WIND ENERGY SYSTEMS - Cd3wd

Chapter 4—Wind Turbine Power 4–22
and the gustiness of the wind. We will assume that the power output can be adequately
described by the model of Eqs. 21, although more sophisticated models might be necessary in
rare cases. We now want to combine the variation in output power with wind speed with the
variation in wind speed at a site to find the average power P e,ave that would be expected from
a given turbine at a given site. The average power output of a turbine is a very important
parameter of a wind energy system since it determines the total energy production and the
total income. It is a much better indicator of economics than the rated power, which can
easily be chosen at too large a value.
The average power output from a wind turbine is the power produced at each wind speed
times the fraction of the time that wind speed is experienced, integrated over all possible wind
speeds.
In integral form, this is
P e,ave =
∫ ∞
0
P e f(u)du W (23)
where f(u) is a probability density function of wind speeds. We shall use the Weibull distribution
f(u) = k ( u
c c
) [
k−1 ( ) ]
u
k
exp
c
−
(24)
as described in Chapter 2.
Substituting Eqs. 21 and 24 into Eq. 23 yields
P e,ave =
∫ uR
u c
(a + bu k )f(u)du + P eR
∫ uF
u R
f(u)du W (25)
There are two distinct integrals in Eq. 25 which need to be integrated. One has the
integrand u k f(u) and the other has the integrand f(u). The integration can be accomplished
best by making the change in variable
( ) u k
x =
(26)
c
The differential dx is then given by
( ) u k−1 ( ) u
dx = k d
c c
(27)
The two distinct integrals of Eq. 25 can therefore be written as
Wind Energy Systems by Dr. Gary L. Johnson November 21, 2001