Internati
o
nal
Journal of P
o
wer Elect
roni
cs an
d
Drive
S
y
ste
m
(I
JPE
D
S)
V
o
l.
6, N
o
. 1
,
Mar
c
h
20
15
,
pp
. 10
0
~
10
8
I
S
SN
: 208
8-8
6
9
4
1
00
Jo
urn
a
l
h
o
me
pa
ge
: h
ttp
://iaesjo
u
r
na
l.com/
o
n
lin
e/ind
e
x.ph
p
/
IJPEDS
Line and Grid Impedance Impact
on the P
e
rf
orm
a
nces of a
Parall
el Connect
ed M
o
dular Inverter Sys
t
em
Tahar
Z
e
bbadji,
S
e
ddik H
a
dji and Rachi
d
Ib
tioue
n
Labora
t
oire
de
R
echer
che
en
El
ec
tr
otechnique, LR
E, Eco
l
e
N
a
tion
a
le
Poly
techn
i
que d’Alger
,
ENP,
Algeria
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Dec 8, 2014
Rev
i
sed
Jan 22, 201
5
Accepte
d
Fe
b 4, 2015
W
ith the ris
i
ng fuel cos
t
, in
cre
a
s
ing
demand of
power and the concerns for
global climate change, th
e use o
f
clean
energ
y
make
the connection of power
electronics build
ing bloc
in th
e h
eart
of
the
current resear
ch. The
high outpu
t
current app
lications
make
the par
a
llel connection
of modular inverters to be a
solution for the use of low po
wer building block inverters where the outpu
t
power cannot be handled b
y
a single inver
t
er co
nfiguration. In this context,
averag
e-m
odelin
g us
ing average
phas
e
–leg te
ch
nique al
lows
the
n
-paral
le
l
connected
inver
t
ers to b
e
an
aly
z
ed a
ccur
a
tely
an
d rapid
l
y
withou
t requ
iring
the
complexity
o
f
the full switch
e
d inve
r
t
er
topolo
g
y
. Th
e ob
tain
ed
analy
t
ical
solution along
with the
equiv
a
lent circ
uit model makes easier the design of
the
control loop
. The an
aly
t
ical s
o
lution of
th
e
n
-
p
aral
lel
conn
ect
ed inv
e
rters
shows the impact of
th
e line an
d gr
id impedance
on the perfor
m
ance of the
overall s
y
stem.
The impact of
this coup
ling has
to be investig
ated such that
the m
a
in
fea
t
ure
of par
a
ll
eling
i
nverters
is guar
a
nte
e
d and
tha
t
the inv
e
rt
er
m
ode of operat
i
on will not b
e
com
p
rom
i
sed. The m
a
in ad
vantag
e of
paralleling inverters can b
e
lost
for a cer
t
ain
coupling
impedance
considerations
.
Keyword:
Ave
r
a
g
e phase
-l
eg
t
e
c
hni
que
C
o
u
p
l
i
n
g Im
pedance
Im
pact
M
o
d
u
lar i
nve
rt
er
Parallel con
n
e
ctio
n
Perform
a
nces
Copyright ©
201
5 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
Tahar
Ze
bba
d
j
i
,
Laboratoi
r
e de Recherc
h
e
e
n
Electrotechnique, L.R.E
,
Ecol
e
Nat
i
onal
e
Pol
y
t
ech
ni
q
u
e
d
’
Al
ger,
10
, Ave
n
ue Ha
cene
B
a
di
,
B
P
18
2,
El
-
H
ar
rac
h
, 1
6
2
0
0
, Al
gi
ers, Al
ge
ri
a.
Em
a
il: tah
a
r.zeb
b
a
dj
i@g
.
en
p.ed
u.d
z
1.
INTRODUCTION
In t
h
e rece
nt
y
ears,a ra
pi
d
gr
owt
h
i
n
t
h
e pr
o
duc
tion of electrical energy
fr
om
renewabl
e ene
r
gy
sou
r
ces
has
be
en m
a
de.
Thi
s
has l
e
d
t
o
a
gr
eat
devel
opm
ent in the
power electroni
cs a
r
e
a
.
Indeed, the
out
put
of eac
h
rene
w
a
bl
e ene
r
gy
s
o
u
r
ce s
h
oul
d
be c
o
n
n
ect
ed
t
o
a s
p
ecific
conve
r
ter
or i
nve
rter s
u
c
h
t
h
at the
co
nv
erted
en
erg
y
will b
e
availab
l
e
at th
e consu
m
er lev
e
l [1
], [2
].
In ord
e
r t
o
m
eet th
e
g
r
o
w
t
h
o
f
power d
e
m
a
n
d
of
t
h
e i
n
du
stry,
research in
po
wer electron
ics still
need
s t
o
fi
n
d
s
o
l
u
t
i
o
ns t
o
t
h
e
ener
gy
co
n
v
er
s
i
on at
hi
g
h
po
wer l
e
vel
s
[
3
]
,
[4]
.
K
n
o
w
i
n
g t
h
at
t
h
e c
o
nve
rs
i
on
o
f
el
ect
ri
cal
ener
gy
uses
p
o
w
e
r el
ect
ro
ni
cs
com
pone
nt
s,i
nve
rt
ers
base
d o
n
t
h
e
s
e s
w
i
t
c
hes s
h
o
w
bet
t
e
r
per
f
o
r
m
a
nces if t
h
ey
are used
at
hi
gh swi
t
c
h
i
ng f
r
eq
ue
nci
e
s
[
5
]
,
wi
t
h
l
o
w
st
resses.
H
o
we
ver
,
t
h
e swi
t
c
h
e
s ca
n
wi
t
h
st
an
d o
n
l
y
l
i
m
i
t
e
d val
u
es
of cu
rre
nt
an
d v
o
l
t
a
ge.
Al
t
h
o
u
g
h
,
t
h
e p
o
w
er rat
i
n
gs
of
po
we
r sem
i
cond
uct
o
r
devices
have i
n
crease
d
c
onsi
d
era
b
ly since the intr
odu
ction
of th
e first
co
mmercial s
w
itch
,
th
e m
a
x
i
m
u
m
po
we
r rat
i
n
gs
m
a
y
be l
i
m
i
t
e
d by
t
e
c
hni
cal
or e
c
o
n
o
m
i
cal
consi
d
erat
i
o
n
s
. Th
ere
f
o
r
e,
f
o
r a
hi
ghe
r
p
o
we
r
trans
f
er,
a degradation of
efficiency
with
t
h
e in
crease
of
th
e switch
i
ng
fre
que
ncy
occ
u
rs
. Un
fo
rtu
n
a
t
ely
,
to
achieve a
n
acc
eptable efficiency while increasing s
w
itc
hi
ng
fre
quency, t
h
e trends e
nha
nce active rese
arch
on
vari
ous
t
y
pes
of c
o
nne
ct
ed i
nve
rt
er m
o
d
u
l
e
s suc
h
t
h
at
j
u
st
a fract
i
o
n
o
f
t
h
e
o
u
t
p
ut
p
o
we
r
of
t
h
e sy
st
em
i
s
handled
by each m
odule. S
o
at high leve
l, com
p
lete
power converte
rs
are develope
d around the
powe
r
m
odul
es wi
t
h
e
i
t
h
er
paral
l
e
l
o
r
se
ri
es co
n
n
ec
t
i
on.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
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SN
:
208
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9
4
Li
ne an
d Gri
d
Im
pe
da
nce Im
pact
o
n
t
h
e
Per
f
orm
a
n
ces of
a
Par
a
l
l
e
l
C
o
nn
ect
ed
M
o
d
u
l
a
r
…
(
T
ah
ar Zeb
b
a
d
j
i
)
10
1
If t
h
e system
i
s
op
erating
i
n
p
a
rallel, th
e total o
u
t
pu
t
p
o
wer can
b
e
p
a
rtitio
n
e
d in
a way th
at the
ove
rall efficiency can be
ke
pt as
hi
g
h
as
p
o
ssi
bl
e,
t
h
e
n
i
t
has t
o
be cl
ari
f
i
e
d at
whi
c
h
poi
nt
t
h
e
n
u
m
b
er
n
of
p
a
rallel inv
e
rters
h
a
s to b
e
increased in
o
r
der to
ac
hie
v
e maxim
u
m
efficiency
[6
].
In
th
is stud
y,
n
i
d
entical
inve
rters are c
o
nnected i
n
pa
rallel both at t
h
e input a
n
d
ou
tpu
t
, so
th
at th
e ou
tp
ut current and power
m
eet
hi
g
h
er
re
qui
r
e
m
e
nt
s appl
i
cat
i
ons
. T
h
i
s
co
n
n
ect
i
o
n
i
s
m
a
de by
m
ean of t
h
e
di
ffe
rent
l
i
n
e im
pedance
s
t
o
t
h
e
gri
d
t
r
ou
g
h
a
g
r
i
d
i
m
pedance
.
Thi
s
c
o
upl
i
n
g
al
on
g
wi
t
h
t
h
e
num
ber
n
o
f
inv
e
rters to b
e
co
nn
ected in p
a
rallel
i
n
fl
ue
nce
gr
eat
l
y
on t
h
e
per
f
o
r
m
a
nces of
t
h
e
ove
ral
l
ci
rc
ui
t
.
The st
udy
foc
u
ses o
n
t
h
e
de
v
e
l
opm
ent
of t
h
e paral
l
e
l
co
n
n
ect
i
on
of i
nve
r
t
ers w
h
i
c
h i
s
o
f
t
e
n
use
d
t
o
achi
e
ve p
o
w
er
l
e
vel
s
bey
o
n
d
t
h
e capaci
t
y
of t
h
e hi
g
h
p
o
w
e
r avai
l
a
bl
e fr
om
a si
ngl
e conve
nt
i
o
nal
st
ru
ct
ure:
su
ch
a system
co
nstitu
tes a sp
ecial conn
ectio
n of
bu
ild
ing
b
l
o
c
k
s
t
o
p
r
ov
id
e a h
i
g
h
l
y
reliab
l
e system
[7
],
[8
]
.
The
obt
ai
ned
s
t
ruct
u
r
e i
s
ca
pa
bl
e o
f
del
i
v
eri
n
g
hi
g
h
o
u
t
p
ut
c
u
r
r
ent
.
The
r
e
f
o
r
e,
paral
l
e
l
co
n
n
ect
i
o
n
o
f
i
n
ve
rt
ers
has
bec
o
m
e
a desi
ra
bl
e s
o
l
u
t
i
on,
p
a
rt
i
c
ul
arl
y
i
n
area
s
whe
r
e a
hi
g
h
dem
a
nd
f
o
r
e
n
er
gy
i
s
re
qui
red
wi
t
h
hi
g
h
o
u
t
p
u
t
curren
t
. Parallelin
g
is u
s
ed
to
ach
iev
e
th
e follo
wing
ch
aracteristics: red
u
n
d
a
n
c
y, flexib
ility,
standa
rdization of low
powe
r com
pone
nts
ratings, reliability, size reduction,
low c
o
st
of m
a
intenance, etc.
Although the
m
odular m
u
ltilevel converte
rs[9]
offers
s
o
m
e
m
a
jor a
d
va
ntages
as to re
duce
the am
plitude
of
harm
onics inj
e
cted into the
load, t
h
e a
b
ility to work at
low fre
nque
ny
with acce
ptable
e
f
ficiency etc.
but the
stan
d
a
rtisatio
n
an
d flex
ib
ility may b
e
co
m
p
ro
m
i
sed
.
There
f
ore,
a c
o
m
p
rehe
nsi
v
e
st
udy
of t
h
e
gl
obal
be
havi
or
of t
h
e ci
rc
ui
t
i
s
co
nsi
d
e
r
ed
.
M
odel
i
n
g
i
s
m
o
re t
h
an nec
e
ssary
f
o
r a
q
u
i
c
k a
nd m
e
t
hodi
cal
st
u
d
y
o
f
t
h
e st
ea
dy
an
d dy
nam
i
c st
ates.
Se
veral
m
odel
i
n
g
t
echni
q
u
es
can
t
h
en
be
u
s
ed
.
The t
e
c
hni
que
use
d
i
n
t
h
i
s
w
o
r
k
i
s
t
h
eave
r
a
g
e
pha
se-l
eg
t
e
chni
que
[
1
1]
,
whi
c
h
descri
bes a si
m
p
le averaging m
odel to be
accurately
and ra
pidly sim
u
lated without the nee
d
for the full
swi
t
c
he
d m
ode
l
s
.
B
e
si
de t
h
ese a
dva
nt
age
s
, p
a
r
a
l
l
e
l
connect
i
o
n o
f
i
n
vert
ers presents
undesirable constraint
s that can be
stated as c
u
rre
n
t unbalance
,
instability due
to t
h
e inte
rac
tion
of th
e
differe
n
t m
odules
, circ
ulating c
u
rrent
bet
w
ee
n m
odul
es [1
0]
(det
e
r
i
o
rat
i
on
of t
h
e ef
fi
ci
ency
an
d f
o
rm
fact
or of t
h
e out
put
c
u
r
r
en
t
s
),sy
nc
hr
o
n
i
zat
i
o
n
pr
o
b
l
e
m
of t
h
e
out
put
c
u
rre
nt
s
,
et
c.
In t
h
i
s
fi
rst
p
a
per
,
we p
r
ese
n
t
a
m
odul
ar
i
nve
rt
er arc
h
i
t
ect
ure i
n
w
h
i
c
h an
n
p
a
rallel-con
n
ected
inve
rters syste
m
is tied to a
n
infinite
gri
d
vi
a line and
grid im
pedances
.
B
ecause of
the ti
me varying as
pect of
th
e system
, tra
n
sfo
r
m
a
tio
n
in
d
-
q
f
r
am
e i
s
requi
red t
o
s
o
l
v
e f
o
r
t
h
e
gen
e
ral
sol
u
t
i
on i
nde
pe
nde
nt
l
y
of t
h
e
circu
it p
a
ram
e
ters.
Th
en
, a si
m
p
lified
av
erag
e equ
i
v
a
le
n
t
m
o
d
e
l is d
e
riv
e
d wh
ere analytical so
lu
tio
n of
di
ffe
re
nt
t
r
an
sf
er f
u
nct
i
o
n
s
ca
n
be
a
n
alyzed
whate
v
er the
num
b
er
n
o
f
inverters is. Th
e
op
en loo
p
stab
il
ity fo
r
th
e
n
-pa
r
allel-connected inverters is
analyzed with
respect t
o
th
e diff
ere
n
t p
a
ram
e
ters of
the
circu
it.Fin
a
lly,an
a
lytical resu
lts are g
i
v
e
n to
sh
ow t
h
e effects o
f
t
h
e co
up
lin
g im
p
e
d
a
n
ce
an
d th
e
n
u
m
b
e
r
n
of
i
nve
rt
ers t
o
be
co
nnect
e
d
i
n
paral
l
e
l
o
n
t
h
e
pe
rf
orm
a
n
ces
of the
system. T
h
en a c
r
iterio
n
is stated
t
o
first
respect t
h
e m
ode
of
operation of the ove
r
all circuit an
d se
cond t
o
gua
r
antee the m
a
in
p
u
r
po
se of
p
a
r
a
l
l
elin
g
i
nve
rt
er m
o
d
u
l
e
s.
2.
PHASE-LEG
ANALYSIS OF THE
N PA
RALLEL
MODUL
AR CONNECTED INVERTER
The ave
r
age
phase-le
g techni
que is one of t
h
e esse
n
tial tech
n
i
q
u
e
s u
s
ed
in
th
e an
alysis o
f
switch
e
d
m
ode po
wer
c
o
n
v
e
r
si
o
n
[
1
1]
-[
12]
.
It
al
l
o
w
s
t
h
e s
w
i
t
c
hed
sy
st
em
t
o
be
desc
ri
be
d by
a si
m
p
l
e
averagi
n
g
circu
it m
o
d
e
l, wh
ich
th
en
en
ab
les its precise and
rap
i
d
simu
latio
n
witho
u
t
th
e
n
e
ed
for fu
ll
switch
e
d
inv
e
rter
m
odels. The
n
,
the inverte
r
s
w
itches can be
replaced
by a
function re
pre
s
enting thei
r a
v
era
g
e
value.
The structure is com
posed
of
n
id
en
tical p
a
rallel co
nn
ected
inv
e
rt
er
s at
bot
ht
he i
n
p
u
t
and
o
u
t
p
u
t
sides. It supplies a special
load, i.e. a three pha
se
infi
nite grid cha
r
acterized
by
a gri
d
i
n
duct
a
n
ce (
c
b
a
g
L
L
L
L
) a
n
d a
gri
d
re
sistance (
c
b
a
g
R
R
R
R
) a
n
d
a m
a
xim
u
m
li
ne t
o
ne
ut
ral
vol
t
a
ge
am
pl
i
t
ud
e
equal
t
o
E
. Such a l
o
ad is
conside
r
ed to s
h
o
w
th
at, ev
enwith
a
p
a
rticu
l
ar case
(infin
ite g
r
i
d
); t
h
e m
o
d
e
l u
s
ed
gi
ves sat
i
s
fact
ory
res
u
l
t
s
.T
h
e
n
-parallel inverter s
h
are
s
the sam
e
DC
link,
whic
h can be connected to the
out
put
o
f
a
p
h
o
t
ov
ol
t
a
i
c
or
wi
nd
ene
r
gy
sy
st
em
.
Each inve
rter
is connected to the i
n
fi
nite gr
i
d
b
y
th
e
mean
s of th
e lin
e i
m
p
e
d
a
nce wh
ich
is
ch
aracterized
b
y
a p
a
ssiv
e
fi
rst o
r
d
e
r filter (
k
j
L
,
,
k
j
R
,
), where “
j”
d
e
sign
s th
e phase lin
e (
a
, b
or
c)
an
d
“
k”
designs t
h
e i
n
verter num
b
er. Figure
1 s
h
ows
t
h
esy
s
t
e
m
st
ruct
ure
o
f
t
h
e
n
-parallel connected
inverters
co
nsid
er
ed
i
n
t
h
is stud
y.
In
th
e curren
t
-b
id
irection
a
l switch
b
a
sed
inv
e
rters,
the a
v
erage m
odel of the phase leg
has a voltage
source
in
one s
i
de a
n
d a c
u
rre
n
t s
o
urce i
n
t
h
e ot
her side
and
whe
r
e
i
d
is de
fined as t
h
e
dut
y cycle of the
top
switch. T
h
e most widely applied PW
M technique for t
h
e
th
ree
p
h
a
se vo
ltag
e
source inverter is th
e sine p
u
l
se
m
odul
at
i
on [
1
3]
. The a
v
era
g
i
n
g f
o
r t
h
e t
h
ree
-
p
h
ase i
n
v
e
rt
er i
s
base
d on t
h
e p
h
ase-t
o
-
p
hase ave
r
a
g
i
n
g i
n
whi
c
h t
h
e
com
m
on
m
ode c
o
m
ponent
s a
r
e i
n
t
e
nt
i
o
nal
l
y
ne
gl
ect
ed.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-86
94
I
J
PED
S
Vo
l. 6
,
No
. 1
,
Mar
c
h
2
015
:
10
0
–
10
8
10
2
Figu
re
1.
Circu
it structu
r
e f
o
r
the
n
-parallel c
o
nnected inverters
Using the phas
e-leg technique, the electrical equations
of t
h
e di
f
f
ere
n
t
ph
ases du
ri
n
g
o
n
e
swi
t
c
hi
ng
peri
od
are gi
v
e
n by
E
quat
i
o
n (1
):
n
k
a
c
ci
ci
n
k
ak
ck
g
ak
ck
g
k
k
ai
ai
ci
ci
n
k
c
b
bi
bi
n
k
ck
bk
g
ck
bk
g
k
k
ci
ci
bi
bi
n
k
b
a
ai
ai
n
k
bk
ak
g
bk
ak
g
k
k
bi
bi
ai
ai
n
k
n
k
n
k
ck
k
bk
k
ak
k
in
g
in
in
e
e
i
R
i
i
R
dt
d
dt
di
L
V
d
d
dt
di
L
dt
di
L
e
e
i
R
i
i
R
dt
di
dt
di
L
V
d
d
dt
di
L
dt
di
L
e
e
i
R
i
i
R
dt
di
dt
di
L
V
d
d
dt
di
L
dt
di
L
i
d
i
d
i
d
i
dt
dV
C
V
v
dt
di
L
11
2
3
3
11
3
1
3
11
1
3
2
3
11
1
3
1
3
2
3
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
(1)
Whe
r
e:
-
L
in
and
i
ak
are
resp
ectiv
ely th
e in
pu
t i
nductance and the
phas
e line
a
current
of the
k
th
i
nve
r
t
er.
-
C
is t
h
e
DC s
i
de capa
c
itance
.
-
e
a
, e
b
and
e
c
rep
r
esen
t th
e three ph
ase lin
e to
n
e
u
t
ral vo
ltag
e
so
f t
h
e infin
i
te g
r
i
d
.
-
d
3k-2
,
d
3k-1
,
d
3k
represe
n
t res
p
e
c
tively the dut
y
cycl
es of t
h
e
pha
se a, b and
cof the
k
th
inv
e
rter.
W
i
t
h
sinu
so
id
al PW
M, th
e duty cycl
es are v
a
ried
sinu
so
id
ally in
syn
c
h
r
onis
m
with
th
e ac lin
e. Th
e
syste
m
is assu
med
to
b
e
p
e
rfectly b
a
lan
ced
. Th
e set of th
e ab
ov
e eq
u
a
tion
s
can
b
e
written
in
th
e
state sp
ace
fo
rm
[14]
,
[
15]
o
f
3
n
+
2
or
de
r.
This will lead t
o
a set
of a
com
p
lex nonline
a
r ti
m
e
varying a
v
era
g
ed state
space system of
equations
t
h
at
desc
ri
bes
t
h
e o
v
eral
l
ci
r
c
ui
t
be
havi
or.
It
i
s
necessa
ry
t
o
m
a
ke a change
o
f
co
o
r
di
nat
e
s t
o
c
o
n
v
e
r
t
a
c
sin
u
s
o
i
d
a
l
q
u
a
n
tities to
d
c
quan
tities p
r
ior to
th
e averag
e
p
r
o
cess. Th
e
referen
ce fram
e i
n
wh
ich
the averag
ed
state space exhibits a ti
me
in
variant syste
m
of equations is chosen su
ch as
a
m
u
lti-phase syste
m
appears as a
statio
n
a
ry on
e
in
a coo
r
d
i
n
a
te ro
tating
at th
e sam
e
in
stan
tan
e
ou
s
v
e
lo
city [16
]
. Th
is is d
o
n
e
u
s
ing
th
e Park
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
Li
ne an
d Gri
d
Im
pe
da
nce Im
pact
o
n
t
h
e
Per
f
orm
a
n
ces of
a
Par
a
l
l
e
l
C
o
nn
ect
ed
M
o
d
u
l
a
r
…
(
T
ah
ar Zeb
b
a
d
j
i
)
10
3
tran
sform
.
Fo
r
th
e sak
e
o
f
sim
p
l
i
city, th
e
n
i
nve
rt
ers
al
o
n
g
wi
t
h
t
h
e
l
i
n
e
an
d
gri
d
i
m
pedance
s
are
c
o
n
s
i
d
ere
d
id
en
tical su
ch
t
h
at:
g
c
b
a
l
ci
bi
ai
g
c
b
a
l
ci
bi
ai
L
L
L
L
L
L
L
L
R
R
R
R
R
R
R
R
(
2
)
Doi
ng s
o
, t
h
e t
r
ans
f
orm
e
d set
of t
h
e p
r
evi
o
u
s
equ
a
t
i
ons i
n
t
h
e r
o
t
a
t
i
ng
dq
co
ord
i
n
a
tes can
b
e
written
as fo
llo
w:
dq
dq
dq
dq
B
X
A
X
(
3
)
Whe
r
e t
h
e m
a
trices
dq
A
,
dq
B
and
dq
X
h
a
v
e
th
e fo
llowing
represen
tatio
n
:
L
R
L
d
L
R
L
d
L
R
L
d
L
R
L
d
C
d
C
d
C
d
C
d
C
L
A
qn
dn
q
d
qn
dn
q
d
in
dq
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
1
1
1
1
(
4
)
L
e
L
e
L
e
L
e
L
v
B
q
d
q
d
in
g
t
dq
0
(
5
)
qn
dn
q
d
in
t
dq
i
i
i
i
V
i
X
1
1
(6)
Whe
r
e:
g
l
g
l
i
m
qi
i
m
di
L
n
L
L
R
n
R
R
d
d
d
d
i
i
sin
2
3
cos
2
3
(
7
)
Whe
r
e
i
is the
phase a
n
gle bet
w
een the
output voltage
of e
v
ery sing
le inv
e
rter an
d th
e i
n
fin
ite g
r
i
d
v
o
ltag
e
,
i
m
d
i
s
t
h
e m
odul
at
i
o
n
i
n
dex
o
f
t
h
e d
u
t
y
cy
cl
esand
,
d
e
and
q
e
are
respectively t
h
e forward a
n
d
backwa
rd
com
pone
nt
s
of
t
h
e t
h
ree
p
h
as
e i
n
fi
ni
t
e
g
r
i
d
l
i
ne-t
o
-
l
i
n
e
v
o
l
t
a
ge.
R
e
ferri
ng t
o
t
h
e
m
a
t
r
i
ces gi
ven i
n
Eq
uat
i
o
n
(4) an
d
(6), one can re
write
the
electrical equations
in
th
e
d
-
q
fram
e. From
these equations, th
e a
v
era
g
e circ
uit m
odel of the
n
-p
arallel conn
ected
inv
e
rters with
d
i
fferen
t
p
a
rameterscan
b
e
deriv
e
d
.
Th
erefo
r
e
on
e can
o
b
tain
th
e av
erage circu
it m
o
d
e
l wh
ich
will hav
e
th
e
sam
e
general
r
e
prese
n
t
a
t
i
o
n a
s
t
h
e
o
n
e
deri
v
e
d
usi
n
g t
h
e a
v
era
g
e c
o
nnect
ion c
o
e
fficient
reflecte
d
to the DC
or AC
si
de [1
7]
-[
1
9
]
.
If al
l
the i
nve
rters a
r
e identical and have the same
d
di
d
d
and
q
qi
d
d
, the average
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-86
94
I
J
PED
S
Vo
l. 6
,
No
. 1
,
Mar
c
h
2
015
:
10
0
–
10
8
10
4
eq
u
i
v
a
len
t
circu
it
m
o
d
e
l o
f
Fig
u
re 3
can
eith
er
b
e
reflected
to
th
e
DC sid
e
or th
e u
tility
dq
s
i
d
e
.
T
h
e f
i
c
tiv
e
tran
sform
e
r [20
]
shown in Fi
g
u
re
3
m
o
d
e
ls on
ly th
e
primary curren
t
and
v
o
ltag
e
fro
m th
e
DC t
o
th
e u
tility
sid
e
with a turn
ratio of
3
d
d
fo
r t
h
e
di
rect
com
pone
nt
a
nd
3
q
d
f
o
r
t
h
e i
n
di
rect
co
m
ponent
.
The
st
eady
st
at
e
resp
o
n
se ca
n b
e
obt
ai
ne
d i
n
t
h
e
d
-
q
f
r
am
e. The i
n
verse t
r
an
sfo
r
m
can be a
ppl
i
e
d t
o
o
b
t
a
i
n
t
h
e t
i
m
e respons
e o
f
an
y d
e
sir
e
d state.
Fi
gu
re
3.
A
si
m
p
li
fi
ed ave
r
a
g
e e
qui
val
e
nt
c
i
rcui
t
m
odel
Fr
o
m
th
e above cir
c
u
it,
o
n
e
can
an
alyze the in
pu
t im
p
e
d
a
n
ce, t
h
e
d
i
f
f
e
r
e
n
t
tr
an
sf
er
fu
n
c
tion
s
, th
e
o
u
t
p
u
t
im
p
e
d
a
n
ce seen
fro
m
th
e DC sid
e
, etc., and
th
en
the stab
ility an
al
ysis o
f
th
e
op
en
loo
p
ci
rcu
it
can
b
e
per
f
o
r
m
e
d. Fr
om
t
h
e equi
v
a
l
e
nt
ci
rcui
t
m
odel
,
one c
a
n de
ri
ve a
n
y
t
r
ans
f
er
fu
nc
t
i
on t
h
at
need
s t
o
b
e
investigate
d
.T
he a
n
alytical expre
ssi
on of the cha
r
acteristic equation
n
D
gi
ven
by
:
)
(
(
2
2
2
2
2
3
4
2
L
R
CL
L
LRs
CL
s
L
CL
D
in
in
in
n
)
(
2
)
8
/
3
2
(
)
8
/
3
2
2
2
2
2
2
L
R
s
d
L
L
R
s
Ld
L
m
in
m
in
(8
)
The location of the ze
ros
of
n
D
(
w
hat
e
ver
t
h
e
n
u
m
b
er
n
of pa
rallel-connected i
nve
rt
er i
s
);
w
h
i
c
h a
r
e
fun
c
tion
s
o
f
the p
a
ram
e
ters o
f
th
e circu
it, d
e
termin
es th
e st
ab
ility o
f
th
e ov
erall stru
cture.
Th
e tran
sfer fu
n
c
tion
o
f
t
h
e in
pu
t curren
t
with
re
sp
ect t
o
th
e inpu
t voltag
e
an
d ou
tpu
t
vo
ltag
e
is
gi
ve
n by
:
n
d
m
g
m
m
in
D
s
E
L
R
sL
d
s
v
Rd
s
L
R
C
Ld
CLRs
s
CL
s
i
)
(
)
sin(
)
cos(
)
cos(
(
2
2
)
(
)
8
3
))
(
8
3
(
2
[(
)
(
2
2
2
2
2
2
2
3
2
(9
)
No
te also
th
at
th
e po
sitio
n of
th
e zero
s
an
d po
les of
th
e g
i
ven
tran
sfer
functio
n
d
e
p
e
nd
s o
n
t
h
e p
a
ram
e
t
e
rs
of
the inve
rters
which are
L
d
C
R
L
m
in
,
,
,
,
. If all
the zeros of t
h
e
characteristic
equation are l
o
cated in the l
e
ft half
s
-pl
a
ne, t
h
e st
eady
st
at
e i
nput
cur
r
ent
in
I
and
t
h
e phase
out
put
cu
rre
nt
a
I
i
n
t
h
e gri
d
ca
n b
e
gi
ven
by
t
h
e
fol
l
o
wi
n
g
e
x
pr
essi
ons:
2
2
)
cos(
2
3
8
/
3
Z
Z
d
E
Rv
d
I
m
g
m
in
(
1
0
)
)
6
cos(
)
6
cos(
2
1
t
V
t
v
d
Z
I
m
g
m
a
(1
1)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
Li
ne an
d Gri
d
Im
pe
da
nce Im
pact
o
n
t
h
e
Per
f
orm
a
n
ces of
a
Par
a
l
l
e
l
C
o
nn
ect
ed
M
o
d
u
l
a
r
…
(
T
ah
ar Zeb
b
a
d
j
i
)
10
5
Whe
r
e
Z
and
θ
are re
spect
i
v
el
y
t
h
e m
a
gni
t
u
de a
n
d
t
h
e
ar
g
u
m
e
nt
of t
h
e e
qui
val
e
nt
c
o
up
l
i
ng i
m
pedanc
e
defi
ned
as
bei
n
g e
qual
t
o
jL
R
.
From
Equat
i
on
(1
0) a
n
d (
1
1)
,
t
h
e co
u
p
l
i
ng i
m
pedance pl
ay
s an i
m
port
a
nt
rol
e
i
n
b
o
t
h
t
h
e t
r
a
n
si
ent
and steady sta
t
e varia
b
les of the ov
e
r
al
l
ci
rcui
t
.
T
h
i
s
c
o
upl
i
n
g i
m
peda
nce
depe
n
d
s
d
i
rect
l
y
on t
h
e
gri
d
im
pedance a
n
d
just
a f
r
act
i
o
n of t
h
e l
i
n
e i
m
pedance
.
T
h
e ab
ove a
n
al
y
t
i
cal
equat
i
o
ns
desc
ri
be t
h
e st
ea
dy
st
at
e
in
pu
t cu
rren
t
of th
e
n
inv
e
rters conn
ected in
p
a
rallel wh
atever th
e valu
e
of
th
e
n
u
m
b
e
r
n
i
s
. T
h
e
performances
o
f
th
e
o
v
e
rall circu
it can
th
en
b
e
an
al
yzed
with
resp
ect to all th
e tran
sf
e
r
fu
nct
i
o
n pa
ra
m
e
t
e
rs by
m
e
ans o
f
any m
a
the
m
atical tool.
3
.
RESULTS AND
INTERPRETATIONS
For a
gi
ve
n va
l
u
e of
g
m
in
v
d
L
,
,
,
and
C
, the
characteristic equation s
h
ows clearly that the location
of ze
ro
s de
pe
n
d
s o
n
t
h
e val
u
e of
n
,
R
and
L
. Fo
r a
nu
ll coup
lin
g
resistan
ce, th
e four ze
ros are l
o
cated
on t
h
e
im
agi
n
ary
axi
s
. Thi
s
cont
ri
b
u
t
e
s t
o
an un
s
t
abl
e
sy
st
em
.
To j
u
st
i
f
y
wh
at
i
s
st
at
ed above
, o
n
e can sol
v
e a
num
erical exa
m
ple as given in Ta
ble
1.
For a t
w
o
-
para
l
l
e
l
connect
e
d
i
nve
rt
ers a
nd
n
u
l
l
co
upling re
sistance, the c
h
aracteristic e
quation
has
four zeros l
o
c
a
ted on t
h
e i
m
aginary a
x
is
of the
s
dom
ai
n:
t
h
i
s
m
a
kes t
h
e sy
st
em
unst
a
bl
e
(s
12
=
±
j154
,
03)
and
(s
34
= ±
j407
, 9
2
)
.Therefore
, for a s
t
able open loop system
,
a coupling
resistance has t
o
be inserted i
n
the circuit. The
m
ove of thes
e zeros fr
om
the im
agi
n
ary
axi
s
i
s
m
o
re rel
e
vant
f
o
r t
h
e c
a
se of t
h
e va
ri
at
i
on of
th
e grid resistan
ce
rath
er th
an
that of t
h
e line
resistance
.
Figure 4 s
h
ows the step respons
e of the a
v
e
r
age i
nput current for the case where the
gri
d
im
pedance
is co
n
s
i
d
er
ed
to
b
e
nu
ll. Th
e
b
a
se cur
r
en
t is tak
e
n
to
b
e
equ
a
l to
th
e input cu
r
r
e
n
t
fo
r
an
ind
i
v
i
du
al inv
e
r
t
er
with
a
nu
ll grid i
m
p
e
d
a
n
c
e: this g
i
v
e
s a
base
avera
g
e i
n
put c
u
rrent e
q
ual to
35A.
Th
is last fig
u
re sh
ows th
at an
in
crease in
th
e
line resista
n
ce for the cas
e of two pa
rallel-connecte
d
inve
rter will induce a
decreas
e of the
stea
dy
state input current. T
h
is de
c
r
ease m
i
ght cha
nge c
o
m
p
letely the
m
ode of ope
ra
t
i
on of t
h
e
hal
l
st
ruct
ur
e. I
n
t
h
i
s
case, if the line resist
ance is greater than0.223
Ω
, the overal
l
syste
m
wo
rk
s i
n
th
e rectifier
m
o
d
e
rath
er than
th
e inv
e
rter
m
o
d
e
.
Table 1. Param
e
ters
For
The
Analyzed
E
x
a
m
ple
Input Voltage
Input f
ilter
Inverter
specifications
2
,
6
.
0
,
6
n
d
m
Line
para
m
e
te
rs
1
.
0
,
340
l
l
R
H
L
The infini
te
grid
para
m
e
te
rs
kHz
f
H
L
V
E
g
50
,
170
,
2
220
05
.
0
g
R
For a larger line or grid
resist
an
ce, t
h
e st
ep
r
e
sp
onse
o
f
t
h
e
i
n
p
u
t
cu
rre
nt
s
h
o
w
s
bet
t
e
r
pe
rf
orm
a
nces.
This e
n
sures
a stable system with
accept
a
ble pe
rform
a
nces
but at the
detrim
ental
of t
h
e ove
r
all circuit
efficiency. The
steady state input c
u
rre
nt de
c
r
eases as
the li
ne
or gr
id
resis
t
ance inc
r
eases
(Fi
g
ure
4).
The inc
r
ease
of the num
b
er
n
o
f
the inv
e
rters to
b
e
co
nn
ected
in
p
a
rallel (fro
m
1
to
3
)
lets
th
e syste
m
(see fi
gure
5)
reaches
an ave
r
age i
n
put c
u
rrent
greate
r
than the
bas
e
cu
rre
nt but at a
rate which does not
co
m
p
ly with
the p
a
rallelin
g
p
r
in
cip
l
e. Th
is is m
a
in
ely d
u
e
to
th
e presen
ce
o
f
th
e
g
r
i
d
im
p
e
d
a
n
c
e.
Fig
u
re 6
illu
st
rates th
e effects o
f
b
o
t
h
th
e line resistan
ce
wh
ile k
eep
i
n
g
t
h
e g
r
i
d
resistance co
nstan
t
and t
h
e grid re
sistance while keepi
ng
t
h
e line resistance constant. Fi
rst, th
e
m
ode of ope
r
ation
of the
overall
ci
rcui
t
can
be
ei
t
h
er i
n
t
h
e i
n
vert
er m
ode
or
i
n
t
h
e rect
i
f
i
e
r
m
ode de
pe
ndi
ng
o
n
t
h
e
val
u
e of t
h
e l
i
n
e o
r
gri
d
resistance.
For
the case (a
), t
h
e inve
rter m
o
deis
obtaine
d
for a grid
resista
n
ce sm
aller than 0.0617
Ω
w
h
i
l
e for
the case (b), this sam
e
m
ode is obtaine
d
for a line resista
n
ce sm
a
ller than 0.1224
Ω
. Se
cond, for t
h
e i
nve
rter
m
ode, t
h
e
rat
e
of
va
ri
at
i
on
o
f
t
h
e i
n
put
c
u
rre
nt
i
s
greater for the case
(a
) rath
er th
an
th
e case (b
).
Fi
gu
re
7 s
h
o
w
s t
h
at
f
o
r
t
h
e i
nve
rt
er m
ode
o
f
o
p
e
r
a
tio
n, the v
a
lu
e of th
e
g
r
i
d
ind
u
ctan
ce for t
h
e case
(a) has t
o
be
g
r
eat
er t
h
an
0.
0
0
0
1
3
5
H
m
eanewhi
l
e
fo
r t
h
e case (b) t
h
e l
i
n
e i
nduct
a
nce h
a
s t
o
be great
e
r
t
h
an
V
v
g
400
mF
C
mH
L
in
5
;
5
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-86
94
I
J
PED
S
Vo
l. 6
,
No
. 1
,
Mar
c
h
2
015
:
10
0
–
10
8
10
6
0
.
0
002
7H
. The r
a
te of
v
a
r
i
ati
o
n of
th
e inp
u
t cu
rren
t is
greater fo
r t
h
e ca
se of the
grid
inductance
va
riation
rather tha
n
t
h
e
line inducta
n
ce
va
riation.
Fi
gu
re 4.
St
ep
resp
o
n
se of
the
ave
r
age
input
current
(p.u) for
two
parallel-con
n
ect
ed
inv
e
rters wi
th
nu
ll
gri
d
re
sistance
and di
ffe
rent
values
of line
re
sistance:
5
.
0
a
,
2
.
0
b
,
1
.
0
c
,
05
.
0
d
Fig
u
r
e
5
.
Step
r
e
spon
se
o
f
th
e in
pu
t cu
rr
en
t (p
.u
) fo
r
a gi
ve
n
n
u
m
b
er
of
pa
ral
l
e
l
-
co
nnect
e
d
i
n
ve
rt
ers:
n
=1;
b-
n=
2 a
n
d c-
n
=
3
Figure
6. Line
and grid resista
n
ce e
ffects
on t
h
e
steady state input c
u
rre
nt (p.u): a
)
variation of
g
R
, b)
vari
at
i
o
n of
l
R
Fi
gu
re
7.
Li
ne
and
g
r
i
d
i
n
duct
a
nce e
ffect
s
o
n
t
h
e
g
R
steady state input c
u
rre
nt (p.u): a
)
variation of
g
L
, b)
vari
at
i
o
n of
l
L
.
Fig
u
re 8
illu
strates h
o
w
th
e syste
m
can
lo
se th
e
m
a
in
ad
v
a
n
t
ag
e of p
a
rallelin
g
in
v
e
rters:
an
arb
itrary
v
a
lu
e
of th
e grid
im
p
e
d
a
n
ce l
e
ts a li
m
ited
in
crease in th
e in
pu
t cu
rren
t
.
If th
e grid
i
m
pedance
i
s
t
a
ke
n
t
o
b
e
equal t
o
ze
ro, t
h
e input curre
nt
with
resp
ect t
o
th
e nu
m
b
er
n
of inv
e
rters to b
e
con
n
ected
i
n
p
a
rallel in
creases
in
a lin
ear m
a
n
n
e
r.
Oth
e
rwise, th
is in
creas
e is no
n
lin
ear and
tend
s to
b
e
limited
as
n
in
creases. Th
is is
because of the
reflected im
pedance
see
n
from
the DC side which inc
r
ea
ses nonlinearly
with the number
n
of
in
v
e
rters to
b
e
co
nn
ected
in
p
a
rallel. Ho
wev
e
r, for th
e ca
se (d), the increase of the re
flected im
pedance is
l
i
n
ear. Th
us
, f
o
r a m
a
xim
u
m
po
wer t
r
a
n
sfe
r
fr
om
t
h
e DC
si
de t
o
t
h
e gri
d
, t
h
e g
r
i
d
i
m
pedance s
h
oul
d be as
sm
a
ll as possi
ble.
For
a five parallel-connected
inve
rters
,
the
curve
(d)
shows a
50
0%inc
r
ease in the i
n
put c
u
rre
nt
whi
l
e
t
h
e c
u
r
v
e (a) s
h
o
w
s
on
l
y
a 142% i
n
c
r
ease i
n
i
t
.
I
n
t
h
e case (a)
,
t
h
e gri
d
i
m
peda
nce has
d
r
am
at
i
cal
ly
decrease
d
the i
n
put power s
u
ch
th
at th
e
fiv
e
p
a
rallel-conn
ected
in
v
e
rter st
ru
ct
u
r
e is no
t ev
en
equ
i
v
a
len
t
to
a
t
w
o pa
ral
l
e
l
-
co
nnect
e
d
i
n
ve
rt
ers wi
t
h
nul
l
gri
d
i
m
pedance
.
Th
is m
a
k
e
s th
e ch
o
i
ce o
f
th
e co
up
lin
g
im
p
e
d
a
n
ce a priority in
an
n
parallel-connected
inve
rters. T
h
erefore
,
t
h
e co
n
n
ect
i
o
n
po
rt
of a
pa
ral
l
e
l
m
odul
a
r
st
r
u
ct
ure i
s
a
key
p
o
i
n
t
i
n
t
h
e
desi
gn
co
nsi
d
erat
i
o
ns.
0
0.
0
5
0.
1
0.
15
-10
-5
0
5
10
15
20
25
Ti
m
e
(
s
e
c
)
A
v
er
ag
e i
n
pu
t cu
r
r
e
n
t
(
p
.
u
)
d
c
b
a
0
0.
02
0.
04
0.
06
0.
08
0.
1
-2
0
2
4
6
8
Ti
m
e
(
s
e
c
)
A
v
erage i
nput
c
u
r
r
ent
(p.
u
)
b
c
a
0
0.
2
0.
4
0.
6
0.
8
1
-3
-2
-1
0
1
2
3
R
e
si
st
a
n
c
e
(
O
h
m
)
A
v
er
ag
e
i
n
p
u
t
c
u
r
r
e
n
t
(
p
.
u
)
a
b
0
0.
002
0.
004
0.
006
0.
008
0.
01
-2
-1
0
1
2
3
4
5
I
n
d
u
c
t
an
ce
(
H
e
n
r
y
)
A
v
e
r
ag
e in
p
u
t cu
r
r
e
n
t
(
p
.
u
)
a
b
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
Li
ne an
d Gri
d
Im
pe
da
nce Im
pact
o
n
t
h
e
Per
f
orm
a
n
ces of
a
Par
a
l
l
e
l
C
o
nn
ect
ed
M
o
d
u
l
a
r
…
(
T
ah
ar Zeb
b
a
d
j
i
)
10
7
Figure
8. Average input curre
n
t(p.
u)
wi
t
h
res
p
ect
t
o
t
h
e
num
ber
n
o
f
p
a
rallel co
nn
ected
inverters
with
di
ffe
re
nt
gri
d
i
m
pedance
)
,
(
g
g
L
R
:a-
05
.
0
,
mH
17
.
0
;b
-
02
.
0
,
mH
068
.
0
; c-
01
.
0
,
mH
034
.
0
d-
0
,
0
mH
4. CO
N
C
L
U
S
I
ON
The
pha
se l
e
g
t
echni
q
u
e a
p
pl
i
e
d t
o
t
h
e
n
-pa
r
al
l
e
l
con
n
ect
e
d
i
n
vert
e
r
s
gi
v
e
s, o
n
one
ha
n
d
, a cl
ose
d
form
so
lu
tio
n fo
r th
e
3
n
+2 order system
whatever t
h
e
num
ber
n
of t
h
e inv
e
rters to b
e
co
nn
ected in p
a
rallel is.
On th
e o
t
h
e
r
h
a
nd
, th
e ob
tain
ed sim
p
lifie
d
av
erag
e equiv
a
len
t
circu
it m
o
d
e
l can be u
s
ed to
d
e
riv
e
t
h
e
di
ffe
re
nt
t
r
a
n
s
f
er f
unct
i
o
ns
of
t
h
e o
v
eral
l
ci
rcui
t
.
Th
is allo
ws th
e an
alysiso
f
th
e
o
p
e
n
l
o
o
p
syste
m
per
f
o
r
m
a
nces of t
h
e ci
rc
ui
t
wi
t
h
a
preci
se
det
e
rm
i
n
at
i
o
n
of t
h
e
pol
es
and z
e
r
o
s l
o
c
a
t
i
on
ofa
n
y
t
r
ansfe
r
fun
c
tion
o
f
th
e system
. Th
eir po
sitio
n is cl
osely lin
k
e
d to
th
e
v
a
lu
e
of the coup
lin
g imp
e
d
a
n
ce, th
e nu
m
b
er
n
o
f
i
nve
rt
ers c
o
n
n
ect
ed i
n
pa
ral
l
e
l
and t
h
e di
ffe
re
nt
para
m
e
t
e
rs of t
h
e ci
rcui
t
.
T
h
e val
u
e of t
h
e l
i
n
e
and
gri
d
resi
st
ance pl
a
y
s an im
port
a
nt
rol
e
f
o
r t
h
e perf
o
r
m
a
n
ces at th
e d
e
trimen
tal o
f
the syste
m
efficiency.
Furt
herm
ore, t
h
e va
ri
at
i
on
o
f
t
h
e c
o
u
p
l
i
n
g
im
pedance
t
h
at can
be ei
ther affect
ed
b
y
th
e lin
e
or grid
im
pedance m
a
y
co
m
p
l
e
t
e
l
y
chan
ge t
h
e m
ode of o
p
erat
i
o
n
of the
global circuit and th
en t
h
e purpose of such a
circuit can be
c
o
m
p
rom
i
sed.
The inc
r
ease
of the
num
b
er
n
of t
h
e invert
ers to
be c
o
nnected in
parall
el
m
a
y not always allow a
lin
e
a
r
in
cr
e
a
s
e
o
f
th
e
av
er
a
g
e
in
pu
t cu
rren
t: t
h
is is m
a
in
ly d
u
e
to th
e equ
i
valen
t
coup
lin
g
i
m
p
e
d
a
n
c
e seen
b
y
th
e
n
p
a
rallel in
v
e
rters con
n
e
cted
to
th
e sp
ecial lo
ad
(inf
inite g
r
id
). Howev
e
r, if th
e
g
r
i
d
im
p
e
d
a
n
ce i
s
to
o
sm
a
ll co
m
p
ared to the line impeda
nce,
t
h
en t
h
e inc
r
ease of t
h
e num
b
er of i
nve
rters t
o
be
connected i
n
parallel
ten
d
s
t
o
b
e
close to
a lin
ear in
crease of th
e
in
pu
t curren
t
.
Th
is im
p
o
r
tan
t
resu
lt im
p
o
s
es th
at fo
r a p
a
rallel-
con
n
ect
ed i
n
ve
rt
er, t
h
e c
o
n
n
e
c
t
i
on p
o
i
n
t
of t
h
e di
ffe
rent
m
o
d
u
l
e
s s
h
o
u
l
d
be as cl
ose a
s
pos
si
bl
e t
o
t
h
e
gri
d
.
Th
is
will g
u
a
ran
t
ee th
e m
a
in
adv
a
n
t
ag
e of
p
a
rallelin
g inve
rters th
at is t
h
e lin
ear in
crease of power
as th
e
n
u
m
b
e
r
o
f
inv
e
rters con
n
ected in
p
a
rallel is i
n
creased
;
o
t
h
e
rwise, t
h
is prin
cip
l
e will b
e
com
p
ro
mised
.
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BIOGRAP
HI
ES
OF AUTH
ORS
Tahar
Z
e
bbadj
i
receiv
e
d h
i
s
Engineer Diplo
m
a from the
Ecole Nationa
le
Polytechn
i
que
d’Alger
in 1984 and then obtained the Master deg
r
ee
from the University
of Color
a
do, Boulder
,
USA in 1987.He is currently
a senior lecturer in
the depar
t
ment of
electrical
enegineering
at the
Ecole Nationale Polytechnique
,
ENP
, Algi
ers
.
He is
a
m
e
m
b
er of th
e res
e
ach
te
am
of th
e
Laboratoire de Recherche en
Electrotechniq
u
e,
ENP,
Algi
e
r
s.His area
of
inter
e
t
is the
modelling and control
of
power
ele
c
troni
cs
conv
erters
.
Se
ddik H
a
dji
re
ceiv
e
d the degree of Ingénieur in
Electrical Engin
eering in 1979 from
the Ecol
e
Nationale Polytechniqu
e
of Alg
i
ers (
ENP-
Alger
), the M
.
S
c
.
(En
g
) in El
ectron
i
c
and El
ectr
i
c
a
l
Engineering in
1
986 from the University
of B
i
rmigham, UK (within the Power
Electronics an
d
Transportation Sy
stems – PETS Group (1983–1986) and the Ph.D. degree
in the same field in
2007 from ENP-Alger. He worked as a Lectur
er
and as
a Senior
Lecturer
(1987
–2009) at
the
University
of B
é
jaïa wher
e he carried ou
t re
sear
ch work (1991–2009) with the
Laboratoire d
e
Recher
che en
Technolog
ie Ind
u
s
trielle
et
de
l’In
formation-LTII (
2000–2009). He is currently
a
Professor with the
Eco
l
e Pr
épar
atoir
e
en S
c
ien
c
es
et
Techn
i
ques
of Algiers (
E
PST-Alger)
and
an As
s
o
ciat
e D
i
rec
t
or of
res
e
a
r
ch with
ENP
-
Alger. His
r
e
s
e
arch
inter
e
s
t
s
i
n
clude
el
ec
tri
c
traction, power
factor corr
ecti
on, activ
e
fi
lt
ers, PWM converters
and PWM m
u
ltilevel
converters, and PV
and
Wind
en
erg
y
conv
ersion
s
y
stems.
Rachid Ibtioue
n
receiv
e
d the
P
HD degree in ele
c
tri
cal
engi
neering from
Ecole Nationale
Polyt
echn
i
que (
E
NP)
, Algiers,
Algeria, and
Inst
itut Nationa
l Pol
y
te
chnique de L
o
rraine
, N
a
nc
y,
France, in 1993.
He integrated th
e
Groupe de Recherche en Electr
o
techn
i
que et Electroniqu
e de
Nancy
, Nancy
,
f
r
om 1988 to 199
3. From 2005 to
2013, h
e
was
th
e Director
of
the
Laboratoire
de R
echer
ch
e
e
n
Ele
c
tr
ot
echni
que
at
ENP. He
is the
Head of
the Dep
a
rtm
e
n
t
of E
l
e
c
tri
c
a
l
Engine
ering
at
ENP
.
He is
curr
entl
y
a P
r
ofes
s
o
r and th
e As
s
o
ciated Dir
e
c
t
or of
Res
earch
wit
h
ENP
.
His
curr
en
t res
e
arch
int
e
res
t
s
inc
l
ude
modeling electric s
y
s
t
ems and drives.
Evaluation Warning : The document was created with Spire.PDF for Python.