Indonesi
an
Journa
l
of El
ect
ri
cal Engineer
ing
an
d
Comp
ut
er
Scie
nce
Vo
l.
1
3
,
No.
1
,
Jan
uar
y
201
9
,
pp.
331
~
338
IS
S
N: 25
02
-
4752, DO
I: 10
.11
591/ijeecs
.v1
3
.i
1
.pp
331
-
338
331
Journ
al h
om
e
page
:
http:
//
ia
es
core.c
om/j
ourn
als/i
ndex.
ph
p/ij
eecs
A
statisti
ca
l
jacobi
an appli
cation fo
r power
system
op
timizati
on
of voltag
e s
t
abilit
y
Raja
Ma
s
ood
Larik
1
, Mo
hd.Waz
ir
M
us
taf
a
2
, M
anoj
Ku
mar P
anjw
ani
3
1
Depa
rtment of
El
e
ct
ri
ca
l
Eng
in
ee
ring
NED Uni
ver
sit
y
of
Eng
in
ee
ring
and
Tech
nolog
y
Sindh,
Pakista
n
1
,2
School
of El
ectrical
Engi
n
ee
rin
g,
Univer
si
ti T
ek
nologi
Ma
lay
sia
Skudai,
Johor
Bharu
Mal
a
y
s
ia 8
1310
3
Depa
rtment of
Ene
rg
y
S
y
stems
Engi
ne
eri
ng,
Su
kkur
IBA Univers
ity
Pakist
an
Art
ic
le
In
f
o
ABSTR
A
CT
Art
ic
le
history:
Re
cei
ved
J
un
4
, 2018
Re
vised
A
ug
22
, 2
018
Accepte
d
Se
p
8
, 2
018
Despite
a
tre
m
e
ndous
deve
lopment
in
opti
m
al
power
flow
(OPF
),
owing
to
the
obvious
co
m
ple
xity
,
non
-
l
ine
ar
ity
and
u
nwiel
d
y
size
o
f
the
la
rg
e
int
er
conne
c
te
d
p
ower
s
y
st
ems
,
seve
ra
l
probl
ems
remai
n
un
answere
d
in
th
e
exi
sting
m
et
hod
s
of
OP
F.
Seiz
i
ng
spec
if
ic
topics
for
m
axi
m
izi
ng
voltage
stabi
lit
y
m
arg
i
n
and
it
s
imp
le
m
ent
a
ti
on,
a
det
a
il
ed
li
t
erature
surv
e
y
discussing
the
exi
sting
m
et
ho
ds
of
soluti
on
and
the
ir
dr
a
wbac
ks
are
pre
sente
d
in
this
rese
arc
h
.
The
phenomenon
of
volt
ag
e
collap
se
in
power
s
y
stems
,
m
et
ho
ds
to
inve
stig
ate
voltage
col
l
ap
se,
and
m
et
hod
s
rel
ated
to
volt
ag
e
stabi
l
ity
are
bri
efly
surve
y
ed
.
Finall
y
,
the
stud
y
pre
sen
ts
a
stat
ist
ic
a
l
m
et
hod
for
ana
l
y
z
ing
a
power
sy
stem
through
eigenva
lu
e
ana
l
y
si
s
in
rel
at
io
n
to
the
singula
r
val
ues
of
the
loa
d
flow
Jac
obia
n.
Future
stud
y
m
ay
foc
us
on
cha
nges
in the
or
ie
s in
conj
un
ctio
n
with la
rg
e
pow
er
s
y
s
te
m
s.
Ke
yw
or
d
s
:
Op
ti
m
al
p
ow
e
r
f
lo
w
Op
ti
m
iz
ation
Power sy
ste
m
Vo
lt
age
stabil
it
y
Copyright
©
201
9
Instit
ut
e
o
f Ad
vanc
ed
Engi
n
ee
r
ing
and
S
cienc
e
.
All right
s
reserve
d
.
Corres
pond
in
g
Aut
h
or
:
Ra
j
a Ma
s
ood Larik
,
Dep
a
rtm
ent o
f El
ect
rical
En
gi
neer
i
ng N
E
D
,
Un
i
ver
sit
y o
f En
gin
eeri
ng a
nd Tec
hnol
og
y
Sindh
,
Pa
kistan
.
Em
a
il
:
r
m
la
rik
@g
m
ai
l.co
m
1.
INTROD
U
CTION
Fo
r
t
he
pa
st
de
cades,
powe
r
s
yst
e
m
op
tim
izati
on
te
ch
niqu
es
hav
e
bee
n
s
ubj
ect
to
m
any
stud
ie
s
f
or
plan
ning
a
nd
s
trat
egy
de
velo
pm
ent
[
1
,
2
]
.
Op
ti
m
al
Po
wer
Flo
w
(
OPF)
re
fer
s
t
o
a
sta
nda
rd
te
rm
that
def
ines
a
com
pr
ehe
ns
ive
set
of
ob
sta
cl
e
s
in
wh
ic
h
rese
arch
e
r
purs
ues
achievin
g
one
or
m
or
e
obj
ect
ive
functi
ons
wh
il
e
sat
isfyi
ng
c
onstrai
nts
dicta
te
d
by
op
e
rati
o
nal
an
d
ph
ysi
cal
restrict
ion
s
of
t
he
el
ect
ric
netw
ork.
Th
e
O
P
F
pro
blem
s
req
uire
the
determ
inati
on
of
the
op
ti
m
al
se
tt
i
ng
s
of
co
ntr
ol
var
ia
bles
s
ub
j
ect
to
the
operati
ng
const
raints
s
uc
h
that
the
ope
r
at
ing
ob
j
ect
ive
s
are
opti
m
iz
e
d.
As
a
res
ult
of
the
c
onti
nu
ou
s
resea
rch
e
ffor
ts
ov
e
r
the
la
st
de
cades,
t
he
O
PF
al
gorithm
s
hav
e
sig
nifica
ntly
m
at
ur
ed
a
longside
de
velop
m
ents
in
the
oth
e
r
areas
of
te
ch
nolog
y.
Mo
de
rn
OP
F
al
gorithm
s
cov
e
r
bot
h
re
al
and
reacti
ve
power
dispa
tc
h
an
d
ca
n
so
l
ve
ver
y
la
rg
e
a
nd
com
plex
form
ul
at
i
on
s
in
a
relat
iv
el
y
sh
ort
tim
e.
The
volt
age
st
abili
ty
ph
en
ome
na
are
disc
us
s
ed
in
sm
art g
rid
f
ra
m
ewo
r
k
[
3
,
4
]
and
are
doc
ume
nted
a
s a si
gn
i
ficant
prob
le
m
for
a
sh
el
te
re
d sy
stem
o
per
at
i
on.
As
the
ne
w
c
oncer
ns
li
ke
ass
essm
ent
and
im
pr
ov
em
ent
of
volt
age
sta
bil
it
y
m
arg
in
(
VSM
)
bo
t
her
t
he
powe
r
en
gine
e
rs,
t
he
O
PF
al
gorithm
s
are constantl
y re
view
ed
a
nd n
e
wer
m
et
ho
ds
are e
volve
d
in
a c
on
t
inuous
effor
t
to
ad
dr
e
ss
these
new
c
on
ce
r
ns
a
nd
be
tt
er
the
existi
ng
m
et
hods
.
I
n
this
orbit
of
OP
F
pro
blem
s,
real
powe
r
sc
hedul
ing
is
a
n
inte
gr
al
par
t.
The
act
ivit
ie
s
of
op
ti
m
al
real
po
we
r
sc
he
d
uli
ng
of
a
powe
r
syst
e
m
assum
e
sign
ific
ance
in
view
of
their
ove
rb
e
arin
g
fina
ncial
and
op
e
rati
ona
l
i
m
pl
ic
at
ion
s.
W
it
h
increasi
ng
fu
el
costs
an
d
decli
ning
capit
al
in
vestm
ents,
the
econom
ic
s
of
r
eal
power
sc
he
du
li
ng
hav
e
a
t
rem
end
ous
ef
f
ect
on
the
prof
it
able
and
r
el
ia
ble
operati
on
of
a
powe
r
syst
e
m
.
Pr
oper
real
powe
r
sche
duli
ng
has
a
wide
-
ra
ng
i
ng
eff
ect
on
the
operati
on
an
d
con
t
ro
l
of
powe
r
syst
e
m
.
Vo
lt
age
secu
rity
and
vo
lt
age
sta
bili
ty
of
a
po
we
r
syst
e
m
are prof
oundly
aff
ect
ed by re
al
p
ower sc
he
duli
ng. S
yst
e
m
gen
e
rati
on co
st
an
d VSM ar
e a
m
on
g
m
any that can
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2502
-
4752
Ind
on
esi
a
n
J
E
le
c Eng &
Co
m
p
Sci,
Vo
l.
1
3
, N
o.
1
,
Ja
nu
a
ry
201
9
:
331
–
338
332
be
im
pr
oved
by
eff
ect
ive
real
powe
r
sc
he
du
l
ing
.
Se
ns
or
te
c
hnologies
for
pro
per
est
im
a
ti
on
of
volt
age
c
urre
nt
and po
wer p
res
ented, l
ow
vo
lt
age
ride
t
hroug
h
L
VRT
was p
rop
os
ed
.
The
re
view
of
the
e
xisti
ng
m
et
ho
ds
f
or
t
he
powe
r
syst
e
m
op
ti
m
iz
a
tio
n
s
hows
that
sig
nificant
pro
gr
ess
has
be
en
m
ade
in
the
de
velo
pm
ent
of
reli
able
and
e
ff
ic
ie
nt
m
et
ho
ds.
The
s
e
m
et
ho
ds
a
dd
ress
the
pro
blem
and
use
a
m
ulti
tud
e
of
f
or
m
ulati
ons
an
d
s
olu
ti
on
te
chn
iq
ues
.
Howev
e
r,
t
her
e
ar
e
sti
ll
so
m
e
con
cer
ns
hav
e
not
bee
n
ade
qu
at
el
y
ta
ckled
su
c
h
a
s
consi
der
i
ng
ge
ner
at
io
n
c
os
t
m
ini
m
iz
at
ion
and
tra
ns
m
issio
n
loss
m
ini
m
iz
at
ion
al
ong
with
volt
age
sta
bili
ty
marg
i
n.
Th
us,
th
is
stud
y
aim
s
t
o
ad
dr
es
s
the
issue
in
acc
ord
ance
with
gen
e
rati
on c
os
t a
nd trans
m
issi
on
los
s
m
ini
m
iz
at
ion
with
resp
ect
t
o vo
lt
age
stabil
it
y
m
arg
in.
This
pap
e
r
org
anizes
as
fo
ll
ows.
T
he
li
te
rat
ur
e
sect
ion
i
nvol
ves
powe
r
syst
e
m
ov
er
view,
optim
al
real
po
wer
sch
edu
li
ng,
a
nd
S
cheduli
ng
for
vo
lt
age
sta
bili
ty
.
The
ne
xt
se
ct
ion
de
vo
te
s
to
OP
F
opti
m
izati
on
te
chn
iq
ues
a
nd
powe
r
syst
em
cl
assifi
cat
ion
s
.
Lat
er
in
t
his
pap
e
r,
t
he
po
wer
flo
w
Jac
obia
n
discusse
d
al
ong
with
researc
h
c
on
cl
us
io
n
a
nd
po
te
ntial
r
esear
ch opp
or
t
un
it
y.
2.
LIT
ERATUR
E REVIE
W
Power
Syst
em
is
a
la
rg
e
an
d
com
plex
ph
ysi
cal
syst
e
m
.
It
has
pr
ob
a
bly
the
la
rg
est
nu
m
ber
of
interact
ing
el
em
ents
of
va
ryi
ng
de
grees
of
com
plexiti
es.
I
t
has
e
norm
ou
s
capit
al
inve
stm
ent
m
aking
it
one
of
the
costli
est
physi
cal
syst
e
m
s.
It
s
pans
gi
ga
ntic
ge
ogra
ph
ic
al
exp
a
ns
e
m
aking
it
a
ph
ysi
cal
ly
la
rg
e
syst
e
m
.
The
powe
r
sys
tem
is
a
dyna
m
ic
syst
e
m
.
It
enco
m
passes
ge
ner
at
io
n,
tra
nsm
issi
on
,
a
nd
distrib
ution
of
el
ect
ric
powe
r
to
m
il
lio
ns
of
c
us
tom
ers
delive
rin
g
bill
ion
s
of
jo
ules
of
e
nergy.
I
ts
i
m
po
rtance
is
par
am
o
un
t
to
the
so
ci
et
y
in
al
l
ways.
Its
dy
na
m
ic
natur
e
an
d
sta
bili
ty
are
causes
f
or
great
con
cer
n.
K
undur
in
pro
pose
d
a
def
i
niti
on
a
nd
cl
assifi
cat
ion
f
or powe
r
syst
e
m
stability
as sh
ow
n
in
Fig
ure
1.
Figure
1
.
P
ow
e
r
syst
em
sta
bility
Power
syst
em
sta
bili
ty
is
de
fine
d
as
“P
ower
syst
em
st
abili
ty
is
the
abili
ty
of
a
n
e
le
ct
ric
powe
r
syst
e
m
,
fo
r
a
gi
ven
i
niti
al
op
e
rati
ng
co
ndit
ion
,
t
o
reg
ai
n
a
s
ta
te
of
operati
ng
e
qu
il
ib
rium
after
bein
g
s
ub
j
ect
ed
to
a
ph
ysi
cal
di
sturb
a
nce,
wit
h
m
os
t
syst
e
m
va
riables
bounde
d
s
o
t
hat
pract
ic
al
ly
the
entire
syst
em
rem
ai
ns
intac
t.”
The
m
entione
d
powe
r
syst
e
m
sta
bili
ty
def
in
it
ion
is
from
t
he
ge
ner
al
dynam
ic
syst
e
m
vi
ewpoint
.
I
n
the
ab
ove
def
i
niti
on
,
se
ver
al
aspects
of
po
w
er
syst
em
insta
bili
ty
are
co
nt
ai
ned
.
T
hese
a
sp
ect
s
a
re
el
eg
antly
identifie
d
a
nd
represe
nted
i
n
w
ork
.
F
ur
t
her,
opti
m
a
l
po
we
r
fl
ow,
in
essence,
deal
s
with
gen
e
rat
ing
a
nd
routin
g
power
thr
ough
the
tra
ns
m
issi
on
syst
e
m
to
m
ee
t
cho
se
n
ob
j
ect
ive
s
an
d
a
dhere
t
o
operati
ng
c
on
strai
nts
[
5
]
. Som
e o
f
th
ese
co
ns
trai
nts
are e
xp
la
ine
d h
ere.
3.
OPTIM
AL
R
EAL PO
WER
SCHED
ULI
NG
Fu
el
cost
m
ini
m
iz
at
ion
is
pr
im
aril
y
an
op
er
at
ion
al
pro
ble
m
.
This
is
us
ually
ref
err
e
d
to
as
econom
ic
disp
at
c
h.
To
m
ini
m
iz
e
the
f
uel
c
os
t,
it
ne
eds
t
o
c
om
pr
ehend
the
f
uel
cost
c
urves
f
or
al
l
par
ti
cular
powe
r
gen
e
rati
ng
c
om
po
nen
ts
i
n
th
e
syst
e
m
.
A
preci
se
cos
t
c
urv
es
il
lustrati
on
m
ay
inv
olv
e
a
piecewise
poly
no
m
ia
l
Evaluation Warning : The document was created with Spire.PDF for Python.
Ind
on
esi
a
n
J
E
le
c Eng &
Co
m
p
Sci
IS
S
N:
25
02
-
4752
A stati
sti
cal ja
cob
i
an ap
plica
ti
on
fo
r
po
we
r
syste
m op
ti
miz
ation of
v
olta
ge
sta
bili
ty
(
Raja
M
asood
La
ri
k
)
333
form
or
can
be
est
im
at
ed
t
hro
ugh
oth
e
r
appr
oach
es
,
w
it
h
com
m
on
ones
bein
g:
1.
Piec
ewise
li
ne
ar,
2.
Qu
a
drat
ic
, 3. C
ub
ic
,
and
4. Pi
ecewise
qu
a
dr
a
ti
c.
The
Linear
a
ppr
oxim
a
ti
on
is
not
c
omm
on
l
y
us
e
d
wh
il
e
t
he
piecewise
li
near
f
or
m
is
use
d
i
n
m
any
pro
du
ct
io
n
-
gra
de
li
near
pr
ogr
a
m
m
ing
a
pp
li
c
at
ion
s.
The
qu
adr
at
ic
a
pprox
i
m
ation
is
us
e
d
in
m
os
t
nonl
inear
pro
gr
am
m
ing
app
li
cat
io
ns
.
C
on
t
ro
l
var
ia
bles
are
usual
ly
t
he
in
de
pende
nt
var
ia
bles
i
n
an
OP
F
pro
ble
m
and
they
inclu
de 1.
A
ct
ive
powe
r gene
rati
on, 2. Gene
rato
r bu
s
vo
lt
age
s, 3. Tr
ansfo
rm
er tap
r
at
ios,
4. P
hase
-
sh
ifte
r
ang
le
s
,
a
nd
5.
Sett
ing
s
of
s
hunt
capaci
t
or
s
a
nd
in
du
ct
or
s
.
The
a
ppli
cat
ion
of
t
he
a
fore
m
entioned
c
om
po
nen
ts
as co
ntr
ol v
a
ri
ables
would p
r
ov
i
de
the
b
e
st (
le
ast
ex
pe
ns
i
ve
)
re
su
lt
.
Ther
e
a
re
so
m
e
assum
ption
s
need
to
be
m
a
de
in
m
od
el
ing
of
the
obj
ect
ives
an
d
co
ns
trai
nts
w
here
fu
el
c
os
t
cu
rv
e
s
are
sm
oo
th
a
nd
quad
rati
c
in
nat
ur
e;
on
ly
act
ive
powe
r
gen
e
rati
ons
ar
e
con
t
ro
ll
ed
for
cost
m
ini
m
iz
at
ion
and
tra
ns
f
or
m
er
ta
ps
,
vo
lt
age
of
ge
ne
rato
rs,
sh
unt
capaci
to
r
set
ti
ng
s,
an
d
inducto
r
posit
io
ns
are
held
at
their
nom
inal
set
values
thr
ough
ou
t
the
op
ti
m
iz
a
tio
n
ass
um
ing
P
–
Q
dec
oupling
;
cu
rr
e
nt
flo
ws
ar
e
con
t
ro
ll
ed
us
in
g
volt
age
an
d
ph
a
se
an
gle
re
stric
ti
on
acr
os
s
the
li
nes
and
finall
y
con
ti
ng
ency
const
rain
ts
are
neg
le
ct
e
d.
4.
SC
HE
DU
LI
N
G
U
SING
VOL
TAGE ST
A
BIL
ITY MA
R
GIN
Vo
lt
age
sta
bili
ty
assess
m
ent
and
e
nh
a
ncem
ent
is
a
per
ple
xing
issue
in
the
fiel
d
of
po
wer
syst
em
s
eng
i
neer
i
ng.
It
has
gaine
d
m
or
e
at
te
ntio
n
once
powe
r
syst
e
m
s
op
erati
ons
beco
m
e
m
or
e
com
plex
and
m
any
lim
it
at
ion
s
sti
ll
exist.
It
has
be
en
m
any
ye
ar
s
since
A
ngle
sta
bili
ty
becom
es
the
m
ai
n
issue
withi
n
th
e
fiel
d.
Be
cause
of
det
erior
at
in
g
in
ve
st
m
ents
in
power
facil
it
ie
s
and
a
novel
tra
ns
m
issi
on
syst
e
m
in
the
1980s,
th
e
powe
r
syst
em
s
w
e
re
ov
e
rloa
de
d,
ca
us
i
ng ph
eno
m
ena kn
own
as
volt
age i
nst
abili
ty
[
6
]
.
Since
the
n,
th
e
i
m
po
rtance
of
reacti
ve
po
wer
in
upholdi
ng
a
ppr
opriat
e
vo
lt
age
le
vel
thr
oughout
the
power
syst
e
m
beco
m
e
a
m
a
in
con
c
e
rn
of
e
nginee
rs.
T
her
e
are
s
om
e
vo
lt
age
c
ollapse
inci
dent
s
set
the
init
ia
ti
ve
to
stud
y
volt
age
s
ta
bili
ty
and
te
chn
i
qu
e
s
to
pr
e
ven
t
furthe
r
oc
currence
.
O
rigi
nally
,
vo
lt
ag
e
sta
bili
ty
stud
ie
d
in
a
sta
ti
c
fash
io
n
wh
il
e
th
e
a
pp
li
cat
ion
of
m
achines,
e
xcite
rs,
an
d
ta
p
cha
nge
rs
is
dynam
ic
s
an
d
ca
n
be
c
ha
ng
e
d
by loa
d flo
w
c
on
si
der
a
bly
[
7
]
.
Me
thods
that
determ
ine
the
distance
of
the
c
urren
t
operati
ng
sta
te
f
ro
m
the
point
of
V
oltage
Coll
apse
(V
C
)
usual
ly
ex
pr
e
ss
this
distanc
e
in
te
rm
s
of
t
he
a
dd
it
io
nal
MVA
/
M
W
/
MVAR
loa
d
t
hat
the
syst
e
m
is
capab
le
of
s
upplyi
ng
be
for
e
it
e
ncou
nters
VC.
This
a
dd
it
io
na
l
loading
is
r
efer
red
t
o
as
S
ecur
it
y
Ma
rg
in
(S
M)
.
In
this
m
et
ho
d,
wh
il
e
sche
du
li
ng
the
ge
nerat
or
s
f
or
op
ti
m
u
m
real
po
w
er
outp
uts
the
VS
M
m
axi
m
iz
ation
i
s also ta
ken int
o
acc
ount as
one
of the
pr
im
ary o
bject
ives
[
8
]
.
5.
SC
HE
DU
LI
N
G FO
R ONLI
NE APPL
IC
A
TIONS
As
a
resu
lt
of
a
lot
of
var
ia
bl
es
and
co
ns
trai
nts,
the
siz
e
of
the
op
ti
m
iz
at
i
on
prob
le
m
beco
m
es
la
rg
e
in
siz
e
a
nd
the
co
nv
e
ntio
nal
op
ti
m
iz
ation
m
et
ho
ds
face
pro
blem
s
in
ter
m
s
of
com
pu
t
at
ion
al
ti
m
e.
Sinc
e
the
com
pu
ta
ti
on
al
tim
e
has
been
consi
der
a
bly
increase
d
du
e
t
o
pro
blem
co
m
plexit
y,
the
op
e
rati
onal
en
gin
ee
rs
face a
pro
blem
in
on
li
ne
sc
he
du
li
ng.
Ther
e
f
or
e,
one
can
easi
ly
see
that
there
is
an
urgen
t
nee
d
to
de
velo
p
powe
rful
m
et
ho
ds
that
c
a
n
op
ti
m
iz
e the
po
we
r
syst
em
f
or
r
eal
po
wer
ge
ner
at
io
n
in
real
tim
e. I
n
view of v
oltage stab
il
ity i
ssu
es, it wil
l be
ben
e
fici
al
if
t
hese
newer
m
et
hods
acc
omm
od
at
e
vo
lt
ag
e
sta
bili
ty
m
a
xim
iz
at
ion
as
an
obj
ect
ive
.
Othe
r
i
m
po
rtant
secu
rity
const
raints
su
c
h
a
s
li
ne
overl
oad
i
n
g
,
volt
age
m
agn
it
ude
lim
it
s
and
oth
ers
m
us
t
al
so
be
a
par
t
of
any
propose
d
on
li
ne
real
po
wer
s
cheduli
ng
sc
hem
e.
In
t
he
f
ollow
i
ng
sect
ion,
OP
F
opti
m
iz
at
ion
te
chn
iq
ues
are
discusse
d.
6.
OPF O
PTIMI
Z
ATION
TE
CHNIQ
UES
Ther
e
a
re
m
a
ny
te
chn
i
qu
es
to
op
ti
m
iz
e
t
he
powe
r
fl
ow
w
hich
ca
n
be
cl
assifi
ed
i
nto
di
ff
e
ren
t
cat
egories
as
de
fine
d
in
[
9
]
.
Nonlinea
r
pro
gram
m
ing
(
NLP)
is
one
of
the
te
chn
iq
ues
that
deals
with
pr
oble
m
s
involvin
g
nonl
inear
obj
ect
iv
es
an
d
c
onstr
ai
nts.
T
he
li
m
it
at
ion
s
m
ay
i
nvolv
e
issues
li
ke
eq
ualit
y
and
/
or
ineq
ualit
y
for
m
ula
ti
on
s.
Se
ver
al
m
et
ho
ds
su
c
h
as
Se
quentia
l
Un
c
on
strai
ned
Mi
ni
m
iz
at
ion
Tec
h
ni
qu
e
(S
UMT
),
L
ag
r
ang
e
m
ulti
plier
base
d
m
et
ho
d
ca
n
be
us
ed
to
s
olv
e
O
PF
pro
blem
s
[
10
]
.
T
hese
a
ssu
m
e
nonlinea
r obje
ct
ives and c
ons
trai
nts.
The
Q
ua
dr
at
ic
Pr
og
ram
m
ing
is
a
disti
nctive
form
of
no
nli
ne
ar
pro
gr
am
m
i
ng
te
ch
nique
whose
m
ain
pur
po
se
is
qu
adr
at
ic
a
nd
ha
s
li
near
c
ons
trai
nts.
Qu
a
si
-
New
t
on
an
d
sensiti
vity
-
bas
ed
m
e
tho
ds
c
an
be
e
m
plo
ye
d
f
or
so
lvi
ng
real
powe
r
on
li
ne
O
PF
pr
ob
le
m
s
[
11
]
.
Li
near
Program
m
ing
(L
P)
[
12
,
13
]
is
ano
t
he
r
com
m
on
ly
appl
ie
d
te
chn
i
qu
e
to
unra
vel
va
rio
us
li
near
/
nonlinear
powe
r
syst
e
m
op
tim
i
zat
ion
com
plica
ti
on
s
li
ke
transm
issio
n
plan
ning,
s
ecur
it
y
co
ns
tra
ined
disp
at
c
h,
gen
e
ral
OPF,
e
m
erg
ency
c
on
t
ro
l,
et
c.
A
set
of
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2502
-
4752
Ind
on
esi
a
n
J
E
le
c Eng &
Co
m
p
Sci,
Vo
l.
1
3
, N
o.
1
,
Ja
nu
a
ry
201
9
:
331
–
338
334
app
li
ca
ti
ons
known
as
Va
riant
s
of
sim
plex
ba
sed
m
et
ho
dol
og
ie
s
is
com
m
on
ly
em
plo
ye
d
f
or
res
olv
in
g
t
he
L
P
com
plica
ti
on
s.
The
i
nterio
r
po
i
nt
m
et
ho
d
(IPM
)
was
dev
el
op
e
d
i
n
[
14
]
.
It
stu
nned
the
ope
rati
on
resea
rc
h
com
m
un
it
y
si
nce
the
sc
hem
e
so
lves
the
li
near
pro
gr
am
m
ing
pro
blem
m
uch
faster
t
han
t
he
co
nve
ntion
al
si
m
plex
m
et
ho
d.
The
exte
ns
i
on
of
t
he
i
nteri
or
point
m
et
hod
ca
n
be
a
ppli
ed
to
NLP
an
d
QP
pro
blem
s
and
ha
s
sh
ow
n
s
uperi
or
qu
al
it
ie
s
an
d
prom
isi
ng
re
s
ults.
T
he
i
nteri
or
point
m
et
hod,
al
t
hough
re
centl
y
introd
uc
ed
to
powe
r
syst
em
optim
iz
at
ion
,
has
de
velo
pe
d
as
a
well
-
sui
te
d
m
et
ho
d
a
nd
unceasi
ng
l
y
m
od
ifie
d
f
or
m
any
powe
r
syst
em
op
ti
m
iz
ation
gl
it
ches.
I
nterio
r
poi
nt
m
et
ho
d
for
nonlinea
r
pro
gr
am
m
ing
f
or
m
ulati
on
has
bee
n
widely
ap
plied
to
a
ns
we
r
t
he
OPF
iss
ues.
This
optim
izati
on
al
go
rith
m
req
uires
t
he
so
luti
on
of
a
set
of
nonlinea
r
e
qu
a
ti
on
s t
o ob
ta
in
the opti
m
al
so
l
ution o
f
t
he power
n
et
work e
qu
at
io
ns.
In
t
he
case
of
a
non
-
m
on
otonic
so
l
ution
surface,
t
rad
it
io
na
l
op
ti
m
iz
a
ti
on
ap
proac
hes
r
e
m
ai
n
ver
y
delic
at
e
and
t
he
y
of
te
n
a
ddre
ss
a
local
opti
m
u
m
reso
luti
on
of
the
syst
em
.
Hen
ce,
ther
e
is
a
nee
d
to
i
ntr
oduc
e
a
ne
w,
m
or
e
gen
e
ral
a
nd
re
li
able
al
gorith
m
s
that
can
ta
ckle
no
n
-
c
onve
x
s
olu
ti
on
surfaces
.
E
vo
l
ution
a
r
y
Pr
og
ram
m
ing
(
EP)
is
one
of
t
he
m
ai
n
te
chn
iqu
e
s
that
can
be
us
ed
i
n
this
m
ann
er
[
15
]
.
T
he
EP
te
c
hn
i
que
is
a
stochastic
op
ti
m
iz
at
ion
m
et
ho
d
w
hich
em
plo
ys
the
m
echan
ic
s
of
e
vo
l
ution
to
yi
el
d
id
eal
reso
l
ution
for
the
pro
blem
in
han
d.
This
ca
n
be
done
by
devel
op
in
g
a
po
pula
ti
on
of
ca
ndidate
so
l
utio
ns
on
t
he
way
to
the
ov
e
rall
opti
m
i
zat
ion
by
em
plo
yi
ng
a
n
e
voluti
on
op
e
rat
or
an
d
a
sel
ect
ion
sc
hem
e.
The
EP
m
et
hod
is
pr
e
dom
inantly
fitt
ed
to
non
-
m
onotonic s
olu
ti
on surfa
ces
where m
any local
m
ini
m
a exist.
EP
te
ch
nique
was
s
ugge
ste
d
f
or
t
he
de
velop
m
ent
of
the
finite
-
sta
te
m
achines
in
ord
er
to
s
olve
pr
e
d
ic
ti
on
ta
s
ks.
The
re
are
m
any
adjustm
ents,
au
gm
entat
ion
s,
a
nd
e
xe
cutions
ap
plied
an
d
exam
ined
.
EP
te
chn
iq
ue has
been exte
nd
e
d t
o
treat
r
eal
-
val
ued ob
j
ect
ive fun
ct
io
ns
a
nd
oth
er
data str
uct
ur
es
. T
ran
s
f
orm
at
ion
is
fr
e
qu
e
ntly
app
li
ed
by
us
in
g
a
new
ca
su
a
l
nu
m
b
er
or
a
vecto
r
w
hile
us
in
g
cl
assic
al
EP
(CE
P)
[
16
]
to
a
par
e
nt.
T
he
disti
nction
le
vel
of
t
he
Ga
us
sia
n
m
utati
on
is
orga
nized
us
in
g
sta
nda
rd
de
vi
at
ion
or
c
omm
on
ly
known
as “st
ra
te
gy p
a
ram
et
er
” in the
evol
ution
a
ry sea
rch.
The
EP
al
go
rithm
beg
ins
by
consi
der
i
ng
a
s
et
of
possible
P
so
luti
ons
.
Le
t
the
m
be
ref
er
red
to
as
X1
to
XP
w
he
re
X1
is
the
fir
st
of
s
uch
s
olu
ti
ons
an
d
X
P
is
the
Pth
s
olu
ti
on
.
Using
this
set
of
P
so
l
utio
ns
X1
to
XP
,
an
oth
e
r
se
t
of
P
s
olu
ti
on
s
XP
+
1
t
o
X2P
is
ge
ne
rated
thr
ough
a
ra
ndom
process.
Fr
om
this
set
of
2P
so
luti
ons
X
1
t
o
X
2P
,
P
so
lut
ion
s
th
at
sat
isf
y
al
l
the
con
st
r
ai
nts
and
ha
ving
the
best
obj
e
ct
ive
functi
on
values
are
ch
os
en
.
T
he
sel
ect
ed
set
of
P
so
luti
ons
is
her
ea
fter
ref
er
red
to
a
s
X1
to
X
P.
W
it
h
an
ap
pr
opriat
e
te
rm
inati
on
cri
te
rion,
it
is
determ
ined
wh
et
he
r
the
op
ti
m
um
so
luti
on
has
been
reac
hed.
Othe
rw
ise
,
t
he
set
of
ste
ps
descr
i
bed ab
ov
e
is re
pea
te
d
unti
l t
he o
pt
i
m
u
m
is reache
d.
7.
POWER
S
YST
EMS S
T
ABI
LIT
Y
CL
ASS
IFIC
AT
IO
NS
7.1
.
Rotor
A
ng
le
S
t
ab
il
ity
An
interc
onne
ct
ed
po
wer
sy
stem
is
ro
to
r
a
ng
le
sta
ble
if
i
t
is
able
to
retai
n
sync
hro
nism
of
al
l
it
s
synch
ron
ous
m
achines
a
fter
be
ing
s
ubj
ect
ed
to
a
disruptio
n.
The
eq
uili
br
iu
m
between
el
ect
ro
m
agn
et
ic
tor
que
and
m
echan
ic
a
l
to
rque
at
each
sync
hrono
us
m
achine
in
the
syst
e
m
m
us
t
be
m
a
intai
ned
after
the
distu
r
ban
ce
[
17
]
.
Wh
e
n
th
e
r
oto
r
a
ng
le
swings
of
s
om
e
gen
e
rato
rs
increa
se
a
nd
eve
ntu
al
ly
re
su
lt
in
t
heir
l
os
s
of
synch
ronism
w
it
h
oth
e
r gen
er
at
or
s,
ins
ta
bili
t
y occu
rs. The
re
al
p
owe
r
outp
ut of a
sy
nchr
onous m
achine
var
ie
s
pro
portion
al
ly
to
a
f
un
ct
io
n
of
it
s
ro
t
or
a
ng
l
e
change.
I
nput
m
echan
ic
al
torque
a
nd
t
he
outp
ut
el
ect
ro
m
agn
et
i
c
tor
qu
e
of a
ge
ne
rator bala
nce
each
oth
e
r
in
st
eady sta
te
and
it
s sp
eed
r
em
ain
s c
onsta
nt.
On
pe
rtu
rb
at
io
n
of
t
he
syst
e
m
by
a
load
c
hange,
the
l
oa
ding
of
t
he
ge
ner
at
or
s
(elect
rical
tor
qu
e
)
changes
a
nd
he
nce,
they
s
pe
ed
up
or
retar
d
[
18
]
.
T
his
cha
ng
e
i
n
s
peed
m
ay
no
t
be
uni
form
as
the
cha
ng
e
i
n
the
load
re
flec
ts
un
e
ven
ly
on
the
gen
e
rato
rs
.
Ov
e
r
tim
e,
the
syst
e
m
os
ci
l
l
at
es
and
fi
nd
s
a
new
e
qu
il
ibr
ium
wh
e
rein
al
l
the
gen
erat
ors
ha
ve
their
m
ec
han
ic
al
an
d
el
ect
rical
torq
ue
s
balance
d.
I
n
certai
n
cases,
du
e
to
sever
al
under
l
yi
ng
reas
ons,
these
o
sci
ll
at
io
ns
incre
ase
an
d
pull
ou
t
s
ome
m
achines
fro
m
synchr
onis
m
.
Ther
e
are
so
m
e
te
xts
that
deal
with
t
his
pro
blem
.
The
s
olu
ti
on
sc
hem
es
fo
r
c
ontr
ols
ha
ve
al
s
o
be
en
res
earc
hed
an
d
ci
te
d
in
7.2
.
Fre
quen
cy
S
t
ab
il
ity
The
f
re
qu
e
ncy
of
a
powe
r
s
yst
e
m
need
s
to
be
m
ai
ntained
at
the
s
peci
fied
value
[
19
]
.
Re
li
abili
t
y
crit
erion
of
a
powe
r
syst
e
m
would
restrict
the
m
axi
m
u
m
dev
ia
ti
on
from
the
set
fr
eq
ue
ncy
in
te
rm
s
of
a
nu
m
ber
of
acc
um
ulate
d
cy
cl
es
of
dev
ia
ti
on
per
day.
Co
ns
t
antly
var
yi
ng
powe
r
dem
and
in
th
e
power
s
yst
e
m
requires
t
he
gove
r
nor
c
on
tr
ol
s
to
ad
just
th
e
outp
ut
of
th
e
ge
ner
at
ors
s
uch
that
they
m
ai
ntain
set
frequ
e
ncy
[
20
]
.
Howe
ver,
a
rap
i
d
c
hange
in
loa
d
or
su
dde
n
l
os
s
of
load
du
e
to
equ
i
pm
ent
fail
ur
e
w
ould
res
ult
i
n
fr
e
qu
e
ncy
exc
ur
si
on
s
.
These
hav
e
to
be
c
on
t
ro
ll
ed
with
a
m
ini
m
u
m
swing
in
fr
e
quency
an
d
m
i
nim
u
m
dro
p/ad
d
of
cy
cl
es
ov
e
r
the
entire
day.
I
n
essence,
fr
e
qu
ency
sta
bili
ty
is
def
ine
d
by
as
“a
capa
bili
ty
of
a
powe
r
syst
e
m
to
upho
l
d
ste
ady
fr
e
qu
e
nc
y
fo
ll
ow
i
ng
a
sever
e
syst
em
up
set
resu
l
ti
ng
in
a
signi
fican
t
Evaluation Warning : The document was created with Spire.PDF for Python.
Ind
on
esi
a
n
J
E
le
c Eng &
Co
m
p
Sci
IS
S
N:
25
02
-
4752
A stati
sti
cal ja
cob
i
an ap
plica
ti
on
fo
r
po
we
r
syste
m op
ti
miz
ation of
v
olta
ge
sta
bili
ty
(
Raja
M
asood
La
ri
k
)
335
i
m
balance
bet
ween
ge
ner
at
i
on
a
nd
loa
d.
”
Su
sta
ine
d
f
re
quency
s
wings
cause
loa
ds
to
be
s
hed
or
ge
ner
at
or
s
cutbac
ks
.
T
hes
e
ha
ve
far
-
reac
hing
im
plica
ti
o
ns
on
the
healt
hy
ope
rati
on
of
powe
r
syst
e
m
s.
F
reque
ncy
s
wing
s
al
te
r
power
fl
ow
s
[
21
]
,
loa
d
pro
file
base
d
up
on
l
oad
c
har
act
erist
ic
s,
transm
issi
on
ne
twork
m
at
ric
es
a
nd
vo
lt
age
s
am
ongs
t
oth
e
r.
It
c
auses
un
desira
ble
act
io
ns
s
uc
h
as
init
ia
ti
on
of
vo
lt
age/f
reque
ncy
pr
otect
io
n
te
rm
ed
as
ov
er
-
flu
xing
protec
ti
on
an
d
oth
e
r
s
relat
ed
to
boil
er
dynam
i
cs
[
22
]
.
Lar
ge
po
wer
syst
em
s
u
se
the
fr
e
qu
e
ncy
as
a
n
im
po
rtant
cri
te
rion
for
s
plit
t
ing
the
syst
em
into
isl
an
ds
f
or
su
sta
ine
d
an
d
reli
able
operati
on
i
n
tim
es
of
em
erg
ency.
F
reque
ncy
sta
bili
ty
he
nce
relat
es
to
stu
dy
of
th
e
powe
r
syst
em
w
hile
ex
pe
rienci
ng
fr
e
qu
e
ncy
s
wi
ng
s
to
dete
rm
i
ne
wh
et
her
the
syst
e
m
would
be
sta
ble
ta
kin
g
int
o
c
on
si
de
rati
on
s
a
m
yri
ad
of
protect
ive a
nd
op
e
rati
onal
crit
eria.
7.3
.
Vo
l
tage
St
abi
li
ty
Vo
lt
age
sta
bili
ty
of
a
powe
r
syst
e
m
ref
le
ct
s
it
s
abili
ty
to
s
us
ta
in
the
load
by
hav
in
g
ste
ady
vo
lt
age
s
after
a
disturb
ance.
T
he
re
ar
e
severa
l
aspe
ct
s
of
t
he
po
w
er
syst
em
that
interplay
in
t
his
issue
[
23
]
.
These
include
sl
ow
act
ing
volt
age
con
tr
ol
de
vic
es
li
ke
ta
p
set
ti
ng
of
tra
ns
f
orm
ers,
set
ti
ng
of
s
hunt
cap
a
ci
tors,
set
ti
ng
of
s
hunt
reactors
a
nd
su
c
h
disc
rete
de
vices
[
24
]
.
T
he
se
al
so
inclu
de
fast
act
ing
de
vices
li
ke
ge
ne
rator
fiel
d
excit
at
ion
con
tr
ol
an
d
li
m
it
ers,
sta
ti
c
r
eact
ive
power
so
urce
co
ntr
ollers,
unifie
d
po
wer
fl
ow
c
ontr
oller
(its
su
bcate
gor
ie
s)
and
s
uc
h
oth
e
rs
[
25
]
.
T
he
se
con
tr
ollers
try
to
cor
rect
the
load
bus
volt
ages
an
d
m
ai
ntain
them
within
an
acce
ptable
r
ang
e
.
S
om
e
of
these
al
so
ha
ve
s
hort
-
te
rm
ov
e
rloa
d
facil
it
ie
s
li
ke
the
ge
ner
at
or
fiel
d
cu
rr
e
nt
li
m
it
ers
[
26
]
.
D
ue
to
dif
fer
e
nt
tim
e
scal
e
op
erati
on
c
ha
racteri
sti
cs
of
t
h
es
e
de
vices,
the
y
m
ay
m
anifest
volt
age
i
ns
ta
bili
ty
.
So
m
et
i
m
es,
the
colla
ps
e
of
a
li
ne
cause
s
a
c
ascadin
g
e
ff
ect
that
le
ad
s
to
tr
ipp
i
ng
of o
t
her
li
ne
s a
nd ev
e
ntu
al
syst
e
m
co
ll
apse
[
27
]
.
Fu
rt
her,
w
hile
viewin
g
the
tra
ns
m
issi
on
in
Figure
2,
ther
e
are
two
co
ndit
ion
s
in
the
sta
ti
c
view
that
con
t
rib
utes
to
vo
lt
age
colla
ps
e.
Both
of
thes
e
per
ta
in
to
la
c
k
of
reacti
ve
powe
r.
Th
us
th
e
two
m
ajor
cau
ses
of
vo
lt
age
i
ns
ta
bi
li
ty
are
the
lack
of
reacti
ve
power
due
to
high
I
2
X
l
os
s
es
in
hea
vily
l
oad
e
d
li
ne
s
a
nd
the
dynam
ic
nature
of
the
po
we
r
syst
em
.
Vo
lt
age
sta
bili
ty
is
la
rg
el
y
cat
eg
or
iz
e
d
int
o
tw
o
ty
pes
base
d
on
it
s
sever
it
y.
Figure
2
.
Tr
a
nsm
issi
on
Line
These
c
har
act
e
rizat
ion
s
al
lo
w
the
dete
rm
inatio
n
of
m
od
el
ing
acc
ur
acy
a
nd
ty
pe.
It
al
s
o
he
lps
in
t
he
determ
inati
on
of
the
kind
of
si
m
ulati
on
req
ui
red.
The
fi
rst
cat
egorizat
ion
is
the
la
r
ge
-
scal
e
disturbanc
e
vo
lt
age
sta
bili
ty
.
Wh
en
the
s
yst
e
m
enco
unt
ers
la
r
g
e
distu
rb
a
nces
s
uc
h
as
loss
of
ge
ne
rati
on
s
,
sym
m
et
ric
fau
lt
s,
a
nd
ot
her
s
,
la
rg
e
-
sca
le
disturbance
vo
lt
age
sta
bili
ty
deter
m
ines
wh
et
her
the
syst
e
m
wo
uld
retai
n
appr
opriat
e
volt
ages
at
al
l
the
bu
s
es.
T
his d
et
erm
inati
on
requires
a non
-
li
ne
ar
analy
sis
of the
p
ow
e
r
syst
e
m
.
It
al
so
re
quires
t
he
a
dequate
m
od
el
in
g
of
slo
w
an
d
fast
act
ing
vo
lt
a
ge
co
ntr
ol
de
vices
s
uch
as
ta
p
c
ha
ng
e
rs
,
gen
e
rato
r
fiel
d
current
li
m
i
ter
s,
a
nd
oth
e
rs.
The
sec
ond
c
har
act
erist
ic
re
la
te
s
to
the
S
m
al
l
-
distur
ba
nc
es
a
nd
their
ef
fects
on
vo
lt
age
sta
bili
ty
[
28
]
.
It
trie
s
to
assess
the
s
tructu
re’
s
ca
pa
bili
ty
to
keep
s
ta
ble
vo
lt
a
ges
on
c
e
expose
d
to
m
i
nor
distress
es
su
c
h
as
increm
ental
changes
in
syst
e
m
lo
ad.
In
this
case,
the
cha
racteri
sti
cs
of
loads
,
co
ntinuou
s
c
ontrols
,
and
discrete
con
t
ro
ls
at
a
giv
e
n
instan
t
of
tim
e
influ
ence
the
re
sp
ons
e
char
act
e
risti
cs.
Ba
sed
upon
th
e
tim
e
fr
a
m
es,
assessm
ent
m
e
thods
can
be
w
orke
d
ou
t
.
The
se
tim
e
fr
am
es
and
their im
pl
ic
at
io
ns
for powe
r
s
yst
e
m
stud
ie
s a
re elab
or
at
e
d
i
n
the
foll
owin
g.
Shor
t
-
te
rm
vo
l
ta
ge
sta
bili
ty
is
the
first
cat
ego
ry.
T
his
cat
ego
ry
pri
ncipall
y
con
side
rs
the
existe
nce
of
a
s
m
al
l
tim
e
fr
a
m
e
fo
r
the
ph
ysi
cal
pr
ocess.
All
the
dev
ic
e
m
od
el
s
that
are
si
m
ple
y
et
a
dequatel
y
rep
r
esent
the
com
plexities
require
d
f
or
the
stud
y
m
us
t
be
include
d.
Pr
im
arily
tho
se
el
e
m
ents
and
con
tr
ol
act
ions
of
th
e
powe
r
syst
em
t
hat
ca
n
act
i
n
t
his
ti
m
e
fr
am
e
are
c
onsidere
d
for
m
od
el
in
g.
Loa
ds
t
hat
are
var
yi
ng
with
re
sp
ect
to the v
olta
ge m
us
t al
so
b
e a
dequatel
y m
od
el
ed.
Lo
ng
-
te
rm
volt
age
sta
bili
ty
include
s
m
o
deling
a
nd
ad
equ
at
e
represe
ntati
on
of
sl
ower
act
in
g
equ
i
pm
ent
su
c
h
as
ta
p
-
c
ha
ngin
g
tra
nsfo
r
m
ers,
the
rm
os
ta
ti
cal
l
y
con
tr
olled
loa
ds,
a
nd
ge
ne
rator
current
lim
it
ers.
A
pe
r
iod
of
se
ver
al
m
inu
te
s
is
re
quire
d
f
or
the
long
-
te
rm
si
m
u
la
ti
on
to
so
l
ve
these
case
s.
I
n
the
se
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2502
-
4752
Ind
on
esi
a
n
J
E
le
c Eng &
Co
m
p
Sci,
Vo
l.
1
3
, N
o.
1
,
Ja
nu
a
ry
201
9
:
331
–
338
336
cases,
the
i
nter
act
ion
of
sl
ow
act
ing
c
on
tr
ol
li
ke
ta
p
cha
ng
ers
an
d
ge
ner
a
tor
ov
e
re
xcita
ti
on
li
m
i
te
rs
play
s
an
i
m
po
rtant
r
ole.
The
ef
fect
of
load
recovery
after
fall
of
vo
lt
age
and
su
c
h
char
act
erist
ic
s
al
so
play
de
fining
ro
le
s.
Thei
r
a
de
qu
at
e
m
od
el
ing
is
al
so
of
par
am
ount
im
portance
.
I
n
t
hi
s
pap
e
r
m
et
ho
d
of
a
naly
sis
of
the
vo
lt
age
sta
bili
ty
of
the
po
wer sy
stem
is
the
pri
m
ary
con
cer
n.
T
he
sta
ti
c
m
e
asur
e
s
a
re
a
dequate
as
they
n
e
ed
t
o
be
inclu
ded
in the
op
ti
m
iz
a
ti
on
schem
e.
Thus,
the
fo
ll
owin
g
sect
ion
pr
ov
i
des
sta
ti
sti
cs
in
reg
ar
ds
to
f
ulf
il
l
the
researc
h o
bj
ect
ive.
8.
POWER
FLO
W JA
COBI
A
N
In
our
opti
m
iz
at
ion
sc
hem
es,
there
is
a
nee
d
to
ide
ntify
a
m
easur
e
that
giv
es
the
volt
age
sta
bili
t
y
inf
or
m
at
ion
of
the
powe
r
syst
e
m
.
This
nee
ds
to
be
relat
ed
to
the
co
ntr
ol
var
ia
bles
of
th
e
po
wer
syst
em
.
The
singular
value
s
of
a
m
at
rix
A
are
th
e
squ
are
r
oo
ts
of
t
he
ei
genvalues
of
AT
A.
T
hus
,
as
ei
ge
nv
al
ue
s
are
adequate
in
dic
at
or
s
of
volt
age
sta
b
il
it
y,
so
are
t
he
sin
gula
r
values
.
H
ence,
i
n
this
s
ect
ion
,
a
relat
ion
s
hi
p
betwee
n
si
ngul
ar
val
ues
of
the
loa
d
flo
w
J
acob
ia
n
is
der
i
ved
in
te
rm
s
of
t
he
sta
te
s
of
the
po
wer
s
yst
e
m
,
nam
ely the bus
volt
age m
agn
i
tud
es
and t
he b
us
phase a
ng
le
s.
The
inc
rem
ent
al
c
hange
in
any
singular
value
of
the
loa
d
flo
w
Jaco
bian
[
29
]
is
char
act
erized
by
th
e
increm
ental
ch
ang
e
in
the
sta
te
of
the
po
we
r
syst
em
.
The
equ
at
io
ns
are
descr
i
bed
in
f
ollow
i
ngs.
A
pply
ing
Singular
V
al
ue
D
ec
om
po
sit
ion
(S
V
D
),
l
oad
flo
w
Jac
ob
ia
n,
[
]
is equ
at
ed
.
[
]
= [
SU
]
[
Σ]
[
SV
]
t
(
1
)
The
values
of
[S
U]
a
nd
[SV]
wer
e
basical
ly
sh
owe
d
ort
hogo
nal
sin
gu
la
r
vect
or
m
at
rices;
wh
e
rea
s
[Σ]
rep
res
ents
a
diagonal
m
a
t
rix
co
ntainin
g
the
singular
va
lues.
Be
arin
g
in
m
ind
a
m
ino
r
trepidati
on
in
the
sta
te
, Δδ
a
nd ΔV
ca
n be
pr
ese
nted
a
s:
[
+
,
+
]
= [
SU
+Δ
S
U
]
[
Σ+Δ
Σ]
[S
V+
ΔSV]
(2)
The
L
HS
of
th
e
la
te
r
eq
uatio
n
is
e
xten
ded
t
hro
ugh
Tay
lo
r’s
series.
The
fi
rst
orde
r
te
rm
of
t
he
se
ries
com
pr
isi
ng
t
he
load
flo
w
H
essia
n
[
]
(Lo
a
d
fl
ow
Hessia
n
is
the
first
order
de
rivati
ve
of
Loa
d
flo
w
Ja
co
bian) is re
serv
e
d w
hen ig
norin
g
t
he hig
he
r order
term
s.
The follo
wing
equ
at
io
n
is
r
es
ulted.
[
+
,
+
]
-
[
]
= [
]
[
]
(3)
Using
eq
uatio
n
(3
)
t
o
cha
racteri
ze
LHS
of
th
e
equ
at
io
n
(
2),
exp
a
ndin
g
RH
S
and
preser
vi
ng
on
ly
the
first
order t
erm
s,
e
qu
at
io
n(3
)
i
s r
e
-
w
ritt
en
as
[
]
[
]
= [
Δ
S
U][
Σ
]
[
SV
]
t+
[SU]
[
ΔΣ
]
[S
V]t
+[S
U][
Σ
][
Δ
SV
]
t
(4)
Applyi
ng
or
t
hogo
nalit
y
con
s
trai
nts
on
the
updated
le
ft
and
rig
ht
singular
vect
or
m
at
rices,
the
fo
ll
owin
g
e
qua
ti
on
s a
re
rev
eal
ed:
I
=
[SU+
Δ
SU
]
[S
U+
Δ
SU
]
t
(5)
I
=
[SV+Δ
SV
]
[S
V+Δ
SV
]
t
(6)
By
exp
a
ndin
g
the
a
bove
e
qu
at
io
ns
w
hile
igno
rin
g
th
e
seco
nd
or
de
r
te
rm
s
and
us
in
g
t
he
or
t
hogonalit
y p
roper
ty
of [
S
U
]
an
d [S
V]
have the
fo
ll
owin
gs:
SN
=
[
S
U]t
[
Δ
S
U] = −[
Δ
S
U]t
[
SU
]
(7)
SM = [
Δ
S
V]t
[
SV
]
= −
[SV]
t[
Δ
SV
]
(8)
Accor
ding
to
Venkates
h
B.,
et
.
al
.
(
2000
),
t
he
values
of
[
SN
]
a
nd
[S
M]
as
dia
gonal
el
e
m
ents
set
to
be
ze
ro
s
.
Pr
e
-
m
ul
ti
plyi
ng
an
d
post
-
m
ulti
pl
yi
ng
e
quat
ion
(4)
with
[SU]
t
an
d
[SV]
res
pe
ct
ively
,
one
ge
ts
the
fo
ll
owin
g
:
Evaluation Warning : The document was created with Spire.PDF for Python.
Ind
on
esi
a
n
J
E
le
c Eng &
Co
m
p
Sci
IS
S
N:
25
02
-
4752
A stati
sti
cal ja
cob
i
an ap
plica
ti
on
fo
r
po
we
r
syste
m op
ti
miz
ation of
v
olta
ge
sta
bili
ty
(
Raja
M
asood
La
ri
k
)
337
[S
U]t
[
]
[
]
[
]
=
[S
N][
Σ
]
+[
ΔΣ
]
+[
Σ
]
[S
M]
(9)
Since
diag
onal
el
em
ents
of
[
SN
]
[Σ]
a
nd
[Σ
]
[S
M]
con
ta
in
zero
s
,
the
diag
on
al
el
em
ents
of
eq
uatio
n
(4)
a
re
ob
ta
ine
d
in
the
f
ollow
i
ng
:
=[
ΔΣ
]
ii
=[[
SU
]
t[
]
[
]
[
]
]
ii
(10)
Abo
ve
e
qu
at
io
n
e
xpresses
t
he
increm
ental
changes
i
n
the
singular
val
ue
s
of
the
l
oad
fl
ow
Jaco
bian
(volta
ge
sta
bili
ty
m
arg
in
in
ot
her
words),
re
pr
ese
nted
by
a
set
of
fe
w
le
ast
singular
val
ues
of
t
he
loa
d
flo
w
Jaco
bian,
in
te
rm
s
of
the
c
ha
ng
e
s
in
the
inc
rem
e
ntal
syst
e
m
var
ia
bles.
T
he
a
bove
relat
ion
ca
n
be
gain
fu
ll
y
us
e
d
f
or
op
ti
m
iz
at
ion
m
et
ho
ds
in
powe
r
syst
e
m
s.
Eigen
value
analy
sis
in
r
el
at
ion
to
the
s
ingular
values
of
th
e
load
flo
w
Jaco
bian
with
the
sta
te
of
the
syst
e
m
.
These
re
la
ti
on
sh
i
ps
a
re
ex
plo
it
ed
f
or
m
axi
m
iz
ation
of
the
vo
lt
age
sta
bili
t
y
m
arg
in.
The
stud
y
offers
a
pr
act
ic
al
so
lut
ion
th
rou
gh
w
hich
pr
opos
e
d
te
chn
iq
ue
ca
n
adopt
diff
e
re
nt m
od
el
s su
c
h
as
an A
rtific
ia
l Neural
N
et
w
ork
m
od
el
[
30
,
31
]
f
or
op
ti
m
al
sch
edul
ing
.
9.
ACKN
OWLE
DGME
NT
The
a
utho
rs
w
ou
l
d
li
ke
to
ac
knowle
dge
t
he
facil
it
ie
s
pro
vid
ed
by
U
niv
e
r
sit
i
Teknolo
gi
Ma
la
ysi
a
fo
r
the
acc
om
plishm
ent
of
this
w
ork
a
nd
Mi
nist
ry
of
E
ducat
io
n
(MoE
)
of
M
al
ay
sia
fo
r
the
ir
fi
nan
ci
al
s
uppo
rt
unde
r
vo
te
nu
m
ber
GUP
U
TM
17H
10.
RM
L
is
al
so
than
kful
to
N
ED
U
niv
e
rsity
of
E
nginee
ring
a
nd
Tech
no
l
og
y
Si
ndh,
Pa
kistan
for
prov
i
ding
f
inancial
assist
a
nce
by
H
um
an
Re
source
De
ve
lop
m
ent
unde
r
the
schem
e
“Stren
gth
e
ning
of
N
ED
Un
i
versi
ty
of
E
ng
i
neer
i
ng
an
d
Tech
nolo
gy,
Me
ga
-
M3”
of
t
he
Higher
Ed
ucati
on Co
m
m
iss
ion
(H
E
C), P
a
kistan
.
10.
FUTU
RE
RE
SEARCH
AN
D CO
NC
L
USIONS
Op
ti
m
al
po
we
r
flo
w
te
ch
niq
ue
s
are
bei
ng
co
ntin
uous
l
y
i
m
pr
ov
e
d
to
su
it
the
ne
wly
e
m
erg
in
g
chall
enges
f
ac
ed
by
po
wer
e
ng
i
neer
i
ng
co
m
m
un
it
y.
In
t
hi
s
purs
uit,
on
e
or
m
or
e
featu
r
e
nee
ds
t
o
be
t
ackled
t
o
i
m
pr
ove
the
volt
age
sta
bili
ty
m
arg
in
in
po
wer
syst
em
s
wh
e
r
eas
op
ti
m
al
ly
sched
ulin
g
powe
r
syst
em
s.
As
su
c
h,
seve
ral
s
chem
es
hav
e
be
en
pro
posed
i
n
the
li
te
ratur
e
fo
r
th
e
analy
sis
of
powe
r
syst
e
m
to
est
i
m
a
te
the
vo
lt
age
sta
bili
ty
of
a
powe
r
s
yst
e
m
.
This
pap
er
the
oret
ic
al
l
y
con
trib
utes
t
o
de
velo
ping
a
new
OP
F
s
ol
utio
n
m
et
ho
d
res
ponsi
ve
for
operat
ion
al
ap
plica
ti
on
of
power
s
yst
e
m
s
ta
bili
ty
and
ad
diti
ona
l
kn
owle
dge
on
the
Jaco
bian
m
od
el
and
it
s
appl
ic
at
ion
in
that
essence.
A
ddit
ion
al
ly
,
dif
f
eren
t
opti
m
iz
at
ion
te
ch
nique
s
hav
e
rev
ie
wed
the
aspects
with
r
espect
to
volt
age
sta
bili
ty
.
Power
syst
e
m
sta
bili
ty
cl
as
sific
at
ion
s
a
re
al
s
o
introd
uced f
or
dev
el
op
i
ng a
prom
isi
ng
opti
m
iz
ing
sc
hem
e fo
r
power syst
e
m
s.
The
stu
dy
the
n
presents
a
s
ta
ti
sti
ca
l
m
e
tho
d
for
analy
zi
ng
a
po
wer
s
yst
e
m
throu
gh
ei
gen
va
lue
analy
sis
in
relat
ion
to
the
singular
values
of
the
loa
d
fl
ow
Jac
obia
n
with
the
sta
te
of
the
syst
em
.
These
relat
ion
s
hip
s
a
re
ex
plo
it
ed
f
or
m
axi
m
iz
ati
on
of
th
e
volt
age
sta
bili
ty
m
arg
in.
T
he
s
tud
y
offe
rs
a
pr
act
ic
a
l
so
luti
on
t
hroug
h
wh
ic
h
pro
posed
te
c
hn
i
qu
e
can
a
dopt
diff
e
ren
t
m
od
el
s
s
u
ch
as
an
A
rtific
ia
l
Neural
Ne
twor
k
m
od
el
f
or opti
m
al
sch
ed
uling.
Fu
tu
re
w
ork
m
ay
be
unde
rtak
en
in
se
ver
al
di
recti
on
s.
On
e
of
the
pri
m
ary
con
ce
r
ns
t
hat
a
re
e
vo
l
ving
in
the
a
rea
of
powe
r
syst
em
op
ti
m
iz
ation
is
the
intr
oducti
on
of
m
ark
et
s
tructu
re.
This
br
i
ng
s
ab
out
s
ever
al
changes
in
t
he
or
ie
s
that
we
re
dev
el
op
e
d,
ini
ti
al
l
y
fo
r
ve
rtic
al
ly
integrated
powe
r
syst
e
m
s
in
this
researc
h.
I
f
gen
e
rati
on,
tra
ns
m
issi
on
,
an
d
distri
bu
ti
on
are
gove
r
ned
by
diff
e
re
nt
com
pan
ie
s,
th
e
form
ulati
on
of
th
e
obj
ect
ive f
unct
ion
a
nd
co
ns
tra
ints
will
becom
e
com
plex
and
he
nce f
urt
he
r
wor
k
has
to b
e
unde
rtake
n
to so
lv
e
the OPF.
W
it
h
reg
a
rd to t
he
a
pp
li
cabil
it
y of
the
pr
opos
e
d
s
chem
e to sm
a
ll system
s,
the onli
ne
im
ple
m
e
ntati
on
m
et
ho
d
nee
ds
to
be
te
ste
d
throu
gh
ne
w
te
chn
i
qu
e
s,
f
or
e
xam
ple,
S
CADA
(
Super
vis
or
y
co
ntr
ol
and
data
acqu
isi
ti
on
)
and PM
Us (P
has
or Mea
surem
e
nt Unit
s)
i
n
cas
e of lar
ge
a
nd
com
plex
powe
r
syst
em
s.
REFERE
NCE
S
[1]
D.
P.
Kothar
i,
"P
ower
sy
stem
opti
m
iz
ation,
"
in
Computati
onal
I
nte
lligen
ce
and
Signal
Proce
ss
i
n
g
(
CISP
)
,
2012
2nd
Nati
onal
Con
fe
r
enc
e
on
,
2012,
p
p.
18
-
21
.
[2]
G.
Le
vitin,
A.
Li
snianski
,
and
D.
El
m
aki
s,
"S
truc
ture
opt
imiza
t
ion
of
power
sy
st
em
with
di
ffe
ren
t
r
edundant
el
ements,
"
Elec
t
ric
Pow
er
Syst
e
ms
Re
search,
vo
l.
43
,
no
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