Environmental Engineering Reference
In-Depth Information
i
2
) is the output voltage
of the inverter. The plant
P
can then be described by the state equation
P
is the tracking error
e
=
e
u
=
u
ref
−
u
o
, where
u
o
=
u
c
+
R
d
(
i
1
−
x
=
Ax
+
B
1
w
+
B
2
u
(5.1)
and the output equation
y
=
e
=
C
1
x
+
D
1
w
+
D
2
u
(5.2)
with
⎡
⎤
R
f
+
R
d
R
d
L
f
1
L
f
−
−
⎣
⎦
L
f
R
g
+
R
d
R
d
L
g
1
L
g
−
A
=
,
L
g
1
C
f
1
C
f
−
0
⎡
⎣
⎤
⎦
,
⎡
⎣
⎤
⎦
,
1
L
f
0
0
00
1
L
g
−
0
B
1
=
B
2
=
00
C
1
=
−
1
,
R
d
R
d
−
D
1
=
01
,
D
2
=
0
.
The corresponding plant transfer function is then
A B
1
B
2
C
1
D
1
D
2
=
D
1
D
2
+
A
)
−
1
B
1
B
2
=
P
C
1
(
sI
−
.
(5.3)
5.2.2 Frequency-adaptive Internal Model M
The internal model
M
, shown in Figure 5.2, is infinite dimensional and consists of a low-pass
filter
W
(
s
)
ω
c
s
+
ω
c
cascaded with a delay line
e
−
τ
d
s
. It is capable of generating periodic signals
of a given fundamental period
=
τ
d
so it is capable of tracking periodic references and rejecting
periodic disturbances having the same period. In order to improve the performance of the
controller, the delay time
τ
d
used in the internal model
M
should be slightly less than the
fundamental period
τ
(Weiss and Hafele 1999), and is chosen as
1
ω
c
,
τ
d
=
τ
−
(5.4)
where
ω
c
is the cut-off frequency of the low-pass filter
W
.
The internal model has a very high gain at frequencies pre-defined by the internal model
delay line; see Figure 5.4(a). When the actual grid frequency
f
varies, its performance
is degraded. This problem could be solved by changing the delay time
τ
d
with respect to
the grid frequency. However, following the discrete-time implementation and low sampling
frequency used (e.g. 5 kHz), it is impossible to implement the adaptive delay time without
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