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d
d
θ
d
d
θ
Ts
=+
K
F
1
and
Ts
=
K
θ
−
F
2
(13)
1
1
1
1
2
2
2
2
KK
*
Ts
*
Ts
θθθ
=−,
K
=
1
2
and
T
=
1
2
(14)
eq
1
2
eq
eq
KK
+
Ts
+
Ts
1
2
1
2
F
d
θ
d
eq
TK
=
θ
+
(15)
eq
eq
eq
eq
d
d
θ
eq
=−
ωω
(16)
t
r
dθ
eq
=−
ωω
(17)
t
r
dt
where
H
,
Hg
are the pu turbine and generator inertia respectively.
J
and
J
are the
inertia in kgm
2
.
T
is the electromechanical torque developed by the induction
generator,
T
is the pu mechanical torque applied to the turbine by the wind derived
from (6).
Ts
,
e
T
are the torques developed by the shaft at the low speed side,
torque developed by the shaft at the high speed side and the equivalent torque
developed by the shafts respectively.
Ts
,
1
2
and ω are the pu turbine and generator rotor
speed.
K
,
K
and
e
K
are shaft stiffness at low speed side, shaft stiffness at high
speed side and the total shaft stiffness.
F
,
F
and
e
F
are the damping coefficient of
the shaft at the low speed side, high speed side and the equivalent damping
coefficient of the shaft respectively.
ω
t
θ and
e
θ are the angle of twist of the shaft at
low speed, high speed and the equivalent angle of twist of the shaft respectively.
θ
,
1
2
n
is
the number of pole pairs,
n
is the gear ratio,
ω
is
P
is the generator active power,
2
π
f
where
f
is the frequency (Hz).
2.4
Pitch Angle Controller Model
Pitch angle controller mainly serves the purpose of limiting the generated power to
the rated power at time of high wind speed. It also limits the speed of the generator
during heavy disturbances. The pitch controller based on PI is given by (18) [13]
d
dt
β
1
(
)
=
ββ
−
ref
τ
s
(18)
k
=+
(
)
i
β
k
P
−
P
ref
P
ref
m
s
where
β
is the reference pitch control,
k
and
k
are the proportional and integral
ref
P
parameters of the PI controller,
P
is the reference turbine power.
ref
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