Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
V
o
l.
6, N
o
. 5
,
O
c
tob
e
r
201
6, p
p
. 2
225
~223
8
I
S
SN
: 208
8-8
7
0
8
,
D
O
I
:
10.115
91
/ij
ece.v6
i
5.1
081
8
2
225
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 Novel Approach to PID Contro
l
l
er Design for Im
provem
ent
of Transient Stability and Volt
age Regulation of Nonlinear
Power S
y
st
em
Rekh
a Ch
au
d
h
ar
y
1
,
Arun
K
u
mar
Sin
g
h
1
,
S
a
lig
ra
m
Ag
ra
wa
l
2
1
Electr
i
cal
Engg
. Deptt., NIT Jamshedpur, India
2
RVSCET, Jam
s
hedpur, Ind
i
a
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Apr 12, 2016
Rev
i
sed
Ju
l 13
,
20
16
Accepte
d
J
u
l 30, 2016
In this paper,
a novel design
met
hod for determining the o
p
timal PID
controller p
a
ram
e
ters for
non-lin
ear pow
er s
y
s
t
em using the particle swarm
optim
izat
ion (P
S
O
) algorithm
is
pres
ented
.
The d
i
rec
t
feedb
ack l
i
n
eari
zat
ion
(DFL) techn
i
que is used
to
linear
i
ze
the nonlinear
s
y
stem for
com
puting th
e
PID (DFL-PID)
controller
parameters. B
y
taking
an example of single
machine inf
i
nite
bus (SMIB) power s
y
st
em it has
been shown that
PSO based
P
I
D controller
s
t
abili
zes
th
e s
y
s
t
em
and r
e
s
t
ores
the pr
e-fa
ult s
y
s
t
em
perform
ance
aft
e
r fau
lt
is
cl
ear
ed and
lin
e is
r
e
s
t
ored.
Th
e per
f
orm
a
nce o
f
this
contro
lled
s
y
s
t
em
is
com
p
ared wi
th th
e
perform
ance
o
f
DF
L-s
t
ate
feedba
ck con
t
rol
l
ed power s
y
s
t
e
m
. It ha
s been shown that th
e per
f
ormance o
f
DF
L-P
I
D controlled s
y
s
t
em
is
superior as
com
p
ared to DF
L-s
t
a
t
e feedb
ack
controll
ed s
y
ste
m
. For sim
u
latio
n MATLAB 7 s
o
ftware
is used.
Keyword:
Direct fee
d
bac
k
linea
rization
Particle swarm op
ti
m
i
zatio
n
PI
D con
t
ro
ller
Stab
ility
State feedbac
k
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
:
Rek
h
a
Ch
au
dhary,
Depa
rtem
ent of Electrical a
nd Co
m
p
u
t
er
Engin
eer
ing
,
N I
T J
a
m
s
hedpu
r,
I
ndi
a.
Em
a
il: rch
y
7
2
@g
m
a
il.co
m
1.
INTRODUCTION
During the
pas
t
decades
, the
non
linea
r cont
rol techniques
have m
a
de gre
a
t adva
ncem
ent. Electrical
powe
r system
is an exam
ple of
nonlinea
r s
y
ste
m
[1]
whose
control has attracted
a
lot of researche
r
s.
The
maj
o
r
p
r
o
b
l
em
o
f
electrical p
o
w
er system is
to
m
a
in
tain
tr
an
sien
t stab
ility an
d
vo
ltag
e
regu
latio
n
fo
ll
o
w
i
n
g
t
h
e occ
u
r
r
ence
of s
u
d
d
e
n
di
st
ur
bance s
u
c
h
as faul
t
.
T
h
e
exci
t
a
t
i
on sy
s
t
em
of t
h
e ge
nerat
o
r c
ont
rol
s
t
h
e
t
e
rm
i
n
al
vol
t
a
ge an
d m
a
i
n
t
a
i
n
s i
t
at con
s
t
a
nt
pre
-
di
st
u
r
ba
nce o
p
erat
i
ng
poi
nt
[2]
.
To achi
e
ve t
h
e pre-
d
i
stu
r
b
a
n
ce
p
a
ram
e
ters o
f
the syste
m
, it is
equ
i
pp
ed wit
h
au
to
m
a
tic v
o
ltag
e
regu
lato
r
(AVR
) to
su
stain
v
o
ltag
e
v
a
riati
o
n
an
d
po
wer syste
m
stab
il
iz
er
(PSS)
to
p
r
o
v
i
d
e
o
s
cillatio
n
d
a
m
p
in
g
[3]. Th
e con
v
e
n
t
io
n
a
l
AVR/PS
S
are
designe
d
according to the sm
all perturba
ti
on linea
rized
m
odel [4]-[5].
This
m
e
thod
suffers
fro
m
th
e li
mita
tio
n
th
at it
is v
a
lid
fo
r sm
all
p
a
ram
e
tric
change a
nd he
nce
not
ef
fec
tiv
e to
larg
e d
i
sturban
ces.
In
case
o
f
large d
i
sturb
a
n
ce,
th
e lo
ad
an
g
l
e
ch
ang
e
s wh
ich m
a
y resu
lt in
o
s
cillatio
n
an
d th
e system
set
tles at
n
e
w
op
erating
p
o
i
n
t
or th
e l
o
ad
ang
l
e
o
s
cillatio
n
m
a
y b
e
i
n
creasi
n
g con
t
in
uo
usly with
ti
m
e
an
d
fin
a
ll
y th
e
syste
m
lo
ses syn
c
hr
on
ism
.
N
o
n
lin
ear
ex
citatio
n
co
n
t
ro
l
o
f
p
o
w
e
r
g
e
n
e
r
a
tio
n
equ
i
p
m
en
t is r
e
por
ted
[
6
]-[7
] in
whi
c
h i
m
prove
d m
e
t
hodol
ogy
fo
r p
o
w
e
r sy
st
em
dam
p
i
ng c
ont
rol
l
e
r a
nd
P
SS desi
gn
f
o
r i
n
t
e
rc
on
nect
ed
po
we
r
syste
m
h
a
s b
e
en
co
n
s
i
d
ered
. Co
-o
rd
inated
co
n
t
ro
l
for tran
sien
t stab
ility
enh
a
n
cem
en
t h
a
s
b
e
en
repo
rted
i
n
recent yea
r
s [8].
PID
con
t
ro
ller is th
e on
e which
is wid
e
ly
u
s
ed
in
th
e i
n
d
u
s
t
r
y b
ecau
s
e o
f
its sim
p
le
stru
cture an
d
robu
st
p
e
rfo
r
man
ce
un
d
e
r wid
e
rang
e v
a
riatio
n
o
f
op
eratin
g
con
d
ition
s
. Th
e con
v
e
n
tio
n
a
l
PI and
PID
cont
rol
l
e
rs a
r
e
i
n
effi
ci
e
n
t
and
sl
ow i
n
ha
ndl
i
ng sy
st
em
non
-l
i
n
eari
t
y
[9]
.
Gene
ral
l
y
t
h
e gai
n
s
of t
h
e co
nt
r
o
l
l
e
r
are tune
d eithe
r
by (i) m
a
nual (ii)
Ziegler Nichol
’s m
e
thod
and (iii) s
o
ft
ware m
e
thod. System
s that require a
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE
Vo
l. 6
,
N
o
. 5
,
O
c
tob
e
r
20
16
:
222
5
–
22
38
2
226
very
fast
ri
se t
i
m
e
and /
o
r ze
r
o
ove
rs
ho
ot
ca
nn
ot
be
t
u
ned
wi
t
h
Zi
egl
e
r
N
i
chol
’s
m
e
t
hod
. Fo
r t
h
ese rea
s
ons
, it
is d
e
sirab
l
e to add
n
e
w feat
u
r
es to
con
t
ro
ller to
im
p
r
ov
e th
e co
n
t
ro
ller p
e
rfo
r
m
a
n
ce.
Artificial in
tellig
en
ce
(A
I) t
ech
ni
q
u
e
s
such as ne
ur
al
net
w
or
k,
fuz
z
y
sy
st
em
and neu
r
al
-
f
uzzy
h
a
ve bee
n
wi
del
y
appl
i
e
d fo
r p
r
o
p
e
r
tu
n
i
ng
o
f
PID co
n
t
ro
ller [10
]
-[1
2
]
.
Recen
tly
d
e
sign
of in
t
e
llig
en
t PID co
n
t
ro
ller
for AVR system
h
a
s b
e
en
car
r
i
ed ou
t
[
13].
In
recent years
a num
ber of s
t
ochastic optim
iza
tion
m
e
th
ods has been developed,
am
ong
t
h
ese
PS
O
is one
of the
powe
rful m
e
thod for s
o
lving
optim
i
zation problem
which is
use
d
here.
It was devel
ope
d
through
sim
u
l
a
t
i
on of
a sim
p
l
i
f
i
e
d soci
al
sy
st
em
and
has bee
n
f
o
u
n
d
t
o
be r
o
bust
i
n
s
o
l
v
i
n
g co
nt
i
n
uo
us
no
nl
i
n
ea
r
o
p
tim
izat
io
n
prob
lem
s
[1
4
]
-[1
8
]
. Th
e PSO tech
n
i
qu
e ca
n
g
e
n
e
rate a h
i
gh
qu
ality s
o
lu
tion
with
sh
orter
calculation tim
e and
stable c
o
nve
r
ge
nce c
h
aracteristic th
a
n
othe
r st
ocha
stic
m
e
thods
[14]-[16]. Kee
p
ing i
n
v
i
ew
of ab
ov
e
facts, it is
p
r
opo
sed
to im
p
l
e
m
en
t PSO to
fi
nd
the
PI
D co
n
t
roller param
e
ters fo
r p
o
we
r
s
y
ste
m
un
de
rg
oi
n
g
l
a
r
g
e di
st
ur
ba
nce /
faul
t
.
2.
PARTICLE SWARM OPTIMIZ
A
TION
PSO
is a popu
latio
n
b
a
sed
sto
c
h
a
stic appr
o
a
ch
fo
r
so
lvin
g
co
n
tinuo
u
s
an
d
d
i
scr
e
te
o
p
tim
izat
io
n
pr
o
b
l
e
m
.
In t
h
i
s
pape
r, t
h
e P
S
O
m
e
t
hod i
s
u
s
ed f
o
r c
ont
rol
l
er desi
g
n
o
f
p
o
we
r sy
st
em
.
Hence a
bri
e
f
r
e
vi
ew
of t
h
e m
e
thod is prese
n
te
d here. T
h
e P
S
O
m
e
thod
wa
s
first introduce
d
by Ke
nnedy a
nd
Ebe
r
hart in 1995
[1
4]
. It
i
s
a popul
at
i
o
n base
d
evol
ut
i
o
na
ry
heu
r
i
s
t
i
c
opt
i
m
i
zat
i
on t
echni
q
u
e de
vel
o
ped
on t
h
e b
a
si
s of
soci
al
beha
vi
o
u
r
o
f
b
i
rds fl
oc
ki
n
g
i
n
searc
h
of
fo
o
d
an
d fi
s
h
sc
h
ool
i
n
g.
The m
e
t
h
o
d
ha
s bee
n
fo
un
d t
o
be
ro
bust
i
n
so
lv
i
n
g
prob
lem
s
with
n
o
n
lin
earity, non
-d
i
fferen
tiab
ility an
d m
u
ltip
le
o
p
tim
a. Th
e
main
featu
r
e
o
f
t
h
is
m
e
t
hod i
s
t
h
at
i
t
can be
easi
l
y
im
pl
em
ent
e
d an
d
has st
abl
e
c
o
nve
r
g
ence c
h
aract
eri
s
t
i
c
wi
t
h
g
o
o
d
com
put
at
i
onal
effi
ci
ency
[
1
5]
. A s
u
r
v
ey
of t
h
e m
e
t
hod
and i
t
s
ap
pl
i
cat
i
on t
o
p
o
w
er sy
st
em
pro
b
l
e
m
s
i
s
avai
l
a
bl
e i
n
re
fere
nce [
1
6]
-[
1
7
]
.
PS
O
has
b
een
used
f
o
r
o
p
t
i
m
u
m
desi
g
n
of
PI
D c
o
nt
ro
l
l
e
r i
n
A
V
R
s
y
st
em
[1
8]
, w
h
er
e pa
r
a
m
e
t
e
rs of c
o
n
t
rol
l
e
r are t
une
d t
h
ro
u
gh P
S
O. The m
o
st attractive feat
ure
of t
h
e PSO m
e
thod is
its si
m
p
l
i
city
o
f
ap
p
licatio
n
an
d
it in
vo
lv
es o
n
l
y two
equations (1) a
nd
(2) associated
with two
vect
ors the
p
o
s
ition
X and
v
e
lo
city V. The m
e
th
o
d
is
d
e
scrib
e
d
in brief as fo
llows:
Each pa
rticle in PSO
flies in
the searc
h
s
p
ace with
velocity V
whic
h is dynam
i
ca
lly adjusted
according to its own flying e
xpe
rience a
nd i
t
s com
p
anion’s
flying experie
n
ce.
Eac
h
parti
c
le keeps track of its
co-ordinate in the pr
oblem
space, whic
h is associat
ed with the best solution
it has achieve
d so fa
r. The
n
u
m
b
e
r of v
a
riab
les o
f
th
e
o
p
tim
izat
io
n
prob
lem
is assu
med
as ‘m
’. F
o
r th
e so
lu
tion
of th
is p
r
oble
m
a
p
opu
latio
n
/
swarm
o
f
‘n’ p
a
rticles is assu
med
wh
ere ea
ch
p
a
rticle rep
r
esen
ts a feasib
le so
lu
tion in
m
-
dim
e
nsional problem
space. The
position
and v
e
l
o
city
of
p
a
rticle is rep
r
esented
as:
,
,………,
and
,
,…
……,
res
p
ectivel
y.
Th
e so
lu
tion
of th
is
p
r
o
b
l
em
is go
vern
ed b
y
t
h
e
fo
llowing
t
w
o equ
a
tion
s
:
(1
)
(2
)
The si
gnifica
nce of t
h
e term
s use
d
i
n
e
q
ua
t
i
on (
1
) an
d
(2
) are:
and
are t
w
o
po
sitiv
e con
s
tan
t
s
gene
ral
l
y
assu
m
e
d a val
u
e
o
f
2;
and
are t
w
o ra
nd
om
l
y
generat
e
d n
u
m
b
ers wi
t
h
i
n
a ra
nge
of
[
0
&
1
]
;
is th
e b
e
st
p
o
sitio
n
of
p
a
rti
c
le i;
is
th
e b
e
st p
a
rticle
po
sitio
n b
a
sed
o
n
o
v
e
rall
swarm
’
s
ex
p
e
rien
ce
(h
av
ing
op
tim
al c
o
st ou
t of th
e co
m
p
lete swarm
)
if th
e
p
a
rticle h
a
s th
e
b
e
st
co
st th
en
,
,
………,
; k
is t
h
e iteratio
n coun
t.
Fo
r iterativ
e so
lu
tion
o
f
th
e p
r
ob
lem
in
iti
al ran
d
o
m
v
a
l
u
e o
f
th
e swarm p
o
s
itio
n
X is assu
m
e
d
k
eep
i
n
g in
v
i
ew th
e equ
a
lity an
d in
eq
u
a
lity co
nstrain
t
s
o
f
th
e
v
a
riab
les.
In
itial v
a
lu
e
o
f
th
e swarm
v
e
lo
city
i
s
assum
e
d at
ran
d
o
m
bou
nd
ed
bet
w
ee
n m
a
xi
m
u
m
and
m
i
nim
u
m
val
u
e of
vel
o
ci
t
y
,
of eac
h
p
a
rticle. In
itial v
a
lu
e of
is assu
m
e
d
as in
it
ial v
a
lu
e
o
f
the po
sitio
n
. The v
e
lo
city an
d po
sitio
n are
i
m
proved using
equation
(1) and (2)
re
spect
ively at each it
eration
keepi
n
g
in vie
w
of position a
nd
vel
o
city
con
s
t
r
ai
nt
s.
T
h
e i
t
e
rat
i
v
e
pr
o
cess st
o
p
s
as
per
t
h
e
st
o
ppi
ng
cri
t
e
ri
on
d
e
fi
ne
d
whi
c
h
can
be t
h
e m
a
xi
m
u
m
num
ber
of
i
t
e
r
a
t
i
ons as
si
g
n
ed
o
r
ot
her
cri
t
e
ri
on
.
3.
SYSTE
M
REPRESENT
A
T
I
ON
AND PROBLEM FORMUL
ATION
The p
r
o
b
l
e
m
o
f
po
we
r sy
st
em
cont
rol
has
been c
onsi
d
ere
d
i
n
t
h
i
s
sect
i
on. The
po
wer
sy
st
em
i
s
a
larg
e
co
m
p
lex
n
e
two
r
k
represen
ted
b
y
n
o
n
lin
ear
m
a
th
em
a
t
ical
m
o
d
e
l. For con
t
ro
llin
g an
d op
eratio
n of th
is
sy
st
em
a sim
p
li
fi
ed p
o
we
r sy
st
em
m
odel
i
s
con
s
i
d
ere
d
i
n
Fi
gu
re 1
.
I
n
p
o
w
er sy
st
em
dynam
i
c st
udy
, t
h
e m
o
st
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
A No
vel App
r
oa
ch to
PID C
o
n
t
ro
ller
Design fo
r
Imp
r
o
v
emen
t o
f
Tra
n
s
ient S
t
ab
ility .... (Rekha
Chau
dha
ry)
2
227
im
port
a
nt
c
o
m
p
o
n
e
n
t
i
s
t
h
e s
y
nch
r
o
n
ous
ge
nerat
o
r
wi
t
h
its asso
ciated
excitatio
n
syste
m
an
d
its co
n
t
rol [3
]
.
Al
t
h
o
u
gh t
h
e
d
y
n
am
i
c
behavi
ou
r o
f
a sy
nch
r
on
o
u
s ge
nerat
o
r i
n
p
o
we
r sy
st
em
i
s
very
com
p
li
cat
ed un
d
e
r faul
t
co
nd
itio
n
d
u
e
to
non
lin
eariti
es su
ch as t
h
e m
a
g
n
e
tic sa
turatio
n, a
classical si
m
p
lified
th
ird ord
e
r
d
y
n
a
mi
c
gene
rat
o
r m
o
d
e
l
i
s
n
o
rm
al
l
y
use
d
fo
r e
x
ci
t
a
t
i
on c
o
nt
rol
.
T
h
e cl
assi
cal
dy
nam
i
cal
m
odel o
f
a
SM
IB
p
o
w
e
r
sy
st
em
i
s
descr
i
bed
bel
o
w:
Fi
gu
re
1.
Sc
he
m
a
t
i
c
m
odel
of
SM
IB
sy
st
em
The
p
o
we
r sy
s
t
em
i
s
represe
n
t
e
d
by
fol
l
o
w
i
ng
dy
nam
i
cal
equat
i
o
ns
wi
t
h
st
an
dar
d
ass
u
m
p
ti
ons a
n
d
nom
encl
at
ure (
A
p
p
e
ndi
x):
Mechanical dynam
i
cal
equations:
∆
(3
)
∆
(4
)
Gene
rato
r Dy
n
a
m
i
cal
e
quation (Electrical):
́
(5
)
Turbine
Dy
na
mical equation (Mecha
n
ical):
(6
)
Turbine
val
v
e cont
rol (Mecha
n
ical):
(7
)
El
ect
ri
cal
ope
r
a
t
i
onal
E
quat
i
o
ns:
(8
)
(
9
)
(1
0)
(1
1)
(1
2)
(1
3)
2
1/2
(1
4)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE
Vo
l. 6
,
N
o
. 5
,
O
c
tob
e
r
20
16
:
222
5
–
22
38
2
228
The
po
wer
sy
st
em
nonl
i
n
ea
r
dy
nam
i
cal
m
o
del
i
s
re
prese
n
t
e
d by
e
q
uat
i
o
n
s
(3
) t
o
(
7
).
Eq
uat
i
on
(
3
) t
o
(5
) re
pre
s
ent
s
el
ect
ri
cal
dy
nam
i
cs and eq
ua
t
i
on (
6
)
–
(7
) represe
n
ts m
e
c
h
anical dy
namics. Equations
(8) t
o
(1
4) a
r
e t
h
e o
p
e
rat
i
onal
eq
uat
i
ons
. Un
de
r sm
al
l
di
st
urba
nce
t
h
e sy
st
em
i
s
cont
rol
l
e
d
on t
h
e basi
s o
f
l
i
n
e
a
ri
zed
m
o
d
e
l (ab
o
u
t
th
e op
erating
po
in
t). For larg
e d
i
stu
r
b
a
n
ce lik
e fau
lt o
n
th
e
syste
m
, if th
e c
o
n
t
ro
ller is d
e
sig
n
e
d
o
n
th
e
b
a
sis of lin
earized
m
o
d
e
l th
e con
t
ro
lled
syste
m
m
a
y
n
o
t
wo
rk
satisfacto
r
ily o
r
m
a
y lead
to
in
stab
ility.
Th
e o
t
h
e
r
p
r
o
b
le
m
is
th
at th
e
termin
al v
o
ltage o
f
th
e g
e
ne
ra
tor is also affe
cted by large disturba
nce a
n
d it is
an
im
p
o
r
tan
t
param
e
ter o
f
th
e po
wer system
to
b
e
m
a
in
tain
ed
co
nstan
t
for satisfacto
r
y
operatio
n.
The problem
c
a
n be stated as
: ‘T
h
e
prob
lem is
to
d
e
sign PID co
n
t
ro
ller in
th
e ex
citatio
n
loo
p
and
p
r
op
ortio
n
a
l co
n
t
ro
ller in
gov
erno
r loop
wh
o
s
e op
ti
m
a
l
p
a
ram
e
ters are co
m
p
u
t
ed
wit
h
th
e help
of
PSO
alg
o
rith
m
th
at
will i
m
p
r
o
v
e
bo
th
th
e t
r
an
sien
t stab
ility an
d termin
al v
o
ltag
e
sim
u
ltan
e
o
u
sly fo
r
p
o
wer syste
m
un
de
rg
oi
n
g
l
a
r
g
e
di
st
ur
ba
nce
suc
h
as
fa
ul
t
’
. I
n
t
h
i
s
pa
per
DFL
t
ech
ni
q
u
e
i
s
use
d
t
o
c
ont
rol
t
h
e n
o
n
l
i
n
ear
sy
st
em
as di
scusse
d i
n
t
h
e
ne
xt
sect
i
o
n.
4.
DFL TEC
HN
IQUE
FO
R P
O
WE
R SYST
EM SOLUTION
The stability and
volta
ge regulation
proble
m
of powe
r syst
em
unde
r fault condition is stated in the
p
r
ev
iou
s
secti
o
n. Th
e so
l
u
tio
n
o
f
t
h
is pr
ob
lem
is co
n
s
ider
ed
in
t
h
is sectio
n
.
Th
e non-
lin
ear
co
n
t
r
o
ll
er
is a
d
y
n
a
m
i
c DFL co
m
p
en
sato
r
th
ro
ugh
th
e excitatio
n
lo
op
to cancel the
non-
linearity and a robust fee
dbac
k
co
n
t
ro
l wh
ich
g
u
a
ran
t
ee th
e asy
m
p
t
o
tic stab
ility & v
o
ltag
e
regu
latio
n. To
i
m
p
r
ov
e th
e
p
e
rfo
r
m
a
n
ce o
f
po
wer
system
under
fault condition DFL tec
hni
que [19] is useful m
e
thod fo
r powe
r system
non-linear
cont
rol
th
ro
ugh
ex
citatio
n
l
o
op
. In
DFL co
n
t
ro
ller
d
e
sign
t
h
e
n
onlin
ear term
s o
f
th
e
p
o
wer syste
m
are co
m
p
en
sated
by
m
easurable
output feedback [2
0]. In recent years design of obse
rver ba
sed
dyna
m
i
c controller has
em
erged
as a
p
o
we
rf
ul
t
o
ol
fo
r t
h
e
sy
st
em
havi
n
g
un
-m
easurabl
e
o
u
t
p
ut
va
ri
abl
e
s
[2
1]
-[
2
2
]
.
For
i
m
pl
em
ent
a
t
i
on o
f
DFL s
t
at
e/
out
p
u
t
va
ri
ables
should be
m
easurable
. Since
i
n
eq
uat
i
on
(5
)
is p
h
y
sically un
-m
easu
r
ab
le,
it is eli
m
in
ated
b
y
d
i
fferen
tiatin
g
equ
a
tio
n fo
r
(
e
qu
atio
n
(
1
0
)
)
an
d u
s
i
ng
o
t
h
e
r
n
ecessary su
bstitu
tio
n
o
n
e
can write:
′
(1
5)
Equ
a
tio
n (1
5),
after carrying
o
u
t
t
h
e
re
qu
ired
su
bstitu
tio
n
can
b
e
written
as:
∆
∆
+
(1
6)
W
h
er
e
=
′
′
′
(1
7)
Since
,
a
r
e m
easura
b
le,
can
b
e
com
put
e
d
us
i
n
g
o
p
e
r
at
i
onal
eq
uat
i
o
ns.
,
are a
v
ailable
by
direct
m
e
a
s
urem
ent,
he
nce the
c
o
m
p
ensating la
w
is
p
r
actically realizab
le wh
ich
is
rep
r
ese
n
t
e
d by
equat
i
o
n (
1
8
)
.
′
′
(1
8)
Th
e m
o
d
e
l (3)
to
(5
) is th
erefo
r
e lin
earized
an
d it is
re
pres
ent
e
d
by
fol
l
o
wi
n
g
e
quat
i
o
ns
(
1
9
)
t
o
(
2
1)
∆
(1
9)
∆
(2
0)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
A No
vel App
r
oa
ch to
PID C
o
n
t
ro
ller
Design fo
r
Imp
r
o
v
emen
t o
f
Tra
n
s
ient S
t
ab
ility .... (Rekha
Chau
dha
ry)
2
229
∆
∆
+
(2
1)
Whe
r
e
is th
e
new inpu
t rep
r
esen
ted b
y
equ
a
tio
n
(17
)
.
Fo
r im
p
l
e
m
en
t
a
tio
n
of t
h
e ex
citatio
n
con
t
ro
ller,
is th
e
co
n
t
ro
lled inpu
t
g
i
v
e
n
t
o
t
h
e syste
m
th
ro
ugh
excitatio
n
.
Wh
en
com
p
o
s
ite co
n
t
roller (ex
c
itation
an
d gov
erno
r co
n
t
ro
l) is im
p
l
e
m
en
ted
th
en
bo
th
th
ro
ugh
ex
citatio
n
and
th
ro
ugh
gov
ern
i
ng
m
ech
an
ism actin
g
sim
u
lt
an
eou
s
ly. Th
e d
e
tail ab
ou
t
th
e
cont
rol
l
e
r i
m
pl
em
ent
a
t
i
on i
s
d
i
scusse
d
next
.
4.
1.
PID c
o
ntr
o
ller implementation
It is assu
m
e
d
t
h
at th
e po
wer syste
m
is o
p
e
ratin
g
at n
o
rm
al
co
nd
itio
n
b
e
fore th
e in
cep
ti
on
of fau
lt. It
is req
u
ire
d
to
desig
n
a P
I
D c
ont
roller
fo
r a
fault o
n
t
h
e power system
so that the
con
t
rolled
syste
m
is
stab
le
an
d
attain
s t
o
th
e pre-fau
lt o
p
eratin
g
con
d
itio
n
after
fa
ult is
cleared.
The
block
diagram
represe
n
tation for t
h
e
PID con
t
ro
ller
i
m
p
l
e
m
en
ted
on
th
e lin
earized
sy
st
em
has b
een
prese
n
t
e
d
i
n
Fi
gu
re
2.
Figure
2. Powe
r system
lineari
zed m
odel wit
h
PID controller
In t
h
is fi
gu
re
is n
e
w inpu
t linearized
p
a
rt
of
th
e con
t
ro
ller.
is th
e non
lin
ear con
t
ro
l inpu
t
to
th
e
syste
m
co
m
posed of
and the
measured
va
riable as in equation
(18).
X(t)
is the state vari
able of t
h
e syste
m
.
The PI
D
c
o
nt
r
o
l
l
e
r i
s
desi
gn
ed t
o
t
r
ack t
h
e desi
red o
u
t
p
ut
val
u
es. T
h
e
i
nput
gi
ve
n t
o
t
h
e PI
D con
t
rol
l
e
r
considere
d
here is error i
n
s
p
eed
as th
is quan
tity is easily
measu
r
ab
le and
realizab
le. Th
e m
a
th
e
m
at
ic
al
m
odel
l
i
ng f
o
r
t
h
e P
I
D
c
ont
r
o
l
l
er t
o
be
desi
gn
ed i
s
di
scu
ssed
bel
o
w:
(2
2)
(2
3)
R
e
pl
aci
ng
i
n
t
e
gral
t
e
rm
by
an
d
deri
vat
i
v
e
t
e
rm
wi
t
h
t
h
e e
quat
i
o
n
(
2
0)
,
eq
u
a
tion
(2
3) will
b
e
written
as:
(2
4)
(2
5)
Sub
s
titu
tin
g the v
a
lu
e of
f
r
o
m
equat
i
on
(2
5
)
,
∆
in
equ
a
tio
n (2
2)
on
e can
write
∆
∆
+
1
′
∆
1
′
1
(2
6)
The values of coefficients
,
and
are to
b
e
ob
tain
ed
with
t
h
e h
e
l
p
of PSO. So
m
e
ti
mes
it is
o
b
s
erv
e
d th
at
ex
citatio
n
co
ntro
l alon
e stabilizes th
e syste
m
an
d
regu
lates eith
er lo
ad ang
l
e or termin
al
u
f
(t
)
Direct Feedback
L
i
nearized
M
odel
X(
t
)
v
f
(
t
)
PID
Co
n
t
ro
ller
Co
m
p
ensatingL
aw
Nonlinear
M
odel
Ref
e
rence
+
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE
Vo
l. 6
,
N
o
. 5
,
O
c
tob
e
r
20
16
:
222
5
–
22
38
2
230
vol
t
a
ge
. I
f
bot
h t
h
e l
o
ad
an
gl
e an
d t
e
rm
inal
v
o
l
t
a
ge
ha
s t
o
be
re
gul
a
t
ed t
h
e
n
b
o
t
h
exci
t
a
t
i
on as
wel
l
as
go
ve
rn
or
co
nt
r
o
l
ha
s t
o
be
i
m
pl
em
ent
e
d sim
u
l
t
a
neo
u
sl
y
,
w
h
i
c
h i
s
co
nsi
d
e
r
ed
ne
xt
.
Th
e m
ech
an
ical g
o
v
e
rn
i
n
g eq
u
a
tion
s
are
written
as
in equ
a
tio
n (6
) and (7). Po
wer con
t
ro
l inp
u
t
,
i
s
t
h
e
out
put
of
g
o
v
er
n
o
r
co
nt
r
o
l
.
It
i
s
p
r
o
pos
ed
t
o
be
re
al
i
zed by
p
r
op
ort
i
o
nal
c
o
nt
ro
l
.
T
h
e
rel
a
t
i
o
n
bet
w
ee
n c
ont
ro
l
l
e
r gai
n
and
i
s
gi
ven
by
e
q
u
a
t
i
on
(2
7)
.
(2
7)
The l
a
y
o
u
t
f
o
r
PSO
o
p
t
i
m
i
zed PI
D co
nt
r
o
l
l
e
r
desi
g
n
has
bee
n
s
h
o
w
n i
n
Fi
g
u
re
3.
Er
ro
r
is th
e inp
u
t
to
th
e
PID contro
ller, t
h
e
ou
tp
u
t
of th
e con
t
ro
ller is
in
case of ex
citatio
n
co
n
t
ro
l and
pl
us
fo
r
bot
h
ex
citatio
n
and
g
o
v
e
rno
r
con
t
ro
l. Th
is is i
n
pu
t to
t
h
e
p
o
wer system
to
b
e
co
n
t
ro
lled. The op
ti
m
u
m
v
a
lu
es
for
,
,
and
ar
e obtain
e
d
thro
ugh th
e PSO algor
ith
m
.
Th
e b
l
ock
w
ith
rep
r
esen
ts the
ob
ject
i
v
e
fu
nct
i
on as i
n
t
e
gral
squa
re er
ro
r f
o
r c
o
m
put
i
ng
t
h
e PI
D co
nt
r
o
l
l
e
r pa
ram
e
t
e
rs by
PS
O i
n
fi
g
u
re
3
.
whe
r
e
is act
u
a
l ou
t
p
u
t
wh
ich
is m
o
n
itored
an
d
i
s
t
h
e d
e
si
red val
u
e of
varia
b
le (pre
-fault value
)
. T
h
ere are t
h
ree s
t
ates:
,
and
s
o
error in
eac
h state is:
t
;
t
0
;
t
So
t
h
e to
tal i
n
teg
r
al squ
a
re error is
represen
ted
b
y
equ
a
tion
(4.29
)
(2
8)
Fi
gu
re 3.
B
l
oc
k di
ag
ram
for PSO
base
d PI
D
c
ont
rol
l
e
r
The o
p
t
i
m
al
v
a
l
u
es of co
nt
r
o
l
l
e
r param
e
t
e
rs of t
h
e exci
t
a
t
i
on/
g
o
v
er
no
r cont
rol
can be
obt
ai
ne
d b
y
appl
y
i
n
g
PS
O (m
i
n
im
i
z
i
ng object
i
v
e fu
nct
i
o
n
)
. T
h
e com
p
l
e
t
e
proce
d
u
r
e i
s
expl
ai
ne
d by
t
a
ki
ng an e
x
a
m
pl
e of
SM
IB
p
o
w
er
s
y
st
em
i
n
next
s
ect
i
on.
5.
SYSTE
M
SI
MUL
A
TIO
N
AND RES
U
L
T
S
For si
m
u
l
a
t
i
on SM
IB
sy
st
em
is consi
d
ere
d
as
sho
w
n i
n
Fi
gu
re 1, t
h
e ge
ner
a
t
o
r pa
ram
e
t
e
r
s
have be
e
n
t
a
ken f
r
o
m
W
a
ng et
al
[2
0]
as i
ndi
cat
ed
b
e
l
o
w:
(sat
urat
i
on
phe
n
o
m
e
na has n
o
t
bee
n
con
s
i
d
ere
d
). S
y
st
em
p
a
ram
e
ters u
s
ed
in
t
h
e sim
u
latio
n
stud
ies are:
= 1.
86
3,
= 0.
25
7,
= 0
.
127
,
= 6.
9,
= 0.4853
,
H
=
4
,
D =
5,
= 1,
= 1,
= 1,
R
=
0.
05
,
=0
.2
,
=2.0,
= 1.
71
2,
ω
s
= 314
.1
59
. Th
e ph
ysical lim
i
t
of
ex
citatio
n vo
ltag
e
is: -3
≤
.
≤
6
.
Th
e
n
o
r
m
al op
er
ating
po
in
t
of th
e pow
er system
is:
= 47
,
=
0.
45
p.
u,
= 1.
0
p.
u a
n
d
(relative sp
eed
)
=
0
.
A symmetrical
th
ree-ph
ase sho
r
t circu
it fau
lt o
c
cu
rr
ing
on
on
e of t
h
e tran
smissio
n
lin
e is
co
nsid
ered
.
Two cases
of
fault seque
n
ce i
s
conside
r
e
d
here:
Case I: Pe
rm
anent type
(fa
ult
cleared)
Stag
e
1
:
th
e syste
m
is in
a pre
-fa
ult steady-st
ate
Stage
2: a
fault
occ
u
rs
at tim
e
0 sec
Stag
e
3
:
th
e fau
lt is rem
o
v
e
d
b
y
op
en
ing
th
e break
ers
o
f
the fau
lted
lin
e at ti
m
e
0
.
1
5
sec
Stag
e
4
:
th
e syste
m
is in
a post-fau
lt state.
+
e
+
/
Controller para
m
eter
updated
b
y
PSO at
ever
y
it
er
ation
Linearized
Model
-
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
A No
vel App
r
oa
ch to
PID C
o
n
t
ro
ller
Design fo
r
Imp
r
o
v
emen
t o
f
Tra
n
s
ient S
t
ab
ility .... (Rekha
Chau
dha
ry)
2
231
Case II:
Tem
pora
r
y type
(fa
ult cleared a
n
d line rest
ore
d
)
Stag
e
1
:
th
e syste
m
is in
a pre
-fa
ult steady-st
ate
Stage
2: a
fault
occ
u
rs
at tim
e
0 sec
Stag
e
3
:
th
e fau
lt is rem
o
v
e
d
b
y
op
en
ing
th
e break
ers
o
f
the fau
lted
lin
e at ti
m
e
0
.
1
5
sec
Stag
e
4
:
th
e syste
m
is in
a post-fau
lt state.
Stage
5: the
syste
m
has bee
n
restored at time 1.3
sec.
Fo
r carrying
o
u
t
th
e sim
u
l
a
tio
n
,
th
e fau
l
t is co
n
s
id
ered
at a lo
catio
n
λ
= 0.
2 o
n
one
of t
h
e
tran
sm
issio
n
lin
e,
wh
ich
sign
i
f
ies th
at th
e
fau
lt is at a
distance of
20
% of the line from
the ge
ne
rator bus. T
h
e
PID co
n
t
ro
ller is design
ed to ach
iev
e
the re
qui
red system
perform
a
nce i.e
the
pre-fault
value
of l
o
ad
angle,
om
ega and el
ect
ri
cal
powe
r
. Fo
r carry
i
n
g
out
t
h
e de
si
g
n
o
f
PI
D co
nt
rol
l
e
r
based
o
n
exci
t
a
t
i
on c
ont
rol
,
perm
anent type of
fault
(cas
e I)
ha
s b
e
en
co
nsid
ered
and
for d
e
si
g
n
i
ng
t
h
e
co
n
t
ro
ller with
ex
citatio
n
an
d
go
ve
rn
or
co
nt
r
o
l
si
m
u
l
t
a
neou
sl
y
,
t
e
m
pora
r
y
t
y
pe o
f
fa
ul
t
(c
ase II
) i
s
c
o
nsi
d
ere
d
.
F
o
r ca
rr
y
i
ng
out
si
m
u
l
a
t
i
on,
th
e con
tinu
o
u
s
ti
m
e
m
o
d
e
l is d
i
scretized
b
y
first ord
e
r app
r
ox
im
a
tio
n
with
sam
p
lin
g ti
m
e
o
f
0
.
0
1
sec.
C
ont
i
n
u
ous
t
i
m
e sy
st
em
:
Discrete tim
e s
y
ste
m
:
1
∆
1
∆
.
∆
.
1
wh
er
e
.
∆
,
.
∆
and
∆
is th
e sam
p
l
i
n
g
tim
e
in
terv
al.
5.
1.
Simula
ti
o
n
Results
5.
1.
1.
Unc
o
ntr
o
lled Sys
t
em
Th
e
u
n
c
on
tro
lled
r
e
spo
n
se of th
e pow
er
syste
m
af
ter
in
cep
tio
n
of
is pr
esen
ted
i
n
Figur
e 4(
a)
and
4(
b)
. Fi
g
u
r
e 4
(
a)
prese
n
t
s
t
h
e va
ri
at
i
on
o
f
p
o
w
er a
n
gle with tim
e. T
h
e gra
ph
shows that the a
ngle i
s
in
creasing
contin
u
o
u
s
ly wh
ich
lead
s t
o
in
st
ab
ility. Fig
u
r
e
4
(
b
)
shows th
e v
a
riation
of term
in
al v
o
ltag
e
with
ti
m
e
. Th
e vo
ltag
e
pro
f
ile is oscillato
ry an
d
co
n
tinuo
usly
decreasing
with
ti
m
e
. Th
u
s
from th
ese two
resu
lts it
can
b
e
inferred th
at th
e
un
co
ntro
lled
system
is un
st
ab
le an
d con
t
ro
ller is
need
ed to
stabilize the system
.
In
t
h
e n
e
x
t
sub
-
section
,
PID co
n
t
ro
l thro
ug
h
ex
citatio
n
h
a
s
b
een
con
s
id
ered
fo
r
stab
ilizin
g
the
sy
st
em
, PSO a
l
go
ri
t
h
m
has b
een
used
t
o
c
o
m
put
e t
h
e o
p
t
i
m
u
m
val
u
es of
PI
D c
ont
r
o
l
l
e
r pa
ram
e
t
e
rs
,
and
.
T
h
e
si
m
u
l
a
t
i
on has bee
n
carri
ed
o
u
t
fo
r di
ffe
re
nt
t
y
pes of
co
nt
r
o
l
l
e
rs:
P,
P
I
, PD
an
d PID
co
nt
r
o
l
l
e
rs
.
Figure
4(a)
. P
o
we
r a
ngle
res
p
ons
e
Figure
4(
b)
. Te
rm
in
al voltage re
spons
e
5.
2.
Excita
tio
n
Co
ntr
o
l
The t
h
ree
c
ontr
o
ller
param
e
ters:
,
and
are
t
o
be c
o
m
put
ed
u
s
i
n
g
PS
O
f
o
r
e
x
ci
t
a
t
i
on c
o
nt
r
o
l
(gi
v
en
by
equ
a
t
i
on (
2
8
)
. T
h
e po
pul
at
i
o
n si
ze i
s
assum
e
d
as fo
ur
whi
c
h i
s
hi
ghe
r t
h
a
n
num
ber o
f
va
r
i
abl
e
s
(th
r
ee) [23
]
. Fo
r im
p
l
e
m
en
ta
tio
n
of PSO i
n
itial p
o
s
itio
n
(co
n
t
ro
ller
p
a
rameters) and
v
e
lo
city
m
a
trices
(wit
h
vel
o
ci
t
y
l
i
m
i
t
of
±2
) ha
s be
en ass
u
m
e
d. For
up
dat
i
n
g t
h
e part
i
c
l
e
p
o
si
t
i
on,
dy
nam
i
c
PSO
wi
t
h
wei
ght
e
d
0
10
0
20
0
300
40
0
50
0
600
0.
8
1
1.
2
1.
4
1.
6
1.
8
2
pl
ot
f
o
r
f
aul
t
del
t
a
0
100
200
300
400
500
600
0.
7
0.
75
0.
8
0.
85
0.
9
0.
95
1
pl
ot
f
o
r t
e
rm
i
nal
v
o
l
t
a
ge (
un
c
ont
rol
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE
Vo
l. 6
,
N
o
. 5
,
O
c
tob
e
r
20
16
:
222
5
–
22
38
2
232
in
ertia h
a
s
b
e
en
con
s
i
d
ered
.
Term
in
atio
n
o
f
iterativ
e
pr
oce
ss has
bee
n
ass
u
m
e
d t
o
3
00
(
m
axim
u
m
num
ber
o
f
iteratio
n
s
).
Fo
r
p
a
ram
e
ter id
en
tificatio
n of PID co
n
t
ro
ller,
n
u
m
b
e
r of
v
a
riab
les
is th
ree.
In
itial v
a
lu
e of
p
opu
latio
n is assu
m
e
d
k
e
ep
ing
in v
i
ew t
h
e
p
a
ram
e
ter cons
train
t
as m
e
n
tio
n
e
d
in tab
l
e
1
an
d th
e v
e
l
o
city is
in
itialized
k
eep
ing
in
v
i
ew t
h
e v
e
lo
city li
mit. Fo
r in
itial
i
zatio
n
o
f
th
e iteratio
n
,
th
e initial
v
a
lu
e o
f
pb
est
is
co
nsid
ered
t
o
be in
itial v
a
lu
e
o
f
pop
u
l
ation
.
Tabl
e
1. R
a
n
g
e
o
f
C
o
nt
r
o
l
l
e
r
Param
e
t
e
rs
C
o
n
t
r
o
l
l
e
r
P
a
r
a
me
t
e
r
M
i
n
i
mu
m
V
a
l
u
e
M
a
x
i
mu
m V
a
l
u
e
K
0 300
K
0 300
K
0 50
K
0 100
The
pr
op
ose
d
no
nl
i
n
ea
r co
nt
rol
al
g
o
ri
t
h
m
is im
pl
em
ent
e
d and t
h
e
param
e
t
e
rs f
o
r
di
f
f
er
ent
t
y
pes o
f
cont
rol
sc
hem
e
s P, PI
, P
D
and
PI
D are c
o
m
put
ed usi
n
g
PSO al
go
ri
t
h
m
.
The c
ont
rol
l
e
d res
p
onse
wi
t
h
t
h
ese
cont
rol sc
hem
e
s are
sim
u
lated
and a
r
e
prese
n
ted as
follows:
5.
2.
1.
Sys
t
em res
ponse with
P c
o
ntroller
In th
is case
num
b
e
r o
f
v
a
riable is on
ly
o
n
e
(
)
.
Thr
oug
h PSO
al
g
o
r
ith
m
,
valu
e
o
f
i
s
com
put
e
d
whi
c
h c
o
m
e
s out
t
o
be
3.
70
3.
The
gra
p
hs
fo
r
p
o
we
r a
ngl
e
(
F
i
g
u
r
e
5(a
)
)
an
d t
e
rm
i
n
al
vol
t
a
ge (
F
i
g
ure
5
(
b)
) are
p
l
o
tted
.
Th
ese
g
r
aph
s
show that th
e an
g
l
e is in
creasi
n
g
con
tin
uou
sly with
oscillato
ry n
a
tu
re an
d
th
e
vo
ltag
e
is
also
o
s
cillating
with
m
a
g
n
i
t
u
d
e
d
e
creasing
.
It can
b
e
in
ferred
th
at
with
P con
t
ro
l,
th
e syste
m
can
no
t b
e
stab
ilized
and
i
t
will g
o
ou
t
o
f
step
.
Fi
g
u
r
e
5(a
)
.
Po
wer
a
ngl
e
r
e
sp
onse
f
o
r
P c
ont
rol
Fi
gu
re
5(
b
)
. Te
rm
i
n
al
vol
t
a
ge
resp
o
n
se
fo
r
P
c
o
n
t
r
o
l
l
e
r
5.
2.
2.
Response
for
PI contr
o
ller
In
t
h
is case, nu
m
b
er of
v
a
ri
ab
les is: two
(
). T
h
r
o
ug
h
PS
O al
g
o
ri
t
h
m
,
val
u
e o
f
and
are com
put
ed
whi
c
h com
e
s out
t
o
be
3.
70
3 an
d
0.It
i
s
o
b
ser
v
e
d
t
h
at
t
h
e cont
rol
l
e
d
re
spo
n
se i
s
si
m
i
lar t
o
th
e P con
t
ro
ller as abo
v
e
. So
t
h
e PI con
t
ro
l is
also
no
t su
itable for co
n
t
ro
llin
g th
is system
.
5.
2.
3.
Resp
onse
for
PD c
o
n
t
roller
In th
is case,
nu
m
b
er
o
f
v
a
riab
les is:
two
(
)
.
Th
ro
ugh PSO algo
r
ith
m
,
v
a
l
u
e
of
an
d
are c
o
m
put
ed
whi
c
h c
o
m
e
s out
t
o
be
20
4.
43
a
n
d
5.
46
6.
Fi
gu
re
6
(
a)
a
n
d
6
(
b
)
p
r
esent
s
t
h
e
res
u
l
t
s
fo
r
P
D
cont
roller, it is seen that the powe
r
angle is increasi
n
g
co
n
t
i
n
uou
sly (slo
wl
y) with
ti
m
e
an
d
vo
ltag
e
is ten
d
i
ng
to
ward
s stead
y
state v
a
lu
e slowly. It is o
b
s
erv
e
d
th
at th
e
oscillatio
n
h
a
v
e
b
een
elim
in
at
ed
in
th
is case. It is
g
i
v
i
ng
better resu
lt th
an
P an
d
PI con
t
ro
ls. Howev
e
r it
i
s
t
o
poi
nt
out
t
h
at
as l
o
ad angl
e i
s
i
n
cre
a
si
ng
co
n
tinuo
usly
with
ti
m
e
th
e syste
m
will fin
a
lly g
o
o
u
t
o
f
st
ep
in
d
u
e
co
urse o
f
ti
m
e
. Hen
ce PD con
t
ro
l
is als
o
not
sui
t
a
bl
e.
0
100
200
30
0
40
0
50
0
60
0
0.
8
0.
9
1
1.
1
1.
2
1.
3
1.
4
1.
5
1.
6
1.
7
1.
8
di
s
c
r
et
e
t
i
m
e
i
n
t
e
rv
a
l
d
e
l
L
Q
/
del
ps
o
C
o
nt
rol
l
ed d
e
l
by
del
f
l
a
t
and P
S
O
del
P
S
O
del
f
l
a
t
0
100
200
30
0
40
0
50
0
60
0
0.
75
0.
8
0.
85
0.
9
0.
95
1
1.
05
1.
1
P
l
ot
f
o
r
t
e
rm
i
n
a
l
v
o
l
t
ag
e
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
A No
vel App
r
oa
ch to
PID C
o
n
t
ro
ller
Design fo
r
Imp
r
o
v
emen
t o
f
Tra
n
s
ient S
t
ab
ility .... (Rekha
Chau
dha
ry)
2
233
Fi
gu
re
6(a
)
.
P
o
wer
an
gl
e re
sp
ons
e
Fi
gu
re
6(
b)
. Te
rm
i
n
al
vol
t
a
ge
res
p
on
se
5.
2.
4.
Response
for
PID c
o
ntr
o
ller
In th
is case,
num
b
e
r o
f
v
a
riables is: th
ree (
,
). Through
PSO algorithm
,
values
of
t
h
ese
param
e
t
e
rs are
com
put
ed
w
h
i
c
h c
o
m
e
s out
t
o
be
= 169
.3
7,
=1
39
.7
7
an
d
= 6.08
.
W
i
th
th
e
s
e
v
a
l
u
es
of
co
nt
r
o
l
l
e
r
p
a
ram
e
t
e
rs t
h
e c
ont
rol
l
e
d
res
p
o
n
se
of
t
h
e
sy
st
em
i
s
obt
ai
ned
an
d s
h
ow
n
i
n
Fi
gu
re
7(
a)
an
d
7(
b
)
for power an
gle an
d
term
in
al v
o
ltag
e
respectiv
ely. Th
e respon
se sh
ows th
at: (i) th
e co
n
t
ro
lled
syst
e
m
is
stable as t
h
e load a
ngle
and
voltage
bo
th
are stab
ilized
. (ii) It is
ob
serv
ed
th
at th
e
po
wer
an
g
l
e h
a
s stab
i
lized
n
ear th
e set
poin
t
(p
re-fau
lt valu
e)
wh
ereas
th
e vo
ltag
e
settles at a v
a
lu
e
1
.
0
4
p.u (h
igher th
an
t
h
e
p
r
e-fau
lt
val
u
e 1
.
0
p.
u).
The abo
v
e re
spo
n
se s
h
o
w
s
t
h
at
:
am
ong t
h
e fou
r
co
nt
r
o
l
l
e
rs (P, P
I
, P
D
and PI
D
)
di
sc
usse
d,
only the
P
I
D c
ont
rol sc
hem
e
stabilizes
the
powe
r system
unde
r
fault c
o
ndition.
It is to
po
in
t
ou
t th
at if vo
ltag
e
h
a
s also
t
o
b
e
regu
lated
in ad
d
ition
to
t
h
e lo
ad
ang
l
e then
gov
erno
r
co
n
t
ro
l
h
a
s t
o
b
e
im
p
l
e
m
en
te
d
in add
ition
to ex
citatio
n contro
l.
It is
p
r
esen
ted
b
e
low.
5.
3.
Ex
cit
a
t
i
on plus go
verno
r
C
o
nt
ro
l
5.
3.
1.
Response
for
PID c
o
ntr
o
ller
Ex
citatio
n p
l
us gov
erno
r con
t
ro
l h
a
s t
w
o
in
pu
ts
n
a
m
e
ly
and
(e
qu
at
i
o
n
(
2
5)
a
n
d
(
2
8)
)
act
i
ng si
m
u
l
t
a
neo
u
sl
y
.
I
n
de
si
gni
ng t
h
i
s
co
nt
r
o
l
l
e
r f
o
u
r
p
a
ram
e
t
e
rs:
,
,
and
(
g
i
v
en
b
y
eq
uation
(2
5)
an
d
(2
8)
)
are t
o
be c
o
m
put
e
d
.
Here
n
u
m
ber of
vari
a
b
l
e
s i
s
fo
ur
an
d
t
h
e p
o
pul
at
i
o
n si
ze i
s
ass
u
m
e
d as
fi
ve hi
g
h
er
t
h
a
n
num
ber of
va
ri
abl
e
s [2
3]
.
The c
ont
rol
l
e
r
param
e
t
e
rs co
m
put
ed t
h
r
o
u
g
h
PS
O a
r
e:
= 11
1.
5
5
,
=1
85
.1
65
,
=5
.04
6
and
= 51
.0
6. Th
e respo
n
s
e fo
r pow
er
ang
l
e and ter
m
in
al
vol
t
a
ge
has
been
o
b
t
a
i
n
ed
u
s
i
n
g t
h
ese c
ont
rol
l
e
r
param
e
t
e
rs an
d pl
ot
t
e
d i
n
Fi
gu
re
8(a
)
an
d
8(
b)
. It
has
be
en o
b
se
rv
ed t
h
at
by
a
ddi
ng
po
we
r co
nt
r
o
l
i
n
p
u
t
,
t
h
e
per
f
o
r
m
a
nce o
f
t
h
e
sy
st
em
i
m
prove
d
f
u
rt
her
.
T
h
e
l
o
a
d
a
ngl
e
an
d
t
e
rm
inal
vol
t
a
ge
t
r
a
c
ks t
o
t
h
e
p
r
e-
fau
lt co
nd
itio
n
in
th
is case.
5.
3.
2.
Error c
o
mpar
ision - exci
tation c
o
ntr
o
l
an
d exci
tation
pl
us
governor c
o
ntrol
The er
ro
r bet
w
een t
h
e set
p
o
i
n
t
of l
o
ad a
ngl
e and t
h
e
PI
D
cont
rol
l
e
d sy
st
em
respo
n
se i
s
pl
ot
t
e
d i
n
Figure 9(a) al
so the error between
th
e termin
al v
o
ltag
e
(pre-fau
lt v
a
l
u
e) and
th
e termin
al v
o
ltag
e
o
f
th
e
cont
rol
l
e
d sy
st
em
i
s
co
m
put
ed an
d pl
ot
t
e
d i
n
Fi
g
u
re
9(
b
)
fo
r t
w
o cases e
x
ci
t
a
t
i
on co
nt
r
o
l
o
n
l
y
and e
x
ci
t
a
t
i
on
pl
us
g
o
v
er
no
r
cont
rol
.
Inference
:
F
o
r com
p
ari
s
i
o
n t
h
e err
o
r
bet
w
een set
-
poi
nt
and c
o
nt
rol
l
e
d r
e
sp
onse
fo
r t
h
i
r
d a
nd
fi
ft
h
or
der
PI
D c
o
n
t
rol
l
e
d sy
st
em
has
been
pl
ot
t
e
d i
n
Fi
g
u
re
9
(
a) a
n
d
9
(
b
)
.
I
t
sho
w
s t
h
at
i
n
t
h
e
pl
ot
of e
r
r
o
r i
n
po
we
r an
gl
e w
i
t
h
t
i
m
e
(Fi
gur
e 9(a
)) a
nd e
r
r
o
r i
n
t
e
rm
i
n
al v
o
ltag
e
with
time (Fig
ure
9(a
)). T
h
e error
has been
m
u
ch
r
e
du
ced in
case of
PID
co
n
t
r
o
ller
w
ith
b
o
t
h
ex
ci
tatio
n
an
d gover
n
o
r
con
t
ro
l
co
m
p
ar
e to
excitatio
n
cont
rol
onl
y
.
I
n
t
h
e
next
s
u
b
-
sect
i
on,
DFL
-
s
t
at
e feedbac
k
c
ont
rol
l
e
r
has b
een di
sc
usse
d i
n
b
r
i
e
f f
o
r com
p
ari
n
g
its
resu
lts with
p
r
op
o
s
ed
PID co
n
t
ro
ller.
0
100
20
0
300
40
0
50
0
60
0
0.
82
0.
8
2
2
0.
8
2
4
0.
8
2
6
0.
8
2
8
0.
83
0.
8
3
2
0.
8
3
4
0.
8
3
6
d
i
s
c
r
et
e t
i
m
e
i
n
t
e
r
v
al
del
LQ
/
d
el
ps
o
C
o
nt
rol
l
ed
de
l
b
y
d
e
l
f
l
a
t
an
d
P
S
O
d
e
l
PSO
d
e
lf
la
t
0
100
200
30
0
40
0
50
0
60
0
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
1.
1
P
l
ot
f
o
r
t
e
rm
i
n
a
l
v
o
l
t
ag
e
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE
Vo
l. 6
,
N
o
. 5
,
O
c
tob
e
r
20
16
:
222
5
–
22
38
2
234
Figure
7(a
)
.
P
o
we
r a
n
gle res
p
onse
Fi
gure
7(
b).
Term
inal voltage
response
Figure
8(a
)
.
Power
angle
re
s
p
o
n
se
Fi
gu
re 8(
b
)
.
Te
rm
i
n
al
vol
t
a
ge resp
o
n
se
Fi
gu
re
9(a
)
.
P
o
wer
an
gl
e re
sp
ons
e
Fi
gu
re
9(
b
)
. Te
rm
i
n
al
vol
t
a
ge
resp
o
n
se
5.
4.
DFL-s
t
ate fee
dbac
k
Contr
o
ller
5.
4.
1.
Response
for
controller
DFL
-
st
at
e fee
dbac
k
c
ont
rol
l
er desi
gn
fo
r t
h
e exci
t
a
t
i
o
n pl
us
go
ve
rn
or
cont
rol
has be
en di
sc
usse
d
here i
n
bri
e
f.
For
D
F
L-
st
at
e feed
bac
k
c
o
nt
rol
l
e
r
desi
gn
p
r
o
b
l
e
m
,
t
h
e pr
op
o
r
t
i
onal
e
r
r
o
r i
n
st
at
e i
s
us
ed
f
o
r
feed
bac
k
t
o
ge
nerat
e
t
h
e c
ont
rol
si
gnal
a
s
s
h
ow
n
bel
o
w:
Ex
citatio
n
con
t
ro
l si
g
n
a
l
(
:
(
2
9)
0
10
0
200
300
400
50
0
60
0
0.
8
2
0.
82
2
0.
82
4
0.
82
6
0.
82
8
0.
8
3
0.
83
2
0.
83
4
0.
83
6
pl
ot
f
o
r
del
,
P
S
O
i
m
p
l
em
en
t
e
d
0
100
200
300
400
500
600
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
1.
1
1.
2
1.
3
P
l
ot
f
o
r
t
e
rm
i
nal
v
o
l
t
age
0
100
200
300
400
500
600
0.
82
0.
821
0.
822
0.
823
0.
824
0.
825
0.
826
0.
827
di
s
c
r
et
e t
i
m
e
i
n
t
e
rv
al
de
l
L
Q
/
de
l
p
s
o
C
ont
rol
l
ed del
by
d
e
l
f
l
a
t
and
P
S
O
del
P
S
O
del
f
l
at
0
100
200
300
400
500
60
0
0.
75
0.
8
0.
85
0.
9
0.
95
1
1.
05
1.
1
1.
15
P
l
o
t
f
o
r
t
e
rm
i
nal
v
o
l
t
age
0
100
200
300
400
500
600
0
0.
002
0.
004
0.
006
0.
008
0.
01
0.
012
0.
014
0.
016
0.
018
A
ngl
e E
rror
pl
ot
f
o
r P
I
D
D
i
s
r
et
e t
i
m
e
i
n
t
e
r
v
al
A
n
g
l
e
error
e
xci
t
a
t
i
o
n
ex
c
i
t
a
t
i
on+
gov
ernor
0
100
20
0
300
40
0
500
600
0
1
2
3
4
5
6
V
o
l
t
a
ge E
r
ror
pl
ot
f
o
r
P
I
D
Di
s
r
et
e
t
im
e
in
te
rv
a
l
V
o
l
t
ag
e
er
r
o
r
e
x
c
i
ta
ti
o
n
ex
c
i
t
a
t
i
o
n
+
gov
ernor
Evaluation Warning : The document was created with Spire.PDF for Python.