Environmental Engineering Reference
In-Depth Information
used which is not subject to parameter variations. The reader is referred to [
14
,
15
]
for a detailed description of a method for rotor current monitoring, where the
DFIG model is used and the parameter variations are accounted for.
The following sections successively address modeling of a balanced three-
phase system, residual generation for three-phase signals, and application to the
monitoring of stator voltages and currents for a wind-driven DFIG.
10.5.1 Model of a Balanced Three-Phase System
Consider a sinusoidal signal with amplitude M
o
, frequency x
o
and phase /
o
,
represented by:
y
ð
t
Þ¼
M
o
sin
ð
x
o
t
þ
/
o
Þ
ð
10
:
25
Þ
This particular signal can be modeled by the following state-space representation
¼
x
1
ð
t
Þ
x
2
ð
t
Þ
0
x
o
x
1
ð
t
Þ
x
2
ð
t
Þ
ð
10
:
26
Þ
x
o
0
|{z}
A
o
|{z}
x
ð
t
Þ
x
1
ð
t
Þ
x
2
ð
t
Þ
y
ð
t
Þ¼
1
½
|{z}
C
o
ð
10
:
27
Þ
where x
ð
0
Þ¼½
M
o
sin
ð
/
o
Þ;
M
o
cos
ð
/
o
Þ
T
is the initial state.
The idea behind the modelling of a sinusoidal signal is now used to model a
three-phase balanced system. Consider a balanced three-phase sinusoidal electric
system (current or voltage). All the signals have identical amplitudes (M) and
frequencies (x
e
), and their mutual phase shift is 2p
=
3. They can be described by:
y
a
ð
t
Þ¼
M sin x
e
ð
t
Þ
t
þ
/
a
ð
Þ
ð
10
:
28a
Þ
y
b
ð
t
Þ¼
M sin x
e
ð
t
Þ
t
þ
/
a
2p
3
ð
10
:
28b
Þ
y
c
ð
t
Þ¼
M sin x
e
ð
t
Þ
t
þ
/
a
þ
2p
3
ð
10
:
28c
Þ
where /
a
is the initial phase of y
a
ð
t
Þ
. Notice that although we consider the same
frequency for all the signals, the value of x
e
can be time-varying as explicitly
indicated above. Since the system in Eqs.
10.28a
-
10.28c
is balanced, the sum-
mation of the three signals must be equal to zero:
