Inter
national
J
our
nal
of
P
o
wer
Electr
onics
and
Dri
v
e
System
(IJPEDS)
V
ol.
17,
No.
3,
September
2026,
pp.
1620
∼
1629
ISSN:
2088-8694,
DOI:
10.11591/ijpeds.v17.i3.pp1620-1629
❒
1620
L
yapuno
v
contr
oller
design
f
or
a
fuel
cell
electric
v
ehicle
po
wer
ed
by
a
thr
ee-le
v
el
oating
capacitor
boost
con
v
erter
Chaouqi
Aouadi
1,2
,
Abdelmajid
Abouloifa
1
,
Meriem
Aourir
3
,
Ibtissam
Lachkar
1
1
ESE
Lab,
ENSEM
of
Casablanca,
Uni
v
ersity
Hassan
II,
Casablanca,
Morocco
2
Department
of
Electrical
Engineering,
ENSEM
of
Casablanca,
Uni
v
ersity
Hassan
II,
Casablanca,
Morocco
3
LIME
Lab,
ISGA
Casablanca,
Edv
antis
Higher
Education,
Casablanca,
Morocco
Article
Inf
o
Article
history:
Recei
v
ed
Aug
9,
2025
Re
vised
Mar
24,
2026
Accepted
Apr
23,
2026
K
eyw
ords:
Boost
con
v
erter
Electric
v
ehicle
Flying
capacitor
L
yapuno
v
theorem
Nonlinear
controller
Stability
analysis
ABSTRA
CT
Electric
v
ehicles
(EVs)
ha
v
e
under
gone
considerable
progress
f
acilitated
by
the
adoption
of
contemporary
technologies,
such
as
inno
v
ati
v
e
ener
gy
sources
lik
e
the
fuel
cell
(FC)
in
conjunction
with
no
v
el
po
wer
con
v
erter
architectures.
In
this
conte
xt,
this
w
ork
addresses
the
problem
of
controlling
a
three-le
v
el
ying
capacitor
boost
con
v
erter
(FCBC)
po
wered
by
a
fuel
cell.
The
con
v
erter
is
link
ed
at
its
output
to
an
im
pedance
modelling
the
DC/A
C
con
v
erter
and
a
brushless
motor
inte
grated
into
an
FC-EV
.
V
ariations
in
load
demand
and
po
wer
pro
vided
by
the
fuel
cell
are
an
issue
to
be
considered,
alongside
v
oltage
balancing,
which
in
v
olv
es
maintaining
balanced
v
oltages
between
ying
capacitors,
as
an
imbalance
can
lead
to
une
v
en
component
v
oltage
and
reduced
con
v
erter
ef
cienc
y
.
T
o
solv
e
these
issues,
a
nonlinear
controller
is
de
v
eloped
making
use
of
the
L
yapuno
v
approach
which
ensures
the
proper
operation
of
the
system
by
continuously
adjusting
control
actions
based
on
the
system
dynamics.
Indeed,
the
re
gulator
comprises
three
loops:
one
dedicated
to
balancing
the
v
oltage
across
the
ying
capacitor
,
and
tw
o
nested
loops;
The
inner
loop
controls
the
inductor
current,
while
the
outer
loop
re
gulates
the
output
v
oltage.
This
is
an
open
access
article
under
the
CC
BY
-SA
license
.
Corresponding
A
uthor:
Chaouqi
Aouadi
ESE
Lab,
ENSEM
of
Casablanca,
Uni
v
ersity
Hassan
II
Casablanca,
Morocco
Email:
chaouqi.aouadi@uni
vh2c.ma
1.
INTR
ODUCTION
Sustainable
ener
gy
sources,
including
pho
t
o
v
olt
aic
(PV)
ener
gy
and
h
ydrogen
ener
gy
,
ha
v
e
become
a
crucial
el
ement
in
social
de
v
elopment.
The
need
for
clean
ener
gy-based
systems
is
currently
gro
wing
considerably
[1]–[3],
particularly
in
k
e
y
sectors
such
as
industry
and
transport.
The
latter
has
under
gone
a
remarkable
e
v
olution
with
t
h
e
deplo
yment
of
electric
v
ehicles
(EV),
replaci
ng
con
v
entional
internal
comb
ustion
engine
v
ehicles
kno
wn
for
their
substantial
emissions
of
greenhouse
g
ases.
Electric
v
ehicles
are
systematically
cate
gorized
based
on
the
specic
ener
gy
source
that
po
wers
their
dri
v
e
system
[4].
When
the
automobile
is
po
wered
e
xclusi
v
ely
by
a
battery
,
it
is
referred
to
as
a
battery
electric
v
ehicle
(BEV).
If,
in
addition
to
the
battery
,
the
electric
v
ehicle
incorporat
es
a
fuel
cell
or
internal
comb
ustion
engine,
it
is
classied
as
a
fuel
cell
electric
v
ehicle
(FCEV)
or
h
ybrid
electric
v
ehicle
(HEV),
respecti
v
ely
.
Ener
gy
source,
dri
v
etrain
system,
and
po
wer
electronics
with
its
control
unit
are
the
essential
parts
of
an
EV
[5].
Fuel
cell
(FC)
is
one
of
the
ener
gy
sources
being
promoted
to
be
used
alongside
the
batteries
in
FCEV
.
The
v
oltage
supplied
by
proton
e
xchange
membrane
fuel
cells
(PEMFCs)
inte
grated
into
EVs
is
relati
v
ely
lo
w
.
T
o
o
v
ercome
this
issue,
a
J
ournal
homepage:
http://ijpeds.iaescor
e
.com
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
❒
1621
DC/DC
con
v
ersion
interf
ace
w
as
added
to
increase
the
po
wer
at
the
terminals
of
the
FC
[6].
Man
y
DC/DC
con
v
erter
topologies
are
studied
in
the
literature
[7]–[10],
and
the
y
can
be
cate
gorized
into
tw
o
groups.
The
rst
group
comprises
classical
con
v
erters
such
as
the
boost
con
v
erter
and
the
b
uck
con
v
erter
.
The
second
class
includes
multile
v
el
po
wer
con
v
erters
[9],
[11],
which
of
fer
a
high
v
oltage
g
ain
[12]–[14]
without
using
a
high
duty
c
ycle.
This
paper
focuses
on
the
study
of
the
inte
gration
of
a
three-stage
ying
capacitor
boost
con
v
erter
(FCBC)
[15]–[17]
in
a
fuel
cell
electric
v
ehicle.
The
FCBC
is
a
v
ariant
of
a
multi-le
v
el
boost
con
v
erter
that
of
fers
se
v
eral
adv
antages
compared
with
a
con
v
entional
boost
chopper
[17].
One
o
f
the
k
e
y
benets
i
s
a
signicant
increase
in
v
oltage
amplication
due
to
the
oating
capacitor
that
stores
ener
gy
and
contrib
utes
to
the
amplication
and
transfer
of
ener
gy
from
a
PEM
FC
source
to
the
load.
It
also
of
fers
a
small
input
inductance,
which
contrib
utes
to
reducing
its
size
[18].
The
FCBC
is
suitable
for
adapting
current
and
v
oltage
le
v
els
from
a
lo
w-v
oltage
source,
b
ut
its
control
remains
a
challenge
due
to
the
zero
dynamics
[19].
Alongside
re
gulating
the
FCBC
output
v
oltage,
it
is
important
to
consider
stabilizing
the
oating
capaci
tor
v
oltage.
Inadequate
stabilization
results
in
high
v
oltage
stress
across
po
wer
switches
and
increased
po
wer
losses
[15].
T
o
address
this
issue,
se
v
eral
methods
for
stabilizing
oating
capacitor
v
oltage
ha
v
e
been
proposed
[19]–[22].
The
primary
purpose
of
this
paper
is
to
propose
a
controller
for
a
three-le
v
el
FCBC
used
in
FCEV
that
achie
v
es
tw
o
objecti
v
es:
–
CO1:
controlling
the
v
oltage
across
the
load;
this
objecti
v
e
is
accomplished
through
tw
o
cascaded
loops:
an
internal
loop
re
gulates
the
inductor
current
while
an
e
xternal
loop
controls
the
output
v
oltage.
–
CO2:
stabilizing
the
ying
capacitor
v
oltage;
this
objecti
v
e
is
realized
using
a
single
control
loop
that
enforces
the
ying
capacitor
v
oltage
to
follo
w
its
reference.
This
study
contrib
utes
to
the
literature
as
follo
ws:
i)
Designing
a
nonlinear
controller
based
on
lar
ge
signal
non-linear
model,
unlik
e
other
w
orks
that
use
linear
methods
for
system
modeling
and
controller
synthesis
[17];
and
ii)
The
global
stability
of
the
system
is
demonstrated
by
mathematical
tools
and
v
alidated
numerically
by
simulations,
which
conrm
the
rob
ustness
and
ef
fecti
v
eness
of
the
controllers
ag
ainst
disturbances
and
load
v
ariation.
The
remaining
part
of
this
w
ork
is
or
g
anized
as
follo
ws:
i)
The
second
section
focuses
on
sys
tem
description
and
mathematical
modeling;
ii)
Section
3
is
dedicated
to
controller
design;
iii)
Section
4
is
de
v
oted
to
stability
analysis;
i
v)
A
sim
u
l
ation
w
as
then
carried
out
to
v
alidate
the
controller’
s
ef
fecti
v
eness
in
section
5;
and
v)
A
conclusion
at
the
end
of
the
paper
.
2.
SYSTEM
DESCRIPTION
AND
MODELLING
The
h
ybrid
v
ehicle
sho
wn
in
Figure
1
can
be
dri
v
en
by
tw
o
ener
gy
chains
[23]
and
[24].
The
rst
consists
of
a
lithium
battery
associated
with
a
bidirectional
DC/DC
con
v
erter
,
while
second
consists
of
a
fuel
cell
connected
to
a
three-le
v
el
boost
con
v
erter
presented
in
Figure
2.
Both
ener
gy
sources
can
pro
vide
po
wer
to
the
v
ehicle’
s
dri
v
e
system,
which
consists
of
a
DC/A
C
con
v
erter
and
a
brushless
motor
.
The
fuel
cell-boost
con
v
erter
association
is
subject
of
the
study
presented
in
this
w
ork.
Indeed,
from
a
topological
point
of
vie
w
,
the
boost
con
v
erter
used
in
this
st
ud
y
considerably
reduces
the
v
oltage
stress
on
the
semiconductors,
while
from
a
control
point
of
vie
w
,
the
system
presents
a
challenge
due
to
its
comple
x,
non-linear
,
and
h
ybrid
model.
2.1.
Thr
ee
le
v
el
FCBC
modelling
Figure
2
illustrates
the
diagram
of
three-le
v
el
boost
con
v
erter
,
po
wered
by
a
fuel
cell
and
supplying
an
impedance
Z
L
which
is
the
do
wnstream
circuit
formed
by
the
brushless
motor
-con
v
erter
combination.
The
circuit
includes
tw
o
unidirectional
current
semiconductors
(
D
1
,
D
2
)
with
spontaneous
switching
and
tw
o
bidirectional
current
semiconductors
(
S
1
,
S
2
)
with
controlled
switching
by
tw
o
binary
signals
(
µ
1
,
µ
2
)
which
are
generated
by
phase
shifted
PWM
(PSPWM)
strate
gy
.
The
con
v
erter
dynamics
is
represented
by
the
well-kno
wn
a
v
erage
model
(1)–(3).
where
v
in
,
i
L
,
v
c
and
v
o
denote
re
specti
v
ely
,
input
v
oltage,
current
through
inductance
L,
the
v
oltage
across
oating
capacitor
C
and
output
v
oltage.
The
signals
(
µ
1
,
µ
2
)
are
the
switched
control
inputs.
Control
la
w
is
de
v
eloped
on
the
basis
of
the
model
gi
v
en
in
the
(4)–(6),
by
respecting
the
follo
wing
v
ariable
change;
x
1
=
i
L
,
x
2
=
v
c
and
x
3
=
v
o
.
L
˙
x
1
=
v
in
+
(
u
1
−
u
2
)
x
2
−
(1
−
u
2
)
x
3
(1)
L
yapuno
v
contr
oller
design
for
a
fuel
cell
electric
vehicle
power
ed
by
a
thr
ee-le
vel
...
(Chaouqi
Aouadi)
Evaluation Warning : The document was created with Spire.PDF for Python.
1622
❒
ISSN:
2088-8694
C
˙
x
2
=
−
(
u
1
−
u
2
)
x
1
(2)
C
o
˙
x
3
=
−
i
Z
+
(1
−
u
2
)
x
1
(3)
Figure
1.
Fuel
cell
electric
v
ehicle
Figure
2.
Three-le
v
el
ying
capacitor
boost
con
v
erter
3.
CONTR
OLLER
DESIGN
This
section
is
de
v
oted
to
the
synthesis
of
a
non-linear
state
feedback
controller
to
simult
aneously
achie
v
e
the
abo
v
e-mentioned
objecti
v
es.
The
controller
t
o
be
de
v
eloped
has
a
cascade
structure
with
nested
loops.
The
inner
control
la
ws
re
gulate
the
v
oltage
x
2
of
the
oating
capacitor
and
the
current
x
1
at
the
con
v
erter
input.
The
e
xternal
control
la
w
guarantees
the
re
gulation
of
the
output
v
oltage.
Direct
re
gulation
of
this
latter
is
not
possible,
as
the
system
has
unstable
zero
dynamics.
3.1.
Inner
r
egulator
The
internal
re
gulator
consists
of
tw
o
control
la
ws.
The
rst
re
gulates
the
v
oltage
of
the
oating
capacitor
to
reduce
v
oltage
stress
on
the
semi-conductors,
while
the
second
re
gulates
the
fuel
cell’
s
current
output
to
a
v
oid
the
problem
of
unstable
zero
dynamics.
Both
control
subsystems
e
xhibit
initial
coupling,
which
will
be
addressed
by
decoupling
in
the
re
gulator
to
be
de
v
eloped.
3.1.1.
Fuel
cell’
s
curr
ent
contr
ol
Due
to
the
non-minimum
phase
characteristics
of
t
he
system,
the
output
v
oltage
x
3
is
indirectly
re
gulated
by
controlling
the
current
x
1
.
The
controller
is
made
up
of
tw
o
nested
loops;
An
inner
control
l
oo
p
is
used
to
re
gulate
the
current
x
1
,
while
an
outer
control
loop
is
established
to
achie
v
e
the
re
gulation
of
the
output
v
oltage
x
3
and
to
pro
vide
a
current
reference
x
∗
1
.
Based
on
subsystem
(1),
an
error
between
current
x
1
and
its
reference
x
∗
1
can
be
dened
as
z
1
=
L
(
x
1
−
x
∗
1
)
.
Using
subsystem
(1),
the
dynamics
of
z
L
is
gi
v
en
as
(4).
˙
z
1
=
v
in
+
(
u
1
−
u
2
)
x
2
−
(1
−
u
2
)
x
3
−
r
x
1
−
L
˙
x
∗
1
(4)
T
o
guaranty
the
asymptotic
stability
of
the
system
one
dened
a
positi
v
e
L
yapuno
v
function
v
2
=
1
2
z
2
1
,
it’
s
dynamic
is
gi
v
en
by
(5).
˙
v
2
=
z
1
(
v
in
+
(
u
1
−
u
2
)
x
2
−
(1
−
u
2
)
x
3
−
r
x
1
−
L
˙
x
∗
1
)
(5)
T
o
mak
e
the
L
yapuno
v
function
dynamics
ne
g
ati
v
e,
one
chooses
the
(6).
v
in
+
(
u
1
−
u
2
)
x
2
−
(1
−
u
2
)
x
3
−
r
x
1
−
L
˙
x
∗
1
=
−
k
L
z
1
(6)
W
ith
k
L
is
a
positi
v
e
design
parameter
.
Re
writing
(6),
the
input
control
u
2
is
gi
v
en
by
the
(7).
u
2
=
1
x
3
−
x
2
(
x
3
−
v
in
−
u
1
x
2
+
r
x
1
+
L
˙
x
∗
1
−
k
L
z
1
)
(7)
Hence,
in
(4)
and
(5)
become:
˙
z
1
=
−
k
L
z
1
and
˙
v
1
=
−
k
L
z
2
1
<
0
.
Int
J
Po
w
Elec
&
Dri
Syst,
V
ol.
17,
No.
3,
September
2026:
1620–1629
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
❒
1623
3.1.2.
Flying
capacitor
v
oltage
r
egulation
F
or
the
FCBC
to
w
ork
correctly
,
it
is
crucial
to
maintain
the
v
oltage
across
FC
at
half
of
the
output
v
oltage
x
3
,
re
g
ardless
of
the
operating
conditions
of
the
system.
Theref
ore,
L
yapuno
v’
s
theorem
is
used
to
design
the
output
v
oltage
controller
x
2
.
T
aking
the
tracking
error
as
the
dif
ference
between
the
actual
signal
x
2
and
the
desired
reference
x
∗
2
=
x
1
2
.
z
2
=
C
(
x
2
−
x
∗
2
)
(8)
Using
the
subsystem
(2),
the
error
dynamics
z
c
can
be
e
xpressed
as
(9).
˙
z
2
=
−
(
u
1
−
u
2
)
x
1
−
C
˙
x
∗
2
(9)
Re
g
arding
the
L
yapuno
v
function
v
1
=
1
2
z
2
2
and
its
deri
v
ati
v
e
˙
v
1
=
z
2
˙
z
2
,
the
choice
˙
z
2
=
−
K
C
z
2
,
mak
es
the
L
yapuno
v
function
v
1
to
be
ne
g
ati
v
e
as
gi
v
en
by
(10).
˙
v
1
=
−
K
C
z
2
2
(10)
Where
K
C
ϵ
R
+
is
the
adjustment
parameter
.
Replacing
the
dynamic
of
z
2
by
its
e
xpression,
the
control
la
w
u
1
for
the
subsystem
(2)
is
constructed
as
(11).
u
1
=
1
x
1
(
K
C
z
2
−
C
˙
x
∗
2
)
+
u
2
(11)
Solving
the
system
of
equations
formed
by
(7)
and
(11)
leads
to
the
control
la
ws
(12)
and
(13).
u
1
=
1
+
(
K
C
z
2
−
C
˙
x
∗
2
)
x
3
−
x
2
x
1
x
3
+
1
x
3
(
r
x
1
−
v
in
+
L
˙
x
∗
1
−
k
L
z
1
)
(12)
u
2
=
1
+
1
x
3
−
v
in
−
x
2
x
1
(
K
C
z
2
−
C
˙
x
∗
2
)
+
r
x
1
+
L
˙
x
∗
1
−
k
L
z
1
(13)
3.2.
Outer
v
oltage
r
egulator
The
purpose
of
this
loop
is
to
force
the
squared
output
v
olt
age
to
track
a
desired
signal
x
∗
3
2
.
T
o
achie
v
e
this,
a
PI-type
re
gulator
is
used.
Its
output
will
serv
e
as
the
reference
signal
for
the
inner
loop.
Assumption
1:
i)
It
is
assumed
that
the
state
v
ariables
x
1
and
x
2
e
xhibit
f
aster
dynamics
compared
to
x
3
.
Consequently
,
the
y
will
quickly
reach
their
respecti
v
e
reference
v
alues
x
∗
1
and
x
∗
2
=
x
3
2
,
leading
to
the
con
v
er
gence
of
errors
(
z
1
,
z
2
)
to
zero;
ii)
The
dynamics
of
the
state
v
ariables
x
1
and
x
2
are
assumed
to
be
ne
gligible,
implying
that
u
1
=
u
2
;
and
iii)
The
parasitic
parameter
r
is
assumed
to
be
small.
T
o
design
this
loop,
a
relationship
is
establis
hed
between
the
DC
output
v
oltage
x
3
and
the
virtual
control
input
x
∗
1
.
By
substituting
the
control
la
w
u
2
from
(11)
into
(3)
and
considering
Assumption
1,
one
gets
(14).
C
o
2
dx
2
3
dt
=
−
x
3
i
Z
+
v
in
x
∗
1
(14)
The
controller
input
of
the
system
(3)
is
chosen
as
(15).
x
∗
1
=
α
(
t
)
+
x
3
i
Z
v
in
(15)
Where:
α
(
t
)
=
k
p
z
3
+
1
τ
i
z
4
(16)
is
a
signal
gi
v
en
by
the
PI-controller
.
W
ith
in
(17)
and
(18).
z
3
=
(
x
∗
3
)
2
−
x
2
3
(17)
L
yapuno
v
contr
oller
design
for
a
fuel
cell
electric
vehicle
power
ed
by
a
thr
ee-le
vel
...
(Chaouqi
Aouadi)
Evaluation Warning : The document was created with Spire.PDF for Python.
1624
❒
ISSN:
2088-8694
z
4
=
z
3
dτ
(18)
The
closed
loop
transfer
function
of
the
system
(23)
is
gi
v
en
by
(19).
H
(
p
)
=
1
+
τ
i
p
1
+
τ
i
p
+
τ
i
C
o
2
k
p
p
2
(19)
Where
τ
i
,
k
p
are
the
design
parameters
to
be
determined.
Comparing
the
transfer
function
(19)
with
the
standard
second-order
transfer
function,
the
PI
parameters
can
be
obtained
as:
τ
i
=
2
m
ω
0
,
k
p
=
mC
o
ω
0
.
where
m
and
ω
0
are
the
damping
coef
cient
and
natural
frequenc
y
of
the
system,
respecti
v
ely
.
T
o
comply
with
Assumption
1,
the
parameters
m
and
ω
0
are
selected
such
that
the
dynamics
of
the
inner
loop
are
twice
as
f
ast
as
the
dynamics
of
the
outer
loop.
4.
ST
ABILITY
AN
AL
YSIS
Stability
theory
is
fundamental
to
dynamical
systems
theory
and
essential
for
e
v
aluating
the
characteristics
of
a
closed-loop
system
with
it
s
controller
.
V
arious
techniques
and
theories
e
xist
in
the
literature
for
analyzing
system
stability
[25],
[26],
including
the
indirect
L
yapuno
v
method
and
the
Routh-Hurwitz
criterion.
Based
on
these
theorems,
we
will
demonstrate
that
the
designed
controllers
meet
the
dened
control
objecti
v
es.
The
dynamic
beha
vior
of
the
feedback
control
system
can
be
described
by
the
state
v
ector
(20)
in
the
z
1
z
2
z
3
z
4
coordinates:
˙
Z
=
˙
z
1
˙
z
2
˙
z
3
˙
z
4
T
(20)
where,
˙
z
1
=
−
k
L
z
1
,
˙
z
2
=
−
K
C
z
2
,
˙
z
3
=
−
c
1
z
1
+
c
2
z
2
+
c
3
z
3
+
c
4
z
4
−
c
5
z
2
1
+
c
6
z
2
2
−
c
7
−
˙
x
2
3
,
˙
z
4
=
z
3
and
c
1
=
−
2
v
in
LC
o
−
2
r
x
∗
1
LC
o
+
k
L
x
∗
1
C
o
,
c
2
=
2
K
C
x
∗
2
C
o
,
c
3
=
2
k
p
C
o
,
c
4
=
2
k
p
C
o
τ
i
,
c
5
=
−
2
−
r
C
o
L
2
+
k
L
C
o
L
,
c
6
=
2
K
C
C
C
o
,
c
7
=
2
r
C
o
x
∗
1
2
The
system
(20)
is
nonlinear
and
has
the
equilibrium
points
gi
v
en
by
(21).
Z
0
=
0
0
0
τ
i
r
k
p
x
∗
1
2
T
(21)
T
o
analyze
the
stability
of
the
equilibrium
Z
0
indirect
L
yapuno
v
technique
ha
v
e
been
used
[27].
This
in
v
olv
es
checking
whether
the
Jacobian
matrix
(22)
of
the
state
model
(20),
at
Z
0
is
Hurwitz.
J
=
⎛
⎜
⎜
⎝
−
c
2
0
0
0
0
−
c
1
0
0
c
1
0
−
c
2
0
−
c
3
1
−
c
4
0
⎞
⎟
⎟
⎠
(22)
The
equilibrium
Z
0
will
be
asymptotically
stable
if
the
Jacobian
matrix
(22)
is
Hurwitz.
This
requires
pro
ving
that
the
real
part
of
the
eigen
v
alues
of
the
follo
wing
characteristic
equation
is
ne
g
ati
v
e
R
e
(
λ
i
)
<
0
[26]:
π
J
=
λ
4
+
a
0
λ
3
+
a
1
λ
2
+
a
2
λ
+
a
3
(23)
where,
a
0
=
c
1
+
c
2
+
2
k
p
C
o
,
a
1
=
c
1
c
2
+
2
k
p
c
2
C
o
+
2
k
p
C
o
τ
i
+
2
k
p
c
1
C
o
a
2
=
2
k
p
K
C
C
o
τ
i
+
2
k
p
k
L
K
C
C
o
+
2
k
p
k
L
C
o
τ
i
,
a
3
=
2
k
p
K
C
k
L
C
o
τ
i
Using
Routh’
s
algebraic
criterion,
it
appears
that
all
zeros
of
the
characteristic
polynomial
in
(23)
ha
v
e
ne
g
ati
v
e
real
parts
if
the
follo
wing
conditions
(
a
0
a
1
−
a
2
a
0
>
0
,
a
0
(
a
1
a
2
−
a
0
a
3
)
a
0
a
1
−
a
2
>
0
,
a
3
>
0
)
are
satised.
Gi
v
en
these
conditions,
the
matrix
J
is
a
Hurwitz
matrix.
According
to
the
indirect
L
yapuno
v
method,
the
equilibrium
Z
0
of
the
system
(29)
is
asymptotically
stable.
Int
J
Po
w
Elec
&
Dri
Syst,
V
ol.
17,
No.
3,
September
2026:
1620–1629
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
❒
1625
5.
SIMULA
TION
RESUL
TS
The
controller
performance
w
as
v
alidated
through
simulations
of
the
po
wer
system
and
the
control
unit,
as
illustrated
in
Figure
3
using
MA
TLAB-Simulink.
The
parameters
for
the
entire
system
used
in
the
simulation
are
pro
vided
in
T
ables
1
and
2.
Three
test
protocols
were
emplo
yed
for
the
simulation:
i)
Normal
operation
mode:
Initially
,
the
system’
s
performance
w
as
e
v
aluated
under
normal
operating
conditions,
assuming
no
changes
to
the
o
v
erall
system
parameters;
ii)
Output
v
oltage
reference
V
ariation:
In
the
second
test,
a
v
ariation
in
the
output
v
oltage
reference
w
as
introduced
to
assess
the
system’
s
response
to
changes
in
the
desired
output
v
oltage;
and
iii)
Load
v
ariation:
In
the
nal
test,
a
load
v
ariation
w
as
applied
to
e
v
aluate
the
rob
ustness
of
the
system.
E
q
17 & 18
E
q
12& 13
Figure
3.
System
structure
in
MA
TLAB/Simulink
T
able
1.
System
parameters
Item
Symbol
V
alue
Unit
DC
source
E
100
V
Current
inductance
L
500
H
P
arasitic
resistance
r
2
Ω
Floating
capacitor
C
f
c
100
µ
F
Output
capacitor
C
o
100
µ
F
Load
R
40
Ω
Switching
frequenc
y
f
sw
20
k
H
z
T
able
2.
Controller
parameters
Item
Symbol
V
alue
Unit
PI
coef
cients
ω
0
312
rad/s
m
4
−
K
p
0.833
mA.V
−
2
τ
i
0.833
A.V
−
2
.s
−
1
Output
v
oltage
re
gulator
coef
cient
k
L
18000
−
Floating
capacitor
v
oltage
re
gulator
coef
cient
k
c
18000
−
Routh
Hurwitz
coef
cients
a
0
66100
−
a
1
1
.
0706
×
10
9
−
a
2
1
.
0706
×
10
11
−
a
3
1
.
0640
×
10
13
−
5.1.
Operation
under
normal
mode
The
system’
s
response
in
normal
operation
is
depicted
in
Figures
4–6.
These
gures
illustrate
that
the
inductance
current
i
L
,
ying
capacitor
v
oltage
v
c
and
the
output
v
oltage
v
o
,
are
well
re
gulated
to
their
reference
v
alues.
Where,
Figure
4
sho
ws
the
inductance
current
tracking
its
reference
v
alue
with
minimal
de
viation,
indicating
that
the
inner
control
loop
ef
fecti
v
ely
maintains
the
desired
current
le
v
el.
This
precise
control
ensures
that
the
current
supplied
to
the
load
rema
ins
stable,
contrib
uting
to
the
o
v
erall
stability
of
the
L
yapuno
v
contr
oller
design
for
a
fuel
cell
electric
vehicle
power
ed
by
a
thr
ee-le
vel
...
(Chaouqi
Aouadi)
Evaluation Warning : The document was created with Spire.PDF for Python.
1626
❒
ISSN:
2088-8694
po
wer
system.
Figure
5
sho
ws
the
v
oltage
across
t
he
ying
capacitor
.
The
plot
demonstrates
that
the
ying
capacitor
v
oltage
is
consistently
re
gulated,
reecting
the
controller’
s
ability
to
manage
the
v
oltage
stres
s
on
the
semiconductors
ef
fecti
v
ely
.
Proper
re
gulation
of
the
ying
capacitor
v
ol
tage
is
crucial
for
the
longe
vity
and
reliability
of
the
po
wer
-electronic
components
within
the
s
ystem.
Figure
6
highlights
the
output
v
oltage
v
o
,
sho
wing
its
stable
tracking
of
the
reference
v
oltage.
0
0.005
0.01
0.015
0.02
Time (S)
10
20
30
40
50
Current (A)
i
L
i
L
*
Figure
4.
Inductance
current
i
L
and
its
reference
0
0.005
0.01
0.015
0.02
Time (s)
0
20
40
60
80
Voltage (V)
v
C
v
C
*
Figure
5.
FC
v
oltage
v
c
and
its
reference
5.2.
Rob
ustness
tests
under
parameter
output
v
oltage
r
efer
ence
v
ariation
Figures
7–9
sho
w
the
simulation
results
for
a
step
change
in
the
reference
output
v
oltage
v
o
.
The
reference
v
oltage
is
altered
from
100
V
to
140
V
at
time
t
1
=
20
ms
and
from
140
V
back
to
100
V
at
t
2
=
40
ms
.
Despite
these
abrupt
changes,
the
gures
demonstrate
that
the
signals
i
L
,
v
c
and
v
o
closely
follo
w
their
respecti
v
e
reference
v
alues,
Figure
7
illustrates
the
response
of
the
inductance
current
i
L
,
which
adapts
rapidly
to
the
ne
w
reference
v
alues.
The
control
system
ef
fecti
v
ely
manages
the
transit
ion,
ensuring
that
the
current
remains
stable
and
within
the
desired
range.
This
rapid
adjustment
indicates
the
inner
loop’
s
rob
ustness
and
its
ability
to
handle
sudden
changes
without
signicant
o
v
ershot
or
oscillation.
Figure
8
sho
ws
the
v
oltage
across
the
ying
capacitor
v
c
.
The
plot
re
v
eals
that
the
ying
capacitor
v
oltage
quickly
stabilizes
after
each
change
in
the
reference
output
v
oltage,
highlighting
the
controller’
s
ef
cienc
y
in
managing
v
oltage
stress.
This
capability
is
crucial
for
maintaining
the
reliability
of
the
po
wer
electronic
components,
as
it
pre
v
ents
e
xcessi
v
e
v
oltage
v
ariations
that
could
potentially
harm
the
system.
Figure
9
focuses
on
the
output
v
oltage
v
o
,
demonstrating
its
precise
tracking
of
the
reference
v
oltage
throughout
the
changes.
The
controller
successfully
adjusts
the
output
v
oltage
to
the
ne
w
reference
v
alues
with
minimal
delay
.
0
0.005
0.01
0.015
0.02
Time (s)
0
50
100
150
Voltage (V)
v
o
v
o
*
Figure
6.
Output
v
oltage
v
o
and
its
reference
0
0.02
0.04
0.06
Times (S)
0
10
20
30
40
50
Current (A)
i
L
i
L
*
Figure
7.
Inductance
current
i
L
and
its
reference
0
0.01
0.02
0.03
0.04
0.05
0.06
Times (S)
0
20
40
60
Voltage (V)
v
c
v
c
*
Figure
8.
FC
v
oltage
v
c
and
its
reference
0
0.01
0.02
0.03
0.04
0.05
0.06
Time (S)
0
50
100
150
200
Voltage (V)
v
o
v
o
*
Figure
9.
FC
v
oltage
v
c
and
its
reference
Int
J
Po
w
Elec
&
Dri
Syst,
V
ol.
17,
No.
3,
September
2026:
1620–1629
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
❒
1627
5.3.
Rob
ustness
tests
under
load
v
ariation
In
the
third
test,
the
load
under
goes
v
ariation
from
40
Ω
to
20
Ω
at
t
1
=
20
ms
and
from
20
Ω
to
10
Ω
at
the
time
t
2
=
40
ms
.
The
results
of
the
test
are
sho
wn
in
Figures
10
and
11.
These
gures
demonstr
ate
that
the
proposed
controller
successfully
maintains
the
inductance
current
i
L
and
the
output
v
oltage
v
c
at
their
reference
v
alues
despite
the
signicant
change
in
resisti
v
e
load.
Figure
10
displays
the
response
of
the
inductance
current
i
L
.
Ev
en
with
the
sudden
decrease
in
load
resistance,
the
inductance
current
closely
follo
ws
its
reference
v
alue,
sho
wcasing
the
controller’
s
ability
to
adapt
t
o
changes
in
the
load.
Figure
11
sho
ws
the
ying
capacitor
v
oltage
v
c
.
The
plot
indicates
that
v
c
stays
balanced
despite
the
changes
in
load
resistance,
highlighti
n
g
the
control
ler’
s
capability
to
maintain
v
oltage
stability
across
the
ying
capacitor
.
Figure
10.
Inductance
current
i
L
and
its
reference
under
parameter
v
ariations
0
0.01
0.02
0.03
0.04
0.05
0.06
Time (s)
0
20
40
60
80
Voltage (V)
v
c
v
c
*
Figure
11.
FC
v
oltage
v
c
and
its
reference
under
parameter
v
ariations
6.
CONCLUSION
This
w
ork
presents
the
control
of
a
three-le
v
el
ying
capacitor
boost
con
v
erter
inte
grated
into
a
h
ybrid
electric
v
ehicle
po
wered
by
a
fuel
cell.
The
proposed
controller
features
an
inter
nal
loop
des
igned
to
re
gulate
the
input
current
and
balance
the
oating
capacitor
v
oltage,
de
v
eloped
using
the
L
yapuno
v
technique.
Additionally
,
an
e
xternal
loop
re
gulates
the
output
v
oltage
and
pro
vides
a
suitable
reference
for
the
inductor
current.
The
stability
analysis
conducted
in
this
w
ork
has
v
eried
that
the
control
objecti
v
es
are
ef
fecti
v
ely
achie
v
ed.
Simulation
results
ha
v
e
v
alidated
the
controllers
rob
ustness
ag
ainst
load
v
ariations
and
changes
in
the
output
v
oltage
reference,
these
ndings
corroborate
the
theoretical
results
and
demonstrate
the
practical
ef
fecti
v
eness
of
the
proposed
control
strate
gy
.
In
conclusion,
the
de
v
eloped
controller
impro
v
es
the
operational
stability
and
ef
cienc
y
of
the
three-le
v
el
ying
capacitor
boost
con
v
erter
in
a
h
ybrid
electric
v
ehicle
conte
xt.
The
study
contrib
utes
v
aluable
insights
into
adv
anced
control
techniques
for
po
wer
electronics
applications,
pa
ving
the
w
ay
for
further
adv
ancements
in
h
ybrid
v
ehicle
technologies.
FUNDING
INFORMA
TION
This
research
recei
v
ed
no
e
xternal
funding.
A
UTHOR
CONTRIB
UTIONS
ST
A
TEMENT
This
journal
uses
the
Contrib
utor
Roles
T
axonomy
(CRediT)
to
recognize
indi
vidual
author
contrib
utions,
reduce
authorship
disputes,
and
f
acilitate
collaboration.
Name
of
A
uthor
C
M
So
V
a
F
o
I
R
D
O
E
V
i
Su
P
Fu
Chaouqi
Aouadi
✓
✓
✓
✓
✓
✓
✓
✓
Abdelmajid
Abouloif
a
✓
✓
✓
✓
✓
✓
✓
Meriem
Aourir
✓
✓
✓
✓
✓
Ibtissam
Lachkar
✓
✓
✓
✓
C
:
C
onceptualization
I
:
I
n
v
estig
ation
V
i
:
V
i
sualization
M
:
M
ethodology
R
:
R
esources
Su
:
Su
pervision
So
:
So
ftw
are
D
:
D
ata
Curation
P
:
P
roject
Administration
V
a
:
V
a
lidation
O
:
Writing
-
O
riginal
Draft
Fu
:
Fu
nding
Acquisition
F
o
:
F
o
rmal
Analysis
E
:
Writing
-
Re
vie
w
&
E
diting
L
yapuno
v
contr
oller
design
for
a
fuel
cell
electric
vehicle
power
ed
by
a
thr
ee-le
vel
...
(Chaouqi
Aouadi)
Evaluation Warning : The document was created with Spire.PDF for Python.
1628
❒
ISSN:
2088-8694
CONFLICT
OF
INTEREST
ST
A
TEMENT
The
authors
declare
no
conict
of
interest.
D
A
T
A
A
V
AILABILITY
Data
supporting
the
ndings
of
this
study
are
a
v
ailable
in
the
article.
More
information
is
a
v
ailable
from
the
corresponding
authors
upon
reasonable
request.
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BIOGRAPHIES
OF
A
UTHORS
Chaouqi
Aouadi
holds
his
Ph.D.
de
gree
in
control
engine
ering
and
rene
w
able
ener
gies
in
2018
and
a
Master
de
gree
in
information
processing
in
2014
from
the
F
aculty
of
Science
Ben
M’
sick,
Hassan
II
Uni
v
ersity
.
He
is
currently
profess
or
with
the
department
of
Electrical
Engineering
at
National
High
school
of
electricity
and
Mechanics
(ENSEM)
at
Hassan
II
uni
v
ersity
of
Casablanca,
Morocco.
His
research
areas
of
i
nterest
include
po
wer
con
v
erter
,
nonlinear
control
and
articial
intelligent.
He
can
be
contacted
at
email:
c.aouadi@ensem.ac.ma
or
chaouqi.aouadi@uni
vh2c.ma.
Abdelmajid
Abouloifa
recei
v
ed
the
Aggre
g
ation
of
Electrical
Engineering
from
Normal
High
School,
Rabat,
Morocco,
in
1999,
and
Ph.D.
de
gree
in
control
engineering
from
the
Uni
v
ersity
of
Caen
Basse-Normandie,
Caen,
France
and
from
Moham
med
V
Uni
v
ersity
,
Rabat,
Moroc
co
in
2008.He
is
currently
Professor
at
the
National
High
School
for
Electricity
and
mechanics
(ENSEM),
Uni
v
ersity
Hassan
II,
Casablanca,Morocco.
His
research
interests
include
high
frequenc
y
po
wer
con
v
erter
topologies,
po
wer
-f
actor
-correction
techniques,
po
wer
supplies,
and
nonlinear
control.
He
can
be
contacted
at
email:
a.abouloif
a@gmail.com.
Meriem
Aourir
recei
v
ed
her
Ph.D.
de
gree
in
in
the
eld
of
automatic,
rene
w
able
ener
gy
,
Po
wer
electronics
in
2021
and
she
has
got
master
de
gree
in
information
processing
from
F
aculty
of
Science
Ben
M’
sik,
Uni
v
ersity
of
Hassan
II
Casablanca
in
2015.
Currently
,
she
is
a
permanent
professor
in
Higher
Institute
of
engineering
and
b
usines.
She
can
be
contacted
at
email:
meriem.aourir@gmail.com.
Ibtissam
Lachkar
recei
v
ed
the
graduate
de
gree
from
the
Normal
High
School
of
T
echnical
Education,
Rabat,
Morocco,
in
1995,
the
de
gree
in
high
depth
studies
from
the
Mohammadia
School
of
Engineers,
Rabat,
in
2005,
and
the
Ph.D.
de
gree
from
the
Mohammadia
School
of
Engineers.
Currently
,
she
is
a
professor
at
the
National
High
School
for
Electricity
and
mechanics
(ENSEM),
Hassan
II
Uni
v
ersity
of
Casablanca.
Her
research
interests
include
modeling,
nonlinear
control,
po
wer
con
v
erters,
uninterruptible
po
wer
supplies,
and
photo
v
oltaic
systems
and
drones.
She
can
be
contacted
at
email:
lachkaribtissam@gmail.com.
L
yapuno
v
contr
oller
design
for
a
fuel
cell
electric
vehicle
power
ed
by
a
thr
ee-le
vel
...
(Chaouqi
Aouadi)
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