WIND ENERGY SYSTEMS - Cd3wd

Chapter 4—Wind Turbine Power 4–23
∫
f(u)du =
∫
e −x dx = −e −x (28)
∫
u k f(u)du =
∫
c k (
u
k
c k )
f(u)du
= c k ∫
xe −x dx = −c k (x +1)e −x (29)
When we substitute the limits of integration into Eq. 25, and reduce to the minimum
number of terms, the result is
{ [
exp[−(uc /c) k ] − exp[−(u R /c) k ( ) ]}
]
k uF
P e,ave = P eR
(u R /c) k − (u c /c) k − exp −
c
W (30)
We now have an equation which shows the effects of cut-in, rated, and furling speeds
on the average power production of a turbine. For a given wind regime with known c and
k parameters, we can select u c , u R , and u F to maximize the average power, and thereby
maximize the total energy production. There are relationships among u c , u R ,andu F which
must be considered, however, if realistic results are to be expected. The wind must contain
enough power at the cut-in speed to overcome all the system losses. A cut-in speed of 0.5u R
would imply that the gearbox and generator losses at cut-in are the fraction (0.5)3 = 0.125 of
rated power. A cut-in speed of 0.4u R implies that the losses in that case are the fraction (0.4) 3
= 0.064 of rated power. It would take a rather efficient generator and gearbox combination
to have losses less than 6.4 percent of rated power while losses of 12.5 percent would indicate
a rather mediocre design. We would expect then that u c would almost always lie in the range
between 0.4 and 0.5u R .
Commercial wind turbines typically have furling speeds between 20 and 25 m/s and rated
wind speeds between 10 and 15 m/s. A furling speed of twice the rated speed means that
the turbine control system is able to maintain a constant power output over an eight to one
range of wind power input. This is quite an engineering challenge. This design difficulty plus
the difficulty of building wind turbines which can survive operation in wind speeds greater
than perhaps 25 m/s means that the furling speed will not normally be above 2u R , unless u R
happens to be chosen unusually low for a special application.
We can see from this discussion that selecting a rated wind speed u R is an important
part of wind turbine design. This selection basically determines the cut-in speed and also
imposes certain constraints on the furling speed. As stated earlier, we want to select u R so
Wind Energy Systems by Dr. Gary L. Johnson November 21, 2001