Power curves for wind turbines - Wind Power Generation and Wind Turbine Design - page 598

2

r p D

( 2)

3

Pu

wind ()

= u

24

where r is the air density.
This stream-tube expansion shows that c p ( u ) has a physical limit called Betz
limit such that c p ( u )

0.593 [2, 5]. Regardless of its design, a wind turbine
can thus extract at most 59.3% of the wind energy. Figure 2 shows the power coef-
fi cient as a function of a = (1

≤ 16/27 ≈

u downstream / u upstream ), the axial fl ow induction factor
a gives the ratio of speed lost by the wind. The Betz limit corresponds to the
maximum power a wind turbine can extract, when a = 1/3 [2]. This result is
obtained for an actuator disc. The more complex shape of a real wind turbine cer-
tainly brings a lower limit for c p . This physical limit is due to the stream-tube
expansion induced by the presence of the turbine, i.e. by distorting the wind fi eld,
a wind turbine sets a limit for the energy availability. Criticism of this approach is
given in [6, 7], leading to a less well defi ned upper limit of c p .
Although it is based on a simplifi ed approach, the Betz limit is a widely used
and accepted value. The power coeffi cients of modern commercial wind turbines
reach values of 0.45 and more. Physical aspects that limit the value of the power
coeffi cient are, e.g. the fi nite number of blades and losses due to the drag and stall
effects of the blades [2, 8].
Joining eqns (1) and (2), the theoretical power curve reads

−

2

rp D

(3 )

3

P () u = cuP () () u = cu () 24 u

theoretical p wind p

P theoretical ( u ), or more simply P ( u ) is the electrical power output and u is the input
wind speed ( u upstream ), i.e. a power curve is roughly characterized by a cubic

Figure 2: Power coeffi cient c p as a function of the axial fl ow induction factor a .