Publishers version - DTU Orbit

ΣVar(y) FF+FB /ΣVar(y) FB
0.55
0.5
0.45
0.4
0.35
0.3
0.25
Sum of Variance Components
r = 35 m (55% R), w/ filter
r = 35 m (55% R), w/o filter
r = 41 m (65% R), w/ filter
r = 41 m (65% R), w/o filter
r = 47 m (75% R), w/ filter
r = 47 m (75% R), w/o filter
0.2
40 60 80 100 120 140 160 180
d (m)
Figure 147: The sum of the variance terms of blade root bending moment Var(y0) +
1
2
Var(yv)+ 1
2 Var(yh) with feedforward control normalized by the sum of the variance terms
with feedback only for a few scan radii r vs. preview distance d. The solid curves represent
the variance with minimum mean square error prefiltering and the dashed curves show the
performance without prefiltering (Hpre = 1). Measurement coherence is calculated using the
ZephIR CW lidar model. The transfer functions are computed for the NREL 5-MW turbine
at rated wind speed 13 m/s.
variances in Eq. (241) can be calculated using
Var(y0) =
Var(yv) =
Var(yh) =
∞
0
∞
0
∞
0
|Ty0uhh (f)|2 Suhhuhh (f) 1−γ 2 uhhûhh (f) df, (242a)
|Tyv∆v(f)| 2
S∆v∆v (f) 1−γ 2
∆v ˆ
(f) df, (242b)
∆v
|Tyh∆h (f)|2 S∆h∆h (f)
1−γ 2
∆h ˆ
(f) df. (242c)
∆h
In order to find the optimal r and d that satisfy Eq. (241), the variance terms in Eq. (242)
are calculated for a number of scan radii and preview distances. Figure 147 shows the sum of
the variance terms for scan radii r = 35 m (55% blade span), r = 41 m (65% blade span),
and r = 47 m (75% blade span) plotted as a function of preview distance d. The variance
is normalized by the sum of the variance terms without feedforward control (calculated using
Eq. (220)). The solid curves indicate the variance achieved with the ideal feedforward controller
and the optimal prefilter while the dashed curves represent the variance as a result of
ideal feedforward control without prefiltering. The wind field model is comprised of the von
Karman spectral model with wind evolution described in Section 10.5.3.
The results in Fig. 147 reveal that the variance of blade root bending moment is minimized
using optimal measurement prefiltering with a scan radius r = 41 m (65% blade span) and
a preview distance d = 120 m (almost one rotor diameter). This lidar geometry achieves
a bending moment variance that is less than 25% of the value when feedforward control is
not used. Without prefiltering, the variance is roughly 28% of the value with feedback only.
Although the analysis in Fig. 139 showed that measurements at 70% blade span provide
the highest correlation with blade effective wind speed, a scan radius of 65% blade span is
optimal for this scenario, largely because of the implicit penalty on large scan radii caused
by lidar geometry errors. Measurement performance suffers for preview distances less than
120 m because of additional errors caused by large measurement angles, as explained in
Section 10.5.2. For preview distances beyond 120 m, on the other hand, the severity of wind
evolution increases, as discussed in Section 10.5.3, causing measurement coherence to drop.
210 DTU Wind Energy-E-Report-0029(EN)