Internati
o
nal
Journal of P
o
wer Elect
roni
cs an
d
Drive
S
y
ste
m
(I
JPE
D
S)
Vol
.
4
,
No
. 2,
J
une
2
0
1
4
,
pp
. 17
3~
19
1
I
S
SN
: 208
8-8
6
9
4
1
73
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
Application of Backstepping to
the
V
i
rtu
a
l Flux Di
r
ect
P
o
wer
Contr
o
l of Five-Level
Thr
e
e-Ph
ase Shunt
Active Power
Filter
Bo
uzidi Manso
u
r
*,**
,
Bo
ua
f
i
a
Sa
b
e
r
**
, Bouzidi Ali
***
, Benaiss
a
Abdelk
ader
**
, B
a
rk
at
Said
****
*
Department of
Electronics and
Comm
unications, Faculty
of
New Techno
logi
es
of Information
and Communication
Unive
r
sity
Ka
sdi Me
rba
h
,
Ouargl
a, Alg
e
ri
a.
** Departmen
t
o
f
Electr
i
cal
Engineering
,
Fa
culty
of Engin
eerin
g,
University
of
Dj
ilal
i
Liab
es
, S
i
d
i
Bel
Abbes
,
Alg
e
ria
,
Intell
igen
t Contr
o
l E
l
ec
troni
c Po
wer S
y
st
em
labo
rator
y
(I
.C.
E
.P.S
)
*** Departmen
t
of Electr
i
cal
Eng
i
neer
ing, Faculty
of Techno
log
y
,
University
of
Hadj Lakhdar
Batn
a
**** Departmen
t
of
Electr
i
cal
En
gineer
ing, Faculty
of
Eng
i
neering
,
M’sila University
, M’sila, Alger
i
a.
Article Info
A
B
STRAC
T
Article histo
r
y:
Received
Ja
n 11, 2014
Rev
i
sed
Mar
13
, 20
14
Accepted
Mar 26, 2014
This paper proposes a virtual
flux direct power control-space ve
cto
r
modulation combined with back
stepping
control for three-phase five-lev
el
neutral point clamped s
hunt activ
e power filter.
The main g
o
al of
th
e
proposed activ
e filtering s
y
st
em
is to
elim
inat
e t
h
e unwanted har
m
onics and
com
p
ens
a
te fun
d
am
ental r
eac
tiv
e power
drawn from the nonlinear loads. In
this stud
y,
the
voltag
e
-bal
anci
n
g
contro
l of
fou
r
split
dc
cap
ac
itors of
the
five-l
evel act
ive
fi
lter is achieved using five-lev
el space vector modulatio
n
with balancing strateg
y
based
on
the
effectiv
e us
e of th
e redund
ant switching
states of the inv
e
rter voltage vectors.
The obtained results showed that, th
e
proposed m
u
ltil
evel
shunt activ
e power f
ilt
er w
ith backstepp
i
ng
contro
l can
produce a sinusoidal supply
cu
rrent w
ith low harmonic distortion and in
phase with
th
e line voltage.
Keyword:
Five-level shu
n
t
active power
filter
Direct power co
ntrol
Virtual flux concept
Back
step
ping c
ontrol
Multilev
e
l space vector
m
o
d
u
lation
Copyright ©
201
4 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
:
B
ouzi
d
i
M
a
ns
o
u
r
,
Depa
rt
m
e
nt
of
El
ect
roni
cs
an
d C
o
m
m
uni
cati
ons
,
Facul
t
y
o
f
Ne
w Tec
h
nol
ogi
e
s
o
f
I
n
fo
rm
at
i
o
n a
n
d
C
o
m
m
uni
cat
i
on
Uni
v
ersity Ka
s
d
i Merbah,
Ouargla, 30000,
Alge
ria
Depa
rtem
ent of Elect
ri
cal
E
n
gi
nee
r
i
n
g,
Faculty of E
n
gineering,
Un
i
v
ersity of
Dj
ilali Liab
es,
Sid
i
Bel Abb
e
s,
Si
di
B
e
l
A
b
bes
2
2
0
0
0
,
Al
geri
a
Em
ail: bouzidi
.
m
.
28@gm
ail.com
1.
INTRODUCTION
No
wa
day
s
t
h
e
p
o
we
r el
ect
r
o
ni
c eq
ui
pm
ents are
wi
del
y
u
s
ed i
n
di
st
ri
b
u
t
i
on
net
w
or
ks
whi
c
h act
as
n
o
n
lin
ear lo
ads. Man
y
p
o
wer q
u
a
lity d
i
stu
r
ban
ces su
ch
as h
a
rm
o
n
i
cs po
llu
tio
n, un
b
a
lanced
lo
ad
cu
rren
ts, and
reactive powe
r
problem
s
ar
e caused
by the nonlinear l
o
ads; as a res
u
lt poor
powe
r factor,
weakening
efficien
cy, ov
er
h
eatin
g of m
o
to
rs and
tran
sf
o
r
m
e
rs, m
a
lfu
n
ctio
n
o
f
sensitiv
e
d
e
v
i
ces etc.
To
o
v
e
rco
m
e t
h
e aforem
en
tio
n
e
d prob
lem
s
p
a
ssiv
e
filters
can
b
e
used
t
o
co
m
p
en
sate some o
f
th
em
.
Ho
we
ver
,
b
u
l
k
passi
ve com
pone
nt
s, se
ri
es and
paral
l
e
l
res
ona
nce an
d a fi
xed c
o
m
p
ensat
i
on cha
r
act
eri
s
t
i
c
are
th
e m
a
in
d
r
awb
acks of p
a
ssiv
e filters
[1
].
Th
erefo
r
e, t
o
g
i
v
e
an
effectiv
e so
l
u
tion
fo
r h
a
rm
o
n
i
cs con
cern
s
sev
e
ral activ
e
p
o
wer filter (APF) t
o
po
log
i
es h
a
v
e
b
e
en
p
r
op
o
s
ed. Th
ey aimed
no
t on
ly fo
r curren
t
or
vo
ltag
e
co
m
p
en
satio
n
b
u
t
also
for
vo
ltag
e
d
i
p
s
, flicker an
d im
b
a
lan
ce [2
].
Am
ong va
rious active filter configurations
, the shun
t active power filters
(SAP
F) ha
ve
a num
ber of
adva
ntage
s
[3]. Com
p
ared
with the
seri
es
and
hy
bri
d
co
nfi
g
u
r
at
i
o
n
s
, t
h
e S
A
PF
d
o
n
o
t
nee
d
a
n
a
d
di
t
i
onal
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-86
94
I
J
PED
S
Vo
l. 4
,
No
. 2
,
Jun
e
2
014
:
17
3
–
19
1
17
4
cou
p
l
i
n
g t
r
a
n
s
f
o
r
m
e
r and
r
e
qui
re m
u
ch
l
e
ss pr
ot
ec
tion a
nd
switchgear. T
h
ey operate as t
h
re
e-phase
cont
rol
l
e
d h
a
r
m
oni
c current
sour
ces an
d are not
af
fect
ed by
harm
oni
c di
st
ort
i
o
ns i
n
su
ppl
y
v
o
l
t
a
ges. I
n
gene
ral
,
t
h
e
rat
i
ngs
of
sh
u
n
t
a
c
t
i
v
e p
o
we
r fi
l
t
ers are
base
d
o
n
t
h
e m
a
gni
t
u
d
e
s of
com
p
ens
a
t
i
ng c
u
r
r
ent
a
nd t
h
e
co
rresp
ond
ing
filter termin
al v
o
ltag
e
. Fo
r med
i
u
m
to
h
i
g
h
p
o
wer app
licatio
n
s
th
e m
u
ltil
ev
el con
v
e
rters are
the m
o
st attractive techno
logy. Indeed, m
u
ltilevel conve
rt
ers ha
ve
shown som
e
signifi
cant advanta
g
e
s
ove
r
trad
itio
n
a
l t
w
o-lev
e
l co
nv
erte
rs [4
]-[7
]
. Th
e
main
ad
v
a
n
t
ages of th
e
m
u
ltil
ev
el conv
erter
are a sm
aller o
u
t
put
voltage ste
p
, lowe
r ha
rm
onic com
ponents, a
better el
ectrom
a
gnetic com
p
atibility and lowe
r switchi
ng losses
[4]-[7]. In the
recent ti
m
e
the use of
m
u
lti
le
vel inverte
r
s is prevailing in
medium
-voltage active powe
r
filters
with
ou
t
u
s
ing
a cou
p
ling
t
r
ansform
e
r [8
]-[11
].
Vari
ous
co
nt
r
o
l
st
rat
e
gi
es ha
v
e
bee
n
pr
op
ose
d
t
o
co
nt
r
o
l
t
h
e SA
PF,
suc
h
as hy
st
eresi
s
b
a
nd
cu
rre
nt
co
n
t
ro
l (HBCC) [12
]
, vo
ltage o
r
ien
t
ed
con
t
ro
l (VOC
)
[13]. An
o
t
h
e
r in
terestin
g
em
erg
i
n
g
co
n
t
ro
l tech
n
i
q
u
e
cal
l
e
d di
rect
p
o
we
r c
ont
r
o
l
(
D
PC
)
has
bee
n
i
nvest
i
g
at
e
d
[
13]
,
[1
4]
.
DPC
i
s
based
o
n
t
h
e i
n
st
ant
a
ne
o
u
s
act
i
v
e
an
d reactive
po
wer con
t
ro
l lo
op
s. It
is
d
e
velo
p
e
d
an
alogo
u
s
ly
with th
e well-k
nown
direct to
rqu
e
con
t
ro
l
(DTC
) use
d
f
o
r ad
j
u
st
abl
e
s
p
eed
d
r
i
v
es
. I
n
DPC
,
t
h
ere
are no
i
n
t
e
r
n
al
cu
rre
nt
l
o
op
s
or
m
odul
at
o
r
bl
oc
k
because the conv
erter switching states are selected
via
a
switching table.
Alt
h
ough DPC has many
adva
nt
age
s
, so
m
e
di
sadva
nt
ages o
f
t
h
i
s
co
n
t
rol
t
echni
que
are hi
g
h
ri
p
p
l
e
cont
ent
i
n
t
h
e
sy
st
em
curre
n
t
, hi
g
h
ripp
le in
th
e
co
mman
d
e
d
activ
e an
d
reactiv
e p
o
wer,
variable
switchi
ng fre
quency,
and requires a
high
swi
t
c
hi
n
g
fre
q
u
ency
[
15]
.
The DPC
can
be fo
rce
d
t
o
op
erat
e at
a const
a
nt
fre
que
ncy
by
usi
n
g space
vect
or m
odul
a
t
i
on (S
VM
)
to synt
hesize t
h
e s
p
ace-vect
or
voltage
dem
a
nde
d
by th
e
s
w
itching ta
ble
[15],
[1
6].
Although classica
l DPC
uses t
h
e
fi
x
n
u
m
ber of
vect
o
r
s pre
s
ent
i
n
t
h
i
s
t
a
bl
e, m
o
re v
ect
ors ca
n
be a
r
bi
t
r
a
r
i
l
y
gene
r
a
t
e
d by
usi
n
g
SVM
.
In th
is
way, the ripp
le in
current can be
re
duced
[15].
Recent de
velopm
ents have popularize
d the
virtual
fl
ux
(VF) conce
p
t, which ass
u
m
e
s that both the
g
r
i
d
an
d
co
nv
erter’s lin
e filter b
e
h
a
v
e
as an AC
m
o
to
r [17]. On
e o
f
th
e
main
ad
v
a
n
t
ages o
f
th
is ap
pro
ach
is
th
at it is less s
e
n
s
itiv
e to
lin
e-vo
ltag
e
variatio
n
s
t
h
an
o
t
h
e
r
ap
pro
ach
es. Th
e v
i
rt
u
a
l
flux
d
i
rect po
wer co
n
t
ro
l
(VF
D
PC
) is a
n
ada
p
tation
o
f
t
h
e
DPC t
o
a
V
F
re
fere
nce
fra
m
e
[1
7]
.
In t
h
i
s
pa
per
,
a no
nl
i
n
ear co
nt
rol
st
rat
e
gy
ba
sed o
n
t
h
e bac
k
st
ep
pi
n
g
asso
ci
at
ed t
o
VFD
P
C
-
S
V
M
i
s
ap
p
lied to
t
h
ree-ph
ase
fiv
e
-lev
el sh
un
t
p
o
wer activ
e
filter in
th
e aim
to
i
m
p
r
ov
e its p
e
rform
a
n
ces.
In t
h
e
prese
n
t
st
udy
, i
t
i
s
s
h
o
w
n
vi
a si
m
u
l
a
ti
on
res
u
l
t
s
t
h
at
t
h
e
pr
op
ose
d
back
st
ep
pi
n
g
c
ont
rol
l
e
r
ha
s
h
i
gh
p
e
rfo
r
m
a
n
ce bo
th
in
th
e tran
sien
t and
in
th
e stead
y state operations
. The line curre
n
ts are ve
ry close to
sin
u
s
o
i
d
a
l
w
a
vef
o
r
m
s, a go
od con
t
ro
l of
the
D
C
-b
u
s
vo
ltage is ob
tain
ed
, an
d un
ity po
w
e
r f
actor
o
p
e
r
a
tio
n is
achi
e
ve
d.
2.
SHUNT
AP
F
CO
NFIG
U
R
A
TIO
N
2.1. Sys
t
em
Description
c
v
a
v
b
v
4
C
1
a
S
2
a
S
1
2
3
4
4
C
v
1
a
S
2
a
S
3
a
S
4
a
S
3
a
S
4
a
S
1
b
S
2
b
S
1
b
S
3
b
S
4
b
S
3
b
S
4
b
S
1
c
S
2
c
S
1
c
S
2
c
S
3
c
S
4
c
S
3
c
S
4
c
S
0
3
C
v
2
C
v
1
C
v
4
i
3
i
2
i
1
i
0
i
2
b
S
a
b
c
Fa
i
Fb
i
Fc
i
Fa
u
Fb
u
Fc
u
3
C
2
C
1
C
dc
v
P
CC
4
C
i
3
C
i
2
C
i
1
C
i
F
R
F
L
F
R
F
L
F
R
F
L
s
R
s
R
s
R
L
L
s
c
i
s
b
i
s
a
i
s
L
s
L
s
L
L
L
L
R
L
a
i
Lb
i
Lc
i
L
R
L
R
L
L
s
c
e
s
b
e
s
a
e
l
L
l
R
No
n
l
i
n
e
a
r
Lo
a
d
Fig
u
re
1
.
Fiv
e
-lev
el shun
t activ
e
p
o
wer filter
co
nfigu
r
ation
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
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S
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6
9
4
Application of
Backsteppi
ng to the
Virtual F
l
ux Direct P
o
w
e
r
Control
of F
i
ve-Level… (Bouzi
d
i M
a
nsour)
17
5
The st
r
u
ct
u
r
e i
n
Fi
g
u
re
1,
des
c
ri
bes t
h
e
pr
o
p
o
se
d
SAPF
bas
e
d on a three-phase fi
ve-le
v
el VSC. T
h
e
SAP
F
co
nsi
s
t
s
of t
h
ree pri
n
ci
pal
part
s, t
h
e t
h
ree
-
phase c
o
nverte
r, four capacitors
(
C
1
,
C
2
,
C
3
, and
C
4
) and t
h
e
sm
oot
hi
ng i
n
d
u
ct
ances
L
F
. T
h
e converte
r is use
d
to c
h
arge and t
o
d
i
sch
a
rg
e th
e cap
a
cito
rs to
p
r
ov
id
e t
h
e
requ
ired
co
m
p
en
sating
cu
rrent.
The ca
pacitors
are
use
d
to
st
ore
e
n
ergy and t
h
e inductances
L
F
are
use
d
t
o
sm
ooth and
decrease the
r
i
pp
les of
t
h
e
har
m
o
n
i
c cu
rr
en
ts in
j
ected
b
y
SAPF [9
].
Th
e m
a
in
tas
k
o
f
th
e SAPF is to
redu
ce h
a
rm
onic curre
nts and to ensure re
active power
co
m
p
en
satio
n.
Id
eally, th
e SAPF n
e
ed
s to
g
e
n
e
rate
j
u
st en
ou
gh
reactiv
e an
d
h
a
rm
o
n
i
c cu
rren
t to
co
m
p
en
sate
th
e non
lin
ear lo
ad h
a
rm
o
n
i
c in
th
e lin
e.
Th
e
resu
lting
to
tal
cu
rren
t
drawn
fro
m
th
e AC main
is sinu
so
idal.
2.
2. M
o
del
i
n
g
of
the
PW
M fi
ve-l
e
v
el
i
n
ver
t
er
The t
o
pol
ogy
o
f
t
h
e t
h
ree
-
p
h
a
s
e fi
ve-l
e
v
el
N
P
C
i
nvert
e
r
i
s
sho
w
n al
so i
n
Fi
gu
re 1.
Here
,
x
v
and
F
x
i
,
x
=
a
,
b
,
c
,
represen
t th
e po
in
t of co
mm
o
n
coup
lin
g
(PC
C
) vo
ltag
e
s and
AC sid
e
curren
ts, resp
ectively.
R
F
is a lin
e resist
an
ce th
at m
o
dels th
e p
a
rasitic resistiv
e effects o
f
t
h
e inducto
r
L
F
. T
h
e c
a
pacitances
of input
capacitors
are
assum
e
equal
C
1
=
C
2
=
C
3
=
C
4
=
C
. Fo
r a
net
dc-
s
i
d
e
vol
t
a
ge
of
v
dc
, each
cap
acito
r
voltag
e
is
id
eally
v
Cj
=
v
dc
/4
,
j
= 1,..
.,
4 and eac
h ge
ne
r
a
t
e
d p
h
ase v
o
l
t
a
ge
u
Fx
,
x
=
a
,
b
,
c
,
h
a
s fi
v
e
lev
e
ls with
respect to
dc-si
d
e re
fere
n
ce poi
nt
0.
Th
e swi
t
c
hi
n
g
st
a
t
es and t
h
e res
u
ltant phase
voltages are list
e
d in Ta
ble 1,
whe
r
e
state co
nd
itio
ns 1
and
0
i
n
d
i
cate ON and
OFF switch
statu
s
, resp
ectiv
ely.
Tabl
e
1.
Swi
t
c
hi
n
g
st
at
es
of
a
fi
ve
-l
evel
i
nve
rt
er
Switching
state
1
S
2
S
3
S
4
S
1
S
2
S
3
S
4
S
Phase
voltage
4
1 1
1 1
0 0
0
0
dc
v
3
0 1
1 1
1 0
0
0
3/
4
dc
v
2
0 0
1 1
1 1
0
0
/2
dc
v
1
0 0
0 1
1 1
1
0
/4
dc
v
0
0 0
0 0
1 1
1
1
0
The s
w
i
t
c
hi
n
g
fu
nct
i
o
ns
of
t
h
e fi
ve
l
e
vel
i
n
v
e
rt
er
of
Fi
g
u
re
1,
are e
xpress
e
d as
:
44
3
2
1
3
432
1
24
3
2
1
14
3
2
1
0
432
1
,
o
u
xx
x
x
x
xx
x
x
x
xx
x
x
x
xx
x
x
x
xx
x
x
x
FS
S
S
S
FS
S
S
S
FS
S
S
S
x
a
b
c
FS
S
S
S
FS
S
S
S
(1
)
Referri
ng all of the
voltages
to the
lower
DC-link voltage level
(“
0” refe
rence
)
,
each output voltage
c
onsis
t
s
of
co
nt
ri
b
u
t
i
o
n
s
by
a
det
e
rm
i
n
at
e num
ber
of
con
s
ecut
i
v
e
ca
paci
t
o
rs:
4
0
00
,
,
,
i
j
xx
j
C
ji
uF
v
x
a
b
c
(2)
Wh
en
b
a
lanced
d
i
stribu
tio
n of th
e DC
-lin
k vo
ltag
e
am
o
n
g
t
h
e cap
acito
rs i
s
assu
m
e
d
:
4
0
0
,
,
,
4
dc
xx
j
j
v
uj
F
x
a
b
c
(3)
Th
e lin
e to
lin
e vo
ltag
e
is
g
i
v
e
n
b
y
:
00
00
00
ab
a
b
bc
b
c
ca
c
a
uu
u
uu
u
uu
u
(4)
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I
S
SN
:
2
088
-86
94
I
J
PED
S
Vo
l. 4
,
No
. 2
,
Jun
e
2
014
:
17
3
–
19
1
17
6
The ex
pressio
n
s
of inst
an
tan
e
ous in
vert
er phas
e
o
u
t
p
u
t
v
o
lt
ages
are gi
v
e
n by
:
1
3
F
aa
b
c
a
F
bb
c
a
b
F
cc
a
b
c
uu
u
uu
u
uu
u
(5)
2.3. Mathematical
m
o
del of th
ree-phase
five levels
SAP
F
Th
e m
a
th
e
m
ati
cal eq
u
a
ti
o
n
s
wh
ich
g
o
v
e
rn th
e
b
e
h
a
v
i
our
of th
e ac-si
d
e
o
f
shu
n
t
activ
e
filter are:
1
()
1
()
1
()
Fa
F
aa
F
F
a
F
Fb
F
bb
F
F
b
F
Fc
F
cc
F
F
c
F
di
uv
R
i
dt
L
di
uv
R
i
dt
L
di
uv
R
i
dt
L
(6)
Whe
r
e,
,,
,
Fi
ui
a
b
c
represent voltage
s of
th
e SA
PF
.
By transform
i
n
g
(6) in statio
nary fram
es, it follows that:
1
()
1
()
F
FF
F
F
F
FF
F
F
di
uv
R
i
dt
L
di
uv
R
i
dt
L
(7)
Whe
r
e,
v
and
v
are the PCC
voltages in the stationary
α
-
β
co
or
dinates.
F
i
and
F
i
are
α
-
β
c
o
m
pone
nt
s
of
AC
c
u
rrents of
SAPF.
F
u
and
F
u
are
α
-
β
c
o
m
p
o
n
ents
o
f
AC
si
de voltage
s of SAP
F
.
The
DC
si
de
of the f
ilt
er ca
n be
expressed
as:
1
234
()
dc
CC
C
C
dv
d
vv
v
v
dt
dt
(8)
Equa
ti
on (
8
) c
a
n a
l
so
be
writte
n as:
12
3
4
1
()
dc
CC
C
C
dv
ii
i
i
dt
C
(9)
Whe
r
e
i
Cj
( j =1
,2
,3
,4
) is th
e
cu
rrent t
h
rough ca
pacitor
C
j
.
The eq
uatio
n of DC
side (
9
) can
be rel
a
ted to the A
C
side by
the followi
n
g
po
wer
-
bala
nce
relations
hip:
1
1
22
3
3
44
C
C
CC
C
C
CC
a
F
a
b
F
b
c
F
c
vi
v
i
v
i
v
i
v
i
v
i
v
i
(1
0)
If
we a
ssum
e
that the c
a
pacit
o
r
v
o
ltages a
r
e
balance
d
, t
h
e e
quatio
n
(
9
)
bec
o
m
e
s:
1
()
dc
aF
a
b
F
b
c
F
c
eq
d
c
dv
vi
v
i
v
i
dt
C
v
(11)
Whe
r
e:
/4
eq
CC
.
Equation (11)
can
be e
x
press
e
d as:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PEDS
I
S
SN:
208
8-8
6
9
4
Application of
Backsteppi
ng to the
Virtual F
l
ux Direct P
o
w
e
r
Control
of F
i
ve-Level… (Bouzi
d
i M
a
nsour)
17
7
dc
F
eq
dv
p
dt
C
(12)
Whe
r
e:
p
F
is the instantaneous
active power of SAPF.
F
FF
pv
i
v
i
(13)
3.
NO
NLINE
A
R
VIRT
UAL
FLUX
BASED
DIRE
CT
PO
WER
CO
NT
ROL
The
bac
k
step
p
i
ng
VF
DPC
-
S
V
M
c
ontr
o
l s
t
rategy
m
a
in schem
e
is pre
s
ented
in
Fig
u
re
2
.
T
h
e
no
nlinea
r loa
d
and
SAP
F
cu
rr
ents are se
nse
d
usin
g to
w cu
rr
ent sens
or
s loc
a
ted in p
h
ases
(a) a
nd
(b
), a
n
d the
esti
m
a
ted
PCC
virtual flu
x
co
m
ponents
,
P
CC
P
C
C
are
use
d
for the powers estim
ation (
p
L
,
q
L
and
p
F
,
q
F
)
and
bac
k
ste
ppi
ng
p
o
w
er
co
ntr
o
ller.
PC
C
c
v
b
v
a
v
F
re
f
u
F
re
f
u
L
q
PC
C
F
ab
c
i
ab
c
S
F
ab
c
i
F
p
F
q
F
i
L
ab
c
i
3
C
v
2
C
v
1
C
v
L
i
L
i
L
p
F
re
f
p
F
re
f
p
L
p
1
C
v
2
C
v
C
C
C
C
L
c
i
L
b
i
L
a
i
s
R
s
R
s
R
L
L
F
R
F
R
F
R
F
L
F
L
F
L
s
c
i
s
b
i
s
a
i
s
L
s
L
s
L
L
L
L
R
L
R
L
R
F
a
i
F
c
i
F
b
i
L
L
s
c
e
s
b
e
s
a
e
l
L
l
R
ab
c
4
C
v
3
C
v
0
Fre
f
q
ab
c
S
ab
c
dc
v
dc
r
e
f
v
PCC
Figu
re
2.
B
ack
steppi
ng
V
F
D
P
C
-
S
V
M
sc
he
m
e
of
five
-leve
l
SAP
F
The active
power
p
L
is delivered t
o
the
high pass filter
(HPF)
to
ob
tai
n
t
h
e a
lter
n
at
e v
a
lu
es, whi
c
h
final
l
y
are us
ed as
com
p
ensatin
g c
o
m
ponen
t
. T
h
e re
a
c
ti
ve
pow
er
q
L
can
be
de
li
vered
to
th
e
H
PF or
direc
tly
to
th
e in
pu
t of th
e
bac
k
ste
p
p
i
n
g
pow
er co
ntr
o
ll
er de
pen
d
i
ng o
n
c
o
m
p
ensati
o
n
requ
ir
em
ents
(co
m
pensat
ion of higher harm
onics or c
o
m
p
ensat
i
on
of hi
gher har
m
onics and r
eact
ive power at t
h
e
s
a
me
t
i
me
)
.
The re
fere
nce a
c
tive po
wer
p
Fr
e
f
(gene
r
ated
by the outer
nonlin
ear
DC vol
t
age controller) and
reactive powe
r
re
fere
nce
q
Fref
(set to zero for unity power
factor)
va
lues are c
o
m
p
ared
with esti
m
a
ted
instantaneous
p
F
and
q
F
va
lues au
gm
ented
by
their c
o
rre
s
p
onding al
ternate loa
d
powe
rs
p
L
an
d
q
L
,
respectively. T
h
e errors are delivere
d
to th
e backste
ppi
ng
po
wer co
ntr
o
l
l
er, whic
h elim
inates steady state
error. The
out
put signals from
back
stepping powe
r controller are us
ed
fo
r switchi
ng s
i
gnals ge
ne
ration
by
a
five-le
v
el s
p
ac
e vector m
o
dulator
[13].
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN:
2
088
-86
94
I
J
PEDS
Vo
l. 4
,
No
. 2
,
Jun
e
2
014
:
17
3
–
19
1
17
8
3.
1. PC
C vi
rt
ual
fl
ux
esti
m
a
t
o
r
The pri
n
ci
pl
e
of V
F
is based on ass
u
m
p
tio
n th
at th
e vol
ta
ges im
posed by
the li
n
e
pow
er in
com
b
inati
on
with the AC side i
n
duct
ors can be cons
id
ered as qu
ant
i
ties rela
te
d to
a virtua
l AC m
o
tor,
where
R
F
and
L
F
represent
the st
at
or resi
stanc
e
and l
e
a
k
ag
e inducta
n
ce of t
h
e virt
u
a
l
m
o
tor [17].
With
the defi
ni
ti
ons
:
PC
C
vdt
(14)
In
ge
neral
R
F
c
a
n
be
neglecte
d
a
n
d
the
v
o
ltage
v
ca
n
be
express
e
d as in st
ationary
α
-
c
o
or
dinates a
s
f
o
l
l
ows:
F
FF
F
FF
di
vu
L
dt
di
vu
L
dt
(15)
The i
n
tegr
ated
of
b
o
th
sides
o
f
(
1
5)
gi
ves:
P
CC
F
F
F
P
CC
F
F
F
ud
t
L
i
ud
t
L
i
(16)
The m
easure
d
line c
u
rrents
,
F
F
ii
and the
estim
a
t
ed
virt
u
a
l f
l
ux co
m
p
on
en
ts
,
P
CC
P
C
C
are
use
d
f
o
r S
A
PF
po
we
r estim
ation
.
3.
2. Ac
ti
ve an
d
reac
t
ive
powers estimator
Using the m
e
a
s
ure
d
curre
nt and the estim
a
t
ed P
CC virtual flux, the est
i
m
a
ted active
and re
active
po
we
rs ca
n
be
descri
bed
by
t
h
e f
o
llowi
ng
f
o
r
m
ulas [17]
:
a)
Active
filter powers are:
()
()
FP
C
C
F
P
C
C
F
F
P
CC
F
P
CC
F
pi
i
qi
i
(17)
b)
Nonlinear load powers are:
()
()
LP
C
C
L
P
C
C
L
LP
C
C
L
P
C
C
L
pi
i
qi
i
(
1
8
)
Whe
r
e:
ω
is
the angular frequency.
B
o
th
po
wer
es
tim
a
tion eq
uations
ar
e sim
p
le to calculate a
nd
d
o
not
req
u
i
re the c
o
m
putation
of th
e
current de
rivatives.
3.
3. Ac
ti
ve an
d
reac
t
i
v
e
p
o
w
er
model
ba
sed
o
n
vi
rtu
a
l
fl
ux
The bac
k
ste
ppi
ng
po
we
r co
ntroller is base
d
on in
st
antane
o
u
s p
o
w
er tim
e
deri
vative be
h
a
vio
r
. Fr
om
Eq
uation
(
1
7
)
,
the de
rivative
s
of
active a
n
d
r
eactive p
o
w
e
rs
are
give
n
by
:
FP
C
C
PC
C
F
F
F
P
CC
F
P
CC
PC
C
F
PC
C
F
F
F
P
CC
F
P
CC
di
d
dd
i
dp
ii
dt
dt
d
t
d
t
dt
dd
i
dd
i
dq
ii
dt
dt
d
t
d
t
dt
(1
9)
For three-phase balanced
syste
m
, the following re
lations
can be written:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PEDS
I
S
SN:
208
8-8
6
9
4
Application of
Backsteppi
ng to the
Virtual F
l
ux Direct P
o
w
e
r
Control
of F
i
ve-Level… (Bouzi
d
i M
a
nsour)
17
9
,
,
PC
C
P
CC
P
C
C
PC
C
PC
C
P
C
C
v
d
dt
d
v
dt
(2
0)
R
e
placing
(
7
)
,
(2
0)
into
(
1
9),
po
we
r
derivati
ves ca
n
be e
x
p
r
esse
d as:
PRC
F
F
F
F
PRC
F
F
PR
C
F
F
PRC
F
F
PR
C
F
FF
F
P
R
C
F
F
PRC
FF
P
R
C
F
F
dp
qR
i
u
dt
L
Ri
u
L
dq
pR
i
u
dt
L
Ri
u
L
(2
1)
3.4. B
a
cks
t
epping c
o
ntr
o
llers design
Backstepping i
s
a system
a
tic
and rec
u
rsi
v
e
design
m
e
thod
o
l
ogy
f
o
r n
o
n
lin
ear fee
d
back
c
ont
rol.
This
approach
has
e
m
erged as powerful tools
for stabilizi
ng
nonlinear system
s both fo
r tracking and
regulation
pu
r
poses
[
18]
.
The b
ackste
p
p
i
ng alg
o
r
ithm
takes ad
va
ntag
e
of the i
d
ea that certain
vari
ables can
be used a
s
virtual c
ontr
o
l
s
to m
a
ke the
ori
g
inal hi
gh
o
r
de
r sy
stem
sim
p
le, thus the final co
ntr
o
l o
u
tp
uts can
be
deri
ved
step by step t
h
rough suita
ble Lyap
unov functions ens
u
ri
ng
global stab
ility. This control m
e
thod
ha
s been
successfully applied on a gr
owing collection of
plant.
Howeve
r, fe
w
pa
pers a
r
e de
vot
ed to the bac
k
steppi
ng
cont
rol o
f
p
o
w
er electr
onic
s
con
v
e
r
ters [
19]
-
[
20]
. I
n
th
e follo
win
g
, t
h
e bac
k
step
pin
g
desi
gn
pr
oc
edu
r
e is
applied to fi
ve
-level shunt a
c
tive power
fi
lter. A
s
c
h
o
s
e
n
i
n
Fi
gu
re
2,
the c
o
ntr
o
l st
rategy
is
base
d
o
n
a
cascade str
u
ct
ure
,
nam
e
ly
, the
out
put
of
o
u
ter
volta
ge lo
op
is use
d
as
refe
rence
sig
n
a
l in the in
ne
r
p
o
we
r
loop.
The ap
pr
oac
h
ado
p
ted
herei
n
design
s by
br
eakin
g d
o
w
n
a com
p
lex nonl
inear sy
stem
into sm
aller
sub
-
sy
stem
s, then
desig
n
in
g
cont
rol Ly
ap
u
n
o
v
f
u
nctions
and
virtual co
n
t
rols f
o
r these
sub
-
sy
stem
s. In o
r
de
r
to de
sign the
cont
rol al
gorithm
fo
r a
c
tive
powe
r
filter
with the
aid
of
bac
k
steppi
ng m
e
thod,
nonlinear
di
ff
erential Equation (12) and
(
2
1
)
m
u
st be
p
o
rtio
ne
d
in three SISO
subsy
s
te
m
s
at the following
form
:
()
;
1
,
2
,
3
kk
kf
k
g
k
k
kk
k
Lh
L
h
u
yh
k
(2
2)
Whe
r
e
k
,
k
u
and
k
y
represe
n
t state,
co
ntr
o
l in
put
and
o
u
tp
ut
of
k
th
system
, respectively.
k
f
and
k
g
are
sm
ooth fields, and
k
h
is a s
m
ooth scalar function. The term
k
fk
Lh
stands f
o
r the
Lie derivative
of
k
h
with
respect t
o
k
f
, sim
i
larly
k
g
k
Lh
.
By identifying
the first subsys
tem
,
based
o
n
equatio
n
(
1
2
)
,
with
(2
2)
, it ca
n
be y
i
eld:
1
1
11
1
1
1
1
,
,
y
,
0
1
dc
F
d
c
f
g
eq
d
c
vu
p
h
v
L
h
Lh
Cv
(2
3)
The second and the third subsyste
m
s
are built ba
sed on
the active
and r
eactive derivative
(21).
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I
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I
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PEDS
Vo
l.
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,
No
.
2
,
Jun
e
2
014
:
17
3
–
19
1
18
0
The i
d
entification
of
(
21)
with (2
2)
leads to
:
2
2
3
22
22
2
2
33
33
3
33
,
,1
,
,1
FF
P
C
C
F
P
C
C
F
F
Fg
f
F
P
C
C
F
FP
C
C
P
C
C
F
FP
C
C
F
F
F
PC
C
F
PC
C
F
F
Fg
fF
P
C
C
F
F
P
C
C
P
C
C
F
F
F
pu
u
u
u
L
yh
p
L
h
Lh
q
R
i
R
i
L
qu
u
u
u
L
yh
q
L
h
Lh
p
R
i
R
i
L
PCC
(2
4)
In following sections the bac
k
steppi
ng
m
e
thod
will be used for
devel
opi
ng the
dc
voltage
and p
o
w
e
r
cont
rollers
.
3.
4.
1. D
C
Vol
t
age
C
o
n
t
r
o
l
l
er
Sy
nthe
si
s
In
or
der to e
n
sure that the S
A
PF
ope
rates e
ffectively it i
s
i
m
portant to
m
a
intain the
dc
capacit
o
r
voltage at a c
onsta
nt desire
d val
u
e. T
h
e
back
step
pin
g
dc
voltage controller sets the active power of the
inverter to regulate the
dc
v
o
ltage ba
sed
o
n
i
t
s refe
rence
v
a
lue c
ove
rin
g
t
h
e inve
rter l
o
sse
s.
The
purpose of this cont
rol
is to achieve t
h
e
dc
v
o
ltage
refe
rence
,
s
o
t
h
e fi
rst trac
king
er
ro
r is
defi
ned
as:
11
1
d
zy
y
(2
5)
Whe
r
e:
1
d
d
cr
ef
yv
Differentiating (25)
with
respect to tim
e, it i
s
obtained that:
11
11
1
1
f
gF
r
e
f
d
zL
h
L
h
p
y
(2
6)
The
ca
ndidate function of Lyap
uno
v is ch
o
s
en
as:
2
11
1
2
Vz
(2
7)
Th
e
d
e
r
i
v
a
tiv
e
o
f
th
e Lyap
unov
f
u
n
c
tion
is ex
pr
essed
as:
11
11
11
1
1
1
()
fg
F
r
e
f
d
Vz
z
z
L
h
L
h
p
y
(2
8)
To
guara
n
tee t
h
e Lya
p
unov s
t
ability
, the control law is chosen as:
1
1
11
1
1
1
f
d
Fr
ef
g
kz
L
h
y
p
Lh
(2
9)
Whe
r
e,
k
1
is a
positive
co
nsta
nt.
3.
4.
2.
Power Contr
o
ller
Synthesis
Using the
bac
k
steppi
ng a
p
proach,
one can
s
y
nthesize the c
ont
rol la
w forc
ing t
h
e active
and
reactive
p
o
wer
s
to fo
llow th
e d
e
sir
e
d po
wer
s
.
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PEDS
I
S
SN:
208
8-8
6
9
4
Application of
Backsteppi
ng to the
Virtual F
l
ux Direct P
o
w
e
r
Control
of F
i
ve-Level… (Bouzi
d
i M
a
nsour)
18
1
For the
fi
rst step the follow
ing tracki
n
g-errors are c
o
nside
r
e
d
:
22
2
33
3
d
d
zy
y
zy
y
(3
0)
Whe
r
e:
2_
3_
d
F
r
e
f
t
ot
al
d
F
r
e
f
t
ot
al
yp
yq
Whe
r
e:
_
_
F
r
e
f
t
ot
al
F
r
e
f
L
F
r
e
f
to
ta
l
F
re
f
pp
p
qq
(3
1)
The
res
u
lting e
r
r
o
r
dy
nam
i
cs
equatio
n
c
a
n
b
e
ex
pres
sed
as:
22
33
22
2
2
33
3
3
fg
F
r
e
f
d
fg
F
r
e
f
d
zL
h
L
h
u
y
zL
h
L
h
u
y
(3
2)
The c
h
osen
Ly
apu
n
o
v
f
u
nctio
n
s a
r
e
give
n
by
the
follo
win
g
exp
r
essi
ons:
2
22
2
33
1
2
1
2
Vz
Vz
(3
3)
Th
e tim
e
d
e
r
i
vativ
es of
Lyapu
nov
f
u
n
c
tion
s
23
an
d
VV
are give
n by
:
22
33
22
2
2
2
2
2
33
3
3
3
3
3
()
()
f
gF
r
e
f
d
f
gF
r
e
f
d
Vz
z
z
L
h
L
h
u
y
Vz
z
z
L
h
L
h
u
y
(3
4)
In
or
der to m
a
ke ne
gative the de
rivatives
of
Lyapunov functions, the interm
ediate
cont
rol laws
F
re
f
u
and
F
re
f
u
ar
e pr
opo
sed
in th
e
fo
llowing
eq
u
a
tion
:
2
2
3
3
22
2
2
2
33
3
3
3
f
d
Fr
e
f
g
f
d
Fr
e
f
g
kz
L
h
y
u
Lh
kz
L
h
y
u
Lh
(3
5)
Whe
r
e,
k
2
and
k
3
are positive constants.
The
relation be
tween t
h
e inte
rm
ediate
and
fi
nal co
ntr
o
l law
s
is gi
ven
by
:
F
ref
F
r
e
f
F
ref
F
r
e
f
uu
D
uu
(3
6)
Whe
r
e:
P
R
C
PRC
PRC
PRC
F
D
L
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I
S
SN:
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-86
94
I
J
PEDS
Vo
l.
4
,
No
.
2
,
Jun
e
2
014
:
17
3
–
19
1
18
2
The
D
m
a
trix determinant is
different to zero,
so
the
final
co
ntr
o
l laws a
r
e
give
n as:
1
F
re
f
F
re
f
F
re
f
F
re
f
uu
D
uu
(3
7)
3.
5.
Sp
ace
Vec
t
or
M
o
d
u
lati
o
n
w
i
th
DC
-
C
a
p
a
c
ito
r
Vo
ltages
Ba
lancing Stra
teg
y
In
the
five
-lev
el NPC
to
pol
o
g
y
,
t
h
e
voltag
e
s o
f
t
h
e
four se
ries-c
onnected
dc-l
ink ca
pacit
o
rs
m
u
st be
confine
d
t
o
v
dc
/4 to ta
ke a
d
v
a
ntage
o
f
the
inve
rter.
T
h
e
d
c
-v
oltage
bac
k
steppi
ng
co
ntr
o
l re
g
u
lates o
n
l
y
the
total dc voltage. For this
reason, th
e
dc-c
apa
c
itor v
o
ltages a
r
e ke
pt eq
uals
usin
g fi
ve-le
v
e
l
SVP
W
M
that
takes
adva
ntage
s
of r
e
du
n
d
ant switc
h
ing states to c
o
unteract t
h
e
dc
v
o
ltages
dri
f
t p
h
en
om
enon
[
21]
.
The five
-level
SVP
W
M
tec
hni
que ca
n ap
pr
o
x
im
at
e the
refere
nce vol
t
age vector, c
o
m
puted by
back
step
pin
g
p
o
we
r c
ont
rolle
r
,
usin
g the
ne
arest three
ve
c
t
ors. They a
r
e
selected to m
i
nim
i
ze the ene
r
gy
of
the dc
-ca
p
acito
r
voltages
[
2
1]
.
Figure 3 re
pre
s
ents the spac
e vector states for th
e fi
ve-le
v
el inverte
r
th
er
e are 125 s
w
itching-state
v
ector
s.
App
l
yin
g
Clark
’
s tran
sfor
m
a
t
i
o
n
to
all co
m
b
in
atio
n
s
o
f
ou
tpu
t
v
o
l
tag
e
s associated
with
the 125
switching-state vectors
res
u
lts in
60
nonzero voltage
s
p
ace vectors.
Projection of t
h
e vect
ors
on
αβ
co
or
dinates
fo
rm
s a four
-lay
er hexa
g
on
centere
d at the origi
n
o
f
th
e
αβ
plane
(Fig
u
r
e
3),
an
d ze
r
o
-
voltage
ve
ctor
s
are l
o
cated at t
h
e
ori
g
in
of the
plane
.
1
st
Se
c
t
or
1
1
f
g
F
re
f
u
m
o
n
l
j
k
1
s
u
a
b
c
d
e
h
i
1
1
F
re
f
u
1
2
F
re
f
u
2
s
u
400
440
410
420
430
411
300
42
1
31
0
431
320
441
330
422
311
200
43
2
321
210
442
331
220
433
322
211
100
443
332
221
110
000
111
222
333
444
343
232
121
010
344
233
122
011
434
323
212
101
334
223
112
001
342
242
231
120
131
020
243
132
021
244
133
022
42
3
312
201
234
123
012
424
313
202
32
4
21
3
102
224
113
002
341
241
141
030
130
230
142
031
143
032
144
033
134
023
41
2
301
413
302
41
4
30
3
124
013
314
203
214
103
114
003
041
340
240
140
04
0
04
2
043
044
03
4
024
401
402
40
3
404
304
204
104
004
014
2
nd
Sect
o
r
3
rd
Sect
or
4
th
S
ect
or
5
th
S
ect
o
r
6
th
S
ect
or
F
re
f
u
F
re
f
u
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
10
1
9
1
11
1
12
1
13
1
14
1
15
1
16
Figu
re
3.
S
p
ac
e v
o
ltage
vecto
r
s f
o
r a
five
-level in
verte
r
3.5.1. De
terminati
on of th
e Space
Vec
t
or Location
The s
p
ace
vect
or l
o
cation is determ
in
ed in t
w
o steps
.
T
h
e
first step
determ
ines the sect
or
num
b
er of
whe
r
e t
h
e
vector lies. T
h
e sec
o
nd ste
p
determ
ines
the triangle in
which
t
h
e vector lies
[22].
Step 1:
Secto
r
num
ber
c
o
m
p
u
t
ation.
The refe
ren
ce voltage
ve
ctor
m
a
gnitude
a
n
d its angle are
determ
ined
from
:
22
F
ref
F
ref
F
r
e
f
uu
u
and
1
ta
n
2
Fr
e
f
Fr
e
f
u
u
(3
8)
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