Environmental Engineering Reference
In-Depth Information
Fig. 6.9
Three-mass drive
train model of a wind system
T wind turbine; B gear box;
G generator
R
4
L
J
=
2
π
K
=
G
R
4
L
;
2
π
(6.50)
Note that the inertia values of two disks in series are added,
J
=
J
1
+
J
2
, while
the spring constant of two cylindrical shaft elements in series follows 1
/
K
= 1
/
K
1
+
1
/
K
2
. The differential equation assigned to Fig. 6.9 becomes:
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
+
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
+
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
=
⎡
⎣
⎤
⎦
J
T
ϕ
T
ϕ
B
ϕ
G
D
T
ϕ
T
ϕ
B
ϕ
G
K
TB
−
K
TB
−
K
TB
(
K
TB
+
K
BG
)
−
K
BG
−
K
BG
ϕ
T
ϕ
B
ϕ
G
T
T
J
B
D
B
J
G
D
G
K
BG
T
G
In this model damping effects in the steel shafts and in couplings are neglected,
and only viscous damping losses are considered. When a gear-box is present, all
quantities have to be referred to either the low-speed side or the high-speed side.
Normally it is the latter, i.e. the generator side. The parameters of the turbine side
must be referred to the generator side using the speed ratio; as example the equiva-
lent turbine inertia is calculated as:
J
T
=
ω
G
ω
2
J
T
T
A parameter example of realistic wind energy systems shows that the relation
J
T
:
J
B
:
J
G
is in the region of 12:0, 6:1. In conclusion a two-mass system may be
used as reduced order model. Then the gear box inertia
J
B
is mostly halved, with the
halves added to either side, on the other hand simulation results recommend adding
the total gear-box value to the generator.
The two-mass model improves considerably the one-mass representation; it al-
lows for determining one mechanical eigenfrequency and estimation of transient
torques stressing shafts and couplings. Therefore its properties are discussed shortly.
Defining the difference
−
ϕ
G
as the torsion angle, a second order linear
differential equation can be established:
ϕ
=
ϕ
T
+
D
1
J
T
+
K
1
J
T
1
J
G
1
J
G
=
T
T
J
T
+
T
G
J
G
ϕ
¨
+
ϕ
˙
+
ϕ
(6.51)
2
dv
0
0
ν
The two-mass system features a damping coefficient
d
and a characteristic fre-
quency
ν
0
. Since damping values of the shaft train are low, the response to a step
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