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Applied
Soft
Computing
47
(2016)
494–514
Contents
lists
available
at
ScienceDirect
Applied
Soft
Computing
j
ourna
l
ho
me
page:
www.elsevier.com/locate
/asoc
A
modified
MOEA/D
approach
to
the
solution
of
multi-objective
optimal
power
flow
problem
Jingrui
Zhang
a,b,∗
,
Qinghui
Tang
a
,
Po
Li
a
,
Daxiang
Deng
a
,
Yalin
Chen
a
a
Department
of
Instrumental
&
Electrical
Engineering,
School
of
Aerospace
Engineering,
Xiamen
University,
Xiamen
361005,
China
b
Shenzhen
Research
Institute
of
Xiamen
University,
Shenzhen
518063,
China
a
r
t
i
c
l
e
i
n
f
o
Article
history:
Received
17
January
2016
Received
in
revised
form
3
May
2016
Accepted
17
June
2016
Available
online
23
June
2016
Keywords:
Multi-objective
optimization
Optimal
power
flow
MOEA/D
MOPSO
NSGA-II
a
b
s
t
r
a
c
t
This
study
presents
a
modified
multi-objective
evolutionary
algorithm
based
decomposition
(MOEA/D)
approach
to
solve
the
optimal
power
flow
(OPF)
problem
with
multiple
and
competing
objectives.
The
multi-objective
OPF
considers
the
total
fuel
cost,
the
emissions,
the
power
losses
and
the
voltage
magnitude
deviations
as
the
objective
functions.
In
the
proposed
MOEA/D,
a
modified
Tchebycheff
decom-
position
method
is
introduced
as
the
decomposition
approach
in
order
to
obtain
uniformly
distributed
Pareto-Optimal
solutions
on
each
objective
space.
In
addition,
an
efficiency
mixed
constraint
handling
mechanism
is
introduced
to
enhance
the
feasibility
of
the
final
Pareto
solutions
obtained.
The
mecha-
nism
employs
both
repair
strategy
and
penalty
function
to
handle
the
various
complex
constraints
of
the
MOOPF
problem.
Furthermore,
a
fuzzy
membership
approach
to
select
the
best
compromise
solu-
tion
from
the
obtained
Pareto-Optimal
solutions
is
also
integrated.
The
standard
IEEE
30-bus
test
system
with
seven
different
cases
is
considered
to
verify
the
performance
of
the
proposed
approach.
The
obtained
results
are
compared
with
those
in
the
literatures
and
the
comparisons
confirm
the
effectiveness
and
the
performance
of
the
proposed
algorithm.
©
2016
Elsevier
B.V.
All
rights
reserved.
1.
Introduction
The
optimal
power
flow
(OPF)
problem
plays
one
of
the
most
important
roles
in
the
operation
of
modern
power
systems
and
has
received
a
lot
of
interest
over
the
years
[1].
It
involves
the
dispatching
or
setting
for
all
the
generator
real
powers,
the
gen-
erator
bus
voltages,
the
tap
ratios
of
transformers
and
the
reactive
power
generations
of
VAR
sources.
Simultaneously,
the
generator
reactive
powers,
the
load
bus
voltages
and
the
power
flow
of
net-
work
lines
should
also
be
calculated
through
the
solving
of
OPF
problems.
The
objective
function
of
OPF
problem
was
usually
for-
mulated
as
minimizing
the
total
fuel
cost
in
past
years.
However,
voltage
instability
is
now
emerging
as
a
new
challenge
for
power
system
planning
and
operation
[2]
due
to
the
continuous
growth
in
the
demand
of
electricity
with
unmatched
generation
and
trans-
mission
capacity
expansion.
At
the
same
time,
insufficient
reactive
power
sources
produce
large
transmission
losses.
More
and
more
concerns
to
environmental
problem
make
it
necessary
to
consider
the
emission
as
one
of
the
objectives
instead
of
constraints.
In
such
∗
Corresponding
author
at:
Department
of
Instrumental
&
Electrical
Engineering,
School
of
Aerospace
Engineering,
Xiamen
University,
Xiamen
361005,
China.
E-mail
addresses:
zjrhust@gmail.com,
zjrhust@163.com
(J.
Zhang).
situations,
it
is
necessary
to
consider
emissions,
voltage
magnitude
deviations
and
transmission
losses
as
part
of
the
objective
func-
tions
of
the
OPF
problem.
Hence,
the
OPF
problem
has
become
a
multi-objective
optimization
problem.
Multi-objective
optimal
power
flow
(MOOPF)
is
a
large-scale
highly
constrained
non-linear
optimization
problem.
It
is
very
dif-
ficult
to
solve
the
problem
and
the
solving
of
it
involves
special
efficiency
solution
techniques.
In
past
decades,
many
heuristic
methods
have
been
proposed
aiming
at
solving
multi-objective
OPF
problems.
These
intelligent
solution
methods
are
usually
divided
into
two
kinds
according
to
whether
the
preference
information
of
a
decision-maker
is
obtained
before
the
solution.
If
the
prefer-
ence
information
can
be
obtained,
then
the
MOOPF
problem
can
be
transformed
into
an
optimization
problem
with
only
one
objective
function
through
weighted
sum
[3,4],
fuzzy
membership
functions
[5],
etc.
However,
this
type
of
solution
just
identifies
one
Pareto
solution
at
one
time.
If
the
preference
information
changes,
one
has
to
run
the
solution
approach
once
again
with
the
changed
prefer-
ence
information.
If
the
preference
information
of
a
decision-maker
can’t
be
obtained
before
the
solution,
one
had
to
address
the
whole
Pareto
solutions
directly.
Then
the
decision-maker
chooses
one
of
the
obtained
Pareto
candidates
as
the
most
preferred
solution
based
on
his/her
preference
information.
This
posterior
approach
which
searches
the
whole
Pareto
optimal
solutions
before
multi-criterion
http://dx.doi.org/10.1016/j.asoc.2016.06.022
1568-4946/©
2016
Elsevier
B.V.
All
rights
reserved.
![](https://csdnimg.cn/release/download_crawler_static/16111795/bg2.jpg)
J.
Zhang
et
al.
/
Applied
Soft
Computing
47
(2016)
494–514
495
decision
[6]
represents
the
main
trend
for
solving
multi-objective
optimizations.
This
type
of
method
tries
to
obtain
a
good
represen-
tation
of
the
Pareto
Front
(PF)
to
present
to
a
decision
maker.
In
recent
years,
the
posterior
multi-objective
evolutionary
algo-
rithms
(MOEAs)
have
been
well
investigated.
Many
novel
and
effective
MOEAs
have
been
proposed
constantly.
Among
these
methods,
NSGA-II
[7,8],
multi-objective
particle
swarm
optimiza-
tion
[9],
immune
algorithm
[10,11],
memetic
algorithm
[12]
are
popular
developed
and
applied
in
many
fields.
The
optimal
power
flow
problem
as
one
of
the
famous
multi-objective
optimizations
also
attracts
many
scholars.
Some
MOEAs
such
as
multi-objective
differential
evolutionary
[13–18],
artificial
bee
colony
algo-
rithm
[5,19–22],
multi-objective
adaptive
immune
algorithm
[10],
enhanced
genetic
algorithm
[23],
NSGA-II
[7],
multi-objective
PSO
[24,25],
quasi-oppositional
biogeography-based
optimization
[26],
multi-objective
harmony
search
algorithm
[27],
modi-
fied
shuffle
frog
leaping
algorithm
[28,29],
gravitational
search
algorithm
[1,30–34]
,
multi-objective
modified
imperialist
compet-
itive
algorithm
[35,36],
multi-hive
bee
foraging
algorithm
[37],
teaching-learning
based
optimization
algorithm
[2,38],
multi-
objective
solution
Q()
learning
[39],
etc.,
have
been
proposed
aiming
at
the
solution
of
MOOPF
problems.
However,
the
above
methods
have
to
do
some
more
efforts
in
order
to
approach
to
the
true
Pareto-Optimal
Front
and
obtain
the
diversity
of
the
solutions.
On
the
other
hand,
the
preference
information
of
a
decision-maker
is
neither
prior
nor
posterior
in
practical
multi-objective
optimiza-
tion
problems
(MOPs).
Reference
point
which
consists
of
aspiration
levels
reflecting
preference
values
for
the
objective
functions
is
one
of
the
most
important
ways
to
provide
preference
information.
Also,
it
is
a
natural
way
of
expressing
the
preference
information
as
a
tar-
get
the
decision-maker
is
hoping
to
obtain
[6].
For
this
type
of
preference
information
expression,
Zhang
and
Li
[40]
proposed
a
multi-objective
evolutionary
algorithm
based
on
decomposi-
tion
(MOEA/D)
approach
in
2007.
This
approach
decomposes
a
multi-objective
optimization
problem
(MOP)
into
a
number
of
scalar
single-objective
optimization
problems
using
uniformly
dis-
tributed
preference
directions
with
the
same
reference
point.
In
addition,
the
use
of
weight
design
methods
and
the
neighbor
infor-
mation
in
the
MOEA/D
approach
naturally
guarantee
the
diversity
of
the
obtained
solutions.
One
can
just
focus
on
how
to
approach
the
true
Pareto-Optimal
Front
within
the
approach.
MOEA/D
is
a
simple
yet
efficient
MOEA
and
has
been
ded-
icated
to
knapsack
problem
[41–44],
job
shop
scheduling
[45],
traveling
salesman
problem
[46–49],
test
task
scheduling
prob-
lem
[50],
antenna
array
synthesis
[49],
wireless
sensor
networks
[51],
portfolio
management
[52]
and
reservoir
flood
control
[6].
The
experimental
and
practical
results
all
show
perfect
performance
of
the
MOEA/D.
The
MOEA/D’s
successful
solution
to
the
famous
multi-objective
optimizations
above
will
attract
more
and
more
applications
in
engineering
and
scientific
areas.
The
aim
of
this
work
is
to
propose
a
modified
MOEA/D
approach
to
solve
the
MOOPF
problem.
To
our
best
knowledge,
this
may
be
the
first
try
of
MOEA/D
to
the
solution
of
MOOPF
problem.
In
order
to
obtain
uniformly
distributed
Pareto-Optimal
solutions
on
each
objective
space,
a
modified
Tchebycheff
decomposition
approach
instead
of
traditional
Tchebycheff
decomposition
one
is
introduced.
To
enhance
the
feasibility
of
the
final
Pareto-Optimal
solutions
obtained,
an
efficiency
mixed
constraint
handling
mech-
anism
is
employed.
The
mechanism
employs
both
repair
strategy
and
penalty
function
to
handle
the
various
complex
constraints
of
the
MOOPF
problem.
The
modified
MOEA/D
approach
also
inte-
grates
a
fuzzy
membership
approach
to
select
the
best
compromise
solution.
The
performance
of
comparison
experiments
with
the
popular
NSGA-II
and
multi-objective
particle
swarm
optimization
(MOPSO)
on
several
test
cases
shows
the
superiority
of
the
pro-
posed
modified
MOEA/D
approach.
The
rest
of
this
paper
is
organized
as
follows.
Section
2
formu-
lates
the
general
multi-objective
optimization
problem.
Section
3
presents
the
mathematical
formulation
of
the
multi-objective
OPF
problem.
Section
4
introduces
the
general
framework
of
the
mod-
ified
MOEA/D.
In
Section
5,
the
implementation
of
the
modified
MOEA/D
for
the
multi-objective
OPF
problem
is
present.
Simula-
tion
studies
and
comparisons
with
other
approaches
are
given
in
Section
6.
Finally,
conclusions
are
made
in
Section
7.
2.
Multi-objective
optimization
problem
In
general,
a
multi-objective
optimization
problem
consists
of
multiple
conflicting
objectives
to
be
optimized
simultaneously,
while
satisfying
a
range
of
equality
and
inequality
constraints
[30].
The
multi-objective
optimization
problems
have
been
usually
for-
mulated
as
follows.
minimizeF
=
(
f
1
(
x
)
,
f
2
(
x
)
,
·
·
·,
f
M
(
x
))
,
x
=
[
x
1
,
x
2
,
·
·
·,
x
n
]
subjecttog
j
(
x
)
≥
0,
j
=
1,
2,
·
·
·,
G
h
k
(
x
)
=
0,
k
=
1,
2,
·
·
·,
H
x
min
i
≤
x
i
≤
x
max
i
,
i
=
1,
2,
·
·
·,
n
where
f
i
(
x
)
represents
the
ith
objective
function
of
the
optimization
problem,
x
is
a
decision
vector
that
consists
of
n
decision
variables,
G
and
H
represent
the
numbers
of
inequality
constraints
and
equality
constraints,
respectively.
In
single
objective
optimizations,
the
optimal
solution
is
usually
unique,
while
the
optimal
solution
for
multi-objective
opti-
mizations
is
often
a
set
due
to
the
conflict
between
different
objective
functions.
It
is
necessary
to
understand
the
concepts
of
Pareto-Dominance,
Pareto-Optimal,
Pareto-Optimal
Set,
and
Pareto-Optimal
Front
in
order
to
solve
the
optimization
problem.
The
detailed
definitions
for
these
concepts
please
refer
to
[53].
Within
the
definitions,
a
multi-objective
optimization
problem
can
be
seen
as
the
method
of
looking
for
the
Pareto-Optimal
solutions
or
approaching
the
Pareto-Optimal
Front
(PF).
3.
Multi-objective
OPF
problem
The
optimal
power
flow
has
been
considered
as
one
of
the
well-known
multi-objective
optimization
problems.
It
aims
at
opti-
mizing
some
selected
objective
function
simultaneously
through
the
optimal
setting
of
control
variables
while
satisfying
various
constraints.
The
control
variables
usually
consist
of
the
real
power
outputs
from
all
the
generators,
the
voltage
magnitudes
from
all
the
generator
buses,
the
tap
ratios
of
transformers
and
the
reac-
tive
power
outputs
of
all
VAR
sources.
The
multi-objective
optimal
power
flow
problem
is
difficult
to
solve,
because
it
must
consider
various
complicated
operation
and
system
constraints.
Some
of
the
constraints
are
nonlinear
and
time
coupling.
It
is
very
difficult
to
satisfy
these
various
constraints.
Here
in
the
rest
of
this
section,
some
of
the
popular
objective
functions
and
constraints
considered
for
the
multi-objective
optimal
power
flow
problem
are
reformu-
lated.
3.1.
Objective
functions
3.1.1.
Minimization
of
total
fuel
cost
It
is
important
to
find
an
optimal
operation
scheme
to
generate
electricity
with
minimum
costs
with
the
rise
in
fuel
prices
and
the
increased
power
loads.
The
total
fuel
cost
($/h)
of
generating
units
![](https://csdnimg.cn/release/download_crawler_static/16111795/bg3.jpg)
496
J.
Zhang
et
al.
/
Applied
Soft
Computing
47
(2016)
494–514
considering
the
valve
loading
effects
can
be
expressed
as
follows
[5].
f
cost
=
N
g
i=1
a
i
+
b
i
P
gi
+
c
i
P
2
gi
+
|d
i
×
sin
e
i
×
P
min
gi
−
P
gi
|
(1)
where
a
i
,
b
i
,
c
i
,
d
i
and
e
i
are
the
cost
coefficients
of
the
ith
generator,
P
gi
is
the
real
power
output
of
thermal
unit
i,
and
P
min
gi
represents
its
minimum
value.
3.1.2.
Minimization
of
total
active
power
losses
The
series
resistance
and
the
shunt
conductance
of
transmission
lines
will
produce
active
power
losses
and
more
power
losses
will
increase
the
generated
power
costs
[20].
Here
the
total
active
power
losses
are
considered
as
one
of
the
objective
functions,
and
it
is
expressed
as
follows.
f
loss
=
N
L
k=1
G
k
V
2
i
+
V
2
j
−
2V
i
V
j
cos
ij
(2)
where
N
L
is
the
total
branch
of
the
power
systems
considered,
G
k
is
the
conductance
of
the
kth
branch,
V
i
and
V
j
are
the
voltage
magni-
tudes
of
terminal
buses
of
branch
k,
and
ij
is
phase
angles
difference
between
them.
3.1.3.
Minimization
of
total
emission
More
and
more
countries
or
regions
are
concerned
about
envi-
ronmental
protection
due
to
the
increasingly
serious
problem
of
air
pollutions.
It
is
necessary
to
reduce
the
emissions
of
atmospheric
pollutants
caused
by
the
thermal
generation
units
[20].
The
total
emission
(ton/h)
of
the
atmospheric
pollutants
such
as
SO
x
and
NO
x
caused
by
fossil-fueled
thermal
generation
facilities
can
be
expressed
as:
f
emission
=
N
g
i=1
˛
i
+
ˇ
i
P
gi
+
i
P
2
gi
+
i
e
i
P
gi
(3)
where
˛
i
,
ˇ
i
,
i
,
i
and
i
are
the
emission
constant
coefficients
of
the
ith
unit.
3.1.4.
Minimization
of
total
voltage
magnitude
deviations
Bus
voltage
is
one
of
the
most
important
indices
for
secure
oper-
ation
and
voltage
quality.
The
objectives
without
voltage
indices
may
result
in
a
feasible
solution
that
has
an
unattractive
voltage
profile
[20].
The
objective
function
considered
here
is
to
minimize
the
voltage
magnitude
deviations
(VMD)
of
all
the
load
bus
from
1
per
unit
(p.u.)
[1].
The
objective
function
can
be
defined
as
follows:
f
VMD
=
Npq
i=1
|
V
i
−
V
reference
|
(4)
where
f
VMD
is
the
total
voltage
magnitude
deviations;
N
pq
is
the
total
number
of
PQ
bus
and
V
reference
is
the
reference
voltage
magnitude
in
p.u.,
here
V
reference
=
1.0.
3.2.
Constraints
The
MOOPF
problem
is
a
non-linear
optimization
problem
which
determines
the
optimal
control
variables
for
minimizing
the
some
objectives
subject
to
several
equality
and
inequality
con-
straints
[27].
The
selected
equality
and
inequality
constraints
are
formulated
as
follows.
3.2.1.
Equality
constraints
The
equality
constraints
of
MOOPF
problem
are
typical
load
flow
equation
and
for
each
bus
i,
they
can
be
described
as
follows.
0
=
P
G
i
−
P
D
i
−
V
i
N
b
j=1
V
j
G
ij
cos
ij
+
B
ij
sin
ij
i
=
1,
2,
.
.
.,
N
b
(5)
0
=
Q
G
i
−
Q
D
i
−
V
i
N
b
j=1
V
j
G
ij
sin
ij
−
B
ij
cos
ij
i
=
1,
2,
.
.
.,
N
b
(6)
where
P
G
i
andQ
G
i
are
active
and
reactive
power
injection
at
bus
i,
P
Di
andQ
Di
are
active
and
reactive
power
demand
at
bus
i
and
N
b
is
the
total
bus
of
the
power
system.
3.2.2.
Inequality
constraints
The
MOOPF
is
optimized
subject
to
the
following
inequality
con-
straints
of
the
power
systems.
(1)
Generator
constraints:
generator
active
power
output
P
g
,
generator
reactive
power
output
Q
g
,
and
generator
voltage
magnitude
V
g
are
restricted
by
their
lower
and
upper
limits
as
follows.
P
min
gi
≤
P
gi
≤
P
max
gi
i
=
1,
2,
·
·
·,
N
g
Q
min
gi
≤
Q
gi
≤
Q
max
gi
i
=
1,
2,
·
·
·,
N
g
V
min
gi
≤
V
gi
≤
V
max
gi
i
=
1,
2,
·
·
·,
N
g
(2)
Transformer
constraints:
transformer
taps
are
restricted
by
their
minimum
and
maximum
setting
limits.
min
i
≤
i
≤
max
i
i
=
1,
2,
·
·
·,
N
T
where
N
T
is
the
number
of
on-load
voltage
regulating
transformer
in
the
power
systems.
(3)
Switchable
VAR
sources:
the
switchable
VAR
sources
have
restrictions
as
follows.
Q
min
ci
≤
Q
ci
≤
Q
max
ci
i
=
1,
2,
·
·
·,
N
C
where
N
c
is
the
number
of
switchable
VAR
sources.
(4)
Security
constraints:
these
include
the
limits
of
voltage
magni-
tudes
on
load
bus
and
the
limits
of
transmission
line
flow.
V
min
Li
≤
V
Li
≤
V
max
Li
i
=
1,
2,
·
·
·,
N
pq
where
N
pq
is
the
number
of
load
bus.
S
Li
≤
S
max
Li
i
=
1,
2,
·
·
·,
N
L
where
N
L
is
the
number
of
transmission
lines.
4.
General
framework
of
MOEA/D
In
this
section,
the
general
idea
of
MOEA/D
algorithm
used
for
addressing
the
multi-objective
optimization
problem
is
described.
As
indicated
in
[40],
the
objectives
of
many
practical
MOPs
are
often
contradicting
with
each
other,
usually
no
point
in
feasible
space
can
minimize
all
the
objectives
simultaneously.
Actually,
addressing
all
the
non-dominated
or
non-inferior
solutions
has
been
becom-
ing
the
purpose
of
the
multi-objective
optimization
problem.
In
order
to
find
the
non-dominated
solutions,
MOEA/D
decomposes
a
multi-objective
optimization
problem
into
a
set
of
single
objec-
tive
optimization
sub-problems
using
the
weighted
sum
approach,
the
Tchebycheff
approach
and
the
boundary
intersection
approach
[54]
.
These
sub-problems
are
then
optimized
concurrently
and
collaboratively
by
evolving
population
of
solutions
using
an
evo-
lutionary
algorithm
(EA).
The
sub-problem
optimization
uses
the
information,
mainly
from
its
neighboring
sub-problem
while
the
neighborhood
relations
among
these
sub-problems
are
defined
based
on
the
distances
between
their
aggregation
coefficient
vec-
tors.
![](https://csdnimg.cn/release/download_crawler_static/16111795/bg4.jpg)
J.
Zhang
et
al.
/
Applied
Soft
Computing
47
(2016)
494–514
497
The
approximation
of
the
Parent-Optimal
Front
(PF)
of
a
MOP
can
be
decomposed
into
N
scalar
optimization
sub-problems
in
MOEA/D.
As
described
in
Ref.
[40],
any
decomposition
approach
can
serve
for
converting
approximation
of
the
PF
of
MOP
into
a
number
of
single
objective
optimization
problems.
Here
the
Tchebycheff
approach
is
employed
to
describe
the
main
framework
of
MOEA/D
for
the
solution
of
MOPs.
The
jth
single
objective
optimization
sub-
problem
is
g
te
x|
j
,
z
∗
=
max
1≤i≤m
j
i
|f
i
(
x
)
−
z
∗
i
|
(7)
where
j
=
j
1
,
j
2
,
·
·
·,
j
m
T
.
MOEA/D
attempts
to
optimize
these
N
scalar
optimization
sub-problems
simultaneously
instead
of
solv-
ing
MOP
problem
directly
in
a
single
run.
In
Zhang’s
approach,
the
neighborhood
of
j
is
constituted
of
the
T
closest
weight
vectors
in
1
,
2
,
·
·
·,
N
,
and
the
neighborhood
of
the
jth
subproblem
consists
of
all
the
sub-problems
with
the
weight
vectors
from
the
neighborhood
of
j
.
The
population
is
composed
of
the
best
solu-
tions
found
so
far
for
each
sub-problem.
Only
the
current
solutions
to
its
neighboring
sub-problems
are
exploited
for
optimizing
a
sub-
problem
in
MOEA/D.
The
general
main
framework
of
MOEA/D
is
described
as
Fig.
1
shown.
5.
Modified
MOEA/D
approach
to
MOOPF
problem
The
modified
MOEA/D
approach
to
the
solution
of
the
MOOPF
problem
is
discussed
and
implemented
in
this
section.
The
modified
Tchebycheff
decomposition
approach,
mixed
constraints
handling
mechanism,
and
the
fuzzy
selection
method
of
the
best
compro-
mise
solution
are
first
introduced
and
then
the
detailed
steps
to
the
solution
of
MOOPF
problem
are
present.
5.1.
Modified
tchebycheff
decomposition
approach
As
we
know,
the
measurement
units
and
the
value
ranges
of
objective
functions
for
the
OPF
problem
are
different.
This
condi-
tion
will
make
each
objective
function
has
different
weightiness
and
this
will
result
in
converging
to
some
certain
region.
Zhang
Fig.
1.
The
general
main
framework
of
MOEA/D.
and
Li
[40]
have
proposed
the
Tchebycheff
approach
to
decrease
the
effects
of
different
unit
measurements
and
value
ranges,
but
this
is
not
enough.
Here,
we
introduce
a
modified
Tchebycheff
approach
to
overcome
the
above
lacks
for
the
solution
of
MOOPF
problem.
Table
1
Parameters
setting
for
different
cases.
Method
Parameters
Case
1
Case
2
Case
3
Case
4
Case
5
Case
6
Case
7
MOEA/D
Population
Size
100
100
100
100
300
300
300
Niche
Size
20
20
20
20
20
20
20
Maximum
Iteration
500
500
500
500
500
500
500
F
0.5
0.5
0.5
0.5
0.5
0.5
0.5
Crossover
Probability
0.5
0.5
0.5
0.5
0.5
0.5
0.5
NSGA-
II
Population
Size
–
100
100
100
300
300
300
Maximum
Iteration
–
500
500
500
500
500
500
Mutation
Distribution
Index
–
20
20
20
20
20
20
Crossover
Distribution
Index
–
20
20
20
20
20
20
MOPSO
Population
Size
–
100
100
100
300
300
300
Repository
Size
–
100
100
100
300
300
300
Maximum
Iteration
–
500
500
500
500
500
500
Grid
Inflation
Parameter
–
0.1
0.1
0.1
0.1
0.1
0.1
Number
of
Grids
per
Dimension
–
10
10
10
10
10
10
Table
2
Obtained
results
when
optimizing
each
single
objective
optimization
independently.
Optimizing
Objective Fuel
Cost
($/h)
VMD
(p.u.)
Power
Loss
(MW)
Emission
(ton/h)
Min
Fuel
Cost
799.0263
1.7679
8.6163
0.3663
Min
VMD
849.9159
0.0936
7.0740
0.2835
Min
Power
Loss
967.0704
2.0298
2.8518
0.2072
Min
Emission
943.5519
2.0196
2.9800
0.2047
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