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r
ea
s
i
n
g
e
l
e
c
t
r
ic
i
t
y
d
e
m
a
n
d
a
n
d
u
n
m
a
n
a
g
e
d
i
n
t
e
g
r
a
t
i
o
n
o
f
d
e
c
e
n
t
r
a
l
i
z
e
d
r
e
s
o
u
r
c
es
a
r
e
p
u
tt
i
n
g
u
n
p
r
e
c
e
d
e
n
t
e
d
p
r
es
s
u
r
e
o
n
a
g
i
n
g
a
n
d
u
n
d
e
r
s
iz
e
d
i
n
f
r
as
tr
u
c
t
u
r
e
s
[
4
]
.
T
h
is
is
r
e
f
l
e
c
t
e
d
i
n
v
o
l
t
a
g
e
u
n
b
a
l
a
n
ce
,
v
o
l
t
a
g
e
d
r
o
p
s
,
l
i
n
e
c
o
n
g
e
s
ti
o
n
,
a
n
d
w
a
v
e
f
o
r
m
d
is
t
o
r
ti
o
n
[
5
]
.
I
n
d
e
v
e
l
o
p
i
n
g
c
o
u
n
t
r
i
e
s
,
t
h
e
p
r
o
b
l
e
m
i
s
c
o
m
p
o
u
n
d
e
d
b
y
l
o
w
i
n
v
e
s
t
m
e
n
t
p
o
t
e
n
t
i
a
l
a
n
d
t
h
e
l
a
c
k
o
f
a
n
e
f
f
e
c
t
i
v
e
r
e
g
u
l
at
i
o
n
s
y
s
te
m
[
6
]
–
[
8
]
.
T
h
e
c
o
n
v
e
n
t
i
o
n
a
l
p
a
s
s
i
v
e
m
a
n
a
g
e
m
e
n
t
a
p
p
r
o
a
c
h
i
s
n
o
l
o
n
g
e
r
s
u
f
f
i
c
i
e
n
t
,
a
n
d
t
h
e
t
r
a
n
s
i
t
i
o
n
t
o
w
a
r
d
s
t
h
e
ac
t
i
v
e
a
n
d
i
n
t
e
l
li
g
en
t
o
p
e
r
a
t
i
o
n
o
f
L
V
d
i
s
t
r
i
b
u
t
i
o
n
n
e
t
w
o
r
k
s
is
a
p
r
e
s
s
i
n
g
n
e
e
d
[
9
]
.
I
n
th
is
r
eg
a
r
d
,
t
h
e
o
p
tim
al
p
o
wer
f
lo
w
(
OPF)
f
r
a
m
ewo
r
k
h
as
b
ee
n
estab
lis
h
ed
as
an
o
p
ti
m
al
to
o
l
in
th
e
s
tr
ateg
ic
m
an
a
g
em
en
t
o
f
elec
tr
ical
n
etwo
r
k
s
.
I
n
itially
,
it
was
p
r
im
ar
ily
u
s
ed
to
o
p
ti
m
ize
tr
an
s
m
is
s
io
n
s
y
s
tem
s
,
with
th
e
m
ain
o
b
jectiv
e
o
f
r
e
d
u
cin
g
p
r
o
d
u
ctio
n
co
s
ts
.
T
o
d
ay
,
its
s
co
p
e
h
as b
r
o
a
d
e
n
ed
:
it
is
n
o
w
u
s
ed
in
th
e
m
an
ag
em
e
n
t
o
f
d
is
tr
ib
u
tio
n
n
etwo
r
k
s
,
wh
er
e
it
h
el
p
s
to
lim
it
tech
n
ical
lo
s
s
es,
im
p
r
o
v
e
v
o
ltag
e
q
u
ality
,
an
d
m
ax
im
ize
t
h
e
in
teg
r
atio
n
o
f
r
en
ewa
b
le
en
er
g
ies.
R
ec
en
t
r
esear
ch
h
as
b
ee
n
f
o
c
u
s
ed
o
n
u
n
b
alan
ce
d
th
r
ee
-
p
h
a
s
e
OPF
f
o
r
m
u
latio
n
s
,
as
th
e
y
o
f
f
er
m
o
r
e
ac
cu
r
ate
m
o
d
els
o
f
t
h
e
L
V
n
etwo
r
k
’
s
b
eh
a
v
io
r
.
T
h
ese
m
o
d
els
m
ay
s
im
u
ltan
eo
u
s
ly
m
in
im
ize
p
o
wer
l
o
s
s
es,
im
p
r
o
v
e
v
o
ltag
e
p
r
o
f
ile,
a
n
d
r
ed
u
ce
v
o
ltag
e
u
n
b
alan
ce
f
ac
to
r
(
VUF)
,
all
u
n
d
er
p
er
-
p
h
ase
v
o
ltag
e
an
d
th
er
m
al
co
n
s
tr
ain
ts
[
1
0
]
,
[
1
1
]
.
New
d
ev
elo
p
m
e
n
ts
,
s
u
ch
as
c
o
n
v
e
x
r
elax
atio
n
s
(
Seco
n
d
-
o
r
d
er
co
n
e
p
r
o
g
r
am
m
in
g
(
SOC
P)
an
d
s
em
id
ef
in
ite
p
r
o
g
r
am
m
in
g
(
SDP)
)
,
m
ix
ed
-
in
te
g
er
ap
p
r
o
ac
h
es,
an
d
m
etah
eu
r
is
tics
,
h
av
e
g
r
ea
tly
im
p
r
o
v
e
d
th
e
co
m
p
u
tatio
n
o
f
u
n
b
alan
ce
d
th
r
ee
-
p
h
ase
OPF
m
o
d
els
[
1
2
]
,
[
1
3
]
.
Ad
d
itio
n
ally
,
n
ew
an
d
im
p
r
o
v
e
d
o
p
e
n
-
s
o
u
r
ce
to
o
ls
,
s
u
ch
as
Pan
d
ap
o
wer
,
h
av
e
b
ee
n
d
ev
elo
p
ed
f
o
r
u
n
b
alan
ce
d
th
r
ee
-
p
h
ase
OPF,
en
co
u
r
a
g
in
g
r
e
p
r
o
d
u
cib
ilit
y
an
d
ap
p
licab
ilit
y
o
f
r
esear
ch
r
esu
lts
[
1
0
]
,
[
1
4
]
.
Am
o
n
g
th
e
av
ailab
le
OPF
p
latf
o
r
m
s
,
Pan
d
ap
o
wer
h
as
g
ain
ed
co
n
s
id
er
a
b
le
atten
tio
n
d
u
e
to
its
o
p
en
-
s
o
u
r
ce
ar
c
h
itectu
r
e,
ex
ten
s
iv
e
v
alid
atio
n
a
g
ain
s
t
estab
lis
h
ed
p
o
wer
s
y
s
tem
to
o
ls
,
an
d
s
u
p
p
o
r
t
f
o
r
th
r
ee
-
p
h
ase
u
n
b
alan
ce
d
n
etwo
r
k
a
n
aly
s
is
.
C
o
m
p
ar
ed
with
m
eta
h
eu
r
is
tic
o
p
tim
izatio
n
tech
n
iq
u
es,
wh
i
ch
o
f
ten
r
eq
u
ir
e
ex
ten
s
iv
e
p
a
r
am
eter
tu
n
in
g
an
d
m
ay
n
o
t
g
u
ar
an
tee
c
o
n
v
e
r
g
e
n
ce
,
th
e
in
ter
io
r
p
o
in
t
m
eth
o
d
im
p
lem
en
te
d
in
Pan
d
a
p
o
wer
o
f
f
er
s
r
o
b
u
s
t
co
n
v
er
g
en
ce
p
r
o
p
er
ties
an
d
c
o
m
p
u
tatio
n
al
ef
f
icien
c
y
.
T
h
ese
ch
ar
ac
ter
is
tics
m
ak
e
it
p
ar
ti
cu
lar
ly
s
u
itab
le
f
o
r
p
r
ac
tical
ap
p
licatio
n
s
in
v
o
l
v
in
g
r
ea
l
lo
w
-
v
o
ltag
e
d
is
tr
ib
u
ti
o
n
n
etwo
r
k
s
.
Mo
r
e
r
ec
e
n
t
d
e
v
elo
p
m
en
ts
,
s
u
ch
as
th
e
in
cr
ea
s
in
g
d
eg
r
ee
o
f
u
n
ce
r
tain
ty
ass
o
ciate
d
with
r
e
n
ewa
b
le
en
er
g
y
,
h
av
e
led
t
o
th
e
ap
p
licatio
n
o
f
ar
tific
ial
in
tellig
en
t (
AI
)
tech
n
i
q
u
es,
s
u
ch
as d
ee
p
r
ei
n
f
o
r
ce
m
en
t le
ar
n
in
g
(
DR
L
)
,
f
o
r
o
p
tim
izatio
n
p
u
r
p
o
s
es
[
2
]
.
S
e
v
e
r
a
l
a
p
p
r
o
a
c
h
e
s
h
a
v
e
b
e
e
n
p
r
o
p
o
s
e
d
t
o
m
i
t
i
g
a
t
e
p
h
a
s
e
i
m
b
a
l
a
n
c
e
i
n
l
o
w
-
v
o
l
t
a
g
e
d
i
s
t
r
i
b
u
ti
o
n
n
e
t
w
o
r
k
s
.
T
r
a
d
i
ti
o
n
a
l
m
e
t
h
o
d
s
m
a
i
n
l
y
r
e
l
y
o
n
m
a
n
u
a
l
p
h
a
s
e
r
e
a
ll
o
c
a
t
i
o
n
,
w
h
e
r
e
c
u
s
t
o
m
e
r
s
a
r
e
p
h
y
s
i
c
al
l
y
t
r
a
n
s
f
e
r
r
e
d
f
r
o
m
o
v
e
r
l
o
a
d
e
d
p
h
a
s
e
s
t
o
u
n
d
e
r
u
t
i
li
z
e
d
o
n
es
[
1
5
]
.
A
l
t
h
o
u
g
h
e
f
f
e
c
t
i
v
e
i
n
s
o
m
e
c
a
s
es
,
t
h
i
s
a
p
p
r
o
a
c
h
i
s
l
a
b
o
r
-
i
n
t
e
n
s
i
v
e
,
ti
m
e
-
c
o
n
s
u
m
i
n
g
,
a
n
d
o
f
t
e
n
u
n
s
u
i
ta
b
l
e
f
o
r
r
a
p
i
d
l
y
e
v
o
l
v
i
n
g
u
r
b
a
n
n
e
t
w
o
r
k
s
[
2
]
.
M
o
r
e
r
e
ce
n
t
l
y
,
a
u
t
o
m
a
t
i
c
p
h
a
s
e
s
wi
t
c
h
i
n
g
t
ec
h
n
o
l
o
g
i
e
s
a
n
d
i
n
t
e
l
li
g
e
n
t
p
h
a
s
e
b
a
l
a
n
c
i
n
g
s
y
s
t
e
m
s
h
a
v
e
b
e
e
n
i
n
t
r
o
d
u
c
e
d
t
o
d
y
n
a
m
i
c
a
l
l
y
r
e
d
i
s
t
r
i
b
u
t
e
l
o
a
d
s
a
m
o
n
g
p
h
a
s
e
s
.
H
o
w
e
v
e
r
,
t
h
es
e
s
o
l
u
ti
o
n
s
g
e
n
e
r
a
ll
y
r
e
q
u
i
r
e
a
d
d
i
t
i
o
n
a
l
h
a
r
d
w
a
r
e
i
n
v
e
s
t
m
e
n
ts
a
n
d
a
d
v
a
n
c
e
d
c
o
m
m
u
n
i
c
a
t
i
o
n
i
n
f
r
a
s
t
r
u
c
t
u
r
e
s
[
1
6
]
.
C
o
n
s
e
q
u
e
n
t
l
y
,
o
p
t
i
m
i
z
a
tio
n
-
b
a
s
e
d
a
p
p
r
o
a
c
h
e
s
s
u
c
h
a
s
O
PF
h
a
v
e
e
m
e
r
g
e
d
a
s
a
t
t
r
a
ct
i
v
e
a
l
t
e
r
n
at
i
v
es
,
s
i
n
ce
t
h
e
y
c
a
n
d
e
t
e
r
m
i
n
e
o
p
t
i
m
a
l
p
h
a
s
e
a
l
l
o
c
a
ti
o
n
s
t
r
a
t
e
g
i
e
s
w
h
i
l
e
s
i
m
u
lt
a
n
e
o
u
s
l
y
c
o
n
s
i
d
e
r
i
n
g
v
o
l
t
a
g
e
li
m
i
ts
,
t
h
er
m
a
l
c
o
n
s
t
r
ai
n
t
s
,
a
n
d
p
o
w
e
r
l
o
s
s
e
s
.
Am
o
n
g
th
e
a
v
ailab
le
OPF
p
la
tf
o
r
m
s
,
Pan
d
a
p
o
wer
h
as
g
ain
e
d
co
n
s
id
er
ab
le
atten
tio
n
d
u
e
to
its
o
p
e
n
-
s
o
u
r
ce
ar
c
h
itectu
r
e,
e
x
ten
s
iv
e
v
alid
atio
n
a
g
ain
s
t
estab
lis
h
ed
p
o
wer
s
y
s
tem
to
o
ls
,
an
d
s
u
p
p
o
r
t
f
o
r
u
n
b
alan
ce
d
th
r
ee
-
p
h
ase
n
etwo
r
k
an
aly
s
is
.
C
o
m
p
ar
ed
with
m
etah
e
u
r
is
tic
ap
p
r
o
ac
h
es,
wh
ich
o
f
ten
r
eq
u
ir
e
ex
te
n
s
iv
e
p
ar
am
eter
tu
n
i
n
g
an
d
m
ay
n
o
t
g
u
ar
an
tee
co
n
v
er
g
en
ce
,
th
e
i
n
ter
io
r
p
o
i
n
t
m
eth
o
d
im
p
lem
e
n
ted
in
Pan
d
ap
o
wer
p
r
o
v
id
es
r
o
b
u
s
t
co
n
v
er
g
e
n
ce
p
r
o
p
er
ties
,
co
m
p
u
tatio
n
al
e
f
f
icien
cy
,
an
d
r
ep
r
o
d
u
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ak
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it
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itab
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n
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ase
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in
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u
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e,
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r
t
h
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er
ica,
o
r
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a,
wh
er
e
ad
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a
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ce
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m
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ito
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g
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f
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astru
ct
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r
es a
n
d
h
ig
h
-
q
u
ality
n
etwo
r
k
d
atab
as
es a
r
e
av
ailab
le.
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n
c
o
n
tr
ast,
v
er
y
lim
ited
r
esear
ch
h
as
in
v
esti
g
ated
th
e
ap
p
licati
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n
o
f
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tech
n
iq
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es
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lo
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n
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in
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b
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ar
an
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r
ica,
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is
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tem
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o
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ten
ch
ar
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te
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ized
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ad
ed
f
ee
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er
s
,
i
n
f
o
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m
al
co
n
n
ec
tio
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s
,
s
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e
p
h
ase
im
b
alan
ce
,
lim
ited
o
p
er
atio
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a
l
d
ata,
an
d
ag
in
g
in
f
r
astru
ctu
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es.
C
o
n
s
eq
u
en
tly
,
th
e
p
r
ac
tical
ef
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ec
tiv
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o
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ad
v
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h
n
iq
u
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n
d
er
s
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ch
o
p
e
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atin
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n
d
itio
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s
r
em
ain
s
lar
g
ely
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n
ex
p
lo
r
ed
.
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h
is
s
t
u
d
y
c
o
n
t
r
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b
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tes
to
e
x
i
s
tin
g
l
ite
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a
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e
in
th
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ee
m
ai
n
w
ay
s
.
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r
s
t
,
it
p
r
es
en
ts
o
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e
o
f
th
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f
ir
s
t
ap
p
li
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s
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n
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al
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ce
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th
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p
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f
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a
m
ew
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v
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e
d
is
t
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b
u
t
io
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n
et
wo
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k
i
n
t
h
e
Dem
o
c
r
atic
R
e
p
u
b
li
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o
f
t
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e
C
o
n
g
o
.
S
ec
o
n
d
,
th
e
n
etw
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k
m
o
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el
is
p
a
r
am
ete
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d
u
s
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g
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iel
d
m
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r
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e
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ts
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l
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cte
d
o
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f
o
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m
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g
a
r
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is
tic
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ep
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es
en
tat
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o
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o
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e
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at
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o
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d
iti
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.
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h
ir
d
,
th
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s
tu
d
y
q
u
a
n
t
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e
v
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t
h
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m
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ct
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v
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l
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cti
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d
e
r
h
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v
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ly
l
o
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d
e
d
c
o
n
d
iti
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s
.
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o
ac
h
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e
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t
h
is
o
b
j
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ti
v
e
,
an
u
n
b
ala
n
c
ed
t
h
r
ee
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ase
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p
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et
wo
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k
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as
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i
,
DR
C
.
T
h
e
o
p
tim
iz
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p
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l
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f
o
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m
u
la
te
d
as
a
m
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in
te
g
e
r
n
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l
in
ea
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p
r
o
g
r
am
m
i
n
g
(
M
I
N
L
P)
p
r
o
b
l
em
a
n
d
s
o
l
v
ed
u
s
i
n
g
th
e
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8
7
0
8
I
n
t J E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
16
,
No
.
5
,
Octo
b
e
r
20
26
:
2
3
0
4
-
2
3
2
0
2306
in
t
er
io
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p
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m
et
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p
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p
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wi
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t
r
ai
n
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.
2.
M
E
T
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D
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h
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y
was
ca
r
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ied
o
u
t
in
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u
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i,
th
e
p
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p
ital
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th
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o
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th
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s
t
D
em
o
cr
atic
R
ep
u
b
lic
o
f
th
e
C
o
n
g
o
'
s
H
au
t
-
Kata
n
g
a
p
r
o
v
in
ce
.
r
an
k
e
d
as
th
e
n
atio
n
's
s
ec
o
n
d
-
m
o
s
t
p
o
p
u
lo
u
s
an
d
ec
o
n
o
m
ically
s
ig
n
if
ica
n
t
city
.
T
h
e
city
is
p
lace
d
at
ab
o
u
t
1
1
°2
0
′
to
1
2
°0
0
′
s
o
u
th
latitu
d
e
a
n
d
2
7
°1
0
′
to
2
7
°2
0
′
ea
s
t
lo
n
g
itu
d
e,
c
u
r
r
en
tly
g
o
in
g
th
r
o
u
g
h
r
ap
id
u
r
b
a
n
izatio
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an
d
r
o
b
u
s
t
p
o
p
u
latio
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g
r
o
wth
im
p
ly
in
g
m
o
r
e
lo
a
d
o
n
its
elec
tr
ical
n
etwo
r
k
[
1
7
]
.
Kam
alo
n
d
o
,
o
n
e
o
f
th
e
city
'
s
s
ev
en
ad
m
in
is
tr
ativ
e
co
m
m
u
n
es,
was
s
elec
ted
as
th
e
s
tu
d
y
a
r
ea
b
ec
a
u
s
e
o
f
its
h
ig
h
r
ate
o
f
elec
tr
if
icatio
n
[
8
]
,
[
1
8
]
an
d
th
e
f
r
eq
u
e
n
cy
o
f
illi
c
it
co
n
n
ec
tio
n
s
t
h
at
h
av
e
b
ee
n
d
o
cu
m
en
ted
n
ea
r
th
e
MV
/LV
s
u
b
s
tatio
n
o
n
B
ab
em
b
a
an
d
Kar
av
ia
Av
en
u
e
s
.
W
ith
an
ar
ea
o
f
r
o
u
g
h
ly
1
.
3
5
k
m
²,
Kam
alo
n
d
o
is
th
e
s
m
allest
co
m
m
u
n
e
in
L
u
b
u
m
b
ash
i
an
d
is
s
itu
ated
in
th
e
city
'
s
s
o
u
th
er
n
r
eg
io
n
as sh
o
wn
in
Fig
u
r
e
1
.
2
.
1
.
Descript
io
n o
f
t
he
B
a
bem
ba
s
ub
s
t
a
t
io
n (
ca
s
e
s
t
ud
y
)
T
h
e
elec
tr
ical
in
f
r
astru
ctu
r
e
u
n
d
er
s
tu
d
y
c
o
n
s
is
ts
o
f
a
6
3
0
k
VA
tr
an
s
f
o
r
m
er
,
c
o
n
n
ec
ted
i
n
a
Dy
n
1
1
v
ec
to
r
g
r
o
u
p
in
g
.
I
t
is
eq
u
ip
p
e
d
with
a
1
2
5
0
A
cir
cu
it
b
r
ea
k
e
r
,
co
r
r
esp
o
n
d
in
g
to
a
n
o
m
in
al
cu
r
r
en
t
o
f
9
0
7
A
at
a
n
o
m
in
al
v
o
ltag
e
o
f
4
0
0
V.
T
wo
f
ee
d
er
s
,
d
esig
n
ated
N
R
1
5
(
Am
ato
an
d
Sy
n
tesk
in
)
,
p
r
o
v
id
e
p
o
wer
an
d
co
n
n
ec
tio
n
t
o
th
e
u
p
s
tr
ea
m
n
etwo
r
k
.
T
h
is
s
u
b
s
tatio
n
d
ir
ec
t
ly
s
u
p
p
lies
f
iv
e
d
is
tr
ib
u
tio
n
f
ee
d
er
s
to
p
r
o
v
id
e
a
lo
ca
l lo
w
-
v
o
ltag
e
s
er
v
ice,
as sh
o
wn
in
Fig
u
r
e
2
.
Fig
u
r
e
1
.
L
o
ca
tio
n
o
f
th
e
c
o
m
m
u
n
e
Kam
alo
n
d
o
Fig
u
r
e
2
.
L
a
y
o
u
t
o
f
th
e
L
V
m
ain
s
witch
b
o
ar
d
o
f
th
e
B
ab
em
b
a
s
u
b
s
tatio
n
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2088
-
8
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0
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I
mp
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o
p
tima
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lity in
a
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…
(
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Mil
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2307
2
.
2
.
Da
t
a
c
o
llect
i
o
n
T
o
s
u
p
p
o
r
t
th
e
m
o
d
elin
g
a
n
d
s
im
u
latio
n
o
f
OPF
in
an
u
n
b
alan
ce
d
t
h
r
ee
-
p
h
ase
co
n
tex
t,
a
m
ea
s
u
r
em
en
t
ca
m
p
ai
g
n
was
ca
r
r
ied
o
u
t
o
n
th
e
L
V
d
is
tr
ib
u
tio
n
n
etwo
r
k
u
n
d
er
an
al
y
s
is
.
T
h
e
m
ea
s
u
r
ed
p
ar
am
eter
s
in
clu
d
e
cu
r
r
en
ts
,
v
o
ltag
es,
ac
tiv
e
an
d
r
ea
ctiv
e
p
o
wer
s
,
as
well
as
r
esis
tan
ce
an
d
r
ea
ctan
ce
.
Data
co
llectio
n
was
p
er
f
o
r
m
ed
b
o
t
h
d
ay
a
n
d
n
ig
h
t,
d
u
r
in
g
th
e
r
eg
io
n
’
s
two
m
ain
s
ea
s
o
n
s
(
d
r
y
an
d
r
ain
y
)
,
o
v
e
r
a
f
o
u
r
-
m
o
n
th
p
er
io
d
f
r
o
m
Sep
t
em
b
er
to
Dec
em
b
er
2
0
2
4
.
M
ea
s
u
r
em
en
ts
wer
e
tak
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u
s
in
g
a
C
h
au
v
in
A
r
n
o
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x
Qu
alis
tar
s
er
ies
8
3
3
1
p
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n
etwo
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k
an
aly
ze
r
as
s
h
o
wn
in
Fig
u
r
e
3
,
p
lace
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at
two
s
tr
ate
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ic
lev
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at
th
e
L
V
m
ain
s
witch
b
o
ar
d
(
L
VM
S)
o
f
th
e
s
u
b
s
tatio
n
to
ca
p
t
u
r
e
o
v
er
all
f
lo
ws,
an
d
at
th
e
en
d
o
f
th
e
lin
e
o
f
ea
ch
f
ee
d
er
.
Fig
u
r
e
3
.
Me
asu
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em
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t d
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ice
(
C
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Ar
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8
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Data
lik
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tan
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,
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e,
len
g
th
,
an
d
m
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x
im
u
m
allo
wab
le
cu
r
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e
n
t
ar
e
u
s
ed
t
o
d
e
s
cr
ib
e
th
e
elec
tr
ical
lin
es.
T
h
ese
elem
en
ts
allo
w
p
r
ec
is
e
m
o
d
elin
g
o
f
th
e
n
etwo
r
k
.
Pan
d
ap
o
wer
is
th
en
u
s
ed
to
ca
lcu
lat
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th
e
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p
ed
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o
f
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h
co
m
p
o
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en
t.
T
h
ese
im
p
ed
a
n
ce
s
ar
e
in
teg
r
ated
in
to
th
e
Z
0
1
2
m
atr
ix
,
wh
ich
co
m
p
iles
th
e
ze
r
o
,
p
o
s
itiv
e,
an
d
n
e
g
ativ
e
s
eq
u
en
ce
v
alu
es.
Usi
n
g
th
e
tr
an
s
f
o
r
m
atio
n
m
atr
ix
an
d
its
in
v
er
s
e,
th
ese
d
ata
ar
e
co
n
v
e
r
ted
to
o
b
tain
th
e
th
r
ee
-
p
h
ase
im
p
ed
a
n
ce
m
atr
ix
f
o
r
ea
ch
f
ee
d
e
r
[
1
6
]
,
[
1
8
]
.
T
ab
le
1
s
u
m
m
ar
izes
th
e
elec
tr
ical
q
u
a
n
titi
es
m
ea
s
u
r
ed
at
th
e
h
ea
d
o
f
ea
ch
f
ee
d
er
an
d
co
n
s
titu
tes
th
e
b
asis
f
o
r
th
e
ca
lib
r
atio
n
o
f
th
e
OPF m
o
d
el.
T
h
ese
m
ea
s
u
r
em
en
ts
s
er
v
e
th
r
ee
m
ain
p
u
r
p
o
s
es:
f
ir
s
t,
to
in
itialize
th
e
lo
ad
f
lo
w
o
p
er
at
in
g
p
o
i
n
t;
s
ec
o
n
d
,
to
ca
lib
r
ate
n
o
d
al
p
o
wer
in
jectio
n
s
b
ased
o
n
r
ea
l
-
wo
r
ld
d
em
a
n
d
;
a
n
d
f
i
n
ally
,
to
q
u
a
n
tify
p
h
ase
im
b
alan
ce
s
an
d
n
e
u
tr
al
cu
r
r
e
n
t
lev
els.
T
h
e
ex
is
ten
ce
o
f
s
ig
n
if
ican
t
n
eu
tr
al
cu
r
r
en
ts
ju
s
tifie
s
th
e
ad
o
p
tio
n
o
f
a
f
o
u
r
-
wir
e
u
n
b
ala
n
ce
d
m
o
d
el
t
o
ac
cu
r
ately
r
ep
r
esen
t th
e
n
et
wo
r
k
’
s
p
h
y
s
ics.
T
ab
le
1
.
Me
asu
r
e
d
elec
tr
ical
q
u
an
titi
es a
t f
ee
d
er
h
ea
d
s
F
e
e
d
e
r
1
2
3
4
5
P
A
(
k
W
)
2
0
0
0
45
2
2
5
1
5
0
P
B
(
k
W
)
2
8
5
1
5
0
46
2
3
5
1
7
0
P
C
(
k
W
)
2
0
5
1
7
0
0
1
3
8
0
V
A
(
V
)
1
9
0
1
9
9
2
3
2
2
0
5
2
1
4
.
8
V
B
(
V
)
1
9
8
2
0
1
2
2
3
2
1
5
2
1
3
.
4
V
C
(
V
)
2
0
6
2
1
7
2
1
9
1
6
5
2
1
5
.
4
I
A
(
A
)
3
4
0
0
65
3
5
0
2
6
1
.
6
I
B
(
A
)
3
7
0
2
9
0
55
3
6
0
2
5
9
.
8
I
C
(
A
)
3
4
0
2
4
0
0
2
1
0
0
I
N
(
A
)
-
1
5
0
20
1
8
0
1
5
3
.
9
2
.
3
.
E
lect
rica
l
lin
e
pa
ra
m
et
er
s
T
h
e
p
h
y
s
ical
ch
ar
a
cter
is
tics
o
f
ea
ch
f
ee
d
e
r
ar
e
s
u
m
m
a
r
ized
in
T
ab
le
2
,
wh
ich
p
r
o
v
id
es
th
e
f
u
n
d
am
e
n
tal
elec
tr
ical
p
ar
am
eter
s
r
eq
u
ir
ed
f
o
r
ac
c
u
r
ate
n
etwo
r
k
m
o
d
elin
g
.
E
ac
h
b
r
an
c
h
is
d
ef
in
ed
b
y
its
len
g
th
(
)
,
co
n
d
u
cto
r
cr
o
s
s
-
s
ec
tio
n
(
2
)
,
r
esis
tan
ce
(
Ω
)
,
an
d
r
ea
ctan
ce
(
Ω
)
,
T
h
ese
p
ar
am
eter
s
ar
e
ess
en
tial
f
o
r
co
n
s
tr
u
ctin
g
th
e
s
er
ies
im
p
e
d
an
c
e
o
f
th
e
lin
es
an
d
d
ir
ec
tly
i
n
f
l
u
en
ce
th
e
elec
tr
ical
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8
7
0
8
I
n
t J E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
16
,
No
.
5
,
Octo
b
e
r
20
26
:
2
3
0
4
-
2
3
2
0
2308
b
eh
av
io
r
o
f
th
e
d
is
tr
ib
u
tio
n
s
y
s
tem
.
I
n
p
ar
ticu
lar
,
t
h
ey
d
et
er
m
in
e
th
e
m
ag
n
itu
d
e
o
f
v
o
lt
ag
e
d
r
o
p
s
alo
n
g
th
e
f
ee
d
er
s
,
g
o
v
er
n
t
h
e
d
is
tr
ib
u
tio
n
o
f
ac
tiv
e
an
d
r
ea
ctiv
e
p
o
we
r
f
lo
ws,
a
n
d
d
ef
i
n
e
th
e
th
er
m
al
lo
ad
in
g
lim
its
o
f
th
e
co
n
d
u
cto
r
s
u
n
d
er
o
p
er
atin
g
co
n
d
itio
n
s
.
T
ab
le
2
.
E
lectr
ical
p
a
r
am
eter
s
o
f
L
V
f
ee
d
e
r
s
F
e
e
d
e
r
N
o
d
e
Le
n
g
t
h
(
m)
C
r
o
ss
-
se
c
t
i
o
n
(
mm
2
)
R
(
Ω
)
X
(
Ω
)
1
7
6
0
2
.
7
9
1
1
0
0
.
1
5
0
.
0
5
2
12
5
2
4
.
2
8
1
1
0
0
.
1
3
0
.
0
4
3
14
1
1
1
.
9
3
35
0
.
0
9
0
.
0
1
4
18
4
2
0
.
0
2
1
1
0
0
.
1
1
0
.
0
3
5
28
1
1
4
5
.
2
8
1
1
0
0
.
2
9
0
.
0
9
2
.
4
.
M
et
ho
do
lo
g
y
:
F
o
r
m
ula
t
io
n
o
f
t
he
un
ba
la
nced
t
hree
-
ph
a
s
e
O
P
F
pro
blem
2
.
4
.
1
.
P
hy
s
ica
l
m
o
delin
g
o
f
t
he
un
ba
la
nced
L
V
net
wo
rk
L
V
n
etwo
r
k
s
f
r
eq
u
e
n
tly
h
av
e
s
in
g
le
-
p
h
ase
lo
ad
s
an
d
s
ig
n
i
f
ican
t
asy
m
m
etr
y
,
wh
ich
ca
n
r
esu
lt
in
p
h
ase
im
b
alan
ce
s
,
n
e
u
tr
al
cu
r
r
en
ts
,
th
e
ap
p
ea
r
an
ce
o
f
n
o
n
-
ze
r
o
s
eq
u
e
n
ce
co
m
p
o
n
e
n
ts
in
ac
co
r
d
a
n
ce
with
[
1
9
]
.
T
o
ac
cu
r
ately
ca
p
tu
r
e
th
ese
p
h
y
s
ical
p
h
e
n
o
m
e
n
a,
an
u
n
b
alan
ce
d
f
o
u
r
-
wir
e
t
h
r
ee
-
p
h
ase
m
o
d
el
is
ad
o
p
ted
,
wh
ich
ex
p
licitly
in
clu
d
es th
e
ef
f
ec
t
o
f
th
e
n
eu
tr
al
c
o
n
d
u
ct
o
r
.
a.
Sy
m
m
etr
ical
co
m
p
o
n
en
t tr
a
n
s
f
o
r
m
atio
n
T
h
e
f
o
r
m
alis
m
o
f
s
y
m
m
etr
ic
al
co
m
p
o
n
en
ts
(
p
o
s
itiv
e,
n
eg
ativ
e,
ze
r
o
)
is
in
t
r
o
d
u
ce
d
to
d
ec
o
m
p
o
s
e
an
y
u
n
b
alan
ce
d
v
o
ltag
e
o
r
c
u
r
r
en
t
v
ec
to
r
in
t
o
th
r
ee
i
n
d
e
p
en
d
en
t
b
alan
ce
d
n
etwo
r
k
s
b
ased
o
n
th
e
m
o
d
el
p
r
o
p
o
s
ed
by
[
2
0
]
,
[
2
1
]
.
L
et:
012
=
[
0
1
2
]
,
012
=
[
0
1
2
]
(
1
)
T
h
e
tr
an
s
f
o
r
m
atio
n
to
t
h
e
d
o
m
ain
(
ab
c)
is
ex
p
r
ess
ed
as:
=
[
1
1
1
1
2
1
2
]
(
2
)
wh
er
e:
=
120
°
T
h
e
in
v
er
s
e
tr
a
n
s
f
o
r
m
atio
n
is
g
iv
en
b
y
:
=
012
,
012
=
−
1
,
(
3
)
with
r
ep
r
esen
tin
g
t
h
e
v
o
ltag
e
o
r
cu
r
r
en
t v
ec
to
r
s
f
o
llo
win
g
th
e
ap
p
r
o
ac
h
d
e
v
elo
p
ed
in
[
2
2
]
,
[
2
3
]
.
b.
I
m
p
ed
an
ce
in
s
y
m
m
etr
ic
c
o
m
p
o
n
en
ts
T
h
e
s
y
m
m
etr
ic
im
p
e
d
an
ce
m
a
tr
ix
is
p
r
esen
ted
d
iag
o
n
ally
as:
012
=
[
0
0
0
0
1
0
0
0
2
]
(
4
)
wh
er
e
0
: Z
er
o
s
eq
u
en
ce
im
p
ed
a
n
ce
,
as
s
o
ciate
d
with
th
e
r
etu
r
n
th
r
o
u
g
h
n
eu
tr
al/g
r
o
u
n
d
;
1
: Po
s
itiv
e
s
eq
u
en
ce
im
p
ed
an
ce
,
d
o
m
in
a
n
t c
o
m
p
o
n
e
n
t in
b
ala
n
ce
d
o
p
er
atio
n
;
2
: N
eg
ativ
e
s
eq
u
en
ce
im
p
e
d
an
c
e,
r
elate
d
to
im
b
alan
ce
.
T
h
e
tr
an
s
f
o
r
m
atio
n
to
t
h
e
p
h
as
e
im
p
ed
an
ce
m
atr
ix
(
)
is
wr
itten
as:
=
012
−
1
(
5
)
T
h
is
o
p
er
atio
n
m
ak
es e
x
p
licit
th
e
co
u
p
lin
g
s
b
etwe
en
p
h
ases
an
d
with
th
e
n
eu
tr
al
in
th
e
o
v
e
r
all
m
o
d
el.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2088
-
8
7
0
8
I
mp
a
ct
o
f
o
p
tima
l p
o
w
er flo
w
o
n
p
o
w
er q
u
a
lity in
a
l
o
w
vo
lta
g
e
th
r
ee
-
p
h
a
s
e
…
(
Da
vid
Mil
a
mb
o
K
a
s
u
mb
a
)
2309
2
.
4
.
2
.
Dec
is
io
n
v
a
ria
bles
Ph
ase
r
ec
o
n
f
ig
u
r
atio
n
is
m
o
d
e
led
b
y
a
b
in
ar
y
v
ar
iab
le:
ø
∈
{
0
,
1
}
∀
,
∀
ø
{
,
,
}
,
∀
(
6
)
s
u
ch
as:
ø
∈
Ø
o
u
Ø
=
{
,
,
}
,
ø
,
=
{
1
,
0
,
ℎ
ℎ
ø
,
ℎ
,
wh
er
e:
∈
: set o
f
n
o
d
es;
ø
∈
{
,
,
}
: p
h
ases
;
∈
: tim
e
.
Ph
ase
r
ec
o
n
f
ig
u
r
atio
n
is
m
o
d
eled
b
y
a
b
in
ar
y
v
ar
iab
le
f
o
ll
o
win
g
[
2
4
]
o
v
e
r
th
e
tem
p
o
r
a
l
h
o
r
izo
n
.
T
h
is
v
ar
iab
le
allo
ws
f
o
r
d
y
n
a
m
ic
ad
ju
s
tm
en
t
o
f
lo
ad
allo
ca
tio
n
to
p
h
ases
to
m
in
im
ize
lo
s
s
es
an
d
im
b
alan
ce
[
2
5
]
.
2
.
4
.
3
.
O
bje
ct
iv
e
f
un
ct
io
n
T
h
e
o
b
jectiv
e
is
to
m
in
im
ize
to
tal
ac
tiv
e
lo
s
s
es
o
v
er
th
e
tim
e
h
o
r
izo
n
T
.
R
esis
t
iv
e
lo
s
s
es
ar
e
p
r
o
p
o
r
tio
n
al
to
×
|
|
2
ac
co
r
d
in
g
to
J
o
u
le'
s
law,
th
u
s
:
min
∑
∑
∑
,
ø
|
,
ø
,
|
2
∈
ø
∈
{
,
,
}
(
,
)
∈
,
(
7
)
wh
er
e:
:
s
et
o
f
lin
es;
,
ø
:
r
esis
tan
ce
o
f
lin
e
(
,
)
o
f
p
h
ase
ø
;
,
ø
,
:
cu
r
r
e
n
t
o
n
p
h
ase
ø
at
tim
e
t
.
T
h
is
n
o
n
-
lin
ea
r
f
o
r
m
u
latio
n
is
ch
ar
ac
ter
is
tic
o
f
th
e
p
r
o
b
lem
s
MI
NL
P
ac
co
r
d
in
g
to
th
e
class
ic
f
o
r
m
u
latio
n
o
f
[
2
6
]
.
2
.
4
.
4
.
AC
po
wer
f
lo
w
equa
t
i
o
ns
Activ
e
an
d
r
ea
ctiv
e
f
l
u
x
es
ar
e
d
ef
in
e
d
b
ased
o
n
th
e
r
ea
l
(
)
an
d
im
ag
in
a
r
y
(
)
co
m
p
o
n
e
n
ts
o
f
th
e
v
o
ltag
es:
a.
Activ
e
p
o
wer
,
ø
,
=
,
ø
,
∑
(
,
ø
,
ø
,
−
,
ø
,
ø
,
)
+
,
ø
,
∈
∑
(
,
ø
,
ø
,
−
,
ø
,
ø
,
)
∈
∙
(
8
)
b
.
R
ea
ctiv
e
p
o
wer
,
ø
,
=
,
ø
,
∑
(
,
ø
,
ø
,
−
,
ø
,
ø
,
)
+
,
ø
,
∈
∑
(
,
ø
,
ø
,
−
,
ø
,
ø
,
)
∙
∈
(
9
)
W
h
er
e:
,
ø
,
,
ø
: c
o
n
d
u
ctan
c
e
et
s
u
s
ce
p
tan
ce
d
u
ter
m
e
ad
m
ittan
ce
s
n
o
d
ales;
: set o
f
n
o
d
es c
o
n
n
ec
te
d
to
i.
T
h
ese
eq
u
atio
n
s
r
ep
r
esen
t
th
e
co
m
p
lete
f
o
r
m
s
o
f
t
h
e
n
o
n
lin
e
ar
AC
p
o
wer
f
lo
w
in
a
n
u
n
b
al
an
ce
d
th
r
ee
-
p
h
ase
s
y
s
tem
[
1
0
]
.
2
.
4
.
5
.
O
pera
t
io
na
l
c
o
ns
t
ra
ints
a.
Po
wer
b
alan
ce
At
ea
ch
n
o
d
e
we
h
av
e:
,
ø
,
+
∑
ℎ
,
,
ø
,
=
,
ø
,
+
∑
ℎ
,
,
ø
,
ℎ
∈
,
ø
|
,
ø
,
|
2
ℎ
∈
,
∀
,
ø
∈
∅
,
∈
(
1
0
)
w
h
er
e:
,
ø
,
=
ac
tiv
e
p
o
wer
g
en
e
r
at
ed
/in
jecte
d
at
th
e
n
o
d
e
n
;
,
ø
,
=
ac
tiv
e
p
o
wer
a
b
s
o
r
b
e
d
b
y
th
e
l
o
ad
at
th
e
n
o
d
e
n
;
ℎ
∈
=
s
et
o
f
b
r
an
c
h
es e
n
ter
in
g
n
o
d
e
n
;
a
n
d
ℎ
∈
=
s
et
o
f
b
r
a
n
ch
es lea
v
in
g
n
o
d
e
n.
A
s
im
ilar
f
o
r
m
u
latio
n
h
o
ld
s
f
o
r
r
ea
ctiv
e
p
o
wer
(
Q)
.
T
h
e
n
o
d
al
b
alan
ce
is
en
f
o
r
ce
d
th
r
o
u
g
h
Kir
ch
h
o
f
f
’
s
cu
r
r
en
t
law
(
KC
L
)
,
en
s
u
r
in
g
co
n
s
is
ten
cy
b
et
wee
n
p
o
wer
g
e
n
er
atio
n
,
lo
ad
d
em
an
d
,
an
d
p
o
wer
f
lo
ws
th
r
o
u
g
h
o
u
t
th
e
n
etwo
r
k
.
T
h
e
p
r
o
p
o
s
ed
OPF
f
r
a
m
e
wo
r
k
r
elies
o
n
th
e
AC
p
o
wer
f
lo
w
eq
u
atio
n
s
to
m
ain
tain
th
is
b
alan
ce
at
ea
ch
n
o
d
e
wh
ile
ac
c
u
r
ately
r
e
p
r
es
en
tin
g
th
e
elec
tr
ical
b
e
h
av
io
r
o
f
th
e
u
n
b
alan
ce
d
th
r
ee
-
p
h
ase
l
o
w
-
v
o
ltag
e
s
y
s
tem
.
T
o
en
s
u
r
e
co
m
p
u
tatio
n
al
ef
f
icien
c
y
an
d
r
o
b
u
s
t
co
n
v
er
g
en
ce
,
th
e
o
p
tim
izatio
n
p
r
o
ce
s
s
is
im
p
lem
en
ted
u
s
in
g
th
e
in
ter
io
r
p
o
i
n
t
m
eth
o
d
a
v
ailab
le
in
Pan
d
a
p
o
wer
.
Activ
e
p
o
wer
lo
s
s
es
ar
e
ex
p
licitly
co
n
s
id
er
ed
in
th
e
o
b
jectiv
e
f
u
n
ctio
n
an
d
e
v
alu
ated
f
r
o
m
f
ee
d
er
r
e
s
is
tan
ce
s
an
d
p
h
ase
cu
r
r
en
ts
,
en
s
u
r
in
g
a
p
h
y
s
ically
co
n
s
is
ten
t
r
ep
r
esen
tatio
n
o
f
n
etwo
r
k
lo
s
s
es
wh
ile
p
r
eser
v
in
g
th
e
tr
ac
tab
ilit
y
o
f
th
e
o
p
tim
izatio
n
p
r
o
b
le
m
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8
7
0
8
I
n
t J E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
16
,
No
.
5
,
Octo
b
e
r
20
26
:
2
3
0
4
-
2
3
2
0
2310
b
.
Vo
ltag
e
lim
its
T
o
g
u
a
r
an
tee
ac
ce
p
tab
le
v
o
lta
g
es,
th
e
v
o
ltag
e
c
o
n
s
tr
ain
t is e
x
p
r
ess
ed
b
y
:
(
)
2
≤
(
,
ø
,
)
2
+
(
,
ø
,
)
2
≤
(
)
2
(
1
1
)
T
y
p
ically
,
with
=
0
.
95
et
=
1
.
05
ac
co
r
d
in
g
to
th
e
s
tan
d
ar
d
E
N
5
0
1
6
0
[
2
6
]
.
c.
T
h
er
m
al
lim
its
E
ac
h
lin
e
m
u
s
t c
o
m
p
ly
with
:
,
ø
,
2
+
,
ø
,
2
≤
(
S
ij
,
ø
m
ax
)
2
(1
2
)
wh
er
e
is
th
e
ap
p
ar
en
t h
ea
t c
a
p
ac
ity
o
f
lin
e
.
d
.
Vo
ltag
e
u
n
b
alan
ce
f
ac
to
r
(
VUF)
T
h
e
u
p
p
er
an
d
lo
wer
lim
its
o
f
th
e
VUF
ar
e
d
ef
in
ed
as f
o
llo
w
s
:
|
,
−
|
|
,
+
|
≤
,
∀
,
∀
(1
3
)
wh
er
e
+
an
d
−
ar
e
th
e
p
o
s
itiv
e
an
d
n
eg
ativ
e
s
eq
u
e
n
ce
s
o
f
th
e
v
o
l
tag
es
[
2
1
]
,
[
2
6
]
.
e.
Sin
g
le
ass
ig
n
m
en
t
E
ac
h
lo
ad
ca
n
o
n
l
y
b
e
c
o
n
n
ec
t
ed
to
o
n
e
ø
at
an
y
tim
e:
∑
,
ø
,
=
1
∙
ø
∈
{
,
,
}
(
14)
2
.
5
.
Sim
ula
t
io
n
f
ra
m
ew
o
rk
T
h
e
u
n
b
ala
n
ce
d
th
r
ee
-
p
h
ase
OPF
m
o
d
el
was
im
p
lem
en
ted
in
Pan
d
ap
o
wer
[
1
4
]
,
[
1
7
]
,
[
1
9
]
–
[
2
2
]
,
[
2
5
]
–
[
2
9
]
u
s
in
g
r
ea
l
f
ield
m
e
asu
r
em
en
ts
co
llected
f
r
o
m
th
e
Kam
alo
n
d
o
l
o
w
-
v
o
ltag
e
d
i
s
tr
ib
u
tio
n
n
etwo
r
k
.
T
h
e
n
etwo
r
k
was
m
o
d
eled
as
a
f
o
u
r
-
wir
e
s
y
s
tem
with
ex
p
licit
r
ep
r
esen
tatio
n
o
f
th
e
n
eu
tr
al
co
n
d
u
ct
o
r
an
d
p
h
ase
im
p
ed
an
ce
m
atr
ices
(
Z
a
bc
)
d
er
iv
ed
f
r
o
m
th
e
elec
tr
ical
p
ar
am
eter
s
r
ep
o
r
ted
in
T
ab
le
2
.
T
h
is
m
o
d
elin
g
ap
p
r
o
ac
h
en
ab
les
th
e
ac
c
u
r
ate
r
ep
r
esen
tatio
n
o
f
ze
r
o
-
s
eq
u
en
ce
cu
r
r
en
ts
,
n
e
u
tr
al
co
n
d
u
cto
r
lo
ad
in
g
,
a
n
d
p
h
ase
asy
m
m
etr
ies co
m
m
o
n
ly
o
b
s
er
v
ed
in
u
r
b
an
lo
w
-
v
o
lta
g
e
f
ee
d
er
s
.
Simu
latio
n
s
wer
e
co
n
d
u
cte
d
o
v
er
a
2
4
-
h
o
u
r
h
o
r
izo
n
u
s
in
g
h
o
u
r
ly
lo
ad
p
r
o
f
iles
d
er
i
v
ed
f
r
o
m
f
ield
m
ea
s
u
r
em
en
ts
.
T
h
e
o
p
tim
izati
o
n
p
r
o
b
lem
was
f
o
r
m
u
lated
a
s
a
MI
NL
P
m
o
d
el
a
n
d
s
o
l
v
ed
u
s
in
g
th
e
in
ter
io
r
p
o
in
t
m
eth
o
d
im
p
lem
e
n
ted
in
Pan
d
ap
o
wer
.
T
h
is
s
o
lv
er
was
s
elec
ted
b
ec
au
s
e
o
f
its
r
o
b
u
s
tn
ess
,
co
m
p
u
tatio
n
a
l
ef
f
icien
cy
,
a
n
d
p
r
o
v
en
ca
p
a
b
i
lity
to
h
an
d
le
lar
g
e
-
s
ca
le
n
o
n
l
in
ea
r
OPF
p
r
o
b
lem
s
wh
ile
e
n
s
u
r
in
g
r
e
p
r
o
d
u
cib
le
r
esu
lts
.
T
h
e
o
p
tim
izatio
n
p
r
o
c
ess
aim
ed
to
m
in
im
ize
t
o
tal
ac
tiv
e
p
o
wer
lo
s
s
es
wh
ile
s
atis
f
y
in
g
p
o
wer
b
alan
ce
eq
u
atio
n
s
,
v
o
ltag
e
c
o
n
s
tr
ain
ts
,
th
er
m
al
l
o
ad
in
g
lim
its
,
an
d
p
h
ase
u
n
b
ala
n
ce
r
estrictio
n
s
.
V
o
ltag
e
m
ag
n
itu
d
es
wer
e
m
ain
tain
ed
with
in
±
5
%
o
f
th
eir
n
o
m
in
al
v
alu
es in
ac
c
o
r
d
an
ce
with
t
h
e
E
N
5
0
1
6
0
s
tan
d
ar
d
.
T
h
e
av
er
a
g
e
c
o
m
p
u
tatio
n
tim
e
r
eq
u
ir
e
d
to
s
o
lv
e
t
h
e
2
4
-
h
o
u
r
OPF
p
r
o
b
lem
r
an
g
e
d
b
etw
ee
n
4
5
an
d
6
0
s
ec
o
n
d
s
o
n
a
s
tan
d
a
r
d
w
o
r
k
s
tatio
n
eq
u
ip
p
e
d
with
a
n
I
n
tel
C
o
r
e
p
r
o
ce
s
s
o
r
a
n
d
1
6
GB
o
f
R
AM
.
T
h
is
co
m
p
u
tatio
n
al
p
e
r
f
o
r
m
an
ce
d
em
o
n
s
tr
ates
th
e
p
r
ac
tical
ap
p
licab
ilit
y
o
f
th
e
p
r
o
p
o
s
ed
MI
NL
P
-
b
ased
OPF
f
r
am
ewo
r
k
f
o
r
o
p
er
ati
o
n
al
p
la
n
n
in
g
a
n
d
p
o
wer
q
u
ality
ass
es
s
m
en
t in
lo
w
-
v
o
ltag
e
d
is
tr
ib
u
t
io
n
n
etwo
r
k
s
.
2
.
6
.
O
ptim
iza
t
io
n a
nd
c
o
ns
t
ra
in
t
s
T
h
e
o
b
jectiv
e
f
u
n
ctio
n
(
5
)
m
i
n
im
izes
to
tal
ac
tiv
e
p
o
wer
lo
s
s
es
s
u
b
ject
to
th
e
th
e
r
m
al,
v
o
ltag
e,
an
d
u
n
b
alan
ce
co
n
s
tr
ain
ts
(
8)
–
(
1
2
)
.
T
h
ese
in
cl
u
d
ed
:
i
)
lin
e
lo
a
d
in
g
lim
its
to
p
r
ev
e
n
t
th
er
m
a
l
o
v
er
ca
p
ac
ity
(1
0
)
,
ii)
v
o
ltag
e
b
o
u
n
d
s
with
in
[
0
.
9
5
,
1
.
0
5
]
p
u
(
9
)
,
a
n
d
iii)
a
m
ax
im
u
m
VUF
en
s
u
r
in
g
p
h
a
s
e
s
y
m
m
etr
y
(1
1
)
.
Pan
d
ap
o
wer
in
teg
r
ates
th
ese
co
n
s
tr
ain
ts
in
to
an
o
p
tim
izatio
n
r
o
u
tin
e,
iter
ativ
ely
a
d
ju
s
tin
g
p
h
ase
v
o
ltag
es
an
d
in
jectio
n
s
to
m
in
im
ize
lo
s
s
es a
n
d
r
esto
r
e
n
etwo
r
k
b
alan
ce
.
2
.
7
.
M
o
del
v
a
lid
a
t
io
n a
nd
perf
o
rm
a
nce
a
s
s
ess
m
ent
Pan
d
ap
o
wer
s
im
u
latio
n
r
esu
lts
wer
e
co
m
p
ar
ed
with
m
ea
s
u
r
ed
o
p
er
atio
n
al
d
ata
in
o
r
d
er
to
v
alid
ate
th
e
m
o
d
el
as
s
h
o
wn
in
T
ab
le
1
.
T
h
e
m
o
d
el'
s
ac
cu
r
ac
y
was
co
n
f
ir
m
ed
b
y
th
e
m
ea
n
ab
s
o
lu
te
er
r
o
r
b
etwe
en
s
im
u
lated
an
d
o
b
s
er
v
e
d
v
o
lta
g
es,
wh
ich
s
tay
ed
b
elo
w
2
%
.
T
h
e
OPF
lo
wer
ed
th
e
VUF
f
r
o
m
8
%
to
3
%,
d
ec
r
ea
s
ed
ac
tiv
e
lo
s
s
es
b
y
ap
p
r
o
x
im
ately
3
0
%,
an
d
en
h
a
n
c
ed
m
in
im
u
m
v
o
lta
g
e
lev
els
(
1
9
8
V
to
2
1
0
V)
an
d
o
v
er
v
o
ltag
es
(
2
3
0
V
to
2
2
6
V
)
.
T
h
ese
r
esu
lts
d
em
o
n
s
tr
ate
th
at
th
e
p
r
o
p
o
s
ed
u
n
b
ala
n
ce
d
OPF
f
o
r
m
u
latio
n
is
b
o
th
co
m
p
u
tatio
n
ally
s
tab
le
an
d
o
p
er
atio
n
ally
r
ea
lis
tic,
o
f
f
er
in
g
a
v
alu
ab
le
f
r
am
ew
o
r
k
f
o
r
im
p
r
o
v
in
g
p
o
we
r
q
u
ality
in
L
V
n
etwo
r
k
s
o
f
d
ev
elo
p
in
g
r
e
g
io
n
s
s
u
ch
as L
u
b
u
m
b
ash
i.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2088
-
8
7
0
8
I
mp
a
ct
o
f
o
p
tima
l p
o
w
er flo
w
o
n
p
o
w
er q
u
a
lity in
a
l
o
w
vo
lta
g
e
th
r
ee
-
p
h
a
s
e
…
(
Da
vid
Mil
a
mb
o
K
a
s
u
mb
a
)
2311
3.
RE
SU
L
T
S
3
.
1
.
Cha
ra
ct
er
iza
t
io
n o
f
lo
a
d be
ha
v
io
r
per
F
ee
der
T
h
e
h
o
u
r
ly
ev
o
lu
tio
n
o
f
ac
tiv
e
p
o
wer
co
n
s
u
m
p
tio
n
o
v
er
a
2
4
-
h
o
u
r
p
er
io
d
ac
r
o
s
s
th
e
f
iv
e
f
ee
d
er
s
o
f
th
e
s
tu
d
ied
L
V
n
etwo
r
k
is
d
ep
icted
in
Fig
u
r
e
4
.
T
h
e
d
etailed
h
o
u
r
l
y
ac
tiv
e
p
o
wer
lo
ad
p
r
o
f
ile
f
o
r
ea
ch
f
ee
d
er
is
s
h
o
wn
s
eq
u
en
tially
in
F
ig
u
r
e
4
(
a)
th
r
o
u
g
h
Fig
u
r
e
4
(
e)
.
T
h
e
Kam
alo
n
d
o
d
is
tr
ib
u
tio
n
s
y
s
tem
's
o
p
er
atio
n
al
f
r
ag
ilit
y
u
n
d
er
h
i
g
h
lo
a
d
in
g
co
n
d
itio
n
s
is
r
ef
lecte
d
in
t
h
e
r
esu
lts
,
wh
ich
s
h
o
w
s
ig
n
if
ican
t
tem
p
o
r
al
a
n
d
s
tr
u
ctu
r
al
h
eter
o
g
en
eity
.
Feed
er
2
'
s
lo
ad
p
r
o
f
ile
is
co
n
s
ta
n
tly
h
ig
h
an
d
u
n
p
r
e
d
ictab
le,
with
p
h
ase
p
o
wer
s
r
ea
ch
in
g
1
5
0
-
1
7
0
k
W
.
T
h
is
i
m
p
lies
th
e
ex
is
ten
ce
o
f
c
o
m
m
er
cial
ac
tiv
ity
,
d
en
s
e
r
esid
e
n
tial
ag
g
r
e
g
atio
n
,
o
r
n
ea
r
ly
co
n
s
tan
t
o
p
e
r
atio
n
o
f
i
n
s
titu
tio
n
al
co
n
s
u
m
er
s
.
Feed
er
3
'
s
to
tal
p
h
ase
lo
ad
s
ar
e
s
til
l
less
th
an
4
0
k
W
,
wh
ich
is
ty
p
ical
o
f
a
r
esid
en
tial
ar
ea
with
litt
le
tr
af
f
ic,
an
d
i
ts
p
o
wer
lev
els
ar
e
s
ig
n
if
ican
t
ly
lo
wer
th
an
t
h
o
s
e
o
f
th
e
o
t
h
er
f
ee
d
er
s
.
(
a)
(
b
)
(
c)
(
d
)
(
e)
Fig
u
r
e
4
.
Ho
u
r
ly
lo
a
d
ev
o
lu
tio
n
o
f
f
e
ed
er
s
1
to
5
(
a
)
d
ep
ar
t 1
,
(
b
)
d
ep
ar
t
2
,
(
c)
d
ep
a
r
t 3
,
(
d
)
d
ep
ar
t 4
,
an
d
(
e)
d
ep
ar
t
5
Feed
er
s
1
,
4
,
an
d
5
s
h
o
w
h
ig
h
lev
els
o
f
p
o
wer
a
n
d
h
ig
h
lev
els
o
f
tem
p
o
r
al
v
ar
ia
b
ilit
y
.
Fe
ed
er
5
ca
n
h
an
d
le
m
o
r
e
th
an
1
5
0
k
W
o
f
lo
ad
o
n
two
p
h
ases
an
d
Feed
e
r
s
1
an
d
4
ex
ce
ed
2
0
0
k
W
in
p
ea
k
p
h
ase
p
o
wer
.
T
h
er
e
ar
e
n
o
tab
le
d
r
o
p
s
an
d
d
is
co
n
tin
u
ities
f
o
r
th
ese
f
ee
d
er
s
f
r
o
m
1
0
:0
0
to
1
8
:0
0
.
Feed
er
5
n
ea
r
ly
lo
s
es
o
n
e
p
h
ase
en
tire
ly
i
n
th
e
late
af
te
r
n
o
o
n
.
Feed
er
4
h
as
two
n
o
tab
le
in
ter
r
u
p
tio
n
s
o
f
th
e
l
o
ad
an
d
Feed
er
1
h
as
a
s
ig
n
if
ican
t
d
r
o
p
i
n
th
e
m
id
d
le
o
f
th
e
d
ay
.
I
n
an
alr
ea
d
y
s
tr
a
in
ed
L
V
n
etwo
r
k
,
s
u
c
h
p
atter
n
s
ar
e
in
d
icativ
e
o
f
lo
ad
s
h
ed
d
in
g
p
r
o
ce
d
u
r
es,
p
r
o
tectio
n
-
tr
ig
g
er
e
d
d
is
co
n
n
ec
tio
n
s
,
o
r
u
n
s
tab
le
d
o
wn
s
tr
ea
m
c
o
n
n
ec
tio
n
s
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8
7
0
8
I
n
t J E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
16
,
No
.
5
,
Octo
b
e
r
20
26
:
2
3
0
4
-
2
3
2
0
2312
Sev
er
e
im
b
alan
ce
s
with
in
an
d
b
etwe
en
f
ee
d
er
s
ca
n
b
e
s
ee
n
b
y
clo
s
ely
e
x
am
in
in
g
th
e
p
h
ase
-
wis
e
cu
r
v
es.
W
h
ile
all
th
r
ee
p
h
ases
s
till
p
r
o
d
u
ce
co
n
s
id
er
a
b
le
n
eu
tr
al
cu
r
r
e
n
ts
,
with
Ph
ase
A
p
r
o
d
u
cin
g
c
u
r
r
e
n
ts
ab
o
u
t
2
0
k
W
less
th
an
o
th
er
s
,
Ph
ase
B
s
till
d
o
m
in
ates
Feed
e
r
1
with
p
o
wer
s
g
r
e
ater
th
a
n
2
1
0
k
W
.
T
h
e
f
ee
d
er
ty
p
e
o
f
Feed
e
r
2
is
q
u
asi
-
m
o
n
o
p
h
asic,
wh
er
e
Ph
ase
A
is
d
ea
ctiv
ated
,
an
d
c
o
m
p
lete
lo
a
d
is
p
lace
d
o
n
Ph
ases
B
an
d
C
,
r
esu
ltin
g
in
ex
tr
em
e
lo
ad
asy
m
m
etr
y
.
Feed
er
3
s
h
o
ws
co
n
s
tan
t
u
n
d
er
u
tili
za
tio
n
o
f
Ph
ase
C
,
wh
ile
u
n
d
er
u
tili
za
tio
n
o
f
Feed
er
4
an
d
5
alo
n
g
with
s
tr
o
n
g
p
h
ase
d
o
m
in
an
ce
an
d
u
n
ev
e
n
p
h
a
s
e
r
ed
u
ctio
n
s
ca
u
s
e
v
o
ltag
e
u
n
b
alan
ce
a
n
d
in
c
r
ea
s
ed
th
er
m
al
s
tr
ess
.
3.
2
.
P
ha
s
e
i
m
ba
la
nce
a
nd
v
o
lt
a
g
e
beha
v
io
r
in
t
he
L
V
net
wo
rk
Fig
u
r
e
5
p
r
o
v
id
es
an
in
teg
r
at
ed
o
v
er
v
iew
o
f
th
e
c
u
r
r
en
t,
v
o
ltag
e,
an
d
VUF
d
is
tr
ib
u
tio
n
s
with
in
th
e
a
n
a
l
y
z
e
d
L
V
n
e
t
w
o
r
k
,
h
i
g
h
li
g
h
t
i
n
g
t
h
e
m
a
g
n
it
u
d
e
a
n
d
o
p
e
r
at
i
o
n
a
l
c
o
n
s
e
q
u
e
n
c
e
s
o
f
p
h
a
s
e
im
b
a
l
a
n
c
e
o
n
p
o
w
e
r
q
u
a
l
i
t
y
.
T
h
e
o
b
s
e
r
v
e
d
p
a
t
t
e
r
n
s
r
e
v
e
a
l
p
e
r
s
is
t
e
n
t
as
y
m
m
e
t
r
i
es
in
b
o
t
h
c
u
r
r
e
n
t
a
n
d
v
o
l
t
a
g
e
b
e
h
a
v
i
o
r
,
s
y
m
p
t
o
m
a
t
i
c
o
f
h
i
g
h
l
y
u
n
e
v
e
n
l
o
a
d
a
l
l
o
c
at
i
o
n
a
n
d
w
e
a
k
n
e
t
w
o
r
k
m
es
h
i
n
g
in
t
h
e
K
a
m
al
o
n
d
o
d
i
s
t
r
i
b
u
ti
o
n
s
y
s
t
e
m
.
F
ig
u
r
e
5
(
a
)
,
th
e
th
r
ee
p
h
ase
cu
r
r
en
ts
ar
e
in
itially
well
b
alan
ce
d
at
th
e
s
u
b
s
tatio
n
,
co
n
f
ir
m
in
g
a
co
r
r
ec
t p
h
ase
d
is
tr
ib
u
tio
n
at
th
e
s
o
u
r
ce
.
H
o
wev
er
,
a
p
r
o
n
o
u
n
ce
d
im
b
alan
ce
em
er
g
es
i
m
m
ed
iately
alo
n
g
th
e
d
o
wn
s
t
r
ea
m
f
ee
d
e
r
s
.
Fro
m
Feed
er
1
an
d
b
ey
o
n
d
,
th
e
p
h
a
s
e
cu
r
r
en
ts
ar
e
b
etwe
en
2
6
0
–
3
7
0
A
an
d
th
e
n
eu
tr
al
cu
r
r
en
t
ex
ce
ed
3
4
0
A.
I
n
s
o
m
e
ca
s
es,
th
e
n
eu
tr
al
c
u
r
r
e
n
t
ca
n
b
ec
o
m
e
co
m
p
ar
ab
le
t
o
o
r
h
ig
h
er
t
h
an
t
h
e
p
h
ase
cu
r
r
en
ts
.
Feed
er
s
2
a
n
d
3
also
s
h
o
w
h
ig
h
n
e
u
tr
al
cu
r
r
e
n
t,
d
esp
ite
ca
r
r
y
in
g
a
lo
wer
to
tal
p
o
wer
.
T
h
is
s
h
o
ws
th
at
th
e
im
b
alan
ce
is
s
tr
u
ctu
r
al
an
d
n
o
t
ju
s
t
s
tr
ictly
p
r
o
p
o
r
tio
n
al
to
th
e
m
ag
n
itu
d
e
o
f
th
e
lo
a
d
.
Feed
er
s
4
a
n
d
5
f
u
r
th
er
ex
ac
er
b
ate
th
is
co
n
d
itio
n
th
r
o
u
g
h
s
tr
o
n
g
p
h
ase
d
o
m
in
an
ce
,
p
ar
ticu
lar
l
y
o
n
Ph
ase
B
,
an
d
n
ea
r
ab
s
en
ce
o
f
Ph
ase
C
o
n
ce
r
tain
b
r
an
ch
es.
Fig
u
r
e
5
(
b
)
s
h
o
ws
h
o
w
p
h
ase
-
to
-
n
eu
tr
al
v
o
ltag
es
v
ar
y
ac
r
o
s
s
th
e
n
etwo
r
k
.
Su
b
s
tatio
n
an
d
n
o
d
es
1
4
an
d
2
8
p
r
esen
t
v
o
ltag
es
th
at
ar
e
with
in
th
e
ac
ce
p
tab
le
r
an
g
e
.
Ho
wev
e
r
,
n
o
d
es
7
,
1
2
,
a
n
d
1
8
s
h
o
w
cr
itical
d
ev
iatio
n
s
.
No
d
e
1
2
s
h
o
ws
th
e
g
r
ea
test
d
e
g
r
ad
ati
o
n
,
c
h
ar
ac
ter
ized
b
y
p
h
ase
A
v
o
ltag
e,
a
n
d
well
b
elo
w
r
eg
u
lato
r
y
lim
its
,
alo
n
g
with
s
ig
n
if
ican
t
d
ev
iatio
n
s
in
p
h
ases
B
an
d
C
.
Fig
u
r
e
5
(
c
)
s
h
o
ws
th
e
VUF
at
ea
ch
n
o
d
e,
wh
ich
ad
d
s
a
q
u
an
titativ
e
d
im
en
s
io
n
to
o
u
r
o
b
s
er
v
atio
n
s
.
W
h
ile
m
o
s
t
n
o
d
es
h
av
e
VUFs
b
elo
w
3
%,
wh
ich
is
ac
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Evaluation Warning : The document was created with Spire.PDF for Python.
I
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a
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Fig
u
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h
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Ph
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u
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J
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3
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ates
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ate
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ase
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atin
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o
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u
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7
(
a
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,
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ee
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itially
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ated
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7
(
b
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ase
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ase
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lo
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is
illu
s
tr
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in
Fig
u
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es
7
(
c
)
an
d
7
(
e
)
.
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ef
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e
o
p
tim
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,
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f
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v
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f
ee
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s
,
p
ar
ticu
lar
ly
Feed
er
s
1
an
d
5
.
Fo
llo
win
g
o
p
tim
izatio
n
,
lo
s
s
es
wer
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r
ed
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ce
d
b
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ap
p
r
o
x
im
ately
3
0
%
–
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0
%,
with
th
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est
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ed
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cc
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r
r
in
g
o
n
f
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s
ca
r
r
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in
g
th
e
h
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g
h
est
cu
r
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en
ts
.
T
h
e
s
p
atial
h
ea
t
m
ap
f
u
r
th
e
r
co
n
f
ir
m
s
a
s
ig
n
if
ican
t
d
ec
r
ea
s
e
in
lo
s
s
co
n
ce
n
tr
atio
n
th
r
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g
h
o
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th
e
n
etwo
r
k
,
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em
o
n
s
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atin
g
th
e
ab
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o
f
OPF
to
r
ed
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tr
ib
u
te
p
o
wer
f
lo
ws
an
d
r
ed
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ce
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esis
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d
is
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ip
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n
d
er
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n
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ce
d
o
p
e
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atin
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co
n
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itio
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s
.
Fig
u
r
e
7
(
d
)
s
u
m
m
ar
izes
th
e
m
ain
p
er
f
o
r
m
an
ce
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n
d
icato
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s
b
ef
o
r
e
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n
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af
ter
o
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cu
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r
en
t
lo
a
d
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,
ac
tiv
e
lo
s
s
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d
v
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u
n
b
alan
ce
.
T
h
e
r
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lts
in
d
icate
th
at
th
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p
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p
o
s
ed
OPF
f
r
am
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k
s
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cc
ess
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u
lly
im
p
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o
v
es
n
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k
p
er
f
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m
a
n
ce
with
o
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t
in
tr
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ci
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g
a
d
d
itio
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al
o
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ts
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g
e
v
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latio
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s
.
Fin
ally
,
Fig
u
r
e
7
(
f
)
p
r
esen
ts
th
e
ev
o
lu
tio
n
o
f
th
e
VUF.
A
s
ev
er
e
im
b
alan
ce
was
o
b
s
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e
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n
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2
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o
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e
o
p
tim
izatio
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,
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e
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p
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o
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im
at
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m
ain
ly
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u
e
to
a
p
h
a
s
e
-
o
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tag
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o
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d
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in
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tr
e
m
e
lo
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m
m
etr
y
.
Af
ter
OPF
im
p
lem
en
tatio
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,
th
e
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n
th
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ee
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er
was
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ap
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o
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ately
1
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ile
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e
r
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ain
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s
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ited
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w
5
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T
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s
tr
ate
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p
ab
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tially
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m
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d
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all
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ality
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r
o
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g
h
o
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t t
h
e
d
is
tr
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tio
n
n
et
wo
r
k
.
Evaluation Warning : The document was created with Spire.PDF for Python.