Inter
national
J
our
nal
of
Electrical
and
Computer
Engineering
(IJECE)
V
ol.
11,
No.
3,
June
2021,
pp.
2027
2044
ISSN:
2088-8708,
DOI:
10.11591/ijece.v11i3.pp2027-2044
r
2027
Sliding-mode
contr
oller
f
or
a
photo
v
oltaic
system
based
on
a
´
Cuk
con
v
erter
Carlos
Andr
´
es
Ramos-P
aja
1
,
Daniel
Gonz
´
alez-Motoya
2
,
J
uan
P
ablo
V
illegas-Ceballos
3
,
Ser
gio
Ignacio
Ser
na-Gar
c
´
es
4
,
Roberto
Giral
5
1
F
acultad
de
Minas,
Uni
v
ersidad
Nacional
de
Colombia,
Colombia
2,3,4
Departamento
de
Electr
´
onica
y
T
elecomunicaciones,
Instituto
T
ecnol
´
ogico
Metropolitano,
Colombia
5
Departament
d’
Engin
yeria
Electr
`
onica,
El
`
ectrica
i
Autom
`
atica,
Uni
v
ersitat
Ro
vira
i
V
ir
gili,
Spain
Article
Inf
o
Article
history:
Recei
v
ed
Apr
24,
2020
Re
vised
Sep
19,
2020
Accepted
Oct
6,
2020
K
eyw
ords:
´
Cuk
con
v
erter
Photo
v
oltaic
DMPPT
Sliding
mode
control
W
ide
range
v
oltage
ABSTRA
CT
The
wide
range
of
step-up
and
step-do
wn
input-output
v
oltage
cha
racteristic
of
the
´
Cuk
con
v
erter
mak
es
it
a
good
candidate
to
interf
ace
photo
v
oltaic
arrays
in
both
clas-
sical
and
distrib
uted
maximum
po
wer
point
tracking
systems.
Because
its
tw
o
inductor
structure,
´
Cuk
con
v
erters
ha
v
e
continuous
input
and
output
currents,
which
reduce
the
additional
filtering
elements
usually
required
for
interf
acing
dc/dc
con
v
erter
topolo-
gies.
Ho
we
v
er
,
PV
systems
based
on
´
Cuk
con
v
erters
usually
do
not
pro
vide
formal
proofs
of
global
stability
under
realistic
conditions,
which
mak
es
im
possible
to
en-
sure
a
s
afe
operation
of
the
PV
installation.
Therefore,
this
paper
proposes
a
high-
performance
sliding-mode
c
ontroller
for
PV
systems
based
on
´
Cuk
con
v
erters,
which
re
gulates
the
PV
v
oltage
in
agreement
with
the
commands
imposed
by
a
MPPT
algo-
rithm,
rejecting
both
load
and
en
vironmental
perturbations,
and
ensuring
global
stabil-
ity
for
real
operation
conditions.
Finally
,
the
performance
of
the
re
gulated
PV
system
is
tested
using
both
simulations
and
e
xperiments.
This
is
an
open
access
article
under
the
CC
BY
-SA
license
.
Corresponding
A
uthor:
Juan
P
ablo
V
ille
g
as-Ceballos
Departamento
de
Electr
´
onica
y
T
elecomunicaciones
Instituto
T
ecnol
´
ogico
Metropolitano
Calle
73
No.
76A
-
354,
Medell
´
ın,
Colombia
Email:
juan
ville
g
as@itm.edu.co
1.
INTR
ODUCTION
The
de
v
elopment
of
PV
systems
is
an
acti
v
e
topic
due
to
the
use
of
those
generation
systems
in
man
y
types
of
electrical
installations:
i)
lar
ge
po
wer
plants
[1],
ii)
small
installations
for
both
grid-tie
and
stand-alone
applications
[2–4],
and
iii)
industrial
installations.
In
some
cases,
it
is
necessary
to
connect
the
PV
installation
to
the
A
C-grid
and
in
other
cases
to
stand
alone
loads
or
DC-b
uses.
The
first
ones
usually
adopt
step-up
con-
v
erters
to
match
the
lo
w-v
oltage
pro
vided
by
the
PV
modules
with
the
high-v
oltage
required
by
commercial
grid-connected
in
v
erters
[5–10].
Instead,
the
second
systems
ha
v
e
more
di
v
erse
requirements;
for
e
xample,
PV
systems
interacting
with
stand-alone
A
C
loads
usually
adopt
st
ep-up
con
v
erters
[5,
11–13],
b
ut
PV
sys-
tems
designed
for
battery
char
ging
or
to
interacting
with
DC
loads
often
use
step-do
wn
con
v
erters
[14,
15].
Figure
1
depicts
some
of
those
types
of
PV
systems:
T
ype
I
uses
a
step-up
dc/dc
con
v
erter
to
perform
the
max-
imum
po
wer
point
tracking
(MPPT)
control
on
the
PV
source,
which
ensures
the
e
xtraction
of
the
maximum
po
wer
from
the
PV
string.
Then,
a
dc/ac
con
v
erter
(in
v
erter)
injects
such
a
PV
po
wer
into
the
grid;
in
this
type
of
PV
system
usually
v
pv
<
v
dc
.
T
ype
II
has
a
simil
ar
structure,
b
ut
the
dc/ac
con
v
erter
interacts
with
an
J
ournal
homepage:
http://ijece
.iaescor
e
.com
Evaluation Warning : The document was created with Spire.PDF for Python.
2028
r
ISSN:
2088-8708
A
C
load,
which
requieres
a
dif
ferent
control
algorithm
for
the
in
v
erter
.
T
ype
III
does
not
use
dc/ac
con
v
erters;
instead,
the
step-do
wn
dc/dc
con
v
erter
interacts
with
a
battery
or
DC
load,
hence
v
pv
>
v
dc
.
dc
/
dc
grid
dc
/
ac
PV string
MPPT
converter
Grid
interface
dc
/
dc
dc
/
ac
PV string
MPPT
converter
A
C load
interface
A
C
load
dc
/
dc
Battery
and DC
load
PV string
MPPT
converter
Low voltage
load
DMPPT string
T
ype I
dc
/
dc
dc
/
dc
dc
/
dc
+
v
pv
-
+
v
pv
-
+
v
pv
-
+
v
dc
-
+
v
dc
-
+
v
dc
-
+
v
dc
-
T
ype II
T
ype III
T
ype IV
dc
/
ac
A
C
interface
grid or
A
C load
Figure
1.
T
ypes
of
PV
systems
Moreo
v
er
,
there
e
xists
a
no
v
el
PV
system
structure
named
distrib
uted
maximum
po
wer
point
track-
ing
(DMPPT)
[6,
16–18],
which
is
designed
to
reduce
the
detrimental
impact
of
the
partial
shading:
such
a
phenomenon
triggers
the
acti
v
ation
of
the
bypass
diodes
installed
in
commercial
PV
panels,
which
forces
the
shaded
modules
to
operate
at
the
second
quadrant
(with
ne
g
ati
v
e
v
oltage);
this
condition
produces
a
reduction
in
t
he
po
wer
production
and
a
health
de
gradation
on
the
shaded
PV
modules
[13].
Those
DMPPT
systems
ha
v
e
dc/dc
con
v
erters
isolating
each
PV
module,
hence
each
PV
module
operates
at
the
corresponding
maximum
po
wer
point
(MPP)
independent
of
the
shading
condition
of
the
other
modules.
This
type
of
PV
system,
de-
picted
in
Figure
1
as
T
ype
IV
,
e
xhibits
a
series-connection
of
the
dc/dc
con
v
erters
outputs
to
produce
the
high
v
oltage
requir
ed
by
the
dc/ac
con
v
erter
.
Such
a
series-connection
cause
s
that
the
dc/dc
c
o
n
v
erters
must
to
share
the
v
oltage
imposed
at
the
input
of
the
dc/ac
con
v
erter
.
Therefore,
with
a
l
o
w
number
of
PV
modules,
each
dc/dc
con
v
erter
must
support
an
output
v
oltage
hi
gher
than
the
input
v
oltage,
hence
requiring
a
step-up
v
oltage
operation.
Instead,
with
a
high
number
of
PV
modules,
each
dc/dc
con
v
erter
must
withstand
an
output
v
oltage
lo
wer
than
the
input
v
oltage,
hence
requiring
a
step-do
wn
v
oltage
operation.
Finally
,
it
is
also
possible
that,
for
a
particular
number
of
PV
modules,
each
dc/dc
con
v
erter
will
e
xhibit
the
same
v
oltage
at
both
input
and
output
terminals.
Therefore,
PV
systems
require
both
step-up
and
step-do
wn
dc/dc
con
v
erters
depending
on
the
ap-
plication.
Moreo
v
er
,
DMPPT
systems
require
dc/dc
con
v
erters
able
to
support
both
step-up/do
wn
operation
conditions.
This
type
of
problem
has
been
addressed
using
t
he
´
Cuk
con
v
erter
[19–21],
which
can
operate
with
output
v
oltages
lo
wer
,
equal,
or
higher
than
the
input
v
oltage.
Moreo
v
er
,
the
´
Cuk
con
v
erter
pro
vides
contin-
uous
input
and
output
current,
which
reduces
t
h
e
size
of
both
the
input
and
output
filters
required
to
pro
vide
high-quality
po
wer
to
the
load.
Those
characteristics
ha
v
e
made
the
´
Cuk
con
v
erter
a
suitable
option
to
design
PV
syst
ems.
F
or
e
xampl
e,
the
w
orks
reported
in
[19,
21–25]
propose
PV
systems
based
on
´
Cuk
con
v
erters
de-
Int
J
Elec
&
Comp
Eng,
V
ol.
11,
No.
3,
June
2021
:
2027
–
2044
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
2029
signed
to
supply
resisti
v
e
loads.
Ho
we
v
er
,
such
a
design
choice
is
not
applicable
to
grid-connected
systems,
in
which
the
con
v
erter
load
is
the
dc-link
between
the
MPPT
con
v
erter
and
the
in
v
erter
.
Another
problem
of
those
solutions
is
the
lack
of
a
capacitor
between
the
PV
module
and
the
con
v
erter
,
hence
all
the
switching
ripple
of
the
con
v
erter
is
propag
ated
to
the
PV
module,
which
produces
additional
po
wer
losses.
Finally
,
the
PV
v
oltage
is
not
re
gulated,
which
allo
ws
the
propag
ation
of
the
perturbations
present
at
the
dc-link
into
the
PV
module
terminals.
In
contrast,
in
[23,
25]
is
adopted
a
MPPT
fuzzy
controller
that
could
mitig
ate,
in
some
de
gree,
the
ef
fects
of
those
perturbations,
b
ut
formal
stability
analyses
are
not
gi
v
en,
hence
it
is
not
a
safe
solution.
A
similar
approach
w
as
proposed
in
[26],
b
ut
the
PV
module
is
modeled
as
a
v
oltage
source,
which
significantly
reduces
the
v
alidity
of
the
analyses;
and
similar
to
the
pre
vious
cases,
the
PV
v
oltage
is
not
re
gulated,
thus
de
grading
the
MPPT
performance
when
perturbations
occur
[17].
A
PV
system
with
similar
limitations
w
as
proposed
in
[27]
for
injecting
po
wer
to
the
grid.
As
in
the
pre
vious
cases,
the
load
of
the
´
Cuk
con
v
erter
is
considered
as
a
constant
resistance,
which
does
not
accurately
represent
the
input
impedance
of
an
in
v
erter
.
Moreo
v
er
,
the
mathematical
analyzes
consider
the
PV
module
as
a
constant
v
oltage
source,
whi
ch
is
a
strong
simplific
ation.
Therefore,
both
modeling
consi
derations
are
not
realistic
enough
for
grid-connected
systems.
A
similar
approach
w
as
used
in
[28],
where
the
load
is
considered
as
a
constante
resistance.
In
this
case
the
PV
module
equations
are
described,
b
ut
those
e
xpressions
are
not
used
in
the
Cuk
con
v
erter
analysis.
Finally
,
such
a
paper
does
not
considers
an
y
dynamic
analysis
or
formal
stability
proof,
hence
it
is
not
demonstrated
the
system
safe
operation.
A
w
ork
representing
the
source
and
load
of
the
´
Cuk
con
v
erter
with
realistic
models
is
reported
in
[29].
The
system
is
formed
by
both
wind
and
PV
generators,
and
it
is
controlled
with
an
MPPT
based
on
an
ant
colon
y
optimization
algorithm
and
a
fuzzy
controller
for
a
three-phase
in
v
erter
.
The
MPPT
operates
at
the
´
Cuk
con
v
erter
for
tracking
the
maximum
po
wer
point,
b
ut
no
formal
analysis
of
the
stability
is
pro
vided.
In
f
act,
the
´
Cuk
con
v
erter
is
modeled
with
static
equations
without
accounting
for
the
dynamic
beha
vior
of
the
po
wer
system,
which
mak
es
dif
ficult
to
guarantee
both
the
global
st
ability
of
the
system
and
the
rejection
of
perturbations
caused
by
the
in
v
erter
or
en
vironmental
conditions.
On
the
other
hand,
the
w
ork
reported
in
[30]
impro
v
es
the
PV
systems
by
controlling
the
´
Cuk
con
v
erter
using
a
sliding-mode
controller
.
Such
a
control
approach
pro
vides
f
ast
dynamic
responses
and
rob
ustness
to
grid-connected
perturbations.
Ho
we
v
er
,
the
mathematical
analyses
of
that
w
ork
adopt
inaccurate
models
for
both
the
load
and
the
PV
source:
a
resisti
v
e
load
and
v
oltage
source,
respecti
v
ely
.
Moreo
v
er
,
the
conditions
for
global
stability
are
not
discussed.
The
´
Cuk
con
v
erter
has
also
been
used
for
designing
industrial
PV
applications,
such
as
the
w
a-
ter
pumping
systems
reported
in
[31–33].
The
de
vices
de
v
eloped
in
[31,
32]
implement
an
incremental-
conductance
MPPT
algorithm,
b
ut
no
formal
re
gulation
of
the
PV
v
oltage
is
implemented.
Therefore,
the
perturbations
in
the
motor
could
spread
to
both
the
PV
module
and
the
MPPT
controller
.
In
[33],
in
addition
to
the
abo
v
e
limitations,
a
fuzzy
MPPT
controller
w
as
designed
considering
a
linearized
model
of
the
´
Cuk
con
v
erter
,
which
mak
es
dif
ficult
to
guarantee
the
global
stability
of
the
system
at
multiple
operation
points.
The
´
Cuk
con
v
erter
has
been
also
used
for
DMPPT
applications
with
se
v
eral
limitations.
F
or
e
xample,
the
w
ork
reported
in
[34]
demonstrates
the
adv
antages,
in
terms
of
po
wer
production,
of
a
DMPPT
system
based
on
´
Cuk
con
v
erters.
Ho
we
v
er
,
such
a
w
ork
does
not
consider
an
y
control
system
to
compensate
the
load
perturbations,
hence
it
is
not
e
vident
that
such
a
s
y
s
tem
will
be
stable
under
real
operation
condition.
Similarly
,
the
w
ork
reported
in
[35]
proposed
a
DMPPT
system
with
PI
v
oltage
controllers,
which
are
unable
t
o
ensure
global
stability
.
Moreo
v
er
,
the
PV
module
is
represented
by
a
The
v
enin
equi
v
alent,
hence
the
analyses
are
v
alid
just
at
a
gi
v
en
operating
point.
Finally
,
the
w
ork
adopts
a
cascade
MPPT
algorithm
based
on
the
perturb-and-
observ
e
(P&O)
algorithm.
Another
DMPPT
approach,
bas
ed
on
the
current
equalization
technique,
is
proposed
in
[36,
37].
The
main
disadv
antage
of
such
a
technique
is
the
lack
of
v
oltage
con
v
ersion
ratio,
since
the
string
v
oltage
is
imposed
by
the
MPP
v
oltage
of
the
modules.
The
w
ork
in
[36]
adopts
a
classical
programmed
current
control,
while
the
w
ork
in
[37]
does
not
consider
an
y
formal
controller
.
Theref
ore,
both
approaches
are
not
rob
ust
to
perturbations
generated
by
a
grid-connected
in
v
erter
.
In
terms
of
PV
systems
control,
´
Cuk
con
v
erters
ha
v
e
been
re
gulated
using
both
linear
and
non-linear
techniques.
That
is
the
case
of
the
´
Cuk
con
v
erter
used
for
the
micro-in
v
erter
reported
in
[38],
which
adopts
a
PI
controller
and
a
cascade
MPPT
algorithm.
Ho
we
v
er
,
in
that
w
ork
the
load
i
s
modeled
with
a
constant
impedance,
which
is
not
accurate
to
represent
a
grid-connected
po
wer
system.
Another
approach
w
as
pre-
sented
in
[39],
which
uses
the
jump-parameter
li
near
-optimal-control
technique
to
re
gulate
the
´
Cuk
con
v
erter
.
Unfortunately
,
as
in
the
pre
vi
ous
cases,
the
load
and
PV
model
are
simplified
to
a
resistance
and
v
oltage
source,
respecti
v
ely
.
PV
systems
based
on
´
Cuk
con
v
erters
ha
v
e
been
also
controlled
using
more
comple
x
control
tech-
Sliding-mode
contr
oller
for
a
photo
voltaic
system
based
on
a...
(Carlos
Andr
´
es
Ramos-P
aja)
Evaluation Warning : The document was created with Spire.PDF for Python.
2030
r
ISSN:
2088-8708
niques,
such
as
the
model
predicti
v
e
controller
reported
in
[40].
The
main
problems
of
such
a
solution
are
the
inaccurate
PV
source
model
(constant
v
oltage)
and
the
particular
load
model
(constant
resistance),
which
are
not
realistic
for
practical
applications.
Moreo
v
er
,
the
is
not
a
formal
stability
analysis,
hence
a
safe
operation
is
not
demonstrated.
In
conclusion,
the
foll
o
wi
ng
problems
are
present
in
the
PV
systems
based
on
´
Cuk
con
v
erters
reported
in
literature:
Some
solutions
adopt
non-realistic
PV
source
and
load
models
for
the
con
v
erter
analysis,
hence
the
system
performance
and
stability
are
not
ensured
Other
solutions
do
not
consider
the
con
v
erter
re
gulation
using
a
high-bandwidth
controllers
able
to
miti-
g
ate
perturbations.
Therefore,
load
perturbations
occurring
in
grid-conned
PV
systems
[41]
and
DMPPT
systems
[17,
13]
will
be
transferred
to
the
source
terminals,
which
could
lead
the
MPPT
controller
to
an
unstable
and
dangerous
operation
[42]
The
solutions
accounting
for
high-bandwidth
controllers
do
not
pro
vide
a
global
stability
proof.
Some
of
those
solutions
are
based
on
linear
controllers,
which
are
not
able
to
ensure
the
same
performance
and
stability
in
all
the
operation
range
[43,
44],
other
solutions
do
not
pro
vide
an
y
stability
analysis.
Therefore,
no
stable
and
safe
operation
is
ensured
This
paper
pro
vides
a
solution
solving
all
those
problems,
a
non-linear
controller
ensuring
global
stability
in
all
the
operation
range
of
the
´
Cuk
based
PV
system,
which
is
aimed
at
rejecting
both
the
load
and
en
vironmental
perturbati
ons
to
ensure
a
stable
and
safe
operation.
Such
a
controller
is
designed
us
ing
realistic
PV
source
and
load
models
for
the
con
v
erter
analysis,
which
ensures
the
solution
applicability
to
stand-alone,
grid-connected
and
DMPPT
PV
systems.
The
proposed
con
v
erter
analysis
tak
es
into
account
the
ef
fect
of
the
non-linear
beha
vior
of
the
PV
panel
into
the
PV
system,
and
the
adopted
load
model
is
in
agreement
with
the
electrical
beha
vior
of
a
capaciti
v
e
dc-link
re
gulat
ed
by
an
additional
de
vice.
The
proposed
controller
is
based
on
the
sliding-mode
theory
,
which
has
been
successfully
used
[43]
to
de
v
elop
a
non-linear
controller
for
a
PV
system
based
on
a
boost
con
v
erter
.
Finally
,
the
transv
ersality
,
reachability
and
equi
v
alent
control
conditions
of
the
sliding-mode
controller
are
analyzed
in
detail
to
pro
vide
a
global
stability
proof.
The
remain
of
the
paper
is
or
g
anized
as
follo
ws:
Section
2
presents
the
analysis
of
the
PV
system
and
the
design
of
the
proposed
sliding-mode
controller
,
describing
also
the
stability
conditions
.
Section
3
presents
the
performance
e
v
aluation
of
the
system,
under
realistic
conditions,
using
detailed
circuital
simulations.
Then,
section
4
presents
the
e
xperimental
v
alidation
of
the
solution
using
a
proof-of-concept
prototype.
Finall
y
,
the
conclusions
of
the
w
ork
close
the
paper
.
2.
DESIGN
OF
THE
PR
OPOSED
SLIDING-MODE
CONTR
OLLER
The
circuital
scheme
of
the
PV
system
based
on
the
´
Cuk
con
v
erter
is
presented
in
Figure
2.
Such
a
scheme
considers
the
PV
string
lumped
into
a
single
PV
model
scaled
in
current,
which
is
a
widely
adopted
model
[44].
This
modeling
strate
gy
enables
to
generalize
the
system
analysis
for
an
y
size
of
the
PV
array
,
i.e.
lar
ge
arrays
used
in
PV
systems
T
ype
I,
II
and
III
or
the
single
modules
used
in
PV
systems
T
ype
IV
.
C
pv
L
1
u
v
pv
-
+
i
L1
i
L2
v
dc
+
-
+
-
i
Cpv
i
pv
i
ci
C
i
v
ci
+
-
-
+
L
2
MPPT
SMC
v
r
u
v
pv
i
pv
i
Cpv
Figure
2.
Circuital
scheme
of
the
PV
system
based
on
a
´
Cuk
con
v
erter
The
model
also
considers
the
output
v
oltage
v
dc
re
gulated
by
an
e
xternal
de
vice:
in
systems
T
ype
I,
II
and
IV
the
grid-connected
in
v
erter
usually
re
gulates
the
dc
v
oltage
v
dc
at
the
input
terminals
[17,
44],
while
Int
J
Elec
&
Comp
Eng,
V
ol.
11,
No.
3,
June
2021
:
2027
–
2044
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
2031
in
systems
T
ype
III
the
battery
imposes
the
v
dc
v
oltage
[14,
43].
The
´
Cuk
con
v
erter
pro
vides
in
v
erse
polarities
to
the
input
terminals
with
respect
to
the
output
v
oltage,
hence
the
PV
string
(or
module)
is
connected
as
it
is
sho
wn
in
Figure
2.
2.1.
Mathematical
model
of
the
PV
system
The
´
Cuk
con
v
erter
is
controlled
using
a
binary
signal
u
,
which
defines
the
state
of
the
MOSFET
(
u
=
1
is
closed
and
u
=
0
is
open;
its
complementary
signal
is
u
=
1
u
).
The
signal
u
is
generated
by
a
sliding-mode
controller
(SMC),
which
follo
ws
the
command
v
r
pro
vided
by
a
MPPT
controller
.
In
this
cascade
structure
the
MPPT
controller
defines
the
optimal
PV
v
oltage
needed
to
e
xtract
the
MPP
po
wer
from
the
PV
source,
while
the
SMC
ensures
the
s
table
operation
of
the
PV
system
in
such
an
optimal
PV
v
oltage,
rejecting
also
perturbations.
Applying
the
flux
balance
principle
i
n
the
inductors
and
the
char
ge
balance
principle
in
the
capacitors
[45],
switched
dif
ferential
equations
are
obtained
is
being
as
(1)-(4):
C
pv
dv
pv
dt
=
i
pv
i
L
1
(1)
C
i
dv
ci
dt
=
i
L
1
u
i
L
2
u
(2)
L
1
di
L
1
dt
=
v
pv
v
ci
u
(3)
L
2
di
L
2
dt
=
v
ci
u
v
b
(4)
Then,
the
a
v
eraged
dif
ferential
equations
are
obtained
by
a
v
eraging
the
switched
equations
within
the
switched
period
[45]
is
being
as
(5)-(8):
C
pv
dv
pv
dt
=
i
pv
i
L
1
(5)
C
i
dv
ci
dt
=
i
L
1
(1
d
)
i
L
2
d
(6)
L
1
di
L
1
dt
=
v
pv
v
ci
(1
d
)
(7)
L
2
di
L
2
dt
=
v
ci
d
v
b
(8)
In
the
pre
vious
equations
d
represents
the
duty-c
ycle
of
the
con
v
erter
.
Finally
,
by
applying
the
steady-
state
analysis
of
dc/dc
con
v
erters
[45],
i.e.
considering
the
deri
v
ati
v
es
of
(5)-(8)
equal
to
zero,
static
e
xpressions
are
obtained
as
being
as
(9)-(12):
I
pv
=
I
L
1
(9)
I
L
1
=
I
L
2
D
(1
D
)
(10)
V
ci
=
V
pv
(1
D
)
(11)
V
ci
=
V
b
D
(12)
The
pre
vious
equations
use
capital
v
ariables
to
represent
steady-state
v
alues.
Sliding-mode
contr
oller
for
a
photo
voltaic
system
based
on
a...
(Carlos
Andr
´
es
Ramos-P
aja)
Evaluation Warning : The document was created with Spire.PDF for Python.
2032
r
ISSN:
2088-8708
2.2.
Contr
oller
surface
and
stability
The
main
objecti
v
e
of
the
sliding-mode
controller
(SMC)
is
to
impose
the
PV
v
oltage
v
alue
defined
by
the
MPPT
controller
[41,
44,
46].
Therefore,
the
SMC
must
to
dri
v
e
the
´
Cuk
semiconductors
to
reject
the
perturbations
generated
by
both
the
load
and
the
en
vironment
al
conditions.
This
paper
proposes
the
switching
function
and
the
sliding
surf
ace
gi
v
en
in
(13)
and
(14),
respecti
v
ely
.
=
K
(
v
pv
v
r
)
i
C
pv
(13)
=
f
=
0
g
(14)
In
(13),
K
is
a
constant
that
must
be
designed
to
impose
the
desired
dynamic
beha
vior
to
the
PV
v
oltage.
Moreo
v
er
,
i
C
pv
represents
the
current
of
the
capacitor
C
pv
that
defines
the
PV
v
oltage,
while
v
r
is
the
reference
imposed
by
the
MPPT
controller
.
T
o
ensure
both
the
viability
and
stability
of
the
sliding
surf
ace
(14)
it
is
needed
to
perform
tw
o
tests
[17],
transv
ersality
condition
and
reachability
conditions.
It
must
be
noted
that
the
equi
v
alent
control
condition,
in
the
conte
xt
of
dc/dc
con
v
erters,
is
equi
v
alent
to
the
reachabil
ity
conditions
as
it
w
as
demonstrated
by
Sira-Ramirez
in
[17].
The
pre
vious
analyses
require
the
deri
v
ati
v
e
of
the
switching
function
(13)
is
being
as
(15):
d
dt
=
K
dv
pv
dt
di
pv
dt
+
di
L
1
dt
K
dv
r
dt
(15)
Such
a
deri
v
ati
v
e
tak
es
into
account
that
i
C
pv
=
i
pv
{
i
L
1
.
Then,
replacing
the
dif
ferential
(1)
and
(3)
into
(15)
leads
to
the
e
xplicit
e
xpression
gi
v
en
in
(16).
d
dt
=
K
i
pv
i
L
1
dt
di
pv
dt
+
v
pv
v
ci
u
L
1
K
dv
r
dt
(16)
The
deri
v
ati
v
es
of
both
PV
current
di
pv
dt
and
reference
v
oltage
dv
r
dt
are
discussed
afterw
ards.
2.2.1.
T
ransv
ersality
condition
The
transv
ersality
condition
v
erifies
the
presence
of
the
control
signal
u
into
the
switching
function
deri
v
ati
v
e,
which
is
needed
to
modify
the
trajectory
of
the
system
state-v
ariables.
The
mathematical
represen-
tation
of
this
condition
is
gi
v
en
in
(17).
d
du
d
dt
6
=
0
(17)
Replacing
e
xpression
(16)
into
(17)
leads
to
e
xpression
(18):
d
du
d
dt
=
v
ci
L
1
>
0
(18)
Since
both
v
ci
and
L
1
are
al
w
ays
positi
v
e,
as
it
is
demonstrated
in
(12),
the
proposed
SMC
fulfills
the
transv
er
-
sality
condition.
2.2.2.
Reachability
conditions
The
reachability
conditions
v
erify
the
capability
of
the
SMC
to
con
v
er
ge
into
the
desired
sliding
surf
ace
(14).
The
reachability
conditions
are
the
follo
wing
ones:
i)
if
the
system
operates
under
the
surf
ace,
the
switching
function
deri
v
ati
v
e
must
be
positi
v
e
to
reach
the
surf
ace;
and
ii)
if
the
system
operates
abo
v
e
the
surf
ace,
the
switching
function
deri
v
ati
v
e
must
be
ne
g
ati
v
e
to
reach
the
surf
ace
[17].
Moreo
v
er
,
the
sign
of
the
transv
ersality
condition
also
af
fects
the
reachability
conditions:
a
positi
v
e
transv
ersality
v
alue
implies
that
u
=
1
produces
a
positi
v
e
v
alue
of
d
dt
;
while
a
ne
g
ati
v
e
transv
ersality
v
alue
implies
that
u
=
1
produces
a
ne
g
ati
v
e
v
alue
of
d
dt
[17].
Ther
efore,
taking
into
account
that
(18)
is
a
positi
v
e
v
alue,
the
follo
wing
reachability
conditions
must
be
fulfilled:
Int
J
Elec
&
Comp
Eng,
V
ol.
11,
No.
3,
June
2021
:
2027
–
2044
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
2033
lim
!
0
d
dt
u
=1
>
0
(19)
lim
!
0
+
d
dt
u
=0
<
0
(20)
Replacing
(16)
into
(19)
leads
to:
K
i
pv
i
L
1
C
pv
di
pv
dt
+
v
pv
L
1
K
dv
r
dt
>
0
(21)
Ho
we
v
er
,
the
small-ripple
approximation
principle
[45]
establishes
that
the
capacitor
current,
in
a
second-order
filter
,
corresponds
to
the
current
ripple
in
the
inductor
.
Examining
the
ci
rcuital
scheme
of
Figure
2
re
v
eals
that
the
node
connecting
the
PV
source,
the
capacitor
C
pv
and
the
inductor
L
1
form
a
second
order
filter
.
This
is
v
erified
by
(9),
which
confirms
that
the
steady-state
current
of
both
the
PV
source
and
the
inductor
L
1
are
equal,
hence
the
steady-state
v
alue
of
C
pv
current
is
cero.
Therefore,
the
only
component
present
into
the
current
of
C
pv
is
the
ripple
of
L
1
.
Under
the
light
of
the
pre
vious
analysis,
the
C
pv
current
is
gi
v
en
in
(22),
where
i
L
1
(
t
)
is
the
time-
v
arying
current
ripple
in
L
1
.
The
maximum
and
minimum
v
alues
of
i
L
1
(
t
)
correspond
to
the
current
ripple
magnitude
in
L
1
,
which
is
denoted
by
i
L
1
.
i
C
pv
=
i
pv
i
L
1
=
i
L
1
(
t
)
i
L
1
<
i
L
1
(
t
)
<
i
L
1
(22)
Finally
,
e
xpression
(21)
is
modified
by
considering
that
i
C
pv
=
i
pv
i
L
1
,
replacing
also
the
v
alue
obtained
in
(22)
for
i
C
pv
,
to
define
a
first
limitation
on
dv
r
dt
:
dv
r
dt
>
1
K
di
pv
dt
v
pv
L
1
K
i
L
1
C
pv
(23)
Therefore,
dv
r
dt
must
to
fulfill
e
xpression
(23)
to
guarantee
the
surf
ace
reachability
.
The
e
v
aluation
of
such
a
limit
must
be
performed
at
the
w
orst-case
conditions,
i.e.
the
most
restricti
v
e
v
alues:
di
pv
dt
=
max
di
pv
dt
>
0
i
L
1
=
max(
i
L
1
)
=
i
L
1
>
0
)
(24)
Similarly
,
the
second
reachability
condit
ion
must
be
also
analyzed.
F
or
that
case,
e
xpression
(16)
is
replaced
into
(20)
is
being
as
(25):
K
i
pv
i
L
1
C
pv
di
pv
dt
+
v
pv
v
ci
L
1
K
dv
r
dt
>
0
(25)
Expression
(24)
is
also
modified
by
considering
that
i
C
pv
=
i
pv
i
L
1
and
the
i
C
pv
v
alue
gi
v
en
in
(22)
is
being
as
(26):
dv
r
dt
<
1
K
di
pv
dt
v
pv
v
ci
L
1
K
i
L
1
C
pv
(26)
Then,
dv
r
dt
must
to
also
fulfill
e
xpression
(26)
to
guarant
ee
the
surf
ace
reachability
.
The
e
v
aluation
of
such
a
second
limit
must
be
performed
at
the
w
orst-case
conditions,
i.e.
the
most
restricti
v
e
v
alues:
di
pv
dt
=
min
di
pv
dt
<
0
i
L
1
=
min(
i
L
1
)
=
i
L
1
<
0
)
(27)
Sliding-mode
contr
oller
for
a
photo
voltaic
system
based
on
a...
(Carlos
Andr
´
es
Ramos-P
aja)
Evaluation Warning : The document was created with Spire.PDF for Python.
2034
r
ISSN:
2088-8708
In
conclusion,
the
reference
signal
pro
vided
by
the
MPPT
algorithm
must
to
fulfill
the
dynamic
re-
strictions
gi
v
en
by
(23)-(24)
and
(26)-(27)
to
ensure
the
reachability
conditions.
In
practice,
those
dynamic
restrictions
correspond
to
a
sle
w-rate
limitation
of
the
reference
signal.
2.2.3.
Closed-loop
dynamics
In
dc/dc
con
v
erter
applications,
the
stability
of
a
SMC
is
ensured
by
both
the
transv
ersality
and
reacha-
bility
conditions;
this
taking
into
account
that
the
equi
v
alent
control
condition
is
equi
v
alent
to
both
reachability
conditions
[17].
Such
stability
ensures
the
e
xistence
of
the
sliding-mode,
which
imposes
a
system
trajectory
parallel
to
the
surf
ace
and
with
a
null
steady-state
error
[17].
Those
conditions
are
formalized
is
being
as
(28):
=
0
;
d
dt
=
0
(28)
The
pre
vious
conditions
correspond
to
the
closed-loop
beha
vior
of
the
PV
system
under
the
action
of
the
SMC.
Replacing
e
xpressions
(1)
and
(13)
into
=
0
,
and
taking
into
account
that
C
pv
dv
pv
dt
=
i
C
pv
=
i
pv
i
pv
,
the
closed-loop
e
xpression
is
obtained
is
being
as
(29):
dv
pv
dt
=
K
C
pv
(
v
pv
v
r
)
(29)
Such
an
e
xpression
defines
the
closed-loop
beha
vior
of
the
PV
v
oltage.
Therefore,
(29)
is
used
in
the
follo
wing
section
to
calculate
a
v
alue
of
K
that
imposes
the
desired
beha
vior
to
the
PV
system.
2.3.
Design
of
the
dynamic
r
esponse
Expression
(29)
put
into
e
vidence
that
the
closed-loop
beha
vior
of
the
PV
v
oltage
is
equi
v
alent
to
a
linear
system.
Hence,
e
xpression
(29)
is
analyzed
in
Laplace
domain
as
being
as
(30):
v
pv
(
s
)
v
r
(
s
)
=
1
1
C
pv
K
s
(30)
The
most
commonly
used
MPPT
algorithm
for
PV
systems
is
the
perturb
and
observ
e
(P&O)
solution
[13],
which
produces
step-lik
e
changes
on
the
PV
v
oltage
to
track
the
optimal
v
alue
that
maximizes
the
po
wer
production.
The
P&O
is
characterized
by
tw
o
parame
ters:
the
perturbation
size
po
and
period
T
a
.
Therefore,
the
dynamic
response
of
the
PV
v
oltage
corresponds
to
a
first-order
step
response,
in
which
the
step
magnitude
is
equal
to
po
:
v
pv
(
s
)
=
po
1
s
1
s
K
C
pv
!
(31)
From
the
pre
vious
e
xpression
it
is
noted
that
a
stable
(ne
g
ati
v
e)
pole
is
obtained
only
for
ne
g
ati
v
e
v
alues
of
K
.
Hence,
the
restriction
must
be
fulfilled
as
being
as
(32):
K
<
0
(32)
T
ranslating
e
xpression
(31)
to
the
time
domain:
v
pv
=
po
1
e
K
C
pv
t
(33)
Then,
e
xpression
(33)
describes
the
closed-loop
beha
vior
of
the
PV
v
oltage
under
the
action
of
both
the
SMC
and
the
P&O
algorithm.
Ho
we
v
er
,
the
P&O
requires
a
stable
PV
po
wer
to
detect
the
best
direction
of
the
ne
xt
perturbation,
as
it
is
e
xplained
in
[13].
This
means
that
the
perturbation
period
T
a
must
be
longer
than
the
settling-time
t
s
of
the
PV
system,
otherwise
the
P&O
controller
wil
l
be
unstable;
such
an
analysis
can
be
found
in
[13]:
Int
J
Elec
&
Comp
Eng,
V
ol.
11,
No.
3,
June
2021
:
2027
–
2044
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
2035
T
a
>
t
s
(34)
Therefore,
K
must
be
designed
t
o
ensure
that
condition
(34)
is
fulfilled.
This
is
done
by
calculating
the
settling
time
of
the
PV
v
oltage
from
(33),
which
is
gi
v
en
in
(35)
for
the
classical
band
of
2%:
t
s
=
4
C
pv
K
(35)
Finally
,
replacing
(35)
into
(34)
gi
v
es:
K
4
C
pv
T
a
(36)
In
conclusi
on,
K
must
to
fulfill
the
limit
imposed
by
(36)
to
ensure
a
stable
PV
system.
Ho
we
v
er
,
the
v
alue
of
K
could
be
tuned
to
reduce
the
settling-time
of
the
PV
system,
which
e
v
entually
mak
es
possible
to
reduce
the
P&O
period,
hence
pro
viding
a
f
aster
tracking
of
the
optimal
PV
v
oltage.
2.4.
Practical
implementation
Theoretical
SMCs
impose
infinite
switching
frequenc
y
[14],
which
could
destro
y
the
se
miconductor
de
vices.
Therefore,
SMCs
for
dc/dc
con
v
erters
are
implemented
using
h
ysteresis
comparators
to
limit
the
switching
frequenc
y
[14].
The
h
ysteresis
band
H
is
placed
around
the
desired
v
alue
of
the
proposed
sliding-
surf
ace
to
limit
the
switching
frequenc
y
,
which
transforms
(14)
is
being
as
(37):
=
fj
j
<
H
g
(37)
Considering
the
reachability
conditions
gi
v
en
in
(19)
and
(20),
the
control
la
w
needed
to
dri
v
e
the
system
into
the
h
ysteresis
band
is
being
as
(38):
u
=
1
if
<
H
u
=
0
if
>
+
H
(38)
The
switching
circuit
designed
to
implement
the
control
la
w
gi
v
en
in
(38)
is
depicted
in
Figure
3,
which
is
based
on
tw
o
comparators
and
a
Set-Reset
Flip-Flop.
In
addition,
Figure
3
also
presents
the
calculation
of
the
switching
function
from
the
measured
v
ariables:
the
PV
v
oltage
and
C
pv
current.
Finally
,
the
figure
also
depicts
the
digital
implementation
of
both
the
P&O
algorithm
and
the
sle
w-rate
limitation
(SR)
for
the
v
oltage
reference,
which
is
needed
to
fulfill
the
reachability
conditions
(23)-(24)
and
(26)-(27).
+
-
+
-
-H
+H
Ѱ
S
R
Q
u
Flip-Flop
k
v
pv
v
r
+
-
i
Cpv
-
+
P&O
SR
v
po
v
pv
i
pv
Switching cir
cuit
Digital
pr
ocessor
Figure
3.
Implementation
of
the
SMC
Moreo
v
er
,
from
e
xpressions
(13)
and
(37)
it
is
noted
that
H
<
<
H
,
which
corresponds
to
H
<
K
(
v
pv
v
r
)
i
C
pv
<
H
.
Ho
we
v
er
,
under
a
correct
operation
of
the
SMC,
the
PV
v
oltage
is
equal
to
the
reference
v
alue,
i.e.
v
pv
=
v
r
,
which
leads
to
practical
condition
is
being
as
(39):
Sliding-mode
contr
oller
for
a
photo
voltaic
system
based
on
a...
(Carlos
Andr
´
es
Ramos-P
aja)
Evaluation Warning : The document was created with Spire.PDF for Python.
2036
r
ISSN:
2088-8708
H
<
i
C
pv
=
i
L
1
(
t
)
<
H
(39)
Finally
,
in
section
2.2.2
the
maximum
and
minimum
v
alues
of
i
L
1
(
t
)
where
cal
culated
as
i
L
1
and
i
L
1
,
respecti
v
ely
.
But
e
xpression
(39)
also
reports
the
maximum
and
minimum
v
alues
of
i
L
1
(
t
)
,
which
corresponds
to
H
and
H
,
respecti
v
ely
.
Therefore,
the
magnitude
of
the
current
ripple
in
the
inductor
L
1
(and
capacitor
C
pv
)
is
equal
to
H
,
and
the
limit
v
alues
of
i
L
1
(
t
)
are:
min
(
i
L
1
)
=
i
L
1
=
H
max
(
i
L
1
)
=
i
L
1
=
H
(40)
In
practice,
the
swit
ching
circuit
can
be
implemented
using
a
TS-555
inte
grated
circuit
as
it
is
de-
scribed
in
[14],
or
using
the
inte
grated
circuit
C
D
4043
N
as
it
will
be
presented
in
section
4.
Similarly
,
the
calculation
of
could
be
implemented
with
operational
amplifiers,
digital
processors,
or
with
a
combination
of
both
circuits.
Finally
,
the
sle
w-rate
limitation
and
P&O
algorithm
can
be
inte
grated
into
a
single
program
inside
a
digital
microprocessor
as
it
is
reported
in
[14].
3.
RESUL
TS
AND
DISCUSSION
T
o
test
the
performance
of
the
proposed
SMC,
the
follo
wing
parameters
are
considered:
C
pv
=
C
i
=
44
F
,
L
1
=
L
2
=
100
H
,
v
b
=
18
V
.
Moreo
v
er
,
H
=
500
mA
w
as
selected
to
ensure
a
maximum
switching
frequenc
y
equal
to
85
k
H
z
.
The
PV
system
considers
a
BP585
PV
panel
[17]
as
PV
source,
which
is
modeled
using
the
ideal
single
diode
model
gi
v
en
in
(41)
[44],
where
S
represents
the
solar
irradiance
measured
in
W
=m
2
,
with
the
follo
wing
parameters:
A
=
896
:
8
nA
,
B
=
0
:
7029
V
1
and
k
s
=
(5
=
1000)
A=
(
W
=m
2
)
.
i
pv
=
i
sc
A
e
B
v
pv
;
i
sc
=
k
s
S
(41)
Figure
4
presents
the
polarization
and
po
wer
curv
es
of
the
BP585
PV
panel
under
fi
v
e
dif
ferent
ir
-
radiance
conditions.
The
figure
also
depicts
the
MPP
at
each
irradiance
v
alue.
Such
data
put
into
e
vidence
the
requirement
of
using
a
MPPT
algorithm
t
o
track
the
MPP
under
time-v
arying
en
vironmental
conditions.
As
described
before,
this
paper
adopts
the
P&O
algorithm,
which
parameters
were
calculated
follo
wing
the
procedure
proposed
in
[17],
obtaining
T
a
=
1
ms
and
po
=
0
:
3
V
.
0
5
10
15
20
v
pv
[V]
0
1
2
3
4
5
i
pv
[A]
0
5
10
15
20
v
pv
[V]
0
17
34
51
68
85
p
pv
[W]
1000 W/m
2
800 W/m
2
600 W/m
2
400 W/m
2
MPP
200 W/m
2
MPP
1000 W/m
2
200 W/m
2
Figure
4.
Polarization
and
po
wer
curv
es
of
the
BP585
PV
panel
The
first
step,
for
the
PV
system
design,
is
to
apply
(35)-(37)
to
impose
a
settling-time
t
s
equal
to
500
s
,
which
is
shorter
than
T
a
,
obtaining
K
=
352
mA=V
.
Moreo
v
er
,
to
test
the
dynamic
restrictions
Int
J
Elec
&
Comp
Eng,
V
ol.
11,
No.
3,
June
2021
:
2027
–
2044
Evaluation Warning : The document was created with Spire.PDF for Python.