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Wind field modelling

The wind simulation and modelling can be divided into an ambient field model, which describes the wind field if no turbines where present and a wake model which describes the turbines effect on the ambient wind field. The following description is relevant for both SimWindFarm, expect where stated otherwise, i.e., where there are differences a subsection titled SWF Taylor will describe the original version and SWF No Taylor will describe the new version.

SWF Taylor: In this version of SWF both ambient field model and wake model assume Taylor's frozen turbulence hypothesis for inviscid flow [Davidson] to be true. This greatly simplifies the effort of generating an ambient wind field, and provides relatively simple equations for the wake effect models.

SWF No Taylor: In this version of SWF the wake model assumes Taylor's frozen turbulence hypothesis for inviscid flow [Davidson] to be true. This assumption is, however, not made in the ambient wind field model. The coherence between two points separated by a distance downwind will be one if Taylor's frozen turbulence hypothesis is assumed, i.e., the wind at these points will be identical except for a time delay. This is not realistic, which is the reason for introducing a version without the assumption.

Ambient wind field

The ambient wind in a wind farm is usually described by spectral matrices describing the wind speed variation at a number of points in the wind farm and their relation using the method described in [Veers]. The wind model assumes a constant mean wind speed and a zero lateral mean wind speed, i.e. the mean wind direction is constant in the longitudinal direction.

SWF Taylor

Due to the frozen turbulence assumption in this version it is only necessary to simulate stochastic processes in the first line in the lateral direction in the park. These velocities are then assumed to travel with the average wind speed in the longitudinal direction. In this way the wind speeds at downwind grid points will eventually be generated.

SWF No Taylor

lateral wind component, more or less, is only used for wake meandering and is therefore generated using taylor - The lateral component, which is the largest part of wind turbines see, is generated for x point at each turbine according to the spectrum described next. where the lateral wind is implemented as a number of stochastic processes, one for each point.

Turbulence Spectrum

The wind field is generated according to the recommendations in IEC 61400-3 concerning offshore turbines, which state that for non-site specific wind conditions the parameter values in IEC 61400-1 (2005) can be used.

The spectrum used is the Kaimal spectrum

     S_k(f)=sigma_k^2frac{4tfrac{L_k}{U}}{(1+6ftfrac{L_k}{U})^{tfrac{5}{3}}}

where L_k is velocity component integral scale parameter, f is the frequency in Hertz, k denotes the velocity component, U is the hub height mean wind speed and sigma_k is the variance determined by the turbulence intensity, T_i, given by

 begin{aligned}   sigma_x&=T_i(tfrac{3}{4}U+5.6)    sigma_y&=0.8sigma_x end{aligned}

The coherence between two point separated by distance l is given as

   label{eq:coherence}   C_k(f)=e^{-c_kftfrac{l}{U}}

where c_k is the coherence parameter, which depends on the separation direction. In SWF three coherence parameters are used, c_{xx} is used for the coherence of longitudinal wind speed component between points separated by a longitudinal distance, c_{xy} is used for the coherence of longitudinal wind speed component between points separated by a lateral distance, and c_{yy} is used for the coherence of lateral wind speed component between points separated by a lateral distance.

As the hub height is assumed to be above 60m then L_x=340.2 and L_y=113.4 and according to [Kristensen & Jensen] c_{xy} and c_{yy} can be set to 7.1 and 4.2 respectively.

SWF Taylor: In this version only c_{xy} and c_{yy} is needed for wind generation as Taylor's frozen turbulence is assumed and only point separated by a lateral displacement is needed. Notice that the simple wind field model using the frozen turbulence assumption has a coherence of 1 in the longitudinal direction, which does not conform to IEC 61400-1, and care should thus be taken not to design controllers that rely on this fact.

SWF No Taylor: Generating wind in this version is based on [Sørensen et. al.]. The coherence between each turbine is needed and as turbines need not be displaced only in the lateral or longitudinal direction the following equation is used

   label{eq:coherencent}   c_{rc}(alpha) =sqrt{left(c_{xx}cos{alpha}right)^2+left(c_{xy}sin{alpha}right)^2}

where alpha is the angle between wind direction and a line between the two turbine r and turbine c . Furthermore, the delay from turbine to turbine is needed, which can be calculated as

   label{eq:delay}   tau_{rc} =frac{d_{rc}cos{alpha}}{U_0}

where d is the distance between turbines, alpha the angle between turbines, and U_0 is the mean wind speed. Now the cross spectrum between turbines can be found from

   label{eq:spectrum2}   S_{rc}(f) = C_{rc}(f)sqrt{S_{rr}(f)S_{cc}(f)}exp{left(-j2pi ftau_{rc}right)}

where S_{rc} is the cross spectrum between turbine r and turbine c, C_{rc} is the coherence, S_{rr} is the auto spectrum at turbine r, S_{cc} is the auto spectrum at turbine c, and tau_{rc} is the time delay from turbine r to turbine c.

Wake effects

It has been shown in [Larsen et al.] that a good approximation of the meandering is to consider the wake center as a passive tracer which moves downwind with the mean wind speed. It is, therefore, possible to rank turbines relative to each other as being either downwind or upwind.

For wake effect calculations at a given turbine it is only necessary to consider upwind turbines, and as this relationship is fixed it considerably simplifies the calculations.

SWF considers three wake effects; deficit, expansion and center, where wake deficit is a measure of the decrease in downwind wind speed, wake expansion describes the size of the downwind area affected by the wake and wake center defines the lateral position (meandering) of the wake area, see the figure below and [Larsen et al.].

Wind turbine wake illustration

Expressions for wake deficit, center and expansion was developed in [Frandsen et al., Jensen]. The wake center, expansion and deficit at a given point p downwind from a turbine at time t_1 is defined by the wind field at the turbine and its coefficient of thrust at the time of tracer release, t_0,

   label{eq:t0}   t_0=t_1-tfrac{d}{U}

where d is the longitudinal distance between the upwind turbine and p.

Wake Expansion

SWF Taylor: In this version of SWF in particular we use the equation below from [Jensen] to calculate the wake expansion radius, which is a simplified model independent of C_T.

 begin{aligned}   e_i(d)=2Rsqrt{1+tfrac{d}{4R}}=sqrt{4R^2+dR},   label{eq:expand} end{aligned}

where R is the rotor radius.

SWF No Taylor: In this version of SWF, we use the equation below from [Frandsen et al.] to calculate the wake expansion (radius of the wake, e_i(d))

 begin{aligned} e_i(d) = tfrac{1}{2}left(beta^{frac{k}{2}} +alphafrac{d}{D_0}right)^{frac{1}{k}}D_0, end{aligned}

where D_0 is the rotor diameter, alpha and k are parameters which here are set to 0.5 and 2, respectively. Furthermore,

 begin{aligned} beta = frac{1 + sqrt{1-C_T}}{2sqrt{1-C_T}}. end{aligned}

Wake Center

The wake center is computed as a passive tracer, such that the center at time t_k is a function of the center at time t_{k-1} and the average lateral wind speed, over the wake area,

 begin{aligned}   &W_i(d,t_{k-1})={(x,y)|x=d wedge yin W_y} notag    &W_y=[w_i(d,t_{k-1})-e_i(d);w_i(d,t_{k-1})+e_i(d)]   &w_i(d,t_k)=w_i(d,t_{k-1})+bar{v}_{y,i}left(d,W_i(d,t_{k-1})right)   label{eq:centerline} end{aligned}

where w_i(d,t_k) is the wake center at time k, W_i(d,t_{k-1}) is the wake area distance d from the turbine at time t_{k-1} and bar{v}_{y,i}left(d,W_i(d,t_{k-1})right) is the average lateral wind speed over the wake area.

Wake Deficit

SWF Taylor: The deficit from turbine i at distance d can according to [Jensen] be approximated as

   label{eq:deficit}   dU_i(d) = frac{U(d)}{U_0} = 1-2a_i(t_0)left(tfrac{R}{R+kappa d}right)^2

where dU_i is the wind deficit, kappa = 0.1, U(d) is the wind speed distance d down wind, U_0 is the ambient wind speed.a_i is the induction factor of turbine i at time t_0 and R is the radius of the rotor.

Instead of using the induction factor, a more realistic result is obtained by using the simulated thrust coefficient. The thrust coefficient is given by

   label{eq:thrust}   C_T = 4a_i(1-a_i)

According to [Frandsen et al.], the above expressions can be approximated by

   label{eq:deficit1}   dU_i(d)approx 1-tfrac{1}{2}C_{T,i}(t_0)left(1+tfrac{d}{4 R}right)^{-1}

where C_{T,i}(t_0) is the coefficient of thrust of turbine i at time t_0.

SWF No Taylor:In this version of SWF the deficit behind a single turbine is given by the equation below from [Frandsen et al.]

   label{eq:deficit2} c = frac{U(d)}{U_0} approx 1-frac{C_{T}}{2}frac{D^2_{0}}{D^2(d)}

where U(d) is the wind speed distance d down wind, U_0 is the ambient wind speed, C_T is the thrust coefficient, D_0 is the rotor diameter, and D(d) is the wake diameter distance d down wind.

Wake Merging

SWF Taylor: Using equations the equations for meandering, expansion and deficit it is possible to calculate the wake contributions for each turbine and this enables us to calculate the actual wind speed at any point in the farm as

 label{eq:merge}  v_x(x,y)=u_x(x,y)Pi_{i in I}a_i(d_i)

where I contains the indices of all turbines where the point (x,y) is in its wake area, yin W_i(x-P_{x,i}), with P_{x,i} being the longitudinal position of the i'th turbine.

SWF No Taylor:In this version of SWF a slightly differenc approach is taken. In [Frandsen et al.] a row of turbines is considered and the deficit at turbine n+1 can be described as

 label{eq:merge2}  c_{n+1} = 1-left(frac{D^2_n}{D^2_{n+1}}left(1-c_nright) + frac{C_{T_n}}{2}frac{D^2_R}{D^2_{n+1}}c_nright)

where c_{n+1} = tfrac{U_{n+1}}{U_0} is the deficit at turbine n+1, D_n is the wake diameter at tubine n, D_{n+1} is the diameter at turbine n+1, c_n is the deficit at turbine n, C_{T_n} is the thrust coefficient of turbine n and D_R is the diameter of the rotor.

Model of Added Turbulence Intensity

SWF No Taylor: The model of added turbulence intensity which is used here is the one suggested in the IEC 61400-1 [International Electrotechnical Commission] which comes from [Sten T Frandsen] and is given by

 label{eq:AddedTurbulenceIntensity}   I_{add,j}= frac{sigma_{add,j}}{U_j}=  frac{1}{1.5 + 0.8frac{s_{ij}}{sqrt{C_{T,i}}}}

where sigma_{add,j} is the added standard deviation in the wind field at the jth WT which is in wake; U_j is the effective wind speed at the jth WT; s_{ij} is the spacing in rotor diameters between the wake generating WT and the WT in wake; C_{T,i} is the thrust coefficient of the wake generating WT.

Turbulence
Davidson, P.A.
Oxford University Press, 2004
[BibTeX]

@book{Davidson:2004,
  author = {P. A. Davidson},
  title = {Turbulence},
  publisher = {Oxford University Press},
  year = {2004}
}

Analytical Modelling of Wind Speed Deficit in Large Offshore Wind Farms
Frandsen, S., Barthelmie, R., Pryor, S., Rathmann, O., Larsen, S and Højstrup, J
Wind Energy, 2006, Vol. 9, pp. 39-53
[BibTeX]

@article{Frandsen:2006,
  author = {Sten Frandsen and Rebecca Barthelmie and Sara Pryor and Ole Rathmann and Søren Larsen and Jørgen Højstrup},
  title = {Analytical Modelling of Wind Speed Deficit in Large Offshore Wind Farms},
  journal = {Wind Energy},
  year = {2006},
  volume = {9},
  pages = {39--53}
}

Aeolus Toolbox for Dynamic Wind Farm Model, Simulation and Control
Grunnet, J.D., Soltani, M., Knudsen, T., Kragelund, M. and Bak, T.
In Proc. of the 2010 European Wind Energy Conference, 2010
[BibTeX]

@conference{grunnet:2010,
  author = {Jacob Deleuran Grunnet and Mohsen Soltani and Torben Knudsen and Martin Kragelund and Thomas Bak},
  title = {Aeolus Toolbox for Dynamic Wind Farm Model, Simulation and Control},
  booktitle = {Proc. of the 2010 European Wind Energy Conference},
  year = {2010}
}

A note on wind generator interaction
Jensen, N.
Risø National Laboratory, 1983
[BibTeX]

@techreport{Jensen:1983,
  author = {N. Jensen},
  title = {A note on wind generator interaction},
  institution = {Risø National Laboratory},
  year = {1983}
}

Definition of a 5-MW Reference Wind Turbine for Offshore System Development
Jonkman, J., Butterfield, S., Musial, W. and Scott, G.
National Renewable Energy Laboratory, 2009
[BibTeX]

@techreport{NREL5MW,
  author = {J. Jonkman and S. Butterfield and W. Musial and G. Scott},
  title = {Definition of a 5-MW Reference Wind Turbine for Offshore System Development},
  institution = {National Renewable Energy Laboratory},
  url = {http://www.nrel.gov/wind/pdfs/38060.pdf},
  year = {2009}
}

Lateral coherence in isotropic turbulence and in the natural wind
Kristensen, L. and Jensen, N.O.
Boundary-Layer Meteorology, November, 1979, Vol. 17, pp. 353-373
[BibTeX] [DOI]

@article{kristensen:1979,
  author = {Kristensen, L. and Jensen, N.~O.},
  title = {Lateral coherence in isotropic turbulence and in the natural wind},
  journal = {Boundary-Layer Meteorology},
  year = {1979},
  volume = {17},
  pages = {353-373},
  doi = {http://dx.doi.org/10.1007/BF00117924}
}

Wake Meandering: A Pragmatic Approach
Larsen, G.C., Madsen, H.A., Thomsen, K. and Larsen, T.J.
Wind Energy, 2008, Vol. 11, pp. 377-395
[BibTeX]

@article{Larsen:2008b,
  author = {Gunner C. Larsen and Helge Aa. Madsen and Kenneeth Thomsen and Torben J. Larsen},
  title = {Wake Meandering: A Pragmatic Approach},
  journal = {Wind Energy},
  year = {2008},
  volume = {11},
  pages = {377--395}
}

Modeling and Simulation of Offshore Wind Farms for Farm Level Control
Soltani, M., Knudsen, T. and Bak, T.
In European Offshore Wind Conference and Exhibition (EOW) 2009, 2009
[BibTeX]

@inproceedings{mohsen:2009c,
  author = {Mohsen Soltani and Torben Knudsen and Thomas Bak},
  title = {Modeling and Simulation of Offshore Wind Farms for Farm Level Control},
  booktitle = {European Offshore Wind Conference and Exhibition (EOW) 2009},
  year = {2009}
}

Three-Dimensional Wind Simulation
Veers, P.S.
Sandia National Laboratories, 1988 (SAND88-0152 UC-261)
[BibTeX]

@techreport{Veers:1988,
  author = {Paul S. Veers},
  title = {Three-Dimensional Wind Simulation},
  institution = {Sandia National Laboratories},
  year = {1988},
  number = {SAND88-0152 UC-261}
}

Wind Models for Simulation of Power Fluctuations from Wind Farms
Sørensen, P., Hansen, A., AndrĂ©, P., Rosas, C.
Journal of Wind Engineering and Industrial Aerodynamics, 2002
[BibTeX]

@Article{Soerensen02,
  author = 	 {Poul Sørensen and Anca D. Hansen and Pedro AndrĂ© and Carvalho Rosas},
  title = 	 {Wind Models for Simulation of Power Fluctuations from Wind Farms},
  journal = 	 {Journal of Wind Engineering and Industrial Aerodynamics},
  year = 	 2002,
  volume = 	 90,
  pages = 	 {1381-1402}
}