Wind Turbines - Wind Energy Systems for Electric Power Generation - page 18

Due to the cubic relationship between power and wind velocity the maximum of
e i ( v i ) is seen at significantly larger values v i than the maximum of h i ( v i ). Figure 2.9
gives an example, using a class width of Δ v = 0 . 5m / s.

2.3.3 Power and Torque Characteristics

The power delivered by a wind turbine is a function of tip speed ratio (see Fig. 2.3)
and hence depends on wind velocity and rotational speed. Figure 2.10 shows a nor-
malized representation of power and torque of a rotor with fixed blade position over
speed, with wind velocity values as parameter. In the example the rated wind veloc-
ity is 12 m/s. The design is such that at approximately v = 8m / s the tip speed ratio
is optimal, λ opt . The power maxima are indicated by the cubic function of v .

From the power characteristic it is seen that P / P N = 1at n / n N = 1. Measures for
power limitation at higher wind speeds are not considered in the graphs.
The graphs are analytically derived from a power coefficient curve, see Fig. 2.3.
When c p (

λ =

) is given as an empirical function, e.g. in form of a look-up table, then
the curves shown in Fig. 2.8 can be calculated by the following algorithm for a
specific design.
Let

λ

- λ A the design tip speed ratio and c pA = c p ( λ A ) the optimum power coefficient of

the rotor,

- λ N the tip speed ratio for rated condition, with c p , N = c p ( λ N ),

- v N the wind speed for rated condition.
The ratio

λ A can be chosen. In case of a constant speed system for operation at
n / n N = 1 the wind speed at which c p is optimum is calculated by:

λ N /

v A = λ N

λ A v N

(2.13)

To calculate power and torque curves, selected parameter wind speeds v i are
chosen. The following equations are normalized to give P, T and n referred to rated
values.

Fig. 2.10 Power and torque characteristics vs. rotational speed ( v N = 12 m / s)