Inter
national
J
our
nal
of
P
o
wer
Electr
onics
and
Dri
v
e
Systems
(IJPEDS)
V
ol.
12,
No.
4,
December
2021,
pp.
2349
2357
ISSN:
2088-8694,
DOI:
10.11591/ijpeds.v12.i4.pp2349-2357
r
2349
Lo
w
switching
fr
equency
modulation
f
or
generalized
thr
ee-phase
multile
v
el
in
v
erters
gear
ed
to
ward
Grid
Codes
compliance
Mohammed
Setti,
Mohamed
Cherkaoui
Engineering
for
Smart
and
Sustainable
Systems
Research
Center
,
Mohammed
V
Uni
v
ersity
,
Rabat,
Morocco
Article
Inf
o
Article
history:
Recei
v
ed
Jun
26,
2021
Re
vised
Sep
21,
2021
Accepted
Sep
28,
2021
K
eyw
ords:
Nearest
le
v
el
control
Nonlinear
constrained
optimization
Po
wer
quality
grid
codes
Reduced
switch
Three-phase
cascaded
multile
v
el
in
v
erter
ABSTRA
CT
In
this
paper
,
a
generalized
three-phase
multile
v
el
po
wer
in
v
erter
(MLI)
structure
is
proposed
under
asymmetric
configurations.
The
operating
mode
and
the
switching
combinations
are
briefly
e
xposed
according
to
the
parity
of
the
number
of
direct
cur
-
rent
(DC)
v
oltage
sources
in
use.
Subsequently
,
the
proposed
topology
is
e
v
aluated
in
terms
of
commonl
y
used
f
actors
and
then
benchmark
ed
ag
ainst
some
of
the
state-of-
the-art
cascaded
MLIs
featuring
multiple
DC
v
ol
tage
sources
(MDCS-CMLIs)
while
putting
emphasis
on
the
reduction
of
po
wer
switching
de
vices.
Moreo
v
er
,
a
ne
w
near
-
est
le
v
el
control
(NLC)-based
modulation
technique
is
designed
for
the
purpose
of
better
comply
with
some
quali
ty
grid
codes,
namely
the
European
EN
50160
and
the
International
IEC
61000-2-12.
The
identification
of
the
optimal
control
thresholds
is
realized
by
a
cons
trained
optimization
algorithm
(
e
.g
.
,
particle
sw
arm
optimization
(PSO))
which
is
implemented
in
p
ython
script
and
v
alidated
through
SIMULINK
f
ast
fourier
transform
(FFT)
analysis
tool.
Lastly
,
the
harmonic
performance
of
the
pro-
posed
technique
is
compared
side-by-si
de
with
that
of
the
c
on
v
ent
ional
NLC
scheme
and
e
xhibits
significant
reduction
in
harmonic
distortion.
This
is
an
open
access
article
under
the
CC
BY
-SA
license
.
Corresponding
A
uthor:
Mohammed
Setti
Mohammadia
School
of
Engineers
Ibn-Sina
A
v
enue,
765,
Rabat,
Morocco
Email:
mohammedsetti@research.emi.ac.ma
1.
INTR
ODUCTION
In
medium-v
oltage
(MV)
and
medium-to
high-po
wer
le
v
el
con
v
ersion
field,
multile
v
el
in
v
erters
(MLIs)
are
considered
to
be
the
flagship
solution
for
grid-related
applications
including
rene
w
able
ener
gy
sources
in-
te
gration,
high
v
oltage
DC
transmission/fle
xible
A
C
transmission
system
(HVDC/F
A
CTS),
shunt
acti
v
e
po
wer
filter
(APF),
and
static
synchronous
compensator
(ST
A
TCOM).
Moreo
v
er
,
the
y
are
reno
wned
as
one
of
the
most
ef
fecti
v
e
w
ay
to
synthesize
high
resolution
A
C
po
wer
endo
wed
with
attracti
v
e
features
such
as
lo
w
harmonic
distortion,
high
po
wer
handling,
and
best
po
wer
con
v
ersion
ef
ficienc
y
to
name
a
fe
w
[1].
Despite
all
of
the
aforementioned
benefits,
MLI
topologi
es
require
a
relati
v
ely
lar
ge
number
of
po
wer
semiconductor
and
passi
v
e
components.
T
o
address
this
issue,
a
lot
of
research
are
led
to
w
ard
the
reduction
of
the
po
wer
components
count
so
as
to
enhance
both
the
o
v
erall
po
wer
ef
ficienc
y
and
the
system
reliability
[2]-[4].
In
practice,
cascaded
MLI
structures
with
multiple
DC
sources
(MDCS-CMLIs)
are
re
g
arded
to
be
the
most
authoritati
v
e
MLIs
since
t
he
y
ha
v
e
been
emplo
yed
in
some
rene
w
able
ener
gy
systems
for
more
than
a
decade.
The
y
can
be
set
either
in
symmetric
or
in
asymmetric
configuration
[1],
[5].
Asymmetric
nature
of
DC
J
ournal
homepage:
http://ijpeds.iaescor
e
.com
Evaluation Warning : The document was created with Spire.PDF for Python.
2350
r
ISSN:
2088-8694
sources
w
ould
bring
more
le
v
els
to
the
output
w
a
v
eforms
which
positi
v
ely
impact
their
harmonic
performance.
F
or
this
reason,
only
asymmetric
structures
are
considered
throughout
this
paper
.
Lo
w
switching
frequenc
y
modulations
(LSFM)
found
their
w
ay
in
the
field
of
MLI
structures
with
the
main
objecti
v
e
of
alle
viating
the
switching
losses
incurred
by
the
relati
v
e
high
number
of
po
wer
switches
[6],
[7].
No
w
adays,
there
is
only
one
well-established
LSFM
scheme
geared
to
w
ard
grid-tied
applications
kno
wn
as
selecti
v
e
harmonic
mitig
ation
(SHM)
which
in
turn
is
akin
to
the
selecti
v
e
harmonic
elimination
(SHE)
modulation
t
echnique
[8]-[11].
Ho
we
v
er
,
SHE/SHM
has
some
limitations
that
could
be
o
v
ercome
by
NLC
[5],
[12],
especially
those
related
to
dynamic
performance
under
real-time
operation
and
computational
ef
fort
required
while
using
lookup
tables
or
curv
e
fitting
polynomials
or
an
y
initial
guessing
and
iterati
v
e
routines
[13],
[14].
The
ne
xt
section
proposes
a
generalized
three-phase
MLI
topology
under
dif
ferent
parity
of
k
DC
v
oltage
sources.
T
o
e
v
aluate
performance,
the
in
v
erter
structure
is
analyzed
in
terms
of
some
commonly
used
f
actors
and
compared
to
other
state-of-the-art
topologies.
Afterw
ards,
in
section
3,
a
ne
w
NLC-based
mod-
ulation
is
de
v
eloped
for
the
purpose
of
better
meeting
the
grid
quality
requirements
of
both
EN
50160
and
IEC
61000-2-12
codes.
In
order
to
achie
v
e
that,
some
mathematical
formulations
based
on
fourier
series
e
x-
pansion
are
shortly
e
xposed.
Ne
xt,
the
formulation
of
the
optimization
problem
is
gi
v
en
as
a
system
of
nonlinear
inequalities
considering
each
indi
vidual
harmonic
limit
L
h
and
the
total
harmonic
distortion
(THD)
of
a
gi
v
en
line-to-line
output
v
oltage
v
`
(
!
t
)
.
Then,
the
particle
sw
arm
optimization
(PSO)
algorithm
[15]
is
called
to
determine
the
appropriate
switching
angles
i
depending
on
the
v
oltage
ratio
r
and
the
thresholds
i
.
Finally
,
the
harmonic
performance
of
the
proposed
control
tec
hn
i
que
is
benchmark
ed
side-by-side
ag
ainst
that
of
the
con
v
entional
NLC
scheme.
Results
and
dis
cussion
are
carried
out
while
rele
v
ant
conclusions
are
highlighted
at
the
end
of
this
paper
.
2.
ASSESSMENT
OF
THE
PR
OPOSED
MLI
T
OPOLOGY
2.1.
Basic
unit
cell
MDCS-CMLI
structur
e
The
proposed
three-phase
MLI
topology
is
depicted
by
Figure
1
where
the
basic
unit
cell
is
shaded
in
light
color
.
As
it
can
be
seen,
each
single-phase
cell
is
made
up
from
tw
o
v
oltage
sources,
four
unidirectional
(
S
1
;
1
,
S
1
;
2
,
S
3
;
1
,
S
3
;
2
)
and
tw
o
bidirectional
po
wer
switches
(
S
2
;
1
,
S
2
;
2
).
Se
v
eral
unit
cells
can
be
strung
to-
gether
in
series
so
as
to
synthesize
high
resolution
output
v
oltages.
A
comparati
v
e
study
with
detailed
operating
modes
for
both
5-le
v
el
symmetric
and
7-le
v
el
asymmetric
configurations
has
already
done
in
[16],
so
it
will
not
be
discussed
further
e
xcept
for
some
reminiscent
results
required
when
designing
the
generalized
structure.
Figure
1.
Proposed
generalized
three-phase
multile
v
el
po
wer
in
v
erter
topology
.
Basic
unit
cell
is
shaded
in
which
bidirectional
switches
are
illustrated
by
tw
o
back-to-back
N-channel
po
wer
MOSFETs
The
focus
of
the
present
w
ork
w
as
mainly
dri
v
en
to
w
ard
grid-connected
MLIs
for
rene
w
able
ener
gy
inte
gration.
Asymmetric
nature
of
the
discrete
DC
sources
w
ould
be
of
great
benefit
in
enhancing
po
wer
quality
required
by
the
grid
code.
In
each
cell,
a
binary
v
oltage
ratio
w
as
used
(
i.e
.
,
r
=
V
2
=V
1
=
2
)
and
for
the
sak
e
of
ef
ficienc
y
,
bidirectional
switches
were
made
of
tw
o
back-to-back
N-channel
po
wer
MOSFETs.
T
w
o
aspects
of
importance
should
be
at
the
forefront
when
designing
MLI
topologies:
total
standing
v
oltage
(TSV)
and
total
acti
v
e
switches
(T
AS).
The
former
is
defined
as
the
sum
of
the
highest
v
oltage
stress
Int
J
Po
w
Elec
&
Dri
Syst,
V
ol.
12,
No.
4,
December
2021
:
2349
–
2357
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
r
2351
across
each
po
wer
switch
while
the
latter
is
the
total
number
of
conducting
MOSFETs
o
v
er
one
time
period:
TSV
=
4(
V
1
+
V
2
)
+
2
V
2
=
16
V
1
T
AS
=
32
MOSFETs
:
(1)
2.2.
Extended
MDCS-CMLI
structur
e
The
basic
unit
cell
within
the
generalized
structure
of
Figure
1
can
be
e
xtended
in
tw
o
dif
ferent
w
ays
according
to
the
parity
of
the
k
number
of
DC
v
oltage
sources.
F
or
each
phase,
e
v
ery
unit
cell
is
connected
together
in
series
so
as
to
generate
high-resolution
output
w
a
v
eforms
by
virtue
of
arithmetic
combinations
of
the
DC
po
wer
sources
where
their
p.u.
magnitudes
V
k
feature
septenary
configurations
as
follo
ws:
8
m
2
N
;
V
k
=V
1
=
(
7
(
k
1)
=
2
;
k
=
2
m
1
2
(7
k
=
2
1
)
;
k
=
2
m;
(2)
where
m
represents
the
number
of
basic
unit
cells
being
connected.
In
order
to
b
uild
up
all
the
a
v
ailable
n
le
v
els
and
guarantee
e
v
ery
time
that
an
y
of
the
2
m
DC
v
oltage
sources
will
ne
v
er
be
short-circuited,
then
the
follo
wing
general
rule
should
be
applied:
S
i;j
=
f
0
;
1
g
2
m
Y
j
=1
3
X
i
=1
S
i;j
=
1
;
(3)
S
i;j
correspond
to
the
enhancement-mode
NMOS,
so
‘0’
and
‘1’
denote,
respecti
v
ely
,
the
OFF-
and
ON-state.
In
addition,
the
a
v
ailable
switch
combinations
for
the
generalized
single-phase
MLI
are
tab
ulated
in
T
able
1.
T
able
1.
Switching
combinations
of
the
generalized
single-phase
MLI
under
asymmetric
operation
P
E
R
-
U
N
I
T
O
U
T
P
U
T
V
O
L
T
A
G
E
L
E
V
E
L
S
(
V
k
=V
1
)
1
7
m
2
24
10
3
2
1
0
+1
+2
+3
+10
+24
7
m
1
2
S
1
;
1
ON
ON
ON
ON
OFF
ON
ON
OFF
OFF
OFF
OFF
OFF
OFF
S
1
;
2
OFF
OFF
OFF
OFF
OFF
OFF
ON
ON
OFF
ON
ON
ON
ON
S
1
;
3
ON
ON
ON
ON
ON
ON
ON
ON
ON
ON
OFF
OFF
OFF
S
1
;
4
OFF
OFF
OFF
ON
ON
ON
ON
ON
ON
ON
ON
ON
ON
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
S
1
;
2
m
1
ON
ON
ON
ON
ON
ON
ON
ON
ON
ON
ON
ON
OFF
S
1
;
2
m
OFF
ON
ON
ON
ON
ON
ON
ON
ON
ON
ON
ON
ON
S
2
;
1
OFF
OFF
OFF
OFF
ON
OFF
OFF
ON
OFF
OFF
OFF
OFF
OFF
S
2
;
2
OFF
OFF
OFF
OFF
OFF
ON
OFF
OFF
ON
OFF
OFF
OFF
OFF
S
2
;
3
OFF
OFF
OFF
OFF
OFF
OFF
OFF
OFF
OFF
OFF
ON
OFF
OFF
S
2
;
4
OFF
OFF
ON
OFF
OFF
OFF
OFF
OFF
OFF
OFF
OFF
OFF
OFF
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
S
2
;
2
m
1
OFF
OFF
OFF
OFF
OFF
OFF
OFF
OFF
OFF
OFF
OFF
OFF
OFF
S
2
;
2
m
OFF
OFF
OFF
OFF
OFF
OFF
OFF
OFF
OFF
OFF
OFF
OFF
OFF
S
3
;
1
OFF
OFF
OFF
OFF
OFF
OFF
OFF
OFF
ON
ON
ON
ON
ON
S
3
;
2
ON
ON
ON
ON
ON
OFF
OFF
OFF
OFF
OFF
OFF
OFF
OFF
S
3
;
3
OFF
OFF
OFF
OFF
OFF
OFF
OFF
OFF
OFF
OFF
OFF
ON
ON
S
3
;
4
ON
ON
OFF
OFF
OFF
OFF
OFF
OFF
OFF
OFF
OFF
OFF
OFF
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
S
3
;
2
m
1
OFF
OFF
OFF
OFF
OFF
OFF
OFF
OFF
OFF
OFF
OFF
OFF
ON
S
3
;
2
m
ON
OFF
OFF
OFF
OFF
OFF
OFF
OFF
OFF
OFF
OFF
OFF
OFF
Depending
on
the
parity
of
k
number
of
DC
v
oltage
sources,
the
equations
of
the
number
of
switches
(
N
sw
),
g
ate
dri
v
ers
(
N
g
d
)
and
total
standing
v
oltage
(
TSV
)
for
the
proposed
n
-le
v
el
po
wer
in
v
erter
are
gi
v
en
Low
switc
hing
fr
equency
modulation
for
g
ener
alized
thr
ee-phase
multile
vel
in
verter
s...
(Mohammed
Setti)
Evaluation Warning : The document was created with Spire.PDF for Python.
2352
r
ISSN:
2088-8694
as
sho
wn
in:
8
m
2
N
j
k
=
2
m;
8
m
2
N
j
k
=
2
m
1
;
n
=
7
k
=
2
N
sw
=
8
log
7
(
n
)
N
g
d
=
6
log
7
(
n
)
TSV
=
P
k
i
=1
16
V
i
=
3
=
8(
n
1)
V
1
=
3
:
n
=
3
(7
(
k
1)
=
2
)
N
sw
=
8
log
7
(
n=
3)
+
4
N
g
d
=
6
log
7
(
n=
3)
+
4
TSV
=
P
k
1
i
=1
16
V
i
=
3
+
4
V
k
=
(20
n
24)
V
1
=
9
:
(4)
It
is
w
orth
noting
that
the
use
of
the
last
tw
o
bidirectional
po
wer
switches
(
i.e
.
,
S
2
;
2
m
1
and
S
2
;
2
m
,
m
2
)
is
not
justified
when
k
is
odd;
only
unidirectional
are
needed.
F
u
r
thermore,
and
in
order
to
estimate
the
o
v
erall
cost-per
-le
v
el
in
v
olv
ed
in
creating
an
n
-le
v
el
in
v
erter
structure,
the
cost
function
$(
n;
)
is
tak
en
into
consideration
with
the
assumption
that
all
the
switches
in
use
share
the
same
po
wer
rating
[17],
[18].
Depending
on
its
v
alues
(greater
or
less
than
unity),
the
weight
coef
ficient
is
used
to
emphasize
either
the
TSV
pu
or
the
total
number
of
required
electric
components:
$(
n;
)
=
(
N
sw
+
N
g
d
+
[
N
c
+
N
d
]
+
TSV
pu
)
n
1
k
;
(5)
where
the
per
-unit
TSV
is
defined
as:
TSV
pu
=
TSV
=
P
k
i
=1
V
i
.
Ho
we
v
er
,
topologies
with
po
wer
diodes
N
d
and/or
electrolytic
capacitors
N
c
are
not
considered
here
since
the
y
are
generally
lossy
and/or
b
ulk
y
.
2.3.
Comparati
v
e
study
with
r
ecent
MDCS-CMLIs
A
comparati
v
e
study
of
some
cutting-edge
MDCS-CMLIs
is
performed
in
order
to
sho
w
the
m
erits
of
the
proposed
structure
[19]-[22].
T
ypically
,
the
n
is
limited
to
50
le
v
els,
which
w
ould
be
f
ar
enough
for
an
y
grid-tied
applica
tions
tar
geting
either
single-
or
three-phase
in
v
erters.
As
can
be
seen
from
Figure
2,
the
proposed
topology
requires
the
least
number
of
k
(
n
)
DC
sources
with
moderate
N
sw
(
n
)
,
N
g
d
(
n
)
and
p.u.
TSV
(
n
)
,
resulting
in
the
lo
west
cost
function
$(
n;
)
for
both
cases
(
i.e
.
,
1
and
>
1
)
re
g
ardless
the
parity
of
k
.
Figure
2.
Comparison
of
the
proposed
asymmetric
topology
to
some
state-of-the-art
MDCS-CMLIs
3.
PR
OPOSED
CONTR
OL
STRA
TEGIES
FOR
GRID-TIED
MLI
T
OPOLOGIES
3.1.
Mathematical
f
ormulation
of
the
pr
oposed
MLI
output
v
oltage
wa
v
ef
orm
The
output
v
oltage
w
a
v
eform
of
the
proposed
in
v
erter
dri
v
en
by
an
NLC
modulation
is
similar
to
a
staircase
sine
w
a
v
e
function
with
equal
riser
(
r
=
2
),
and
because
of
its
odd
quarter
-w
a
v
e
symmetry
property
,
the
general
form
of
the
F
ourier
coef
ficient
b
h
is
simplified
as
sho
wn
in
with
a
h
=
b
2
h
=
0
for
all
harmonics
h
:
b
h
=
4
V
1
Z
2
1
sin
h
d
+
Z
3
2
r
sin
h
d
+
Z
4
3
(
r
+
1)
sin
h
d
+
+
Z
=
2
n
1
2
(
r
+
1)
log
7
(
n
)
X
m
=2
(7
m
1
+
1)
sin
h
d
(6)
Int
J
Po
w
Elec
&
Dri
Syst,
V
ol.
12,
No.
4,
December
2021
:
2349
–
2357
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
r
2353
The
fourier
series
of
the
phase
v
oltage
v
(
!
t
)
can
then
be
written
as
a
function
of
the
v
oltage
ratio
r
and
the
switching
angles
i
:
v
(
!
t
)
=
1
X
h
=1
b
h
sin
h!
t
=
4
V
1
h
1
X
h
=1
cos
h
1
+
(
r
1)
cos
h
2
+
cos
h
3
+
(5
2
r
)
cos
h
4
+
cos
h
5
+
(
r
1)
cos
h
6
+
+
cos
h
(
n
1)
=
2
sin
h!
t
(
h
:
odd
harmonics
)
(7)
where
i
is
defined
by
(8),
taking
into
account
the
number
of
a
v
ailable
le
v
els
n
and
the
thresholds
i
.
The
introduction
of
the
latter
paramet
ers
allo
ws
to
distinguish
tw
o
operating
modes
re
g
arding
the
symmetry
or
asymmetry
of
the
thresholds
in
which
the
magnitudes
of
i
are,
respecti
v
ely
,
equal
or
not
necessarily
equal.
Thus,
con
v
entional
NLC
modulation
can
be
re-established
simply
by
setting
all
symmetrical
thresholds
to
1.
8
i
2
f
1
;
2
;
:
:
:
;
(
n
1)
=
2
g
9
2
R
(
n
1)
=
2
+
:
0
1
2
;
:
:
:
;
i
=
arcsin
2
i
(
i
0
:
5)
n
1
2
(8)
The
generalized
form
of
the
modulation
inde
x
m
a
(
r
;
)
is
gi
v
en
is
being
as:
m
a
(
r
;
)
,
^
v
;
1
max(
^
v
;
1
)
=
^
v
;
1
(
r
;
)
2(
n
+
2
r
5)
V
1
(9)
where
^
v
;
1
(
r
;
)
and
max(
^
v
;
1
)
represent,
respecti
v
ely
,
the
amplitude
of
the
fundamental
component
of
v
(
!
t
)
and
its
maximum
by
which
all
i
are
zeroed
out.
3.2.
Constrained
optimization
pr
oblem
f
or
grid-tied
applications
Po
wer
quality
grid
codes
usually
specify
an
upper
limit
L
h
for
each
specific
h
a
rmonic
h
and
indicate
the
maximum
accepted
distortion
which,
con
v
entionally
,
considers
all
harmonics
up
to
the
40th.
F
or
instance,
the
signal
quality
requirements
of
both
the
European
and
the
International
grid
codes,
namely
EN
50160
and
IEC
61000-2-12,
are
tab
ulated
in
T
able
2,
where
the
limited
v
oltage
THD
is
set
to
8%.
It
is
w
orth
noting
that
the
maximum
harmonic
orders
required
by
EN
50160
and
IEC
61000-2-12
are
25th
and
50th,
respecti
v
ely
[23],
[24].
T
able
2.
Indi
vidual
harmonic
limits
according
to
codes
EN
50160
and
IEC
61000-2-12
in
MV
netw
orks
EN
50160
(
h
25)
O
D
D
H
A
R
M
O
N
I
C
S
(
Non-multiple
of
3
)
O
D
D
H
A
R
M
O
N
I
C
S
(
Multiple
of
3
)
E
V
E
N
H
A
R
M
O
N
I
C
S
h
L
h
(%)
h
L
h
(%)
h
L
h
(%)
5
6
3
5
2
2
7
5
9
1.5
4
1
11
3.5
15
0.5
6
0.5
13
3
21
0.5
17
2
19
1.5
IEC
61000-2-12
(
h
50)
O
D
D
H
A
R
M
O
N
I
C
S
(
Non-multiple
of
3
)
O
D
D
H
A
R
M
O
N
I
C
S
(
Multiple
of
3
)
E
V
E
N
H
A
R
M
O
N
I
C
S
h
L
h
(%)
h
L
h
(%)
h
L
h
(%)
5
6
3
5
2
2
7
5
9
1.5
4
1
11
3.5
15
0.4
6
0.5
13
3
21
0.3
8
0.5
17
2
27
0.2
10
0.5
19
38.59
=h
0.27
12
2.5
=h
+
0.25
T
o
seek
t
he
appropriate
v
oltage
ratio
r
and
all
the
thresholds
i
(
i.e
.
,
the
switching
angles
i
)
that
w
ould
ha
v
e
compliance
with
the
aforementioned
quality
grid
re
gulations,
a
constrained
o
pt
imization
problem
whose
cost
function
CF(
r
;
)
is
formulated
is
being
as:
CF(
r
;
)
CF(
r
;
1
;
2
;
3
;
:
:
:
;
(
n
1)
=
2
)
=
min
r
;
THD
40
v
`
(
!
t
)
s:t:
8
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
:
"
f
^
v
`;
1
(
r
;
)
2
p
3
V
1
(
n
+
2
r
5)
m
a
(
r
;
)
(1
)
L
h
4
p
3
V
1
h
^
v
`;
1
(
r
;
)
cos
h
1
+
(
r
1)
cos
h
2
+
cos
h
3
+
+
cos
h
(
n
1)
=
2
8%
THD
40
v
`
(
!
t
)
=
v
u
u
t
37
X
h
=5
;
7
;:::
^
v
2
`;h
(
r
;
)
^
v
2
`;
1
(
r
;
)
(10)
Low
switc
hing
fr
equency
modulation
for
g
ener
alized
thr
ee-phase
multile
vel
in
verter
s...
(Mohammed
Setti)
Evaluation Warning : The document was created with Spire.PDF for Python.
2354
r
ISSN:
2088-8694
where
is
a
safety
mar
gin;
v
`
and
v
`;h
are,
respecti
v
ely
,
the
line-to-line
output
v
oltage
and
its
specific
non-
triplen
odd
harmonics
under
three-phase
balanced
condition.
The
y
are
interrelated
and
can
be
written
as
sho
wn
in:
v
`
(
!
t
)
=
p
3
1
X
h
=1
b
h
cos(
h!
t
+
'
h
)
=
1
X
h
=1
v
`;h
(
r
;
)
(
h
:
non-triplen
odd
harmonics
)
(11)
3.3.
Assessment
and
implementation
of
the
pr
oposed
modulation
technique
MLIs
with
higher
number
of
le
v
els
are
more
lik
ely
to
meet
grid
quality
requirements.
Ho
we
v
er
,
the
y
significantly
suf
fer
from
lo
w
po
wer
ef
ficienc
y
,
b
ulk
y
footprint,
high
cost
and
comple
x
design
and
control.
F
or
instance,
the
proposed
in
v
erter
structure
with
21
l
e
v
el
s
may
be
emplo
yed,
b
ut
instead
it
w
ould
be
more
advisable
to
use
that
with
7
le
v
els
tuned
by
a
small
passi
v
e
filter
since
the
number
of
po
wer
switches
N
sw
(
n
)
,
DC
sources
k
(
n
)
and
TSV
(
n
)
are
reduced
by
f
actors
of
1.75,
2,
and
2.75,
respecti
v
ely
.
Figure
3
depicts
the
inte
gration
of
the
proposed
7-le
v
el
po
wer
in
v
erter
into
the
medium-v
oltage
(MV)
po
wer
distrib
ution
system.
P
S
f
r
a
g
r
e
p
l
a
c
e
m
e
n
t
s
C
f
L
f
L
f
L
g
P
O
W
E
R
I
N
V
E
R
T
E
R
L
C
L
-
F
I
L
T
E
R
P
C
C
L
O
A
D
(
Z
`
)
L
I
N
E
F
R
E
Q
.
T
R
A
N
S
F
O
.
S
W
G
R
I
D
(
2
0
/
2
2
k
V
)
Figure
3.
Inte
gration
of
the
proposed
7-le
v
el
po
wer
in
v
erter
into
medium-v
oltage
po
wer
distrib
ution
netw
orks
Henceforth,
all
the
abo
v
e
generalized
equations
are
called
to
deal
with
only
one
symmetrical
threshold
under
-NLC
S
modulation
or
with
three
asymmetrical
thresholds
under
-NLC
A
.
The
domain
of
each
of
these
thresholds
depend
upon
the
definition
gi
v
en
by
(8),
where
in
the
first
case
2
[0
;
1
:
2]
and
in
the
second
case
1
2
[0
;
6]
,
2
2
[0
;
2]
and
3
2
[0
;
1
:
2]
such
that
1
3
2
5
3
.
The
major
hindrance
of
solving
the
constrained
optimization
problem
arises
from
the
challenge
of
e
xploring
a
wide
search
space
in
order
to
locate
feasible
solutions
f
i
;
r
g
that
gi
v
e
best
harmonic
profiles
of
the
output
v
oltage.
So,
with
the
intention
to
enhance
the
solutions
quali
ty
and
speed
up
the
processing
time
of
the
used
e
v
olutionary
algorithm,
only
potential
feasible
solutions
with
THD
40
8%
are
k
ept
and
then
plotted
in
Figures
4
(a)
and
(b)
for
both
-
NLC
S
and
-NLC
A
modulation
strate
gies,
respecti
v
ely
.
Moreo
v
er
,
it
can
be
seen
in
Figure
4
(c)
that
MLIs
dri
v
en
by
con
v
entional
NLC
(
=
1
)
are
out
of
the
allo
wed
range
and
those
with
binary
configuration
(
r
=
2
)
e
xhibit
the
lo
west
THD.
The
last
result
has
already
been
demonstrated
in
[25].
Figure
4.
Potential
feasible
solutions
to
the
constrained
optimization
problem
under:
(a)
-NLC
S
and
(b)
-NLC
A
(with
r
=
2
)
modulations.
(c)
Best
harmonic
performance
with
respect
to
v
oltage
ratio
r
Int
J
Po
w
Elec
&
Dri
Syst,
V
ol.
12,
No.
4,
December
2021
:
2349
–
2357
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
r
2355
In
consequence,
the
binary
configuration
is
chosen
which
additionally
narro
w
do
wn
the
search
space.
The
best
possible
solutions
f
i
g
to
the
non-linear
constrained
optimization
problem
of
(11)
are
determined
using
an
implementation
of
PSO
in
Python.
The
v
alidation
is
done
by
the
algorithm
of
Figure
5
under
SIMULINK.
Figure
6
plots
the
modulating
signals,
phase-
and
line-output
v
oltages
under
con
v
entional,
sym-
metrical
and
asymmetrical
NLC
modulations,
whereas
Figure
7
indicates,
side-by-side,
the
harmonic
contents
of
the
respecti
v
e
line-output
v
oltages
along
with
the
limits
imposed
by
European
and
International
Grid
Codes.
Noticeable
enhancements
in
terms
of
harmonic
distortion
are
directly
obtained
by
the
proposed
-NLC
S
and
-NLC
A
modulations,
particularl
y
the
THD
reductions
are
of
about
34%
and
43%
with
respect
to
the
con
v
en-
tional
NLC
when
tar
geti
ng
full
EN
50160
compliance
with
10%
of
safety
mar
gin,
and
about
30%
and
41%
when
tar
geting
IEC
61000-2-12
Grid
Code.
It
is
w
orth
to
mention
that
all
the
lo
wer
order
harmonics
are
can-
celed
out
whereas
fe
w
of
those
with
higher
order
can
be
mitig
ated
by
introducing
small
lo
w-pass
filters.
These
results
are
summarized
within
T
able
3
in
which
best
feasible
solutions
are
recorded.
Figure
5.
NLC-based
algorithm
for
the
implementation
of
the
proposed
7-le
v
el
modulation
schemes
Figure
6.
Modulating
signals,
phase-and
line-output
v
oltages
using
NLC,
-NLC
S
and
-NLC
A
modulations
Figure
7.
Harmonic
contents
of
the
line-output
v
oltages
using
NLC,
-NLC
S
and
-NLC
A
modulations
Low
switc
hing
fr
equency
modulation
for
g
ener
alized
thr
ee-phase
multile
vel
in
verter
s...
(Mohammed
Setti)
Evaluation Warning : The document was created with Spire.PDF for Python.
2356
r
ISSN:
2088-8694
T
able
3.
Salient
results
of
the
proposed
modulation
schemes
under
some
best
feasible
solutions
M
O
D
U
L
A
T
I
O
N
T
H
R
E
S
H
O
L
D
S
T
H
D
G
R
I
D
C
O
D
E
C
O
M
P
L
I
A
N
C
E
N
O
T
E
S
S
C
H
E
M
E
1
2
3
(%)
EN
50160
IEC
61000-2-12
(
Remaining
harmonics
)
Con
v
.
NLC
1.00
1.00
1.00
8.81
No
No
13th–19th,
25th,
29th,
35th–41st
-NLC
S
0.55
0.55
0.55
5.83
Y
es
y
N/A
y
Safety
mar
gin
=
10%
0.52
0.52
0.52
6.17
Y
es
No
31st,
35th
and
37th
-NLC
A
0.61
0.56
0.68
5.01
Y
es
y
N/A
y
Safety
mar
gin
=
10%
0.52
0.55
0.68
5.15
Y
es
No
35th
and
37th
4.
CONCLUSION
In
this
article,
a
generalized
three-phase
cascaded
mult
ile
v
el
in
v
erter
topology
with
reduced
po
wer
switching
de
vices
w
as
presented.
The
asymmetric
nature
of
its
odd/e
v
en
DC
po
wer
sources
featuring
a
septe-
nary
relationship
in
v
oltage
magnitude
allo
ws
to
synthesize
high-resolution
output
w
a
v
eforms.
Subsequently
,
comparison
to
some
of
the
cutting-edge
MDCS-CMLIs
were
carried
out.
It
has
sho
wn
that
the
proposed
struc-
ture
accomplishes
the
lo
west
cost-per
-le
v
el
within
all
le
v
els
re
g
ardless
the
v
alue
of
the
weight
coef
ficient
.
Moreo
v
er
,
a
ne
w
NLC-based
modulation
strate
gy
w
as
suggested
with
a
tw
ofold
aim:
firstly
,
to
comply
with
tw
o
European
and
International
po
wer
quality
grid
codes,
respecti
v
ely
,
EN
50160
and
IEC
61000-2-12;
and
secondly
,
to
pro
vide
an
alternati
v
e
pathw
ay
to
o
v
ercome
the
intrinsic
dra
wbacks
inherited
from
SHE/SHM
techniques.
The
obtained
results
within
SIMULINK
FFT
analysis
tool
clearly
demonstrated
the
outperfor
-
mance
of
the
proposed
modulation
with
(a)
symmetrical
thresholds
i
o
v
er
the
con
v
entional
NLC,
in
which
the
optim
al
feasible
solutions
i
to
the
constrained
optimization
problem
w
as
determined
by
a
bio-inspired
algorithm
(
e
.g
.
,
PSO)
described
in
p
ython
programming
language.
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(Mohammed
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