Inter
national
J
our
nal
of
Electrical
and
Computer
Engineering
(IJECE)
V
ol.
12,
No.
1,
February
2022,
pp.
82
91
ISSN:
2088-8708,
DOI:
10.11591/ijece.v12i1.pp82-91
r
82
P
arametric
estimation
in
photo
v
oltaic
modules
using
the
cr
o
w
sear
ch
algorithm
Oscar
Danilo
Montoya
1,2
,
Carlos
Alberto
Ram
´
ır
ez-V
anegas
3
,
Luis
F
er
nando
Grisales-Nor
e
˜
na
4
1
F
acultad
de
Ingenier
´
ıa,
Uni
v
ersidad
Distrital
Francisco
Jos
´
e
de
Caldas,
Bogot
´
a
D.C.,
Colombia
2
Laboratorio
Inteligente
de
Ener
g
´
ıa,
Uni
v
ersidad
T
ecnol
´
ogica
de
Bol
´
ıv
ar
,
Cartagena,
Colombia
3
F
acultad
de
Ciencias
B
´
asicas,
Uni
v
ersidad
T
ecnol
´
ogica
de
Pereira,
Pereira,
Colombia
4
Departamento
de
Electromec
´
anica
y
Mecratr
´
onica,
Instituto
T
ecnol
´
ogico
Metropolitano,
Medell
´
ın,
Colombia
Article
Inf
o
Article
history:
Recei
v
ed
Feb
1,
2021
Re
vised
May
20,
2021
Accepted
Jun
16,
2021
K
eyw
ords:
Cro
w
search
algorithm
Manuf
acturer
information
Metaheuristic
optimization
P
arametric
estimation
Photo
v
oltaic
modules
Single-diode
model
ABSTRA
CT
The
problem
of
parametric
estimation
in
photo
v
oltaic
(PV)
modules
considering
man-
uf
acturer
information
is
addressed
in
this
research
from
the
perspecti
v
e
of
combinato-
rial
optimization.
W
ith
the
data
sheet
pro
vided
by
the
PV
manuf
acture
r
,
a
non-linear
non-con
v
e
x
optimization
problem
is
formulated
tha
t
contains
information
re
g
arding
maximum
po
wer
,
open-ci
rcuit,
and
short-circuit
points.
T
o
estimate
the
three
param-
eters
of
the
PV
model
(i.e.,
the
ideality
diode
f
actor
(
a
)
and
the
parallel
and
series
resistances
(
R
p
and
R
s
)),
the
cro
w
search
algorithm
(CSA
)
is
emplo
yed,
which
is
a
metaheuristic
optimi
zation
technique
inspired
by
the
beha
vior
of
the
cro
ws
searching
food
deposits.
The
CSA
allo
ws
the
e
xploration
and
e
xploitation
of
the
solution
space
through
a
simple
e
v
olution
rule
deri
v
ed
from
the
classical
PSO
method.
Numerical
simulations
re
v
eal
the
ef
fecti
v
eness
and
rob
ustness
of
the
CSA
to
est
imate
these
pa-
rameters
with
objecti
v
e
function
v
alues
lo
wer
t
han
1
10
28
and
processing
times
less
than
2
s.
All
the
numerical
simulations
were
de
v
eloped
in
MA
TLAB
2020
a
and
compared
with
the
sine-cosine
and
v
orte
x
search
algorithms
recently
reported
in
the
literature.
This
is
an
open
access
article
under
the
CC
BY
-SA
license
.
Corresponding
A
uthor:
Oscar
Danilo
Monto
ya
F
acultad
de
Ingenier
´
ıa,
Pro
yecto
Curricular
de
Ingenier
´
ıa
El
´
ectrica,
Uni
v
ersidad
Distrital
Francisco
Jos
´
e
de
Caldas
Cra
7
#
40B-53,
Bogot
´
a
D.C.,
Colombia
Email:
odmonto
yag@udistrital.edu.co
1.
INTR
ODUCTION
The
presence
of
photo
v
oltaic
(PV)
sources
has
increased
rapidly
in
the
past
tw
o
decades
in
lo
w
,
medium
and
high-v
oltage
le
v
els,
and
their
accelerated
de
v
elopment
has
decreased
their
production,
mainte-
nance,
and
operati
v
e
costs
[1]-[3].
Moreo
v
er
,
these
rene
w
able
ener
gy
resources
h
a
v
e
reduced
the
ener
gy
pur
-
chase
costs
in
urban
areas
and
greenhouse
g
as
emiss
ions
in
rural
netw
orks
po
wered
by
diesel
generators
[4].
The
inte
gration
of
these
PV
sources
into
electrical
grids
generally
requires
the
po
wer
electronic
con
v
erters
to
manage
their
ener
gy
production
to
maximize
the
producer’
s
profit
[5].
This
ener
gy
management
is
achie
v
ed
through
linear
and
non-linear
control
strate
gies
applied
to
find
and
maintain
the
operation
of
the
PV
module
in
the
maximum
po
wer
point
(MPP)
[6]-[8].
In
the
literature,
dif
ferent
models
are
utilized
to
represent
the
PV
modules,
which
are
composed
of
one,
tw
o,
or
three
diodes.
Each
one
of
them
has
multiple
parameters
that
ha
v
e
to
be
found
prior
to
determining
the
panel
beha
vior
re
g
arding
current,
v
oltage,
and
po
wer
outputs
[2],
[9].
The
most
accepted
model
to
represent
the
PV
module
is
the
single-diode
representation.
In
this
model,
we
need
J
ournal
homepage:
http://ijece
.iaescor
e
.com
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
83
to
find
three
parameters
of
the
PV
module
that
are
associated
with
the
ideality
diode
f
actor
and
the
series
and
parallel
resistances
[10].
T
o
find
these
parameters,
in
the
current
literat
ure,
se
v
eral
optimization
techniques
are
a
v
ailable
that
use
the
data
sheet
information
pro
vided
by
the
panel
manuf
acturer
,
where
three
main
operati
v
e
points
can
be
highlighted:
i)
open-circuit
point,
ii)
short-circuit
point,
and
iii)
MPP
.
These
points
formulate
a
non-linear
non-con
v
e
x
optimization
problem
to
determine
the
best
combination
of
the
model
parameters
to
represent
the
complete
electrical
beha
vior
of
the
PV
module
[11],
[12].
In
the
literature,
methods
such
as
the
adapti
v
e
dif
ferential
e
v
olution
algorithm
[5],
the
sine-cosi
ne
algorithm
[10],
the
analytical
method
[13],
the
genetic
algorithm
[14],
the
cuck
oo
search
algorithm
[15],
the
least
square
method
[16],
the
electrom
agnetic-lik
e
algori
thm
[17],
and
the
grasshopper
optimization
algorithm
[18],
[19],
among
others,
can
be
found.
It
should
be
noted
that
this
re
vision
demonstrates
that
the
non-linear
,
non-con
v
e
x
nature
of
the
parametric
estimation
problem
mak
es
the
appli
cation
of
po
werful
optimization
tech-
niques
necessary
to
find
the
optimal
parameters
that
ensure
the
correct
operation
of
the
PV
modules’
equi
v
alent
electric
circuit.
Based
on
this
state-of-the-art
re
vision,
we
propose
a
ne
w
optimization
algorithm
to
estimate
the
elec-
trical
parameters
of
the
PV
module
using
the
single-diode
model
representation.
The
proposed
algorithm
corresponds
to
the
cro
w
search
algorithm
(CSA),
which
has
not
been
pre
viously
applied
to
this
problem
using
the
PV
data
sheet
with
the
main
adv
antage
that
only
four
parameters
can
be
tuned.
Moreo
v
er
,
numerical
results
demonstrate
the
v
alues
of
objecti
v
e
functions
that
are
lo
wer
than
1
10
28
,
which
are
clearly
better
than
the
results
reported
in
[5]
and
[10],
where
the
v
alues
of
objecti
v
e
functions
were
1
10
12
and
1
10
15
,
respec-
ti
v
ely
.
An
additional
adv
antage
of
the
proposed
approach
is
that
it
reaches
the
optimal
solution
in
less
than
2
s
by
ensuring
optimal
funding
through
the
non-parametric
W
ilcoxon
test.
It
is
important
to
mention
that
the
CSA
has
pre
viously
been
reported
in
[2]
to
determine
the
parameters
of
the
PV
modules.
Ho
we
v
er
,
the
authors
of
[2]
focus
their
study
on
an
optimization
model
that
tak
es
into
account
only
the
po
wer
tracking
error
,
considering
v
ariations
in
the
temperature
and
irradiance
inputs.
It
dif
fers
from
our
proposal
since
we
are
w
orking
directly
with
the
manuf
acturer
nameplate
in
which
three
operati
v
e
points
are
considered
to
determine
the
general
model
of
the
PV
system,
which
correspond
to
short-circuit
point,
open-circuit
point,
and
MPP
.
The
remainder
of
this
paper
is
or
g
anized
as
follo
ws.
Section
2
presents
the
formulation
of
the
para-
metric
estimation
problem
in
PV
modules
considering
the
data
sheet
information
pro
vided
by
the
manuf
acturer
.
Section
3
presents
the
general
description
of
the
proposed
CSA.
Section
4
presents
the
primary
characteristics
of
the
test
system
and
the
computational
v
alidation
features.
Section
5
sho
ws
the
numerical
v
alidation
of
the
proposed
methodology
and
its
analysis
and
discussion.
Finally
,
section
6
presents
the
main
concluding
remarks
deri
v
ed
from
this
research
as
well
as
some
possible
future
w
orks.
2.
OPTIMIZA
TION
MODEL
The
nomenclature
of
the
mathematical
optimization
model
presented
for
the
parametric
estimation
in
PV
modules
using
a
single-diode
model
has
been
listed
belo
w:
V
ariables
and
parameters
I
pv
:
Define
the
photoelectric
current
a
:
Ideality
f
actor
of
the
diode
I
0
:
In
v
erse
saturation
current
in
the
PV
module
N
c
:
Number
of
PV
cells
connected
in
series
I
mpp
:
Maximum
po
wer
point
current
a
min
:
Lo
wer
bound
of
the
ideality
diode
f
actor
I
sc
:
Short-circuit
current
a
max
:
Upper
bound
of
the
ideality
diode
f
actor
V
mpp
:
Maximum
po
wer
point
v
oltage
R
min
p
:
Lo
wer
bound
of
the
parallel
resistance
V
oc
:
Open-circuit
v
oltage
R
max
p
:
Upper
bound
of
the
parallel
resistance
R
s
:
Series
equi
v
alent
resistance
R
min
s
:
Lo
wer
bound
of
the
series
resistance
R
p
:
P
arallel
equi
v
alent
resistance
R
max
s
:
Upper
bound
of
the
series
resistance
q
:
Electron
char
ge
(e.g.,
1
:
60217646
10
19
Coulomb)
T
:
Absolute
temperature
in
the
diode
union
(
273
:
15
+
25
K
elvin)
k
:
Boltzmann
constant
(
1
:
38064852
10
23
Joules/K
elvin)
The
parametric
estimation
problem
in
PV
modules
considering
N
c
cells
connected
in
series
is
de-
v
eloped
based
on
their
ideal
model
using
a
single-diode
representation
[10].
The
schematic
modeling
of
the
P
ar
ametric
estimation
in
photo
voltaic
modules
using
the
cr
ow
sear
c
h
algorithm
(Oscar
Danilo
Montoya)
Evaluation Warning : The document was created with Spire.PDF for Python.
84
r
ISSN:
2088-8708
single-diode
model,
that
is,
the
electrical
circuit
equi
v
alent,
has
been
presented
in
Figure
1.
Figure
1.
Equi
v
alent
circuit
of
a
photo
v
oltaic
module
T
o
analyze
PV
modules
with
the
electrical
circuit
equi
v
alent
depicted
in
Figure
1,
the
e
xponential
relation
is
used
to
define
the
output
v
oltage
and
current
as
in
(1)
[20],
[21].
I
=
I
pv
I
0
exp
q
V
+
R
s
I
ak
N
c
T
1
V
+
R
s
I
R
p
:
(1)
W
ith
the
objecti
v
e
to
determine
all
the
parameters
of
the
PV
module,
that
is,
the
series
and
parallel
resistances
and
the
diode
ideality
f
actor
,
the
three
main
operati
v
e
points
pro
vided
by
the
module
manuf
acturer
ha
v
e
been
considered.
These
operati
v
e
points
include:
i)
open-circuit
point,
ii
)
short-circuit
point,
and
iii
)
MPP
.
The
analysis
of
each
one
of
these
points
has
been
presented
belo
w
.
2.1.
Open-cir
cuit
operati
v
e
point
The
open-circuit
operati
v
e
point
of
the
PV
module
presented
in
Figure
1
implies
that
the
v
oltage
in
its
terminals
is
V
=
V
oc
with
a
null
current
flo
w
through
them,
that
is,
I
=
0
.
W
ith
this
operati
v
e
condition,
it
is
possible
to
obtain
an
e
xpression
for
I
pv
from
(1)
as
(2):
I
pv
=
I
0
exp
q
V
oc
ak
N
c
T
1
V
oc
R
p
:
(2)
2.2.
Short-cir
cuit
operati
v
e
point
The
second
operati
v
e
point
pro
vided
by
the
PV
m
odu
l
e
manuf
acturer
corresponds
to
the
s
hort-circuit
scenario
at
the
terminals
of
the
module,
which
implies
that
I
=
I
sc
and
V
=
0
.
W
ith
these
operati
v
e
conditions,
in
(1)
assumes
in
(2):
I
sc
=
I
pv
I
0
exp
q
R
s
I
sc
ak
N
c
T
1
R
s
I
sc
R
p
:
(3)
No
w
,
if
we
combine
(2)
and
(3),
and
some
algebraic
manipulations
are
made,
the
follo
wing
equation
for
the
in
v
erse
saturation
current
is
deri
v
ed:
I
0
=
I
sc
+
R
s
I
sc
R
p
+
V
oc
R
p
exp
q
V
oc
ak
N
c
T
exp
q
R
s
I
sc
ak
N
c
T
:
(4)
It
should
be
noted
that
if
we
substitute
(4)
in
(2),
we
can
obtain
a
general
representation
of
the
PV
current
as
(5):
I
pv
=
I
sc
+
R
s
I
sc
R
p
+
V
oc
R
p
h
exp
q
V
oc
ak
N
c
T
1
i
exp
q
V
oc
ak
N
c
T
exp
q
R
s
I
sc
ak
N
c
T
V
oc
R
p
:
(5)
2.3.
Maximum
po
wer
point
In
the
information
pro
vided
by
the
PV
module
manuf
acturer
,
an
operational
point
named
MPP
is
a
v
ailable.
This
point
presents
the
information
re
g
arding
the
maximum
possible
po
wer
transferred
from
the
panel
to
the
system
with
which
this
is
interconnected,
that
is,
(
I
mpp
;
V
mpp
)
.
If
this
point
is
substituted
in
(1),
a
general
equation
for
I
mpp
is
found
as
(6):
I
mpp
=
I
pv
I
0
h
exp
q
V
mpp
+
R
s
I
ak
N
c
T
1
i
V
mpp
+
R
s
I
mpp
R
p
:
!
:
(6)
Int
J
Elec
&
Comp
Eng,
V
ol.
12,
No.
1,
February
2022:
82–91
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
85
2.4.
Optimization
model
Based
on
the
information
pro
vided
by
the
PV
module
manuf
acturer
,
that
is,
the
three
aforement
ioned
operati
v
e
points,
it
is
possible
to
formulate
an
optimization
model
that
allo
ws
the
estimation
of
the
parameters
of
the
electrical
equi
v
alent
circuit,
that
is,
series
and
parallel
resistances
and
t
he
ideality
diode
f
actor
using
a
single-objecti
v
e
formulation
that
minimizes
the
mean
square
error
between
the
manuf
acturer
data
and
the
calculated
v
alues.
The
general
structure
of
the
objecti
v
e
function
assumes
the
follo
wing
form:
min
z
=
E
2
oc
+
E
2
sc
+
E
2
mpp
;
(7)
where
E
oc
=
I
0
exp
q
V
oc
ak
N
c
T
1
V
R
p
I
pv
;
(8)
E
sc
=
I
pv
I
0
exp
q
R
s
I
sc
ak
N
c
T
1
R
s
I
sc
R
p
I
sc
;
(9)
E
mpp
=
I
pv
I
0
h
exp
q
V
mpp
+
R
s
I
ak
N
c
T
1
i
V
mpp
+
R
s
I
mpp
R
p
!
I
mpp
:
(10)
It
should
be
noted
that
to
find
the
v
alue
of
the
objecti
v
e
function
defi
n
e
d
in
(7),
it
is
necessary
to
kno
w
the
v
alues
of
the
parameters
a
,
R
s
,
and
R
p
(decision
v
ariables)
in
conjunction
with
the
simultaneous
solution
of
the
(4)
and
(5)
for
the
in
v
erse
saturation
and
the
PV
current.
T
o
complete
the
optimization
model
for
parametric
estimation
in
PV
modules
considering
manuf
acturer
data,
we
assign
the
lo
wer
and
upper
bounds
for
the
decision
v
ariables
as
in
(11)
[10]:
a
min
a
a
max
;
R
min
p
R
p
R
max
p
;
R
min
s
R
s
R
max
s
:
(11)
It
should
be
observ
ed
that
the
propos
ed
optimization
model
defined
from
(7)
to
(11)
added
with
the
equality
constraints
(4)
and
(5)
corresponds
to
a
continuous
non-linear
non-con
v
e
x
op
t
imization
problem.
It
implies
that
multiple
solutions
for
the
parameters
a
,
R
s
,
and
R
p
can
e
xist
with
the
same
numerical
perfor
-
mance,
that
is,
multimodal
optimization
beha
vior
[22].
Due
to
this
reason,
to
reach
an
adequate
solution
with
minimal
computational
ef
fort,
t
h
i
s
paper
proposes
the
application
of
the
methaeuristic
optimization
technique
kno
wn
as
CSA
[23],
[24],
which
has
not
been
pre
viously
reported
in
the
literature
to
address
the
parametric
es-
timation
probl
em
in
P
V
modules.
This
is
the
main
contrib
ution
of
this
research
based
on
its
e
xcellent
numerical
performance.
In
the
ne
xt
section,
we
will
present
the
CSA
and
its
application
of
the
studied
problem.
3.
CR
O
W
SEARCH
ALGORITHM
The
CSA
is
a
recently
de
v
eloped
combinatorial
optimization
technique
to
solv
e
continuous
non-linear
comple
x
and
non-con
v
e
x
optimizati
on
problems
with
multiple
constraints
[25],
[26].
This
technique
belongs
to
the
f
amily
of
the
bio-inspired
optimization
deri
v
ed
from
the
con
v
entional
particle
sw
arm
optimization
(PSO)
methodology
.
It
is
commonly
kno
wn
that
cro
ws
observ
e
other
birds
to
kno
w
where
the
y
hide
their
food
in
order
to
steal
it
once
the
o
wner
lea
v
es.
In
the
case
a
cro
w
commits
thie
v
ery
,
this
cro
w
will
tak
e
additional
precautions
such
as
mo
ving
hiding
places
to
reduce
the
possibility
of
being
a
future
victim
[27].
In
f
act,
the
y
use
their
o
wn
e
xperience
of
ha
ving
been
a
thief
to
forecast
the
beha
vior
of
possible
thie
v
es
and
can
determine
the
safest
course
to
protect
their
caches
from
being
pilfered.
The
main
characteristics
of
the
cro
ws
are:
i)
the
y
li
v
e
in
flocks;
ii)
the
y
can
memorize
the
position
of
their
hiding
places;
iii)
the
y
can
follo
w
each
other
to
commit
thie
v
ery;
and
i
v)
the
y
can
protect
their
caches
from
being
pilfered
by
a
probability
.
In
this
paper
,
we
present
the
mathematical
adaptation
of
this
beha
vior
to
solv
e
comple
x
optimization
problems
as
originally
proposed
in
[23].
The
primary
steps
in
the
implementation
of
the
CSA
ha
v
e
been
discussed
belo
w
.
3.1.
Initialization
of
the
pr
oblem
and
selection
of
the
adjustable
parameters
The
optimization
problem
is
defined,
that
is,
the
optimization
model
(7)
to
(11)
is
added
with
the
equality
constraints
(4)
and
(5).
Then,
the
adjustable
parameters
of
the
CSA
are
s
elected
,
that
is,
the
flock
size
n
,
the
maximum
number
of
iterations
t
max
,
the
flight
length
f
l
,
and
the
a
w
areness
probability
A
p
.
It
should
be
P
ar
ametric
estimation
in
photo
voltaic
modules
using
the
cr
ow
sear
c
h
algorithm
(Oscar
Danilo
Montoya)
Evaluation Warning : The document was created with Spire.PDF for Python.
86
r
ISSN:
2088-8708
noted
that
the
selection
of
these
parameters
is
heuristic
and
depends
on
the
kno
wledge
that
the
programmer
has
about
the
optimization
problem
under
study
.
These
are
typically
selected
using
multiple
simulations
to
identify
the
best
trade-of
f
between
response
quality
and
processing
times.
3.2.
Initial
position
and
memory
of
the
cr
o
ws
In
a
d
-dimensional
s
pace,
all
the
n
cro
ws
are
initially
positioned
as
the
members
of
the
flock.
It
is
w
orth
mentioning
that
each
cro
w
represents
a
feasible
solution
of
the
optimization
problem
that
is
composed
by
d
decision
v
ariables.
Cro
ws
=
2
6
6
6
4
x
11
x
12
x
1
d
x
21
x
22
x
2
d
.
.
.
.
.
.
.
.
.
.
.
.
x
n
1
x
n
2
x
nd
3
7
7
7
5
(12)
It
should
be
noted
that
the
generation
of
each
component
of
the
cro
w
i
associated
with
the
v
ariable
j
tak
es
the
follo
wing
form:
x
ij
=
x
min
j
+
r
x
max
j
x
min
j
;
8
i
=
1
;
2
;
:::;
n
j
=
1
;
2
;
:::;
d
(13)
where,
r
is
a
random
number
between
0
and
1
generated
with
a
normal
distrib
ution.
In
the
be
ginning
of
the
search
process,
t
he
memory
of
each
cro
w
is
initialized.
Ho
we
v
er
,
in
the
i
nitial
iteration,
the
cro
ws
ha
v
e
no
search
e
xperience.
Then,
it
is
assumed
that
the
y
ha
v
e
hidden
their
foods
at
their
initial
positions.
Memory
=
2
6
6
6
4
m
11
m
12
m
1
d
m
21
m
22
m
2
d
.
.
.
.
.
.
.
.
.
.
.
.
m
n
1
m
n
2
m
nd
3
7
7
7
5
(14)
3.3.
Fitness
function
e
v
aluation
As
a
con
v
entional
metaheuristic
optimization
algorithm,
the
CSA
w
orks
with
a
fitness
function
instead
of
the
original
objecti
v
e
function
of
the
problem
to
deal
with
infeasibilities
in
the
solution
space
[28].
Ho
we
v
er
,
in
the
case
of
the
parametric
estimation
of
PV
modules,
due
to
the
structure
of
the
optimization
model
,
it
is
possible
to
directly
e
v
aluate
the
objecti
v
e
function
for
each
cro
w
.
3.4.
Generation
of
the
new
position
f
or
each
cr
o
w
The
generation
of
ne
w
positions
for
the
cro
ws
in
the
search
space
proceeds
as
follo
ws:
consider
that
the
cro
w
i
w
ants
to
pass
to
a
ne
w
position.
F
or
this
goal,
this
cro
w
randomly
selects
one
of
the
flock
cro
ws
(e.g.,
cro
w
k
)
to
follo
w
in
order
to
disco
v
er
the
position
where
the
y
ha
v
e
hidden
their
food
(i.e.,
m
k
).
F
ollo
wing
this
procedure,
the
ne
w
position
of
the
cro
w
i
is
obtained
as
(16):
x
t
+1
i
=
x
t
i
+
r
i
f
i;t
l
(
m
t
k
x
t
i
)
;
r
k
A
k
;t
p
a
random
position
(see
(13))
;
otherwise
(15)
where
r
j
is
a
random
number
with
uniform
distrib
ution
between
0
and
1,
and
A
k
;t
p
denotes
the
a
w
areness
probability
of
cro
w
k
at
iteration
t
.
It
should
be
observ
ed
that
this
procedure
is
repeated
for
all
the
cro
ws.
It
is
w
orth
mentioning
that
each
cro
w
generated
with
(15)
is
re
vised
if
the
lo
wer
and
upper
bounds
of
the
decision
v
ariables
are
fulfilled.
In
the
case
that
the
cro
w
i
violates
these
bounds,
in
(13)
is
used
to
correct
it.
3.5.
Ev
aluation
of
the
objecti
v
e
function
and
updating
of
the
memories
F
or
each
one
of
the
positions
of
the
cro
ws,
the
objecti
v
e
function
is
e
v
aluated,
which
is
used
to
update
their
memories
as
(16):
m
t
+1
i
=
x
t
+1
i
;
z
x
t
+1
i
is
better
than
z
(
x
t
i
)
m
t
i
;
otherwise
(16)
It
is
observ
ed
that
if
the
fitness
function
v
alue
of
the
ne
w
position
of
the
cro
w
i
is
better
than
the
fitness
function
v
alue
of
the
memorized
position,
the
cro
w
updates
its
memory
by
the
ne
w
position.
”Better”
in
the
case
of
parametric
estimation
in
PV
modules
implies
”lo
wer”.
Int
J
Elec
&
Comp
Eng,
V
ol.
12,
No.
1,
February
2022:
82–91
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
87
3.6.
Check
termination
criterion
The
steps
defined
in
subsections
3.4.
and
3.5.
are
repeated
until
the
maximum
number
of
iter
ations
is
reached
(i.e.,
t
max
).
When
the
termination
criterion
is
met,
the
best
position
of
the
memory
in
terms
of
the
objecti
v
e
function
v
alue
(i.e.,
the
minimum
v
alue)
is
reported
as
the
solution
of
the
optimization
problem.
3.7.
Implementation
of
the
CSA
Algorithm
1
presents
the
general
pseudo-code
that
defines
the
steps
necessary
to
implement
the
CSA
to
solv
e
the
problem
of
parametric
estimation
in
PV
modules
[23].
Algorithm
1:
Application
of
the
cro
w
search
algorithm
to
the
problem
of
the
parametric
estimation
in
PV
modules
Data:
Read
the
data
of
the
PV
module
pro
vided
by
the
manuf
acturer;
Define
the
algorithm
parameters;
Randomly
initialize
the
position
of
a
flock
of
n
cro
ws
in
the
search
space;
Ev
aluate
the
fitness
function
for
all
the
cro
ws;
Initialize
the
memory
of
each
cro
w;
f
or
t
=
1
:
t
max
do
f
or
i
=
1
:
n
do
Randomly
select
one
of
the
cro
ws
to
follo
w
(e.g.,
cro
w
k
);
Define
an
a
w
areness
probability;
if
r
k
A
k
;t
p
then
x
t
+1
i
=
x
t
i
+
r
i
f
i;t
l
(
m
t
k
x
t
i
)
;
else
x
t
+1
i
=
a
random
position
(apply
Eq.
(13));
end
Check
the
feasibility
of
ne
w
positions;
Ev
aluate
the
ne
w
position
of
the
cro
ws;
Update
the
memory
of
cro
ws;
end
end
Result:
Return
the
best
solution
stored
in
the
memory
of
the
cro
ws
4.
TEST
SYSTEM
AND
COMPUT
A
TION
AL
V
ALID
A
TION
The
implementation
of
the
proposed
CSA
to
the
problem
of
the
parametric
estimation
in
PV
m
o
dul
es
represented
with
its
single-diode
model
has
considered
the
information
pro
vided
by
the
manuf
acturer
of
the
K
y-
ocera
KC200GT
[10]
as
sho
wn
in
T
able
1,
in
the
softw
are
MA
TLAB
2020
a
using
a
desk
computer
INTEL(R)
Core(TM)
i
5
3550
3
:
5
-GHz,
8
GB
of
RAM
v
ersion
64-bit
with
Microsoft
W
indo
ws
7
Professional.
T
o
v
alidate
the
ef
fecti
v
eness
of
the
CSA
to
find
a
high-quali
ty
solution
to
the
problem
of
the
para-
metric
estimation
in
PV
modules,
a
population
of
20
indi
viduals
w
as
considered,
that
is,
n
=
20
,
with
100000
consecuti
v
e
iterations;
an
a
w
arenes
s
probability
fix
ed
as
0
:
75
,
and
a
v
ariable
fly
length
defined
by
the
follo
w-
ing
rule
f
l
=
2
r
and
(1
t=t
max
)
.
Moreo
v
er
,
the
lo
wer
and
upper
bounds
for
the
decision
v
ariables
are
0
:
5
a
2
,
0
:
001
R
p
1
,
and
50
R
s
200
,
which
ha
v
e
been
tak
en
from
[10].
T
able
1.
K
yocera
KC200GT
manuf
acturer
information
(T
ak
en
from
[5])
P
arameter
Symbol
V
alue
Open-circuit
v
oltage
V
oc
32.900
V
T
emperature
coef
ficient
for
V
oc
K
V
oc
-0.123
V/
o
C
Short-circuit
current
I
sc
8.210
A
T
emperature
coef
ficient
for
I
sc
K
I
sc
3.180
10
3
A/
o
C
V
oltage
on
the
MPP
V
mpp
26.300
V
Number
of
cell
in
series
N
c
54
Current
on
the
MPP
I
mpp
7.610
A
P
ar
ametric
estimation
in
photo
voltaic
modules
using
the
cr
ow
sear
c
h
algorithm
(Oscar
Danilo
Montoya)
Evaluation Warning : The document was created with Spire.PDF for Python.
88
r
ISSN:
2088-8708
5.
NUMERICAL
RESUL
TS
The
application
of
the
CSA
to
the
problem
of
parametric
estimation
in
PV
modules
produced
the
results
reported
in
T
able
2,
where
the
best
10
solutions
are
presented
after
100
consecuti
v
e
e
v
aluations.
From
results
in
T
able
2,
we
can
observ
e
that:
i)
the
v
alue
of
the
objecti
v
e
function
related
with
the
mean
square
error
(see
(7))
that
e
v
aluates
the
error
re
g
arding
the
open-circuit
point,
short-circuit
point,
and
MPP
pro
vided
by
the
manuf
acturer
of
the
PV
module
and
the
c
alculated
v
alues
using
the
single-diode
model
are
lo
wer
(i.e.,
better)
than
1
10
29
,
which
can
be
considered
null
for
an
y
practical
implementation.
In
this
conte
xt,
as
mentioned
in
[5],
all
the
parameters
represent
optimal
solutions,
moreo
v
er
,
these
impro
v
e
the
concl
u
s
ion
reported
in
[10]
wherein
v
alues
lo
wer
than
15
were
considered
optimal;
ii)
the
solutions
in
the
range
from
3
to
8
present
the
same
objecti
v
e
function
v
alue,
that
is,
7
:
8886
10
31
,
which
confirm
the
multimodal
nature
of
the
problem
of
the
parametric
estimation
in
PV
modules
since
there
are
dif
ferent
combinations
of
the
decision
v
ariables
that
ha
v
e
the
same
numerical
performance;
iii)
the
electrical
parameter
that
presents
more
v
ariations
along
the
optimal
solutions
is
the
parallel
resistance
since
the
minimum
v
alue
reached
for
this
parameters
is
found
in
the
solution
10
with
a
v
alue
of
55
:
0001
and
the
maximum
v
alue
is
found
in
the
solution
5
with
a
v
alue
of
188
:
3342
,
that
is,
a
dif
ference
superi
o
r
than
120
between
both
solutions;
and
i
v)
the
a
v
erage
processing
times
reported
by
the
CSA
to
find
the
numerical
results
reported
in
T
able
2
w
as
about
1
:
80
s
with
a
standard
de
viation
of
0
:
20
s,
which
demonstrates
the
ef
ficienc
y
of
the
CSA
to
find
the
global
optimal
solution.
In
Figure
2
is
presented
the
V
I
curv
e
of
the
PV
module
for
each
one
of
the
ten
solutions
reached
by
the
CSA
and
presented
in
T
able
2.
These
curv
es
were
obtained
making
a
sweep
in
the
v
oltage
v
ariable
from
0
to
V
oc
in
steps
of
0
:
10
V
by
solving
the
(1)
for
all
the
combinations
o
f
a
,
R
s
and
R
p
parameters
of
the
single-diode
model
of
the
PV
module
presented
in
Figure
1.
From
the
numerical
results
presented
in
Figure
1,
it
can
be
noted
that
the
points
P
1
,
P
2
,
and
P
3
correspond
to
the
open-circuit
point,
MPP
,
and
short-circuit
operati
v
e
points,
which
confirms
that
the
information
pro
vided
by
the
PV
module
manuf
acturer
is
suf
ficient
to
estimate
with
minimum
errors
the
complete
beha
vior
of
the
panel
in
all
its
operati
v
e
range;
moreo
v
er
,
when
the
W
ilcoxon
test
w
as
applied
for
ten
independent
samples,
each
one
of
them
with
10
optimal
solutions,
a
mean
v
alue
for
p
of
about
p
of
0
:
5486
with
a
v
alue
of
h
=
0
w
as
obtained;
this
implies
that
the
null
h
ypothesis
of
the
W
ilcoxon
test
is
confirmed,
and
therefore,
the
analyzed
samples
present
the
same
median
wit
h
a
significance
le
v
el
of
100
%,
which
demonstrates
that
the
CSA
has
the
ability
to
find
the
global
optimal
solution
at
each
e
v
aluations
with
100
consecuti
v
e
search
through
the
solution
space.
T
able
2.
T
en
best
results
reached
by
the
CSA
N
o
a
R
s
(
)
R
p
(
)
f
f
1
0.65018
0.3938
56.8600
0
2
0.57158
0.5199
81.3492
0
3
0.59211
0.4996
76.4901
7.8886
10
31
4
0.64397
0.5071
113.2315
7.8886
10
31
5
0.50549
0.6038
188.3341
7.8886
10
31
6
0.51566
0.5436
74.2629
7.8886
10
31
7
0.58256
0.5137
81.6568
7.8886
10
31
8
0.60683
0.5451
159.1443
7.8886
10
31
9
0.67940
0.3646
55.8052
2.2877
10
29
10
0.67623
0.3605
55.0001
2.5243
10
29
5.1.
Comparison
with
combinatorial
methods
In
this
section,
we
present
the
comparati
v
e
results
among
the
proposed
CSA
and
dif
ferent
combina-
torial
methods
that
solv
e
optimization
problems
in
the
continuous
domain.
These
methods
are:
v
orte
x
search
algorithm
[12],
sine-cosine
algorithm
[10],
PSO
[29],
and
genetic
algorithm
[30].
F
or
each
one
of
these
com-
parati
v
e
methods,
100
consecuti
v
e
e
v
aluations
were
made
to
obtain
the
best
10
results
reported
in
T
able
3.
From
results
listed
in
T
able
3
it
is
possible
to
observ
e
that:
The
CSA
and
the
VSA
optimization
approaches
present
the
best
numerical
performance
with
respect
to
the
objecti
v
e
function
v
alues
lo
wer
than
10
28
and
10
25
,
respecti
v
ely
.
The
solutions
found
by
the
SCA
and
the
CGA
methods
can
also
be
considered
optimal,
since
the
objecti
v
e
function
is
in
practical
terms
null.
Ho
we
v
er
,
we
can
mention
that
the
SCA
presents
a
better
numerical
performance
when
compared
with
the
CGA
as
w
as
demonstrated
in
[10],
since
the
best
objecti
v
e
function
is
9
:
7192
10
28
for
the
SCA
and
3
:
2207
10
12
for
the
CGA,
which
implies
a
dif
ference
higher
than
1
10
5
between
them.
Int
J
Elec
&
Comp
Eng,
V
ol.
12,
No.
1,
February
2022:
82–91
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
89
The
list
of
dif
ferent
solutions
reported
in
3
confirms
the
non-linear
non-con
v
e
x
nature
of
the
problem
of
the
parametric
estimation
in
PV
systems,
since
the
CGA
and
the
SCA
are
stuck
in
local
optimums,
while
the
PSO,
the
VSA,
and
the
proposed
CSA
present
a
better
numerical
performance
reaching
high-quality
objecti
v
e
function
v
alues.
T
o
complement
the
analysis
among
the
CSA
and
the
comparati
v
e
metaheuristic
m
ethods,
the
parameters
for
the
optimal
solutions
are
listed
in
T
able
4
with
label
number
1
presented
in
T
able
3.
Figure
2.
V
I
curv
e
obtained
by
solving
(1)
for
the
parameters
presented
in
T
able
2
T
able
3.
T
en
best
solutions
reported
by
each
comparati
v
e
method
N
o
CSA
VSA
SCA
PSO
CGA
1
0
0
9.7192
10
18
2.5243
10
29
3.2207
10
12
2
0
0
1.0823
10
17
1.0097
10
28
5.1049
10
11
3
7.8886
10
31
0
1.3799
10
16
1.6155
10
27
2.3014
10
10
4
7.8886
10
31
0
1.9364
10
16
2.5243
10
27
2.7546
10
10
5
7.8886
10
31
7.8886
10
29
3.5309
10
16
6.0647
10
27
3.9197
10
10
6
7.8886
10
31
1.9248
10
28
5.2122
10
16
2.9181
10
26
5.1134
10
10
7
7.8886
10
31
1.3067
10
26
5.4738
10
16
3.5498
10
26
5.2351
10
10
8
7.8886
10
31
1.3354
10
26
8.4393
10
16
5.0553
10
26
6.0393
10
10
9
2.2877
10
29
2.9614
10
26
9.3922
10
16
7.6361
10
26
6.2646
10
10
10
2.5243
10
29
4.8872
10
26
1.5123
10
16
1.4485
10
25
6.8811
10
10
T
able
4.
Optimal
solutions
reported
by
the
proposed
and
comparati
v
e
methods
Method
a
R
s
(
)
R
p
(
)
CSA
0.650181877806710
0.393806884684579
56.8600309871423
VSA
0.502572297672421
0.505917172241395
57.6920080257133
PSO
0.681740933460645
0.508343460036893
170.884395321487
SCA
0.917140758347724
0.146864384999908
52.6718647012995
CGA
0.985461664181760
0.234102605899478
70.5159926098178
6.
CONCLUSION
In
this
research,
the
CSA
w
as
implemented
to
find
the
optimal
parameter
combination
to
represent
PV
modules
with
its
single-diode
model.
Numerical
results
demonstrated
that
this
algorithm
finds
solutions
with
v
alues
lo
wer
than
1
10
28
re
g
arding
the
objecti
v
e
function
v
alue
after
100
consecuti
v
e
e
v
aluati
ons,
which
were
better
in
comparison
with
the
classical
metaheuristic
methods
used
to
solv
e
this
problem;
these
methods
were
the
VSA,
SCA,
PSO,
and
CGA
respecti
v
ely
.
The
first
10
solutions
reached
by
the
CSA
confirm
that
the
problem
of
the
parametric
estimation
in
PV
modules
is
a
multimodal
non-linear
optimization
problem
with
dif
ferent
combinations
of
the
decision
v
ariables
that
present
the
same
numerical
performance.
Re
g
arding
the
processing
times,
the
proposed
CSA
tak
es
about
1
:
80
s
to
find
the
optimal
solution
of
the
studied
problem
with
the
main
adv
antage
being
that
based
on
t
he
W
ilcoxon
test,
after
100
consecuti
v
e
e
v
aluations,
the
possibility
P
ar
ametric
estimation
in
photo
voltaic
modules
using
the
cr
ow
sear
c
h
algorithm
(Oscar
Danilo
Montoya)
Evaluation Warning : The document was created with Spire.PDF for Python.
90
r
ISSN:
2088-8708
of
finding
the
global
optimum
is
ensured.
Moreo
v
er
,
the
CSA
is
easily
implementable
in
an
y
programming
language
with
only
4
parameters
to
be
tuned.
In
the
future,
it
will
be
possible
to
de
v
elop
the
follo
wing
re-
search
w
orks:
i)
to
e
xte
n
d
the
proposed
CSA
to
the
parametric
estimation
in
induction
motors
and
distrib
ution
transformers
that
are
modeled
with
non-linear
non-con
v
e
x
optimization
models
and
ii)
to
apply
the
proposed
optimization
model
to
the
estimation
of
parameters
in
PV
modules
considering
real
measures
of
v
oltages
and
currents
including
v
ariable
weather
conditions.
A
CKNO
WLEDGMENTS
This
w
ork
w
as
supported
in
part
by
the
Centro
de
In
v
estig
aci
´
on
and
Desarrollo
Cient
´
ıfico
de
la
Uni-
v
ersidad
Distrital
Francisco
Jos
´
e
de
Caldas
under
grant
1643-12-2020
associated
with
the
project:
“Desarrollo
de
una
metodolog
´
ıa
de
optimizaci
´
on
para
la
gesti
´
on
´
optima
de
recursos
ener
g
´
eticos
distrib
uidos
en
redes
de
dis-
trib
uci
´
on
de
ener
g
´
ıa
el
´
ectrica.
”
and
in
part
by
the
Direcci
´
on
de
In
v
estig
aciones
de
la
Uni
v
ersidad
T
ecnol
´
ogica
de
Bol
´
ıv
ar
under
grant
PS2020002
associated
with
the
project:
“Ubicaci
´
on
´
optima
de
bancos
de
capacitores
de
paso
fijo
en
redes
el
´
ectricas
de
distrib
uci
´
on
para
reducci
´
on
de
costos
and
p
´
erdidas
de
ener
g
´
ıa:
Aplicaci
´
on
de
m
´
etodos
e
xactos
and
metaheur
´
ısticos.
”
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