I
nte
rna
t
io
na
l J
o
urna
l o
f
Appl
ied P
o
wer
E
ng
i
neer
ing
(
I
J
AP
E
)
Vo
l.
14
,
No
.
4
,
Dec
em
b
er
20
25
,
p
p
.
9
2
3
~
9
3
3
I
SS
N:
2252
-
8
7
9
2
,
DOI
:
1
0
.
1
1
5
9
1
/ijap
e
.
v
14
.
i
4
.
pp
923
-
933
923
J
o
ur
na
l ho
m
ep
a
g
e
:
h
ttp
:
//
ija
p
e.
ia
esco
r
e.
co
m/
Three
-
pha
se
po
w
er f
lo
w so
lu
tion o
f
a
ctive dis
tribu
ti
o
n net
wo
rk
using
t
rust
-
regi
o
n
method
Rudy
G
ia
nto
,
M
.
I
qb
a
l A
rsy
a
d,
M
a
na
g
a
m
R
a
j
a
g
u
k
g
uk
D
e
p
a
r
t
me
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t
o
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El
e
c
t
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i
c
a
l
En
g
i
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e
r
i
n
g
,
Ta
n
j
u
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p
u
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a
U
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i
v
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si
t
y
,
P
o
n
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a
n
a
k
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I
n
d
o
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i
a
Art
icle
I
nfo
AB
S
T
RAC
T
A
r
ticle
his
to
r
y:
R
ec
eiv
ed
Sep
2
9
,
2
0
2
4
R
ev
is
ed
Au
g
1
3
,
2
0
2
5
Acc
ep
ted
Oct
1
6
,
2
0
2
5
Distrib
u
ti
o
n
sy
ste
m
s
o
r
n
e
two
r
k
s
a
re
in
h
e
re
n
tl
y
u
n
b
a
lan
c
e
d
.
As
a
re
su
lt
,
sin
g
le
-
p
h
a
se
p
o
we
r
flo
w
m
e
th
o
d
s
a
re
g
e
n
e
ra
ll
y
n
o
lo
n
g
e
r
v
a
li
d
fo
r
su
c
h
sy
ste
m
s.
Th
e
re
fo
re
,
t
o
o
b
tain
a
c
c
u
ra
te
re
su
lt
s,
u
n
b
a
lan
c
e
d
sy
ste
m
s
sh
o
u
ld
b
e
a
n
a
ly
z
e
d
u
sin
g
t
h
re
e
-
p
h
a
se
p
o
we
r
fl
o
w
m
e
th
o
d
s,
wh
ich
a
re
fa
r
m
o
re
c
o
m
p
li
c
a
ted
t
h
a
n
th
e
sin
g
le
-
p
h
a
se
m
e
th
o
d
s.
M
o
re
o
v
e
r,
a
t
p
r
e
se
n
t,
t
h
e
p
e
n
e
tratio
n
o
f
d
istri
b
u
ted
g
e
n
e
ra
ti
o
n
(
DG
)
in
th
e
d
istr
ib
u
ti
o
n
n
e
t
wo
rk
h
a
s
sig
n
ifi
c
a
n
t
ly
i
n
c
re
a
se
d
.
DG
in
teg
ra
ti
o
n
will
in
c
re
a
se
th
e
c
o
m
p
li
c
a
ti
o
n
o
f
th
e
pow
e
r
flo
w
a
n
a
l
y
sis
a
s
it
c
h
a
n
g
e
s
th
e
n
e
two
rk
'
s
b
a
sic
c
o
n
fig
u
ra
ti
o
n
fr
o
m
p
a
ss
iv
e
to
a
c
ti
v
e
sy
ste
m
.
Th
is
c
o
m
p
u
tati
o
n
a
l
b
u
r
d
e
n
will
sig
n
ifi
c
a
n
tl
y
b
e
h
ig
h
e
r
if
t
h
e
p
o
we
r
flo
w
c
a
lcu
lat
io
n
h
a
s
to
b
e
c
o
n
d
u
c
ted
se
v
e
ra
l
t
ime
s
(fo
r
e
x
a
m
p
le,
i
n
fe
e
d
e
r
re
c
o
n
fi
g
u
ra
t
io
n
s
o
r
se
rv
ice
re
sto
ra
ti
o
n
s)
.
T
h
is
p
a
p
e
r
in
v
e
stig
a
tes
th
e
u
ti
li
z
a
ti
o
n
o
f
th
e
t
ru
st
-
re
g
io
n
m
e
th
o
d
i
n
o
b
tain
i
n
g
t
h
e
so
lu
ti
o
n
to
t
h
e
th
re
e
-
p
h
a
se
p
o
we
r
flo
w
p
ro
b
lem
o
f
a
n
a
c
ti
v
e
d
istri
b
u
t
io
n
n
e
t
wo
rk
(i.
e
.
,
d
istri
b
u
ti
o
n
n
e
two
r
k
e
m
b
e
d
d
e
d
wit
h
DG
).
Tru
st
-
re
g
i
o
n
c
o
m
p
u
tatio
n
a
lg
o
rit
h
m
is
r
o
b
u
st
a
n
d
p
o
we
rfu
l
sin
c
e
th
e
o
p
ti
m
iza
ti
o
n
tec
h
n
i
q
u
e
is
e
m
p
lo
y
e
d
in
fi
n
d
i
n
g
n
e
w
so
l
u
ti
o
n
s
in
th
e
it
e
ra
ti
o
n
p
ro
c
e
ss
.
Re
su
lt
s
o
b
tain
e
d
f
ro
m
th
re
e
re
p
re
se
n
tativ
e
u
n
b
a
lan
c
e
d
d
istr
ib
u
ti
o
n
n
e
two
rk
s
(i.
e
.
,
1
0
-
n
o
d
e
,
1
9
-
n
o
d
e
,
a
n
d
25
-
n
o
d
e
n
e
two
rk
s) v
e
rify
t
h
e
v
a
li
d
it
y
o
f
t
h
e
p
ro
p
o
se
d
m
e
th
o
d
.
Th
e
e
ffe
c
ts
o
f
DG
in
sta
ll
a
ti
o
n
o
n
d
istri
b
u
ti
o
n
n
e
two
rk
ste
a
d
y
-
sta
te
p
e
rfo
rm
a
n
c
e
s
a
re
a
lso
in
v
e
stig
a
te
d
in
th
e
p
re
se
n
t
p
a
p
e
r.
K
ey
w
o
r
d
s
:
Activ
e
d
is
tr
ib
u
tio
n
n
etwo
r
k
Dis
tr
ib
u
ted
g
en
er
atio
n
Po
wer
f
lo
w
T
r
u
s
t
-
r
eg
io
n
m
eth
o
d
Un
b
alan
ce
d
d
is
tr
ib
u
tio
n
s
y
s
tem
T
h
is i
s
a
n
o
p
e
n
a
c
c
e
ss
a
rticle
u
n
d
e
r th
e
CC B
Y
-
SA
li
c
e
n
se
.
C
o
r
r
e
s
p
o
nd
ing
A
uth
o
r
:
R
u
d
y
Gian
to
Dep
ar
tm
en
t o
f
E
lectr
ical
E
n
g
i
n
ee
r
in
g
,
T
an
ju
n
g
p
u
r
a
Un
iv
er
s
ity
Pro
f
.
Dr
.
H.
Had
ar
i N
awa
wi
Stre
et
,
Po
n
tian
ak
7
8
1
2
4
,
I
n
d
o
n
esia
E
m
ail:
r
u
d
y
.
g
ian
to
@
ee
.
u
n
tan
.
ac
.
id
1.
I
NT
RO
D
UCT
I
O
N
T
h
e
ab
ilit
y
to
p
r
o
v
id
e
v
alid
i
n
f
o
r
m
atio
n
ab
o
u
t
s
y
s
tem
s
tead
y
-
s
tate
co
n
d
itio
n
s
at
v
ar
io
u
s
l
o
ad
in
g
an
d
g
en
er
atio
n
lev
els
is
n
ec
ess
ar
y
f
o
r
t
h
e
s
ec
u
r
e
o
p
er
atio
n
an
d
p
r
o
p
er
p
lan
n
in
g
o
f
an
elec
tr
ic
d
is
tr
ib
u
tio
n
s
y
s
tem
.
T
h
e
in
f
o
r
m
atio
n
ab
o
u
t
s
y
s
tem
s
tead
y
-
s
tate
c
o
n
d
itio
n
s
is
u
s
u
ally
co
llected
b
ased
o
n
th
e
r
es
u
lts
o
f
p
o
wer
f
lo
w
s
tu
d
ies.
T
h
e
im
p
o
r
tan
ce
o
f
th
i
s
in
f
o
r
m
atio
n
ca
n
b
e
ex
p
lain
ed
as
f
o
llo
ws.
Fo
r
e
x
am
p
le,
in
t
h
e
o
p
er
atio
n
al
s
tag
e,
if
p
o
wer
f
lo
w
r
esu
lts
in
d
icate
th
at
s
o
m
e
elec
tr
ical
q
u
an
titi
es
v
io
late
th
eir
l
im
its
,
th
en
s
o
m
e
co
r
r
ec
tiv
e
ac
tio
n
s
ar
e
n
ee
d
ed
to
b
r
in
g
th
e
q
u
an
ti
ties
b
ac
k
to
th
eir
allo
wab
le
li
m
its
.
T
h
is
p
r
o
ce
s
s
n
ee
d
s
a
v
al
id
an
d
r
o
b
u
s
t
p
o
wer
f
lo
w
s
o
lu
tio
n
m
eth
o
d
f
o
r
v
ar
io
u
s
s
y
s
tem
lo
ad
in
g
s
,
g
en
e
r
atio
n
s
,
an
d
co
n
f
ig
u
r
atio
n
s
.
C
h
ar
ac
ter
is
tics
o
f
an
elec
tr
ic
d
is
tr
ib
u
tio
n
n
etwo
r
k
ar
e
u
s
u
ally
d
if
f
er
en
t
f
r
o
m
th
o
s
e
o
f
an
elec
tr
ic
tr
an
s
m
is
s
io
n
n
etwo
r
k
.
T
h
ese
s
p
ec
ial
ch
ar
ac
ter
is
tics
o
f
th
e
d
is
tr
ib
u
tio
n
n
etwo
r
k
m
a
k
e
th
e
co
n
v
en
tio
n
al
p
o
wer
f
lo
w
s
o
lu
tio
n
tech
n
i
q
u
es
d
if
f
i
cu
lt
o
r
im
p
o
s
s
ib
le
to
b
e
a
p
p
li
ed
[
1
]
,
[
2
]
.
Mo
r
eo
v
e
r
,
t
h
e
as
s
u
m
p
tio
n
s
th
at
th
e
s
y
s
tem
is
alwa
y
s
b
alan
ce
d
an
d
ca
n
b
e
a
n
aly
ze
d
u
s
in
g
a
s
in
g
le
-
p
h
ase
m
o
d
el
ar
e
also
n
o
t
ap
p
licab
le
to
a
d
is
tr
ib
u
tio
n
n
etwo
r
k
.
T
h
e
u
n
b
alan
ce
in
a
d
is
tr
ib
u
tio
n
n
etwo
r
k
is
u
s
u
ally
o
f
a
h
i
g
h
er
le
v
el,
a
n
d
th
er
e
f
o
r
e
ca
n
n
o
t
b
e
n
eg
lecte
d
.
C
o
n
s
eq
u
e
n
tly
,
th
e
s
in
g
le
-
p
h
ase
m
o
d
el
is
n
o
lo
n
g
er
v
alid
f
o
r
th
e
d
is
tr
ib
u
tio
n
s
y
s
tem
,
an
d
th
er
ef
o
r
e,
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
2
5
2
-
8
7
9
2
I
n
t J Ap
p
l Po
wer
E
n
g
,
Vo
l.
14
,
No
.
4
,
Dec
em
b
er
20
25
:
9
2
3
-
933
924
th
e
s
y
s
tem
s
h
o
u
ld
b
e
a
n
aly
z
ed
u
s
in
g
a
th
r
ee
-
p
h
ase
m
o
d
e
l.
T
h
is
will
co
m
p
licate
th
e
p
r
o
ce
s
s
o
f
co
llectin
g
in
f
o
r
m
atio
n
ab
o
u
t
s
y
s
tem
s
te
ad
y
-
s
tate
co
n
d
itio
n
s
,
as
th
r
ee
-
p
h
ase
p
o
wer
f
lo
w
s
tu
d
ies
h
av
e
to
b
e
u
s
ed
in
th
e
p
r
o
ce
s
s
[
3
]
-
[6
]
.
At
p
r
esen
t,
th
e
p
en
etr
atio
n
o
f
d
is
tr
ib
u
ted
g
en
er
atio
n
(
DG
)
in
th
e
d
is
tr
ib
u
tio
n
n
etwo
r
k
h
as
s
ig
n
if
ican
tly
in
cr
ea
s
ed
.
I
n
c
o
n
tr
ast
to
co
n
v
e
n
tio
n
al
p
o
wer
g
en
er
atio
n
,
wh
i
ch
is
u
s
u
ally
lar
g
e
an
d
c
o
n
n
ec
t
ed
to
a
tr
an
s
m
is
s
io
n
n
etwo
r
k
,
DG
is
a
s
m
all
-
s
ized
p
o
wer
g
en
er
atio
n
an
d
is
g
en
er
ally
co
n
n
ec
ted
to
a
d
is
tr
ib
u
tio
n
n
etwo
r
k
[
7
]
-
[
1
2
]
.
Sin
ce
DG
is
ca
p
ab
le
o
f
s
u
p
p
ly
in
g
o
r
in
jectin
g
elec
tr
ical
p
o
w
er
in
to
t
h
e
d
is
tr
ib
u
tio
n
n
etwo
r
k
,
it will c
h
an
g
e
th
e
n
etwo
r
k
'
s
b
asic
co
n
f
ig
u
r
atio
n
f
r
o
m
a
p
ass
iv
e
to
an
ac
tiv
e
s
y
s
tem
.
T
h
is
will
also
in
cr
ea
s
e
th
e
co
m
p
u
tatio
n
al
b
u
r
d
e
n
in
p
o
wer
f
lo
w
s
tu
d
ie
s
.
T
h
is
co
m
p
u
tatio
n
al
b
u
r
d
e
n
will
s
ig
n
if
ican
tly
in
cr
ea
s
e
if
th
e
p
o
wer
f
lo
w
ca
lcu
latio
n
h
as
to
b
e
c
o
n
d
u
cted
s
ev
er
al
tim
es
(
f
o
r
e
x
am
p
le,
i
n
f
ee
d
er
r
ec
o
n
f
ig
u
r
at
io
n
s
o
r
s
er
v
ice
r
esto
r
atio
n
s
)
[
1
3
]
-
[
1
5
]
.
Sev
er
al
m
eth
o
d
s
f
o
r
i
n
co
r
p
o
r
atin
g
DG
in
to
p
o
wer
f
lo
w
an
aly
s
is
h
av
e
b
ee
n
p
r
o
p
o
s
ed
b
y
s
o
m
e
r
esear
ch
er
s
[
1
6
]
-
[
2
5
]
.
I
n
[
1
6
]
-
[
2
0
]
,
n
o
d
e
-
b
ased
tech
n
i
q
u
es
,
i.e
.
,
New
to
n
-
R
ap
h
s
o
n
(
NR
)
m
eth
o
d
s
,
h
av
e
b
ee
n
u
s
ed
to
f
in
d
th
e
s
o
lu
tio
n
t
o
th
e
p
o
wer
f
lo
w
p
r
o
b
lem
.
T
h
e
i
m
p
ac
t
o
f
th
e
i
n
teg
r
atio
n
o
f
r
e
n
ewa
b
le
en
er
g
y
DG
(
i.e
.
,
win
d
tu
r
b
i
n
e
g
en
er
at
o
r
)
i
n
an
elec
tr
ic
n
etw
o
r
k
h
as
b
ee
n
in
v
esti
g
ated
in
[
1
6
]
.
T
h
e
p
o
w
er
f
lo
w
p
r
o
b
le
m
in
[
1
6
]
h
as
b
ee
n
s
o
lv
ed
u
s
in
g
th
e
NR
m
eth
o
d
.
I
t
h
as
also
b
ee
n
f
o
u
n
d
in
[
1
6
]
th
at
th
e
in
teg
r
atio
n
o
f
DG
ca
u
s
es
s
o
m
e
v
o
ltag
e
f
lu
ctu
atio
n
s
,
an
d
th
e
f
lu
ctu
atio
n
s
ca
n
b
e
m
in
im
ized
af
ter
th
e
in
jec
tio
n
o
f
r
ea
ct
iv
e
p
o
wer
.
I
n
[
1
7
]
,
an
ap
p
r
o
ac
h
n
am
ed
co
m
p
lex
p
er
-
u
n
it
(
C
PU
)
n
o
r
m
aliza
tio
n
h
as
b
ee
n
p
r
o
p
o
s
ed
.
T
h
e
p
r
o
p
o
s
ed
C
PU
n
o
r
m
aliza
tio
n
a
p
p
r
o
ac
h
is
in
te
n
d
ed
to
im
p
r
o
v
e
th
e
p
er
f
o
r
m
a
n
ce
o
f
t
h
e
NR
-
b
ased
m
eth
o
d
(
i.e
.
,
f
ast
d
ec
o
u
p
led
p
o
wer
f
lo
w
m
eth
o
d
)
in
s
o
lv
in
g
th
e
p
o
wer
f
lo
w
p
r
o
b
lem
o
f
a
d
is
tr
ib
u
tio
n
s
y
s
tem
.
Hig
h
p
en
etr
atio
n
o
f
DGs,
m
o
r
e
in
ter
c
o
n
n
ec
tio
n
am
o
n
g
d
is
tr
ib
u
tio
n
f
ee
d
e
r
s
,
an
d
d
is
tr
ib
u
tio
n
n
etwo
r
k
h
i
g
h
R
/X
r
atio
s
h
av
e
also
b
ee
n
co
n
s
id
er
ed
in
t
h
e
m
eth
o
d
[
1
7
]
.
T
h
e
d
e
v
elo
p
m
e
n
t
o
f
a
n
esti
m
ated
s
o
lu
tio
n
to
a
s
et
o
f
n
o
n
lin
ea
r
eq
u
atio
n
s
th
at
r
ep
r
esen
t
elec
tr
ic
p
o
wer
d
is
tr
ib
u
tio
n
n
e
two
r
k
s
h
as
b
ee
n
d
is
cu
s
s
ed
in
[
1
8
]
.
T
h
e
esti
m
atio
n
in
[
1
8
]
h
as
b
ee
n
m
ad
e
in
ter
m
s
o
f
lin
ea
r
a
p
p
r
o
x
im
ati
o
n
o
f
a
ctiv
e
an
d
r
ea
ctiv
e
p
o
wer
d
em
an
d
s
at
PQ b
u
s
es.
T
h
e
p
r
o
p
o
s
ed
m
o
d
el
[
1
8
]
ca
n
also
b
e
u
s
ed
as a
to
o
l f
o
r
m
o
n
ito
r
in
g
an
d
co
n
tr
o
llin
g
th
e
d
esig
n
o
f
t
h
e
elec
tr
ic
p
o
we
r
d
is
tr
ib
u
tio
n
g
r
id
.
Ser
ee
ter
et
a
l.
[
1
9
]
ex
p
lain
ed
th
at
s
o
m
e
n
ew
v
er
s
io
n
s
o
f
th
e
New
to
n
p
o
wer
f
l
o
w
m
eth
o
d
f
o
r
s
o
lv
in
g
th
e
th
r
ee
-
p
h
ase
p
o
wer
f
lo
w
p
r
o
b
lem
h
av
e
b
ee
n
d
ev
el
o
p
ed
.
T
h
e
th
r
ee
-
p
h
ase
p
o
wer
f
lo
w
p
r
o
b
lem
is
f
o
r
m
u
lated
in
ter
m
s
o
f
p
o
wer
an
d
cu
r
r
en
t
m
is
m
atch
f
u
n
ctio
n
s
.
T
o
in
v
e
s
tig
ate
th
e
co
n
v
er
g
en
ce
p
r
o
p
e
r
ty
o
f
t
h
e
p
r
o
p
o
s
ed
m
eth
o
d
,
d
if
f
er
e
n
t
lo
ad
in
g
co
n
d
itio
n
s
an
d
d
is
tr
ib
u
tio
n
n
etwo
r
k
R
/X
r
a
tio
s
h
av
e
b
ee
n
co
n
s
i
d
er
ed
in
[
1
9
]
.
T
h
e
p
r
o
p
o
s
ed
m
eth
o
d
[
1
9
]
h
as
al
s
o
b
ee
n
im
p
lem
e
n
ted
a
n
d
co
m
p
ar
ed
with
th
e
b
ac
k
war
d
/f
o
r
war
d
s
wee
p
(
B
FS
)
alg
o
r
ith
m
f
o
r
b
alan
ce
d
an
d
u
n
b
alan
ce
d
elec
tr
ic
p
o
wer
d
is
tr
ib
u
tio
n
s
y
s
tem
s
.
A
p
o
wer
f
lo
w
m
eth
o
d
s
p
ec
ial
f
o
r
r
ec
o
n
f
ig
u
r
atio
n
a
n
aly
s
is
in
a
d
is
tr
i
b
u
tio
n
s
y
s
tem
is
p
r
o
p
o
s
ed
in
[
2
0
]
.
T
h
e
p
r
o
p
o
s
ed
m
eth
o
d
h
as
b
ee
n
b
ased
o
n
th
e
ex
ten
d
e
d
f
ast
d
ec
o
u
p
led
m
eth
o
d
.
A
C
PU
n
o
r
m
aliza
tio
n
tech
n
iq
u
e
is
also
ad
o
p
ted
i
n
[
2
0
]
t
o
d
ea
l
with
th
e
h
ig
h
R
/X
r
atio
n
etwo
r
k
s
.
Ho
w
ev
er
,
in
m
an
y
ca
s
es,
th
e
d
ir
ec
t
ap
p
licatio
n
o
f
NR
-
b
ased
m
et
h
o
d
s
to
d
is
tr
ib
u
ti
o
n
n
etwo
r
k
s
as
p
r
o
p
o
s
ed
i
n
[
1
6
]
-
[
20
]
ca
n
ca
u
s
e
c
o
n
v
e
r
g
en
ce
p
r
o
b
lem
s
,
an
d
in
tu
r
n
,
th
e
s
o
l
u
tio
n
is
d
if
f
icu
lt
o
r
im
p
o
s
s
ib
le
to
o
b
tain
.
Alth
o
u
g
h
s
o
m
e
tech
n
iq
u
es
h
av
e
b
ee
n
d
ev
elo
p
e
d
to
im
p
r
o
v
e
th
e
co
n
v
er
g
en
ce
o
f
th
e
m
eth
o
d
s
,
th
e
NR
-
b
ased
m
et
h
o
d
s
s
till
r
eq
u
ir
e
a
r
elativ
ely
lo
n
g
co
m
p
u
tatio
n
tim
e
[1
]
,
[
2]
.
I
n
[
2
]
,
[
2
1
]
-
[
2
5
]
,
b
r
a
n
ch
-
b
ased
tech
n
iq
u
es,
i.e
.
,
B
FS
m
eth
o
d
s
,
h
av
e
b
ee
n
em
p
l
o
y
ed
to
f
i
n
d
th
e
s
o
lu
tio
n
to
th
e
p
o
wer
f
lo
w
p
r
o
b
lem
o
f
a
d
is
tr
ib
u
tio
n
s
y
s
tem
.
A
p
o
wer
f
lo
w
m
eth
o
d
th
at
is
d
e
r
iv
ed
b
ased
o
n
th
e
co
n
v
en
tio
n
al
B
FS
tech
n
iq
u
e
f
o
r
a
n
u
n
b
alan
ce
d
d
is
tr
ib
u
tio
n
n
etwo
r
k
is
p
r
o
p
o
s
ed
in
[
2
]
.
Un
lik
e
th
e
co
n
v
en
tio
n
a
l
B
FS
,
wh
ich
em
p
lo
y
s
two
d
if
f
e
r
en
t
s
tep
s
(
b
ac
k
war
d
an
d
f
o
r
w
ar
d
s
wee
p
s
)
to
ca
lcu
late
b
u
s
v
o
ltag
es,
th
e
m
eth
o
d
in
[
2
]
u
s
es
a
s
in
g
le
s
tep
to
ca
lcu
late
th
e
b
u
s
v
o
ltag
es.
T
h
is
s
in
g
le
-
s
tep
ca
lcu
latio
n
is
ca
r
r
ied
o
u
t
th
r
o
u
g
h
t
h
e
u
s
e
o
f
th
e
l
o
ad
-
im
p
ed
an
ce
m
atr
ix
tech
n
iq
u
e
.
I
n
[
2
1
]
,
a
n
im
p
r
o
v
ed
B
FS
p
o
wer
f
lo
w
al
g
o
r
ith
m
s
u
itab
le
f
o
r
th
r
ee
-
p
h
ase
r
ad
ial
d
is
tr
ib
u
tio
n
s
y
s
te
m
s
is
p
r
esen
ted
.
I
n
th
e
m
eth
o
d
[
2
1
]
,
Kir
c
h
h
o
f
f
v
o
ltag
e
a
n
d
c
u
r
r
en
t
laws
ar
e
u
s
ed
in
th
e
b
ac
k
war
d
s
wee
p
,
an
d
th
e
lin
ea
r
p
r
o
p
o
r
tio
n
al
p
r
in
cip
le
is
em
p
lo
y
e
d
in
t
h
e
f
o
r
war
d
s
wee
p
.
T
h
e
s
o
f
twar
e
p
ac
k
ag
e
ca
lled
d
is
tr
ib
u
tio
n
s
y
s
tem
s
p
o
wer
f
lo
w
an
aly
s
is
p
ac
k
ag
e
(
DSPF
AP
)
f
o
r
p
o
wer
f
lo
w
an
aly
s
is
h
as
b
ee
n
d
ev
elo
p
e
d
in
[
2
2
]
.
Dif
f
er
e
n
t
B
FS
-
b
ased
alg
o
r
ith
m
s
ca
n
b
e
f
a
cilitated
in
th
e
s
o
f
twar
e
p
ac
k
a
g
e.
Als
o
,
th
e
e
f
f
ec
ts
o
f
v
o
ltag
e
-
d
ep
e
n
d
en
t l
o
ad
an
d
DGs o
n
p
o
wer
f
l
o
w
s
o
lu
tio
n
s
ca
n
b
e
in
v
esti
g
ated
u
s
in
g
th
e
s
o
f
twar
e
p
ac
k
a
g
e
.
T
h
e
r
esu
lt
[
2
3
],
a
tech
n
i
q
u
e
f
o
r
h
a
n
d
lin
g
PV
n
o
d
es,
i.e
.
,
n
o
d
es
wh
er
e
DGs
with
co
n
s
tan
t
v
o
ltag
e
co
n
tr
o
l
ar
e
c
o
n
n
ec
te
d
,
i
s
p
r
o
p
o
s
ed
.
T
h
e
tech
n
iq
u
e
[
2
3
]
is
b
a
s
ed
o
n
lo
o
p
an
aly
s
is
an
d
in
c
o
r
p
o
r
ated
in
t
h
e
FB
S
fra
m
ewo
r
k
.
I
t
h
as
b
ee
n
f
o
u
n
d
in
[
2
3
]
th
at
th
e
co
n
v
er
g
e
n
ce
p
r
o
p
er
ty
o
f
th
e
p
r
o
p
o
s
ed
tech
n
i
q
u
e
is
s
ati
s
f
ac
to
r
y
f
o
r
a
wid
e
r
an
g
e
o
f
lin
e
R
/X
r
a
tio
s
an
d
s
ev
er
al
co
n
n
ec
ted
PV
n
o
d
es.
T
h
e
tech
n
iq
u
e
ca
lled
q
u
ad
r
atic
-
b
ased
B
FS
(
QB
B
F
S
)
m
eth
o
d
f
o
r
d
is
tr
ib
u
tio
n
s
y
s
tem
p
o
wer
f
lo
w
a
n
aly
s
is
is
p
r
o
p
o
s
ed
in
[
2
4
]
.
T
h
e
m
eth
o
d
ca
n
ac
co
m
m
o
d
ate
m
u
lti
-
p
h
ase
la
ter
als,
d
if
f
er
en
t
lo
a
d
ty
p
es,
ca
p
ac
ito
r
s
,
d
is
tr
ib
u
tio
n
t
r
an
s
f
o
r
m
er
s
,
an
d
DGs.
I
m
p
r
o
v
em
en
t
to
th
e
co
n
v
e
n
tio
n
al
B
FS
p
o
wer
f
lo
w
alg
o
r
ith
m
f
o
r
u
n
b
alan
ce
d
th
r
ee
-
p
h
ase
d
is
tr
ib
u
tio
n
n
etwo
r
k
s
h
as
b
ee
n
p
r
o
p
o
s
ed
in
[
2
5
]
.
I
n
th
e
alg
o
r
ith
m
[
1
8
]
,
th
e
n
etwo
r
k
is
r
ep
r
esen
ted
as
a
tr
ee
-
lik
e
s
tr
u
ctu
r
e,
an
d
th
er
ef
o
r
e,
th
e
in
cid
en
ce
m
atr
i
x
co
n
ce
p
t
is
n
o
t
u
s
ed
in
th
e
alg
o
r
ith
m
.
W
ith
th
is
n
etwo
r
k
tr
ee
-
lik
e
s
tr
u
ct
u
r
e
r
ep
r
esen
tatio
n
,
t
h
e
u
s
e
o
f
r
e
d
u
n
d
a
n
t
co
m
p
u
tatio
n
s
an
d
th
e
s
to
r
in
g
o
f
u
n
n
ec
ess
ar
y
d
at
a
ca
n
b
e
a
v
o
id
e
d
.
Ho
wev
er
,
B
FS
-
b
as
ed
m
eth
o
d
s
as
p
r
o
p
o
s
ed
in
[
2
]
,
[
21
]
-
[
25
]
r
eq
u
ir
e
s
o
m
e
d
if
f
icu
lt
an
d
s
p
ec
ial
tr
ea
tm
en
ts
in
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J Ap
p
l Po
wer
E
n
g
I
SS
N:
2252
-
8
7
9
2
Th
r
ee
-
p
h
a
s
e
p
o
w
er flo
w
s
o
lu
tio
n
o
f a
ctive
d
is
tr
ib
u
tio
n
n
etw
o
r
k
u
s
in
g
…
(
R
u
d
y
Gia
n
to
)
925
b
r
an
ch
n
u
m
b
e
r
in
g
a
n
d
b
u
s
ar
r
an
g
em
en
t
o
f
th
e
s
y
s
tem
.
Mo
r
e
o
v
er
,
th
e
B
FS
-
b
ased
m
eth
o
d
s
ar
e
n
o
t
d
esig
n
e
d
to
s
o
lv
e
th
e
p
o
wer
f
lo
w
p
r
o
b
lem
o
f
m
esh
ed
a
n
d
/o
r
lo
o
p
e
d
d
is
tr
i
b
u
tio
n
n
etwo
r
k
s
[
1
]
,
[
9
]
,
[
17
]
.
R
ec
en
tly
,
th
e
tr
u
s
t
-
r
eg
io
n
m
et
h
o
d
h
as
b
ee
n
p
r
o
p
o
s
ed
as
an
alter
n
ativ
e
to
th
e
n
o
d
e
-
b
ased
tech
n
iq
u
es
o
r
th
e
b
r
a
n
ch
-
b
ased
tec
h
n
iq
u
e
s
f
o
r
s
o
lv
in
g
a
s
et
o
f
n
o
n
lin
ea
r
eq
u
atio
n
s
ar
is
in
g
f
r
o
m
th
e
p
o
we
r
f
lo
w
p
r
o
b
lem
f
o
r
m
u
latio
n
[
1
]
,
[
5
]
,
[
9
]
,
[
26
]
,
[
2
7
]
.
I
n
[
1
]
,
[
5
]
,
t
h
e
tr
u
s
t
-
r
eg
i
o
n
m
eth
o
d
h
as su
cc
ess
f
u
lly
b
ee
n
ap
p
lied
to
o
b
tain
th
e
p
o
wer
f
l
o
w
s
o
lu
tio
n
o
f
a
d
is
tr
ib
u
tio
n
n
etwo
r
k
with
o
u
t
DG
.
I
n
[
9
]
,
th
e
tr
u
s
t
-
r
e
g
io
n
m
eth
o
d
h
as
also
b
ee
n
ap
p
lied
in
a
b
ala
n
ce
d
d
is
tr
ib
u
tio
n
n
etwo
r
k
em
b
e
d
d
ed
wit
h
DG.
T
h
is
p
ap
er
in
v
esti
g
ates
th
e
ex
ten
s
io
n
an
d
ap
p
licatio
n
o
f
th
e
t
r
u
s
t
-
r
eg
io
n
m
eth
o
d
to
s
o
lv
e
th
e
p
o
wer
f
lo
w
p
r
o
b
lem
o
f
an
u
n
b
alan
ce
d
d
is
tr
ib
u
tio
n
n
etwo
r
k
em
b
ed
d
e
d
with
DG
(
i
.
e.
,
ac
tiv
e
d
is
tr
ib
u
tio
n
n
etwo
r
k
)
.
R
esu
lts
o
b
t
ain
ed
f
r
o
m
th
r
ee
r
ep
r
ese
n
tativ
e
d
is
tr
ib
u
tio
n
n
etwo
r
k
s
(
i.e
.
,
1
0
-
n
o
d
e
,
1
9
-
n
o
d
e,
an
d
2
5
-
n
o
d
e
n
etwo
r
k
s
)
ar
e
also
p
r
esen
ted
in
th
e
p
r
esen
t
p
ap
er
.
T
h
e
e
f
f
ec
ts
o
f
DG
in
s
tallatio
n
o
n
d
is
tr
ib
u
tio
n
n
etwo
r
k
s
tead
y
-
s
tate
p
er
f
o
r
m
an
ce
s
ar
e
also
in
v
esti
g
ated
an
d
p
r
esen
t
ed
in
th
e
p
r
esen
t p
ap
er
.
T
h
e
r
est
o
f
th
e
p
a
p
er
will
b
e
o
r
g
an
ized
as
f
o
llo
ws.
Sectio
n
2
d
is
cu
s
s
es
th
e
f
o
r
m
u
latio
n
o
f
th
e
th
r
ee
-
p
h
ase
p
o
wer
f
lo
w
p
r
o
b
lem
o
f
th
e
d
is
tr
ib
u
tio
n
n
etwo
r
k
.
Secti
o
n
3
ad
d
r
ess
es
th
e
f
o
r
m
u
latio
n
o
f
th
e
th
r
ee
-
p
h
ase
p
o
wer
f
lo
w
p
r
o
b
lem
o
f
t
h
e
d
is
tr
ib
u
tio
n
n
etwo
r
k
em
b
e
d
d
ed
with
DGs
(
ac
tiv
e
d
is
t
r
ib
u
tio
n
n
etwo
r
k
s
)
.
T
h
e
u
n
b
alan
ce
d
d
is
tr
ib
u
tio
n
p
o
wer
f
lo
w
s
o
lu
tio
n
u
s
in
g
th
e
p
r
o
p
o
s
ed
tr
u
s
t
-
r
eg
io
n
m
eth
o
d
is
ex
p
lain
ed
i
n
s
ec
tio
n
4
.
C
ase
s
tu
d
ies
to
ev
alu
ate
th
e
p
er
f
o
r
m
an
ce
o
f
th
e
p
r
o
p
o
s
ed
tr
u
s
t
-
r
eg
io
n
m
et
h
o
d
i
n
s
o
lv
in
g
t
h
e
th
r
ee
-
p
h
ase
lo
ad
f
lo
w
p
r
o
b
lem
o
f
an
ac
tiv
e
d
is
t
r
ib
u
tio
n
n
etwo
r
k
ar
e
g
iv
e
n
in
s
ec
tio
n
5
.
Sectio
n
6
g
iv
es
s
o
m
e
co
n
clu
s
io
n
s
o
f
th
e
p
r
esen
t w
o
r
k
.
2.
T
H
RE
E
-
P
H
ASE
P
O
W
E
R
F
L
O
W
ANA
L
YS
I
S O
F
DIS
T
RIB
U
T
I
O
N
N
E
T
WO
RK
T
h
e
r
elatio
n
s
h
ip
a
m
o
n
g
th
e
n
o
d
al
q
u
a
n
titi
es
o
f
a
th
r
ee
-
p
h
a
s
e
p
o
wer
s
y
s
tem
ca
n
b
e
f
o
r
m
u
lated
.
T
h
e
f
o
r
m
u
latio
n
is
ex
p
r
ess
ed
as f
o
l
lo
ws [
5
]
,
[
2
8
]
.
−
=
0
(
1
)
I
n
(
1
)
,
I
abc
an
d
V
abc
ar
e
v
ec
to
r
s
o
f
n
o
d
al
cu
r
r
e
n
ts
an
d
v
o
lta
g
es,
r
esp
ec
tiv
ely
,
an
d
Y
ab
c
is
s
y
s
tem
ad
m
ittan
ce
m
atr
ix
.
T
h
e
v
ec
to
r
s
/m
atr
ix
I
abc
,
Y
abc
,
an
d
V
abc
in
(
1
)
h
av
e
th
e
f
o
r
m
s
as f
o
llo
ws
:
=
[
1
2
⋮
]
;
=
[
1
2
⋮
]
;
=
[
11
12
⋯
1
21
22
⋯
2
⋮
⋮
⋱
⋮
1
2
⋯
]
(
2
)
wh
er
e:
=
[
]
;
=
[
]
;
=
[
]
;
=
[
]
(3
)
B
ec
au
s
e
o
f
th
e
ch
ar
ac
ter
is
tics
o
f
th
e
elec
tr
ical
p
o
wer
s
y
s
tem
,
th
e
co
n
d
itio
n
o
f
ea
ch
n
o
d
e
is
u
s
u
ally
d
ef
in
ed
in
ter
m
s
o
f
n
o
d
al
p
o
wer
r
ath
er
th
an
n
o
d
al
cu
r
r
en
t.
I
n
ter
m
s
o
f
n
o
d
al
p
o
wer
(
an
d
n
o
d
al
v
o
ltag
e)
,
th
e
n
o
d
al
cu
r
r
en
t c
a
n
b
e
e
x
p
r
ess
ed
as
(
4
)
.
=
{
[
(
)
]
−
1
(
−
)
}
∗
(
4
)
I
n
(
4
)
,
S
G
abc
an
d
S
L
abc
ar
e
v
ec
to
r
s
o
f
g
en
er
atio
n
an
d
lo
a
d
p
o
we
r
s
,
r
esp
ec
tiv
ely
.
T
h
e
v
ec
to
r
s
S
G
abc
an
d
S
L
abc
in
(
4
)
h
av
e
th
e
f
o
r
m
s
as f
o
llo
ws:
=
[
1
2
⋮
]
;
=
[
1
2
⋮
]
(5
)
wh
er
e:
=
[
]
;
=
[
]
(6
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
2
5
2
-
8
7
9
2
I
n
t J Ap
p
l Po
wer
E
n
g
,
Vo
l.
14
,
No
.
4
,
Dec
em
b
er
20
25
:
9
2
3
-
933
926
On
u
s
in
g
(
4
)
in
(
1
)
,
t
h
e
r
elatio
n
s
h
ip
am
o
n
g
th
e
n
o
d
al
q
u
an
tit
ies o
f
th
r
ee
-
p
h
ase
p
o
wer
s
y
s
te
m
b
ec
o
m
es
(
7
)
.
{
[
(
)
]
−
1
(
−
)
}
∗
−
=
0
(7
)
T
h
e
s
et
o
f
n
o
n
lin
ea
r
in
(7
)
is
th
e
f
o
r
m
u
latio
n
o
f
a
th
r
ee
-
p
h
ase
p
o
wer
f
lo
w
p
r
o
b
lem
o
f
a
n
elec
tr
ical
p
o
wer
s
y
s
tem
in
te
r
m
s
o
f
n
o
d
al
q
u
an
titi
es.
T
h
is
s
et
o
f
n
o
n
lin
ea
r
eq
u
atio
n
s
is
th
en
s
o
lv
e
d
s
im
u
ltan
eo
u
s
ly
to
o
b
tain
th
e
s
o
lu
tio
n
to
th
e
p
o
we
r
f
lo
w
p
r
o
b
lem
.
I
t
is
t
o
b
e
n
o
te
d
th
at
a
co
n
v
en
ti
o
n
al
d
is
tr
ib
u
tio
n
n
etwo
r
k
u
s
u
ally
h
as
o
n
ly
o
n
e
p
o
wer
s
o
u
r
ce
n
o
d
e
o
r
s
u
b
s
tatio
n
(
SS
)
n
o
d
e
,
a
n
d
th
e
v
o
ltag
e
at
th
is
n
o
d
e
is
n
o
r
m
ally
s
p
ec
if
ied
.
B
ec
au
s
e
it
h
as
o
n
ly
o
n
e
s
o
u
r
ce
n
o
d
e,
p
o
wer
g
e
n
er
atio
n
s
at
th
e
r
em
ain
in
g
n
o
d
es
(
lo
a
d
n
o
d
es)
ca
n
b
e
co
n
s
id
er
ed
to
b
e
all
ze
r
o
s
.
T
a
b
le
1
s
u
m
m
ar
izes
all
o
f
th
e
k
n
o
wn
(
s
p
ec
if
ied
)
an
d
u
n
k
n
o
wn
(
to
b
e
ca
lcu
lated
)
q
u
an
titi
es
o
f
th
e
th
r
ee
-
p
h
ase
d
is
tr
ib
u
tio
n
s
y
s
tem
p
o
wer
f
lo
w
f
o
r
m
u
lat
io
n
ex
p
r
ess
ed
in
(
7
)
.
T
ab
le
1
.
Kn
o
wn
an
d
u
n
k
n
o
w
n
q
u
an
titi
es
N
o
d
e
K
n
o
w
n
q
u
a
n
t
i
t
i
e
s
U
n
k
n
o
w
n
q
u
a
n
t
i
t
i
e
s
SS
;
;
=
[
1
∠
0
°
1
∠
−
12
0
°
1
∠
12
0
°
]
Lo
a
d
;
;
=
[
0
0
0
]
3.
T
H
RE
E
-
P
H
ASE
P
O
W
E
R
F
L
O
W
ANA
L
YS
I
S O
F
DIS
T
RIB
U
T
I
O
N
N
E
T
WO
RK
I
t
h
as
b
ee
n
d
is
cu
s
s
ed
ea
r
lier
th
at
p
en
etr
atio
n
o
f
DG
in
th
e
d
is
tr
ib
u
tio
n
n
etwo
r
k
will
ch
an
g
e
th
e
n
etwo
r
k
'
s
b
asic
co
n
f
ig
u
r
atio
n
f
r
o
m
a
p
ass
iv
e
to
an
ac
tiv
e
s
y
s
tem
.
T
h
is
ch
an
g
e
is
ex
p
ec
ted
s
in
ce
DG
is
ca
p
ab
le
o
f
s
u
p
p
ly
i
n
g
o
r
in
jectin
g
elec
t
r
ical
p
o
wer
in
to
t
h
e
d
is
tr
ib
u
ti
o
n
n
etwo
r
k
.
T
h
e
f
ir
s
t
s
tep
in
an
aly
zin
g
th
e
p
o
wer
f
lo
w
o
f
th
e
d
is
tr
ib
u
tio
n
n
etwo
r
k
co
n
tain
in
g
DG
is
to
d
ev
elo
p
a
p
r
o
p
er
m
o
d
el
o
f
th
e
DG
s
u
itab
le
f
o
r
th
e
p
o
wer
f
lo
w
an
aly
s
is
.
I
n
[
9
]
,
[
29
]
-
[
31
]
,
th
r
ee
m
o
d
els
o
f
DG
f
o
r
s
tead
y
-
s
tate
o
r
p
o
we
r
f
lo
w
an
aly
s
is
h
av
e
b
ee
n
p
r
o
p
o
s
ed
.
T
h
e
p
r
o
p
o
s
ed
DG
m
o
d
els ar
e
as f
o
llo
ws:
˗
C
o
n
s
tan
t
-
PF
m
o
d
el,
˗
C
o
n
s
tan
t
-
Q
m
o
d
el,
an
d
˗
C
o
n
s
tan
t
-
V
m
o
d
el.
wh
er
e
PF
,
Q
,
an
d
V
s
tan
d
f
o
r
p
o
wer
f
ac
to
r
,
r
ea
ctiv
e
-
p
o
wer
,
an
d
v
o
lta
g
e,
r
esp
ec
tiv
ely
.
I
t
h
as
also
b
ee
n
m
en
tio
n
e
d
in
[
9
]
,
[
29
]
-
[
31
]
th
at
i
n
p
o
wer
f
lo
w
an
aly
s
is
,
th
e
n
o
d
es
wh
e
r
e
DGs
o
f
Mo
d
el
1
o
r
2
ar
e
co
n
n
ec
ted
c
an
b
e
co
n
s
id
er
e
d
as
lo
ad
n
o
d
e
s
with
n
eg
ativ
e
p
o
wer
.
Alter
n
ativ
ely
,
th
ey
ca
n
b
e
co
n
s
id
er
ed
as
g
e
n
er
atio
n
n
o
d
e
s
with
co
n
s
tan
t
ac
tiv
e
a
n
d
r
ea
ctiv
e
p
o
wer
s
.
On
th
e
o
th
er
h
a
n
d
,
t
h
e
n
o
d
es
wh
er
e
DGs
o
f
Mo
d
el
3
ar
e
c
o
n
n
ec
t
ed
ca
n
b
e
t
h
o
u
g
h
t
o
f
as
PV
n
o
d
es,
i.e
.
,
g
en
er
atio
n
n
o
d
es
with
co
n
s
tan
t
ac
tiv
e
p
o
wer
s
an
d
v
o
ltag
e
m
a
g
n
itu
d
e
s
.
I
t
is
to
b
e
n
o
ted
th
at
th
e
DG
o
f
Mo
d
el
3
h
as
th
e
ca
p
ab
ilit
y
o
f
r
eg
u
latin
g
v
o
lta
g
e
m
ag
n
itu
d
e
an
d
ac
tiv
e
p
o
wer
o
u
tp
u
t.
Sin
ce
m
o
s
t o
f
t
h
e
DGs d
o
n
o
t h
av
e
th
ese
ca
p
ab
ilit
ies,
o
n
ly
DGs
o
f
Mo
d
el
1
o
r
2
ar
e
co
n
s
id
er
ed
in
th
e
p
r
esen
t
wo
r
k
.
Ho
wev
e
r
,
if
n
ee
d
ed
,
DGs
o
f
Mo
d
el
3
ca
n
also
b
e
in
clu
d
ed
in
th
e
f
o
r
m
u
latio
n
in
a
s
tr
aig
h
tf
o
r
war
d
m
an
n
e
r
.
Fro
m
th
e
ab
o
v
e
e
x
p
lan
atio
n
,
i
t
is
clea
r
th
at
f
o
r
m
u
latio
n
(
7
)
i
s
also
ap
p
licab
le
to
a
p
o
wer
s
y
s
tem
with
DGs.
H
o
w
e
v
e
r
,
t
h
e
e
le
c
t
r
i
ca
l
q
u
a
n
t
i
t
i
e
s
(
k
n
o
w
n
a
n
d
u
n
k
n
o
w
n
)
i
n
t
h
e
f
o
r
m
u
l
a
t
i
o
n
a
r
e
s
l
i
g
h
t
l
y
d
i
f
f
e
r
e
n
t
.
T
a
b
l
e
2
l
is
ts
a
l
l
o
f
t
h
e
q
u
a
n
ti
t
ies
i
n
t
h
e
p
o
w
e
r
f
l
o
w
f
o
r
m
u
l
a
t
i
o
n
o
f
t
h
e
d
i
s
t
r
i
b
u
ti
o
n
n
e
t
w
o
r
k
em
b
e
d
d
e
d
w
i
t
h
D
Gs
.
T
ab
le
2
.
Kn
o
wn
an
d
u
n
k
n
o
w
n
q
u
an
titi
es (
ac
tiv
e
d
is
tr
ib
u
tio
n
n
etwo
r
k
)
N
o
d
e
K
n
o
w
n
q
u
a
n
t
i
t
i
e
s
U
n
k
n
o
w
n
q
u
a
n
t
i
t
i
e
s
SS
;
;
=
[
1
∠
0
°
1
∠
−
12
0
°
1
∠
12
0
°
]
Lo
a
d
;
;
=
[
0
0
0
]
DG
;
;
=
4.
P
O
WE
R
F
L
O
W
SO
L
U
T
I
O
N
M
E
T
H
O
D
As
d
is
cu
s
s
ed
in
s
ec
t
io
n
2
,
th
e
f
o
r
m
u
lati
o
n
o
f
th
e
p
o
wer
f
lo
w
p
r
o
b
lem
(
7
)
h
as
th
e
f
o
r
m
o
f
n
o
n
lin
ea
r
eq
u
atio
n
s
.
I
t
is
also
ac
k
n
o
wled
g
ed
th
at
t
h
e
s
o
lu
tio
n
to
t
h
is
ty
p
e
o
f
e
q
u
atio
n
is
u
s
u
ally
o
b
tain
e
d
b
y
u
s
in
g
iter
ativ
e
tech
n
iq
u
es.
NR
m
eth
o
d
is
p
r
o
b
ab
ly
o
n
e
o
f
th
e
m
o
s
t
p
o
p
u
lar
iter
ativ
e
tech
n
iq
u
es
f
o
r
tr
an
s
m
is
s
io
n
n
etwo
r
k
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J Ap
p
l Po
wer
E
n
g
I
SS
N:
2252
-
8
7
9
2
Th
r
ee
-
p
h
a
s
e
p
o
w
er flo
w
s
o
lu
tio
n
o
f a
ctive
d
is
tr
ib
u
tio
n
n
etw
o
r
k
u
s
in
g
…
(
R
u
d
y
Gia
n
to
)
927
ap
p
licatio
n
s
.
R
ec
en
tly
,
an
o
t
h
er
iter
ativ
e
ap
p
r
o
ac
h
,
i.e
.
,
th
e
tr
u
s
t
-
r
eg
io
n
m
eth
o
d
,
h
as
b
ee
n
p
r
o
p
o
s
ed
as
an
al
ter
n
ativ
e
to
th
e
NR
m
eth
o
d
[
1
]
,
[
5
]
,
[
9
]
,
[
26
]
,
[
2
7
]
.
As
also
m
en
tio
n
ed
in
[
1
]
,
[
5
]
,
[
9
]
,
th
is
o
p
tim
izatio
n
-
b
ased
m
eth
o
d
is
v
er
y
s
u
itab
le
f
o
r
d
is
tr
ib
u
tio
n
n
etwo
r
k
a
p
p
licatio
n
s
.
T
h
is
s
ec
tio
n
b
r
ief
ly
d
is
cu
s
s
es
th
e
d
is
tr
ib
u
tio
n
p
o
wer
f
lo
w
s
o
lu
tio
n
u
s
in
g
th
e
tr
u
s
t
-
r
eg
io
n
m
eth
o
d
.
A
m
o
r
e
d
etailed
ex
p
la
n
atio
n
o
f
th
e
m
eth
o
d
ca
n
b
e
f
o
u
n
d
i
n
[
1
]
,
[
5
]
,
[
9
]
.
Su
p
p
o
s
e
th
at
t
h
e
g
en
e
r
al
n
o
n
lin
ea
r
eq
u
atio
n
s
to
b
e
s
o
lv
ed
ar
e
ar
r
a
n
g
ed
in
v
ec
to
r
f
u
n
ctio
n
f
o
r
m
a
s
(
8
)
.
(
)
=
[
1
(
1
,
2
,
…
,
)
2
(
1
,
2
,
…
,
)
⋮
(
1
,
2
,
…
,
)
]
=
0
(8
)
W
h
er
e
x
i
s
th
e
v
ec
to
r
o
f
u
n
k
n
o
wn
q
u
a
n
titi
es to
b
e
ca
lcu
late
d
(
o
r
t
o
b
e
d
eter
m
in
ed
)
.
I
n
th
e
iter
ativ
e
a
p
p
r
o
ac
h
,
t
h
e
v
ec
to
r
x
is
ca
lcu
lated
c
o
n
s
ec
u
tiv
ely
.
T
h
e
f
ir
s
t
s
tep
in
th
e
ca
lcu
latio
n
p
r
o
ce
s
s
is
to
p
r
o
v
i
d
e
an
in
itial
esti
m
atio
n
f
o
r
x
.
T
h
is
in
itial e
s
tim
atio
n
is
th
en
iter
ativ
ely
co
r
r
ec
ted
u
s
in
g
(
9
)
.
(
+
1
)
=
(
)
+
(
)
(9
)
W
h
er
e
k
is
th
e
co
n
s
ec
u
tiv
e
o
r
iter
atio
n
n
u
m
b
er
a
n
d
d
is
th
e
v
ec
to
r
o
f
c
o
r
r
ec
tio
n
f
ac
to
r
s
.
T
h
e
ab
o
v
e
c
alcu
latio
n
p
r
o
ce
s
s
is
s
to
p
p
ed
wh
en
a
s
o
lu
tio
n
with
th
e
d
esire
d
ac
cu
r
ac
y
h
as b
ee
n
o
b
tain
ed
.
I
t
is
to
b
e
n
o
ted
th
at
in
th
e
N
R
m
eth
o
d
,
th
e
v
ec
to
r
o
f
co
r
r
e
ctio
n
f
ac
to
r
s
d
is
d
eter
m
in
ed
d
ir
ec
tly
b
y
s
o
lv
in
g
a
s
et
o
f
lin
ea
r
eq
u
atio
n
s
.
Ho
wev
er
,
in
th
e
tr
u
s
t
-
r
eg
i
o
n
m
eth
o
d
,
d
is
co
m
p
u
ted
b
y
u
s
in
g
a
co
n
s
tr
ain
ed
o
p
tim
izatio
n
tech
n
i
q
u
e
as
(
1
0
)
[1
]
,
[
5
]
,
[
9
]
,
[
26
]
,
[
27]
.
(
)
‖
1
2
ℎ
(
(
)
)
+
(
)
(
(
)
)
+
1
2
(
)
(
(
)
)
(
)
‖
2
s
ubj
e
c
t
to:
‖
(
)
‖
≤
(
)
(
1
0
)
W
h
er
e
(
k)
>
0
is
r
a
d
iu
s
o
f
th
e
tr
u
s
t
-
r
eg
io
n
.
I
t
ca
n
b
e
s
ee
n
f
r
o
m
(
1
0
)
th
at
a
co
n
s
tr
ain
t
is
im
p
o
s
ed
o
n
d
(
k
)
to
g
u
ar
an
tee
a
v
alid
s
o
lu
tio
n
to
t
h
e
v
ec
to
r
d
(
k
)
.
I
n
o
t
h
er
wo
r
d
s
,
in
th
e
s
o
lu
tio
n
p
r
o
ce
s
s
,
th
e
v
e
cto
r
d
(
k
)
is
co
n
f
in
ed
with
in
a
ce
r
tain
r
eg
i
o
n
(
tr
u
s
t
-
r
eg
io
n
)
.
Als
o
,
in
(
1
0
)
,
h
,
g
,
a
n
d
H
ar
e
ca
lcu
lated
u
s
in
g
(
1
1
)
-
(
1
3
)
.
ℎ
(
(
)
)
=
(
(
)
)
(
(
)
)
(
1
1
)
(
(
)
)
=
(
(
)
)
(
(
)
)
(
1
2
)
(
(
)
)
=
(
(
)
)
(
(
)
)
(
1
3
)
Ma
tr
ix
J
(
x
)
in
(
1
2
)
an
d
(
1
3
)
is
wid
ely
k
n
o
wn
as
th
e
J
ac
o
b
ian
m
atr
ix
o
f
F
(
x
).
E
lem
en
ts
o
f
th
is
m
atr
ix
ar
e
d
eter
m
in
ed
u
s
in
g
p
a
r
tial
d
er
iv
ativ
es
o
f
th
e
in
d
iv
id
u
al
f
u
n
ctio
n
s
in
F
(
x
).
T
h
e
f
o
r
m
u
latio
n
f
o
r
th
e
J
ac
o
b
ia
n
m
atr
ix
ca
n
b
e
wr
itten
as
(
1
4
)
.
(
)
=
[
1
1
1
2
⋯
1
2
1
2
2
⋯
2
⋮
⋮
⋱
⋮
1
2
⋯
]
(
1
4
)
I
t
is
to
b
e
n
o
ted
t
h
at
th
e
NR
m
eth
o
d
ca
n
s
o
m
etim
es
f
ail
to
o
b
tain
a
v
alid
s
o
lu
tio
n
,
esp
ec
iall
y
wh
en
th
e
J
ac
o
b
ian
m
atr
ix
is
s
in
g
u
lar
o
r
ill
-
co
n
d
itio
n
ed
.
T
h
e
s
in
g
u
lar
J
ac
o
b
ian
m
atr
ix
will
co
m
p
licate
t
h
e
co
m
p
u
tatio
n
o
f
co
r
r
ec
tio
n
f
ac
to
r
s
,
a
n
d
i
n
m
a
n
y
ca
s
es,
i
t
is
im
p
o
s
s
ib
le
to
b
e
ca
r
r
ied
o
u
t.
A
d
if
f
er
e
n
t
ap
p
r
o
ac
h
is
em
p
lo
y
ed
to
d
eter
m
in
e
th
e
co
r
r
ec
tio
n
f
ac
to
r
s
in
th
e
tr
u
s
t
-
r
eg
io
n
m
eth
o
d
.
I
t
is
d
eter
m
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Evaluation Warning : The document was created with Spire.PDF for Python.
I
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.
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k
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3
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33
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6
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.
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I
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[
1
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[
1
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