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Dec
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er
20
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p
p
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h
ttp
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//
ija
p
e.
ia
esco
r
e.
co
m/
O
ptimi
ze
the
po
si
tion o
f
t
he dis
trib
uted
g
ene
ra
tor a
n
d capa
citor
ba
nk in th
e dis
tri
buted
g
rid
t
o
min
imize th
e genera
ti
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n cos
t
Ng
o
c
An Lu
u,
Din
h Chu
ng
P
ha
n
F
a
c
u
l
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y
o
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l
e
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t
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v
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t
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i
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a
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a
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i
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t
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Art
icle
I
nfo
AB
S
T
RAC
T
A
r
ticle
his
to
r
y:
R
ec
eiv
ed
Dec
2
1
,
2
0
2
4
R
ev
is
ed
Au
g
1
9
,
2
0
2
5
Acc
ep
ted
Oct
1
6
,
2
0
2
5
In
t
h
is
p
a
p
e
r,
we
f
o
c
u
s
o
n
d
e
term
in
in
g
t
h
e
o
p
t
ima
l
p
o
siti
o
n
a
n
d
siz
e
o
f
m
u
lt
i
-
d
istri
b
u
ted
g
e
n
e
ra
t
o
rs
a
n
d
c
a
p
a
c
it
o
r
b
a
n
k
s
t
o
m
in
imiz
e
th
e
g
e
n
e
ra
ti
o
n
c
o
st
o
f
a
d
istri
b
u
ted
g
rid
.
T
h
e
o
p
ti
m
a
l
p
o
siti
o
n
a
n
d
siz
e
o
f
d
istri
b
u
ted
g
e
n
e
ra
to
rs
a
n
d
c
a
p
a
c
it
o
r
b
a
n
k
s
a
re
d
e
term
in
e
d
u
si
n
g
a
h
y
b
rid
o
f
c
o
n
v
e
n
ti
o
n
a
l
l
o
ss
se
n
si
ti
v
it
y
fa
c
to
r
a
n
d
a
n
imp
r
o
v
e
d
o
n
e
.
Th
e
p
ro
p
o
se
d
a
lg
o
rit
h
m
h
a
s
two
sta
g
e
s.
F
o
r
e
a
c
h
d
istri
b
u
ted
g
e
n
e
ra
to
r
,
we
p
ri
o
rit
ize
it
s
p
o
siti
o
n
a
n
d
siz
e
.
Afte
r
th
a
t,
we
fin
d
th
e
o
p
ti
m
a
l
p
o
siti
o
n
a
n
d
siz
e
o
f
th
e
c
a
p
a
c
it
o
r
b
a
n
k
s
c
o
rre
sp
o
n
d
i
n
g
t
o
t
h
is
d
istri
b
u
ted
g
e
n
e
ra
to
r
i
n
st
a
ll
a
ti
o
n
to
m
in
imiz
e
th
e
p
o
we
r
l
o
ss
.
Afte
r
c
o
n
sid
e
ri
n
g
a
ll
d
istri
b
u
ted
g
e
n
e
ra
to
rs,
t
h
e
o
p
ti
m
a
l
n
u
m
b
e
r,
p
o
siti
o
n
,
a
n
d
siz
e
o
f
th
e
d
istri
b
u
te
d
g
e
n
e
r
a
to
rs
a
n
d
c
a
p
a
c
it
o
r
b
a
n
k
s a
re
d
e
term
in
e
d
b
a
se
d
o
n
th
e
m
in
im
u
m
g
e
n
e
ra
ti
o
n
c
o
st v
a
lu
e
.
Th
is
id
e
a
is
d
e
v
e
lo
p
e
d
i
n
M
AT
LAB
a
n
d
v
e
rifi
e
d
v
ia
sa
m
p
le
d
istri
b
u
te
d
g
rid
s,
i
n
c
lu
d
in
g
th
e
IEE
E
-
6
9
b
u
s
a
n
d
I
EE
E
-
8
5
b
u
s.
Th
e
v
e
rif
y
in
g
r
e
su
lt
s
a
re
e
v
a
lu
a
ted
a
n
d
a
n
a
ly
z
e
d
.
By
c
o
m
p
a
rin
g
t
h
o
se
re
su
lt
s
to
th
o
se
o
f
o
t
h
e
r
m
e
th
o
d
s
,
t
h
e
p
e
rf
o
rm
a
n
c
e
o
f
t
h
e
n
e
wly
i
n
tro
d
u
c
e
d
m
e
th
o
d
is p
r
o
v
e
n
.
K
ey
w
o
r
d
s
:
C
ap
ac
ito
r
b
an
k
Dis
tr
ib
u
ted
g
en
er
ato
r
Dis
tr
ib
u
ted
g
r
id
Gen
er
atio
n
co
s
t
Po
wer
lo
s
s
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
:
Ng
o
c
An
L
u
u
Facu
lty
o
f
E
lectr
ical
E
n
g
in
ee
r
in
g
,
T
h
e
U
n
iv
er
s
ity
o
f
Dan
an
g
-
Un
iv
er
s
ity
o
f
Scien
ce
an
d
T
ec
h
n
o
lo
g
y
5
4
Ng
u
y
en
L
u
o
n
g
B
an
g
Stre
et
,
L
ien
C
h
ieu
W
ar
d
,
Da
Nan
g
,
Vietn
am
E
m
ail: ln
an
@
d
u
t.u
d
n
.
v
n
1.
I
NT
RO
D
UCT
I
O
N
W
ith
th
e
d
ev
elo
p
m
e
n
t
o
f
r
en
ewa
b
le
en
er
g
ies,
s
u
ch
as
win
d
an
d
s
o
lar
,
d
is
tr
ib
u
ted
g
en
e
r
ato
r
s
h
av
e
b
ec
o
m
e
m
o
r
e
p
o
p
u
lar
in
p
o
wer
s
y
s
tem
s
,
e
s
p
ec
ially
in
d
i
s
tr
ib
u
ted
g
r
id
s
.
T
h
ese
d
is
tr
ib
u
ted
g
en
er
ato
r
s
ca
n
im
p
ac
t o
p
er
atio
n
al
in
d
ices,
s
u
ch
as p
o
wer
lo
s
s
,
g
en
er
atio
n
c
o
s
t,
an
d
s
o
o
n
.
T
h
ese
in
d
ices c
an
b
ec
o
m
e
wo
r
s
e
if
DG’
s
p
o
s
itio
n
is
n
o
t
s
u
itab
le.
T
h
er
ef
o
r
e,
th
e
d
eter
m
in
atio
n
o
f
d
is
tr
ib
u
ted
g
en
er
ato
r
s
(
DG)
o
p
tim
al
p
lace
m
en
t
is
im
p
o
r
tan
t.
Un
til
n
o
w,
t
h
er
e
h
av
e
b
ee
n
m
an
y
p
r
o
p
o
s
ed
m
eth
o
d
s
t
o
ta
ck
le
th
e
p
o
wer
lo
s
s
is
s
u
e
[
1
]
-
[
1
5
]
.
So
m
e
au
th
o
r
s
f
o
cu
s
ed
o
n
u
s
in
g
DG
s
o
lely
[
1
]
-
[
3
]
o
r
a
ca
p
ac
ito
r
b
an
k
(
C
B
)
s
o
lely
[
4
]
.
T
h
e
co
m
b
in
atio
n
o
f
DG
an
d
o
th
er
m
eth
o
d
s
was
also
in
tr
o
d
u
ce
d
.
I
n
th
e
r
ef
er
e
n
ce
s
[
5
]
,
[
6
]
,
DGs
ar
e
ass
o
ciate
d
with
g
r
id
r
ec
o
n
f
ig
u
r
atio
n
;
th
e
d
is
ad
v
an
tag
e
o
f
th
is
co
m
b
in
atio
n
is
th
at
it
r
eq
u
ir
e
s
a
h
ig
h
in
v
esti
g
atio
n
t
o
r
ec
o
n
f
ig
u
r
e,
o
r
th
is
co
m
b
in
atio
n
ca
n
n
o
t
ap
p
ly
t
o
a
tie
-
d
is
tr
ib
u
ted
g
r
id
.
T
h
e
co
m
b
in
atio
n
o
f
r
ea
ctiv
e
p
o
wer
co
m
p
en
s
atio
n
eq
u
ip
m
en
t
a
n
d
DG
is
q
u
ite
p
o
p
u
lar
[
7
]
-
[
1
5
]
.
T
h
e
r
esu
lt
[
1
5
]
,
a
STAT
C
OM
wa
s
s
u
g
g
ested
,
wh
ile
in
[
7
]
-
[
1
4
]
,
C
B
s
o
r
s
h
u
n
t
ca
p
ac
ito
r
s
wer
e
u
s
ed
.
Gen
er
ally
,
STAT
C
OM
ca
n
b
e
v
er
y
e
f
f
icien
t
in
v
o
ltag
e
q
u
ality
s
u
p
p
o
r
t
b
u
t is m
o
r
e
e
x
p
en
s
iv
e
th
a
n
a
C
B
o
f
th
e
s
am
e
s
ize.
T
h
er
ef
o
r
e,
th
e
co
m
b
in
atio
n
o
f
DG
an
d
C
B
i
s
q
u
ite
p
o
p
u
la
r
.
Ma
n
y
alg
o
r
ith
m
s
wer
e
s
u
g
g
es
ted
to
d
eter
m
in
e
th
e
p
o
s
itio
n
o
f
DG
an
d
C
B
[
7
]
-
[
1
3
]
.
Gen
e
r
ally
,
th
ese
alg
o
r
ith
m
s
ar
e
d
iv
id
ed
in
to
t
h
r
ee
g
r
o
u
p
s
,
in
clu
d
in
g
th
e
c
o
n
v
en
tio
n
al
alg
o
r
ith
m
s
[
1
6
]
,
th
e
h
eu
r
is
tic
-
b
ased
alg
o
r
ith
m
s
[
8
]
,
[
9
]
,
an
d
th
e
h
y
b
r
id
alg
o
r
ith
m
s
[
1
0
]
,
[
1
4
]
,
[
1
7
]
.
T
h
ese
alg
o
r
ith
m
s
ca
n
ap
p
ly
to
two
class
e
s
o
f
o
b
jectiv
e
f
u
n
ctio
n
s
:
s
in
g
le
an
d
m
u
lti
-
co
s
t
f
u
n
ctio
n
s
.
C
o
n
ce
r
n
in
g
th
e
co
s
t
f
u
n
ctio
n
,
th
e
r
ed
u
ctio
n
o
f
p
o
we
r
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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p
l Po
wer
E
n
g
I
SS
N:
2252
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8
7
9
2
Op
timiz
e
th
e
p
o
s
itio
n
o
f th
e
d
i
s
tr
ib
u
ted
g
en
era
to
r
a
n
d
c
a
p
a
c
ito
r
b
a
n
k
in
th
e
d
is
tr
ib
u
ted
…
(
N
g
o
c
A
n
Lu
u
)
971
lo
s
s
o
r
en
e
r
g
y
lo
s
s
was
th
e
m
ain
an
d
p
o
p
u
lar
o
b
jectiv
e.
Sti
ll,
s
o
m
e
r
esear
ch
er
s
co
n
s
id
er
e
d
o
th
e
r
o
b
jectiv
es
s
u
ch
as
v
o
ltag
e
in
d
ex
,
b
en
ef
it,
an
d
s
o
o
n
[
1
7
]
.
Dep
e
n
d
in
g
o
n
th
e
ap
p
lied
alg
o
r
ith
m
,
t
h
e
r
esu
lts
m
ay
b
e
d
if
f
er
en
t,
n
o
m
atter
th
e
co
s
t
f
u
n
ctio
n
.
T
h
e
p
e
r
ce
n
tag
e
o
f
p
o
wer
r
ed
u
ctio
n
is
n
o
r
m
all
y
u
s
ed
to
co
m
p
ar
e
alg
o
r
ith
m
s
to
g
eth
e
r
.
No
r
m
ally
,
if
th
e
in
s
talled
p
o
wer
o
f
DG
an
d
C
B
is
h
ig
h
,
th
e
p
o
wer
l
o
s
s
m
ay
b
e
lo
w,
an
d
h
en
ce
,
th
e
p
er
ce
n
tag
e
o
f
p
o
w
er
r
ed
u
ctio
n
is
h
ig
h
er
.
Ho
we
v
er
,
wh
e
n
we
i
n
s
tall
a
DG
a
n
d
C
B
co
m
b
in
atio
n
with
a
h
ig
h
er
ca
p
ac
ity
,
t
h
e
an
n
u
al
o
p
e
r
atio
n
c
o
s
t
will
b
e
h
i
g
h
er
.
T
h
er
ef
o
r
e,
th
e
h
ig
h
e
r
p
o
wer
lo
s
s
d
o
es
n
o
t
m
ea
n
th
at
we
ca
n
o
b
tain
b
en
ef
its
.
I
n
th
is
r
esear
ch
,
we
s
u
g
g
est
a
n
alg
o
r
ith
m
t
o
id
en
tify
t
h
e
b
es
t
lo
ca
tio
n
an
d
ca
p
ac
ity
o
f
DG
s
an
d
C
B
s
.
T
h
e
o
p
tim
izatio
n
g
o
al
f
o
cu
s
es
o
n
m
in
im
izin
g
th
e
g
e
n
er
atio
n
co
s
t,
b
u
t
we
s
till
ac
h
iev
e
lo
wer
lo
s
s
es
w
ith
in
th
e
d
is
tr
ib
u
tio
n
n
etwo
r
k
.
T
h
e
p
r
o
p
o
s
ed
alg
o
r
ith
m
is
d
ev
elo
p
e
d
f
r
o
m
th
e
lo
s
s
s
en
s
itiv
ity
f
ac
to
r
(
L
SF
)
.
Un
li
k
e
m
an
y
p
r
e
v
io
u
s
wo
r
k
s
th
at
f
o
c
u
s
m
ain
ly
o
n
p
o
wer
lo
s
s
o
r
v
o
ltag
e
p
r
o
f
ile,
o
u
r
r
esear
ch
em
p
h
asizes
g
en
er
atio
n
co
s
t,
in
clu
d
in
g
b
o
t
h
DG
an
d
C
B
in
v
estme
n
t,
u
s
in
g
an
ef
f
ic
ien
t
s
tep
-
wis
e
an
aly
tical
ap
p
r
o
ac
h
.
W
e
u
tili
ze
th
e
MA
T
L
AB
s
cr
ip
tin
g
en
v
ir
o
n
m
en
t
to
ex
ec
u
te
th
e
alg
o
r
ith
m
,
a
n
d
we
em
p
l
o
y
th
e
I
E
E
E
-
6
9
b
u
s
an
d
I
E
E
E
-
8
5
b
u
s
p
o
wer
s
y
s
tem
s
f
o
r
v
e
r
if
icatio
n
.
T
h
e
f
in
d
in
g
s
ar
e
e
v
alu
ated
a
n
d
ju
x
ta
p
o
s
ed
with
o
th
e
r
ex
is
tin
g
m
eth
o
d
s
.
2.
P
RO
B
L
E
M
ST
A
T
E
M
E
NT
AND
L
O
S
S SE
N
SI
T
I
V
I
T
Y
F
ACTOR
I
n
th
is
s
tu
d
y
,
we
d
o
n
o
t
m
ain
l
y
f
o
cu
s
o
n
th
e
p
o
wer
l
o
s
s
m
in
im
izatio
n
,
b
u
t
we
f
o
cu
s
o
n
m
in
im
izin
g
th
e
g
en
er
atio
n
c
o
s
t
in
th
e
d
is
tr
ib
u
tio
n
s
y
s
tem
.
T
h
is
co
s
t
in
cl
u
d
es
th
e
en
e
r
g
y
co
s
t
f
r
o
m
b
o
t
h
DGs
an
d
th
e
g
r
i
d
,
an
d
th
e
in
v
estme
n
t
in
C
B
s
.
C
o
m
p
ar
ed
to
m
a
n
y
p
r
e
v
io
u
s
s
tu
d
ies
th
at
o
n
ly
m
in
im
ize
p
o
wer
lo
s
s
,
th
i
s
o
b
jectiv
e
b
etter
r
ef
lects
th
e
ec
o
n
o
m
ic
p
er
f
o
r
m
an
ce
o
f
t
h
e
s
y
s
tem
.
T
o
o
b
tain
th
is
g
o
al,
we
u
s
e
an
an
aly
tical
m
eth
o
d
,
wh
ich
is
n
am
ed
th
e
im
p
r
o
v
e
d
lo
s
s
s
en
s
itiv
ity
f
ac
to
r
(
I
L
SF
)
,
to
d
eter
m
in
e
th
e
o
p
tim
al
in
s
tallatio
n
s
ite
an
d
r
atin
g
o
f
DGs
an
d
C
B
s
.
T
h
is
m
eth
o
d
d
o
es
n
o
t
r
eq
u
ir
e
p
o
p
u
latio
n
-
b
ased
o
r
h
eu
r
is
tic
alg
o
r
ith
m
s
,
s
o
it
is
f
aster
an
d
ea
s
ier
to
im
p
lem
en
t.
T
o
o
u
r
k
n
o
wled
g
e,
th
is
o
p
tim
al
p
r
o
b
lem
an
d
s
o
lu
tio
n
m
eth
o
d
h
a
v
e
n
o
t
b
ee
n
u
s
ed
in
s
im
ilar
wo
r
k
s
.
2
.
1
.
G
ener
a
t
io
n c
o
s
t
T
h
e
g
en
er
atio
n
co
s
t
is
d
ef
in
ed
as
th
e
co
s
t
to
s
u
p
p
ly
ele
ctr
icity
to
lo
ad
s
p
er
h
o
u
r
.
T
h
is
co
s
t
i
s
co
m
p
u
ted
f
r
o
m
th
e
i
n
v
estme
n
t
an
d
th
e
en
e
r
g
y
s
ellin
g
c
o
s
ts
f
r
o
m
elec
tr
ical
s
o
u
r
ce
s
in
th
e
g
r
id
.
W
ith
an
ex
is
tin
g
g
r
id
,
th
e
in
v
estme
n
t
c
o
s
t
is
alm
o
s
t
co
n
s
tan
t.
Hen
ce
,
th
e
g
en
er
atio
n
c
o
s
t
is
o
n
ly
r
e
lian
t
o
n
th
e
en
er
g
y
q
u
an
tity
r
ec
eiv
ed
f
r
o
m
s
o
u
r
c
es,
th
e
n
ew
co
m
p
o
n
en
ts
’
co
s
t.
Hen
ce
,
th
e
g
en
er
atio
n
co
s
t
is
s
im
p
lifie
d
as
(
1
)
-
(
4
)
[
1
7
]
.
=
+
+
(
1
)
=
(
2
)
=
+
×
(
3
)
=
×
(
4
)
W
h
er
e,
is
th
e
p
r
ice
o
f
a
k
W
h
f
r
o
m
DG;
,
,
an
d
ar
e
co
n
s
tan
t
co
s
t,
life
tim
e,
an
d
in
v
estme
n
t
o
f
a
k
VAr
o
f
C
B
;
is
th
e
n
u
m
b
er
o
f
n
o
d
es
wh
er
e
th
e
C
B
is
in
s
t
alled
;
is
th
e
s
ellin
g
p
r
ice
o
f
1
k
W
h
f
r
o
m
th
e
g
r
id
;
is
th
e
p
o
wer
s
u
p
p
lied
f
r
o
m
th
e
g
r
i
d
;
an
d
an
d
ar
e
th
e
DG
p
o
wer
g
e
n
er
ati
o
n
a
n
d
C
B
ca
p
ac
ity
.
2
.
2
.
O
pti
m
iza
t
io
n pro
blem
T
h
e
o
b
jectiv
e
is
to
m
in
im
ize
th
e
g
en
e
r
atio
n
c
o
s
t
o
f
th
e
g
r
i
d
.
Acc
o
r
d
in
g
ly
,
th
e
o
p
tim
izatio
n
p
r
o
b
lem
is
d
ef
in
ed
as:
=
+
+
→
(
5
)
≤
≤
(
6
)
≤
(
7
)
0
.
8
≤
≤
1
(
8
)
0
≤
∑
,
=
1
≤
(
9
)
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
7
0
-
979
972
0
≤
∑
,
=
1
≤
−
∑
,
=
1
(
1
0
)
∆
≤
∆
−
1
(
1
1
)
wh
er
e,
is
th
e
cu
r
r
en
t
o
n
th
e
ℎ
lin
e;
is
th
e
v
o
ltag
e
at
th
e
ℎ
n
o
d
e
;
an
d
ar
e
ac
tiv
e
an
d
r
ea
ctiv
e
p
o
wer
d
em
an
d
s
in
th
e
g
r
id
;
,
,
an
d
ar
e
ac
tiv
e,
r
ea
ct
iv
e
p
o
wer
,
an
d
p
o
wer
f
ac
to
r
o
f
DG;
∆
is
th
e
p
o
wer
lo
s
s
at
th
e
ℎ
iter
atio
n
;
,
,
an
d
ar
e
th
e
b
o
u
n
d
ed
v
a
lu
es
o
f
n
o
d
e
v
o
ltag
e
an
d
cu
r
r
en
t o
n
th
e
ℎ
lin
e;
an
d
a
r
e
t
h
e
m
a
x
i
m
u
m
n
u
m
b
e
r
o
f
D
G
a
n
d
C
B
,
r
e
s
p
ec
t
i
v
e
l
y
.
N
o
t
e
d
t
h
a
t
:
,
=
min
(
,
−
1
,
0
.
95
)
(
1
2
)
2
.
3
.
L
o
s
s
s
ens
it
iv
it
y
f
a
ct
o
r
2
.
3
.
1
.
Co
nv
ent
io
na
l LS
F
m
et
ho
d
W
e
co
n
s
id
er
th
e
s
im
p
lifie
d
tr
ee
-
s
tr
u
ctu
r
ed
g
r
i
d
in
Fig
u
r
e
1
,
wh
er
e
̇
is
th
e
to
tal
ap
p
ar
en
t
p
o
wer
o
f
lo
ad
s
in
b
r
an
ch
es c
o
n
n
ec
ted
t
o
th
e
ℎ
n
o
d
e
a
n
d
̇
is
th
e
ap
p
ar
en
t p
o
wer
in
jecte
d
in
to
th
e
ℎ
n
o
d
e.
T
h
e
p
o
we
r
lo
s
s
f
r
o
m
th
e
s
o
u
r
ce
to
th
e
ℎ
n
o
d
e
ca
u
s
ed
b
y
th
e
p
o
wer
̇
ca
n
b
e
s
im
p
lifie
d
as
(
1
3
)
.
∆
=
2
∑
−
1
2
=
2
=
(
2
+
2
)
∑
−
1
2
=
2
(
1
3
)
W
h
er
e,
−
1
is
th
e
r
esis
tan
ce
o
f
t
h
e
lin
e
f
r
o
m
th
e
(
−
1
)
ℎ
n
o
d
e
to
th
e
ℎ
n
o
d
e
a
n
d
is
th
e
ℎ
n
o
d
e
’
s
v
o
ltag
e.
T
h
e
L
SF
v
alu
e
at
th
e
ℎ
n
o
d
e
v
e
r
s
u
s
ac
tiv
e
p
o
wer
(
,
)
an
d
v
er
s
u
s
r
ea
ctiv
e
p
o
wer
(
,
)
is
co
m
p
u
ted
as
(
1
4
)
an
d
(
1
5
)
.
,
=
2
∑
∑
2
=
2
(
1
4
)
,
=
2
∑
∑
2
=
2
(
1
5
)
T
o
o
b
tain
∆
in
(
1
3
)
e
q
u
al
to
ze
r
o
,
we
s
h
o
u
ld
in
s
tall D
G
at
th
e
ℎ
n
o
d
e
s
u
ch
th
at
:
,
=
∑
(
1
6
)
,
=
∑
.
(
1
7
)
Fig
u
r
e
1
.
A
s
am
p
le
o
f
th
e
d
is
tr
ib
u
ted
g
r
i
d
2
.
3
.
2
.
I
m
pro
v
e
m
ent
o
f
L
SF
m
et
ho
d
T
h
e
im
p
r
o
v
em
en
t
o
f
L
SF
(
I
L
SF
)
was
d
ev
elo
p
ed
f
r
o
m
L
SF
,
an
d
th
e
I
L
SF
d
etail
was
d
escr
ib
ed
in
[
1
8
]
.
T
h
e
I
L
SF
v
alu
e
o
f
ac
tiv
e
p
o
wer
an
d
r
ea
ctiv
e
p
o
wer
at
t
h
e
ℎ
n
o
d
e
is
co
m
p
u
ted
as (
1
8
)
a
n
d
(
1
9
)
.
,
=
2
∑
∑
2
−
1
=
2
(
1
8
)
,
=
2
∑
∑
2
−
1
=
2
(
1
9
)
T
h
e
o
p
tim
al
ac
tiv
e
an
d
r
ea
cti
v
e
p
o
wer
o
f
DG
at
th
e
ℎ
n
o
d
e
t
o
m
in
im
ize
th
e
p
o
we
r
lo
s
s
is
d
ef
in
ed
as
(
2
0
)
an
d
(
2
1
)
.
,
=
∑
∑
2
−
1
=
2
(
∑
−
1
2
=
2
)
−
1
(
2
0
)
,
=
∑
∑
2
−
1
=
2
(
∑
−
1
2
=
2
)
−
1
.
(
2
1
)
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
Op
timiz
e
th
e
p
o
s
itio
n
o
f th
e
d
i
s
tr
ib
u
ted
g
en
era
to
r
a
n
d
c
a
p
a
c
ito
r
b
a
n
k
in
th
e
d
is
tr
ib
u
ted
…
(
N
g
o
c
A
n
Lu
u
)
973
3.
P
RO
P
O
SE
D
AL
G
O
R
I
T
H
M
Un
lik
e
s
im
u
ltan
eo
u
s
o
p
tim
iza
tio
n
ap
p
r
o
ac
h
es
as
B
SA
[
7
]
an
d
SS
A
[
8
]
,
th
e
p
r
o
p
o
s
ed
m
et
h
o
d
ap
p
lies
a
s
tep
-
wis
e
o
p
tim
izatio
n
m
eth
o
d
t
h
at
f
ir
s
t
d
eter
m
in
es
th
e
o
p
tim
al
p
lace
m
e
n
t
an
d
s
ize
o
f
DGs,
f
o
llo
wed
b
y
th
at
o
f
C
B
s
.
T
h
is
ap
p
r
o
ac
h
is
m
o
tiv
ated
b
y
th
e
f
o
llo
win
g
co
n
s
id
er
atio
n
s
.
Firstl
y
,
DGs
an
d
C
B
s
af
f
ec
t
th
e
d
is
tr
ib
u
tio
n
n
etwo
r
k
in
f
u
n
d
a
m
en
tally
d
if
f
er
en
t
way
s
.
W
h
ile
DGs
ca
n
s
u
p
p
ly
b
o
th
ac
tiv
e
an
d
r
ea
ctiv
e
p
o
wer
,
C
B
s
o
n
ly
p
r
o
v
id
e
r
ea
ctiv
e
co
m
p
en
s
atio
n
.
I
f
b
o
th
ar
e
o
p
tim
ized
s
im
u
ltan
eo
u
s
ly
,
th
e
alg
o
r
ith
m
m
ay
in
s
tall
lar
g
e
ca
p
ac
ito
r
s
with
o
u
t
f
u
lly
u
tili
zin
g
th
e
DGs'
r
ea
ctiv
e
p
o
wer
ca
p
ab
ilit
y
,
lead
in
g
to
in
ef
f
icien
cy
.
Seco
n
d
ly
,
o
u
r
m
eth
o
d
u
s
es
a
d
eter
m
in
i
s
tic
an
aly
tical
ap
p
r
o
ac
h
b
y
u
s
in
g
I
L
SF
,
r
ath
e
r
th
a
n
m
etah
eu
r
is
tic
alg
o
r
ith
m
s
.
T
h
is
h
elp
s
av
o
i
d
th
e
n
ee
d
to
s
et
u
p
a
p
o
p
u
latio
n
,
a
d
ju
s
t
p
a
r
am
eter
s
,
o
r
ev
alu
ate
f
itn
ess
m
an
y
tim
es,
m
a
k
in
g
th
e
m
eth
o
d
s
im
p
ler
an
d
f
aster
to
r
u
n
.
T
h
e
r
ef
o
r
e
,
alth
o
u
g
h
s
im
u
ltan
eo
u
s
o
p
tim
izatio
n
u
s
in
g
m
etah
eu
r
is
tic
alg
o
r
ith
m
s
m
a
y
wo
r
k
well
in
o
th
er
s
tu
d
ies,
th
e
s
tep
-
wis
e
m
eth
o
d
u
s
ed
in
th
is
p
ap
er
is
r
e
aso
n
ab
le
an
d
tak
es
ad
v
an
tag
e
o
f
th
e
s
p
ec
if
ic
s
tr
en
g
th
s
o
f
th
e
I
L
SF
ap
p
r
o
ac
h
.
T
o
en
s
u
r
e
m
in
im
al
g
e
n
er
atio
n
co
s
t
in
th
e
g
r
id
,
an
alg
o
r
i
th
m
is
p
r
o
p
o
s
ed
in
Fig
u
r
e
2
.
I
n
th
is
alg
o
r
ith
m
,
t
h
e
ca
lcu
latio
n
o
f
ea
ch
DG’
s
o
p
tim
al
p
o
s
itio
n
a
n
d
s
ize
is
alwa
y
s
p
r
io
r
itized
o
v
er
th
at
o
f
C
B
s
to
u
tili
ze
th
e
r
ea
ctiv
e
p
o
wer
s
u
p
p
o
r
ted
b
y
DG.
I
t
m
ea
n
s
af
ter
d
eter
m
in
in
g
th
e
ℎ
DG,
we
s
tar
t
to
d
eter
m
in
e
th
e
o
p
tim
al
in
s
tallatio
n
s
ite
an
d
r
atin
g
o
f
C
B
s
to
o
b
tain
th
e
lo
west
g
en
er
atio
n
co
s
t,
a
n
d
th
e
n
m
o
v
e
to
th
e
n
ex
t
DG.
T
h
e
C
B
n
u
m
b
er
,
p
o
s
itio
n
,
an
d
s
ize
o
f
C
B
s
ar
e
d
ep
en
d
e
n
t o
n
th
e
in
s
talled
DG
n
u
m
b
e
r
.
Her
e,
we
co
m
p
ar
e
th
e
r
esu
lt
o
f
th
e
L
SF
m
eth
o
d
t
o
th
e
I
L
SF
m
eth
o
d
to
id
e
n
tify
th
e
b
est
p
o
s
itio
n
an
d
p
o
wer
r
atin
g
o
f
DG
o
r
C
B
.
T
h
is
alg
o
r
ith
m
is
s
h
o
wn
in
Fig
u
r
e
2
.
Fig
u
r
e
2
.
Alg
o
r
ith
m
to
d
eter
m
in
e
th
e
o
p
tim
al
p
o
s
itio
n
an
d
s
ize
o
f
b
o
th
DG
an
d
C
B
Step
1
:
R
ea
d
in
g
d
ata,
in
clu
d
in
g
th
e
g
r
id
p
ar
am
ete
r
s
,
; c
o
s
ts
o
f
DG,
C
B
,
an
d
th
e
co
n
n
ec
ted
g
r
id
;
an
d
r
u
n
th
e
p
o
wer
f
lo
w.
W
e
s
tar
t
th
e
f
ir
s
t
D
G,
d
=1
.
Step
2
:
C
alcu
latin
g
(
1
4
)
a
n
d
(
1
8
)
at
all
n
o
d
es.
W
e
ch
o
s
e
th
e
b
est
p
o
s
itio
n
f
o
r
t
h
e
ℎ
DG
b
ased
o
n
an
d
,
an
d
we
d
ete
r
m
in
e
DG’
s
s
ize
(
,
f
r
o
m
(
1
6
)
,
(
1
7
)
,
a
n
d
,
f
r
o
m
(
2
0
)
,
(
2
1
)
)
at
t
h
ese
n
o
d
es.
Du
e
to
th
e
lim
itatio
n
o
f
r
ea
ctiv
e
p
o
wer
f
r
o
m
DG,
th
e
o
p
tim
al
r
ea
ctiv
e
p
o
wer
o
f
DG
is
d
eter
m
in
ed
b
y
th
e
allo
wab
le
p
o
wer
f
ac
to
r
as
(
2
2
)
.
,
=
(
,
)
(
22)
Step
3
:
R
u
n
n
in
g
p
o
wer
f
lo
w
with
th
e
ex
is
ten
ce
o
f
th
i
s
D
G
to
o
b
tain
th
e
p
o
wer
lo
s
s
,
∆
an
d
∆
,
co
r
r
esp
o
n
d
in
g
t
o
th
e
L
SF
m
et
h
o
d
an
d
I
L
SF
m
eth
o
d
,
r
esp
ec
t
iv
ely
.
T
h
e
e
f
f
icien
cy
f
ac
to
r
o
f
DG
in
s
tallatio
n
is
:
,
=
∆
−
1
−
∆
an
d
,
=
∆
−
1
−
∆
(
2
3
)
wh
er
e,
∆
−
1
is
th
e
p
o
wer
lo
s
s
b
ef
o
r
e
in
s
tallin
g
th
e
ℎ
DG
f
o
r
ea
c
h
m
eth
o
d
.
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
7
0
-
979
974
Step
4
:
Der
iv
in
g
th
e
o
p
tim
al
i
n
s
tallatio
n
s
ite
an
d
s
ize
o
f
D
G.
I
f
,
>
,
,
th
e
b
est
in
s
tallatio
n
s
it
e
an
d
s
ize
o
f
DG
co
m
e
f
r
o
m
t
h
e
L
SF
m
eth
o
d
;
o
th
er
wis
e,
t
h
ey
co
m
e
f
r
o
m
th
e
I
L
SF
m
e
th
o
d
.
W
e
s
to
r
e
th
e
p
o
s
itio
n
an
d
r
atin
g
o
f
DG
in
th
e
s
et
ℵ
an
d
we
u
s
e
th
e
p
o
wer
l
o
s
s
,
∆
,
co
r
r
esp
o
n
d
in
g
to
t
h
is
ca
s
e
in
th
e
n
ex
t
s
tep
s
.
Step
5
:
T
esti
n
g
th
e
s
to
p
co
n
d
itio
n
.
I
f
∆
≤
∆
−
1
,
≤
,
Step
6
is
u
s
ed
;
o
th
er
wis
e,
Step
1
3
is
d
o
n
e.
Step
6
:
Up
d
atin
g
th
e
g
r
id
d
ata
b
y
ad
d
in
g
t
h
e
ℎ
DG
i
n
.
W
e
s
tar
t
th
e
f
ir
s
t
C
B
,
=
1
,
an
d
we
s
et
∆
−
1
=
∆
−
,
−
=
−
,
=
,
an
d
r
u
n
p
o
wer
f
lo
w
.
Step
7
:
C
alcu
latin
g
(
1
5
)
an
d
(
1
9
)
at
all
n
o
d
es.
W
e
ch
o
o
s
e
th
e
b
est
n
o
d
e
t
o
in
s
tal
l
C
B
an
d
its
s
ize
f
o
r
t
h
e
L
SF
m
eth
o
d
,
(
1
7
)
an
d
th
e
I
L
SF
m
et
h
o
d
,
(
21
)
.
Step
8
:
R
u
n
n
in
g
p
o
wer
f
lo
w
a
f
ter
ad
d
i
n
g
th
is
C
B
f
o
r
ea
ch
m
eth
o
d
t
o
o
b
tain
th
e
p
o
wer
lo
s
s
,
∆
an
d
∆
.
W
e
co
m
p
u
te
th
e
ef
f
icien
t
f
ac
t
o
r
as
(
2
4
)
,
wh
er
e
∆
−
1
is
a
p
o
wer
lo
s
s
b
ef
o
r
e
in
s
tallin
g
th
e
ℎ
CB
.
,
=
∆
−
1
−
∆
an
d
,
=
∆
−
1
−
∆
(
24)
Step
9
:
Der
iv
i
n
g
t
h
e
o
p
tim
al
p
o
s
itio
n
an
d
s
ize
o
f
C
B
.
I
f
,
≥
,
,
th
e
o
p
tim
al
p
o
s
itio
n
an
d
s
ize
o
f
th
is
C
B
co
m
e
f
r
o
m
th
e
L
SF
m
eth
o
d
;
o
th
er
wis
e,
we
u
s
e
th
e
I
L
SF
m
eth
o
d
.
W
e
u
s
e
th
e
p
o
wer
lo
s
s
,
∆
,
co
r
r
esp
o
n
d
in
g
t
o
th
is
ca
s
e
in
th
e
n
ex
t step
s
.
Step
1
0
:
If
∆
≤
∆
−
1
,
≤
,
≤
−
1
,
Step
1
1
is
d
o
n
e
; o
th
er
wis
e,
we
m
o
v
e
t
o
Step
1
2
.
Step
1
1
:
Up
d
atin
g
th
e
ℎ
C
B
in
th
en
r
u
n
th
e
p
o
wer
f
l
o
w.
W
e
s
et
c=
c+
1
,
an
d
th
en
Step
7
is
r
e
tu
r
n
ed
.
Step
1
2
:
W
e
s
et
=
−
1
,
d
=d
+1
,
u
s
e
to
r
u
n
t
h
e
p
o
wer
f
lo
w,
a
n
d
th
e
n
ex
ec
u
te
Step
2
ag
ain
.
Step
1
3
:
Der
iv
in
g
th
e
o
p
tim
al
DG
an
d
C
B
n
u
m
b
er
s
f
r
o
m
min
{
1
,
2
,
…
,
−
1
}
.
4.
RE
SU
L
T
S AN
D
D
I
SCU
SS
I
O
N
T
o
v
er
if
y
th
e
p
r
o
p
o
s
ed
alg
o
r
i
th
m
,
we
u
s
e
th
e
I
E
E
E
6
9
b
u
s
an
d
I
E
E
E
-
8
5
b
u
s
d
is
tr
ib
u
ted
s
y
s
tem
as
s
h
o
wn
in
Fig
u
r
e
3
.
T
h
e
to
tal
l
o
ad
in
th
e
I
E
E
E
-
6
9
b
u
s
g
r
i
d
is
3
8
0
1
.
9
k
W
an
d
2
6
9
4
.
1
k
VAr
,
wh
ile
in
th
e
I
E
E
E
-
8
5
b
u
s
g
r
id
,
it
is
2
5
7
0
.
3
k
W
an
d
2
6
2
2
.
1
k
VAr
.
T
h
ese
g
r
id
s
’
d
ata
ar
e
lis
ted
in
[
1
9
]
,
[
2
0
]
.
W
e
s
u
p
p
o
s
e
th
at
th
e
en
er
g
y
p
r
ice
f
r
o
m
th
e
g
r
id
a
n
d
DG
ar
e
4
9
$
/MWh
an
d
5
1
.
4
5
$
/MWh
,
r
esp
ec
tiv
ely
;
th
e
C
B
in
v
estme
n
t
an
d
its
co
n
s
tan
t a
r
e
0
.
3
5
$
/
k
VAr
/y
ea
r
an
d
1
0
0
$
/y
ea
r
.
(
a)
(
b
)
Fig
u
r
e
3
.
T
h
e
co
n
f
ig
u
r
atio
n
o
f
th
e
s
am
p
le
d
is
tr
ib
u
tio
n
s
y
s
te
m
: (
a)
I
E
E
E
6
9
b
u
s
an
d
(
b
)
I
E
E
E
8
5
b
u
s
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
Op
timiz
e
th
e
p
o
s
itio
n
o
f th
e
d
i
s
tr
ib
u
ted
g
en
era
to
r
a
n
d
c
a
p
a
c
ito
r
b
a
n
k
in
th
e
d
is
tr
ib
u
ted
…
(
N
g
o
c
A
n
Lu
u
)
975
4
.
1
.
IEEE
-
6
9
bu
s
di
s
t
ribute
d g
rid
W
ith
th
e
I
E
E
E
-
6
9
b
u
s
s
y
s
tem
,
if
th
e
m
ax
im
u
m
DG
n
u
m
b
er
i
s
4
,
we
o
b
tain
th
e
r
esu
lts
in
T
a
b
le
1
an
d
Fig
u
r
e
4
.
C
lear
ly
,
with
th
e
p
r
o
p
o
s
ed
alg
o
r
ith
m
,
we
o
n
ly
in
s
tall
1
DG
at
th
e
6
1
s
t
n
o
d
e
(
1
5
6
2
k
W
)
a
n
d
3
C
B
s
at
th
e
1
6
th
,
6
4
th
,
an
d
1
7
th
n
o
d
es
(
with
2
2
5
,
2
0
4
,
an
d
1
9
5
k
VAr
,
r
esp
ec
tiv
ely
)
.
W
ith
t
h
is
in
s
tallatio
n
,
th
e
p
o
wer
lo
s
s
an
d
th
e
g
en
e
r
atio
n
co
s
t
ar
e
r
ed
u
ce
d
s
ig
n
if
ican
tly
,
an
d
th
e
m
in
im
u
m
n
o
d
e
v
o
lta
g
e
in
th
is
n
etwo
r
k
is
elev
ated
.
A
n
o
tab
le
p
o
wer
l
o
s
s
r
ed
u
ctio
n
is
o
b
s
er
v
ed
,
f
r
o
m
2
2
5
k
W
to
2
0
.
4
5
k
W
,
an
d
t
h
e
g
e
n
er
atio
n
co
s
t
is
cu
t f
r
o
m
1
9
7
.
3
$
/h
to
1
9
1
.
1
9
6
$
/h
.
T
h
e
v
o
ltag
e
at
n
o
d
es f
r
o
m
th
e
5
0
th
n
o
d
e
to
th
e
6
9
th
n
o
d
e
is
o
v
er
9
8
%,
an
d
th
e
m
in
im
u
m
v
o
ltag
e
in
th
e
g
r
id
is
ar
o
u
n
d
9
8
% a
t th
e
2
7
th
n
o
d
e,
wh
ile
in
th
e
b
ase
ca
s
e,
th
e
m
in
im
u
m
v
o
ltag
e
is
9
1
.
0
2
% a
t th
e
6
5
th
n
o
d
e.
T
ab
le
1
.
R
esu
lts
o
f
ap
p
ly
i
n
g
t
h
e
p
r
o
p
o
s
ed
alg
o
r
ith
m
to
th
e
I
E
E
E
-
6
9
b
u
s
C
a
se
D
G
si
z
e
(
n
o
d
e
)
(
k
W
)
pf
(
%)
C
B
s
i
z
e
(
n
o
d
e
)
(
k
V
A
r
)
∆
(
k
W
)
,
(
%)
C
o
s
t
(
$
/
h
)
B
a
se
2
2
5
9
1
.
0
2
1
9
7
.
3
P
r
o
p
o
se
d
m
e
t
h
o
d
1
5
6
2
(
6
1
)
81
1
9
5
(
1
7
)
2
0
4
(
6
4
)
2
2
5
(
1
6
)
2
0
.
4
5
9
7
.
7
3
1
9
1
.
1
9
6
Fig
u
r
e
4
.
T
h
e
v
o
lta
g
e
at
n
o
d
es in
th
e
I
E
E
E
-
6
9
b
u
s
af
ter
i
n
s
tallin
g
DG
an
d
C
B
s
T
o
clar
if
y
th
e
ab
o
v
e
r
esu
lts
,
t
h
e
r
esu
lts
o
f
all
ca
s
es
ar
e
s
h
o
wn
in
Fig
u
r
e
5
.
Fig
u
r
e
5
(
a
)
in
d
icate
s
th
e
p
o
wer
l
o
s
t
an
d
th
e
g
en
e
r
atio
n
co
s
t
wh
en
we
in
s
tall
4
DGs
s
tep
b
y
s
tep
.
Ob
v
io
u
s
ly
,
a
f
ter
i
n
s
tallin
g
th
e
1
st
DG,
th
e
p
o
wer
lo
s
s
is
r
ed
u
ce
d
f
r
o
m
2
2
5
k
W
to
2
6
.
7
3
7
k
W
,
an
d
its
ef
f
icien
cy
is
ar
o
u
n
d
0
.
056
%
/
wh
ile
af
ter
in
s
tallin
g
th
e
4
th
DG,
th
e
p
o
w
er
lo
s
s
is
7
.
0
1
7
k
W
an
d
th
e
e
f
f
icien
cy
is
ar
o
u
n
d
0
.
0
4
3
%
/
.
C
o
n
ce
r
n
in
g
th
e
g
en
er
atio
n
c
o
s
t,
af
ter
in
s
tallin
g
th
e
1
st
DG,
th
e
g
en
er
atio
n
co
s
t
is
th
e
lo
west,
1
9
1
.
4
3
5
$
/
h
.
B
y
ad
d
in
g
m
o
r
e
DGs,
th
e
g
en
er
atio
n
c
o
s
t
in
cr
ea
s
es.
Fig
u
r
e
5
(
b
)
in
d
icate
s
th
e
r
esu
lts
in
th
e
ca
s
e
o
f
b
o
th
DG
an
d
C
B
in
s
tallatio
n
.
Fro
m
t
h
is
f
ig
u
r
e,
wh
en
th
e
DG
n
u
m
b
e
r
is
h
i
g
h
er
,
th
e
C
B
n
u
m
b
e
r
is
lo
wer
,
a
n
d
in
th
e
ca
s
e
o
f
4
DGs,
n
o
n
e
C
B
ar
e
s
u
g
g
ested
.
Ob
v
io
u
s
ly
,
b
y
a
d
d
in
g
C
B
,
b
o
th
p
o
wer
lo
s
s
an
d
g
en
e
r
atio
n
c
o
s
t
ar
e
lo
wer
th
an
th
o
s
e
in
Fig
u
r
e
5
(
a
)
.
Ho
we
v
er
,
th
e
h
ig
h
er
th
e
DG
n
u
m
b
er
,
th
e
h
ig
h
er
th
e
g
en
er
atio
n
co
s
t.
T
h
is
in
cr
ea
s
e
co
m
es
f
r
o
m
t
h
e
in
c
r
ea
s
e
in
th
e
DG
co
s
t.
T
h
er
ef
o
r
e,
we
s
h
o
u
ld
in
s
tall
a
DG
an
d
3
C
B
as
T
ab
le
1
to
o
b
tain
th
e
lo
west g
en
er
atio
n
co
s
t.
(
a)
(
b
)
Fig
u
r
e
5
.
Po
wer
l
o
s
s
an
d
g
en
e
r
atio
n
co
s
t a
f
ter
in
s
tallin
g
DG
an
d
C
B
: (
a)
On
ly
DG
an
d
(
b
)
b
o
th
C
B
an
d
DG
T
o
co
m
p
ar
e
to
o
th
er
r
esear
ch
,
two
ca
s
es
o
f
p
o
wer
f
ac
to
r
(
=
100%
an
d
=
80%
)
an
d
two
ca
s
es
o
f
th
e
DG
n
u
m
b
er
(
=
1
=
3
)
ar
e
u
s
ed
.
T
h
e
co
m
p
ar
is
o
n
r
esu
lts
ar
e
s
h
o
wn
in
T
ab
le
2
[
7
]
-
[
9
]
,
[
2
1
]
,
[
2
2
]
.
C
lear
ly
,
with
o
u
r
alg
o
r
ith
m
,
th
e
p
o
wer
lo
s
s
ca
n
n
o
t
b
e
co
m
p
ar
ed
to
o
th
e
r
s
b
ec
au
s
e
th
e
to
tal
ca
p
ac
ity
o
f
DG
an
d
C
B
in
t
h
e
p
r
o
p
o
s
ed
alg
o
r
ith
m
is
lo
wer
th
an
th
at
o
f
o
th
er
s
.
Ho
we
v
er
,
th
e
in
v
estme
n
t
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
7
0
-
979
976
ef
f
ec
tiv
en
ess
o
f
th
e
i
n
tr
o
d
u
ce
d
s
ch
em
e
s
u
r
p
ass
es
o
th
er
s
.
F
o
r
ex
a
m
p
le,
in
th
e
ca
s
e
o
f
3
DGs
with
=
100%
,
in
[
2
2
]
,
af
ter
in
s
tallin
g
2
5
4
7
k
W
o
f
DG
an
d
1
7
9
7
k
VAr
o
f
C
B
,
th
e
s
y
s
tem
lo
s
s
is
as
lo
w
as
4
.
2
6
3
k
W
,
b
u
t
th
e
g
en
er
ati
o
n
co
s
t
is
1
9
2
.
8
8
3
$
/
h
,
wh
ich
is
h
ig
h
er
t
h
a
n
o
u
r
alg
o
r
ith
m
.
L
ik
el
y
,
in
th
e
ca
s
e
o
f
1
DG
with
=
80%
,
with
th
e
p
r
o
p
o
s
ed
alg
o
r
ith
m
,
th
e
p
o
wer
lo
s
s
an
d
g
en
er
ati
o
n
co
s
t
ar
e
2
0
.
4
5
k
W
an
d
1
9
1
.
1
9
6
$
/h
,
wh
ile
with
th
e
AB
C
alg
o
r
ith
m
,
th
e
d
ata
is
1
8
.
5
5
1
k
W
a
n
d
1
9
1
.
8
1
7
$
/h
.
T
h
is
m
ea
n
s
t
h
at
m
y
alg
o
r
ith
m
is
m
o
r
e
ef
f
icien
t.
T
ab
le
2
.
C
o
m
p
a
r
is
o
n
b
etwe
en
th
e
p
r
o
p
o
s
ed
m
eth
o
d
a
n
d
o
t
h
e
r
m
eth
o
d
s
M
e
t
h
o
d
D
G
si
z
e
i
n
k
W
(
n
o
d
e
/
p
f
)
C
B
s
i
z
e
i
n
k
V
A
r
(
n
o
d
e
)
∆
i
n
k
W
C
o
s
t
i
n
$
/
h
A
B
C
[
2
1
]
1
8
7
0
(
6
1
/
0
.
8
5
)
3
0
0
(
1
8
)
1
8
.
5
5
1
1
9
1
.
8
1
7
P
r
o
p
o
se
1
5
6
2
(
6
1
/
0
.
8
1
)
2
2
5
(
1
6
)
2
0
4
(
6
4
)
1
9
5
(
1
7
)
2
0
.
4
5
1
9
1
.
1
9
6
B
S
A
[
7
]
2
9
4
(
1
9
/
0
.
8
6
6
)
2
1
9
(
2
2
/
0
.
8
6
6
)
1
7
6
8
(
6
1
/
0
.
8
6
6
)
4
5
0
(
7
)
3
0
0
(
2
)
1
5
0
(
3
)
7
.
6
0
4
1
9
2
.
3
4
4
SSA
[
8
]
3
5
8
(
1
9
/
N
A
)
5
1
8
(
1
0
/
N
A
)
1
6
7
3
.
5
(
6
1
/
N
A
)
6
0
0
(
1
1
)
6
0
0
(
4
8
)
2
0
0
(
6
0
)
4
.
8
5
3
1
9
2
.
9
5
2
P
r
o
p
o
se
1
5
6
2
(
6
1
/
0
.
8
1
)
3
4
2
(
1
6
/
0
.
8
3
)
1
8
0
(
2
7
/
0
.
9
6
)
2
0
4
(
6
4
)
8
.
2
5
6
1
9
1
.
8
3
1
A
B
C
[
2
1
]
1
8
0
0
(
6
1
/
1
)
1
3
5
0
(
6
1
)
2
3
.
2
8
2
1
9
1
.
9
3
7
P
r
o
p
o
se
1
5
6
2
(
6
1
/
1
)
1
1
1
6
(
6
1
)
2
2
5
(
1
6
)
2
0
4
(
6
4
)
1
9
5
(
1
7
)
2
0
.
4
4
6
1
9
1
.
2
7
1
W
C
A
[
9
]
5
4
0
.
8
(
1
7
/
1
)
2
0
0
0
(
6
1
/
1
)
1
1
5
9
.
2
(
6
9
/
1
)
1
1
8
7
.
9
(
2
)
1
2
3
7
.
3
(
6
2
)
2
6
9
.
7
(
6
9
)
3
3
.
3
3
9
1
9
7
.
1
8
4
R
e
f
.
[
2
2
]
5
0
4
(
1
1
/
1
)
3
7
6
(
1
7
/
1
)
1
6
6
7
(
6
1
/
1
)
1
1
9
3
(
6
1
)
3
6
7
(
1
1
)
2
3
7
(
2
0
)
4
.
2
6
3
2
1
9
2
.
8
8
3
P
r
o
p
o
se
1
5
6
2
(
6
1
/
1
)
3
4
2
(
1
6
/
1
)
1
8
0
(
2
7
/
1
)
1
1
1
5
(
6
1
)
2
2
5
(
1
6
)
2
0
4
(
6
4
)
8
.
7
4
7
1
9
1
.
9
5
4
4
.
2
.
IEEE
-
8
5
bu
s
di
s
t
ribute
d g
rid
B
y
ap
p
ly
in
g
th
e
p
r
o
p
o
s
ed
al
g
o
r
ith
m
to
th
e
I
E
E
E
-
8
5
b
u
s
g
r
id
,
we
ca
n
g
et
r
esu
lts
in
T
ab
le
3
an
d
Fig
u
r
e
6
.
Ob
v
io
u
s
ly
,
with
2
DGs
an
d
6
C
B
s
as
T
ab
le
3
,
b
o
th
th
e
p
o
wer
lo
s
s
an
d
th
e
g
en
er
atio
n
co
s
t
ar
e
r
ed
u
ce
d
s
ig
n
if
ican
tly
,
an
d
th
e
n
o
d
es’
v
o
ltag
e
in
th
e
g
r
id
b
ec
o
m
es
f
lat.
A
co
n
s
id
er
ab
le
d
r
o
p
in
p
o
wer
lo
s
s
es
is
o
b
s
er
v
ed
,
f
r
o
m
3
1
4
.
5
3
7
k
W
to
4
5
.
7
6
0
k
W
,
an
d
th
e
g
e
n
er
at
io
n
co
s
t
is
cu
t
d
o
wn
b
y
ab
o
u
t
1
0
$
/
h
.
T
h
e
n
o
d
es’
v
o
ltag
e
is
f
r
o
m
9
7
.
7
1
% to
1
.
0
1
%.
T
ab
le
3
.
R
esu
lts
as a
p
p
ly
in
g
t
h
e
p
r
o
p
o
s
ed
alg
o
r
ith
m
to
th
e
I
E
E
E
-
8
5
b
u
s
g
r
id
C
a
se
D
G
si
z
e
(
n
o
d
e
)
(
k
W
)
pf
(
%)
C
B
s
i
z
e
(
n
o
d
e
)
(
k
V
A
r
)
∆
(
k
W
)
,
(
%)
C
o
s
t
(
$
/
h
)
B
a
se
3
1
4
.
5
3
7
8
7
.
4
3
1
4
1
.
3
5
7
P
r
o
p
o
se
d
m
e
t
h
o
d
7
9
4
(
5
4
)
6
1
7
(
7
6
)
8
6
8
2
1
1
6
7
(
8
)
9
9
(
8
4
)
4
8
(
4
7
)
6
2
(
1
5
)
6
6
(
2
2
)
5
9
(
2
1
)
4
5
.
7
6
0
9
7
.
7
1
1
3
1
.
7
9
8
Fig
u
r
e
6
.
Vo
ltag
e
at
n
o
d
es in
t
h
e
I
E
E
E
-
8
5
b
u
s
g
r
id
T
o
clar
if
y
th
e
ab
o
v
e
r
esu
lt,
t
h
e
ca
s
e
o
f
=
4
is
u
s
ed
an
d
th
e
d
etailed
r
esu
lts
ar
e
s
h
o
wn
in
Fig
u
r
e
7
,
in
wh
ich
Fig
u
r
e
7
(
a
)
r
ep
r
esen
ts
th
e
ca
s
e
o
f
DG
with
o
u
t
C
B
an
d
Fig
u
r
e
7
(
b
)
r
ep
r
esen
ts
th
e
ca
s
e
o
f
DG
with
C
B
s
.
Fig
u
r
e
7
(
a
)
s
h
o
ws
th
at
th
e
p
o
wer
lo
s
s
an
d
th
e
g
en
er
atio
n
co
s
t
ar
e
r
ed
u
ce
d
with
th
e
in
cr
ea
s
e
in
th
e
DG
n
u
m
b
er
.
Ob
v
io
u
s
ly
,
a
f
ter
in
s
tallin
g
th
e
4
th
DG
(
7
9
4
k
W
,
6
1
7
k
W
,
3
4
8
k
W
,
an
d
1
8
7
k
W
at
th
e
5
4
th
,
76
th
,
8
4
th
,
an
d
6
2
nd
n
o
d
es,
r
esp
ec
tiv
ely
)
,
th
e
s
y
s
tem
lo
s
s
is
d
ec
r
ea
s
ed
s
ig
n
if
ica
n
tly
f
r
o
m
3
1
4
.
5
3
7
k
W
to
5
4
.
3
6
3
k
W
,
an
d
t
h
e
g
e
n
er
atio
n
co
s
t
is
cu
t
d
o
wn
f
r
o
m
1
4
1
.
3
5
7
$
/
h
to
1
3
3
.
3
7
6
$
/
h
.
Fig
u
r
e
7
(
b
)
in
d
icate
s
th
at
th
e
co
m
b
in
atio
n
o
f
DG
an
d
C
B
s
will
r
ed
u
ce
th
e
p
o
wer
lo
s
s
,
b
u
t
th
e
g
en
er
atio
n
co
s
t
in
cr
ea
s
es
ag
ain
wh
en
we
u
s
e
m
o
r
e
th
an
2
DGs.
Fo
r
ex
a
m
p
le,
with
3
DGs
(
7
9
4
k
W
at
th
e
5
4
th
n
o
d
e,
6
1
7
k
W
at
th
e
7
6
th
n
o
d
e,
3
4
8
k
W
at
th
e
8
4
th
n
o
d
e)
an
d
5
C
B
s
(
9
0
6
k
VAr
,
6
5
k
VAr
,
5
6
k
VA
r
,
7
3
k
VAr
,
an
d
8
9
k
VAr
at
t
h
e
8
th
,
4
7
th
,
4
3
rd
,
22
nd
,
an
d
2
0
th
n
o
d
es,
r
esp
ec
tiv
ely
)
,
th
e
p
o
wer
l
o
s
s
is
3
6
.
2
6
4
k
W
b
u
t
th
e
g
e
n
er
atio
n
c
o
s
t
is
1
3
2
.
1
5
6
$
/h
wh
ich
is
h
ig
h
er
th
an
t
h
e
d
ata
in
T
ab
le
3
.
T
h
er
ef
o
r
e,
th
e
o
p
tim
al
r
esu
lt
is
th
e
ca
s
e
o
f
2
DGs
an
d
6
C
B
s
,
a
s
T
ab
le
3
.
T
o
co
m
p
ar
e
th
e
p
r
o
p
o
s
ed
alg
o
r
ith
m
an
d
o
th
er
s
,
h
er
e
we
u
s
e
th
e
ca
s
e
o
f
DG
w
ith
u
n
ity
p
o
wer
f
ac
to
r
;
th
e
v
alu
e
o
f
an
d
ar
e
s
et
b
a
s
ed
o
n
th
e
c
o
m
p
a
r
ed
r
ef
er
e
n
ce
s
,
an
d
we
r
elax
th
e
co
n
d
itio
n
o
f
g
en
er
atio
n
co
s
t
(
s
tep
1
3
in
Fig
u
r
e
2
)
.
C
o
m
p
ar
is
o
n
r
esu
lts
a
r
e
s
h
o
wn
in
T
ab
le
4
[
2
3
]
-
[
2
5
]
.
Fro
m
T
ab
le
4
,
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
Op
timiz
e
th
e
p
o
s
itio
n
o
f th
e
d
i
s
tr
ib
u
ted
g
en
era
to
r
a
n
d
c
a
p
a
c
ito
r
b
a
n
k
in
th
e
d
is
tr
ib
u
ted
…
(
N
g
o
c
A
n
Lu
u
)
977
th
e
ca
s
e
o
f
s
o
le
DG,
with
t
h
e
p
r
o
p
o
s
ed
alg
o
r
ith
m
,
th
e
p
o
wer
lo
s
s
m
ay
b
e
h
ig
h
er
t
h
an
o
th
er
s
,
b
u
t
th
e
g
en
er
atio
n
co
s
t
is
alwa
y
s
lo
wer
th
an
o
th
er
s
.
T
a
k
e
th
e
SA
alg
o
r
ith
m
[
2
4
]
with
=
2
f
o
r
e
x
a
m
p
le,
th
e
p
o
wer
l
o
s
s
is
1
7
0
k
W
l
o
wer
t
h
an
1
7
7
.
6
3
7
k
W
o
f
th
e
p
r
o
p
o
s
ed
alg
o
r
ith
m
,
b
u
t
t
h
e
g
en
er
at
io
n
co
s
t
is
1
3
9
.
6
3
7
$
/h
,
h
i
g
h
er
t
h
an
1
3
8
.
1
0
3
$
/h
o
f
th
e
p
r
o
p
o
s
ed
alg
o
r
ith
m
.
T
h
is
is
ex
p
lain
ed
b
y
th
e
lo
w
er
DG
s
ize
in
th
e
p
r
o
p
o
s
ed
alg
o
r
ith
m
.
I
n
th
e
ca
s
e
o
f
DG
an
d
C
B
,
th
e
p
o
wer
lo
s
s
an
d
th
e
g
en
er
atio
n
co
s
t
ar
e
alwa
y
s
lo
wer
th
an
th
o
s
e
o
f
o
th
er
s
.
Ob
v
io
u
s
ly
,
with
=
=
3
,
th
e
d
ata
with
th
e
in
tr
o
d
u
c
ed
s
ch
em
e
is
4
9
.
5
7
0
k
W
an
d
1
3
2
.
8
6
4
$
/h
,
wh
ich
a
r
e
lo
wer
th
an
7
3
.
2
4
k
W
an
d
1
3
5
.
0
2
7
$
/
h
o
f
th
e
GABC
s
ch
em
e
[
2
5
]
.
T
h
is
p
r
o
v
es
t
h
at
th
e
p
r
o
p
o
s
ed
alg
o
r
ith
m
is
m
o
r
e
e
f
f
icien
t th
an
o
th
e
r
s
.
(
a)
(
b
)
Fig
u
r
e
7
.
Po
wer
l
o
s
s
an
d
g
en
e
r
atio
n
co
s
t a
s
: (
a)
o
n
ly
DG
an
d
(
b
)
c
o
m
b
i
n
atio
n
o
f
C
B
an
d
DG
T
ab
le
4
.
C
o
m
p
a
r
in
g
th
e
p
r
o
p
o
s
ed
alg
o
r
ith
m
to
o
th
er
s
in
t
h
e
ca
s
e
o
f
th
e
u
n
ity
p
o
wer
f
ac
to
r
&
M
e
t
h
o
d
D
G
si
z
e
i
n
k
W
(
n
o
d
e
)
C
B
s
i
z
e
i
n
k
V
A
r
(
n
o
d
e
)
∆
in
kW
C
o
s
t
i
n
$
/
h
1
&0
W
O
A
[
2
3
]
9
4
6
.
3
(
5
5
)
2
2
4
.
0
4
9
1
3
9
.
2
4
1
P
r
o
p
o
se
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9
4
(
5
4
)
2
2
0
.
2
7
9
1
3
8
.
6
8
3
2
&0
S
A
[
2
4
]
5
9
1
.
2
(
3
6
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1
5
9
7
.
5
(
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7
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1
3
9
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6
3
7
P
r
o
p
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se
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9
4
(
5
4
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1
7
(
7
6
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1
7
7
.
6
3
7
1
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8
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1
0
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S
A
[
2
4
]
3
2
1
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1
(
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9
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1
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2
(
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3
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4
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3
(
9
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1
6
6
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1
3
8
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P
r
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9
4
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7
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4
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1
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1
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6
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A
B
C
[
2
5
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8
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0
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1
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6
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3
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1
4
P
r
o
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se
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9
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(
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4
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(
8
)
1
0
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.
9
1
9
1
3
2
.
9
6
4
2
&2
G
A
B
C
[
2
5
]
8
5
1
(
3
6
)
1
3
4
9
(
5
6
)
6
0
0
(
5
3
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5
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(
4
6
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6
.
3
4
1
3
5
.
6
4
8
P
r
o
p
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se
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9
4
(
5
4
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6
1
7
(
7
6
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0
6
8
(
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1
2
(
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0
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8
.
4
6
5
1
3
2
.
4
3
9
3
&3
G
A
B
C
[
2
5
]
5
7
4
(
3
6
)
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0
4
(
5
6
)
4
2
6
(
5
4
)
3
0
0
(
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3
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5
0
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0
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5
4
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7
3
.
2
4
1
3
5
.
0
2
7
P
r
o
p
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se
7
9
4
(
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1
7
(
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6
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3
4
8
(
8
4
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2
0
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8
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0
)
3
3
(
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2
)
4
9
.
5
7
0
1
3
2
.
8
6
4
5.
CO
NCLU
SI
O
N
T
h
is
p
ap
er
p
r
o
p
o
s
ed
an
al
g
o
r
i
th
m
to
m
in
im
ize
th
e
g
en
e
r
atio
n
co
s
t
an
d
r
e
d
u
ce
t
h
e
p
o
wer
l
o
s
s
in
th
e
g
r
id
b
y
d
eter
m
in
in
g
th
e
o
p
tim
al
p
o
s
itio
n
,
s
ize,
an
d
p
o
wer
f
a
cto
r
o
f
DGs
an
d
th
e
o
p
tim
al
p
o
s
itio
n
an
d
s
ize
o
f
C
B
s
.
T
h
e
alg
o
r
ith
m
is
d
ev
elo
p
ed
f
r
o
m
th
e
lo
s
s
s
en
s
itiv
ity
f
ac
to
r
.
B
y
ap
p
ly
in
g
th
is
alg
o
r
it
h
m
to
th
e
I
E
E
E
-
69
b
u
s
an
d
I
E
E
E
-
8
5
b
u
s
d
is
tr
ib
u
ted
g
r
id
,
th
e
o
p
tim
al
p
o
s
itio
n
,
s
ize,
an
d
p
o
wer
f
ac
to
r
o
f
D
Gs
an
d
C
B
s
in
ea
ch
g
r
id
ar
e
d
eter
m
in
e
d
,
th
e
p
o
w
er
lo
s
s
in
th
e
g
r
id
is
r
e
d
u
ce
d
s
ig
n
if
ican
tly
,
an
d
th
e
g
en
er
ati
o
n
co
s
t
is
m
in
im
al.
C
o
m
p
ar
ed
to
o
th
er
r
esear
ch
,
with
th
e
p
r
o
p
o
s
ed
alg
o
r
ith
m
,
th
e
g
en
er
atio
n
co
s
t
is
alwa
y
s
lo
wer
th
an
o
th
er
s
.
T
h
is
is
th
e
ef
f
icien
cy
o
f
th
e
p
r
o
p
o
s
ed
alg
o
r
ith
m
.
I
n
th
e
f
u
tu
r
e,
th
is
m
eth
o
d
ca
n
b
e
ex
ten
d
ed
to
ap
p
ly
to
r
en
ewa
b
le
s
o
u
r
ce
s
with
u
n
c
er
tain
ty
,
s
u
ch
as
win
d
g
en
e
r
ato
r
s
,
s
o
lar
s
y
s
tem
s
,
o
r
co
m
b
in
ed
with
o
th
e
r
tech
n
iq
u
es to
s
o
lv
e
m
o
r
e
c
o
m
p
lex
p
r
o
b
lem
s
.
F
UNDING
I
NF
O
R
M
A
T
I
O
N
Au
th
o
r
s
s
tate
n
o
f
u
n
d
in
g
in
v
o
lv
ed
.
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
7
0
-
979
978
AUTHO
R
CO
NT
RI
B
UT
I
O
NS ST
A
T
E
M
E
N
T
T
h
is
jo
u
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c
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m
.
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