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Coherence
1
0.8
0.6
0.4
0.2
0
10 −3
Estimation of Blade Effective Wind Speeds, r = 41 m
10 −2
10 −1
Frequency (Hz)
10 0
d = 30 m
d = 90 m
d = 150 m
d = 30 m, w/ Lidar
d = 90 m, w/ Lidar
d = 150 m, w/ Lidar
Figure 144: Coherence between lidar measurements and blade effective wind speed for a hubmounted
lidar with a scan radius of 41 m for three different preview distances. Geometry
errors and wind evolution are included. The solid curves represent measurement coherence
without lidar range weighting while the dashed curves include range weighting.
10.6 Lidar Measurement Example: Hub Height and Shear
Components
In the previous section, the measurement quality between a lidar measurement and blade
effective wind speed at a fixed location was investigated. In reality, the blades of a turbine
rotate through the wind field and it is the total contribution from all blades comprising the
rotor that affects the turbine. For a three-bladed turbine, such as the NREL 5-MW model, the
wind speeds felt bythe rotating blades can be described as the sum ofacollectivecomponent,
which all three blades experience; a linear vertical shear component, which describes the
gradientofthewindspeedsintheverticalz direction;andalinearhorizontalshearcomponent,
which describes the wind speed gradient in the horizontal y direction. These collective and
shear components describe the u component of wind speed in the rotor plane.
The three blade effective wind speeds, using the weighting function Wb(r) defined in
Eq. (232), can be written in terms of collective and shear components as
⎡ ⎤ ⎡
ublade,1 1 Bcos(ψ) −Bsin(ψ)
⎣ublade,2⎦
= ⎣ 1 Bcos
ublade,3
ψ + 2π
2π
3 −Bsin ψ + 3
1 Bcos ψ + 4π
⎤⎡
⎤
uhh
⎦⎣∆v
⎦
(237)
4π
3 −Bsin ψ + 3
∆h
where ψ is the azimuth angle in the rotor plane corresponding to blade number one, uhh is
the wind speed at hub height, ∆v is the difference in the wind speed across the rotor disk in
the vertical direction normalized by the mean wind speed U, ∆h is the difference in the wind
speed across the rotor in the horizontal direction normalized by the mean wind speed, and
B = U
2R
R
0
Wb(r)rdr. (238)
Azimuth angle is defined such that ψ = 0◦ corresponds to the top of rotation (in the z
direction) and ψ = 90◦ corresponds to the left side of the rotor disk when facing upwind
(in the −y direction). By inverting Eq. (237), the collective and shear components can be
described as a function of the three blade effective wind speeds as
⎡
⎣
uhh
∆v
∆h
⎤
⎡
⎦ = ⎣
− 2
3B
1
3
1
3
2
3B cos(ψ) 2
3B cosψ + 2π
2
3 3B cosψ + 4π
3
2 sin(ψ) −3B sinψ + 2π
2
3 −3B sinψ + 4π
3
1
3
⎤⎡
⎦⎣
ublade,1
ublade,2
ublade,3
⎤
⎦. (239)
Using the frequency domain description of the wind field with defined power spectra and
spatial coherence functions, the spectrum of the wind experienced by a blade rotating through
the wind field can be calculated, as explained in Simley and Pao (2013a). The method in
DTU Wind Energy-E-Report-0029(EN) 207