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Normalized W(F l ,R l )
1
0.8
0.6
0.4
0.2
F l = 50 m F l = 125 m F l = 200 m Pulsed
0
0 50 100 150
Range R (m)
l
200 250 300
Figure 140: Normalized range weighting function W (Fℓ,Rℓ) for a CW lidar similar to the
ZephIR 300 at three different focus distances along with the range weighting function for a
pulsed lidar similar to the Windcube WLS7.
where the velocity vector u = [u,v,w] is defined such that the u velocity is in the x direction,
v is in the y direction, and w is in the z direction (see Fig. 133). Hub height is represented
by h. The minus signs appear in Eq. (234) because the measured line-of-sight velocity is the
projection of the velocity vector onto the direction from the measurement point to the lidar,
opposite from the lidar look direction. W (Fℓ,Rℓ) is the range weighting function with focus
distance Fℓ and range along the lidar beam Rℓ as arguments.
The range weighting function for a CW lidar becomes broader as the focus distance increases
(the Full width at half maximum of W (Fℓ,Rℓ) scales with the square of the focus
distance Fℓ) (Frehlich and Kavaya 1991; Slinger and Harris 2012). W (Fℓ,Rℓ) is plotted as
a function of range along the lidar beam for three different focus distances in Fig. 140. Also
included in Fig. 140 is the range weighting function for a pulsed lidar similar to the Windcube
WLS7 (Frehlich et al. 2006; Mikkelsen 2009). Note that the pulsed lidar range weighting
function does not vary with measurement distance. Additional details on CW and pulsed
range weighting functions can be found in Simley et al. (2013c).
10.5.2 Geometry Errors
As mentioned in Section 10.4.1, the u component of the wind speed is of interest for wind
turbine control. An estimate of the u component is formed by dividing the line-of-sight
measurement in Eq. (234) by −ℓx, yielding
û = − 1
ℓx uwt,los
= uwt + ℓy
ℓx vwt + ℓz
ℓx wwt.
(236)
As revealed in Eq. (236), detection of the v and w wind speed components causes a degradation
in the estimate of the u component. For a fixed scan radius r, short preview distances d
will result in large measurement angles, causing significant measurement error due to detection
of the v and w components (Simley et al., 2013c). As preview distance increases, the ℓy
and ℓz
ℓx
terms decrease and error caused by measurement geometry will decrease. Figure 141
contains coherence curves for a hub-mounted lidar without range weighting measuring at a
scan radius of r = 41 m for different preview distances. The coherence curves represent the
correlation between the lidar measurement and the wind speed at a single point downstream
of the measurement, so that only geometry errors are included. As preview distance increases,
the measurement coherence increases due to less contribution from the v and w components
in Eq. (236).
204 DTU Wind Energy-E-Report-0029(EN)
ℓx