065101.Google Scholar
Luenberger, D. G. 2005 Linear and Nonlinear Programming, 2nd edn. Kluwer Academic.Google Scholar
Manwell, J. F., McGowan, J. G. & Rogers, A. L. 2002 Wind Energy Explained: Theory, Design and Application. Wiley.CrossRefGoogle Scholar
Markfort, C. D., Zhang, W. & Porté-Agel, F. 2012 Turbulent flow and scalar transport through and over aligned and staggered wind farms. J. Turbul. 13 (33), 1–36.Google Scholar
Mason, P. J. & Thomson, T. J. 1992 Stochastic backscatter in large-eddy simulations of boundary layers. J. Fluid Mech. 242, 51–78.Google Scholar
Meneveau, C. 2012 The top-down model of wind farm boundary layers and its applications. J. Turbul. 13, 7,1–12.Google Scholar
Meyers, J. & Meneveau, C.2010 Large eddy simulations of large wind-turbine arrays in the atmospheric boundary layer. In Proceedings of the 48th AIAA Aerospace Sciences Meeting, Including the New Horizons Forum and Aerospace Exposition, Orlando, FL, AIAA-2010-827.Google Scholar
Meyers, J. & Meneveau, C. 2012 Optimal turbine spacing in fully developed wind-farm boundary layers. Wind Energy 15, 305–317.Google Scholar
Meyers, J. & Meneveau, C. 2013 Flow visualization using momentum and energy transport tubes and applications to turbulent flow in wind farms. J. Fluid Mech. 715, 335–358.Google Scholar
Meyers, J. & Sagaut, P. 2006 On the model coefficients for the standard and the variational multi-scale Smagorinsky model. J. Fluid Mech. 569, 287–319.Google Scholar
Meyers, J. & Sagaut, P. 2007 Evaluation of Smagorinsky variants in large-eddy simulations of wall-resolved plane channel flows. Phys. Fluids 19, 095105.Google Scholar
Mikkelsen, R.2003 Actuator disc methods applied to wind turbines. PhD dissertation, Department of Mechanical Engineering, Technical University of Denmark.Google Scholar
Moeng, C.-H. 1984 A large-eddy simulation model for the study of planetary boundary-layer turbulence. J. Atmos. Sci. 41, 2052–2062.2.0.CO;2>CrossRefGoogle Scholar
Newman, B. G. 1976 The spacing of wind turbines in large arrays. Energy Convers. 16, 169–179.Google Scholar
Newman, J., Lebron, J., Meneveau, C. & Castillo, L. 2013 Streamwise development of the wind turbine boundary layer over a model wind turbine array. Phys. Fluids 25, 085108.Google Scholar
Nocedal, J. & Wright, S. J. 2006 Numerical Optimization, 2nd edn. Springer.Google Scholar
Pao, L. Y. & Johnson, K. E. 2009 A tutorial on the dynamics and control of wind turbines and wind farms. In Proceedings of the 2009 American Control Conference, pp. 2076–2089. IEEE.Google Scholar
Pironneau, O. 1974 On optimum design in fluid mechanics. J. Fluid Mech. 64, 97–110.Google Scholar
Press, W. H., Teukolsky, S. A., Vetterling, W. T. & Flannery, B. P. 1996 Numerical Recipes in FORTRAN77: The Art of Scientific Computing, 2nd edn. Cambridge University Press.Google Scholar
Rathmann, O., Frandsen, S. & Barthelmie, R.2007 Wake modelling for intermediate and large wind farms. In European Wind Energy Conference and Exhibition, Milan, Italy, BL3.199.Google Scholar
Rawlings, J. B. & Mayne, D. Q. 2008 Model Predictive Control: Theory and Design. Nob Hill.Google Scholar
Sanderse, B., van der Pijl, S. P. & Koren, B. 2011 Review of computational fluid dynamics for wind turbine wake aerodynamics. Wind Energy 14, 799–819.CrossRefGoogle Scholar
Schumann, U. 1975 Subgrid scale model for finite difference simulations of turbulent flows in plane channels and annuli. J. Comput. Phys. 18, 376–404.Google Scholar
Smagorinsky, J. 1963 General circulation experiments with the primitive equations: I. The basic experiment. Mon. Weath. Rev. 91 (3), 99–165.2.3.CO;2>CrossRefGoogle Scholar
Soleimanzadeh, M., Wisniewski, R. & Kanev, S. 2012 An optimization framework for load and power distribution in wind farms. J. Wind Engng Ind. Aerodyn. 107–108, 256–262.CrossRefGoogle Scholar
Spruce, C. J.1993 Simulation and control of windfarms. PhD dissertation, Department of Engineering Science, University of Oxford.Google Scholar
Stevens, R. J. A. M. 2015 Dependence of optimal wind-turbine spacing on wind-farm length. Wind Energy 1–9 (submitted).Google Scholar
Stull, R. B. 1988 An Introduction to Boundary Layer Meteorology. Kluwer.Google Scholar
Tennekes, H. & Lumley, J. L. 1972 A First Course in Turbulence. MIT Press.CrossRefGoogle Scholar
Tröltzsch, F. 2010 Optimal Control of Partial Differential Equations: Theory, Methods, and Applications. American Mathematical Society.Google Scholar
Verstappen, R. W. C. P. & Veldman, A. E. P. 2003 Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187, 343–368.Google Scholar
Wei, M. & Freund, J. B. 2006 A noise-controlled free shear flow. J. Fluid Mech. 546, 123–152.CrossRefGoogle Scholar
Wu, Y.-T. & Porté-Agel, F. 2011 Large-eddy simulation of wind-turbine wakes: evaluation of turbine parametrisations. Boundary-Layer Meteorol. 138, 345–366.Google Scholar
Yang, X., Kang, S. & Sotiropoulos, F. 2012 Computational study and modeling of turbine spacing effects in infinite aligned wind farms. Phys. Fluids 24, 115107.CrossRefGoogle Scholar
Yang, D., Meneveau, C. & Shen, L. 2014a Effect of swells on offshore wind energy harvesting – a large-eddy simulation study. J. Renew. Energy 70, 11–23.CrossRefGoogle Scholar
Yang, D., Meneveau, C. & Shen, L. 2014b Large-eddy simulation of off-shore wind farm. J. Renew. Energy 26, 025101.Google Scholar
Zilitinkevich, S. S. 1989 Velocity profiles, the resistance law and the dissipation rate of mean flow kinetic energy in a neutrally and stably stratified planetary boundary layer. Boundary-Layer Meteorol. 46 (4), 367–387.Google Scholar