WIND ENERGY SYSTEMS - Cd3wd

Chapter 2—Wind Characteristics 2–37
We may think of this value as the true or actual power in the wind if the probabilities
p(u i ) are determined from the actual wind speed data.
If we model the actual wind data by a probability density function f(u), then the average
power in the wind is
¯P w = 1 ∫ ∞
2 ρA u 3 f(u)du W (45)
0
It can be shown[13] that when f(u) is the Weibull density function, the average power is
¯P w = ρAū3 Γ(1 + 3/k)
2[Γ(1 + 1/k)] 3 W (46)
If the Weibull density function fits the actual wind data exactly, then the power in the
wind predicted by Eq. 46 will be the same as that predicted by Eq. 44. The greater the
difference between the values obtained from these two equations, the poorer is the fit of the
Weibull density function to the actual data.
Actually, wind speeds outside some range are of little use to a practical wind turbine.
There is inadequate power to spin the turbine below perhaps 5 or 6 m/s and the turbine may
reach its rated power at 12 m/s. Excess wind power is spilled or wasted above this speed so
the turbine output power can be maintained at a constant value. Therefore, the quality of fit
between the actual data and the Weibull model is more important within this range than over
all wind speeds. We shall consider some numerical examples of these fits later in the chapter.
The function u 3 f(u) starts at zero at u = 0, reaches a peak value at some wind speed
u me , and finally returns to zero at large values of u. The yearly energy production at wind
speed u i is the power in the wind times the fraction of time that power is observed times the
number of hours in the year. The wind speed u me is the speed which produces more energy
(the product of power and time) than any other wind speed. Therefore, the maximum energy
obtained from any one wind speed is
W max = 1 2 ρAu3 mef(u me )(8760) (47)
The turbine should be designed so this wind speed with maximum energy content is
included in its best operating wind speed range. Some applications will even require the
turbine to be designed with a rated wind speed equal to this maximum energy wind speed.
We therefore want to find the wind speed u me . This can be found by multiplying Eq. eq:2.30
by u 3 , setting the derivative equal to zero, and solving for u. After a moderate amount of
algebra the result can be shown to be
( ) k +2 1/k
u me = c
m/s (48)
k
Wind Energy Systems by Dr. Gary L. Johnson November 20, 2001