Environmental Engineering Reference
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(a)
(b)
C p
Betz limit (0.593)
C p max
N = 5 blades
C pmax ( N )
...
N = 4 blades
β k
β k+1
...
N = 3 blades
λ
λ
λ
opt
Fig. 14.11 a Typical aerodynamic power coefficient C p as a function of k and b. b Maximum
power coefficient C pmax for different number of blades N at a given b [ 2 ]
effect of the number of blades N in the maximum rotor power coefficient C pmax for
a classical wind turbine with 25 \ C L /C D \? is shown in ( 14.10 ),
2
3
2
1
k opt 8
20
1 : 32 þ
4
5
0 : 57k opt
C pmax ¼ 0 : 593 k opt
k opt þ
ð 14 : 10 Þ
C L
C D
1 þ 1
2N
N 2 = 3
where N is the number of blades, C L is the rotor lift coefficient and C D is the rotor
drag coefficient (see also [ 2 ], Chap. 12). Figure 14.11 b shows some calculations of
C pmax using this expression with C L /C D = 76 and for N = 3, 4 and 5
14.3.3 Dynamics of Rotor Speed Versus Torque, Pitch Angle
and Wind Velocity Variation
Figure 14.12 shows a variable-speed pitch-controlled wind turbine with a
mechanical (gearbox) drive-train. A shaft connects a large inertia rotor at one end
(blades) with a gearbox, which is coupled to a generator at the other end. The wind
applies an aerodynamic torque T r to the rotor, which is connected to the low-speed
shaft of the gearbox. At the other end of the drive-train, the generator, with a
power converter, applies an antagonistic electrical torque T g on the high-speed
shaft of the gearbox. The rotor presents a moment of inertia I r . The shaft has a
torsional stiffness coefficient K s and a viscous damping coefficient B s . The gen-
erator shows a moment of inertia I g . The rotor angle is h r , the rotor speed is
X r = dh r /dt, and the generator angle is h g . Also, w is the yaw angle error (nacelle-
wind angle) and b is the pitch angle (blades).
The excitation current, I x , is introduced in the rotor, and the active and reactive
power, respectively P and Q, are supplied to the grid. f, U and / are the frequency,
voltage, and power factor at the grid connection point respectively.
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