I
nte
rna
t
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na
l J
o
urna
l o
f
Appl
ied P
o
wer
E
ng
i
neer
ing
(
I
J
AP
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)
Vo
l.
14
,
No
.
4
,
Dec
em
b
er
20
25
,
p
p
.
8
7
9
~
8
9
2
I
SS
N:
2252
-
8
7
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2
,
DOI
:
1
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.
1
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/ijap
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.
v
14
.
i
4
.
pp
879
-
892
879
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o
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h
ttp
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//
ija
p
e.
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Im
pro
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he ad
a
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of
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Art
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I
nfo
AB
S
T
RAC
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A
r
ticle
his
to
r
y:
R
ec
eiv
ed
Oct
1
6
,
2
0
2
3
R
ev
is
ed
Au
g
7
,
2
0
2
5
Acc
ep
ted
Oct
1
6
,
2
0
2
5
As
m
o
re
a
n
d
m
o
re
non
-
li
n
e
a
r
lo
a
d
s a
re
u
se
d
in
in
d
u
strial
a
p
p
li
c
a
ti
o
n
s,
p
o
we
r
q
u
a
li
t
y
p
r
o
b
lem
s
b
e
c
o
m
e
m
o
re
se
rio
u
s
.
T
o
a
d
d
re
ss
th
is
c
h
a
ll
e
n
g
e
,
a
ro
b
u
st
n
o
n
li
n
e
a
r
c
o
n
t
ro
l
stra
teg
y
is
in
tr
o
d
u
c
e
d
u
si
n
g
a
n
a
c
ti
v
e
p
o
we
r
fil
te
r
(APF
)
to
e
n
h
a
n
c
e
th
e
p
o
we
r
q
u
a
li
t
y
o
f
th
e
th
re
e
-
p
h
a
se
n
e
u
tral
v
o
lt
a
g
e
.
T
h
e
sy
ste
m
e
m
p
lo
y
s
a
c
o
n
tr
o
l
a
lg
o
rit
h
m
t
a
il
o
re
d
fo
r
a
th
re
e
-
p
h
a
se
sp
li
t
-
c
a
p
a
c
it
o
r
in
v
e
rter,
w
h
ich
e
li
m
i
n
a
tes
h
i
g
h
-
o
rd
e
r
h
a
rm
o
n
ics
t
h
ro
u
g
h
a
v
o
lt
a
g
e
so
u
rc
e
in
v
e
rter
(VSI)
e
q
u
i
p
p
e
d
wit
h
a
n
LCL
fil
ter.
Th
e
g
rid
-
si
d
e
c
o
m
p
o
n
e
n
ts
o
f
t
h
e
LCL
fil
ter
a
re
in
c
o
r
p
o
ra
te
d
in
t
o
a
slid
i
n
g
m
o
d
e
c
o
n
tro
l
fra
m
e
wo
rk
to
m
in
imiz
e
o
sc
il
latio
n
s
wh
il
e
m
a
in
tai
n
in
g
p
e
rfo
rm
a
n
c
e
.
Ad
d
it
i
o
n
a
ll
y
,
th
e
d
-
q
-
0
tran
sf
o
rm
a
ti
o
n
with
in
th
e
s
y
n
c
h
ro
n
o
u
s
re
fe
re
n
c
e
fra
m
e
is
a
p
p
li
e
d
to
e
ffe
c
ti
v
e
ly
m
a
n
a
g
e
t
h
e
se
c
o
n
d
h
a
rm
o
n
ic
c
o
m
p
o
n
e
n
t
.
I
n
a
d
d
it
i
o
n
,
th
e
li
n
e
a
r
fe
e
d
b
a
c
k
in
p
u
t
-
o
u
tp
u
t
sl
id
i
n
g
m
o
d
e
fa
c
il
it
a
tes
th
e
c
o
n
tr
o
l
sy
ste
m
.
Th
is
sy
ste
m
c
a
n
h
e
lp
d
e
c
re
a
se
to
tal
h
a
rm
o
n
ic
d
isto
rt
io
n
(THD)
to
m
e
e
t
IEE
E
-
5
1
9
sta
n
d
a
rd
s.
Th
is
m
e
th
o
d
d
e
m
o
n
st
ra
tes
it
s
e
ffe
c
ti
v
e
n
e
ss
th
r
o
u
g
h
s
imu
latio
n
re
su
lt
s,
re
d
u
c
in
g
THD
t
o
les
s
t
h
a
n
5
%
a
n
d
d
e
fe
a
ti
n
g
p
re
v
io
u
s
m
e
th
o
d
s
d
e
sp
it
e
stil
l
u
sin
g
sim
p
le al
g
o
r
it
h
m
s
.
K
ey
w
o
r
d
s
:
Activ
e
p
o
wer
f
ilter
Ad
ap
tiv
e
co
n
tr
o
l
Po
wer
q
u
ality
Sli
d
in
g
m
o
d
e
Sli
p
co
n
tr
o
l
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
:
L
em
in
h
T
h
ie
n
Hu
y
n
h
Facu
lty
o
f
E
n
g
in
ee
r
in
g
an
d
T
e
c
hn
o
lo
g
y
,
Saig
o
n
Un
iv
er
s
ity
2
7
3
An
D
u
o
n
g
Vu
o
n
g
S
tr
ee
t,
W
ar
d
C
h
o
Qu
an
,
Ho
C
h
i
Min
h
C
ity
,
Vietn
am
E
m
ail:
lem
in
h
th
ien
.
h
u
y
n
h
@
s
g
u
.
ed
u
.
v
n
1.
I
NT
RO
D
UCT
I
O
N
I
n
to
d
a
y
’
s
in
d
u
s
tr
ial
lan
d
s
ca
p
e,
th
e
in
cr
ea
s
in
g
u
s
e
o
f
n
o
n
li
n
ea
r
lo
ad
s
h
as
a
s
ig
n
if
ican
t
i
m
p
ac
t
o
n
p
o
wer
q
u
ality
.
T
h
ese
s
y
s
tem
s
ar
e
r
eq
u
ir
ed
to
co
m
p
l
y
with
s
tan
d
ar
d
s
s
u
c
h
as
I
E
E
E
5
1
9
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2
0
1
4
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I
E
C
6
1
0
0
0
-
3
-
2
,
an
d
I
E
C
6
1
0
0
0
-
3
-
4
[
1
]
.
W
ith
ad
v
an
ce
m
e
n
ts
in
m
icr
o
el
ec
tr
o
n
ic
tech
n
o
lo
g
y
,
s
u
ch
lo
ad
s
p
r
o
d
u
ce
n
o
n
-
s
in
u
s
o
id
al
cu
r
r
e
n
ts
th
at
in
tr
o
d
u
ce
h
ar
m
o
n
ics
in
to
t
h
e
p
o
wer
s
y
s
tem
,
m
ak
in
g
h
ig
h
-
q
u
ality
p
o
wer
s
u
p
p
lies
ess
en
tial.
Activ
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p
o
wer
f
ilter
s
(
APFs
)
n
o
t
o
n
ly
im
p
r
o
v
e
p
o
wer
q
u
alit
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b
u
t
also
s
er
v
e
as
a
n
ef
f
ec
tiv
e
s
o
lu
tio
n
f
o
r
m
itig
atin
g
h
a
r
m
o
n
ics
an
d
o
th
er
d
is
tu
r
b
an
ce
s
.
As
a
r
esu
l
t,
APF
co
n
tr
o
ller
s
h
av
e
r
ap
i
d
ly
b
ec
o
m
e
a
wid
ely
ad
o
p
ted
to
o
l
f
o
r
o
p
tim
izin
g
p
o
wer
q
u
ality
.
Fo
r
i
n
s
tan
ce
,
a
d
ed
icate
d
th
r
ee
-
p
h
ase
in
v
er
ter
with
a
cu
r
r
en
t
co
n
tr
o
ller
h
as
b
ee
n
d
ev
elo
p
ed
to
en
h
an
ce
p
o
wer
q
u
ality
f
o
r
u
n
b
alan
ce
d
n
o
n
lin
ea
r
lo
ad
s
,
el
im
in
atin
g
th
e
n
ee
d
f
o
r
an
ad
d
itio
n
al
s
witch
in
g
b
r
a
n
ch
to
r
e
g
u
late
n
e
u
tr
al
cu
r
r
en
t
[
2
]
.
L
ev
er
ag
in
g
v
o
ltag
e
s
o
u
r
ce
i
n
v
er
ter
s
(
VSI
s
)
,
APFs
en
h
a
n
ce
p
o
wer
q
u
ality
b
y
r
esh
a
p
in
g
in
p
u
t
h
ar
m
o
n
ic
r
ef
e
r
en
ce
s
ig
n
als
[
3
]
,
[
4
]
.
C
o
m
p
lex
n
o
n
lin
ea
r
l
o
ad
a
p
p
licatio
n
s
—
s
u
ch
as
el
ec
tr
ic
tr
an
s
p
o
r
tatio
n
s
y
s
tem
s
an
d
s
o
lar
p
o
we
r
in
s
tallatio
n
s
—
o
f
ten
d
em
an
d
h
ig
h
l
y
s
tab
le
an
d
clea
n
p
o
wer
s
u
p
p
lies
[
5
]
-
[
9
]
.
T
h
ese
lo
ad
s
m
ay
r
esu
lt
in
eith
er
b
alan
ce
d
o
r
u
n
b
alan
ce
d
th
r
e
e
-
p
h
ase
co
n
d
itio
n
s
th
at
f
ail
to
m
ee
t
s
tan
d
a
r
d
r
eq
u
ir
em
e
n
ts
,
lead
in
g
to
th
e
em
er
g
en
ce
o
f
ze
r
o
-
s
eq
u
en
ce
an
d
r
ev
e
r
s
e
-
s
eq
u
en
ce
cu
r
r
en
t
s
.
T
h
ese
n
eg
ativ
e
-
s
eq
u
en
ce
co
m
p
o
n
en
ts
ca
n
ca
u
s
e
m
o
to
r
o
v
er
h
ea
tin
g
,
tr
an
s
f
o
r
m
er
s
atu
r
atio
n
,
p
o
wer
o
u
ta
g
es,
an
d
co
m
p
r
o
m
is
e
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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N
:
2
2
5
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-
8
7
9
2
I
n
t J Ap
p
l Po
wer
E
n
g
,
Vo
l.
14
,
No
.
4
,
Dec
em
b
er
20
25
:
8
7
9
-
892
880
th
e
s
af
ety
o
f
n
eu
tr
al
lin
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lv
es
u
s
in
g
a
∆/Y
tr
an
s
f
o
r
m
er
,
wh
er
e
th
e
∆
s
id
e
co
n
n
ec
ts
to
th
e
VS
I
an
d
th
e
Y
s
id
e
to
th
e
lo
ad
[
8
]
-
[
1
1
]
.
Ho
wev
e
r
,
th
is
ap
p
r
o
a
ch
r
eq
u
ir
es
a
b
u
lk
y
,
h
ea
v
y
,
a
n
d
c
o
s
tly
tr
an
s
f
o
r
m
e
r
.
An
alter
n
ativ
e
tech
n
iq
u
e
u
tili
ze
s
th
r
ee
-
p
h
ase
s
p
lit
-
ca
p
ac
ito
r
VSI
s
an
d
f
o
u
r
-
p
in
VSI
s
,
co
m
p
r
is
in
g
a
to
tal
o
f
1
6
s
witch
es.
I
n
[
1
2
]
,
[
1
3
]
,
t
h
e
f
o
u
r
-
p
in
i
n
v
er
ter
d
esig
n
in
clu
d
es
two
ad
d
itio
n
al
s
witch
es,
wh
ich
ar
e
ess
en
tial f
o
r
im
p
lem
en
tin
g
c
o
m
p
lex
3
D
s
p
ac
e
v
ec
t
o
r
m
o
d
u
l
atio
n
.
A
th
r
ee
-
p
h
ase,
f
o
u
r
-
wir
e
s
y
s
tem
ca
n
also
b
e
d
er
iv
ed
f
r
o
m
a
co
n
v
en
tio
n
al
th
r
ee
-
p
h
ase,
th
r
ee
-
wir
e
VSI
b
y
in
co
r
p
o
r
ati
n
g
a
DC
v
o
lt
ag
e
d
iv
i
d
er
.
I
n
th
is
co
n
f
ig
u
r
atio
n
,
th
e
n
eu
tr
al
p
o
in
t ser
v
es a
s
th
e
f
o
u
r
th
wir
e,
en
a
b
lin
g
s
p
ac
e
v
ec
to
r
m
o
d
u
latio
n
.
T
h
is
s
etu
p
allo
ws
f
o
r
o
u
tp
u
t v
o
ltag
e
r
e
g
u
latio
n
a
n
d
f
ac
ilit
ates th
e
f
lo
w
o
f
ze
r
o
-
s
eq
u
en
ce
cu
r
r
en
t th
r
o
u
g
h
th
e
n
eu
tr
al
co
n
d
u
cto
r
.
R
ec
en
t
r
esear
ch
h
as
f
o
cu
s
e
d
o
n
m
ain
tain
i
n
g
s
tab
le
o
u
tp
u
t
f
r
o
m
t
h
r
ee
-
p
h
ase
in
v
e
r
ter
s
u
n
d
er
u
n
b
alan
ce
d
lo
ad
co
n
d
itio
n
s
.
Fo
r
ex
am
p
le
,
C
h
en
et
a
l
.
[
1
4
]
em
p
lo
y
ed
d
ea
d
b
ea
t
co
n
tr
o
l
with
cu
r
r
e
n
t
an
d
v
o
ltag
e
f
ee
d
b
ac
k
to
co
m
p
en
s
ate
f
o
r
co
il
lo
s
s
es,
th
o
u
g
h
th
i
s
m
eth
o
d
d
o
es
n
o
t
ad
d
r
ess
h
ar
m
o
n
ic
d
is
to
r
tio
n
.
Gen
g
et
a
l
.
[
1
5
]
in
tr
o
d
u
ce
d
r
e
p
etitiv
e
co
n
tr
o
l
to
s
u
p
p
r
ess
p
e
r
io
d
ic
d
is
tu
r
b
an
ce
s
in
n
o
n
lin
e
ar
s
y
s
tem
s
,
b
u
t
its
s
lo
w
r
esp
o
n
s
e
an
d
lim
ited
a
p
p
licab
ilit
y
p
o
s
e
ch
allen
g
es.
Oth
er
co
n
tr
o
l
s
tr
ateg
ies,
s
u
ch
as
s
lid
in
g
m
o
d
e
c
o
n
tr
o
l,
h
av
e
d
e
m
o
n
s
tr
ated
im
p
r
o
v
ed
APF
p
er
f
o
r
m
an
ce
u
n
d
e
r
u
n
ce
r
tain
g
r
id
co
n
d
itio
n
s
[
3
]
,
[
1
1
]
,
[
1
6
]
-
[
2
0
]
.
Dis
cr
ete
s
lid
in
g
m
o
d
e
co
n
tr
o
l
o
f
f
er
s
ex
ce
llen
t
d
y
n
am
ic
r
esp
o
n
s
e,
r
ed
u
ce
d
s
en
s
itiv
ity
to
lo
ad
v
ar
iatio
n
s
,
an
d
r
o
b
u
s
t
s
ta
b
ilit
y
.
T
o
m
an
ag
e
asy
m
m
etr
ic
th
r
ee
-
p
h
ase
s
ig
n
a
ls
,
th
e
s
y
m
m
etr
ic
co
n
s
tan
t
-
d
i
s
tr
ib
u
tio
n
m
eth
o
d
is
u
s
ed
in
co
n
ju
n
ctio
n
with
a
p
r
o
p
o
r
tio
n
al
-
in
te
g
r
al
(
PI)
co
n
t
r
o
ller
to
b
alan
ce
in
v
er
ter
o
u
tp
u
t
v
o
ltag
e
[
2
1
]
.
Ho
wev
er
,
PI
co
n
tr
o
ller
s
in
cr
ea
s
e
co
m
p
u
tatio
n
al
c
o
m
p
lex
ity
,
m
a
k
in
g
th
em
m
o
r
e
s
u
itab
le
f
o
r
n
o
n
lin
ea
r
an
d
u
n
b
alan
ce
d
lo
ad
s
ce
n
ar
io
s
[
2
2
]
.
T
h
is
p
ap
er
p
r
esen
ts
a
d
etail
ed
in
v
esti
g
atio
n
in
to
th
e
d
e
v
elo
p
m
en
t
an
d
im
p
lem
en
tati
o
n
o
f
th
e
lin
ea
r
izatio
n
f
ee
d
b
ac
k
in
p
u
t
-
o
u
tp
u
t
s
lid
in
g
m
o
d
e
(
L
F
-
I
OSM)
co
n
tr
o
l
m
eth
o
d
f
o
r
APFs
.
E
x
ten
s
iv
e
s
im
u
latio
n
r
esu
lts
v
alid
ate
th
e
ef
f
ec
tiv
e
n
ess
o
f
th
e
p
r
o
p
o
s
ed
ap
p
r
o
a
ch
ac
r
o
s
s
d
iv
e
r
s
e
o
p
e
r
atin
g
co
n
d
itio
n
s
.
R
ec
en
t
s
tu
d
ies
[
3
]
-
[
7
]
p
r
o
v
id
e
a
s
o
lid
f
o
u
n
d
atio
n
f
o
r
th
is
wo
r
k
.
Fig
u
r
e
1
illu
s
tr
ates
th
e
s
ch
em
atic
o
f
th
e
ac
tiv
e
p
o
wer
f
ilter
s
y
s
tem
.
Fig
u
r
e
1
.
Sli
d
in
g
m
o
d
e
co
n
tr
o
l f
o
r
th
r
ee
-
p
h
ase
ac
tiv
e
p
o
wer
f
ilter
s
y
s
tem
s
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J Ap
p
l Po
wer
E
n
g
I
SS
N:
2252
-
8
7
9
2
I
mp
r
o
vin
g
th
e
a
d
a
p
ta
b
ilit
y
o
f
an
a
ctive
p
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er filter
u
s
in
g
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r
iz
a
tio
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… (
Lemin
h
Th
ien
Hu
yn
h
)
881
Ass
u
m
in
g
th
e
th
r
ee
-
p
h
ase
g
e
n
er
ato
r
is
u
n
b
alan
ce
d
,
Kir
ch
h
o
f
f
’
s
f
ir
s
t
law
ca
n
b
e
ap
p
lied
to
th
e
s
y
s
tem
.
Usi
n
g
th
is
law,
th
e
s
in
g
le
-
p
h
ase
cu
r
r
en
t c
an
b
e
ca
lcu
lated
as sh
o
wn
in
(
1
)
.
=
−
(
1
)
L
et
i
ga
r
ep
r
esen
t
th
e
cu
r
r
en
t
i
n
p
h
ase
a
,
i
La
th
e
lo
ad
c
u
r
r
e
n
t
in
p
h
ase
a
,
an
d
i
Fa
th
e
cu
r
r
en
t
in
jecte
d
b
y
th
e
ac
tiv
e
p
o
wer
f
ilter
(
APF)
in
to
th
e
g
r
id
in
p
h
ase
a
.
T
h
e
A
PF
g
en
er
ates
cu
r
r
en
ts
i
Fa
,
i
Fb
,
an
d
i
Fc
th
at
s
atis
f
y
Kir
ch
h
o
f
f
’
s
f
ir
s
t
law,
as
d
escr
ib
ed
in
(
1
)
-
(
3
)
.
T
h
ese
APF
-
g
en
er
ated
c
u
r
r
en
ts
e
f
f
ec
tiv
ely
el
im
in
ate
m
o
s
t
o
f
th
e
h
ig
h
-
o
r
d
er
h
ar
m
o
n
ics
p
r
o
d
u
c
ed
b
y
n
o
n
lin
ea
r
lo
a
d
s
.
Ap
p
l
y
in
g
th
e
s
am
e
p
r
in
ci
p
le
to
p
h
ases
b
an
d
c
,
th
e
co
r
r
esp
o
n
d
in
g
c
u
r
r
e
n
ts
f
o
r
th
e
s
e
p
h
ases
ar
e
also
d
eter
m
in
ed
u
s
in
g
(
2
)
an
d
(
3
)
.
=
−
(
2
)
=
−
(
3
)
I
n
a
f
o
u
r
-
wir
e
s
y
s
tem
with
u
n
b
alan
ce
d
lo
ad
s
,
th
e
th
r
ee
-
p
h
ase
p
o
wer
s
o
u
r
ce
b
ec
o
m
es
d
is
to
r
ted
b
y
h
ar
m
o
n
ics,
lead
in
g
to
r
e
d
u
ce
d
p
o
wer
q
u
ality
an
d
p
o
ten
tial
d
am
ag
e
to
elec
tr
ical
an
d
elec
tr
o
n
ic
d
ev
ices.
T
h
es
e
h
ar
m
o
n
ics ca
n
b
e
r
e
p
r
esen
ted
b
y
(
4
)
,
wh
er
e
n
d
e
n
o
tes th
e
h
a
r
m
o
n
ic
o
r
d
er
[
2
3
]
.
(
)
=
√
2
0
(
+
0
)
+
√
2
+
(
+
+
)
+
√
2
−
(
+
−
)
(
)
=
√
2
0
(
+
0
)
+
√
2
+
(
+
+
−
2
3
)
+
√
2
−
(
+
−
+
2
3
)
(
)
=
√
2
0
(
+
0
)
+
√
2
+
(
+
+
+
2
3
)
+
√
2
−
(
+
−
−
2
3
)
(
4
)
T
h
e
p
r
im
ar
y
o
b
jectiv
e
o
f
th
e
ac
tiv
e
p
o
wer
f
ilter
(
APF)
s
y
s
tem
is
to
r
ed
u
ce
to
tal
h
ar
m
o
n
i
c
d
is
to
r
tio
n
(
T
HD)
.
No
n
-
lin
ea
r
lo
a
d
s
g
en
er
ate
h
ig
h
-
f
r
eq
u
en
cy
co
m
p
o
n
en
ts
th
at
co
n
tr
ib
u
te
to
T
HD
,
wh
ich
is
ca
lled
THD
sm
.
Acc
o
r
d
in
g
to
I
E
E
E
Std
.
5
1
9
,
th
e
THD
sm
o
f
th
e
s
o
u
r
ce
cu
r
r
en
t
s
h
o
u
l
d
b
e
k
ep
t
b
elo
w
5
%.
THD
sm
q
u
an
tifie
s
th
e
h
ar
m
o
n
ic
d
is
to
r
tio
n
b
y
ca
lcu
latin
g
t
h
e
r
atio
o
f
th
e
s
u
m
o
f
all
h
ar
m
o
n
ic
cu
r
r
e
n
ts
(
I
₂,
I
₃,
I
₄,
.
.
.
,
I
ₖ
)
to
th
e
f
u
n
d
am
en
tal
c
u
r
r
en
t
(
I₁
)
,
ex
p
r
ess
ed
as a
p
er
ce
n
tag
e,
as
illu
s
tr
ated
in
(
4
)
.
=
√
2
2
+
3
2
+
4
2
+
.
.
.
+
2
1
2
.
100%
(
5
)
T
h
is
p
ap
er
in
tr
o
d
u
ce
s
an
ad
v
an
ce
d
n
o
n
lin
ea
r
c
o
n
tr
o
l
s
tr
ateg
y
f
o
r
a
th
r
ee
-
p
h
ase
v
o
ltag
e
s
o
u
r
ce
in
v
er
ter
(
VSI
)
,
u
tili
zin
g
in
p
u
t
–
s
lip
f
ee
d
b
ac
k
to
ac
h
iev
e
o
u
tp
u
t
lin
ea
r
izatio
n
u
n
d
er
b
o
th
b
alan
ce
d
an
d
u
n
b
alan
ce
d
n
o
n
lin
ea
r
lo
ad
co
n
d
itio
n
s
.
T
o
s
tr
ea
m
lin
e
c
o
m
p
u
tatio
n
an
d
r
e
d
u
ce
h
ar
d
wa
r
e
co
m
p
lex
ity
,
s
lid
in
g
m
o
d
e
co
n
tr
o
l
th
e
o
r
y
is
em
p
lo
y
ed
.
T
h
e
p
r
o
p
o
s
ed
m
eth
o
d
ef
f
ec
tiv
ely
s
u
p
p
r
ess
es
h
ig
h
-
o
r
d
er
h
a
r
m
o
n
ics
an
d
lo
wer
s
to
tal
h
ar
m
o
n
ic
d
is
to
r
ti
o
n
(
T
HD)
in
n
o
n
lin
ea
r
lo
a
d
s
,
as v
alid
ated
b
y
s
im
u
latio
n
r
es
u
lts
.
T
h
e
s
tr
u
ctu
r
e
o
f
th
e
p
ap
e
r
is
as
f
o
llo
ws:
s
ec
tio
n
2
p
r
esen
ts
th
e
m
at
h
em
ati
ca
l
m
o
d
elin
g
,
s
ec
tio
n
3
d
etai
ls
th
e
in
p
u
t
–
o
u
t
p
u
t
lin
ea
r
izatio
n
f
ee
d
b
ac
k
s
y
s
tem
m
o
d
el,
s
ec
tio
n
4
s
h
o
wca
s
es
th
e
s
im
u
latio
n
o
u
tco
m
es,
an
d
s
ec
tio
n
5
p
r
o
v
id
es
d
is
cu
s
s
io
n
an
d
co
n
cl
u
s
io
n
s
.
2.
M
O
DE
L
-
B
A
SE
D
SYS
T
E
M
ANALY
SI
S
Fig
u
r
e
2
illu
s
tr
ates
th
e
th
r
ee
-
p
h
ase
ac
tiv
e
p
o
wer
f
ilter
(
APF)
co
n
f
ig
u
r
ed
with
a
s
p
li
t
-
ca
p
ac
ito
r
n
eu
tr
al
p
o
in
t
v
o
ltag
e
s
o
u
r
ce
i
n
v
er
ter
(
VSI
)
to
p
o
lo
g
y
.
I
n
th
e
d
-
q
-
0
c
o
o
r
d
in
ate
s
y
s
tem
[
2
4
]
,
[
2
5
]
,
ac
c
o
r
d
i
n
g
t
o
th
e
u
n
b
ala
n
ce
d
lo
a
d
s
,
th
e
ze
r
o
-
s
eq
u
en
ce
co
m
p
o
n
en
ts
ar
e
g
iv
en
b
y
(
6
)
a
n
d
(
7
)
.
̇
=
−
−
,
̇
0
=
(
0
−
0
)
(
+
3
)
(
6
)
̇
=
(
−
)
−
,
̇
0
=
(
0
−
0
)
(
7
)
Her
e,
L
g
d
en
o
tes
th
e
g
r
id
-
s
i
d
e
in
d
u
cto
r
in
in
d
u
ct
o
r
–
ca
p
a
cito
r
–
in
d
u
ct
o
r
(
L
–
C
–
L
)
f
ilter
co
n
f
ig
u
r
atio
n
;
L
n
r
ep
r
esen
ts
th
e
n
eu
tr
al
wir
e
f
ilt
er
;
C
f
i
s
th
e
f
ilter
ca
p
ac
ito
r
;
v
dq
an
d
v
0
ar
e
th
e
o
u
tp
u
t
v
o
ltag
e
co
m
p
o
n
en
ts
o
f
th
e
VSI
in
th
e
d
–
q
–
0
r
ef
er
en
ce
f
r
am
e;
i
d
,
i
q
,
an
d
i
0
a
r
e
th
e
tr
a
n
s
f
o
r
m
ed
s
witch
in
g
c
u
r
r
e
n
t
c
o
m
p
o
n
en
ts
d
er
iv
ed
f
r
o
m
i
Fj
(
wh
er
e
j=a
,
b
,
c
)
i
n
to
t
h
e
d
–
q
–
0
co
o
r
d
i
n
ate
s
y
s
tem
;
i
ld
,
i
lq
,
an
d
i
l0
ar
e
th
e
lo
a
d
cu
r
r
e
n
t
co
m
p
o
n
en
ts
;
an
d
ω
is
th
e
an
g
u
lar
f
r
eq
u
en
c
y
o
f
th
e
p
o
wer
s
o
u
r
ce
.
B
ased
o
n
(
6
)
an
d
(
7
)
,
th
e
s
y
s
tem
’
s
s
tate
-
s
p
ac
e
m
o
d
el
ca
n
b
e
f
o
r
m
u
lated
as
(
8
)
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
2
5
2
-
8
7
9
2
I
n
t J Ap
p
l Po
wer
E
n
g
,
Vo
l.
14
,
No
.
4
,
Dec
em
b
er
20
25
:
8
7
9
-
892
882
̇
=
+
−
(
8
)
I
n
w
h
ich
:
=
[
,
,
,
,
,
]
is
th
e
s
tate
v
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to
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=
[
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,
]
is
th
e
in
p
u
t
f
r
o
m
th
e
s
o
u
r
c
e
,
an
d
=
[
0
,
0
,
0
,
,
,
]
is
th
e
lo
ad
cu
r
r
en
t
.
an
d
ar
e
th
e
co
ef
f
icien
t
m
atr
ices
e
x
tr
ac
ted
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r
o
m
th
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ig
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al
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p
r
ess
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n
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=
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−
1
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1
an
d
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(
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1
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−
1
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0
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0
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,
wh
er
e
A
ω
co
n
tain
s
ele
m
en
ts
r
elate
d
to
ω
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A
L
co
n
tain
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elem
en
ts
r
elate
d
t
o
−
1
,
A
n
elem
en
t
c
o
n
tain
s
a
s
in
g
le
elem
en
t r
elate
d
to
(
+
3
)
−
1
,
an
d
A
C
c
o
n
tain
s
elem
en
ts
r
elate
d
to
−
1
.
S
T
S
A
P
F
L
f
C
f
C
f
C
f
v
gc
v
gb
v
ga
T
h
r
e
e
-
p
h
a
s
e
S
o
u
r
c
e
Ma
i
n
b
u
s
Ma
i
n
b
u
s
Ma
i
n
b
u
s
C
C
L
f
L
f
L
n
v
l
a
bc
a
b
c
d
q
0
v
l
d
q0
a
b
c
d
q
0
v
lc
v
lb
v
la
a
b
c
d
q
0
a
i
b
i
c
i
d
c1
d
c2
V
dc
+
-
3D
SVP
W
M
abc
dq0
0
0
s
+
1
m
in
u
1
m
a
x
u
1
eq
u
1
S
2
m
in
u
2
m
a
x
u
3
m
in
u
3
m
a
x
u
2
S
3
S
3
eq
u
2
eq
u
1
S
2
S
3
S
v
a
v
b
v
c
1
u
2
u
3
u
−
+
−
+
−
+
*
lde
vV
=
*
0
lqe
v
=
*
lq
e
v
*
lde
v
*
ln
0
v
=
*
ln
v
i
l
d
q0
i
dq0
L
ga
L
gb
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N
o
n
-
L
i
n
e
a
r
L
o
a
d
n
s
m
_3
u
s
m
_2
u
s
m
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u
1
s
2
s
3
s
0
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+
0
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n
Fa
i
Fb
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Fc
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n
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h
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h
u
s
m
_1
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s
m
_2
,
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n
d
u
s
m
_3
f
o
r
m
E
q
.
2
9
;
s
1
,
s
2
,
a
n
d
s
3
f
r
o
m
E
q
.
2
0
Fig
u
r
e
2
.
L
a
y
o
u
t
o
f
th
r
ee
-
p
h
as
e
APF u
s
in
g
s
lid
in
g
m
o
d
e
3.
F
E
E
D
B
ACK
R
E
G
UL
A
T
I
O
N
WI
T
H
I
N
T
H
E
SL
I
DI
NG
M
O
DE
F
RA
M
E
WO
RK
3
.
1
.
Co
ntr
o
l o
f
inp
ut
a
nd
o
utput
us
ing
f
ee
db
a
ck
lin
ea
riz
a
t
io
n
T
h
e
lin
ea
r
m
u
lti
-
in
p
u
t
m
u
lti
-
o
u
tp
u
t
(
MI
MO
)
ac
ce
s
s
co
n
tr
o
ll
er
was
p
r
o
p
o
s
ed
to
elim
in
ate
n
o
n
lin
ea
r
b
eh
av
io
r
in
th
e
s
im
u
latio
n
m
o
d
el
[
2
6
]
,
[
2
7
]
.
T
h
e
f
o
llo
win
g
eq
u
atio
n
s
r
ep
r
esen
t
th
e
co
r
r
esp
o
n
d
in
g
MI
MO
co
n
tr
o
l f
r
am
ewo
r
k
:
̇
=
(
)
+
.
(
9
)
=
ℎ
(
)
(
1
0
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J Ap
p
l Po
wer
E
n
g
I
SS
N:
2252
-
8
7
9
2
I
mp
r
o
vin
g
th
e
a
d
a
p
ta
b
ilit
y
o
f
an
a
ctive
p
o
w
er filter
u
s
in
g
li
n
ea
r
iz
a
tio
n
… (
Lemin
h
Th
ien
Hu
yn
h
)
883
wh
er
e
x
sm
d
en
o
tes
th
e
s
tate
v
ec
to
r
,
u
sm
r
ep
r
esen
ts
th
e
co
n
t
r
o
l
in
p
u
t,
y
sm
is
th
e
s
y
s
tem
o
u
tp
u
t,
f
an
d
g
ar
e
s
m
o
o
th
m
atr
i
x
f
u
n
ctio
n
s
,
an
d
h
is
a
s
m
o
o
th
s
ca
lar
f
u
n
ctio
n
.
T
h
e
d
y
n
am
ic
b
eh
av
i
o
r
o
f
th
e
VSI
,
as
d
ef
in
ed
in
(
1
0
)
,
is
f
o
r
m
u
lated
th
r
o
u
g
h
(
9
)
to
(
1
2
)
[
2
7
]
.
=
[
0
]
;
=
[
]
;
=
[
]
=
1
.
(
1
,
1
,
+
3
)
(
1
1
)
(
)
=
[
∏
(
)
=
1
,
6
]
,
ℎ
ℎ
:
1
(
)
=
−
,
2
(
)
=
−
−
,
3
(
)
=
(
+
3
)
4
(
)
=
−
+
,
5
(
)
=
−
−
,
6
(
)
=
−
(
1
2
)
T
o
cr
ea
te
a
d
ir
ec
t
m
ap
p
i
n
g
b
etwe
en
th
e
o
u
tp
u
ts
y
sm_i
f
o
r
i=1
,
2,
3
an
d
t
h
e
in
p
u
ts
u
sm_i
f
o
r
i=1
,
2,
3
,
ea
ch
o
u
tp
u
t is sy
s
tem
atica
lly
v
ar
ied
u
n
til a
co
r
r
esp
o
n
d
i
n
g
co
n
tr
o
l
in
p
u
t is tr
ig
g
er
e
d
.
[
̈
_
1
̈
_
2
̈
_
3
]
=
(
)
+
(
)
[
_
1
_
2
_
3
]
(
1
3
)
T
h
e
co
n
tr
o
l stra
teg
y
is
d
ef
in
e
d
as:
(
)
(
)
12
12
1
l
n
l
n
2
1
(
)
.
2
(
3
)
d
g
f
l
d
l
d
l
q
dd
s
m
d
q
q
g
f
l
q
l
d
l
d
f
gn
L
C
v
i
i
i
A
x
i
L
C
v
i
i
C
vi
LL
−
−
−
=
+
−
−
−
=
−
−
=
+
−
+
−
=
−
+
(
1
4
)
−
1
(
)
=
.
(
1
,
1
,
(
1
+
3
.
−
1
)
)
(
1
5
)
w
h
er
e
:
[
(
)
]
=
1
(
+
3
)
2
3
≠
0
(
1
6
)
T
o
s
atis
f
y
th
e
r
eq
u
ir
em
en
ts
o
f
d
y
n
am
ic
s
y
s
tem
s
,
th
e
ex
p
ec
ted
v
alu
es
o
f
d
y
n
am
ic
r
esp
o
n
s
es
ar
e
r
eq
u
ir
ed
.
T
h
ese
v
alu
es c
an
b
e
d
eter
m
in
ed
u
s
in
g
(
1
7
)
.
[
1
2
3
]
=
[
̈
_
1
−
11
̇
1
−
12
1
̈
_
2
−
21
̇
1
−
22
1
̈
_
3
−
31
̇
1
−
32
1
]
(
1
7
)
I
n
th
is
c
o
n
tex
t,
1
=
1
∗
−
1
,
2
=
2
∗
−
2
,
an
d
3
=
3
∗
−
3
,
wh
er
e
1
∗
,
2
∗
,
a
n
d
3
∗
r
ep
r
esen
t
th
e
r
ef
er
e
n
ce
v
al
u
e
s
co
r
r
esp
o
n
d
in
g
to
y
sm1
,
y
sm
2
,
an
d
y
sm
3
,
r
esp
ec
tiv
ely
.
T
h
e
er
r
o
r
d
y
n
a
m
ics
ar
e
d
escr
ib
ed
in
(
1
3
)
th
r
o
u
g
h
(
1
9
)
.
̈
1
+
11
̇
1
+
12
1
=
0
̈
2
+
21
̇
1
+
22
1
=
0
̈
3
+
31
̇
1
+
32
1
=
0
(
1
8
)
T
h
e
er
r
o
r
d
y
n
am
ics
ar
e
en
s
u
r
ed
wh
en
th
e
g
ain
c
o
ef
f
icien
ts
η
11
,
η
12
,
η
21
,
η
22
,
η
31
,
an
d
η
32
ar
e
all
p
o
s
itiv
e.
T
h
e
co
n
tr
o
ller
in
p
u
t i
n
(
1
3
)
r
ef
lects th
e
co
n
ce
p
t o
f
i
n
teg
r
ated
co
n
tr
o
l a
s
(
1
9
)
.
[
_
]
=
.
[
]
|
[
_
]
=
[
_
1
_
2
_
3
]
,
[
]
=
[
1
2
3
]
(
1
9
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
2
5
2
-
8
7
9
2
I
n
t J Ap
p
l Po
wer
E
n
g
,
Vo
l.
14
,
No
.
4
,
Dec
em
b
er
20
25
:
8
7
9
-
892
884
I
n
wh
ich
:
(
)
(
)
(
)
(
)
(
)
(
)
1
1
2
1
1
11
1
1
2
1
1
22
1
11
3
3
l
n
l
n
2
.
.
.
2
.
.
.
1
3
(
3
)
C
.
q
f
g
f
ld
ld
f
lq
f
d
f
g
f
lq
lq
f
ld
f
n
g
g
n
f
f
v
i
C
L
C
v
i
C
i
C
v
i
C
L
C
v
i
C
i
C
L
L
v
L
L
v
i
C
−
−
−
−
−
−
−
−
−
−−
=
−
+
+
+
+
=
+
+
+
+
−
=
+
+
+
+
A
lth
o
u
g
h
th
e
v
o
ltag
es
v
ld
,
v
lq
,
an
d
v
ln
ar
e
lin
ea
r
ized
,
im
p
lem
en
tin
g
a
d
ig
ital
p
r
o
ce
s
s
o
r
ca
p
ab
le
o
f
h
an
d
lin
g
co
m
p
lex
o
p
er
atio
n
s
lik
e
d
er
iv
ativ
es
an
d
d
if
f
er
en
tials
r
em
ain
s
ch
allen
g
in
g
.
As
a
r
esu
lt,
th
e
d
er
iv
ativ
es
o
f
th
e
elec
tr
ical
cu
r
r
en
ts
i
ld
, i
lq
,
an
d
i
ln
co
n
tin
u
e
to
em
er
g
e
with
in
th
e
cir
cu
it.
T
o
ad
d
r
ess
th
is
i
s
s
u
e
,
a
co
n
tr
o
ller
b
ased
o
n
s
lid
in
g
m
o
d
e
t
h
eo
r
y
is
p
r
o
p
o
s
ed
.
T
h
is
ap
p
r
o
ac
h
en
a
b
les
th
e
d
esig
n
o
f
a
li
n
ea
r
ized
f
ee
d
b
ac
k
in
p
u
t
-
o
u
tp
u
t
co
n
tr
o
l
s
y
s
tem
u
s
in
g
c
o
n
s
is
ten
t
co
m
m
u
n
icatio
n
an
d
weig
h
t
in
g
p
a
r
am
eter
s
,
s
im
p
lify
in
g
i
m
p
lem
en
tatio
n
[
1
3
]
,
[
2
8
]
,
[
2
9
]
.
3
.
2
.
Sli
din
g
m
o
de
co
ntr
o
ller
ba
s
ed
o
n linea
rize
d in
pu
t
-
o
utput
f
ee
db
a
ck
As
a
r
esu
lt
o
f
f
a
u
lts
in
th
e
in
d
ir
ec
t
v
o
ltag
e
d
ev
ice
,
th
e
s
lid
in
g
m
o
d
e
c
o
n
tr
o
ller
s
u
r
f
ac
es
ar
e
af
f
ec
ted
.
T
h
ese
s
u
r
f
ac
es a
r
e
d
eter
m
i
n
ed
u
s
in
g
(
2
0
)
.
1
=
̇
1
+
11
1
+
12
∫
1
2
=
̇
2
+
21
2
+
22
∫
2
3
=
̇
3
+
31
3
+
32
∫
3
(
2
0
)
I
f
th
e
s
y
s
tem
s
tates
o
p
er
ate
al
o
n
g
a
s
p
ec
if
ic
s
u
r
f
ac
e,
th
e
n
1
=
2
=
3
=
0
an
d
(
2
0
)
ca
n
b
e
r
ef
o
r
m
u
lated
i
n
th
e
f
o
llo
win
g
m
atr
ix
f
o
r
m
:
̇
1
=
̇
2
=
̇
3
=
0
.
̈
1
=
−
11
̇
1
−
12
1
̈
2
=
−
21
̇
2
−
22
2
̈
3
=
−
31
̇
3
−
32
3
(
2
1
)
T
h
e
(
2
1
)
en
s
u
r
es
th
at
th
e
s
y
s
tem
s
tates
v
ld
,
v
lq
,
an
d
v
ln
will
co
n
v
er
g
e
to
th
eir
r
ef
er
e
n
ce
p
o
wer
v
alu
es
wh
en
co
n
s
tr
ain
ed
to
th
e
ze
r
o
s
lid
in
g
s
u
r
f
ac
e
[
3
0
]
.
On
th
is
s
u
r
f
ac
e,
wh
er
e
=
̇
=
0
,
th
e
eq
u
iv
alen
t
co
n
tr
o
l
ac
ts
as
a
co
n
v
en
tio
n
al
co
n
tr
o
ller
th
at
f
ac
ilit
ates
tr
ajec
to
r
y
tr
ac
k
in
g
.
T
h
e
e
q
u
iv
alen
t
co
n
t
r
o
l
law
is
d
ef
in
e
d
as
(
2
2
)
.
̇
1
=
̈
1
+
11
̇
1
+
12
1
̇
2
=
̈
2
+
21
̇
2
+
22
2
̇
3
=
̈
3
+
31
̇
3
+
32
3
(
2
2
)
W
h
er
e
v
1
,
v
2
,
an
d
v
3
co
r
r
esp
o
n
d
to
th
e
n
ewly
d
ef
in
e
d
in
p
u
ts
o
f
th
e
s
y
s
tem
,
as illu
s
tr
ated
in
(
2
3
)
.
1
=
̈
∗
+
11
̇
1
+
12
1
2
=
̈
∗
+
21
̇
2
+
22
2
3
=
̈
∗
0
+
31
̇
3
+
32
3
(
2
3
)
Ultim
ately
,
th
e
eq
u
iv
ale
n
t
co
n
tr
o
l
p
h
ase
is
r
ea
lized
b
y
en
f
o
r
cin
g
t
h
e
co
n
d
itio
n
̇
1
=
̇
2
=
̇
3
=
0
.
T
h
e
co
r
r
esp
o
n
d
i
n
g
co
n
tr
o
l e
x
p
r
ess
io
n
is
g
iv
en
b
y
(
2
4
)
.
(
)
(
)
(
)
(
)
1
1
2
1
1
_
1
1
1
1
2
1
1
_
2
2
1
1
_
3
3
l
n
l
n
2
.
.
.
2
.
.
.
(
3
)
C
(
3
)
C
.
s
m
e
q
g
f
q
f
g
f
ld
ld
f
lq
f
s
m
e
q
g
f
d
f
g
f
lq
lq
f
ld
f
s
m
e
q
g
n
f
g
n
f
f
u
L
C
v
i
C
L
C
v
i
C
i
C
u
L
C
v
i
C
L
C
v
i
C
i
C
u
L
L
v
L
L
v
i
C
−
−
−
−
−
−
−
−
−
−
=
−
+
+
+
+
=
+
+
+
+
−
=
+
+
+
+
(
2
4
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J Ap
p
l Po
wer
E
n
g
I
SS
N:
2252
-
8
7
9
2
I
mp
r
o
vin
g
th
e
a
d
a
p
ta
b
ilit
y
o
f
an
a
ctive
p
o
w
er filter
u
s
in
g
li
n
ea
r
iz
a
tio
n
… (
Lemin
h
Th
ien
Hu
yn
h
)
885
I
t
is
im
p
o
r
tan
t
to
en
s
u
r
e
t
h
at
th
e
v
alu
es
o
b
tain
ed
f
o
r
t
h
e
eq
u
iv
alen
t
co
n
tr
o
l
alig
n
with
(
1
9
)
.
T
o
r
eg
u
late
th
e
s
tate
v
ar
iab
les o
n
th
e
s
lid
in
g
s
u
r
f
ac
e,
th
e
c
o
n
d
itio
n
s
s
1
= s
2
=
s
3
=
0
an
d
γ
1
>
0
,
γ
2
>
0
,
γ
3
>
0
m
u
s
t b
e
s
atis
f
ied
.
If
s
1
, s
2
, s
3
≠
0
,
th
e
co
n
tr
o
l stra
teg
y
is
d
ef
in
ed
as
(
2
5
)
.
_
1
=
_
1
+
1
(
1
)
_
2
=
_
2
+
2
(
2
)
_
3
=
_
3
+
3
(
3
)
(
2
5
)
Su
b
s
titu
tin
g
(
2
3
)
in
to
(
2
0
)
y
ield
s
th
e
f
o
llo
win
g
s
et
o
f
(
2
6
)
.
̇
1
=
−
1
(
1
)
̇
2
=
−
2
(
2
)
̇
3
=
−
3
(
3
)
(
2
6
)
T
o
ad
d
r
ess
th
e
c
o
m
p
lex
ity
o
f
th
e
c
o
n
tr
o
ller
ar
is
in
g
f
r
o
m
th
e
u
s
e
o
f
d
is
co
n
tin
u
o
u
s
f
u
n
ctio
n
s
,
u
sm
-
sw
,
ea
ch
s
lid
in
g
s
u
r
f
ac
e
o
n
ly
n
ee
d
s
to
r
e
g
u
late
th
e
s
p
ee
d
t
o
r
e
ac
h
th
e
a
v
er
ag
e
s
witch
in
g
le
v
el
th
r
o
u
g
h
a
b
asic
L
C
L
f
ilter
[
6
]
.
T
h
e
s
y
s
tem
’
s
s
tab
ilit
y
an
d
r
o
b
u
s
tn
ess
ca
n
b
e
ev
alu
ated
u
s
in
g
th
e
L
y
a
p
u
n
o
v
f
u
n
ctio
n
,
as
p
r
o
p
o
s
ed
b
y
Slo
tin
e
a
n
d
L
i
[
2
7
]
,
d
ef
i
n
ed
as
(
2
7
)
.
̄
_
=
0
+
0
−
w_
i
=
{
0
+
0
_
+
,
>
0
0
+
0
_
−
,
<
0
(
2
7
)
Her
e,
ω
0
=
2
π
f
0
r
ad
/s
r
e
p
r
esen
ts
th
e
cu
t
-
o
f
f
f
r
e
q
u
en
c
y
o
f
t
h
e
p
ass
iv
e
f
ilter
.
T
o
p
r
e
v
en
t
s
y
s
tem
d
elay
s
an
d
o
s
cillatio
n
s
,
th
e
cu
t
-
o
f
f
f
r
eq
u
e
n
cy
m
u
s
t
b
e
ca
r
e
f
u
lly
c
h
o
s
en
—
n
o
t
ex
ce
s
s
iv
ely
h
ig
h
o
r
lo
w
.
Acc
o
r
d
i
n
g
ly
,
f
0
is
s
et
to
2
k
Hz
t
o
ef
f
ec
tiv
ely
s
u
p
p
r
ess
o
s
cillatio
n
s
,
ev
en
w
h
en
th
e
v
al
u
es
o
f
u
+
sm_1max
,
u
+
sm_1min
,
u
+
sm_2max
,
u
+
sm_2min
,
u
+
sm_3max
,
an
d
u
+
sm_3
min
r
ea
ch
th
e
s
tead
y
-
s
tate
co
n
t
r
o
l
in
p
u
ts
(
u
sm_1st
,
u
sm_
2
st
,
u
sm_3st
)
.
T
h
e
s
tead
y
-
s
tate
co
n
d
itio
n
is
d
e
f
in
ed
in
(2
8
)
.
_
1
=
(
2
−
)
.
(
−
1
+
2
)
−
1
_
2
=
(
2
−
)
.
[
−
1
+
2
]
−
1
_
3
=
0
(
2
8
)
T
h
e
v
alu
es
o
f
u
+
sm_1max
an
d
u
+
sm_1min
ar
e
d
ef
in
ed
as
u
+
sm_ima
x
=
u
sm_ist
+
∆
i
an
d
u
+
sm_imin
=
u
sm_ist
-
∆
i
,
r
esp
ec
tiv
ely
,
wh
er
e
∆
i
is
a
co
n
s
tan
t
in
tr
o
d
u
ce
d
to
p
r
eser
v
e
s
y
s
tem
s
tab
il
ity
.
C
o
n
s
eq
u
en
tly
,
th
e
ex
ten
t
o
f
s
y
s
tem
o
s
cillatio
n
is
d
ir
ec
tly
in
f
lu
en
ce
d
b
y
t
h
e
m
ag
n
itu
d
e
o
f
∆
i
.
I
n
s
u
m
m
ar
y
,
th
e
in
p
u
t
f
u
n
ctio
n
s
in
(
2
5
)
ar
e
m
o
d
if
ied
as
(
2
9
)
,
in
wh
ich
t
h
e
co
ef
f
icien
ts
γ
1
,
γ
2
,
a
n
d
γ
3
en
s
u
r
e
th
e
ex
is
ten
ce
o
f
m
o
d
e
s
lid
in
g
o
n
th
e
s
u
r
f
ac
e
.
_
1
=
̄
_
1
+
1
(
1
)
_
2
=
̄
_
2
+
2
(
2
)
_
3
=
̄
_
3
+
3
(
3
)
(
2
9
)
Fig
u
r
e
2
p
r
esen
ts
th
e
s
tr
u
ctu
r
e
o
f
t
h
e
lin
ea
r
ize
d
in
p
u
t
f
ee
d
b
ac
k
–
s
lid
in
g
m
o
d
e
o
u
tp
u
t
co
n
tr
o
l,
f
o
cu
s
in
g
o
n
th
e
r
ef
er
en
ce
v
al
u
e
alo
n
g
th
e
d
-
a
x
is
(
v
*
ld
)
.
W
h
en
th
e
v
o
ltag
e
s
o
u
r
ce
in
v
e
r
ter
(
VSI
)
d
eliv
e
r
s
a
b
alan
ce
d
th
r
ee
-
p
h
ase
o
u
tp
u
t,
th
e
r
e
m
ain
in
g
r
e
f
er
en
ce
v
al
u
es a
r
e
s
et
to
ze
r
o
.
4.
SI
M
UL
A
T
E
D
P
E
RF
O
R
M
A
NCE
E
ac
h
co
n
tr
o
ller
is
test
ed
th
r
o
u
g
h
two
s
im
u
latio
n
r
u
n
s
in
c
o
r
p
o
r
atin
g
a
n
ac
tiv
e
p
o
wer
f
ilter
.
T
h
e
f
ir
s
t
s
ce
n
ar
io
em
p
lo
y
s
PI
co
n
tr
o
l
,
wh
ile
th
e
s
ec
o
n
d
u
tili
ze
s
s
lid
in
g
m
o
d
e
(
SM)
c
o
n
tr
o
l.
T
o
e
v
alu
ate
th
e
ef
f
ec
tiv
en
ess
o
f
th
e
p
r
o
p
o
s
ed
ap
p
r
o
ac
h
,
s
im
u
latio
n
s
ar
e
co
n
d
u
cted
u
s
in
g
th
e
PS
I
M
s
o
f
twar
e,
tar
g
etin
g
b
o
t
h
u
n
b
alan
ce
d
an
d
n
o
n
lin
ea
r
lo
a
d
co
n
d
itio
n
s
.
T
h
e
in
p
u
t
s
o
u
r
ce
is
a
th
r
ee
-
p
h
ase
AC
g
r
i
d
(
3
8
0
V,
5
0
Hz)
,
an
d
th
e
in
v
er
ter
o
p
er
ates
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ile
T
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le
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p
r
esen
ts
th
e
s
im
u
latio
n
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tc
o
m
es f
o
r
b
o
th
PI
a
n
d
SM
co
n
tr
o
l stra
teg
ies.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
2
5
2
-
8
7
9
2
I
n
t J Ap
p
l Po
wer
E
n
g
,
Vo
l.
14
,
No
.
4
,
Dec
em
b
er
20
25
:
8
7
9
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886
T
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1
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T
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ig
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T
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le
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.
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co
m
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cr
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s
d
if
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t lo
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d
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tr
o
ller
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Lo
a
d
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y
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e
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o
n
t
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l
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e
TH
D
[
%]
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d
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1
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l
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6
6
T
h
e
s
im
u
latio
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tco
m
es
d
e
p
icted
in
Fig
u
r
es
3
th
r
o
u
g
h
8
r
e
v
ea
l
s
ev
er
al
k
ey
o
b
s
er
v
atio
n
s
:
f
ir
s
t,
t
h
e
p
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ase
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ad
v
o
ltag
e
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em
ai
n
s
co
n
s
is
ten
tly
b
alan
ce
d
u
n
d
e
r
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n
tr
o
l.
Seco
n
d
,
s
y
s
tem
cu
r
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en
ts
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er
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to
Kir
ch
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o
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f
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r
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en
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ws,
as d
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o
n
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tr
ated
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y
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1
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(
3
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.
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th
e
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o
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ed
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n
tr
o
l m
eth
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ac
h
iev
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wer
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tal
h
ar
m
o
n
ic
d
is
to
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tio
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(
T
HD)
in
th
e
p
h
ase
l
o
ad
c
u
r
r
e
n
ts
co
m
p
ar
ed
to
PI
co
n
t
r
o
l,
ac
r
o
s
s
b
o
th
n
o
n
lin
ea
r
an
d
u
n
b
alan
ce
d
n
o
n
lin
ea
r
lo
a
d
co
n
d
itio
n
s
.
T
h
e
s
im
u
latio
n
r
esu
lts
em
p
h
a
s
ize
th
e
b
e
h
av
io
r
o
f
th
e
th
r
ee
-
p
h
ase
g
r
id
a
n
d
lo
ad
cu
r
r
en
ts
.
Fig
u
r
es
4
an
d
7
illu
s
tr
ate
th
e
v
o
ltag
e
o
f
p
h
ase
a
(
v
F_aN
)
an
d
lin
e
a
(
v
F_ab
)
o
f
th
e
VSI
u
n
d
er
d
if
f
e
r
e
n
t
lo
ad
co
n
d
itio
n
s
.
T
ab
le
3
h
ig
h
lig
h
ts
th
e
s
ig
n
if
ican
t
d
if
f
er
en
ce
in
T
HD
b
etwe
en
th
e
two
co
n
tr
o
ller
s
wh
en
s
u
b
jecte
d
to
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u
n
b
alan
ce
d
lin
ea
r
lo
ad
.
Sp
ec
if
ically
,
th
e
PI
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n
tr
o
ller
r
e
d
u
c
ed
th
e
a
-
p
h
ase
lo
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ltag
e
T
HD
f
r
o
m
2
.
1
5
%
to
1
.
9
7
%,
wh
ile
th
e
SM
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n
tr
o
ll
er
ac
h
iev
ed
a
g
r
ea
ter
r
ed
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ctio
n
f
r
o
m
2
.
1
2
%
to
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5
%,
as
s
h
o
wn
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Fig
u
r
es
5
an
d
8
.
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n
c
o
m
p
ar
is
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n
,
T
ab
le
2
s
h
o
ws
th
at
u
n
d
er
u
n
b
alan
ce
d
n
o
n
lin
ea
r
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ad
co
n
d
itio
n
s
—
p
ar
tic
u
lar
ly
with
p
h
ase
b
im
b
alan
ce
—
th
e
T
HD
o
f
ib
is
n
o
tab
ly
h
ig
h
e
r
th
a
n
in
o
th
e
r
c
ases
.
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wev
er
,
with
t
h
e
SM
c
o
n
tr
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ller
,
it
r
em
ai
n
s
b
elo
w
5
%,
as
d
e
p
icted
in
Fig
u
r
e
8
.
T
h
is
lo
wer
T
HD
c
o
n
f
ir
m
s
th
at
th
e
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n
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o
l
m
eth
o
d
o
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er
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o
r
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s
th
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n
tr
o
ller
in
m
an
ag
i
n
g
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o
th
n
o
n
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ea
r
an
d
u
n
b
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ce
d
n
o
n
lin
ea
r
lo
ad
s
.
Fig
u
r
e
3
.
T
r
an
s
ien
t b
eh
a
v
io
r
o
f
th
e
p
r
o
p
o
s
ed
SM
co
n
tr
o
ller
u
n
d
er
n
o
n
lin
ea
r
lo
ad
i
n
g
: g
r
i
d
cu
r
r
en
t (
i
_gj
)
,
f
ilter
cu
r
r
en
t (
i
_Fj
)
,
an
d
l
o
ad
cu
r
r
en
t
(
i
_Lj
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J Ap
p
l Po
wer
E
n
g
I
SS
N:
2252
-
8
7
9
2
I
mp
r
o
vin
g
th
e
a
d
a
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ta
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ilit
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887
Fig
u
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4
.
T
r
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s
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t r
esp
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e
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f
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h
ase
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a
v
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n
d
er
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a
d
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s
in
g
t
h
e
p
r
o
p
o
s
ed
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n
tr
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ller
Fig
u
r
e
5
.
Dy
n
am
ic
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e
h
av
io
r
o
f
cu
r
r
e
n
t T
HD
u
n
d
e
r
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o
n
lin
ea
r
lo
ad
u
s
in
g
t
h
e
p
r
o
p
o
s
ed
SM
c
o
n
tr
o
ller
Fig
u
r
e
6
.
R
esp
o
n
s
e
o
f
g
r
id
,
f
il
ter
,
an
d
lo
a
d
cu
r
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en
ts
(
i
_gj
,
i
_Fj
,
i
_Lj
)
u
n
d
er
u
n
b
alan
ce
d
n
o
n
lin
ea
r
lo
ad
u
s
in
g
th
e
p
r
o
p
o
s
ed
SM
co
n
tr
o
ller
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
2
5
2
-
8
7
9
2
I
n
t J Ap
p
l Po
wer
E
n
g
,
Vo
l.
14
,
No
.
4
,
Dec
em
b
er
20
25
:
8
7
9
-
892
888
Fig
u
r
e
7
.
Vo
ltag
e
r
esp
o
n
s
e
at
VSI
p
h
ase
-
a
u
n
d
er
a
n
u
n
b
alan
ce
d
n
o
n
lin
ea
r
lo
ad
with
th
e
p
r
o
p
o
s
ed
s
lid
in
g
m
o
d
e
co
n
tr
o
ller
Fig
u
r
es
7
an
d
8
d
ep
ict
th
e
r
es
p
o
n
s
e
s
ig
n
als
o
f
i_
gb
.
I
n
itially
,
h
ig
h
s
ig
n
al
lev
els
m
ay
ad
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er
s
ely
im
p
ac
t
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ice
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er
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o
r
m
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ce
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h
o
wev
er
,
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is
is
s
u
e
ca
n
b
e
m
itig
ated
u
s
in
g
a
d
elay
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er
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witch
.
Acc
o
r
d
in
g
t
o
L
y
ap
u
n
o
v
s
tab
ilit
y
th
eo
r
y
[
2
7
]
a
n
d
th
e
5
%
s
tan
d
ar
d
co
n
tr
o
l
cr
iter
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n
,
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s
tab
le
i_
gb
r
esp
o
n
s
e
ex
ce
ed
in
g
8
5
%
is
ac
h
iev
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at
ap
p
r
o
x
im
ately
3
.
0
6
×
1
0
⁻²
s
ec
o
n
d
s
f
r
o
m
s
tar
tu
p
—
eq
u
i
v
alen
t
to
ar
o
u
n
d
1
.
5
cy
cles
o
f
th
e
f
u
n
d
am
en
tal
f
r
eq
u
e
n
cy
—
as
illu
s
tr
ated
in
Fig
u
r
e
6
.
T
h
e
d
e
v
iatio
n
b
etwe
en
th
e
r
ef
e
r
en
ce
s
ig
n
al
i_
sine_b
an
d
th
e
ac
tu
al
r
esp
o
n
s
e
i_
gb
is
0
.
7
7
A
f
o
r
t
h
e
SM
co
n
tr
o
ller
an
d
5
.
3
A
f
o
r
th
e
PI
co
n
tr
o
ller
,
i
n
d
icatin
g
s
u
p
er
i
o
r
tr
ac
k
in
g
p
er
f
o
r
m
an
ce
o
f
th
e
SM
m
eth
o
d
.
I
n
ac
co
r
d
a
n
ce
with
th
e
I
E
E
E
1
9
9
2
s
tan
d
a
r
d
s
f
o
r
to
tal
h
a
r
m
o
n
ic
d
is
to
r
tio
n
(
T
HD)
,
o
v
er
s
h
o
o
t
an
d
its
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r
r
esp
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n
d
in
g
r
esp
o
n
s
e
tim
e
ar
e
k
ey
p
er
f
o
r
m
an
ce
in
d
icato
r
s
in
co
n
tr
o
l
s
y
s
tem
s
—
esp
ec
ia
lly
in
ac
tiv
e
p
o
wer
f
ilter
(
APF)
ap
p
licatio
n
s
d
ea
lin
g
with
u
n
b
alan
ce
d
n
o
n
lin
ea
r
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ad
s
.
B
y
ap
p
ly
in
g
L
y
ap
u
n
o
v
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u
n
ctio
n
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ese
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ar
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eter
s
wer
e
p
r
ec
is
ely
ca
l
cu
lated
an
d
ar
e
illu
s
tr
ated
in
F
ig
u
r
es
9
an
d
10
.
T
h
e
p
r
o
p
o
s
ed
s
lid
in
g
m
o
d
e
(
SM)
co
n
tr
o
l
m
eth
o
d
was
ev
alu
ated
ac
r
o
s
s
a
lo
ad
p
o
wer
r
an
g
e
o
f
5
to
4
8
k
W
,
as
s
h
o
wn
in
Fig
u
r
e
11
.
T
h
e
r
esu
lts
d
em
o
n
s
tr
ate
a
s
ig
n
if
ican
t
r
e
d
u
ctio
n
in
T
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HD
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