In
te
r
n
ation
a
l Jou
rn
al
o
f Po
we
r
Elec
tron
ic
s an
d
D
r
ive S
y
stem
(IJ
PED
S
)
Vol.
10, No.
1, Mar
ch 2019,
pp.
19~26
IS
S
N
: 2088-
86
94,
D
O
I
:
10.11
5
9
1
/ij
ped
s
.
v10
.
i
1.pp
1
9
-2
6
19
Jou
rn
a
l
h
o
me
pa
ge
:
ht
tp:
//i
a
e
score
.
com
/
j
o
u
r
na
l
s
/
i
n
d
e
x
.
p
hp/IJ
PED
S
Design and im
ple
m
entation of no
nlinear integral backstepping
contr
o
l strategy for single
-
ph
ase grid
-connected VSI
Mo
ussa
R
edd
a
k
, A
bdelo
ua
hed
Mes
b
a
h
i, Ay
o
u
b
No
u
a
i
t
i, Abdelma
j
id B
er
da
i
D
e
pa
rtme
nt
o
f
El
e
c
t
rica
l
Eng
i
ne
eri
ng,
Univer
sit
y
o
f H
a
ssan II
Casab
lan
ca, M
o
r
o
cco
Art
i
cl
e In
fo
ABSTRACT
A
r
tic
le hist
o
r
y
:
R
e
c
e
i
v
e
d
Sep
1
3
,
2
018
Re
vise
d N
ov
8
, 2018
Ac
ce
p
t
ed
No
v
2
2
,
2
018
Th
is
p
ap
er
p
re
s
e
nt
s
the
des
i
gn
a
n
d
t
he
i
m
p
l
e
m
e
n
t
atio
n
o
f
a
n
o
n
l
ine
a
r
integral
back
st
eppi
ng
c
on
tro
l
s
t
r
at
egy
f
o
r
a
s
i
n
g
le-p
h
a
s
e
g
ri
d-co
nn
ect
ed
v
ol
ta
ge
so
urce
i
n
v
e
rter
(
V
S
I).
Th
e
propo
sed
con
t
ro
l
s
t
rat
e
gy
aim
s
t
o
re
solve
the
pro
b
l
e
m
o
f
t
he
s
i
n
u
s
oidal
re
f
e
rence
current
t
rackin
g,
t
o
h
a
ve
a
r
apid
t
ransi
e
n
t
resp
on
se
n
eeded
t
o
d
e
al
w
it
h
t
h
e
trans
i
en
t
inc
i
den
t
s
uch
as
g
ri
d
voltag
e
d
rops
and
t
o
g
en
erat
e
int
o
t
h
e
g
rid
a
curr
en
t
with
f
ewer
h
arm
o
n
i
cs
.
T
hes
e
pro
p
ert
i
es
a
re
v
erifi
e
d
b
y
s
eekin
g
the
d
i
rectio
n,
t
he
a
m
p
litude
a
n
d
t
h
e
tran
si
ent
res
p
o
n
s
e
o
f
t
h
e
i
n
ject
ed
g
rid
current
.
A
n
e
xp
erim
en
ta
l
VSI
pro
t
o
t
yp
e
is
u
s
e
d
t
o
v
al
id
ate
t
h
e
ef
f
ecti
v
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s
o
f
t
h
e
p
rop
o
s
e
d
c
on
trolle
r,
b
a
sing
o
n
a
digita
l
sig
n
a
l
p
r
o
c
e
s
sor
DSP
m
i
c
r
o
c
on
trolle
r.
T
he
a
ch
ieved
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e
s
u
l
t
s
s
h
o
w
t
h
a
t
t
h
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-
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a
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e
r
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e
s
s
a
t
i
s
f
y
t
h
e
g
r
i
d
current
injection cond
i
t
ions.
K
eyw
ord
s
:
DS
P
E
xpe
rime
n
t
a
l
V
SI
Gri
d
-co
n
n
ect
ed
VS
I
Inte
gra
l
bac
kst
e
ppi
ng
co
n
t
ro
l
Ra
pid
trans
i
e
n
t
Co
pyri
gh
t © 2
019 In
stit
u
t
e
of Advanced
En
gi
neeri
n
g
an
d
S
c
ien
ce.
All
rights
res
e
rv
ed.
Corres
pon
d
i
n
g
Au
th
or:
Mo
ussa
R
ed
da
k,
D
e
pa
rtme
nt
o
f
El
e
c
t
rica
l
Eng
i
ne
eri
ng,
Un
iv
e
r
sit
y
of
Ha
ssa
n
II
C
as
ab
l
a
n
c
a,
7 km
El
Ja
dida
Roa
d,
P
ostal
addr
ess
:
8
11
8,
O
asis,
C
a
sab
l
a
n
ca
,
M
or
occ
o
.
Em
ail:
moussa
.
r
edda
k@
gma
il.c
o
m
1.
I
N
TR
OD
U
C
TI
O
N
G
r
i
d
-
c
on
nec
t
e
d
V
S
I
a
re
m
ost
l
y
use
d
a
s
a
pa
rt
o
f
co
nvers
i
on
sys
t
em
s
int
e
rfac
es
i
n
d
i
s
t
rib
u
t
e
d
p
ow
er
gene
ra
ti
o
n
,
w
h
ic
h
is
b
a
s
e
d
o
n
rene
w
a
b
l
e
e
n
erg
i
es
l
i
k
e
w
i
n
d
a
n
d
PV
e
n
e
rgy
s
y
ste
m
s.
G
ener
ally,
t
h
es
e
in
t
e
rfa
ces
h
a
v
e
be
e
n
i
nt
r
o
d
u
c
e
d
be
tw
ee
n
the
r
e
new
a
b
l
e
e
n
e
r
gy
s
o
urc
e
s
a
n
d
the
gri
d
i
n
order
t
o
c
o
n
v
er
t
a
D
C
voltage
t
o
an
A
C
f
o
rm
[
1]-[5].
T
o
ensure
a
s
i
nus
oidal
output
c
u
rren
t
,
usua
ll
y,
a
f
ilte
r
is
u
se
d.
S
o,
t
h
e
perform
ance
s
of
t
he
se
d
e
v
i
c
e
s
can
b
e
typ
i
c
a
l
l
y
e
v
a
l
uat
e
d
b
y
c
he
ckin
g
t
h
e
en
e
r
gy
e
f
f
i
c
i
e
n
c
y
,
t
h
e
t
ot
al
harm
on
ic
d
i
s
t
o
rtion
(TH
D
)
, and t
h
e
trans
i
e
n
t re
spo
n
se
.
F
o
r
t
h
i
s
k
in
d
o
f
i
nv
e
r
t
e
rs,
dif
f
e
re
nt
l
in
ea
r
a
n
d
no
nli
n
ea
r
c
o
nt
ro
l
s
t
r
a
te
g
i
es
h
ave
bee
n
p
r
o
pose
d
t
o
all
o
w
a
gr
ea
t
t
r
ac
ki
n
g
o
f
a
s
i
nus
oi
da
l
r
e
fe
re
nc
e.
A
mong
th
em
,
t
he
l
i
n
ea
r
P
I
c
ontro
l
sy
ste
m
h
a
s
b
ee
n
u
s
e
d
t
o
regu
la
t
e
t
he
i
n
j
ec
te
d
gri
d
c
u
rre
nt
.
Thi
s
c
o
n
t
r
o
l
ler
d
o
es
n’t
inc
rea
s
e
c
o
mputa
t
io
n
i
n
t
he
s
ystem
a
l
gor
ithm
,
w
h
ic
h
m
a
ke
s
its
i
m
p
lem
e
nt
ati
o
n
ea
si
e
r
f
or
t
he
w
h
o
l
e
c
ontro
l
sys
te
m
.
N
eve
r
thele
ss,
I
n
spite
o
f
t
h
ese
adva
n
t
a
g
e
s
,
th
e
P
I
c
ontro
l
sys
t
em
h
as
m
any
lim
it
a
t
i
ons,
e
s
pe
c
i
a
l
l
y
a
g
a
i
n
s
t
n
o
n
l
i
n
e
a
r
s
y
s
t
e
m
s
[
6
]
.
T
h
i
s
con
t
ro
l
l
er
c
a
n
’t
d
ea
l
per
f
ec
t
l
y
wit
h
s
in
us
o
i
d
a
l
r
e
fe
renc
es
w
ith
ou
t
usin
g
grid
v
olta
g
e
f
e
e
d
-forw
a
rd,
w
h
i
c
h
c
a
n
be
c
o
n
s
i
derin
g
as
a
w
eakne
ss
in
g
r
i
d-c
o
n
n
ected
i
nv
erter
s
.
S
o
,
t
o
alle
v
i
a
t
e
the
dr
aw
bac
k
s
o
f
t
he
P
I
c
ontr
o
l
syste
m
,
PR
c
o
n
tr
ol
l
e
r
is
u
sed
a
s
a
n
al
terna
t
i
v
e
so
l
u
ti
on
to
t
ra
ck
s
in
uso
i
da
l
re
fer
e
nce
s
,
and
w
i
t
h
a
n
ad
di
tio
na
l
resona
nt
t
erm
,
t
h
i
s
c
o
nt
r
o
l
l
e
r
c
a
n
su
p
p
r
e
ss
som
e
s
pec
i
fi
c
ha
rm
o
ni
c
compone
n
t
[7]-
[
9
].
T
o
get
m
o
re
perform
ance
s,
o
t
h
er
a
p
p
roac
he
s
l
i
ke
n
o
n
l
i
n
e
a
r
contr
o
l
s
t
r
a
t
e
gie
s,
c
ons
ide
r
ed
a
s
n
o
n
-
s
e
l
e
c
ti
ve
t
ec
hn
iq
ue
s,
f
or
g
r
id
-c
o
n
n
ect
ed
i
nv
ert
e
rs
h
a
v
e
b
e
en
r
e
p
o
r
te
d
i
n
t
h
e
l
it
e
r
ature
.
In
[
1
0
],
a
c
ompa
rat
i
ve
s
tu
dy
be
tw
ee
n
s
l
i
d
in
g
m
o
d
e
c
o
n
t
r
o
l
(
S
M
C
)
s
t
r
a
t
e
g
y
w
i
t
h
t
h
e
P
I
c
o
n
t
r
o
l
l
e
r
f
o
r
c
u
r
r
e
n
t
in
ve
rte
r
h
ave
b
e
e
n
a
ddre
ssi
ng.
T
h
i
s
stu
dy
show
s
t
h
e
e
ffe
cti
v
e
n
ess
of
t
h
e
S
MC
t
ha
t
g
i
v
e
s
go
o
d
t
ra
ck
in
g
a
n
d
f
ast
t
r
an
si
en
t
re
sponse
c
o
n
t
ra
ry
t
o
t
h
e
P
I
con
t
ro
l
l
er.
Thi
s
r
a
p
id
d
y
n
a
m
i
c
r
espo
nse
is
n
e
e
de
d
to
o
verc
ome
s
o
m
e
t
r
ans
i
en
t
i
n
c
i
d
e
nt
s
su
c
h
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s
a
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r
an
si
en
t
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN: 2088-
8694
I
nt
J
P
ow
Elec
& Dr
i
S
y
st, Vol. 10,
N
o.
1, Mar
c
h 2
0
1
9
:
19
–
26
20
low
v
o
l
t
a
g
e.
A
lso,
i
n
[1
1]
,
[12]
,
the
pre
d
ic
t
i
ve
c
on
tro
l
s
t
r
a
t
e
g
y
i
s
pro
p
o
s
e
d
t
o
r
eg
ula
t
e
the
gr
id
c
ur
ren
t
o
f
a
gri
d
-con
ne
c
t
e
d
inver
t
e
r
.
The
s
y
nc
h
r
on
iz
ati
on
of
t
he
c
urr
e
nt
o
ut
p
u
t
of
t
he
i
nvert
er
w
it
h
t
h
e
u
til
i
t
y
g
r
id
i
s
co
nsi
d
ere
d
a
s
an
impor
ta
nt
c
ha
l
l
e
n
ge.
In
o
rde
r
t
o
ach
ie
ve
t
his
o
p
er
at
ion,
d
i
g
i
t
a
l
a
n
d
ana
l
og
P
LL
a
re
i
nv
est
i
g
at
ed
i
n
ma
n
y
stud
ies.
W
ith
a
nal
o
g
P
LL, so
m
e
e
l
e
c
tro
n
i
c dev
i
ces
are
a
d
d
e
d
t
o
the
c
i
rc
ui
t
,
w
h
i
ch
incr
e
ases the
com
p
l
ex
i
t
y o
f
the
s
y
ste
m
a
n
d
the
c
o
ntro
l
pa
rt.
Bu
t
w
ith
d
igi
t
a
l
P
LL,
easy
imp
le
me
nta
t
i
on,
m
ore
pe
rfo
r
m
a
nc
e,
a
nd
lo
w
c
o
st
w
ill
be no
t
e
d
t
o
the s
y
stem
[1
3
],
[
14].
Rec
e
n
t
l
y,
t
he
b
a
c
k
s
te
p
p
i
n
g
c
o
n
t
r
o
l
s
t
rate
gy
h
as
o
b
t
a
i
ne
d
a
trem
e
n
do
us
a
t
t
e
nti
o
n
a
s
a
s
ol
ut
i
o
n
fo
r
con
t
ro
l
of
non
l
i
nea
r
s
ys
tem
s
[
15],
[1
6
]
.
The
idea
b
e
h
in
d
th
is
p
roce
d
u
re
i
s
to
d
e
v
ise
the
sys
t
em
t
o
su
bs
y
s
t
e
ms
o
f
t
h
e
f
i
r
st
o
rder,
a
n
d
by
a
dop
ti
n
g
t
h
e
L
y
a
pu
nov
f
un
cti
o
n
t
o
e
n
sur
e
t
he
s
tab
i
li
ty
o
f
t
h
e
w
h
o
l
e
s
y
s
t
em
,
t
h
e
con
t
ro
l
law
is
d
eri
v
e
d
f
rom
d
i
ffere
n
t
s
t
a
g
e
s
tha
t
g
ive
a
v
i
r
t
ua
l
co
n
t
ro
l
la
w
for
t
h
e
s
ubs
yste
m.
S
o,
by
go
ing
st
e
p
by
ste
p
,
th
e
globa
l con
t
ro
l la
w
for
the
l
a
st
s
ubsy
s
tem
is
obta
i
ned.
I
n
o
r
d
e
r
t
o
ge
t
a
fas
t
d
yna
m
i
c
re
spo
n
se
o
f
the
in
j
e
c
t
e
d
g
ri
d
cu
rre
n
t
a
nd
t
o
t
r
ack
a
s
in
usoi
da
l
refere
nce
pric
i
s
el
y,
t
h
i
s
pa
pe
r
prop
ose
s
a
n
o
n
l
i
n
e
ar
b
ac
k
s
t
e
pp
i
ng
c
o
n
t
ro
l
s
t
ra
t
e
gy
w
i
t
h
a
n
in
tegra
l
a
c
t
i
o
n
for
a
gri
d
-con
ne
c
t
e
d
V
S
I
p
rot
o
ty
pe
.
The
ex
perim
e
nta
l
p
ro
to
t
ype
c
on
sis
t
s
o
f:
a
f
ull
bri
dge
i
nve
r
t
e
r
,
an
L
-fil
ter
tha
t
is
a
d
o
p
t
e
d
t
o
a
l
l
o
w
a
re
aso
n
ab
le
h
arm
o
n
i
c
cur
r
en
t
a
t
t
e
nua
ti
on.
T
h
e
pr
op
ose
d
c
o
n
tr
ol
s
tra
t
e
gy
i
s
v
e
r
ifi
e
d
expe
r
i
me
n
t
a
lly
unde
r c
o
nside
r
ati
on o
f
g
ri
d cur
r
ent c
o
nd
it
io
ns.
2.
SY
S
T
E
M
D
ESCRIPTIO
N AND
M
ODE
L
ING
2.1.
S
ystem
d
e
scr
i
p
t
ion
F
i
gure
1
sh
ow
s
the
c
i
r
c
ui
t
o
f
t
he
u
se
d
P
W
M
V
S
I
w
h
ich
i
s
f
orm
e
d
by
:
four
p
ow
er
s
w
i
t
c
hes
(S
1,
S
2,
S
3
,
and
S
4
)
se
le
cted
f
r
o
m
fa
s
t
I
G
B
T
de
vic
e
s
,
a
low
pass
f
il
t
e
r
com
p
ose
d
by
L-fi
l
t
e
r
(
Lf)
pl
a
c
e
d
b
e
t
w
een
t
he
in
verter
a
n
d
t
h
e
g
rid
(v
g).
The
i
n
pu
t
vo
l
t
a
g
e
is
o
bta
i
ned
from
a
d
c
pow
e
r
s
upp
ly
(
V
)
t
o
e
ns
ure
a
c
onst
a
n
t
voltage leve
l
.
Figure
1.
D
escripti
o
n
of t
h
e
VSI
syste
m
The
pow
er
s
w
i
tche
s
o
f
t
he
V
S
I
a
r
e
c
ontro
ll
ed
u
si
n
g
c
omp
a
rati
ve
blo
c
s
of
t
w
o
s
i
n
u
s
oi
da
l
re
fe
ren
c
es
signa
l
s
(
a
nd
) sh
i
f
ted by
18
0° w
ith a h
i
g
h-
fr
e
que
nc
y t
r
i
a
n
g
u
l
a
r c
a
rr
i
e
r sig
n
al (
),
a
s show
n i
n
F
ig
ure
2.
T
he
p
oss
i
ble
sw
itch
i
ng is e
sta
b
l
i
s
h
ed
by
ta
ki
n
g
in
t
o
acc
o
u
n
t
t
he lo
g
i
c
ope
rat
i
o
n
i
n Ta
b
l
e
1.
Figure
2.
D
escripti
o
n
of t
h
e
VSI
syste
m
Evaluation Warning : The document was created with Spire.PDF for Python.
Int J
P
o
w
El
e
c
&
D
ri S
yst
I
S
S
N
:
2088-
86
94
D
e
sig
n
a
nd im
plem
e
n
ta
ti
o
n
o
f
no
n
l
i
n
e
a
r i
n
te
gr
al
b
a
cks
tep
p
i
n
g c
o
n
t
r
o
l st
ra
t
e
gy
for..
.
(Mous
sa Re
dd
a
k
)
21
Tabl
e 1
.
S
w
i
t
c
h
i
ng
l
ogi
c
f
o
r p
o
w
e
r
swi
t
c
h
e
s
L
ogic
op
e
r
a
tion
O
n
s
wi
t
c
h
No
t S
1
S1
S2
S3
No
t S
3
S
4
2.2.
V
S
I
syste
m m
o
d
e
ll
in
g
A
c
cordi
n
g
t
o
K
i
r
chh
o
ff’
s
l
a
w
s
a
pplie
d
t
o
t
he
a
bo
ve
c
irc
u
i
t
s
h
o
w
s
i
n
F
ig
ure
1,
a
nd
o
w
in
g
to
t
he
symm
etry pro
p
e
r
t
y o
f
the
pos
i
t
i
v
e-
ha
lf a
n
d
n
e
g
at
ive
-
ha
lf pe
r
i
o
d
in t
h
e un
i
p
o
l
ar P
WM
s
w
itc
h
i
ng (t
hre
e
-l
eve
l
)
,
the
dy
nam
i
c
e
qua
t
i
o
n
r
ela
t
i
n
g
be
tw
ee
n
the
inje
c
t
e
d
c
ur
rent
a
n
d
gri
d
v
ol
ta
ge
a
n
d
t
hat
d
e
fine
s
ys
tem
m
o
d
e
l
i
s
gi
ve
n a
s
:
L
D
V
v
(
1
)
Where
D
re
p
rese
nt
t
he
d
ut
y c
y
c
l
e of t
he
s
w
i
t
c
h
e
s S
1
a
nd
S
4
dur
ing
o
n
e
sw
it
chi
n
g pe
ri
od
.
B
y
c
ons
i
d
e
r
in
g
the
de
fi
n
i
t
i
o
n
o
f
th
is
d
ut
y
cycle
(
m
odu
la
t
i
o
n
i
n
de
x)
D
v
V
a
nd
t
h
e
inve
r
t
er
dy
nam
i
c
K
V
V
,
t
h
e
gri
d
c
urre
nt
d
y
n
am
i
c
c
an be
d
e
r
i
ve
d a
s
:
ı
K
v
v
(
2
)
Where,
v
i
s
the si
nus
o
i
da
l
co
nt
rol si
gna
l,
a
nd
V
i
s
t
h
e
a
m
pl
itu
de
of
t
h
e
tria
n
g
u
lar
ca
rrier
signal.
3.
GRI
D
C
URRE
NT CONTR
OL
UNDER
BACKS
TEPPING WITH
INT
E
GRAL
ACTIO
N
3.1.
T
h
e
ob
jectif
o
f the
control
st
rate
g
y
Th
e
pu
rp
o
s
e
s
b
ehi
n
d
of
i
nt
e
g
ra
l
b
ack
st
eppi
ng
c
on
t
r
o
l
o
f
t
h
e
Th
i
s
P
W
M
V
S
I
connec
t
e
d
t
o
t
h
e
gri
d
inc
l
ude
t
hree
m
ain
o
b
jec
t
ive
s
:
The
first
on
e
is
t
o
in
jec
t
a
s
i
n
us
oida
l
c
u
rrent
(
i
g
)
in
t
o
t
h
e
g
r
i
d
w
i
th
r
es
pec
t
in
g
of
t
he
p
ow
er
f
a
c
tor
cor
r
ec
tio
n
(P
F
C
),
i
n
oth
e
r
w
o
rds,
t
he
i
nj
e
c
te
d
gri
d
c
u
r
r
e
nt
a
nd
the
g
r
id
v
o
l
t
a
ge
s
h
oul
d
b
e
in p
hase
(
0.9
cos
φ
1
).
Th
e
sec
ond p
u
rp
o
s
e i
s
to ha
ve
a
r
api
d
d
y
n
a
m
ic
of
the i
n
jec
t
e
d gri
d
c
urre
nt
w
i
t
h
a
TH
D
less
tha
n
5
%
.
T
he
t
hi
r
d
purp
o
se
i
s
t
h
e
c
o
n
t
ro
l
sta
b
il
ity
o
f
t
he
g
ri
d-c
onne
c
t
ed
V
S
I
w
hic
h
i
nc
rea
s
es
t
he
rob
u
st
n
e
ss o
f
t
he
w
ho
le
s
yste
m
.
A
s
s
how
n
in
F
i
gur
e
3,
t
he
c
on
tro
l
s
tra
t
e
g
y
o
f
t
he
i
n
t
e
g
ra
l
bac
k
st
e
p
pi
ng
o
f
th
e
g
r
i
d
-
c
o
nn
e
c
t
e
d
i
s
ac
hi
e
v
e
d
b
y
u
s
i
ng
the
P
L
L
bl
oc
t
ha
t
g
i
ve
s
the
r
e
fe
renc
e
c
u
rre
nt
,
the
c
o
ntr
o
l
bl
oc
k,
a
nd
the
u
n
i
p
olar
P
WM
sw
it
c
h
i
n
g
me
t
h
od.
F
i
gur
e 3.
Co
n
t
r
ol
s
t
r
ateg
y de
s
c
rip
tio
n
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN: 2088-
8694
I
nt
J
P
ow
Elec
& Dr
i
S
y
st, Vol. 10,
N
o.
1, Mar
c
h 2
0
1
9
:
19
–
26
22
3.2.
PLL
b
l
oc
k
A
ll
P
h
ase
l
o
c
k
e
d
l
oops
(
P
LL)
h
ave
the
ma
i
n
s
t
r
uc
t
u
re
i
n
sp
ite
o
f
t
h
e
i
r
d
i
vers
itie
s.
T
h
i
s
st
r
u
ct
ure
i
s
com
p
o
s
ed
o
f
a
p
h
ase
de
tec
t
or
,
a
lo
o
p
f
il
ter
,
a
nd
a
v
o
lta
ge
c
on
t
roll
ed
o
sc
i
lla
t
o
r.
T
h
e
d
if
fe
ren
c
e
b
e
t
w
een
P
LLs
is
l
oca
t
ed
a
t
t
h
eir
p
h
ase
de
te
ctor.
I
n
t
h
i
s
paper
,
t
he
u
se
d
P
L
L
,
that
i
s
ba
se
d
on
p
ow
e
r
c
a
l
c
u
l
a
t
i
on,
i
s
in
ves
tiga
t
e
d
in [13]
, and its s
truct
u
re
is de
pi
c
t
e
d
in F
i
g
u
r
e
4
.
W
h
e
r
e
v_
g
is
t
he
g
r
i
d v
o
l
t
a
g
e
tha
t
i
s a
s
su
me
d
b
y
a
p
u
r
e
si
nus
oi
d
a
l
i
n
pu
t
,
ω
_
f
f
is
t
h
e
n
om
in
al
v
alu
e
o
f
t
h
e
fr
equ
en
c
y
,
a
n
d
θ
̂
r
eflec
t
s
t
h
e
e
s
t
i
m
a
t
e
d
vo
lta
ge a
n
g
l
e
.
F
i
gur
e 4.
G
r
i
d ang
l
e
est
i
m
a
t
i
o
n
by
usi
n
g P
L
L bl
ock
F
o
r
t
h
e
sa
k
e
o
f
t
h
e
s
i
m
p
l
i
c
ity,
be
fore
t
he l
oop
fi
lter,
a
s
i
n
us
oi
da
l
mu
lt
i
p
lie
r
bl
ock
is
u
se
d i
n
t
his
P
L
L
t
o
f
ul
fi
ll
t
h
e
p
h
a
se
a
ng
l
e
d
ete
c
t
o
r
rol
e
.
Th
eref
o
r
e,
t
hi
s
bl
o
c
k
m
ake
s
t
he
p
ar
ticu
l
a
r
it
y
of
t
he
p
r
opose
d
P
LL.
Ac
co
rd
ing
t
o
Fi
g
u
r
e
3
,
Th
e
f
ic
t
iti
ou
s po
wer i
s
mat
h
e
mati
cal
l
y
g
ive
n
a
s:
p
v
∗i
s
(3)
B
y
subs
t
itu
t
i
ng
t
he ter
ms
V
a
nd is,
t
h
is pow
e
r
w
i
l
l be
:
p
V
c
o
s
θ
s
i
n
θ
(4)
p
sin
θ
θ
sin
θ
θ
(
5
)
B
y
t
ak
i
n
g
in
to
a
c
c
o
u
n
t
t
he
o
u
t
p
u
t
o
f
t
he
p
hase
d
e
t
ect
or
e
qua
l
t
o
z
e
ro,
t
h
e
mu
lt
i
p
lica
tio
n
o
f
g
ri
d
vo
lta
ge,
w
h
ic
h
ser
v
es
a
s
t
h
e
in
p
u
t
,
w
it
h
t
h
e
fi
c
t
iti
o
u
s
curr
ent
i
g
ives
t
w
o
t
er
ms:
the
firs
t
on
e
is
a
s
m
a
ll
d
c
term
, and the
s
e
c
o
n
d
o
ne c
o
n
t
a
in
s
a
d
o
u
b
l
e freque
ncy. T
o k
e
ep u
p
the
dc
p
a
r
t,
a low
p
as
s
f
i
l
t
er
i
s
u
s
ed.
I
n
t
h
i
s
c
a
s
e
,
b
y
p
a
s
si
ng
t
h
r
ou
gh
t
h
e
V
CO
b
lo
ck
,
th
e
e
s
ti
mat
e
d
g
r
i
d
vol
t
age
a
n
gl
e
w
i
ll
b
e
e
qua
l
t
o
t
he
moni
t
o
red
an
gl
e.
3.3.
Grid
c
u
r
ren
t con
t
rol u
n
d
e
r
in
t
e
gra
l
b
ac
k
s
tep
p
i
n
g
c
on
tr
ol
R
e
gard
in
g
the
stab
il
ity
o
f
V
g
ett
i
ng
from
a
c
on
sta
n
t
pow
e
r
s
up
p
l
y,
t
he
b
ac
kste
p
p
in
g
proce
d
u
re
w
ill
be de
s
i
g
ne
d in o
ne
s
t
e
p (one
s
ubs
yste
m
of t
he
g
ri
d c
u
rrent
d
yn
ami
c
i
s c
o
nsid
e
r
ed
).
The
gri
d
c
urre
nt
e
r
r
or w
i
t
h
th
e
int
e
gral
act
i
o
n ca
n be
def
ine
d
as:
z
i
i
K
i
i
(
6
)
Where
K
i
s
a
cons
tan
t
par
am
et
er
a
nd
i
i
s t
h
e cu
rre
n
t
re
f
e
r
en
ce
.
The
i
n
tegra
l
a
ctio
n
is
a
d
d
e
d
t
o
ha
ve
m
ore
robus
t
n
ess
o
f
t
he
c
o
n
tro
ller,
e
specia
lly
t
o
have
a
g
o
o
d
trac
k
i
n
g
w
ith
zer
o er
ror. The
der
iva
t
i
v
e
of
t
h
e
above
ex
p
r
e
s
s
io
n
i
s
g
ive
n
b
y the
fo
ll
ow
in
g
equat
i
o
n
:
z
ı
ı
K
i
i
(7)
S
ubsti
t
u
ti
ng
th
e
dynam
i
c e
q
u
a
ti
on
of
i
cur
ren
t
in
2, the
er
r
or
d
yna
mic
z
ca
n
be
gi
ve
n a
s
:
z
v
v
ı
K
i
i
(8)
Evaluation Warning : The document was created with Spire.PDF for Python.
Int J
P
o
w
El
e
c
&
D
ri S
yst
I
S
S
N
:
2088-
86
94
D
e
sig
n
a
nd im
plem
e
n
ta
ti
o
n
o
f
no
n
l
i
n
e
a
r i
n
te
gr
al
b
a
cks
tep
p
i
n
g c
o
n
t
r
o
l st
ra
t
e
gy
for..
.
(Mous
sa Re
dd
a
k
)
23
To
e
ns
ure
the sys
t
em
stabi
l
ity
a
nd t
o
de
f
i
n
e
the
con
t
ro
l la
w
,
a
Lya
pu
no
v
fu
nct
i
on c
a
n b
e
d
efine
d
a
s:
V
z
(9)
I
t
s
dy
nam
i
c
is gi
v
e
n
a
s:
V
z
z
(1
0
)
So,
this
d
eriv
at
ive
ca
n b
e
n
e
g
at
ive
by i
mp
os
in
g:
z
K
z
(1
1
)
Where
K
i
s a co
n
s
t
a
n
t
p
a
r
amet
er.
A
nd
t
h
e c
o
n
t
r
o
l
l
a
w
c
a
n
be c
o
n
cl
u
d
ed a
s
:
v
ı
v
K
i
i
K
z
(1
2
)
I
n
th
i
s
case
,
t
he
Lya
pun
o
v
f
u
n
ct
i
on der
i
va
ti
ve
i
s:
V
K
z
(
1
3
)
4.
EXPE
RIMENTAL
VALID
AT
ION
4.1.
Exp
e
r
i
m
e
n
t
al
t
est
s
w
ith
in
tegra
l
b
ac
kst
e
p
p
in
g
con
t
rol str
ategy
To
v
a
l
i
d
a
t
e
the
pro
pose
d
n
on
linea
r
in
t
e
gral
b
ac
k
s
t
e
p
p
in
g
c
ontro
l
l
er,
an
e
xper
i
m
e
nta
l
p
rot
o
t
y
pe
o
f
a
gri
d
-con
ne
c
t
e
d
V
S
I
s
ystem
is
u
sed,
a
nd
i
t
s
d
escri
p
t
i
on
i
s
dep
i
c
ted
in
F
igure
5.
T
he
v
o
lta
ge
V
i
s
obtai
n
e
d
from
a
v
ariab
l
e
D
C
pow
e
r
s
up
pl
y
of
6
3V
/
3
A
.
T
he
f
ul
l
brid
g
e
P
WM
i
n
ve
rt
er
i
s
drive
n
b
y
2.
5
kH
z
si
gna
l
s
t
ha
t
are
gener
a
te
d
by
a
di
gi
tal
s
i
gna
l
pr
o
c
ess
o
r
D
S
P.
T
his
lat
t
e
r
i
s
a
D
S
P
T
MS
2
0
F
2
8
3
7
9
D
L
aunc
hpa
d
w
i
t
h
a
cloc
k
o
f
2
00
MH
z,
m
a
nufac
tu
r
e
d
by
Te
xas
Instrum
e
nt.
The
pro
g
ra
m
o
f
the
c
o
n
t
r
o
l
sy
stem
t
ha
t
is
u
p
l
oa
de
d
on
t
h
e
D
S
P
is
car
ried
o
u
t
b
y
us
i
n
g
t
h
e
em
bedde
d
c
o
der
to
o
l
o
f
M
at
la
b
/
Si
m
u
l
i
n
k
,
on
w
h
ich
is
a
P
WM
b
loc
is
i
m
p
l
e
m
en
t
e
d
.
T
o
b
e
a
d
a
pt
e
d
w
i
t
h
A/
D
c
onve
rs
i
o
n
blo
c
k
,
t
h
e
m
ea
su
re
d
gri
d
v
olt
a
g
e
a
nd
t
h
e
inj
e
c
t
ed
c
u
rre
n
t
by
H
a
ll
E
f
fec
t
s
e
n
sor
s
a
re
i
m
p
lem
e
n
t
e
d
o
n
circu
it
b
o
ard.
T
he
v
a
l
u
e
of
t
he
L
-f
i
l
ter
is
3
0mH
a
nd
it
is
s
e
l
e
c
t
ed
for
ha
rmon
ic
a
t
t
e
n
ua
t
i
o
n
.
A
n
a
uto
t
ra
nsfor
m
e
r
i
s
a
d
de
d
betw
ee
n
t
he
g
rid
a
n
d
the
L-
fil
t
er
s
o
t
h
a
t
t
he
R
MS
val
u
e
o
f
gr
i
d
v
o
lta
ge
is 3
6
V
.
F
i
gure
5. T
h
e
use
d
expe
rime
nta
l
t
es
t
be
nc
h
I
n
t
h
i
s
sec
t
io
n,
t
he
m
ai
n
o
b
je
c
tive
of
t
he
w
h
o
l
e
s
ys
tem
is
t
o
i
n
j
e
c
t
int
o
t
h
e
g
rid
a
c
u
r
r
en
t
w
i
t
h
R
MS
val
u
e
of
1
A
.
T
he
u
nfi
l
t
e
re
d
o
u
t
p
ut
v
olta
ge
o
f
t
h
e
i
n
v
e
r
t
er
a
nd
t
h
e
in
j
ect
ed
c
u
rre
n
t
r
e
s
po
ns
es
a
re
d
e
p
i
c
t
e
d
i
n
F
i
gure
6a
,
b
y
u
s
i
n
g
t
he
r
e
f
er
e
n
ce
c
urre
n
t
t
hat
is
g
e
n
era
t
e
d
b
y
t
h
e
P
LL
bl
oc
k.
A
ccor
d
ing
t
o
t
hi
s
fig
u
r
e,
i
t
sho
u
l
d
b
e
n
o
t
e
d
t
ha
t
t
h
e
se
ns
or
c
ur
rent
d
e
l
ive
r
s
2
1
0
m
V
for
e
a
c
h
1A
R
MS
a
n
d
t
he
i
n
t
egra
l
bac
k
s
t
e
p
p
i
ng
con
t
ro
l
l
er
t
rac
k
p
e
r
fec
t
ly
t
he
s
i
nus
o
i
da
l
r
e
fe
renc
e
in
phase
a
n
d
am
pl
itu
de.
F
i
g
u
re
6
b
s
h
o
w
s
t
h
e
F
F
T
sp
ec
t
r
um
of
t
h
e
u
nf
ilter
e
d
o
u
t
pu
t
v
o
l
t
a
ge
o
f
t
h
e
in
v
e
rter
a
nd
the
gr
id
i
nj
e
c
t
e
d
cu
rre
n
t;
i
t
can
b
e
ob
serv
ed
t
h
a
t
th
e
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN: 2088-
8694
I
nt
J
P
ow
Elec
& Dr
i
S
y
st, Vol. 10,
N
o.
1, Mar
c
h 2
0
1
9
:
19
–
26
24
ob
ta
ine
d
T
H
D
i
s
a
c
c
e
p
t
ab
le,
w
h
ich
c
o
n
f
irm
tha
t
u
n
d
er
i
nt
e
g
ral
b
a
c
k
s
te
pp
i
ng
c
o
n
t
ro
l
of
t
he
g
rid-c
o
n
n
e
c
t
e
d
VSI
with L-
f
ilt
er
,
fewe
r ha
rmon
i
c
s ar
e
present.
(a)
(b)
F
i
gure
6.
(
a) The
un
f
il
t
e
red
v
o
lta
ge ou
t
p
u
t o
f
t
he
i
n
v
erte
r (ch
1)
a
n
d
the
i
nje
c
te
d c
u
rre
n
t
i
g (c
h4),
(b) F
F
T
spectr
u
m
un
de
r i
n
te
g
r
a
l
bac
kste
p
p
i
n
g c
o
n
t
ro
l
F
i
gure 7
i
l
l
u
s
tr
ate
s
the res
po
n
s
es
o
f
t
h
e i
n
je
c
t
ed curre
n
t
a
n
d
the
gr
id v
olta
ge
, w
h
ich i
t
s R
M
S va
lue i
s
1A
. A
c
c
o
rdin
g
to
t
h
i
s
fig
u
re
,
the
s
ync
hr
o
n
i
z
a
ti
on m
ode
i
s rea
c
h
ed un
d
er inte
g
ral ba
cks
t
e
p
p
i
ng
c
o
n
t
ro
l, w
hic
h
con
f
irm
s
t
he
e
ffe
c
t
i
v
enes
s
o
f
t
he
c
o
n
t
r
o
l
strat
e
gy
a
n
d
t
h
e
P
LL
bl
oc
k.
T
h
i
s
a
c
hie
v
e
d
p
er
for
m
anc
e
t
ha
t
re
fl
ec
t
s
the
c
o
n
d
i
t
i
on
o
f
P
F
C
i
s
he
l
p
f
u
l
to
f
u
l
fi
ll
t
h
e
grid-c
o
nne
cte
d
c
on
di
t
i
on
o
f
VSI.
C
o
n
t
r
oll
e
r
g
a
i
n
s
are
:
K
_e=
1
2
and
K
_
i=0.2.
F
i
gur
e 7.
G
r
i
d vo
lta
ge (
ch
1)
a
nd t
h
e
i
n
j
e
cte
d
c
urre
nt
i
g (c
h4
)
under
in
tegra
l
b
ac
kste
p
p
i
n
g c
o
n
t
r
o
l
4.2.
C
o
mp
arati
v
e
Exp
erime
n
t
al
R
esu
l
ts
To
s
ho
w
th
e
imp
o
rt
an
c
e
o
f
t
h
e
p
r
o
p
o
s
ed
i
nt
eg
ral
ba
ck
st
e
p
p
i
ng
c
o
n
t
ro
l,
a
c
om
par
a
t
i
ve
e
xper
i
m
e
nta
l
tes
t
s
betw
e
e
n
t
his
la
t
t
er
a
nd
a
P
I
c
on
t
r
ol
s
y
s
t
e
m
ar
e
ex
pose
d
i
n
t
h
i
s
s
ec
ti
o
n
.
For
t
h
is
p
urp
o
s
e
,
th
e
bac
k
ste
p
pi
n
g
c
on
t
r
o
ller
bl
ock
is
s
ubs
t
itu
ted
by a
P
I
c
o
n
tro
l
s
y
stem
bl
o
ck.
F
i
rst
l
y,
t
he
b
lo
ck
o
f
P
I
c
on
tro
l
l
e
r
is
i
na
pt
t
o
re
gu
l
a
te
t
h
e
g
r
i
d
c
urre
nt
u
nd
e
r
t
he
s
i
n
usoi
da
l
re
fer
e
nce
,
it
n
ee
d
t
h
e
gr
i
d
v
o
lta
ge
f
e
e
d
forw
a
r
d,
w
hic
h
can
b
e
cons
ide
r
ed
as
d
ra
wb
ac
k
fo
r
th
e
g
r
id
-con
n
ect
ed
a
p
pli
c
at
ion
and
whi
c
h
l
ead
t
o
dou
bt
c
on
c
e
r
n
i
ng
t
h
e
l
i
n
ea
rit
y
o
f
t
he
c
o
n
tr
ol
ler
i
n
t
his
ca
se
.
O
n
t
he
o
t
h
e
r
h
a
n
d,
t
o
a
c
c
e
s
s
th
e
tran
si
e
n
t
resp
on
se
o
f
t
h
e
PI
b
lo
ck
c
ont
ro
lle
r
a
g
a
inst
t
he
b
ac
kste
p
p
i
n
g
con
t
ro
l
l
e
r
,
the
c
u
rre
n
t
r
e
f
e
r
e
n
c
e
i
s
v
a
r
i
e
d
a
s
a
s
t
e
p
.
U
s
i
n
g
P
I
c
o
n
t
r
o
l
s
y
s
t
e
m
b
l
o
c
,
a
n
d
un
der
the
sam
e
c
ond
it
i
ons,
th
e
in
j
e
c
t
e
d
g
ri
d
curr
ent
a
n
d
th
e
u
n
fil
t
e
r
ed
v
olta
ge
o
ut
put
o
f
t
h
e
V
S
I
w
i
th
t
he
ir
F
F
T
s
pec
t
r
u
m
s
a
re
ill
us
t
r
ate
d
i
n
F
i
gur
e
8.
I
t
ca
n
be
n
o
t
ed
t
ha
t,
c
om
par
a
t
i
ve
l
y
w
it
h
the
i
n
tegr
a
l
b
a
c
k
st
e
p
p
i
ng
c
on
trol,
the
TH
D
is
i
ncr
ease
d
,
a
nd
th
e
des
i
red
i
n
j
e
cte
d
c
urr
e
nt
v
a
l
u
e
i
s
less
prec
ise
,
w
hi
c
h
i
s
c
ons
id
ere
d
a
s
a
we
a
k
n
e
s
s
o
f
t
h
e
c
o
nt
ro
l
sys
t
em
robustn
e
ss for
the
g
r
i
d
-c
onn
ect
ed
VSI.
Evaluation Warning : The document was created with Spire.PDF for Python.
Int J
P
o
w
El
e
c
&
D
ri S
yst
I
S
S
N
:
2088-
86
94
D
e
sig
n
a
nd im
plem
e
n
ta
ti
o
n
o
f
no
n
l
i
n
e
a
r i
n
te
gr
al
b
a
cks
tep
p
i
n
g c
o
n
t
r
o
l st
ra
t
e
gy
for..
.
(Mous
sa Re
dd
a
k
)
25
(a)
(b)
F
i
gure
8.
(a
)
The un
f
il
tere
d v
o
l
t
a
g
e
out
p
u
t o
f
the
i
n
v
er
ter
(ch
1)
a
n
d
i
n
j
ec
te
d gri
d
c
urre
nt (c
h4); (b)
FF
T
spec
trum
,
unde
r P
I
contro
l
F
i
gure
9
gi
ves
the
gri
d
v
ol
ta
ge
a
nd
t
h
e
i
n
je
cted
c
urr
e
n
t;
it
c
a
n
be
s
ee
n
tha
t
t
he
P
F
C
c
on
d
iti
on
i
s
respe
c
te
d,
t
he
i
n
j
ec
te
d
gr
id
c
urr
e
nt
i
s
n
o
t
p
erfectl
y
p
re
ci
s
e
(
213.
1mV
)
a
g
a
i
n
st
i
nte
g
ra
l
bac
k
ste
p
pi
n
g
c
o
n
tro
l
(210.2m
V
)
.
F
i
gure
9.
T
he
g
ri
d
vo
l
t
a
g
e (
c
h1)
a
n
d
the
i
n
j
e
c
ted
curre
n
t
(c
h4)
u
nder
P
I
con
t
r
o
l
A
l
so,
a
c
c
o
r
d
in
g
to
F
ig
ure
1
0
t
ha
t
g
i
ve
s
t
h
e
gr
id
c
urren
t
r
e
s
pon
ses
of
P
I
co
nt
rol
sy
st
e
m
a
nd
i
nt
e
g
ra
l
bac
k
ste
p
pi
n
g
c
on
tro
l
ler,
i
t
ca
n
be
s
ee
n
tha
t
u
n
d
er
t
he
l
a
t
te
r
i
n
Fi
gu
re
1
0
(
a),
t
h
e
t
r
a
n
si
en
t
re
sp
on
se
o
f
th
e
in
j
e
c
t
e
d
c
ur
rent
i
s
mor
e
r
a
p
id
c
ompa
rat
i
ve
ly
w
ith
t
he
P
I
b
l
ock
in
F
ig
ure
10(
b),
and
t
h
e
gri
d
i
n
j
e
c
t
ed
c
urren
t
in t
he
case
of
i
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c
k
s
t
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p
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co
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ac
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ma
gnit
ude
s
t
e
p re
fer
e
nce
w
ith
i
n
s
ome
m
i
l
l
iseco
n
d
s.
Re
ga
rdi
ng
the
e
x
p
e
rim
e
nta
l
r
esul
ts,
an
d
i
n
t
e
r
m
s
o
f
c
o
mpa
r
ison,
i
t
c
a
n
be
c
on
c
l
uded
t
h
at
t
h
e
pro
pose
d
i
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t
e
g
ra
l
bac
k
s
t
ep
p
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g
co
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t
rol
de
si
gne
d
f
o
r
t
h
e
gr
id-c
o
n
n
ecte
d
V
SI
o
wns
exc
e
l
l
e
n
t
pro
p
e
rties
i
n
clu
d
i
n
g
robust
c
ont
ro
l
p
e
rfo
rman
c
e
,
a
c
c
e
p
t
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bl
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THD
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o
t
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d
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f
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r
a
p
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d
d
y
n
a
m
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c
o
f
the c
u
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n
t
i
nje
c
ted i
n
to
t
he
g
r
i
d.
(a)
(b)
F
i
gure
1
0
. G
rid vo
l
t
a
g
e
(ch
1
)
and
grid
c
urr
e
nt r
esp
o
n
s
e
s
w
it
h v
ar
i
a
tio
n
of
the
g
ri
d cur
r
ent r
e
fe
renc
e
(ch4):
(a)
un
der
in
te
gra
l
b
ack
ste
p
pi
n
g
c
ontr
o
l
(b)
un
de
r P
I
c
ontro
l
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN: 2088-
8694
I
nt
J
P
ow
Elec
& Dr
i
S
y
st, Vol. 10,
N
o.
1, Mar
c
h 2
0
1
9
:
19
–
26
26
5.
CONCL
U
S
ION
I
n
t
hi
s
p
a
pe
r,
a
n
on
l
i
ne
ar
c
on
tro
l
s
tra
t
e
g
y
based
on
i
n
tegra
l
b
acks
t
e
p
p
i
ng
t
ec
h
n
i
q
ue
f
or
g
r
i
d-
con
n
ec
ted
V
S
I
i
s
d
e
s
i
gne
d
a
nd
im
p
l
em
ente
d
o
n
D
S
P
board.
Thi
s
c
o
nfigur
a
t
i
on
c
i
rcu
i
t
w
ith
t
he
p
ro
pos
e
d
con
t
ro
l
l
er
i
s
de
si
gna
te
d
for
d
i
stri
bu
t
e
d
p
o
w
e
r
ge
nera
tio
n
of
a
rene
w
a
ble
e
n
e
r
g
y
s
o
u
rce
,
a
nd
t
o
o
v
er
co
m
e
t
he
sinus
o
i
da
l
re
fe
r
e
nce
trac
k
i
n
g
.
The
sta
b
i
l
i
t
y
of
t
he
c
o
n
tro
l
s
ys
tem
is
v
eri
f
ied
by
L
y
a
p
u
n
ov
the
o
ry
p
roc
e
dure
,
and
the
d
ynam
i
c m
odel
o
f
t
he
V
S
I
is w
e
l
l
d
escrip
ted
an
d ve
rifie
d
also
u
nd
er
t
he
c
on
t
r
ol
s
y
s
tem
.
Base
d
on
the
e
x
p
e
r
i
me
nta
l
r
esu
l
ts,
the
pro
pos
ed
i
nte
g
ra
l
ba
c
k
st
e
ppi
ng
c
o
n
t
rol
can
i
mp
ro
v
e
t
he
perform
ance
a
nd
r
o
bus
tne
s
s
of
V
S
I
s
ystem
,
w
it
h
a
go
o
d
s
i
nus
o
i
da
l
reference
trac
k
i
ng.
Mo
re
ov
er,
t
h
e
ra
pi
d
dy
nam
i
c
t
r
a
n
si
ent
resp
o
n
se
o
f
the
i
n
j
e
cte
d
c
urr
e
nt
i
s
nee
d
e
d
f
o
r
t
h
e
c
l
e
a
n
d
i
s
t
r
i
b
u
t
e
d
p
o
w
e
r
e
n
e
r
g
y
,
w
h
i
c
h
i
s
c
o
nsid
e
r
ed
a
s
a
f
e
at
u
r
e
of
t
he
a
p
p
l
i
e
d
c
ontro
l
st
ra
t
e
gy
.
A
l
so
,
t
h
e
cu
rre
nt
h
a
r
mo
ni
c
di
st
o
r
t
i
on
r
a
t
io
i
s
wel
l
am
eliora
ted
an
d
t
h
e
pr
oc
ess o
f
g
rid c
u
rren
t
i
njec
tio
n
i
s
s
tric
t
l
y
r
espec
t
e
d
u
nder
t
h
e
con
d
itio
n o
f
PF
C
.
REFE
RENCES
[1]
E.
K
outro
ulis
e
t
F
.
B
l
aabj
e
rg
,
“
M
e
t
h
o
d
o
l
o
g
y
f
or
t
he
o
p
t
imal
d
es
ign
of
t
rans
f
o
rm
erles
s
g
rid
-
conn
ected
P
V
inverters
,”
IE
T Po
wer E
l
ectr
o
n
.
,
vo
l. 5
, no
8,
p.
1
4
9
1
, 20
1
2
.
[2]
M
.
R
ed
dak,
A
.
Berdai
,
A
.
G
ourm
a
,
et
J
.
Bo
uk
hero
uaa,
“
A
n
I
mp
rov
e
d
Co
nt
rol
S
t
rat
e
gy
us
in
g
RS
C
o
f
t
he
W
ind
Turbine Based o
n
DFI
G
f
o
r Grid
H
arm
o
n
i
c
Current
s
M
i
ti
gatio
n.
”
IJR
E
R
,
M
a
rch,
v
ol
8
,
n
o
3
,
20
18.
[3]
S
.
S
harm
a
et
B
.
S
i
ngh,
“
Co
nt
rol
of
p
erman
e
nt
m
agn
e
t
s
ynch
r
on
ou
s
g
enerato
r
-based
s
t
a
nd
-alon
e
w
i
n
d
energ
y
con
v
ersi
on
sy
s
t
e
m
,
”
I
E
T
Power
El
ectron
.
,
vo
l
. 5
, n
o 8
, p
.
1
5
1
9
, 2
01
2.
[4]
Y.
W
.
K
ean,
A
.
R
am
asam
y
,
S
.
Su
kum
ar,
et
M
.
M
a
rsadek
,
“
A
da
p
t
i
v
e
Contro
l
l
ers
f
o
r
E
n
h
a
ncem
ent
of
S
t
a
nd-Alo
n
e
Hyb
r
id
S
ys
te
m
Pe
r
f
or
ma
n
c
e
,
”
In
ter
n
a
t
io
nal Jo
ur
na
l of Power
Electr
o
n
i
cs
and
D
r
i
v
e
Sys
t
em (
I
JPED
S)
,
v
o
l
.
9,
n
o 3,
p.
8
,
201
8.
[5]
A
.
P
a
n
d
a
,
M
.
K
.
P
a
t
h
a
k
,
e
t
S
.
P
.
S
r
i
v
a
s
t
a
v
a
,
“
A
s
i
n
g
l
e
p
h
a
s
e
p
h
o
t
ov
oltai
c
i
n
v
erter
con
t
rol
for
gri
d
c
on
nect
ed
sy
st
e
m
,
”
S
pri
nger,
S
adh
a
na,
v
ol.
4
1
,
n
o
1,
2
01
6.
[6]
I.
S
e
tiaw
a
n,
M
.
F
acta,
A
.
P
r
i
y
adi
,
e
t
M.
H
.
P
u
rnom
o,
“
In
vestig
a
ti
on
o
f
S
y
mm
etrical
O
ptim
um
P
I
Con
t
roller
bas
e
d
on
Plan
t
an
d
F
eed
back
Lin
earization
i
n
G
rid- Ti
e
Inv
erter S
y
ste
ms
,”
IJRER
,
vol7,
no
3,
2
0
17.
[7]
H.
A
t
t
ia,
“Evaluation
of
S
e
l
ectiv
e
Harm
on
ic
E
l
i
m
i
na
ti
on
P
u
l
s
e
W
idth
M
o
dulatio
n
Tech
niq
u
e
f
o
r
U
n
ip
o
l
ar
S
i
ngle-
P
h
as
e
H-Bri
dge
I
nv
e
r
t
e
r:
C
o
m
parat
i
ve
S
tud
y
,
”
Int
e
rnational
J
o
urnal
o
f
P
o
we
r
Electroni
cs and Dr
ive
Syst
em
(I
J
P
ED
S
)
,
vol.
9
,
n
o
3,
p
.
9
,
2
018.
[8]
S.
S
a
limin,
S.
A
.
Zulki
f
li,
e
t
M.
A
r
m
stron
g
,
“
Re
du
c
t
io
n
in
C
urr
ent
T
H
D
of
G
rid
P
a
ra
l
l
el
I
nvert
ers
Using
Rand
om
ized
P
R
Co
ntrol,
”
Inter
n
a
tio
nal Jo
ur
n
a
l
o
f
P
o
wer
Elect
ro
n
i
cs
an
d
Drive Sys
t
em
(IJPED
S
)
,
v
o
l.
8
,
no
1
,
p
.
290,
2
017.
[9]
R.
T
eod
o
res
c
u,
F
.
Blaab
jerg,
M.
L
iserre,
et
P
.
C.
L
o
h
,
“
Propo
r
t
i
onal-
r
esonant cont
rollers
and
filt
er
s f
o
r gr
i
d
-
con
n
ected
vol
t
a
g
e-s
our
ce con
vert
e
rs
,
”
IE
EE
Proc.
-
E
l
ectr
.
Po
w
e
r
,
v
o
l.
1
5
3
,
no
5
, p
.
75
0, 2
00
6
[10]
J.
A
.
Cort
ajarena,
O
.
Baram
bon
es,
P
.
A
lk
ort
a
,
et
J
.
D
e
M
arcos
,
“
S
lid
i
n
g
m
o
de
c
o
n
t
r
ol
o
f
gri
d
-tied
si
ngle-ph
ase
in
verter
i
n
a
ph
o
t
ov
olt
a
ic M
PP
T app
l
i
catio
n
,
”
Sol.
Ener
gy
,
v
o
l
.
15
5
, p.
7
9
3
-
80
4,
2
01
7.
[11]
S.
S
a
limin,
S.
A
.
Z
u
lki
f
li
,
et
a
l
M.
A
r
m
stron
g
,
“Reduction
in
C
urrent
T
HD
of
G
r
i
d
Paral
l
el
I
nverters
Usi
ng
Rand
om
ized
P
R
Cont
ro
l
,
”
Inter
n
a
t
i
onal
Jou
r
n
a
l
o
f
Power
E
l
ectron
i
cs
a
n
d
D
r
i
v
e
Sys
t
em
(
I
JPE
D
S)
,
vo
l.
8
,
n
o
1,
p
.
29
0,
M
ar
201
7
.
[12]
K.
R
a
m
e
s
hk
um
a
r
,
V.
I
n
d
r
a
g
a
n
dh
i,
K
.
Pa
la
n
i
sa
my
,
e
t
T
.
Arun
ku
m
a
ri
.
“
M
od
el
P
red
i
cti
v
e
Current
C
o
n
t
r
ol
o
f
S
i
ngle
P
h
as
e S
h
u
n
t
Ac
tive
Po
wer Fi
lt
er,”
En
erg
y
Pr
oced
ia
, v
ol
.
1
1
7
, p
. 6
5
8
-6
6
5
, 20
1
7
.
[13]
S
.
N
e
t
al
C
.
Lak
s
h
m
in
a
r
ay
ana,
“
P
e
rf
o
r
m
a
nce
E
n
h
a
ncem
e
n
t
in
A
cti
ve
P
ower
F
il
ter
(APF)
by
FPGA
Impl
e
m
entation,
”
Int
e
rn
at
ion
a
l Jour
na
l of
Electri
cal
a
n
d
Comp
utin
g En
g
i
n
eeri
n
g
(
I
JE
CE)
,
v
o
l
.
8
,
n
o
2
,
p
.
6
89
,
20
18
.
[14]
T.
T
hacker,
D
.
B
oroy
evi
c
h,
R
.
Burg
os,
et
F
.
W
a
ng,
“
Ph
ase-Locked
L
oo
p
No
is
e
Red
u
ct
ion
v
i
a
P
h
a
s
e
Detector
Im
pl
e
m
entation
f
o
r
Sing
l
e
-Phase
Sys
t
e
ms,”
I
E
E
E
Tran
s.
In
d.
Ele
c
t
ro
n
, vo
l
.
5
8
,
no
6
, p.
24
8
2
-2
49
0,
2
01
1.
[15]
M.
A
o
u
r
i
r
,
A.
A
b
o
u
l
oi
f
a
, I
.
L
a
chk
a
r, A. Ha
m
d
o
u
n
,
F
.
G
iri,
e
t
F.
C
un
y,
“
Nonli
n
ea
r Co
ntr
o
l
of
P
V
S
y
st
em Co
nnect
ed
to
S
i
ngle P
hase
G
r
i
d
t
h
r
oug
h Ha
l
f
Bridg
e
Po
wer
In
vert
e
r
,” IFAC-
P
a
p., vo
l
. 50
,
n
o
1
,
p
.
74
1-7
4
6
,
2
0
1
7
.
[16]
R.-J.
W
a
i,
C
.-Y
.
L
in
,
H.
-
N
.
Hu
a
n
g,
e
t
W
.
-C
.
W
u
,
“Design
o
f
b
ack
s
t
ep
pi
ng
co
nt
rol
f
o
r
h
i
gh
-perform
an
ce
in
verter
with
st
a
nd
-a
l
o
n
e
a
nd
g
ri
d-co
nnect
ed
p
ow
er-sup
p
l
y
mo
des,
”
I
E
T
Pow
e
r
El
ect
r
o
n
,
v
o
l. 6
,
n
o
4
,
p
. 75
2
-7
62
, 20
1
3
.
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