Engineering wake models - commonalities
The engineering calculation of wake effects employs a systematic approach where each turbine is considered in turn in order of increasing axial displacement downstream. By this method, the first turbine considered is not subjected to wake effects. The first turbine’s incident wind speed, the thrust coefficient and the tip-speed ratio are calculated. Its wake is then modelled, as described below, and the parameters which describe its wake are stored. The effect of this wake on all turbines downstream can then be modelled. If any of the downstream turbines fall within the wake of this turbine, the velocity and turbulence incident on these turbines can be determined, solely due to this upstream turbine being considered. As the calculation progresses through the turbines, the incident wind speed on the turbine is the sum of the wake and topographic effects as described in the energy calculation section.
There are three engineering wake models available within WindFarmer;
Modified PARK model based on the method presented by Jensen and Katic [1].
Eddy Viscosity model based on work conducted by Ainslie [2] [3]
Due to the complexity of the wake directly behind the rotor, all models are initiated from two diameters (2D) downstream. This is assumed to be the distance where pressure gradients no longer dominate the flow. If a turbine is within this limit, the program resets the axial distance offset to a value of two rotor diameters.
For a comparison of the model results with that of real wind farms please refer to the WindFarmer White Paper.
Wake superposition
For each turbine downstream of the turbine under consideration, the program determines the axial displacement assuming rotational symmetry of the wake. The wake width and the wind speed at this displacement are then calculated. The turbines affected by the wake may not be totally in the wake so the percentage cover of the turbine’s rotor in the wake is determined. If the whole rotor is within the wake, then the turbine wind speed is set as Uw. If some of the rotor is outside the wake, the wind speed at the turbine is the sum of Uw and the upstream velocity of the turbine creating the wake multiplied by the relative percentages of rotor cover.
If the turbine under consideration is in the wake of another turbine, the initial wake velocity deficit is corrected from the incident rotor wind speed to the free stream wind speed. This correction is necessary in order to ensure that at distances far downstream, the wake wind speed will recover to the free stream value rather than that incident on the rotor. Therefore, the initial centre line velocity Uwi is scaled by the ratio of average influx velocity Ui and free upstream wind velocity according to the following formula:
$$U_{w} \ = \ \left( \frac{U_{0}}{U_{i}} \right) U_{wi} $$
To combine the wakes of two wind turbines onto a third turbine the overall wake effect is taken as the largest wind speed deficit and other smaller wake effects are neglected. This methodology is based on the results of the assessment of measured data from a number of wind farms.
Furthermore, where multiple turbine types are present, the model takes into account any variation in hub height and rotor diameter.
Wake profile integration
The incident wind speed at the target turbine is the rotor averaged wind speed.
Where:
D: is the diameter of the target turbine (in meters)
Bwake: is the width of the wake from the upstream turbine at the target turbine (in diameters of target turbine)
offset: distance of the target turbine from the centre line of the wake (in meters)
rad: radial distance from the centre line of wake (in diameters of target turbine)
U0: free stream wind speed (in m/s)
Uc: waked wind speed at the centre line at the target turbine location (in m/s)
Uw: rotor averaged wind speed at the target turbine (in m/s)
Considering that the wake at the target turbine has a gaussian shape, Uw can be calculated by:
$$ U_{w}=\int^{rad_{1}}_{rad{0}} U_{0}+(U_{c}-U_{0})e^{-3.56\left( \frac{rad}{B_{wake}} \right)^2} drad$$
With rad0 to rad1 being the rotor diameter. WindFarmer has two methods to solve the integral and calculate the rotor averaged wind speed: using the Simpsons numerical method or an analytical solution.
Simpsons method
This numerical method divides the region [rad0, rad1] into evenly spaced slices and sums up the quadrature of the several slices to obtain the solution.
The simpsons rule uses a higher accuracy quadrature by carrying out the integration for every pair of slices together.
This method is used in WindFarmer Analyst 1.2.2.1. It is available in later versions for continuity but it is no longer the default option.
Analytical method
From the above equation we can obtain the following expression:
$$U_{w} = U_{0} + \frac{(U_{c}-U_{0})}{A}\frac{\sqrt{\pi}}{2}(erf(A \cdot rad_{B})-erf(A \cdot rad_{A})), -B_{wake} \leq rad_{A} \leq rad_{B} \leq B_{wake}$$
Where:
$$A = \frac{\sqrt{3.56}}{B_{wake}}$$
and erf(z) is the gaussian error function. To obtain a solution to this function, WindFarmer uses a polynomial approximation.
This method is faster and produces results close to the Simpsons method, it is not available in the classic energy calculation.