T
E
L
KO
M
N
I
KA
T
e
lec
om
m
u
n
icat
ion
,
Com
p
u
t
i
n
g,
E
lec
t
r
on
ics
an
d
Cont
r
ol
Vol.
18
,
No.
3
,
J
une
2020
,
pp.
119
5
~
120
2
I
S
S
N:
1693
-
6930
,
a
c
c
r
e
dit
e
d
F
ir
s
t
G
r
a
de
by
Ke
me
nr
is
tekdikti
,
De
c
r
e
e
No:
21/E
/KP
T
/2018
DO
I
:
10.
12928/
T
E
L
KO
M
NI
KA
.
v18i3.
15113
1195
Jou
r
n
al
h
omepage
:
ht
tp:
//
jour
nal.
uad
.
ac
.
id/
index
.
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Duy
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Hu
n
g
Ha
2
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i
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r
an
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u
an
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y
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h
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h
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ai
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an
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1
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l
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y
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ech
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v
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o
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i
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V
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et
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am
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re
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mmu
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earc
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o
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v
ers
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o
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t
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V
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et
n
am
Ar
t
icle
I
n
f
o
AB
S
T
RA
CT
A
r
ti
c
le
h
is
tor
y
:
R
e
c
e
ived
De
c
31
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2019
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e
vis
e
d
J
a
n
21
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2020
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c
e
pted
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e
b
7
,
2020
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h
e
u
n
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al
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ced
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o
d
e,
n
e
g
at
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v
e
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zer
o
s
e
q
u
e
n
ce,
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ari
a
t
i
o
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o
f
r
eal
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cau
s
e
d
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y
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h
e
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l
i
n
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o
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u
n
b
a
l
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ce
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o
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d
s
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as
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h
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ran
s
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s
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i
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r
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al
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ect
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c
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n
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o
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n
me
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h
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d
s
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r
o
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n
g
as
y
mmet
r
y
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h
e
cri
t
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ca
l
i
s
s
u
e
i
n
reac
t
i
v
e
p
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m
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en
s
at
i
o
n
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s
t
h
e
o
p
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i
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l
cal
c
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l
a
t
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m
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en
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at
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o
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al
u
es
t
h
a
t
i
s
ex
t
reme
l
y
d
i
ff
i
cu
l
t
i
n
co
mp
l
ex
c
i
rcu
i
t
s
.
W
e
p
ro
p
o
s
ed
a
n
o
v
e
l
a
p
p
r
o
ach
t
o
o
v
e
rc
o
me
t
h
es
e
d
i
ff
i
cu
l
t
i
es
b
y
p
r
o
v
i
d
i
n
g
t
h
e
creat
i
o
n
o
f
n
e
w
an
a
l
y
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i
ca
l
co
n
n
ect
i
o
n
s
o
f
t
h
e
s
t
ead
y
-
s
t
a
t
e
mo
d
e
p
arame
t
ers
(v
o
l
t
ag
e
s
,
cu
rren
t
s
)
d
e
p
en
d
s
o
n
t
h
e
co
n
t
r
o
l
l
ed
p
arame
t
er
fo
r
t
h
e
arb
i
t
rar
y
ci
rcu
i
t
s
.
T
h
e
b
a
s
e
o
f
o
u
r
ap
p
ro
ac
h
t
o
react
i
v
e
p
o
w
er
co
m
p
en
s
at
i
o
n
i
s
t
h
e
f
ract
i
o
n
al
-
p
o
l
y
n
o
m
i
al
fu
n
ct
i
o
n
s
.
W
e
p
res
e
n
t
a
n
ew
d
e
s
cri
p
t
i
o
n
o
f
t
h
e
b
e
h
av
i
o
r
o
f
v
o
l
t
ag
e
s
an
d
c
u
rren
t
s
d
ep
e
n
d
i
n
g
o
n
t
h
e
co
n
t
ro
l
l
e
d
p
aramet
er
s
o
f
t
h
e
reac
t
i
v
e
p
o
w
er
co
m
p
en
s
at
i
o
n
d
e
v
i
ce
s
,
an
d
w
e
p
ro
v
e
i
t
s
effect
i
v
e
n
es
s
.
K
e
y
w
o
r
d
s
:
F
r
a
c
ti
ona
l
-
polynom
ial
f
unc
t
ion
M
e
s
h
c
ur
r
e
nt
method
Node
volt
a
ge
method
Optim
iza
ti
on
R
e
gulable
pa
r
a
mete
r
S
ymm
e
tr
iza
ti
on
T
hr
e
e
-
pha
s
e
s
ys
t
e
ms
Th
i
s
i
s
a
n
o
p
en
a
c
ces
s
a
r
t
i
c
l
e
u
n
d
e
r
t
h
e
CC
B
Y
-
SA
l
i
ce
n
s
e
.
C
or
r
e
s
pon
din
g
A
u
th
or
:
Duy
-
Hung
Ha
,
W
ir
e
les
s
C
omm
unica
ti
ons
R
e
s
e
a
r
c
h
Gr
oup,
F
a
c
ult
y
of
E
lec
tr
ica
l
a
nd
E
lec
tr
onics
E
nginee
r
ing,
T
on
Duc
T
ha
ng
Unive
r
s
it
y,
Ho
C
hi
M
inh
C
it
y,
Vi
e
tnam
.
E
mail:
ha
duyhung@tdt
u
.
e
du.
vn
1.
I
NT
RODU
C
T
I
ON
T
he
incr
e
a
s
e
in
the
us
e
of
nonli
ne
a
r
de
vice
s
a
nd
the
unba
lanc
e
of
c
ons
umpt
ion,
in
ge
ne
r
a
l
,
a
r
e
the
c
a
us
e
s
of
a
s
ymm
e
tr
ic
ope
r
a
ti
ng
modes
in
the
powe
r
s
upply
s
ys
tem.
T
he
s
e
de
vice
s
c
a
u
s
e
da
mage
to
the
s
ys
tem
,
e
lec
tr
ica
l
e
quipm
e
nt,
a
nd
e
ne
r
gy
los
s
e
.
C
ons
e
que
ntl
y,
ove
r
c
omi
ng
a
s
ymm
e
tr
y,
whic
h
c
a
n
be
a
c
c
ompl
is
he
d
with
many
methods
,
a
lwa
ys
oc
c
upi
e
s
a
n
im
po
r
tant
plac
e
in
the
s
tudy
of
i
t.
T
he
r
e
a
r
e
s
e
ve
r
a
l
ge
ne
r
a
l
a
nd
popula
r
methods
s
uc
h
a
s
r
e
dis
tr
ibut
ion
of
loads
a
t
pha
s
e
s
,
the
us
e
of
r
e
a
c
ti
ve
powe
r
ge
ne
r
a
tor
s
or
s
pe
c
ial
tr
a
ns
f
or
mer
s
,
the
us
e
of
e
quipm
e
nt
s
tatic
r
e
a
c
ti
ve
powe
r
(
F
AC
T
S
[
1
-
10
]
.
One
of
the
mos
t
i
mpor
tant
is
s
ue
s
of
thes
e
methods
is
the
opti
mal
c
a
lcula
ti
on
of
c
ompens
a
ti
ng
va
lues
.
And
in
ge
ne
r
a
l
,
thi
s
c
a
lcula
ti
on
is
inf
ini
tely
c
ompl
e
x
.
I
t
c
a
n
be
lea
d
to
the
li
mi
tatio
n
of
de
s
c
r
ibi
ng
the
r
e
lations
hip
be
twe
e
n
s
tea
dy
-
s
t
a
te
mode
pa
r
a
mete
r
s
a
nd
the
r
e
gul
a
ble
pa
r
a
mete
r
s
o
f
the
c
o
mpens
a
tor
s
.
I
n
th
is
pa
pe
r
,
we
pr
opos
e
a
nove
l
a
pp
r
oa
c
h
to
ove
r
c
ome
that
dif
f
iculty
f
or
methods
us
ing
s
tatic
c
ompens
a
ti
on
de
vice
s
a
nd
it
c
ould
a
ls
o
be
e
xt
e
nde
d
to
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N
:
1693
-
6930
T
E
L
KO
M
NI
KA
T
e
lec
omm
un
C
omput
E
l
C
ontr
o
l
,
Vol.
18
,
No
.
3
,
J
une
2020:
119
5
-
120
2
1196
the
methods
by
whic
h
it
us
e
s
the
S
ync
hr
onous
ge
ne
r
a
tor
s
.
B
e
c
a
us
e
,
in
pr
inciple
,
s
ync
hr
onous
c
omp
e
ns
a
ti
on
ge
ne
r
a
tor
s
c
a
n
ge
ne
r
a
te
or
a
bs
or
b
r
e
a
c
ti
ve
powe
r
a
nd
withi
n
a
c
e
r
tain
li
mi
t
,
it
c
a
n
be
c
onve
r
ted
to
the
e
q
uivale
nt
of
s
tatic
-
c
ompens
a
ti
ng
de
vice
s
[
11]
.
T
his
pr
oblem
c
a
n
be
s
olve
by
pr
ovid
ing
a
li
nk
be
t
we
e
n
the
pa
r
a
mete
r
s
of
the
s
tea
dy
-
s
tate
mode
a
nd
the
c
ontr
ol
pa
r
a
mete
r
s
of
the
c
ompens
a
tor
.
T
he
r
e
la
ti
ons
hip
is
de
s
c
r
ibed
by
the
f
r
a
c
ti
ona
l
-
polynom
ial
f
unc
ti
on,
whic
h
de
s
c
r
ibes
the
va
r
iation
o
f
volt
a
ge
s
a
nd
c
u
r
r
e
nts
a
c
c
or
ding
to
r
e
gulable
pa
r
a
mete
r
s
[
12
-
21]
.
I
n
S
e
c
ti
on
I
I
,
we
will
pr
e
s
e
nt
the
pr
oblem
that
is
the
a
ns
we
r
to
how
to
ge
t
the
f
unc
ti
on
,
a
s
mentioned
e
a
r
li
e
r
,
a
l
ong
with
the
c
ompar
is
on
of
it
s
pr
e
c
is
ion
thr
ough
a
n
e
xa
mpl
e
.
I
n
S
e
c
ti
on
I
I
I
,
we
will
pr
e
s
e
nt
s
ome
r
e
s
ult
s
that
ha
ve
be
e
n
made
f
or
op
ti
mi
z
ing
the
e
lec
tr
ica
l
s
ys
tem
of
a
glas
s
f
a
c
tor
y
that
ope
r
a
tes
in
t
he
a
s
ymm
e
tr
ic
mode.
2.
F
RA
CT
I
ONAL
-
P
OL
YN
OMI
A
L
F
UN
CT
I
ONS
2.
1.
No
d
e
volt
age
s
m
e
t
h
od
W
e
c
ons
ider
a
thr
e
e
-
pha
s
e
c
ir
c
uit
c
ons
is
ti
ng
of
(
n
+
1)
node
s
a
nd
m
(
n+
1
<
m)
s
o
we
ha
ve
the
matr
ix
1
2
m
=
A
1
11
a
12
a
…
1
m
a
2
21
a
22
a
…
2
m
a
…
…
…
…
…
n
1
n
a
2
n
a
…
nm
a
whe
r
e
1
;
1
;
1
;
ij
a
i
n
j
m
=
=
=
node
;
1
ij
a
=−
–
e
nter
s
;
0
ij
a
=
Ve
c
tor
of
the
c
onduc
tanc
e
of
the
br
a
nc
he
s
is
(
)
12
d
i
a
g
,
,
m
Y
Y
Y
=
Y
Ve
c
tor
s
c
ur
r
e
nt
a
nd
e
lec
tr
omot
ive
f
o
r
c
e
s
our
c
e
s
a
r
e
give
a
s
(
)
12
,
,
.
t
m
J
J
J
=
J
(
)
12
,
,
.
t
m
E
E
E
=
E
T
he
node
volt
a
ge
e
qua
ti
ons
a
r
e
f
or
mul
a
ted
a
s
in
[
4
,
5]
(
)
0
t
=
−
+
A
Y
A
U
J
Y
E
(
1
)
whe
r
e
(
)
0
1
2
,
,
t
n
U
U
U
=
U
–
ve
c
tor
o
f
the
node
volt
a
ge
s
.
He
r
e
t
=
AYA
B
is
the
matr
ix
of
the
a
ggr
e
ga
te
c
onduc
tanc
e
,
th
e
n
the
ve
c
tor
e
quivale
nt
c
ur
r
e
nt
s
o
ur
c
e
s
+=
J
Y
E
C
c
a
n
be
r
e
wr
i
tt
e
n
a
s
the
f
oll
owing
1
…
i
…
n
B
=
1
1
,
1
B
…
1,
i
B
…
1,
n
B
…
…
…
…
…
…
i
,1
i
B
…
,
ii
B
…
,
in
B
…
…
…
…
…
…
n
,1
n
B
…
,
ni
B
…
,
nn
B
a
nd
(
)
1
,
,
,
,
t
in
C
C
C
=
C
whe
r
e
,
,
1
i
j
k
n
=
(
1)
be
c
omes
0
=
B
U
C
T
he
node
volt
a
ge
s
c
a
n
be
f
or
mul
a
ted
a
s
d
e
t
d
e
t
i
i
U
=
B
B
Evaluation Warning : The document was created with Spire.PDF for Python.
T
E
L
KO
M
NI
KA
T
e
lec
omm
un
C
omput
E
l
C
ontr
o
l
N
ov
e
l
de
pe
nde
nc
ies
of
c
ur
r
e
nts
and
v
olt
age
s
in
po
w
e
r
s
y
s
tem
s
teady
s
tat
e
mode
on
…
(
P
hu
T
r
an
T
in
)
1197
T
he
matr
ix
de
ter
mi
na
nts
o
f
B
a
n
d
i
B
a
r
e
de
f
ined
a
s
f
oll
ows
0
1
2
3
4
5
6
7
de
t
a
a
x
a
y
a
z
a
x
y
a
x
z
a
y
z
a
x
y
z
=
+
+
+
+
+
+
+
B
0
1
2
3
4
5
6
7
de
t
i
i
i
i
i
i
i
i
i
b
b
x
b
y
b
z
b
x
y
b
x
z
b
y
z
b
x
y
z
=
+
+
+
+
+
+
+
B
.
T
a
ke
thes
e
two
e
qua
ti
ons
divi
de
d
by
0
a
a
nd
de
noted
by
0
/
;
1
7
pp
a
a
p
=
=
a
nd
,
0
,
/
;
0
7
q
i
q
i
b
a
c
q
=
=
,
we
got
:
0
1
2
3
4
5
6
7
1
2
3
4
5
6
7
d
e
t
d
e
t
1
i
i
i
i
i
i
i
i
i
i
c
c
x
c
y
c
z
c
x
y
c
x
z
c
y
z
c
x
y
z
U
x
y
z
x
y
x
z
y
z
x
y
z
+
+
+
+
+
+
+
=
=
=
+
+
+
+
+
+
+
B
B
(
2
)
whe
r
e
,
c
oe
f
f
icie
nts
07
cc
và
17
a
r
e
c
ompl
e
x
numbe
r
s
;
x
,
y
,
a
nd
z
a
r
e
r
e
a
l
number
s
;
1
in
=
.
T
he
c
ur
r
e
nt
f
low
in
that
b
r
a
nc
h
f
r
om
k
to
j
is
e
qua
l
to
:
(
)
.
i
k
j
i
i
I
U
U
E
Y
=
−
+
(
3
)
the
c
ur
r
e
nts
in
the
ge
ne
r
a
l
f
or
a
ll
br
a
nc
he
s
a
r
e
a
s
f
o
ll
ows
:
0
1
2
3
4
5
6
7
1
2
3
4
5
6
7
1
i
i
i
i
i
i
i
i
i
d
d
x
d
y
d
z
d
x
y
d
x
z
d
y
z
d
x
y
z
I
x
y
z
x
y
x
z
y
z
x
y
z
+
+
+
+
+
+
+
=
+
+
+
+
+
+
+
(
4
)
it
c
a
n
be
s
e
e
n
that
in
(
3)
,
the
c
omponent
in
pa
r
e
nthes
e
s
is
in
the
f
or
m
o
f
(
2)
,
whic
h
is
the
vo
lt
a
ge
on
the
c
ons
umpt
ion
load
of
the
i
-
th
b
r
a
nc
h.
W
e
labe
l
,
k
j
i
b
r
i
U
U
E
U
−
+
=
.
I
n
the
c
a
lcula
ti
on
of
a
ll
the
c
ur
r
e
nts
in
the
b
r
a
nc
he
s
of
the
c
ir
c
uit
in
(
3)
we
obtaine
d
the
pr
ope
r
ti
e
s
that
will
be
us
e
d
late
r
f
or
f
indi
ng
the
c
oe
f
f
icie
nts
of
th
e
f
unc
ti
ons
(
2)
a
nd
(
4)
,
a
s
f
oll
ows
:
I
f
x
(
Ohm
)
is
c
onne
c
ted
in
pa
r
a
ll
e
l
with
i
-
th
br
a
nc
h,
a
nd
we
labe
l
(
)
1
/
1
/
i
i
i
Y
Y
j
x
j
Y
x
j
x
=
+
=
+
,
wh
e
r
e
2
1
j
=−
;
i
Y
–
c
ompl
e
x
c
onduc
tanc
e
of
i
-
th
br
a
nc
h.
then,
,
br
i
U
a
s
f
oll
ows
:
(
)
0
1
2
3
,
1
2
3
4
5
6
7
1
,
1
i
i
i
i
i
b
r
i
jx
e
e
y
e
z
e
y
z
jY
x
U
x
y
z
x
y
x
z
y
z
x
y
z
+
+
+
+
=
+
+
+
+
+
+
+
a
nd
ther
e
f
or
e
:
(
)
0
1
2
3
1
2
3
4
5
6
7
1
i
i
i
i
i
e
e
x
e
z
e
x
z
I
x
y
z
x
y
x
z
y
z
x
y
z
+
+
+
=
+
+
+
+
+
+
+
(
5
)
s
im
il
a
r
ly
,
i
f
y
(
or
z
)
(
Ohm)
is
c
onne
c
ted
in
pa
r
a
ll
e
l
with
i
-
th
b
r
a
nc
h.
(
)
0
1
2
3
1
2
3
4
5
6
7
1
i
i
i
i
i
e
e
x
e
y
e
x
y
I
x
y
z
x
y
x
z
y
z
x
y
z
+
+
+
=
+
+
+
+
+
+
+
(
6
)
or
(
)
0
1
2
3
1
2
3
4
5
6
7
1
i
i
i
i
i
e
e
y
e
z
e
y
z
I
x
y
z
x
y
x
z
y
z
x
y
z
+
+
+
=
+
+
+
+
+
+
+
(
7
)
If
x
(
y
or
z
)
(
Ohm)
is
c
onne
c
ted
in
s
e
r
ial
with
i
-
th
br
a
nc
h,
a
nd
we
labe
l
(
)
1
/
1
/
i
i
i
Y
Y
j
x
j
x
Y
=
+
=
+
,
then,
,
br
i
U
a
s
f
oll
ows
:
(
)
0
2
3
6
,
1
2
3
4
5
6
7
1
,
1
i
i
i
i
b
r
i
jx
f
f
y
f
z
f
y
z
jx
U
x
y
z
x
y
x
z
y
z
x
y
z
+
+
+
+
=
+
+
+
+
+
+
+
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N
:
1693
-
6930
T
E
L
KO
M
NI
KA
T
e
lec
omm
un
C
omput
E
l
C
ontr
o
l
,
Vol.
18
,
No
.
3
,
J
une
2020:
119
5
-
120
2
1198
a
nd
(
)
0
2
3
6
1
2
3
4
5
6
7
1
i
i
i
i
i
f
f
y
f
z
f
y
z
I
x
y
z
x
y
x
z
y
z
x
y
z
+
+
+
=
+
+
+
+
+
+
+
(
8
)
(
)
0
2
3
6
1
2
3
4
5
6
7
1
i
i
i
i
i
f
f
x
f
z
f
x
z
I
x
y
z
x
y
x
z
y
z
x
y
z
+
+
+
=
+
+
+
+
+
+
+
(
9
)
(
)
0
2
3
6
1
2
3
4
5
6
7
1
i
i
i
i
i
f
f
x
f
y
f
x
y
I
x
y
z
x
y
x
z
y
z
x
y
z
+
+
+
=
+
+
+
+
+
+
+
(
10
)
to
f
ind
a
ll
the
c
oe
f
f
icie
nts
of
f
unc
ti
ons
(
2)
a
nd
(
4)
,
f
i
r
s
t,
we
ne
e
d
to
s
olve
a
li
ne
a
r
a
lgeb
r
a
ic
s
ys
tem
of
15
e
qua
ti
ons
a
nd
then
s
olve
the
e
qua
ti
on
s
ys
tems
of
8
e
qua
ti
ons
.
How
e
ve
r
,
by
a
na
lyzing
the
c
ur
r
e
nt
f
low
in
the
br
a
nc
h
with
the
c
ompens
a
ti
ng
de
vice
s
,
the
nu
mber
of
e
qua
ti
ons
o
f
the
s
ys
tems
de
c
r
e
a
s
ing,
r
e
s
pe
c
ti
ve
ly,
is
11
a
nd
8.
2.
2.
M
e
s
h
c
u
r
r
e
n
t
m
e
t
h
od
W
he
n
we
a
na
lyze
d
a
s
im
il
a
r
c
ir
c
uit
by
the
mes
h
c
u
r
r
e
nts
method
[
11
-
15]
:
(
)
t
l
l
s
=+
A
Z
A
I
Z
J
E
or
l
s
l
=
B
I
C
whe
r
e
,
l
A
–
matr
ix
of
mes
h
c
ur
r
e
nts
method,
it
s
s
ize
(
(
)
1)
m
n
m
−+
;
Z
–
diagona
l
matr
ix
of
the
r
e
s
is
tor
s
of
the
br
a
nc
he
s
;
s
=+
I
I
J
–
ve
c
tor
tot
a
l
e
lec
tr
ic
c
u
r
r
e
nts
of
the
br
a
nc
he
s
;
t
l
l
l
=
A
Z
A
B
;
l
+=
Z
J
E
C
.
T
he
c
ur
r
e
nt
of
the
i
-
th
mes
h
(
)
(
)
11
i
m
n
=
−
+
is
a
s
f
ol
lows
:
0
1
2
3
4
5
6
7
1
2
3
4
5
6
7
d
e
t
d
e
t
1
l
i
i
i
i
i
i
i
i
i
si
l
g
g
x
g
y
g
z
g
x
y
g
x
z
g
y
z
g
x
y
z
I
x
y
z
x
y
x
z
y
z
x
y
z
+
+
+
+
+
+
+
==
+
+
+
+
+
+
+
B
B
(
11
)
whe
r
e
c
oe
f
f
icie
nts
7
gg
a
r
e
c
ompl
e
x
number
s
.
17
in
thi
s
c
a
s
e
,
ha
s
the
s
a
me
va
lue
a
s
the
c
oe
f
f
icie
nts
17
,
it
mea
ns
:
0
1
2
3
4
5
6
7
1
2
3
4
5
6
7
1
i
i
i
i
i
i
i
i
si
d
d
x
d
y
d
z
d
x
y
d
x
z
d
y
z
d
x
y
z
I
x
y
z
x
y
x
z
y
z
x
y
z
+
+
+
+
+
+
+
==
+
+
+
+
+
+
+
.
T
he
c
ur
r
e
nt
f
lows
in
j
-
th
br
a
nc
h
(
)
1
jm
=
c
a
n
be
f
ound
by
s
ome
s
im
ple
c
a
lcula
ti
ons
a
nd
tr
a
ns
f
or
mations
f
r
om
ve
c
to
r
s
I
:
0
1
2
3
4
5
6
7
1
2
3
4
5
6
7
1
j
j
j
j
j
j
j
i
j
e
e
x
e
y
e
z
e
x
y
e
x
z
e
y
z
e
x
y
z
I
x
y
z
x
y
x
z
y
z
x
y
z
+
+
+
+
+
+
+
=
+
+
+
+
+
+
+
(
12
)
by
a
na
lyzing
s
im
il
a
r
in
the
p
r
e
vious
s
e
c
ti
on,
we
a
ls
o
ge
t
the
s
a
me
r
e
s
ult
s
a
s
(
5
-
10)
.
2.
3.
Ot
h
e
r
c
irc
u
it
a
n
alys
is
m
e
t
h
od
s
S
im
il
a
r
r
e
s
ult
s
we
r
e
a
ls
o
obtaine
d
us
ing
the
meth
od
of
loop
c
ur
r
e
nts
,
e
quivale
nt
tr
a
ns
f
or
mations
of
the
c
ir
c
uit
[
11
-
15]
.
2.
4.
Ot
h
e
r
c
as
e
s
of
f
r
ac
t
ion
a
l
-
p
olyn
o
m
ial
f
u
n
c
t
i
on
s
B
y
a
na
lyzing
the
c
ir
c
uit
a
s
in
s
e
c
ti
on,
A,
whe
n
on
ly
one
a
nd
two
c
ompens
a
ti
on
de
v
ice
s
we
r
e
us
e
d,
we
got:
Evaluation Warning : The document was created with Spire.PDF for Python.
T
E
L
KO
M
NI
KA
T
e
lec
omm
un
C
omput
E
l
C
ontr
o
l
N
ov
e
l
de
pe
nde
nc
ies
of
c
ur
r
e
nts
and
v
olt
age
s
in
po
w
e
r
s
y
s
tem
s
teady
s
tat
e
mode
on
…
(
P
hu
T
r
an
T
in
)
1199
01
1
01
1
1
1
ii
i
ii
i
a
a
x
U
x
b
b
x
I
x
+
=
+
+
=
+
(
13)
a
nd
0
1
2
3
1
2
3
0
1
2
3
1
2
3
1
1
i
i
i
i
i
i
i
i
i
i
a
a
x
a
y
a
x
y
U
x
y
x
y
b
b
x
b
y
b
x
y
I
x
y
x
y
+
+
+
=
+
+
+
+
+
+
=
+
+
+
(
14)
c
oe
f
f
icie
nts
03
aa
;
13
a
r
e
c
ompl
e
x
number
s
.
T
he
s
e
r
e
s
ult
s
c
a
n
a
ls
o
be
de
r
ived
f
r
om
(
2)
a
nd
(
4
)
.
As
s
umi
ng
that,
we
dis
c
onne
c
t
the
c
ompens
a
tor
(
z
-
Ohm)
out
of
the
c
ir
c
uit
,
whic
h
wa
s
in
pa
r
a
ll
e
l
,
it
mea
ns
z
→
∞
,
then
,
(
)
(
)
0
1
2
3
4
5
6
7
1
2
3
4
5
6
7
0
1
2
4
3
5
6
7
1
2
4
3
5
6
7
1
1
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
c
c
x
c
y
c
z
c
x
y
c
x
z
c
y
z
c
x
y
z
jz
U
x
y
z
x
y
x
z
y
z
x
y
z
jz
c
c
x
c
y
c
x
y
j
c
c
x
c
y
c
x
y
jz
x
y
x
y
j
x
y
x
y
jz
+
+
+
+
+
+
+
=
+
+
+
+
+
+
+
+
+
+
−
+
+
+
=
+
+
+
−
+
+
+
3
5
6
7
3
5
6
7
i
i
i
i
i
c
c
x
c
y
c
x
y
U
x
y
x
y
+
+
+
=
+
+
+
be
c
a
us
e
0
1
2
4
1
2
4
1
0
;
0
i
i
i
i
c
c
x
c
y
c
x
y
x
y
x
y
j
z
j
z
+
+
+
+
+
+
==
if
we
c
onti
nue
to
dis
c
onne
c
t
the
c
ompens
a
tor
out
o
f
the
c
ir
c
uit
(
y
-
Ohm)
,
whic
h
c
onne
c
ted
in
pa
r
a
ll
e
l
,
it
mea
ns
y
→
∞
,
then
,
(
)
(
)
3
5
6
7
3
5
67
67
3
5
6
7
3
5
67
67
i
i
i
i
i
i
ii
ii
i
c
c
x
c
y
c
x
y
c
c
x
j
c
c
x
c
c
x
jy
jy
U
x
y
x
y
x
x
jx
jy
jy
+
+
+
+
−+
+
=
=
=
+
+
+
+
+
−+
the
s
a
me
f
or
the
c
ur
r
e
nts
a
nd
in
the
c
a
s
e
of
c
omp
e
ns
a
tor
s
a
r
e
c
onne
c
ted
in
s
e
r
ies
.
T
hus
,
in
thi
s
s
e
c
ti
on,
we
s
how
how
we
got
the
f
r
a
c
ti
ona
l
-
polynom
ial
f
unc
ti
ons
[
22
-
27]
.
3.
NU
M
E
RI
C
AL
RE
S
UL
T
S
AN
D
DI
S
CU
S
S
I
ON
3.
1.
T
e
s
t
in
g
Ne
xt,
we
c
ompar
e
the
dif
f
e
r
e
nc
e
be
twe
e
n
the
r
e
s
ult
s
of
the
c
a
lcula
ti
on
of
the
c
ur
r
e
nt
a
nd
volt
a
ge
b
y
the
pr
opos
e
d
f
unc
ti
on
a
nd
by
the
us
ua
l
s
olut
ion.
F
or
the
c
ir
c
uit
de
s
c
r
ibed
in
F
ig
ur
e
1
(
in
the
c
a
s
e
of
two
c
ompens
a
ti
on
de
vice
s
a
r
e
c
onne
c
ted
in
s
e
r
ies
)
.
N
ote
that
Va
lues
x
1
a
nd
x
2
c
a
n
be
ne
ga
ti
ve
(
c
a
pa
c
it
ive)
or
pos
it
ive
(
inductive)
.
L
oa
d
1
a
nd
load
2
in
the
ge
ne
r
a
l
c
a
s
e
c
a
n
be
in
a
tr
iangula
r
c
onne
c
ti
on
or
s
tar
(
with/
without
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N
:
1693
-
6930
T
E
L
KO
M
NI
KA
T
e
lec
omm
un
C
omput
E
l
C
ontr
o
l
,
Vol.
18
,
No
.
3
,
J
une
2020:
119
5
-
120
2
1200
ne
utr
a
l
wi
r
e
)
.
T
o
f
ind
a
ll
the
c
oe
f
f
icie
nts
of
f
unc
ti
ons
(
14)
,
f
i
r
s
t,
we
ne
e
d
to
s
olve
a
l
inea
r
a
lgebr
a
ic
s
ys
tem
of
7
e
qua
ti
ons
a
nd
then
s
olve
the
e
qua
ti
on
s
ys
tems
of
4
e
qu
a
ti
ons
.
How
e
ve
r
,
i
f
the
a
r
gument
is
the
s
a
me
a
s
to
ge
t
the
(
5
-
10)
,
the
number
of
e
qua
ti
ons
of
the
s
ys
tems
de
c
r
e
a
s
ing,
r
e
s
pe
c
ti
ve
ly,
is
5
a
nd
4.
F
r
om
ther
e
we
ge
t
the
f
unc
ti
ons
that
de
s
c
r
ibe
the
de
pe
nde
nc
ies
of
vol
tage
s
a
nd
c
ur
r
e
nts
on
the
r
e
gulable
pa
r
a
met
e
r
s
,
we
labe
led
(
)
,
1
2
,
i
p
r
o
p
o
s
e
U
x
x
a
nd
(
)
,
1
2
,
i
p
r
o
p
o
s
e
I
x
x
.
T
o
f
ind
the
c
ur
r
e
nt
a
nd
volt
a
ge
of
the
i
-
th
br
a
nc
h
a
t
the
(
)
12
,
xx
,
jus
t
put
1
x
a
nd
2
x
in
the
f
unc
ti
ons
(
)
,
1
2
,
i
p
r
o
p
o
s
e
U
x
x
a
nd
(
)
,
1
2
,
i
p
r
o
p
o
s
e
I
x
x
.
T
he
c
or
r
e
c
t
c
ur
r
e
nts
a
nd
volt
a
ge
s
c
a
n
be
f
ound
s
o
lvi
ng
the
c
ir
c
uit
whe
n
given
(
)
12
,
xx
,
we
labe
led
(
)
,
1
2
,
i
c
o
r
r
e
c
t
U
x
x
a
nd
(
)
,
1
2
,
i
c
o
r
r
e
c
t
I
x
x
.
T
he
di
f
f
e
r
e
nc
e
be
twe
e
n
the
two
r
e
s
ult
s
that
we
r
e
mentioned
a
bove
a
s
s
hown
in
F
igur
e
s
2
a
nd
3
.
T
he
di
f
f
e
r
e
nc
e
be
twe
e
n
the
two
r
e
s
ult
s
o
f
the
c
a
s
e
of
one
c
ompens
a
ti
on
d
e
vice
is
c
onne
c
ted
in
the
s
e
r
ial
wa
s
s
hown
in
F
igur
e
4.
I
n
t
he
c
a
s
e
s
of
thr
e
e
c
ompens
a
tor
s
a
r
e
c
onne
c
ted
in
s
e
r
ial
or
o
f
one
/t
wo/thr
e
e
or
mo
r
e
c
ompens
a
ti
on
de
vice
(
s
)
is
(
a
r
e
)
c
onne
c
ted
in
pa
r
a
l
lel
a
r
e
a
ls
o
tes
ted
a
nd
g
e
ne
r
a
ll
y,
the
dif
f
e
r
e
nc
e
is
ti
ny,
a
ppr
oxim
a
tely
10
-
7
%.
F
igur
e
1.
M
ode
li
ng
of
e
lec
tr
ica
l
s
ys
tems
F
igur
e
2
.
T
he
dif
f
e
r
e
nc
e
be
twe
e
n
the
c
ur
r
e
nts
F
igur
e
3.
T
he
dif
f
e
r
e
nc
e
be
twe
e
n
the
volt
a
ge
s
3.
2.
App
li
c
a
t
ion
T
he
pr
opos
e
d
f
r
a
c
ti
ona
l
-
polynom
ial
f
unc
ti
on
ha
s
be
e
n
a
ppli
e
d
to
opti
mi
z
ing
the
e
lec
tr
ica
l
s
ys
tem
of
the
glas
s
f
a
c
tor
y
ope
r
a
ti
ng
in
a
s
ymm
e
tr
ic
mode
,
w
hich
wa
s
mentioned
in
the
pr
e
vious
a
r
ti
c
le
[
15
]
.
I
n
F
igur
e
5
is
one
of
the
r
e
s
ult
s
us
ing
the
pr
opos
e
d
f
unc
ti
on
f
or
opti
mal
c
a
lcula
ti
on,
in
whic
h
c
a
s
e
we
us
e
only
two
c
ompens
a
tor
s
.
I
t
c
a
n
be
s
e
e
n
that
the
c
ur
r
e
nts
a
nd
volt
a
ge
s
ha
ve
be
e
n
s
igni
f
ica
ntl
y
im
p
r
ove
d
c
om
pa
r
e
d
to
S
y
st
e
m
Lo
a
d
1
L
o
a
d
2
A
1
B
1
C
1
A
2
B
2
C
2
D
2
D
1
jx
2
=
j
ω
L
2
jx
1
=
j
ω
L
1
Evaluation Warning : The document was created with Spire.PDF for Python.
T
E
L
KO
M
NI
KA
T
e
lec
omm
un
C
omput
E
l
C
ontr
o
l
N
ov
e
l
de
pe
nde
nc
ies
of
c
ur
r
e
nts
and
v
olt
age
s
in
po
w
e
r
s
y
s
tem
s
teady
s
tat
e
mode
on
…
(
P
hu
T
r
an
T
in
)
1201
F
igur
e
6
.
T
oge
ther
wi
th
the
r
e
s
ult
s
hown
in
F
igur
e
7
a
nd
the
r
e
s
ult
s
mentioned
in
the
pr
e
vious
a
r
ti
c
le
s
,
a
ll
us
e
the
pr
opos
e
d
f
unc
ti
on
in
the
opti
mi
z
a
ti
on
ha
s
pr
ov
e
d
it
s
e
f
f
e
c
ti
ve
ne
s
s
.
F
igur
e
4.
T
he
dif
f
e
r
e
nc
e
be
twe
e
n
the
c
ur
r
e
nts
F
igur
e
5.
T
he
dif
f
e
r
e
nc
e
be
twe
e
n
the
volt
a
ge
s
F
igur
e
6.
B
e
f
or
e
c
ompens
a
ti
on
F
igur
e
7.
Af
ter
c
ompens
a
ti
on
4.
CONC
L
USI
ON
T
he
main
is
s
ue
of
thi
s
pa
pe
r
that
we
w
ould
li
ke
to
e
mphas
ize
is
the
f
indi
ng
of
the
f
r
a
c
ti
ona
l
-
polynom
ial
f
unc
ti
on
that
de
s
c
r
ibes
the
va
r
iation
of
volt
a
ge
a
nd
c
ur
r
e
nt
a
c
c
or
ding
to
the
r
e
gulable
pa
r
a
mete
r
s
o
f
the
c
ompens
a
tor
s
.
T
h
is
pr
opos
a
l
c
a
n
be
a
ppli
e
d
to
the
opti
mal
c
omput
a
ti
on
of
r
e
a
c
ti
ve
powe
r
c
ompens
a
ti
on
s
ys
tems
that
us
e
s
tati
c
VA
R
c
ompens
a
tor
s
a
nd
the
a
bil
it
y
of
e
xtens
ion
f
or
a
f
e
w
other
e
xc
e
pti
ona
l
c
a
s
e
s
.
T
he
int
r
oduc
ti
on
of
a
f
unc
ti
on
de
s
c
r
ibi
ng
the
f
unda
menta
l
qua
nti
ti
e
s
of
the
e
lec
tr
ica
l
s
ys
tems
(
volt
a
ge
a
nd
c
u
r
r
e
nt)
in
the
de
pe
nde
nc
ies
on
the
va
lue
o
f
the
c
ompens
a
tor
in
t
he
ge
ne
r
a
l
c
a
s
e
is
of
c
ons
ider
a
ble
s
igni
f
ica
nc
e
,
whic
h
make
s
the
c
a
lcula
ti
on
mor
e
c
onve
nient
a
nd
quicke
r
.
RE
F
E
RE
NC
E
S
[1
]
So
n
g
Y
.
H
.
,
J
o
h
n
s
A
.
,
“
Fl
ex
i
b
l
e
A
C
T
ra
n
s
m
i
s
s
i
o
n
Sy
s
t
e
ms
(FA
CT
S),
”
IE
E
E
,
1
9
9
9
.
[2
]
H
i
n
g
o
ran
i
N
arai
n
,
G
.
,
L
as
zl
o
G
y
u
g
y
i
,
“
U
n
d
ers
t
an
d
i
n
g
FA
CT
S
:
Co
n
ce
p
t
s
an
d
T
ech
n
o
l
o
g
y
o
f
Fl
ex
i
b
l
e
A
C
T
ran
s
mi
s
s
i
o
n
Sy
s
t
ems
,
”
W
i
l
ey
-
I
E
E
E
P
r
es
s
,
1
9
9
9
.
[3
]
Co
at
e
s
D
.
,
"
FA
CT
S:
A
T
ran
s
mi
s
s
i
o
n
U
t
i
l
i
t
y
Pers
p
ec
t
i
v
e
,
"
IE
E
Co
l
l
o
q
u
i
u
m
F
l
exi
b
l
e
A
C
Tr
a
n
s
m
i
s
s
i
o
n
S
y
s
t
em
t
h
e
F
A
CTS
,
1
9
9
8
.
[4
]
X
i
a
o
P.
Z
.
,
"
FA
CT
S
-
D
ev
i
ces
a
n
d
A
p
p
l
i
ca
t
i
o
n
s
,
"
F
l
ex
i
b
l
e
A
C
Tr
a
n
s
m
i
s
s
i
o
n
S
ys
t
em
s
:
M
o
d
e
l
l
i
n
g
a
n
d
Co
n
t
r
o
l
P
o
wer
S
ys
t
em
s
,
pp
. 1
-
3
0
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Evaluation Warning : The document was created with Spire.PDF for Python.
I
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C
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l
,
Vol.
18
,
No
.
3
,
J
une
2020:
119
5
-
120
2
1202
[6
]
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“
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5
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3
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G
rai
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J
.
J.
,
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D.
, “
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4
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5
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6
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8
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9
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0
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2
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,
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3
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4
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i
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.
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Po
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[2
5
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G
al
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-
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2
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.
[2
6
]
Mah
mo
u
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,
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arar,
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h
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2
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.
[2
7
]
V
.
Q
.
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