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2
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p
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[
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e
co
n
t
r
ib
u
tio
n
o
f
t
h
e
d
i
v
i
s
io
n
o
f
t
h
e
lo
ad
cu
r
r
en
t
is
in
ac
co
r
d
an
ce
w
it
h
th
e
cr
o
s
s
-
s
ec
tio
n
al
ar
ea
o
f
co
n
d
u
cto
r
s
.
T
h
is
co
n
ce
p
t
is
ex
p
r
ess
ed
b
y
t
h
e
co
m
p
o
n
e
n
ts
o
f
a
li
n
e
i
n
to
t
w
o
p
ar
allel
lin
e
s
as
eq
u
iv
ale
n
t
l
in
e
r
esis
ta
n
ce
.
C
h
i
n
g
[
1
2
]
d
ev
is
ed
a
m
e
th
o
d
o
f
t
r
an
s
m
is
s
io
n
s
u
p
p
le
m
en
t
c
h
a
r
g
e
allo
ca
tio
n
b
ased
o
n
th
e
to
p
o
lo
g
ical
an
al
y
s
is
o
f
p
o
w
er
f
lo
w
s
an
d
D
C
l
in
ea
r
ized
s
y
s
te
m
m
o
d
el.
T
h
is
m
et
h
o
d
is
ap
p
lied
b
y
ass
u
m
in
g
t
h
er
e
is
a
li
n
e
b
et
wee
n
th
e
t
w
o
e
n
d
s
o
f
th
e
s
h
u
n
t
ca
p
ac
ito
r
,
w
h
ic
h
is
e
x
e
m
p
lifi
ed
as
th
e
f
ictitio
u
s
lo
ad
b
u
s
.
Kaig
u
e
Xie
[
1
3
]
p
r
o
p
o
s
ed
an
an
al
y
tical
m
o
d
el
an
d
alg
o
r
ith
m
b
ased
o
n
th
e
p
o
w
er
f
lo
w
,
u
s
i
n
g
E
x
te
n
d
ed
I
n
cid
en
ce
M
atr
ix
(
E
I
M)
.
I
n
th
is
m
et
h
o
d
,
f
ir
s
tl
y
,
a
co
n
s
tr
u
ctio
n
m
atr
i
x
E
I
M
is
d
ef
i
n
ed
as
Ma
tr
ix
A
,
f
o
llo
w
ed
b
y
d
e
f
in
i
n
g
th
e
lo
s
s
le
s
s
p
o
w
er
n
et
w
o
r
k
(
L
P
N)
th
at
p
r
o
v
id
es
t
h
e
i
n
f
o
r
m
at
io
n
ab
o
u
t
b
u
s
in
cid
en
ce
r
elatio
n
s
an
d
d
ir
ec
tio
n
o
f
ac
tiv
e
p
o
w
er
in
b
r
an
c
h
f
lo
w
.
Y.
A
.
A
ltat
u
r
k
i
[
1
4
]
p
r
o
p
o
s
ed
th
e
u
s
e
o
f
C
u
r
r
en
t
A
d
j
u
s
t
m
en
t
Facto
r
(
C
AF)
m
et
h
o
d
to
allo
ca
te
r
ea
l
an
d
r
ea
ctiv
e
p
o
w
er
d
u
e
to
s
y
s
t
e
m
li
n
ea
r
it
y
.
I
n
r
ef
.
[
1
5
]
S.
No
j
en
g
,
p
r
o
p
o
s
ed
d
et
er
m
in
e
o
f
p
o
w
er
f
lo
w
b
ased
o
n
th
e
p
o
w
er
f
ac
to
r
ap
r
r
o
ac
h
.
T
h
e
r
esu
lt
o
f
t
h
i
s
m
et
h
o
d
h
ad
p
r
o
v
en
it
s
v
alid
it
y
a
n
d
co
n
s
is
te
n
c
y
.
Ho
w
ev
er
,
t
h
is
m
e
th
o
d
f
ailed
to
co
n
s
id
er
th
e
p
o
w
er
f
ac
to
r
o
f
th
e
lo
ad
,
th
o
u
g
h
it
is
o
n
e
o
f
th
e
f
ac
to
r
s
ass
o
ciate
d
w
it
h
th
e
tr
an
s
m
is
s
io
n
tr
a
n
s
f
er
ca
p
ab
ilit
y
.
I
n
g
en
er
al,
th
e
p
o
w
er
f
ac
to
r
d
o
es n
o
t c
h
an
g
e
t
h
e
ac
ti
v
e
p
o
w
er
,
b
u
t it
ca
n
af
f
ec
t p
o
w
er
q
u
a
lit
y
.
T
h
e
im
p
ac
t
o
f
t
h
e
lo
ad
ch
ar
ac
ter
is
tic
(
f
o
r
in
s
tan
ce
;
in
d
u
s
tr
ie
s
lo
ad
,
etc.
)
d
e
p
en
d
s
o
n
th
e
p
o
w
er
f
lo
w
co
n
tr
ib
u
tio
n
o
f
ea
ch
g
e
n
e
r
ato
r
to
th
e
lo
ad
in
ea
ch
tr
an
s
m
is
s
io
n
li
n
e.
Ho
w
ev
er
,
s
e
v
er
al
m
et
h
o
d
s
h
a
v
e
b
ee
n
u
s
ed
to
th
e
p
o
w
er
f
lo
w
tr
ac
in
g
w
it
h
o
u
t t
h
e
co
n
s
id
er
atio
n
o
f
l
o
ad
ch
ar
ac
ter
is
tic.
T
h
is
p
ap
er
is
o
r
g
an
ized
as
f
o
ll
o
w
s
:
Sectio
n
I
I
d
escr
ib
es th
e
c
u
r
r
en
t D
i
s
tr
ib
u
t
io
n
f
ac
to
r
m
et
h
o
d
,
w
h
il
e
Sectio
n
I
I
I
d
escr
ib
es
th
e
p
r
o
p
o
s
ed
m
et
h
o
d
b
ased
o
n
th
e
lo
ad
p
o
w
er
f
ac
to
r
.
A
p
p
licati
o
n
o
f
th
e
p
r
o
p
o
s
ed
m
et
h
o
d
o
n
th
e
I
E
E
E
-
3
0
b
u
s
s
y
s
te
m
an
d
d
is
c
u
s
s
io
n
ar
e
p
r
esen
ted
in
Sectio
n
I
V.
Fin
all
y
,
S
ec
tio
n
VI
co
n
clu
d
es
th
is
s
t
u
d
y
p
ap
er
.
2.
DIS
T
RIB
U
T
I
O
N
F
AC
T
O
R
M
E
T
H
O
D
T
h
e
Dis
tr
ib
u
tio
n
f
ac
to
r
u
s
es
t
h
e
A
f
ac
to
r
to
d
eter
m
i
n
e
t
h
e
p
o
w
er
f
lo
w
i
n
ea
c
h
li
n
e,
t
h
r
o
u
g
h
a
D
C
lin
e
f
lo
w
ap
p
r
o
x
i
m
at
io
n
.
T
h
is
p
r
in
cip
le
u
s
es
t
h
e
to
p
o
lo
g
y
f
ac
to
r
d
is
tr
ib
u
tio
n
b
y
i
m
p
le
m
en
tin
g
a
m
atr
i
x
o
f
Gen
er
atio
n
S
h
i
f
t
Dis
tr
ib
u
tio
n
Facto
r
(
GSDF),
f
ac
to
r
s
Gen
er
alize
Gen
er
atio
n
Dis
tr
ib
u
tio
n
Facto
r
(
GGDF)
an
d
Gen
er
alize
d
L
o
ad
Dis
tr
ib
u
t
io
n
Facto
r
(
G
L
DF)
to
d
eter
m
in
e
th
e
co
n
tr
ib
u
tio
n
o
f
t
h
e
g
e
n
er
ato
r
o
r
th
e
lo
ad
o
n
ea
ch
lin
e.
A
Facto
r
s
ar
e
d
ef
in
ed
as a
s
en
s
itiv
it
y
f
ac
to
r
,
ca
lcu
lated
as
f
o
ll
o
w
s
:
(
1
)
Ma
tr
ix
A
also
is
d
e
f
in
ed
as Ge
n
er
aliza
tio
n
Sh
i
f
t D
i
s
tr
ib
u
t
io
n
Facto
r
(
GSDF).
W
h
er
e
:
Z
ik
=
R
ea
ctan
ce
o
f
li
n
e
f
r
o
m
b
u
s
i to
u
s
er
k
Z
jk
=
R
ea
ctan
ce
o
f
li
n
e
f
r
o
m
b
u
s
j
to
u
s
er
k
X
ij
=
R
ea
ctan
ce
o
f
li
n
e
f
r
o
m
b
u
s
i to
b
u
s
j
Fo
r
d
eter
m
in
in
g
t
h
e
lo
ad
co
n
tr
ib
u
tio
n
to
t
h
e
li
n
e
f
lo
w
,
t
h
e
Gen
er
alize
d
L
o
ad
Di
s
tr
ib
u
ti
o
n
Facto
r
(
C
f
ac
to
r
)
ca
n
b
e
u
s
ed
,
as f
o
ll
o
w
s
:
(
2
)
E
q
u
atio
n
(
2
)
ca
n
also
b
e
d
ef
in
ed
as
a
C
f
ac
to
r
,
w
h
er
e
F
i
-
j
is
p
o
w
er
f
lo
w
in
li
n
e
i
-
j
,
L
k
is
lo
ad
-
k
,
R
i
s
R
ef
er
e
n
ce
b
u
s
a
n
d
k
a
s
b
u
s
-
k
.
T
h
e
Dis
tr
ib
u
t
io
n
Facto
r
i
n
th
e
ex
is
ti
n
g
m
e
th
o
d
s
,
e
x
p
r
ess
ed
as
E
q
u
atio
n
(
1
)
to
(
2
)
,
o
n
ly
s
h
o
w
s
t
h
e
co
n
tr
ib
u
tio
n
o
f
t
h
e
p
o
w
er
f
lo
w
to
ea
c
h
g
en
er
ato
r
an
d
lo
ad
b
ased
o
n
th
e
li
n
e
i
m
p
ed
an
ce
.
T
h
is
m
et
h
o
d
,
h
o
w
e
v
er
,
o
n
l
y
co
n
s
id
er
s
t
h
e
l
i
n
e
p
ar
a
m
ete
r
s
(
lin
e
r
ea
cta
n
ce
)
,
t
h
e
p
o
w
er
in
j
ec
ted
b
y
th
e
ij
jk
ik
k
j
i
x
z
z
A
)
(
,
k
k
R
k
k
k
j
i
j
i
R
j
i
L
L
A
F
C
.
,
)
,
(
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
5
0
2
-
4752
I
n
d
o
n
esia
n
J
E
lec
E
n
g
&
C
o
m
p
Sci,
Vo
l
.
11
,
No
.
3
,
Sep
tem
b
er
201
8
:
1
2
3
6
–
12
42
1238
g
en
er
ato
r
s
a
n
d
th
e
lo
ad
co
n
s
u
m
p
tio
n
,
w
it
h
o
u
t
co
n
s
id
er
in
g
t
h
e
ch
ar
ac
ter
is
tics
o
f
ea
c
h
lo
ad
.
I
n
r
ea
lity
,
it
d
o
e
s
n
o
t r
ef
lect
t
h
e
ch
ar
ac
ter
is
tics
o
f
th
e
lo
ad
(
ie.
lo
ad
p
o
w
er
f
ac
t
o
r
)
w
h
ic
h
ca
n
a
f
f
ec
t th
e
tr
a
n
s
m
is
s
io
n
li
n
e
u
s
a
g
e.
3.
P
RO
P
O
SE
D
M
E
T
H
O
D
P
o
o
r
p
o
w
er
f
ac
to
r
w
i
ll
ca
u
s
e
a
d
ec
r
ea
s
e
in
tr
an
s
m
is
s
io
n
p
o
w
er
tr
a
n
s
f
er
ca
p
ab
ilit
y
.
Fi
g
u
r
e
1
s
h
o
w
s
a
s
in
g
le
li
n
e
d
iag
r
a
m
,
w
it
h
th
e
r
esis
ta
n
ce
ig
n
o
r
ed
.
Fig
u
r
e
1
.
Sin
g
le
li
n
e
d
iag
r
a
m
T
h
e
lin
e
i
m
p
ed
an
ce
ca
n
b
e
ex
p
r
ess
ed
as f
o
llo
w
s
:
(
3
)
T
h
e
r
ea
l p
o
w
er
P
ca
n
b
e
ex
p
r
e
s
s
ed
as f
o
llo
w
s
:
(
4
)
W
h
er
e:
=
A
n
g
le
b
et
w
e
n
th
e
r
ea
ctiv
e
co
m
p
o
n
e
n
t
w
it
h
ac
ti
v
e
co
m
p
o
n
en
in
li
n
e
w
h
er
e
V
an
d
I
a
r
e
th
e
v
o
ltag
e
an
d
cu
r
r
en
t,
r
esp
ec
tiv
e
l
y
.
Z
l
=
L
i
n
e
i
m
p
ed
a
n
ce
W
ith
as
s
u
m
p
tio
n
th
a
t
Xl
i
s
li
n
e
r
ea
ctan
ce
o
f
tr
a
n
s
m
is
s
io
n
l
in
e
b
ased
o
n
r
e
f
er
en
ce
p
o
w
er
f
ac
to
r
s
,
i.e
(
co
s
p
h
i=0
.
8
)
.
Hen
ce
,
b
y
ass
u
m
i
n
g
R
l=0
,
w
e
ca
n
o
b
tain
:
(
5
)
Ass
u
m
in
g
t
h
at
t
h
e
Vb
(
o
ld
)
an
d
Vb
(
n
e
w
)
ar
e
v
o
ltag
e
s
i
n
b
u
s
as
co
n
s
ta
n
t
b
ef
o
r
e
a
n
d
af
ter
i
m
p
r
o
v
in
g
th
e
lo
ad
p
o
w
er
f
ac
to
r
,
th
e
f
o
ll
o
w
i
n
g
E
q
u
a
tio
n
is
f
o
r
m
u
lated
:
(
6
)
T
h
er
ef
o
r
e:
(
7
)
W
ith
co
n
s
id
er
atio
n
o
f
v
al
u
e
f
r
o
m
Fi
g
u
r
e
1
,
E
q
u
atio
n
7
ca
n
b
e
ex
p
r
ess
ed
as f
o
llo
w
s
:
(
8
)
Af
ter
eli
m
i
n
ati
n
g
t
h
e
p
o
w
er
f
l
o
w
i
n
P
(ne
w
)
an
d
P
(old)
,
E
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I
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CASE
S
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0
1
6
M
W
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
d
o
n
esia
n
J
E
lec
E
n
g
&
C
o
m
p
Sci
I
SS
N:
2502
-
4752
Dis
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ased
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in
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f
air
m
a
n
n
er
.
RE
F
E
R
E
NC
E
S
[1
]
S
y
a
rif
u
d
d
in
No
jen
g
,
e
t.
a
l.
Ha
rm
o
n
ic
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o
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Co
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tri
b
u
ti
o
n
f
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th
e
T
r
a
n
s
m
issio
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o
ss
A
ll
o
c
a
ti
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n
in
De
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late
d
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e
rg
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rk
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t:
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Ne
w
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c
h
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m
e
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r
In
d
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stry
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u
p
p
ly
.
J
.
El
e
c
t.
En
g
i
n
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e
rin
g
T
e
c
h
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lo
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y
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v
o
l
1
0
,
n
o
.
1
,
2
0
1
5
,
:
1
-
7
.
[2
]
Biale
k
,
J.
T
ra
c
in
g
th
e
f
lo
w
o
f
e
le
c
tri
c
it
y
.
IEE
Pro
c
e
e
d
in
g
s
-
Ge
n
e
ra
ti
o
n
T
ra
n
sm
issio
n
a
n
d
Distri
b
u
ti
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n
,
1
9
9
6
.
1
4
3
(
4
)
:
3
1
3
-
3
2
0
.
[3
]
Hu
g
h
Ru
d
n
ick
,
M
a
n
u
e
l
S
o
t
o
.
R
o
d
rig
o
P
a
lm
a
.
Us
e
o
f
s
y
ste
m
a
p
p
ro
a
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s
f
o
r
t
ra
n
sm
issio
n
o
p
e
n
a
c
c
e
s
s
p
ricin
g
.
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e
c
trica
l
Po
we
r a
n
d
En
e
rg
y
S
y
st
e
ms
,
1
9
9
9
(2
1
)
:
1
2
5
–
1
3
5
.
[4
]
D.
Kirsc
h
e
n
,
R.
A
ll
a
n
,
a
n
d
G
.
S
tr
b
a
c
.
Co
n
tr
ib
u
ti
o
n
s o
f
in
d
iv
id
u
a
l
g
e
n
e
ra
to
rs t
o
l
o
a
d
s an
d
f
lo
w
s.
IEE
E
T
ra
n
s.
Po
we
r
S
y
ste
m
,
v
o
l.
1
2
,
1
9
9
7
:
5
2
-
60.
[5
]
F
e
li
x
F
.
W
u
,
Y.
Ni,
P
.
W
e
i.
.
P
o
w
e
r
tran
s
f
e
r
a
ll
o
c
a
ti
o
n
f
o
r
o
p
e
n
a
c
c
e
s
s
u
sin
g
g
ra
p
h
th
e
o
ry
–
F
u
n
d
a
m
e
n
tals
a
n
d
a
p
p
li
c
a
ti
o
n
s in
sy
ste
m
s
w
it
h
o
u
t
lo
o
p
f
lo
w
s.
IEE
E
T
ra
n
s.P
o
we
r S
y
st
e
ms
,
2
0
0
0
,
v
o
l
.
1
5
,
n
o
.
3
:
9
2
3
-
9
2
9
.
[6
]
Yu
X
iao
,
P
e
n
g
W
a
n
g
.
T
ra
c
i
n
g
n
o
d
a
l
ma
rk
e
t
p
o
we
r
u
sin
g
p
ro
p
o
rt
io
n
a
l
t
h
re
e
me
th
o
d
”
,
P
o
w
e
r
S
y
st
e
m
s
Co
n
f
e
re
n
c
e
a
n
d
Ex
p
o
siti
o
n
,
2
0
0
4
.
IEE
E
P
ES
.
[7
]
F
e
rd
in
a
n
d
G
u
b
in
a
,
Da
v
id
G
rg
i
a
n
d
Iv
o
Ba
n
i.
A
M
e
th
o
d
f
o
r
De
ter
m
in
in
g
th
e
G
e
n
e
ra
to
rs’
S
h
a
re
in
a
Co
n
s
u
m
e
r
L
o
a
d
.
IEE
E
T
r
a
n
s
a
c
ti
o
n
o
n
P
o
we
r S
y
ste
m,
v
o
l.
1
5
,
n
o
.
4
,
2
0
0
0
:
1
3
7
6
-
1
3
8
1
.
[8
]
P
a
n
to
s,
M
.
,
V
e
rb
ic,
G
.
,
&
G
u
b
in
a
.
F
.
M
o
d
if
ied
to
p
o
l
o
g
ica
l
g
e
n
e
ra
ti
o
n
a
n
d
lo
a
d
d
istri
b
u
ti
o
n
f
a
c
to
rs.
IEE
E
T
ra
n
sa
c
ti
o
n
s
o
n
Po
we
r
S
y
ste
ms
,
2
0
(
4
)2
0
0
5
:
1
9
9
8
-
2
0
0
5
.
[9
]
L
i
m
,
V
.
S
.
C
.
,
M
c
Do
n
a
ld
,
J.
D.
F
.
,
&
S
a
h
a
,
T
.
K.
De
v
e
lo
p
m
e
n
t
o
f
a
n
e
w
lo
ss
a
ll
o
c
a
ti
o
n
m
e
th
o
d
f
o
r
a
h
y
b
rid
e
lec
tri
c
it
y
m
a
r
k
e
t
u
sin
g
g
ra
p
h
th
e
o
ry
.
El
e
c
tric P
o
we
r S
y
ste
ms
Res
e
a
rc
h
,
2
0
0
9
.
[1
0
]
S
h
a
re
e
f
,
Hu
ss
a
in
e
t
a
l.
Re
a
l
a
n
d
Re
a
c
ti
v
e
P
o
w
e
r
T
ra
n
s
f
e
r
Allo
c
a
ti
o
n
Util
izin
g
M
o
d
if
ied
No
d
a
l
Eq
u
a
ti
o
n
s.
In
ter
n
a
t
io
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J
o
u
rn
a
l
o
f
Eme
rg
i
n
g
El
e
c
tric P
o
we
r
S
y
ste
ms
:
V
o
l
.
9
:
Iss
u
e
6
.
[1
1
]
A
b
d
e
lk
a
d
e
r,
S
.
De
ter
m
in
in
g
g
e
n
e
ra
to
rs'
c
o
n
tri
b
u
ti
o
n
t
o
l
o
a
d
s
a
n
d
li
n
e
f
lo
w
s
&
lo
ss
e
s
c
o
n
sid
e
ri
n
g
lo
o
p
f
lo
w
s.
In
ter
n
a
t
io
n
a
l
J
o
u
rn
a
l
o
f
E
lec
trica
l
Po
we
r
&
En
e
rg
y
S
y
ste
ms
,
3
0
(
6
-
7
),
2
0
0
8
:
3
6
8
-
3
7
5
.
[1
2
]
Ch
in
g
-
T
z
o
n
g
S
u
,
Ji
-
Ho
r
n
g
L
ia
w
.
Co
m
p
lex
p
o
w
e
r
f
lo
w
tra
c
in
g
c
o
n
sid
e
rin
g
c
o
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v
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c
ti
o
n
li
n
e
s u
sin
g
n
o
m
in
a
l
-
T
m
o
d
e
l.
In
tern
a
ti
o
n
a
l
Jo
u
rn
a
l
o
f
El
e
c
tri
c
a
l
P
o
w
e
r
&
En
e
rg
y
,
V
o
l
u
m
e
2
9
,
Iss
u
e
1
,
2
0
0
7
:
2
8
–
3
5
.
[1
3
]
Ka
ig
u
i
X
ie,
Jia
q
i
Zh
o
u
,
W
e
n
y
u
a
n
L
i.
A
n
a
l
y
ti
c
a
l
m
o
d
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l
a
n
d
a
lg
o
rit
h
m
f
o
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tra
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in
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a
c
ti
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e
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o
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r
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w
b
a
se
d
o
n
ex
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d
e
d
in
c
i
d
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c
e
m
a
tri
x
.
El
e
c
tr
ic P
o
w
e
r
S
y
ste
m
s Re
se
a
rc
h
,
7
9
,
2
0
0
9
:
3
9
9
–
4
0
5
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
5
0
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-
4752
I
n
d
o
n
esia
n
J
E
lec
E
n
g
&
C
o
m
p
Sci,
Vo
l
.
11
,
No
.
3
,
Sep
tem
b
er
201
8
:
1
2
3
6
–
12
42
1242
[1
4
]
A
lt
u
rk
i,
Y.A
.
T
o
w
a
rd
s
re
a
c
ti
v
e
p
o
w
e
r
m
a
r
k
e
ts.
G
e
n
e
ra
ti
o
n
,
T
ra
n
s
m
is
sio
n
&
Distrib
u
ti
o
n
,
IET
,
V
o
l
u
m
e
:
2
,
Iss
u
e
:
4
,
2
0
0
8
,
p
p
.
5
1
6
–
5
2
9
.
[1
5
]
S
y
a
rif
u
d
d
in
No
jen
g
e
t.
a
l.
Im
p
ro
v
in
g
th
e
M
W
M
il
e
M
e
th
o
d
Us
in
g
th
e
P
o
w
e
r
F
a
c
to
r
A
p
p
ro
a
c
h
f
o
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P
ricin
g
in
th
e
T
ra
n
s
m
issio
n
s L
in
e
S
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rv
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s.
IEE
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T
ra
n
sa
c
ti
o
n
s o
n
P
o
we
r S
y
ste
m,
v
o
l.
2
9
,
n
o
.
5
,
2
0
1
4
:
2
0
4
2
-
2
0
4
8
.
Evaluation Warning : The document was created with Spire.PDF for Python.