WIND ENERGY SYSTEMS - Cd3wd

Chapter 2—Wind Characteristics 2–34
(
ū = cΓ 1+ 1 )
k
(35)
Published tables that are available for the gamma function Γ(y) are only given for 1 ≤ y ≤ 2.
If an argument y lies outside this range, the recursive relation
must be used. If y is an integer,
Γ(y +1)=yΓ(y) (36)
Γ(y +1)=y! =y(y − 1)(y − 2) ···(1) (37)
The factorial y! is implemented on the more powerful hand calculators. The argument y
is not restricted to an integer, so the quantity computed is actually Γ(y +1). Thismay be
the most convenient way of calculating the gamma function in many situations.
Normally, the wind data collected at a site will be used to directly calculate the mean
speed ū. We then want to find c and k from the data. A good estimate for c can be obtained
quickly from Eq. 35 by considering the function c/ū as a function of k which is given in Fig. 17.
For values of k below unity, the ratio c/ū decreases rapidly. For k above 1.5 and less than 3
or 4, however, the ratio c/ū is essentially a constant, with a value of about 1.12. This means
that the scale parameter is directly proportional to the mean wind speed for this range of k.
c =1.12ū (1.5 ≤ k ≤ 3.0) (38)
Most good wind regimes will have the shape parameter k in this range, so this estimate
of c in terms of u will have wide application.
It can be shown by substitution that the Weibull distribution function F (u) which satisfies
Eq. 27, and also meets the other requirements of a distribution function, i.e. F (0) = 0 and
F (∞) =1,is
[ ( ) ]
u
k
F (u) =1− exp −
c
(39)
The variance of the Weibull density function can be shown to be
(
σ 2 = c
[Γ
2 1+ 2 ) (
− Γ 2 1+ 1 )] [ ]
Γ(1 + 2/k)
=(ū) 2
k
k
Γ 2 (1 + 1/k) − 1
(40)
The probability of the wind speed u being equal to or greater than u a is
Wind Energy Systems by Dr. Gary L. Johnson November 20, 2001