Environmental Engineering Reference
In-Depth Information
3.11 WIND SPEED DISTRIBUTIONS
If data are not available, then the wind speeds can be predicted from one or two parameters. A num-
ber of different distributions have been tried, but only two are in general use, Rayleigh and Weibull
distributions. These distributions give poor estimates of power for low mean wind speed situations.
At higher wind speeds, both give adequate estimates for many locations; however, for those regions
with steady winds, such as the trade winds, the Weibull distribution is better. The Rayleigh distribu-
tion is simpler because it depends only on the mean wind speed.
The Rayleigh distribution is
§
¶
2
¤
¦
¥
³
µ
´
$
P
v
v
P
4
v
v
¨
¨
·
·
Fv
()
v
exp
(3.16)
2
2
a
a
©
¸
where
F
(
v
) frequency of occurrence associated with each wind speed,
v
, which is at the center of
Δ
v
; Δ
v
width of class or bin; and
v
a
average wind speed (same as mean wind speed)
The wind speed histogram for 1 year can be calculated from 8,760 *
F
(
v
).
The Rayleigh frequency is calculated for two different values,
v
3 m/s and
v
9 m/s, with
v
a
8 m/s
and Δ
v
2 m/s:
§
2
¶
¤
¦
¥
p
3
8
p
3
8
³
µ
´
¨
¨
·
·
0.11
F
(3)
2
exp
0.147
e
0.132
2
2
4
©
¸
§
¶
2
p
9
8
p
¤
¦
¥
9
8
³
µ
´
¨
¨
·
·
0.994
F
(9)
2
exp
0.44 e
0.164
2
2
4
©
¸
Note:
As a check, the sum of the frequencies (probabilities) should be close to 1. If not, you have
made a mistake. Also, the curve will be smoother for smaller bin widths; however, 1 m/s will suffice.
For large bin widths, the wind speed histogram might have to be renormalized by bin value * 8,760/
(sum of observations).
The Weibull distribution is characterized by two parameters, the shape parameter,
k
(dimension-
less), and the scale parameter,
c
(m/s). The Rayleigh distribution is a special case of the Weibull
distribution where
k
2. For regions of the trade winds where the winds are fairly steady, the shape
factor may be as high as 4 to 5. For most sites in Europe and the United States,
k
varies between
1.8 and 2.4.
§
¶
k
1
k
¤
¦
¥
³
µ
´
¤
¦
¥
³
µ
´
v
k
c
v
c
v
c
¨
¨
·
·
(3.17)
Fv
()
$
exp
©
¸
In many parts of the world the wind speed data are sparse. If only the average wind speed by day
or month is known, then the average values and deviation of the average values are used to estimate
the two parameters. Rohatgi and Nelson [9, chap. 9] give details on estimating the Weibull param-
eters by three methods: a plot of c and k from log-log paper, analysis of standard deviations, and
analysis of the energy pattern factor.
A higher
k
value means wind speeds are peaked around the average wind speed (
Figure 3.17
)
.
The values in the graph were calculated for a mean wind speed of 6 m/s for the Rayleigh distribution
and
c
6 m/s and
k
3 for the Weibull distribution, and both used a bin width of 1 m/s.
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