Environmental Engineering Reference
In-Depth Information
(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.