Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
V
o
l.
6, N
o
. 3
,
Ju
n
e
201
6, p
p
. 1
305
~ 13
18
I
S
SN
: 208
8-8
7
0
8
,
D
O
I
:
10.115
91
/ij
ece.v6
i
3.1
016
3
1
305
Jo
urn
a
l
h
o
me
pa
ge
: h
ttp
://iaesjo
u
r
na
l.com/
o
n
lin
e/ind
e
x.ph
p
/
IJECE
Voltage Instability of Initiation
Fault Duration as Influenced by
Nodes S
h
ort Ci
rcuit Levels NSCL
Y
o
ussef Mo
bara
k
1,2
, M
a
hm
oud
Hus
s
ein
2
1
Departem
ent
of
El
ectr
i
c
a
l
Engin
eering
,
F
a
cult
y
o
f
Engin
eer
ing
,
Rabigh, King Abd
u
laziz Un
iv
ers
i
t
y
,
S
a
udi
Arabi
a
2
Departem
ent
of
El
ectr
i
c
a
l
Engin
eering
,
F
a
cult
y
o
f
Energ
y
Eng
i
ne
ering,
As
wan Univers
i
t
y
,
Eg
yp
t
Article Info
A
B
STRAC
T
Article histo
r
y:
Received
Ja
n 20, 2016
Rev
i
sed
Mar
12
, 20
16
Accepted
Mar 29, 2016
The oc
currenc
e
of voltage
insta
b
ilit
ies or volt
a
ge coll
apses de
pend on the
duration of
the
persistence of
th
e fault
and
on the ty
pe of fau
l
t, some faults
lead
to vol
tage
instabil
iti
es, oth
e
rs lead
to vo
lt
age co
ll
apse.
E
v
alua
tion of
fault durations causing occurren
ce of
volta
ge
ins
t
abil
iti
es is
the
m
a
in goal
of
this paper. Th
is paper searches
for th
e effect
of nodes short
circu
it lev
e
ls
NSCL and its
duration p
e
riods
initiati
on of v
o
ltag
e
instab
ili
t
y
,
at
ce
rta
i
n
loads buses. The fault which leads to vo
ltage ins
t
ability
is found to be short
circu
its at certain nodes clear
ed
without
an
y
v
a
riation
in
the tr
ansmission
s
y
s
t
em
el
em
ents
, th
e pos
t-fau
lt
conditions
wi
ll
be the
s
a
m
e
as
t
h
e pre-f
a
ul
t
conditions
. The
power sy
stem dy
namic
simulation program is d
e
velop
e
d for
d
y
nam
i
c
an
al
ysi
s
of voltag
e
sta
b
ilit
y.
Mode
ls for loads,
in
this stud
y a
r
e
induction motor
s
with three d
i
fferen
t
shaft
mechanical lo
ads, constant
impedance C
Z
loads, constan
t
current
CI loads
and constant po
wer CP loads
are us
ed, as
th
e
y
d
e
pic
t
the b
e
havior of m
o
s
t
power s
y
s
t
em
loads
.
Th
e
influen
ce of th
e transmission n
e
twork impedan
ces, which
are
nearly
th
e
inverse of the NSCL, on the fau
lt dura
tion which lead to the occurrence of
voltag
e
inst
abil
ities
,
ar
e stud
ied
and
evalu
a
ted
using v
a
rious loa
d
representations
.
Keyword:
Power system
stab
ility
Vo
ltag
e
in
stab
ility
Vo
ltag
e
co
llapse
No
des
sh
ort
ci
r
c
ui
t
l
e
vel
s
Diffe
re
nt loa
d
s
Copyright ©
201
6 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
Yo
usse
f M
oba
rak
,
Electrical Engi
neeri
n
g De
part
ment,
Facu
lty of
En
gin
eer
ing
,
Rab
i
g
h
,
K
i
ng
A
bdulaziz U
n
iver
sit
y
,
Rab
i
gh
, 21
911
,
Saudi Arab
ia.
Em
a
il: yso
l
i
m
a
n
@k
au
.ed
u
.sa
1.
INTRODUCTION
Vo
ltag
e
in
stabilit
ies o
r
v
o
ltag
e
co
llap
s
es
dep
e
nd
o
n
th
e
du
ration
o
f
t
h
e
p
e
rsisten
ce of
th
e fau
lt and
o
n
th
e typ
e
o
f
fau
lt, so
m
e
fau
lts lead
t
o
v
o
ltag
e
in
stab
ilities,
o
t
h
e
rs lead to
v
o
ltag
e
co
llap
s
e. Ev
al
u
a
tio
n
of
fau
lt du
ration
s
cau
sing
o
c
cu
rren
ce of v
o
ltage in
stab
ilities o
r
co
llap
s
e is d
i
scu
ssed
in
[1
]-[3
]
. Vo
ltag
e
collap
s
e
has
recently gained i
n
creasi
n
g attention, t
h
is phe
n
om
enon is cha
r
act
e
r
ized by progressive fall
in voltage
m
a
gni
t
ude at
a
part
i
c
ul
ar l
o
ca
t
i
on fi
nal
l
y
spr
ead o
u
t
i
n
th
e
n
e
two
r
k
cau
s
ing
a co
m
p
lete syste
m
v
o
ltag
e
failu
re
o
r
a
b
l
acko
u
t
[4
]-[6
]. Th
e
phen
o
m
en
on
h
a
s
b
een attribu
t
ed m
a
in
ly to
th
e
in
ab
ility o
f
power system
to
meet a
certain load
de
m
a
nd of reac
tive power
[
7
]
-
[
9
]
.
The ef
fec
t
s of t
h
e exci
t
a
t
i
on sy
st
em
cont
rol
pa
ram
e
ters o
n
fau
lt duratio
n
cau
sing
v
o
ltage in
stab
ility in
itiatio
n
[10
]
.
Also
, th
e effects o
f
t
h
e ex
citatio
n
system
co
n
t
rol
p
a
ram
e
ters o
n
fau
lt du
ration cau
sing
to
reach
vo
ltag
e
collap
s
e in
itiatio
n
[1
1
]
.
Vo
ltage stab
ility
is t
i
g
h
tly
related
to lo
ad
ch
aracteristics an
d to
vo
ltag
e
mag
n
itu
d
e
s [1
2]
. It
i
s
di
rect
l
y
depe
n
d
ent
o
n
l
o
a
d
be
havi
o
r
wi
t
h
v
o
ltag
e
and
freq
u
e
n
c
y v
a
riatio
n
s
[13
]
. Th
e ex
act d
e
tectio
n
o
f
th
e
o
c
cu
rrence o
f
v
o
ltag
e
i
n
stab
ility p
h
e
no
m
e
n
a
i
s
m
a
i
n
l
y
depe
nde
nt
on
t
h
e c
o
r
r
ect
ness
o
f
t
h
e l
o
a
d
s
po
wer/v
o
ltag
e
and
reactiv
e po
wer/vo
ltag
e
relatio
ns [1
4
]
-
[15
]
. Recen
tly it h
a
s b
e
en su
gg
ested
to rep
r
esen
t t
h
es
e
relatio
n
s
in
p
o
l
yn
o
m
ial fo
rm
s
with certain
o
r
d
e
rs,
wh
ich
led
to
co
in
cid
e
n
ce
o
f
resu
lts of vo
ltag
e
stab
ility
cri
t
eria [1
6
]
. Th
e lo
ad
po
wer/vo
ltag
e
ch
aracteristics
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E
V
o
l
.
6,
No
. 3,
J
u
ne 2
0
1
6
:
13
0
5
– 13
18
1
306
can be de
ri
ve
d
from
t
h
e l
o
ad/
t
i
m
e
and
vo
ltag
e
/ti
m
e
ch
aracteristics, u
s
u
a
lly r
ecorded in a
ll power stations and
t
r
ans
f
o
r
m
e
r su
bst
a
t
i
ons
[
1
7]
-[
20]
.
The be
st
m
e
t
hod
fo
r l
o
a
d
re
p
r
esent
a
t
i
o
n at
any
l
o
ad
bu
s i
s
t
o
o
b
ser
v
e a
nd
reco
r
d
t
h
e
chan
ge i
n
t
h
e
activ
e an
d
reactiv
e p
o
wer when
vo
ltag
e
an
d freq
u
e
n
c
y ar
e
ch
ang
e
d
.
Th
e
av
ailab
ility o
f
n
e
w co
m
p
u
t
ers with
larg
e
facilities
h
a
s en
cou
r
ag
ed
th
e
n
e
w lo
ad p
o
wers repres
en
tatio
n
t
o
b
e
estab
lish
e
d. It
is b
a
sed on
p
r
actical
or
ex
peri
m
e
nt
al
m
easurem
ent
s
a
n
d
ext
r
a
pol
at
i
o
n t
e
c
h
n
i
que
t
o
det
e
r
m
i
n
e t
h
e a
p
p
r
o
p
ri
at
e
p
o
l
y
n
o
m
i
al
ex
pon
en
ts and after wh
ich th
e
v
o
ltag
e
stab
ility resu
lts re
st un
ch
ang
e
d or
b
e
co
in
ci
den
t
[21
]
-[22
].
Usu
a
lly
ti
m
e
reco
rds fo
r
po
wer, reactiv
e po
wer/vo
l
t
ag
e are
read
ily av
ailab
l
e b
y
g
e
ttin
g th
e
valu
es of t
h
e
po
wer,
reactive
powe
r and
volta
ge a
t
each
discrete
tim
e
inte
rval [23].
T
h
e polynom
ial
lo
ad models
for induction
m
o
tors with various m
echanical lo
ads are
di
spl
a
y
e
d
[2
4]
-[
25]
. T
h
e pac
k
age
of t
h
e characteristics
of the
in
du
ctio
n m
o
t
o
r
with
no
rmalized
P/V and
Q/
V ch
arac
teristics with
co
nstan
t
m
ech
an
ical lo
ad torqu
e
(T=consta
n
t). Mechanical
load
t
o
r
q
ue as f
u
nct
i
on i
n
spee
d (T
αω
), and
mechanical load
to
rqu
e
as fu
nctio
n
in
the square
of t
h
e spee
d (T
αω
2) a
r
e co
nsi
d
e
r
ed [
26]
-
[
28]
.
Usi
n
g ext
r
ap
ol
at
i
on t
ech
ni
q
u
e
, t
h
e eq
uat
i
o
n
s
wer
e
d
e
r
i
v
e
d fo
r th
e m
o
to
r
pow
er
s
an
d r
eactiv
e
po
w
e
r
s
as a
fu
nctio
n
o
f
th
e lo
ad
term
in
al vo
ltag
e
s. Usi
n
g
d
i
fferen
t
po
we
rs o
f
t
h
e
pol
y
n
o
m
i
al
s
of t
h
e
po
we
r/
v
o
l
t
a
ge (P/
V
) a
nd t
h
e react
i
v
e po
wer/
vol
t
a
g
e
(Q/
V
), t
h
e
d
e
ri
ve
d
m
odels [29]-[30]. The effect
o
f
th
e
p
eak
inru
sh
cu
rren
ts on
th
e fa
u
lt d
u
ratio
n
vo
ltag
e
co
llap
s
e,
wh
en
after
certain steady state is reach
ed, the line between two buses
is ope
ned,
with diffe
rent laggi
ng a
nd lea
d
ing loads
po
we
r fact
ors
[
31]
-
[
33]
.
2.
P
O
W
E
R SY
STEM
MO
D
E
L REPRE
S
ENTATION
In
o
r
d
e
r to
st
u
d
y
th
e critical clearin
g
ti
me fo
r
vo
ltage in
stab
ility, an
d
t
o
p
e
rform co
m
p
arativ
e
st
udi
es,
vari
o
u
s
t
y
pes of l
o
a
d
s s
u
ch as C
Z
, C
I
, C
P
l
o
a
d
s, a
nd t
h
ree
t
y
pes of i
n
d
u
c
t
i
on m
o
t
o
r l
o
a
d
s are
co
nsid
ered
[3
4]-[35
]
. A
m
u
lt
i-m
ach
in
e
po
wer sy
st
em
,
whi
c
h
c
o
nsi
s
t
s
of 9
-
b
u
ses
,
3-
m
achi
n
es of West
e
r
n
States Co
ord
i
natin
g
C
o
un
cil
(WSCC) in Un
ited
States is
use
d
i
n
t
h
e a
n
al
y
s
i
s
an
d e
v
al
uat
i
on
o
f
t
h
e
faul
t
d
u
ration
for vo
ltag
e
in
stab
il
ities in
it
iatio
n
o
r
wh
at can b
e
called
th
e critical cleari
n
g
tim
e fo
r vo
ltag
e
in
stab
ility [3
6
]
. Also, we search
es abou
t th
e fau
lt du
ration
wh
ich
cau
se initiatio
n
o
f
v
o
l
t
a
g
e
in
stab
ility.
Th
e
i
n
fl
ue
nce
of t
h
e t
r
ansm
i
ssi
on sy
st
em
im
ped
a
nces,
w
h
i
c
h r
e
prese
n
t
t
h
e
re
ci
pr
ocal
of
t
h
e
sy
st
em
nodes
sho
r
t
-
circu
it lev
e
ls NSCL is stu
d
i
ed
in
th
is p
a
p
e
r. As th
is
syst
e
m
will b
e
u
s
ed
th
rou
gho
u
t
th
e stud
y its
mo
d
e
ling
an
d so
lu
tion
tech
n
i
q
u
e
are g
i
v
e
n in
d
e
tails in
th
is sectio
n, t
o
g
e
t
h
er with its d
a
ta and
its prin
cip
a
l equ
a
tio
n
s
.
2.
1.
L
oad
s Ma
the
m
ati
c
al
M
o
del
l
i
n
g:
In t
h
i
s
p
a
pe
r,
we c
once
r
n
e
d
wi
t
h
t
h
e
st
u
d
y
of t
h
e
q
u
a
si static lo
ad
s
wh
ich
it con
s
ists
o
f
h
eating
and
l
i
ght
i
ng e
q
ui
p
m
ent
’
s. Al
s
o
, t
h
ree
quasi
st
at
i
c
m
odel
s
are u
s
ed
for ind
u
c
tio
n
m
o
to
r lo
ad
rep
r
esen
tation
.
The
com
m
onl
y
used re
prese
n
t
a
t
i
o
ns o
f
st
a
tic lo
ad
s are eith
er C
Z
to
g
r
o
und
,
CI and CP. These load m
odels are
gi
ve
n i
n
t
h
e f
o
l
l
o
wi
ng
, w
h
e
r
e P
L
, Q
L
and
V
L
are t
h
e loa
d
active
powe
r, reacti
v
e power, and
bus
voltage
,
resp
ectiv
ely
(th
e
ir i
n
itial v
a
lues are
P
Lo
, Q
Lo
and
V
Lo
).
C
onst
a
nt
Im
pedance
M
o
del
(
C
Z):
,
(
1
)
C
onst
a
nt
C
u
rre
nt
M
o
del
(C
I):
,
(
2
)
C
onst
a
nt
P
o
we
r
M
o
del
(C
P):
,
(
3
)
Three m
odels
are use
d
for induction m
o
tor
load, a
n
d the a
c
tive and
react
ive powe
r ha
s
expresse
d by
t
h
e 4t
h
or
de
r pol
y
n
o
m
i
al
s r
e
prese
n
t
a
t
i
o
n [3
7]
:
In
duct
i
o
n
m
o
t
o
rs wi
t
h
const
a
nt
m
e
chani
cal
l
o
a
d
s
t
o
r
q
u
e
(T=Co
n
stant
)
:
8.02
23.23
2
3
.
2
7
8
.
7
9
0
.
7
6
(
4
)
44.76
148
.56
179
.72
9
2
.
94
16.11
(
5
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Vo
ltag
e
In
st
a
b
i
lity o
f
In
itia
tion
Fau
lt Du
ra
ti
o
n
a
s
In
flu
en
ced
b
y
N
o
d
e
s Sho
rt Circu
it .... (You
ssef Mob
a
r
a
k
)
1
307
Indu
ctio
n m
o
to
rs
with
m
ech
anical lo
ad
t
o
rq
ue propo
rtion
a
l
to
sp
eed (T
αω
):
64.67
233
.03
268
.03
161
.67
3
3
(
6
)
32.56
142
.
22
273.
50
175.
67
4
8
.
83
(
7
)
Indu
ctio
n m
o
to
rs
with
m
ech
anical lo
ad
t
o
rq
ue propo
rtion
a
l
to
sq
u
a
re
o
f
speed
(T
αω
2
):
75.97
342
.0
627
.97
598
.5
309
.9
81.57
9
.
2
3
(8
)
422
.04
1900
.0
3488
.7
325
.0
1721
.67
5
3
.
15
4
6
.
74
(9
)
2.
2.
Generato
rs Ma
thematica
l Modelling
:
Fo
r tran
sien
t stab
ility s
t
u
d
i
es, th
ere are
man
y
m
o
d
e
ls to
rep
r
esen
t
th
e syn
c
hronou
s g
e
n
e
rat
o
r
dy
nam
i
cs [38]
-[
40]
. T
h
e o
n
e
of t
h
ese m
odel
s
used i
n
t
h
i
s
st
udy
i
s
t
h
e one
-a
xi
s
m
ode
l
.
In t
h
i
s
m
odel
t
h
e
transient e
ffec
t
s are acc
ount
ed for,
while
the s
ubt
ra
nsi
e
nt effects are ne
glected.
The tra
n
sie
n
t effect
s
dom
inated by t
h
e
rot
o
r ci
rcuit
s
, whic
h ar
e the field
circu
it i
n
th
e d-ax
is and
an
eq
u
i
v
a
lent circu
it in
t
h
e
q
-
ax
is
fo
rm
ed by
t
h
e sol
i
d
r
o
t
o
r i
s
n
e
gl
ect
ed. Fi
gu
r
e
1 sh
ows t
h
e gene
rat
o
r m
odel
i
n
bl
ock
di
a
g
ram
form
. The
m
odel
equat
i
o
ns
are
s
u
m
m
a
ri
zed bel
o
w
[
41]
-
[
43]
.
Fi
gu
re 1.
Sy
nc
hr
o
n
o
u
s ge
nera
t
o
r
m
odel
For eac
h
gene
rator electrical
equatio
ns
are q-axis stator
e
quation,
(10)
(11)
The p
o
we
r s
y
st
em
consi
d
ered co
nsi
s
t
i
n
g of
n sy
nc
hr
o
n
o
u
s ge
ne
rat
o
r
s
feedi
n
g
t
h
ro
ug
h a
t
r
ansm
i
ssi
on n
e
t
w
o
r
k
,
a
nu
m
b
er of l
o
ads
.
The
sy
st
em
m
o
ti
on
usi
n
g
one
-a
xi
s ge
ne
r
a
t
o
r m
odel
,
u
nde
r
a
di
st
ur
ba
nce, t
h
e f
o
l
l
o
wi
ng
set
o
f
di
ffe
rent
i
a
l
equat
i
o
ns
desc
ri
be
d f
o
r eac
h
gene
rat
o
r
[4
4]
-
[
4
7
]
:
2
(
1
2
)
(
1
3
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
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08
IJEC
E
V
o
l
.
6,
No
. 3,
J
u
ne 2
0
1
6
:
13
0
5
– 13
18
1
308
(
1
4
)
Whe
r
e:
δ
, f
0
,
ω
, H,
D, and
E’q
i
are th
e ro
tor ang
l
e, in
itial
freq
u
e
n
c
y, sp
eed
d
e
v
i
atio
n, in
ertia constan
t
,
m
echani
cal
da
m
p
i
ng coef
fi
ci
ent
,
an
d t
h
e q
-
axi
s
v
o
l
t
a
ge c
o
m
ponent
,
resp
ect
i
v
el
y
.
P
m
, P
e
, E
fd
, I
d
, a
n
d
T
’
d0
are
t
h
e ge
ne
rat
o
r
i
n
p
u
t
m
echani
cal
po
wer
,
out
p
u
t
el
ect
ri
cal
p
o
w
er
, exci
t
a
t
i
o
n
v
o
l
t
a
ge,
d-a
x
i
s
cu
rre
nt
com
pone
nt
,
an
d
d
-
ax
is transien
t op
en
-circu
it ti
m
e
co
n
s
tan
t
,
resp
ectiv
ely. Fin
a
lly,
X
d
and X
’
d
are
the
d-axis
and tra
n
sient
d-a
x
is reactanc
e
’s. Note
that, P
e
is com
pute
d
as:
(
1
5
)
Whe
r
e, E
’
d
is th
e d-ax
is co
m
p
on
en
t of th
e
vo
ltag
e
E
’
, a
n
d I
q
i
s
t
h
e q
-
axi
s
com
pone
nt
o
f
t
h
e ge
ne
rat
o
r c
u
r
r
ent
,
whi
c
h
i
s
gi
ve
n as:
(
1
6
)
Whe
r
e,
E
’
is t
h
e voltage
be
hind the
reacta
n
ce
X
’
d
, and
V is t
h
e
g
e
n
e
rator term
in
al v
o
ltag
e
.
2.
3.
Multi-mac
h
ine Aggre
g
ate
d
Sys
t
em:
Th
e in
terrelatio
n
o
f
th
e system ele
m
en
ts is
sh
own
in Figure 2. The three
sy
ste
m
solution syste
m
are
t
h
e dy
nam
i
c sim
u
l
a
t
i
on pr
og
ram
,
t
h
e net
w
or
k re
duct
i
o
n
pr
o
g
ram
and t
h
e l
o
ad fl
ow
pr
o
g
ram
wi
t
h
l
o
ads
rep
r
ese
n
t
e
d
by
pol
y
nom
i
a
l
model
s
. T
h
ey
sh
oul
d
be us
ed i
n
se
que
nce t
o
sol
v
e t
h
e sy
st
e
m
equat
i
ons.
A fl
o
w
ch
art
o
f
m
u
lti-
mach
in
e power syste
m
s written
in Ma
tlab
Too
l
Box
,
is sh
own in
Figu
re 3.
Fi
gu
re
2.
Si
m
u
l
a
t
i
on p
o
w
er
s
y
st
em
pro
g
ram
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Vo
ltag
e
In
st
a
b
i
lity o
f
In
itia
tion
Fau
lt Du
ra
ti
o
n
a
s
In
flu
en
ced
b
y
N
o
d
e
s Sho
rt Circu
it .... (You
ssef Mob
a
r
a
k
)
1
309
Fig
u
re
3
.
Flow ch
art of
d
i
g
ital sim
u
lat
i
o
n
o
f
m
u
l
ti-
m
ach
in
e p
o
wer system
Th
e
d
y
n
a
m
i
c s
i
m
u
latio
n
program
read
s th
e
n
o
d
a
l in
itial con
d
ition
s
, equ
i
valen
t
m
a
trix
, the g
e
n
e
rator,
exci
t
a
t
i
on, t
u
rb
i
n
e-
go
ver
n
or a
nd
fi
nal
l
y
t
h
e t
y
pe of l
o
a
d
s
da
t
a
. The o
u
t
p
ut
of t
h
e
dy
nam
i
c sim
u
l
a
t
i
on pr
o
g
ram
i
s
t
h
e sy
st
em
ope
n l
o
o
p
per
f
o
r
m
a
nce [4
8]
-[
49]
.
The
sy
s
t
em
perf
orm
a
nce i
s
det
e
rm
ined
by
s
o
l
v
i
n
g t
h
e
mach
in
e m
a
th
ematical
m
o
d
e
ls
to
g
e
t
h
er with th
e co
nstr
ai
nt
s i
m
posed
by
t
h
e
net
w
or
k.
The
n
o
n
-
l
i
n
ear
m
achi
n
e
m
odels are solved
num
e
rically usi
ng t
h
e ap
pr
o
p
ri
at
e i
n
t
e
g
r
at
i
on t
e
c
hni
que.
Once t
h
e load-fl
ow st
udy has
been
per
f
o
r
m
e
d an
d al
l
dat
a
have
been c
o
n
v
ert
e
d t
o
a co
m
m
on po
wer
base, t
h
e
next
st
eps i
n
t
h
e si
m
u
l
a
t
i
on
are initial condition calc
u
lations
of ev
e
r
y dynam
ic com
ponent
represe
n
ted,
a
nd re
ducti
on of
the network
in
or
der
t
o
el
i
m
i
n
at
e al
l
no
des t
h
at
have
zer
o i
n
j
ect
i
on c
u
r
r
e
n
t
s
.
2.
4.
Initia
l Conditi
o
n
Ca
lculations:
Fro
m
all th
e dyn
amic
m
o
d
e
ls in
vo
l
v
ed in
t
h
is typ
e
o
f
simu
latio
n
,
th
e
electric altern
ators is th
e
on
e
req
u
i
r
i
n
g m
o
re cal
cul
a
t
i
ons si
nce t
h
e rest
of
t
h
e dy
nam
i
c com
pone
nt
s are
gi
ven by
a bl
ock
di
agr
a
m
,
in t
h
e
initial condition calculations
are m
a
de by sim
p
ly setting
to zero a
n
y term
containi
ng a
derivative term
[50].
Using
th
e
ph
aso
r
d
i
agram
s
h
own
in
Fi
g
u
re 4
th
e i
n
itial
co
nd
ition
fo
r th
e syn
c
h
r
o
n
o
u
s
g
e
n
e
rat
o
r
can
be
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E
V
o
l
.
6,
No
. 3,
J
u
ne 2
0
1
6
:
13
0
5
– 13
18
1
310
calcu
lated
as fo
llo
ws: At th
e ter
m
in
al sid
e
o
f
ev
ery alternato
r
, th
e
k
nown
v
a
riab
les g
i
v
e
n
b
y
a lo
ad
-flow
st
udy
, are:
Te
rm
i
n
al
vol
t
a
ge V
a
, an
d ge
ne
rat
e
d act
i
v
e p
o
we
r P, a
nd r
eact
i
v
e po
wer
Q. k
n
o
w
i
n
g
t
h
ese
v
a
riab
les, it is
easy to
calcu
late th
e arm
a
tu
re curren
t
of
ev
ery u
n
it an
d th
e
co
rr
esp
ond
ing
p
o
w
e
r
f
act
o
r
by:
∗
(17)
Whe
r
e: I
a
i
s
t
h
e arm
a
t
u
re cu
rr
ent
p
h
as
or
, a
n
d
res
o
l
v
e i
n
t
o
co
m
ponent
s
wi
t
h
V
a
as
a
refe
rence:
cos
∅
,
sin
∅
(
1
8
)
Fig
u
re
4
.
Altern
ator in
itial con
d
ition
p
h
a
sor
d
i
agram
The phasor
E
qa
i
n
Fi
gu
re
(4
),
i
s
gi
ven
by
:
(
1
9
)
T
h
en
th
e a
n
g
l
e (
δ
-
β
)
i
s
gi
ve
n by
:
(
2
0
)
W
h
er
e
ang
l
e
δ
is th
e
p
o
s
ition
o
f
th
e
ro
t
o
r. Then
, calcu
late the term
in
al v
o
ltag
e
d
-
q
co
m
p
on
en
ts:
sin
,
cos
(
2
1
)
Al
so,
t
h
e
arm
a
ture
cu
rre
nt
d-
q
com
pone
nt
s:
sin
∅
,
cos
∅
(
2
2
)
The field volta
ge of
the
m
achine
from
the st
ator si
de ca
n
be calculated
by
:
(
2
3
)
Th
e in
itial p
o
sitio
n
o
f
th
e ro
tor is calcu
lated
b
y
:
δ
=
γ
+
β
. Calcu
l
ate th
e in
itial
v
a
lu
e o
f
th
e state
v
a
riab
le of t
h
e
altern
ator m
o
del u
s
ing
t
h
e
q
u
an
tities j
u
st
calcu
lated
.
2.
5.
Netw
o
r
k Red
u
ctio
n
On
ce th
e
bu
s ad
m
i
ttan
ce
m
a
trix
rep
r
esen
ting th
e
n
e
two
r
k
has b
e
en
b
u
ilt an
d its en
tries
org
a
n
i
zed as
gene
rat
o
r n
o
d
e
s
, n
o
n
-
l
i
n
ear l
o
ad
no
des a
n
d
ot
her
no
des
,
i
t
i
s
necessary
t
o
el
im
i
n
at
e
t
hose n
o
d
es wi
t
h
zer
o
cu
rren
t inj
ectio
n
s
. Th
e
p
r
o
c
ed
ure to
ob
tain
th
e redu
ced
n
e
two
r
k
is as fo
llows: Assume th
e Y
BUS
matrix
is
p
a
rtitio
n
e
d
i
n
t
h
e
fo
llowing
way:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Vo
ltag
e
In
st
a
b
i
lity o
f
In
itia
tion
Fau
lt Du
ra
ti
o
n
a
s
In
flu
en
ced
b
y
N
o
d
e
s Sho
rt Circu
it .... (You
ssef Mob
a
r
a
k
)
1
311
0
(
2
4
)
Whe
r
e:
G m
e
ans ge
nerat
o
r n
ode
s, L
m
eans no
n-l
i
n
ea
r l
o
a
d
n
odes
,
R
m
e
ans rem
a
i
n
i
ng no
des
.
El
im
i
n
at
e t
h
e
rem
a
ining nodes by successi
ve eli
m
ination procedure:
(
2
5
)
Whe
r
e:
After a
ddi
ng t
h
e internal impeda
nce of each ge
nerat
o
r
to new a
d
m
i
ttance
m
a
trix,
the e
quation (25) becom
e
s
re-a
rra
nge
d a
s
follo
ws:
(
2
6
)
Whe
r
e just Y
’
GG
change to Y
”
GG
, and E
’
G
m
e
ans the
voltage
behi
nd the tra
n
sient reacta
n
c
e
. The e
quation (26)
can
b
e
re-arrang
e
d as fo
llow
s
, th
is is b
ecau
s
e th
e in
itial
lo
ad cu
rren
t at lo
ad b
u
ses are know
n, an
d
it is
u
s
ed
to
d
e
term
in
e th
e lo
ad vo
ltag
e
at lo
ad bu
ses
res
p
ectively at the
next
step.
∗
(
2
7
)
Whe
r
e:
Y
*
GG
New ad
m
ittan
ce m
a
trix
, K
GL
, H
LG
N
o
n-
di
m
e
nsi
o
nal
m
a
t
r
i
c
es, Z
LL
L
o
a
d
i
m
pedance m
a
tri
x
.
The
equi
val
e
nt
m
a
tri
x
fo
r t
h
e e
n
t
i
r
e n
e
t
w
or
k i
s
r
e
prese
n
t
e
d
by
t
h
e si
ngl
e l
i
n
e
di
ag
ram
of p
o
w
er
sy
st
em
operat
i
n
g
at th
e no
m
i
n
a
l lo
ad
ing
con
d
iti
o
n
is fou
n
d
as:
∗
(
2
8
)
Whe
r
e:
The sy
st
em
di
ffere
nt
i
a
l
eq
uat
i
ons
are
de
pi
ct
s
:
,
0
(
2
9
)
,
(
3
0
)
Net
w
or
k ge
ner
a
l
feat
ur
es:
(31)
cos
sin
sin
cos
or
cos
s
in
sin
cos
(
3
2
)
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I
S
SN
:
2
088
-87
08
IJEC
E
V
o
l
.
6,
No
. 3,
J
u
ne 2
0
1
6
:
13
0
5
– 13
18
1
312
Also. steady
-
state equati
ons
s
o
lution:
(33)
(
3
4
)
(
3
5
)
cos
sin
sin
cos
cos
s
in
sin
cos
(36)
(
3
7
)
∗
∗
(38)
0
(
3
9
)
(
4
0
)
3.
STUDIED SYSTEM
A si
ngl
e l
i
n
e i
m
pedance di
ag
ram
of t
h
e sy
stem
i
s
shown i
n
Fi
g
u
re 5
,
w
h
ere t
h
e sy
st
em
i
s
basi
cal
l
y
com
posed
of t
h
ree
ge
nerat
i
n
g u
n
i
t
s
an
d
three lo
ad
s, lo
ad
A, lo
ad
B, an
d
load C are l
o
c
a
ted at bus
es #4,
#5,
an
d
#6
,
resp
ectiv
ely. Un
it one is h
y
d
r
o
e
lect
ric, wh
ile
units two a
n
d thre
e are steam
dri
v
en
gene
rat
o
r
s
. The
sy
st
em
i
s
oper
a
t
e
d no
rm
al
ly
fo
r 1 sec. be
fo
r
e
a t
h
ree phase
short
ci
rc
ui
t
faul
t
occur
s
at
bus
# 9. T
h
e l
o
ads are
rep
r
ese
n
t
e
d
by
C
Z
, C
I
, C
P
l
o
ads,
an
d i
n
d
u
c
t
i
on m
o
t
o
r
l
o
ad
s with
t
h
ree
typ
e
s o
f
sh
aft mech
an
ical
lo
ad
s (IM
load (T=c
o
n
sta
n
t) suc
h
as (c
o
nvey
o
rs
), IM
load (
T
αω
) s
u
c
h
as (reci
p
r
oca
t
i
ng p
u
m
p
s), and IM
l
o
a
d
(T
αω
2
)
su
ch
as (cen
trifug
al pu
m
p
s)) m
o
d
e
ls. Th
e syste
m
fau
lt so
lu
tion
is ob
tain
ed
b
y
in
t
e
g
r
ating
th
e syste
m
di
ffe
re
nt
i
a
l
equat
i
o
n
s
an
d so
l
v
i
ng t
h
e sy
st
em
al
gebrai
c
equ
a
tio
ns du
ri
n
g
th
e ti
m
e
s
i
m
u
latio
n
b
y
in
tegration
t
echni
q
u
e
usi
n
g t
r
a
p
ezoi
d
al
m
e
t
hod, t
h
e t
i
m
e i
n
t
e
rval
i
s
Δ
t=0
.
001
sec.. Before fau
lt, th
e h
ybrid
m
a
trix
i
s
calculated a
n
d
fed i
n
to the sta
b
ility
evaluation program
for each
tra
n
sm
iss
i
on system
i
m
p
e
dance
s
are
taken as
1
5
0
%
, 12
0
%
, 1
0
0
%
(b
ase case), 80
%, and
5
0
% o
f
th
ei
r no
m
i
n
a
l v
a
lu
es. Th
is will b
e
referred
to
in
th
e tex
t
b
y
NSCL (67%,
83%, 100% (base case),
125%, and 200%), re
sp
ectiv
ely. A lo
ad
flow st
u
d
y
is p
e
rformed
for
each
of those
NSCL a
n
d the
resu
lts a
r
e s
hown in
Figure
6.
Fi
gu
re
5.
St
u
d
i
e
d sy
st
em
si
ng
l
e
l
i
n
e di
a
g
ram
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISS
N
:
2088-8708
Vo
ltag
e
In
st
a
b
i
lity o
f
In
itia
tion
Fau
lt Du
ra
ti
o
n
a
s
In
flu
en
ced
b
y
N
o
d
e
s Sho
rt Circu
it .... (You
ssef Mob
a
r
a
k
)
1
313
Figure
6. A loa
d
flow study is
pe
rform
e
d
for each of
those NSCL%
4.
RESULTS
A
N
D
DI
SC
US
S
I
ON
4.
1.
Fa
ult Duration fo
r
Vo
l
t
ag
e
Insta
b
ility a
s
Influenced by
Va
ry
ing
o
f
the NSCL:
Th
is will b
e
referred
t
o
in
t
h
e stud
y b
y
NS
CL (6
7
%
,
8
3
%, 100
% (b
ase case),
12
5%, an
d 200
%),
resp
ectiv
ely. Tab
l
e (1
) ind
i
cate th
e fau
lt duratio
n
wh
ich
cau
s
e vo
ltag
e
i
n
stab
ility wh
en
a th
ree-ph
ase sh
ort-
circuit occ
u
rs
at node
(9) at
t=1 sec., with
the NSCL
va
ri
es by
a ce
rt
ai
n
perce
n
t
a
ge
s
u
ch
as (
6
7%,
83
%,
10
0%
, 12
5%
, and
20
0%
) Th
ese vari
at
i
o
n
s
are ap
pl
i
e
d w
h
en
the loads are CZ load, CI
load, CP loa
d
, at all
load
buses, thi
s
is beca
use i
n
cr
easi
n
g the
NSCL m
eans
m
o
re parallel lines and m
o
re
corridors for
powe
r
flo
w
. Als
o
, IM
load (T=c
o
n
st
ant),
IM
load
(
T
αω
) an
d IM
l
o
ad
(T
αω
2
)
at lo
ad
b
u
s
(4) on
ly with
th
e load
at
lo
ad
bu
ses (
5
)
an
d
(6
)
h
a
v
i
ng
CZ
lo
ad
s. T
h
e syste
m
cannot operate sta
b
ly
w
h
en
a
ll lo
ad
s
ar
e
a
s
su
me
d
p
u
r
e
in
du
ctio
n m
o
to
r lo
ad
s.
Tho
s
e n
e
twork
s
are con
s
id
ered
strong
powe
r sys
t
e
m
s. They allow longe
r
faul
t tim
e
s
b
e
fo
re anno
uncin
g
vo
ltag
e
i
n
stab
ility states.
Tab
l
e
1
.
Effect of NSCL
on
t
h
e
fau
lt
d
u
ration
for
v
o
ltag
e
instab
ility at d
i
fferen
t l
o
ad
s
%
NSCL
Fault duration for voltage instability (m
sec
.
)
CZ loads
CI loads
CP loads
IM lo
ad
T
=
constant
IM T
αω
I.M
.
lo
ad
T
αω
2
67%
291
273
220
253
262
268
83%
300
281
237
274
279
285
100
304
285
243
279
290
293
125
307
289
284
284
294
302
200
316
299
260
296
298
310
Deno
ting
th
e fau
lt du
ration
wh
ich
cau
s
e
vo
ltag
e
in
stab
ility in
itiat
i
o
n
b
y
TFau
lt an
d the p
e
rcen
tag
e
NSC
L
, t
h
e
pol
y
nom
i
a
l
equat
i
on
o
f
t
h
i
r
d
or
der
fo
u
n
d
by
M
A
TLAB
p
r
o
g
ram
i
ndi
cat
e t
h
e rel
a
t
i
o
n
be
t
w
ee
n
Fau
lt duratio
n
wh
ich
cau
s
e
vo
ltag
e
instab
ility an
d
p
e
rcen
tag
e
o
f
NSC
L
Th
ey are d
i
sp
layed
:
Fo
r CZ l
o
ad
s at all lo
ad
bu
ses:
193
.77
243
.71
174
.74
4
1
.
72
For CI l
o
a
d
s at
all load
buses:
195
.78
188
.18
N
128
.77
3
0
.
24
Fo
r CP l
o
ads at all lo
ad
bu
ses:
3
8
.
00
459
.12
331.
64
78.79
For
IM
l
o
a
d
at
no
de
(
4
),
n
o
d
e
s
(
5
)
an
d
(6
) a
r
e co
nst
a
nt
i
m
pedance
l
o
a
d
s:
Fo
r I
M
lo
ad
(T=co
n
stan
t)
25.90
581.
12
428
.66
102
.82
Fo
r I
M
lo
ad
(T
αω
):
64.47
487
.
06
338
.57
7
6
.
7
1
Fo
r I
M
lo
ad
(T
αω
2):
119
.28
253
.07
225
.94
48.
79
Whe
r
e: TFa
u
lt is
m
easured by
m
s
ec., and these equations
h
a
v
e
so
m
e
erro
rs
with
resp
ect to
ex
act v
a
lu
es
of
T
Fault
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E
V
o
l
.
6,
No
. 3,
J
u
ne 2
0
1
6
:
13
0
5
– 13
18
1
314
4.
2.
Time Respons
e
for Loads
Voltages
with
a
Three Phase
Shor
t Circ
uit
on Node
# 9,
as
Influe
nced
by Varying of the NSCL
In
th
is section
,
in
d
i
cates th
e time resp
on
se for lo
ad
bu
ses
v
o
ltag
e
s at lo
ad
bu
ses wit
h
d
i
fferen
t v
a
l
u
es
o
f
N
S
C
L
s
u
c
h
a
s
(
6
7
%
a
n
d
2
0
0
%
)
,
w
h
e
n
(
C
Z
,
a
n
d
C
P
) loads are conne
c
ted at all load buses
. Als
o
, IM loa
d
(T=co
n
sta
n
t), a
nd
IM
load (T
αω
2) loa
d
s are
connected at load
bus # 4
w
ith
th
e lo
ad
s at lo
ad
bu
ses
#
5
,
an
d
#
6
h
a
v
i
ng
CZ l
o
ad
s CZ.
W
h
en
a 3-
ph
ase shor
t-
cir
c
u
it
f
a
u
lt is o
c
cu
rr
ed
at
bus # 9 at t=1 sec., an
d lasts
u
n
t
i
l
th
e
v
o
ltag
e
instab
i
lity
o
ccu
r, with sev
e
ral n
e
twork
variatio
ns. Fig
u
re 7
ind
i
cates th
e ti
m
e
resp
on
se fo
r load
b
u
s
es
v
o
ltag
e
s,
wh
en th
e lo
ad
s at all lo
ad bu
ses are
CZ
lo
ad
, with
th
e
NSCL
are 6
7
%,
and
2
00%.
(
a
)
N
S
CL=67%, CZ l
o
ad
s, an
d TC=290
m
s
ec
(
e
)
N
S
CL=67%, CP l
o
ads, an
d TC=219
m
s
ec
(
b
)
N
S
CL=6
7%, CZ l
o
ad
s, an
d TC=291
m
s
ec
(
f
)
NSCL=6
7%, CP l
o
ads, an
d TC=220
m
s
ec
(
c
)
N
S
CL=20
0
%, CZ l
o
ad
s, an
d TC=315
m
s
ec
(
g
)
N
S
CL=2
00%, CP l
o
ads, an
d TC=259
m
s
ec
(
d
)
N
S
CL=2
00%, CZ l
o
ad
s, an
d TC=316
m
s
ec
(
h
)
N
S
CL=2
00%, CP l
o
ads, an
d C=260
m
s
ec
Fi
gu
re
7.
Ti
m
e
res
p
o
n
se
f
o
r
l
o
ad
s
vol
t
a
ges
wi
t
h
a t
h
ree
p
h
a
se sh
o
r
t
ci
rcui
t
on
n
o
d
e
#
9,
as i
n
fl
uence
d
b
y
vary
i
n
g of
t
h
e NSC
L
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