Int
ern
at
i
onal
Journ
al of
P
ower E
le
ctr
on
i
cs a
n
d
Drive
S
ystem
(I
J
PE
D
S
)
Vo
l.
11
,
No.
3
,
Septem
be
r 2020
, pp.
135
0
~
13
59
IS
S
N:
20
88
-
8694
,
DOI: 10
.11
591/
ij
peds
.
v11.i
3
.
pp
135
0
-
13
59
1350
Journ
al h
om
e
page
:
http:
//
ij
pe
ds
.i
aescore.c
om
Transie
nt st
ab
ilit
y enhanc
ement i
n the
presenc
e of o
ptimall
y
placed D
-
FACTS
Omar
K
amil
Dahh
am Alaz
zaw
i,
Must
afa
A
hme
d
Na
yy
ef, Y
as
ir
Abd
ulhafedh
Ahm
ed
Depa
rtment
o
f
E
le
c
tri
c
al E
ngin
eering,
Univ
ersit
y
of
Anbar, Ira
q
Art
ic
le
In
f
o
ABSTR
A
CT
Art
ic
le
history:
Re
cei
ved
Feb
20
, 20
20
Re
vised
A
pr
4
,
20
20
Accepte
d
Apr
27
, 20
20
The
power
f
low
in
power
sys
te
m
ca
n
b
e
cont
ro
lle
d
in
an
eff
ectiv
e
me
thod
by
using
FA
CTS
devi
c
es
through
tra
nsmiss
ion
l
ine
s.
Distribu
ted
FA
CTS
rep
rese
nt
a
n
ew
way
to
over
come
the
s
e
dra
wba
c
ks.
The
y
are
cos
t
-
eff
e
ct
iv
e,
li
ght
weigh
te
d
,
high
er
p
erf
or
ma
nc
e
and
minimi
z
at
ion
of
e
nvironm
ental
im
pa
ct
.
The
y
ar
e
place
d
in
a
dis
tri
buti
on
me
thod
and
in
sel
ec
t
ed
li
nes
in
th
e
net
work
accord
ing
to
li
n
e
flo
w
stabilit
y
ind
ic
es
ba
sed
on
lo
ad
flow
ca
l
cul
a
ti
ons.
In
thi
s
pape
r
,
distr
i
bute
d
sta
ti
c
seri
es
com
pens
at
or
(DS
SC
)
is
used
in
orde
r
to
at
t
ai
n
the
req
u
ir
ed
power
flow
c
ontrol
by
mod
if
ying
th
e
li
ne
rea
c
ta
n
ce
.
The
y
are
inst
al
l
ed
dir
ec
t
ly
on
the
pr
e
sent
H.V
tra
nsm
ission
li
nes
and
as
a
resul
t
does
not
r
equi
re
H.V
insulation.
The
loa
d
flow
ana
lysis
h
as
bee
n
ca
rr
ie
d
usi
ng
ma
t
l
ab
and
t
he
simul
at
i
on
fo
r
the
purpose
of
stabi
lity
by
using
Pow
er
Wo
rld
Simul
at
or
software
.
Th
e
r
esult
s
show
an
enhance
m
ent
i
n
stabi
lity p
erf
orm
anc
es
of
th
e
sys
t
em
ov
er the
con
vent
ion
al
cont
rol
le
rs.
Ke
yw
or
d
s
:
Distrib
uted FA
CTS
Power flo
w
c
ontr
ol
Power w
or
l
d
si
mu
la
to
r
Weak
e
st b
us
i
de
ntific
at
ion
Power syste
m
sta
bili
ty
enh
a
ncem
ent
This
is an
open
acc
ess arti
cl
e
un
der
the
CC
BY
-
SA
l
ic
ense
.
Corres
pond
in
g
Aut
h
or
:
Oma
r Kamil
Dahh
a
m
Alazza
wi
Dep
a
rtme
nt of
Ele
ct
rical
En
gi
neer
i
ng,
Un
i
ver
sit
y o
f Anba
r,
11, Rama
di,
Iraq.
Emai
l:
om
a
r.d
ahh
a
m@
uoan
ba
r.
e
du.iq
1.
INTROD
U
CTION
In
m
od
e
rn
po
wer
sy
ste
m,
t
he
proces
s
of
ge
ner
at
io
n
a
nd
distribu
ti
on
of
el
ect
rical
energ
y
is
co
mp
le
x
.
It
pro
du
ce
s
ne
w
an
d
diff
ic
ult
chall
eng
es
t
o
the
stu
dy
of
sta
bili
ty
in
or
der
t
o
ens
ure
a
sta
ble
an
d
r
el
ia
ble
op
e
rati
on
of
powe
r
syst
em
in
the
e
ven
t
of
s
udde
n
f
aults
an
d
dist
urba
nces.
F
A
CTS
co
ntr
ollers
offe
r
set
tl
ements
to
these
pr
ob
le
ms
wh
ic
h
a
re
c
ompete
nt
of
pre
vaili
ng
t
he
s
yst
em
paramet
er
s
in
order
t
o
e
nh
a
nce
vo
lt
age
,
ste
a
dy
sta
te
and
t
ransi
ent
sta
bili
ty
of
t
he
netw
ork
[1].
He
nce
it
is
necessa
r
y
to
pro
du
ce
meth
ods
f
or
the
pur
pose
of
op
ti
mal
locat
i
on
of
F
ACTS
dev
ic
es
sat
isfyi
ng
ce
rtai
n
c
onstrai
nts
.
As
a
resu
lt
,
c
onve
nt
ion
al
FA
CTS
ha
ve
va
rio
us
ben
e
fits
an
d
D
-
FA
CT
S
ha
ve
e
xtra
be
nef
it
s
a
nd
wil
l
be
e
xp
la
i
ned
.
D
-
FA
CT
S
represen
t
an
al
te
r
native
dev
ic
e
bas
ed
on
e
ff
ic
ie
nt
cost
re
du
ct
io
n
so
l
ut
ion
t
o
t
he
loa
d
flo
w
pro
blem
[2]
.
T
he
im
pacts
of
so
me
of
the
se
de
vices
on
th
e
sta
bili
ty
of
a
wind
farm,
especial
ly
D
-
S
TATC
OM,
S
VC,
a
nd
F
uzz
y
S
V
C
con
t
ro
ll
er
are
pr
ese
nted
in
[3
].
A
stu
dy
of
e
nh
a
nceme
nt
of
po
wer
qu
al
it
y
by
usi
ng
D
-
F
ACTS
in
distri
bu
te
d
netw
orks
is
pr
opos
e
d
in
[
4].
The
goal
of
pa
per
is
to
ad
dre
ss
the
iss
ue
of
sta
bili
ty
an
d
qual
it
y
of
powe
r
at
the
degree
of
co
m
bin
at
io
n
of
wi
nd
tu
r
bin
e
a
nd
diesel
ge
ne
rator
s
ys
te
ms
bu
i
lt
in
D
-
FA
CT
S
s
ys
te
ms
s
uch
a
D
VR
for
s
us
ta
ina
ble
el
ect
rical
pow
er
gen
e
rati
on.
T
he
im
porta
nc
e
of
F
ACTS
f
or
the
pur
pose
of
c
ompens
at
ion
of
reacti
ve
powe
r
in
t
he
der
e
gula
te
d
po
wer
s
yst
em
is
prese
nted
in
[
5].
It
is
ne
cessar
y
to
co
mp
e
ns
at
e
for
r
eact
ive
powe
r,
bo
os
t
t
he
act
ual
act
i
ve
powe
r
flo
ws
i
n
t
he
el
ect
rical
trans
mis
sion
li
ne,
e
nhance
t
he
e
ff
ic
i
ency
of
transmissi
on,
enh
a
nce
sta
bili
ty
of
s
ys
te
m
and
be
in
a
n
entire
posit
ion
to
econo
mize
the
fo
s
sil
fu
e
ls
fo
r
afterti
me.
A
st
udy
of
a
n
op
ti
mal
locat
io
n
method
bas
e
d
on
util
iz
at
ion
of
the
sensiti
vi
ty
of
t
otal
po
wer
loss
with
res
pect
to
con
t
ro
l
par
am
et
ers
of
de
vice
s
is
pr
ese
nted
i
n
[
6].
A
meth
odol
ogy
for
opti
mal
locat
ion
ba
sed
Evaluation Warning : The document was created with Spire.PDF for Python.
In
t J
P
ow Elec
& Dri S
ys
t
IS
S
N: 20
88
-
8
694
Transient
sta
bili
ty
en
hance
me
nt in
t
he prese
nce
of opti
m
ally …
(
Omar
K
amil
Dahh
am Al
az
z
awi
)
1351
on
the
us
e
of
a
n
e
ff
ic
ie
nt
(PI)
sensiti
vity
i
s
pro
posed
in
[
7]
.
A
n
al
go
rithm
pro
po
se
d
f
or
o
ptimal
placem
ent
o
f
FA
CTS
de
vices
to
get
the
des
ired
vo
lt
age
st
abili
ty
with
op
ti
mal
siz
ing
of
FA
CTS
de
vices
is
pr
ese
nted
i
n
[
8]
.
A
st
ud
y
of
t
he
opti
mal
lo
cat
ion
of
F
A
CTS
dev
ic
es
us
in
g
non
–
dominate
d
s
ort
ing
pa
rtic
le
swar
m
op
ti
miza
ti
on
i
n
Fu
zz
y
f
ram
ewor
k
is
pr
opos
e
d
i
n
[9]
.
F
irefly
opti
miza
ti
on
al
gorithm
f
or
op
ti
miza
ti
on
the
rati
ng
of
UPF
C
is
prese
nted
in
[
10]
.
T
he
opti
mal
locat
io
n
base
d
on
(
L
-
ind
e
x)
a
nd
the
pro
po
se
d
al
gor
it
hm
is
com
par
e
d
with
Ge
netic
Algo
rithm
T
he
pro
blem
of
c
onge
sti
on
w
hich
m
eans
(the
tra
nsmi
ssion
s
ys
te
m
will
op
e
rate
beyo
nd
tra
nsfer
li
mit
s)
is
st
ud
ie
d
in
[
11]
.
D
-
FA
CT
S
are
the
de
velop
e
d
FA
CT
S
t
echnolo
gy
wh
i
ch
a
re
distrib
uted
t
hroug
h
the
tra
nsmi
ssion
li
ne
a
s
a
small
de
vice
betwee
n
th
e
small
sp
a
n
of
tran
smissi
on
li
ne
as
sh
ow
n
in
Fig
ure
1
.
T
he
y
can
off
er
se
ries
c
apacit
ive
or
i
nductive
reacta
nce
in
li
ne
th
ough
SS
T
(sing
le
tur
n
trans
forme
r)
.
A
dva
ntages
of
D
-
F
ACTS
ove
r
F
ACTS
de
vic
es
are
[
12]:
1
-
It
do
e
s
no
t
r
equ
i
re
br
ea
k
i
n l
ine.
2
-
Small
modu
le
s
(10 K
va
a
nd 50
-
60
kg
per m
odule).
3
-
Com
patible
and easi
ly
upgr
adab
le
.
4
-
B
ypasses the
curre
nt s
o no cha
nc
es to
get
dama
ged.
5
-
Co
nv
e
ntio
na
l t
ran
smissi
on li
nes
ca
n be
use
d.
6
-
Distrib
uted o
ve
r
t
he whole
li
ne.
7
-
N
o
e
ff
ect
of sin
gle point
fa
il
ur
e.
8
-
D
oes
no
t
re
qu
ire
mai
ntena
nce,
i
f re
qu
i
red n
ot c
os
tl
y.
The
idea
of
Distribu
te
d
Stat
ic
Synchr
onous
Series
Co
mp
e
ns
at
or
(
DSSC
)
is
to
em
ploy
a
low
-
po
wer
sing
le
phase
i
nverter
, which
a
tt
aches
to
the
t
ran
s
missi
on
c
onduct
or
a
nd
e
ffec
ti
vely
c
on
t
rol
s
the
c
orres
pond
i
ng
trans
fer
im
pe
da
nce a
nd the
ac
ti
ve
co
ntr
ol of
powe
r
fl
ow on
the li
ne
is ac
hi
eved as s
how
n i
n
F
i
gu
re
2
[
13
].
Figure
1. D
-
F
ACTS
dif
fuse i
n
tra
ns
missi
on
li
ne
Figure
2
.
Sc
he
mati
c d
ia
gram
of D
S
SC m
odul
e
2.
PRI
NC
I
PLE
OF OPE
R
ATION
OF
DSS
C
2.1.
Ac
tive
p
ower
flo
w
c
on
c
epts
The
series
el
ements
ha
ve
th
e
hi
gh
e
st
pros
pect
for
powe
r
flo
w
c
ontr
olli
ng
on
tra
ns
mi
ssion
li
nes
.
M
a
gn
it
udes
of
1
and
2
gove
r
n
reacti
ve
an
d
act
ive
powe
r
f
low
in
t
he
li
ne
in
ad
diti
on
to
phase
a
ng
le
diff
e
re
nce, δ
[14]
:
12
=
1
∗
2
∗
si
n
(1)
12
=
2
−
1
∗
2
∗
c
os
(2)
Wh
e
re
represe
nts
the
li
ne
im
ped
a
nce.
S
o,
t
he
ch
an
ge
of
t
his
impe
da
nce
can
be
implem
ented
by
a
series
injec
ti
on
of
pa
ssive
inducti
ve
or
ca
pacit
ive
dev
ic
e
in
t
he
li
ne
.
The
re
dundanc
y
giv
es
f
or
c
onti
nuou
s
op
e
rati
on
wh
e
n
a
fail
ure
occ
ur
s
in
t
he
unit
pro
vid
es
high
reli
abili
ty
an
d
SST
al
lo
ws
t
re
at
ing
high
val
ues
of
fau
lt
c
urren
t.
2.2.
D
-
F
AC
T
ob
j
ect
and
oper
at
i
onal pr
ofi
le
The
act
ive
im
pe
dan
ce
i
njuncti
on
is
produce
d
by
s
ynch
ron
ous
volt
age
s
ou
rce
as
sho
wn
i
n
Fig
ur
e
s
3
and 4
[
15]:
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2088
-
8
694
In
t J
P
ow
Ele
c
&
D
ri
S
ys
t,
V
ol
.
11
, N
o.
3
,
Se
ptembe
r
2020
:
135
0
–
13
59
1352
Figure
3. The
s
yn
c
hro
nous v
ol
ta
ge
source
(
S
VS
)
It h
as
the
f
ollo
wing
functi
ons
:
a. Inject
s an A
.
C v
oltage,
. b. C
on
tr
ols
with
resp
ect
t
o
c. C
hanges
ef
fecti
ve
li
ne
I
mp
e
da
nc
e.
The give
n plot
exp
la
in
it
s fu
nc
ti
on
.
Figure
4. S
yn
c
hro
nous
volt
ag
e sour
ce
phas
or
diagr
a
m
A
sin
gle t
op
ic
sy
m
boli
zes al
l t
he
D
-
F
ACTS
dev
ic
es
on a
li
ne
a
re:
−
Connect
s to
a t
ran
s
missi
on li
ne
.
−
All value
s ar
e
per p
hase.
−
Normall
y,
47
µH
per
m
odule
.
−
assign ma
ximum n
umbe
r of
modu
le
s
.
−
e. Ap
po
i
nt ma
ximum a
nd mi
nimum li
ne
acti
vation
c
urre
nt.
Figure
5
s
how
s
the
D
-
FA
CT
operati
onal
prof
il
e.
Be
lo
w
0
,
the
D
-
F
ACTS
dev
ic
es
are
i
na
ct
ive
an
d
above
the
pro
gr
essi
ve
i
nject
ion o
f
t
he D
-
F
A
CTS de
vices is
at it
s max
im
um val
ue.
Figure
5. D
-
F
ACT
op
e
rati
on
profil
e
The
D
-
FA
CT
modu
le
c
onsist
s
of
a
sin
gle
-
phase
i
nv
e
rter
with
s
mall
rate
d
(10
K
Va)
a
nd
the
wei
gh
t
of
the
mod
ule
in
the
range
of
(
50
-
60)
kg
pe
r
m
odule.
I
f
the
c
urrent
in
the
li
ne
is
great
er
tha
n
t
he
pr
e
determi
ne
d
value,
i
ncr
eas
ing
the
num
be
rs
of
m
odules
to
be
s
witc
he
d
an
d
as
a
re
s
ult
increase
th
e
li
ne
impeda
nce
.
O
ur
ta
r
get
is
to
av
oid
th
e
li
ne
s
f
r
om
ther
mal
ove
r
l
oad.
T
he
co
ntr
ol
al
gorith
m
of
de
v
ic
e
op
e
rati
on is
de
fine
d:
=
−
0
ℎ
−
0
,
(3)
: Li
ne
i
nducta
nce in
je
ct
e
d,
:
Def
i
niti
ve
val
ue
of in
duct
ance
w
it
h
al
l m
od
ules on t
he
li
ne
a
ct
ive.
0
: t
rig
ger
in
g va
lue of c
urre
nt of th
e
m
odule,
ℎ
:
Thermal
li
mit
b
ey
ond w
hich
there is
no inje
ct
ion
.
An exp
on
e
ntial
ly d
eca
ying
est
imat
or
is
nee
de
d
to
mini
mize
interact
io
ns
bet
ween l
ines a
nd
mod
ules:
Evaluation Warning : The document was created with Spire.PDF for Python.
In
t J
P
ow Elec
& Dri S
ys
t
IS
S
N: 20
88
-
8
694
Transient
sta
bili
ty
en
hance
me
nt in
t
he prese
nce
of opti
m
ally …
(
Omar
K
amil
Dahh
am Al
az
z
awi
)
1353
= (
−
) (1
-
−
(
−
0
)
)
+
(4)
Wh
e
re
eq
uals
act
ual
inju
nction re
quest
at e
very s
a
mp
li
ng
mo
me
nt.
2.3.
Simul
at
io
n of
p
ower sy
s
tem u
sin
g po
wer
wo
rl
d
sim
ulator
Power
w
or
l
d
s
imulat
or
is
a
c
om
me
rcial
–
gr
ade
powe
r
s
ys
t
em
a
nalysis
an
d
sim
ulati
on
pa
ckag
e
.
It
is
desig
ne
d
to
si
mu
la
te
el
ect
rical
power
syst
ems
ope
ra
ti
ng
on
a
ti
me
f
r
om
sever
al
min
utes
to
se
ver
al
da
ys
[
16]
.
It h
as
the
give
n ad
van
ta
ges: [1
7]
−
Flexible a
nd ea
sy
for user
s to
interact
w
it
h al
l app
li
cat
io
ns
i
n
the
scr
ee
n.
−
It is a
us
er
-
fr
ie
nd
l
y Gr
a
phic
al
U
se
r
I
nter
face
(
G
U
I)
a
s it
inc
lud
es
small
s
ource
and e
xec
ut
able file
s.
−
Allows d
ynami
c interact
io
n w
it
h
the a
nimate
d powe
r
fl
ow.
−
Pr
ovi
de
sel
ect
ion o
f fou
r
m
od
es of
op
e
rati
on
of D
-
F
ACT.
3.
METHO
D
O
F WEA
KEST
LOA
D
B
US
I
DENTIFI
C
A
TION
Diff
e
re
nt
te
ch
niques
are
pre
sented
in
papers
f
or
wea
ke
st
bus
ide
ntific
at
ion
[
18].
I
n
t
his
pa
per,
weak
e
st
bus
id
entifi
cat
ion
te
c
hn
i
qu
e
implem
ented
is
modif
ie
d
(
)
fact
or
usi
ng
le
ast
m
ea
n
s
qu
a
re
met
hod
(L
M
PI).
3.1.
Iden
tific
ati
on o
f
weakes
t
bu
s usin
g m
od
ifie
d
(
)
fa
c
t
or
(LMPI
)
Con
si
der
t
he
giv
e
n
sim
plifie
d
syst
em
w
hi
ch
co
ns
ist
s
of
a
load
bus
with
the
ven
i
n
ci
rc
uit
to
represe
nt
the
r
est
of
sy
ste
m
as
simple
rad
i
al
feed
t
o
loa
d
wh
ic
h
se
rv
e
s
a
certai
n
big
ne
twork
us
i
ng
s
ing
le
transmissi
on li
ne.
Simpli
fie
d t
hev
e
nin
e
quiv
al
ent circuit
of
local
bus
as
s
how
n
i
n
Fi
gure
6.
Figure
6. Sim
pl
ifie
d
the
ven
i
n equ
i
valent circ
uit o
f
local
bus
The val
ue
of c
urren
t i
s
[19]:
I=
ℎ
√
(
ℎ
∗
c
os
+
∗
c
os
∅
)
²
+
(
ℎ
∗
sin
+
∗
sin
∅
)
²
2
(5)
I=
ℎ
√
ℎ
2
+
2
+
2
ℎ
c
os
(
−
∅
)
2
(6)
wh
e
re
:
ℎ
=
|
ℎ
|
∟Ɵ
,
=
|
|
∟
∅
, T
he r
ecei
vin
g e
nd volt
age
of the
syst
em is g
i
ven by
=
I=
ℎ
√
ℎ
2
+
2
+
2
ℎ
c
os
(
−
∅
)
2
(7)
The
t
otal ap
parent p
ower
prov
ided
i
n
the
loa
d
is
S=
²
=
ℎ
²
ℎ
2
+
2
+
2
ℎ
c
os
(
−
∅
)
(8)
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2088
-
8
694
In
t J
P
ow
Ele
c
&
D
ri
S
ys
t,
V
ol
.
11
, N
o.
3
,
Se
ptembe
r
2020
:
135
0
–
13
59
1354
A plot is
dr
a
w
n betwee
n
the
app
a
re
nt po
we
r
S
an
d
l
oad ad
mit
ta
nce n as s
how
n
in
Fig
ure
7
.
Assumin
g
ℎ
=
1
.
nd
ℎ
=1 p.
u for th
re
e d
if
fer
e
nt
valu
es of it
em
(
−
∅
)
=
70
º
,
100º a
nd 12
0º
.
Figure
7. The
Loa
d Appare
nt
Power v
s. Loa
d
Admit
ta
nce
It
is
cl
ear
f
rom
Fi
gure
7
t
ha
t
the
inc
rease
i
n
l
oad
le
a
ds
to
an
i
ncr
e
menta
l
val
ue
of
l
oad
ad
mit
ta
nce
un
ti
l
reac
hing max
im
um
,
f
or
h
ig
her
l
oad
d
e
man
d,
t
he
act
ion
is dif
fer
e
nt an
d
unsta
ble
l
oa
d
va
riat
ion
is
pr
ese
nt
[20].
We
hav
e
(
)
=
ℎ
²
(
1
−
2
ℎ
2
)
(
1
+
2
ℎ
2
+
2
ℎ
c
os
(
−
∅
)
)
²
(9)
Fo
r
ma
xim
um
load
c
onditi
on,
(
)
=
0
le
ads
to
t
he
s
olu
ti
on
for
c
riti
cal
v
olta
ge
inst
abili
ty whe
n
ℎ
=
(10)
Since
(
)
e
qu
al
s
un
it
y
at
no
l
oa
d
(Y
=
0)
a
nd
e
qu
al
s
zer
o
at
c
riti
cal
maxim
um
loa
d
w
he
n
ℎ
=
,it
is
more
acc
ur
at
e
t
o
us
e
it
for
cal
c
ulati
on
operati
ng
points
a
nd
ide
ntif
y
the
op
e
rati
ng
c
onditi
ons
of
the
sy
ste
m.
P
oin
ts
belo
w
the
ma
xi
mu
m
l
oad
i
ng
represe
nt
the
f
avo
ra
ble
opera
ti
ng
c
onditi
ons
.
The
re
fore,
it
can
be
us
e
d
f
or
the
id
entifi
cat
ion
of
weak
e
st
bus
an
d
as
a
resu
lt
th
e
opti
mal
locat
ion
of
F
ACT
de
vice.
T
he
pro
po
s
ed
value
is:
(
) =
2
−
1
2
−
1
(11)
It
is
neces
sar
y
to
stu
dy
t
he
c
apab
il
it
y
of
ex
tract
i
ng
a
ne
w
in
dicat
or
base
d
on
t
his
facto
r
a
nd
t
hen
determi
ne
the
best
fit
t
o
data.
It
us
es
the
m
et
hod
of
Least
Squa
res
for
e
rror
re
du
ct
io
n
and
obta
in
a
be
st
fit
wh
ic
h
ma
ke
th
e
resu
lt
s
m
or
e accur
at
e
a
nd
it
is
cal
le
d
Least
M
ea
n
Squa
re
P
ow
e
r
I
ndic
at
or
(
L
M
PI). W
e
ha
ve
a
data [
21].
{(
1
,
1
,
)
,
…
…
…
(
,
,
)
}
(12)
The
n
the
er
ror a
sso
ci
at
ed
t
o y
= ax+
b may
be de
fine
d
as:
E(a, b
)
=
∑
(
=
1
-
(a
+b)) ²
(13)
As
the
er
ror
de
pends
on
tw
o
va
riables
(a
a
nd
b)
t
he
n
our
ta
r
get
is
to
cal
cul
at
e
the
val
ues
of
(a
a
nd
b)
su
c
h
that t
o
mi
nimize
the
er
ror.
T
he
refor
e
, it
is require
d
t
hat to calculat
e t
he
v
al
ue
s
of
(a a
nd b) s
uc
h
that
(
)
=
(
∑
²
=
1
∑
=
1
∑
=
1
∑
1
=
1
)
-
¹
(
∑
=
1
∑
=
1
)
(14)
0
0
.
2
0
.
4
0
.
6
0
.
8
1
1
.
2
1
.
4
1
.
6
1
.
8
2
0
0
.
1
0
.
2
0
.
3
0
.
4
0
.
5
0
.
6
0
.
7
0
.
8
0
.
9
1
L
o
a
d
A
d
m
i
t
t
a
n
c
e
(
p
.
u
)
A
p
p
e
a
r
e
n
t
P
o
w
e
r
(
p
.
u
)
A
p
p
e
a
r
e
n
t
L
o
a
d
p
o
w
e
r
v
s
,
L
o
a
d
A
d
m
i
t
t
a
n
c
e
70
120
100
Evaluation Warning : The document was created with Spire.PDF for Python.
In
t J
P
ow Elec
& Dri S
ys
t
IS
S
N: 20
88
-
8
694
Transient
sta
bili
ty
en
hance
me
nt in
t
he prese
nce
of opti
m
ally …
(
Omar
K
amil
Dahh
am Al
az
z
awi
)
1355
The
best
fit
of
values
of
(a
a
nd
b)
can
be
obta
ined
by
s
olv
i
ng
the
(
14)
a
nd
c
omplet
e
the
so
l
ution
in
order
to
cal
cu
la
te
the
L
M
P
I
in
dicat
or.
T
he
bus
with
mi
nimum
LMPI
represe
nts
the
wea
kest
bus
of
th
e
netw
ork.
3.2. R
otor
ang
ular s
tabil
ity
Rotor
a
ngle
sta
bili
ty
ind
ic
at
e
the
capa
bili
ty
of
a
po
wer
sy
s
te
m
to
sta
y
in
s
yn
c
hro
nous
a
fter
a
disturba
nce.
I
f
the
dist
urbance
is
near
the
ge
ne
rator,
t
he
ge
ne
rator
ca
n
c
ollapse
of
po
i
nt
as
it
ha
s
be
en
acce
le
rated du
r
ing
fau
lt
. T
he values
of ti
me f
or s
uch
a
n
in
s
ta
bili
ty is o
ne
t
o
tw
o
sec
onds
and
it
is in s
ha
pe
of a
per
i
od
ic
a
ngul
ar
se
par
at
io
n
due
t
o
los
s
of
s
yn
c
hro
nizin
g
t
orq
ue.
[22
].
T
he
giv
e
n
Fig
ure
8
sho
ws
t
he
r
oto
r
ang
le
res
pons
e
for
t
wo stat
es
[23].
Figure
8. Roto
r
angle res
pons
e to a
t
ran
sie
nt
disturba
nce
In
powe
r
s
ys
t
em
sta
bili
ty
st
udy,
the
s
wi
ng
e
qu
at
io
n
re
presents
the
vari
at
ion
of
ro
t
or
an
gle
δ
(in
el
ect
rical
r
adia
ns
)
w
it
h res
pec
t t
o
ti
me (
i
n
se
conds
)
as
in
dicat
ed
P_
a=P
_m
-
P_
e
= M d2
δ/dt²
(15)
The
s
olu
ti
on
pro
vid
es
t
he
dat
a
need
e
d
for
s
wing
cu
r
ves
an
d
w
he
n
the
ma
chine
is
loa
de
d,
ro
t
or
a
ngle
will
b
e c
hange
d
a
nd an ex
pre
ssion bas
ed
on swin
g [24]
:
dδ/d
t=[
]
t
(16)
4.
RESU
LT
S
AND DI
SCUS
S
ION
4.1.
Te
st
Syst
em (IEEE
9
-
B
us)
with
D
-
F
A
CT
The
IE
EE
-
ni
ne
bus
te
st
netw
ork
is
s
how
n
in
Fig
ure
9
a
nd
the
netw
ork
da
ta
are
giv
e
n
i
n
[
25].
The
netw
ork
is
m
odel
le
d
by
us
in
g
powe
r
w
or
ld
simulat
or
wit
h
t
he
i
nclusion
a
nd
with
ou
t
D
-
F
ACT
t
o
e
xecu
t
e
powe
r
s
ys
te
m
sta
bili
ty stu
dies in
a
ddit
ion
t
o l
oad
-
fl
ow pr
ogra
m fo
r
the
purp
os
e
of w
ea
ke
st b
us
ide
ntifi
cat
ion
.
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2088
-
8
694
In
t J
P
ow
Ele
c
&
D
ri
S
ys
t,
V
ol
.
11
, N
o.
3
,
Se
ptembe
r
2020
:
135
0
–
13
59
1356
Figure
9. IEE
E
-
Ni
ne bu
s
s
ys
te
m
4.2.
Iden
tific
ati
on o
f
weakes
t
bu
s usin
g
(
)
The
cal
culat
io
n
is
base
d
on
loading
the
th
r
ee
loa
d
buses
(5,
7,
9)
from
normal
loa
ding
t
o
(
1.5)
loa
di
ng
in
ste
ps
of 10%
.
In eac
h
ste
p
the
f
act
or
(
)
is cal
culat
ed
acc
ordi
ng to
e
q. (
11),
The res
ults are
li
ste
d
in Ta
ble
1.
Table
1.
(
)
Va
ria
ti
on
of I
EE
E
9
-
Bus
with B
us
Loa
ding
Bu
s No
.
Load
Bu
s Incre
m
e
n
tal L
o
ad
in
g
10%
0
.90
6
20%
0
.90
5
4
30%
40%
50%
5
0
.86
3
4
0
.85
7
1
0
.83
1
7
7
0
.90
1
8
0
.88
5
8
0
.88
5
5
0
.85
0
6
0
.82
9
4
9
0
.84
2
4
0
.79
5
0
.75
4
5
0
.74
4
7
0
.72
3
1
It
is
cl
ear
from
the
ta
ble
gi
ve
n
that
loa
d
bu
s
(9)
is
the
wea
kest
bus
in
t
he
sy
ste
m.
A
gr
a
ph
is
plo
tt
ed
as sho
wn Fig
ure 1
0 rep
rese
nts the rel
at
ion
be
tween
(
)
fact
or
a
nd d
i
ff
e
ren
t l
oa
d bus loa
ding
s.
Figure
10. Vari
at
ion
of
(
)
Fact
or of IEEE
9
-
B
us wit
h
B
us
Loa
di
ng
4.3.
Iden
tific
ati
on o
f
weakes
t
bu
s usin
g m
od
ifie
d
(
)
with
LMPI
The
cal
c
ulati
on
of
(
)
is
en
ha
nc
ed
by
pro
du
c
ing
a
new
in
di
cat
or
f
or
wea
ke
st
bus
i
den
ti
fi
cat
ion
.
The
i
nd
ic
at
or
LMPI
base
d
usi
ng
le
ast
mea
n
s
qu
a
re
meth
od
f
or
the
val
ues
of
(
)
and
th
e
lowe
st
val
ue
s
represe
nts the weake
st b
us
a
s
shown i
n
Ta
bl
e 2
.
Table
2.
L
MPI
V
a
riat
ion
of
I
EEE
9 wit
h
Bu
s Lo
a
ding
Bu
s No
.
Load
Bu
s Incre
m
e
n
tal L
o
ad
in
g
10%
20%
30%
40%
50%
5
0
.91
2
1
0
.89
2
4
0
.87
2
7
0
.85
3
0
.83
3
3
7
0
.90
6
6
0
.88
8
6
0
.87
0
6
0
.85
2
5
0
.83
4
5
9
0
.82
9
6
0.
8
0
0
7
8
0
.77
1
8
0
.74
2
9
0
.71
4
Figure
11
is
pl
otted
w
hich
re
pr
ese
nt
t
he
rel
at
ion
of
LMPI
in
dicat
or
with
di
ff
e
ren
t
loa
d
bu
s
l
oad
i
ng
wh
ic
h
i
nd
ic
at
e
s
the
same
res
ult.
T
his
mean
s
that
t
he
us
e
of
m
od
i
fied
(
)
is
us
e
f
ul
an
d
m
or
e
accu
rate
a
nd
op
ti
mal.
0
0
.
0
5
0
.
1
0
.
1
5
0
.
2
0
.
2
5
0
.
3
0
.
3
5
0
.
4
0
.
4
5
0
.
5
0
.
6
5
0
.
7
0
.
7
5
0
.
8
0
.
8
5
0
.
9
0
.
9
5
1
%
L
o
a
d
B
u
s
I
n
c
r
i
m
e
n
t
a
l
L
o
a
d
i
n
g
d
S
/
d
Y
V
a
r
i
a
t
i
o
n
o
f
d
S
/
d
Y
w
i
t
h
L
o
a
d
C
h
a
n
g
e
o
f
A
l
l
L
o
a
d
B
u
s
e
s
B
u
s
5
B
u
s
7
B
u
s
9
Evaluation Warning : The document was created with Spire.PDF for Python.
In
t J
P
ow Elec
& Dri S
ys
t
IS
S
N: 20
88
-
8
694
Transient
sta
bili
ty
en
hance
me
nt in
t
he prese
nce
of opti
m
ally …
(
Omar
K
amil
Dahh
am Al
az
z
awi
)
1357
Figure
11. Vari
at
ion
of L
M
PI
of
IEE
E
9
-
B
us wit
h
B
us
Loa
di
ng
4.
4
.
IEE
E 9
-
B
us w
i
th
D
-
FA
CT
st
ab
il
ity
s
t
udy usin
g
p
ower
wo
rl
d
sim
ulator
The
e
nhance
m
ent
of
powe
r
sy
ste
m
sta
bili
ty
is
im
pleme
nt
ed
by
inse
rtio
n
of
D
-
F
ACT
in
t
he
li
ne
connecte
d
wea
kest
bus
a
nd
li
ne
(
9
-
4)
is
sel
ect
ed
f
or
D
-
F
ACT
placeme
nt
an
d
the
n
st
ud
ie
d
by
us
i
ng
powe
r
world
simulat
or
program
. T
he
pro
ces
s step
s
are li
ste
d belo
w:
a. L
oad flow i
mp
le
me
ntati
on u
si
ng N
e
wton
Ra
ph
s
on met
hod usi
ng
powe
r
w
orl
d
si
mu
la
t
or.
b.
Ca
lc
ulati
on
the
desi
gn
pa
r
amet
ers
of
D
-
FA
CT
base
d
onli
ne
fl
ow
in
l
ine
(
9
-
4)
an
d
i
mp
le
me
nt
in
D
-
F
ACT
m
odule
on
the
li
ne
as
sho
wn
in
Fi
gure
12
.
The
li
ne
is
l
oa
ded
to
it
s
f
ull
rati
ng
f
or
te
sti
ng
the
s
ys
te
m
to ensu
re
D
-
F
ACT is act
ive
and sele
ct
li
mit
b
ase
d o
n
c
urre
nt ope
rati
on m
od
e
.
Figure
12. D
-
F
ACT
desig
n pa
r
amet
ers
in
li
ne
(4
-
9)
Thr
ee
-
ph
ase
f
a
ult i
n
li
ne
(8
-
9)
a.
Wh
e
n
the
Faul
t i
s clea
red
: A t
hr
ee
phase fa
ul
t i
s ap
plied in
l
ine (8
-
9)
at (
=1
sec.
)
an
d
the
fau
lt
is cle
ared
and
the
sy
ste
m
is
ret
urned
to
it
s
pr
e
vious
sta
te
at
=1.2
6
sec.
Fig
ur
e
13
sho
ws
a
re
du
ct
i
on
in
ro
t
or
an
gle
of
Gen.
#2
be
cause
of
us
i
ng
D
-
F
ACT.
T
he
same
te
sts
are
im
plemente
d
for
Ge
n.
#3
a
nd
give
s
t
he
sa
me
con
cl
us
io
n
as
s
how
n
in
Fig
ure
14.
b.
Wh
e
n
the
Faul
t
in
Line
(
8
-
9)
is
cl
eared
by
Op
e
ning
the
L
ine.
Fig
ure
15
sh
ows
Ge
n
.
#2
ro
to
r
a
ng
le
w
hen
fau
lt
is
cl
ear
ed
by
op
e
ni
ng
th
e
li
ne
at
t_c
r=
1.158
sec.
W
hi
le
Figure
16
s
hows
Ge
n.
#3
ro
t
or
a
ngle
when
fau
lt
is cle
are
d by ope
ning the
li
ne
at
t_c
r=1.
158
s
ec.
c.
Thr
ee
phase
F
ault
in
Bu
s
(
8).
A
nother
ty
pe
of
fa
ult
is
stu
died
w
hich
is
t
he
fa
ult
in
bus
(8).
The
fa
ult
is
cl
eared
by
rec
onnecti
on
of
t
he
br
ea
ke
rs
of
bu
s
(8)
t
o
t
he
netw
ork
with
=1.26
sec.
Fi
gure
17
sho
ws
a
reducti
on in
ro
tor
a
ngle
of
Ge
n.
#2
because
of u
si
ng D
-
F
A
CT. Fig
ure
18
sh
ows
the s
am
e test
f
or
Ge
n.
#3.
0
.
1
0
.
1
5
0
.
2
0
.
2
5
0
.
3
0
.
3
5
0
.
4
0
.
4
5
0
.
5
0
.
7
0
.
7
5
0
.
8
0
.
8
5
0
.
9
0
.
9
5
%
L
o
a
d
B
u
s
I
n
c
r
i
m
e
n
t
a
l
L
o
a
d
i
n
g
L
M
P
I
I
n
d
i
c
a
t
o
r
V
a
r
i
a
t
i
o
n
o
f
L
M
P
I
I
n
d
i
c
a
t
o
r
w
i
t
h
L
o
a
d
C
h
a
n
g
e
f
o
r
L
o
a
d
B
u
s
e
s
B
u
s
5
B
u
s
7
B
u
s
9
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2088
-
8
694
In
t J
P
ow
Ele
c
&
D
ri
S
ys
t,
V
ol
.
11
, N
o.
3
,
Se
ptembe
r
2020
:
135
0
–
13
59
1358
Figure
13. Ge
n.
2 r
otor a
ng
le
wh
e
n 3
-
phase
f
ault
Figure
14. Ge
n.
3 r
otor a
ng
le
wh
e
n 3
-
phase
f
ault
Figure
15. Ge
n.2 r
oto
r
angle
wh
e
n 3
-
phase
Figure
16. Ge
n.3 r
oto
r
angle
wh
e
n 3
-
phase
f
ault
Figure
17. Ge
n.
2 r
otor a
ng
le
wh
e
n
3
-
phase
Figure
18. Ge
n.
3 r
otor a
ng
le
wh
e
n 3
-
phase
f
ault at
bu
s
8 fa
ult at
bus 8
5.
CONCL
US
I
O
NS
The o
ptimal
lo
cat
ion
a
nd desi
gn sett
ing
s
of
D
-
F
ACTS
for t
ran
sie
nt stabil
it
y
is
discu
ssed
. T
he desig
n
values
of
D
-
F
ACT
are
im
ple
mented
us
i
ng
powe
r
w
or
l
d
simula
to
r
softwa
re
an
d
diff
e
rent
fau
lt
s
we
re
a
pp
li
ed
in
diff
e
ren
t
pl
aces
in
the
net
work.
T
he
res
ults
sho
w
a
n
e
nh
a
nceme
nt
i
n
r
otor
a
ng
le
a
nd
po
wer
fl
ow
in
t
he
netw
ork.
T
he
s
la
ck
bus
in
the
netw
ork
is
Ge
n.
#1
a
nd
was
no
t
c
on
si
der
e
d
becau
se
it
do
es
no
t
af
fect
by
th
e
changes
in
the
net
wor
k
due
t
o
it
s
hi
gh
ca
pa
ci
ty,
wh
ic
h
ma
ke
it
sla
ck.
T
he
ti
me
durati
on
was
decr
eas
ed
t
o
3
sec
onds
as
it
is
imp
or
ta
nt
in
s
uch
ty
pe
of
dr
a
wings
with
high
value
s
of
r
otor
a
ng
le
s
°
to
be
s
how
n
in
the dra
wings.
REFERE
NCE
S
[1]
Singhal
,
P.,
Aga
rwal,
S
.
K.
,
&
Kumar
,
N
.
,
"O
ptimzation
of
UP
FC
Control
l
er
Par
am
e
te
rs
Us
ing
Bac
t
eri
a
l
Forag
i
ng
Te
chn
ique
for
E
nhanc
ing
Pow
er
Sys
te
m
Stab
il
ity
,
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Ad
v
ance
ments
in
Te
chnol
ogy
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vo
l.
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,
no.
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2014
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Evaluation Warning : The document was created with Spire.PDF for Python.
In
t J
P
ow Elec
& Dri S
ys
t
IS
S
N: 20
88
-
8
694
Transient
sta
bili
ty
en
hance
me
nt in
t
he prese
nce
of opti
m
ally …
(
Omar
K
amil
Dahh
am Al
az
z
awi
)
1359
[2]
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D.
K
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&
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,
M.
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ew
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D
-
FA
CTS
Control
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nhanc
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Pow
er
Flow
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r
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urnal
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i
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r
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nsa
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SV
C
Control
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r
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y
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ind
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Inte
rnat
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ournal
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r
El
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ct
ronics
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iv
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bbar
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B
ena
ch
ai
ba
,
and
B
.
Maz
ar
i,
"P
owe
r
quality
enha
n
ceme
nt
by
using
D
-
FA
CTS
sys
te
ms
appl
i
ed
to
di
stribut
ed
genera
ti
on
,
"
Inte
rnat
i
onal
Journal
o
f
Pow
er
Elec
tronic
s
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Dr
i
v
e
Syste
ms
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PE
DS
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[5]
Packi
asudha
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ACT
Device
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iv
e
Pow
er
Compe
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n
the
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gu
la
t
e
d
Elec
tr
ical
Pow
er
Envi
ronm
ent
,
"
Inte
rnational
J
ournal
of
Pow
er
El
e
ct
roni
cs
and
Dr
iv
e
Syst
ems
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PE
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el
l
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N
.
,
"O
pti
ma
l
Lo
ca
t
io
n
and
Par
am
e
ter
Sett
ings
of
FA
CTS
Devic
es
for
Enha
nc
ing
Pow
er
Sys
te
m
St
abilit
y
and
Mini
mization
of
Pow
er
Losses
,
"
Int
ernati
onal
Journal
of
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ineering
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en
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Tec
hnology
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[7]
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sh,
Burad
e
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and
Helf
ord,
"O
ptim
al
Lo
ca
t
ion
of
FA
CTS
Devic
e
on
Enha
nc
ing
Sys
te
m
Secur
it
y
,
"
Inte
rnational
Jo
urnal
of
S
ci
en
ti
f
i
c
&
Engi
ne
ering Re
search
,
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,
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012.
[8]
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K.
S.
R
Anj
ane
y
ulu,
"M
a
int
a
ini
n
g
Volta
g
e
Stabil
it
y
by
Opti
ma
l
Loc
a
ti
ng
and
Si
zi
ng
by
Comb
in
ed
Evol
uti
on
ary
Al
gorit
hm
,
"
Int
ernati
onal Journal of
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pli
cations
,
vo
l.
8
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no
.
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,
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.
39
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[9]
M.
Sedighizadeh
,
H.
Fa
ramarz
i
and
S.
Fara
m
ar
zi
,
"O
pti
mal
Lo
ca
t
ion
and
Setti
ng
of
FA
CTS
devi
c
es
Us
ing
Non
-
Domina
t
ed
Sorti
ng
Parti
c
le
Sw
a
rm
Optimization
in
Fuzzy
Fr
a
m
ework
,
"
Inte
rnat
ional
Journal
o
n
Techni
ca
l
and
Phy
sica
l
Probl
e
ms
of
Engi
n
ee
ri
ng
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.
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,
no
.
15
,
pp
.
95
-
107
,
20
13.
[10]
P.
Ramesh
and
M.
Da
ma
dor
a
Reddy,
"Loss
Reduc
ti
on
throu
gh
Optimal
Pla
ce
m
ent
of
Unif
ie
d
Pow
er
–
Flo
w
Control
le
r
Us
in
g
Firefl
y
Algori
thm
,
"
In
te
rnati
o
nal
Journal
of
Adv
anc
ed
Re
sea
rch
in
El
e
ct
ri
ca
l,
El
e
ct
roni
cs
a
nd
Instrum
ent
ati
on
Engi
ne
ering
,
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ol
.
2
,
no
.
10
,
pp
.
4
657
-
4667,
2013
.
[11]
Raj
ase
k
ar
Thot
a
and
Sha
ik
Ha
me
ed
,
"O
ptimal
Locat
ion
of
U
nifi
ed
Pow
er
Fl
ow
Control
l
er
(
UP
FC
)
to
Contr
ol
Acti
ve
Pow
er
Flow
s
and
Conge
stion
Mana
g
em
e
nt
in
a
Tra
nsm
i
ss
ion
Li
nes
by
Us
ing
Fuzzy
Ba
sed
Techni
ques
,
"
Inte
rnational
Jo
urnal
of Emergi
ng
Tr
ends
&Technology in
Com
pute
r Sc
ie
nc
e
,
v
ol.
3
,
no
.
5
,
p
ages
77
-
82,
2014
.
[12]
S.
Noor
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