Environmental Engineering Reference
In-Depth Information
Table 13.1
Parameters of the neutral leg
Parameters
Value
Parameters
Value
L
N
2
.
35 mH
C
N
+
,
C
N
−
1000
μ
F
R
N
0
.
54
f
s
5 kHz
The controller can then be obtained by using the standard
H
∞
control algorithms to guarantee
the stability of the system.
13.2.5 Design Example
The parameters of the neutral leg circuit are given in Table 13.1. The switching frequency is
f
s
=
5. This is to reflect
the case that the neutral current
i
N
weighs more than the noise
n
and the output
z
1
weighs
more than the control signal
u
.
Controllers with different numbers of harmonics compensators (i.e., with different weighting
functions
W
) are designed in this section. Controller
C
1
is designed according to the weighting
function
W
(
s
)
5 kHz. The weighting parameters are chosen to be
ξ
=
0
.
5 and
μ
=
0
.
ζω
K
1
2
s
=
that has a high gain at the fundamental grid frequency
ω
s
2
+
ζω
+
ω
2
2
s
K
1
2
ζω
s
K
3
6
ζω
s
only. Controller
C
3
is designed according to
W
(
s
)
2
,
which has an extra high gain at the 3rd harmonic frequency to further improve the performance
of the controller. Similarly, Controllers
C
5
and
C
7
are designed with weighting functions having
an added high gain at the 5th and 7th harmonic frequencies, respectively. The gains in the
weighting functions are chosen as
K
1
=
=
2
+
s
2
+
2
ζω
s
+
ω
s
2
+
6
ζω
s
+
9
ω
10,
K
3
=
1
.
666,
K
5
=
0
.
5 and
K
7
=
0
.
5 and the
damping factor is chosen as
ζ
=
0
.
005, according to the guidelines reported in (Castilla
et al
.
2009).
Using the MATLAB
R
hinfsyn
algorithm, the
H
∞
controllers
C
1
(
s
),
C
3
(
s
),
C
5
(
s
) and
C
7
(
s
)
are obtained as
09)(
s
2
10
5
)
75
.
7417(
s
−
55
.
+
296
s
+
1
.
393
×
C
1
(
s
)
=
10
4
)
,
(13.9)
(
s
+
1147)(
s
+
82
.
99)(
s
2
+
3
.
14
s
+
9
.
87
×
02)(
s
2
10
5
)(
s
2
10
5
)
100
.
0978(
s
−
87
.
+
295
.
9
s
+
1
.
442
×
−
149
.
6
s
+
6
.
693
×
C
3
(
s
)
=
,
(
s
+
1323)(
s
+
71
.
9)(
s
2
+
3
.
14
s
+
9
.
87
×
10
4
)(
s
2
+
9
.
42
s
+
8
.
883
×
10
5
)
(13.10)
26)(
s
2
10
5
)(
s
2
10
5
)
107
.
6544(
s
−
97
.
+
295
.
3
s
+
1
.
454
×
−
168
.
3
s
+
6
.
659
×
C
5
(
s
)
=
(
s
+
1374)(
s
+
69
.
27)(
s
2
+
3
.
14
s
+
9
.
87
×
10
4
)(
s
2
+
9
.
42
s
+
8
.
883
×
10
5
)
(
s
2
10
6
)
−
106
.
1
s
+
2
.
284
×
×
10
6
)
,
(13.11)
(
s
2
+
15
.
7
s
+
2
.
467
×
.
−
104)(
s
2
+
.
+
.
×
10
5
)(
s
2
−
.
+
.
×
10
5
)
112
5476(
s
294
8
s
1
461
178
5
s
6
666
=
C
7
(
s
)
71)(
s
2
10
4
)(
s
2
10
5
)
(
s
+
1405)(
s
+
67
.
+
3
.
14
s
+
9
.
87
×
+
9
.
42
s
+
8
.
883
×
(
s
2
10
6
)(
s
2
10
6
)
−
122
.
1
s
+
2
.
287
×
−
133
s
+
4
.
579
×
×
10
6
)
.
(13.12)
(
s
2
+
15
.
7
s
+
2
.
467
×
10
6
)(
s
2
+
21
.
98
s
+
4
.
836
×
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