4.2 Static Wind Field Reconstruction
4.2.1 Application to Ground Based Lidars
In this first example the SWE scanning lidar system (see Appendix B) was installed at the Risø Campus tilted by ΘL = 25 deg, scanning the wind close to a met mast with a 3x3 grid trajectory on 5 horizontal planes within 2 s, see Figure 4.9. The center points of the third measurement plane were located close to the ultrasonic anemometers installed on the met mast. For the following investigations, all 10-minute-mean values are used, including low wind speeds and wind directions orthogonal to the main measurement direction.
More details on the experimental setup can be found in [79].
4.2 Static Wind Field Reconstruction 63
Reconstruction in Flat Terrain
If the tilt angle ΘL is known, the wind can be reconstructed in the I-coordinate system, assuming that the wind is homogeneous in planes parallel to the ground. The measurement points are transformed from the L to the I-coordinate system by (3.29). Furthermore, the origin of the L- and the I-coordinate system are set equal.
In a first step, the sonic longitudinal and the vertical wind speed measurements of the center sonic anemometer is compared to the lidar data reconstructed from the two measurement points next to the anemometer, see dots in Figure 4.9. Similar to (3.16), uI and vI is assumed to be constant in each measurement point i, but here wI is neglected:
ui,I vi,I
wi,I
=
uI vI 0
. (4.17)
This wind model of homogeneous flow is combined with the lidar measurement model (3.31) and the line-of-sight wind speeds of the two points with the distancesrL1 andrL2 from the lidar system and next to the center anemometer are simulated by the set of linear equations
"
vlos,1 vlos,2
#
| {z }
m
=
"x1,I
rL1 y1,I
rL1 x2,I
rL2 y2,I
rL2
#
| {z }
A
"
uI vI
#
| {z }
s
. (4.18)
The two-dimensional wind can then be reconstructed simply by a matrix inversion
s=A−1m. (4.19)
This example shows why observations can be considered to be inverse to simulations. The notation as a linear equation system shows that the reconstruction can be done independently of the shape of the scan. Circular scans normally used for lidar measurements are not necessary.
However, matrix A needs to be invertible.
The results of the investigation are shown in Figure 4.10 (top) and Table 4.2. A higher cor-relation between the sonic anemometer and the lidar system is achieved for the longitudinal componentuI compared to the lateral component vI. Due to the values in the inverted matrix A−1, measurement errors in the line-of-sight wind speeds have more effect on vI. An appro-priate measure for robustness can be defined in the following way: The condition number of A describes the worst case factor which transfers relative errors from measurement vector m to the vector of the searched variables s. This approach can be used to optimize the setup of lidar measurements: in this case the condition number could have been reduced from 5.23 to 1, by measuring closer to the met mast, setting x1,I =x2,I =y1,I =−y2,I.
wL,I [m/s]
vL,I [m/s]
uS,I/vS,I/wS,I[m/s]
uL,I [m/s]
uS,I/vS,I/wS,I[m/s]
−10 0 10
−10 0 10
−10 0 10
−10 0 10
−10 0 10
Figure 4.10:Regression between lidar and sonic anemometer measurements in the inertial coordinate system. For the 2D homogeneous flow model (top) and the 3D homogeneous flow model (bottom).
In a second step, the wind model of three-dimensional homogeneous flow (3.16) used in the DBS technique is considered. A third measurement is added to have the number of linear equations equal to the number of unknowns:
vlos,1 vlos,2
vlos,3
| {z }
m
=
x1,I
rL1 y1,I
rL1 z1,I
rL1 x2,I
rL2 y2,I
rL2 z2,I
rL2 x3,I
rL3
y3,I rL3
z3,I rL3
| {z }
A
uI vI wI
| {z }
s
. (4.20)
If the measurement point close to the sonic anemometer is added, the matrixAis not invertible, because the xI and zI of all three points are equal and thus the first and third column are linearly dependent. This can be avoided and observability can be restored by choosing the measurement point above the sonic anemometer. However, the wI component still cannot be observed satisfactorily applying the inversion (4.19), see Figure 4.10 (bottom) and Table 4.2.
Furthermore, the reconstruction of the uI component is negatively affected.
The insufficient results can be assigned to the uncertainties in the wind model. Due to the vertical shear, the assumption of a homogeneous wind vector in all measurement points is unrealistic. Simply neglecting the vertical wind component yields better results for flat terrain.
4.2 Static Wind Field Reconstruction 65
Table 4.2: Linear regression between sonic and reconstructed lidar measurements in the inertial coordinate system.
2D homogeneous flow 3D homogeneous flow (4.17), 2 points (3.16), 3 points uI vI wI uI vI wI slope [-] 1.000 0.933 ∞ 1.112 0.933 -0.034 offset [m/s] 0.006 0.043 -0.076 -0.039 0.043 -0.068 R2 [-] 0.998 0.975 0.000 0.970 0.975 0.207
Reconstruction in Complex Terrain
The investigation above used the knowledge of the tilt angle ΘL and assumed that the flow is homogeneous in planes parallel to the ground. At least two measurement points in each plane are necessary to reconstruct the two-dimensional wind vector in that plane. In a more complex terrain this assumption is not always useful, see Section 4.1.2: the wind will be parallel to a linear slope at lower heights, but parallel to the surface of the earth at higher heights. In a further investigation, the knowledge of the tilt angle is not used and the wind is reconstructed in the L-coordinate system, by including the vertical wind shear in the wind model. Using more than one focus distance to distinguish between shears and inflow angles was proposed in [38] and tested in simulations in [4], ignoring the drift of the shear due to the flow angle.
Here, the 3D inhomogeneous flow model (3.17) is used, which parameterizes the wind speed vector by the effective wind speed v0, by the horizontal and the vertical shears δh and δv, and by the horizontal and the vertical inflow angles αh and αv. For a given set of the wind characteristics v0, αh, αv, δh, δv, the lidar measurement in each point i can be simulated by combining the wind model (3.17) with the lidar measurement model (3.31):
vlos,i = xi,W
rLi (v0+δhyi,W +δvzi,W) with
xi,W
yi,W
zi,W
=TWI(αh, αv)
xi,L
yi,L
zi,L
(4.21)
In this case TWI is used, because the L-system coincides with the I-system. For several measurement points this forms a nonlinear set of equations and an inversion to obtain the wind characteristics from a given set of line-of-sight wind speeds is not directly possible. Here, a numerical inversion for the nonlinear equations can be achieved by solving the least squares minimization problem fornP measurement points
v0,αhmin,αv,δh,δv
nP
X
i=1
vlos,i− xi,W
rLi (v0+δhyi,W+δvzi,W)2
. (4.22)
wL,L [m/s]
vL,L [m/s]
uS,I/vS,I/wS,I[m/s]
uL,L [m/s]
uS,I/vS,I/wS,I[m/s]
−10 0 10
−10 0 10
−10 0 10
−10 0 10
−10 0 10
Figure 4.11: Regression between lidar and sonic anemometer measurements in the lidar coordinate system. For the 3D homogeneous flow model (top) and the 3D inhomogeneous flow model (bottom).
Finally, the wind vector in lidar coordinates can be calculated with the found wind character-istics and the flow model (3.17).
The linear model (3.16), which only accounts for the sloped inflow, and the nonlinear model (3.17) neglecting δh are applied to the data using nP = 12 points (see Figure 4.9) and the least squares method. The coefficient of determination can be improved significantly for thewI component, see Table 4.3. Figure 4.11 shows that it is possible to enhance the measurement of the 3D wind vector in the presence of vertical shear, but it is necessary to investigate under which conditions observability is given and how higher robustness can be obtained.
Table 4.3: Linear regression between sonic and reconstructed lidar measurements in the inertial coordinate system.
3D homogeneous flow 3D inhomogeneous flow (3.16), 12 points (3.17), 12 points uL vL wL uL vL wL slope [-] 0.969 0.900 1.059 0.986 0.929 0.968 offset [m/s] 0.021 0.008 -0.339 0.030 0.020 -0.241 R2 [-] 0.996 0.974 0.699 0.998 0.971 0.951
4.2 Static Wind Field Reconstruction 67