Indonesian
Journal
of
Electrical
Engineering
and
Computer
Science
V
ol.
5,
No
.
1,
J
an
uar
y
2017,
pp
.
19
32
DOI:
10.11591/ijeecs
.v5.i1.pp19-32
19
Optimal
design
of
PID
contr
oller
f
or
load
frequenc
y
contr
ol
using
Harmon
y
sear
c
h
algorithm
D
.
K.
Sambariy
a*
and
Seema
Shrangi
Depar
tment
of
Electr
ical
Engg.,
Rajasthan
T
echnical
Univ
ersity
Ra
w
atbhata
Road,
K
ota,
324010,
India
*Corresponding
author
,
e-mail:
dsambar
iy
a
2003@y
ahoo
.com
Abstract
This
paper
deals
with
soft
computing
technique
,
used
f
or
tuning
PID
controller
.
Controller
tuned
with
a
har
mon
y
search
algor
ithm
is
used
f
or
controlling
the
frequency
and
Tie-line
po
w
er
responses
of
a
non
reheat
tw
o
area
po
w
er
system.
Step
load
per
turbation
has
been
giv
en
in
both
areas
sim
ultan
eously
.
The
dynamic
results
obta
ined
b
y
the
proposed
controller
are
compared
with
PID
controller
of
recent
pub
lished
paper
.
The
per
f
or
mance
of
the
controllers
is
sim
u
lated
using
MA
TLAB/Sim
ulation
softw
are
.
The
results
of
tuned
PID
are
compared
with
con
v
entional
controller
on
the
basis
of
settling-time
,
peak
o
v
er-shoot
and
peak
under-shoot.
Proposed
PID
giv
es
b
etter
results
than
the
con
v
entional
controller
.
The
compar
ativ
e
results
also
tab
ulated
as
a
compar
ativ
e
perf
or
mance
.
K
e
yw
or
ds:
Load
frequency
control
(LFC),
A
utomatic
control
error
(A
CE),
Propor
tional
Intergal
Der
iv
ativ
e
controller
(PID),
Har
mon
y
search
algor
ithm
(HSA).
Cop
yright
c
2016
Institute
of
Ad
v
anced
Engineering
and
Science
.
All
rights
reser
v
ed.
1.
Intr
oduction
P
o
w
er
system
oper
ator
has
a
responsibility
that
adequate
po
w
er
m
ust
deliv
er
to
con-
sumer
.
Reliability
and
economically
m
ust
maintain.
F
or
this
control,
str
ategy
is
needed
[1].
It
means
that
there
should
balance
betw
een
real
and
reactiv
e
po
w
er
.
As
reactiv
e
po
w
er
depends
on
the
v
oltage
and
real
po
w
er
on
the
gener
ation,
it
means
frequency
[2].
The
analysis
f
or
control-
ling
frequency
and
v
oltage
can
be
tak
en
separ
ately
.
In
this
paper
,
automatic
frequency
control
is
tak
en.
As
the
load
is
changed
;
the
frequency
is
changed
to
meet
the
requi
rement
of
the
connected
load
[3].
The
load
is
v
ar
ied
with
f
ast
r
ate
and
slo
w
r
ate
w
e
are
consider
ing
the
f
ast
v
ar
iation;
slo
w
v
ar
iation
is
not
considered.
In
larger
iner
ter-connected
po
w
er
system
tie
line
po
w
er
should
be
maintained
in
a
toler
ab
le
limit
with
respect
to
load
change
[4].
It
becomes
necessar
y
to
regulate
the
input
to
the
turbine
b
y
opening
of
v
alv
e
the
input
can
be
steam
o
r
h
ydro
f
or
h
ydro
alter
nator
.
F
or
this
,
an
efficient
control
str
ategy
is
required.
The
controlling
of
electr
ic
po
w
er
of
the
gener
ator
b
y
this
method
is
automatic
gener
ation
control
(A
GC)
[5,
6].
In
this
paper
,
a
step
disturbance
is
considered,
and
it
is
seen
that
the
frequency
steady
state
error
and
tie
line
steady
sta
te
error
f
ollo
wing
the
step
disturbance
should
be
z
ero
[7].
In
the
ALFC
,
there
are
tw
o
loops
pr
imar
y
loops
and
secondar
y
loop
,
pr
imar
y
loop
is
f
ast,
and
it
is
called
uncontrolled
loop
.
The
secondar
y
loop
is
slo
w
,
which
is
called
controlled
loop
.
Sec-
ondar
y
control
loop
is
done
to
mak
e
the
automatic
control
error
to
z
ero
.
The
con
v
entional
control
str
ategy
f
or
LFC
prob
lems
is
to
be
tak
en
the
integ
r
al
of
the
control
error
as
the
control
signal,
an
integ
r
al
controller
pro
vides
z
ero
steady-state
error
.
Ho
w
e
v
er
,
it
e
xhibits
poor
dynamic
beha
vior
,
to
impro
v
e
the
dynamic
beha
vior
man
y
control
techniques
are
used
such
as
linear
f
eedbac
k,
optimal
control,
and
v
ar
iab
le
str
ucture
control
ha
v
e
been
proposed
[
8].
Man
y
algor
ithms
f
or
controlling
the
con
v
entional
controller
gains
are
proposed.
The
algor
ithms
are
PSO
,
genetic
algor
ithm,
ZN
algor
ithm
[9].
In
this
paper
,
the
har
mon
y
search
algor
ithm
is
used
f
or
tuning
the
free
coef
ficients
of
PID
controller
.
Har
mon
y
search
algor
ithm
uses
a
Meta
heur
istic
approach
to
solv
e
man
y
optimiza-
Receiv
ed
No
v
ember
18,
2016;
Re
vised
December
15,
2016;
Accepted
December
28,
2016
Evaluation Warning : The document was created with Spire.PDF for Python.
20
ISSN:
2502-4752
tion
prob
lems
this
algor
ithm
w
as
inspired
b
y
the
method
used
to
impro
v
e
the
tuning
or
pitches
of
the
m
usic
to
obtain
better
har
mon
y
[10].
Har
mon
y
search
algor
ith
m
w
or
ks
on
the
mimic
king
the
impro
visation
of
m
usic
pla
y
ers
.
As
the
most
str
uctur
al
optimization
methods
are
based
on
mathematical
algor
ithms
that
require
strongly
b
uilt
g
r
adient
inf
or
mation
[11].
The
har
mon
y
search
algor
ithm
does
not
require
initial
v
alues
and
uses
a
r
andom
search
instead
of
a
g
r
adient
search;
so
that
the
der
iv
ativ
e
inf
or
mation
is
unnecessar
y
.
This
algor
ithm
is
conceptual
using
the
m
usical
impro
visation
process
of
searching
f
or
a
perf
ect
state
of
har
mon
y
(i.e
har
mon
y
memor
y)
to
auto-
matically
adjust
par
ameter
v
alues
.
This
algor
ithm
does
not
require
to
fix
e
initial
condition.
This
algor
ithm
is
used
b
y
engineers
to
solv
e
the
optimization
prob
lem
[12].
Boroujeni
et
al
[13],
proposed
w
or
k
f
or
tw
o
are
po
w
er
system.
A
uthor
T
uned
the
PI
b
y
HSA
f
or
quenching
the
de
viation
in
frequency
and
Tie-line
po
w
er
due
to
diff
erent
loading
conditions
.
Compare
the
result
of
tuned
PI
b
y
tuned
IP
.
Sho
w
eff
ectiv
eness
of
both
controllers
in
industr
ies
.
Sambar
iy
a
and
Nath,
2015,
ha
v
e
proposed
the
application
of
par
ticle
s
w
ar
m
optimization
(PSO)
in
optimal
tuning
of
PID
par
ameters
f
or
tw
o-area
load
frequency
po
w
er
system
in
[14].
The
ne
w
NARMA
L2
controller
is
presented
in
[15].
The
fuzzy
logic
based
controller
f
or
m
ulti-area
system
is
presented
in
[16].
The
application
of
adaptiv
e
nero
fuzzy
logic
controller
is
presented
in
[17].
Abedinia
et
al
[18],
proposed
w
or
k
to
solv
e
the
LFC
prob
lem.
By
implementing
a
ne
w
m
ulti-stage
fuzzy
(MSF)
controller
based
on
m
ulti-objectiv
e
Har
mon
y
search
algor
ithm
(MOHSA).
Membership
Function
of
fuzzy
is
designed
automatically
b
y
the
proposed
MOHSA
method.
Com-
pare
results
with
other
controllers
used
f
or
LFC
.
Sanpala
and
V
akula
[19],
in
proposed
w
or
k
tw
o
interconnected
are
with
identical
par
ameters
are
considered
f
or
study
.
The
presented
w
or
k
better
utiliz
ed
the
har
mon
y
search
algor
ithm
to
design
the
optimal
par
ameters
of
Fuzzy-PID
.
Sambar
iy
a
and
Pr
asad
[20],
ha
v
e
designed
a
Fuzzy
logic
po
w
er
system
stabiliz
er
,
whose
par
ameters
are
tuned
b
y
Har
mon
y
search
algor
ithm.
The
P
ar
ameters
of
the
PD
controller
are
considered
as
the
nor
maliz
ed
f
actors
of
FPSS
and
T
uned
using
har
mon
y
search
algor
ithm.
In
this
paper
,
con
v
entional
PID
controllers
are
compared
with
HSA-PID
controller
.
It
is
f
ound
t
hat
tuned
PID
controller
is
better
than
the
con
v
entional
controller
.
The
settling
time
,
o
v
er-
shoot
time
,
steady-state
error
is
reduced
b
y
using
a
proposed
controller
as
compared
to
the
con-
v
ention
controller
.
By
MA
TLAB
softw
are
the
frequency
response
,
tie
line
po
w
er
,
go
v
er
nor
po
w
er
disturbance
response
is
compared
of
both
the
area.
The
paper
is
organiz
ed
in
5
sections
.
The
prob
lem
f
or
m
ulation
with
system
descr
iption,
process
and
objectiv
e
function
is
presented
in
section
2..
The
har
mon
y
search
algor
ithm
used
to
tune
the
par
ameters
of
PID
controller
is
presented
in
section
3..
The
system
responses
with
proposed
PID
controller
in
ter
ms
of
frequency
and
tie-line
po
w
er
are
compared
to
that
of
the
controllers
a
v
ailab
le
in
liter
a
ture
and
mentioned
in
section
4..
The
paper
is
concluded
in
section
5.
and
f
ollo
w
ed
with
appendix,
nomenclature
and
ref
erences
.
2.
Pr
ob
lem
f
orm
ulation
2.1.
System
description
The
system
under
consider
ation
is
a
tw
o-area
po
w
er
system.
It
consists
of
go
v
er
nor
,
non-reheat
turbine
and
load-iner
tia.
The
models
of
the
components
are
solv
ed
f
or
obtaining
state
v
ector
matr
ix
b
y
state
space
modeling
method.
Defining
the
state
v
ar
iab
les
in
ter
ms
of
diff
erential
v
ar
iab
les
of
the
system.
The
state
v
ar
iab
les
as
x
1
;
:::;
x
9
are
defined
b
y
diff
erential
v
ar
iab
les
as
in
Eqn.
1.
The
b
loc
k
diag
r
am
representation
of
the
system
is
sho
wn
in
Fig.
1.
The
u
1
;
u
2
and
IJEECS
V
ol.
5,
No
.
1,
J
an
uar
y
2017
:
19
32
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
2502-4752
21
d
1
=
p
d
1
;
d
2
=
p
d
2
are
representing
the
control
and
disturbance
v
ar
iab
les
.
x
1
=
f
1
x
2
=
p
t
1
x
3
=
p
g
1
x
4
=
f
2
x
5
=
p
t
2
x
6
=
p
g
2
x
7
=
p
tie
12
x
8
=
R
AC
E
1
dt
x
9
=
R
AC
E
2
dt
(1)
R
!
R
s
T
g
s
T
t
!
D
H
s
T
g
!
!
s
T
t
!
!
!
D
H
L
P
m
P
l
P
m
P
!
l
P
x
x
x
u
x
x
x
x
x
u
x
f
f
s
T
!
!
Figure
1.
Bloc
k
Diag
r
am
of
tw
o-area
interconnected
system
State
space
equations
f
or
tw
o
area
are
giv
en
as
f
ollo
wing:
_
x
1
=
D
1
2
H
1
x
1
+
1
2
H
1
x
2
+
1
2
H
1
x
7
+
(
d
1
)
2
H
1
(2)
_
x
2
=
1
T
t
1
x
2
+
1
T
t
1
x
3
(3)
_
x
3
=
1
R
1
T
g
1
x
1
1
T
g
1
x
3
+
1
T
g
1
u
1
(4)
_
x
4
=
D
2
2
H
2
x
4
+
1
2
H
2
x
5
+
1
2
H
2
x
7
d
2
2
H
2
(5)
_
x
5
=
1
T
t
2
x
2
+
1
T
t
2
x
6
(6)
Optimal
design
of
PID
controller
f
or
LFC
using
HS
algor
ithm
(D
.
K.
Sambar
iy
a)
Evaluation Warning : The document was created with Spire.PDF for Python.
22
ISSN:
2502-4752
2
6
6
6
6
6
6
6
6
6
6
6
6
4
_
x
1
_
x
2
_
x
3
_
x
4
_
x
5
_
x
6
_
x
7
_
x
8
_
x
9
3
7
7
7
7
7
7
7
7
7
7
7
7
5
=
2
6
6
6
6
6
6
6
6
6
6
6
6
6
4
D
1
2
H
1
1
2
H
1
0
0
0
0
0
0
0
0
1
T
t
1
1
T
t
1
0
0
0
0
0
0
1
R
1
T
g
1
0
T
g
1
0
0
0
0
0
0
0
0
0
D
2
2
H
2
1
2
H
2
0
1
2
H
2
0
0
0
1
T
t
2
0
0
0
1
T
t
2
0
0
0
0
0
0
1
R
2
T
g
2
0
1
T
g
2
0
0
0
2
T
o
0
0
2
T
o
0
0
0
0
0
B
1
0
0
0
0
0
1
0
0
0
0
0
B
2
0
0
1
0
0
3
7
7
7
7
7
7
7
7
7
7
7
7
7
5
2
6
6
6
6
6
6
6
6
6
6
6
6
4
x
1
x
2
x
3
x
4
x
5
x
6
x
7
x
8
x
9
3
7
7
7
7
7
7
7
7
7
7
7
7
5
(12)
_
x
6
=
1
R
2
T
g
2
x
4
1
T
g
2
x
6
+
1
T
g
2
u
2
(7)
_
x
7
=
2
T
0
x
1
2
T
0
x
4
(8)
_
x
8
=
(
B
1
)
x
1
x
7
(9)
_
x
9
=
(
B
4
)
x
4
+
x
7
(10)
The
Eqns
.
2
-
10
can
be
organiz
ed
as
a
v
ector
matr
ix
as
f
ollo
wing
in
Eqn.
11.
_
x
=
Ax
+
B
u
+
C
d
(11)
where
x
is
state
v
ector
,
u
is
control
v
ector
and
d
is
the
disturbance
v
ector
.
The
complete
matr
ix
representation
is
sho
wn
in
f
ollo
wing
Eqn.
12.
The
tw
o
systems
interconnected
are
sho
wn
in
Fig.
1.
The
Go
v
er
nor
,
turbine
and
load
iner
tia
b
loc
k
are
rep
resented
with
their
time
constant.
The
both
areas
are
interconnected
using
tie-line
.
The
rele
v
ant
data
of
the
system
is
presented
in
the
Appendix.
2.2.
Pr
ocess
Study
of
load
f
requency
control
is
considered
f
or
tw
o
interconnected
area
po
w
er
system.
Step
Disturbance
of
0.01
p
.u
is
giv
en
to
both
areas
.
The
u
1
;
u
2
are
tw
o
controlled
signal
giv
en
to
go
v
er
nor
f
or
controlling
dynamic
response
de
viations
.
Har
mon
y
Search
algor
ithm
is
used
f
or
tuning
t
he
par
ameters
of
PID
controller
.
The
designed
controller
is
used
control
the
frequency
and
tie
-
line
po
w
er
de
viations
in
the
interconnected
area
are
controlled.
The
main
impact
of
HSA-PID
is
to
control
the
tr
ansient
response
of
the
system.
The
tuning
scheme
of
PID
par
ameters
is
sho
wn
in
Fig.
2.
2.3.
Objective
function
The
area
control
error
of
both
of
the
areas
is
sensed
and
the
diff
erence
(i.e
.
error
of
A
CE)
is
considered
as
the
signal
to
be
minimiz
ed
under
optimization.
The
par
ameters
of
both
PID
controllers
are
optimiz
ed
under
optimization
using
HS
algor
ithm
subjected
to
the
upper
and
lo
w
er
bounds
of
the
par
ameters
as
sho
wn
in
Eqn.
13.
K
min
pj
K
pj
K
max
pj
K
min
ij
K
ij
K
max
ij
K
min
d
j
K
d
j
K
max
d
j
(13)
where
,
j
stands
f
or
the
PID
controller
connected
an
area
i.e
j
=
1
;
2
.
These
par
ameters
of
the
PIDs
are
optimiz
ed
using
the
objectiv
e
function
as
sho
wn
in
Eqn.
14.
It
sho
ws
the
minimization
of
integ
r
al
square
error
(ISE)
of
the
area
control
error
f
or
the
sim
ulation
time
T
sim
.
I
S
E
=
T
sim
Z
0
AC
E
1
(
t
)
AC
E
2
(
t
)
dt
(14)
IJEECS
V
ol.
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.
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uar
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:
19
32
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IJEECS
ISSN:
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23
S
Kp
Kd
Pow
er
Sys
tem
Model
Obj
.
Fun
.
Harmony
S
ea
rc
h
Algorithm
+
-
PID
Controller
Kp
S
+
+
+
1
f
D
1
A
C
E
Figure
2.
The
scheme
of
optimization
of
PID
par
ameters
using
har
mon
y
search
algor
ithm
3.
Harmon
y
sear
c
h
algorithm
The
har
mon
y
search
algor
ithm
(HSA)
w
as
de
v
eloped
b
y
Geem.et.al
in
2001
[21].
The
HSA
is
made
efficient
b
y
using
impro
visation
technique
used
in
m
usic
making.
The
m
usician
selects
the
pitches
of
the
instr
ument
f
or
getting
better
state
of
har
mon
y
.
A
m
usician
impro
visation
is
same
as
the
search
process
in
optimization.
Ev
er
y
m
usic
pitch
is
equal
to
the
appreciation
of
beauty
of
quantity
[22].
The
per-f
or
mation
and
efficiency
of
most
of
the
Meta
heur
istic
algor
ithm
depends
on
the
e
xtent
of
balance
betw
een
div
ersification
and
intensification
dur
ing
the
search
process
.
The
e
xploitation
is
the
ability
of
an
algor
ithm
to
e
xploit
the
search
space
in
the
neighborhood
of
the
current
good
solution
using
the
inf
or
mation
already
collected
[12].
The
pitch
adjustment
technique
helps
in
finding
the
best
solution
in
the
har
mon
y
memor
y
[23].
The
har
mon
y
search
algor
ithm
depends
on
the
points
such
as
Har
mon
y
memor
y
siz
e
(HMS),
Har
mon
y
memor
y
consider
ation
r
ate
(HMCR),
when
a
m
usician
is
impro
ving,
he
or
she
has
three
possib
le
choice:(1)
pla
ying
an
y
f
amous
e
xactly
from
his
or
her
memor
y
,(2)
pla
ying
some-
thing
similar
to
the
f
or
a
mentioned
tune(thus
adjusting
the
pitch
slig
htly),(3)
composition
ne
w
or
r
andom
notes
[24,
25].
3.1.
Steps
of
HS
algorithm
Each
ro
w
of
har
mon
y
memor
y
(HM),
consists
of
N
decision
v
ar
iab
les
and
the
fitness
score
!
([
x
1
;
x
2
;
:::;
!
]).
The
HM
is
initializ
ed
with
HMS
r
andomly
gener
ated
solution
v
ectors
[24].
3.1.1.
Initialization
Let
in
an
optimization
prob
lem,
the
objectiv
e
function
is
represented
b
y
Minimization
of
F
(
x
)
,
which
is
subjected
to
x
i
2
X
i
,
i
=
1
;
2
;
3
N
.
Where
,
x
is
the
set
of
design
v
ar
iab
les
(
x
i
),
(
x
L
i
x
i
x
U
i
),
and
N
is
the
count
of
design
v
ar
iab
les
.
Define
the
v
ar
iab
le
limits
as
lo
w
er
(
x
L
i
)
and
upper
(
x
U
i
)
or
x
L
i
x
i
x
U
i
.
Deciding
the
v
alue
of
har
mon
y
memor
y
siz
e
(HMS),
from
the
r
ange
10
H
M
S
100
.
Decide
v
alue
of
HMCR
(har
mon
y
memor
y
consider
ation
r
ate)
within
the
r
ange
0
:
0
H
M
C
R
Optimal
design
of
PID
controller
f
or
LFC
using
HS
algor
ithm
(D
.
K.
Sambar
iy
a)
Evaluation Warning : The document was created with Spire.PDF for Python.
24
ISSN:
2502-4752
1
:
0
.
x
0
i
x
0
i
l
f
x
1
i
;
x
2
i
;
:::;
x
H
M
S
i
:
pr
ob:
H
M
C
R
x
0
i
2
X
i
:
pr
ob:
(1
H
M
C
R
)
(15)
Decide
the
v
alue
of
P
AR
(pitch
adjustment
r
ate)
from
the
r
ange
0
:
0
P
AR
1
:
0
.
x
i
j
=
x
L
j
+
r
and
(0
;
1)
(
x
U
j
x
L
j
)
(16)
x
0
i
y
es
w
ith
pr
obabil
ity
P
AR
no
w
ith
pr
oba
bil
ity
(1
P
AR
)
(17)
Compute
the
step
siz
e
(
b
i
)
as
b
i
=
x
U
i
x
L
i
N
.
Specify
the
maxim
um
limit
of
iter
ation
n
umber
.
3.1.2.
HM
Initiation
The
HM
matr
ix
as
in
Eqn.
18
is
filled
with
r
andomly
gener
ated
possib
le
solution
v
ectors
f
or
HMS
and
is
sor
ted
b
y
the
v
alues
of
the
objectiv
e
function
f
(
x
)
.
H
M
=
2
6
6
6
6
6
4
x
1
1
x
1
2
:::
x
1
N
1
x
1
N
x
2
1
x
2
2
:::
x
2
N
1
x
2
N
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
x
H
M
S
1
1
x
H
M
S
1
2
:::
x
H
M
S
1
N
1
x
H
M
S
1
N
x
H
M
S
1
x
H
M
S
2
:::
x
H
M
S
N
1
x
H
M
S
N
3
7
7
7
7
7
5
)
2
6
6
6
6
6
4
f
(
x
1
)
f
(
x
2
)
.
.
.
f
(
x
H
M
S
1
)
f
(
x
H
M
S
)
3
7
7
7
7
7
5
(18)
3.1.3.
Impr
o
visation
A
Ne
w
Har
mon
y
v
ector
x
0
=
(
x
0
1
;
x
0
2
;
:::;
x
0
n
)
is
gener
ated
based
on
three
cr
iter
ia:
r
andom
selection,
memor
y
consider
ation,
and
pitch
adjustment.
Random
Selection
:
T
o
decide
the
v
alue
of
x
0
1
f
or
the
Ne
w
Har
mon
y
x
0
=
(
x
0
1
;
x
0
2
;
:::;
x
0
n
)
,
the
HS
algor
ithm
r
andomly
selects
a
v
alue
from
r
ange
with
a
probability
of
1
H
M
C
R
.
Memor
y
Consider
ation
:
T
o
decide
the
v
alue
of
x
0
1
,
the
HS
algor
ithm
r
andomly
selects
a
v
alue
from
the
HM
with
a
probability
of
H
M
C
R
,
where
j
=
1
;
2
;
:::;
H
M
S
.
It
can
be
represented
as
in
Eqn.
18.
Pitch
Adjustment
:
Each
element
of
the
Ne
w
HM
v
ector
x
0
=
(
x
0
1
;
x
0
2
;
:::;
x
0
n
)
is
subjected
to
deter
mine
whether
it
should
be
pitch-adjusted
or
not.
The
selected
x
0
i
is
fur
ther
adjusted
b
y
adding
an
amount
to
the
v
alue
with
a
probability
of
P
AR.
The
P
AR
par
ameter
is
called
the
probability
of
pitch
adjustment
and
is
represented
b
y
Eqn.
17
[26].
The
0
no
0
with
probability
1
P
AR
represents
that
the
probability
of
not
adding
an
y
amount.
On
the
other
hand,
if
the
pitch
adjustment
decision
f
or
x
0
i
is
y
es
,
then
it
is
replaced
b
y
x
0
i
x
0
i
bw
;
where
,
bw
is
distance
ban
dwidth
in
case
a
contin
uous
v
ar
iab
le
.
The
pitch
adjustment
is
perf
or
med
on
e
v
er
y
v
ar
iab
le
of
the
Ne
w
HM
v
ector
.
IJEECS
V
ol.
5,
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.
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uar
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:
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32
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25
3.1.4.
Updating
HM
Let
the
HM
v
ector
is
x
0
=
(
x
0
1
;
x
0
2
;
:::;
x
0
n
)
,
which
is
resolv
ed
b
y
minimization
of
objectiv
e
function
is
better
than
the
w
orst
har
mon
y
present
in
the
HM.
Theref
ore
,
the
Ne
w
Har
mon
y
is
inser
ted
into
the
HM,
while
,
the
w
orst
har
mon
y
is
remo
v
ed
from
the
HM.
3.1.5.
Stopping
criterion
If
the
maxim
um
count
of
impro
visations
is
reached
and
the
stopping
cr
iter
ion
as
maxim
um
n
umber
of
iter
ations
is
satisfied,
then
the
process
of
computation
is
ter
minated.
Otherwise
,
go
to
steps
impro
visatio
n
and
updating
HM
as
abo
v
e
to
repeat
the
process
.
The
HS
algor
ithm
is
sho
wn
in
the
Fig.
3.
Start
Initialization
S
e
l
e
ct
H
M
CR
S
e
lect
P
A
R
Com
p
ute
new
H
a
rm
on
y
Modify
H
M
by
new
Harmon
y
Stop
Cri
te
ri
on
U
pda
t
ion
of
HM
Yes
No
S
t
op
Yes
No
Im
prov
isa
ti
on
Figure
3.
W
or
king
of
Har
mon
y
Search
Algor
ithm
4.
Result
and
discussion
4.1.
Optimization
of
PID
parameter
s
The
system
sho
wn
in
Fig.
1
equipped
with
PID
controllers
with
unkno
wn
par
ameters
as
K
p
1
;
2
;
K
i
1
;
2
;
K
d
1
;
2
is
sim
ulated
to
tune
these
par
ameters
using
har
mon
y
search
algor
ithm.
The
prob
lem
of
optimization
is
subjected
to
minimization
of
ISE
based
objectiv
e
function
as
sho
wn
Optimal
design
of
PID
controller
f
or
LFC
using
HS
algor
ithm
(D
.
K.
Sambar
iy
a)
Evaluation Warning : The document was created with Spire.PDF for Python.
26
ISSN:
2502-4752
in
Eqn
.
14
and
the
par
ameter
bounds
as
in
Eqn.
13.
The
optimiz
ed
PID
par
ameters
using
HS
algor
ithm
are
sho
wn
in
T
ab
le
1.
The
plot
of
the
fitness
function
with
iter
ation
counts
is
sho
wn
in
Fig.
4.
It
can
be
seen
that
the
fitness
function
gets
reduced
as
the
iter
ation
count
is
increased
and
around
190
it
becomes
constant.
It
is
the
instant,
where
the
optimal
par
ameters
are
considered.
0
50
100
150
200
0.5
1
1.5
2
2.5
3
x 10
−6
Interations
F
min
Fitness value
Figure
4.
The
v
ar
iation
of
fitness
function
with
iter
ations
using
har
mon
y
search
algor
ithm
T
ab
le
1.
Compar
ison
of
par
ameters
of
PID
controller
in
liter
ature
and
proposed
HSA-PID
controller
Controllers
K
p
1
K
i
1
K
d
1
K
p
2
K
i
2
K
d
2
HSA-PID
(Prop
.)
1.8677
1.9934
1.9099
1.4324
1.9139
1.8247
PSO-PID
[27]
3.0
0.8531
0.35
1.98
0.3919
0.1978
Con
v-PID
[28]
0.1109
0.2742
0.1110
0.0121
0.2019
0.0030
PSO-PID
[5]
22.8070
2.0734
17.4628
22.8070
2.0734
17.4628
BFO
A-PID
[29]
0.1317
0.4873
0.2506
0.1317
0.4873
0.2506
PSO-PID
[30]
0.790
1.4252
0.4652
0.790
1.4252
0.4652
4.2.
P
erf
ormance
comparison
In
this
presented
w
or
k,
a
tw
o
area
interconnected
system
with
non-reheat
turbine
is
con-
sidered
f
or
load
frequency
control
(LFC)
prob
lem.
A
step
load
of
0.01
p
.u
are
giv
en
in
both
areas
,
which
sho
w
rob
ustness
of
the
system.
Gener
ally
,
liter
ature
sur
v
e
y
on
LFC
sho
w
that
authors
con-
sidered
in
gener
al
1%
step
load
per
turbation
in
an
area
is
giv
en
f
or
an
optimal
results
.
In
proposed
w
or
k,
disturbance
is
giv
en
in
both
area.
It
has
been
used
f
or
PID
tuning.
The
HSA-PID
results
are
compared
with
recent
pub
lished
papers
b
y
consider
ing
their
PID
par
ameters
.
F
requency
re-
sponses
of
area-1
and
area-2
are
sho
wn
in
Fig.
5
-
Fig.
6.
The
tie
-
line
po
w
er
response
is
sho
wn
in
Fig.
7.
It
is
seen
that
proposed
controller
results
are
f
ar
better
than
the
con
v
entional
con-
troller
.
The
tie-line
po
w
er
response
of
an
interconnected
area
obtained
b
y
the
proposed
controller
is
compared
with
PID
controller
.
Compar
ison
of
frequency
responses
on
the
basis
of
settling
-
time
,
o
v
er
-
shoot
and
under-shoot
as
giv
en
in
the
tab
ular
f
or
m
in
T
ab
le
2
-
T
ab
le
3
and
the
tie
IJEECS
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.
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uar
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27
-line
compar
ison
is
giv
en
in
T
ab
le
4.
It
is
f
ound
that
HSA
tuned
PID
is
better
than
controllers
in
[5,
27–30].
0
5
10
15
20
25
30
35
−6
−5
−4
−3
−2
−1
0
1
x 10
−4
Time (s)
Frequency deviation
Frequency response of two area−1 system
Prop. (HSA−PID)
PID (Kavya, 2015)
PID (Ikhe, 2013 )
PID (Bassi, 2011)
PID (Ali, 2013)
PID (Dhanlakshmi, 2015)
Figure
5.
F
req
uency
response
of
area-1
obtained
b
y
HSA-PID
and
compared
with
controllers
in
[5,
27–30]
0
5
10
15
20
25
30
−10
−5
0
5
x 10
−4
Time (s)
Frequency deviation
Frequency response of two area−2 system
Prop. (HSA−PID)
PID (Kavya, 2015)
PID (Ikhe, 2013 )
PID (Bassi, 2011)
PID (Ali, 2013)
PID (Dhanlakshmi, 2015)
Figure
6.
F
req
uency
response
of
area-2
obtained
b
y
HSA-PID
and
compared
with
controllers
in
[5,
27–30]
Optimal
design
of
PID
controller
f
or
LFC
using
HS
algor
ithm
(D
.
K.
Sambar
iy
a)
Evaluation Warning : The document was created with Spire.PDF for Python.
28
ISSN:
2502-4752
T
ab
le
2.
Compar
ison
of
frequency
response
of
area-1
of
HSA-PID
compared
with
con
v
entional
PID
Controllers
Settling
time
Ov
er
shoot
Under
shoot
HSA-PID
12
0
:
085
10
6
2
:
03
10
4
PSO-PID
[27]
14.1
7
:
34
10
4
4
:
71
10
4
Con
v-PID
[28]
28.35
-
6
:
51
10
4
PSO-PID
[5]
33.16
2
:
2
10
4
6
:
3
10
4
BFO
A-PID
[29]
17.28
-
3
:
9
10
4
Con
v-PID
[30]
18.32
7
:
34
10
4
3
:
8
10
4
T
ab
le
3.
Compar
ison
of
frequency
response
of
area-2
of
HSA-PID
compared
with
con
v
entional
PID
Controllers
Settling
time
Ov
er
shoot
Under
shoot
HSA-PID
12.2
0
:
31
10
4
3
:
24
10
4
PSO-PID
[27]
13.0
1
:
65
10
4
6
:
32
10
4
Con
v-PID
[28]
27.50
1
:
05
10
4
9
:
70
10
4
PSO-PID
[5]
27.10
4
:
4
10
4
1
:
7
10
4
BFO
A-PID
[29]
16.20
2
:
8
10
5
5
:
4
10
4
Con
v-PID
[30]
28.55
3
:
00
10
4
5
:
5
10
4
T
ab
le
4.
Compar
ision
of
Tie-line
po
w
er
response
of
area-12
of
HSA-PID
compared
with
con
v
en-
tionl
PID
Controllers
Settling
time
Ov
er
shoot
Under
shoot
HSA-PID
26.0
1
:
70
10
4
-
PSO-PID
[27]
27.2
4
:
4
10
4
4
:
4
10
4
Con
v-PID
[28]
43.0
1
:
043
10
4
-
PSO-PID
[5]
-
2
:
25
10
4
-
BFO
A-PID
[29]
29
3
:
87
10
5
0
:
98
10
4
Con
v-PID
[30]
27.0
3
:
35
10
4
1
:
0
10
4
IJEECS
V
ol.
5,
No
.
1,
J
an
uar
y
2017
:
19
32
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