Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
V
o
l.
4, N
o
. 4
,
A
ugu
st
2014
, pp
. 49
8
~
51
1
I
S
SN
: 208
8-8
7
0
8
4
98
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
A MODEL FOR ASSESSMENT OF TRANSIENT STABILITY
OF ELECTRICAL POWER S
Y
STEM
Gani
yu
A.
A
j
e
n
i
k
ok
o,
A
n
th
o
n
y
A.
Ol
a
o
mi
Department o
f
Electronic & Electr
ical
Engin
eerin
g, Ladoke Akintola Univ
ersity
of
Technolog
y
,
Og
bo
m
o
so
, N
i
g
e
r
i
a
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Feb 25, 2014
Rev
i
sed
May 31
, 20
14
Accepted
Jun 16, 2014
The stab
ili
t
y
of
a s
y
st
em
is its a
b
ilit
y
to re
turn t
o
norm
a
l or stab
le oper
a
tio
n
after
having
been subjected
to some fo
rms of disturbances.
A disturbance in
a
power s
y
stem is
a sudden
chang
e
or sequ
ence of
chang
e
s in on
e
or more of
the ph
y
s
i
c
a
l
qua
ntiti
es. In this pa
per, th
e transien
t
react
anc
e
of a sy
nchronous
machine, mechanical input power, kinetic
energ
y
of a rotating bo
d
y
, moment
of iner
tia
,
ang
u
lar
acc
el
erat
io
n, angu
lar d
i
s
p
lac
e
m
e
nt and
the ro
tor
displacement an
gles were used as i
nput parameters for the dev
e
lopment of
the T
r
ans
i
en
t S
t
abili
t
y
m
odel
.
T
h
e m
odel
is
val
i
d
ated
with
a s
i
n
g
le m
ach
ine
s
y
stem
,
a 2-m
a
c
h
ine s
y
s
t
em
and
a m
u
lti-m
a
chin
e s
y
st
em
.
The
r
e
sults of th
e
work showed that th
e single machine
s
y
stem supply
i
ng an
infin
ite bus-bar
fluctu
ates while the 2-machine sy
stem remains unstable thro
ughout th
e
period. Gener
a
t
o
r 3 of the
multi-m
ach
ine s
y
stem
experienc
e
d
the m
o
st
violent swing, p
u
lled out of s
y
n
c
hroni
sm during the first swing thus making
the s
y
s
t
em to b
e
unstable.
The
Transien
t Stab
ility
Model d
e
velo
ped can b
e
used for
effectiv
e planning
a
nd o
p
eration of
power s
y
stems.
Keyword:
Critical clearing tim
e
Electrical power
Gene
rat
o
rs’
sy
nch
r
oni
sm
In
finite b
u
s
-
ba
r
Power system
stab
ility
Rotor angles
Swing c
u
rves
Transien
t stab
ility
Copyright ©
201
4 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
:
Gani
y
u
A
.
Aje
n
i
k
ok
o,
Depa
rt
m
e
nt
of
El
ect
roni
c
& E
l
ect
ri
cal
Engi
n
eeri
n
g,
Lado
k
e
Ak
in
t
o
la Un
i
v
ersity of Tech
no
log
y
,
P.M.B. 40
00
, O
gbo
m
o
so
, N
i
g
e
r
i
a
Em
a
il: aj
eed
o
l
l
a
r@g
m
ail.co
m
1.
INTRODUCTION
Power
syste
m
s are
designe
d
to ope
r
ate in t
h
e stea
dy
and transient states
and are
cha
r
a
c
terized
by
t
r
ansi
ent
s
or di
st
ur
bance
s
[5]
,
[6]
,
[7]
,
[
9
]
.
A
di
st
ur
bance i
n
a powe
r
sy
st
em
i
s
sud
d
en c
h
ange
or se
que
n
ce of
ch
ang
e
s i
n
o
n
e
o
r
m
o
re of the ph
ysical q
u
a
n
tities [2
].
Larg
e
d
i
stu
r
b
a
n
c
es often
refer to sev
e
re
d
i
stu
r
ban
ces,
suc
h
as a
fa
ul
t
on
t
r
a
n
sm
i
ssi
o
n
net
w
or
k, l
o
ss
of
ge
ne
rat
i
o
n
or l
o
ss
o
f
a l
a
r
g
e l
o
a
d
[
11]
,
[
16]
,
[
24]
, a
n
d t
h
at
t
h
e
equat
i
o
ns
desc
ri
bi
n
g
t
h
e
p
o
w
e
r sy
st
em
cannot
be l
i
n
ea
ri
zed f
o
r
anal
y
s
i
s
pu
r
pose
w
h
e
n
su
bject
e
d
t
o
l
a
rge
d
i
stu
r
b
a
n
ces [2
1
]
. Th
e m
a
in
facto
r
co
n
t
ri
bu
tin
g
t
o
th
e tran
sien
t in
stab
ility is
th
e in
su
fficien
t
syn
c
hron
izing
t
o
r
que
d
u
ri
ng
t
h
e
di
st
ur
ba
nce
peri
o
d
i
n
t
h
e
sy
st
em
[12
,
1
3
]
. The
di
st
u
r
ba
nces m
a
y
be ei
t
h
er el
ect
r
o
m
a
gnet
i
c
suc
h
as ove
r-
vol
t
a
ge
, o
v
er
-
c
ur
rent
, a
b
n
o
r
m
a
l
wave sh
apes or electrom
echanical
t
r
an
sien
ts wh
ich
are
co
n
c
ern
e
d
with
th
e stab
ility o
f
th
e
po
wer syste
m
. Th
e rec
o
v
e
ry of a po
wer syste
m
su
bjected
to
a sev
e
re large
di
st
ur
ba
nce i
s
of
i
n
t
e
re
st
t
o
sy
st
em
pl
ann
e
rs a
n
d
o
p
e
r
at
ors
.
Ty
pi
cal
l
y
t
h
e sy
st
em
m
u
st
be
desi
g
n
e
d
a
n
d
o
p
e
rated in
such
a
way th
at
a nu
m
b
er of cred
ib
le co
n
t
i
n
g
e
n
c
ies
do
no
t resu
lt in
failu
re
o
f
qu
ality an
d
co
n
tinu
ity o
f
po
wer supp
ly to
th
e lo
ad
s. Th
i
s
calls
for acc
urat
e cal
cul
a
t
i
on
of t
h
e sy
st
em
dy
nam
i
c behavi
or
,
whic
h include
s
the electro-m
echanical dynam
i
c
characteristics of the
rot
a
t
i
n
g m
achines, ge
ne
rat
o
r
cont
rol
,
static v
a
r co
m
p
ensato
rs, lo
ads, pro
t
ectiv
e sy
ste
m
an
d
o
t
h
e
r co
n
t
ro
ls.
Th
e
stab
ility o
f
th
e p
o
wer system is th
e
ab
ility o
f
th
e
g
e
n
e
rator to
re
m
a
in
in
syn
c
h
r
on
ism
after
a d
i
stu
r
b
a
n
ce
to
th
e system
[10
]
, [1
4
]
,
[15]. Th
e
hol
ding toget
h
er of these
generators are affe
cted th
ro
ugh
th
e po
wer n
e
two
r
k
and
an
y loss in
th
e d
e
grad
atio
n
of
t
h
e net
w
or
k securi
t
y
s
u
ch
a
s
o
u
t
a
ges
,
bl
acko
u
t
s
a
n
d
s
o
o
n
[1]
,
[
33]
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE Vo
l. 4
,
N
o
. 4
,
Au
gu
st 2
014
:
49
8
–
51
1
4
99
Power system
stab
ility is classified
i
n
to
t
h
ree typ
e
s
n
a
m
e
ly
: stead
y-state, t
r
an
sien
t an
d lon
g
term
[3
],
[36
]
, [3
8
]
. Stead
y
-state stab
ility is p
r
i
m
arily
co
n
c
ern
e
d
w
ith
th
e ab
ility o
f
th
e syste
m
g
e
n
e
rat
o
rs t
o
remain
in
sy
nch
r
o
n
i
s
m
aft
e
r m
i
nor di
st
ur
ba
nces suc
h
as gra
dual
l
o
a
d
chan
ges, cha
n
ges i
n
exci
t
a
t
i
o
n, l
i
n
e swi
t
c
hi
ng a
n
d
so
on
.
It is also con
cern
e
d
with
sud
d
e
n
and
l
a
rg
e ch
ang
e
s i
n
th
e netwo
r
k
co
nd
itio
ns su
ch
as broug
h
t
ab
ou
t
b
y
faul
t
s
t
h
e
m
o
st
severe
o
f
w
h
i
c
h i
s
t
h
e t
h
ree
p
h
ase s
h
ort
-
ci
rc
ui
t
[
18]
,
[
25]
,
[
29]
.
Wh
en
a fau
lt occu
rs at th
e termin
als o
f
a sy
n
c
hro
n
o
u
s gen
e
rat
o
r
,
t
h
e
p
o
w
er out
put
of
t
h
e
m
achi
n
e
i
s
g
r
eatly redu
ced
as it is su
pp
l
y
in
g
m
a
in
ly, a
n
ind
u
c
tiv
e ci
r
c
ui
t
[1
8]
, [
22]
,
[2
6]
, [
29]
.
Ho
weve
r, t
h
e i
n
p
u
t
p
o
we
r
to
th
e g
e
n
e
rato
r
fro
m
th
e tu
rb
i
n
e h
a
s
no
time to
ch
ang
e
d
u
ring
th
e short p
e
ri
o
d
of the fau
lt and
th
e ro
t
o
r
endea
v
ou
rs t
o
gai
n
spee
d t
o
s
t
ore t
h
e excess
energy
[
38]
. I
f
t
h
e faul
t
persi
s
t
s
l
ong en
o
u
g
h
, t
h
e r
o
t
o
r ang
l
e wi
l
l
i
n
crease co
nt
i
n
u
o
u
sl
y
and s
y
nch
r
o
n
i
s
m
i
s
l
o
st
. Hence
,
t
h
e t
i
m
e
of op
erat
i
on
of t
h
e
pr
ot
ect
i
on a
n
d
ci
rcui
t
brea
ker
s
i
s
al
l
im
port
a
nt
.
A
n
aut
o
recl
osi
n
g c
i
rcui
t
brea
ke
r ope
ns
whe
n
t
h
e faul
t
i
s
det
e
c
t
ed an
d aut
o
m
a
t
i
cal
l
y
recl
oses a
f
t
e
r a
presc
r
i
b
e
d
pe
r
i
od.
If t
h
e fa
ul
t
persi
s
t
s
, t
h
e ci
rcui
t
b
r
eake
r
r
e
ope
ns a
nd t
h
e
n
recl
oses as
b
e
fo
re.
Th
is is
rep
eated
o
n
c
e m
o
re,
wh
en
i
f
th
e fault still
p
e
rsists,
th
e break
e
r
remain
s op
en [22
]
, [30
]
, [33
]
.
Transien
t stab
i
lity
is j
u
dg
ed
fro
m
th
e n
a
tu
re o
f
th
e swing
cu
rv
es.
If the curv
es settles at th
e pre-fau
l
t
lev
e
l or so
m
e
n
e
w lev
e
l aft
e
r so
m
e
ti
m
e
s
,
th
e system
i
s
stab
le. Howev
e
r, if th
e roto
r ang
l
e in
creases
co
n
tinuo
usly with
ti
m
e
, th
e syste
m
is u
n
s
tab
l
e [3
5
]
,
[3
7
]
.
Lon
g
-term
sta
b
ility fo
rm
s th
e tran
sitio
n
b
e
tween
tr
an
sient stab
ility
an
d
stead
y state st
ab
ility [2
7
]
,
[3
1]
, [3
2]
, [3
4]
.
Whe
n
a
faul
t
occu
rs, t
h
e sy
st
em
m
a
y
becom
e
unst
a
bl
e i
n
t
h
e
dy
nam
i
c pr
ocess a
n
d s
e
parat
e
i
n
t
o
several
pa
rts [4]. Obviously, t
h
e syst
em
is unreliable under t
h
is situation.
Hence, it is im
portant a
n
d neces
sary
to
ev
alu
a
te the syste
m
relia
b
ility b
a
sed
on
bo
th
t
h
e d
y
n
a
m
i
c an
d
static b
e
h
a
v
i
o
r
of th
e system
.
In
real
syste
m
s, tran
sien
t fau
lts con
t
ribu
te to
m
o
st
o
f
th
e to
tal faults [8
]. Th
e sy
ste
m
may also
lo
se stab
ility
u
n
d
e
r
tran
sien
t
fau
lts. It is ev
id
en
t t
h
at th
e
po
wer
q
u
a
lity will
b
e
affected and
ev
en th
e l
o
ad may b
e
sh
ed
[15], [17
]
.
Th
e
reclo
s
i
n
g ti
m
e
to
g
e
th
er with fau
lt duratio
n are
use
d
to incl
ude
t
h
e im
pacts of both transient and
p
e
rm
an
en
t fau
lts [19
]
, [2
3
]
.
In
reliab
ility an
alysis, wh
en
tran
sien
t stab
ility is co
n
s
id
ered, th
e d
i
st
u
r
b
a
nces are
refe
rre
d to
as
p
e
rm
anent fa
ults [
25]
,
[
28]
.
Transien
t stab
ility
an
alysis ca
n
b
e
u
s
ed
fo
r
d
y
n
a
m
i
c an
aly
s
is o
v
e
r tim
e
p
e
riod
fro
m
few seco
nd
s to
few m
i
nut
es depen
d
i
n
g o
n
t
h
e t
i
m
e
const
a
nt
of t
h
e
dy
na
m
i
c phenom
en
on m
odel
e
d.
I
n
t
h
e past
, t
r
a
n
si
ent
stab
ility h
a
s b
e
en
ev
alu
a
ted
usin
g ti
m
e
d
o
m
ain
(TD) app
r
oach
. If th
e syste
m
co
u
l
d
su
rv
i
v
e
for t
h
e
first
swing
,
i.e stab
le in
the first swing
it will g
e
n
e
rally rem
a
in
stab
le in
th
e fo
llowing
swing
s
[4
], [2
7
]
. TD app
r
oach
is
foun
d to
b
e
ti
me co
n
s
u
m
in
g
and
in
efficient for ev
alu
a
ting
stab
ility fo
r a larg
e system
wh
ere th
e syste
m
com
pone
nt
va
r
y
dy
nam
i
cal
l
y
and y
e
t
,
re
peat
ed si
m
u
l
a
t
i
on has t
o
be m
a
de. Thi
s
has enc
o
ura
g
e
d
t
h
e e
x
p
a
nsi
o
n
o
f
v
a
riou
s transien
t stab
ility
assessm
en
ts, su
ch
as Ex
tended
Equ
a
l Area Criterio
n
(
EEAC) [35
]
, [37
]
, Direct
M
e
t
h
o
d
of Ly
apu
n
ov F
u
nct
i
on [
7
]
,
[
9
]
,
t
r
ansi
ent
Ene
r
g
y
Funct
i
o
n (T
EF) [
9
]
,
[1
0]
,
[11]
, De
ci
si
o
n
Tree
Transient Sta
b
ility Method [2]
,
Com
posite El
ectrom
echanical Distance
(C
ED) Method [2] and ot
hers
.
A direct m
e
thod of tra
n
sient s
t
ability analysis of
a
m
u
lti-
m
achine power s
y
stem
using e
x
tension of
EAC
h
a
s
b
e
en propo
sed b
y
[2
].
[1
], [15
]
hav
e
in
d
e
p
t
h d
e
tails o
n
EEAC
m
e
th
od
for mu
lti m
ach
in
e syste
m
t
r
ansi
ent
st
abi
l
i
t
y
. Di
rect
m
e
tho
d
o
f
Ly
ap
un
ov f
u
nct
i
o
n or
TEF has bee
n
used by
[
1
]
,
[
5
]
,
[1
4]
, [
15]
,
[1
8]
,
[2
6]
, [2
7]
. [
4
]
has pr
o
v
i
d
e
d
furt
her ex
pl
a
n
at
i
o
n
s
on t
h
e
conce
p
t
of di
rect
m
e
t
hod o
f
Ly
apu
n
ov /
ener
gy
f
u
n
c
tion
2.
CLAS
SIFI
C
A
T
ION
OF P
O
WER S
Y
STE
M
ST
ABILIT
Y.
I
t
is b
a
sed on
th
e
f
o
llow
i
ng
[4
],
[8
],
[
9
]:
(a)
Th
e
ph
ysical natu
re
of th
e resu
ltin
g m
o
d
e
of in
stab
ility as i
n
d
i
cated
b
y
the m
a
in
syste
m
v
a
riab
le in
wh
ich
instab
ility can
b
e
ob
serv
ed.
(b
)
The si
ze
o
f
t
h
e
di
st
u
r
bance
co
nsi
d
e
r
ed
whi
c
h
i
n
fl
uence
s
t
h
e
m
e
t
hod
of
cal
cul
a
t
i
o
n
an
d
p
r
edi
c
t
i
o
n
o
f
stab
ility.
(c)
The
devi
ces,
p
r
oces
ses, a
nd t
h
e t
i
m
e
span t
h
at
m
u
st
be t
a
ken i
n
t
o
c
o
n
s
i
d
erat
i
o
n i
n
or
d
e
r t
o
assess
stab
ility.
Power
system
is a
highly non-linear
system
that ope
r
ates in
a co
nst
a
nt
l
y
ch
angi
ng
en
vi
r
o
n
m
ent
.
Accord
ing
to [5
], [3
],
[2
], power system
st
a
b
ility can
b
e
classified
i
n
to
t
h
e fo
llo
wi
n
g
:
2
.
1
.
Ro
to
r Ang
l
e
Stability
It is th
e ab
il
ity o
f
in
tercon
n
ected
syn
c
hrono
u
s
m
ach
in
es of a p
o
wer system
t
o
rem
a
in
in
synchronism
.
The sta
b
ility problem
invol
ves
the study
of t
h
e electrom
echani
cal oscillations inhe
rent i
n
powe
r
sy
st
em
s [1]
,
[7]
,
[1
1]
. Ac
c
o
r
d
i
n
g t
o
[
7
]
,
a fu
ndam
e
nt
al
fact
or i
n
t
h
i
s
pr
o
b
l
e
m
is ho
w t
h
e
ou
t
put
s o
f
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
A Mod
e
l for Assessmen
t o
f
Tra
n
s
ien
t
S
t
ab
ilit
y o
f
Electrica
l Po
wer S
y
stem (Ga
n
i
yu
A.
Ajen
iko
k
o
)
50
0
syn
c
hrono
us
mach
in
es v
a
ry
with
resp
ect
to
th
eir
ro
t
o
rs
o
s
cillatio
n
s
. A
brief d
i
scussio
n
o
f
syn
c
h
r
o
nou
s
mach
in
es ch
aracteristics is h
e
lp
fu
l to
d
e
v
e
lop
th
e b
a
sic concep
ts
o
f
stab
ility [23
]
,
[25
]
,
[29
]
.
A sy
n
c
hrono
us m
ach
in
e h
a
s two
essen
tial circu
its: th
e field
,
wh
ich
is on
th
e ro
tors, and
t
h
e
arm
a
t
u
re, whi
c
h i
s
on t
h
e st
at
or
. The fi
el
d
w
i
ndi
n
g
i
s
su
ppl
i
e
d by
di
rect
cur
r
ent
p
o
w
er
wh
ile th
e term
i
n
als o
f
th
e arm
a
tu
re prov
id
e t
h
e lo
ad
power. Th
e
ro
tating
m
a
g
n
e
tic filed
o
f
t
h
e field
wi
n
d
i
ng
ind
u
c
es altern
atin
g
vol
t
a
ge
s w
h
e
n
t
h
e r
o
t
o
r i
s
dri
v
en
by
a p
r
i
m
e
m
over
(t
u
r
bi
n
e
). T
h
e f
r
eq
ue
ncy
o
f
t
h
e i
n
d
u
ced v
o
l
t
a
ges
d
e
pen
d
s
on
t
h
e
s
p
ee
d
o
f
t
h
e
r
o
t
o
r a
n
d
t
h
e
num
ber
of
pol
es
o
f
t
h
e m
achi
n
e.
The
c
h
a
nge
i
n
el
ect
ro
m
a
gnet
i
c
t
o
rq
u
e
o
f
a
sy
nch
r
o
n
o
u
s
m
achi
n
es
f
o
l
l
o
wi
ng
a
pert
ur
bat
i
o
n
can
be
res
o
l
v
ed
i
n
t
o
t
w
o c
o
m
pone
nt
s [
1
7
]
, [2
9]
:
(a)
Sy
nch
r
oni
zi
n
g
t
o
r
que
com
p
o
n
e
nt
, i
n
pha
se
wi
t
h
rot
o
r a
n
gl
e de
vi
at
i
o
n
(b
)
Dam
p
i
ng t
o
r
q
u
e
com
pone
nt
,
i
n
pha
se
wi
t
h
t
h
e spee
d
de
vi
at
i
o
n
.
Syste
m
stabilit
y depe
nds
on t
h
e e
x
is
tence
of both c
o
m
p
one
n
ts
of t
o
rque
for each of t
h
e
synchronous
mach
in
es. Lack
o
f
sufficien
t syn
c
hron
izing
to
rq
u
e
resu
lts
i
n
a p
e
riod
o
r
n
o
n
o
s
cillatio
n
in
stab
ility,
whereas
lack
of
d
a
m
p
in
g
torq
u
e
in
o
s
cillatio
n
in
stab
ility.
As i
n
th
e case
o
f
ro
tor an
g
l
e
stab
ility, it is usefu
l
t
o
cl
assi
fy v
o
ltag
e
stab
ility in
to
th
e
fo
l
l
o
w
ing
sub
categ
ories
[
5
],
[7
],
[9
],
[
1
1
]
.
(a)
Larg
e-d
i
stu
r
b
a
n
ce vo
ltag
e
stab
ility refers to
th
e system
’s ab
ili
ty to
m
a
in
tain
stead
y vo
ltag
e
s
fo
llowing
large d
i
stu
r
b
a
nces su
ch
as syste
m
fau
lts, lo
ss
o
f
g
e
n
e
ration
,
or
circu
it co
n
tingen
c
ies. Th
is
ab
ility
is d
e
termin
ed
b
y
th
e syste
m
an
d
lo
ad
ch
aract
eristics, an
d
th
e in
teractio
n
s
of b
o
t
h co
n
tinu
o
u
s
an
d
d
i
screte con
t
ro
ls an
d
p
r
o
t
ectio
n
s
[1
0
]
, [1
7
]
.
Determ
in
a
tio
n
o
f
larg
e-d
i
stu
r
b
a
n
ce
v
o
ltag
e
stab
ility
req
u
i
r
es t
h
e e
x
am
i
n
at
i
on o
f
t
h
e
no
nl
i
n
ea
r
res
p
o
n
se
o
f
t
h
e
po
wer
sy
st
em
ove
r a
per
i
od
of
t
i
m
e
su
fficien
t
to
cap
t
ure th
e
p
e
rform
a
n
ce an
d in
ter
act
i
o
n
s
of s
u
c
h
d
e
vi
c
e
s as m
o
t
o
rs,
un
de
r l
o
a
d
t
r
ans
f
o
r
m
e
r t
a
p cha
n
ger
s
, a
nd
ge
nerat
o
r
f
i
el
d-cu
r-
rent
l
i
m
i
t
e
rs. The st
udy
peri
od
o
f
i
n
t
e
rest
m
y
ext
e
n
d
fr
om
a few sec
o
nds
t
o
t
e
ns
of
m
i
nut
es [1
1]
,
[1
3]
,
[2
7]
.
(b
)
Sm
a
ll-d
i
stu
r
b
a
n
ce
v
o
ltag
e
stab
ility refers t
o
th
e system
’s ab
ility to
m
a
i
n
tain
stead
y
vo
ltag
e
s
wh
en
su
bj
ected
to
small p
e
rtu
r
b
a
tio
n
s
su
ch
as in
crem
en
tal ch
an
ges in
system
lo
ad
. Th
is
form
o
f
stab
ility is
influe
nce
d
by the characteris
tics of
l
o
ads,
cont
i
n
u
o
u
s
co
nt
r
o
l
s
, an
d di
s
c
ret
e
cont
r
o
l
s
at
a gi
ven
in
stan
t of tim
e.
Th
is con
cep
t i
s
u
s
efu
l
i
n
d
e
term
in
in
g
,
at any in
stan
ts,
ho
w th
e system
v
o
ltag
e
s will
respond to sm
a
ll syste
m
changes.
W
ith a
p
propriat
e ass
u
m
p
tions, system
e
quations
can
be linearize
d
for an
alysis th
ereb
y allo
wi
n
g
co
m
p
u
t
atio
n
of v
a
lu
ab
le sen
s
itiv
ity
in
fo
rm
at
io
n
as in
th
e case o
f
ro
tor
an
g
l
e stab
ility
[34
]
, [36
]
.
2
.
2
.
Vo
lta
g
e
Sta
b
ility
:
Vo
ltag
e
Stab
ility refers to
t
h
e ab
ility o
f
a p
o
wer system
to
main
tain
stead
y v
o
ltag
e
s at all b
u
s
es i
n
th
e syste
m
afte
r b
e
ing
su
bj
ect
ed
to
a d
i
sturban
ce fro
m
a g
i
v
e
n
i
n
itial o
p
e
ratin
g
co
nd
ition
.
It
d
e
p
e
nd
s on
th
e
ab
ility
to
m
a
in
tain
an
d
rest
o
r
e eq
u
ilib
ri
u
m
b
e
tween
lo
ad
de
m
a
n
d
and
lo
ad
sup
p
l
y fro
m
th
e p
o
wer syst
e
m
. In
-
stab
ility th
at
may resu
lt o
c
cu
rs i
n
th
e
form o
f
a pro
g
ressiv
e fall or rise o
f
vo
ltag
e
o
f
so
m
e
b
u
s
es
[9
], [1
0
]
,
[17
]
, [19
]
. A
po
ssib
l
e ou
tco
m
e o
f
vo
ltag
e
s in
stab
ility is
lo
ss o
f
lo
ad
in
an
area,
o
r
tripp
i
ng
of tran
sm
issio
n
l
i
n
es an
d
ot
he
r
el
em
ent
s
by
t
h
ei
r
pr
ot
ect
i
v
e
sy
st
em
s l
eadi
n
g
t
o
c
a
scadi
n
g
out
a
g
es.
L
o
s
s
o
f
sy
nch
r
o
n
i
s
m
of
som
e
generat
o
r
s
m
a
y
resul
t
fr
om
t
h
ese o
u
t
a
g
e
s o
r
fr
om
oper
a
t
i
ng c
o
n
d
i
t
i
o
n
s
t
h
at
vi
ol
at
e fi
el
d cu
rre
nt
l
i
m
i
t
.
(a)
Short-te
rm
voltage stability involve
s
dyna
m
i
cs of
fast a
c
ting loa
d
com
ponents s
u
ch as induction
m
o
to
rs, electron
i
cally co
n
t
ro
lled
lo
ad
s, an
d
HVDC conv
ert
e
rs. Th
e stud
y p
e
ri
o
d
o
f
in
terest is in
th
e
or
der
o
f
se
ve
ra
l
seco
nds
an
d
anal
y
s
i
s
re
qui
r
e
s sol
u
t
i
o
n
o
f
app
r
op
ri
at
e sy
st
em
di
ffere
nt
i
a
l
equat
i
on
s
;
th
is is si
milar
to
an
alysis o
f
ro
t
o
r ang
l
e stab
ility. Dyn
a
mi
c
m
o
d
e
lin
g
o
f
lo
ad
s is often
essen
tial. In
co
n
t
rast to
an
gle stab
ility, sh
o
r
t circu
its n
e
ar lo
ad
s are imp
o
rtan
t.
It is reco
mmen
d
e
d
th
at th
e term
tran
sien
t vo
ltag
e
stab
ility
n
o
t
b
e
u
s
ed
[12
]
, [1
8
]
,
[2
6
]
.
(b
)
Lon
g
-term
v
o
ltag
e
stab
ility
i
n
vo
lv
es slo
w
er actin
g
equ
i
pmen
t su
ch
as tap
-
ch
an
g
i
n
g
tran
sform
e
rs,
th
erm
o
staticall
y
co
n
t
ro
lled
lo
ad
s,
an
d ge
ne
r
a
t
o
r cu
rre
nt
l
i
m
i
t
e
rs. The st
udy
pe
ri
o
d
o
f
i
n
t
e
rest
m
a
y
ext
e
n
d
t
o
se
ve
ral
or m
a
ny
m
i
nut
es, a
n
d l
o
n
g
-t
erm
sim
u
l
a
ti
ons a
r
e re
q
u
i
r
ed f
o
r a
n
al
y
s
i
s
of sy
st
em
d
y
n
a
m
i
c p
e
rform
an
ce [8
], [9
], [19
]
. Stab
i
lity
is u
s
u
a
lly
d
e
term
in
ed
b
y
th
e resu
ltin
g
ou
tag
e
of
eq
u
i
p
m
en
t, rath
er th
an
th
e sev
e
rity o
f
th
e in
itial
d
i
stu
r
b
a
nce. In
stab
ility
is d
u
e
to
th
e lo
ss of lo
ng
-
term eq
u
ilib
ri
u
m
(e.g
, wh
en
lo
ads try t
o
restore th
ei
r p
o
wer b
e
yon
d
th
e cap
a
b
i
lity
o
f
th
e
t
r
ansm
i
ssi
on n
e
t
w
o
r
k a
nd c
o
nnect
e
d
ge
nera
t
i
on)
, p
o
st
-
d
i
s
t
u
r
b
a
n
ce st
ead-
s
t
a
t
e
operat
i
n
g
poi
nt
b
e
i
n
g
sm
a
ll-d
i
stu
r
ban
ce un
stab
le,
o
r
a lack
of at
tractio
n
toward
th
e stab
le po
st d
i
sturb
a
n
c
e eq
u
ilibriu
m
(e.g,
when a re
medial action i
s
applied t
o
o late.) [41].
2
.
3 Frequency
Sta
b
ility
Freq
u
e
n
c
y stabilit
y refers to
th
e ab
ility o
f
a p
o
wer syste
m
t
o
m
a
in
tain
stead
y freq
u
e
n
c
y fo
llo
wi
n
g
a
sev
e
re system up
set resu
lting
in
a sign
ifican
t im
b
a
la
n
ce
b
e
tween
g
e
n
e
ratio
n
and
lo
ad
. It
d
e
p
e
n
d
s o
n
th
e
ab
ility
to
m
a
in
tain
/resto
re equ
ilib
riu
m
b
e
tween
syste
m
g
e
n
e
ration
and
load
, with
m
i
n
i
mu
m
u
n
i
n
t
en
tio
nal lo
ss
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE Vo
l. 4
,
N
o
. 4
,
Au
gu
st 2
014
:
49
8
–
51
1
5
01
o
f
load
[4
],
[6]. In
stab
ility t
h
at m
a
y resu
lt o
ccurs in
t
h
e form
o
f
su
stain
e
d
freq
u
e
n
c
y swin
gs lead
i
n
g
t
o
t
r
i
ppi
ng
o
f
ge
nerat
i
n
g
u
n
i
t
s
an
d/
o
r
l
o
a
d
s.
Seve
re sy
st
e
m
upset
s ge
n
e
ral
l
y
resul
t
i
n
l
a
r
g
e e
x
c
u
rs
i
ons
o
f
fre
que
ncy
,
po
wer fl
ow
s, v
o
l
t
a
ge, an
d ot
h
e
r sy
st
em
vari
abl
e
s, t
h
e
r
eby
i
nvo
ki
ng t
h
e
act
i
ons o
f
p
r
o
cesses
,
co
n
t
ro
ls, an
d pro
t
ection
s
t
h
at
are
n
o
t
m
o
d
e
led
in co
nv
en
tion
a
l tran
sien
t st
ab
ility o
r
v
o
ltag
e
stab
ility stu
d
i
es
[1
9]
, [2
6]
. The
s
e pr
ocesses m
a
y
be very
sl
ow, suc
h
as b
o
i
l
e
r dy
nam
i
cs, or onl
y
t
r
i
g
gere
d fo
r ext
r
em
e sy
st
em
co
nd
itio
ns, such
as vo
lts/Hertz p
r
o
t
ectio
n
trip
p
i
n
g
g
e
n
e
ra
to
rs.
In
larg
e in
tercon
n
ected
po
wer system
s,
th
is
typ
e
o
f
situ
atio
n
is m
o
st co
mmo
n
l
y asso
ciated
with
co
nd
itio
n
s
fo
llowin
g
sp
littin
g
o
f
syste
m
s
in
to
islan
d
s
[1
2]
, [2
4]
.
3.
SINGLE
MACHINE INFINITE BUS
Accord
ing
to
[4
0
]
, sing
le m
a
ch
in
e infin
ite b
u
s
sy
stem
(SMIB) syste
m
is used to
demonst
r
ate the
fund
am
en
tal c
o
n
c
ep
ts and
p
r
i
n
cip
l
es
o
f
tran
sien
t stab
ility wh
en su
bj
ected to
larg
e
d
i
sturban
ces.
To
sim
p
lify
th
e assessm
en
t
o
n
tran
sien
t st
ab
ility, a class
i
cal
m
o
d
e
l o
f
th
e
m
ach
in
e is u
s
ed. Th
e
assum
p
t
i
ons m
a
de a
r
e as
f
o
l
l
o
w [
4
]
,
[3
9]
:
(i)
All m
echanical powe
r i
n
puts a
r
e c
onsta
nt
(ii)
Dam
p
i
ng
or
as
y
n
ch
ro
n
ous
p
o
w
er
i
s
ne
gl
i
g
i
b
l
e
(iii)
Voltage
E
be
hind the t
r
ansie
n
t reactance is
c
onsta
nt.
(iv)
Loa
d
s a
r
e re
presented as c
o
nstant im
pedanc
es.
The e
q
uat
i
o
n
o
f
m
o
t
i
on
or t
h
e
swi
n
g
eq
uat
i
o
n
descri
bi
n
g
t
h
e SM
IB
sy
st
e
m
i
s
as bel
o
w
[
4
]
:
= P
a
= P
m
–
P
e
;
=
ω
(1
)
Whe
r
e P
m
is the constant m
echanical
powe
r
input and
P
e
i
s
t
h
e ge
ne
rat
o
r’s
el
ect
ri
cal
po
w
e
r
out
put
.
4.
EQUAL
A
R
E
A
CRITE
RIO
N
(E
A
C
)
The t
h
ree c
o
n
d
i
t
i
ons:
Pre-
faul
t
,
duri
n
g
-fa
ul
t
,
post
-fa
u
lt conditions are very
significa
nt in the analysis
of E
q
ual
A
r
ea
C
r
i
t
e
ri
on
(E
A
C
). Fa
ul
t
occu
rre
nce o
n
one
o
f
the transmissio
n
lin
es
re
duce
s
the elec
trical
powe
r out
put a
nd accelerates
the rot
o
r an
gle. Syste
m
kinetic energy event
u
ally builds up until it arrives at the
clearing a
n
gle
δ
cl
, with accel
eration a
r
ea
A
acc
accu
m
u
lated.
At this inst
ance, the
exce
ss of electrical
powe
r
out
put
decelerates the rot
o
r a
ngle
until it reaches a
poi
nt
whe
r
e the
pre
v
ious st
ore
d
kinetic energy is t
o
tally
co
nv
erted
i
n
to
p
o
t
en
tial en
er
gy, i.e.
whe
n
t
h
e
area
A
dec
is equ
a
l to
area
A
acc
[3
9]
, [
4
1]
.
5.
LYAPUNOV THEORE
M
Lyap
uno
v’s stab
ility th
eo
rem
[21
]
,
[26
]
states th
at:
Th
e equ
ilib
rium
p
o
i
n
t
o
f
a
d
y
n
a
m
i
c syste
m
d
x
/
d
t
=f(x
)
is stab
le if there ex
ists a co
n
tinuo
usly
d
i
fferen
tiab
l
e
p
o
s
itiv
e d
e
fin
i
t
e
fu
n
c
tion
V(x) su
ch th
at
d
V
/d
t<0
.
If th
e total d
e
riv
a
tiv
e i
s
n
e
g
a
tiv
e, th
en
th
e
eq
u
ilibriu
m
p
o
in
t is said
t
o
b
e
asym
p
t
o
tical
ly
stab
le.
Direct m
e
th
o
d
o
f
Lyapun
ov
fun
c
tion
on
ly requ
ires th
e
kn
owledg
e at the in
stan
t wh
en th
e last o
p
e
ratio
n
is
car
r
i
ed ou
t
[
15], [
39-
41
].
Th
e
po
st-fau
lt
eq
u
a
tion
of a si
m
p
le syste
m
is
M
= P
m
–
P
e
_postf
ault
= P
m
– P
m
a
x_postf
ault
sin
(2
)
Int
e
grat
i
n
g
b
o
t
h
si
des
gi
ve t
h
e sy
st
em
ener
g
y
:
V =
-P
m
(
δ
-
δ
SE
P
)- P
m
a
x_postf
ault
[cos (
δ
)-c
os(
δ
SE
P
)]
(3
)
The c
r
itical energy
V
cr
is ev
al
u
a
ted
whe
r
e
δ
=
δ
UEP
,
ω
=0 as
indicated
in equ
a
tion
(4).
V
cr
= -P
m
(
δ
UEP-
δ
SEP
) -
P
m
a
x_postf
ault
[cos (
δ
UEP
)-
cos(
δ
SEP
)]
(4
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
A Mod
e
l for Assessmen
t o
f
Tra
n
s
ien
t
S
t
ab
ilit
y o
f
Electrica
l Po
wer S
y
stem (Ga
n
i
yu
A.
Ajen
iko
k
o
)
50
2
Th
e stab
ility o
f
th
e system
c
o
u
l
d
b
e
assured
if
δ
<
δ
UEP
or th
e system
’s to
tal en
erg
y
v
i
s
less th
an
t
h
e
critical
energy V
cr
for
δ
SEP
<
δ
<
δ
UEP.
If the system exceeds the un
stable equilibrium
point
δ
UEP,
the syste
m
would
co
n
tinu
e
to
build
-up
th
e k
i
n
e
tic en
erg
y
, which
g
i
v
e
s rise t
o
th
e v
e
lo
city. As a resu
lt, th
e ro
t
o
r an
g
l
e
wou
l
d
accelerate and
machine would lo
se sy
nchroni
sm
[15], [40].
6.
MULTI-MACHINE
SYST
EM
Let u
s
ex
tend
th
e tran
sien
t st
ab
ility assess
men
t
to
th
e
m
u
lt
i-m
ach
in
e syst
e
m
. A classica
l
m
o
d
e
l o
f
th
e
m
ach
in
es is u
s
ed
in
sy
ste
m
with
m
e
chanical powe
r
and electrical powe
r
ass
u
m
e
d t
o
be co
nst
a
nt
th
ro
ugh
ou
t th
e tran
sien
t, and
all lo
ad
s are
m
o
d
e
le
d as
consta
nt im
pedance
.
T
h
e c
o
nductance
G’s and
sub
s
cept
a
nces
B
’
s vary
f
r
om
pre-
, d
u
ri
ng
-,
t
o
post
-
faul
t
sy
st
em
confi
g
u
r
at
i
o
ns. T
h
e
m
o
t
i
on of t
h
e
i
-th
m
achi
n
e o
f
a
m
u
lt
i
-
m
achi
n
e sy
st
em
reduce
d
t
o
ge
ne
rat
o
r
i
n
t
e
r
n
al
n
o
d
es i
s
desc
ri
be
d
by
[2
7]
,
[3
6]
,
[3
7]
.
=
ω
i ;
M
I
=
P
mi
– P
ei
i=
1, 2,
…., n
(5)
W
h
er
e
P
ei
= E
i
2
G
ii
+
∑
cos
P
ei
= E
i
2
G
ii
+
∑
cos
Fo
r
j
=
1
,
2,…., n,
j
≠
i
,
= in
itial op
eratin
g
ro
t
o
r ang
l
e.
By so
lv
ing
th
e n
o
n
-
lin
ear swin
g
eq
u
a
tion
(1
), th
e tran
sien
t stab
ility o
f
a p
o
wer system co
u
l
d
b
e
d
e
term
in
ed
. Ho
wev
e
r d
u
e to
th
e n
on-lin
eari
t
y o
f
th
e d
i
fferen
tial eq
u
a
tions, th
e so
lv
i
n
g
pro
cess is ted
i
o
u
s
and
com
p
licated. Thus the
num
erical inte
rgration m
e
thods
have bee
n
a
ppli
e
d to e
x
am
ine a system
’s stability.
Rotor a
n
gle plot is obtaine
d
to de
term
ine the transient stability. Num
e
ri
cal intergration m
e
thods, s
u
ch as
R
u
n
g
e-
K
u
t
t
a
m
e
t
hods
, are
u
s
ed i
t
e
rat
i
v
el
y
t
o
ap
p
r
o
x
i
m
at
e t
h
e s
o
l
u
t
i
o
n
of
o
r
di
nary
di
ffe
r
e
nt
i
a
l
eq
uat
i
o
n
s
.
7.
E
X
TE
NDE
D EQUAL
A
R
E
A
CRITE
RIO
N
(EE
A
C
)
Ex
tend
ed
equal-area criteri
o
n
(E
EAC)
basically red
u
c
es th
e m
u
lt
i-mach
in
e tran
sien
t stab
ilit
y
assessm
en
t to
t
h
e equ
a
l-area criterio
n
b
y
d
e
co
m
p
o
s
ing
and
ag
greg
atin
g
t
h
e
m
u
lti-
m
ach
in
e syste
m
in
to
a two
-
machine equivalence, and further i
n
to
a si
ngl
e
-
m
achi
n
e infi
ni
t
e
bus (
S
M
I
B
)
eq
ui
val
e
nce [
6
]
,
[3
9]
.
In t
h
i
s
pape
r, only a si
ngle c
r
itical
machine,
presu
m
ed
to m
o
v
e
ap
art fro
m
th
e rest
, is co
n
s
i
d
ered
for sim
p
licit
y.
Th
e
m
u
lti-
m
a
c
h
in
e
system is
d
eco
m
p
o
s
ed
into
a critical
mach
in
e and
(
n
-1) of the rem
a
ining m
achines. The
exp
r
essi
on of r
e
l
a
t
i
v
e
m
o
t
i
on of
t
h
e
cr
itical machine with
respect
to
th
e
remain
in
g
m
ach
in
es in
t
h
e syste
m
i
s
devel
ope
d i
n
[
29]
,
[
36]
.
Th
e fo
llowing
n
o
t
ation
s
were u
s
ed
:
s “critical
m
achines”
a
its equivalent,
aggre
g
ated m
achine
A
th
e set
o
f
all
remain
in
g
m
ach
in
es
Th
e equ
i
v
a
len
t
in
ertia co
efficien
ts:
Ms=inertia coe
fficients
of the
critical
m
achine
M
a
=
∑
Є
; M
total
=
∑
; M =
(6
)
C
e
nt
re
of
an
gl
es (C
O
A
)
c
onc
ept
i
s
used
t
o
m
odel
t
h
e eq
ui
val
e
nt
m
achi
n
e
s
an
d t
h
ei
r m
o
t
i
ons
[
2
]
:
= rotor angle
of critical m
achine
δ
u
= M
a
-1
∑
Є
(7
)
The m
o
tion
of
the critical m
a
chine
a
n
d t
h
e (
n
-
1
)
rem
a
i
n
i
n
g
m
achi
n
es, w
h
i
c
h a
r
e
descri
be
d
by
[
5
]
:
M
s
= P
ms
-P
es
For 1
Є
A
(8
)
M
l
= P
ml
-P
el
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE Vo
l. 4
,
N
o
. 4
,
Au
gu
st 2
014
:
49
8
–
51
1
5
03
Th
e m
o
tio
n
o
f
th
e re
m
a
in
in
g syste
m
A is
i
llu
strated
b
y
th
e to
tal su
m o
f
all th
e
m
o
t
i
o
n
o
f
each
remain
ing
m
achi
n
es, w
h
i
c
h gi
ves
t
h
e fol
l
owi
n
g [
5
]
M
a
=
∑
f
o
r
1
Є
A (
9
)
For
f
u
rt
her
si
m
p
l
i
f
i
cat
i
on,
r
o
t
o
r
an
gl
e
of t
h
e
rem
a
i
n
i
ng m
a
chi
n
es
δ
j
are
m
a
de e
qui
val
e
nt
t
o
δ
a
fo
r
j
Є
A.
Hence
,
δ
s
-
δ
I
=
δ
s
-
δ
a
;
δ
j
-
δ
I =
0
fo
r I
,
j
Є
A
(
1
0
)
The electrical
powe
r c
ontri
bu
t
e
d by
eac
h
sy
st
em
are desc
ri
b
e
d as
[
5
]
,
[
6
]
:
P
es
= E
s
2
Y
ss
cos
(
θ
ss
) +
∑
Є
,
cos
(1
1)
P
el
= E
l
2
Y
ll
co
s(
θ
ll
) +
E
l
E
s
Y
ls
cos(
δ
a
-
δ
s
-
θ
ls
) +
∑
Є
,
cos
(1
2)
To m
odel
e
qui
val
e
nt
SM
IB
s
y
st
em
, t
h
e rot
o
r a
ngl
e i
s
de
fi
n
e
d as
δ
=
δ
s-
δ
a [5
]
The m
o
t
i
on
o
f
t
h
e eq
ui
val
e
nt
SM
IB
sy
st
em
is
M
= P
m
- P
e
(1
3)
W
h
er
e
P
m
= M
total
-1
(M
a
P
ms
-M
s
∑
Є
); P
e
= M
total
-1
(M
a
P
es
-M
s
∑
Є
Now, th
e equ
i
valen
t
SMIB equ
a
tio
n of m
o
tio
n is m
o
d
e
led
as fo
llo
ws [5
]
M
= P
m
- P
e
=P
m
– [P
c
– P
ma
x
sin(
δ
-
v
)]
(1
4)
W
h
er
e
P
e
= P
c
– P
ma
x
sin(
δ
-v
)
=
M
total
-1
(M
a
P
es
– M
s
∑
)
For 1
Є
A
P
ma
x
= M
tot
a
l
-1
[
∑
Є
2
cos
2
V= tan
-1
t
a
n
-
tan
=
∑
Є
∑
Є
=
∑
Є
∑
Є
8.
REVIEW OF
RELATED WORK
In
{32
}
, t
h
e
i
m
p
r
ov
em
en
t o
f
po
wer syst
e
m
tran
sien
t
stab
ility with
static syn
c
h
r
on
ou
s
series
com
p
ensator was prese
n
ted.
Th
e stud
y app
lied
th
e static syn
c
hrono
u
s
series co
m
p
en
sator (SSSC) to
imp
r
ov
e tran
sien
t stab
ility o
f
powe
r system
. The m
a
them
atical
and c
ont
rol
strategy of a
S
SSC is pre
s
ent
e
d to
verify the effect of the
SSSC
on tra
n
sient sta
b
ility. The SSSC is presented
via varia
b
le
voltage injection with asso
ciate trans
f
orm
e
r lea
k
age
reactance a
nd
the voltage source
. The se
ries voltage
i
n
jection m
odel SSSC is m
odeled into powe
r
flow
equat
i
o
n,
w
h
i
c
h i
s
use
d
t
o
d
e
t
e
rm
i
n
e an
d
cont
rol
t
h
e st
r
a
t
e
gy
. T
h
e
wo
rk
uses
m
achine s
p
ee
d de
vi
at
i
on t
o
cont
rol it. T
h
e
swing c
u
rve
of the t
h
ree
pha
s
e fa
ulted power system
with
and without a
SSSC is teste
d
a
n
d
com
p
ared in
various cases. T
h
e swi
ng c
u
rve
of system
without a SSSC i
n
crease
s
m
onotonically and t
hus t
h
e
system
can be conside
r
e
d
unstable,
whe
r
e
as the sw
ing
curve of system
with
a SSSC can be cons
idere
d
stab
le. SSSC can
th
erefore imp
r
ov
e tran
sien
t
stab
ility o
n
power system
.
[11
]
presen
ted a Po
wer Fl
ow an
d Tran
sien
t Stab
ility
Mo
d
e
ls of Facts Co
n
t
ro
llers for vo
ltag
e
and
angle
stability studies. In t
h
e
work, tra
n
sie
n
t stability
and powe
r
flow m
odel of
Thyrist
o
r Controlled R
eactor
(TCR) a
n
d
Vol
t
age Source
Inverter (V
SI)
b
a
sed
Flex
ib
le AC
tran
sm
issio
n
Sy
stem
(FACTS) c
o
ntr
o
llers
we
re
prese
n
t
e
d
.
[31] prese
n
ted a m
e
thod to
im
prove tra
n
si
ent st
ability of powe
r system
by T
h
yristor
Controlled
Phase Shifter
Transform
e
r TCPST). T
h
e mathem
atical
model of power
sy
st
em
equi
pp
ed wi
t
h
a TC
P
S
T was
syste
m
atically
deri
ved. The
param
e
ters of TCPST are
m
o
deled
in
to
power flow equ
a
tion an
d
thu
s
it was u
s
ed
to
determ
in
e co
n
t
ro
l st
rateg
y
. Th
e swing
curv
es of t
h
e t
h
ree ph
ase fau
lted
p
o
wer system
with
an
d wi
th
ou
t a
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
A Mod
e
l for Assessmen
t o
f
Tra
n
s
ien
t
S
t
ab
ilit
y o
f
Electrica
l Po
wer S
y
stem (Ga
n
i
yu
A.
Ajen
iko
k
o
)
50
4
TC
PST are t
e
s
t
ed and c
o
m
p
ared i
n
vari
ous
cases. The s
w
i
ng c
u
r
v
e o
f
sy
st
em
wi
t
hout
a TC
PST i
n
cre
a
ses
m
onot
oni
cal
l
y
and t
h
u
s
t
h
e sy
st
em
can be co
nsi
d
e
r
ed a
s
u
n
s
t
a
bl
e w
h
ereas
t
h
e swi
ng c
u
r
v
es
of sy
st
em
wi
t
h
a
TCPST returns to
stab
le eq
u
ilib
riu
m
p
o
i
nt. Fro
m
th
e si
m
u
la
tio
n
resu
l
t
s, th
e TCPST in
creases tran
sien
t
stab
ility o
f
p
o
wer system
.
In [4
1]
,
t
h
e pr
esent
a
t
i
on of
a
com
p
arative study of the
differe
n
t t
echni
ques in asse
ssing tra
n
sient
stab
ility was c
a
rried
o
u
t
. The p
a
p
e
r
d
i
scu
s
sed
th
e tran
si
en
t stab
ility o
f
a s
m
all
p
o
w
er syste
m
su
b
j
ected
to
l
a
rge
di
st
ur
ba
n
ces vi
a ap
pl
i
cat
i
on o
f
t
i
m
e
do
m
a
i
n
(TD
)
ap
p
r
oac
h
, e
x
t
e
n
d
e
d
eq
ual
area c
r
i
t
e
ri
on (
EEAC
)
an
d
direct
m
e
thod of Lyapunov
f
unction. T
h
ese
three m
e
thod a
r
e used to
determ
ine transient
stability of a sy
stem
.
Studies ha
ve been carrie
d
out on the IEEE
14 Bus syst
e
m
and sim
u
lation assessm
ent can be conducte
d on a
sm
al
l
power s
y
st
em
effect
i
v
el
y
.
In usi
ng
TD ap
pr
oac
h
,
several
si
m
u
l
a
t
i
ons are r
e
q
u
i
r
ed t
o
det
e
rm
ine t
h
e
critical clearing tim
e. EEAC can
determ
ine critical clear
i
ng a
ngl
e t
h
r
o
u
g
h
a si
ngl
e si
m
u
l
a
t
i
on
fo
r a
n
y
nat
u
re
of fault,
a
n
d hence
,
the syste
m
’s
critical
cleari
ng
tim
e
coul
d be
calculated. Direct m
e
thod of
Lyapunov
fun
c
tion
requ
ires o
n
l
y th
e k
n
o
w
led
g
e
at th
e last
in
stan
t o
f
fau
lt clearing
to
d
e
term
in
e tran
sien
t stab
ility. Th
is
m
e
thod is stra
ightforwa
r
d
but com
put
ational requi
rem
e
nts to determ
ine th
e unstable e
quilibrium
point are
si
gni
fi
ca
nt
.
9.
MODEL DE
VELOPMENT
Conssi
der a cla
ssical powe
r sy
st
e
m
sh
ow
n b
e
lo
w
i
n
Fi
g
u
r
e
1.
P
m
12
P
e
(
E
l
e
ct
r
i
ca
l
P
o
w
e
r)
W
Ge
ne
ra
to
r
X
1
R
X
1
g
In
f
i
n
i
te
Bu
s-
Bar
Fi
gu
re
1.
A
C
l
assi
cal Power
System
.
Defi
ne X
1
=X
1
g
+ X
1
R
(1
5)
Whe
r
e:
X
1
g
= tra
n
sient
reactance
of
s
y
nchronous
m
a
chine.
P
m
=
m
echanical input power.
Fr
o
m
Figu
r
e
1
,
in
th
e absen
c
e of
fr
ictio
n and w
i
nd
ag
e losses,
P
m
= P
e
in the
steady state
The ki
net
i
c
e
n
ergy
(K
E) o
f
a rot
a
t
i
n
g bo
dy
i
s
gi
ven
by
:
K.E
=
I
2
j
o
ul
e (1
6)
Whe
r
e
I = m
o
m
e
n
t
o
f
in
ertia
= ang
u
l
ar
v
e
lo
city in
⁄
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE Vo
l. 4
,
N
o
. 4
,
Au
gu
st 2
014
:
49
8
–
51
1
5
05
Or
K.E
=
I
(
)
=
M
j
oul
e
(1
7)
Whe
r
e
M = angu
lar
mo
m
e
n
t
u
m
M = I
Defi
n
e
H = In
ertia con
s
tan
t
o
f
th
e m
ach
in
e
H
=
H =
ω
(1
8)
Whe
r
e
G = m
achine
rating.
M =
ω
(1
9)
W
h
er
e
s
=
2
f
⁄
.
=
360
f ele
c
t -
⁄
.
M
⁄
=
⁄
M =
°
=
°
=
.
=MJ -
⁄
If a
s
u
dde
n
disturba
nce
occurs
in the
system
,
the ge
ne
rator e
xpe
riences
an
accelerating t
o
rque
T
a
= T
m
- T
e
(2
0)
Exp
r
essi
n
g
th
is in
term
s o
f
power;
T
a
= T
m
- T
e
(2
1)
P
a
= T
a
(2
2)
= (I
)
= (I
)
=M
Whe
r
e
is the
angula
r
accele
r
ation.
Ass
u
m
e
that the angula
r
displ
ace
m
e
nt is
, then
θ
=
ω
s
+
δ
(2
3)
Whe
r
e
δ
is t
h
e
rot
o
r
dis
p
lacement angle
.
Differen
tiate eq
u
a
tion
(2
3)
with
resp
ect to time,
(2
4)
Differen
tiate eq
u
a
tion
(2
4)
with
resp
ect to time,
=
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
A Mod
e
l for Assessmen
t o
f
Tra
n
s
ien
t
S
t
ab
ilit
y o
f
Electrica
l Po
wer S
y
stem (Ga
n
i
yu
A.
Ajen
iko
k
o
)
50
6
If t
h
ere
i
s
n
o
di
st
ur
bance
i
n
t
h
e sy
st
em
,
= 0
θ
δ
(
25
)
δ
δ
P
P
P
(2
6)
δ
P
P
′
δ
(2
7)
Whe
r
e
Eg
1
= gene
rator
internal volta
ge
calculate
d be
hind the
tra
n
s
i
ent
reactance
in
p.u..
V=
vol
t
a
ge
at
t
h
e i
n
fi
ni
t
e
bu
s
bar
.
δ
= R
o
t
o
r
di
s
p
l
acem
e
nt
an
gl
e i
n
radi
a
n
s
or
de
gree
.
Equ
a
tio
n (2
7) i
s
kn
own
as th
e Tran
sien
t stabilit
y
m
o
d
e
l.
To s
o
l
v
e
eq
uat
i
on
(
2
7
)
, t
h
e st
ep-
b
y
-
st
e
p
a
p
p
r
oac
h
i
s
use
d
wi
t
h
t
h
e
f
o
l
l
o
w
i
ng
basi
c as
su
m
p
ti
ons.
(i)
The accele
r
ating power
(Pa
)
com
puted at the be
ginn
ing
of an i
n
terval is
constant from
the m
i
ddle
o
f
th
e
p
r
o
ceed
i
n
g
in
terv
al to th
e in
terv
al consid
ered
.
(ii)
Th
e angu
lar
velo
city is co
n
s
tan
t
th
r
oug
hout an
y in
ter
v
al at th
e v
a
lu
e com
p
u
t
ed
f
o
r
th
e
m
i
d
d
l
e o
f
th
e in
terv
al.
In
teg
r
ating
equatio
n
(26
)
twice
over a sm
all interval of tim
e
, yields
Δδ
Δ
(2
8)
δ
Δ
(2
9)
Δ
=
Δδ
(3
0)
Equ
a
tio
n
s
(28
)
an
d
(3
0) g
i
v
e
t
h
e so
lu
tion
of th
e Tran
sien
t stab
ility
m
o
d
e
l
i
n
term
s o
f
ch
an
g
e
in
tim
e
an
d
ro
t
o
r
an
g
l
e
resp
ectively. Th
is m
o
d
e
l is u
s
ed
t
o
assess th
e st
ab
ility lev
e
ls of electrical p
o
wer syste
m
an
d
ind
i
cates
t
h
e be
ha
vi
o
u
r
of
p
o
w
e
r sy
st
e
m
s in the transi
ent state.
Transien
t stab
ility is j
u
dg
ed
fro
m
th
e
n
a
t
u
re of th
e swing
cu
rv
es
wh
ich is a
p
l
o
t
o
f
th
e
ro
t
o
r ang
l
es again
s
t
ti
m
e
. If th
e cu
rv
es settle at th
e pre-fau
lt
lev
e
l o
r
so
m
e
n
e
w lev
e
l aft
e
r so
m
e
ti
m
e
,
th
e system
is
stab
le.
Howe
ver if the
rotor angle i
n
c
r
eases c
o
ntinuous
ly
with
time, th
e
syste
m
is un
stab
le.
10
.
SIMULATION
Based
o
n
th
e d
e
v
e
lop
e
d
m
o
d
e
l an
d
t
h
e assu
m
p
tio
n
s
, a
p
r
o
g
ram
is written
i
n
MATLAB
programming
language
for t
h
e c
o
m
putation of the
m
achines rotor a
n
gles, electrical powe
r, acceleration
po
we
r an
d c
h
a
nge
i
n
rot
o
r
an
gl
es.
A
pl
ot
of
t
h
e swi
n
g
cu
r
v
e fo
r eac
h
of
t
h
e m
achi
n
e sy
st
em
s i
s
do
ne
wi
t
h
t
h
i
s
pr
o
g
ram
m
i
ng l
a
ng
ua
ge.
11
.
N
U
M
E
RICAL R
E
SU
LTS
The res
u
l
t
s
o
f
t
h
e w
o
r
k
i
n
di
ca
t
e
d t
h
at
fo
r t
h
e
si
ngl
e m
achi
n
e sy
st
em
suppl
y
i
ng a
n
i
n
fi
ni
t
e
bus
ba
r, t
h
e
rot
o
r angle increases as the
time in
creases. The cha
n
ge
in rotor angl
es
in
creases with
in
th
e first 0
.
2
500
secon
d
s
wh
ile
it falls after
0
.
6
500
seco
nd
s,
mak
i
n
g
t
h
e syst
e
m
to
b
e
un
st
ab
le with
i
n
th
e first
0
.
6
500
seco
nd
s.
I
t
b
eco
m
e
s st
ab
le b
e
tw
een
0
.
6
500
secon
d
s an
d
1.100
0
secon
d
s
an
d
betw
een
1.100
secon
d
s
an
d
3.250
secon
d
s
, it lo
ses its stab
ilit
y.
Th
e ch
ang
e
in
ro
t
o
r an
g
l
es
as
well as th
e ro
t
o
r an
g
l
es in
crease stead
ily after 0
.
70
seco
nds a
s
t
h
e
t
i
m
e
prog
resse
s as sh
ow
n i
n
Fi
gu
re
2. T
h
e r
e
l
a
t
i
onshi
p bet
w
een el
ect
ri
cal
po
we
r an
d t
i
m
e fo
r a
single m
achine
syste
m
is shown in
Figure
3.
Th
e electr
i
cal p
o
w
e
r
and
acceler
atio
n
p
o
w
e
r
fo
r
th
e
2
-
m
ach
in
e system
in
cr
ease fr
o
m
0
p
.
u
to
0
.
1
192
p.
u an
d
0
p.
u t
o
0
.
7
2
0
0
p
.
u
r
e
spect
i
v
el
y
w
i
t
h
i
n
t
h
e
fi
rst
0.
05
seco
n
d
s
whi
l
e
t
h
e
r
o
t
o
r
angl
es a
n
d c
h
ange i
n
rot
o
r a
n
gl
es i
n
crease
fr
om
0
p.
u
t
o
6.
6
6
7
2
p.
u a
n
d
11
.4
6
p.
u t
o
18
.1
2
7
2
p.
u
res
p
ect
i
v
e
l
y
as wel
l
wi
t
h
i
n
t
h
e
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE Vo
l. 4
,
N
o
. 4
,
Au
gu
st 2
014
:
49
8
–
51
1
5
07
sam
e
time frame. The electrical powe
r
dec
r
ease from
0.1192
p.u to -0.5198
p.u
while the acceleration
powe
r
i
n
creases
fr
om
0.7
2
00
p.
u t
o
1.
51
9
8
p.
u aft
e
r 0.
1
0
0
0
sec
o
n
d
s. The
rot
o
r
angl
es a
nd c
h
ange i
n
r
o
t
o
r a
ngl
e
s
increase stea
dily as the ti
me
progre
sses a
s
illustrated in Fi
gure 4.
While
the electrical powe
r and accel
eration
p
o
w
e
r
f
l
u
c
tu
at
e th
rou
gho
u
t
t
h
e ti
m
e
p
e
r
i
od
as show
n in
Figu
r
e
5.
For t
h
e m
u
lti-
machine system
s, the electri
cal powe
r a
nd acceleration
powe
r fl
uctuate
as the ti
m
e
pr
o
g
resses
wi
t
h
ge
ne
rat
o
r
2
wi
t
h
ave
r
a
g
e v
a
l
u
es o
f
-
1
.
3
80
0 p
.
u a
n
d 2
.
0
9
00
p.
u
respect
i
v
el
y
.
Ge
nerat
o
r 3
has
an a
v
era
g
e el
ectrical power and accelerat
ion
powe
r
of -1.4900 p.u
and 2.16
00
p.u re
spectively
while
gene
rat
o
r 4
has
an ave
r
age el
e
c
t
r
i
cal
and acc
el
erat
i
on
po
we
rs o
f
-
0
.
9
7
0
0
p
.
u a
nd
2.
08
0
0
p.
u res
p
ect
i
v
el
y
wi
t
h
the rotor a
ngle
s
and c
h
ange
in rotor a
ngles
increasi
n
g con
tin
uou
sly w
i
t
h
tim
e. f
o
r
g
e
n
e
r
a
t
o
r
s
2
,
3 an
d 4.
Gene
rat
o
r
2 h
a
s an a
v
era
g
e
chan
ge i
n
r
o
t
o
r an
gl
e a
nd
r
o
t
o
r a
ngl
es
o
f
19
3.
6
1
deg
r
ee
s an
d
78
3.
0
1
deg
r
ees
respectively. Gene
rator
3 has
an
aver
age
change in rotor a
ngles and
ro
t
o
r
ang
l
es of
26
8.17
d
e
g
r
ees and
47
3.
1
2
de
gree
s respect
i
v
el
y
whi
l
e
ge
ne
rat
o
r 4
has an a
v
e
r
age c
h
a
nge i
n
r
o
t
o
r an
gl
es
and
rot
o
r a
n
g
l
es of
31
3.
1
6
deg
r
ees
an
d 47
3.
1
2
de
grees
res
p
ect
i
v
el
y
.
For t
h
e m
u
lti-machine system
,
Generat
o
r 3 expe
rience
d t
h
e
m
o
st
vi
ol
ent
swi
n
g beca
use
i
t
i
s
cl
ose t
o
th
e fau
lt, h
a
s t
h
e larg
est ro
tor an
g
l
e, and
a
fairly lo
w in
er
tia co
nstan
t
,
h
e
nce it is ex
p
ected
to
b
e
th
e first
to
go
unst
a
bl
e,
henc
e un
rel
i
a
bl
e.
D
u
ri
ng t
h
e fi
rst
s
w
i
n
g ge
ner
a
t
o
r
3 p
u
l
l
s
o
u
t
of
sy
nch
r
o
n
i
s
m
and i
t
i
n
di
cat
e t
h
at
t
h
e
sy
st
em
i
s
unst
a
bl
e/
u
n
rel
i
a
bl
e
as sh
o
w
n
i
n
Fi
gu
re
6.
It is also
observed from
the s
w
ing c
u
rves t
h
at th
e fa
ult m
u
st be cleare
d
within a
certai
n
peri
od for
th
e syste
m
to
b
e
ab
le to
reg
a
i
n
its stab
ility o
p
e
rating
st
ate.
Th
e variatio
n
between
electrical p
o
wer an
d
ti
me i
s
sho
w
n i
n
Fi
gu
r
e
7.
Fi
gu
re
2.
S
w
i
n
g c
u
r
v
e
of
a si
ngl
e m
achi
n
e s
y
st
em
.
0
0.5
1
1.5
2
2.5
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
1000
0
T
i
me
(s
ec
o
nds
)
Rotor Angle
(degree
)
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