I
nte
rna
t
io
na
l J
o
urna
l o
f
Appl
ied P
o
w
er
E
ng
ineering
(
I
J
AP
E
)
Vo
l.
9
,
No
.
2
,
A
u
g
u
s
t
2020
,
p
p
.
1
5
3
~1
5
8
I
SS
N:
2252
-
8792
,
DOI
: 1
0
.
1
1
5
9
1
/i
j
ap
e.
v
9
.
i2
.
p
p
1
5
3
-
158
153
J
o
ur
na
l ho
m
ep
a
g
e
:
h
ttp
:
//ij
a
p
e.
ia
esco
r
e.
co
m
M
o
deling
and co
ntrol o
f
4
1
-
lev
el in
v
erte
r using
best
s
w
itching
a
ng
les ca
lcula
tion
m
ethod
Alla
E
dd
i
ne
T
o
u
ba
l M
a
a
m
a
r,
M
’
ha
m
ed
H
ela
i
m
i,
Ra
chi
d T
a
leb,
Abde
la
t
if
G
a
do
u
m
El
e
c
tri
c
a
l
En
g
in
e
e
rin
g
De
p
a
rtm
e
n
t,
L
a
b
o
ra
to
ire G
é
n
ie E
lec
tri
q
u
e
e
t
En
e
rg
ies
Re
n
o
u
v
e
lab
les
(L
G
EE
R),
Ha
ss
ib
a
Be
n
b
o
u
a
li
Un
iv
e
rsity
o
f
Ch
lef
,
A
l
g
e
ria
Art
icle
I
nfo
AB
ST
RAC
T
A
r
ticle
his
to
r
y:
R
ec
eiv
ed
Dec
8
201
9
R
ev
i
s
ed
Feb
1
9
,
2
0
2
0
A
cc
ep
ted
Ma
r
3
,
2
0
2
0
In
th
is
p
a
p
e
r,
a
n
a
ly
sis
a
n
d
m
o
d
e
li
n
g
o
f
a
sin
g
le
-
p
h
a
se
H
-
b
rid
g
e
f
o
rt
y
-
o
n
e
lev
e
l
in
v
e
rter
a
re
c
o
n
si
d
e
re
d
.
T
h
e
c
o
n
tro
l
o
f
p
ro
p
o
se
d
i
n
v
e
rter
b
y
e
q
u
a
l
-
p
h
a
se
a
n
d
h
a
lf
-
h
e
ig
h
t
m
e
t
h
o
d
s
is
im
p
le
m
e
n
ted
.
M
A
TL
A
B
/S
im
u
li
n
k
e
n
v
iro
n
m
e
n
ts
a
re
u
se
d
to
sim
u
la
te
th
e
m
o
d
e
l
a
n
d
sh
o
w
o
b
tain
e
d
re
su
lt
s
o
f
w
a
v
e
f
o
r
m
s
w
it
h
F
F
T
a
n
a
l
y
sis.
Ev
e
n
tu
a
ll
y
,
th
e
to
tal
h
a
rm
o
n
ic
d
isto
rti
o
n
o
b
tai
n
e
d
f
o
r
e
a
c
h
lev
e
l
w
it
h
th
e
tw
o
m
e
th
o
d
s
is
p
re
se
n
ted
,
c
o
m
p
a
ra
ti
v
e
l
y
,
f
o
r
a
c
o
m
p
a
riso
n
.
K
ey
w
o
r
d
s
:
E
q
u
al
-
p
h
a
s
e
m
eth
o
d
Half
-
h
eig
h
t
m
et
h
o
d
MA
T
L
A
B
/
S
i
m
u
li
n
k
Mu
ltil
e
v
el
i
n
v
er
ter
P
o
w
er
elec
tr
o
n
ic
T
h
is i
s
a
n
o
p
e
n
a
c
c
e
ss
a
rticle
u
n
d
e
r th
e
CC B
Y
-
SA
li
c
e
n
se
.
C
o
r
r
e
s
p
o
nd
ing
A
uth
o
r
:
A
lla
E
d
d
in
e
T
o
u
b
al
Ma
a
m
ar
,
E
lectr
ical
E
n
g
i
n
ee
r
i
n
g
Dep
ar
t
m
en
t,
L
ab
o
r
ato
ir
e
Gén
ie
E
lectr
iq
u
e
et
E
n
er
g
ie
s
R
e
n
o
u
v
elab
les (
L
GE
E
R
)
,
Hass
ib
a
B
en
b
o
u
ali
U
n
i
v
er
s
it
y
o
f
C
h
lef
,
B
P
.
7
8
C
,
Ou
led
Far
es 0
2
1
8
0
,
C
h
le
f
,
A
l
g
er
ia.
E
m
ail: a
.
to
u
b
al
m
aa
m
ar
@
u
n
iv
-
ch
lef
.
d
z
1.
I
NT
RO
D
UCT
I
O
N
Uses
o
f
ca
s
ca
d
ed
H
-
b
r
id
g
e
i
n
v
er
ter
h
a
v
e
b
ec
o
m
e
m
o
r
e
p
o
p
u
lar
in
p
o
w
er
elec
tr
o
n
ic
a
p
p
licatio
n
s
b
ec
au
s
e
th
e
s
i
m
p
licit
y
o
f
co
n
t
r
o
l
an
d
th
e
ab
ilit
y
to
g
en
er
ate
h
ig
h
o
u
tp
u
t
v
o
ltag
e
le
v
els
[
1
,
2
]
.
I
n
v
er
ter
s
ca
n
b
e
class
i
f
ied
in
to
t
w
o
t
y
p
e
s
,
v
o
lta
g
e
s
o
u
r
ce
i
n
v
er
te
r
(
VSI
)
an
d
c
u
r
r
en
t
s
o
u
r
ce
in
v
er
ter
(
C
SI)
.
I
f
th
e
D
C
v
o
lta
g
e
i
s
m
ai
n
tai
n
ed
an
d
co
n
s
tan
t,
t
h
e
i
n
v
er
ter
i
s
ca
lled
VSI
o
r
v
o
lta
g
e
f
ed
i
n
v
er
ter
(
VFI
)
,
else
i
f
t
h
e
i
n
p
u
t
c
u
r
r
en
t
i
s
m
ai
n
tai
n
ed
an
d
co
n
s
ta
n
t,
th
e
i
n
v
er
ter
is
ca
l
led
C
SI
o
r
cu
r
r
en
t f
ed
in
v
er
ter
(
C
FI
)
[
3
-
6
]
.
I
n
ap
p
lied
p
o
w
er
e
n
g
i
n
ee
r
i
n
g
f
ield
s
,
th
e
m
u
l
tile
v
el
i
n
v
er
ter
u
s
ed
in
m
a
n
y
ap
p
licatio
n
s
[
7
-
9
]
,
th
er
e
ar
e
th
r
ee
co
n
v
en
t
io
n
al
ca
te
g
o
r
ies
o
f
t
h
e
m
u
l
tile
v
el
i
n
v
er
ter
s
:
ca
s
ca
d
ed
H
-
b
r
id
g
e,
n
e
u
tr
al
p
o
in
t
cla
m
p
ed
an
d
f
l
y
i
n
g
ca
p
ac
ito
r
m
u
l
tile
v
e
l
in
v
er
ter
[
10
-
1
2
]
.
T
h
e
m
u
lt
ile
v
el
i
n
v
er
ter
u
s
ed
f
o
r
I
n
d
u
ctio
n
h
ea
t
in
g
,
T
r
ac
tio
n
s
y
s
te
m
s
,
A
c
tiv
e
f
ilter
i
n
g
,
Mo
t
o
r
d
r
iv
es,
Hig
h
-
v
o
lta
g
e
an
d
Me
d
iu
m
-
v
o
ltag
e
ap
p
licatio
n
s
[
1
3
-
1
4
]
.
T
h
e
p
r
o
b
lem
is
th
e
c
h
o
ice
o
f
t
h
e
s
w
itc
h
i
n
g
an
g
les
r
eq
u
ir
ed
to
co
n
tr
o
l
a
m
u
lti
-
lev
e
l
in
v
er
ter
w
it
h
a
m
i
n
i
m
u
m
T
HD
in
th
e
s
y
s
te
m
,
to
tal
h
ar
m
o
n
ic
d
is
to
r
tio
n
is
to
th
e
r
atio
b
et
w
ee
n
th
e
R
M
S
v
al
u
e
o
f
th
e
s
ig
n
al
h
ar
m
o
n
ics
(
v
o
ltag
e
o
r
cu
r
r
en
t)
an
d
th
e
R
M
S v
al
u
e
o
f
th
e
f
u
n
d
a
m
e
n
tal
f
r
eq
u
e
n
c
y
[
1
5
]
.
T
o
o
v
er
co
m
e
t
h
e
m
en
tio
n
ed
p
r
o
b
lem
a
m
o
d
elin
g
a
n
d
s
i
m
u
la
tio
n
o
f
a
4
1
-
le
v
el
i
n
v
er
ter
u
s
in
g
t
h
e
b
es
t
s
w
itc
h
in
g
an
g
le
s
ca
lcu
latio
n
m
et
h
o
d
h
av
e
b
ee
n
p
r
o
p
o
s
ed
u
s
i
n
g
Si
m
u
li
n
k
/M
A
T
L
A
B
p
r
o
g
r
a
m
,
th
i
s
s
o
lu
tio
n
f
o
r
m
u
ltil
e
v
el
in
v
er
ter
co
n
tr
o
l
b
ased
o
n
tr
ig
o
n
o
m
etr
ic
ca
lc
u
l
atio
n
m
eth
o
d
.
T
h
is
p
ap
er
is
o
r
g
an
ized
a
s
f
o
llo
w
s
,
in
Se
ct
io
n
I
I
m
o
d
eli
n
g
an
d
c
o
n
tr
o
l
o
f
a
m
u
l
tile
v
el
i
n
v
er
ter
u
s
i
n
g
Si
m
u
li
n
k
/M
A
T
L
A
B
h
av
e
b
ee
n
d
is
c
u
s
s
ed
w
it
h
an
an
al
y
s
is
o
f
th
e
t
w
o
co
n
tr
o
l
m
eth
o
d
s
,
eq
u
al
-
p
h
ase
(
E
P
)
m
et
h
o
d
,
h
alf
-
h
eig
h
t
(
HH)
m
et
h
o
d
.
T
h
e
r
esu
lts
ar
e
p
r
esen
ted
an
d
co
m
p
ar
ed
in
Sectio
n
I
I
I
.
Fin
all
y
,
s
o
m
e
co
n
clu
s
io
n
s
ar
e
p
r
esen
ted
i
n
Secti
o
n
I
V.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2252
-
8792
I
n
t J
A
p
p
l
P
o
w
er
E
n
g
,
Vo
l.
9
,
No
.
2
,
A
u
g
u
s
t
2020
:
1
5
3
–
1
5
8
154
2.
M
O
DE
L
I
N
G
AN
D
CO
NT
R
O
L
O
F
M
UL
T
I
L
E
VE
L
I
NV
E
RT
E
R
2
.
1
.
T
he
m
o
del
o
f
m
ultilev
el
i
nv
er
t
er
T
h
e
s
i
m
u
latio
n
m
o
d
el
o
f
a
4
1
-
lev
el
in
v
er
ter
is
s
h
o
w
n
i
n
Fig
u
r
e
1
,
co
n
s
i
s
ts
o
f
ca
s
ca
d
e
d
H
-
b
r
id
g
e
in
v
er
ter
s
,
DC
p
o
w
er
s
u
p
p
lies
an
d
co
n
tr
o
l b
lo
ck
w
i
th
ad
j
u
s
ti
n
g
t
h
e
f
r
eq
u
en
c
y
an
d
s
w
itc
h
in
g
an
g
les.
Fig
u
r
e
1
.
T
h
e
MA
T
L
A
B
/S
i
m
u
lin
k
m
o
d
el
o
f
4
1
-
le
v
el
i
n
v
er
ter
2
.
2
.
Co
ntr
o
l o
f
inv
er
t
er
w
it
h e
qu
a
l
-
ph
a
s
e
m
et
ho
d
S
w
itc
h
in
g
a
n
g
les
w
it
h
t
h
e
eq
u
al
-
p
h
a
s
e
(
EP
)
m
eth
o
d
i
n
d
eg
r
e
es
is
s
h
o
w
n
i
n
T
ab
le
1
.
T
h
e
p
r
in
cip
le
o
f
th
is
m
et
h
o
d
is
th
e
d
i
v
is
io
n
o
f
th
e
p
er
io
d
(
1
8
0
°)
b
y
th
e
i
n
v
er
ter
lev
els
(
m
)
a
n
d
m
u
ltip
lic
ativ
e
t
h
e
r
esu
l
t
b
y
th
e
n
u
m
b
er
o
f
t
h
e
a
n
g
le
(
i)
[
5
]
.
T
h
e
f
o
r
m
u
la
o
f
ca
lc
u
latio
n
is
g
iv
e
n
by
:
T
ab
le
1
.
Sw
itch
in
g
an
g
le
s
w
it
h
th
e
E
P
m
et
h
o
d
in
d
eg
r
ee
s
L
e
v
e
l
A
n
g
l
e
s s
w
i
t
c
h
i
n
g
w
i
t
h
t
h
e
EP
me
t
h
o
d
i
n
d
e
g
r
e
e
s
3
θ1
=
6
0
5
θ1
=
3
6
;
θ2
=
7
2
7
θ1
=
2
5
.
7
1
4
3
;
θ2
=
5
1
.
4
2
8
6
;
θ3
=
7
7
.
1
4
2
9
9
θ1
=
2
0
;
θ2
=
4
0
;
θ
3
=
6
0
;
θ
4
=
8
0
11
θ1
=
1
6
.
3
6
3
6
;
θ2
=
3
2
.
7
2
7
3
;
θ
3
=
4
9
.
0
9
0
9
;
θ4
=
6
5
.
4
5
4
5
;
θ5
=
8
1
.
8
1
8
2
13
θ1
=
1
3
.
8
4
6
2
;
θ2
=
2
7
.
6
9
2
3
;
θ
3
=
4
1
.
5
3
8
5
;
θ
4
=
5
5
.
3
8
4
6
;
θ5
=
6
9
.
2
3
0
8
;
θ
6
=
8
3
.
0
7
6
9
15
θ1
=
1
2
;
θ
2
=
2
4
;
θ3
=
3
6
;
θ
4
=
4
8
;
θ
5
=
6
0
;
θ
6
=
7
2
;
θ7
=
8
4
17
θ1
=
1
0
.
5
8
8
2
;
θ2
=
2
1
.
1
7
6
5
;
θ
3
=
3
1
.
7
6
4
7
;
θ
4
=
4
2
.
3
5
2
9
;
θ5
=
5
2
.
9
4
1
2
;
θ
6
=
6
3
.
5
2
9
4
;
θ
7
=
7
4
.
1
1
7
6
;
θ
8
=
8
4
.
7
0
5
9
19
θ1
=
9
.
4
7
3
7
;
θ
2
=
1
8
.
9
4
7
4
;
θ3
=
2
8
.
4
2
1
1
;
θ4
=
3
7
.
8
9
4
7
;
θ
5
=
4
7
.
3
6
8
4
;
θ6
=
5
6
.
8
4
2
1
;
θ7
=
6
6
.
3
1
5
8
;
θ8
=
7
5
.
7
8
9
5
;
θ
9
=
8
5
.
2
6
3
2
21
θ1
=
8
.
5
7
1
4
;
θ
2
=
1
7
.
1
4
2
9
;
θ3
=
2
5
.
7
1
4
3
;
θ4
=
3
4
.
2
8
5
7
;
θ
5
=
4
2
.
8
5
7
1
;
θ6
=
5
1
.
4
2
8
6
;
θ7
=
6
0
.
0
0
0
0
;
θ
8
=
6
8
.
5
7
1
4
;
θ9
=
7
7
.
1
4
2
9
;
θ1
0
=
8
5
.
7
1
4
3
23
θ1
=
7
.
8
2
6
1
;
θ
2
=
1
5
.
6
5
2
2
;
θ3
=
2
3
.
4
7
8
3
;
θ4
=
3
1
.
3
0
4
3
;
θ
5
=
3
9
.
1
3
0
4
;
θ6
=
4
6
.
9
5
6
5
;
θ7
=
5
4
.
7
8
2
6
;
θ
8
=
6
2
.
6
0
8
7
;
θ9
=
7
0
.
4
3
4
8
;
θ1
0
=
7
8
.
2
6
0
9
;
θ
1
1
=
8
6
.
0
8
7
0
25
θ1
=
7
.
2
0
0
0
;
θ
2
=
1
4
.
4
0
0
0
;
θ3
=
2
1
.
6
0
0
0
;
θ4
=
2
8
.
8
0
0
0
;
θ
5
=
3
6
.
0
0
0
0
;
θ6
=
4
3
.
2
0
0
0
;
θ7
=
5
0
.
4
0
0
0
;
θ
8
=
5
7
.
6
0
0
0
;
θ9
=
6
4
.
8
0
0
0
;
θ1
0
=
7
2
.
0
0
0
0
;
θ
1
1
=
7
9
.
2
0
0
0
;
θ1
2
=
8
6
.
4
0
0
0
27
θ1
=
6
.
6
6
6
7
;
θ
2
=
1
3
.
3
3
3
3
;
θ3
=
2
0
.
0
0
0
0
;
θ4
=
2
6
.
6
6
6
7
;
θ
5
=
3
3
.
3
3
3
3
;
θ6
=
4
0
.
0
0
0
0
;
θ7
=
4
6
.
6
6
6
7
;
θ
8
=
5
3
.
3
3
3
3
;
θ9
=
6
0
.
0
0
0
0
;
θ1
0
=
6
6
.
6
6
6
7
;
θ
1
1
=
7
3
.
3
3
3
3
;
θ1
2
=
8
0
.
0
0
0
0
;
θ
1
3
=
8
6
.
6
6
6
7
29
θ1
=
6
.
2
0
6
9
;
θ
2
=
1
2
.
4
1
3
8
;
θ3
=
1
8
.
6
2
0
7
;
θ4
=
2
4
.
8
2
7
6
;
θ
5
=
3
1
.
0
3
4
5
;
θ6
=
3
7
.
2
4
1
4
;
θ7
=
4
3
.
4
4
8
3
;
θ
8
=
4
9
.
6
5
5
2
;
θ9
=
5
5
.
8
6
2
1
;
θ1
0
=
6
2
.
0
6
9
0
;
θ
1
1
=
6
8
.
2
7
5
9
;
θ1
2
=
7
4
.
4
8
2
8
;
θ
1
3
=
8
0
.
6
8
9
7
;
θ
1
4
=
8
6
.
8
9
6
6
31
θ1
=
5
.
8
0
6
5
;
θ
2
=
1
1
.
6
1
2
9
;
θ3
=
1
7
.
4
1
9
4
;
θ4
=
2
3
.
2
2
5
8
;
θ
5
=
2
9
.
0
3
2
3
;
θ6
=
3
4
.
8
3
8
7
;
θ7
=
4
0
.
6
4
5
2
;
θ
8
=
4
6
.
4
5
1
6
;
θ9
=
5
2
.
2
5
8
1
;
θ1
0
=
5
8
.
0
6
4
5
;
θ
1
1
=
6
3
.
8
7
1
0
;
θ1
2
=
6
9
.
6
7
7
4
;
θ
1
3
=
7
5
.
4
8
3
9
;
θ
1
4
=
8
1
.
2
9
0
3
;
θ1
5
=
8
7
.
0
9
6
8
=
180°
,
=
1
,
2
,
3
,
4
,
…
,
−
1
2
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J
A
p
p
l P
o
w
er
E
n
g
I
SS
N:
2252
-
8792
Mo
d
elin
g
a
n
d
co
n
tr
o
l o
f 4
1
-
leve
l in
ve
r
ter u
s
in
g
b
est s
w
i
tch
i
n
g
a
n
g
les…
(
A
lla
E
d
d
in
e
To
u
b
a
l Ma
a
ma
r
)
155
T
ab
le
1
.
Sw
itch
in
g
an
g
le
s
w
it
h
th
e
E
P
m
et
h
o
d
in
d
eg
r
ee
s
(
c
o
n
t)
L
e
v
e
l
A
n
g
l
e
s s
w
i
t
c
h
i
n
g
w
i
t
h
t
h
e
EP
me
t
h
o
d
i
n
d
e
g
r
e
e
s
33
θ1
=
5
.
4
5
4
5
;
θ
2
=
1
0
.
9
0
9
1
;
θ3
=
1
6
.
3
6
3
6
;
θ4
=
2
1
.
8
1
8
2
;
θ
5
=
2
7
.
2
7
2
7
;
θ6
=
3
2
.
7
2
7
3
;
θ7
=
3
8
.
1
8
1
8
;
θ
8
=
4
3
.
6
3
6
4
;
θ9
=
4
9
.
0
9
0
9
;
θ1
0
=
5
4
.
5
4
5
5
;
θ
1
1
=
6
0
.
0
0
0
0
;
θ1
2
=
6
5
.
4
5
4
5
;
θ
1
3
=
7
0
.
9
0
9
1
;
θ
1
4
=
7
6
.
3
6
3
6
;
θ1
5
=
8
1
.
8
1
8
2
;
θ1
6
=
8
7
.
2
7
2
7
35
θ1
=
5
.
1
4
2
9
;
θ
2
=
1
0
.
2
8
5
7
;
θ3
=
1
5
.
4
2
8
6
;
θ
4
=
2
0
.
5
7
1
4
;
θ
5
=
2
5
.
7
1
4
3
;
θ6
=
3
0
.
8
5
7
1
;
θ7
=
3
6
.
0
0
0
0
;
θ
8
=
4
1
.
1
4
2
9
;
θ9
=
4
6
.
2
8
5
7
;
θ1
0
=
5
1
.
4
2
8
6
;
θ
1
1
=
5
6
.
5
7
1
4
;
θ1
2
=
6
1
.
7
1
4
3
;
θ
1
3
=
6
6
.
8
5
7
1
;
θ
1
4
=
7
2
.
0
0
0
0
;
θ1
5
=
7
7
.
1
4
2
9
;
θ1
6
=
8
2
.
2
8
5
7
;
θ
1
7
=
8
7
.
4
2
8
6
37
θ1
=
4
.
8
6
4
9
;
θ
2
=
9
.
7
2
9
7
;
θ
3
=
1
4
.
5
9
4
6
;
θ
4
=
1
9
.
4
5
9
5
;
θ
5
=
2
4
.
3
2
4
3
;
θ
6
=
2
9
.
1
8
9
2
;
θ
7
=
3
4
.
0
5
4
1
;
θ
8
=
3
8
.
9
1
8
9
;
θ
9
=
4
3
.
7
8
3
8
;
θ1
0
=
4
8
.
6
4
8
6
;
θ
1
1
=
5
3
.
5
1
3
5
;
θ1
2
=
5
8
.
3
7
8
4
;
θ
1
3
=
6
3
.
2
4
3
2
;
θ
1
4
=
6
8
.
1
0
8
1
;
θ1
5
=
7
2
.
9
7
3
0
;
θ
1
6
=
7
7
.
8
3
7
8
;
θ1
7
=
8
2
.
7
0
2
7
;
θ
1
8
=
8
7
.
5
6
7
6
39
θ1
=
4
.
6
1
5
4
;
θ
2
=
9
.
2
3
0
8
;
θ
3
=
1
3
.
8
4
6
2
;
θ
4
=
1
8
.
4
6
1
5
;
θ
5
=
2
3
.
0
7
6
9
;
θ
6
=
2
7
.
6
9
2
3
;
θ
7
=
3
2
.
3
0
7
7
;
θ
8
=
3
6
.
9
2
3
1
;
θ
9
=
4
1
.
5
3
8
5
;
θ1
0
=
4
6
.
1
5
3
8
;
θ
1
1
=
5
0
.
7
6
9
2
;
θ1
2
=
5
5
.
3
8
4
6
;
θ
1
3
=
6
0
.
0
0
0
0
;
θ
1
4
=
6
4
.
6
1
5
4
;
θ1
5
=
6
9
.
2
3
0
8
;
θ1
6
=
7
3
.
8
4
6
2
;
θ
1
7
=
7
8
.
4
6
1
5
;
θ
1
8
=
8
3
.
0
7
6
9
;
θ
1
9
=
8
7
.
6
9
2
3
41
θ1
=
4
.
3
9
0
2
;
θ
2
=
8
.
7
8
0
5
;
θ3
=
1
3
.
1
7
0
7
;
θ4
=
1
7
.
5
6
1
0
;
θ
5
=
2
1
.
9
5
1
2
;
θ6
=
2
6
.
3
4
1
5
;
θ7
=
3
0
.
7
3
1
7
;
θ8
=
3
5
.
1
2
2
0
;
θ9
=
3
9
.
5
1
2
2
;
θ1
0
=
4
3
.
9
0
2
4
;
θ1
1
=
4
8
.
2
9
2
7
;
θ
1
2
=
5
2
.
6
8
2
9
;
θ1
3
=
5
7
.
0
7
3
2
;
θ1
4
=
6
1
.
4
6
3
4
;
θ1
5
=
6
5
.
8
5
3
7
;
θ1
6
=
7
0
.
2
4
3
9
;
θ1
7
=
7
4
.
6
3
4
1
;
θ
1
8
=
7
9
.
0
2
4
4
;
θ
1
9
=
8
3
.
4
1
4
6
;
θ2
0
=
8
7
.
8
0
4
9
2
.
3
.
Co
ntr
o
l o
f
inv
er
t
er
w
it
h
ha
lf
-
heig
ht
m
et
ho
d
S
w
itc
h
in
g
an
g
les
w
it
h
t
h
e
h
al
f
-
h
e
ig
h
t
(
HH)
is
s
h
o
w
n
i
n
T
ab
l
e
2
.
T
h
e
P
r
in
cip
e
o
f
th
e
HH
m
eth
o
d
[
5
]
is
p
r
esen
ted
b
y
t
h
e
f
o
llo
w
i
n
g
f
o
r
m
u
la
o
f
ca
lc
u
latio
n
:
T
ab
le
2
.
Sw
itch
in
g
an
g
le
s
w
it
h
th
e
H
H
m
et
h
o
d
in
d
eg
r
ee
s
L
e
v
e
l
A
n
g
l
e
s
s
w
i
t
c
h
i
n
g
w
i
t
h
HH
m
e
t
h
o
d
i
n
d
e
g
r
e
e
s
3
θ1
=
3
0
5
θ1
=
1
4
.
4
7
7
5
;
θ2
=
4
8
.
5
9
0
4
7
θ1
=
9
.
5
9
4
1
;
θ
2
=
3
0
;
θ3
=
5
6
.
4
4
2
7
9
θ1
=
7
.
1
8
0
8
;
θ
2
=
2
2
.
0
2
4
3
;
θ3
=
3
8
.
6
8
2
2
;
θ4
=
6
1
.
0
4
5
0
11
θ1
=
5
.
7
3
9
2
;
θ
2
=
1
7
.
4
5
7
6
;
θ3
=
3
0
;
θ
4
=
4
4
.
4
2
7
0
;
θ
5
=
6
4
.
1
5
8
1
13
θ1
=
4
.
7
8
0
2
;
θ
2
=
1
4
.
4
7
7
5
;
θ3
=
2
4
.
6
2
4
3
;
θ4
=
3
5
.
6
8
5
3
;
θ
5
=
4
8
.
5
9
0
4
;
θ6
=
6
6
.
4
4
3
5
15
θ1
=
4
.
0
9
6
0
;
θ
2
=
1
2
.
3
7
3
6
;
θ
3
=
2
0
.
9
2
4
8
;
θ
4
=
3
0
;
θ
5
=
4
0
.
0
0
5
2
;
θ
6
=
5
1
.
7
8
6
8
;
θ
7
=
6
8
.
2
1
3
2
17
θ1
=
3
.
5
8
3
3
;
θ
2
=
1
0
.
8
0
6
9
;
θ3
=
1
8
.
2
1
0
0
;
θ4
=
2
5
.
9
4
4
5
;
θ
5
=
3
4
.
2
2
8
9
;
θ6
=
4
3
.
4
3
2
5
;
θ7
=
5
4
.
3
4
0
9
;
θ
8
=
6
9
.
6
3
5
9
19
θ1
=
3
.
1
8
4
7
;
θ
2
=
9
.
5
9
4
1
;
θ
3
=
1
6
.
1
2
7
6
;
θ
4
=
2
2
.
8
8
5
4
;
θ
5
=
3
0
;
θ
6
=
3
7
.
6
6
9
9
;
θ7
=
4
6
.
2
3
8
3
;
θ
8
=
5
6
.
4
4
2
7
;
θ9
=
7
0
.
8
1
1
9
21
θ1
=
2
.
8
6
6
0
;
θ
2
=
8
.
6
2
6
9
;
θ
3
=
1
4
.
4
7
7
5
;
θ
4
=
2
0
.
4
8
7
3
;
θ
5
=
2
6
.
7
4
3
7
;
θ
6
=
3
3
.
3
6
7
0
;
θ
7
=
4
0
.
5
4
1
6
;
θ
8
=
4
8
.
5
9
0
4
;
θ9
=
5
8
.
2
1
1
7
;
θ1
0
=
7
1
.
8
0
5
1
23
θ1
=
2
.
6
0
5
3
;
θ
2
=
7
.
8
3
7
5
;
θ
3
=
1
3
.
1
3
6
6
;
θ
4
=
1
8
.
5
5
3
0
;
θ
5
=
2
4
.
1
4
7
7
;
θ
6
=
3
0
.
0
0
0
0
;
θ
7
=
3
6
.
2
2
1
5
;
θ
8
=
4
2
.
9
8
5
9
;
θ9
=
5
0
.
5
9
9
4
;
θ1
0
=
5
9
.
7
2
7
4
;
θ
1
1
=
7
2
.
6
5
8
6
25
θ1
=
2
.
3
8
8
0
;
θ
2
=
7
.
1
8
0
8
;
θ
3
=
1
2
.
0
2
4
7
;
θ
4
=
1
6
.
9
5
7
8
;
θ
5
=
2
2
.
0
2
4
3
;
θ
6
=
2
7
.
2
7
9
6
;
θ
7
=
3
2
.
7
9
7
2
;
θ
8
=
3
8
.
6
8
2
2
;
θ9
=
4
5
.
0
9
9
5
;
θ1
0
=
5
2
.
3
4
1
5
;
θ
1
1
=
6
1
.
0
4
5
0
;
θ1
2
=
7
3
.
4
0
2
2
27
θ1
=
2
.
2
0
4
2
;
θ
2
=
6
.
6
2
5
8
;
θ
3
=
1
1
.
0
8
7
5
;
θ
4
=
1
5
.
6
1
8
5
;
θ
5
=
2
0
.
2
5
2
2
;
θ
6
=
2
5
.
0
2
9
0
;
θ
7
=
3
0
.
0
0
0
0
;
θ
8
=
3
5
.
2
3
4
4
;
θ9
=
4
0
.
8
3
2
2
;
θ1
0
=
4
6
.
9
5
0
9
;
θ
1
1
=
5
3
.
8
7
1
1
;
θ1
2
=
6
2
.
2
0
4
2
;
θ
1
3
=
7
4
.
0
5
7
6
29
θ1
=
2
.
0
4
6
7
;
θ
2
=
6
.
1
5
0
6
;
θ
3
=
1
0
.
2
8
6
6
;
θ
4
=
1
4
.
4
7
7
5
;
θ
5
=
1
8
.
7
4
9
3
;
θ
6
=
2
3
.
1
3
2
4
;
θ
7
=
2
7
.
6
6
4
0
;
θ
8
=
3
2
.
3
9
2
4
;
θ9
=
3
7
.
3
8
3
2
;
θ1
0
=
4
2
.
7
3
2
1
;
θ
1
1
=
4
8
.
5
9
0
4
;
θ1
2
=
5
5
.
2
2
8
1
;
θ
1
3
=
6
3
.
2
3
4
5
;
θ
1
4
=
7
4
.
6
4
1
1
31
θ1
=
1
.
9
1
0
2
;
θ
2
=
5
.
7
3
9
2
;
θ
3
=
9
.
5
9
4
1
;
θ4
=
1
3
.
4
9
3
4
;
θ5
=
1
7
.
4
5
7
6
;
θ
6
=
2
1
.
5
1
0
2
;
θ
7
=
2
5
.
6
7
9
3
;
θ8
=
3
0
.
0
0
0
0
;
θ
9
=
3
4
.
5
1
8
1
;
θ1
0
=
3
9
.
2
9
6
5
;
θ
1
1
=
4
4
.
4
2
7
0
;
θ
1
2
=
5
0
.
0
5
5
5
;
θ
1
3
=
5
6
.
4
4
2
7
;
θ1
4
=
6
4
.
1
5
8
1
;
θ
1
5
=
7
5
.
1
6
4
9
33
θ1
=
1
.
7
9
0
8
;
θ
2
=
5
.
3
7
9
4
;
θ
3
=
8
.
9
8
9
3
;
θ4
=
1
2
.
6
3
5
6
;
θ5
=
1
6
.
3
3
4
8
;
θ
6
=
2
0
.
1
0
5
5
;
θ
7
=
2
3
.
9
6
9
5
;
θ8
=
2
7
.
9
5
3
2
;
θ
9
=
3
2
.
0
9
0
0
;
θ1
0
=
3
6
.
4
2
3
6
;
θ
1
1
=
4
1
.
0
1
4
5
;
θ
1
2
=
4
5
.
9
5
1
4
;
θ
1
3
=
5
1
.
3
7
5
2
;
θ
1
4
=
5
7
.
5
3
8
3
;
θ
1
5
=
6
4
.
9
9
2
2
;
θ
1
6
=
7
5
.
6
3
8
5
35
θ1
=
1
.
6
8
5
4
;
θ
2
=
5
.
0
6
2
1
;
θ
3
=
8
.
4
5
6
5
;
θ4
=
1
1
.
8
8
1
2
;
θ5
=
1
5
.
3
4
9
5
;
θ
6
=
1
8
.
8
7
6
5
;
θ
7
=
2
2
.
4
7
9
5
;
θ8
=
2
6
.
1
7
9
0
;
θ
9
=
3
0
.
0
0
0
0
;
θ1
0
=
3
3
.
9
7
4
5
;
θ
1
1
=
3
8
.
1
4
4
5
;
θ
1
2
=
4
2
.
5
6
8
5
;
θ
1
3
=
4
7
.
3
3
2
1
;
θ
1
4
=
5
2
.
5
7
2
0
;
θ
1
5
=
5
8
.
5
3
3
0
;
θ
1
6
=
6
5
.
7
5
0
4
;
θ
1
7
=
7
6
.
0
6
9
4
37
θ1
=
1
.
5
9
1
8
;
θ
2
=
4
.
7
8
0
2
;
θ
3
=
7
.
9
8
3
6
;
θ4
=
1
1
.
2
1
2
3
;
θ5
=
1
4
.
4
7
7
5
;
θ
6
=
1
7
.
7
9
1
6
;
θ
7
=
2
1
.
1
6
8
4
;
θ8
=
2
4
.
6
2
4
3
;
θ
9
=
2
8
.
1
7
8
6
;
θ1
0
=
3
1
.
8
5
5
4
;
θ
1
1
=
3
5
.
6
8
5
3
;
θ
1
2
=
3
9
.
7
0
9
0
;
θ
1
3
=
4
3
.
9
8
3
0
;
θ
1
4
=
4
8
.
5
9
0
4
;
θ
1
5
=
5
3
.
6
6
3
9
;
θ1
6
=
5
9
.
4
4
1
6
;
θ
1
7
=
6
6
.
4
4
3
5
;
θ
1
8
=
7
6
.
4
6
3
8
39
θ1
=
1
.
5
0
8
0
;
θ
2
=
4
.
5
2
8
1
;
θ
3
=
7
.
5
6
0
8
;
θ4
=
1
0
.
6
1
5
1
;
θ5
=
1
3
.
7
0
0
2
;
θ
6
=
1
6
.
8
2
6
4
;
θ
7
=
2
0
.
0
0
5
2
;
θ8
=
2
3
.
2
4
9
6
;
θ
9
=
2
6
.
5
7
5
0
;
θ1
0
=
3
0
.
0
0
0
0
;
θ
1
1
=
3
3
.
5
4
7
7
;
θ
1
2
=
3
7
.
2
4
7
8
;
θ
1
3
=
4
1
.
1
3
9
5
;
θ
1
4
=
4
5
.
2
7
7
8
;
θ
1
5
=
4
9
.
7
4
3
4
;
θ1
6
=
5
4
.
6
6
5
5
;
θ
1
7
=
6
0
.
2
7
5
7
;
θ
1
8
=
6
7
.
0
8
0
5
;
θ
1
9
=
7
6
.
8
2
6
4
41
θ1
=
1
.
4
3
2
5
;
θ
2
=
4
.
3
0
1
2
;
θ
3
=
7
.
1
8
0
8
;
θ4
=
1
0
.
0
7
8
7
;
θ5
=
1
3
.
0
0
2
9
;
θ
6
=
1
5
.
9
6
2
0
;
θ7
=
1
8
.
9
6
5
6
;
θ
8
=
2
2
.
0
2
4
3
;
θ
9
=
2
5
.
1
5
0
7
;
θ1
0
=
2
8
.
3
5
9
4
;
θ
1
1
=
3
1
.
6
6
8
2
;
θ1
2
=
3
5
.
0
9
9
6
;
θ
1
3
=
3
8
.
6
8
2
2
;
θ
1
4
=
4
2
.
4
5
4
2
;
θ1
5
=
4
6
.
4
6
8
8
;
θ
1
6
=
5
0
.
8
0
5
0
;
θ1
7
=
5
5
.
5
8
8
5
;
θ1
8
=
6
1
.
0
4
5
0
;
θ
1
9
=
6
7
.
6
6
8
4
;
θ2
0
=
7
7
.
1
6
1
4
.
=
sin
−
1
2
−
1
−
1
2
=
sin
−
1
2
−
1
−
1
,
ℎ
=
1
,
2
,
3
,
4
,
…
,
−
1
2
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2252
-
8792
I
n
t J
A
p
p
l
P
o
w
er
E
n
g
,
Vo
l.
9
,
No
.
2
,
A
u
g
u
s
t
2020
:
1
5
3
–
1
5
8
156
3.
RE
SU
L
T
S AN
D
D
I
SCU
SS
I
O
N
So
m
e
o
b
tain
ed
r
esu
lt
s
ar
e
p
r
esen
ted
in
t
h
i
s
s
es
s
io
n
w
it
h
a
co
m
p
ar
is
o
n
.
Fig
u
r
e
2
s
h
o
w
th
e
o
u
tp
u
t
v
o
ltag
e
w
a
v
e
f
o
r
m
an
d
Fi
g
u
r
e
3
s
h
o
w
t
h
e
h
ar
m
o
n
ic
s
p
ec
tr
u
m
o
f
a
4
1
-
lev
e
l
in
v
er
ter
w
ith
E
P
m
et
h
o
d
.
Fig
u
r
e
4
s
h
o
w
th
e
4
1
-
le
v
el
i
n
v
er
ter
o
u
tp
u
t
w
a
v
e
f
o
r
m
w
it
h
HH
m
e
th
o
d
an
d
Fig
u
r
e
5
s
h
o
w
t
h
e
h
ar
m
o
n
ic
s
p
ec
tr
u
m
an
al
y
s
is
w
i
th
H
H
m
et
h
o
d
o
f
a
4
1
-
lev
el
in
v
er
ter
.
T
ab
le
3
s
h
o
w
s
t
h
e
d
ata
co
m
p
ar
is
o
n
an
d
Fig
u
r
e
6
s
h
o
w
a
g
r
ap
h
ic
co
m
p
ar
i
s
o
n
o
f
T
HDs b
et
w
ee
n
t
h
e
E
P
m
et
h
o
d
an
d
HH
m
et
h
o
d
.
Fig
u
r
e
2
.
T
h
e
o
u
tp
u
t v
o
ltag
e
w
a
v
e
f
o
r
m
o
f
a
4
1
-
le
v
el
in
v
er
ter
w
it
h
E
P
m
et
h
o
d
Fig
u
r
e
3
.
T
HD
A
n
al
y
s
i
s
o
f
4
1
-
L
ev
el
o
u
tp
u
t
v
o
ltag
e
w
it
h
t
h
e
E
P
m
eth
o
d
Fig
u
r
e
4
.
Ou
tp
u
t
v
o
ltag
e
w
a
v
e
f
o
r
m
f
o
r
a
41
-
lev
el
in
v
er
ter
w
it
h
t
h
e
HH
m
eth
o
d
Fig
u
r
e
5
.
T
HD
an
aly
s
is
o
f
4
1
-
lev
el
o
u
tp
u
t v
o
lta
g
e
w
it
h
t
h
e
HH
m
eth
o
d
T
h
e
Fig
u
r
e
2
an
d
Fig
u
r
e
4
ar
e
th
e
o
u
tp
u
t
v
o
ltag
e
w
a
v
ef
o
r
m
s
o
f
th
e
4
1
-
le
v
el
in
v
er
ter
w
ith
th
e
t
w
o
co
n
tr
o
l
m
et
h
o
d
s
,
E
P
m
et
h
o
d
an
d
HH
m
et
h
o
d
,
an
d
th
e
Fi
g
u
r
e
3
an
d
Fi
g
u
r
e
5
ar
e
t
h
e
to
tal
h
ar
m
o
n
ic
d
is
to
r
tio
n
T
HD
o
f
th
e
4
1
-
le
v
el
i
n
v
er
ter
w
i
th
t
h
e
t
w
o
m
et
h
o
d
s
r
es
p
e
ctiv
el
y
.
Fro
m
t
h
e
a
n
al
y
s
i
s
o
f
T
HD
co
m
p
ar
is
o
n
b
et
w
ee
n
th
e
t
w
o
co
n
tr
o
l
m
et
h
o
d
s
in
Fi
g
u
r
e
6
,
w
e
ca
n
s
a
y
t
h
at
t
h
er
e
is
a
d
ec
r
ea
s
e
o
f
T
HD
w
i
th
a
n
i
n
cr
ea
s
e
o
f
in
v
er
ter
lev
el,
an
d
f
o
r
ea
ch
in
v
er
ter
lev
el,
th
e
T
HD
w
it
h
E
P
co
n
tr
o
l
m
et
h
o
d
is
h
i
g
h
er
th
an
t
h
at
o
b
ta
in
ed
w
it
h
HH
co
n
tr
o
l
m
et
h
o
d
.
T
H
D
o
f
th
e
1
7
-
lev
el
in
v
er
ter
w
it
h
H
H
co
n
tr
o
l
m
et
h
o
d
is
d
o
w
n
o
f
5
%,
an
d
w
i
th
E
P
m
et
h
o
d
is
1
8
.
1
5
%.
T
HD
o
f
a
4
1
-
lev
el
i
n
v
er
ter
w
ith
HH
c
o
n
tr
o
l
m
et
h
o
d
is
1
.
9
8
%,
an
d
w
i
th
E
P
m
et
h
o
d
is
1
3
.
9
3
%.
W
e
n
o
te
th
at
th
e
h
ar
m
o
n
ic
v
o
ltag
e
i
s
li
m
ited
b
y
th
e
s
tan
d
ar
d
I
E
E
E
ST
D
5
1
9
[
1
6
]
:
T
h
e
to
tal
h
ar
m
o
n
ic
d
is
to
r
tio
n
o
f
v
o
lta
g
e
T
HDv
<5
%,
a
n
d
co
n
cl
u
d
e
th
at
th
e
HH
m
et
h
o
d
g
iv
e
s
a
g
o
o
d
r
esu
lt
w
it
h
17
-
lev
el
ca
s
ca
d
ed
H
-
b
r
id
g
e
i
n
v
er
ter
.
0
0
.
0
0
2
0
.
0
0
4
0
.
0
0
6
0
.
0
0
8
0
.
0
1
0
.
0
1
2
0
.
0
1
4
0
.
0
1
6
0
.
0
1
8
0
.
0
2
-
2
0
0
-
1
5
0
-
1
0
0
-
5
0
0
50
100
150
200
T
im
e
[
s
]
Volt
age
[
v
]
0
500
1000
1500
0
5
10
15
F
r
e
q
u
e
n
c
y
(
H
z
)
F
u
n
d
a
m
e
n
t
a
l
(
5
0
H
z
)
=
1
5
9
.
8
,
T
H
D
=
1
3
.
9
3
%
M
a
g
(
%
o
f
F
u
n
d
a
m
e
n
t
a
l
)
0
0
.
0
0
2
0
.
0
0
4
0
.
0
0
6
0
.
0
0
8
0
.
0
1
0
.
0
1
2
0
.
0
1
4
0
.
0
1
6
0
.
0
1
8
0
.
0
2
-
2
0
0
-
1
5
0
-
1
0
0
-
5
0
0
50
100
150
200
T
im
e
[
s
]
Volt
age
[
v
]
0
200
400
600
800
1000
1200
1400
0
0
.
5
1
1
.
5
2
2
.
5
F
r
e
q
u
e
n
c
y
(
H
z
)
F
u
n
d
a
m
e
n
t
a
l
(
5
0
H
z
)
=
2
0
0
.
4
,
T
H
D
=
1
.
9
8
%
M
a
g
(
%
o
f
F
u
n
d
a
m
e
n
t
a
l
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J
A
p
p
l P
o
w
er
E
n
g
I
SS
N:
2252
-
8792
Mo
d
elin
g
a
n
d
co
n
tr
o
l o
f 4
1
-
leve
l in
ve
r
ter u
s
in
g
b
est s
w
i
tch
i
n
g
a
n
g
les…
(
A
lla
E
d
d
in
e
To
u
b
a
l Ma
a
ma
r
)
157
T
ab
le
3
.
T
HD
w
it
h
t
w
o
m
eth
o
d
s
,
co
m
p
ar
is
o
n
s
L
e
v
e
l
T
H
D
(
%) w
i
t
h
EP me
t
h
o
d
T
H
D
(
%) w
i
t
h
H
H
me
t
h
o
d
3
7
8
.
5
5
3
1
.
8
7
5
3
8
.
9
6
1
7
.
5
3
7
3
0
.
0
2
1
2
.
0
9
9
2
5
.
8
2
9
.
2
8
11
2
1
.
7
5
7
.
4
8
13
2
0
.
2
4
6
.
3
5
15
1
8
.
7
5
5
.
4
7
17
1
8
.
1
5
4
.
8
2
19
1
7
.
2
7
4
.
3
2
21
1
6
.
4
9
3
.
8
9
23
1
5
.
9
4
3
.
5
5
25
1
5
.
8
6
3
.
2
4
27
1
5
.
1
6
3
.
0
5
29
1
4
.
0
5
2
.
8
0
31
1
4
.
7
4
2
.
6
5
33
1
4
.
4
5
2
.
4
7
35
1
4
.
1
9
2
.
3
6
37
1
4
.
0
7
2
.
2
1
39
1
4
.
1
4
2
.
1
0
41
1
3
.
9
3
1
.
9
8
Fig
u
r
e
6
.
T
HD
co
m
p
ar
is
o
n
at
d
if
f
er
e
n
t le
v
e
ls
w
it
h
t
h
e
t
w
o
co
n
tr
o
l
m
et
h
o
d
s
4.
CO
NCLU
SI
O
N
I
n
t
h
is
p
ap
er
,
co
n
tr
ib
u
tio
n
to
th
e
m
o
d
eli
n
g
an
d
co
n
tr
o
l
o
f
4
1
-
lev
el
i
n
v
er
ter
u
s
i
n
g
b
est
s
w
itc
h
in
g
an
g
le
s
ca
lc
u
latio
n
m
et
h
o
d
is
p
r
esen
ted
w
ith
t
w
o
co
n
tr
o
l
m
et
h
o
d
s
.
Fro
m
s
i
m
u
latio
n
a
n
d
T
HD
co
m
p
ar
ed
r
esu
lt
s
it h
a
s
b
ee
n
o
b
s
er
v
ed
th
at
HH
m
et
h
o
d
is
th
e
b
est
m
et
h
o
d
f
o
r
s
w
itc
h
i
n
g
a
n
g
les t
h
an
t
h
e
E
P
m
e
th
od.
RE
F
E
R
E
NC
E
S
[1
]
H
.
A
k
a
g
i
,
"
M
u
l
t
i
l
e
v
e
l
C
o
n
v
e
r
te
r
s
:
F
u
n
d
a
m
e
n
t
a
l
C
i
r
c
u
i
t
s
a
n
d
S
y
s
tem
s
,
"
i
n
P
r
o
c
e
e
d
i
n
g
s
o
f
t
h
e
I
E
E
E
,
v
o
l
.
1
0
5
,
n
o
.
1
1
,
p
p
.
2
0
4
8
-
2
0
6
5
,
N
o
v
.
2
0
1
7
.
[2
]
Ya
o
su
o
X
u
e
,
L
iu
c
h
e
n
Ch
a
n
g
,
S
re
n
Ba
e
k
h
j
Kja
e
r,
J.
Bo
rd
o
n
a
u
,
a
n
d
T
.
S
h
im
izu
,
"
T
o
p
o
lo
g
ies
o
f
sin
g
le
-
p
h
a
se
in
v
e
rters
f
o
r
sm
a
ll
d
istri
b
u
ted
p
o
w
e
r
g
e
n
e
r
a
to
rs:
a
n
o
v
e
rv
iew
,
"
in
IEE
E
T
ra
n
sa
c
ti
o
n
s
o
n
P
o
we
r
E
l
e
c
tro
n
ics
,
v
o
l.
1
9
,
n
o
.
5
,
p
p
.
1
3
0
5
-
1
3
1
4
,
S
e
p
t.
2
0
0
4
.
[3
]
Bo
ll
a
M
a
d
h
u
su
d
a
n
a
Re
d
d
y
,
Y.
V
.
S
iv
a
Re
d
d
y
,
a
n
d
M
.
Vijay
a
Ku
m
a
r
,
"
M
o
d
e
li
n
g
a
n
d
S
im
u
lati
o
n
o
f
1
2
7
L
e
v
e
l
Op
ti
m
a
l
M
u
lt
il
e
v
e
l
In
v
e
rter
w
it
h
L
o
we
r
Nu
m
b
e
r
o
f
S
w
it
c
h
e
s
a
n
d
M
in
im
u
m
T
H
D,
"
In
ter
n
a
ti
o
n
a
l
J
o
u
rn
a
l
o
f
P
o
we
r
El
e
c
tro
n
ics
a
n
d
Dr
ive
S
y
ste
m
, v
o
l
.
9
,
n
o
.
4
,
p
p
.
1
7
6
5
-
1
7
7
3
,
2
0
1
8
.
[4
]
K.
Dh
i
n
e
sh
k
u
m
a
r,
C.
S
u
b
ra
m
a
n
i
,
A
.
G
e
e
th
a
,
a
n
d
C.
V
im
a
la
,
"
P
e
rf
o
r
m
a
n
c
e
a
n
a
l
y
sis
o
f
P
V
p
o
w
e
re
d
m
u
lt
il
e
v
e
l
in
v
e
rter
,
"
I
n
ter
n
a
ti
o
n
a
l
J
o
u
rn
a
l
o
f
El
e
c
trica
l
a
n
d
Co
mp
u
ter
E
n
g
i
n
e
e
rin
g
,
v
o
l
.
9
,
n
o
.
2
,
p
p
.
7
5
3
-
7
6
0
,
2
0
1
9
.
[5
]
F
.
L
.
L
u
o
a
n
d
H.
Ye
,
"
A
d
v
a
n
c
e
d
DC/A
C
in
v
e
rter
s
a
p
p
li
c
a
ti
o
n
i
n
r
e
n
e
wa
b
le
e
n
e
rg
y
,
"
CRC
Pre
ss
T
a
y
lo
r
&
Fra
n
c
is
Gr
o
u
p
,
2
0
1
3
.
0
5
10
15
20
25
30
35
40
45
0
10
20
30
40
50
60
70
80
X
:
3
Y
:
3
1
.
8
7
T
o
t
a
l
H
a
r
m
o
n
i
c
D
i
s
t
o
r
t
i
o
n
T
H
D
(
%
)
L
e
v
e
l
o
f
I
n
v
e
r
t
e
r
X
:
3
Y
:
7
8
.
5
5
X
:
4
1
Y
:
1
3
.
9
3
X
:
4
1
Y
:
1
.
9
8
X
:
1
9
Y
:
1
7
.
2
7
X
:
5
Y
:
1
7
.
5
3
E
q
u
a
l
P
h
a
s
e
M
e
t
h
o
d
H
a
l
f
H
e
i
g
h
t
M
e
t
h
o
d
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2252
-
8792
I
n
t J
A
p
p
l
P
o
w
er
E
n
g
,
Vo
l.
9
,
No
.
2
,
A
u
g
u
s
t
2020
:
1
5
3
–
1
5
8
158
[6
]
Q.
-
C
.
Z
h
o
n
g
a
n
d
T
.
H
o
r
n
i
k
,
"
C
o
n
t
r
o
l
o
f
p
o
w
e
r
i
n
v
e
r
te
r
s
i
n
r
e
n
e
w
a
b
l
e
e
n
e
rg
y
a
n
d
sm
a
r
t
g
r
i
d
i
n
t
e
g
ra
t
io
n
,
"
N
e
w
Y
o
rk
,
N
Y
,
U
S
A
:
W
i
l
e
y
,
2
0
1
3
.
[7
]
Ja
g
a
b
a
r
S
a
th
ik
M
o
h
d
.
A
li
a
n
d
V
ij
a
y
a
k
u
m
a
r
Krish
n
a
sw
a
m
y
,
"
A
n
a
s
se
ss
m
e
n
t
o
f
re
c
e
n
t
m
u
lt
il
e
v
e
l
in
v
e
rter
to
p
o
lo
g
ies
w
it
h
re
d
u
c
e
d
p
o
w
e
r
e
lec
tro
n
ics
c
o
m
p
o
n
e
n
ts
f
o
r
re
n
e
w
a
b
le
a
p
p
li
c
a
ti
o
n
s
,
"
Ren
e
w
a
b
le
a
n
d
S
u
st
a
in
a
b
le
E
n
e
rg
y
Rev
iews
, v
o
l.
8
2
,
p
a
rt
3
,
p
p
.
3
3
7
9
-
3
3
9
9
,
2
0
1
8
.
[8
]
C.
R.
Ba
lam
u
ru
g
a
n
a
n
d
R.
Be
sn
ra
j
,
"
A
n
a
l
y
sis
o
f
v
a
rio
u
s
c
a
rriers
o
v
e
rlap
p
i
n
g
P
W
M
stra
teg
ies
f
o
r
a
sin
g
le
p
h
a
se
tern
a
ry
m
u
lt
il
e
v
e
l
in
v
e
rte
r
,
"
In
ter
n
a
ti
o
n
a
l
J
o
u
rn
a
l
o
f
A
p
p
l
ied
P
o
we
r E
n
g
i
n
e
e
rin
g
,
v
o
l.
7
,
n
o
.
1
,
p
p
.
2
7
-
39,
2
0
1
8
.
[9
]
S
.
Ch
ik
k
a
m
a
n
d
B.
Ra
n
g
a
n
a
ik
,
"
M
o
d
e
li
n
g
a
n
d
sim
u
latio
n
o
f
a
n
o
v
e
l
th
re
e
-
p
h
a
se
m
u
lt
il
e
v
e
l
in
v
e
rter
w
it
h
in
d
u
c
ti
o
n
m
o
to
r
d
riv
e
,
"
In
ter
n
a
ti
o
n
a
l
J
o
u
rn
a
l
o
f
Res
e
a
rc
h
i
n
A
d
v
e
n
t
T
e
c
h
n
o
l
o
g
y
,
v
o
l
.
2
,
n
o
.
1
1
,
p
p
.
4
0
-
4
4
,
2
0
1
4
.
[1
0
]
M
.
A
ra
v
in
d
a
n
,
V.
Ba
laji,
V
.
S
a
r
a
v
a
n
a
n
a
n
d
M
.
A
ru
m
u
g
a
m
,
"
Ne
u
tral
p
o
i
n
t
c
lam
p
e
d
q
u
a
si
Z
so
u
rc
e
in
v
e
rter
f
o
r
p
h
o
to
v
o
lt
a
ic sy
ste
m
s
,
"
In
ter
n
a
ti
o
n
a
l
J
o
u
rn
a
l
o
f
A
p
p
li
e
d
P
o
we
r E
n
g
in
e
e
rin
g
,
v
o
l.
8
,
n
o
.
3
,
p
p
.
2
7
7
-
2
8
6
,
2
0
1
9
.
[1
1
]
R.
T
a
leb
,
D.
Be
n
y
o
u
c
e
f
,
M
.
He
l
a
im
i,
Z.
Bo
u
d
jem
a
a
,
a
n
d
H.
S
a
id
i
,
"
Ca
sc
a
d
e
d
H
-
b
rid
g
e
a
s
y
m
m
e
tri
c
a
l
se
v
e
n
-
lev
e
l
in
v
e
rter u
sin
g
T
HIP
W
M
f
o
r
h
ig
h
p
o
w
e
r
in
d
u
c
ti
o
n
m
o
to
r
,
"
En
e
rg
y
Pro
c
e
d
ia
,
v
o
l.
7
4
,
p
p
.
8
4
4
–
8
5
3
,
2
0
1
5
.
[1
2
]
M
.
A
h
m
e
d
,
M
.
Ora
b
i,
S
.
G
h
o
n
e
im
,
M
.
A
lh
a
rth
i,
F
.
S
a
lem
,
B.
A
la
m
ri
,
a
n
d
S
.
M
e
k
h
il
e
f
,
"
S
e
lec
ti
v
e
h
a
r
m
o
n
ic
e
li
m
in
a
ti
o
n
m
e
th
o
d
f
o
r
u
n
e
q
u
a
l
D
C
so
u
rc
e
s o
f
m
u
lt
il
e
v
e
l
in
v
e
rters
,
"
Au
to
ma
t
ika
,
v
o
l
.
6
0
,
n
o
.
4
,
p
p
.
3
7
8
-
3
8
4
,
2
0
1
9
.
[1
3
]
A
.
E.
T
o
u
b
a
l
M
a
a
m
a
r,
M
.
He
l
a
im
i,
R.
T
a
leb
,
a
n
d
F
.
Ch
a
b
n
i
,
"
An
a
ly
sis
a
n
d
im
p
le
m
e
n
tatio
n
o
f
h
a
lf
-
b
rid
g
e
se
rie
s
re
so
n
a
n
t
in
v
e
rter
u
si
n
g
A
rd
u
in
o
,
"
ICCEE
’1
8
I
n
ter
n
a
ti
o
n
a
l
C
o
n
fer
e
n
c
e
o
n
C
o
mm
u
n
ica
ti
o
n
s
a
n
d
El
e
c
trica
l
En
g
i
n
e
e
rin
g
El
-
Ou
e
d
,
A
lg
e
ria,
2
0
1
8
.
[1
4
]
M
.
V
ij
e
h
,
M
.
Re
z
a
n
e
jad
,
E.
S
a
m
a
d
a
e
i
,
a
n
d
K.
Be
rti
lsso
n
,
"
A
g
e
n
e
ra
l
re
v
ie
w
o
f
m
u
lt
il
e
v
e
l
in
v
e
rte
rs
b
a
se
d
o
n
m
a
in
su
b
m
o
d
u
les
:
stru
c
t
u
ra
l
p
o
in
t
o
f
v
iew
,
"
in
IEE
E
T
ra
n
sa
c
ti
o
n
s
o
n
P
o
we
r
El
e
c
tro
n
ics
,
v
o
l.
3
4
,
n
o
.
1
0
,
p
p
.
9
4
7
9
-
9
5
0
2
,
Oc
t.
2
0
1
9
.
[1
5
]
A.
E.
T
o
u
b
a
l
M
a
a
m
a
r,
M
.
He
laim
i,
R.
Tale
b
,
H.
M
o
u
lo
u
d
j,
O.
El
a
m
ri
,
a
n
d
A
.
Ga
d
o
u
m
,
"
M
a
th
e
m
a
ti
c
a
l
a
n
a
ly
sis
o
f
N
-
R
a
lg
o
rit
h
m
f
o
r
e
x
p
e
rime
n
tal
im
p
le
m
e
n
tatio
n
o
f
S
HE
P
W
M
c
o
n
tro
l
o
n
sin
g
le
-
p
h
a
se
in
v
e
rter
,
"
In
ter
n
a
ti
o
n
a
l
J
o
u
rn
a
l
o
f
E
n
g
i
n
e
e
rin
g
T
re
n
d
s
a
n
d
T
e
c
h
n
o
l
o
g
y
,
v
o
l.
6
8
,
n
o
.
2
,
p
p
.
9
-
1
6
,
2
0
2
0
.
[1
6
]
S
.
S
.
Ra
n
g
a
ra
jan
,
E.
R.
Co
ll
i
n
s
,
a
n
d
J.
C.
F
o
x
,
"
De
tu
n
i
n
g
o
f
h
a
r
m
o
n
ic res
o
n
a
n
t
m
o
d
e
s in
a
c
c
o
rd
a
n
c
e
w
it
h
IEE
E
5
1
9
sta
n
d
a
rd
in
a
n
e
x
e
m
p
lar
y
n
o
rth
a
m
e
rica
n
d
istri
b
u
ti
o
n
sy
ste
m
w
it
h
P
V
a
n
d
w
in
d
,
"
2
0
1
7
IE
EE
6
t
h
In
ter
n
a
ti
o
n
a
l
Co
n
fer
e
n
c
e
o
n
Ren
e
wa
b
le E
n
e
rg
y
Res
e
a
rc
h
a
n
d
A
p
p
li
c
a
ti
o
n
s (
ICRE
RA
)
, S
a
n
Die
g
o
,
CA
,
2
0
1
7
,
p
p
.
4
3
5
-
4
4
0
.
B
I
O
G
RAP
H
I
E
S O
F
AUTH
O
RS
Alla
Ed
d
in
e
To
u
b
a
l
M
a
a
m
a
r
re
c
e
iv
e
d
th
e
B.
En
g
.
d
e
g
re
e
a
n
d
m
a
ste
r
d
e
g
re
e
in
e
lec
tri
c
a
l
e
n
g
in
e
e
rin
g
f
ro
m
Un
iv
e
rsit
y
o
f
Ak
li
M
o
h
a
n
d
Ou
l
h
a
d
j
Bo
u
ira
(U
A
M
OB),
A
l
g
e
ria,
in
2
0
1
4
a
n
d
2
0
1
6
,
re
sp
e
c
ti
v
e
ly
.
H
e
is
c
u
rre
n
tl
y
w
it
h
th
e
De
p
a
rtm
e
n
t
o
f
El
e
c
tri
c
a
l
En
g
in
e
e
rin
g
a
n
d
(L
G
EE
R)
L
a
b
o
ra
to
ry
,
H
a
ss
ib
a
Be
n
b
o
u
a
li
Un
iv
e
rsit
y
o
f
Ch
le
f
,
A
l
g
e
ria.
His
re
se
a
rc
h
in
tere
sts
in
c
lu
d
e
th
e
d
e
sig
n
,
a
n
a
ly
sis
a
n
d
c
o
n
tr
o
l
o
f
p
o
w
e
r
e
lec
tro
n
ic
c
o
n
v
e
rters
,
e
lec
tr
ic
m
a
c
h
in
e
c
o
n
tro
l,
m
u
lt
il
e
v
e
l
in
v
e
rters
,
a
rti
f
icia
l
n
e
u
ra
l
n
e
tw
o
rk
(AN
Ns
),
ro
b
o
ti
c
s tec
h
n
o
l
o
g
y
,
r
e
n
e
wa
b
le en
e
rg
y
tec
h
n
o
lo
g
ies
.
M
’h
a
m
e
d
H
e
l
a
i
m
i
re
c
e
i
v
e
d
th
e
e
n
g
in
e
e
rin
g
d
e
g
re
e
in
2
0
0
1
,
a
n
d
m
a
g
i
ste
r
d
e
g
re
e
in
e
lec
tri
c
a
l
e
n
g
in
e
e
rin
g
in
2
0
0
4
f
ro
m
Ha
s
sib
a
Be
n
b
o
u
a
li
Un
iv
e
rsit
y
,
Ch
lef
(UH
BC),
A
lg
e
ria,
a
n
d
re
c
e
iv
e
d
th
e
P
h
.
D
.
d
e
g
re
e
in
e
lec
tri
c
a
l
e
n
g
in
e
e
rin
g
f
ro
m
th
e
Un
iv
e
rsit
y
o
f
S
c
ien
c
e
s
a
n
d
T
e
c
h
n
o
lo
g
y
o
f
Ora
n
(UST
O),
Bir
El
Djir,
A
lg
e
ria,
in
2
0
1
4
.
He
j
o
in
e
d
th
e
De
p
a
rt
m
e
n
t
o
f
El
e
c
tri
c
a
l
E
n
g
in
e
e
rin
g
,
Ch
lef
Un
iv
e
rsit
y
,
in
2
0
0
4
.
His
sc
ien
ti
f
ic
w
o
rk
is
re
l
a
ted
to
d
y
n
a
m
ic
m
o
d
e
li
n
g
a
n
d
c
o
n
tr
o
l
o
f
n
o
n
li
n
e
a
r
sy
ste
m
s,
g
e
n
e
ti
c
a
lg
o
rit
h
m
s
a
n
d
a
rti
f
icia
l
in
tell
ig
e
n
c
e
,
i
n
d
u
c
t
io
n
h
e
a
ti
n
g
a
p
p
li
c
a
ti
o
n
s
a
n
d
p
o
w
e
r
e
lec
tro
n
ics
.
Ra
c
h
id
Ta
le
b
re
c
e
iv
e
d
th
e
M
.
S
.
d
e
g
re
e
in
e
lec
tri
c
a
l
e
n
g
in
e
e
rin
g
i
n
2
0
0
4
,
f
ro
m
Ha
ss
ib
a
Be
n
b
o
u
a
li
Un
iv
e
rsity
,
a
n
d
th
e
P
h
.
D.
d
e
g
re
e
in
e
lec
tri
c
a
l
e
n
g
in
e
e
rin
g
in
2
0
1
1
,
f
ro
m
Djil
lali
L
iab
e
s
Un
iv
e
rsit
y
,
A
l
g
e
ria.
He
is
c
u
rre
n
tl
y
a
n
a
ss
o
c
iate
p
ro
f
e
ss
o
r
in
th
e
De
p
a
rtm
e
n
t
o
f
El
e
c
tri
c
a
l
En
g
in
e
e
rin
g
,
Ha
ss
ib
a
Be
n
b
o
u
a
li
Un
iv
e
rsity
,
c
h
le
f
,
A
l
g
e
ria
.
He
is
a
lso
tea
m
lea
d
e
r
o
f
th
e
L
a
b
o
ra
to
ire
G
é
n
ie
El
e
c
tri
q
u
e
e
t
En
e
rg
ies
Re
n
o
u
v
e
lab
les
(L
G
E
ER)
L
a
b
o
ra
to
ry
.
His
re
se
a
rc
h
in
tere
sts
in
c
lu
d
e
p
o
w
e
r
e
lec
tr
o
n
ics
,
a
rti
f
icia
l
in
telli
g
e
n
c
e
c
o
n
tro
l
,
h
e
u
r
isti
c
o
p
ti
m
iza
ti
o
n
a
lg
o
rit
h
m
s,
a
n
d
c
o
n
t
ro
l
th
e
o
ry
o
f
c
o
n
v
e
rters
f
o
r
re
n
e
w
a
b
le en
e
rg
y
s
o
u
rc
e
s
.
Abd
e
la
tif
G
a
d
o
u
m
wa
s
b
o
rn
in
Be
jaia
,
A
lg
e
ria,
in
1
9
8
8
.
Re
c
e
iv
e
d
t
h
e
P
h
.
D.
d
e
g
re
e
in
e
lec
tri
c
a
l
e
n
g
in
e
e
rin
g
in
2
0
1
9
,
f
ro
m
Ha
s
sib
a
Be
n
b
o
u
a
li
Un
iv
e
rsity
o
f
Ch
le
f
,
Ch
le
f
,
A
lg
e
ria.
Hi
s
c
u
rre
n
t
re
se
a
rc
h
in
tere
sts
in
c
lu
d
e
g
a
s
d
isc
h
a
rg
e
s
a
n
d
m
o
d
e
li
n
g
o
f
lo
w
-
p
re
ss
u
re
d
isc
h
a
rg
e
a
n
d
th
e
ir
a
p
p
li
c
a
ti
o
n
s.
Evaluation Warning : The document was created with Spire.PDF for Python.