Page 211 - Wind Energy Handbook

ESTIMATION OF ENERGY CAPTURE 185

rotational speed; the pitch angle should always return to the optimum setting for
highest efficiency. Pitch control regulation is also required in conditions above the
rated wind speed when the rotational speed is kept constant.

4.5 Estimation of Energy Capture

The quantity of energy that can be captured by a wind turbine depends upon the
power versus wind speed characteristic of the turbine and the wind-speed distribu-
tion at the turbine site. The wind-speed distribution at a site can be represented by
the Weibull function: the probability that the wind speed will exceed a value U is

ð
F(U) ¼ e U=cÞ k (4:2)
where c, called the scale factor, is a characteristic speed related to the average wind
speed at the site by

U
c ¼ (4:3)
1
ˆ 1 þ
k

(ˆ being the gamma function and k is a shape parameter, see also Section 2.4). Let
U=U ¼ u a normalized wind speed.
The wind speed distribution density is then the modulus of the derivative of
Equation (4.2) with respect to u:
k
U k 1 U u k
f(u) ¼ k u e c (4:4)
c
i.e., the probability that the wind speed lies between u and u þ äu is f(u)äu.
Alternatively, Equation (4.4) gives the proportion of time for which the wind speed
u will occur.
The performance curve shown in Figure 4.14 is for a turbine designed with an
optimum tip speed ratio of 7. As an example, assume that the turbine is stall-
regulated and operates at a fixed rotational speed at a site where the average wind
speed is 6 m=s and the Weibell shape factor k ¼ 1:8 then, from Equation (4.3), the
scale factor c ¼ 6:75 m=s.
Figure 4.15 shows the K P –1=º curve for the turbine; from inspection of that
curve the tip speed ratio at which stall (maximum power) occurs is 3.7 and the
corresponding C P is 0.22.
The required maximum electrical power of the machine is 500 kW, the transmis-
sion loss is 10 kW, the mean generator efficiency is 90 percent and the availability of
the turbine (amount of time for which it is available to operate when maintenance
and repair time is taken into account) is 98 percent.
The maximum rotor shaft power (aerodynamic power) is then