In
te
r
n
ation
a
l Jou
rn
al
o
f Po
we
r
Elec
tron
ic
s an
d
D
r
ive S
y
stem
(IJ
PED
S
)
V
o
l.
10, N
o.
4, D
e
c
e
m
ber
201
9,
pp.
1833~
18
51
ISSN: 2088-
8694,
DOI
:
10.11591
/ijpeds.
v10.
i
4.pp1833-1851
1
8
33
Jou
rn
a
l
h
o
me
pa
ge
:
ht
tp:
//i
a
e
score
.
com
/
j
o
u
r
na
l
s
/
i
n
d
e
x
.
p
hp/IJ
PED
S
Modular
multilevel converter
(MMC) based STATC
O
M with
vector control and virtual im
pe
dance voltage c
om
pensations
A
.
U. Law
an
1
,
Haid
er
A
.
F.
A
lmu
r
ib
2
,
Jeen
G
.
K
h
or
3
1
,
2,
3
D
e
p
a
rtm
e
n
t
o
f
El
ectrical & Electro
ni
c En
gin
e
erin
g,
U
ni
versit
y
o
f Not
t
ingha
m
,
Malaysia
1
D
e
p
a
rtm
e
n
t
o
f El
ectrical
En
g
i
n
eeri
ng,
Hu
s
s
e
in
i
A
d
am
u F
e
d
e
ral P
o
lytechn
i
c,
N
igeria
Art
i
cl
e In
fo
ABSTRACT
A
r
tic
le hist
o
r
y
:
Re
ce
i
v
e
d
O
c
t
8
,
2018
Re
vise
d N
ov
2
0
,
201
8
Ac
ce
p
t
ed
M
a
y
2
9
,
2
019
Thi
s
p
aper
p
r
e
sent
s
modul
a
r
multi
l
e
vel
convert
e
r
(MM
C
)
wi
t
h
c
ir
cu
lati
ng
curren
t
c
on
tro
l
w
hich
p
ro
vid
e
s
an
i
m
p
ro
ved
balan
ced
cap
aci
to
rs
vo
lt
ages
.
The
c
o
ntrol
is
a
c
h
ie
ve
d
b
y
em
ploy
in
g
c
o
mp
e
n
sa
tion
t
e
c
h
ni
qu
e
s
i
n
t
h
e
ext
e
rnal
a
nd
in
t
e
rnal
c
ont
rol
s
o
f
th
e
MM
C
b
a
sed
st
atic
c
o
m
pen
s
a
tor
(S
TATCO
M
).
P
erf
o
rm
an
ce
variat
io
ns
a
ri
se
d
u
r
in
g
STATCOM
n
o
n
-
idea
l
op
erati
on
wi
th
g
rid
ext
e
rnally
a
nd
d
uri
ng
capaci
to
rs
v
oltag
e
t
r
an
si
ent
s
due
t
o
charg
i
n
g
a
nd
d
ischarg
i
n
g
w
ith
in
t
he
M
M
C
t
h
a
t
resu
lt
i
n
t
h
e
em
er
gen
ce
of
a
vo
lt
age
d
i
sturb
a
nce.
T
h
e
p
ro
po
rt
io
na
l
integ
r
a
l
(
P
I)
co
nt
ro
ll
er
is
u
sually
em
pl
oyed
i
n
t
he
e
x
t
ern
a
l an
d i
n
t
e
rnal
c
on
trol
s fo
r
a f
a
s
t
res
po
n
s
e
and
reacti
v
e
current
c
ont
r
ol
.
In
t
his
paper,
t
he
p
erf
o
rm
anc
e
o
f
the
PI
c
ontr
ol
l
e
r
is
im
p
r
oved
u
s
i
ng
vo
lt
age
com
p
en
sat
i
on
i
n
the
ext
e
rnal
c
on
tro
l
a
n
d
v
i
r
t
u
a
l
im
p
e
dance
i
n
t
h
e
c
ap
a
c
ito
r
vo
lta
ge
c
o
n
t
r
o
l
wi
t
h
in
a
f
i
v
e
-
le
v
e
l
MMC.
The
p
r
o
p
o
s
e
d
c
o
n
t
r
o
l
m
i
n
i
m
i
z
e
s
t
h
e
v
a
r
i
a
t
i
o
n
s
i
n
t
h
e
S
T
A
T
C
O
M
o
p
e
r
a
t
io
ns
w
i
t
h
th
e
gri
d
a
n
d
w
it
h
i
n
t
h
e
MM
C
t
o
p
rovid
e
a
n
enhan
ced
o
ver
a
ll
s
yst
em
response.
K
eyw
ord
s
:
Mu
lt
ile
ve
l c
o
n
v
erter
STATCOM
P
I
contro
l
l
er
Vol
t
a
g
e
com
p
e
n
sa
tio
n
Co
pyri
gh
t © 2
019 In
stitute
of
Advanced
En
gi
neeri
n
g
an
d
S
c
ien
ce.
All
rights
res
e
rv
ed.
Corres
pon
d
i
n
g
Au
th
or:
Ab
du
rrah
m
an
Umar
Lawan
,
D
e
pa
rtem
ent o
f
Elec
t
r
i
c
a
l a
n
d
Com
p
uter
E
n
g
i
ne
er
in
g,
Uni
v
ersi
ty o
f
Notti
n
gham
, Ma
l
ays
i
a Cam
pus,
43
5
00 S
e
me
ny
i
h
.
Ma
la
ysia
.
Em
ail:
kecx
4a
u
a
@
n
ot
tin
g
h
a
m
.e
du.m
y
1.
I
N
TR
OD
U
C
TI
O
N
M
u
lti
l
e
v
e
l
in
ve
rt
er
t
op
olo
g
i
e
s
a
r
e
wi
d
e
l
y
a
cc
ep
t
e
d
as
o
n
e
o
f
t
h
e
m
o
s
t
r
ece
nt
t
ec
hno
lo
g
i
cal
a
d
v
a
n
ces
i
n
p
o
w
er
e
l
e
ct
ro
ni
c
i
ndu
st
ri
es.
Th
e
i
r
u
s
e
s
a
re
m
o
s
t
l
y
i
n
th
e
a
r
ea
of
h
i
gher-
vo
ltage
p
ow
e
r
a
pp
l
i
ca
ti
o
n
s
su
ch
a
s
fle
x
ib
le
A
.C
t
r
a
nsm
i
ss
i
on
sys
t
em
s
(
F
A
C
TS)
a
p
plica
tio
ns,
a
nd
lar
ge
p
o
w
e
r
e
l
ect
ri
c
d
r
iv
e
s
.
Th
es
e
ap
pl
i
cat
i
o
n
s
are
poss
i
ble
d
u
e
to
m
u
l
t
i
leve
l
con
v
er
t
e
rs’
ba
l
a
nc
ed
a
n
d
d
istr
ib
u
t
e
d
v
o
lta
ge
s
tresses
tha
t
a
r
e
s
har
e
d
am
ong
th
e
sw
it
c
h
es.
P
o
w
e
r
qua
lit
y
(
P
Q
)
e
nha
ncem
e
n
t
in
t
he
p
o
w
e
r
s
ystem
d
i
st
ri
but
ion
has
b
e
en
a
t
t
a
i
n
ed
u
si
ng
m
u
l
t
i
l
e
v
e
l
S
T
A
T
C
O
M
s
[
1
–
4
]
.
T
h
e
r
e
a
r
e
m
a
n
y
t
i
m
e
d
o
m
a
i
n
b
a
s
e
d
c
o
n
tr
ol
s
c
h
em
es
s
uc
h
as
d
irec
t
a
n
d
i
n
-
direc
t
c
ontro
l,
pow
e
r
s
y
n
ch
ron
i
za
ti
on,
a
n
d
d
ire
c
t
-qu
a
dr
ature(
d-
q)
v
ec
t
o
r
c
o
ntro
l
m
e
th
o
d
s
a
p
pl
i
e
d
to
t
he
mult
i
l
e
v
el
S
TA
TCO
M
f
or
t
h
e
pow
er
qua
l
ity
e
n
h
a
n
cem
e
n
t
[5–
8
].
A
m
o
ng
t
he
se
s
c
h
em
es,
the
m
o
st
c
om
m
o
n
l
y
em
plo
y
e
d
a
re
t
he
d
i
r
ec
t
con
t
rol
an
d
t
h
e
d-q
ve
c
t
or
c
o
n
tr
ol
m
e
t
h
o
d
s,
e
mplo
ye
d
m
o
stl
y
f
or
r
eac
tive
p
o
w
e
r
(V
A
R
) com
p
ensa
ti
o
n
s i
n
S
T
A
TCO
M
a
pp
l
i
c
a
ti
ons[
5
, 6]
. STA
TCO
M
a
pp
l
ica
t
i
ons for V
A
R
com
pe
nsa
t
ion a
n
d
st
a
b
il
it
y st
u
d
i
e
s
have
bee
n empl
o
y
ed
b
a
s
ed o
n
differ
e
n
t c
o
n
t
ro
l
s a
nd m
o
d
u
l
a
t
i
on
sc
hem
e
s [9–
11]
.
G
e
nera
lly,
t
h
e
r
e
is
t
rade
-o
ff
b
e
t
w
e
e
n
t
he
m
odu
la
tio
n
sc
he
me
s,
t
he
s
w
itc
h
i
n
g
fre
que
ncy,
t
ra
n
s
i
e
nt
respo
n
se
a
n
d
t
he
u
se
o
f
filter
s
f
or
S
TA
TCO
M
V
A
R
c
om
p
e
nsa
t
i
on.
T
he
t
ransie
nt
r
e
s
po
n
s
e
c
a
n
impr
o
v
e
w
i
t
h
hi
gh
sw
i
t
c
h
in
g
fre
q
uenc
y,
b
ut
a
t
t
h
e
e
xpe
nse
of
h
i
g
h
l
o
ss
a
nd
h
a
r
m
o
ni
c
s
.
S
i
nc
e
fi
l
t
e
r
s
su
ch
a
s
i
n
duc
t
o
rs
a
n
d
ca
paci
t
o
rs
a
re
i
n
v
e
r
sel
y
p
r
o
p
o
rt
io
na
l
to
s
w
i
tch
i
ng
fre
que
n
c
y,
t
he
h
igh-s
w
itc
hi
n
g
f
re
q
u
e
ncy
a
l
l
o
w
s
t
h
e
u
se
o
f
smal
l
f
ilt
ers
and
re
sult
s
in
m
ore
ma
rg
i
n
g
ai
n
t
o
b
e
u
s
ed
t
o
co
n
tro
l
t
h
e
V
A
R
a
t
t
h
e
mo
de
rate
f
r
e
qu
enc
i
e
s
.
This
con
t
ro
l
ac
t
i
o
n
w
i
l
l
hel
p
i
n
im
pr
o
v
in
g
t
h
e
tra
n
s
i
en
ts.
Fil
t
ers
d
o
red
u
ce
h
arm
onic
s
i
n
t
h
e
s
y
s
t
e
m
.
H
o
w
e
ver
,
t
he
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
: 208
8-
869
4
I
nt
J
P
ow
Elec
& Dr
i
S
y
st V
ol.
10,
N
o.
4
, Dec
201
9 : 1
8
3
3
–
18
51
1
834
use
o
f
f
i
l
ters
i
n
V
A
R
a
p
p
lic
a
tio
n
resu
l
t
s
i
n
t
he
i
ncre
ase
o
f
t
h
e
sys
t
e
m
r
e
s
p
onse
de
la
y
[1
2].
F
o
r
ins
t
a
n
ce
,
a
selec
t
i
v
e harm
on
ic mod
u
l
at
i
o
n, w
hi
c
h
is a
lo
w
-
fr
equenc
y m
o
d
u
l
a
t
i
on c
o
n
t
r
o
l w
a
s em
pl
oye
d i
n
[1
3
]. This w
a
s
ca
rried
o
u
t
u
s
i
ng
a pr
o
p
o
r
ti
o
n
al
(
P) contr
o
l
l
er
t
o ac
hie
v
e
VAR
com
p
ensa
t
i
o
n
i
n a
n
un
b
a
l
anc
e
d s
y
stem
.
Al
th
oug
h
th
e
c
o
nt
rol
u
s
ed
a
s
e
c
o
nd
-o
rd
e
r
f
il
t
e
r
to
s
up
p
r
es
s
t
h
e
e
f
f
ec
t
s
o
f
di
stor
tio
ns,
t
h
e
pro
p
o
s
e
d
con
t
ro
l
s
t
i
ll
o
f
fe
rs
a
f
ast
r
e
s
p
o
n
se
by
re
ac
hi
n
g
i
ts
s
tea
d
y
-
st
a
te
w
it
hi
n
5
cyles.
H
ow
e
v
er,
this
c
ompe
nsa
tio
n
w
a
s
ac
h
i
e
v
e
d
w
ith
a
h
ig
her
-
sw
it
c
h
in
g
fr
equ
e
nc
y
(
i
.
e
1
0
k
H
z)
t
o
att
a
in
t
he
d
esi
r
e
d
c
u
r
re
n
t
c
ont
rol
b
a
ndwi
d
t
h
.
F
u
r
t
he
rm
o
r
e,
w
hen
t
h
e
P
-
contr
o
l
a
c
t
i
o
n
w
a
s
com
p
are
d
w
ith
t
he
p
r
o
p
o
r
tio
na
l
i
n
te
gra
l
(
P
I
)
contro
l
in
t
he
syste
m
,
the
d
i
s
t
or
ti
o
n
i
n
t
h
e
c
u
r
r
ent
w
a
s
ma
rgi
n
a
lly
r
ed
uce
d
.
A
no
ther
c
o
n
t
ro
l
us
ing
the
c
a
rr
ier
shifte
d
pu
ls
e
w
i
d
t
h
m
udl
a
u
a
tio
n
(CS
P
WM)
w
a
s
c
a
rr
i
e
d
o
u
t
i
n
[
1
4
].
T
his
c
o
n
t
r
o
l
w
as
f
or
a
V
AR
c
ompe
n
s
a
t
i
on
i
n
a
mult
i
l
e
v
el
S
T
A
TCO
M
u
n
d
e
r
d
iffere
nt
o
pe
r
a
ti
ng
c
o
nd
iti
o
n
s.
T
he
c
o
ntr
o
l
pe
rform
anc
e
w
as
e
val
u
a
t
ed
f
r
o
m
ca
paci
t
i
ve
m
od
e
to
i
n
duc
t
i
ve
m
ode.
The
au
t
hor
c
l
a
ime
d
t
o
ha
v
e
r
e
duc
ed
t
he
s
w
itc
h
i
n
g
freque
ncy
by
th
e
use
of
a
n
e
w
r
e
act
ive
c
u
rren
t
r
e
f
er
ence
,
the
c
o
nt
r
o
l
wa
s
ac
h
i
eve
d
u
si
ng
P
-
con
t
ro
l
l
e
r
b
ase
d
d
ec
o
u
p
l
e
d
c
u
r
rent
con
t
ro
l.
H
ow
e
v
e
r
,
there
w
a
s
a
de
la
y
i
n
t
he
s
ys
t
e
m
at the
s
t
a
r
t
up
o
f
th
e
con
t
ro
l
,
w
hi
ch
r
e
a
c
h
e
d
i
t
s
s
t
eady
-
st
at
e
aft
e
r
ap
pro
x
im
atel
y
10
c
ycle
s.
T
hou
g
h
a
t
t
h
e
t
r
an
si
tio
n,
t
he
t
r
ans
i
ent
res
p
on
se
w
a
s
f
ast
.
A
w
el
l
-
t
une
d
PI-
c
o
nt
roll
e
r
c
ould
y
i
e
l
d
re
sp
on
se
w
it
h
ze
ro
o
r mi
n
i
mu
m
st
ead
y
st
a
te
err
or w
i
t
hi
n
a
sm
al
l rise ti
m
e.
H
ow
e
v
er,
P
I
con
t
ro
l
l
ers
may
ha
ve
s
om
e
d
r
aw
ba
cks
w
h
ic
h
i
n
c
l
u
d
e
h
i
gh
s
ettl
in
g
tim
e
a
nd
ove
rs
ho
ots
[9,
15].
It
h
as
a
l
s
o
bee
n
r
e
port
e
d
tha
t
P
I
con
t
ro
l
l
ers
ha
ve
p
o
o
r
t
r
ans
i
en
ts
c
o
m
par
e
d
t
o
o
t
h
e
r
c
o
n
t
r
o
lle
rs
s
uc
h
as
hys
ter
e
sis
a
n
d
ram
p
t
i
m
e
c
o
n
t
rol
l
ers
in
v
ol
t
a
ge
s
our
ce
i
n
ve
rter
(
V
S
I)
S
TATCO
M
s
[
1
5
]
.
F
or
e
xa
mple
i
n
[
16],
P
I
c
ontr
o
l
w
a
s
em
plo
y
e
d
f
or
S
TA
TCO
M
v
o
l
t
a
ge reg
ula
t
io
ns,
bu
t osc
i
l
l
a
t
i
on ha
s o
cc
urre
d
i
n
t
he re
s
po
n
s
e and pr
od
uce
d
p
oor
trans
i
en
ts
t
ho
u
gh
t
h
er
e
was
a
minim
a
l
stea
d
y
-st
a
te
e
rror
in
t
he
c
omm
a
nd
trac
k
i
n
g
.
A
lter
n
at
ive
l
y,
t
he
s
i
m
ple
s
t
appr
oa
ch
i
s
to
u
se
car
rier
s
hi
fted
p
u
l
se
w
id
t
h
m
od
u
l
at
i
o
n
(
C
S
P
W
M
)
w
h
i
c
h
h
a
s
a
m
o
d
e
r
a
t
e
s
w
i
t
c
h
i
n
g
fre
que
nc
y
fe
a
t
ure
a
n
d
a
few
c
ontr
o
l
c
o
m
p
l
i
c
a
ti
o
n
s.
T
h
i
s
c
o
u
l
d
b
e
e
m
ploy
ed
w
i
t
h
p
r
o
port
i
o
n
a
l
co
nt
rol
l
ers
f
o
r
a
dec
r
ea
sed
rise
t
im
e
and
s
t
e
a
dy-s
t
ate
e
rror
[14]
.
H
o
w
e
ve
r,
t
h
i
s
w
o
ul
d
be
a
t
t
h
e
e
xpe
ns
e
of
a
n
i
n
c
r
ea
se
d
over
sho
o
t
com
p
are
d
t
o
pro
p
o
rt
iona
l
in
t
e
gral
(
P
I
)
contro
llers.
Th
is
i
s
p
o
ssibl
e
du
e
to
t
h
e
f
acts
tha
t
a
s
t
h
e
P
-
c
o
n
t
r
o
l
l
e
r
p
u
s
h
e
s
h
a
r
d
e
r
f
o
r
a
g
i
v
e
n
l
e
v
e
l
o
f
a
n
e
r
r
o
r
,
i
t
f
o
r
c
e
s
t
h
e
c
on
t
r
o
l
t
o
resp
ond
q
ui
ck
ly
but
a
l
s
o
t
o
overs
ho
o
t
m
ore
.
F
or
a
n
M
M
C
i
n
ver
t
e
r
s
pe
ci
fica
ll
y,
i
n
c
re
asin
g
t
h
e
s
t
a
c
ke
d
ce
lls
i
nc
rea
s
e
s
t
he
s
w
i
t
c
h
i
n
g
fre
que
nc
y
a
nd
vo
l
t
age
r
i
p
p
l
es
.
D
e
pend
in
g
o
n
a
n
um
ber
o
f
t
he
l
e
v
e
l
s,
t
h
i
s
w
ould,
i
n
tur
n
,
incr
ease
th
e
c
o
n
t
ro
l
com
p
l
i
c
a
t
i
o
ns
s
uc
h
as
t
he
c
a
p
ac
it
ors
v
o
l
t
a
g
e
b
a
l
anc
i
ng
a
n
d
v
o
l
t
a
g
e
di
st
u
r
b
a
n
c
es
w
hi
ch
p
ro
du
ce
h
a
r
mo
n
i
c
rip
p
les.
H
ow
e
v
er,
t
h
e
sw
i
t
c
h
i
n
g
fr
eq
uenc
y
a
n
d
v
o
l
t
a
g
e
r
i
pp
les
i
s
s
ues
cou
l
d
be
r
esol
ve
d
b
y
t
he
u
se
o
f
fu
n
d
am
enta
l
(low
)
freq
u
e
n
c
y
t
ec
hn
i
q
ues
suc
h
a
s
selec
tive
ha
rm
o
n
i
c
e
limi
n
a
t
i
o
n
(S
H
E
)
or
opti
m
ize
d
sw
it
c
h
i
n
g
tec
h
niq
u
e,
b
u
t
w
it
h
an
i
ncre
ase
d
c
ontro
l
com
p
le
x
ity
s
uch
a
s
t
he
m
odu
la
ti
on
a
n
gl
e
ca
lcu
l
a
t
io
n
plus
the i
n
her
e
n
t
di
f
fic
u
lt im
p
l
e
m
enta
t
i
o
n
o
f a
lo
w
sw
itch
i
n
g
te
c
h
n
i
que
[1
7
]
.
An
ot
h
e
r
ch
al
len
g
e
i
n
th
e
MMC
i
s
th
e
st
ack
ed
cell
s
cap
aci
t
o
r
vo
l
ta
ge
b
a
l
a
n
ci
n
g
.
I
f
t
he
p
ow
er
l
osses
we
re
i
d
e
n
t
i
cal
i
n
all
th
e
di
ff
eren
t
H-b
r
i
d
g
e
c
e
l
ls,
th
e
l
o
ss
es
co
ul
d
be
s
har
e
d
un
i
f
orm
l
y
a
m
ong
t
he
c
el
ls
;
th
us,
the
ac
t
i
ve
a
nd
r
eac
ti
ve
p
ow
e
r
w
oul
d
be
d
is
trib
u
t
e
d
e
q
u
a
lly
w
i
t
h
i
n
t
he
M
M
C
.
H
o
w
e
ver
,
b
eca
use
o
f
t
h
e
m
i
s
m
a
t
c
h
a
m
o
n
g
t
h
e
c
o
m
p
o
n
e
n
t
s
e
s
p
e
c
i
a
l
l
y
t
h
e
c
e
l
l
c
a
p
a
c
i
t
o
r
s
,
a
n
d
t
h
e
c
harg
i
n
g
a
n
d
di
schar
g
in
g
dur
in
g
s
w
i
t
c
h
i
n
g
,
t
h
e
c
e
l
l
s
w
o
u
l
d
n
o
t
b
e
h
a
v
e
i
d
e
n
t
i
c
a
l
l
y
,
h
e
n
c
e
r
e
s
u
l
t
i
n
d
i
f
f
er
ent
los
s
e
s
.
Th
is
l
e
a
d
s
t
o
v
o
l
t
a
ge
imba
l
a
nces
a
n
d
h
arm
o
n
i
c
d
i
s
t
or
ti
o
n
s.
T
o
a
voi
d
th
is,
bala
nc
in
g
c
on
tro
l
s
ch
em
es
c
oul
d
be
e
mplo
ye
d
[18].
Thi
s
has
be
en
f
urt
h
er
h
i
g
h
lig
h
t
ed
a
s
pa
rt
o
f
t
h
e
MMC
d
es
i
g
n
and
o
p
e
r
at
io
na
l
c
ons
trai
n
t
s
w
h
ic
h
c
o
ntri
b
u
t
e
t
o
t
h
e
ca
paci
t
o
r vo
lta
ge
rip
ple,
c
urr
e
nt r
ipp
l
e,
a
nd
p
ow
er
loss
de
pe
nd
i
ng o
n
t
he
c
apac
it
or
a
nd i
n
d
u
ct
or
s
i
z
es [1
9
]
.
The
ba
lanc
i
ng
c
ontro
l
ha
s
e
f
fe
cts
on
t
he
p
o
w
er
qua
li
t
y
.
The
vo
l
t
a
g
e
i
mbal
an
c
e
t
h
a
t
c
o
ul
d
ari
s
e
due
to
p
oor
v
o
l
tag
e
b
a
l
a
n
c
i
ng
c
o
ntr
o
l
i
n
jec
t
s
ha
r
m
onic
c
u
r
r
ent
s
i
nt
o
t
h
e
sy
st
e
m
[
1
8
]
.
Haw
et
a
l
.
[
14
]
and
Kaw
et
al.
[
20]
h
a
v
e
s
t
ud
i
e
d
t
h
e
vo
l
t
a
g
e
s
u
pp
ort
b
u
t
d
i
d
n
o
t
c
on
s
i
d
e
r
the
issu
e
of
c
e
ll
bala
nc
in
g
vo
l
t
age
dis
t
ur
b
a
nce
.
T
h
i
s
d
i
s
t
u
r
b
a
n
c
e
c
o
u
l
d
c
o
n
t
r
i
b
u
t
e
t
o
t
h
e
e
m
e
r
g
e
n
c
e
o
f
t
h
e
v
o
l
t
a
ge
d
i
f
fere
nce
be
tw
e
e
n
t
h
e
a
r
ithm
e
ti
c
a
l
ave
r
age
dc
c
ap
ac
i
t
or
v
o
l
t
a
ge
o
f
a
l
l
t
h
e
p
h
as
es,
a
n
d
t
h
e
tota
l
a
r
ithm
e
t
i
ca
l
me
an
d
c
ca
pac
i
t
o
r
vo
lta
ge
f
r
o
m
ea
ch
pha
se.
Th
i
s
c
o
n
s
i
dera
t
i
on
c
o
u
l
d
h
a
v
e
im
pro
v
ed
t
he
s
y
s
t
e
m
pe
rfor
m
a
nc
e
i
n
t
e
r
ms
o
f
the
a
c
tive
pow
er
s
hari
ng
betw
ee
n
the
S
T
A
T
CO
M
a
n
d
t
h
e
gri
d
,
a
nd
i
m
pro
v
e
sys
t
em
s
tab
i
l
i
t
y
.
H
ow
e
v
er,
balanc
in
g
co
ntr
o
l
add
s
com
p
l
i
c
a
t
i
o
ns
by
a
d
di
n
g
m
a
n
y
co
n
t
ro
llers
m
ost
l
y
P
I
-
c
on
t
r
ol
l
e
rs
i
n
t
he
m
odu
la
ti
o
n
c
o
n
tr
ol.
T
h
is
a
d
d
it
i
on
o
f
t
h
e
c
ont
rol
l
e
rs
i
n
th
e
sy
st
e
m
p
ut
s
mo
re
p
re
ssu
re
o
n
t
h
e
digi
t
a
l
i
mpl
e
ment
at
i
o
n
;
t
hu
s
,
r
equ
i
ri
ng
h
ea
vy
d
i
g
i
t
a
l
signa
l
proce
s
s
i
ng.
H
ua
n
g
e
t
a
l
.
[2
1]
p
r
o
p
o
s
e
d
c
on
tro
l
u
s
i
n
g
a
r
e
p
et
i
t
i
v
e
co
ntro
l
l
e
r
t
o
el
imina
t
e
harm
onic
curr
ent
base
d
on
t
h
e
CS
P
W
M.
T
he
t
ra
ns
i
e
nt
w
as
g
oo
d,
h
ow
eve
r
,
t
h
e
prac
ti
c
a
l
im
ple
m
e
n
t
a
tio
n
of
r
e
p
e
tit
iv
e
con
t
ro
l
l
er
r
equ
i
res
c
o
mp
lica
t
ed
s
ta
bil
i
t
y
an
d
des
i
gn
i
n
c
h
oos
in
g
t
h
e
ap
pro
p
riate
low
pass
fi
lt
e
r
s
or
c
omb
fi
l
t
e
r
s
nee
d
e
d
t
o
s
t
ab
i
l
iz
e
the
con
t
ro
lle
r.
L
iao
e
t
a
l.
i
n
[2
2]
h
av
e
i
m
p
r
ov
e
d
t
h
e
b
al
a
n
c
i
ng
c
o
n
t
r
ol
b
ut
t
he
i
ssu
e
of
c
irc
u
la
t
i
n
g
c
ur
rent
r
ip
p
l
es
a
n
d
v
ol
t
a
ge
r
i
p
p
l
es
r
em
ain.
L
i
v
e
s
e
t
a
l
.
[
2
3
]
a
n
d
T
u
e
t
a
l
.
[
2
4
]
h
a
v
e
s
t
u
d
i
e
d
t
h
e
circ
u
l
a
t
i
n
g
c
u
r
r
e
n
t
t
h
a
t
c
o
u
l
d
be
e
mp
lo
ye
d
to
ach
ie
ve
c
a
p
aci
t
o
r
s
vo
l
t
age
ba
la
nc
i
n
g
,
a
nd
sup
p
r
e
ss
t
h
e
circ
u
l
a
t
i
n
g
cur
r
e
nt
b
y
incre
a
s
i
n
g
t
he
i
nduc
t
a
nce
of
t
he
a
r
m
o
f
t
h
e
MM
C.
H
oweve
r
,
t
h
e
vo
l
t
a
g
e
ac
r
o
ss
t
he
in
duc
t
o
r
ar
m
d
e
teri
ora
t
e
s
t
he
p
er
form
anc
e
o
f
the
MM
C
a
s
t
he
i
nd
uc
tor
c
onsum
es
p
a
r
t
of
t
he
a
m
o
u
n
t
of
t
h
e
rea
c
ti
ve
pow
er
(
V
A
R
),
a
nd
t
h
e
c
o
s
t
i
ncr
eas
e
due
t
o
the
in
cr
ease
o
f
the
a
r
m
induc
tor.
B
ergna
e
t
a
l
.
[2
5
]
h
ave
pro
pose
d
e
ner
gy-
ba
sed
ba
l
a
nci
n
g
co
ntr
o
l,
u
si
ng
a
mod
e
l
w
h
ic
h
c
o
n
t
rols
t
he
v
o
lta
ge
s
um
a
nd
vo
l
t
a
g
e
Evaluation Warning : The document was created with Spire.PDF for Python.
Int J
P
o
w
E
l
e
c
&
D
ri S
yst
IS
S
N
:
2088-
86
94
Mo
du
l
a
r
m
u
l
t
i
l
e
v
e
l c
onv
erte
r (
MMC) base
d
ST
AT
CO
M wi
th
vec
to
r
con
t
r
o
l… (
A
bd
urr
a
h
m
an
U
m
a
r L
a
w
an)
1
835
di
ffe
re
nce
be
t
w
een
t
he
u
pp
e
r
a
rm
c
apac
itor
s
a
nd
t
heir
c
or
resp
on
d
i
n
g
l
ow
er
a
rm
capa
c
it
ors
i
n
t
he
t
hr
ee
-
phase
M
M
C.
T
he
c
o
n
tr
ol w
a
s
w
ith
r
espe
ct t
o th
e
ene
r
g
y
store
d
in
t
he a
rm
s
.
I
n
t
h
i
s
pa
pe
r,
f
or
a
n
M
M
C
w
i
th
s
tacke
d
c
el
ls,
lim
i
t
e
d
s
w
i
t
c
hin
g
f
r
e
q
u
enc
y
e
m
p
lo
yed
usi
ng CS
P
W
M
as
i
n
[2
1],
w
oul
d
pro
v
i
d
e
a
suf
f
icie
n
t
c
on
tr
ol
b
an
dw
i
d
th
w
hic
h
w
o
uld
im
prove
t
h
e
r
espo
nse
a
n
d
prov
i
d
e
a
go
o
d
d
am
p
i
ng
e
ffect.
Furthe
r
m
or
e,
t
h
e
i
ncr
eased
n
o.
o
f
t
h
e
stac
ke
d
ce
lls
v
o
lta
ge
v
ar
iat
i
o
n
i
n
t
h
e
m
u
l
tile
v
e
l
w
ill
be
u
se
d
to
i
mpr
o
ve
t
he
s
t
a
bi
l
i
t
y
a
t
b
o
t
h
g
ri
d-in
t
e
rfa
c
e
l
e
ve
l
a
n
d
w
i
th
in
t
he
MMC
i
n
t
erna
l
c
o
n
t
r
o
l.
U
nl
ik
e
in
[
2
1
],
w
he
re
a
l
a
r
ge
s
t
a
b
i
l
i
ty
a
na
lys
i
s
d
u
e
to
t
he
r
e
p
e
t
iti
ve
-c
on
t
r
oll
e
r
para
me
t
e
r
se
l
e
c
t
i
on
wa
s
invo
lve
d
,
a
si
m
p
le
v
irt
u
al
i
m
p
eda
n
ce
b
a
se
d
P
I
c
on
trol
w
i
t
h
s
o
me
qua
lit
ies
o
f
a
low
pa
ss
f
ilter
such
a
s
har
m
onic
s
reduc
tio
n
w
o
u
l
d
be
e
m
p
lo
ye
d
.
T
his
c
o
n
t
ro
l,
u
n
l
ike
in
B
e
g
n
a
e
t
a
l.
[
25],
c
oul
d
be
a
p
p
l
i
e
d
i
n
b
o
t
h
sin
g
l
e-
pha
se
or
t
hr
ee-
p
h
ase
syste
m
,
a
nd
s
til
l,
u
ses
the
d
e
cou
p
l
ed
c
on
tr
ol
.
F
urther
more
,
the
use
o
f
C
SP
WM
w
ith
a
h
igh
in
t
e
gral
g
a
i
n
w
oul
d
e
lim
ina
t
e
t
h
e
ste
a
dy
s
t
a
t
e
er
ror
[26
]
w
h
e
n
e
m
pl
o
y
ed
i
n
the
pro
pos
ed
con
t
ro
l sche
me
.
F
i
gure
1
de
pic
t
s
t
h
e
three
-
p
h
a
se
M
M
C
b
a
s
ed
S
TA
T
C
O
M
s
ys
t
e
m.
I
f
a
n
d
Vc
r
e
t
h
e
M
M
C
b
a
s
ed
STATCOM
th
re
e-p
h
a
se
out
pu
t
cu
rre
nt
a
nd
v
o
lt
a
g
e
res
p
e
c
t
i
v
e
l
y
.
Vf
i
s
t
h
e
g
r
i
d
v
o
lta
g
e
a
t
th
e
po
in
t
of
cou
p
l
i
ng
(P
CC
).
A
pply
i
n
g
K
i
r
c
hhof
f’s
pr
i
n
c
i
p
l
e
o
n
a
ny
le
g/
p
h
ase
i
n
t
h
e
three
-
p
h
ase
sy
ste
m
w
i
t
h
r
e
sp
ect
t
o
no
de
n
,
the
m
a
them
a
tica
l
r
el
ati
o
nshi
p
be
t
w
e
e
n
t
h
e
per
p
h
a
s
e
o
u
t
put
current
i
k
,
the
l
o
w
e
r
and
u
pper
arm
curr
ents (
i
kp
a
nd
i
kn
),
a
nd the c
i
rcu
l
at
i
ng c
u
rrent
i
ci
r
k
a
r
e
g
i
v
en
b
y:
kk
p
k
n
ii
i
(
1
)
1
2
ci
rk
kp
kn
ii
i
(
2
)
+
+
‐
‐
+
‐
+
‐
+
un
c
V
+
‐
up
i
wp
i
vp
i
1
SM
1
SM
1
SM
2
SM
2
SM
2
SM
SM
N
S
M
N
SM
N
cw
p
V
cv
p
V
cu
p
V
r
L
r
L
r
R
r
R
f
R
s
R
s
L
f
L
s
V
GR
I
D
u
i
w
i
v
i
r
L
r
R
r
R
r
R
r
R
U
V
W
r
L
r
L
r
L
ci
r
u
i
ci
r
v
i
ci
r
w
i
1
SM
N
1
SM
N
1
SM
N
2
SM
N
2
SM
N
2
SM
N
cw
n
V
cun
V
cv
n
V
un
i
vn
i
wn
i
SM
2
Vd
c
2
Vd
c
SM
P
CC
c
V
f
V
n
f
I
2
SM
N
2
SM
N
2
SM
N
F
i
gure
1.
G
rid-c
o
n
n
ec
ted
MMC ba
s
e
d
S
TA
TCO
M
i
n t
h
e
t
h
r
ee-pha
se c
o
n
fi
gura
t
i
o
n.
The
s
ub-scr
i
p
t
k
re
presents a
n
y
o
f t
h
e t
h
re
e-pha
se
s (u,
v or w
). Ba
s
ed on (
1
) an
d (
2
), the r
egu
l
at
ion
of
the
c
i
r
c
u
l
at
ing
curre
nt
i
ci
rk
a
nd
t
h
e
o
u
t
pu
t
vo
lt
ag
e
V
ck
c
ou
l
d
b
e
ac
hi
ev
ed
b
y
co
nt
roll
ing
t
h
e
u
p
p
e
r
and
l
o
wer
curr
ents.
A
nd t
h
e up
pe
r a
nd l
o
w
e
r
arm
volta
ges (
V
ckp
a
n
d
V
ck
n
)
given
in
(
3)
a
nd (
4
) r
e
spec
t
i
ve
ly.
1
2
kp
d
c
k
cc
VV
V
(
3
)
1
2
kn
d
c
k
cc
VV
V
(
4
)
V
dk
is t
h
e cumu
la
ti
ve e
qu
i
v
ale
n
t
d
i
re
ct c
urre
nt (D
C
) v
o
lta
ge
a
c
r
os
s t
h
e arm
s
.
L
r
i
s t
h
e a
r
m i
n
du
ct
ance
of
e
a
c
h
ar
m.
R
r
i
s
t
h
e
se
ries
r
esis
t
a
nce
a
ssoc
i
ate
d
w
i
t
h
the
bu
ffer
i
nd
uc
ta
nce
[
2
1
]
.
V
ck
i
s
the
M
M
C
A
C
o
u
t
put
vo
lta
ge de
f
ine
d
a
s:
*
1
sin
2
dc
ck
VV
M
t
(
5
)
Evaluation Warning : The document was created with Spire.PDF for Python.
ISS
N
:
208
8-
8
6
9
4
I
n
t
J Po
w El
ec &
D
ri S
y
s
t
Vo
l. 1
0
,
No
. 4
,
D
e
c
2
0
19
:
18
33 –
1
8
51
1
836
Th
e
(5
)
gi
v
e
s
th
e
out
put
v
o
l
t
a
g
e
und
e
r
i
d
e
a
l
c
o
n
d
i
tio
ns
w
h
e
n
e
a
ch
v
o
lta
ge
o
f
the
s
u
bmod
ule
s
(
S
M
)
c
a
pac
i
tor
is
c
o
n
s
t
an
t
a
nd
e
q
u
a
l
to
t
he
v
al
ue
o
f
V
dk
/N, M
a
nd
I
c
a
r
e
t
he
m
odu
la
ti
on
v
o
l
ta
g
e
a
nd
pea
k
o
u
t
p
u
t
cu
rren
t
r
esp
e
c
t
i
v
ely
.
ω
a
nd
a
r
e
t
he
a
ngu
l
a
r
fr
e
quency
a
n
d
loa
d
a
ngle
r
e
spec
t
i
ve
ly.
The
bas
ic
p
er-
p
h
a
se
c
o
n
t
r
o
l
sc
hem
e
s
w
h
ic
h a
r
e
av
e
r
agi
ng c
o
nt
r
o
l,
ind
i
v
id
ua
l
c
o
ntr
o
l
, and
t
h
e
a
r
m
b
alan
c
i
ng
co
n
trol
u
se (6
)
-
(
9
)
f
o
r
t
h
ei
r
i
m
p
l
e
m
en
t
a
ti
on
,
a
nd
the
c
o
n
t
r
o
l
sc
he
me
c
o
u
l
d
b
e
e
x
t
e
nd
e
d
t
o
t
he
t
hr
ee
-
phase
c
on
f
i
g
u
r
a
tio
n
[
27]
.
The
m
e
a
n
i
n
g
o
f
t
he
s
ymbo
ls
i
n
F
i
gur
e
2-
5
a
r
e
def
i
ne
d
as
1
VV
V
(
6
)
2
kk
p
k
n
CC
C
(
6
)
1
1
VV
(
7
)
kp
ki
N
CC
i
N
(
7
)
2
1
1
VV
(
8
)
kn
k
j
N
CC
jN
N
(
8
)
**
VV
(
9
)
dc
C
Ck
V
N
(
9
)
i
s
the
per
-
p
h
a
s
e
a
v
era
g
e
ca
pa
cit
o
r
vo
lta
ge
i
n
the
cor
r
esp
o
n
d
i
n
g
ph
as
e.
V
kp
C
a
n
d
V
kn
C
a
r
e
t
h
e
a
v
er
a
g
e
capa
c
i
tor
vo
lta
ge
o
f
t
h
e
u
pper
ar
m
a
nd
low
e
r
a
r
m
of
t
he
c
or
r
e
spo
n
d
i
ng
ph
ase
r
e
spec
ti
ve
ly.
N
i
s
t
h
e
num
ber
of
t
he
S
M
i
n
t
he
u
p
p
e
r
a
nd
l
o
w
e
r
a
r
m,
V
ki
C
a
n
d
V
kj
C
a
r
e
t
h
e
v
o
l
t
a
g
e
s
o
f
t
h
e
i
th
a
nd
j
th
S
M
s
i
n
the
u
p
p
e
r
ar
m
and
lo
w
e
r
arm
resp
ectiv
e
l
y
.
*
V
Ck
and
*
cir
k
i
rep
r
esen
t
th
e
referen
ce
vo
ltag
e
and
ref
e
r
e
nc
e
c
i
rcul
a
t
i
ng
c
u
r
r
e
nt
i
n
th
e
c
o
r
r
e
spo
n
d
i
n
g
l
e
g
r
e
spe
c
t
i
ve
l
y
.
F
i
gur
e
2.
B
loc
k
d
ia
gr
a
m
o
f
a
v
er
a
g
ing
co
ntr
o
l
F
i
g
u
r
e
3
.
I
ndiv
i
d
u
a
l
b
a
l
a
n
c
i
ng
c
ont
r
o
l
b
l
oc
k
dia
g
r
a
m
F
i
gur
e
2
de
pi
c
t
s
t
h
e
a
v
e
r
ag
in
g
lo
o
p
d
iagr
a
m
,
this
c
o
n
tr
o
l
p
us
he
s
t
h
e
a
v
er
age
v
o
lta
ge
V
k
C
of
e
ach
pha
se
l
e
g
t
o
fo
l
l
ow
i
ts
r
ef
e
r
enc
e
v
o
l
t
a
ge
*
V
Ck
.
Wh
en
*
VV
Ck
Ck
,
*
cir
k
i
i
nc
r
e
ases
a
nd
f
o
r
c
es
t
he
a
ct
ua
l
ci
rc
u
l
a
tin
g
c
u
rre
n
t
ci
r
k
i
t
o
foll
o
w
i
t
s
r
e
f
ere
n
ce
c
u
rre
n
t
*
cir
k
i
m
a
ki
ng
V
Ck
t
o
trac
k
it
s
referen
c
e
voltag
e
*
V
Ck
.
P
I
co
nt
r
o
l
l
e
r
s
c
an
b
e
emp
l
oy
e
d
i
n
t
h
e
t
r
ac
k
i
ng
, a
n
d
t
h
e
out
put
v
ol
ta
ge
c
ommand
ci
r
k
v
from
t
h
e a
v
er
ag
in
g
c
ont
r
o
l
c
a
n be
p
r
o
du
ce
d for
t
h
e
CS
P
W
M.
T
he
g
a
i
n
s
o
f
t
h
e
P
I
c
o
n
tr
oller
s
whi
c
h
are
1
K
,
2
K
,
3
K
a
n
d
4
K
c
o
u
l
d
b
e
ob
ta
ine
d
b
ase
d
on
in
ter
n
a
l
m
ode
c
on
tr
o
l
t
he
o
r
y
[28]
.
+
‐
F
i
gur
e
4.
A
rm ba
l
a
n
c
i
n
g
c
o
n
t
r
o
l
b
l
oc
k
d
i
agr
a
m
+
‐
+
+
+
‐
+
‐
+
F
i
gur
e
5.
V
olta
ge
c
om
m
a
nds
f
or
t
he
P
WM
b
a
l
anc
i
ng
co
n
t
r
o
l
o
f
t
h
e
MM
C
a
r
m
Evaluation Warning : The document was created with Spire.PDF for Python.
Int J
P
o
w
E
l
e
c
&
D
ri S
yst
IS
S
N
:
2088-
86
94
Mo
du
l
a
r
m
u
l
t
i
l
e
v
e
l c
onv
erte
r (
MMC) base
d
ST
AT
CO
M wi
th
vec
to
r
con
t
r
o
l… (
A
bd
urr
a
h
m
an
U
m
a
r L
a
w
an)
1
837
The
i
n
di
v
i
d
u
a
l
bala
n
c
i
n
g
c
on
t
r
ol
p
us
hes
ea
c
h
S
M
c
a
pac
i
t
o
r
vo
l
t
a
ge ind
iv
i
dua
l
l
y
i
n
e
very
s
ubmo
d
u
le
to
f
ol
low
i
t
s
v
o
ltage
reference
*
V
Ck
,
the
i
n
di
v
i
d
u
a
l
ba
la
nc
in
g
c
ontr
o
l
bl
oc
k
d
i
a
g
ram
is
d
e
p
ic
te
d
in
F
i
g
u
r
e
3
.
The
ob
t
a
in
ed
v
o
l
tage
c
omm
a
nd
f
or
t
his
mod
u
l
at
i
o
n
c
o
n
t
r
o
l
is
kB
v
.
W
h
en
.
*
(,
)
VV
Ck
i
j
Ck
t
h
e
ar
m
cur
r
ent
i
s
cha
r
g
i
ng
t
h
e
S
M
c
a
p
ac
i
t
or,
the
po
lar
i
t
y
o
f
t
h
e
ou
tpu
t
c
urr
e
nt
i
s
pos
iti
ve.
Whe
n
i
t
is
r
e
v
er
sed,
t
he
a
r
m
c
urrent
get
s
t
he
opp
osi
t
e
p
o
l
ari
t
y.
T
h
e
p
ro
port
i
ona
l
con
t
ro
l
l
er
w
it
h
a
gai
n
5
K
c
o
uld
be
e
mp
l
o
y
e
d
.
T
h
e
c
ont
rol
blo
c
k
for
the
ar
m
ba
lanc
i
ng
is
s
h
o
w
n
in
F
i
gure
4
.
This
c
on
tro
l
l
esse
ns
t
he
d
i
s
cr
epanc
i
e
s
i
n
vo
lta
ge
s
be
tw
e
e
n
t
h
e
vo
lta
ge
s
of
u
p
p
e
r
a
nd
l
o
w
e
r
a
r
m.
T
he
vol
ta
ges,
V
kp
C
an
d
V
kn
C
a
re
a
lso
k
now
n
as
t
h
e
a
ve
rage
v
ol
ta
ges
o
f
t
he
pos
it
ive
ar
m
a
nd
ne
ga
t
i
ve
a
rm
r
e
s
pe
cti
v
el
y
.
T
he
p
rop
o
r
t
i
ona
l
co
n
t
ro
ller
6
K
c
o
u
l
d
be
e
mp
l
oye
d.
F
igure
5
show
s
t
h
e
m
a
t
h
em
ati
c
s
for
th
e
overa
ll
vo
l
t
a
g
e
c
o
mm
and
g
e
nera
tio
n
for
t
h
e
CS
P
W
M
i
n
t
he
c
on
tro
l
d
i
a
gram
repr
esenta
t
i
o
n
[
2
9
].
H
oweve
r
,
the
c
i
rc
u
l
a
t
ing
curr
ent
c
o
n
t
r
o
l
b
ase
d
o
n
F
i
g
u
re
2
w
hic
h
i
s
usi
n
g
t
h
e
P
I-contro
l
is
u
s
u
a
l
l
y
i
n
s
ufficie
n
t
i
n
s
uppr
essin
g
t
he
c
i
r
cula
ti
n
g
c
urr
e
nt
[2
1].
Thus,
vo
lta
ge
d
ist
u
rb
a
n
ce
is
i
ntr
o
d
u
ced
o
n
the
S
M
s
a
n
d
i
n
jec
t
s
harm
o
n
ic
c
urr
e
nt
d
ur
ing
n
o
rm
al
ope
rat
i
o
n
.
T
ak
i
n
g
in
a
ny
p
hase-
k
,
the
varia
t
io
n
c
o
u
l
d
b
e
exp
l
a
i
ne
d
by
t
h
e
rela
ti
o
n
shi
p
b
e
t
w
e
e
n
t
he
e
nerg
y
st
ore
d
w
i
t
h
i
n
t
h
e
MM
C
a
r
m
s
.
The
act
ive
e
n
erg
y
s
tor
e
d
i
n
the upper and lower
ra
ms are
a
s
f
o
llows:
2
1
2
kp
C
k
p
WC
v
N
(
10)
2
1
2
kn
C
k
n
WC
v
N
(
11)
Where
C
is
t
he
S
M
ca
pac
i
tanc
e.
F
r
o
m
(10)
a
n
d
(
11),
let
t
h
e
a
c
t
i
ve
v
o
l
tage
d
i
s
tur
b
a
n
ce
w
h
i
c
h
con
t
ri
b
u
tes
t
o
t
he
M
MC
v
ar
i
a
tio
n
be
w
ri
t
t
e
n
i
n
t
e
rm
s
of
t
h
e
D
C
com
p
one
nt
dc
I
a
n
d
harm
o
n
ic
c
om
po
ne
n
t
s
n
i
as
1
1
1s
i
n
22
3
k
Ck
p
d
c
n
n
Ni
vM
t
i
i
d
t
C
(
12)
1
1
1s
i
n
22
3
k
Ck
n
d
c
n
n
Ni
vM
t
i
i
d
t
C
(
13)
Where
kp
vc
a
nd
kp
vc
are
the
v
o
l
t
a
g
e
chan
ge
d
ue
t
o
d
i
s
t
urba
nce
s
i
n
the
up
per
a
n
d
low
er
a
r
m
s
respe
c
t
i
ve
l
y
. The
c
i
r
cu
lat
i
ng
curr
ent i
n
th
e
M
M
C
c
o
u
l
d
be
give
n
a
s
1
1
3
ci
r
d
c
n
n
ii
i
(
14)
F
o
r
ty
p
i
ca
l
S
T
A
T
CO
M
a
p
pli
c
a
tio
ns,
t
h
e
D
C
c
omp
one
nt
dc
I
i
s
z
e
ro
s
inc
e
t
her
e
i
s
no
D
C
l
i
nk
D
C
vo
lta
ge
s
o
u
r
ce.
T
he
h
arm
o
n
i
c
com
p
one
n
t
s
n
i
a
r
e
t
h
e
n
t
h
o
r
d
e
r
h
a
r
m
o
n
i
c
s
o
f
t
h
e
c
i
rc
ul
a
tin
g
c
u
rr
ent.
Th
e
r
efo
r
e, t
h
e
rea
l
o
u
t
put
v
olt
a
ges of t
he
upp
e
r
a
nd low
e
r
arms
c
onsi
d
eri
n
g
the
varia
t
i
o
n c
oul
d
be a
s
1
1s
i
n
2
Ck
p
d
c
C
k
p
VM
t
V
v
(
15)
1
1s
i
n
2
Ckn
d
c
C
kn
VM
t
V
v
(
16)
The
n
cons
i
d
eri
ng
the
var
i
a
t
i
o
n,
the
t
o
t
a
l
out
pu
t v
o
l
t
a
g
e
of t
he
pha
se
l
eg c
o
u
ld
b
e
give
n as
f
ol
low
s
:
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
: 208
8-
869
4
I
nt
J
P
ow
Elec
& Dr
i
S
y
st V
ol.
10,
N
o.
4
, Dec
201
9 : 1
8
3
3
–
18
51
1
838
11
si
n
22
ckp
ckn
d
c
C
k
p
C
kn
C
k
n
C
kp
d
c
kd
VV
V
v
v
M
t
v
v
V
V
(
17)
The
r
ef
ore,
kd
v
i
s
the
v
o
l
ta
ge
v
a
r
iat
i
o
n
b
e
t
w
e
en
t
he
p
ha
se
-le
g
a
nd
the
to
ta
l
S
M
vo
lt
ag
es
(
i
.
e.
dc
V
)
w
h
en
t
he
s
ys
te
m
is
u
n
d
er
nor
m
a
l
m
o
d
u
l
at
io
n.
E
q
u
a
t
i
o
ns
(
6)
–(9)
d
e
s
cri
b
e
t
h
e
c
a
p
ac
itor
v
o
l
t
a
g
e
c
o
nt
r
o
l
an
d
the
a
c
t
i
v
e
p
o
w
e
r
sha
r
in
g
w
i
t
h
in
a
ny
le
g.
I
n
ot
her
w
o
rds,
t
he
a
c
t
i
v
e
pow
er
b
ala
n
ci
n
g
i
s
per-pha
se,
an
d
it
i
s
em
plo
y
e
d
b
y
a
d
jus
t
i
n
g
the
dc
c
ircula
t
i
n
g
c
u
r
rent
w
it
hin
th
e
pha
se
l
eg;
th
u
s
,
i
t
h
a
s
no
i
nf
lu
en
ce
o
n
th
e
t
o
t
a
l
ac
t
i
ve
pow
er
a
bsor
bed
b
y
t
he
S
TA
TCO
M
a
t
the
g
r
id
t
hre
e
-pha
se
i
n
te
rfa
c
e
[1].
T
o
co
ntr
o
l
the
ac
ti
ve
pow
e
r
and
t
h
e
rea
c
ti
v
e
pow
er
a
t
t
h
e
gri
d
P
CC
s
h
o
w
n
i
n
F
ig
ure
1,
t
he
g
ri
d
-
inte
r
f
ac
e
re
fer
e
nce
s
f
or
t
he
i
ns
tan
t
aneo
us
ac
t
i
ve
c
urr
e
n
t
*
fd
I
an
d
th
e
re
ac
t
i
v
e cu
rre
n
t
*
fq
I
coul
d
b
e
ob
t
a
i
ne
d
a
s
foll
o
w
s
:
**
(V
V
)
fd
C
C
IK
(
18)
K
i
s
the
pr
opor
ti
ona
l
c
o
n
t
ro
l
l
e
r
g
ai
n.
*
V
C
i
s
the
r
e
fe
renc
e
D
C
v
o
l
t
a
ge
f
or
t
he
S
M
c
a
pa
c
i
tor
s
.
V
C
is
the a
r
i
t
hme
tica
l
a
ve
rage
ca
p
a
c
it
ors vo
l
t
a
g
e
of
a
l
l
t
he S
M capac
it
ors
i
n
t
he
thre
e
le
g
s
an
d
d
e
fi
ned
as
,,
1
VV
3
k
ku
v
w
CC
(
19)
The
r
eac
ti
ve
c
ur
rent
r
efe
r
e
n
c
e
*
fq
I
co
ul
d
be
a
s
in
[
1
4
],
a
ssu
m
ing
a
ll
the
l
o
ad
r
ea
ct
ive
p
o
w
e
r
i
s
ob
ta
ina
b
le fro
m
the S
T
A
T
CO
M. T
h
i
s
co
ul
d
be
d
efi
n
ed
a
s
*
*
L
fq
Lq
fd
Q
II
V
(
20)
wh
ere
L
Q
a
nd
L
q
I
are
t
h
e
r
eac
tive
lo
ad
pow
er
a
nd
c
urre
nt
r
e
s
pe
cti
v
e
l
y.
fd
V
is
t
he
M
M
C
o
ut
put
vo
lta
ge
i
n
t
h
e
d-
ax
is
u
sua
l
l
y
a
lig
ned
w
ith
t
he
g
rid
v
o
l
t
a
ge
u
s
i
ng
t
h
e
p
ha
se
-l
o
c
k
e
d
loop
.
N
e
g
l
e
c
ti
ng
t
h
e
g
rid
so
u
r
c
e
i
mp
ed
an
c
e
(
s
L
an
d
s
R
)
in
F
ig
u
r
e
1,
t
he
r
e
l
at
ion
s
h
i
p
be
tw
e
e
n
the
gri
d
A
C
v
o
l
t
a
ge
(
f
V
),
a
nd
MMC
base
d
S
T
A
T
CO
M A
C
ou
t
p
u
t
vol
ta
ge
(
c
V
) in the
d
, q
f
orm
s
:
22
2
c
d
fd
fq
fd
fd
rr
r
ff
f
c
q
fq
fd
f
q
f
q
VV
I
I
I
d
LR
L
LR
L
dt
VV
I
I
I
(
21)
I
n
(
21),
fq
I
r
e
p
resent
s
t
h
e
rea
c
tive
c
u
r
r
ent
of
t
he
M
MC
i
n
q-axi
s
,
w
h
ile
f
d
I
i
s
the
act
i
v
e
c
u
rre
nt
i
n
the
d-
ax
is.
cd
V
a
nd
cq
V
a
re
t
he
d
-
a
xi
s
an
d
q-ax
i
s
M
M
C
out
pu
t
vo
l
t
age
res
p
e
c
tive
l
y.
fq
V
i
s
q
-
a
x
i
s
g
r
i
d
vo
lta
ge.
The
MM
C-t
o
-gri
d im
peda
nc
e
i
s
a
s
22
rr
Tf
f
T
T
LR
Z
LR
L
R
(
22)
Usi
n
g
th
e
,
dq
Q
vari
able
i
n
(
23)
a
n
d
(
24)
,
a
nd
ass
u
min
g
t
he
S
T
A
TCO
M
c
urre
nts
tr
ac
k
t
h
ei
r
cu
rre
n
t
re
fe
ren
c
es
a
s i
n
[
14
], t
h
e
M
MC o
utp
u
t
vo
l
t
a
g
e
cou
l
d
be
co
nt
rol
l
ed
u
s
i
ng
(
25).
df
d
c
d
T
f
q
T
f
d
QV
V
L
I
R
I
(
23)
qf
q
c
q
T
f
d
T
f
q
QVV
L
I
R
I
(
24)
Evaluation Warning : The document was created with Spire.PDF for Python.
Int J
P
o
w
E
l
e
c
&
D
ri S
yst
IS
S
N
:
2088-
86
94
Mo
du
l
a
r
m
u
l
t
i
l
e
v
e
l c
onv
erte
r (
MMC) base
d
ST
AT
CO
M wi
th
vec
to
r
con
t
r
o
l… (
A
bd
urr
a
h
m
an
U
m
a
r L
a
w
an)
1
839
**
,
_
(,
)
(
,
)
(,
)
_
(,
)
(
,
)
(,
)
*
*
*
*
2
2
dq
P
i
dq
f
d
q
f
d
q
I
i
d
q
f
d
q
f
dq
T
cd
d
f
d
f
q
T
f
d
fq
T
cq
q
f
q
f
d
T
f
q
fd
QK
I
I
K
I
I
d
t
L
VQ
V
I
I
R
I
L
VQ
V
I
I
R
I
(
25)
2.
T
H
E PROPOSED VIRTUA
L
IMPEDANCE A
ND
VECTO
R
CONT
R
O
L COMPENS
A
T
I
ONS
The
p
r
op
ose
d
c
on
tro
l
i
n
t
ro
du
ce
s
e
x
t
r
a
vo
lta
ge
c
om
pe
nsa
t
i
on
i
n
t
he
e
xte
r
na
l
c
o
n
t
r
o
l
o
f
t
he
M
M
C
w
ith
t
he
g
ri
d
a
nd
a
d
ds
a
n
ex
t
r
a
com
p
ensa
t
i
on
su
p
p
o
r
t
in
t
he
i
n
t
erna
l
c
o
n
t
ro
l
.
T
he
se
t
w
o
c
omp
e
nsa
tio
ns
a
re
em
plo
y
e
d
t
o
form
t
he
o
vera
ll
c
o
n
tro
l
.
U
n
like
[14]
a
n
d
[30]
,
i
n
t
h
is
c
o
n
tr
ol,
vo
l
t
a
ge
c
om
p
e
nsa
tio
n
i
s
con
s
i
d
ere
d
i
n
s
te
ad
o
f
cur
r
ent
w
h
ic
h
has
t
o
p
ass
t
h
ro
ug
h
the
PI-
contr
o
l
l
e
r
s
or
P
-con
trol
lers
a
n
d
a
d
d
s
add
i
tio
na
l
stre
ss
o
n
t
he
c
o
n
t
rolle
rs
dur
in
g
t
h
e
tra
n
s
i
e
n
ts.
A
l
s
o
un
l
i
ke
[
14],
the
ne
gl
i
g
ib
le
a
c
t
ive
c
u
r
r
ent
t
r
a
n
sf
er
i
s
con
s
i
d
ere
d
t
o
enh
a
n
ce
th
e
d
-
a
x
i
s
c
ont
rol
,
a
n
d
e
xt
ra
c
om
pe
n
s
a
t
i
o
n
i
n
t
he
i
n
t
er
na
l
c
o
ntr
o
l
i
s
in
t
r
od
uc
ed
t
o
e
n
ha
nce
t
h
e
c
a
p
aci
t
o
rs
v
olta
ge
b
a
l
a
n
c
i
n
g
.
Th
is
w
o
ul
d
,
i
n
t
u
rn
,
en
h
a
nc
e
t
h
e
act
iv
e
p
o
we
r
bala
nc
i
n
g
w
i
th
i
n
t
he
S
TA
TC
O
M
;
t
h
us,
e
n
h
a
nci
n
g
the
ove
ra
l
l
s
ys
t
e
m
co
ntr
o
l.
P
r
i
nc
iple
s
o
f
t
r
a
ns
ie
nt
i
m
p
ac
ts
em
plo
y
e
d
i
n
[31]
a
n
d
t
he
c
a
p
ac
it
ors
vo
l
t
ag
e
varia
t
io
n
ar
e
sum
m
e
d
u
p
t
o
ge
ne
rate
a
n
extra
trans
i
e
n
t
vo
l
t
ag
e
sup
por
t
t
o
the
P
I
contro
l
l
e
r
s a
t
the
g
r
i
d
i
n
te
rfa
ce
duri
ng t
h
e
tr
ans
i
e
n
ts.
2.1.
E
x
ter
n
al c
on
tr
ol
v
o
l
t
a
ge c
o
m
p
e
n
s
ation
of
t
h
e
MM
C
I
n
F
igure
1,
t
h
e
d
-q
v
o
l
t
a
ge
d
ro
p
w
h
i
c
h
c
o
ntr
i
b
u
t
e
t
o
t
h
e
s
ma
ll
v
ar
iat
i
on
s
betw
ee
n
the
S
T
A
T
CO
M
des
i
red
v
o
lta
g
e
c
om
m
a
nds
(
*
cd
V
a
nd
*
cq
V
)
,
a
nd
t
h
e
S
T
A
T
CO
M
v
o
lta
ges
(
cd
V
a
nd
cq
V
)
,
a
r
e
g
i
v
e
n
i
n
t
h
e
d
-
axi
s
a
nd q-a
x
is
form
s
a
s
*
cd
cd
cd
VV
V
(
26)
*
cq
c
q
cq
VV
V
(
27)
S
i
nce
fa
s
t
s
p
e
e
d
r
espo
nse
w
i
t
h
m
in
im
al
v
a
r
iat
i
on
is
t
he
a
im
o
f
STA
T
C
O
Ms,
t
h
ese
v
o
lta
ge
dev
i
a
t
i
o
ns
(
cd
V
and
cq
V
)
c
oul
d
b
e
s
u
b
t
r
ac
ted
in
f
or
m
of
c
om
pe
ns
atio
n
t
o
m
i
t
i
g
ate
t
h
e
v
a
r
iat
i
ons,
an
d
spee
d
u
p
t
he
c
urre
nt
r
e
s
p
o
n
s
e
duri
n
g
t
h
e
t
r
ans
i
e
n
t
s
.
The
vo
l
t
a
g
e
c
o
mpe
n
sa
tio
n
i
s
d
eri
v
ed
a
s
from
(26)
a
nd
(2
7
)
,
t
h
e
c
o
rres
p
ondi
ng
d
e
s
i
r
ed
S
TA
TC
OM
c
u
rren
t
s
w
o
u
l
d
h
a
v
e
s
ma
l
l
d
evia
t
i
o
n
s
from
t
he
ir
c
orr
e
sp
o
n
d
i
ng
ou
tpu
t
c
urre
n
t
s
,
these
v
aria
ti
o
n
a
re
d
efine
d
a
s
*
fq
fq
fq
II
I
(
28)
*
fd
fd
fd
II
I
(
29)
A
s
s
u
m
e
t
h
e
r
e
s
i
s
t
a
n
c
e
i
n
(
2
2
)
t
o
b
e
z
e
r
o
,
a
n
d
t
h
e
t
o
t
a
l
i
n
t
e
r
f
a
cin
g
i
nd
uc
ta
nc
e
T
L
in
(
2
2
)
w
h
ic
h
a
l
s
o
con
t
ri
b
u
tes
to t
he
S
TA
TCO
M
va
r
iat
i
ons
i
s
mod
i
fie
d
d
ue
t
o
hig
h
-s
wi
t
c
hi
ng [3
1
]
an
d
c
on
sid
e
red
a
s
2
r
Tc
om
f
L
LL
(
30)
To
r
e
duce
the
var
i
a
tio
n
assu
m
i
ng
it
ha
ppe
n
s
dur
in
g
t
h
e
tra
n
s
i
en
t
t
i
m
e
(
i.e
.
t
),
i
n
order
t
o
s
pee
d
u
p
the
res
p
o
n
se
o
f
the
PI
-contro
l
l
e
r
and
re
duce
t
h
e
cur
r
en
t
err
o
r
i
n
(
28)
a
n
d
(
29),
an
a
ddi
t
i
ona
l
tra
n
s
i
e
n
t
v
o
l
t
a
g
e
cou
l
d be
der
iv
e
d
fr
o
m
(21) a
s
*
()
fq
f
q
q
T
com
T
co
m
f
d
II
VL
L
I
t
(
31)
S
i
nce
t
h
e
va
ri
ati
o
n
ha
p
p
e
n
s
dur
in
g
the
tran
sie
n
t
t
i
me
(
i
.
e.
t
)
in
e
ver
y
s
w
i
tc
hin
g
p
er
io
d
[31],
th
e
sw
it
c
h
i
n
g
fre
q
uenc
y
(
i
.e
.
s
f
)
cou
l
d
be
e
s
t
i
m
ated
a
s
th
e
in
ve
rse
o
f
t
he
t
ra
nsi
e
nt
t
i
m
e
for
t
h
e
i
nt
ro
du
c
e
d
com
p
en
sat
i
on,
a
nd t
h
e
com
p
e
n
sa
ti
n
g
vo
l
t
a
ge
in
t
h
e
q-a
x
is
w
ou
ld
th
en
b
e a
s
q
s
Tc
o
m
f
q
Tc
o
m
f
d
Vf
L
I
L
I
(
32)
Evaluation Warning : The document was created with Spire.PDF for Python.
ISS
N
:
208
8-
8
6
9
4
I
n
t
J Po
w El
ec &
D
ri S
y
s
t
Vo
l. 1
0
,
No
. 4
,
D
e
c
2
0
19
:
18
33 –
1
8
51
1
840
Fig
u
r
e 6
.
MMC
b
ased
S
TAT
C
OM syste
m wi
th
t
h
e
p
r
o
pose
d
ext
ern
a
l
a
n
d
i
n
t
er
na
l
co
ntr
o
l
sc
hem
e
s.
S
i
milar
l
y,
f
r
o
m
(
21)
,
the
c
o
r
r
e
spond
i
ng
a
c
t
i
v
e
vo
l
t
age
c
o
m
p
en
sa
ti
on
w
o
ul
d
be
d
e
r
i
v
ed
a
s
:
(
)
(3
3
)
fd
d
T
co
m
T
co
m
f
d
T
c
o
m
f
q
I
d
VL
R
I
L
I
dt
(
33)
S
i
milarly
(
3
3
)
is simpl
i
fied
furth
er as
:
()
(
3
4
)
d
s
T
c
o
m
fd
T
c
o
m
fd
T
c
o
m
fq
Vf
L
I
R
I
L
I
(
34)
D
u
r
i
n
g
t
he
t
r
a
n
s
ie
nts,
t
he
v
o
l
t
a
ge
d
r
o
p
va
r
i
a
b
le
s (
i
.
e
.
,,
,
Tc
o
m
fd
Tc
o
m
fd
Tc
o
m
fq
RI
L
I
RI
an
d
Tc
o
m
f
q
LI
in
(
34)
a
n
d
(
32)
c
a
n
b
e
co
nsi
d
er
e
d
a
s
n
on-
tr
ans
i
e
n
t
ter
m
s,
w
hose
var
i
a
t
i
o
ns
c
a
n
b
e
a
d
eq
ua
te
ly
c
on
tro
lle
d
b
y
t
he
P
I
c
o
n
t
r
o
ller
s
;
th
us,
ig
nor
e
d
.
I
n
t
he
o
t
h
er
h
a
n
d,
(
17)
s
h
o
w
s
h
ow
t
he v
olta
ge
v
a
r
ia
ti
o
n
kd
V
c
ou
l
d
c
o
n
t
r
i
bu
te to the
emerg
e
n
c
e
of
a
s
ma
l
l
a
cti
v
e
p
o
w
er
d
if
f
e
ren
c
e
bet
w
een
t
h
e
t
h
r
ee
-
p
h
a
s
e
ac
t
i
ve
p
ow
er
a
t
the
gr
i
d
,
and
t
h
e
sum
m
a
tion
o
f
t
he
t
hr
ee
a
ve
r
a
g
e
p
e
r
-
phase
a
c
t
i
v
e
p
o
w
e
r
c
ont
r
o
l
l
e
d
w
i
t
h
i
n
t
he
M
M
C
u
si
n
g
F
i
gur
e
2
-
5.
I
n
o
the
r
w
o
r
d
s
,
the
co
n
t
r
i
b
u
t
i
on
of
t
he
i
nter
na
l
ac
tive
vo
l
t
a
g
e
d
i
s
turb
a
n
ce
p
ro
du
ce
d
in
e
ach
leg
/
pha
se
e
xp
l
a
ine
d
i
n
S
e
cti
o
n
I
I
[21]
i
s
co
n
s
i
d
er
ed.
F
i
gur
e
7.
V
olt
a
ge
v
ar
ia
ti
o
n
d
-
q
c
om
po
ne
nt
s
Th
us,
the
s
e
var
i
at
i
o
n
s
p
r
o
duc
e
the
c
a
p
ac
i
t
or
v
o
l
tage
d
-
q
v
ar
iat
i
o
n
c
o
mpo
n
e
nts
s
how
n
i
n
F
igur
e
7.
I
n
ot
her
w
o
r
d
s,
t
h
e
a
r
i
t
h
me
t
i
c
a
l
a
ver
a
ge
d
c
c
a
p
ac
it
or
v
o
l
ta
ge
o
f
a
ll
t
h
e
pha
ses,
i
.e.
V
C
,
and
t
h
e
ar
i
t
hm
et
ica
l
me
an
d
c
ca
p
a
c
i
to
r
vo
lt
ag
e
of
e
ac
h
ph
a
s
e
wil
l
p
r
odu
ce
vol
t
a
g
e
v
a
r
i
a
t
ion
d-
q
com
p
one
n
t
s
a
s
s
h
o
w
n
i
n
F
i
g
u
r
e
7.
Evaluation Warning : The document was created with Spire.PDF for Python.
Int J
P
o
w
E
l
e
c
&
D
ri S
yst
IS
S
N
:
2088-
86
94
Mo
du
l
a
r
m
u
l
t
i
l
e
v
e
l c
onv
erte
r (
MMC) base
d
ST
AT
CO
M wi
th
vec
to
r
con
t
r
o
l… (
A
bd
urr
a
h
m
an
U
m
a
r L
a
w
an)
1
841
U
s
in
g
the
d-
q
t
r
ansfor
m,
it
is
possi
b
l
e
from
t
he
v
o
l
t
a
ge
d
iffer
ence
s
V
,
Ck
ku
V
Ck
kv
a
n
d
V
Ck
kw
t
o
ob
ta
in
t
he
d
-q
v
a
r
ia
tio
n
co
m
pone
n
t
s
Cd
v
a
n
d
.
Cq
v
Th
e
an
gl
e
r
e
p
res
e
nt
s
t
h
e
ex
t
e
nt
o
f
t
h
e
v
o
lt
age
varia
t
i
o
n
a
m
on
g
t
h
e
S
M
c
apa
c
itor
s
w
ith
r
espec
t
t
o
the
S
T
A
T
CO
M
o
ut
put
c
u
rrents
in
t
he
d
-q
fram
e
,
and
is
gi
ve
n a
s
11
ta
n
t
a
n
fq
C
q
fd
Cd
Iv
I
v
(
35)
Th
e
n
t
h
e
c
o
rres
p
ondi
ng
t
o
t
al
v
o
l
t
a
g
e
v
a
r
i
a
t
i
o
n
i
n
t
h
e
d
-q
f
ra
me
g
i
v
e
n
i
n
(3
2)
a
n
d
(
3
4
)
,
coul
d
be
modi
fie
d
w
i
t
h
re
spe
c
t t
o
th
e
v
ol
t
a
ge va
r
ia
ti
o
n
a
m
ong t
h
e
S
M
c
ap
a
c
itor
s
,
and g
i
ven a
s
()
(
)
()
(
)
dc
om
s
T
c
o
m
f
d
C
d
qc
om
s
T
c
o
m
f
q
C
q
vf
L
I
v
vf
L
I
v
(
36)
The
o
p
p
o
s
i
ng
si
gna
ls
dc
om
v
a
n
d
qc
om
v
i
n
(
36)
a
re
t
o
be
c
on
si
dere
d
a
s
t
he
c
om
pe
nsa
t
i
ng
sig
n
a
l
s
to
t
h
e
P
I
c
ont
r
o
l
a
c
t
ions
i
n
t
h
e
d-q
con
t
ro
l.
T
he
c
o
m
pen
s
a
t
i
n
g
v
a
riab
l
es
dc
om
v
and
qc
om
v
a
r
e
t
hen
ad
de
d
t
o
t
he
o
u
t
p
u
t
s
of
t
he
d
,
q-P
I
c
on
tro
l
l
e
rs
(
d
Q
a
n
d
q
Q
)
in
t
he
v
olta
ge
r
efere
n
ce
e
qua
tio
n
g
i
ve
n
i
n
(
2
5
).
T
h
i
s
w
i
l
l
g
e
ne
rate
t
he
enha
nc
e
d
c
om
pen
s
ate
d
v
o
lta
ge
r
e
f
er
ence
(
*
cd
k
V
a
n
d
*
cq
k
V
)
gi
ve
n
i
n
(
37).
F
r
om
(
3
6
),
onc
e
the
s
t
e
a
dy-
st
a
t
e is rea
che
d
,
the capa
c
i
t
o
rs vo
lta
ge var
iat
i
ons
Cd
v
a
nd
Cq
v
i
n F
i
g
u
re
7 and t
h
e c
u
rre
nt
var
ia
ti
o
n
s
fd
I
and
fq
I
c
ea
se
to pr
evai
l.
T
his
w
ill
ma
ke the
com
pen
s
at
i
on t
o
c
ease
to
exi
s
t
.
*
*
*
*
2
2
T
cd
d
d
co
m
f
d
f
q
T
f
d
fq
T
cq
q
q
c
o
m
f
q
f
d
T
f
q
fd
L
VQ
v
V
I
I
R
I
L
VQ
v
V
I
I
R
I
(
37)
2.2.
MM
C in
t
e
r
n
al con
tr
ol
u
sin
g
v
i
r
tu
al
i
m
p
ed
an
c
e
b
ased
com
p
e
n
s
ation
The
v
o
lta
ge v
a
r
ia
tio
n
from
w
i
th
in t
he MM
C
di
s
cuss
ed in Se
ctio
n
II,
i
s a
volta
ge
r
i
p
p
l
e a
n
d p
r
o
d
u
ces
the
D
C
c
irc
u
l
a
t
i
n
g
c
ur
rent
ci
r
k
i
w
ith
r
ip
ple
c
u
rre
nt
c
omp
o
n
e
n
ts
[
3
2
]
.
T
he
u
se
o
f
t
h
e
P
I-c
ontro
l
lers
i
n
th
e
in
t
e
rna
l
c
o
n
tro
l
o
f
t
h
e
M
M
C
is
o
ne
o
f
t
h
e
m
o
s
t
c
om
m
o
n
s
o
l
u
tio
ns
t
o
o
b
t
a
i
n
a
z
e
r
o
s
t
a
t
i
c
e
r
r
o
r
i
n
c
o
n
t
r
o
l
l
i
n
g
the
dc
c
irc
u
l
a
ti
ng
c
urr
e
n
t
.
H
o
w
e
ver,
P
I-c
ontr
o
lle
rs
h
a
v
e
t
h
e
sm
a
l
l
e
f
fec
t
i
n
s
uppr
ess
i
n
g
t
he
h
ar
mon
i
c
v
o
l
t
a
g
e
un
less
a
h
ig
h-ga
in
h
i
g
h-
orde
r
con
t
ro
llers
s
uch
as
r
eso
n
a
n
t
c
o
n
t
ro
l
l
ers
are
em
pl
oye
d
w
i
th
t
he
P
I-c
on
trolle
rs.
Ba
se
d o
n
t
he i
nte
g
ra
l fe
a
t
ure of reso
n
an
t co
ntr
o
lle
rs, t
he
s
up
p
re
ss
io
n o
f
t
he low
-
o
rder har
monic v
o
l
ta
g
e
c
ou
ld
b
e
a
c
h
i
e
v
e
d
.
T
h
i
s
con
t
ro
l
i
s
m
o
s
tl
y
a
c
hi
ev
e
d
u
si
n
g
se
cond-o
r
d
e
r
r
esona
nt
c
on
tro
l
lers
f
or
t
he
l
ow
h
arm
o
n
i
cs.
To c
ance
l
hi
g
h
-
o
rder
har
moni
cs,
large
-
order
re
sona
te
c
on
tro
l
l
e
r
s ha
v
e
to
be
e
mploye
d.
A
nd
m
a
ny
o
f
t
he
m
ha
ve
t
o
b
e
c
a
s
cade
d
f
or
b
e
tter
harm
on
ics
el
imi
n
a
tio
n
[
2
1
]
.
No
t
e
,
t
h
e
b
a
nd
widt
h
of
t
he
c
i
r
cu
lat
i
ng
c
ur
ren
t
c
on
t
r
o
l
ler
s
a
re
v
ery
sma
ll
d
u
e
to
t
h
e
l
i
m
it
ed
s
amp
l
e
an
d
p
r
o
cess
i
n
g
f
req
u
e
nc
y
in
M
M
C
s
;
th
us,
inc
r
ea
sing
t
he
o
rde
r
o
f
t
h
e
re
son
a
n
t
c
o
n
tr
ol
l
e
r
c
o
u
ld
n
o
t
b
e
feas
i
b
le
[
21].
A
n
o
t
her
a
p
pr
oac
h
t
o
reduc
ing
t
h
e
i
m
pacts
o
f
t
he
v
o
l
tage
h
a
r
m
o
nic
s
i
s
t
h
e
use
of
t
he
r
e
p
et
it
i
v
e
co
nt
rol
l
er.
H
o
we
v
e
r,
t
h
e
r
ep
e
t
i
t
i
ve
con
t
ro
l
l
er
d
esi
gn
re
quire
s
a
com
p
le
x
st
a
b
il
it
y
a
n
a
l
ys
i
s
on
t
h
e
co
n
t
rol
l
er-
f
i
lte
r
des
i
g
n
.
More
over
,
t
he
implem
e
n
ta
t
i
o
n
o
f
repe
ti
t
i
v
e
c
on
tro
l
l
e
r
requ
ire
s
a
h
e
a
vy
d
i
gi
ta
l
si
gn
al
p
roce
ssi
n
g
.
F
ro
m
Se
ct
i
o
n
II,
t
h
e
a
c
t
ive
vo
lta
ge
d
isturb
a
n
ce
kd
v
c
on
tri
but
e
s
t
o
t
h
e
S
T
A
T
CO
M
d
i
s
t
or
ti
on
i
n
e
ach
p
h
a
se
l
e
g
,
b
y
g
e
nera
tin
g
vo
l
t
a
g
e
varia
t
i
o
n
be
tw
ee
n
t
h
e
le
g
and
t
h
e
to
ta
l
S
M
vol
ta
ge
s
(i.
e
.
dc
V
).
I
f
the
var
i
at
i
on
c
o
u
l
d
b
e
m
i
n
im
ize
d
,
b
y
in
j
e
c
t
ion o
f
an exter
n
a
l
virt
u
a
l
c
urrent o
pp
osi
ng the
c
i
r
c
ula
t
i
on c
u
rr
ent, the vo
l
tage
di
s
turb
a
n
ce impa
c
t
s
c
oul
d
be
m
i
t
i
g
a
t
e
d
;
t
hus,
the
t
o
ta
l
o
u
tp
ut
v
olta
ge
w
ou
ld
e
q
u
at
e
t
h
e
t
o
t
a
l
S
Ms
v
ol
t
a
ges.
A
nd
t
h
e
v
o
l
t
a
ge
d
ist
o
rt
i
o
ns
w
o
u
l
d bec
o
me
m
i
n
im
a
l
.
Th
is
s
e
c
t
i
on
co
ns
iders the i
m
p
r
ove
d
m
ode
l
of
c
irc
u
lat
i
on
c
u
r
r
e
n
t
de
ve
l
ope
d
in [2
1
],
w
hic
h
c
ons
i
d
ers
the
pr
esenc
e
o
f
t
h
e
v
o
lta
ge
d
is
t
u
r
b
a
n
ce
i
n
t
h
e
cl
ose
d
l
o
o
p
.
A
ls
o,
i
t
em
p
l
oys
t
he
v
irt
u
a
l
i
m
p
eda
n
c
e
c
o
n
cep
t
base
d
o
n
d
ro
o
p
-m
eth
od
to
i
ncre
ase
t
h
e
M
M
C
i
n
t
e
r
na
l
i
m
pe
dance
a
n
d
l
e
ss
e
n
t
h
e
a
mo
unt
o
f
t
h
e
unwa
n
t
e
d
circ
u
l
a
t
i
on c
u
rre
nt [1
2
].
To che
c
k t
h
e i
n
fl
uen
c
e
of t
he v
irt
u
a
l
i
m
p
e
d
a
n
ce
in
ensuri
n
g an i
m
p
rove
d st
e
a
d
y-sta
t
e
and
d
yna
mic
p
e
r
f
orm
a
nc
e, th
e
P
I-c
ontrol
l
e
r
in
F
i
g
u
re 8
w
i
l
l ha
v
e
p
k
as the
pr
o
p
o
rt
i
ona
l
c
o
effi
c
i
e
n
t,
i
k
a
s
t
h
e
Evaluation Warning : The document was created with Spire.PDF for Python.
ISS
N
:
208
8-
8
6
9
4
I
n
t
J Po
w El
ec &
D
ri S
y
s
t
Vo
l. 1
0
,
No
. 4
,
D
e
c
2
0
19
:
18
33 –
1
8
51
1
842
in
te
gr
a
l
c
oef
f
ic
ien
t
,
a
n
d
ha
ve
t
he
o
pe
n-
l
oop
tr
a
n
s
f
er
f
unc
t
i
o
n
as
()
.
i
Gs
T
h
e
refore,
th
e
dyn
amic
b
e
h
avio
r
of
the
cl
ose
d
-
l
o
o
p
w
oul
d
be
a
s
fo
ll
ow
s
:
*
__
__
__
()
(
)
(
)
()
(
3
8
)
2(
)
kd
ci
rk
ci
rc
pi
i
i
rr
p
i
i
i
rr
p
i
i
i
ks
k
i
s
V
s
is
Ls
R
k
s
k
Ls
R
k
s
k
(
38)
F
r
om
(
38)
,
it
cou
l
d
be
s
ee
n
t
h
at
t
he
i
n
t
e
r
na
l
cir
c
u
it
c
o
u
l
d
b
e
m
odele
d
in
t
he
e
qu
i
v
al
en
t
f
o
r
m
a
s
*
()
()
()
.
(
)
(
3
9
)
()
kd
cir
c
cir
k
o
Vs
is
g
s
i
s
Zs
(
39)
()
gs
is
t
he
c
irc
u
la
tin
g
cur
r
en
t
re
fer
e
nce
-
t
o
-m
easure
d
c
irc
u
l
a
ti
ng
c
u
r
r
ent
tr
a
n
sf
er
f
unc
t
i
o
n
,
and
()
o
Zs
i
s
t
he
im
p
eda
n
c
e
tra
nsfer
fu
n
c
t
i
o
n
of t
h
e
cl
o
s
ed-l
oo
p sys
t
e
m
.
F
i
gur
e
8.
I
m
p
r
ove
d
c
i
r
c
u
l
a
t
i
n
g
cur
r
ent
co
ntr
o
l
u
s
in
g
con
v
e
n
t
i
o
n
al P
I co
n
t
r
o
ller
Cons
i
d
er
ing
(
3
9)
,
i
t
i
s
c
l
e
a
r
t
h
at
t
he
i
m
p
e
d
a
n
ce
()
o
Zs
a
ffe
c
t
s
t
h
e
c
i
rc
u
l
at
ing
cur
r
ent
c
o
ntr
o
l
.
T
o
i
m
p
o
s
e
a
d
e
s
i
r
e
d
i
m
p
e
d
a
n
c
e
()
v
Zs
and
r
e
duc
e
t
h
e
impa
c
t
o
f
the
vo
l
t
a
g
e
di
stur
ba
nce
f
o
r
an
i
m
p
r
o
ve
d
st
ab
ili
ty
,
th
e
cu
rre
n
t
re
f
e
ren
c
e
*
()
cir
k
is
c
o
uld
be
o
bta
i
ne
d
f
r
o
m
a
n
e
w
desir
e
d
c
u
r
r
ent
r
e
fe
r
e
nc
e
*
'
()
ci
r
k
is
as
a
f
u
n
c
tio
n
of
t
he
v
o
lta
ge
d
is
t
u
r
b
anc
e
.
This
c
o
u
l
d
be
a
c
h
ie
ve
d
by
t
h
e
dr
o
o
p
-
m
e
t
hod
as
**
'
()
()
()
(
4
0
)
()
kd
ci
r
k
ci
r
k
v
Vs
is
i
s
Zs
(
40)
wh
ere
()
v
Zs
is
t
he
v
ir
tu
a
l
i
mpe
d
a
n
ce
,
and
c
o
uld
be
obta
i
ne
d
by
ta
ki
n
g
t
he
t
i
m
e
der
i
va
ti
ve
o
f
a
v
i
r
t
ua
l
i
ndu
c
t
an
ce
vir
L
(i.e
.
()
vi
r
v
Zs
s
L
w
h
ere
s is the
L
ap
lac
e
opera
tor)
.
P
u
tt
i
n
g
(
4
0)
i
nt
o
(
3
9)
,
the
d
y
na
mic
e
qua
t
i
o
n
o
f
t
h
e
i
n
t
e
r
n
al
c
u
r
r
e
nt
-
l
o
op
of
t
he
M
M
C
c
o
u
l
d
b
e
m
odi
f
i
ed
a
s
fo
l
l
ow
s
:
*
'
()
(
(
)
(
)
)
()
()
.
(
)
(
)
(
4
1
)
()
()
ci
r
c
ci
rk
ov
kd
vo
gs
Z
s
Z
s
is
g
s
i
s
V
s
Zs
Z
s
(
41)
()
(
)
()
()
(
(
)
(
)
vo
ov
ov
Zs
Z
s
Zs
gs
Z
s
Z
s
is
t
he
a
ve
r
a
g
e
c
apac
itor
vo
lta
ge
c
o
n
tr
o
l
t
ota
l
i
m
p
e
n
den
c
e
tr
ans
fer
f
u
n
c
tio
n.
T
he
s
tab
i
lit
y eff
ects of
t
h
i
s
v
i
r
t
ua
l
i
m
pedanc
e
i
n
t
h
e
clo
s
ed
l
oo
p
sy
s
t
em
a
re
d
escrib
ed
i
n
Section
III-
C.
I
t
has
bee
n
r
epor
te
d
tha
t
t
he
u
se
o
f
t
h
e
d
e
r
i
va
tive
of
v
ir
tu
al
i
mp
ed
an
ce
()
vi
r
Zv
s
s
L
i
n
c
r
e
a
s
e
s
t
h
e
t
o
t
a
l
i
n
t
e
rn
a
l
i
mp
eda
n
ce
,
a
n
d
re
duce
s
t
h
e
c
i
r
cul
a
t
i
on
o
f
un
want
ed
c
u
r
r
e
nt
[
33
]
.
H
ow
e
v
er
,
wit
h
t
he
p
r
e
senc
e
o
f
no
ise,
e
s
p
ec
ial
l
y
in
t
he
e
x
p
e
r
ime
n
t
a
l
o
p
era
t
i
ons
o
f
the
M
M
C
,
t
h
e
de
ri
v
a
ti
v
e
t
er
m
am
pli
f
ies
the
n
o
i
se
i
n
t
h
e
sys
t
em
[
1
2
].
T
o
min
i
miz
e
t
h
e
a
m
p
l
i
fie
d
n
o
i
se
u
s
u
a
l
ly
g
en
er
ated
by
the
der
i
va
t
i
ve
v
ir
t
u
al
i
mpe
d
ance
,
a
low
pa
ss
f
i
l
t
e
r
(
LP
F
)
c
oul
d
be
e
mplo
ye
d
a
l
o
n
g
w
it
h
the
de
r
i
va
tive
b
ased
i
mp
ed
an
ce
.
Ho
wev
e
r,
i
t
ha
s
b
een
r
e
por
te
d
t
h
at
t
he
L
P
F
can
i
ntr
o
d
u
ce
a
n
a
d
d
i
tio
na
l
de
la
y
w
h
i
c
h
d
e
p
e
nds
on
the
type
s
of
t
he
o
p
e
r
a
t
i
ng
pa
r
a
m
e
ter
s
s
uc
h
a
s
t
he
l
oa
ds
a
nd
f
i
l
t
er
s
i
n
t
he
s
ystem
[
2
1]
.
I
n
t
he
o
t
h
er
h
a
nd,
t
he
v
ir
tu
al
t
r
a
ns
i
e
nt
i
m
p
e
d
anc
e
c
once
p
t
co
u
l
d
be
e
m
p
l
o
yed
t
o
g
e
n
er
ate
o
ppos
in
g
v
i
r
t
ua
l
c
u
r
r
ent
s
i
gna
ls
(
i.
e.
cv
i
r
i
a
nd
Lvi
r
i
)
wi
th
t
he
s
ta
b
i
l
i
t
y
e
nha
nce
m
e
n
t
w
i
t
h
o
u
t
t
he
i
n
c
r
e
ase
in
t
he
s
yste
m
dela
y.
F
r
o
m
(
41)
,
it
c
o
uld
be
s
ee
n
t
h
a
t
t
he
e
n
h
anc
e
d
t
ota
l
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