Environmental Engineering Reference
In-Depth Information
The equation of motion describes the behavior of the rotor due to variations in load
torque
T
L
and/or excitation voltage
u
p
. In the following equation
˙
is the velocity
describing the deviation from the steady-state load angle. A damping torque is as-
sumed to be proportional to
˙
ϑ
with the constant parameter
K
D
, representing friction
and windage losses, and partly rotor losses due to rotor eddy-currents.
ϑ
˙
J
z
p
d
dt
=
d
dt
˙
˙
+
K
D
ϑ
=
T
el
+
T
L
where
ϑ
=
z
p
Ω
−
ω
s
(6.38)
Equations (6.25) to (6.27) form a non-linear second-order differential equation
system.
J
denotes the resulting moment of inertia of the whole rotating system,
calculated by taking gear transmissions into account.
For generator operation at constant terminal voltage, the model allows to deter-
mine oscillations due to changes in load torque and/or excitation voltage. When a
constant
e
p
is assumed, the load angle
ϑ
(
t
) is the only state variable.
6.2.3.3 Model for Small Deviations from Steady State
Linearization of the system (6.26), (6.27) for small deviations
Δϑ
from a steady-
state operation point, with deviations
Δ
T
as the input variable leads to a second-
order system
τ
J
ω
s
Δ
+
K
D
ω
s
P
N
+
T
s
0
ω
s
P
N
=
ω
s
¨
˙
ϑ
Δ
ϑ
Δϑ
P
N
Δ
T
L
(6.39)
where
τ
J
is the unit acceleration time according to (6.16), and
T
s0
is the synchroniz-
ing torque at the operation point characterized by the steady-state load angle
ϑ
0
:
T
s
0
=
d
T
el
d
U
s
·
U
s
1
cos(2
ϑ
0
)
(6.40)
e
p
X
d
3
z
p
2
1
X
q
=
−
·
cos
ϑ
0
−
X
d
−
ϑ
ω
s
ϑ
=
ϑ
0
The steady-state load angle may be determined for a machine at synchronous
speed, operating at active power
P
and reactive power
Q
, with
Q
positive when
supplying reactive power (over-excitation):
atan
P
/
3
U
2
2
X
q
+
Q
ϑ
0
=
−
The linear differential equation may be expressed in the conventional form
2
s
=
ω
¨
˙
2
0
Δ
ϑ
+ 2
δΔ
ϑ
+
ν
Δϑ
τ
J
P
N
Δ
T
L
Its characteristic equation is:
p
2
2
0
Δϑ
+
p
2
δΔϑ
+
ν
Δϑ
= 0
(6.41)
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