Inter
national
J
our
nal
of
P
o
wer
Electr
onics
and
Dri
v
e
Systems
(IJPEDS)
V
ol.
12,
No.
2,
Jun
2021,
pp.
858
∼
869
ISSN:
2088-8694,
DOI:
10.11591/ijpeds.v12.i2.pp858-869
❒
858
An
efcient
pr
edicti
v
e
curr
ent
contr
oller
with
adapti
v
e
parameter
estimation
in
3-
Φ
in
v
erter
Haddar
Mabr
ouk,
Allaoua
Boumediene
Smart
Grids
&
Rene
w
able
Ener
gy
(SGRE)
Laboratory
,
Uni
v
ersity
of
T
ahri
Mohammed,
Bechar
,
Algeria
Article
Inf
o
Article
history:
Recei
v
ed
Dec
1,
2020
Re
vised
Feb
14,
2021
Accepted
Mar
19,
2021
K
eyw
ords:
Filter
Grid-connected
In
v
erter
MRAS
observ
er
Predicti
v
e
control
ABSTRA
CT
In
this
paper
,
a
detail
design
and
descri
ption
of
a
predicti
v
e
current
control
scheme
are
adopted
for
three-phase
grid-connected
tw
o-le
v
el
in
v
erter
and
its
application
in
wind
ener
gy
con
v
ersion
sys
tems.
Despite
its
adv
antages,
the
predicti
v
e
current
controller
is
v
ery
sensiti
v
e
to
parameter
v
ariations
which
could
e
v
entually
af
fected
on
system
sta-
bility
.
T
o
solv
e
this
problem,
an
estimation
technique
proposed
to
identify
the
v
alue
of
harmonic
lter
parameter
based
on
Model
reference
adapti
v
e
system
(MRAS).
L
ya-
puno
v
stability
theory
is
selected
to
guarantee
a
rob
ust
adaptation
and
stable
response
o
v
er
lar
ge
system
parameter
v
ariation.
The
simulation
results
sho
ws
the
ef
cienc
y
of
the
proposed
techniques
to
impro
v
e
the
current
tracking
performance.
This
is
an
open
access
article
under
the
CC
BY
-SA
license
.
Corresponding
A
uthor:
Haddar
Mabrouk
Department
of
Electrical
Engineering
SGRE
Laboratory
,
Uni
v
ersity
of
T
ahri
Mohammed,
Bechar
,
Algeria
B.P
417
route
k
enadsa
08000,
Bechar
,
Algeria
Email:
haddar
.mabrouk@uni
v-bechar
.dz
1.
INTR
ODUCTION
In
recent
years,
the
grid-connected
in
v
erters
ha
v
e
made
a
giant
strides
in
the
v
arious
applications
of
industries
and
rene
w
able
ener
gies
,
that
embody
by
introduce
a
ne
w
concepts
of
adv
anced
control
in
detriment
to
other
con
v
entional
controls.
The
grid-connected
in
v
erters
is
widely
used
in
wind
ener
gy
systems.
The
simplied
structure
of
the
grid-connected
in
v
erters
is
illustrated
in
Figure
1.
Where,
it’
s
possible
to
replace
the
maximum
amount
of
po
wer
that
can
be
withdra
wn
from
the
wind
turbine,
generator
,
and
rectier
o
wed
through
the
in
v
erter
by
adequate
v
ariable
DC-current
source
without
causing
an
y
damage
in
system
caracteristic
[1]-[6].
(a)
(b)
Figure
1.
Grid-connected
in
v
erter
in
a
wind
ener
gy
concersion
system,
(a)
General
structure,
(b)
Simplied
structure
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
❒
859
On
the
other
side,
The
progress
of
the
micro
processor
unit
allo
wed
the
possibility
to
impro
v
e
the
controllers
design
with
v
ery
short
sampling
time
and
good
performance.
The
model
predicti
v
e
control
(
MPC)
is
one
of
the
most
important
miles
tones
of
de
v
elopment,
as
it
recently
occupied
a
lar
ge
part
of
studies
and
the
researchers’
ef
fort
as
promise
control
methods
because
of
its:
(1
simple
theory
concept,
(2
easy
to
im-
plement
(3
e
xible
and
promising
for
digital
control,
(4
f
ast
dynamic
response,
[2]
,[3].
Ho
we
v
er
,
despite
of
its
adv
antages,
the
MPC
is
sensi
ti
v
e
to
the
system
parameter
v
ariations
during
the
operating
conditions.
This
results
of
perturbations
inuences
on
the
o
v
erall
stability
of
the
system,
which
mak
es
MPC
need
to
adaptation
or
estimation
algorithm
[7].
The
MPC
applied
for
po
wer
con
v
erters
and
electrical
dri
v
es
include
tw
o
main
cate
gories:
Continuous
Control
Set
MPC
with
a
modulation
stage
and
Finite
Control
Set
MPC
(FCS-MPC)
without
a
modulation
stage
[8],
[9].
In
FCS-MPC,
the
control
objecti
v
es
in
cost
function
may
be
either
main
or
additional
[7]-[11],
the
rst
one
using
ph
ysics
quantities
such
as
current,
v
oltage,
po
wer
,
torque,
etc.
It
e
v
alu-
ates
error
tracking
between
an
y
predicted
v
ariable
v
alue
and
its
reference.
While
the
second
one
is
considering
as
sub-objecti
v
es
control
for
e
xample
switching
frequenc
y
reduce.
The
additional
control
objecti
v
e
multiplying
by
suitable
v
alues
of
weighting
f
actors
witch’
s
reecting
its
importance
in
optimization
criteria.
The
algorithm
of
FCS-MPC
minimizes
the
cost
function
and
applies
the
generated
switching
signal
at
end
each
the
sampling
time
directly
on
po
wer
con
v
erter
[11],
[12].
The
rob
ustness
of
FSC-MPC
ag
ainst
parameter
v
ariations
still
major
concern
in
man
y
studies
[12].
Addressing
the
issue,
a
man
y
control
strate
gies
ha
v
e
been
proposed
to
estimate
the
parameters
v
ariable
in
po
wer
con
v
erter
applications.
In
[9],
an
analytical
approach
is
proposed
to
e
xam
the
inuence
of
model
parametric
uncertainties
on
the
prediction
of
FSC-MPC
error
for
current
control
in
three
phase
tw
o-le
v
el
in
v
erter
.
In
[13],
the
authors
proposed
adapti
v
e
observ
er
to
accurately
estimate
tw
o
v
ariable:
speed
and
ux,
this
observ
er
based
encoderless
FCS-PTC
(predicti
v
e
torque
control)
helps
k
eeping
the
stability
for
induction
machine.
The
Model
Reference
Adapti
v
e
System
(MRAS)
observ
er
is
used
in
[14],
where
the
authors
estimate
line
inductance
to
impro
v
e
rob
ustness
for
sensorless
predicti
v
e
control
of
Acti
v
e
front
End
(AFE)
rectiers.
In
[15],
a
control
scheme
based
on
a
computationally
nite-set
model
predicti
v
e
po
wer
control
(FS-MPPC)
for
grid-connected
photo
v
oltaic
systems
is
proposed,
the
stability
of
(FS-MPPC)
ag
ainst
inductance
v
ariation
is
v
eried
by
means
of
no
v
el
online
nite-set
model
inductance
estimati
on
technique.
In
[16],
an
e
xtended
Kalman
lter
(EKF)
is
proposed
to
estimate
the
system
parameter
(
lter
grid
impedance)
for
AFE.
Ho
we
v
er
,
the
MRAS
technique
has
been
appro
v
ed
to
be
a
po
werful
tool
for
parameter
estimation,
the
MRAS
is
widely
used
in
po
wer
electronics
application
and
motor
dri
v
es
system
to
mainly
estimate:
(1
the
lter
parameter
[16],
(2)
the
machine
state
v
ariables
(speed,
ux,...)
for
sensorless
dri
v
e
systems
[18],
[19],...etc,
b
ut
there
are
limited
researches
based
on
MRAS
for
grid-connected
in
v
erter
[16].
The
MRAS
consists
of
tw
o
models,
the
reference
one
and
the
adapti
v
e
one.
The
dif
ference
between
the
outputs
of
these
tw
o
models
is
then
used
in
an
adaptation
mechanism
such
as
lyapuno
v
theory
to
adjust
the
parameters
in
the
adapti
v
e
model
until
the
response
of
the
main
tw
o
models
become
consistent
and
the
tracking
error
con
v
er
ges
to
zero
[19],
[20].
On
this
light,
the
performance
of
predicti
v
e
current
controller
for
the
On
this
light,
the
perf
ormance
of
predicti
v
e
current
controller
for
the
three-phase
grid-connected
tw
o-le
v
el
in
v
erter
entirely
depends
on
its
mathematical
model
as
wel
l
as
the
accurac
y
of
the
parameters
that
may
assumed
in
theory
constant
and
kno
wn
b
ut
pract
ically
not
so.
In
this
paper
,
the
lter
parameters
may
easily
af
fected
due
to
the
heat
risen,
magnetic
saturation
and
system
lifetime,
etc.
This
paper
presents
a
structure
that
combines
tw
o
types
of
controller;
the
rst
one
is
a
FCS-MPC
witch
generate
a
switching
states
for
t
he
grid-connected
in
v
erter
b
ut
it
is
inuenced
by
parameters
v
ariation,
this
could
yield
ne
g
ati
v
e
results
as
inaccurate
prediction
of
the
fut
ure
beha
vior
due
to
an
incorrect
system
model
and
so
the
selection
of
the
i
ncorrect
switching
states.
The
second
is
a
classic
which
relies
on
the
MRAS
observ
er
to
cope
with
the
aforementioned
dra
wback
through
a
parameter
estimation
method
based
on
L
yapuno
v
stability
theory
.
Therefore,
it
is
possible
to
a
v
oid
the
deterioration
of
the
control
performance.
There
are
a
fe
w
researches
report
on
the
applic
ation
the
MRAS
observ
er
to
grid-connected
in
v
erter
[17].
So,
this
no
v
el
structure
brings
great
adv
antages
such
as
simple
control
scheme,
No
inner
control
loop,
and
No
modulator
.
Additionally
,
although
the
proposed
estimation
t
echnique
is
classic,
it
can
mak
e
the
system
operate
rob
ustly
and
independently
from
mathematical
model.
This
paper
is
arranged
as
follo
ws:
Section
(2
discusses
the
system
structure
and
modeling.
Section
(3
and
section
(4,
present
the
theory
behind
the
v
oltage
oriented
v
ector
and
the
predicti
v
e
current
control
strate
gy
,
respecti
v
ely
.
The
identication
of
the
lter
parameter
based
on
MRAS
observ
er
are
presented
is
gi
v
en
in
Section
(5.
Section
(6
contains
The
simulation
results
and
analysis,
and
nally
conclusions
are
discussed
in
Section
(7.
An
ef
cient
pr
edictive
curr
ent
contr
oller
with
adaptive
par
ameter
estimation...
(Haddar
Mabr
ouk)
Evaluation Warning : The document was created with Spire.PDF for Python.
860
❒
ISSN:
2088-8694
2.
SYSTEM
STR
UCTURE
AND
MODELING
The
con
v
entional
po
wer
circuit
of
a
three-phase
grid-connected
tw
o-le
v
el
in
v
erter
applied
to
wind
ener
gy
is
modeled
as
sho
wn
in
Figure
2.
Where,
a
harmonic
lter
(
r
g
,
L
g
)
tak
es
place
between
the
grid
and
the
in
v
erter
.
The
models
of
the
three-phase
in
v
erter
and
grid
current
dynamic
are
briey
described
in
the
follo
wing
subsections..
2.1.
In
v
erter
model
The
tw
o-le
v
el
in
v
erter
is
composed
of
six
bidirectional
switches.
Since
there
are
three
phases
with
tw
o
operating
modes
in
each
phase,
the
in
v
erter
is
able
to
generate
only
2
3
=
8
states
as
dif
ferent
possible
output
v
oltages.
The
operating
modes
of
the
in
v
erter
are
summarized
as
follo
ws
[2]:
S
x
+
¯
S
x
=
1
for
x
∈
{
a,
b,
c
}
(1)
Figure
2.
T
opology
of
grid-connected
tw
o-le
v
el
in
v
erter
in
wind
ener
gy
con
v
ersion
systems
By
means
of
the
space
v
ector
tool,
The
output
v
oltage
generated
by
t
he
in
v
erter
can
e
xpressed
in
terms
of
the
dif
ferent
switching
states
by
v
αi
v
β
i
=
V
dc
2
3
1
−
1
2
−
1
2
0
√
3
2
−
√
3
2
S
a
S
b
S
c
(2)
Where
V
dc
is
DC
link
v
oltage,
v
αi
and
v
β
i
are
α
and
β
component
of
the
in
v
erter
output
v
oltage
v
ectors.
Figure
3
sho
ws
the
eight
output
in
v
erter
v
ol
tage
v
ectors
in
space
v
ectors
corresponding
the
all
possible
switching
states
which
including
six
dif
ferent
acti
v
e
v
oltage
v
ectors
(
V
1
to
V
6
)
and
tw
o
other
zero
v
ectors
(
V
0
and
V
7
).
Figure
3.
Space
v
ectors
generated
by
the
in
v
erter
The
in
v
erter
v
oltage
component
e
xpressed
in
the
dq
-frame
rotating
at
angle
θ
g
can
be
related
to
the
α
β
component
by
v
di
v
q
i
=
cos
θ
g
sin
θ
g
−
sin
θ
g
cos
θ
g
v
αi
v
β
i
(3)
Where
v
di
and
v
q
i
are
dq
-axis
components
of
the
in
v
erter
output
v
oltage.
Int
J
Po
w
Elec
&
Dri
Syst,
V
ol.
12,
No.
2,
Jun
2021
:
858
–
869
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
❒
861
2.2.
Grid
curr
ent
dynamic
model
The
continuous-time
grid
current
dynamic
equation
of
three-phase
grid-tied
in
v
erter
can
be
written
in
the
dq
-frame
as
[1]-[4]
di
dg
dt
=
−
r
g
L
g
i
dg
+
ω
g
i
q
g
+
1
L
g
(
v
di
−
v
dg
)
di
q
g
dt
=
−
ω
g
i
dg
−
r
g
L
g
i
q
g
+
1
L
g
(
v
q
i
−
v
q
g
)
(4)
Where
v
dg
and
v
q
g
are
dq
-axis
components
of
the
grid
v
oltage
and
ω
g
is
the
angular
fraquenc
y
of
the
grid,
r
g
lter
resistance
and
L
g
is
lter
inductance.
The
grid
acti
v
e
and
reacti
v
e
po
wers
are
e
xpressed
by
[19]:
P
g
=
3
2
{
v
dg
i
dg
+
v
q
g
i
q
g
}
Q
g
=
3
2
{
v
q
g
i
dg
−
v
dg
i
q
g
}
(5)
3.
V
OL
T
A
GE
ORIENTED
VECT
OR
(V
OC)
In
the
literature,
the
phase-lock
ed
loop
(PLL)
is
needed
not
only
to
nd
the
grid
v
oltage
angle
θ
g
for
the
grid
synchronization,
b
ut
also
to
realize
the
v
oltage
oriented
control
(V
OC).
Where,
the
dq
-axis
components
of
the
grid
v
oltage
is
oriented
to
be
align
only
with
its
d-axis
component
and
eliminate
the
second
component.
That
means,
the
acti
v
e
po
wer
equation
and
reacti
v
e
po
wer
equation
are
simplied
to
[20]
P
g
=
3
2
v
dg
i
dg
Q
g
=
−
3
2
v
dg
i
q
g
(6)
As
described
pre
viously
,
It
is
noted
that
the
d
-axis
current
reference
is
i
∗
dg
related
to
the
grid
acti
v
e
po
wer
.
It
can
be
adjusted
through
PI
controller
which
maintains
the
measured
DC-link
v
oltage
v
dc
at
its
gi
v
en
reference
v
alue
v
∗
dc
,
we
can
write
i
∗
dg
=
(
k
p
+
k
i
s
)(
v
∗
dc
−
v
dc
)
(7)
The
q
-axis
current
reference
i
∗
q
g
can
be
calculated
from
reference
reacti
v
e
po
wer
of
the
grid
Q
∗
g
as
i
∗
q
g
(
k
)
=
−
Q
∗
g
1
.
5
v
∗
dg
(8)
4.
PREDICTIVE
CURRENT
CONTR
OL
(PCC)
According
to
the
8
switching
possible
states
for
tw
o-le
v
el
in
v
erter
,
the
predicti
v
e
current
control
ler
e
xploit
the
discrete-time
model
with
one-step
for
grid-connected
in
v
erter
to
predict
at
the
ne
xt
instant
the
future
beha
vior
of
the
controlled
grid
curr
ent.
It
consists
of
thr
ee
major
subsystems:
e
xtrapolation
of
reference
currents,
predicti
v
e
model,
and
cost
function
minimization.
At
each
sampling
time,
cost
function
v
alues
are
calculated
for
all
of
possible
commutation
states,
based
on
predened
crit
erion.
Then,
the
smallest
v
alue
of
the
cost
function
will
be
used
to
determine
the
optimal
switching
state
applied
to
the
in
v
erter
at
the
ne
xt
period
[2].
Approximating
the
grid
current
deri
v
ati
v
es
by
[1]-[3]
di
g
dt
≈
i
g
(
k
+
1)
−
i
g
(
k
)
T
s
(9)
An
ef
cient
pr
edictive
curr
ent
contr
oller
with
adaptive
par
ameter
estimation...
(Haddar
Mabr
ouk)
Evaluation Warning : The document was created with Spire.PDF for Python.
862
❒
ISSN:
2088-8694
The
discrete-time
model
on
dq
-frame
of
grid
currents
at
(
k
+
1)
state
is
e
xpressed
as
follo
ws:
i
dg
(
k
+
1)
i
q
q
(
k
+
1)
=
1
−
r
g
T
s
/L
g
0
0
1
−
r
g
T
s
/L
g
i
dg
(
k
)
i
q
g
(
k
)
+
L
g
0
0
L
g
v
di
(
k
)
−
v
dg
(
k
)
v
q
i
(
k
)
−
v
q
g
(
k
)
(10)
Ho
we
v
er
,
the
future
reference
of
the
grid
current
i
∗
g
,dq
(
k
+
1)
can
be
estimated
by
means
of
current
and
pre
vious
v
alue
of
the
reference
current
with
the
help
of
second-order
Lagrange
e
xtrapolation
as
follo
ws
[2],
[3].
i
∗
dg
(
k
+
1)
i
∗
q
q
(
k
+
1)
=
2
i
dg
(
k
)
i
q
g
(
k
)
−
i
∗
dg
(
k
−
1)
i
∗
q
q
(
k
−
1)
(11)
In
t
his
paper
,
the
rst
tar
get
of
the
predicti
v
e
current
controller
is
to
achie
v
e
the
smallest
current
error
between
the
e
xtrapolated
reference
currents
and
the
predicted
current.
The
second
tar
get
is
to
reduce
signicantly
the
switching
frequenc
y
by
adjusting
the
commutation
number
between
tw
o
successi
v
e
sampling
instants.
In
situations
where
the
switching
loses
are
important.
In
brief,
These
all
objecti
v
es
can
be
e
xpressed
in
the
form
of
a
cost
function
g
to
be
mi
nimized.
The
cost
function
summarizes
the
desired
beha
vior
of
the
in
v
erter
can
be
obtained
using
the
error
squared
such
as
:
g
=
i
∗
dg
(
k
+
1)
−
i
dg
(
k
+
1)
2
+
i
∗
q
g
(
k
+
1)
−
i
q
g
(
k
+
1)
2
+
λ
sw
(
S
(
k
)
−
S
(
k
−
1))
2
(12)
Where
λ
sw
is
weighting
f
actors
for
switching
frequenc
y
reduction,
S
(
k
)
and
S
(
k
−
1)
are
the
present
switching
state
and
past
applied
switching
state
respecti
v
ely
.
Figure
4
sho
ws
the
proposed
predicti
v
e
current
controller
for
the
grid-connected
tw
o-le
v
el
in
v
erter
.
In
this
scheme,
getting
the
angle
θ
g
from
the
measured
grid
v
oltage
is
done
via
PLL
method
to
mak
e
v
dg
equal
to
V
g
and
also
to
transforme
the
v
ariables
measured
(
v
g
and
i
g
)
from
abc
-frame
to
dq
-frame.
Once
reference
grid
currents
i
∗
g
,dq
(
k
)
is
calculated,
the
PCC
procedure
is
established
to
select
the
optimal
switching
signal
applied
to
the
in
v
erter
.
Moreo
v
er
,
the
scheme
in
v
olv
e
s
the
MRAS
observ
er
to
impro
v
e
the
rob
usrness
of
PCC
by
identifying
the
lter
parameter
.
Figure
4.
Proposed
predicti
v
e
current
controller
scheme
for
grid-connected
tw
o-le
v
el
in
v
erter
.
5.
IDENTIFICA
TION
OF
THE
HARMONIC
FIL
TER
P
ARAMETERS
B
ASED
ON
MRAS
In
the
real
time,
Both
v
alues
of
harmonic
lter
parameters
(inductance
and
resistance)
link
ed
between
the
in
v
erter
and
the
grid
are
changed
under
the
operating
conditions
that
reduce
system
ef
cienc
y
in
term
of
current
quality
.
So
that,
these
parameters
are
estimated
based
on
the
model
reference
adapti
v
e
system
(MRAS).
Int
J
Po
w
Elec
&
Dri
Syst,
V
ol.
12,
No.
2,
Jun
2021
:
858
–
869
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
❒
863
Figure
5
presents
the
basic
struct
ure
of
MRAS,
wich
consi
sts
of
three
parts:
a
reference
model
,
adapti
v
e
model,
and
an
adapti
v
e
controller
.
Figure
5.
Basic
structure
of
MRAS
The
equation
of
state
for
reference
model:
(4)
⇔
d
i
g
dt
=
A
i
g
+
B
u
(13)
Where
i
g
=
i
dg
i
q
g
T
,
A
=
α
ω
g
−
ω
g
α
,
α
=
−
r
g
L
g
,
B
=
−
1
L
g
,
u
=
u
d
u
q
T
,
u
d
=
v
di
−
v
dg
,
u
q
=
v
q
i
−
v
q
g
The
adapti
v
e
model
is
gi
v
en
as
follo
ws:
d
ˆ
i
g
dt
=
A
ˆ
i
g
+
B
u
(14)
Where
ˆ
i
g
=
ˆ
i
dg
ˆ
i
q
g
T
,
ˆ
A
=
ˆ
α
ω
g
−
ω
g
ˆ
α
,
ˆ
α
=
−
ˆ
r
g
ˆ
L
g
,
ˆ
B
=
−
1
ˆ
L
g
,
u
=
u
d
u
q
T
i
g
and
ˆ
i
g
are
the
output
current
of
reference
model
and
adapti
v
e
model
respecti
v
ely
.
The
measured
grid
current
is
tuning
as
the
reference
model
witch
is
strongly
reliant
on
the
rated
lter
parameters
(
r
g
and
L
g
),
and
the
estimated
grid
current
is
tuning
as
the
adapti
v
e
model
witch
uses
a
v
alue
of
estimated
lter
parameters
(
ˆ
r
g
and
ˆ
L
g
),
and
then,
the
output
current
error
of
tw
o
models
is
used
by
adapti
v
e
controller
to
adjust
the
parameter
in
adapti
v
e
controller
until
the
identication
becomes
asymptotically
stable
and
the
current
error
approaches
zero
as
small
as
possible.
The
current
error
is
dened
as
e
=
i
g
−
ˆ
i
g
=
[
e
d
e
q
]
T
,
the
error
state
equation
can
be
written:
˙
e
=
A
e
+
(
A
−
ˆ
A
)
ˆ
i
g
+
(
B
−
ˆ
B
)
r
(15)
Letting
aI
=
(
A
−
ˆ
A
)
,
b
=
(
B
−
ˆ
B
)
=
1
L
g
−
1
ˆ
L
g
,
Z
=
[
ˆ
i
g
u
]
T
and
∆
T
=
[
a
b
]
T
The
(15)
can
be
written
as
˙
e
=
A
e
+
∆
T
Z
(16)
Considering
a
L
yapuno
v
function
candidate
as
follo
w:
V
(
e
,
t
)
=
1
2
(
e
T
P
e
+
∆
T
Z
∆)
(17)
Where
P
=
I
2
×
2
and
A
=
λ
1
0
0
λ
2
−
1
λ
1
and
λ
2
are
arithmetic
number
.
Starting
from
L
yapuno
v
stability
concept,
the
L
yapuno
v
function
must
respond
the
follo
wing
three
necessary
conditions
to
ensure
that
the
identication
is
asymptotically
stable
[22]-[25]:
1)
V
(
e
,
t
)
>
0
,
An
ef
cient
pr
edictive
curr
ent
contr
oller
with
adaptive
par
ameter
estimation...
(Haddar
Mabr
ouk)
Evaluation Warning : The document was created with Spire.PDF for Python.
864
❒
ISSN:
2088-8694
2)
˙
V
(
e
,
t
)
<
0
,
3)
V
(
e
,
t
)
−
→
8
as
|
e
|
−
→
8
.
It
is
e
vident
that
the
rst
and
third
condit
ions
are
met.
The
second
condition
can
be
discussed
as
follo
ws:
˙
V
(
e
,
t
)
=
1
2
(
˙
e
T
P
e
+
e
T
P
˙
e
+
˙
∆
T
Z
∆
+
∆
T
Z
˙
∆)
=
1
2
(
e
T
(
P
A
+
P
A
T
)
e
+
a
(
ˆ
i
d
e
d
+
ˆ
i
q
e
q
+
˙
a
λ
1
)
+
b
(
u
d
e
d
+
u
q
e
q
+
˙
b
λ
2
)
(18)
F
or
P
A
+
P
A
T
=
2
α
I
<
0
it
is
clear
that
e
T
(
P
A
+
P
A
T
)
is
ne
g
ati
v
e
denite.
The
term
˙
V
(
e,
t
)
must
be
ne
g
ati
v
e
denite,
this
yields
˙
a
λ
1
+
ˆ
i
dg
(
i
dg
−
ˆ
i
dg
)
+
ˆ
i
q
g
(
i
q
g
−
ˆ
i
q
g
)
=
0
(19)
˙
b
λ
2
+
u
d
(
i
dg
−
ˆ
i
dg
)
+
u
q
(
i
q
g
−
ˆ
i
q
g
)
=
0
(20)
The
adapti
v
e
la
w
can
be
easily
e
xpressed
as
follo
wing:
ˆ
r
g
ˆ
L
g
=
r
g
L
g
−
λ
1
Z
t
0
n
ˆ
i
dg
(
i
dg
−
ˆ
i
dg
)
+
ˆ
i
q
g
(
i
q
g
−
ˆ
i
q
g
)
o
dt
(21)
1
ˆ
L
g
=
1
L
g
+
λ
2
Z
t
0
n
u
d
(
i
dg
−
ˆ
i
dg
)
+
u
q
(
i
q
g
−
ˆ
i
q
g
)
o
dt
(22)
6.
SIMULA
TION
RESUL
TS
T
o
e
v
aluate
the
ef
cienc
y
of
the
considered
predicti
v
e
current
control
model
with
MRAS
oberv
er
for
grid
connected
tw
o-le
v
el
in
v
erter
under
v
arious
v
alues
of
parameter
lter
,
the
whole
simulation
studies
are
implemented
by
means
of
MA
TLAB/simulink
tools.
The
main
parameters
are
indicated
in
T
able
1.
T
able
1.
Simulation
parameters
P
arameter
Nomenclature
V
alue
Rated
Line-to-Line
V
oltage
(rms)
[V]
V
g
690
Rated
current
(rms)
[A]
I
g
627.6
Rated
acti
v
e
po
wer
[kw]
P
g
750
DC-link
v
oltage
[V]
V
dc
1220
DC-link
capacitor
[
µ
F]
C
dc
16714
Filter
resistance
[m
Ω
]
r
g
95.25
Filter
inductance
[mH]
L
g
0.3368
Sampling
time
[
µ
s]
f
s
20
Frequenc
y
of
the
grid
[Hz]
f
g
50
The
results
presented
in
three
scenario:
•
Ideal
Case
operation:
In
this
case
,
the
lter
parameter
are
setting
on
rated
v
alues
•
Non-ideal
Case
operation
without
MRAS:
In
this
case
,
the
parameter
lter
are
changed
to
sho
w
the
inuence
of
parameter
v
ariation
on
whole
performance
of
system.
•
Non-ideal
Case
operation
with
MRAS:
In
this
case
,
the
MRAS
observ
er
is
used
to
estimate
t
he
lter
parameter
and
to
enhance
the
performance
of
system
Int
J
Po
w
Elec
&
Dri
Syst,
V
ol.
12,
No.
2,
Jun
2021
:
858
–
869
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
❒
865
6.1.
Ideal
case
operation
The
performance
of
the
proposed
predicti
v
e
current
controller
scheme
for
three-phase
gri
d-connected
in
v
erters
considering
the
rated
v
alue
of
lter
parameter
is
tested
by
applying
a
DC-link
v
oltage
reference
V
∗
dc
=
1220
V
,
and
a
zero
reference
po
wer
reacti
v
e
Q
∗
g
=
0
to
unitary
the
po
wer
f
actor
.
Figure
6
(a)
sho
ws
the
weighting
f
actor
impact
λ
sw
on
the
grid
current
quality
via
T
otal
Harmonic
Distortion
(THD)
and
the
a
v
erage
switching
frequenc
y
f
sw
.
It’
s
clearly
noted
when
the
weighting
f
actor
is
increasing,
the
of
the
grid
current
is
changed
progress
i
v
ely
from
1.94
to
3.57
%,
while
the
a
v
erage
switching
frequenc
y
is
reduced
considerably
from
8.86
to
3.56
[KHz]
.
F
or
this
reason,
the
weighting
f
actor
can
be
adjusting
at
1700
for
operating
the
in
v
erter
at
f
sw
=
3.703
[KHz]
and
for
ha
ving
a
suitable
v
alue
of
THD
at
3.30
%.
(a)
(b)
(c)
(d)
Figure
6.
Simulation
results
for
case
ideal
operation,
(a)
THD
%
and
f
sw
v
ersus
λ
sw
v
ariation,
(b)
DC-link
v
oltage
[V],
(c)
dq
-axis
grid
current
[A]
(peak),
(d)
w
a
v
eforms
of
phase-a
grid
current
i
ag
[A]
(peak)
.
Figure
6
(b)
sho
ws
the
obtained
simulation
result
where
the
DC-link
v
oltage
is
controlled
to
maint
ain
at
its
reference
v
alue
1200
V
and
it
is
stable
during
steady
state
operation.
This
simulated
response
is
obtained
with
a
PI
controller
ha
ving
a
k
p
=
6
.
90
and
a
k
i
=
1
.
952
10
3
.
The
dq
-axis
grid
current
components
either
the
reference
i
∗
g
,dq
or
the
measured
i
g
,dq
are
tracking
each
other
perfectly
as
sho
wn
in
Figure
6
(c).
Consequently
,
The
amplitude
of
the
reference
d
-axis
grid
current
component
i
∗
dg
created
from
the
v
oltage
PI
controller
is
equal
to
887.5
A
(peak).
In
addition,
the
grid-connected
in
v
erter
is
control
led
to
supply
zero
reacti
v
e
po
wer
corresponding
the
zero
reference
q
-axis
grid
current.
Figure
6
(d)
illustrates
the
w
a
v
eform
of
a-phase
grid
current,
where
the
peak
v
alue
of
i
ag
is
equal
to
i
dg
.
6.2.
Non-ideal
case
operation
without
MRAS
It’
s
the
case
where
the
v
ariation
in
lter
parameter
has
been
considered
as
follo
ws:
•
Filter
resistance
and
inductance
are
50
%
lo
wer
than
its
nominal
v
a
lue
•
Filter
resistance
and
inductance
is
also
100
%
of
its
nominal
v
alue
•
Filter
resistance
and
inductance
are
150
%
higher
than
its
nominal
v
al
ue
These
v
alues
are
applying
at
the
follo
wing
moments
0.1
and
0.2
(s)
respecti
v
ely
.
Figure
7
sho
ws
the
ef
fect
of
parameters
v
ariation
of
both
r
g
and
L
g
on
the
o
v
erall
performance
of
the
controller
as
well
as
its
stability
.
Under
these
conditions,
it
is
clear
to
note
the
tracking
errors
between
the
measured
v
alues
of
i
dg
,
i
q
g
,
and
V
dc
and
their
references,
which
leads
to
unbalanced
v
alue
of
THD
especially
when
both
the
resistance
and
the
inductance
are
higher
than
the
nominal
v
alues
as
presented
in
Figure
8.
An
ef
cient
pr
edictive
curr
ent
contr
oller
with
adaptive
par
ameter
estimation...
(Haddar
Mabr
ouk)
Evaluation Warning : The document was created with Spire.PDF for Python.
866
❒
ISSN:
2088-8694
(a)
(b)
Figure
7.
Simulation
results
for
case
non-ideal
operation
without
MRAS,
(a)
DC-link
v
oltage
[V],
(b)
dq
-axis
grid
current
[A]
(peak).
(a)
(b)
(c)
Figure
8.
Simulation
results
e
xplains
the
inuence
of
lter
parameters
v
ariation
on
the
performance
of
PCC
r
g
and
L
g
are:
(a)
50
%
lo
wer
than
its
nominal
v
alue
,
(b)
aquals
to
nominal
v
alue,
(c)
150
%
higher
than
its
nominal
v
alue.
6.3.
Non-ideal
case
operation
with
MRAS
Figure
9
sho
ws
the
dynamics
performance
of
PCC
scheme
when
the
MRAS
observ
er
based
on
lya-
puno
v
function
is
enable
under
parameters
v
ariation.
The
adapti
v
e
g
ains
are
tuned
using
a
tuning
method
(trial
and
error)
to
gi
v
e
suitable
performance.
After
some
tuning,
the
adapti
v
e
g
ains
are
setting
as
follo
wing:
λ
1
=
0
.
625
and
λ
2
=
6
.
795
.
the
simulation
results
consists
of
tw
o
parts:
in
the
rst
one,
the
lter
parameters
are
increasing
and
switching
from
50
%
of
its
nominal
v
alue
to
nominal
v
alue,
and
in
the
second
one
the
lter
parameters
are
decreasing
and
switching
from
150
%
of
its
nominal
v
alue
to
nominal
v
al
ue.
Int
J
Po
w
Elec
&
Dri
Syst,
V
ol.
12,
No.
2,
Jun
2021
:
858
–
869
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
❒
867
(a)
(b)
Figure
9.
Simulation
results
for
R
L
lter
parameters
estimation
based
on
MRAS
on
serv
er
(a)
starts
from
50
%
of
its
nominal
v
alue
to
nominal
v
alue;
(b)
sta
rts
from
150
%
of
its
nominal
v
alue
to
nominal
v
alue
The
switching
is
applied
at
the
moment
0.15
(s).
In
the
both
simul
ation,
the
MRAS
observ
er
track
and
mak
e
the
estimated
v
alues
con
v
er
ges
to
parameter
v
ariation.
Moreo
v
er
,
the
estimated
current
ˆ
i
g
and
the
mea-
sured
current
i
g
tracks
each
other
.
Despite
lar
ge
error
at
the
same
time
where
the
switching
is
applied.
the
error
current
e
con
v
er
ges
to
zero.
It
is
w
orth
noting
that
the
estimate
v
alue
of
the
lter
resistance
is
more
accurate
than
the
estimated
v
alue
of
the
lter
inductance
with
steady-state
estimation
error
in
range
of
1
.
763
10
−
4
and
0
.
1
%
respecti
v
ely
.
T
o
sho
w
the
rob
ustness
of
PCC
controller
with
a
MRAS
observ
er
for
a
three-phase
grid-connected
in
v
erters,
a
comparison
of
performance
can
be
done
through
tw
o
cases:
The
rst
is
in
agreement
with
the
second
scenario,
where
the
lter
parameters
are
changed
in
the
absence
of
a
MRAS
observ
er
,
while
PCC
controller
is
operating
on
the
normal
v
alues.
As
for
the
second
case,
which
corresponds
to
the
third
scenario,
the
change
of
the
lter
parameters
are
in
the
presence
of
the
MRAS
observ
er
,
which
estimates
the
lter
parameters
so
that
the
PCC
controller
uses
them.
The
comparison
w
as
done
by
measuring
THD
and
f
sw
,
as
sho
wn
in
Figure
10.
Although
the
lter
parameters
are
changed,
the
MRAS
observ
er
helps
the
PCC
controller
to
select
a
switching
states
that
corresponds
to
the
minimal
v
alues
of
f
sw
with
slight
dif
ference
in
THD.
(a)
(b)
Figure
10.
Comparison
of
system
performance
by
measuring;
(a)
THD
%;
(b)
f
sw
[KHz].
An
ef
cient
pr
edictive
curr
ent
contr
oller
with
adaptive
par
ameter
estimation...
(Haddar
Mabr
ouk)
Evaluation Warning : The document was created with Spire.PDF for Python.