Int
ern
at
i
onal
Journ
al of
P
ower E
le
ctr
on
i
cs a
n
d
Drive
S
ystem
s
(
IJ
PEDS
)
Vo
l.
12
,
No.
1
,
M
a
r 202
1
, p
p.
304
~
313
IS
S
N:
20
88
-
8694
,
DOI: 10
.11
591/
ij
peds
.
v12.i
1
.
pp304
-
313
304
Journ
al h
om
e
page
:
http:
//
ij
pe
ds
.i
aescore.c
om
Study of
th
e oper
ation of t
he outp
ut filte
r of a sin
gle
-
phase
series a
ctive p
ow
er filt
er
Miha
il
Antche
v
1
, Va
njo
Gou
rgouli
tsov
2
,
and Hris
to Ant
chev
3
1
Sect
ion
Pow
er El
e
ct
roni
cs,
Fa
c
ult
y
of
Elec
tronic
Eng
ine
e
ring an
d
Technol
ogi
es,
Te
chn
ic
a
l
Univ
e
rsity
,
Sofi
a
,
Bul
gar
ia
2
Coll
ege of
Ene
r
gy
and El
ec
tron
i
cs,
T
ec
hn
ic
a
l
Un
ive
rsity
,
Sofia
,
B
ulga
ri
a
3
Sect
ion
Elec
t
rical Engi
ne
eri
ng
a
nd
Elec
tron
ic
s,
Depa
rtment
o
f M
et
al
lu
rgical
Technol
ogi
es,
El
e
c
tri
c
al
Engi
n
ee
rin
g
and
El
e
ct
roni
cs,
Fa
c
ult
y
of
Met
al
lur
gy
and
Ma
thema
ti
c
al
Sc
ie
n
ce
,
Un
ive
rsity
of
Ch
emical
T
ec
hno
logi
e
s a
nd
Met
allurgy
,
Sofia,
Bu
lga
ri
a
Art
ic
le
In
f
o
ABSTR
A
CT
Art
ic
le
history:
Re
cei
ved
A
ug
28, 202
0
Re
vised
Jan
19
,2021
Accepte
d
Fe
b 2
,
2021
In
th
e
art
i
cle
is
pre
sente
d
a
sing
le
-
phase
seri
es
a
ct
iv
e
power
f
il
t
e
r
(SF
SA
P
F)
designe
d
to
co
mpe
nsat
e
th
e
to
ta
l
h
arm
oni
c
dis
tort
ion
(
THD)
fac
tor
of
the
supply
net
work
volt
ag
e.
Par
ti
cu
l
ar
attention
is
pa
id
to
th
e
work
o
f
the
outpu
t
ca
pa
ci
tor
of
the
passive
fi
lt
er
,
an
d
depe
nd
encie
s
f
or
the
design
of
the
f
il
t
er
at
hystere
sis
vol
ta
g
e
con
trol
are
pr
ese
nte
d
.
The
v
alue
of
the
highe
r
har
mon
ic
s
of
th
e
cur
r
ent
t
hrough
th
e
filte
r
c
apaci
tor
is
d
et
er
mi
ned
.
A
m
at
he
ma
t
ic
a
l
rel
a
ti
onship
b
etw
ee
n
th
e
voltag
e
ripp
le
of
the
ca
pa
ci
tor
,
it
s
v
a
lue
and
the
sw
it
chi
ng
fr
equ
enc
y
is
der
ive
d
.
As
a
result
o
f
the
adv
anc
ed
rese
ar
ch
a
me
thodol
ogy
fo
r
design
ing
the
output
filter
of
a
sing
le
-
ph
ase
s
eri
a
l
active
power
filter
has
bee
n
proposed
.
The
r
esult
s
fro
m
co
mput
e
r
sim
ula
ti
on
and
expe
ri
me
n
t
ar
e a
lso gi
ven
.
Ke
yw
or
d
s
:
Hy
ste
resis
vo
lt
age c
on
t
ro
l
Ou
t
pu
t ca
pacit
or
Series act
ive
pow
e
r fil
te
r
This
is an
open
acc
ess arti
cl
e
un
der
the
CC
BY
-
SA
l
ic
ense
.
Corres
pond
in
g
Aut
h
or
:
M
ihail
An
tc
he
v
Sect
ion
Powe
r El
ect
ro
nics
, F
a
culty
of Elec
tr
on
ic
En
gin
ee
ri
ng and
Tech
no
log
ie
s
Tech
nical
Univers
it
y
–
Sofia
1000, So
fia, B
ulg
a
ria
Emai
l:
an
tc
he
v@t
u
-
sofia
.bg
1.
INTROD
U
CTION
The
s
hunt
an
d
series
act
ive
powe
r
filt
ers
a
re
an
ef
fecti
ve
and
per
s
pecti
ve
mea
ns
f
or
so
lvi
ng
the
pro
blems
with
the
qu
al
it
y
of
e
le
ct
rical
powe
r
[1
]
-
[
3].
Most
commo
nly
,
se
r
ie
s
act
ive
pow
er
filt
ers
(S
A
P
F)
a
re
us
e
d
to
im
pro
ve
the
t
otal
ha
rm
on
ic
distor
ti
on
(T
HD)
factor
of
the
s
uppl
y
net
work
vol
ta
ge,
us
ed
to
powe
r
sensiti
ve
c
onsume
rs
[
4
]
-
[
11].
In
their
co
ntr
ol
sy
ste
ms
a
re
use
d:
sli
ding
m
ode
c
ontrol
[
12]
,
f
u
zz
y
l
og
ic
c
on
t
ro
l
[13
]
,
[
14]
dea
db
eat
c
ontr
ol
[
15],
monit
ori
ng
of
filt
er
cap
aci
tor
c
urren
t
[16],
hy
ste
resis
co
ntr
ol
of
th
e
filt
e
r
capaci
tor
volt
age
or
the
l
oad
vo
lt
age
[
17
]
,
[
18].
Ne
wer
co
ntr
ol
meth
ods
are
al
so
use
d,
su
c
h
as:
base
d
on
pe
r
-
ph
a
se
cu
rr
e
nt
cal
culat
ion
[
19]
a
nd
ne
ur
al
netw
ork
co
nt
ro
l
[
20
]
,
[
21
].
Partic
ular
at
te
ntion
is
paid
to
the
accurate s
ync
hro
nizat
ion wit
h t
he
s
ource
volt
age,
a
s a
n ov
e
r
view o
f
the
ma
in meth
ods is
done
in [
22].
T
he
li
te
ratur
e
study
re
gardi
ng
the
feat
ur
es
of
the
outp
ut
f
il
te
r
sh
ows
tha
t
t
her
e
are
c
ompa
rati
vely
few
e
r
stu
dies dea
li
ng
with
th
e
de
sig
n
featu
re
s o
f
the
o
ut
put passive
filt
er.
In
[
23]
is
a
naly
zed
a n
e
w
seri
es
-
par
al
le
l
-
re
sona
nt
filt
er
a
s
a
part of
a shu
nt
a
ct
ive
po
wer
filt
er.
I
n
[
24]
t
here
is
a
n
anal
ys
is
of
t
he
operati
on
of
filt
er
in
t
he
co
nf
i
gurati
on
al
so
of
a
s
hunt
act
ive
power
f
il
te
r.
Taki
ng
in
to
acc
ount
the
high
fr
e
qu
e
nc
y
modu
la
ti
on,
th
e
values
of
t
he
current
th
rough
the
filt
er
capaci
tor
as
w
el
l
as
through
the
DC
capaci
tor
are
def
i
ned.
Evaluation Warning : The document was created with Spire.PDF for Python.
In
t J
P
ow Elec
& Dri S
ys
t
IS
S
N: 20
88
-
8
694
Study
of the
op
eratio
n of the
outp
ut fi
lt
er o
f a
sin
gle
-
p
hase s
eri
es a
ct
iv
e
powe
r fi
lt
er (
Mi
hail
An
tc
hev
)
305
Ther
e
is
not
a
co
mp
le
te
stu
dy
of
the
oper
at
ion
of
t
he
pa
ssive
filt
er
a
s
par
t
of
a
S
AP
F
with
hy
ste
resis
co
nt
ro
l
of
t
he
filt
er
capaci
tor
volt
age.
T
his
woul
d
ha
ve
al
lo
we
d
the
prese
ntat
ion
of
a
met
hodolo
gy
for
it
s
desig
n.
Figure
1
s
how
s
the
blo
c
k
dia
gr
a
m
of
a
SF
S
AP
F
.
Its
pur
po
se
is
to
pro
du
c
e
su
c
h
a
volt
ag
e
of
the
seco
nd
a
ry
wi
ndin
g
of
t
he
tra
ns
f
ormer
Tr
,
to
com
pe
ns
at
e
th
e
highe
r
har
m
onic
s
of
t
he
vo
lt
age
of
the
po
wer
so
urce
AC
,
so
that t
he
l
oad volt
age
has
a
si
nu
s
oi
dal
sh
a
pe
.
Figure
1.
Bl
oc
k diag
ram
of a
S
PSAPF.
Ther
e
a
re
tw
o
featu
res
of
t
he
outp
ut
ca
pac
it
or
w
ork:
1.
The
volt
age
on
t
he
ca
pacit
or
c
on
ta
i
ns
higher
harmo
ni
cs,
wh
ic
h
must
be
in
a
ntip
hase
to
their
r
especti
ve
ones
,
c
on
ta
i
ned
in
the
s
upply
ne
twork
vo
lt
age
cu
r
ve.
These
ha
rm
on
i
cs
deter
mine
t
he
on
l
y
on
e
co
mpon
e
nt
of
t
he
root
m
ean
square
(R
M
S)
va
lue
o
f
the
cu
rr
e
nt
th
r
ough
the
capa
ci
tor.
In
ad
diti
on,
the
cu
rr
e
nt
al
so
c
on
ta
in
s
a
hi
gh
-
f
reque
ncy
co
mpo
nent
as
a
resu
lt
of
hyste
r
esi
s
co
ntr
ol.
2.
In
hy
ste
resis
c
on
t
ro
l,
the
va
lu
e
of
the
capaci
t
or
an
d
the
out
pu
t
in
du
ct
a
nce
dep
e
nd
on the
hy
ste
resis
par
a
mete
rs
a
nd the
switc
hing
fr
e
quenc
y o
f
the
pow
e
r de
vices.
T
he
prob
le
m
is
that
there
is
no
co
mpre
hensi
ve
stu
dy
of
the
sp
eci
fie
d
feat
ur
es
of
the
op
e
rati
on
of
th
e
capaci
tor
of
t
he
outp
ut
filt
er
.
T
there
is
no
com
pr
e
he
ns
ive
metol
ogy
for
filt
er
desig
n.
The
pur
pose
of
this
arti
cl
e
is
to
pre
sent
a
stu
dy
of
these
feat
ur
es
,
w
hich
is
la
c
ki
ng
in
the
s
peci
al
iz
ed
li
te
rature.
T
he
nove
lt
y
in
the
work
is
i
n
determini
ng
the
r
el
at
ion
sh
i
p
of
the
va
lue
of
t
he
ca
pacit
or
with
th
e
R
M
S
valu
e
of
the
higher
harmo
nic
cu
rrent
an
d
volt
ag
e
ripp
le
.
A
s
a
resu
lt
,
a
ne
w
method
ology
f
or
the
desig
n
of
t
he
outp
ut
f
il
te
r
has
been p
r
opos
e
d.
In
t
he
sec
ond
par
t,
a
mathe
mati
cal
descr
i
ption
is
ma
de,
an
d
are
offe
re
d
e
xpressio
ns
i
n
acc
ordance
with
t
he
tw
o
de
scribe
d
featu
r
es.
T
he
res
ults
f
rom
the
c
ompu
te
r
sim
ulati
on
a
re
pr
e
sente
d
i
n
the
sec
ond
par
t.
The
t
hird
pa
rt
s
hows
t
he
e
xper
imenta
l res
ults
of the
w
ork of
the SP
SAPF
w
it
h
li
near
a
nd
nonlinea
r
l
oad.
2.
RESEA
R
CH MET
HO
D
2.1.
Mathem
ati
cal
descripti
on
It
shou
l
d
be
note
d,
t
hat
the
curves
in
t
he
f
orm
of
the
s
upply
netw
ork
volt
age
ha
ve
dif
fer
e
nt
sh
a
pe
and
c
ha
racter,
so
it
is
neces
sary
t
o
acce
pt
a
certai
n
ap
pro
ximati
on
of
this
form
.
For
the
pur
poses
of
th
e
analysis,
c
onne
ct
ed
with
the
first
featu
re
(
det
ermini
ng
t
he
RMS
value
of
t
he
current
th
rou
gh
t
he
capaci
to
r)
,
i
s
acce
pted
a
n
a
ppr
o
ximati
on
of
the
c
urve
of
the
sup
ply
ne
twork
vo
lt
ag
e
with
trapez
oi
dal
sh
a
pe,
s
hown
i
n
Figure
2.
It
gets
cl
os
e
t
o
a
com
mon
c
ase
in
pr
act
ic
e
-
loa
d
of
the
netw
ork
with
sin
gle
-
phase
br
i
dg
e
un
c
ontr
olled
re
ct
ifie
rs,
w
hic
h
are
no
n
-
li
nea
r
l
oads
an
d
c
ons
ume
cu
rr
e
nt
a
r
ound
t
he
ma
xim
um
s of
the n
et
work
vo
lt
age
.
As
a
re
su
lt
, voltage
drops a
r
ound it
s max
im
um
value.
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2088
-
8
694
In
t J
P
ow
Ele
c
&
D
ri
S
ys
t,
V
ol
.
12
, N
o.
1
,
Ma
rch
20
21
:
304
–
313
306
Figure
2. A
ppr
ox
imat
io
n o
f
t
he
c
urve
of the
supp
l
y netw
ork vo
lt
age
.
The dec
ompo
si
ti
on
of su
c
h
a
trap
ez
oid
al
f
unct
ion
in
F
ourie
r
se
ries
is
known:
(
)
=
4
.
ℎ
.
.
(
)
.
(
)
+
4
.
ℎ
.
∑
1
(
2
.
−
1
)
2
(
2
.
−
1
)
.
.
(
2
.
−
1
)
.
∞
=
2
(
1
)
Ther
e
f
or
e,
for t
he
T
H
D
fact
or,
w
e
obta
in:
=
√
∑
2
∞
=
2
1
=
√
∑
1
(
2
.
−
1
)
4
2
(
2
.
−
1
)
∞
=
2
(2)
Figure
3
s
how
s
the
c
orrespo
ndin
g
gr
a
ph
ic
al
dep
e
ndence
f
or
the
i
nterval
0.3
≤
φ
≤
1.0
ra
d.
T
he
s
um
is l
imi
te
d
to
=
25
in
accor
da
nce
with
[25
]
,
[
26]
.
Figure
3. G
raphical
d
e
pe
nd
e
nc
e acco
rd
i
ng to
expres
sio
n
(
2)
In
this
wa
y,
it
is
possible
to
determi
ne
t
he
amplit
ude
valu
es
of
the
ha
rm
on
ic
s
of
the
vo
lt
age
on
th
e
ou
t
pu
t
ca
pacit
or
of
the
SA
P
F
.
The
se
ha
rm
on
ic
s
,
a
dded
i
n
antip
hase
to
their
res
pecti
ve
on
es
t
o
t
he
c
urve
of
Figure
2, w
ould ma
ke
t
he v
oltage on t
he
l
oa
d
si
nu
s
oid
al
wi
th freq
ue
ncy of
the
first
harmo
nic.
=
4
.
ℎ
.
.
1
(
2
.
−
1
)
2
.
(
2
.
−
1
)
.
(3)
for
=
2
,
3
…
…
Evaluation Warning : The document was created with Spire.PDF for Python.
In
t J
P
ow Elec
& Dri S
ys
t
IS
S
N: 20
88
-
8
694
Study
of the
op
eratio
n of the
outp
ut fi
lt
er o
f a
sin
gle
-
p
hase s
eri
es a
ct
iv
e
powe
r fi
lt
er (
Mi
hail
An
tc
hev
)
307
The
maxim
um
valu
e
of
t
he
current
thr
ough
the
ca
pacit
or
of
t
he
outp
ut
filt
er
f
or
eac
h
of
the
s
e
harmo
nics
will
b
e:
=
.
(
2
.
−
1
)
.
.
(4)
for
=
2
,
3
…
…
Con
si
der
i
ng all
the
harmo
nics,
the R
M
S
valu
e of the
curre
nt
thro
ugh
t
he
fil
te
r
capaci
to
r
will
b
e:
=
2
.
.
ℎ
.
.
√
2
.
.
√
∑
1
(
2
.
−
1
)
2
,
2
(
2
.
−
1
)
.
∞
=
2
(5)
The
rati
o
of
the
RM
S
val
ue
of
the
c
urre
nt
to
the
capaci
t
or
value
at
th
e
fr
e
qu
e
nc
y
of
the
supp
l
y
vo
lt
age
=
50
is:
=
200
.
√
2
.
ℎ
√
∑
1
(
2
.
−
1
)
2
.
2
(
2
.
−
1
)
∞
=
2
.
(6)
Figure
4
sho
ws
a
set
of
f
eat
ures
ref
le
ct
in
g
th
e
above
dep
e
ndence
at
par
a
m
et
er
ℎ
.
The
sum
is
li
mit
ed
to
=
25
in acc
orda
nc
e w
it
h
[
25, 26].
Figure
4.
Set
of f
eat
ur
es
acc
ordi
ng to
e
xpres
sion (
6)
Dep
e
ndin
g
on
(6)
t
he
value
of
the
ca
pacit
or
is
in
dicat
ed
a
nd
the
refo
re
it
is
necessa
r
y
t
o
pe
rform
a
n
analysis to det
ermine t
his v
al
ue.
I
n
c
onnecti
on
with the s
ec
ond
feat
ur
e
(de
te
rmin
in
g
the
va
lue of the capac
it
or
at
hystere
sis c
on
t
ro
l)
, th
e
ti
m
e d
ia
grams
s
hown in Fi
gure
5
are
us
ed
.
Figure
5. Time
d
ia
grams
for h
ys
te
resis c
ontr
ol of t
he vo
lt
ag
e on ca
pacit
or
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2088
-
8
694
In
t J
P
ow
Ele
c
&
D
ri
S
ys
t,
V
ol
.
12
, N
o.
1
,
Ma
rch
20
21
:
304
–
313
308
The
ex
a
minati
on
is
ma
de
w
it
hin
one
s
witc
hing
pe
rio
d
=
5
−
1
of
the
po
wer
de
vices.
T
his
per
i
od
is
sig
nif
ic
antly
sho
rter
than
t
he
hal
f
-
pe
rio
d
of
the
ne
twork
volt
age.
At
m
om
e
nt
=
1
th
e
transisto
rs
from
this
diag
on
al
of
the
br
i
dg
e
powe
r
ci
r
cuit
from
Fi
gure
1
a
re
switc
hed
on,
f
or
w
hich
the
volt
age
decr
ease
s,
a
nd
at
m
om
e
nt
=
3
−
for
wh
ic
h
it
i
ncrea
ses.
Since
t
he
filt
er
is
a
se
cond
-
orde
r
unit
,
after
the
m
om
e
nt
=
1
th
ere
is
a
certai
n
increase
in
volt
age,
a
nd
a
fter
the
m
om
e
nt
=
3
-
a
dec
rease.
Pro
vi
ded
,
that t
he
volt
ag
e rip
ple
on
t
he c
apacit
or
is
ne
glect
ed,
it
can be ass
um
e
d
tha
t t
he
current th
rou
gh
the i
nduc
tor
−
changes
i
n
bo
th
inter
vals
ac
cordin
g
to
a
li
near
la
w.
At
these
inter
vals,
vo
lt
age
with
a
ppr
ox
imat
el
y
const
ant
value
2
is
ap
plied
to
for
t
he
i
nterval
−
=
3
−
1
,
an
d
1
f
or
t
he
i
nter
val
=
5
−
3
.
Fo
r
the
sho
rt
durati
on
of the
pe
rio
d
, t
he
c
hanges
of cu
rr
e
nt
in both
d
i
recti
ons a
re e
qu
al
,
th
eref
or
e:
(
−
)
.
=
(
+
)
.
(
−
)
(7)
Af
te
r
sim
plific
at
ion
, it
fo
ll
ow
s
(
8)
:
=
2
.
−
1
(8)
wh
e
re
=
(9)
Ther
e
f
or
e,
if t
he
r
efe
re
nce c
urve
for
the
volt
a
ge of
the ca
pac
it
or
(
)
=
(
)
is k
now
n,
then
the law
of c
hange
of
(
)
is al
so
know
n.
(
)
=
(
)
2
.
+
1
2
(10)
At
hyste
resis
con
t
ro
l,
t
his
r
efere
nce
c
urv
e
is
monit
or
e
d
with
a
de
finite
hyste
resis
,
w
her
e
t
he
capaci
tor
volt
age
al
so
co
ntain
s
high
-
f
reque
nc
y
ri
pp
le
s
,
wh
i
ch
ar
e
res
ult
of
the
s
witc
hing
with
high
f
re
quenc
y
.
Ba
se
d
on
the
se
rip
ples,
a
wa
y
is
s
ought
to
determi
ne
the
value
of
t
he
capaci
to
r
.
If
the
sta
rt
of
the
coor
din
at
e s
ys
te
m is trans
la
te
d
at
the mome
nt
2
, th
e c
ha
ng
e
o
f
t
he
cap
aci
to
r
volt
age
in th
e interval
2
−
3
can
be fo
und, i
e.
one
par
t
of th
e pu
lsa
ti
on
:
|
∆
′
|
=
1
.
∫
|
∆
|
−
−
2
0
.
.
=
1
8
.
.
|
∆
|
.
(
−
)
(11)
The
re
st
of
t
he
pu
lsa
ti
on
ca
n
be
f
ound
w
hen
t
ran
sla
ti
ng
the
sta
rt
of
th
e
coor
din
at
e
s
ys
te
m
at
the
mo
me
nt
3
:
|
∆
′′
|
=
1
.
∫
|
∆
|
2
0
.
.
=
1
8
.
.
|
∆
|
.
(12)
by
summi
ng (1
1) an
d (12
),
t
he
who
le
pulsa
ti
on can
b
e
fo
und:
|
∆
|
=
|
∆
′
|
+
|
∆
′′
|
=
1
8
.
.
|
∆
|
.
1
(13)
or
|
∆
|
|
∆
|
=
1
8
.
.
1
(14)
The
f
ollo
wing
desig
n
se
qu
e
nc
e
may
be
rec
ommen
de
d:
The
Total
Harmo
ni
c
Distortio
n
(
THD)
fa
ct
or
of
the
s
upply
ne
twork
volt
age
a
nd
the
ma
xi
mu
m
value
of
the
loa
d
cu
rr
e
nt
ar
e
known
.
1.
with
know
n
Total
Ha
rm
on
i
c
Distorti
on
(T
HD)
fact
or
of
the
s
upply
net
work
volt
age
,
from
Fig
ur
e
3
is
determi
ned
the
value
of
the
a
ngle
.
2.
Е
xpres
sion
(2)
a
nd
va
lue
of
ℎ
(to
a
ppr
ox
imat
e
t
he
s
uppl
y
vo
lt
age
c
urve)
are
us
e
d
to
Evaluation Warning : The document was created with Spire.PDF for Python.
In
t J
P
ow Elec
& Dri S
ys
t
IS
S
N: 20
88
-
8
694
Study
of the
op
eratio
n of the
outp
ut fi
lt
er o
f a
sin
gle
-
p
hase s
eri
es a
ct
iv
e
powe
r fi
lt
er (
Mi
hail
An
tc
hev
)
309
determi
ne
the
RMS
value
of
the
hi
gh
e
r
harmo
nics.
Ba
se
d
on
(
14),
the
va
lue
of
t
he
ca
pa
ci
tor
is
deter
m
ined
by
ass
um
i
ng
va
lues
for
∆
(as
a
pe
rcen
ta
ge
of
the
ma
xim
um
value
of
t
he
l
oa
d
c
urre
nt
)
an
d
∆
(
as
a
per
ce
ntage
of
t
he
R
M
S
va
lue
of
the
higher
ha
rm
on
ic
s
)
.
3.
The
R
M
S
val
ue
of
the
capaci
tor
c
urre
nt
at
the
value
of
ℎ
f
rom
it
em
2.
is
dete
rmin
e
d
f
r
om
Fig
ure
4.
To
this
valu
e
i
s
a
dd
e
d
the
cu
rr
e
nt
with
t
he
high
switc
hing
f
requen
c
y
=
20
.
4.
T
he
value
of
the
inducta
nce
i
s
determi
ned,
so
that
the
re
s
on
a
nt
fr
e
qu
e
nc
y
of
t
he
filt
er
to
be
higher
tha
n
the
fr
e
quenc
y
of
t
he
highest
harmo
nic
bein
g
c
ompen
sat
ed,
a
nd
lowe
r
tha
n
t
he swit
chin
g fr
e
quenc
y
. 5. T
he
value o
f
the
hy
ste
resis
as a p
e
rcen
ta
ge of
∆
is
determi
ned.
Example:
It
is
known
a
s
a
r
esult
of
meas
ureme
nt
:
Total
Harmo
nic
Dist
or
ti
on
(
TH
D)
f
act
or
of
t
he
su
ppl
y
net
wor
k
volt
age
=
17%
and
maxim
um
value
of
the
lo
ad
current
=
10
.
A
ppr
ox
imat
e
value
of
ℎ
=
290
.
1)
Fo
r
=
17%
f
r
om Fig
ure
3
is
repor
te
d
=
0
.
5
.
2)
Fr
om
(
2)
a
nd
(
1)
the
R
M
S
va
lue
of
the
high
er
harmo
nics
a
t
ℎ
=
145
is
determi
ne
d
(the
tra
ns
f
or
mer
Tr
increases
the
vo
lt
age
of
t
he
ca
pacit
or
2
ti
mes)
−
.
=
0
.
082
→
4
.
145
.
0
,
5
.
0
.
082
≈
30
.
∆
=
0
.
6
→
[
(
1
−
2%
)
.
30
]
,
∆
=
1
→
[
(
5
−
10%
)
.
2
.
]
is
acce
pted
,
a
nd
t
he
value
of
the
capaci
t
or
at
fr
e
qu
e
nc
y
=
20
=
10
is deter
mine
d from
(
14).
3)
Fr
om
Fi
gure
4
is
determi
ne
d
=
165
.
T
o
this
value
is
ad
de
d
the
cu
rr
e
nt
with
the
hi
gh
switc
hing
f
requen
c
y
=
2
0
.
The
m
aximum
val
ue
of
the
first
hi
gh
fr
e
quenc
y
vo
lt
age
ha
rm
onic
is
equ
al
to
half
t
he
rip
ple
∆
.
T
hus,
the
RMS
va
lue
of
the
c
ur
ren
t
t
hro
ugh
t
he
ca
pacit
or
with
high
fr
e
qu
e
nc
y
is
∆
2
.
√
2
.
.
2
.
.
.
=
0
.
6
√
2
.
.
20
.
10
3
.
10
.
10
−
6
=
266
.
Or
i
n
ge
ner
al
=
165
+
2
6
6
mA
≈
4
3
0
mA
.
4)
The
value
of
the
in
duct
ance
is
relat
ed
to
the
determi
ned
value
of
the
ca
pacit
or
.
T
o
a
void
possibl
e
resona
nce
ph
e
nomena
on
t
he
one
hand,
it
is
necessa
r
y
th
e
res
on
a
nt
fr
e
quenc
y
of
the
fi
lt
er
to
be
high
er
than
the
f
re
quency
of
the
hig
he
st
har
m
onic
,
wh
ic
h
is
c
ompe
ns
at
ed
(i
n
this
case
the
25th
).
O
n
the
ot
her
hand,
it
m
us
t
be
si
gn
ific
a
ntly
lo
we
r
tha
n
the
s
witc
hing
f
reque
ncy
of
t
he
powe
r
de
vices
(in
this
case
20
kH
z
).
F
or
th
es
e
reas
ons,
an
a
ppr
opriat
e
val
ue
is
500
.
At
this
case
the
res
onant
f
re
qu
e
nc
y
of
is
appr
ox
imat
el
y
2.2
5
kHz.
T
he
RM
S
value
of
the
cu
rr
e
nt
thr
ough
the
in
du
ct
a
nce
is
a
su
m
of
t
he
R
MS
values
of
t
he
c
urren
t
of
the
ca
pacit
or
an
d
t
he
load
,
al
s
o
ta
kin
g
into
acc
ount
the
tra
ns
f
orm
at
ion
c
oeffici
ent
of the
outp
ut tr
ansfo
rmer
−
=
2
.
√
2
+
≈
14
.
57
.
5)
Hy
ste
resis
determinati
on.
T
he
dete
rmin
at
i
on
of
t
he
hyst
eresis
val
ue
use
d
to
m
onit
or
the
cap
aci
tor
vo
lt
age
is
do
ne
on
t
he
ba
se
of
the
acc
epted
value
f
or
∆
.
It
is
rec
om
me
nded
2
.
=
[
(
40
−
50%
)
∆
]
.
I
n
case
of
(
0
.
24
−
0
.
3
)
.
2.2.
Co
m
pu
ter sim
ulat
i
on
A
stu
dy
was p
erforme
d b
y
co
mputer sim
ulati
on
of
the
oper
at
ion
of the
S
F
SA
P
F at the th
us
set init
ia
l
data
a
nd
certai
n
valu
es
in
the
ab
ove
e
xam
pl
e.
T
his
is
do
ne
with
the
ORC
AD
-
PS
PI
CE
pro
gr
a
m.
Fig
ure
6
a
nd
Figure
7
sho
w
the
res
ults
w
he
n
a
li
near
l
oa
d
(acti
ve
-
in
duc
ti
ve)
is
co
nnec
te
d
to
the
pow
er
sup
ply
netw
ork.
Figure
8
sho
w
s
the
res
ults
f
or
a
non
-
li
nea
r
loa
d
f
or
the
powe
r
s
uppl
y
netw
ork
-
sin
gle
-
phase
unc
on
tr
olled
br
i
dg
e
r
ect
ifie
r
w
it
h
act
iv
e
-
ca
pacit
ive loa
d.
Figure
6. V
oltage
of the
powe
r
s
upply net
work (a
nd of t
he
l
oad) a
nd loa
d
c
urren
t i
n
the
a
bse
nce
of an
S
A
PF.
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2088
-
8
694
In
t J
P
ow
Ele
c
&
D
ri
S
ys
t,
V
ol
.
12
, N
o.
1
,
Ma
rch
20
21
:
304
–
313
310
Figure
7. Loa
d v
oltage a
nd lo
ad
c
urren
t
with
SAPF
on.
Figure
8. Re
su
l
ts of the
operat
ion
of the
S
APF with
no
n
-
li
ne
ar lo
a
d
3.
RESU
LT
S
AND DI
SCUS
S
ION
The
res
ults
of
the
co
mputer
simulat
io
n
ha
ve
bee
n
c
onfir
med
ex
pe
rime
ntall
y.
The
c
orres
pondin
g
os
ci
ll
ograms
a
re
sho
wn
in
Figure
9
a
nd
Fig
ur
e
10.
Th
e
cu
rr
e
nt
in
Fig
ur
e
9
an
d
Fig
ure
10
was
monit
ored
wit
h
a
100:
1
c
urre
nt
prob
e
.
T
he
re
su
lt
s
of
the
ex
per
ime
ntal
stu
dy
co
nf
i
rm
t
he
res
ults
obta
ine
d
f
r
om
t
he
c
omp
uter
simulat
ion.
Fi
gure
11
s
how
s
the
os
ci
ll
ogr
ams
wit
h
s
ub
s
ta
ntial
ly
nonli
near
loa
d
-
AC
regulat
or
with
phase
co
nt
ro
l.
(a)
(b)
Figure
9. Re
su
l
ts at
acti
ve
-
in
duct
ive lo
ad:
(
a
)
load
volt
age
a
nd loa
d
c
urren
t
w
it
hout
SAPF
;
(
b
)
loa
d v
oltage
and loa
d
c
urre
nt w
it
h S
AP
F
on.
Evaluation Warning : The document was created with Spire.PDF for Python.
In
t J
P
ow Elec
& Dri S
ys
t
IS
S
N: 20
88
-
8
694
Study
of the
op
eratio
n of the
outp
ut fi
lt
er o
f a
sin
gle
-
p
hase s
eri
es a
ct
iv
e
powe
r fi
lt
er (
Mi
hail
An
tc
hev
)
311
(a)
(b)
Figure
10. Res
ults at
no
nlinea
r
loa
d
-
bri
dge r
ect
ifie
r
with
RC
load
:
(
a
)
loa
d vo
lt
age
and
load cu
rr
e
nt
without
SA
P
F;
(
b
)
loa
d v
oltage a
nd lo
ad
c
urren
t
with
SAPF
on
Figure
1
1
. Res
ults at
no
nlinea
r
loa
d
-
AC
regu
la
tor wit
h ph
as
e co
ntr
ol
:
(
a
)
load
volt
age
an
d
loa
d
c
urre
nt
without S
APF;
(
b
)
l
oad volt
ag
e an
d
loa
d
c
urr
ent w
it
h
S
AP
F
on
4.
C
O
NC
L
US
I
O
NS
The
co
ntri
bu
ti
on
of
t
his
stu
dy
is:
1)
Deter
minin
g
the
value
of
the
c
urr
ent
of
t
he
hi
gher
harmo
nic
thr
ough
the
ca
pacit
or
of
t
he
ou
t
pu
t
filt
er
-
e
xpressi
on
(6)
a
nd
Fi
gure
4.
2
)
D
et
ermi
ning
t
he
volt
age
rip
ple
on
the
capaci
to
r
dep
e
ndin
g
on
it
s
value
an
d
s
witc
hing
f
requ
ency
-
e
xpressio
n
(
14).
3
)
O
n
this
basis
offe
rin
g
a
method
ology f
or d
esi
gn
i
ng th
e outp
ut
filt
er.
The
res
ults
pr
esented
in
t
his
arti
cl
e
make
it
po
ssi
ble
to
determi
ne
the
par
a
m
et
ers
of
the
outp
ut
capaci
tor
as
w
el
l
as
t
he
outp
ut
i
nductance
of
a
series
act
i
ve
pow
er
filt
er
wit
h
hyste
resi
s
c
on
t
ro
l.
T
he
resu
lt
s
wer
e
c
onfir
me
d
by
c
ompu
te
r
simulat
io
n
a
nd
e
xperime
nt
al
te
sti
ng
an
d
sh
ow
s
uffici
ent
f
or
the
pra
ct
ic
a
l
pur
po
ses
acc
uracy
.
ACKN
OWLE
DGE
MENTS
The
a
uthor/s
w
ou
l
d
li
ke
to
tha
nk
t
he
Re
sea
rc
h
a
nd
De
velo
pme
nt
Sect
or
at
the
Tech
nical
Un
i
ver
sit
y
-
So
fia
for t
he financial
s
upport
.
REFERE
NCE
S
[1]
Akagi
H.,
“Ne
w
Tre
nds
in
Act
ive
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ers
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er
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ioni
ng
,
”
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E
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ans.
On
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v
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la
ss
ic
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l
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cent
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ct
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ctive
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lity
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nt
,
”
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lassical
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c
ent
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r
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ystem
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on
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d
emic
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ess,
Еl
sev
ie
r
,
p
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eISBN:
97801281
24420,
2018.
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2088
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8
694
In
t J
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ow
Ele
c
&
D
ri
S
ys
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-
keng
Chi
ang
,
Bor
-
r
en
Li
n
,
Kuan
-
wei
Wu
,
“
Serie
s
a
ct
iv
e
p
ower
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t
er
for
c
urre
nt
har
moni
c
and
loa
d
vo
lt
ag
e
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ti
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”
Proceedi
ngs
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th
e
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-
IEE
E
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g
ion
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nfe
renc
e
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DO
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aya
k
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esh
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“
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educ
t
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on
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oni
c
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vo
lt
ag
e
s
ag
co
mpe
nsa
ti
o
n
by
ser
ie
s
ac
t
i
ve
power
fil
t
er
,
”
Pr
oce
ed
ings
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the
2017
Inte
rnatio
nal
Confe
ren
ce
on
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Manage
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ati
on
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ero
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,
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ne
z,
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ves
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ne
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tive
Pow
er
Filt
er
s
for
Li
ne
Con
dit
ioni
ng
:
a
Cr
itical
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lu
at
ion
,
”
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EE
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ansacti
ons
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er
Del
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vo
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325
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[7]
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.
and
Y.
-
L
.
Hou,
“Si
ngle
-
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Int
eg
rat
ed
Pow
er
Qu
al
it
y
Comp
ensa
t
or
Based
on
Ca
pac
i
tor
-
Clamped
Configura
ti
on
,
”
IEE
E
Tr
ansacti
o
ns on
Industrial Elect
ronics
,
vol
.
49,
no.
1,
pp.
1
73
-
185,
2002
.
[8]
Chen
G.,
Zh
.
Lu
,
Zh.
Qi
an
and
F
.
Zh.
Peng
,
“A
New
Seria
l
Hybr
id
Acti
v
e
Pow
er
Filt
er
Us
ing
Con
trol
la
b
le
Curr
ent
Source
,
”
IE
EE 3
3
rd
Annual, PES
C
,
vol
.
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,
pp
.
36
4
-
368,
2002
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[9]
Darwish
M.K,
“
Single
-
Chip
Inv
ert
er
for
Acti
v
e
Filt
ers,”
El
e
ct
ro
nic
s
Let
te
rs
26
th
September
,
vo
l.
38,
no.
20
,
pp
.
1159
-
1160,
200
2.
[10]
Pan
Z
.
,
F.Z
Pen
g
and
S.
Wa
ng
,
“Power
Fa
ct
or
Corre
ction
Us
in
g
a
Seri
es
Act
iv
e
Fil
te
r
,
”
IEEE
Tr
ansacti
ons.
on
Powe
r E
le
c
troni
cs
,
vol
.
20
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no
.
1
,
pp
.
148
-
153
,
2
005.
[11]
Chang
G.W.
a
nd
T.
-
Ch
.
Shee
n,
“A
Comp
ar
at
iv
e
Study
of
Acti
ve
Pow
er
Filt
er
R
efe
r
en
ce
Comp
ensa
ti
o
n
Approac
hes,
”
P
ower
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n
ee
rin
g
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r Me
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ndez
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N.
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z
and
V
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ena
s,
“
Slidi
ng
Mode
C
ontrol
for
A
S
i
ngle
Phase
Act
i
ve
Pow
er
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te
r
,”
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r E
le
c
troni
cs
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I
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rnational
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[13]
Jain
S.K.
,
P.
A
gra
wal
and
H.O.
Gupt
a,
“Fuzz
y
Logic
Con
trol
l
ed
Shunt
Acti
v
e
Pow
er
Filt
er
fo
r
Pow
er
Qual
it
y
Improve
m
ent
,
”
I
EE
Proceedi
ngs
El
e
ct
ri
c
Pow
er
Appl
ic
a
ti
ons
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vo
l.
149
,
no
.
5
,
pp
.
317
-
328,
2002
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[14]
La
xmi
A.
J.,
G.
T.
R
.
D
as
and
K.
U.
R
ao,
“
Rol
e
o
f
PI
and
Fuzzy
Control
le
r
in
Un
ifi
ed
Pow
er
Qua
li
ty
Condit
ion
er,”
AR
PN
Journal
o
f
Eng
ine
ering
an
d
Applied
S
cienc
es
,
vol
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2
,
no
.
2
,
pp.
1
–
10
,
2007
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[15]
La
n
Li,
Ha
i
Che
n,
and
Xiaox
ia
Cao,
“De
adb
eat
Control
Te
chn
iq
ue
Appli
ed
to
A
ct
iv
e
Pow
er
Fi
lter,
”
Informati
cs
in
Control,
Au
tomati
on
and
Robo
ti
cs
,
Spring
er,
v
ol.
2
,
pp.
75
-
83
,
2019,
DOI
htt
ps://
do
i
.
org/10
.
1007/978
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3
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642
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25992
-
0_10
.
[16]
Gela
n
Zhu
;
Ga
ng
Wa
ng
;
D.
H
ua
,
“Sli
ding
m
ode
cont
ro
l
with
var
ia
b
le
stru
ct
u
re
of
seri
es
acti
ve
power
fi
lt
er
,
”
Proce
ed
inc
s
of
th
e
2009
IE
EE
Powe
r
&
En
ergy
Socie
ty
Gene
ral
M
ee
t
ing
,
Ca
lga
ry
,
C
ana
d
a,
DO
I:
10.
1109/PES.20
09.
5275325
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Fan,
X.,
Zha
ng
,
D.,
Sun,
Q.
,
et
a
l.
,
“Hyster
esis
Curre
nt
Contro
l
S
tra
t
egy
for
Three
-
phase
Thre
e
-
wire
Act
ive
Pow
e
r
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er
,
”
Aut
omat
i
on
of El
e
ct
ri
c
Po
wer
Syste
m
,
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no
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[18]
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v
M
.
H.,
M.P.
Petkov
a,
A.T
.
Kos
tov,
“
Hys
te
resis
-
cur
re
nt
con
trol
of
sin
gle
-
phase
shunt
ac
t
ive
power
f
il
t
er
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req
uen
cy li
mitation
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”
Proc
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dings o
f
th
e co
nfe
renc
e
PE
S
20
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,
Cl
ea
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[19]
Jianc
heng
Song
;
Fe
iya
n
T
ia
n
;
Yunguang
Gao
;
L
ij
un
Zh
eng
,
“
A
com
posi
te
cur
ren
t
cont
ro
l
met
hod
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th
e
three
phase
shunt
act
ive
power
fi
lt
e
r
,
”
Proceedi
ngs
of
the
2016
I
EE
E
2nd
Annu
al
South
ern
Po
wer
Elec
troni
cs
Confe
renc
e
(S
P
EC)
Auckl
and,
New Z
e
al
and
,
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10.
1109/SP
EC.
2016.
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[20]
You
-
hua
Jiang
;
Yong
-
wei
C
hen
,
“
N
eur
al
Network
Contr
ol
Techni
ques
of
Hybrid
Active
Pow
er
Filt
e
r,
”
Proce
ed
ings
of
the
2009
Inte
r
nati
onal
Con
fe
r
enc
e
on
Artifi
cial
Intelli
g
ence
and
Computati
o
nal
Intelli
g
ence
,
Shanghai
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DO
I:
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.
1109/
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[21]
El
mitwall
y
A
.
,
S
.
Abdelka
d
er
an
d
M.
El
-
Kat
eb,
“
Neura
l
Network
Control
le
d
Three
-
Phase
Four
-
W
ire
Shunt
Act
ive
Pow
er
Filt
er
,
”
I
EE
Proceedi
ngs
Gene
ration
Tr
an
sm
ission and
Distributi
on
,
vol. 1
47,
no
.
2
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87
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92,
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B.
Gonz
at
t
i
;
S.
C.
Ferre
ira
;
Ca
rlos
H.
da
Silv
a
;
L
.
E.
Borges
d
a
Silva
;
G.
La
m
ber
t
-
Torr
es
;
L
.
G.,
“Fer
n
an
PLL
-
le
ss
con
trol
str
at
egy
app
li
ed
t
o
Hybrid
Acti
v
e
Ser
ie
s
Pow
er
Filt
er,”
Proceedi
ngs
o
f
the
10th
IE
EE
/I
A
S
Inte
rnational
Co
nfe
renc
e
on
Indu
stry
Applications
,
Forta
leza
,
Bra
z
il
,
DOI
:
10
.
1109/
IND
US
C
ON
.
20
12.
6453300
[23]
Jingyang
Fang
;
Guochun
Xiao
;
Xu
Yang
;
Yi
Tang
,
“Pa
ramet
er
Design
of
a
Nov
el
Seri
es
-
Para
l
lel
-
Resonant
LCL
Filt
er
for
Single
-
Phase
Half
-
Br
id
ge
Ac
ti
ve
Pow
er
Filt
ers
,
”
I
EE
E
T
rans
act
ions
on
P
ower
Elec
tronic
s
,
vo
l.
32,
no
.
1
,
2017,
DO
I:
10
.
1
109/T
PEL
.
2016.
2532961
[24]
Plat
ek
,
T
.
,
“Ana
lysis
of
Rippl
e
Curre
nt
in
the
Capa
c
it
ors
of
A
ct
iv
e
Pow
er
Filt
ers,
”
Ene
rgies
,
vol.
12
,
no.
449
3,
2019
.
[25]
Copper
Deve
lop
me
nt
As
soci
at
io
n,
Volt
age
Dist
urba
nce
s,
Stand
ard
EN50160
-
V
olt
ag
e
Char
ac
t
er
isti
cs
in
Public
Distribut
ion
Sys
te
m,
from
htt
ps:
//
copp
e
ral
lianc
e.
org
.
uk/
uploa
ds/2018/03
/542
-
standa
rd
-
en
-
50160
-
volt
ag
e
-
cha
ra
cteri
sti
cs
-
i
n.
pdf
.
[26]
Schnei
der
Elec
tr
ic
SA
,
Harm
oni
c
mi
t
igation, Solu
ti
on
Handbook
,
June,
2009
.
Evaluation Warning : The document was created with Spire.PDF for Python.
In
t J
P
ow Elec
& Dri S
ys
t
IS
S
N: 20
88
-
8
694
Study
of the
op
eratio
n of the
outp
ut fi
lt
er o
f a
sin
gle
-
p
hase s
eri
es a
ct
iv
e
powe
r fi
lt
er (
Mi
hail
An
tc
hev
)
313
BIOGR
AP
HI
ES OF
A
UTH
ORS
Mi
hail
An
tche
v
,
was
born
in
Sofia,
Bulg
ari
a
,
i
n
1955.
He
awa
rde
d
Maste
r
d
eg
ree
at
T
ec
hni
cal
Univer
sity
–
Sof
ia
,
Bu
lga
r
ia
,
Fac
ult
y
of
Rad
ioele
ct
roni
cs,
in
198
1.
Th
e
PhD
deg
ree
r
ec
e
ive
d
in
1990
in
Industri
al
elec
tron
ic
s
at
Te
chn
ic
a
l
Univ
e
rsity
–
Sofia.
Th
e
Ds
c
d
egr
e
e
r
ecei
ved
in
2010
in
El
e
cro
nic
converte
rs
at
Techni
c
al
Univer
si
ty
–
Sofia.
He
is
P
rofe
ss
or
in
Powe
r
elec
tron
ic
s,
Facul
ty
of
el
e
ct
r
onic
eng
ine
er
ing
and
technolog
ies
,
T
ec
hn
ical
Uni
ver
sity
–
Sofia
.
He
works
in
th
e
fie
ld
of
power
e
le
c
troni
c
conv
erters
with
im
prov
ed
en
erg
y
indi
c
a
tors,
as
w
el
l
as
of
dynami
c
and
cont
rol
sys
te
m
o
f
power elect
ron
ic
conve
rt
ers.
E
-
ma
il:
ant
ch
ev@t
u
-
sofia.
bg
Van
jo
G
o
u
r
g
o
u
l
i
t
s
o
v
,
was
bo
rn
in
Vidin,
B
ulga
ri
a,
in
195
1.
He
gra
du
ate
d
in
Polytechnic
Instit
ute
of
Donetsk,
Ukr
ai
ne
.
The
PhD
d
egr
e
e
re
ceive
d
in
2
009
in
Industrial
e
lectr
oni
cs
a
t
Te
chn
ic
a
l
Unive
rsity
–
Sofia
.
He
is
As
socia
t
e
p
rofe
ss
or
at
Co
llege
of
ene
rg
y
a
nd
e
le
c
tronics,
Te
chn
ic
a
l
univ
ersit
y
–
Sofia.
He
works
in
th
e
field
of
power
el
e
ct
roni
cs,
d
igi
t
al
and
mi
cro
pro
ce
ss
or
t
ec
h
nolog
ie
s,
mic
roe
lectr
oni
cs.
E
-
mail:
vtg_ot
k
@abv.
bg
Hr
isto
An
tche
v
,
was
born
in
S
ofia
,
Bulga
r
ia,
i
n
1981.
He
awa
rde
d
Master
deg
ree
at
Techni
c
al
Univer
sity
–
So
fia
,
Bulga
r
ia
in
spec
i
al
i
za
t
ion
“Power
Elec
tro
nic
s”
in
2004.
The
PhD
degr
e
e
rec
e
ive
d
in
2012
in
Industri
al
ele
ct
roni
cs
a
t
Tech
nic
a
l
Univ
ersit
y
–
Sofia
.
He
is
C
hie
f
assistant
a
t
Univer
sity
of
c
hem
i
ca
l
t
ec
hno
l
ogy
and
m
etall
u
rgy.
He
works
i
n
th
e
f
ie
ld
of
d
esign
of
power
el
e
ct
roni
c converter
s
E
-
mai
l:
hristo_antche
v@uc
tm.e
d
u
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