Page 51 - Fluid Power Engineering

Properties of W ind 29

Empirically, it has been observed that wind speed in most locations is
a Weibull distribution. Furthermore, the value of k is approximately
2 for most wind profiles.
A few observations are in order for Weibull as a distribution for
windspeed.Ask increases,theheightofthecurveincreases,whichim-
plies that the curve becomes narrower with a smaller tail—probability
drops quickly at higher wind speeds. This can lead to lower energy
at higher values of k. That is, if two wind speed distributions have
the same average wind speed, but different value of k, say, k = 2 and
k= 3, the latter case will yield lower energy.

Mean and Mode of Weibull Distribution for Wind Speed
The mode of a Weibull distribution is:

1
k − 1
/ K
Mode = A (3-2)
k
1

Mean = ¯v = A 1 + (3-3)
k

2
2 2 1 2 1 + k
2
2
Variance = σ = c 1 + − 1 + = ¯v − 1
k k 2 1 + 1
k
(3-4)
where (x) is the gamma function; if x is an integer, then (x) = x!.
Table 3-1 contains sample values of the mean and mode.
K A Mean Mode
1 8 8.000 0
1.5 8 7.222 3.846
2 8 7.090 5.657
2.5 8 7.098 6.522
3 8 7.144 6.989
1 10 10.000 0
1.5 10 9.027 4.807
2 10 8.862 7.071
2.5 10 8.873 8.152
3 10 8.930 8.736

TABLE 3-1 Mean and Mode of a Weibull Distribution for Different Shape
and Scale Factors