Wind field: unsteady wake simulations

The wind field used for the test cases of this chapter reproduces the wake of a wind turbine with a rotor diameterDof 62 m and a hub-heightz_HH of 62 m. To run the simulations, an unsteady wind field is calculated with the Parallelised LES Model (PALM) (Raasch et al., 2001). An actuator line approach (Troldborg, 2008) is included to simulate the wind turbine. For the experiments, a ten-minute wind field is generated on a 4 m resolved grid with a temporal resolution of 0.4 s.

The boundary conditions were chosen in order to simulate a neutral atmosphere with a roughness lengthz₀=0.05 m and a friction velocityu_∗=0.5 m s⁻¹. Considering the hub-height, this setting generated an average inflow wind speeduHH=9 m s⁻¹, a wind direction of 270° and a turbulence intensityT I of 8 %.

The profiles of the wind speed deficit in the wake used in the following exercises were calculated sub-tracting the streamwise wind componentudownstream of the wind turbine from the corresponding mean values on the cross-section at the inlet of the LES domain.

2.4.2 Measuring the average vertical profile of the wind speed in the wake of a wind turbine

The features of lidar measurements (volume averaging of the laser beam and the projection of the wind speed vector on its radial wind component) are a source of error in non-homogeneous flows (Clifton et al., 2018). Wind turbine wakes belong to this category of flow. In this section, simulations of lidar wind speed measurements in the wake of a wind turbine show how the effects of the lidar measuring principle can be influenced by varying the experimental approach. The results provide also an indication about which strategy should be preferred to minimise the errors.

Simulation setup

The objective of the first simulated experiment is to measure the vertical profile of the mean wind velocity in the wake of a wind turbine. The wind field described in Section 2.4.1 is the virtual environment of the simulations. The profile includes nine heights from 40 m to 120 m every 10 m and is aligned to the centre line of the turbine rotor at a downstream distance of 2.5 D.

Four cases are considered in the simulations:

- ReferenceThe three wind speed componentsu,v andwalong the streamwise, cross-stream and vertical directionsx,y andz, respectively, are interpolated at the target points from the simulated wind field.

22 2.4. LIDAR SIMULATIONS OF WAKE MEASUREMENTS

Figure 2.3: Possible approaches for lidar measurements of a wind turbine wake profile at the reference points: Doppler beam swinging (DBS) (a); concurring, synchronised measurements (multi-lidar) with lidars near to the target (b) and further away from it (c).

- Doppler beam swinging (DBS) A lidar is installed on the ground beneath the target points. The wind vector is evaluated by applying Eq. 2.6 to the radial wind speed component measured on conical scans composed by four radial wind speed components. The elevation angle of the beams is 60°; the azimuth angle varies with 90° steps and is alternatively parallel touandv. This configuration minimises the heterogeneity of the wind field sampled by the four beams. The scanning pattern is plotted in Fig. 2.3a.

- Multi-lidar 1 Three lidars are installed near the target profile and perform concurrent, synchronised measurements with high elevation angles: Two lidars are∼0.67 D downstream of the profile, at opposite sides, separated by 2 D in the cross-stream direction; the third lidar is between the turbine and the profile,∼1.83 D downstream of the former and∼0.64 D upstream of the latter. The wind vector is calculated with the matrix inversion method of Eq. 2.9. Figure 2.3b represents schematically this measurement strategy.

- Multi-lidar 2 Three lidars are installed near the turbine and, as in the previous case, perform concurrent, synchronised measurements with low elevation angles: Two lidars are at opposite sides of the turbine separated by 2 D in the cross-stream direction; the third lidar is on the nacelle of the turbine. The wind vector is calculated again with the matrix inversion method of Eq. 2.9. The trajectory of these measurements is illustrated in Fig. 2.3c.

To isolate and study the effect of the volume and spatial average, the measurement of the full profile is simulated every 2.3 s for all the four cases.

Results

The four approaches are compared in relation to the vertical profiles of mean components of the wind velocity vector (see Fig. 2.4). In this respect, Fig. 2.5 displays the deviations

∆u = ulidar−uref

∆v = vlidar−vref

∆w = w_lidar−w_ref

(2.13)

where the reference and the lidar simulated wind velocity components are indicated with the subscriptsref andlidar, respectively. The statistics of these deviations are collected in Table 2.1 to provide an overview of the results.

Lidar simulations for the design of wake measurement campaigns 23

When the streamwise wind component is addressed, the results of the simulations for the DBS and Multi-lidar 1 configurations look very similar (see Fig. 2.4). The former has a slightly smaller deviation from the reference case. Differently, simulations with the approach Multi-lidar 2 have results much closer to the reference. These points can be better quantified in Fig. 2.5a, where the deviation of the different approaches from the reference is represented.

The orientation and the position of the sample volumes in the simulated measurements are a reasonable explanation of the different results provided by the considered approaches. These two factors have a relevant influence on the volume average in wind fields with high spatial gradients as wakes. In the DBS, the wind field is calculated from radial wind speed components averaged over sample volumes centred at different positions in space (e.g. at opposite directions the volumes have a∼104 m distance at the heightz∼90 m). This is not the case for the multi-lidar measurements, in which the sample volumes of the radial wind speed components concur at the same position eliminating the influence of the spatial averaging and reducing the deviation from the reference.

Another aspect to consider is that, in the multi-lidar configurations, the vertical extension of the sample volumes depends on the elevation angle, while the sample volumes of the DBS approach have it fixed. This fact is particularly important considering that significant vertical gradients of the streamwise wind component are expected at the target points. In the DBS and Multi-lidar 1 simulations, the vertical extension of the sample volume is very similar (∼26 m for the first one and in average∼25 m for the second one ); differently, the vertical extension of the sample volumes in the Multi-lidar 2 simulations is less than a half of the one in the other two cases (∼11 m in average ). In this respect, the aforementioned similar deviation from the reference could be due to the vertical extension of the probe volume.

Concerning the cross-stream wind speed component v (see Fig. 2.4b and Fig. 2.5b), the two multi-lidar configurations can resolve very well the variation from positive to negative values given by the rotation of the wake. This is not the case for the simulation of the DBS measurements, which capture this trend only at the lower heights, nevertheless with a larger deviation from the reference. In fact,vreaches fast the free-stream level (0 m s⁻¹) without changing sign because the sample volumes of the cross-stream beams of the DBS cone move towards and eventually cross the edges of the wake for increasing heights.

Figure 2.4: Measurements of a wind turbine mean wake profile simulated with the approaches of Fig. 2.3 and compared to the corresponding reference values.

24 2.4. LIDAR SIMULATIONS OF WAKE MEASUREMENTS

Figure 2.5: Absolute deviation from the reference of the wind speed components calculated implementing the approaches of Fig. 2.3 into lidar wake measurements simulations.

Table 2.1: Statistics of the deviation from the reference of the wind speed components calculated from lidar measurements simulated for different approaches within a wind turbine wake.

|∆u|[m s⁻¹] ∆u[m s⁻¹] |∆v|[m s⁻¹] ∆v[m s⁻¹] avg st. dev min max avg st. dev min max DBS 0.32 0.15 -0.54 0.43 0.30 0.28 -0.76 0.00 Multi-lidar 1 0.22 0.13 -0.40 0.39 0.03 0.02 -0.04 0.07 Multi-lidar 2 0.02 0.01 -0.02 0.05 0.03 0.02 -0.03 0.07

Approach |∆w|[m s⁻¹] ∆w[m s⁻¹] avg st. dev min max DBS 0.05 0.04 -0.12 0.06 Multi-lidar 1 0.02 0.02 -0.02 0.06 Multi-lidar 2 0.15 0.12 -0.31 0.29

The profiles of the vertical wind speed component w in Fig. 2.4b provide another example of how the orientation and the position of lidar the sample volume affects the reconstruction of the wind vector. In this case, the Multi-lidar 2 simulations have the worst match with the reference because the difference between the elevations of the concurring beams is relatively small (especially above hub-height) and the system matrixMe_R of the wind field reconstruction model is almost singular. In relation to the Multi-lidar 2 approach, the DBS has lower deviations from the reference, but Fig. 2.4 shows that it still does not correlate with the reference very well. The Multi-lidar 1 configuration mitigates best the limits of lidar measurements for the vertical wind component and has results very close to the reference.

Given the low contribution of the vertical wind component to the line of sight wind speed measured by the lidar at very low elevations in the configuration Multi-lidar 2, it is reasonable to assume that similar results in term of precision could be achieved also with two lidars only.

Lidar simulations for the design of wake measurement campaigns 25

Conclusions

The measurement principles of wind lidar limit their use in non-homogeneous flows. Three different measurement approaches have been investigated with lidar simulations in the wake of a wind turbine to find out their advantages and disadvantages.

The common DBS approach has the highest average absolute error with respect to the reference because of the average over steep volumes with strong spatial variation of the wind field. The error is also a consequence of combining radial measurements in the free-flow and in the wake.

Concurrent sample volumes have better performances which vary depending on the inclination of the radial direction. The horizontal wind speed can be estimated better from small elevation angles. However, larger deviations of the vertical wind component from the reference are observed for such configurations.

From these results, a multi-lidar configuration with the devices located at a large distance from the target points should be chosen for measurements of the vertical profile of the average horizontal wind speed at the centre of the wake. In this situation, two devices could probably achieve similar results as three and might reduce the campaign efforts.