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o
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i
a
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o
u
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l
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a
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C
o
m
p
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V
ol
.
8,
N
o.
1,
O
c
t
ob
er
20
17
,
pp
.
1
~
8
D
O
I
:
10.
115
91/
i
j
eec
s
.
v
8
.i
1
.
pp
1
-
8
1
R
ec
ei
v
ed
J
une
1
9
,
20
1
7
;
R
ev
i
s
ed
A
ug
us
t
15
,
20
1
7
;
A
c
c
ept
ed
S
e
pt
e
mber
3,
2
017
O
n
-
lin
e
A
s
sess
m
en
t
o
f
V
o
lt
ag
e
S
t
ab
ili
t
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u
si
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g
S
y
n
c
hr
oph
a
s
or
Te
c
h
nol
og
y
S
at
y
en
d
r
a
P
r
at
ap
S
i
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h
*
1
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S.
P.
S
i
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g
h
2
D
epar
t
m
ent
of
E
l
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c
t
r
i
c
a
l
E
ng
i
n
eer
i
ng
,
I
n
di
an
I
n
s
t
i
t
ut
e o
f
T
ec
h
nol
ogy
(BH
U
) Va
r
a
n
a
s
i
V
ar
ana
s
i
,
2210
05,
I
ndi
a
*
C
or
r
es
po
ndi
ng a
ut
hor
,
e
-
m
ai
l
:
s
ps
i
ng
h.
r
s
.
eee@
i
i
t
b
hu.
a
c
.
i
n
A
b
st
r
act
S
er
i
e
s
of
bl
ac
k
out
s
enc
ount
e
r
ed
i
n
r
e
c
en
t
y
ear
s
i
n
p
ow
er
s
y
s
t
em
ha
v
e
bee
n
o
c
c
ur
r
ed
b
ec
au
s
e
ei
t
her
of
v
ol
t
a
ge or
ang
l
e i
ns
t
a
bi
l
i
t
y
or
bot
h t
o
get
h
er
w
as
not
det
e
c
t
ed w
i
t
h
i
n t
i
m
e and pr
o
gr
es
s
i
v
e v
ol
t
a
ge
or
angl
e i
n
s
t
a
bi
l
i
t
y
f
ur
t
her
de
gr
aded
t
he
s
y
s
t
em
c
o
ndi
t
i
on,
bec
a
us
e
of
i
n
c
r
ea
s
e
i
n l
o
ad
i
ng.
T
hi
s
pape
r
pr
es
e
nt
s
t
he
r
eal
-
t
i
m
e as
s
e
s
s
m
ent
m
et
hodo
l
og
y
of
v
o
l
t
ag
e s
t
abi
l
i
t
y
us
i
ng P
has
or
M
eas
ur
e
m
ent
U
ni
t
(
P
MU
)
w
i
t
h obs
e
r
v
a
bi
l
i
t
y
of
l
o
ad bu
s
e
s
onl
y
i
n pow
er
net
w
or
k
.
P
M
U
s
ar
e pl
a
c
ed at
s
t
r
at
eg
i
c
a
l
l
y
obt
ai
n
ed l
o
c
at
i
on
s
uc
h
t
hat
m
i
ni
m
um
num
ber
of
P
M
U
’
s
c
an
m
ak
e
al
l
l
o
ad
bu
s
es
ob
s
er
v
ab
l
e.
D
at
a
ob
t
ai
n
e
d
by
P
M
U
’
s
ar
e
us
ed
f
or
v
ol
t
a
ge s
t
ab
i
l
i
t
y
a
s
s
e
s
s
m
ent
w
i
t
h t
h
e h
e
l
p of
s
u
c
c
e
s
s
i
v
e
c
han
ge i
n t
he
ang
l
e of
b
us
v
ol
t
a
ge w
i
t
h
r
es
pe
c
t
t
o i
n
c
r
em
ent
a
l
l
o
ad,
w
hi
c
h i
s
u
s
ed as
o
n
-
l
i
ne v
ol
t
age s
t
ab
i
l
i
t
y
pr
e
di
c
t
or
(
V
S
P
)
.
T
he r
ea
l
-
t
i
m
e
v
ol
t
age
pha
s
or
s
obt
ai
ne
d
b
y
P
M
U
’
s
ar
e
us
ed
as
r
eal
t
i
m
e
v
ol
t
age
s
t
ab
i
l
i
t
y
i
ndi
c
at
or
.
T
h
e
c
as
e
s
t
u
dy
h
as
been
c
ar
r
i
ed o
ut
o
n I
E
E
E
-
14
b
us
s
y
s
t
em
and
I
E
E
E
-
30
bus
s
y
s
t
em
s
t
o dem
on
s
t
r
a
t
e t
h
e r
e
s
u
l
t
s
.
Ke
y
w
o
rd
s
:
v
ol
t
a
ge s
t
ab
i
l
i
t
y
,
v
ol
t
ag
e
s
t
abi
l
i
t
y
pr
e
di
c
t
or
,
pha
s
or
m
eas
ur
em
ent
u
ni
t
,
opt
i
m
al
pl
ac
em
ent
C
o
p
y
r
i
g
h
t
©
2
01
7
I
n
s
t
i
t
u
t
e
o
f
A
d
v
a
n
c
e
d
E
n
g
i
n
e
e
r
i
n
g
a
n
d
S
c
i
e
n
c
e
.
A
l
l
r
i
g
h
t
s r
es
er
ved
.
1
.
I
n
tr
o
d
u
c
ti
o
n
T
he pow
er
s
y
s
t
em
i
s
c
r
i
t
i
c
al
i
nt
er
dep
end
ent
i
nf
r
as
t
r
uc
t
ur
e and i
t
i
s
nec
es
s
ar
y
t
o t
r
ans
m
i
t
el
ec
t
r
i
c
a
l
po
w
er
f
r
o
m
gene
r
at
i
on
t
o
l
oad
e
nd r
el
i
ab
l
y
,
k
eepi
ng
g
ener
a
t
i
o
n t
o
l
oa
d b
al
a
nc
e
at
ev
er
y
t
i
m
e,
s
uc
h t
hat
i
t
s
per
f
or
m
anc
e s
houl
d b
e ef
f
ec
t
i
v
e.
A
l
s
o,
po
w
er
s
y
s
t
e
m
s
uppor
t
s
t
he
w
or
l
d
ec
on
om
y
t
o
a
gr
eat
ex
t
ent
.
S
o
,
t
h
e
s
t
ab
i
l
i
t
y
an
d
s
ec
ur
i
t
y
of
e
l
ec
t
r
i
c
p
o
w
er
net
w
or
k
i
s
an
i
m
por
t
ant
i
s
s
ue
f
or
pl
ann
i
ng a
nd
op
er
at
i
on
eng
i
n
ee
r
[
1]
.
F
or
t
h
i
s
pur
pos
e,
i
t
i
s
i
m
por
t
ant
t
o
ana
l
y
z
e
t
he
s
y
s
t
em
f
or
ev
er
y
i
ns
t
a
nt
of
t
i
m
e,
s
o
us
e
of
r
eal
-
t
i
m
e
m
eas
ur
e
m
ent
s
ar
e
nec
es
s
ar
y
.
A
S
y
nc
hr
o
phas
or
b
as
ed
w
i
de
ar
ea m
eas
ur
em
ent
s
y
s
t
em
(
W
A
MS
)
gi
v
es
t
he r
ea
l
-
t
im
e
m
eas
ur
e
m
ent
s
of
v
ol
t
ag
e
and
c
ur
r
e
nt
p
has
or
s
w
i
t
h
t
he
he
l
p
of
P
MU
s
.
T
h
es
e
P
MU
s
ar
e
s
t
r
at
egi
c
a
l
l
y
pl
ac
ed
at
di
f
f
er
ent
l
oc
at
i
o
ns
i
n t
he
po
w
er
n
et
w
or
k
and
gi
v
es
t
he
v
ol
t
ag
e a
nd c
ur
r
e
nt
phas
or
s
of
t
he
bus
w
h
er
e
i
t
i
s
i
ns
t
a
l
l
ed.
V
ol
t
ag
e
s
t
ab
i
l
i
t
y
i
s
one
of
t
he
m
os
t
i
m
por
t
ant
ar
eas
f
or
eng
i
ne
er
s
t
o
m
ai
nt
ai
n t
he o
per
at
i
on of
po
w
e
r
s
y
s
t
em
w
i
t
h
i
n c
ont
r
ac
t
u
al
an
d s
t
ead
y
v
ol
t
age l
i
m
i
t
s
bef
or
e
and
af
t
er
an
y
d
i
s
t
ur
banc
es
.
T
hes
e
d
i
s
t
ur
ba
nc
e
s
ar
e,
m
a
y
b
e,
s
ud
de
n
l
os
s
of
gener
at
i
o
n
or
l
i
nes
,
or
c
ha
ng
i
ng
l
o
ads
,
w
hi
c
h
af
f
ec
t
s
t
he
oper
at
i
n
g
po
i
nt
of
s
y
s
t
em
and
f
r
eq
uenc
y
.
S
o,
i
t
i
s
nec
es
s
ar
y
t
o r
ap
i
dl
y
m
oni
t
o
r
and adj
us
t
t
o s
y
s
t
em
c
hanges
t
o at
t
a
i
n a n
e
w
oper
at
i
ng p
oi
n
t
or
an
equ
i
l
i
br
i
um
poi
nt
k
eepi
ng g
ener
at
i
o
n t
o l
o
ad b
al
a
nc
e.
T
hi
s
abi
l
i
t
y
of
s
y
s
t
em
i
s
t
he goa
l
of
v
ol
t
age
s
t
abi
l
i
t
y
as
s
es
s
m
ent
and c
o
nt
r
ol
[
2]
.
S
er
i
es
of
bl
ac
k
out
s
enc
ou
n
t
er
ed
i
n
r
ec
ent
y
e
ar
s
i
n
po
w
er
s
y
s
t
em
,
ha
v
e
bee
n
oc
c
ur
r
ed
bec
aus
e
ei
t
her
of
v
o
l
t
a
ge
o
r
ang
l
e
i
ns
t
abi
l
i
t
y
or
b
ot
h
t
o
get
h
er
w
as
not
det
ec
t
ed
w
i
t
hi
n
t
i
m
e
an
d
pr
ogr
es
s
i
v
e
v
o
l
t
ag
e
or
an
gl
e
i
ns
t
abi
l
i
t
y
f
ur
t
h
er
d
egr
aded
t
h
e s
y
s
t
em
c
ondi
t
i
o
n,
b
ec
aus
e
of
i
nc
r
eas
e
i
n
l
o
ad
i
ng
[
3,
4]
.
S
o,
s
y
nc
hr
o
ni
z
ed
ph
as
or
m
eas
ur
e
m
ent
s
ar
e
v
er
y
i
m
por
t
ant
f
or
w
i
d
e
ar
ea m
eas
ur
e
m
ent
s
y
s
t
em
s
us
ed i
n a
dv
anc
ed
po
w
er
s
y
s
t
em
m
oni
t
or
i
ng,
pr
o
t
ec
t
i
on,
an
d c
ont
r
o
l
app
l
i
c
at
i
ons
.
P
h
as
or
m
eas
ur
em
ent
uni
t
(
P
MU
)
b
ec
o
m
e
s
m
or
e and m
or
e at
t
r
ac
t
i
v
e
t
o p
o
w
er
en
g
i
ne
er
s
bec
aus
e i
t
c
an
pr
ov
i
de t
i
m
e s
y
nc
hr
on
i
z
ed
m
eas
ur
e
m
ent
s
of
v
ol
t
age
and c
ur
r
ent
s
phas
or
[
5
]
.
S
y
nc
hr
oni
z
a
t
i
on i
s
ac
hi
e
v
e
d b
y
s
am
e
-
t
i
m
e s
a
m
pl
i
ng of
v
ol
t
ag
e an
d c
ur
r
ent
w
av
ef
or
m
s
us
i
ng t
i
m
i
ng s
i
g
nal
s
f
r
om
t
he G
l
oba
l
P
os
i
t
i
o
ni
n
g S
y
s
t
em
S
at
e
l
l
i
t
e
(
G
P
S
)
.
T
h
e pr
es
ent
and
pos
s
i
b
l
e
f
ut
ur
e
a
ppl
i
c
a
t
i
ons
of
phas
or
m
eas
ur
em
ent
uni
t
s
ha
v
e
be
en
w
el
l
d
oc
um
ent
ed
[
6]
.
S
e
v
er
a
l
a
l
g
or
i
t
hm
s
and
a
ppr
oac
h
es
ha
v
e
be
en
pu
bl
i
s
h
ed
i
n
t
h
e
l
i
t
er
at
ur
e
f
or
t
he
op
t
i
m
al
pl
ac
em
ent
of
P
M
U
s
i
n
po
w
er
s
y
s
t
em
.
I
ni
t
i
at
i
ng
w
or
k
i
n P
MU
de
v
e
l
op
m
ent
a
nd
ut
i
l
i
z
at
i
on
i
s
d
on
e
b
y
P
hadk
e
et
al
.
[
5,
7]
.
I
n
r
ef
er
enc
e
[
8]
,
a s
t
r
at
egi
c
P
MU
pl
ac
em
ent
al
g
or
i
t
hm
i
s
dev
e
l
op
ed
t
o
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN
:
25
02
-
4
752
I
J
E
EC
S
V
o
l.
8
,
N
o.
1
,
O
c
t
o
ber
20
17
:
1
–
8
2
i
m
pr
ov
e t
he
ba
d da
t
a pr
o
c
es
s
i
ng c
apa
bi
l
i
t
y
of
s
t
at
e
es
t
i
m
at
i
on b
y
t
ak
i
ng ad
v
a
nt
ag
e of
P
M
U
t
ec
hno
l
og
y
.
P
r
o
v
i
di
n
g
s
el
e
c
t
ed
bus
es
w
i
t
h
P
MU
s
c
an
m
a
k
e
t
he
ent
i
r
e
s
y
s
t
em
obs
er
v
ab
l
e
.
T
he
aut
h
or
s
i
n [
9,
10
]
de
v
e
l
o
pe
d an
opt
i
m
al
pl
ac
em
ent
al
gor
i
t
hm
f
or
P
MU
s
b
y
us
i
n
g
i
nt
e
ger
l
i
n
ear
pr
ogr
am
m
i
ng.
Mor
e
ov
er
,
m
odel
i
ng
of
z
er
o
i
nj
ec
t
i
on
bus
es
has
bee
n pr
es
ent
ed
i
n
[
10
]
.
I
n
r
ef
er
enc
e
[
11]
,
a
ge
ner
a
l
i
z
e
d
i
nt
eg
er
l
i
ne
ar
pr
ogr
am
m
i
ng
f
or
m
ul
at
i
o
n
an
d
s
ol
ut
i
o
n
appr
oac
h
f
or
pl
ac
em
ent
of
P
MU
s
i
s
pr
o
pos
ed.
C
om
m
uni
c
at
i
o
n I
nf
r
as
t
r
uc
t
ur
e an
d I
s
l
an
di
n
g
bas
ed
opt
i
m
al
pl
ac
em
ent
of
P
MU
s
ha
v
e
been
d
oc
um
ent
ed
i
n
[
12
-
14]
an
d
[
15,
16]
r
es
pec
t
i
v
e
l
y
.
V
o
l
t
a
ge
s
t
a
b
i
lit
y
r
a
n
k
in
g
f
or
O
P
P
h
as
be
en
pr
opos
e
d
i
n
[
17
]
.
A
ut
h
or
s
i
n
[
1
8]
pr
es
ent
ed
a
m
ul
t
i
obj
ec
t
i
v
e
f
unc
t
i
on t
o
opt
i
m
i
z
e t
he
n
um
ber
of
P
MU
s
and m
ax
i
m
i
z
e
t
he
obs
er
v
ab
i
l
i
t
y
b
y
us
i
ng
B
i
nar
y
G
r
av
i
t
at
i
ona
l
S
e
ar
c
h A
l
g
or
i
t
hm
(
B
G
S
A
)
.
T
o v
er
i
f
y
t
he ef
f
ec
t
i
v
en
es
s
of
B
G
S
A
,
d
i
f
f
er
ent
c
on
t
i
n
genc
y
h
av
e
bee
n c
o
ns
i
der
e
d
i
n [
18]
.
I
n
t
h
i
s
p
aper
,
a
v
o
l
t
ag
e s
t
a
bi
l
i
t
y
i
n
dex
has
bee
n
pr
opos
e
d as
an
ap
pl
i
c
at
i
on
of
P
MU
s
.
S
ev
er
al
v
ol
t
age
s
t
abi
l
i
t
y
i
nd
ex
ha
v
e
bee
n p
ubl
i
s
hed
i
n
t
he
l
i
t
er
at
ur
e.
A
s
t
at
i
c
v
ol
t
age
s
t
abi
l
i
t
y
i
nd
ex
h
as
be
en
pr
opos
ed
i
n [
1
9]
w
hi
c
h
has
been
us
ed
t
h
e
s
y
nc
hr
op
has
or
t
ec
h
no
l
og
y
t
o ear
l
y
det
ec
t
i
o
n of
i
m
pen
di
n
g v
ol
t
age
i
ns
t
ab
i
l
i
t
y
.
I
n
[
20]
,
a
s
i
m
pl
e
ne
w
t
o
n r
aphs
o
n al
gor
i
t
hm
bas
ed v
ol
t
ag
e s
t
abi
l
i
t
y
a
na
l
y
s
i
s
has
b
een pr
es
e
nt
e
d t
o i
de
nt
i
f
y
t
h
e
beha
v
i
or
of
pow
er
s
y
s
t
e
m
under
v
ar
i
o
us
v
ol
t
a
ge
s
t
abi
l
i
t
y
c
on
di
t
i
o
ns
.
A
ut
h
or
s
i
n
[
21,
22
]
di
s
c
us
s
ed t
h
e s
om
e
m
or
e i
ndex
r
eg
ar
di
ng
v
o
l
t
a
ge s
t
a
bi
l
i
t
y
.
I
n t
h
i
s
pa
per
,
P
MU
s
ar
e
p
l
ac
ed
at
s
t
r
at
egi
c
a
l
l
y
obt
a
i
ned
l
oc
at
i
on
s
uc
h t
h
at
m
i
ni
m
u
m
num
ber
of
P
MU
’
s
c
an
m
ak
e
al
l
l
oa
d
bus
es
obs
er
v
ab
l
e
an
d
pr
o
v
i
de
t
he
o
n
-
l
i
ne
v
ol
t
a
ge
phas
or
s
t
o
pr
e
di
c
t
t
he
V
S
P
.
F
or
t
h
i
s
pur
p
os
e,
a
f
or
m
ul
at
i
on
of
opt
i
m
al
P
MU
p
l
ac
em
ent
pr
obl
em
i
s
done
b
y
m
i
x
ed i
nt
eg
er
l
i
n
ear
pr
ogr
am
m
i
ng (
MI
LP
)
.
T
hes
e P
MU
s
pr
o
v
i
d
e v
o
l
t
a
ge ph
a
s
or
s
of
al
l
l
oa
d
bus
es
at
v
er
y
s
m
al
l
i
nt
er
v
a
l
s
.
T
hus
t
he
r
at
e
of
c
h
ang
e
of
v
ol
t
age
an
gl
e
d
ue
t
o
c
hange
i
n
l
oad
w
i
t
h r
es
p
ec
t
t
o s
uc
c
es
s
i
v
e
i
nt
er
v
a
l
s
of
t
i
m
e and t
hi
s
c
an be us
e
d as
t
h
e V
S
P
t
o
es
t
i
m
at
e t
he
v
o
l
t
ag
e
s
t
ab
i
l
i
t
y
.
A
s
t
he
P
M
U
gi
v
es
r
ea
l
t
i
m
e
v
ol
t
age
a
nd
c
ur
r
ent
phas
or
s
,
t
hus
v
ol
t
a
ge
s
t
a
bi
l
i
t
y
pr
edi
c
t
or
c
a
n b
e us
e
d
f
or
on
-
l
i
ne v
o
l
t
a
ge
s
t
abi
l
i
t
y
pr
e
di
c
t
i
on.
T
he v
o
l
t
ag
e
s
t
a
bi
l
i
t
y
as
s
es
s
m
ent
pr
obl
em
us
i
ng P
MU
i
s
m
or
e ef
f
i
c
i
ent
and c
a
n be
us
ed
i
n pr
ac
t
i
c
e.
T
he m
ai
n c
ont
r
i
but
i
o
ns
of
t
hi
s
paper
ar
e as
u
nder
:
a)
P
r
opos
e
d
an
a
l
g
or
i
t
hm
t
o
opt
i
m
i
z
e
t
he
n
um
ber
of
P
MU
s
and
m
ax
i
m
i
z
e
t
he
o
bs
e
r
v
ab
i
l
i
t
y
of
t
he
l
oa
d bus
es
of
t
he p
o
w
e
r
s
y
s
t
em
.
b)
P
r
opos
e
d
an
o
n
-
l
i
ne
v
ol
t
a
ge
s
t
abi
l
i
t
y
pr
e
di
c
t
or
m
et
hod
w
h
i
c
h
m
oni
t
or
t
he
s
y
s
t
em
and
pr
ev
e
nt
t
he b
l
ac
k
out
s
i
n
po
w
er
s
y
s
t
em
.
T
he
paper
i
s
s
y
s
t
em
i
z
e
d
as
f
ol
l
o
w
s
:
S
ec
t
i
o
n
2
s
t
at
es
t
he
pr
ob
l
em
f
or
m
ul
at
i
on
of
m
i
ni
m
u
m
P
MU
p
l
ac
em
ent
and
v
ol
t
ag
e
s
t
a
bi
l
i
t
y
pr
e
di
c
t
i
on.
S
ec
t
i
on
3
ex
p
l
ai
n
s
t
he
s
o
l
ut
i
o
n
m
et
hodol
o
g
y
of
pr
opos
e
d
m
et
hod.
F
i
na
l
l
y
,
t
he
t
es
t
r
e
s
ul
t
s
ar
e
gi
v
e
n i
n S
ec
t
i
on 4
,
and
S
ec
t
i
o
n
5
c
onc
l
ud
es
t
he
pap
er
.
2.
P
r
o
b
l
em
F
o
r
m
u
l
a
ti
o
n
o
f
P
r
o
p
o
s
e
d
M
e
th
o
d
2.
1.
O
p
ti
m
a
l
P
M
U
P
l
a
c
e
m
e
n
t M
e
th
o
d
o
l
o
g
y
I
n
t
hi
s
pap
er
,
a
nov
el
obj
e
c
t
i
v
e
f
unc
t
i
o
n
has
bee
n
pr
o
pos
ed
t
o
op
t
i
m
i
z
e
t
he
n
um
ber
of
P
MU
s
and pr
o
v
i
des
t
h
e m
ax
i
m
u
m
obs
er
v
abi
l
i
t
y
of
t
he s
y
s
t
em
.
T
he pr
opos
ed obj
ec
t
i
v
e
f
unc
t
i
on c
an
be
w
r
i
t
t
en as
f
ol
l
o
w
s
:
M
in
i
m
i
ze
1
1
(*
)
n
n
i
ii
i
i
c
l
i
p
z
wz
c
NC
c
=
=
=
+
∑∑
(
1)
S
ubj
ec
t
t
o:
f
=
A.
Z
≥
1
(
2)
C
onn
ec
t
i
v
i
t
y
m
at
r
i
x
(
A
)
de
f
i
nes
t
he
i
nt
er
c
o
nnec
t
i
on
o
f
s
y
s
t
em
bus
es
b
y
t
r
ans
m
i
s
s
i
on
l
i
n
es
.
T
he ent
r
i
es
i
n
A
ar
e d
ef
i
ned
as
f
ol
l
o
w
s
:
1 i
f
1
i
f
a
n
d
a
r
e
c
o
n
n
e
c
t
e
d
0
ot
he
r
w
i
s
e
i=
j
A
ij
ij
=
(
3)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
E
EC
S
IS
S
N
:
2
502
-
4
752
On
-
l
i
n
e A
s
s
es
s
me
nt
of
V
o
l
t
age
S
t
a
bi
l
i
t
y
us
i
ng
S
y
nc
hr
o
phas
or
...
(
S
at
y
e
ndr
a
P
r
at
a
p S
i
ng
h
)
3
w
her
e
z
i
i
s
t
he
el
em
ent
s
of
v
ec
t
or
Z
,
w
h
i
c
h
r
epr
es
e
nt
s
t
he
s
t
at
us
of
t
he
i
ns
t
a
l
l
a
t
i
o
n
of
a
P
MU
at
b
u
s
i.
if
z
i
=
1,
i
t
m
eans
P
MU
i
s
i
ns
t
al
l
e
d at
b
us
i
,
ot
her
w
i
s
e
z
i
=
0.
p
i
s
t
h
e t
ot
al
n
um
ber
of
PM
U
s
.
c
cl
an
d
c
p
r
epr
es
e
nt
t
he c
os
t
of
c
hann
el
a
n
d c
os
t
of
P
MU
r
es
pec
t
i
v
e
l
y
.
NC
i
is
t
h
e
num
ber
s
of
c
hanne
l
at
bu
s
i
.
I
n r
ef
er
enc
e [
1
1
]
,
aut
hor
s
us
ed t
h
e m
ul
t
i
obj
ec
t
i
v
e f
unc
t
i
on t
o
opt
i
m
i
z
e
t
he
num
ber
of
P
MU
s
h
av
i
n
g
m
ax
i
m
u
m
obs
er
v
a
bi
l
i
t
y
.
I
n
t
h
i
s
pa
pe
r
,
equa
t
i
o
n
(
1)
pr
ov
i
d
es
bo
t
h obj
ec
t
i
v
es
i
n a s
i
ng
l
e obj
ec
t
i
v
e f
unc
t
i
on
w
h
i
c
h i
s
t
he a
dv
an
t
a
ge
ov
er
pr
e
v
i
ous
m
et
hods
.
I
n equ
at
i
on
(1
),
w
i
i
s
t
he
w
e
i
g
ht
f
ac
t
or
and el
em
ent
of
c
ol
um
n v
ec
t
or
W
,
w
hi
c
h
r
epr
es
ent
s
t
he
i
n
v
er
s
e of
t
he c
os
t
of
P
MU
w
i
t
h r
es
pec
t
t
o num
ber
of
c
hannel
s
(
br
a
nc
hes
)
c
onnec
t
e
d t
o
bus
i
.
w
i
i
s
def
i
ne
d as
f
ol
l
o
w
s
:
1
(*
)
w
i
c
NC
c
p
i
c
l
=
+
(
4)
If
W
i
s
uni
t
y
v
ec
t
or
m
at
r
i
x
,
i
t
m
eans
w
e
i
g
ht
s
of
al
l
t
he
b
us
es
ar
e s
am
e,
and
t
h
i
s
v
a
l
ue
of
W
has
been us
e
d f
or
w
i
t
h
o
ut
c
ons
i
der
i
ng m
ax
i
m
u
m
obs
er
v
ab
i
l
i
t
y
pos
i
t
i
on.
2.
2
.
V
o
l
ta
g
e
S
ta
b
i
l
i
t
y
P
r
e
d
i
c
to
r
(V
S
P
)
T
he v
ol
t
age s
t
abi
l
i
t
y
pr
ob
l
em
i
s
m
o
s
t
l
y
r
eac
t
i
v
e po
w
er
r
el
at
ed pr
o
bl
em
.
T
he r
eac
t
i
v
e
po
w
er
def
i
c
i
t
i
n s
t
r
es
s
ed p
o
w
er
s
y
s
t
em
,
ei
t
her
b
y
i
nc
apab
i
l
i
t
y
of
v
ol
t
age s
o
ur
c
e t
o i
nj
ec
t
t
he
r
equi
r
e
d
r
eac
t
i
v
e
p
o
w
er
t
o
t
he
s
y
s
t
em
or
b
y
i
n
ab
i
l
i
t
y
of
t
r
ans
m
i
s
s
i
on
l
i
n
e
t
o
t
r
ans
f
er
t
he
dem
anded r
eac
t
i
v
e po
w
er
t
o t
he
w
e
ak
bus
es
,
c
aus
es
v
ol
t
ag
e i
ns
t
abi
l
i
t
y
i
n po
w
er
s
y
s
t
em
.
I
f
dur
i
n
g
an
y
on
e
of
t
hes
e
c
ondi
t
i
o
ns
,
an
y
c
h
ang
e
w
i
l
l
o
c
c
ur
s
i
n
t
he
s
y
s
t
em
,
l
i
k
e
i
nc
r
eas
e
i
n
l
o
ad
,
t
r
ans
m
i
s
s
i
on
l
i
ne
out
a
ge o
r
out
age
of
gener
at
i
o
n,
f
ur
t
her
degr
ades
t
h
e v
ol
t
age
s
t
abi
l
i
t
y
of
t
he
s
y
s
t
em
and
m
a
y
c
aus
e d
e
c
l
i
ne
i
n v
ol
t
ag
e at
l
o
ad b
u
s
.
T
he s
uc
c
es
s
i
v
e degr
a
da
t
i
on
i
n s
y
s
t
em
c
ondi
t
i
o
n m
a
y
c
a
us
e v
ol
t
a
ge c
ol
l
a
ps
e,
a
l
s
o.
S
o,
i
t
i
s
i
m
por
t
ant
t
o m
oni
t
or
v
o
l
t
a
ge
of
al
l
l
oad
bus
es
.
Let
t
he
l
oa
d a
t
a
n
y
bus
c
h
a
nged
an
d b
ec
aus
e
of
t
hi
s
t
he
v
o
l
t
ag
e
V
jk
at
an
y
l
o
ad
b
us
j
at
i
ns
t
an
t
k
i
s
c
hanged t
o
V
j
(k+
1)
at
(
k
+
1)
th
i
ns
t
ant
.
S
o,
t
he s
uc
c
es
s
i
v
e c
h
ang
e i
n v
o
l
t
a
ge
w
i
t
h
r
es
pec
t
t
o a
s
uc
c
es
s
i
v
e
t
i
m
e s
t
ep
due
t
o c
ha
nge
i
n
l
oa
d i
s
e
qua
l
t
o
:
(
)
(
)
1
1
()
(
1)
VV
jk
j
t
k
k
S
k
t
k
j
−
+
=
+
−
+
(
5)
w
her
e
S
j
(k+
1
)
i
s
s
uc
c
es
s
i
v
e
c
hange
i
n
v
o
l
t
a
ges
f
r
o
m
i
ns
t
ant
‘
k
’
t
o
‘
(
k
+
1)
’
at
bus
‘
j
’
w
i
t
h
r
es
pec
t
t
o
c
hange
i
n t
i
m
e of
t
he s
y
s
t
e
m
at
t
hes
e i
ns
t
ant
s
.
W
hen l
oad i
s
f
ur
t
her
c
hange
d at
n
ex
t
i
ns
t
ant
i
.
e.
at
i
ns
t
an
t
(
k
+
2)
t
h
en
S
j
(k+
1
)
i
s
c
hang
ed
t
o
S
j
(k+
2
)
due
t
o
c
hange
i
n
v
o
l
t
a
ge
V
j
at
bus
j
.
N
um
er
at
or
,
(t
(k+
1)
-
t
k
)
i
s
t
h
e
s
uc
c
es
s
i
v
e
t
i
m
e
s
t
ep.
T
he
c
han
ge
i
n
s
uc
c
es
s
i
v
e
c
ha
nge
i
n
v
ol
t
a
ge
a
ng
l
e
w
i
t
h
r
es
pec
t
t
o t
i
m
e due t
o c
ha
n
ge i
n
l
oa
d g
i
v
es
t
h
e
v
ol
t
ag
e s
t
ab
i
l
i
t
y
pr
e
di
c
t
or
(
V
S
P
)
f
or
as
s
es
s
m
ent
of
v
ol
t
age s
t
abi
l
i
t
y
of
t
he s
y
s
t
em
.
V
S
P
s
hou
l
d
be c
a
l
c
ul
at
ed
at
a
l
l
l
oa
d b
us
es
of
t
he net
w
or
k
and
i
s
gi
v
e
n b
y
,
(
)
(
)
1
t
a
n
(
1
1
)
/2
S
jk
V
S
P
jk
π
+
+
−
−
=
(
6)
I
n
e
quat
i
o
n
(
6)
,
num
er
at
or
i
s
i
n
r
ad
i
an
an
d
i
t
’
s
m
ax
i
m
um
v
al
ue
i
s
(
/2
π
−
)
.
T
her
ef
or
e,
t
o
f
i
nd
ou
t
t
he V
S
P
w
i
t
h
i
n
0
to
1
, (
/2
π
−
)
i
s
us
ed i
n de
nom
i
nat
or
.
N
o
w
t
he v
al
ue of
V
S
P
i
n
eq
uat
i
o
n (
6)
v
ar
y
bet
w
e
en 0 t
o 1.
T
he m
ax
i
m
u
m
v
al
ue of
V
S
P
at
i
ns
t
ant
(
k
+
1)
gi
v
es
t
he pr
e
di
c
t
i
on of
v
o
l
t
ag
e
s
t
abi
l
i
t
y
of
t
h
e ne
t
w
or
k
at
t
h
at
i
ns
t
ant
.
(
)
(
)
(
)
ma
x
11
V
S
P
V
S
P
k
jk
=
++
(
7)
T
hus
,
t
hi
s
v
al
u
e of
V
S
P
(
k
+
1)
i
nd
i
c
at
es
t
he pr
ox
i
m
i
t
y
of
v
ol
t
a
ge c
ol
l
aps
e
.
W
hen t
he v
a
l
ue
of
VS
P
(k+
1)
i
s
c
l
os
er
t
o
one
t
he
n t
he
s
y
s
t
em
i
s
m
or
e near
e
r
t
o t
he
i
ns
t
abi
l
i
t
y
po
i
nt
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN
:
25
02
-
4
752
I
J
E
EC
S
V
o
l.
8
,
N
o.
1
,
O
c
t
o
ber
20
17
:
1
–
8
4
3
.
S
o
l
u
ti
o
n
M
e
th
o
d
o
l
o
g
y
3.
1.
P
M
U
P
l
acem
en
t
I
n
or
d
er
t
o
f
or
m
t
he
c
ons
t
r
ai
nt
s
et
,
t
he
bi
n
ar
y
c
o
nnec
t
i
v
i
t
y
m
at
r
i
x
A
,
w
h
os
e
e
nt
r
i
es
ar
e
def
i
ned
i
n (
3)
,
w
i
l
l
be f
or
m
ed f
i
r
s
t
.
Mat
r
i
x
A
c
an be d
i
r
ec
t
l
y
obt
a
i
n
ed f
r
o
m
t
he bus
adm
i
t
t
anc
e
m
at
r
i
x
b
y
t
r
a
ns
f
or
m
i
ng i
t
s
e
nt
r
i
es
i
nt
o
bi
nar
y
f
or
m
.
F
i
gur
e 1.
5
-
b
us
s
y
s
t
em
C
ons
i
d
er
t
he 5
-
bus
s
y
s
t
em
and i
t
s
m
eas
ur
em
ent
c
onf
i
gur
at
i
o
n s
ho
w
n ab
ov
e.
B
u
i
l
d
i
n
g
t
he
A
m
at
r
i
x
f
or
t
he 5
-
bus
s
y
s
t
em
of
F
i
gur
e 1
y
i
e
l
ds
:
00000
00000
00000
000
11
000
11
A
=
(
8)
T
he c
ons
t
r
ai
nt
s
f
or
t
hi
s
c
as
e c
an b
e f
or
m
ed as
:
(
)
1
5
4
1
5
4
x
x
fX
x
x
+
≥
=
+
≥
(
9)
T
he us
e of
1 i
n t
he r
i
ght
hand s
i
de of
t
he i
neq
ua
l
i
t
y
ens
ur
es
t
hat
at
l
eas
t
o
ne
of
t
he
v
ar
i
ab
l
es
a
ppe
ar
i
n
g
i
n
t
hi
s
w
i
l
l
be
non
-
z
er
o.
T
he
c
ons
t
r
ai
nt
f
4
≥
1
i
m
pl
i
es
t
h
at
at
l
eas
t
o
ne
P
M
U
m
u
s
t
be pl
ac
ed
at
e
i
t
h
er
on
e of
bus
es
4
or
5 (
or
b
ot
h)
i
n or
der
t
o m
a
k
e bus
4 o
bs
e
r
v
ab
l
e.
3.
2.
V
o
l
t
ag
e S
t
ab
i
l
i
t
y
P
r
ed
i
ct
o
r
F
or
t
he 5
-
bus
s
y
s
t
em
s
how
n i
n
F
i
g
ur
e 1 t
h
er
e ar
e t
w
o
l
oad bus
es
(
bus
4 and bus
5)
and
t
hr
ee
gen
er
at
or
bus
es
(
bus
1,
s
l
ac
k
bus
,
and
bus
2
an
d bus
3)
.
S
o f
or
t
he
l
o
ad
bu
s
4 an
d b
us
5,
t
he v
a
l
u
es
of
V
S
P
hav
e b
een ex
am
i
ned us
i
n
g equ
at
i
on (
10)
and (
11)
as
s
ho
w
n bel
o
w
,
f
r
o
m
E
qu
at
i
on (
5)
,
(
)
(
)
4
41
41
1
VV
k
k
S
k
t
k
k
t
−
+
=
+
−
+
(
10)
A
nd
,
(
)
(
)
5
51
51
1
VV
k
k
S
k
k
t
t
k
−
+
=
+
−
+
(
11)
E
qu
at
i
on (
10)
an
d (
11)
r
epr
es
ent
t
he r
at
e of
c
hange of
v
ol
t
age a
t
bus
w
i
t
h r
es
pec
t
t
o t
i
m
e.
N
ow
f
r
o
m
equat
i
o
n (
6)
,
V
S
P
at
b
us
4 an
d b
us
5 ar
e
as
f
ol
l
o
w
s
:
(
)
1
t
a
n
(
)
41
4
(
/
2)
S
k
V
S
P
π
−
+
=
−
, a
nd
(
12)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
E
EC
S
IS
S
N
:
2
502
-
4
752
On
-
l
i
n
e A
s
s
es
s
me
nt
of
V
o
l
t
age
S
t
a
bi
l
i
t
y
us
i
ng
S
y
nc
hr
o
phas
or
...
(
S
at
y
e
ndr
a
P
r
at
a
p S
i
ng
h
)
5
(
)
1
t
a
n
(
)
51
5
(
/
2)
S
k
V
S
P
π
−
+
=
−
(
13)
F
ro
m
E
qu
at
i
on (
7)
,
m
os
t
c
r
i
t
i
c
al
bus
of
t
he
5
-
b
us
s
y
s
t
em
has
be
en c
a
l
c
u
l
at
e
d,
w
hi
c
h
i
s
as
f
o
llo
w
s
:
(
)
ma
x
,
5
4
V
S
P
V
S
P
V
S
P
=
(
14)
Max
i
m
u
m
v
al
ue
of
V
S
P
r
ep
r
es
ent
s
t
he m
os
t
c
r
i
t
i
c
al
b
u
s
i
n t
he
s
y
s
t
em
w
h
i
c
h m
ov
es
t
o
w
ar
ds
t
h
e
i
ns
t
ab
i
l
i
t
y
of
t
he
s
y
s
t
em
.
4
. V
o
lt
a
g
e
S
t
a
b
ilit
y
P
r
e
d
ic
t
o
r
T
he pr
opos
ed f
or
m
ul
at
i
on
has
been t
es
t
ed
on I
E
E
E
-
14 b
us
and I
E
E
E
-
30
b
us
t
es
t
s
y
s
t
em
s
.
H
ow
e
v
er
,
MI
LP
und
er
M
A
T
LA
B
h
as
be
e
n
us
ed
t
o
s
ol
v
e
t
h
e
pr
op
os
ed
pr
obl
em
ex
pr
es
s
ed b
y
e
quat
i
o
ns
(
1)
and (
2)
.
I
E
E
E
1
4
-
bus
and I
EEE
3
0
-
b
us
s
y
s
t
em
s
ar
e s
how
n i
n
F
i
gur
e
2
&
F
i
g
ur
e 3
.
T
he I
nf
or
m
at
i
on of
t
he
I
E
E
E
14
-
bus
and
I
E
E
E
30
-
b
us
t
es
t
s
y
s
t
em
s
and
l
oa
d bus
es
of
r
es
pec
t
i
v
e
s
y
s
t
em
s
ar
e gi
v
en i
n t
he T
abl
e 1.
T
he r
es
ul
t
s
f
or
P
MU
pl
ac
em
ent
hav
e
been
di
s
p
l
a
y
e
d i
n T
abl
es
2
.
I
n t
hi
s
p
aper
,
v
o
l
t
a
ge s
t
ab
i
l
i
t
y
pr
e
di
c
t
or
m
et
hod d
i
s
c
us
s
ed i
n s
ec
t
i
on I
I
I
i
s
us
ed
t
o
c
hec
k
t
he
v
o
l
t
ag
e s
t
ab
i
l
i
t
y
of
di
f
f
er
ent
l
oa
d bus
es
.
T
he
ef
f
ec
t
of
l
oad
i
ng
at
an
y
l
oad
bus
i
s
as
s
es
s
ed at
i
nc
r
eas
i
ng t
he
l
oa
d
o
n
t
hat
b
us
i
t
s
e
l
f
,
t
o
i
t
s
ne
i
g
hbor
i
ng
bus
c
onn
ec
t
t
o
t
he
b
us
und
er
c
ons
i
der
a
t
i
o
n
an
d
t
o
t
he
ot
her
w
eak
l
oad
bus
es
us
i
n
g
V
S
P
k
pr
ed
i
c
t
or
.
T
he
r
es
ul
t
s
ar
e
s
how
n
i
n
F
i
gur
e
4 t
o
F
i
g
ur
e 1
0.
F
ig
u
r
e 2.
I
E
EE 1
4
-
bus
t
es
t
s
y
s
t
em
F
i
gur
e 3.
I
E
EE 3
0
-
bus
t
es
t
s
y
s
t
em
T
abl
e 1.
S
y
s
t
em
i
nf
or
m
at
i
on of
I
E
E
E
bus
t
es
t
s
y
s
t
em
s
T
e
s
t
S
y
s
t
e
m
N
um
ber
o
f
br
anc
hes
N
um
ber
o
f
l
oad bus
e
s
Load bus
es
I
EE
E 1
4
-
bus
20
9
4,
5,
7,
9
,
10,
11,
12
,
13,
14
I
EE
E 3
0
-
bus
41
24
3,
4,
6,
7
,
9,
10,
12
,
14,
15,
16
,
17,
18,
19
,
20,
21,
22,
23,
24,
25,
26
,
27,
28,
29
,
30
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN
:
25
02
-
4
752
I
J
E
EC
S
V
o
l.
8
,
N
o.
1
,
O
c
t
o
ber
20
17
:
1
–
8
6
T
abl
e 2.
S
i
m
ul
at
i
on
r
es
ul
t
s
f
or
t
he t
es
t
s
y
s
t
em
s
S
y
s
t
em
N
um
ber
o
f
P
M
U
s
Loc
at
i
on
o
f
P
M
U
s
I
EE
E 1
4
-
bus
3
4,
10,
13
I
EE
E 3
0
-
bus
8
4,
6,
10,
12,
18
,
24,
25,
27
F
i
gur
e
4 s
ho
w
s
t
he
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t
ag
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at
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on
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he c
r
i
t
i
c
al
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of
I
E
E
E
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-
b
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o
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nc
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ac
t
i
v
e ad
r
eac
t
i
v
e
po
w
er
)
at
b
us
11
.
A
s
s
ho
w
n i
n F
i
gur
e
5,
v
a
l
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of
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S
P
i
s
v
ar
y
i
n
g f
r
om
a v
a
l
ue
ne
ar
er
t
o
z
er
o t
o
o
ne.
T
he v
ol
t
age
of
bus
11
i
s
a
ppr
oac
h
i
n
g t
o t
h
e c
o
l
l
a
ps
e
poi
n
t
,
due
t
o
c
ont
i
n
uous
i
n
c
r
eas
e
i
n
l
oa
d
at
bus
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t
he
v
a
l
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of
pr
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t
or
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S
P
c
hanges
f
r
o
m
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o
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96.
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ar
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at
i
on i
n
pr
ed
i
c
t
or
V
S
P
i
s
s
l
o
w
i
n
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gi
n
ni
n
g
as
c
om
par
i
s
on w
i
t
h t
he v
ar
i
at
i
on i
n
pr
edi
c
t
or
n
ear
er
t
o t
he c
o
l
l
aps
e po
i
nt
.
F
i
gur
e 6 s
ho
w
s
t
he c
l
os
er
v
i
e
w
o
f
V
S
P
of
F
i
gur
e 5.
A
ga
i
n
F
i
gur
e
7
s
ho
w
s
t
he
v
ol
t
ag
e v
ar
i
a
t
i
o
n
at
s
om
e c
r
i
t
i
c
a
l
b
us
es
of
I
E
E
E
14
-
bus
s
y
s
t
em
due t
o
i
nc
r
eas
e i
n l
o
ads
at
a
l
l
t
he l
oad b
us
es
.
I
t
i
s
c
l
ear
f
r
om
F
i
gur
e 7 t
hat
t
he r
at
e
of
c
hange
of
v
ol
t
age
i
s
m
ax
i
m
u
m
at
bus
5.
T
h
er
ef
or
e v
al
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V
S
P
at
b
us
5 i
s
appr
oac
hes
near
er
t
o 1 w
h
i
c
h i
s
s
ho
w
n
i
n
F
i
gur
e
8
an
d
F
i
g
ur
e
9
s
ho
w
s
t
he
c
l
os
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ar
i
at
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on
of
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S
P
t
o
bet
t
er
un
d
er
s
t
andi
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he
v
ar
i
at
i
on
at
s
om
e of
t
he c
r
i
t
i
c
al
bus
es
.
F
i
gur
e 4.
C
r
i
t
i
c
a
l
b
us
v
o
l
t
a
g
es
due
t
o
i
nc
r
eas
e
l
oa
d at
bus
1
1
F
i
gur
e 5
.
V
S
P
p
l
ot
f
or
l
oad
i
nc
r
eas
e at
bus
1
1
F
i
gur
e 6
.
C
l
os
er
v
i
e
w
of
V
S
P
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F
i
g
ur
e 5
F
i
gur
e 7
.
C
r
i
t
i
c
a
l
b
us
v
o
l
t
a
g
es
due
t
o
i
nc
r
eas
e
l
oa
ds
at
a
l
l
t
he
l
o
ad
bus
es
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
E
EC
S
IS
S
N
:
2
502
-
4
752
On
-
l
i
n
e A
s
s
es
s
me
nt
of
V
o
l
t
age
S
t
a
bi
l
i
t
y
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i
ng
S
y
nc
hr
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at
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ndr
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at
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p S
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7
F
i
gur
e 8
.
V
S
P
p
l
ot
f
or
l
oad
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nc
r
eas
e at
al
l
t
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e
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F
i
gur
e 9.
C
l
os
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v
i
e
w
of
V
S
P
of
F
i
g
ur
e 8
T
he v
ar
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at
i
o
n of
v
o
l
t
a
ge of
bus
19
w
i
t
h
t
he
i
nc
r
eas
e
i
n
l
oa
di
n
g of
bus
19
,
i
n
I
E
E
E
-
30 bus
s
y
s
t
em
,
i
s
s
how
n
i
n
F
i
g
ur
e
10.
Mor
eo
v
er
,
t
h
e
v
ar
i
at
i
o
n
i
n
v
a
l
ues
of
V
S
P
of
bus
e
s
-
14,
-
18,
-
19
and
-
20
ar
e
a
l
s
o
s
ho
w
n
i
n
t
he
F
i
g
ur
e
10.
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he
c
hang
e
i
n
t
he
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al
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of
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S
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i
s
f
as
t
es
t
f
or
bus
19
and
r
es
p
ec
t
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v
e
l
y
l
o
w
er
f
or
ot
her
bus
es
as
b
us
20
,
bu
s
18
a
nd
b
us
14
as
t
h
e
l
oa
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n
g
of
bus
1
9
i
nc
r
eas
es
.
I
t
s
ho
w
s
t
h
at
t
h
e
nei
ghb
or
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n
g
bus
es
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l
i
k
e
bus
20
or
bus
18,
of
bus
19
i
s
m
uc
h
m
or
e
af
f
ec
t
ed t
han t
he f
ar
bus
e
s
as
bus
14
b
y
i
nc
r
eas
e i
n l
o
ad
i
ng
of
bus
19
.
F
i
gur
e
11 s
h
o
w
s
t
h
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l
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on
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n
v
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t
age
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S
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as
s
ho
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n
i
n F
i
gur
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F
i
gur
e 10
.
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ha
nge
i
n
v
ol
t
a
ge/
V
S
P
v
s
t
i
m
e at
B
us
19
,
-
14,
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18 a
nd
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2
0 f
or
I
E
E
E
30
-
bus
t
es
t
s
y
s
t
em
F
i
gur
e 11.
C
l
os
er
v
i
e
w
of
F
i
gur
e 1
0
5
.
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o
n
c
l
u
s
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o
n
T
hi
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paper
pr
opos
e
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i
m
pl
e a
l
gor
i
t
hm
of
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i
m
al
pl
a
c
e
m
ent
of
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MU
s
i
n po
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er
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y
s
t
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t
o
pr
o
v
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de
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h
e
m
ax
i
m
u
m
obs
er
v
ab
i
l
i
t
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of
l
oa
d
bus
es
of
t
he
net
w
or
k
f
or
on
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lin
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v
o
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g
e
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t
a
b
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lit
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es
s
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ent
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he
O
P
P
pr
o
bl
em
i
s
f
or
m
ul
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us
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ng
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o
pol
og
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e
d
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l
g
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t
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d
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ol
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ed
us
i
ng
m
i
x
ed i
nt
eger
l
i
ne
ar
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ogr
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m
m
i
ng.
T
he v
ol
t
a
ge s
t
ab
i
l
i
t
y
as
s
es
s
m
ent
pr
obl
em
i
s
f
or
m
ul
at
ed as
V
S
P
a
nd s
o
l
v
ed
us
i
ng
MA
T
LA
B
pr
o
gr
am
m
i
ng.
S
i
m
ul
at
i
on r
es
u
l
t
s
on
I
E
E
E
1
4
-
bus
and I
E
E
E
30
-
bus
t
es
t
s
y
s
t
em
s
i
ndi
c
at
e t
h
at
t
he pr
opos
e
d p
l
ac
em
ent
m
et
hod
w
i
t
h
v
ol
t
ag
e s
t
ab
i
l
i
t
y
as
s
es
s
m
ent
i
s
s
at
i
s
f
ac
t
or
i
l
y
pr
o
v
i
des
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s
er
v
ab
l
e s
y
s
t
em
m
eas
ur
e
m
ent
s
w
i
t
h m
i
ni
m
u
m
nu
m
ber
of
P
MU
s
and,
al
s
o,
t
he
i
nd
ex
t
o
det
er
m
i
ne t
he
s
t
ab
i
l
i
t
y
of
po
w
er
net
w
or
k
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN
:
25
02
-
4
752
I
J
E
EC
S
V
o
l.
8
,
N
o.
1
,
O
c
t
o
ber
20
17
:
1
–
8
8
R
ef
er
en
ces
[1
]
I
E
E
E
/
C
I
G
R
E
,
J
oi
nt
T
as
k
F
or
c
e o
n S
t
abi
l
i
t
y
T
er
m
s
a
nd D
ef
i
ni
t
i
on
s
.
D
ef
i
n
i
t
i
on
and
c
l
as
s
i
f
i
c
at
i
on
of
pow
er
s
y
s
t
em
s
t
a
bi
l
i
t
y
.
I
EEE T
ra
n
s
.
P
o
w
e
r Sy
s
t
e
m
.
2004
;
19
(
3)
:
13
87
–
1
401.
[2
]
M
or
i
s
on
G
K
,
G
ao
B
,
K
undur
P
.
V
ol
t
age
s
t
ab
i
l
i
t
y
ana
l
y
s
i
s
u
s
i
ng
s
t
at
i
c
an
d
dy
nam
i
c
appr
o
ac
he
s
,
I
EEE
T
r
ans
.
P
ow
er
S
y
s
t
em
.
199
3;
8
(
3)
:
11
59
–
1
171.
[3
]
K
undur
P
.
P
ow
er
S
y
s
t
em
S
t
ab
i
l
i
t
y
and
C
ont
r
ol
,
E
P
R
I
P
ow
er
S
y
s
t
em
E
ng
i
neer
i
ng
S
er
i
es
,
Mc
G
r
a
w
-
H
ill:
1994.
[4
]
C
hak
r
ab
or
t
y
NC,
B
aner
j
i
A,
B
i
s
wa
s
SK.
S
ur
v
e
y
on
m
aj
or
bl
ac
k
out
s
an
al
y
s
i
s
and
pr
ev
e
nt
i
o
n
m
et
hodol
ogi
es
.
M
i
c
ha
el
F
ar
a
d
ay
I
E
T
I
nt
er
nat
i
ona
l
S
u
m
m
i
t
.
K
ol
k
a
t
a.
2
015;
297
-
302
.
[5
]
P
hadk
e
A
G
,
T
hor
p
J
S
,
K
ar
i
m
i
K
J
.
S
t
at
e
E
s
t
i
m
at
i
on
w
i
t
h
P
ha
s
or
M
eas
ur
em
ent
s
.
I
EEE
T
ra
n
s
a
c
t
i
o
n
s
o
n
Po
w
e
r Sy
s
t
e
m
s
.
1986
;
1(
1)
:
2
33
-
241
.
[6
]
E
P
R
I
F
i
nal
R
e
p.
A
s
s
es
s
m
ent
of
A
ppl
i
c
at
i
on
s
an
d B
enef
i
t
s
of
P
has
or
M
eas
ur
em
ent
T
e
c
hno
l
og
y
i
n
Po
w
e
r Sy
s
t
e
m
s
.
G
E
P
ow
er
S
y
s
t
.
E
n
g.
,
1
997.
[7
]
P
hadk
e
A
G
.
S
y
nc
hr
on
i
z
ed
ph
as
or
m
ea
s
ur
e
m
ent
s
i
n
pow
er
s
y
s
t
em
s
,
I
E
E
E
C
om
put
er
A
pp
l
i
c
a
t
i
o
ns
i
n
P
ow
er
.
19
93;
6(
2)
:
1
0
-
15.
[8
]
X
u
B,
Ab
u
r A.
O
bs
er
v
ab
i
l
i
t
y
ana
l
y
s
i
s
an
d m
eas
ur
em
e
nt
pl
a
c
em
ent
f
or
s
y
s
t
em
s
wi
t
h
P
M
Us
.
P
r
oc
ee
di
n
gs
o
f
20
04 I
E
E
E
P
E
S
C
onf
er
e
nc
e
and
E
x
pos
i
t
i
on
,
2004;
2:
9
43
-
9
46.
[9
]
S
i
ngh S
P
,
S
i
ngh S
P
.
O
pt
i
m
al
P
M
U
P
l
ac
em
ent
i
n P
ow
er
S
y
s
t
em
C
on
s
i
d
er
i
n
g t
he M
eas
ur
em
e
nt
R
edund
anc
y
,
I
nt
.
J
.
of
A
dv
anc
es
i
n E
l
e
c
t
r
o
ni
c
and
E
l
e
c
t
r
i
c
E
ngi
ne
er
i
n
g
.
2
014;
4(
6)
:
593
-
59
8.
[
10]
D
eebi
g
a K
,
H
us
s
ai
n A
R
.
O
pt
i
m
al
P
l
ac
e
m
en
t
of
P
ha
s
or
M
eas
ur
e
m
en
t
U
ni
t
f
or
B
e
t
t
er
P
o
w
er
S
y
s
t
em
O
bs
er
v
abi
l
i
t
y
,
I
n
done
s
i
an J
our
nal
o
f
E
l
e
c
t
r
i
c
al
E
ngi
neer
i
ng
.
2
015;
1
4(
2)
:
199
-
204.
[
11]
E
ns
ha
ee
A
,
H
oos
h
m
an
d
R
A
,
F
es
har
ak
i
F
H
.
A
new
m
et
hod
f
or
o
pt
i
m
al
pl
ac
e
m
e
n
t
of
phas
or
m
ea
s
ur
e
m
ent
uni
t
s
t
o
m
ai
nt
ai
n
f
ul
l
n
et
w
or
k
ob
s
er
v
abi
l
i
t
y
u
nder
v
ar
i
ou
s
c
ont
i
nge
nc
i
es
,
E
l
ec
t
.
P
ow
er
S
y
s
t. R
e
s
.
201
2;
89
:
1
–
10.
[
12]
M
oham
m
adi
M
B
,
H
oos
hm
a
nd
R
A
,
F
es
har
a
k
i
F
H
.
A
N
ew
A
ppr
oa
c
h f
or
O
pt
i
m
a
l
P
l
ac
em
e
nt
of
P
M
U
s
and T
hei
r
R
eq
ui
r
ed
C
o
m
m
u
ni
c
at
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o
n I
nf
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a
s
t
r
u
c
t
ur
e
i
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der
t
o M
i
ni
m
i
z
e t
he C
os
t
of
t
he
W
A
M
S
.
I
EEE
T
r
ans
.
on S
m
ar
t
G
r
i
d
.
2
016;
7(
1)
:
84
-
93.
[
13]
S
i
ngh S
P
,
S
i
ngh S
P
.
O
p
t
i
m
al
C
os
t
W
i
d
e A
r
ea M
eas
ur
em
e
nt
S
y
s
t
em
i
nc
or
por
at
i
n
g C
om
m
uni
c
at
i
o
n
In
fr
a
s
tr
u
c
tu
r
e
.
I
E
T
G
ener
at
i
on,
T
r
ans
m
i
s
s
i
on &
D
i
s
t
r
i
but
i
on
.
2
017;
1
1(
11)
:
28
14
–
282
1.
[
14]
Si
n
g
h
SP
,
Si
n
g
h
SP.
S
y
n
c
hr
o
ni
z
ed P
M
U
P
l
ac
em
ent
I
nc
or
p
or
at
i
n
g C
om
m
uni
c
at
i
on I
nf
r
a
s
t
r
uc
t
ur
e
.
7t
h
I
ndi
a
I
nt
er
nat
i
ona
l
C
onf
er
en
c
e
on P
ow
er
E
l
e
c
t
r
on
i
c
s
(
I
I
C
P
E
-
2016)
.
P
at
i
al
a.
201
6.
[
15]
H
uang
L,
S
un
Y
,
X
u
J
,
G
ao
W
,
Z
h
ang
J
,
W
u
Z
.
O
pt
i
m
a
l
P
M
U
P
l
ac
em
en
t
C
ons
i
der
i
ng
C
ont
r
ol
l
e
d
I
s
l
a
ndi
n
g of
P
ow
er
S
y
s
t
em
.
I
E
E
E
T
r
ans
.
on P
ow
er
S
y
s
t
.
201
4;
29(
2
)
:
74
2
-
75
5.
[
16]
Si
n
g
h
SP,
Si
n
g
h
SP.
I
s
l
and
i
ng
bas
ed o
pt
i
m
al
p
l
a
c
em
ent
of
p
has
or
m
ea
s
ur
em
en
t
uni
t
u
s
i
ng
M
I
LP
. 7
th
P
ow
er
I
ndi
a I
nt
er
n
at
i
o
nal
C
onf
er
enc
e (
P
I
I
C
O
N
)
.
B
i
k
aner
.
20
1
6.
[
17]
S
i
ngh
S
P
,
T
hak
ur
A
K
,
S
i
n
gh
S
P
.
P
M
U
P
l
ac
em
e
nt
f
or
M
ax
i
m
um
O
bs
e
r
v
abi
l
i
t
y
of
P
ow
er
S
y
s
t
em
und
er
C
ont
i
ng
en
c
i
e
s
.
E
n
er
gy
P
r
oc
ed
i
a
.
2
017;
117(
C
)
:
893
-
900.
[
18]
Si
n
g
h
SP,
Si
n
g
h
SP.
A
M
u
l
t
i
-
o
bj
e
c
t
i
v
e
P
M
U
P
l
ac
em
e
n
t
M
et
hod
i
n
P
ow
er
S
y
s
t
em
v
i
a
B
i
nar
y
G
r
a
vi
t
a
t
i
o
na
l
S
ea
r
c
h A
l
g
or
i
t
hm
.
E
l
ec
t
r
i
c
P
ow
er
C
om
po
nent
s
and S
y
s
t
em
s
.
20
17.
[
19]
R
e
d
d
y
C
VVSB,
Sri
v
a
s
t
a
v
a
SC
,
C
h
a
k
ra
b
a
rt
i
S.
A
n i
m
pr
ov
ed
s
t
at
i
c
v
ol
t
a
ge s
t
abi
l
i
t
y
i
nde
x
us
i
ng
s
y
n
c
hr
o
pha
s
or
m
eas
ur
em
ent
s
f
or
ear
l
y
det
e
c
t
i
on of
i
m
pe
ndi
ng
v
ol
t
age i
n
s
t
a
bi
l
i
t
y
.
N
at
i
onal
P
ow
er
S
y
s
t
em
C
on
f
er
e
nc
e (
N
P
S
C
)
.
V
ar
ana
s
i
.
2012
.
[
20]
P
av
i
t
hr
en P
,
R
a
m
a
n R
R
,
N
ai
r
P
,
N
i
t
h
i
y
anant
han K
.
V
ol
t
ag
e S
t
ab
i
l
i
t
y
A
nal
y
s
i
s
a
nd S
t
ab
i
l
i
t
y
I
m
pr
ov
e
m
ent
of
P
ow
er
S
y
s
t
em
.
I
nt
.
J
our
nal
of
E
l
e
c
t
r
i
c
al
an
d
C
om
put
er
E
ng
i
ne
er
i
n
g
.
20
1
5;
5(
2)
:
1
89
-
197.
[
21]
G
l
a
v
i
c
M
,
C
u
ts
e
m
T
V
.
A
s
hor
t
s
ur
v
e
y
of
m
et
hod
s
f
or
v
o
l
t
ag
e
i
ns
t
a
bi
l
i
t
y
d
et
e
c
t
i
o
n
.
2
0
1
1
I
EEE
Po
w
e
r
and E
n
er
gy
S
oc
i
et
y
G
ener
al
M
eet
i
n
g.
S
a
n D
i
eg
o.
2
011;
1
-
8.
[
22]
G
oh H
.
et
al
.
P
ow
er
s
t
ab
i
l
i
t
y
m
oni
t
or
i
ng ba
s
ed o
n v
ol
t
a
g
e i
ns
t
a
bi
l
i
t
y
pr
edi
c
t
i
on ap
pr
oa
c
h t
hr
ough w
i
de
ar
ea s
y
s
t
em
.
A
m
er
i
c
an
J
our
na
l
of
A
p
pl
i
ed S
c
i
en
c
e
s
.
2
014;
1
1(
5)
:
7
17
-
73
1.
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