Inter
national
J
our
nal
of
Electrical
and
Computer
Engineering
(IJECE)
V
ol.
10,
No.
2,
April
2020,
pp.
1187
1199
ISSN:
2088-8708,
DOI:
10.11591/ijece.v10i2.pp1187-1199
r
1187
Antlion
optimization
algorithm
f
or
optimal
non-smooth
economic
load
dispatch
Thanh
Pham
V
an
1
,
V
´
acla
v
Sn
´
a
ˇ
sel
2
,
Thang
T
rung
Nguy
en
3
1
F
aculty
of
Electrical
Engineering
and
Computer
Science,
T
echnical
Uni
v
ersity
of
Ostra
v
a,
Czech
Republic
1
European
Cooperation
Center
,
T
on
Duc
Thang
Uni
v
ersity
,
V
iet
Nam
2
T
echnical
Uni
v
ersity
of
Ostra
v
a,
Czech
Republic
3
Po
wer
System
Optimization
Research
Group,
F
aculty
of
Electrical
and
Electronics
Engineering,
T
on
Duc
Thang
Uni
v
ersity
,
V
iet
Nam
Article
Inf
o
Article
history:
Recei
v
ed
May
26,
2019
Re
vised
Oct
5,
2019
Accepted
Oct
11,
2019
K
eyw
ords:
Antlion
optimization
Multi
fuel
sources
Ramp
rate
Spinning
reserv
e
V
alv
e
point
loading
ef
fects
ABSTRA
CT
This
paper
presents
applications
of
Antlion
optimization
algorithm
(ALO)
for
han-
dling
optimal
economic
load
dispatch
(OELD)
problems.
Electricity
generation
cost
minimization
by
controlling
po
wer
output
of
all
a
v
ailable
generating
units
is
a
major
goal
of
the
problem.
ALO
is
a
metaheuristic
algorithm
based
on
the
hunting
process
of
Antlions.
The
ef
fect
of
ALO
is
in
v
estig
ated
by
solving
a
10-unit
system.
Each
studied
case
has
dif
ferent
objecti
v
e
function
and
comple
x
le
v
el
of
restraints.
Three
test
cases
are
emplo
yed
and
arranged
according
to
the
comple
x
le
v
el
in
which
the
first
one
only
considers
multi
fuel
sources
while
the
second
case
is
more
complicated
by
taking
v
alv
e
point
loading
ef
fects
into
account.
And,
the
third
case
is
the
highest
challenge
to
ALO
since
the
v
alv
e
ef
fects
together
with
ramp
rate
limits,
prohibited
operating
zones
and
spinning
reserv
e
constraints
are
tak
en
into
consideration.
The
comparisons
of
the
result
obtained
by
ALO
and
other
ones
indicate
the
ALO
algorithm
is
more
potential
than
most
methods
on
the
solution,
the
stabilization,
and
the
con
v
er
gence
v
elocity
.
Therefore,
the
ALO
method
is
an
ef
fecti
v
e
and
promising
tool
for
systems
with
multi
fuel
sources
and
considering
complicated
constraints.
Copyright
c
2020
Insitute
of
Advanced
Engineeering
and
Science
.
All
rights
r
eserved.
Corresponding
A
uthor:
Thang
T
rung
Nguyen,
Po
wer
System
Optimization
Research
Group,
F
aculty
of
Electrical
and
Electronics
Engineering,
T
on
Duc
Thang
Uni
v
ersity
,
19
Nguyen
Huu
Tho
street,
T
an
Phong
w
ard,
District
7,
Ho
Chi
Minh
City
,
V
iet
Nam.
Email:
nguyentrungthang@tdtu.edu.vn
1.
INTR
ODUCTION
Minimizing
electricity
generation
fuel
cost
in
thermal
po
wer
plants
(TPPs)
is
e
xtremely
import
ant
because
it
accounts
for
a
high
rate
of
total
electricity
generation
cost.
So,
the
OELD
problem
has
been
widely
applied
for
this
purpose.
So
f
ar
solutions
which
ha
v
e
been
just
achie
v
ed
by
the
OELD
problem
is
to
decide
the
po
wer
output
of
each
thermal
generating
unit
(TGU)
so
that
the
electricity
generation
fuel
cost
can
decrease
as
much
as
possible.
In
addition,
the
OELD
problem
also
tak
es
man
y
constraints
into
account.
The
constraints
are
po
wer
balance,
spinning
reserv
e,
po
wer
output
limits
of
generators,
prohibited
operating
zones,
and
ramp
rate
limits.
Furthermore,
fuel
consuming
characteristics
of
TGU
such
as
multi
fuel
sources
and
v
alv
e
point
loading
ef
fects
are
also
considered
as
main
issues
in
the
OELD
problem.
The
OELD
problem
has
attracted
man
y
researchers
because
of
its
importance
in
using
fuel
for
the
TPPs
reasonably
.
T
w
o
main
method
groups
ha
v
e
used
by
man
y
authors
including
traditional
numerical
methods
and
modern
methods
based
on
artificial
intelligence.
J
ournal
homepage:
http://ijece
.iaescor
e
.com/inde
x.php/IJECE
Evaluation Warning : The document was created with Spire.PDF for Python.
1188
r
ISSN:
2088-8708
T
raditional
numerical
methods
ha
v
e
been
applied
to
handle
the
OELD
problem
such
as
Lagrangian
relaxation
(LR)
method
[1],
Linear
programming
techniques
(LPT)
[2],
F
ast
Ne
wton
raphson
(FNR)
method
[3].
Among
the
methods,
LR
is
one
of
the
earliest
methods
which
has
been
applied
to
systems
with
a
quadratic
fuel
cost
function.
The
constraint
s
of
the
problem
were
rather
simple
such
as
po
wer
balance
considering
trans-
mission
losses,
v
oltage
limitations,
and
generation
limits.
This
method
w
as
also
tested
on
the
10-unit
system
and
achie
v
ed
results
also
met
the
requirements
of
technology
at
that
time.
The
LPT
method
w
as
combination
of
the
Lagrangian
method
and
linear
program
(LP)
method
in
which
duty
of
the
LP
method
w
as
to
linearize
non-linear
functions
and
the
Lagrangian
method
w
as
used
as
usual.
Therefore,
when
there
were
man
y
non-
linear
constraints,
this
method
w
ould
f
ace
errors
due
to
linearization.
In
general,
all
of
the
traditional
numerical
methods
only
ha
v
e
considered
basic
constraints
of
the
OELD
problem.
Furthermore,
these
methods
had
to
tak
e
the
partial
deri
v
ati
v
e.
Consequently
,
traditional
numerical
methods
will
ha
v
e
some
restrictions
if
the
y
handle
the
OELD
problem
with
comple
x
constraints.
Unlik
e
traditional
numerical
methods
abo
v
e,
the
modern
methods
ha
v
e
been
proposed
to
handle
the
OELD
problem
more
successfully
including
ANN-based
methods
(ANN:
artificial
neural
netw
ork)
and
metaheuristic-based
methods.
ANN-based
methods
are
comprised
of
Hopfield
neural
netw
ork
(HNN)
[4],
Adapti
v
e
Hopfield
neural
netw
orks
(AHNN)
[5],
Augmented
Lagrange
Hopfield
netw
ork
(ALHN)
[6]
and
Enhanced
Augmented
Lagrange
Hopfield
netw
ork
(EALHN)
[7].
Metaheuristic-based
methods
ha
v
e
been
widely
and
more
successfully
handling
OELD
problem.
Dif
ferential
e
v
olution
algorithm
(DEA)
[8],
Quantum
Ev
olutionary
Algorithm
(QEA)
[9],
Hybrid
inte
ger
coded
dif
ferential
e
v
olution
–
dynamic
programming
(HICDE-DP)
[10],
Impro
v
ed
dif
ferential
e
v
olution
algorithm
(IDEA)
[11],
Colonial
competi-
ti
v
e
dif
ferential
e
v
olution
(CCDE)
[12],
Stud
dif
ferential
e
v
olution
(SDE)
[13]
and
Hybrid
dif
ferential
e
v
o-
lution
and
Lagrange
theory
(HDE-LH)
[14].
Real-coded
genetic
algorithm
(RCGA)
and
Impro
v
ed
RCGA
(IRCGA)
[17].
Hybrid
real
coded
genetic
algorithm
(HRCGA)
[15],
P
article
sw
arm
optimization
(PSO)
[16],
Modified
particle
sw
arm
optimization
(MPSO)
[18],
Quantum-inspired
particle
sw
arm
optimization
(QIPSO)
[19],
Distrib
uted
sobol
PSO
and
T
ab
u
Search
algorithm
(TSA)
(DSPSO-TSA)
[20],
Fuzzy
and
self
adapti
v
e
particle
sw
arm
optimization
(FSAPSO)
[21],
-P
article
sw
arm
optimization
(
-PSO)
[22],
Cuck
oo
search
algorithm
(CSA)
[23],
[24],
Impro
v
ed
CSA
(ICSA)
[25],
Modified
CSA
(MCSA)
[26],
Kill
herd
algorithm
(KHA)
[27],
Impro
v
ed
KHA
(IKHA)
[28],
Artificial
immune
systems
(AIS)
[29],
Biogeograph
y-
based
optim
ization
(BBO)
[30],
Chaotic
firefly
algorithm
(CF
A)
[31],
Gre
y
w
olf
optimizer
(GW
O)
[32],
[33],
Crisscross
optimization
(CO)
[34],
Exchange
mark
et
algorithm
(EMA)
[35],
ALO
[36],
[37],
Impro
v
ed
fire-
fly
algorithm
(IF
A)
[38],
Whale
Optimization
Algorithm
(W
O
A)
[39],
Cro
w
search
algorithm
(CrSA)
[40].
Among
the
DEA
method
and
its
dif
ferent
v
ersions,
SDE
[13]
is
the
best
method.
This
method
w
as
created
by
using
stud
crosso
v
er
operator
with
intent
to
restrict
the
search
around
lo
w
quality
solutions.
The
most
complicated
conditions
that
this
method
has
solv
ed
include
multi
fuel
sources
and
v
alv
e
point
loading
ef
fects.
The
results
sho
w
that
the
SDE
method
has
pro
vided
the
highest
quality
results
compared
to
all
other
methods
including
the
GA
and
TSA
methods
in
[20]
and
HDE-LH
in
[14];
ho
we
v
er
,
more
complicated
constraints
lik
e
ramp
rate,
spinning
reserv
e,
and
prohibited
operating
zones
were
not
tak
en
into
account
for
challenging
the
method.
The
traditional
GA
method
w
as
dif
ficult
to
solv
e
OELD
problem,
b
ut
its
v
ariants
ha
v
e
been
applied
to
this
problem
usefully
.
IRCGA
[17]
w
as
the
strongest
method
in
GA
f
amily
.
In
this
method,
an
ef
ficient
real-coded
genetic
algorithm
(RCGA)
with
arithmetic-a
v
erage-bound
crosso
v
er
and
w
a
v
elet
mutation
w
as
presented.
The
solv
ed
system
by
method
consists
of
10
TGUs
with
v
alv
e
point
loading
ef
fects,
multi
fuel
sources,
ramp
rate
limits,
prohibited
operating
zones,
and
spi
nning
reserv
e.
It
has
pro
v
e
n
to
be
the
m
ost
ef
fec-
ti
v
e
when
comparing
to
other
methods
including
PSO,
ne
w
PSO,
DEA,
an
impro
v
ed
genetic
algorithm
(IGA).
DSPSO–TSA
[20]
is
better
than
se
v
eral
methods
such
as
TSA,
GA,
PSO,
and
other
PSO
v
ariants
b
ut
it
w
as
only
considering
multi
fuel
sources
and
v
alv
e
point
loading
ef
fects
b
ut
po
wer
loss
constraints,
prohibited
oper
-
ating
zones
as
well
as
other
complicated
constraints
were
not
consist
of.
The
IF
A
method
in
[38]
seems
to
be
the
best
method
since
it
w
as
applied
for
solving
systems
with
the
most
complicated
constraints
including
both
constraints
considered
in
mentioned
w
ork
and
all
const
raints
in
transmission
po
wer
netw
orks.
All
t
est
cases
could
demonstrate
the
outstanding
performance
of
the
method.
As
a
ne
w
approach
method,
the
ALO
method
w
as
introduced
the
first
time
by
Mirjalili
in
2015
[41].
Unlik
e
the
other
algorithms,
ALO
has
tw
o
population
sets
are
an
Ant
colon
y
and
an
Antlion
colon
y
.
According
to
paper
[41],
the
ALO
method
has
handled
se
v
eral
mathematical
functions
and
some
engineering
problems
such
as
three
classical
engineering
problems
including
three-bar
truss
design,
cantile
v
er
beam
design,
and
gear
train
design.
The
ALO
method
also
has
been
proposed
for
handling
the
OELD
problem.
F
or
e
xample,
in
[36]
the
OELD
problem
has
handled
with
simple
constraints
Int
J
Elec
&
Comp
Eng,
V
ol.
10,
No.
2,
April
2020
:
1187
–
1199
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
1189
such
as
po
wer
balance
and
generator
limits
constraint,
or
in
[37]
the
method
has
tested
on
small-scale
po
wer
systems
considering
v
alv
e
point
ef
fects.
The
ALO
method
has
sho
wn
its
potential
search
including
se
v
eral
systems
of
the
OELD
problem
as
in
[36],
[37].
Ho
we
v
er
,
the
comple
x
le
v
el
of
considered
systems
w
as
not
lar
ge
a
n
d
complicated
enough
to
decide
its
performance.
So,
in
order
to
clarify
further
for
the
ef
ficienc
y
of
ALO
need
to
be
more
research.
In
this
paper
,
the
ALO
method
has
been
applied
for
handling
the
OELD
problem
with
the
most
con-
straints
and
dif
ferent
fuel
consuming
characteristics.
The
set
of
constraints
is
po
wer
balance,
spinning
reserv
e,
po
wer
output
limits
of
the
TGU,
prohibited
operating
zones
and
ramp
rate
limits.
Fuel
consuming
characteris-
tics
are
directly
related
to
objecti
v
e
functions
such
as
piece
wise
quadratic
functions
and
non-con
v
e
x
piece
wise
function.
The
method
has
been
tested
on
three
study
cases
and
obtained
results
ha
v
e
been
compared
to
other
methods
for
in
v
estig
ation
of
the
ALO
method.
2.
PR
OBLEM
FORMULA
TION
2.1.
Major
objecti
v
e
of
the
pr
oblem
In
operation
process
of
TPPs
using
fossil
fuels,
electricity
generation
cost
is
required
to
be
optim
ized.
It
can
be
mathematically
formulated
by:
F
=
n
X
s
=1
F
s
(
P
s
)
(1)
where
n
is
number
of
TGUs;
F
s
(
P
s
)
is
the
fuel
cost
function
of
the
s
th
TGU.
When
the
system
has
one
fuel
type,
the
fuel
cost
function
of
the
TGU
can
be
presented
in
a
s
ingle
quadratic
form
as
follo
ws:
F
s
(
P
s
)
=
s
P
2
s
+
s
P
s
+
s
(2)
where
s
,
s
,
s
are
the
cost
coef
ficients
of
the
s
th
TGU;
and
P
s
is
po
wer
output
of
the
s
th
TGU.
In
the
case
of
the
multi
fuel
sources,
the
fuel
cost
function
of
each
generator
should
be
represented
by
a
piece
wise
quadratic
function
as
sho
wn
in
(3).
Ho
we
v
er
,
with
the
case
of
v
alv
e
ef
fect
s,
the
cost
function
is
more
complicated
as
gi
v
en
in
(4):
F
s
(
P
s
)
=
8
>
>
>
<
>
>
>
:
s
1
P
2
s
+
s
1
P
s
+
s
1
;
fuel
1
s
2
P
2
s
+
s
2
P
s
+
s
2
;
fuel
2
:::
sm
P
2
s
+
sm
P
s
+
sm
;
fuel
m
(3)
F
s
(
P
s
)
=
8
>
>
>
<
>
>
>
:
s
1
P
2
s
+
s
1
P
s
+
s
1
+
j
s
1
x
sin
(
s
1
(
P
s;min
P
s
))
j
;
fuel
1
s
2
P
2
s
+
s
2
P
s
+
s
2
+
j
s
2
x
sin
(
s
2
(
P
s;min
P
s
))
j
;
fuel
2
:::
sm
P
2
s
+
sm
P
s
+
sm
+
j
sm
x
sin
(
sm
(
P
s;min
P
s
))
j
;
fuel
m
:
(4)
where
sm
,
sm
,
sm
,
sm
,
sm
are
the
cost
coef
ficients
of
the
s
th
TGU;
m
is
number
of
the
fuel
types;
and
P
s;min
is
the
minimum
po
wer
output
of
the
s
th
TGU.
2.2.
Constraints
of
po
wer
system
and
generator
2.2.1.
Real
po
wer
balance
The
total
po
wer
generation
should
meet
the
total
load
demand
P
demand
plus
transmission
losses
P
l
oss
as
the
follo
wing
rule:
n
X
s
=1
P
s
=
P
demand
+
P
l
oss
(5)
where
P
l
oss
is
calculated
by
using
Kron’
s
formula
belo
w:
P
l
oss
=
n
X
s
=1
n
X
h
=1
P
s
B
sh
P
h
+
n
X
s
=1
B
0
s
P
s
+
B
00
(6)
where
B
sh
,
B
0
s
,
and
B
00
are
loss
coef
ficients.
Antlion
optimization
algorithm
for
optimal
non-smooth...
(Thanh
Pham
V
an)
Evaluation Warning : The document was created with Spire.PDF for Python.
1190
r
ISSN:
2088-8708
2.2.2.
Spinning
r
eser
v
e
constraint
All
TGUs
are
required
that
total
acti
v
e
po
wer
reserv
e
of
them
should
be
more
than
or
equal
to
the
lar
gest
generating
unit.
The
constraint
requires
total
acti
v
e
po
wer
reserv
e
of
al
l
units
P
r
s
must
be
at
least
equal
to
the
po
wer
system
requirement
P
pr
.
n
X
s
=1
P
r
s
P
pr
(7)
2.2.3.
Generating
capacity
constraint
The
po
wer
output
of
each
generator
must
not
e
xceed
its
operating
limits
des
cribed
by
the
follo
wing
rule:
P
s;min
P
s
P
s;max
;
for
s
=
1,
2,
...
n
(8)
where
P
s;max
is
the
highest
acceptable
w
orking
po
wer
of
the
s
th
TGU.
2.2.4.
Ramp
rate
constraint
Because
each
TGU
cannot
change
its
po
wer
output
with
a
high
step
compared
to
its
pre
vious
genera-
tion.
Thus,
tw
o
major
conditions
are
added
as
the
follo
wing
inequalities:
P
s
P
0
s
P
r
u
s
for
the
case
of
increasing
po
wer
(9)
P
0
s
P
s
P
r
d
s
for
the
case
of
decreasing
po
wer
(10)
where
P
0
s
is
initial
po
wer
from
the
pre
vious
operating
hour
of
generating
unit;
P
r
u
s
and
P
r
d
s
are
ramp
up
limit
and
ramp
do
wn
limit
of
the
s
th
TGU.
2.2.5.
Pr
ohibited
operating
zones
Due
to
engineering
reason
that
generating
units
must
a
v
oid
operating
in
se
v
eral
operating
zones
as
sho
wn
in
the
follo
wing
model:
P
min
s;h
P
s
P
max
s;h
(11)
where
P
min
s;h
and
P
max
s;h
are
the
minimum
and
maximum
po
wer
output
of
the
s
th
TGU
in
the
h
th
prohibited
operating
zone.
3.
ANTLION
OPTIMIZA
TION
ALGORITHM
Initialization:
In
initialization
of
ALO
algorithm,
the
population
of
Antlions
is
randomly
produced
within
the
upper
and
lo
wer
limitations
as
the
follo
wing
model:
ALO
d
=
C
V
min
+
r
and
(
C
V
max
C
V
min
)
;
d
=
1,...,
N
pop
(12)
where
N
pop
is
the
population
size,
and
C
V
max
and
C
V
min
are
maximum
and
minimum
limitations
of
control
v
ariables.
Random
walk
of
Ant:
The
mo
v
ement
direction
of
Ants
does
not
follo
w
an
y
rules
and
it
is
also
a
random
w
alk
as
described
in
the
follo
wing
model:
R
W
C
I
=
[
0
;
1
X
C
I
=1
(2
C
I
1)
;
2
X
C
I
=1
(2
C
I
1)
;
3
X
C
I
=1
(2
C
I
1)
;
:
:
:
;
G
max
X
C
I
=1
(2
C
I
1)
]
(13)
where
CI
is
an
ordinal
number
of
the
current
iteration;
G
max
is
the
maximum
number
of
iterations;
C
I
is
considered
as
a
mo
ving
f
actor;
and
calculated
by:
C
I
=
(
1
if
0
:
5
0
otherwise
(14)
Int
J
Elec
&
Comp
Eng,
V
ol.
10,
No.
2,
April
2020
:
1187
–
1199
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
1191
The
r
estricted
space
of
Ant:
The
acti
v
e
radius
of
the
j
th
Ant
w
ould
be
more
and
more
decreased
adapti
v
ely
when
the
current
number
of
iterations
is
increased.
F
or
mathematically
modeling
this
beha
vior
,
the
follo
wing
equations
are
used:
c
j
;C
I
=
C
V
min
j
and
d
j
;C
I
=
C
V
max
j
(15)
where
c
j
;C
I
and
d
j
;C
I
are
do
wn
and
up
limitations
in
the
acti
v
e
range
of
the
j
th
Ant
at
iteration
CI;
and
is
a
f
actor
obtained
by:
=
10
C
I
G
max
(16)
where
is
a
constant
defined
based
on
the
current
iteration
CI
(
=
2
when
CI
>
0.1*
G
max
,
=
3
when
CI
>
0.5*
G
max
,
=
4
when
CI
>
0.75*
G
max
,
=
5
when
CI
>
0.9*
G
max
,
=
6
when
CI
>
0.95*
G
max
)
Sliding
Ant
to
ward
Antlion:
The
range
of
acti
vity
of
Ant
is
af
fected
by
beha
vior
of
shooting
sands
of
Antlion.
This
made
Ant
sliding
to
the
bottom
of
the
trap
where
the
massi
v
e
ja
w
w
as
w
aiting
to
catch
pre
y
.
T
o
describe
the
assumption,
the
tw
o
follo
wing
equations
are
necessary:
X
min
j
;C
I
=
ALO
j
;C
I
+
c
j
;C
I
and
X
max
j
;C
I
=
ALO
j
;C
I
+
d
j
;C
I
(17)
where
ALO
j
;C
I
is
the
position
of
the
j
th
Antlion
at
the
C
I
th
iteration;
X
max
j
;C
I
and
X
min
j
;C
I
are
corresponding
to
ne
wly
updated
upper
and
lo
wer
limits
of
control
v
ariables
included
in
the
position
of
Antlions.
The
mo
v
ement
e
v
ery
Ant:
The
mo
v
ement
of
ants
is
corresponding
to
the
determination
of
search
zones
since
random
w
alk
in
(13)
only
produces
an
updated
st
ep
size
that
is
not
related
to
ne
w
solutions.
The
random
w
alk
position
of
ant
can
be
updated
by
the
follo
wing
model:
X
j
;C
I
=
(
R
W
C
I
R
W
min
)
x
(
X
max
j
;C
I
X
min
j
;C
I
)
R
W
max
R
W
min
+
X
min
j
;C
I
(18)
where
R
W
max
and
R
W
min
are
the
minimum
and
maximum
v
alues
of
R
W
C
I
respecti
v
ely
.
As
a
result,
the
real
position
of
Ant
is
updated
by
using
tw
o
random
w
alks
around
X
j
;C
I
and
the
current
best
solution.
The
purpose
is
to
use
information
e
xchange
between
tw
o
other
positions.
The
real
position
is
obtained
by:
Ant
j
;C
I
=
X
R
W
j
;C
I
+
Gbest
R
W
C
I
2
(19)
where
X
R
W
j
;C
I
is
a
ne
w
solution
around
X
j
;C
I
by
using
random
w
alk;
Gbest
R
W
C
I
is
a
ne
w
solution
nearby
the
best
current
solution
Gbest
C
I
.
The
phase
of
catching
pr
ey:
In
the
process,
the
assumption
is
that
the
action
of
catching
pre
y
happens
when
Ants
goes
inside
sand.
The
follo
wing
equation
is
proposed
in
this
re
g
ard:
Antl
ion
j
;C
I
=
Ant
j
;C
I
if
F
F
(
Ant
j
;C
I
)
<
F
F
(
Antl
ion
j
;C
I
)
(20)
where
F
F
(
Ant
j
;C
I
)
and
F
F
(
Antl
ion
j
;C
I
)
are
the
fitness
function
of
Ant
j
;C
I
and
Antl
ion
j
;C
I
respecti
v
ely
.
All
steps
ALO
method
has
been
summarized
as
Figure
1.
4.
IMPLEMENT
A
TION
OF
THE
ALO
ALGORITHM
FOR
OELD
PR
OBLEMS
4.1.
V
ariables
of
each
indi
vidual
of
the
algorithm
The
position
of
each
Antlion
includes
all
control
v
ariables
and
is
i
nitialized
within
limits
as
the
fol-
lo
wing
model:
X
d
=
C
V
min
+
r
and
(
C
V
max
C
V
min
)
;
d
=
1
;
:::;
N
pop
(21)
where
C
V
min
=
f
P
1
;min
;
:::;
P
n
1
;min
g
and
C
V
max
=
f
P
1
;max
;
:::;
P
n
1
;max
g
(22)
As
a
result,
the
po
wer
output
P
n;d
is
obtained
by
equation
(23)
follo
wing:
P
n;d
=
P
demand
+
P
l
oss
n
1
X
i
=1
P
i;d
(23)
Antlion
optimization
algorithm
for
optimal
non-smooth...
(Thanh
Pham
V
an)
Evaluation Warning : The document was created with Spire.PDF for Python.
1192
r
ISSN:
2088-8708
Start
Initializing
the
population
of
Antlions
by
using
(12)
Calculating
fitness
function
for
all
Antlions
Determine
the
best
Antlions
and
set
CI
=
1
Determine
R
W
C
I
using
(13)
Calculate
restrict
space
for
Ants
using
(15)
Determine
lo
wer
and
uper
limits
for
position
of
Ants
using
(17)
Find
ne
w
position
of
Ants
using
(18)
and
(19)
Ev
aluate
such
ne
w
position
by
calculate
fitness
function
Compare
Antlion
and
ne
w
Ant
to
update
ne
w
Antlion
by
using
(20)
Determine
the
best
Antlion
CI
=
G
max
CI
=
CI
+
1
no
Stop
yes
Figure
1.
The
flo
wchart
of
ALO
algorithm
4.2.
Punishment
of
dependent
v
ariable
violations
In
process
of
optimization,
as
P
n;d
is
outside
upper
and
lo
wer
bounds,
it
is
penalized
and
determined
by:
C
ost
P
un
=
8
>
<
>
:
P
n;d
P
n;max
if
P
n;d
>
P
n;max
P
n;min
P
n;d
if
P
n;d
<
P
n;min
0
if
P
n;min
P
n;d
P
n;max
(24)
4.3.
Compatible
function
The
compatible
function
is
added
to
the
product
of
the
square
of
punishment
v
alue
and
a
punishment
f
actor
(
F
a
),
as
follo
wing
equation:
C
F
d
=
n
X
i
=1
F
i
(
P
i
)
+
F
a
(
C
ost
P
un
)
2
(25)
5.
NUMERICAL
RESUL
TS
The
ef
ficienc
y
proposed
method
is
judged
in
this
section.
The
ALO
algorithm
has
been
tested
on
the
10-unit
system
with
constraints
of
the
po
wer
net
w
ork
and
the
generators,
and
dif
ferent
fuel
consuming
characteristics
of
thermal
units.
The
detail
is
as
follo
ws:
(a)
Case
1:
10-unit
po
wer
system
using
multi
fuel
sources
and
without
v
alv
e
point
loading
ef
fects.
(b)
Case
2:
10-unit
po
wer
system
using
multi
fuel
sources
and
v
alv
e
point
loading
ef
fects.
Int
J
Elec
&
Comp
Eng,
V
ol.
10,
No.
2,
April
2020
:
1187
–
1199
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
1193
(c)
Case
3:
10-unit
po
wer
system
using
multi
fuel
sources,
v
alv
e
point
loading
ef
fects
and
complicated
constraints
such
as
spinning
reserv
e,
prohibited
operating
zones
and
ramp
rate
limits.
Data
which
are
used
for
the
three
cases
is
tak
en
from
[42].
In
addition,
The
ALO
algorithm
has
been
coded
in
Matlab
platform
and
personal
computer
with
processor
2.0
GHz
and
Ram
of
2.0
Gb
.
5.1.
In
v
estigating
impact
of
contr
ol
parameters
on
obtained
r
esults
Three
cases
abo
v
e
of
the
OELD
problem
ha
v
e
been
e
x
ecuted
by
the
ALO
algori
thm
to
i
n
v
est
ig
ate
the
impact
of
dif
ferent
v
alues
of
cont
rol
parameters
in
the
ef
fecti
v
eness,
rob
ustness,
and
stability
of
the
search
process
of
ALO.
P
arameters
ha
v
e
been
used
in
the
in
v
estig
ation
are
population
size
(
N
pop
)
and
the
number
of
iterations
(
G
max
).
5.1.1.
Case
1:
10-unit
po
wer
system
using
multi
fuel
sour
ces
and
without
v
alv
e
point
loading
effects
There
are
four
studied
subcases
with
four
load
cases
from
2,400
to
2,700
MW
with
a
change
of
100
MW
.
The
obtained
results
with
respect
to
the
minimum
fuel
cost
for
100
trial
runs
with
dif
ferent
cases
of
N
pop
and
G
max
ha
v
e
been
reported
in
T
able
1.
Experimentation
has
been
di
vided
into
tw
o
parts.
In
the
first
part,
the
population
size
has
been
k
ept
at
N
pop
=
10
and
the
number
of
iterations
has
been
changed
as
the
right
first
column
of
t
he
T
able
1.
While
the
population
size
has
been
k
ept
at
N
pop
=
20
and
the
number
of
iterations
as
the
T
able
1
in
the
second
part.
Observing
the
table
can
see
that
the
same
v
alue
of
N
pop
,
when
G
max
rises,
there
is
a
corresponding
decline
in
the
minimum
fuel
cos
t.
When
the
minimum
fuel
cost
equals
481.7226
$
=h
in
both
of
tw
o
parts
then
it
is
impossible
to
decrease
although
G
max
still
increases.
In
the
first
part
at
the
first
six
ro
ws
of
T
able
1,
the
best
c
osts
of
subcase
1.1,
subcase
1.3
and
subcase
1.4
are
respecti
v
ely
481.7426
$/h,
574.3808
$/h,
623.8092
$/h
corresponding
with
N
pop
=
10,
G
max
=
250.
P
articularly
,
subcase
1.2
get
the
best
cost
at
N
pop
=
10
and
G
max
=
200
with
v
alue
of
526.2388
$/h.
In
the
second
part
at
the
last
four
ro
ws
of
T
able
1
all
four
subcases
reach
the
best
c
o
s
t
at
G
max
=
150.
This
point
out
that
when
the
population
size
is
set
to
higher
v
alue,
the
number
of
iterations
can
be
set
to
lo
wer
v
alue
b
ut
the
best
optimal
solution
can
be
found.
T
able
1.
The
lo
west
cost
($/h)
obtained
from
100
runs
for
dif
ferent
v
alues
of
N
pop
and
G
max
Load
of
Load
of
Load
of
Load
of
N
pop
G
max
2,400
MW
2,500
MW
2,600
MW
2,700
MW
491.2418
533.6331
579.9704
632.5862
10
50
483.0403
526.5162
574.5738
624.9104
10
100
481.7637
526.2820
574.3815
623.8695
10
150
481.7424
526.2388
574.3852
623.8103
10
200
481.7226
526.2388
574.3808
623.8093
10
250
481.7226
526.2388
574.3808
623.8092
10
300
483.5568
526.8289
576.4240
629.4534
20
50
481.7424
526.2494
574.5813
623.8402
20
100
481.7226
526.2388
574.3808
623.8092
20
150
481.7226
526.2388
574.3808
623.8092
20
200
The
results
of
the
distrib
ution
of
the
fuel
costs
for
subcase
1.4
o
v
er
100
trials
are
sho
wn
in
Figure
2.
The
results
of
the
tests
sho
w
that
ALO
can
find
the
best
optimal
solution
for
dif
ferent
setting
of
control
parameters
and
the
de
viation
among
obtained
minimums
is
v
ery
small.
Thus,
ALO
is
stable
and
ef
fecti
v
e
for
the
first
case
with
four
dif
ferent
loads.
5.1.2.
Case
2:
10-unit
po
wer
system
using
multi
fuel
sour
ces
and
v
alv
e
effects
The
second
study
case
only
considers
the
load
demand
of
2,700
MW
.
Meantime,
N
pop
has
bee
n
k
ept
at
the
v
alue
of
20
b
ut
G
max
has
been
adj
u
s
ted
within
8
v
alues
from
50
to
400
with
a
small
change
of
50.
Numbers
yielded
from
the
test
including
minimum
cost,
a
v
erage
cost,
maximum
c
ost
are
presented
in
T
able
2.
As
sho
wn
in
T
able
2,
once
G
max
decrease,
the
fuel
cost
function
will
decrease
in
the
first
fi
v
e
ro
ws.
Ho
we
v
er
,
ro
w
7
indicates
the
fuel
cost
function
incre
ases
unintentionally
although
G
max
increases
equaling
300.
This
is
also
repeated
one
more
time
at
the
last
ro
w
.
The
o
v
ervie
w
on
T
able
2
point
out
that
the
best
cost
of
623.8709
$/h
is
obtained
at
G
max
=
350.
Meanwhile,
the
minimum
cost
at
G
max
=
400
is
higher
than
623.8709
$/h.
Clearly
,
this
phenomenon
has
been
caused
by
the
impact
of
v
alv
e
point
loading
ef
fects
on
the
stability
of
the
ALO
search
process.
The
most
of
the
fuel
costs
for
case
2
o
v
er
100
trials
ha
v
e
distrib
uted
nearby
the
minimum
Antlion
optimization
algorithm
for
optimal
non-smooth...
(Thanh
Pham
V
an)
Evaluation Warning : The document was created with Spire.PDF for Python.
1194
r
ISSN:
2088-8708
v
alue
as
sho
wn
in
Figure
2.
This
sho
ws
that
ALO
has
good
stability
for
solving
the
OELD
problems
on
the
10-unit
system
with
multi
fuel
sources
and
v
alv
e
point
loading
ef
fects.
T
able
2.
Result
obtained
by
ALO
for
case
2
with
dif
ferent
v
alues
of
control
v
ariables
Lo
west
cost
($/h)
A
v
erage
cost
($/h)
Highest
cost
($/h)
N
pop
G
max
629.9271
653.2830
721.8110
20
50
624.1303
638.9358
690.4222
20
100
624.0333
631.6475
653.1102
20
150
623.9216
628.4258
643.9007
20
200
623.8958
625.8331
643.8601
20
250
623.9309
626.1910
644.3901
20
300
623.8709
625.6935
636.3510
20
350
623.8878
625.2053
635.7175
20
400
5.1.3.
Case
3:
10-unit
po
wer
system
using
multi
fuel
sour
ces,
v
alv
e
effects
and
complicated
constraints
In
the
third
studied
case
,
input
parameters
and
obtained
results
are
presented
in
T
able
3.
As
observ
ed
from
T
able
3,
the
minimum
fuel
costs
obtained
by
ALO
can
drop
significantly
from
G
max
=
50
to
G
max
=
400
and
it
reaches
the
best
minimum
at
G
max
=
400
with
the
cost
of
624.3894
$/h.
Ho
we
v
er
,
the
minimum
cannot
be
reduced
since
setting
G
max
to
450
and
500,
corresponding
to
the
cost
of
624.3976
$/h
and
624.4035
$/h.
Clearly
,
the
phenomenon
is
similar
to
that
in
case
2
b
ut
totally
dif
ferent
from
4
s
u
bc
ases
in
case
1.
Ob
viously
,
v
alv
e
point
loading
ef
fects
and
complicated
constraints
ha
v
e
a
highly
significant
impact
on
the
stability
of
ALO.
Figure
2
is
sho
wn
the
fuel
costs
after
100
trials.
The
y
are
w
a
v
ered
between
tw
o
numbers
625
$/h
and
630
$/h.
T
able
3.
Result
obtained
by
ALO
for
case
3
with
dif
ferent
v
alues
of
control
v
ariables
Lo
west
cost
($/h)
A
v
erage
cost
($/h)
Highest
cost
($/h)
N
pop
G
max
625.0314
634.9521
658.0532
30
50
624.6915
628.9171
639.8262
30
100
624.5606
627.2422
637.3534
30
150
624.4409
627.0395
634.9398
30
200
624.4920
626.2745
632.8379
30
250
624.4626
626.3989
630.5901
30
300
624.4287
626.2675
630.5357
30
350
624.3894
625.9337
630.5276
30
400
624.3976
625.7483
630.2841
30
450
624.4035
625.6773
629.0156
30
500
Figure
2.
The
best
cost
of
100
trials
from
sub-case
1.4,
case
2
and
case
3
Int
J
Elec
&
Comp
Eng,
V
ol.
10,
No.
2,
April
2020
:
1187
–
1199
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
1195
5.2.
Result
comparison
obtained
by
ALO
and
other
methods
In
this
section
comparison
of
results
obtained
by
the
ALO
method
and
other
onces
has
been
performed.
In
order
to
in
v
estig
ate
the
real
performance
of
ALO
method,
an
important
f
actor
must
be
concerned
to
be
a
number
of
fitness
function
e
v
aluations
N
f
e
,
which
is
calculated
by:
N
f
e
=
!
N
pop
G
max
(26)
The
ALO
method
has
only
one
ne
w
solution
generation
for
each
iteration,
so
!
is
1
for
ALO.
F
or
other
methods
such
as
F
A
and
IF
A
in
[38],
N
f
e
has
another
model,
that
is:
N
f
e
=
N
pop
(
N
pop
+
1)
G
max
2
(27)
5.2.1.
Comparisons
of
T
est
Case
1
This
section
compares
fuel
cost
obtained
by
ALO
and
dif
ferent
methods
from
four
subcases
of
case
1.
The
best
cost
together
with
N
f
e
are
reported
in
T
able
4.
Numbers
from
T
able
4
point
out
that
the
proposed
method
can
harv
est
optimal
solutions
as
good
as
others.
While
ALHN
[6]
and
EALHN
[7]
are
not
metaheuristic
methods.
These
methods
will
f
ace
to
the
dif
fi
culties
of
applying
for
OELD
problem
with
non-con
v
e
x
fuel
cost
function
and
comple
x
constraints.
ALO
has
an
approximate
minimum
to
most
methods
and
has
better
cost
than
F
A
[38]
for
all
subcases
and
CF
A
[31]
for
subcase
1.4.
Ho
we
v
er
,
as
comparing
N
f
e
v
alues,
ALO
has
tak
en
N
f
e
=
2,500
for
seeking
and
reachi
ng
the
best
optimal
solutions
while
other
methods
ha
v
e
emplo
yed
high
v
alue
of
N
f
e
,
from
5,500
to
156,000.
Namely
,
N
f
e
w
as
respecti
v
ely
set
to
12,000,
30,000
and
156,000
for
DEA,
HRCGA,
and
CF
A.
In
summary
,
ALO
has
found
approximate
or
better
results
than
compared
methods
b
ut
it
has
o
wned
v
ery
f
ast
con
v
er
gence
speed
compared
to
other
ones.
In
s
ummary
,
the
applied
ALO
method
is
really
ef
fecti
v
e
for
case
1
with
discrete
objecti
v
e
function.
T
able
4.
The
lo
west
cost
($/h)
of
case
1
and
compared
methods
Load
(MW)
ALHN
[6]
EALHN
[7]
DEA
[8]
HICDE-
DP
[10]
HRCGA
[15]
CSA
[23]
AIS
[29]
CF
A
[31]
F
A
[38]
IF
A
[38]
ALO
2,400
481.723
481.723
481.723
481.7226
481.7226
-
481.723
-
505.2337
481.7226
481.7226
2,500
526.239
526.239
526.239
526.2388
526.2388
-
526.24
-
580.4417
526.2388
526.2388
2,600
574.381
574.381
574.381
574.3808
574.3808
-
574.381
-
639.287
574.3808
574.3808
2,700
623.809
623.809
623.809
623.809
623.8092
623.8092
623.8092
623.8339
679.9525
623.809
623.8093
N
f
e
-
-
12,000
8,000
30,000
6,000
4,000
156,000
11,000
5,500
2,500
5.2.2.
Comparisons
of
T
est
Case
2
The
section
is
carrying
our
comparison
of
results
from
the
applied
ALO
and
other
methods
for
case
2.
According
to
reported
data
in
T
able
5,
KHA
[27]
is
the
best
one;
ho
we
v
er
,
checking
optimal
solution
reported
in
[27]
sees
that
incorrect
type
of
fuel
w
as
reported
and
fuel
cost
is
much
higher
than
reported
v
alues.
Thus,
KHA
[27]
is
not
a
competitor
of
the
applied
ALO
method.
When
compared
to
accepted
methods
lik
e
GA,
TSA,
PSO
[20],
F
A
and
IF
A
[38],
the
proposed
method
is
better
with
respect
to
the
best
cost.
On
the
contrary
,
the
ALO
method
is
less
ef
fecti
v
e
than
remaining
methods
lik
e
in
DSPSO-TSA
[20],
CSA
[23],
ICSA
[25].
Ho
we
v
er
,
it
should
be
noted
that
CSA
and
ICSA
ha
v
e
used
N
f
e
equaling
10,000
and
12,000
while
that
of
the
applied
ALO
method
w
as
7,000.
Compared
to
GA,
TSA,
and
PSO
in
[20],
ALO
has
better
cost
b
ut
higher
N
f
e
since
those
from
ALO
are
623.8708
and
7,000
while
those
from
these
methods
are
higher
than
624.3045
and
3,000.
Ho
we
v
er
,
results
of
ALO
re
p
or
ted
in
T
able
2
sees
that
ALO
found
the
best
cost
of
624.1303
at
N
pop
=
20
and
G
max
=
100,
corresponding
to
N
f
e
=
2,000.
Thus,
ALO
could
find
better
optimal
solutions
with
f
aster
con
v
er
gence
than
GA,
TSA,
and
PSO
in
[20].
In
summary
,
ALO
has
yielded
better
results
b
ut
its
search
speed
has
been
f
aster
than
these
methods
while
other
ones
with
better
results
than
ALO
were
slo
wer
for
con
v
er
ging
to
the
best
optimal
solution.
In
other
w
ords,
ALO
is
a
promising
method
for
case
2
considering
with
10
units
taking
multi
fuel
sources
and
v
alv
e
point
loading
ef
fects
into
account.
Antlion
optimization
algorithm
for
optimal
non-smooth...
(Thanh
Pham
V
an)
Evaluation Warning : The document was created with Spire.PDF for Python.
1196
r
ISSN:
2088-8708
T
able
5.
The
comparison
of
results
for
case
2
Cost
($/h)
CCDE
[12]
GA
[20]
TSA
[20]
PSO
[20]
DSPSO-
TSA
[20]
CSA
[23]
ICSA
[25]
KHA
[27]
F
A
[38]
IF
A
[38]
ALO
Lo
west
623.8288
624.505
624.3078
624.3045
623.8375
623.8684
623.8684
605.7582
664.5306
623.8768
623.8708
A
v
erage
623.8574
624.7419
635.0623
624.5054
623.8625
623.9495
623.9495
605.8043
675.5344
625.2704
625.6935
Highest
623.8904
624.8169
624.8285
625.9252
623.9001
626.3666
626.3666
605.9426
679.426
629.2765
636.3510
N
f
e
7,000
3,000
3,000
3,000
3,000
10,000
12,000
10,000
11,000
5,500
7,000
5.2.3.
Comparisons
of
T
est
Case
3
The
section
presents
the
com
parison
of
fuel
cost
and
N
f
e
from
the
applied
ALO
and
other
methods
for
case
3.
The
results
obtained
for
case
3
are
gi
v
en
in
T
able
6.
According
to
the
results,
the
ALO
method
is
only
less
ef
fecti
v
e
than
IRCGA
[17]
while
it
is
better
than
all
other
methods;
ho
we
v
er
,
the
applied
ALO
method
has
the
most
ef
fecti
v
e
con
v
er
gence
speed
since
it
has
used
N
f
e
=
12,000
b
ut
that
from
other
w
as
much
higher
.
F
or
instance,
the
v
alue
is
33,000
for
IF
A
[38],
66,000
for
F
A
[38]
and
90,000
for
both
RCGA
and
IRCGA.
Clearly
,
ALO
can
be
f
aster
than
these
methods
approximately
from
3
times
to
8
times.
In
summary
,
ALO
has
found
a
better
optimal
solution
than
three
methods
b
ut
less
ef
fecti
v
e
optimal
solution
than
one
method.
Ho
we
v
er
ALO
is
f
aster
than
all
compared
methods
from
3
times
to
8
times.
Thus,
ALO
is
a
v
ery
ef
fecti
v
e
method
for
the
case,
which
has
multi
fuel
sources,
v
alv
e
point
loading
ef
fects
and
man
y
comple
x
constraints.
The
optimal
solution
obtained
by
the
ALO
algorithm
for
all
cases
ha
v
e
been
presented
in
Appendix.
T
able
6.
The
comparison
of
results
for
case
3
(2,700
MW)
Cost
($/h)
RCGA
[17]
IRCGA
[17]
F
A
[38]
IF
A
[38]
ALO
Lo
west
624.6605
624.355
673.5544
624.4950
624.3894
A
v
erage
625.9201
624.5792
685.2872
625.2647
625.6773
Highest
628.9253
624.7541
699.2855
629.3951
629.0156
N
f
e
90,000
90,000
66,000
33,000
12,000
In
this
section,
it
is
e
xplained
the
results
of
research
and
at
the
same
time
is
gi
v
en
the
comprehensi
v
e
discussion.
Results
can
be
presented
in
figures,
graphs,
tables
and
others
that
mak
e
the
reader
understand
easily
[2,
5].
The
discussion
can
be
made
in
se
v
eral
sub-chapters.
6.
CONCLUSION
In
this
article,
the
proposed
ALO
method
is
ef
fectually
implemented
to
handle
the
OELD
problem.
The
studied
system
has
10
TGUs
with
dif
ferent
types
of
fuel
consuming
characteristic
and
almost
comple
x
operation
constraints
of
the
po
wer
grid
practiced
in
three
tested
cases.
The
method
has
been
pro
v
ed
to
be
stable,
ef
fecti
v
e
and
rob
ust.
The
obtained
results
ha
v
e
been
compared
with
man
y
other
methods.
The
comparison
can
imply
that
the
ALO
method
is
better
than
most
other
methods
in
term
of
lo
wer
fuel
cost
and
smaller
number
of
fitness
e
v
aluations.
Ho
we
v
er
,
ALO
has
not
found
all
better
resul
ts
than
all
methods
for
study
cases,
especially
in
comparison
with
impro
v
ed
v
ersions
of
original
meta-heuristic
algorithms.
Thus,
it
can
conclude
that
ALO
method
can
be
selected
as
an
optimization
tool
for
dealing
with
OELD
problem
b
ut
it
needs
more
impro
v
ements
for
enhancing
optimal
solution
quality
and
con
v
er
ge
speed.
REFERENCES
[1]
Jonathan
F
.
Bard,
”Short-T
erm
Scheduli
n
g
of
Thermal-Electric
Generators
Using
Lagrangian
Relaxation,
”
IEEE
T
ransactions
on
Po
wer
Systems
,
v
ol.
36(5),
pp.
756-766,
1988.
[2]
A.
F
arag,
S.
Al-Baiyat
and
T
.
C.
Cheng,
”Economic
load
dispatch
multiobjecti
v
e
optimization
procedures
using
linear
programming
techniques,
”
IEEE
T
ransactions
on
Po
wer
Systems
,
v
ol.
10(2),
pp.
731-738,
May
1995.
[3]
Jiann-Fuh
Chen
and
Shin-Der
Chen,
”Multiobjecti
v
e
po
wer
dispatch
with
line
flo
w
constraints
using
the
f
ast
Ne
wton-Raphson
method,
”
IEEE
T
ransactions
on
Ener
gy
Con
v
ersion
,
v
ol.
12(1),
pp.
86-93,
March
1997.
Int
J
Elec
&
Comp
Eng,
V
ol.
10,
No.
2,
April
2020
:
1187
–
1199
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