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presented here can be found in [2]. This theoretical approach sets ground for the
further power curve analysis.
In the following derivation, the complexity of turbulence will be set aside so as
to understand the fundamental behavior of a wind turbine. Atmospheric wind has
fi nite time and space structures, more commonly referred to as turbulent struc-
tures. Its statistics display complex properties like unstationarity or intermittency
(such as gusts), whose effects will not be discussed in this section. They repre-
sent active research topics whose detailed analysis is outside the scope of this
introduction, cf. [4]. In this section, a uniform fl ow at steady-state is considered.
Based on the fact that a wind turbine converts the wind power into available
electrical power, one can assume the following relation:
Pu
()
=
c uP
()
()
u
(1)
p
ind
where P wind ( u ) is the power contained in the wind passing with speed u through
the wind turbine, and P ( u ) is the electrical power extracted. The power coeffi cient
c p ( u ) represents the amount of power converted by the wind turbine. Because the
input P wind ( u ) cannot be controlled, improvements in wind power performance
involve increasing the power coeffi cient c p ( u ). Momentum theory can now be
applied to determine this coeffi cient.
Consider a volume of air moving towards the wind turbine, which is modeled as
an actuator disc of diameter D . A stream-tube is defi ned here as the volume of air
that interacts with the turbine (see Fig. 1). The wind is affected by the wind turbine
when crossing its swept area as the turbine extracts part of its energy. The extrac-
tion of kinetic energy accounts for a drop in the wind speed from upstream to
downstream, as shown in Fig. 1. To ensure mass conservation, the stream-tube has
to expand in area downstream, as shown in Fig. 1 [2].
Following this simple analysis, one can estimate the amount of kinetic energy
available for extraction. The wind power P wind ( u ) is derived from momentum
theory for the wind passing with speed u through the rotor of area p D ²/4:
Figure 1 : A visual representation of an airfl ow on a wind turbine. The stream-tube is
affected by the presence of the wind turbine that extracts part of its energy.
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