TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol. 14, No. 2, May 2015, pp. 191 ~ 19
8
DOI: 10.115
9
1
/telkomni
ka.
v
14i2.760
2
191
Re
cei
v
ed Fe
brua
ry 24, 20
15; Re
vised
Ap
ril 18, 201
5; Acce
pted
April 29, 201
5
A Minimax Polynomial Approx
imation Objective
Function Approach for Optimal Design of Power
System Stabilizer by Embedding Particle Swarm
Optimization
Bhan
u Prata
p
Soni*
1
, Akash Saxen
a
2
, Vikas Gupta
1
1
Departme
n
t of Electrical En
gi
neer
ing, Mal
a
vi
ya N
a
tio
nal Ins
t
itute of
T
e
chnolo
g
y
2
S
w
am
i Keshv
ana
nd Institute
of
T
e
chnolo
g
y
, Jaipur, India -
302
01
7
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: er.bpson
i2
01
1@gma
il.com
A
b
st
r
a
ct
T
he pa
per pre
s
ents a nov
el
appr
oach
bas
e
d
on
Mi
ni
max appr
oxi
m
ati
on and
ev
ol
ution
a
r
y
tool
Particle
Swarm
Optimi
z
a
tion (PSO) to fabr
icate the
param
e
ters
of P
o
wer System
Stabili
z
e
rs (PSSs)
for
m
u
lti
m
a
c
h
ine
power system
s
.
The pr
opos
ed approac
h employs PSO algor
i
thm for find the setting of PS
S
para
m
eters. T
h
e w
o
rth
me
ntio
ni
n
g
feat
ure
of
this w
o
rk is t
h
e
formul
a
tion
of
obj
ective fu
ncti
on w
i
th th
e
hel
p
of sw
ing curv
es inter
p
o
l
atio
n. A nov
el tr
ansfor
m
at
i
on
know
n as
min
i
max a
ppr
oxi
m
ation
is us
ed
fo
r
converti
ng the
objectiv
e
into
the poly
n
o
m
i
a
ls of de
gree
one, tw
o and t
h
ree. T
o
const
r
uct the obj
ecti
v
e
function
b
a
sed
on
inter
pol
ati
on s
e
con
d
ord
e
r se
nsit
ivity
a
nalysis
is
perf
o
rmed.
T
h
e p
e
rformanc
e of
th
e
PSSs is tested und
er differ
ent topol
og
ical
chang
es,
op
eratin
g con
d
iti
ons an
d system confi
gur
atio
ns.
Nonlinear sim
u
lation reveals that
proposed PSSs are effectively deal
with local and interarea m
o
des
of
oscillations. PSS design obtain
ed through lower order polynom
i
al ex
pres
sion of objectiv
e function is able
to deal w
i
th the
oscillat
o
ry mo
des efficie
n
tly.
Ke
y
w
ords
:
auto
m
atic v
o
lt
age r
egu
lator
(AVR), min
i
max
ap
proxi
m
ation, mu
ltim
a
c
hin
e
pow
er
system,
particle swarm
optim
i
z
at
ion (P
SO),
power sy
stem
stabili
z
e
r (PSS).
Copy
right
©
2015 In
stitu
t
e o
f
Ad
van
ced
En
g
i
n
eerin
g and
Scien
ce. All
rig
h
t
s reser
ve
d
.
1. Introduc
tion
In the mode
rn po
wer
syst
em high
pe
rforma
nce Automatic Volta
ge Regul
ators (AV
R
s)
(high gai
n and fast resp
onse)
are equipped to
ensure transient stability. AVRs
of high
gain
introduc
e
negative damping in t
he s
y
s
t
em. To provide a c
o
s
t
effec
t
ive c
o
ntrol PSSs
are
employed
with AVRs. The
requi
rem
ent for mod
e
rn ex
citation sy
ste
m
exists in the fact that PSS
and AVR bot
h are dynami
c
ally interlin
ked [1]. The purpo
se of the
paramete
r
d
e
sig
n
is to make
the PSSs provide proper damping for power
sy
stem oscillations. PSS parameter estimation
probl
em i
s
a
n
optimizatio
n problem,
where
aim of
t
he optimi
z
ati
on p
r
ocess i
s
to maximize
the
dampin
g
of the po
we
r net
work. It is qu
ite empiri
cal
to state that there i
s
a tra
deoff betwe
e
n
synchronizi
n
g
torque provi
ded by
the AVR
an
d damping torque prov
ided
by the PSS [2].
Res
e
archers have experimented
with the differe
nt type of objec
tive func
tions
[]. The P
S
S
para
m
eter estimation probl
em ha
s b
een
add
re
sse
d
with different o
p
timization
al
gorithm
s [3
-1
1].
The majo
r co
ntribution
rep
o
rted in literature re
volves
arou
nd ove
r
a
ll system
s dynamic
re
spo
n
s
e
i.e. overshoot
and
settling
time,
co
nverg
ence
cha
r
a
c
teristi
c
s of p
r
o
posed m
e
tho
dology,
soluti
on
quality, time elap
sed a
nd
comp
ari
s
o
n
o
f
the me
thod
ology with co
nventional te
chni
que
s [8-1
1
].
The tra
d
ition
a
l obje
c
tive functio
n
re
po
rted in
literature con
s
id
ered
dam
ping ratios, dampi
ng
frequenci
e
s and weighted
combi
nat
ion
of these to solve the
tuning parameters of PSSs. The
inferio
r
mode
s are
shifted
to D shap
ed
and
fan sh
ap
ed regi
on
s. Sheng
kuan
wang propo
se
d a
new
scale wh
ich drifted ei
g
envalue
s in fan sh
ape
d mode with the t
i
p at damping
ratio [7].
In this
paper,
PSO algorithm [12] is
em
ployed to
s
o
lve PSS parameter es
timation.The
reali
s
a
-
tion of
obje
c
tive fun
c
tion in
st
1
,
nd
2
and
rd
3
ord
e
r p
o
lyno
mials
are
do
ne with th
e h
e
lp of
sen
s
itivities
of derivative
s
and
MATLA
B
cu
rve fitting tool. Fo
r t
e
sting
re
sult
s, the propo
sed
approa
ch i
s
applie
d o
n
t
w
o te
st
ca
se
s of
mult
i
m
a
chi
ne po
we
r system
s.
A
s
sertivene
ss of
prop
osed
me
thodolo
g
y ha
s b
een
teste
d
on
differe
nt type of di
stu
r
ban
ce
s,
lo
a
d
ing
co
nditio
n
s,
and sy
stem configuration
s
.
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046
TELKOM
NI
KA
Vol. 14, No. 2, May 2015 : 191 – 198
192
2. Problem Statement
2.1. Po
w
e
r S
y
stem Model
Formul
ation of
the
power system can
b
e
con
c
lu
ded
and un
de
rsta
nd by followi
ng set of
equatio
ns.
)
,
(
u
x
f
x
(
1
)
Whe
r
e
x
is the state variabl
es,
u
is the vector of input variable.
In the PSS the power sy
stem is
usu
a
lly linearized a
nd o
p
e
r
ating e
qullib
rium a
s
the
study of the
small
di
sturb
ance co
me
s
in
small si
gnal
stability. Equation (1
) ca
n further b
e
tran
sformed a
s
:
u
B
x
A
x
(
2
)
If
n
is the total
no of machines
size of A will be
x
n
n
,
4
4
is
1
4
n
state vector,
whil
e
u
vec
t
or
is
1
pss
n
.
stab
K
sT
sT
1
2
1
1
1
sT
sT
4
3
1
1
sT
sT
s
V
ma
x
s
V
mi
n
s
V
Ga
i
n
S
t
a
b
iliz
e
r
Ga
i
n
W
a
s
hout
r
co
m
p
en
s
a
to
la
g
lea
d
Ph
a
s
e
Figure 1. Structure of
Power System Stabilizer
2.2. PSS Str
u
cture
Conventional
lag lead structure of PSS is show
n in
Figure 1. The st
ructure is used in
this wo
rk whi
c
h ha
s tra
n
sf
er fun
c
tion (3
). Fu
rthe
r the
modern exci
tation system
with AVR an
d
PSS is
s
hown in Figure 2.
U
sT
sT
sT
sT
sT
sT
K
U
i
i
w
w
stabi
i
4
3
2
1
1
1
.
1
1
.
1
(
3
)
The obje
c
tive
function J i
s
to be minimi
zed with the constraints.
2
10
2
2
2
2
1
2
10
2
2
2
2
2
1
1
2
2
10
2
2
2
1
......
.
.
.
.
min
w
w
w
w
w
w
w
w
w
J
k
t
t
t
(4)
Subject to:
max
3
3
min
3
max
1
1
min
1
max
min
i
i
i
i
i
i
stabi
stabi
stabi
T
T
T
T
T
T
K
K
K
(
5
)
The obj
ectiv
e
of the opt
imization i
s
to find the
set of vari
a
b
les
stabi
K
,
i
T
1
,
i
T
3
, fo
r
n
i
,
,
2
,
1
to achieve a
dequ
ate dam
ping in the system.
Here n is the si
ze
of network o
r
tota
l
number
of alt
e
rnators
in s
y
s
t
em.It is
assumed th
at all alternators
have inc
o
rporated with PSSs
.
In this
work
was
h
out time c
o
ns
tant
s
T
w
10
and
2
T
&
4
T
are con
s
idere
d
as
s
05
.
0
. The left over
over pa
ram
e
ters
stab
K
,
1
T
and
3
T
are assum
ed t
o
be the m
o
difiable pa
ra
meters; hen
ce the
number of the parameters for 3 machi
n
es will
be 9 and for 10 machines system
will be 30.
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TELKOM
NIKA
ISSN:
2302-4
046
A Minim
a
x Polyn
o
m
i
al Appro
x
im
ation Obje
ctive Fu
nction App
r
o
a
ch for… (Bh
anu Prata
p
Soni)
193
2.3. Cons
tru
c
tion of O
b
je
ctiv
e Function
The speed deviation based objective functi
on i
s
employed here for the PSS parameter
estimation p
r
oblem. The
steps for
con
s
tructing
the o
b
j
e
ctive functio
n
are a
s
follo
wing.
Step 1.
Initialize the
optimi
z
ation
pro
c
e
s
s, re
ad
syst
e
m
data, sele
ct the co
ntinge
ncie
s/op
erati
ng
con
d
ition
s
‘m’ and simul
a
tion time step
s ’k’.
Step 2.
Initiali
z
e
the PSS data
Step 3.
Simulate the syst
em and sto
r
e the values
of speed d
e
v
iations of the gene
rato
rs for
different faults
.
Step 4.
Apply stoppin
g
crit
erion, if not satisfied then
go to step 2.
Step 5.
End
Figure 2. Modern ex
citatio
n
system
with
AVR & PSS
2.4.
Embedding Objectiv
e Function in Interpolating P
o
ly
nomial
The id
ea i
s
to app
roxima
te obje
c
tive functio
n
)
(
x
f
by a polynomial
)
(
x
p
that gives
a
uniform a
c
cu
rate de
scripti
on in an inte
rval
b
a
,
. Here the
b
a
,
is the interv
al of application of
the certain
di
sturb
a
n
c
e. L
e
t the fun
c
tio
n
)
(
x
f
is a
n
a
p
p
r
oximate
co
n
t
inuou
s fun
c
ti
on o
n
a
n
interval
b
a
,
. This functio
n
with
is
reali
z
ed
th
e set of p
o
lyn
o
mials of
deg
ree
at mo
st
n
and
l
e
t
k
boun
ded fun
c
tion defined
b
a
,
, Minimax app
roximation alg
o
rithm sugg
e
s
ts that maxi
mum erro
r
is minimi
zed [
4
]. The object
i
ve is to find a function
)
(
x
k
to
minimize Equation (6).
)
(
)
(
max
x
k
x
f
J
b
x
a
(
6
)
A sen
s
itivity
analysi
s
is
de
picted in T
abl
e 1 to Table
9 and in
dice
s like first a
nd
se
con
d
derivatives a
r
e calculated
to
ens
ure t
he truthfuln
e
ss
of interp
ol
ation fit. Ho
wever it i
s
worth
mention
he
re that
Che
b
y
shev exp
a
n
s
ion
polyn
o
m
ial of fi
rst
kind
can
clo
s
ely a
p
p
r
oximate
Minimax p
o
lynomial
[4]. T
he p
r
o
p
o
s
ed
wo
rk tran
sfo
r
ms the
tradi
tional o
b
je
ctive functio
n
i
n
to
three types of
polynomial
s
of degre
e
nd
st
2
,
1
an
d
rd
3
order. Th
e para
m
eter tu
ning is do
ne
while
optimizin
g ea
ch polyn
omial
with the help
of PSO.
2.6. Sensitivi
t
y
Anal
y
s
is
To formul
ate
the obje
c
tive function
o
n
the
ba
sis
of spee
d de
viat
ion, the sensitivity
analysi
s
is
required. It is intere
sting to know that how the PSS par
ameter
can be interpolated for
con
s
tru
c
ting t
he obje
c
tive
function. Th
e
objective
of
the sen
s
itivity probl
em is t
o
com
pute the
derivative of the functio
n
. Suppo
se a fun
c
tion:
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046
TELKOM
NI
KA
Vol. 14, No. 2, May 2015 : 191 – 198
194
)
),
(
(
)
(
x
x
u
G
x
f
Sensitivity of
the function
with re
spe
c
t to para
m
eter
x
is given by the followi
ng e
quation:
)
(
)
),
(
(
x
u
x
G
x
u
x
x
u
x
u
x
f
In this work finite method
of differen
c
e
[4]
is used t
o
cal
c
ul
ate the de
rivatives of the
variou
s p
a
ra
meters of PS
S. The an
alysis is
sh
own
in the Ta
ble
1 to Ta
ble 9.
The
ope
rati
ng
points b
e
twe
en the sim
u
la
tion time step
s are sel
e
cte
d
and the se
nsitivity of the
spee
d deviat
i
on
of all 3 g
e
n
e
rato
rs
are
cal
c
ulate
d
. T
he value
s
of
different first order and
se
con
d
o
r
d
e
r
sen
s
itivities a
r
e sh
own in T
able 1 to 9.
Followin
g
point
s are
wo
rth m
ention he
re:
a)
For
the high values of
sim
u
lation
time step
i.e.
whe
n
fault is ge
nerated after a l
ong
wait than
the se
con
d
d
e
rivative of generator 3 for
1
T
is less
sen
s
i
t
ive. It is also
obse
r
ved tha
t
for lower
values of the
simulatio
n
time step the se
con
d
ord
e
r d
e
rivatives of gene
rato
r 3 a
r
e very high.
For
3
T
para
m
et
er the
se
con
d
ord
e
r
deriv
ative of gene
rator
2 is
mo
st se
nsitive.
In fact it
achi
eves hig
hest val
ue. T
he
same
a
n
a
l
ogy is
fo
llowe
d
w
h
en
th
e s
i
mu
la
tion
s
t
e
p
is
de
la
ye
d
and the seco
nd ord
e
r d
e
ri
vative attains highe
r value
s
for gene
rator 2.
b)
Table 1 to T
able 9
sho
w
s the ab
so
l
u
te values
of sensitivities f
o
r
stab
K
.
1
T
and
3
T
. T
he
sen
s
itivity analysis i
s
u
s
e
d
to obtain t
he wei
ght
s for
combi
nati
on of the eff
e
ct of the P
SS
para
m
eters a
nd formin
g the obje
c
tive function
s line
a
r, polynomial 2 and cubi
c p
o
lynomial
s
.
Table 1. Interpolation Fit for
1
T
of Generator 1 (3 M
a
chi
ne System)
i
x
0.6 3.54
6.48
9.42
12.36
15.3 18.2
21.1
24.1
27
30
)
(
i
x
f
12.6727
-22.530817
-320.65
-1166.1
-
2843.
2
-
5636.4
-
9830.1
-
15708
-23556
-33658 -46297
)
(
i
x
f
d
0.497
-40.567
-178.362
-412.888-744.
145
-1172.13-1696.85
-2318.3
-
3036.48-3851.39
-4763.03
)
(
i
x
f
d
2.48325
-30.418 -63.319
-96.221
-129.12
-162.
02
-194.92
-227.82
-260.72
-293.63 -326.53
Table 2. Interpolation Fit for
3
T
of Generator 1 (3 M
a
chi
ne System)
i
x
0.6
3.54 6.48 9.42 12.36
15.3
18.2
21.18
24.12
27.06
30
)
(
i
x
f
12.2603
266.941
2028.19
6771.82
15973.
6
31109.4
53655 85086.2
126879
180509
247451
)
(
i
x
f
d
-2.268 259.183
1022.61
2288.01
4055.39
6324.
74
9096.06
12369.4
16144.6
20421.9
25201.1
)
(
i
x
f
d
3.5594
174.299
345.039
515.778
686.518
857.257
1028 1198.74
1369.48
1540.22
1710.96
Table 3. Interpolation Fit for
stab
K
of Generator 1 (3 M
a
chi
ne System)
i
x
0.6 3.54 6.48
9.42
12.36
15.3
18.24
21.18
24.12
27.06
30
)
(
i
x
f
13.2753
13.5417
13.713
13.8083
13.8464
13.
8465
13.8275
13.8083
13.8081
13.8457
13.9402
)
(
i
x
f
d
0.108
0.073369
0.044
0.0216
0.00542
-0.
004303
-0.00757
-0.0043
0.00528
0.0214
0.043
)
(
i
x
f
d
-0.0131 -0.011
-0.0088
-0.0066
-0.0044
-0.0022 -1.
10E-050.0021
0.00438
0.00658
0.00877
Table 4. Interpolation Fit for
1
T
of Generator 2 (3 M
a
chi
ne System)
i
x
0.6
3.5 6.4 9.4
12
15
18 21.1
24.1
27.06
30
)
(
i
x
f
12.28
9.713
-26.72 -154.1
-429.7
-910.7 -1654
-2717
-4156.68
-6030.23-8394.8
)
(
i
x
f
d
-1.5938
-3.3955 -24.633 -65.
309 -125.42 -204.96
-303.95 -
422.36 -560.23
-717.529-894.26
)
(
i
x
f
d
2.69273
-3.91839-10.5295
-17.1406
-23.7517
-30.3628-36.9739
-43.5851-50.1962
-56.8073-63.4184
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
A Minim
a
x Polyn
o
m
i
al Appro
x
im
ation Obje
ctive Fu
nction App
r
o
a
ch for… (Bh
anu Prata
p
Soni)
195
Table 5. Interpolation Fit for
3
T
of Generator 2 (3 M
a
chi
ne System)
i
x
0.6
3.54 6.48 9.42 12.3
15.3 18.2 21.1
24.1 27
30
)
(
i
x
f
17.2717
19721.2
140499 455544
10000002400000
3500000
55000008200000
1600000019000000
)
(
i
x
f
d
78.8527
18608.3
68836.4
150763
264389 409713
586735
795456 1400000
1300000 1600000
)
(
i
x
f
d
911.611
11693.5
22475.3
33257.2
44039 54820.
9
65602.7
76384.6
87166.4
97948.3
108730
Table 6. Interpolation Fit for
stab
K
of Generator 2 (3 M
a
chi
ne System)
i
x
0.6
3.54 6.48
9.42 12.3
15.3
18.2 21.1 24.1
27
30
)
(
i
x
f
13.7935
13.7092
13.6415
13.5888
13.5495
13.5221
13.5051
13.
4969
13.49513.5006
13.5095
)
(
i
x
f
d
-0.0316 -0.0257 -0.0203
-0.0155 -0.0112
-0.0074 -0.0042
-0.
0014
0.00070.0024
0.00355
)
(
i
x
f
d
0.002092
0.001913
0.001731
0.001555
0.001376
0.
001197
1.02E-3
0.00083
0.00060.00048
0.00030
Table 7. Interpolation Fit for
1
T
of Generator 3 (3 M
a
chi
ne System)
i
x
0.6
3.54 6.48
9.42 12.36
15.
3 18.24
21.18
24.1
27
30
)
(
i
x
f
12.9689
-21.5696
-415.283-1596.16
-3992.19-8031.37
-14141.7-22751.1
-
34287.7-49179.3
-67854
)
(
i
x
f
d
0.811412
-48.5694
-243.525-584.055
-1070.16-1701.84
-
2479.1 -3401.92
-4470.33-5684.31
-7043.86
)
(
i
x
f
d
7.96141
-41.5538
-91.069 -140.584
-190.099-239.615
-
289.13 -338.645
-388.16 -437.676
-487.191
Table 8. Interpolation Fit for
3
T
of Generator 3 (3 M
a
chi
ne System)
i
x
0.6
3.54 6.48
9.42 12.36
15.3
18.24
21.18
24.12
27.06
30
)
(
i
x
f
13.7248
-74.4346-693.471
-2366.1 -5615.03
-10963
-18932.
7
-
30046.8-44828.1
-
63799.3-87483.1
)
(
i
x
f
d
1.03427
-90.639
-360.106
-807.368-1432.42
-2235.27-3215.92
-4374.36-5710.59
-7224.61-8916.43
)
(
i
x
f
d
-0.94429
-61.4185-121.893
-182.367-242.841
-303.315-363.789
-424.264-484.738
-545.212-605.686
Table 9. Interpolation Fit for
stab
K
of Generator 3 (3 M
a
chi
ne System)
i
x
0.6
3.54 6.48
9.42 12.36
15.3
18.24
21.18
24.12
27.06
30
)
(
i
x
f
15.2778
14.9018
14.5859
14.3247
14.1131
13.
946
13.8181
13.7243
13.6593
13.618
13.5951
)
(
i
x
f
d
-0.13867
-0.11738 -0.09785
-0.08011
-0.06410
-0.04987
-0.0374
-0.02671
-0.01778
-0.0106 -0.00521
)
(
i
x
f
d
0.00754
0.006940
0.006340
0.005740
0.005139
0.004539
3.94E-3
0.003338
0.002738
0.00213
0.001537
3. Case 1
:
T
h
ree Mac
h
in
e Po
w
e
r Sy
s
t
em
The case taken over
here
to unde
rsta
n
d
the re
sp
on
se of differen
t
polynomial
s
is the 3
machi
ne 9
-
bu
s syste
m
[13], the minute obse
r
vation
s o
n
the system
sho
w
s
that wi
thout installin
g
PSSs
on generating machine, s
y
s
t
em get
uns
table for variou
s
perturbations
.
Table 10. Ge
neratin
g Co
n
d
itions
Base Case
Case-
A
Case-
B
Case-
C
P Q
P Q
P
Q
P
Q
0.716
0.270
1.527
0.249
1.3283
0.2393
0.5077
0.3029
1.63 0.0665
1.00 -0.003
1.00
-0.006
1.85
0.1134
0.85 -0.108
0.65 -0.117
0.850
-0.12
0.85
-0.094
Table 11. Lo
a
d
ing Conditio
n
s
Base Case
Case-
A
Case-
B
Case-
C
P
Q
P
Q P
Q P
Q
1.25
0.50
0.75
0.39
1.50 0.90
0.65 0.55
0.90
0.30
0.90
0.30
1.20 0.80
0.45 0.35
1.00
0.35
1.00
0.35
1.00 0.50
0.50 0.25
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 14, No. 2, May 2015 : 191 – 198
196
Table 10 an
d Table 11 g
i
ve different operating co
ndition
s relat
ed with gen
e
r
ation a
s
well a
s
load
side. As stated
in literature t
hese ope
ra
tin
g
con
d
ition
s
are con
s
ide
r
e
d
as ha
rd a
s
far
as sy
stem sta
b
ility is con
c
e
r
ned [7].
System re
spo
n
se i
s
judg
ed
with load pat
terns a
nd foll
owin
g pertu
rb
ations:
a)
A 6-cycl
e faul
t disturba
nce at bus 6 at t
he end of line 5-6 with
Ca
se A, Case B. The
fault has be
e
n
clea
red
without trippin
g
b)
A 6-cycl
e faul
t is clea
red b
y
tripping the line 5-6
with Ca
se C.
Figure 4. Speed Deviatio
n of Gene
rator
2
Figure 5. Speed Deviatio
n of Gene
rator
1
Figure 6. Speed Deviatio
n of Gene
rator
3
4. Case 2
:
Ne
w
En
gland
Po
w
e
r Sy
stem
In this
ca
se,
the 10
ma
chin
e 39
bu
s
system [1
4] is
con
s
id
ered.
The
system i
s
comp
aratively larger tha
n
3 gene
rat
o
r syste
m
a
nd dynami
c
as intera
re
a oscillation
is
c
o
ns
ide
r
ed
.
4.1. Test Sy
s
t
em
The
system i
s
te
sted ove
r
different p
e
rt
urbat
io
ns an
d
config
uration
s
which i
s
extremely
hard for system stability [12]
4.2. PSS Design
To d
e
si
gn th
e
propo
sed
PS
S by u
s
ing
mi
nima
x a
pproximation inte
rp
olation
polyn
omials,
three
differe
nt ope
rating
con
d
ition
s
a
nd critic
al lin
e outag
es are co
nsi
d
e
r
ed
whi
c
h
are
the
enormou
s
ly ri
gid from the stability point of view. They can be con
s
id
ered a
s
:
a)
Base Case; No outa
ge of line
b)
Ca
se A; outage of line 22-23
c)
Ca
se B; outage of line 1-3
9
Speed d
e
viation cu
rve of g
enerator 9 i
s
sh
o
w
n to d
e
m
onst
r
ate th
e effectivene
ss
of the
proposed PSSs as it is the nearest wit
h
the f
ault location (li
ne 14-15)
, another speed deviation
curve
of ge
n
e
rato
r 3 i
s
sh
own
as the g
enerator l
o
ca
tion is
also a
key de
rivative co
nsi
d
e
r
ing
the
above given
con
d
ition
s
.
0
1
2
3
4
5
-4
-2
0
2
x 10
-3
ti
m
e
(
s
)
2
(
p
.u
.)
CA
S
E
A
Li
n
e
a
r
Po
l
y
2
Po
l
y
3
0
1
2
3
4
5
-1
0
-5
0
5
x 10
-4
ti
m
e
(
s
)
1
(
p
.u
.
)
CA
S
E
B
Li
ne
a
r
Po
l
y
2
Po
l
y
3
0
1
2
3
4
5
-5
0
5
10
x 1
0
-3
ti
m
e
(
s
)
3
(
p
.u
.)
CA
S
E
C
Li
n
e
a
r
Po
l
y
2
Po
l
y
3
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
A Minim
a
x Polyn
o
m
i
al Appro
x
im
ation Obje
ctive Fu
nction App
r
o
a
ch for… (Bh
anu Prata
p
Soni)
197
Figure 7. Speed Deviatio
n of Gene
rator
9
Figure 8. Speed Deviatio
n of Gene
rator
3
Figure 9. Speed Deviatio
n of Gene
rator
9
4.3. Discus
s
ions
1)
It is observe
d from the
speed
deviatio
n
cu
rves
of d
i
fferent gen
erators f
r
om Fi
gure
(3) to
Figure (6) that PSS des
igned t
h
rough linear pol
ynomial
objective function gives best
solutio
n
a
s
fa
r a
s
the ove
r
all syste
m
st
ability is
con
c
erned.
Ho
wever, it is
wo
rth mentio
n h
e
re
that system gets unstabl
e while
usi
ng
the PSSs parameters obta
ined from either polynom
ial
nd
2
orde
r or p
o
lynomial
rd
3
order.
2)
On Ne
w En
gland Syste
m
, the critical
operating
con
d
ition (Case B) reve
als the
efficacy
of the proposed
linear PSS prominently.
PSS desi
g
ned from ot
her than li
near
polynomial fit sho
w
a po
or
dynamic
re
sp
onse in this o
peratin
g co
nd
ition.
3)
While ob
se
rving
the sp
eed deviation
s
re
lated with
the
polynomial fit
of ord
e
r
st
1
,
nd
2
and
rd
3
; it is observed that th
e maximum e
rro
r te
rm i
s
minimize
d with
usin
g linea
r i.e.
st
1
order
fit.
5. Conclusio
n
Wo
rk
pre
s
e
n
t
ed in thi
s
p
aper is to
transfo
rm the
traditional
ob
jective fun
c
tion into
minimax polynomial
s
. Tabl
e 1 to 9 sho
w
s vari
ou
s in
terpolatio
n st
atistics while
optimizing t
h
e
PSSs
parameter i.e. time c
o
ns
tant
s
and PSSs
Gain
s
s
e
ns
itivity with res
p
ec
t
to the objectiv
e
function. Hi
g
her value
s
o
f
nd
2
derivative
sho
w
s that it pre
s
ent
s a poor fit to the obje
c
tive
function; ho
wever value
s
for
nd
2
derivative
s
is ze
ro in ca
se of linear p
o
lynomial
s
. The techni
que
use
d
for o
p
timization i
s
P
S
O. The re
sp
onse obt
ain
e
d
unde
r different ope
rating
con
d
ition
s
sh
ows
that linear fit is the most suit
able fit for obtaining the
PSSs param
eters. By using linear
objective
function the
small sign
al st
ability can be
enha
nced.
Referen
ces
[1]
Dud
geo
n GJW
,
et al. T
he Eff
e
ctive r
o
le
of
AVR a
nd
PSS
in
po
w
e
r
s
y
stems: frequ
enc
y
Re-s
pons
e
Anal
ys
is.
IEEE Transaction on Power System
.
20
07; 22(
4): 1986-
19
94.
0
2
4
6
8
10
-4
-2
0
2
4
x 10
-3
ti
m
e
(
s
)
9
(
p
.u.)
B
A
SE
C
A
SE
Li
n
e
a
r
Po
l
y
2
Po
l
y
3
0
2
4
6
8
10
-1
0
-5
0
5
x 1
0
-3
ti
m
e
(
s
)
3
(
p
.u
.)
CA
S
E
A
Li
ne
a
r
Po
l
y
3
Po
l
y
2
0
2
4
6
8
10
-4
-2
0
2
4
x 10
-3
tim
e
(
s
)
9
(
p
.u
.)
CA
S
E
B
Li
n
e
a
r
Po
l
y
2
Po
l
y
3
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 14, No. 2, May 2015 : 191 – 198
198
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