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Magnitude Squared
10 8
10 7
10 6
10 5
10 4
|H |
y0 ,u
hh
2
|H |
yv ,Δ
v
2
|H |
yh ,Δ
h
2
Wind to Blade Root Bending Moment
10 −2
10 −1
Frequency (Hz)
Figure 146: Transfer functions from hub height wind speed uhh to the collective component
of flapwise blade root bending moment y0 as well as the transfer functions from vertical
shear ∆v and horizontal shear ∆h to the vertical component yv and horizontal component
yh of blade root bending moment. The transfer functions correspond to the NREL 5-MW
turbine model at above-rated wind speed 13 m/s including feedback control of blade pitch
and generator torque.
torque feedback control (Jonkman et al., 2009) using rotor speed measurements.
Alidarmeasurementscenarioisproposedthatusesthreerotatingmeasurementstoestimate
the three rotating blade effective wind speeds û1, û2, and û3, using Eq. (236). The collective
and shear components of the wind are then estimated from the three lidar measurements
using
⎡
⎡ ⎤
ûhh ⎢
⎣ ˆ∆v ⎦ = ⎢
⎣
ˆ∆h
1
3
1
3
4
3U cos(ψ) 4
3U cosψ + 2π
3
− 4 4
3U sin(ψ) −3U sinψ + 2π
3
1
3
4
3U cosψ + 4π
3
4 −3U sinψ + 4π
3
10 0
⎤
⎡
⎥
⎥⎣
⎦
û1
û2
û3
⎤
⎦. (240)
A measurement scenario based on the lidar scenario in Fig. 133 is investigated for the purpose
of minimizing the variance of the three blade root bending moment components using lidarbased
estimates of the collective and shear wind speed terms and ideal feedforward control
(Eq. (216)). Since the three lidars rotate at the rotor speed to provide measurements of blade
effective wind speed, the only design variables are the scan radius r and preview distance d.
10.6.1 Optimizing the Measurement Scenario
The design problem is formulated to minimize the variance of the blade root bending moment
felt by the three blades. The measurement geometry is to be chosen to minimize the sum of
the variances of the three bending moment components:
(r ∗ ,d ∗ ) = argmin
r,d
Var(y0)+ 1 1
Var(yv)+
2 2 Var(yh)
. (241)
As a blade rotates through the wind field, it experiences the vertical and horizontal components
of bending moment scaled by the cosine and sine of the azimuth angle ψ, respectively.
Since variance is equivalent to the mean square value over time and thus azimuth angle, and
the mean square value of a sinusoid is 1
2
, the factor of 1
2
appears in front of the vertical
and horizontal terms in Eq. (241). Using Eq. (219) from Section 10.2.1 for calculating variance
based on measurement coherence with feedforward control and optimal prefiltering, the
DTU Wind Energy-E-Report-0029(EN) 209