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l
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s
[
9
]
.
H
a
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m
o
n
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c
c
u
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[
1
0
]
.
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c
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m
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s
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q
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[
1
1
]
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[
1
2
]
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p
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[
1
3
]
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4
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[
1
5
]
.
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[
1
6
]
.
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[
1
7
]
.
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(
I
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)
[
1
8
]
b
y
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t
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li
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g
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a
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g
I
R
P
T
[
1
9
]
u
s
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n
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s
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n
ch
r
o
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s
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ef
e
r
en
ce
f
r
am
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(
S
R
F
)
[
2
0
]
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y
t
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n
s
f
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m
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t
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A
C
w
a
v
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s
[
2
1
]
.
H
o
we
v
e
r
,
t
h
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m
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s
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t
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h
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m
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d
s
w
il
l
b
e
[
2
2
]
,
w
h
i
c
h
w
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l
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t
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[
2
3
]
.
V
a
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t
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n
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l
.
[
2
4
]
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n
t
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o
d
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(
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(
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&
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.
S
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.
[
2
5
]
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m
p
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v
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y
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P
C
.
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h
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c
h
a
l
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g
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t
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e
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H
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a
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al
a
n
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es
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i
l
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r
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n
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al
ab
l
e
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o
l
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ti
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f
o
r
c
o
m
p
l
e
x
p
o
w
e
r
s
y
s
te
m
s
.
T
h
e
m
a
i
n
c
o
n
t
r
i
b
u
t
i
o
n
s
a
n
d
n
o
v
e
l
t
y
o
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t
h
i
s
s
t
u
d
y
a
r
e
as
f
o
l
l
o
w
s
:
i)
T
h
i
s
p
a
p
e
r
p
r
es
e
n
ts
a
h
y
b
r
i
d
p
o
w
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l
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m
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lt
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l
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i
n
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t
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s
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n
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C
t
o
r
e
d
u
c
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h
a
r
m
o
n
i
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t
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o
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r
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w
i
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p
o
w
e
r
g
r
i
d
.
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h
e
s
i
m
u
l
a
t
i
o
n
r
e
s
u
l
ts
s
h
o
w
it
s
c
a
p
a
b
i
l
it
y
t
o
l
o
w
e
r
t
h
e
T
HD
b
e
l
o
w
5
%
,
m
ee
t
i
n
g
p
o
w
e
r
q
u
a
l
i
ty
s
t
a
n
d
a
r
d
s
.
i
i
)
T
h
e
p
r
o
p
o
s
e
d
s
y
s
te
m
r
e
d
u
c
e
s
h
a
r
m
o
n
i
c
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r
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t
s
a
n
d
a
d
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s
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l
o
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d
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b
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l
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n
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e
s
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h
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s
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g
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th
o
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t
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l
v
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l
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e
.
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h
is
f
e
a
t
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r
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m
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v
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s
t
h
e
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t
a
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i
t
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n
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f
f
i
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n
c
y
o
f
t
h
e
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o
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r
d
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s
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r
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b
u
t
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o
n
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y
s
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e
m
s
.
i
i
i)
T
h
i
s
s
t
u
d
y
e
x
p
l
o
r
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d
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P
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f
o
r
a
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t
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v
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f
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l
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a
p
p
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c
a
t
i
o
n
s
i
n
m
u
l
ti
l
e
v
el
i
n
v
er
t
e
r
s
,
a
n
a
r
e
a
w
i
t
h
l
i
m
it
e
d
r
e
s
e
a
r
c
h
.
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h
i
s
a
d
v
a
n
c
es
t
h
e
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n
d
e
r
s
t
a
n
d
i
n
g
o
f
M
PC
e
f
f
e
c
t
i
v
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n
es
s
i
n
i
m
p
r
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v
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h
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r
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u
a
l
i
t
y
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n
d
b
a
l
a
n
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n
g
l
o
a
d
s
i
n
c
o
m
p
l
e
x
p
o
w
e
r
s
y
s
t
e
m
s
.
T
h
is
p
ap
er
is
o
r
g
an
ized
in
to
s
ev
er
al
s
ec
tio
n
s
to
s
y
s
tem
ati
ca
lly
d
em
o
n
s
tr
ate
its
co
n
tr
ib
u
tio
n
s
an
d
r
elev
an
ce
.
Sectio
n
1
in
tr
o
d
u
ce
s
th
e
s
ig
n
if
ican
ce
o
f
p
o
w
er
elec
tr
o
n
ics
in
v
ar
i
o
u
s
ap
p
licatio
n
s
an
d
th
e
ch
allen
g
es
p
o
s
ed
b
y
h
ar
m
o
n
i
c
d
is
to
r
tio
n
s
in
p
o
wer
g
r
id
s
,
em
p
h
asizin
g
th
e
n
o
v
elty
o
f
u
s
in
g
a
f
iv
e
-
le
v
el
m
u
ltil
ev
el
in
v
er
ter
with
an
MPC
to
ad
d
r
ess
th
ese
is
s
u
es.
Sectio
n
2
r
ev
iews
ex
is
tin
g
h
ar
m
o
n
ic
m
itig
atio
n
tech
n
iq
u
es,
h
ig
h
lig
h
tin
g
th
e
lim
itatio
n
s
o
f
p
ass
iv
e
f
ilter
s
an
d
tr
ad
itio
n
al
ac
tiv
e
f
ilter
s
,
wh
ile
id
en
tify
in
g
r
esear
ch
g
ap
s
in
ap
p
ly
in
g
MP
C
to
h
y
b
r
id
p
o
wer
f
ilter
s
.
Sectio
n
3
d
escr
ib
es
th
e
r
esear
ch
m
eth
o
d
o
lo
g
y
,
d
etails
th
e
h
y
b
r
id
ac
tiv
e
p
o
wer
f
ilt
er
d
esig
n
,
th
e
u
s
e
o
f
SR
F
th
eo
r
y
f
o
r
r
ef
er
en
ce
cu
r
r
e
n
t
g
en
er
atio
n
,
an
d
t
h
e
im
p
lem
en
tatio
n
o
f
MPC
f
o
r
o
p
tim
izin
g
in
v
er
ter
s
witch
in
g
p
atter
n
s
to
r
ed
u
ce
h
ar
m
o
n
ic
d
is
to
r
tio
n
.
Sectio
n
4
p
r
esen
ts
th
e
s
im
u
latio
n
r
esu
lts
,
d
em
o
n
s
tr
atin
g
th
e
ef
f
ec
tiv
en
ess
o
f
th
e
p
r
o
p
o
s
ed
s
y
s
tem
in
r
ed
u
cin
g
th
e
T
HD
b
elo
w
5
%,
b
alan
cin
g
lo
ad
s
,
an
d
m
in
im
izin
g
n
eu
t
r
al
cu
r
r
en
ts
u
n
d
er
b
o
th
b
alan
ce
d
an
d
u
n
b
alan
ce
d
lo
ad
co
n
d
itio
n
s
,
with
a
d
is
cu
s
s
io
n
co
m
p
ar
in
g
it
t
o
co
n
v
en
tio
n
al
m
eth
o
d
s
.
Fin
ally
,
s
ec
tio
n
5
c
o
n
clu
d
es
th
e
p
a
p
er
,
s
u
m
m
ar
izes th
e
ac
h
iev
em
e
n
ts
,
an
d
s
u
g
g
ests
f
u
tu
r
e
d
ir
ec
tio
n
s
.
2.
M
E
T
H
O
D
A
th
r
ee
-
p
h
ase,
f
o
u
r
-
wir
e
p
o
wer
g
r
id
cir
cu
it
co
n
n
ec
ted
b
y
a
n
o
n
lin
ea
r
lo
a
d
g
en
er
ated
a
h
a
r
m
o
n
ic
cu
r
r
en
t.
T
h
is
cir
cu
it wa
s
th
en
co
n
n
ec
ted
to
a
s
er
ial
-
m
o
u
n
ted
p
ass
iv
e
p
o
wer
f
ilter
with
an
ac
tiv
e
p
o
wer
f
ilter
at
th
e
p
o
in
t
o
f
c
o
m
m
o
n
co
u
p
lin
g
(
PC
C
)
.
I
t
h
as
an
ac
tiv
e
p
o
wer
f
ilter
in
th
e
f
o
r
m
o
f
a
th
r
ee
-
p
h
ase,
f
o
u
r
-
wir
e,
m
u
ltis
tag
e
co
n
v
er
ter
n
etwo
r
k
.
C
u
r
r
en
t
an
d
v
o
ltag
e
s
en
s
o
r
s
ar
e
p
r
o
ce
s
s
ed
u
s
in
g
th
e
SR
F
m
eth
o
d
to
g
en
er
ate
th
e
r
ef
er
e
n
ce
c
u
r
r
e
n
t
u
s
ed
in
t
h
e
MPC
m
eth
o
d
.
T
h
e
f
u
ll
r
an
g
e
o
f
th
ese
n
etwo
r
k
s
is
illu
s
tr
a
ted
in
Fig
u
r
e
1
.
T
h
e
s
im
u
lated
n
etwo
r
k
was
a
th
r
ee
-
p
h
ase,
f
o
u
r
-
wir
e
v
o
ltag
e
n
et
wo
r
k
th
at
ac
c
o
m
m
o
d
ated
th
e
p
r
esen
ce
o
f
a
s
in
g
le
-
p
h
ase
lo
ad
co
n
n
ec
ted
to
a
t
h
r
ee
-
p
h
ase
n
etwo
r
k
.
T
h
ese
s
in
g
le
-
p
h
ase
l
o
ad
s
af
f
ec
t
th
e
lo
ad
b
ala
n
ce
.
T
h
e
m
ag
n
itu
d
e
o
f
th
is
lo
ad
im
b
ala
n
ce
af
f
ec
ted
t
h
e
m
ag
n
itu
d
e
o
f
th
e
n
eu
tr
al
cu
r
r
en
t.
2
.
1
.
M
ultilev
el
co
nv
er
t
er
A
clam
p
ed
d
io
d
e
ca
s
ca
d
e
co
n
v
er
ter
is
a
m
u
ltil
ev
el
in
v
er
ter
th
at
p
r
o
d
u
ce
s
an
o
u
t
p
u
t
v
o
lt
ag
e
with
a
lo
w
h
ar
m
o
n
ic
d
is
to
r
tio
n
(
lo
w
T
HD)
.
T
h
is
in
v
er
ter
u
s
es
a
co
m
b
in
atio
n
o
f
s
em
ico
n
d
u
cto
r
s
witch
es
an
d
s
u
p
p
o
r
tin
g
co
m
p
o
n
en
ts
to
p
r
o
d
u
ce
o
u
tp
u
t
v
o
ltag
e
lev
els
a
p
p
r
o
ac
h
in
g
s
in
u
s
o
id
al
wav
ef
o
r
m
s
.
T
h
is
co
n
v
er
ter
h
as
f
o
u
r
s
em
ico
n
d
u
cto
r
s
witch
es
clam
p
ed
with
a
d
i
o
d
e
o
n
ea
ch
ar
m
.
T
h
e
f
o
u
r
ar
m
s
co
n
s
is
ted
o
f
p
h
ases
a,
b
,
c,
an
d
n
eu
tr
al,
as
s
h
o
wn
in
Fig
u
r
e
2
.
Fo
r
th
e
p
o
wer
s
u
p
p
ly
s
u
p
p
lied
b
y
V
dc
,
th
e
v
o
ltag
e
is
eq
u
al
to
th
e
u
s
e
o
f
C
1
an
d
C
2
with
th
e
s
am
e
v
alu
e
,
s
u
ch
th
at
V
C1
=
V
C2
is
id
ea
l.
T
h
er
e
wer
e
f
o
u
r
I
GB
T
s
in
ea
c
h
ar
m
:
S
1a
,
S
2a
,
S'
1a
,
an
d
S'
2a
.
S
1a
an
d
S'
1a
ar
e
n
eg
ativ
e
with
ea
ch
o
th
e
r
,
an
d
s
o
a
r
e
S
2a
an
d
S'
2a
,
s
o
if
S
1a
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d
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as
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en
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a
s
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Fro
m
th
e
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m
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h
e
th
r
ee
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o
s
s
ib
ilit
ies f
o
r
ea
ch
ar
m
ar
e
lis
ted
in
T
ab
le
1
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J Po
w
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lec
&
Dr
i Sy
s
t
I
SS
N:
2088
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8
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ab
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1
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two
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o
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itiv
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n
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e
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iv
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e
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e
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tch
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n
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itio
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ich
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ea
s
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r
ed
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d
N.
W
h
en
S
2a
is
o
n
,
V
C2
will
b
e
m
ea
s
u
r
ed
b
etwe
en
p
h
ases
a
an
d
N,
an
d
i
f
S
1a
an
d
S
2a
ar
e
o
f
f
,
th
e
v
o
ltag
e
v
alu
e
o
f
p
h
ases
a
t
o
N
will b
e
ze
r
o
.
T
h
e
v
o
ltag
e
g
e
n
er
ated
f
r
o
m
p
h
ase
x
to
p
o
in
t
N
d
ep
e
n
d
s
o
n
th
e
s
witch
in
g
s
ig
n
al
o
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ea
ch
I
GB
T
in
ea
ch
ar
m
; th
is
ca
n
b
e
s
ee
n
f
r
o
m
(
1
)
[
3
7
]
.
[
]
=
1
[
1
1
1
1
]
+
2
[
2
2
2
2
]
(
1
)
W
h
ile
th
e
p
h
ase
v
o
ltag
e
x
to
p
o
in
t n
is
g
en
e
r
ated
f
r
o
m
(
2
)
.
[
]
=
1
[
1
−
1
1
−
1
1
−
1
]
+
2
[
2
−
2
2
−
2
2
−
2
]
(
2
)
3
f
4
W
i
r
e
P
o
w
e
r
G
r
i
d
N
o
n
-
l
i
n
i
e
r
Lo
a
d
R
e
f
e
r
e
n
c
e
G
e
n
e
r
a
t
o
r
C
ur
r
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nt
S
e
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V
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t
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ge
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e
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M
o
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P
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e
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t
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t
r
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l
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e
r
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c
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i
l
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r
P
C
C
a
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C
C
b
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C
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c
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C
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n
Fig
u
r
e
1
.
Po
wer
g
r
id
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c
o
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ec
te
d
n
o
n
li
n
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r
lo
ad
s
with
ac
tiv
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f
ilter
s
P
N
V
dc
0
a
b
c
n
S
1a
S
2a
S
a
S
a
S
1b
S
2b
S
b
S
b
S
c
S
c
S
n
S
n
S
1c
S
2c
S
1n
S
2n
C
1
C
2
Fig
u
r
e
2
.
Netwo
r
k
o
f
th
r
ee
-
p
h
ase,
f
o
u
r
-
wir
e
m
u
ltil
ev
el
co
n
v
er
ter
T
ab
le
1
.
Switch
in
g
s
tate
o
f
m
u
ltil
ev
el
co
n
v
er
ter
S
w
i
t
c
h
i
n
g
si
g
n
a
l
O
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t
p
u
t
v
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l
t
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g
e
S
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S
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1x
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2x
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xN
‘
1
’
‘
1
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‘
0
’
‘
0
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V
C1
+ V
C2
‘
0
’
‘
1
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‘
1
’
‘
0
’
V
C2
‘
0
’
‘
0
’
‘
1
’
‘
1
’
0
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8
6
9
4
I
n
t J Po
w
E
lec
&
Dr
i Sy
s
t
,
Vo
l.
16
,
No
.
2
,
J
u
n
e
20
25
:
9
2
2
-
9
3
5
926
2.
2
.
Ref
er
ence
curr
ent
g
ene
ra
t
io
n
T
h
e
s
y
n
ch
r
o
n
o
u
s
r
ef
er
en
ce
f
r
a
m
e
(
SR
F)
th
eo
r
y
is
em
p
lo
y
e
d
to
tr
an
s
f
o
r
m
t
h
r
ee
-
p
h
ase
cu
r
r
e
n
ts
(
i
a
,
i
b
,
i
c
)
in
to
a
r
o
tatin
g
r
ef
er
en
ce
f
r
a
m
e
(
dq
0
)
u
s
in
g
C
lar
k
an
d
Par
k
tr
a
n
s
f
o
r
m
atio
n
s
.
T
h
is
tr
an
s
f
o
r
m
atio
n
s
im
p
lifie
s
th
e
an
aly
s
is
an
d
co
n
t
r
o
l
o
f
th
r
ee
-
p
h
ase
s
y
s
tem
s
b
y
co
n
v
er
tin
g
th
e
s
in
u
s
o
id
al
wav
ef
o
r
m
s
in
to
DC
co
m
p
o
n
e
n
ts
in
th
e
dq
0
d
o
m
ain
.
T
h
e
p
r
o
c
ess
b
eg
in
s
with
th
e
C
lar
k
tr
an
s
f
o
r
m
atio
n
,
wh
ic
h
p
r
o
jects
th
r
ee
-
p
h
ase
cu
r
r
en
ts
in
to
a
two
-
p
h
ase
s
tatio
n
ar
y
r
ef
er
en
ce
f
r
am
e
(
αβ
)
.
T
h
is
was a
ch
iev
ed
b
y
u
s
in
g
t
h
e
f
o
llo
win
g
eq
u
atio
n
:
[
]
=
[
1
−
1
2
−
1
2
0
√
3
2
−
√
3
2
]
[
]
wh
er
e
i
α
an
d
i
β
r
e
p
r
esen
t
th
e
t
wo
o
r
th
o
g
o
n
al
co
m
p
o
n
en
ts
o
f
th
e
th
r
ee
-
p
h
ase
c
u
r
r
en
t
in
t
h
e
s
tatio
n
ar
y
r
ef
e
r
en
ce
f
r
am
e.
On
ce
th
e
cu
r
r
en
ts
ar
e
in
th
e
αβ
f
r
am
e,
th
e
Par
k
tr
an
s
f
o
r
m
atio
n
is
ap
p
lied
to
r
o
tate
th
is
s
tatio
n
ar
y
f
r
am
e
to
alig
n
it
with
th
e
s
y
n
ch
r
o
n
o
u
s
r
ef
e
r
en
ce
f
r
am
e
(
dq
)
.
T
h
is
alig
n
m
en
t
is
ac
h
ie
v
ed
b
y
u
tili
zin
g
th
e
s
y
s
tem
’
s
in
s
tan
tan
eo
u
s
an
g
u
la
r
p
o
s
itio
n
(
θ
)
to
p
r
o
ject
th
e
c
u
r
r
en
ts
o
n
to
th
e
r
o
tatin
g
dq
-
a
xis
u
s
in
g
:
[
]
=
[
−
]
[
]
h
er
e
,
i
d
r
e
p
r
esen
ts
th
e
d
ir
ec
t
a
x
is
co
m
p
o
n
en
t,
wh
ic
h
alig
n
s
with
th
e
s
y
s
tem
v
o
ltag
e,
an
d
i
q
i_
q
iq
r
ep
r
esen
ts
th
e
q
u
ad
r
atu
r
e
ax
is
c
o
m
p
o
n
en
t
o
r
th
o
g
o
n
al
to
i
d
.
T
h
ese
c
o
m
p
o
n
en
ts
ar
e
n
o
w
DC
s
ig
n
als,
m
a
k
in
g
th
e
m
ea
s
ier
to
f
ilter
an
d
p
r
o
ce
s
s
f
o
r
c
o
n
tr
o
l.
T
h
e
f
in
al
s
tep
is
to
g
en
er
ate
th
e
r
ef
er
en
ce
cu
r
r
en
ts
b
y
f
ilter
in
g
th
e
i
d
an
d
i
q
co
m
p
o
n
en
ts
u
s
in
g
a
lo
w
-
p
ass
f
ilter
to
r
em
o
v
e
h
ar
m
o
n
ic
d
is
to
r
tio
n
.
T
h
e
f
ilter
ed
id
i_
d
id
co
m
p
o
n
en
t
is
f
u
r
th
er
co
r
r
ec
ted
u
s
in
g
a
DC
r
eg
u
lato
r
,
wh
ile
th
e
i
q
an
d
i
0
co
m
p
o
n
en
ts
ar
e
tr
an
s
f
o
r
m
ed
b
ac
k
to
th
e
th
r
ee
-
p
h
ase
ab
c
ab
ca
b
c
d
o
m
ain
v
ia
in
v
er
s
e
Par
k
an
d
C
lar
k
tr
an
s
f
o
r
m
atio
n
s
.
T
h
ese
r
ef
er
e
n
ce
cu
r
r
en
ts
(
i
a
,
ref
,
i
b,
ref
,
i
c,
ref
)
g
u
id
e
th
e
in
v
er
ter
s
witch
in
g
p
atter
n
s
to
in
ject
co
m
p
en
s
atin
g
cu
r
r
e
n
ts
,
en
s
u
r
in
g
h
ar
m
o
n
ic
r
ed
u
ctio
n
a
n
d
a
b
alan
ce
d
s
y
s
te
m
o
p
er
atio
n
.
T
h
e
r
ef
er
en
ce
c
u
r
r
en
ts
to
b
e
u
s
ed
in
th
e
MPC
m
eth
o
d
wer
e
g
en
er
ated
u
s
in
g
th
e
s
y
n
ch
r
o
n
o
u
s
r
ef
er
en
ce
f
r
am
e
(
SR
F)
th
eo
r
y
.
T
h
e
d
ec
o
m
p
o
s
itio
n
d
ia
g
r
am
b
lo
ck
s
to
g
e
n
er
ate
th
e
r
e
f
er
e
n
ce
cu
r
r
e
n
ts
i
a(ref)
,
i
b(ref)
,
an
d
i
c(ref)
ar
e
s
h
o
wn
i
n
Fig
u
r
e
3
.
T
h
e
c
h
an
g
e
o
f
p
h
ase
c
u
r
r
en
t a
b
c
to
d
q
cu
r
r
en
t is d
o
n
e
u
s
in
g
(
3
)
.
[
0
]
=
2
3
[
s
in
(
)
s
in
(
−
2
3
)
s
in
(
+
2
3
)
c
os
(
)
c
os
(
−
2
3
)
c
os
(
+
2
3
)
1
2
1
2
1
2
]
[
]
(
3
)
T
h
e
i
d(ref)
cu
r
r
en
t
was
o
b
tain
ed
v
ia
i
d
f
ilter
in
g
u
s
in
g
a
lo
w
-
p
ass
f
ilter
an
d
co
r
r
ec
tio
n
f
r
o
m
th
e
DC
r
eg
u
lato
r
.
T
h
e
c
u
r
r
en
ts
i
q(ref)
a
n
d
i
0(ref)
ar
e
g
e
n
er
ated
b
y
c
o
n
v
er
tin
g
b
y
-
1
o
f
i
q
an
d
i
0
.
I
t
was
th
en
co
n
v
e
r
ted
ag
ain
to
g
e
n
er
ate
th
e
r
ef
er
en
ce
cu
r
r
en
ts
i
a(ref)
,
i
b(ref)
,
an
d
i
c(ref)
u
s
in
g
(
4
)
.
[
(
)
(
)
c
(
r
ef
)
]
=
[
s
in
(
)
c
os
(
)
1
s
in
(
−
2
3
)
c
os
(
−
2
3
)
1
s
in
(
+
2
3
)
c
os
(
+
2
3
)
1
]
[
(
)
(
)
0
(
)
]
(
4
)
a
bc
dq
P
L
L
i
a
i
b
i
c
v
a
v
b
v
c
L
P
F
-
+
+
-
V
dc
V
r
e
f
PI
dq
a
bc
i
d
i
q
i
0
i
a(
r
e
f
)
i
b(
r
e
f
)
i
c
(
r
e
f
)
w
i
dc
i
d(
r
e
f
)
i
q(
r
e
f
)
i
0(
r
e
f
)
-
1
-
1
Fig
u
r
e
3
.
T
h
e
m
eth
o
d
r
e
f
er
en
c
e
cu
r
r
en
ts
wer
e
g
en
er
ated
u
s
in
g
th
e
SR
F m
eth
o
d
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J Po
w
E
lec
&
Dr
i Sy
s
t
I
SS
N:
2088
-
8
6
9
4
Ha
r
mo
n
ic
r
ed
u
ctio
n
in
p
o
w
er g
r
id
s
:
i
mp
leme
n
tin
g
a
fo
u
r
-
leg
g
ed
mu
ltil
ev
el
in
ve
r
ter
…
(
A
s
ep
A
n
d
a
n
g
)
927
2
.
3
.
M
o
del
predict
iv
e
co
ntr
o
l
T
h
e
MPC
is
ch
o
s
en
f
o
r
g
en
er
atin
g
o
p
tim
al
s
witch
in
g
p
atter
n
s
in
m
u
ltil
ev
el
in
v
er
ter
s
b
ec
au
s
e
o
f
its
ab
ilit
y
to
p
r
ed
ict
f
u
tu
r
e
c
u
r
r
e
n
ts
i
(
k
+1
)
b
ased
o
n
th
e
s
y
s
te
m
p
ar
a
m
eter
s
an
d
ap
p
lied
s
witch
in
g
s
tates.
B
y
lev
er
ag
in
g
a
co
s
t
f
u
n
ctio
n
g
=
∣
i
ref
−
i
(
k
+1
)
∣
,
th
e
MPC
m
in
im
izes
th
e
d
if
f
er
e
n
ce
b
etwe
en
th
e
r
ef
er
e
n
ce
cu
r
r
en
t
(
i
ref
)
an
d
p
r
e
d
icted
c
u
r
r
en
t,
ef
f
ec
tiv
ely
r
ed
u
cin
g
T
HD
an
d
en
h
an
cin
g
p
o
wer
q
u
ality
.
I
ts
f
le
x
ib
ilit
y
allo
ws
it
t
o
ad
ap
t
to
d
y
n
a
m
ic
an
d
u
n
b
alan
ce
d
lo
ad
co
n
d
itio
n
s
,
en
s
u
r
in
g
co
n
s
is
ten
t
p
er
f
o
r
m
an
ce
in
th
r
ee
-
p
h
ase
f
o
u
r
-
wir
e
s
y
s
tem
s
.
Fu
r
th
er
m
o
r
e,
MPC
’
s
co
m
p
u
tatio
n
al
ef
f
icien
cy
o
f
MPC
,
en
ab
led
b
y
a
d
v
an
ce
m
en
ts
in
em
b
ed
d
ed
tech
n
o
lo
g
ies,
m
ak
es
it
p
r
ac
t
ical
f
o
r
r
ea
l
-
tim
e
ap
p
licatio
n
s
.
T
h
e
ca
p
ab
ilit
y
o
f
th
e
al
g
o
r
ith
m
to
h
a
n
d
le
m
u
ltil
ev
el
s
witch
in
g
with
n
u
m
er
o
u
s
co
m
b
in
atio
n
s
(
e.
g
.
,
8
1
f
o
r
a
f
iv
e
-
lev
el
in
v
er
ter
)
en
s
u
r
es
an
o
p
tim
al
v
o
ltag
e
o
u
tp
u
t
wh
ile
m
ain
tai
n
in
g
p
r
ec
is
io
n
.
T
h
ese
ad
v
an
t
ag
es
m
ak
e
MPC
an
id
ea
l
ch
o
ice
f
o
r
im
p
r
o
v
in
g
co
n
tr
o
l a
cc
u
r
ac
y
a
n
d
r
eliab
ilit
y
in
m
u
ltil
ev
el
in
v
er
ter
ap
p
licatio
n
s
.
T
h
e
MPC
alg
o
r
ith
m
u
s
ed
in
th
is
s
tu
d
y
is
as f
o
llo
ws:
˗
I
n
itialize
p
ar
am
eter
s
T
h
e
in
itial
v
alu
e
o
f
i
(
k
)
(
c
u
r
r
en
t
at
tim
e
k
)
is
s
et.
W
e
d
ef
in
e
th
e
n
etwo
r
k
p
ar
am
eter
s
,
in
clu
d
in
g
R
,
L
,
C
,
an
d
th
e
s
am
p
lin
g
tim
e
T
s.
˗
C
alcu
late
r
ef
er
en
ce
cu
r
r
en
t
T
r
an
s
f
o
r
m
th
e
th
r
ee
-
p
h
ase
c
u
r
r
en
t
(
i
a
,
i
b
,
i
c
)
in
to
th
e
dq
0
d
o
m
ain
u
s
in
g
C
lar
k
a
n
d
Par
k
tr
a
n
s
f
o
r
m
atio
n
s
as
in
(
5
)
:
[
0
]
=
T
[
0
]
(
5
)
a
p
p
ly
a
lo
w
-
p
ass
f
ilter
to
i
d
to
r
em
o
v
e
h
ar
m
o
n
ic
co
m
p
o
n
e
n
ts
an
d
g
e
n
er
ate
th
e
r
ef
er
en
ce
cu
r
r
en
t (
i
ref
).
˗
Pre
d
ict
f
u
tu
r
e
c
u
r
r
en
t
Use th
e
s
y
s
tem
eq
u
atio
n
(
b
ase
d
o
n
elec
tr
ical
cir
c
u
it d
y
n
a
m
ics)
as sh
o
wn
in
(
6
)
.
(
+
1
)
=
(
)
+
(
−
−
.
(
)
)
(
6
)
C
u
r
r
en
t
i
(
k
+1
)
is
p
r
ed
icted
u
s
i
n
g
in
v
e
r
ter
v
o
ltag
e
(
v
inv
).
˗
E
v
alu
ate
co
s
t f
u
n
ctio
n
C
o
m
p
u
te
th
e
co
s
t
f
u
n
ctio
n
(
g
)
as
th
e
a
b
s
o
lu
te
d
if
f
er
en
ce
b
etwe
en
th
e
p
r
ed
icted
c
u
r
r
e
n
t
i
(
k
+1
)
an
d
th
e
r
ef
er
en
ce
c
u
r
r
en
t
(
i
ref
)
,
as sh
o
wn
in
(
7
)
.
=
|
−
(
+
1
)
|
(
7
)
˗
Op
tim
ize
s
witch
in
g
p
atter
n
I
ter
ate
all
8
1
p
o
s
s
ib
le
s
witch
in
g
co
m
b
in
atio
n
s
f
o
r
th
e
m
u
ltil
ev
el
in
v
er
te
r
.
Select
th
e
s
witch
in
g
p
atter
n
t
h
at
m
in
im
izes g
.
˗
Gen
er
ate
s
witch
in
g
s
ig
n
al
T
h
e
o
p
tim
al
s
witch
in
g
p
atter
n
is
ap
p
lied
to
th
e
in
v
er
ter
to
p
r
o
d
u
ce
th
e
d
esire
d
v
o
ltag
e
(
v
inv
).
˗
R
ep
ea
t
th
e
p
r
o
ce
s
s
Up
d
ate
i
(
k
)
with
th
e
p
r
ed
icted
v
alu
e
i
(
k
+1
)
.
Step
3
is
co
n
tin
u
ed
f
o
r
t
h
e
n
e
x
t sam
p
lin
g
p
e
r
io
d
.
T
o
g
en
er
ate
a
s
witch
in
g
s
ig
n
al
with
th
e
MPC
m
eth
o
d
,
a
r
ef
er
en
ce
cu
r
r
en
t
d
er
i
v
ed
f
r
o
m
h
ar
m
o
n
ic
s
ig
n
al
ex
tr
ac
tio
n
,
as
s
h
o
wn
i
n
(
4
)
,
an
d
a
p
r
ed
icted
cu
r
r
e
n
t
s
ig
n
al
ar
e
r
eq
u
ir
e
d
.
T
h
e
p
r
ed
ic
ted
cu
r
r
e
n
t
is
b
u
ilt
f
r
o
m
th
e
eq
u
atio
n
o
f
t
h
e
s
y
s
tem
s
er
ies in
th
e
f
o
r
m
o
f
a
clo
s
ed
lo
o
p
,
as sh
o
wn
in
(
8
).
=
+
.
+
1
∫
+
(
8
)
W
h
er
e
v
pccx
is
th
e
v
o
ltag
e
at
th
e
p
o
in
t
o
f
co
m
m
o
n
co
u
p
lin
g
p
h
ase
x
,
L
,
R
,
an
d
C
ar
e
th
e
v
alu
es
o
f
r
esis
tan
ce
,
in
d
u
ctan
ce
,
an
d
ca
p
ac
itan
ce
o
f
th
e
p
ass
iv
e
p
o
wer
f
ilter
,
r
esp
ec
tiv
ely
,
an
d
v
inv
is
th
e
v
alu
e
o
f
th
e
in
v
er
te
r
v
o
ltag
e
ac
co
r
d
in
g
t
o
(
2
)
f
r
o
m
p
h
ase
x
to
n
o
r
a
cc
o
r
d
i
n
g
to
(
9
)
.
=
(
9
)
T
h
e
n
ex
t
p
r
o
ce
s
s
is
to
d
is
cr
etize
(
8
)
,
b
o
th
d
er
iv
ativ
es
an
d
in
teg
r
als,
u
s
in
g
f
o
r
war
d
E
u
ler
s
o
th
at
th
e
p
r
ed
ictio
n
c
u
r
r
en
t
u
s
ed
in
th
e
s
witch
in
g
p
r
o
ce
s
s
is
o
b
tain
ed
as in
(
10
)
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8
6
9
4
I
n
t J Po
w
E
lec
&
Dr
i Sy
s
t
,
Vo
l.
16
,
No
.
2
,
J
u
n
e
20
25
:
9
2
2
-
9
3
5
928
(
+
1
)
=
1
(
+
+
)
(
−
−
(
−
)
(
(
)
)
)
(
10
)
W
h
er
e
i(
k+1
)
is
th
e
p
r
e
d
ictio
n
f
l
o
w
u
s
in
g
th
e
MPC
m
e
th
o
d
.
Ts
is
th
e
tim
e
s
am
p
li
n
g
an
d
i(
k
)
is
th
e
ex
tr
ap
o
latio
n
cu
r
r
en
t
at
tim
e
k
.
T
o
d
eter
m
i
n
e
th
e
b
est
s
witch
in
g
p
atter
n
s
u
c
h
th
at
th
e
p
r
ed
i
cted
cu
r
r
en
t
is
t
h
e
s
am
e
as
th
e
r
e
f
er
en
ce
cu
r
r
en
t,
th
e
co
s
t
f
u
n
ctio
n
is
u
s
ed
,
wh
ic
h
is
th
e
ab
s
o
lu
te
v
alu
e
o
f
th
e
d
if
f
er
en
ce
b
etwe
en
th
e
r
ef
er
e
n
ce
c
u
r
r
e
n
t
an
d
th
e
r
ef
er
en
ce
c
u
r
r
e
n
t,
as
in
(
11
)
.
I
n
o
n
e
s
am
p
lin
g
tim
e,
8
1
iter
atio
n
s
wer
e
p
e
r
f
o
r
m
e
d
to
o
b
tain
th
e
s
m
allest
g
-
v
alu
e
.
T
h
is
s
m
allest
G
p
r
o
d
u
ce
s
a
p
r
ed
ictio
n
v
alu
e
th
at
is
th
e
r
esu
lt
o
f
s
witch
in
g
,
ac
co
r
d
in
g
to
(
2
)
.
=
|
(
)
−
(
+
1
)
|
+
|
(
)
−
(
+
1
)
|
+
|
(
)
−
(
+
1
)
|
+
|
(
)
−
(
+
1
)
|
(
11
)
3.
RE
SU
L
T
S AN
D
D
I
SCU
SS
I
O
N
T
h
e
m
o
d
el
was
test
ed
b
y
s
im
u
latin
g
th
e
cir
cu
its
an
d
eq
u
atio
n
s
u
s
in
g
MA
T
L
AB
Sim
u
lin
k
.
T
h
e
s
im
u
latio
n
h
ad
th
r
ee
m
ain
p
ar
ts
:
a
p
lan
t
with
a
th
r
ee
-
p
h
ase,
f
o
u
r
-
wir
e
s
o
u
r
ce
c
o
n
n
ec
ted
t
o
a
n
o
n
lin
ea
r
lo
a
d
;
r
ef
er
en
ce
c
u
r
r
e
n
t
g
en
e
r
atio
n
u
s
in
g
SR
F
d
er
iv
ed
f
r
o
m
l
o
ad
cu
r
r
en
ts
a
n
d
v
o
ltag
es;
an
d
a
h
y
b
r
id
p
o
wer
f
ilter
co
m
p
r
is
in
g
an
L
C
p
ass
iv
e
f
ilt
er
an
d
a
n
ac
tiv
e
f
ilter
as
a
t
h
r
ee
-
p
h
ase
f
o
u
r
-
wir
e
co
n
v
er
ter
.
Simu
lin
k
was
u
s
ed
to
s
im
u
late
th
e
cir
cu
it
f
o
r
th
e
b
asic
p
ar
am
eter
s
d
escr
ib
ed
in
MA
T
L
AB
.
MPC
p
r
o
g
r
am
m
in
g
is
p
er
f
o
r
m
e
d
u
s
in
g
(
10
)
,
wh
e
r
e
i(
k+1
)
is
th
e
p
r
ed
icte
d
cu
r
r
en
t
o
b
tain
ed
f
r
o
m
t
h
e
in
v
e
r
ter
v
o
ltag
e
v
i
n
v
f
r
o
m
(
2
)
.
T
h
e
p
r
ed
icted
cu
r
r
en
t
m
u
s
t
b
e
clo
s
e
to
th
e
r
ef
er
en
ce
c
u
r
r
e
n
t
to
r
ed
u
ce
h
a
r
m
o
n
y
,
as
d
eter
m
i
n
ed
u
s
in
g
th
e
co
s
t
f
u
n
ctio
n
in
(
11
)
.
T
o
ass
ess
th
e
p
er
f
o
r
m
an
ce
o
f
th
e
m
o
d
el,
test
s
wer
e
co
n
d
u
cted
u
s
in
g
b
alan
ce
d
a
n
d
u
n
b
alan
ce
d
n
o
n
lin
ea
r
lo
a
d
s
.
3
.
1
.
B
a
la
nced
lo
a
d si
m
ula
t
io
n
T
h
e
b
alan
ce
d
lo
ad
test
was
p
er
f
o
r
m
ed
b
y
c
o
n
n
ec
tin
g
a
t
h
r
ee
-
p
h
ase,
f
o
u
r
-
wir
e
elec
tr
ic
al
n
etwo
r
k
with
a
s
in
g
le
-
p
h
ase
r
ec
tifie
r
co
n
n
ec
ted
to
th
e
r
esis
tiv
e
lo
ad
R
an
d
an
in
d
u
cto
r
L
with
a
v
alu
e
o
f
R
a
=
R
b
=
R
c
=
2
5
wh
ile
th
e
in
d
u
cto
r
v
alu
e
is
7
0
m
H.
T
h
e
lo
ad
cir
cu
it
is
illu
s
tr
ated
in
Fig
u
r
e
4
.
Simu
latio
n
s
ar
e
p
er
f
o
r
m
ed
to
ass
ess
th
e
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u
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ates
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icts
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ed
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Fig
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icted
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ilter
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Fig
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r
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ated
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atter
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atin
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ates
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r
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esp
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2
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ip
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ase,
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leav
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ter
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a)
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b
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d
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u
r
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.
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ase
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if
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s
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ated
in
Fig
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8
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with
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ase
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g
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Fig
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8
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,
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ases
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ase
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en
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s
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icted
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Fig
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e
8
(
c)
,
w
h
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s
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r
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m
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s
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m
s
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wn
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Fig
u
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e
8
(
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)
in
d
icate
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f
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ately
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