Geoscience Reference
In-Depth Information
The factor a in (
6.30
), (
6.33
) and (
6.34
) depends on the surface roughness and
the thermal stratification of the boundary layer via (
6.11
). This solution is in the
time domain. It can be converted into the space domain by assuming an average
wind speed over the wake.
The upper frame in Fig.
6.5
shows wake lengths as function of surface
roughness for neutral stability (h/L* = 0) by plotting the third power of R
n
from
(
6.34
). If we define the distance necessary for a recovery of the available power to
95 % of its undisturbed value upstream of the park as wake length, then we see a
wake length of 4 km for rough land surfaces and a wake length of about 18 km for
smooth sea surfaces. Figure
6.5
has been produced for the same park parameters as
Fig.
6.3
. Actually, the results from Fig.
6.3
serve as left boundary conditions for
Fig.
6.5
. The lower frame of Fig.
6.5
demonstrates the strong influence of atmo-
spheric stability on the wake length for an offshore wind park over a smooth sea
surface (z
0
= 0.0001 m). Taking once again the 95 % criterion, the wake length
for very unstable atmospheric conditions is still about 10 km. For very stable
conditions, the wake length is even longer than 30 km. Such long wakes have been
confirmed from satellite observations (Christiansen and Hasager
2005
).
6.4 Application of the Analytical Model with FINO1
Stability Data
The application of the above analytical model to a real wind park needs the
knowledge of the frequency distribution of atmospheric stabilities at the site of the
wind park. We give here an example by using the distribution measured at 80 m
height at the mast FINO1 in the German Bight for the years 2005 and 2006.
Figure
6.6
shows this distribution for the range -2 B z/L
*
B 2. 91.16 % of all
data fall into this range. The highest frequency occurs for the bin -0.15 B z/
L
*
B-0.05. The median of the full distribution is at z/L
*
=-0.11, the median of
the range shown in Fig.
6.6
is z/L
*
=-0.07. Now the above equations for the
reduction of wind speed in the park interior (
6.26
) and the wake length (
6.34
) are
solved for all 41 bins shown in Fig.
6.6
and the resulting values for R
t
and R
n
are
multiplied with the respective frequencies from Fig.
6.6
.
Rebinning the resulting R
t
and R
n
values leads to the distributions shown in
Figs.
6.7
and
6.8
. The top frame in Fig.
6.7
shows the distribution of wind speed
reductions at hub height in the park interior. The most frequent speed reduction R
t
is 0.95, the median is 0.93 and the weighted mean is 0.87. The 90th percentile is
observed at 0.73 and the 95th percentile at 0.65. The lower frame of Fig.
6.7
gives
the resulting reductions in power yield. The most frequent power yield reduction is
0.83, the median is 0.80 and the weighted mean is 0.70. The 90th percentile is
observed at 0.37 and the 95th percentile at 0.24.
Figure
6.8
displays the respective distribution of the wake length. Here, the
wake length has been defined as above as the distance where the power yields have
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