WIND ENERGY SYSTEMS - Cd3wd

Chapter 4—Wind Turbine Power 4–19
by as much as 10 percent under the predicted steady state requirement.
If we choose the 25-kW generator and connect it to the turbine whose shaft power is shown
in Fig. 10, and if a two-stage transmission is assumed, the electrical power output as a function
of wind speed will be as shown in Fig. 15. The shaft power input is also shown for comparison
purposes. It is seen that both the shaft power and the electrical power output increase nearly
linearly with wind speed up to their maximum values. This may seem somewhat surprising
since the power in the wind increases as the cube of the wind speed. It is correct, however,
since the low efficiencies at low wind speeds are responsible for linearizing the power output
curve.
We note in Fig. 15 that the electrical power output rises above zero at a wind speed of
about 5 m/s. This wind speed at which electrical power production starts is called the cut-in
speed u c . The turbine will develop enough mechanical power to rotate itself at slightly lower
speeds, but this wind speed will actually supply all the generator and transmission losses so
useful electrical power can be produced.
Fig. 15 has been developed from actual turbine data and from reasonably complete models
of the transmission and generator. Other turbines, transmissions, and generators will produce
somewhat different curves with approximately the same shape.
It is convenient to define a model for P e thatcanbeusedindiscussinganywindsystem.
The simplest model would use a straight line to describe the variation in output power between
cut-in and rated wind speeds. A straight line describes the output of the Sandia 17-m Darrieus
rather well. We must remember, of course, that other monotonic functions will fit the observed
data nearly as good as a straight line, or perhaps even better for some machines, and may
yield more accurate energy estimates or more convenient analytic results. It will be seen later
that a closed form expression for energy production can be obtained if P e is assumed to vary as
u k between cut-in and rated wind speeds, where k is the Weibull shape parameter. Numerical
integration is required if P e is assumed to vary as u, or in a linear fashion. Therefore, our choice
of a somewhat complicated model will make later computations easier, and perhaps more
accurate, than the choice of the simplest possible model. We therefore define the following
equations for the electrical power output of a model wind turbine[9]:
P e = 0 (u