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Giv
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[
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m
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[
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t
s
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[
4
-
1
8
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.
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Mo
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ev
elo
p
ed
m
at
h
e
m
atica
ll
y
as
cite
d
in
v
ar
io
u
s
w
o
r
k
s
s
u
ch
a
s
[
1
9
-
2
4
]
,
an
d
ad
ap
ted
to
o
u
r
s
y
s
te
m
to
p
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y
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ap
er
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r
o
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id
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m
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d
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ap
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ly
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o
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ased
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m
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d
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s
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ib
u
tio
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y
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te
m
s
[
25
].
T
h
e
r
est
o
f
th
e
p
ap
er
in
o
r
g
an
ized
as
f
o
llo
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s
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n
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r
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DE
SCR
I
P
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YST
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ar
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ief
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m
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2
.
1
.
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a
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er
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ilte
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S
I
n
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er
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d
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co
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p
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g
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r
e
1
.
DC
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D
Vs
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r
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r
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n
t
r
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l
l
e
r
V
i
n
j
Fig
u
r
e
1
.
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g
le
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p
h
ase
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er
ies a
ctiv
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p
o
w
er
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ilter
(
A
P
F
S
)
2
.
2
.
M
o
delin
g
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f
t
he
AP
F
S s
y
s
t
e
m
T
h
is
s
ec
tio
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d
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ls
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h
th
e
m
ath
e
m
atica
l
m
o
d
eli
n
g
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f
t
h
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er
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p
o
w
er
f
ilter
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T
h
e
f
u
ll
s
tr
u
ct
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r
e
o
f
t
h
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n
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e
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ig
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Kir
c
h
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f
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i
s
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p
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lo
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g
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ath
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m
at
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x
p
r
ess
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by
[
5
]
.
T
h
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SA
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F
m
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s
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r
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d
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ter
in
d
u
cto
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cu
r
r
en
t i
L
f
r
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ec
tiv
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y
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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I
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W
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ter
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ter
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r
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3.
CO
NT
RO
L
ST
R
AT
E
G
I
E
S
O
F
T
H
E
SYS
T
E
M
T
h
e
co
n
tr
o
l
s
tr
ateg
ies
u
s
ed
i
n
th
is
w
o
r
k
ar
e
P
I
an
d
B
ac
k
s
tep
p
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No
n
li
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r
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n
tr
o
ller
s
.
T
h
u
s
,
a
co
m
p
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v
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s
t
u
d
y
is
m
ad
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t
w
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m
eth
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d
s
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p
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e
t
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m
et
h
o
d
s
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d
escr
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s
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ar
ately
as f
o
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w
:
3
.
1
.
P
r
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l int
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ntr
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T
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u
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ased
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o
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n
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ated
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w
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I
GB
T
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o
f
th
e
v
o
lt
ag
e
s
o
u
r
ce
in
v
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ter
VSI
[
7
]
.
T
h
e
PI
co
n
tr
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ca
n
b
e
d
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in
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+
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Evaluation Warning : The document was created with Spire.PDF for Python.
I
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p
p
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81
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i
s
co
n
tr
o
l
m
eth
o
d
is
a
r
ec
u
r
s
iv
e
d
esig
n
m
et
h
o
d
o
lo
g
y
th
at
i
n
v
o
l
v
es
a
s
y
s
te
m
atic
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o
n
s
tr
u
ct
io
n
o
f
f
ee
d
b
ac
k
co
n
tr
o
l
la
w
s
a
n
d
ass
o
ciate
d
L
y
ap
u
n
o
v
f
u
n
c
tio
n
s
.
T
h
e
co
n
tr
o
ller
d
esig
n
ca
n
b
e
r
ea
ch
ed
in
a
n
u
m
b
er
o
f
s
tep
s
th
at
d
o
es
n
o
t
ex
ce
ed
th
e
s
y
s
te
m
o
r
d
er
[
25
].
I
n
th
is
p
ar
t
th
e
A
P
FS
s
y
s
te
m
b
ased
o
n
th
e
p
r
o
p
o
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ed
ad
a
p
tiv
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p
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li
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n
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o
ller
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d
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g
le
-
p
h
a
s
e
(
lev
els
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v
o
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g
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o
u
r
ce
in
v
er
te
r
.
Gen
er
al
b
lo
ck
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iag
r
a
m
o
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t
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e
s
y
s
te
m
i
s
s
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o
w
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i
n
F
i
g
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r
e
4.
P
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G
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i
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i
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it
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r
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i
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V
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u
Fig
u
r
e
4
.
Gen
er
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a
m
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f
t
h
e
ad
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tiv
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li
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k
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y
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te
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Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2252
-
8792
I
n
t J
A
p
p
l P
o
w
er
E
n
g
,
Vo
l.
9
,
No
.
1
,
A
p
r
il 2
0
2
0
:
7
8
–
88
82
T
h
e
co
n
tr
o
l
o
b
j
ec
tiv
e
is
to
en
f
o
r
ce
th
e
s
y
s
te
m
to
i
n
j
ec
t
a
c
o
m
p
e
n
s
at
io
n
v
o
lta
g
e
Vc
r
ep
r
esen
ted
b
y
th
e
s
tate
s
p
ac
e
v
ar
iab
le
1
.
T
h
is
i
n
j
ec
ted
v
o
ltag
e
i
s
t
h
e
d
i
f
f
er
en
ce
v
o
lta
g
e
b
et
w
ee
n
t
h
e
r
ef
er
en
ce
Vr
an
d
th
e
in
s
ta
n
ta
n
eo
u
s
s
y
s
te
m
v
o
lt
ag
e
Vs,
t
h
is
d
r
o
p
v
o
ltag
e
ca
n
b
e
d
ef
in
ed
b
y
u
.
I
n
ad
d
itio
n
,
th
e
co
n
tr
o
l
p
r
o
b
lem
ca
u
s
ed
b
y
th
e
n
o
n
m
i
n
i
m
u
m
p
h
ase
n
at
u
r
e
o
f
th
e
p
o
w
er
co
n
v
er
ter
is
s
o
lv
ed
ac
co
r
d
in
g
to
th
e
in
d
ir
ec
t
ap
p
r
o
ac
h
,
w
h
er
e
th
e
d
ir
ec
t
o
u
tp
u
t
v
o
lta
g
e
r
eg
u
latio
n
is
u
n
f
ea
s
ib
le
[
25
-
28
]
.
T
h
e
co
n
tr
o
ller
d
esig
n
i
s
g
iv
e
n
b
elo
w
in
t
w
o
m
ai
n
s
tep
s
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ec
au
s
e
o
f
t
h
e
o
r
d
er
o
f
th
e
s
t
u
d
ied
s
y
s
te
m
(
3
,
4
)
:
S
tep
1:
T
h
e
o
u
tp
u
t
v
o
ltag
e
er
r
o
r
is
d
ef
in
ed
b
y
:
1
̇
=
1
̇
−
̇
.
T
h
en
,
d
ev
iatin
g
1
an
d
s
u
b
s
tit
u
ti
n
g
it
f
r
o
m
(
5
)
,
w
e
g
et
:
1
̇
=
1
̇
−
̇
=
1
2
−
1
−
̇
(
9
)
w
h
er
e
th
e
ter
m
2
⁄
ca
n
b
e
a
v
ir
tu
al
co
n
tr
o
l in
p
u
t.
Fro
m
(
5
)
,
w
e
g
et
th
e
f
o
llo
w
i
n
g
:
1
̇
=
−
1
1
(
1
>
0
)
2
=
−
1
1
+
1
+
̇
w
h
er
e,
alp
h
a
is
a
d
esi
g
n
p
ar
a
m
eter
.
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h
ile
2
⁄
is
n
o
t
an
ef
f
ec
t
iv
e
i
n
p
u
t
f
o
r
th
e
co
n
tr
o
l
d
esig
n
,
t
h
e
(
9
)
is
n
o
t
v
a
l
i
d
f
o
r
al
l
v
a
lu
e
o
f
th
e
tim
e
(
t
>
=
0
)
.
H
en
c
e
,
th
e
d
e
s
i
r
e
d
v
a
lu
e
o
f
th
e
t
e
r
m
2
⁄
c
a
n
b
e
d
e
f
in
e
d
b
y
:
1
=
1
1
+
1
+
̇
(
1
0
)
2
=
2
−
1
(
1
1
)
w
h
e
r
e
i
s
t
h
e
s
t
a
b
i
l
i
z
a
t
i
o
n
f
u
n
c
t
i
o
n
.
T
h
e
n
,
i
f
t
h
e
e
r
r
o
r
o
r
1
=
(
1
−
)
i
s
v
a
n
i
s
h
i
n
g
a
s
y
m
p
t
o
t
i
c
a
l
l
y
,
s
o
t
h
e
c
o
n
t
r
o
l
o
b
j
e
c
t
i
v
e
i
s
r
e
a
c
h
e
d
.
T
h
e
r
e
f
o
r
e
,
r
e
p
l
a
c
i
n
g
2
⁄
i
s
(
5
)
by
(
2
+
1
)
,
a
n
d
t
h
e
n
s
u
b
s
t
i
t
u
t
i
n
g
f
r
o
m
(
10
)
,
w
e
c
a
n
g
e
t
:
1
̇
=
−
1
1
+
2
(
1
2
)
Step
2
:
I
n
t
h
is
s
tep
w
e
d
er
iv
ate
t
h
e
er
r
o
r
2
,
an
d
s
u
b
s
titu
tin
g
f
r
o
m
(
6
)
an
d
(
9
)
,
w
e
g
et
th
e
f
o
llo
w
i
n
g
ex
p
r
ess
io
n
:
2
̇
=
−
1
1
+
1
−
1
̇
(
1
3
)
Fro
m
(
7
)
,
w
e
g
et:
1
̇
=
1
2
1
−
1
2
+
1
+
̈
(
1
4
)
T
h
en
,
s
u
b
s
ti
tu
t
in
g
f
r
o
m
(
10
)
,
w
e
g
et:
2
̇
=
−
1
1
+
1
1
−
1
2
1
+
1
2
+
1
−
̈
(
1
5
)
B
y
c
h
o
o
s
in
g
t
h
e
f
o
llo
w
i
n
g
f
u
n
ct
io
n
=
(
1
2
+
2
2
)
,
w
e
ca
n
d
er
i
v
ate
th
e
s
tate
(
1
,
2
)
to
ze
r
o
,
w
h
er
e
th
e
ti
m
e
v
ar
iatio
n
o
f
t
h
e
f
u
n
ctio
n
V
alo
n
g
th
e
(
1
,
2
)
tr
aj
ec
to
r
y
i
s
ex
p
r
es
s
ed
as f
o
llo
w
:
̇
=
1
1
̇
+
2
2
̇
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J
A
p
p
l P
o
w
er
E
n
g
I
SS
N:
2252
-
8792
S
imu
la
tio
n
a
n
d
ex
p
erimen
ta
l v
a
lid
a
tio
n
o
f sin
g
le
-
p
h
a
s
e
s
erie
s
a
ctive
p
o
w
er
…
(
A
b
d
a
lla
h
B
en
A
b
d
elka
d
er
)
83
̇
=
−
1
1
2
−
2
2
2
+
2
[
1
+
2
2
+
2
̇
]
(
1
6
)
w
h
e
r
e
2
i
s
a
d
e
s
i
g
n
p
a
r
am
e
t
e
r
a
n
d
2
̇
i
s
t
o
b
e
s
u
b
s
t
i
t
u
t
e
d
f
r
o
m
(
1
5
)
.
F
r
o
m
(
1
6
)
,
t
h
e
e
q
u
i
l
i
b
r
i
u
m
(
1
,
2
)
=
(
0
,
0
)
is
g
lo
b
all
y
a
s
y
m
p
to
m
at
icall
y
s
ta
b
le
if
th
e
th
ir
d
ter
m
(
−
1
1
2
)
is
s
e
t
to
ze
r
o
.
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h
er
ef
o
r
e,
w
e
ca
n
g
et
t
h
e
f
o
llo
w
in
g
co
n
tr
o
l la
w
[
25
].
=
1
+
(
2
2
−
1
)
1
−
(
1
+
2
)
2
+
+
̈
(
1
7
)
w
h
er
e
t
h
e
o
u
tp
u
t
f
r
o
m
t
h
e
co
n
tr
o
ller
is
th
e
in
v
er
ter
v
o
lta
g
e
i
s
f
u
n
c
tio
n
i
n
m
o
d
u
la
tio
n
i
n
d
ex
.
4.
RE
SU
L
T
S AN
D
D
I
SCU
SS
I
O
N
I
n
th
i
s
s
ec
tio
n
,
w
e
d
is
c
u
s
s
th
e
an
al
y
tica
l
an
d
ex
p
er
i
m
e
n
tal
r
esu
lts
o
f
t
h
e
d
if
f
er
en
t
co
n
tr
o
l
m
e
th
o
d
s
u
s
ed
in
th
i
s
s
t
u
d
y
.
F
ir
s
t,
we
p
r
esen
t
s
i
m
u
latio
n
d
ata,
a
f
ter
w
e
d
is
c
u
s
s
i
ts
r
es
u
lt
s
.
T
h
u
s
,
w
e
p
r
esen
t
ex
p
er
i
m
e
n
tal
r
es
u
lts
w
i
th
a
d
et
ailed
d
is
cu
s
s
io
n
.
4
.
1
.
Si
m
ula
t
io
n r
esu
lt
s
Si
m
u
latio
n
test
h
as
m
ad
e
in
MA
T
L
A
B
Si
m
u
li
n
k
®
in
o
r
d
er
to
im
p
r
o
v
e
t
h
e
p
er
f
o
r
m
a
n
ce
o
f
th
e
u
s
ed
co
n
tr
o
l
s
tr
ateg
ies
s
u
c
h
a
s
P
I
an
d
B
ac
k
s
tep
p
in
g
No
n
li
n
ea
r
.
T
h
e
s
i
m
u
latio
n
cir
cu
it
i
s
s
h
o
w
n
in
F
ig
u
r
e
5
.
T
h
u
s
,
s
i
m
u
lat
io
n
d
ata
ar
e
s
u
m
m
ar
ize
d
in
th
e
f
o
ll
o
w
i
n
g
T
ab
le
1
.
T
h
e
s
i
m
u
latio
n
r
es
u
lts
o
f
th
e
A
P
FS
ar
e
p
r
esen
ted
in
o
r
d
er
to
im
p
r
o
v
e
its
ca
p
ab
ilit
y
o
f
am
el
io
r
atin
g
th
e
p
o
w
er
q
u
alit
y
.
T
h
e
r
esu
lts
ar
e
d
is
cu
s
s
e
d
s
ep
ar
ately
,
w
h
er
e
a
co
m
p
ar
ativ
e
a
n
al
y
s
e
s
i
s
m
a
d
e
b
et
w
ee
n
t
h
e
t
w
o
co
n
tr
o
lle
r
s
in
th
e
ca
s
h
ar
m
o
n
ic
s
.
Fi
g
u
r
es
6
an
d
7
p
r
ese
n
t
th
e
v
o
lta
g
e
w
a
v
ef
o
r
m
s
o
f
P
I
an
d
th
e
B
ac
k
s
tep
p
in
g
n
o
n
l
in
ea
r
.
T
h
e
T
HD
o
f
th
e
s
o
u
r
ce
v
o
ltag
e
w
as
4
8
.
3
4
%.
As
s
h
o
w
n
,
th
e
A
P
FS
h
a
s
co
m
p
e
n
s
ated
th
e
v
o
ltag
e
h
ar
m
o
n
ic
s
s
u
cc
e
s
s
f
u
ll
y
.
T
h
e
t
w
o
co
n
tr
o
ller
s
h
av
e
p
r
esen
ted
v
ar
io
u
s
ex
ce
lle
n
t
r
esu
lt
s
,
w
h
ich
i
s
s
h
o
w
n
t
h
r
o
u
g
h
a
m
elio
r
ati
n
g
t
h
e
v
o
ltag
e
f
o
r
m
a
n
d
r
ed
u
cin
g
th
e
T
HD
v
al
u
e.
T
h
e
T
HD
h
as
b
ec
o
m
e
4
.
1
5
%
w
it
h
P
I
co
n
tr
o
ller
,
1
.
9
8
%
w
it
h
t
h
e
B
ac
k
s
tep
p
in
g
n
o
n
li
n
ea
r
.
T
h
e
f
o
llo
w
i
n
g
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S
[1
]
A
.
P
ra
sa
i
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d
D.
M
.
Div
a
n
,
“
Zero
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n
sa
c
ti
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In
d
u
stry
Ap
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4
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o
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p
p
.
1
7
7
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[2
]
H.
A
k
a
g
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E.
H.
W
a
tan
a
b
e
,
a
n
d
M
.
A
re
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s,
“
In
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s,
2
0
1
7
.
[3
]
E.
C.
d
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s
S
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to
s,
C.
B.
Ja
c
o
b
in
a
,
J.
A
.
A
.
Dia
s
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a
n
d
N.
Ro
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,
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1
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[4
]
A.
Be
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b
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Ex
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2
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5
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0
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,
2
0
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8
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[5
]
A
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M
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S
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.
[6
]
R.
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a
w
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c
e
,
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P
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a
li
ty
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it
y
:
W
h
e
re
d
o
e
s
th
e
re
sp
o
n
sib
il
it
y
li
e
?
En
e
rg
y
En
g
in
e
e
rin
g
,
”
v
o
l.
1
0
6
,
n
o
.
6
,
p
p
.
2
3
-
3
3
,
2
0
0
9
.
[7
]
H.
A
k
a
g
i
a
n
d
K
.
Is
o
z
a
k
i,
“
A
h
y
b
rid
a
c
ti
v
e
f
il
ter
f
o
r
a
th
re
e
-
p
h
a
se
1
2
-
p
u
lse
d
i
o
d
e
re
c
ti
f
ier
u
se
d
a
s
t
h
e
f
ro
n
t
e
n
d
o
f
a
m
e
d
iu
m
-
v
o
lt
a
g
e
m
o
to
r
d
riv
e
,
”
I
EE
E
tra
n
sa
c
ti
o
n
s
o
n
p
o
we
r E
lec
t
ro
n
ics
,
v
o
l.
2
7
,
n
o
.
1
,
p
p
.
6
9
-
7
7
,
2
0
1
2
.
[8
]
A
.
Ch
a
o
u
i,
J.
P
.
G
a
u
b
e
rt,
F
.
Kri
m
,
a
n
d
G
.
Ch
a
m
p
e
n
o
is,
“
P
I
c
o
n
t
ro
ll
e
d
th
re
e
-
p
h
a
se
sh
u
n
t
a
c
ti
v
e
p
o
w
e
r
f
il
ter
f
o
r
p
o
w
e
r
q
u
a
li
ty
im
p
ro
v
e
m
e
n
t,
”
El
e
c
tric P
o
we
r Co
mp
o
n
e
n
ts
a
n
d
S
y
ste
ms
,
v
o
l.
3
5
,
n
o
.
1
2
,
p
p
.
1
3
3
1
-
1
3
4
4
,
2
0
0
7
.
[9
]
M
.
H.
B
o
ll
e
n
a
n
d
M
.
H.
B
o
ll
e
n
,
“
Un
d
e
rsta
n
d
in
g
p
o
w
e
r
q
u
a
li
ty
p
ro
b
lem
s:
v
o
lt
a
g
e
sa
g
s
a
n
d
i
n
terru
p
ti
o
n
s,”
IEE
E
p
re
ss
Ne
w
Y
o
rk
,
2
0
0
0
.
[1
0
]
A
.
S
a
n
n
in
o
,
M
.
G
.
M
il
ler
,
a
n
d
M
.
H.
Bo
l
len
,
“
Ov
e
rv
ie
w
o
f
v
o
lt
a
g
e
sa
g
m
it
ig
a
ti
o
n
,
”
IEE
E
Po
w
e
r
En
g
in
e
e
rin
g
S
o
c
iety
W
in
ter
M
e
e
ti
n
g
,
.
p
p
.
2
8
7
2
-
2
8
7
8
,
2
0
0
0
.
[1
1
]
D.
J.
Ca
rn
o
v
a
le
a
n
d
T
.
J.
Hro
n
e
k
,
“
P
o
w
e
r
q
u
a
li
ty
so
lu
ti
o
n
s
a
n
d
e
n
e
rg
y
sa
v
in
g
s
-
wh
a
t
is
re
a
l?
En
e
rg
y
En
g
in
e
e
rin
g
,
”
v
o
l.
1
0
6
,
n
o
.
3
,
p
p
.
2
6
-
5
0
,
2
0
0
9
.
[1
2
]
N.
A
th
a
n
a
sia
d
is,
“
P
o
w
e
r
q
u
a
li
ty
so
lu
ti
o
n
s
f
o
r
v
o
lt
a
g
e
sa
g
s
u
sin
g
d
y
n
a
m
i
c
v
o
lt
a
g
e
re
sto
re
rs,”
E
lec
tric
Po
we
r
Co
mp
o
n
e
n
ts
a
n
d
S
y
ste
ms
,
v
o
l.
3
1
,
n
o
.
2
,
p
p
.
1
5
9
-
1
7
0
,
2
0
0
3
.
[1
3
]
J.
G
.
Nie
lse
n
,
M
.
Ne
wm
a
n
,
H.
Nie
lse
n
,
a
n
d
F
.
Blaa
b
jerg
.
C
o
n
tr
o
l
a
n
d
tes
ti
n
g
o
f
a
d
y
n
a
m
ic
v
o
lt
a
g
e
r
e
sto
re
r
(DV
R)
a
t
m
e
d
iu
m
v
o
lt
a
g
e
le
v
e
l,
”
IEE
E
T
ra
n
sa
c
ti
o
n
s
o
n
Po
we
r E
lec
tro
n
ics
,
v
o
l.
1
9
,
n
o
.
3
,
p
p
.
8
0
6
-
8
1
3
,
2
0
0
4
.
[1
4
]
P
.
S
.
Ba
b
u
a
n
d
N.
Ka
m
a
ra
j,
“
P
e
rf
o
rm
a
n
c
e
in
v
e
stig
a
ti
o
n
o
f
d
y
n
a
m
i
c
v
o
lt
a
g
e
re
sto
re
r
u
si
n
g
P
I
a
n
d
f
u
z
z
y
c
o
n
tro
ll
e
r,
”
IEE
E
2
0
1
3
I
n
ter
n
a
t
io
n
a
l
C
o
n
fer
e
n
c
e
o
n
Po
we
r,
E
n
e
rg
y
a
n
d
C
o
n
tro
l
(
ICPE
C),
p
p
.
4
6
7
-
4
7
2
,
2
0
1
3
.
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