## Wind field modellingThe 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.
## Ambient wind fieldThe 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 TaylorDue 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 Taylorlateral 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 SpectrumThe 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 where is velocity component integral scale parameter, is the frequency in Hertz, denotes the velocity component, is the hub height mean wind speed and is the variance determined by the turbulence intensity, , given by The coherence between two point separated by distance is given as where is the coherence parameter, which depends on the separation direction. In SWF three coherence parameters are used, is used for the coherence of longitudinal wind speed component between points separated by a longitudinal distance, is used for the coherence of longitudinal wind speed component between points separated by a lateral distance, and 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 then and and according to [Kristensen & Jensen] and can be set to and respectively.
where is the angle between wind direction and a line between the two turbine and turbine . Furthermore, the delay from turbine to turbine is needed, which can be calculated as where is the distance between turbines, the angle between turbines, and is the mean wind speed. Now the cross spectrum between turbines can be found from where is the cross spectrum between turbine and turbine , is the coherence, is the auto spectrum at turbine , is the auto spectrum at turbine , and is the time delay from turbine to turbine . ## Wake effectsIt 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.]. Expressions for wake deficit, center and expansion was developed in [Frandsen et al., Jensen]. The wake center, expansion and deficit at a given point downwind from a turbine at time is defined by the wind field at the turbine and its coefficient of thrust at the time of tracer release, , where is the longitudinal distance between the upwind turbine and . ## Wake Expansion
where is the rotor radius.
where is the rotor diameter, and are parameters which here are set to and , respectively. Furthermore, ## Wake CenterThe wake center is computed as a passive tracer, such that the center at time is a function of the center at time and the average lateral wind speed, over the wake area, where is the wake center at time , is the wake area distance from the turbine at time and is the average lateral wind speed over the wake area. ## Wake Deficit
where is the wind deficit, , is the wind speed distance down wind, is the ambient wind speed. is the induction factor of turbine at time and 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 According to [Frandsen et al.], the above expressions can be approximated by where is the coefficient of thrust of turbine at time .
where is the wind speed distance down wind, is the ambient wind speed, is the thrust coefficient, is the rotor diameter, and is the wake diameter distance down wind. ## Wake Merging
where contains the indices of all turbines where the point is in its wake area, , with being the longitudinal position of the 'th turbine.
where is the deficit at turbine , is the wake diameter at tubine , is the diameter at turbine , is the deficit at turbine , is the thrust coefficient of turbine and is the diameter of the rotor. ## Model of Added Turbulence Intensity
where is the added standard deviation in the wind field at the th WT which is in wake; is the effective wind speed at the th WT; is the spacing in rotor diameters between the wake generating WT and the WT in wake; is the thrust coefficient of the wake generating WT.
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