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.
3, S
ep 2019,
pp.
1
4
8
3
~1
4
9
9
ISSN: 2088-
8694,
DOI
:
10.11591
/ijpeds.
v10.
i
3.pp1483-1499
1483
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
Enhanced d
ecoupled curren
t control with vo
ltage
C
o
m
p
ensati
on f
or m
odular mul
til
e
vel converter (MMC) based
S
T
ATCOM
Abdu
rra
h
m
a
n
Uma
r Law
a
n
1
,
Haider A
.F.
Almuri
b
2
, Jeen
G
.
K
hor
3
1,
2
,
3
D
e
p
a
rtm
e
n
t
of
El
ectri
cal & Elect
r
oni
c
En
g
i
neeri
ng,
U
niv
e
rsit
y
o
f
Not
tingham,
Malaysia
C
a
m
pu
s, 43500
Semenyih.
Ma
laysi
a
1,
2
Dep
a
rt
m
e
nt
o
f Elect
rical
E
ngineerin
g, H
ussei
n
i Ad
amu
F
e
deral
P
o
l
y
t
e
c
h
ni
c, 1
0
K
a
no
Road,
Kazau
re, Ni
g
eria
Art
i
cl
e In
fo
ABSTRACT
A
r
tic
le hist
o
r
y
:
Re
ce
i
v
e
d
Ju
l
2
2,
201
8
Re
vise
d Jan
1
0
, 2019
Ac
ce
p
t
ed
M
ar 6
, 2
0
19
Th
is
p
ap
e
r
p
rese
n
t
s
t
h
e
ap
plication
of
m
od
u
l
ar
m
ultil
e
vel
con
v
e
rter
(
MMC)
as a st
a
tic com
pen
s
at
or (S
T
ATCO
M
)
f
or react
ive
current
con
tro
l
.
T
he
c
urrent
con
t
ro
l
is
m
ostl
y
ach
iev
e
d
us
in
g
p
r
op
orti
on
al
c
o
n
tro
l
l
e
r,
p
ro
po
rtiona
l
-
integral
(
PI)
controller,
a
nd
h
y
s
t
e
resi
s
c
o
n
t
roller
a
m
ong
other
s
.
P
I
controllers
h
a
ve
t
h
e
a
dv
a
n
t
a
g
e
o
f
lo
w
ha
rm
on
ic
s
a
n
d
sma
l
l
v
a
ria
t
ion
s
.
How
e
v
er,
d
u
e
to
th
e
P
I
c
ontro
ll
er’s
d
ep
enden
c
y
o
n
t
h
e
syst
e
m
p
aram
eters
an
d
als
o
d
u
e
t
o
t
h
e
vari
ati
ons
w
it
h
i
n
t
h
e
MM
C
duri
n
g
cap
aci
to
rs
v
o
l
tag
e
c
on
tro
l
,
va
riation
in
t
he
M
M
C
p
erf
o
rm
ance
d
u
rin
g
t
he
S
TA
T
C
O
M
n
o
n
-i
deal
o
perat
i
ons
o
ccur.
To
m
i
tig
ate
this
,
an
i
m
p
ro
ved
p
e
rform
ance
o
f
M
MC
w
il
l
b
e
p
resen
t
ed
u
s
i
n
g
vect
or-b
a
s
ed
c
o
m
pen
s
atio
n
c
o
ncep
t.
T
h
e
p
ropo
se
d
co
ntro
l
wi
ll
b
e
i
n
t
r
o
d
u
c
e
d
t
o
d
e
p
r
e
s
s
t
h
e
e
f
f
e
c
t
o
f
t
h
e
d
y
n
a
m
i
c
s
o
f
t
h
e
M
M
C
b
a
s
e
d
S
T
A
T
C
O
M
non
-
id
eal
v
ariati
on
s
co
nsi
d
eri
n
g
the
i
m
pact
o
f
th
e
vo
lt
age
dist
urb
a
nce.
T
his
w
i
ll
be
a
chi
e
ved
by
t
he
i
n
t
ro
duct
i
o
n
o
f
v
o
ltage
v
aria
bl
es
t
o
su
btrac
t
the
t
r
ansi
ent
vari
ati
ons
f
rom
P
I
c
on
tro
l
lers
’
o
u
tp
uts
at
t
he
g
rid-i
n
t
e
rf
a
ce;
thu
s
,
i
m
pr
ov
in
g
th
e perf
o
r
m
a
n
ce.
K
eyw
ord
s
:
Mo
du
la
r
Mu
l
tile
ve
l
Con
v
e
r
t
e
r
(MMC
)
STATCOM
Re
ac
t
i
ve
c
urr
e
nt c
o
n
tro
l
Co
pyri
gh
t © 2
019 In
stit
u
t
e
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
,
Unive
r
sit
y
of
Nottingham
,
Malaysia
C
a
m
pu
s
,
43
50
0
S
e
m
e
ny
ih.
M
a
l
ay
si
a
,
Depa
rtem
ent o
f
E
lectr
i
c
a
l a
n
d
C
o
m
p
uter
E
n
g
i
ne
er
in
g,
E-m
a
il:ke
c
x4a
ua
@
n
ott
i
n
g
h
a
m
.edu.
my
1.
I
N
TR
OD
U
C
TI
O
N
The
r
e
are
m
a
ny
t
ype
s
of
d
i
s
t
u
rba
n
c
e
s
w
h
i
c
h
ca
n
a
f
fe
ct
o
r
prec
is
e
l
y
re
du
c
e
t
he
g
r
i
d
po
w
e
r
qual
ity
(P
Q
)
[
1,
2
].
P
ow
er
d
ist
u
rba
n
c
e
s
depe
n
d
i
n
g
o
n
the
i
r
ma
gn
i
t
ude
s,
n
a
t
u
re
o
f
the
e
n
d
use
r
s,
i
n
t
erm
e
diar
y
pow
e
r
dev
i
ce
s
a
m
on
g
o
t
hers
c
an
a
ff
ect
t
he
qua
l
i
t
y
o
f
the
grid
s
ys
t
e
m
si
g
n
i
fica
ntl
y
.
The
de
gra
d
i
n
g
o
f
p
ow
e
r
q
ual
ity
ca
n
be
d
ue
t
o
vo
l
t
age
d
i
s
t
o
r
tions,
sa
g
s
,
v
o
lta
ge
t
r
a
nsie
n
t
s,
s
w
e
l
l
s
,
fli
c
k
e
rs
a
nd
u
nb
a
l
an
c
e
s
in
v
ol
ta
g
e
s,
in
t
e
rrup
t
ions,
and
c
u
r
r
ent
s
p
ik
e
s
a
mong
ot
he
rs.
The
stat
ic
c
om
p
e
n
s
a
t
o
r
s
(
S
T
A
C
O
M
s
)
a
r
e
s
h
u
n
t
d
e
v
i
c
e
s
,
a
n
d
t
h
ei
r
re
ac
ti
v
e
p
o
w
er
(
VAR
)
c
o
mp
e
n
sa
tio
n
s
h
av
e
b
een
u
sed
i
n
m
it
i
ga
tin
g
the
s
e
pow
er
p
ro
b
l
em
s
w
i
t
h
g
o
o
d
trans
i
en
t
a
nd
dy
nam
i
c
p
e
rfo
r
ma
nce
[3–
9
].
S
TA
TCO
M
a
pp
l
i
cat
i
o
ns
f
o
r
VAR
c
o
m
p
e
n
s
a
t
i
o
n
a
n
d
sta
b
ilit
y
stud
ies
ha
ve
b
e
e
n
em
p
l
o
y
e
d
b
ase
d
o
n
d
i
ffer
en
t
co
n
t
rol
schem
e
s
[
10
–1
2].
La
w
a
n
e
t
a
l.
[
1
3
,
14
]
ha
ve
em
plo
y
e
d
t
he
d
irec
t
a
ppr
oa
c
h
t
o
impr
ovi
n
g
t
he
pow
e
r
f
a
c
t
or
s
us
i
ng
M
MC
b
a
s
e
d
S
T
A
TCO
M
s.
T
he
p
ow
e
r
fa
ct
ors
w
e
re
i
m
p
rove
d
w
i
th
g
o
o
d
t
rans
ient
p
erform
ance.
However,
t
he
c
apac
it
or
v
o
lta
ge
d
is
tur
b
a
n
ce
w
a
s
n
o
t
con
s
i
d
ere
d
i
n
t
h
e
c
o
m
p
ensa
ti
on
sc
hem
e
.
Th
is
c
oul
d
ha
ve i
mpr
o
ve
d
t
h
e
t
ra
n
s
i
e
nt
r
e
s
ponse
o
f
t
h
e
s
y
s
t
e
m.
H
a
w
et
a
l.
[
1
2
]
a
nd
K
a
w
e
t
a
l
.
[
15]
p
ro
pos
ed
vol
t
a
ge
s
up
p
o
rt
u
si
ng
S
T
A
T
CO
M
.
I
n
[1
5],
the
r
e
w
e
r
e
goo
d
tra
n
sie
n
t
perform
ance
s
of
t
he
S
TA
T
C
O
M
,
w
h
ic
h
w
a
s
em
plo
y
e
d
u
si
ng
p
r
o
p
o
rt
i
o
na
l
(P
)
c
ontro
ll
er
s.
H
ow
ever
,
t
h
e
iss
u
e
of
i
n
d
i
v
i
d
u
al
c
apac
it
or
c
e
ll
vo
ltage
v
a
r
ia
t
i
on
w
ith
r
e
s
pe
ct
t
o
t
h
e
t
o
ta
l
thre
e-p
h
a
s
e
a
r
ithm
e
tica
l
c
a
p
aci
t
o
r
vo
lta
ge
w
a
s
n
ot
c
onsi
d
er
ed
i
n
t
h
e
com
p
e
n
sat
i
o
n
.
Th
i
s
c
ou
l
d
h
a
v
e
i
m
p
r
o
v
e
d
t
h
e
act
i
v
e
p
o
w
e
r
bala
nc
in
g
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN: 2088-
8694
Int J
P
o
w
El
e
c
&
D
ri S
yst
,
V
ol.
10,
N
o.
3
, S
e
p
2
0
1
9
:
148
3
– 1
499
1
484
betw
ee
n the S
T
A
C
O
M
a
n
d
t
he grid. Mo
r
e
o
ver
,
i
nc
orp
o
rat
i
n
g
t
he v
o
l
t
a
g
e
d
i
st
urba
nce
a
n
d
c
a
p
ac
it
or ba
l
anc
i
n
g
con
t
ro
l i
n
[1
5
] c
o
u
l
d
have
a
d
d
e
d
com
p
lic
a
t
i
o
ns
t
o
t
h
e
sele
ct
i
v
e
ha
r
mo
nic
e
limi
n
a
t
io
n (S
HE) modu
la
t
i
o
n
[1
6
]
.
A
w
e
ll-t
une
d
P
I-contr
o
l
l
er
c
an
y
ie
ld
r
e
s
po
nse
w
i
t
h
z
e
r
o
or
m
in
i
mum
st
e
a
d
y
s
ta
te
e
rror
w
i
t
h
in
a
sma
l
l
ri
se
tim
e.
H
ow
eve
r
,
PI
c
on
tro
lle
rs
m
a
y
h
a
v
e
s
o
m
e
d
ra
w
b
ac
k
s
w
h
ic
h
inc
l
ud
e
s
t
he
h
ig
h
se
t
tli
n
g
t
im
e
[1
7, 18].
It has
a
lso bee
n
r
e
p
o
r
ted tha
t
P
I
contr
o
l
l
er
s ha
ve
po
or t
r
a
ns
ie
n
t
r
e
s
p
onse
c
o
mpa
r
ed t
o o
t
her c
o
m
m
on
con
t
ro
l
l
ers
suc
h
a
s
hys
tere
s
i
s
a
nd
ra
mp
t
im
e
c
o
n
t
r
o
ller
s
i
n
v
o
l
t
a
g
e
s
o
u
r
c
e
i
n
v
e
r
t
e
r
(
V
S
I
)
S
T
A
T
C
O
M
s
[
1
8
]
.
Fo
r
e
x
amp
l
e
in
[
19
],
P
I
co
n
t
ro
l
wa
s
e
m
pl
oyed
fo
r
S
T
ATCO
M
vol
t
a
g
e
re
gu
la
ti
o
n
s,
b
ut
osc
illa
ti
o
n
o
c
c
u
rre
d
i
n
the
resp
o
n
se
a
nd
pro
duce
d
a
p
o
o
r
trans
i
e
n
t
respo
n
se
e
ve
n
th
o
ugh
t
here
w
as
m
i
n
i
m
a
l
s
t
e
a
dy-s
t
ate
er
ror
i
n
t
he
com
m
a
nd
tra
c
k.
A
mir
et
a
l
.
i
n
[
20]
p
re
sen
t
ed
P
I
c
o
nt
r
o
l
l
e
r
i
n
w
hi
c
h
i
t
s
g
a
i
n
w
a
s
obt
a
i
n
e
d
by
a
n
a
u
t
o
ma
t
i
c
gai
n
p
r
o
cess,
p
rop
o
sed to
r
e
d
uce
t
h
e
h
i
gh-
so
ur
ce
impe
da
nc
e
osci
ll
a
t
i
o
n
eff
e
ct
a
n
d
t
o
l
esse
n
th
e
ph
a
s
e-lo
c
k
ed
lo
op
(PLL)
i
n
h
e
rent
d
ela
y
e
ff
ect.
There
was
a
g
o
o
d
s
ta
bil
ity
e
n
ha
nce
m
en
t
due
t
o
re
duc
e
d
o
sci
l
lat
i
on
d
ur
ing
the
stea
d
y
-
s
ta
t
e
p
erforma
n
ce
.
The
c
o
n
t
ro
l
t
e
chn
i
que
w
a
s
f
urt
h
er
e
m
p
loy
e
d
in
[
2
1
]
t
o
r
educ
e
t
h
e
sa
me
e
ffec
t
o
f
t
h
e
P
LL.
H
o
w
ev
e
r
,
ov
erc
u
rre
n
t
th
at
w
as
t
h
r
e
e
t
i
m
es
h
igh
e
r
th
an
t
he
r
a
t
e
d
c
u
r
re
nt
w
as
p
ro
du
ce
d
by
t
he
S
T
A
C
O
M
d
uri
n
g
t
h
e
l
o
ad
s
te
p
c
h
an
ge
.
Th
is
o
verc
urre
nt
i
m
pose
d
t
h
e
us
e
of
i
ncre
ased
r
ated
powe
r
d
e
v
ic
e
s
.
A
n
o
t
her
a
ppr
o
ach
i
s
t
h
e
u
s
e
of
t
he
b
io
l
o
g
i
c
a
lly
s
ha
pe
d
P
I
-con
tr
o
l
lers
[
22
–2
5].
The
o
b
je
cti
v
e
o
f
t
h
i
s
pa
per
is
t
o
i
nt
rod
u
ce
a
si
mpl
e
a
n
d
l
ess
c
o
mp
ut
at
io
na
l
l
y
h
e
a
v
y
a
l
go
rit
h
m
un
l
i
ke
t
he
a
l
g
ori
t
h
ms
i
n
[
2
5]
a
n
d
[
2
3
].
I
t
is
a
l
s
o
,
t
o
e
x
p
l
o
i
t
t
h
e
f
e
a
t
u
r
e
o
f
t
h
e
e
x
t
e
r
n
a
l
c
o
m
p
e
n
s
a
t
i
o
n
o
n
c
o
n
t
ro
l
l
ers
p
r
op
osed
b
y
the
a
u
th
ors
i
n
[
1
2
]
f
or
V
A
R
c
ompe
nsa
t
i
o
n
an
d
c
o
n
t
r
o
l
e
nha
ncem
ent.
S
i
n
c
e
t
he
M
MC
h
as
l
i
t
t
l
e
a
c
ti
ve
pow
er
e
xc
ha
n
g
e
ex
t
e
rna
lly
w
ith
t
he
g
ri
d
for
the
ty
pic
a
l
STA
T
C
O
M
a
ppl
ica
t
io
n
[
2
6
]
,
a
n
i
mpr
ove
d
re
spo
n
se
can
b
e
ac
h
i
ev
ed
b
y
com
p
en
sat
i
ng
bo
t
h
t
he
n
e
g
l
i
gi
b
l
e
act
iv
e
po
w
e
r
and
t
h
e
r
eac
t
i
ve
p
o
w
e
r
.
I
n
ot
her
w
o
rds
,
unl
ike
in
[
1
2
]
,
th
i
s
pro
pose
d
a
l
g
orit
hm
doe
s
n
o
t
sac
r
ifice
the
con
t
ri
b
u
ti
on
o
f
t
h
e
o
f
d-a
x
i
s
c
urr
e
n
t
w
h
i
c
h
i
s
usua
lly
a
ss
um
ed
neg
l
ig
ibl
e
.
F
r
o
m
[
27
]
,
i
n
s
tea
d
o
f
a
d
j
u
s
tin
g
t
h
e
P
I-c
on
t
r
o
l
ler
s
gai
n
s
t
o
achie
ve
i
m
p
ro
ved
res
p
o
n
se
u
n
d
er
di
ffe
re
nt
o
p
e
rati
o
n
s,
a
t
rans
i
e
nt
c
ompe
nsa
t
o
r
t
o
be
d
er
iv
ed
w
i
l
l
be
a
u
g
m
e
nte
d
w
i
t
h
a
c
apac
it
or
v
ol
t
a
ge
in
j
e
c
t
ion
[2
8]
t
o
i
n
tro
d
u
ce
vol
ta
ge
c
om
pe
n
s
at
io
n.
T
h
i
s
c
o
m
p
en
sa
t
i
on
w
i
l
l
e
n
h
a
n
ce
t
he
c
o
n
t
r
o
l
of
t
he
P
I-
con
t
ro
l
l
ers
un
d
e
r
differ
e
n
t
ope
ra
ti
ons.
Th
e
prop
ose
d
a
l
gor
it
hm
w
ill
be
d
eri
v
e
d
i
n
th
i
s
p
aper
t
o
pr
o
duc
e
op
p
o
site v
o
lta
g
e
s
u
p
p
o
rt t
o t
h
e
d,
q-P
I
c
ont
r
o
lle
r
ou
t
p
ut
v
a
r
i
a
b
les
(
d
Q
an
d
q
Q
)
.
T
h
i
s w
ill be i
n respo
n
se t
o
t
h
e
STATCOM
d
-
a
x
is
a
nd
q
-a
xis
v
a
ri
ati
o
n
s
i
n
t
h
e
c
u
rre
n
t
c
o
nt
rol
.
F
u
rther
m
ore,
t
he
i
ntrod
u
c
t
i
o
n
of
v
ol
t
a
ge
c
o
mp
e
n
sa
tio
n
wi
ll
m
iti
g
a
t
e
t
h
e
v
ari
a
t
i
o
n
imp
a
c
t
f
ro
m
t
h
e
g
r
o
w
i
n
g
n
u
m
ber
of
S
M
c
a
p
aci
t
o
rs
i
n
t
h
e
M
M
C
a
p
pli
c
at
ion
s
.
Un
li
k
e
[
2
9
]
,
th
is
p
ap
er
w
ill
no
t
re
p
l
ac
e
th
e
PI
c
o
n
t
r
o
l
l
e
r
s
w
h
i
c
h
ha
v
e
t
h
e
a
dv
a
n
ta
g
e
o
f
l
e
s
s
curr
ent
d
i
s
t
or
ti
on
w
i
t
h
t
he
h
y
s
ter
e
sis-
l
i
ke
c
o
n
tr
ol
l
e
rs
t
hat
e
x
h
i
bi
t
h
i
gh
d
i
stor
ti
o
n
[
2
9
]
.
I
n
s
tea
d
,
t
h
e
pr
op
ose
d
con
t
ro
l
w
i
l
l
i
n
c
orp
o
r
a
te
t
he
a
dva
n
t
a
g
e
of
t
he
u
se
o
f
the
P
I
c
on
t
r
o
l
l
e
r
s
a
n
d
s
u
p
e
r
-
i
m
p
o
s
e
t
h
e
m
w
i
t
h
t
h
e
pro
pose
d
v
ol
ta
ge
com
pen
s
at
o
r
.
2.
MODULAR
M
U
L
T
ILEVEL
C
O
N
VE
RTER (M
M
C)
I
NVERT
ER BAS
E
D
STATCO
M
F
i
gure
1
sh
ow
s
a
n
M
M
C
b
a
s
ed
S
TA
TCO
M
s
ystem
c
o
n
n
e
c
te
d
to
a
p
o
w
er
g
rid.
T
he
a
m
oun
t
of
requ
ire
d
p
ow
e
r
t
ha
t
w
i
ll
p
r
o
duce
the
re
qui
re
d
S
T
A
T
CO
M
c
u
rre
nt
f
I
w
hic
h
f
low
s
t
hr
ou
g
h
t
he
c
ou
p
lin
g
impe
da
nce
f
Z
d
e
p
e
n
ds
o
n
th
e
di
ff
ere
n
ce
i
n
vol
t
a
g
e
b
e
t
ween
t
he
S
TATCOM
A
C
ou
t
put
v
ol
ta
g
e
(
i
.
e
.
c
V
)
and
.
the gr
i
d
v
o
l
ta
g
e
a
t
th
e
po
i
n
t o
f
c
ou
p
lin
g
(PCC) (
i
.
e.
f
V
).
I
f
t
he
s
o
u
r
ce
i
m
pedanc
e
(i.
e
.
sourc
e
i
nd
u
c
t
a
nce
s
L
and
s
ourc
e
resista
n
ce
s
R
)
i
s
a
s
s
u
m
e
d
n
e
g
l
i
g
i
b
l
e
,
the gr
id s
ource
vo
l
t
a
g
e
(
i.e.
s
V
)
w
o
u
ld
b
e th
e s
a
m
e
as
f
V
.
The sys
t
em
v
o
l
tage
re
l
a
t
i
o
ns
hi
p is il
l
us
tra
t
e
d
i
n (
1
)
,
wher
e
r
epre
se
nt
s the
pha
se a
ng
le be
t
w
e
en
t
he
S
TA
TCO
M
volta
ge
(i.
e
.
c
V
)
a
nd
the
gri
d
v
ol
t
a
ge
(
i.e
.
f
V
).
(
1
)
Ba
se
d on
(1)
usi
ng
the
d
i
agra
m
of F
igure.
1 t
he
s
ystem
tran
sfe
r
func
t
i
on
w
ith
a
n
an
gu
lar
fre
q
u
e
n
cy
i
s
gi
ve
n by
:
(
2
)
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
Enha
nc
e
d
de
co
up
le
d c
u
rre
n
t c
o
n
t
r
o
l
wi
t
h
v
o
lta
ge
Com
p
e
n
sat
i
on f
o
r … (
A
bd
urr
a
hm
a
n
U
m
a
r L
a
wa
n)
1
485
+
+
‐
‐
+
‐
+
‐
+
+
‐
up
i
wp
i
vp
i
1
SM
1
SM
1
SM
2
SM
2
SM
2
SM
SM
N
SM
N
S
M
N
r
L
r
L
r
R
r
R
f
R
s
R
s
L
f
L
s
V
GR
ID
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
ru
i
ci
rv
i
ci
r
w
i
1
SM
N
1
SM
N
1
SM
N
2
SM
N
2
SM
N
2
SM
N
un
i
vn
i
wn
i
SM
2
Vdc
2
Vdc
SM
PC
C
c
V
f
V
n
f
I
2
SM
N
2
SM
N
2
SM
N
up
V
vp
V
wp
V
wn
V
vn
V
un
V
F
i
gure.
1.
T
he thre
e
-pha
se M
M
C
base
d
S
TA
TCO
M
c
o
n
n
ecte
d
t
o t
h
e
gr
id
sy
s
t
e
m
In
(
2
)
,
t
h
e
f
q
I
i
s
the
rea
c
t
i
ve
c
urr
e
nt
o
f
the
MM
C
w
h
ile
f
d
I
i
s
th
e
act
iv
e
cu
rre
n
t
.
f
L
a
nd
f
R
a
r
e
t
h
e
cou
p
l
i
ng
i
n
duc
tance
a
n
d
r
e
sista
n
ce
re
spe
c
t
i
ve
ly.
cd
V
i
s
t
h
e
M
M
C
t
hree
p
ha
se
o
u
t
pu
t
vo
l
t
a
g
e
i
n
t
he
d
-a
x
i
s
u
s
ua
ll
y
a
l
i
g
ne
d
with
t
h
e
g
rid
v
o
lt
ag
e
us
ing
t
h
e
ph
a
s
e
-
lo
cked
l
o
op
.
cq
V
i
s
t
h
e
MMC
out
put
v
olt
a
g
e
i
n
t
h
e
q
-
axi
s
.
fd
V
and
fq
V
a
r
e
t
h
e
grid
d
-ax
i
s a
nd q-a
x
is v
o
lta
ge
s
respe
c
t
i
ve
l
y
.
I
n
F
ig
ure.
1
,
t
h
e
per-pha
se
c
u
r
r
e
nt
c
an
b
e
c
ons
ider
ed
a
s
k
i
.
The
s
ub-s
ubscr
i
p
t
k
re
p
r
es
en
t
s
a
n
y
o
f
t
h
e
three
-
p
h
ases
(
u,
v
o
r
w
)
.
The
l
o
w
e
r
and
up
per
arm
c
u
rre
n
ts
a
re
d
e
not
ed
a
s
kp
i
a
n
d
kn
i
respective
l
y
.
The
c
i
rcul
at
in
g
cu
rre
n
t
i
s
d
en
ot
ed
a
s
ci
r
k
i
.
T
h
e
Ck
V
i
s
the
MMC
o
u
t
pu
t
vo
lta
g
e
a
n
d
can
b
e
ob
ta
ine
d
by
c
o
n
t
ro
l
l
i
n
g
the
up
per
arm
vo
lta
ge
kp
V
and
th
e
l
o
w
e
r
a
r
m
volta
ge
.
kn
V
The
dc
V
i
s
the
eq
u
i
va
le
n
t
d
ire
c
t
cur
r
en
t
(D
C)
vo
lta
ge a
cross
the
ar
ms. The
c
um
ula
t
ive
M
M
C
-t
o-gr
id
i
nd
uc
t
a
nce
an
d
resis
t
a
n
ce
ar
e
the
n
defi
ne
d b
y
(
3).
(
3
)
Inte
gra
t
i
n
g (
2
) be
tw
ee
n
th
e
pr
e
s
en
t sam
p
le
(
k) (
i.e
.
sub-sc
ript
k
) a
n
d the
n
e
xt sam
p
l
e (
k
+1)
over
a
gi
ve
n pe
ri
od
(
i.e
.
(,
)
id
q
T
)
durin
g t
h
e
curre
nt co
n
tr
ol
w
il
l gi
ve
:
(
4
)
(
5
)
The
relat
i
onsh
i
p
b
e
tw
ee
n
the r
a
tes of c
ha
n
g
e
o
f
t
he
M
M
C
o
ut
p
u
t
d,
q
c
urrent
s w
i
t
h
tim
e c
a
n
be
expre
s
se
d i
n
te
r
ms
o
f the
var
i
a
b
l
e
s
,1
d
kk
Q
a
n
d
,1
q
kk
Q
gi
ve
n in
(
6)
a
nd (
7
).
(
6
)
(
7
)
The
d
,
q-va
ria
b
les
,1
d
kk
Q
a
nd
,1
q
kk
Q
i
n
(6) and
(7)
can
b
e ob
ta
ine
d
fr
o
m
t
h
e
o
u
t
p
u
t
s
of
P
I
con
t
ro
l
l
ers a
s
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN: 2088-
8694
Int J
P
o
w
El
e
c
&
D
ri S
yst
,
V
ol.
10,
N
o.
3
, S
e
p
2
0
1
9
:
148
3
– 1
499
1
486
(
8
)
The
s
e
d,
q-var
i
a
b
l
e
s
ca
n
be
u
se
d
t
o
o
b
t
a
i
n
the
c
o
r
r
espon
d
i
n
g
d
,q
-
volta
ge
c
om
po
n
e
nt
s
for
th
e
modu
la
ti
o
n
of
the
MM
C d
u
ri
ng t
h
e
sam
e
sample
peri
od
as fol
low
s
:
(
9
)
(
10)
w
h
e
r
e
the
d,q
p
r
o
port
i
ona
l
a
nd
i
n
te
gr
al
c
o
n
t
r
o
l
ler
s
’
ga
ins
_(,
)
P
id
q
K
a
n
d
_(
,
)
I
id
q
K
w
it
h
th
e
s
a
m
p
l
e
rate
(,
)
id
q
T
c
an
b
e obt
aine
d
[12]
r
espe
ctive
l
y
as
(
11)
(
12)
Th
e
PI
c
ont
ro
lle
rs
f
as
t
res
ponse
ma
k
e
t
h
e
S
TATC
OM
outp
u
t
d
,
q
c
u
r
r
ent
references
(
i.e
.
(,
)
1
fd
q
k
I
),
w
ith
t
he
d
,q-
P
I
varia
b
le
s
(
i
.e.
(,
)
,1
dq
kk
Q
)
of
t
he
n
e
x
t
sam
p
le
p
e
r
i
o
d
m
eet
u
p
w
i
t
h
t
he
c
urre
n
t
s
a
m
ple
d,
q
curr
ent r
e
fe
ren
c
es
*
,)
(
q
fd
k
I
w
ith d,
q
-PI
va
riab
les
(,
)
dq
k
Q
res
p
e
c
tiv
el
y
a
s
f
oll
o
ws:
(
13)
(
14)
Eq
uat
i
ons
(
1
3
)
and (
14) c
an furt
h
er
b
e w
r
it
t
e
n
in an
ave
r
a
g
e
fo
rm
as:
(
15)
(
16)
Ex
pan
d
i
n
g
(8)
in se
pa
rate
d
,q
f
orm
s
usi
ng (1
3)a
n
d (14)
gi
v
es:
(
17)
(
18)
D
u
e
to
t
he
a
c
t
i
o
n
o
f
t
he
P
I
c
o
n
t
ro
l
l
ers
i
n
(
17)
a
nd
(1
8).
The
g
ri
d
v
o
l
t
age
an
d
t
h
e
M
M
C
base
d
S
T
A
T
CO
M ou
tp
u
t
vo
l
t
a
g
e
a
re
assum
e
d
con
s
t
a
nt
w
i
t
hi
n
t
h
e
sam
e
p
er
iod,
and
de
fine
d
as:
(
19)
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
Enha
nc
e
d
de
co
up
le
d c
u
rre
n
t c
o
n
t
r
o
l
wi
t
h
v
o
lta
ge
Com
p
e
n
sat
i
on f
o
r … (
A
bd
urr
a
hm
a
n
U
m
a
r L
a
wa
n)
1
487
(20)
P
u
tti
ng (
1
3
)
-(20)
i
n
t
o
(9)
and (
10) gi
v
es
t
he
d,
q
v
o
lta
ge r
efe
r
e
n
ce
s as
(21)
F
u
rtherm
ore
,
t
o
im
p
o
se
t
h
e
a
cti
v
e
po
w
e
r
con
t
ro
l
a
t
t
he
t
hre
e
-p
h
a
s
e
gri
d
i
nt
erf
a
c
e
,
t
h
e
av
e
r
a
g
e
ma
gni
tude
o
f
the
d-a
x
i
s
c
urr
e
nt fr
o
m
the sa
m
p
le
p
eri
ods
,1
kk
c
a
n
b
e
o
b
t
a
i
n
ed
a
s
(
22)
Whe
r
e
vd
T
i
s
the a
v
era
g
e
capa
c
itor
con
t
rol
sa
mpl
i
ng ra
te.
C
is the
SM c
apac
i
t
a
n
ce
as in
[3
0].
D
u
e
t
o
t
h
e
P
I
c
o
n
t
r
o
l
l
e
r
’
s
f
a
s
t
r
e
s
p
o
n
s
e
,
t
h
e
S
T
A
T
C
O
M
o
u
t
p
u
t
d
-
a
x
i
s
c
ur
rent
r
efere
n
ce
(i.e
.
1
fd
k
I
)
w
ith
i
t
s
corr
e
s
p
o
ndi
n
g
thre
e
-
p
h
a
se a
v
e
ra
ge
capa
c
i
t
o
r
v
o
lta
ge
1
V
C
k
o
f
the
ne
x
t
s
am
ple
pe
r
i
o
d
i
n
(
2
2)
m
ee
ts
up
w
i
t
h
t
he c
ur
rent
s
a
m
p
l
e d-
a
x
is curre
nt
r
e
f
er
ence
*
fd
k
I
w
ith i
ts
cor
r
e
s
p
o
n
d
i
ng
r
e
fe
r
e
nc
e
c
a
pa
citor
v
o
l
t
a
g
e
*
V
C
k
as foll
o
ws:
(
23)
(24)
P
u
tt
ing
(2
3)
a
nd (
24) in
t
o (2
2)
give
s
the
i
nsta
nta
n
e
o
u
s
a
c
t
i
v
e
curr
ent r
e
fe
re
nc
e
*
fd
k
I
a
s follow
s
(
25)
K
is
t
he
p
ro
port
i
ona
l
c
o
n
t
ro
l
l
er
g
ai
n
(i.e
.
vd
C
T
).
*
V
C
k
c
a
n
b
e
d
e
fin
e
d
as
t
h
e
r
e
f
eren
ce
DC
v
o
l
t
a
ge
for
the
a
l
l
t
h
re
e-pha
se
S
M
ca
paci
t
o
rs.
V
C
k
c
an
b
e
c
a
l
c
ul
at
ed
a
s
th
e
arit
h
m
e
t
ic
al
a
v
e
rag
e
vol
t
a
g
e
o
f
a
l
l
t
he
SM
capa
c
i
t
o
r
s in t
he
t
hr
ee
leg
s
a
s
(
26)
F
o
r
th
e
rea
c
t
i
v
e
(
V
A
R
)
po
w
e
r
contr
o
l
a
t
t
he
g
rid-
in
ter
f
ac
e,
t
he
r
eactive
curr
ent
referen
c
e
*
fq
k
I
c
a
n
be
a
s in (
27) [1
2
] a
s
sum
i
n
g
al
l
the
loa
d
r
eac
t
i
ve
pow
er
to
b
e
obta
i
nab
l
e
fr
om
the
S
T
A
TCO
M
.
(
27)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN: 2088-
8694
Int J
P
o
w
El
e
c
&
D
ri S
yst
,
V
ol.
10,
N
o.
3
, S
e
p
2
0
1
9
:
148
3
– 1
499
1
488
wher
e
L
Q
and
L
q
I
are
t
h
e
re
ac
t
i
v
e l
o
ad
p
o
w
e
r
a
n
d
curren
t
re
s
p
e
c
t
i
v
el
y
.
Eq
ua
ti
on
(2
1)
c
an
b
e
s
i
mpl
i
fi
ed
i
n
t
e
rm
s
of
t
he
v
a
r
iab
l
es
i
n
(
1
7
)
a
n
d
(
1
8
)
,
(
i
.
e
.
d
k
Q
a
nd
q
k
Q
)
as pa
r
t of t
he
f
ee
d for
w
ard
t
e
rm
s [31],
a
s
fol
l
o
w
s
:
(
28)
(
29)
Th
e
con
t
rol
sc
h
e
me
s
whi
c
h
are
av
erag
ing
co
nt
rol
,
i
nd
ivi
d
u
a
l
con
t
r
o
l
,
an
d
the
a
r
m
ba
la
nci
n
g
c
o
n
t
ro
l
ca
n
b
e
a
p
p
l
i
e
d
us
in
g
(3
0)
a
n
d
F
ig
ure.
2
-4.
The
s
e
c
ontro
ls
c
an
b
e
ext
e
n
d
e
d
t
o
th
e
th
re
e
-
p
h
a
se
c
on
fi
gura
t
ion
[3
4]
.
The
me
ani
ng o
f
t
he v
o
lta
ge
s
y
m
bol
s in F
igur
e.
2
-4 are
as
(
30)
V
Ck
is
t
he
p
e
r-pha
se
a
ve
rage
c
a
p
a
c
i
t
or
v
o
lta
ge
i
n
the
c
o
rre
spon
di
n
g
p
h
a
s
e
.
V
Ck
p
an
d
V
Ck
n
a
r
e
t
h
e
ave
r
age
c
a
p
ac
itor
vo
l
t
ag
e
o
f
t
he
u
p
p
er
a
r
m
a
nd
l
ower
a
rm
o
f
t
h
e
c
orrespo
n
d
i
ng
p
ha
se
r
e
s
pec
tive
l
y.
N
i
s
t
h
e
numbe
r
o
f
t
he
S
M
in
t
he
u
p
p
e
r
a
n
d
l
o
w
e
r
a
r
m,
V
Ck
i
a
n
d
V
Ck
j
a
re
t
he
v
o
l
t
a
ges
of
t
he
i
t
h
a
nd
j
th
S
Ms
i
n
t
h
e
upper
arm
and
lower
arm
res
p
ective
ly
.
*
V
Ck
a
nd
*
ci
rk
i
re
p
r
es
en
t
th
e
ref
e
re
n
c
e
vo
lt
a
g
e
a
nd
reference
circulating
curr
ent i
n
th
e
c
or
respo
n
d
in
g
le
g r
e
spect
iv
el
y
.
+
‐
+
‐
ci
rk
v
ci
r
k
i
*
ci
r
k
i
1
2
+
+
kp
i
kn
i
V
k
C
*
V
Ck
2
(1
,
2
)
1
K
KK
s
4
(3
,
4
)
3
K
KK
s
F
i
gur
e 2.
Bloc
k
dia
gra
m
o
f
aver
ag
in
g co
ntr
o
l
+
‐
(,
)
()
kp
n
sign
i
*
V
Ck
(,
)
V
ki
j
C
5
_
p
bal
KK
(,
)
kB
i
j
v
F
i
gure
3.
Indi
v
i
d
u
a
l
bal
a
n
c
i
ng
contr
o
l b
l
ock
di
a
g
ram
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
Enha
nc
e
d
de
co
up
le
d c
u
rre
n
t c
o
n
t
r
o
l
wi
t
h
v
o
lta
ge
Com
p
e
n
sat
i
on f
o
r … (
A
bd
urr
a
hm
a
n
U
m
a
r L
a
wa
n)
1
489
+
‐
k
i
V
kn
C
V
kp
C
kA
v
6
_
p
ar
m
KK
F
i
gur
e 4.
A
rm bala
n
c
i
n
g
c
o
n
t
rol b
l
oc
k d
i
agr
a
m
c
V
cq
T
f
d
VL
I
s
s
c
V
f
V
Tf
q
L
I
s
s
f
I
f
I
(1
)
PF
(0
)
PF
STA
T
C
O
M
I
ST
AT
COM
I
I
,
()
cf
L
e
adi
n
g
PF
V
V
o
v
e
r
e
x
c
ita
tion
r
e
g
io
n
,
()
cf
La
gg
ing
P
F
V
V
u
n
d
e
r
e
x
c
itatio
n
r
e
g
ion
()
s
t
e
a
d
y
s
tate
s
s
l
i
n
e
f
V
s
s
(
)
T
c
o
m
f
j
L
L
I
(
)
T
c
o
m
f
j
L
L
I
F
i
gure
5.
C
om
bi
ne
d P
h
asor
d
ia
gram
of vo
l
t
a
g
e
so
urc
e
g
rid con
n
e
c
ted
S
T
A
T
CO
Ms for
r
e
g
io
n
I
and
II
opera
tio
ns
F
i
gure
2
de
pic
t
s
t
h
e
a
v
e
r
ag
in
g
co
ntr
o
l
d
i
a
g
r
a
m
,
t
hi
s
co
n
t
rol
p
u
s
h
e
s
t
h
e
a
v
e
r
a
g
e
v
o
l
t
a
g
e
V
Ck
o
f
each
pha
se
l
eg
t
o
fo
l
l
ow
i
ts
r
efe
r
enc
e
v
o
l
ta
ge
*
V
Ck
.
W
h
e
n
*
VV
Ck
Ck
,
*
ci
rk
i
i
n
c
r
e
a
s
e
s
a
n
d
f
o
r
c
e
s
t
h
e
a
c
t
u
a
l
circul
ating
current
cirk
i
t
o
fol
l
ow
i
t
s
r
e
f
er
ence
c
u
rrent
*
ci
rk
i
.
Th
i
s
cause
s
V
Ck
t
o
tra
c
k i
t
s
refere
nce
vo
l
t
a
g
e
*
V
Ck
.
P
I
c
ontro
l
l
ers
ca
n
be
e
m
p
lo
y
e
d
in
t
he
t
ra
ck
in
g,
a
n
d
t
he
o
ut
put
v
o
lta
ge
c
omm
a
nd
ci
r
k
v
fro
m
the
a
v
era
g
in
g
con
t
ro
l
ca
n
be
p
ro
duc
e
d
f
or
t
he
C
S
P
W
M.
T
he
g
a
i
ns
o
f
the
P
I
c
on
t
rollers
w
hich
a
r
e
1
K
,
2
K
,
3
K
a
n
d
4
K
ca
n be
o
b
t
a
i
ne
d ba
se
d on i
n
te
rn
a
l
m
ode c
o
n
tr
ol
t
he
ory [
3
2].
The
i
n
di
vid
u
a
l
b
a
l
anc
i
n
g
c
on
tro
l
p
us
hes
e
a
c
h
S
M
ca
paci
t
o
r
vo
l
t
a
ge
i
nd
ivi
d
ua
ll
y
i
n
e
very
su
b
m
o
dul
e
t
o
f
o
llo
w
it
s
vo
lt
a
g
e
re
fe
ren
c
e
*
V
Ck
.
The
i
n
di
v
i
d
u
a
l
b
alanc
i
ng c
o
ntr
o
l b
l
oc
k d
i
a
g
ram
i
s
dep
ic
ted in
F
i
gure
.
3
.
The obta
i
ne
d
v
o
l
t
a
g
e
com
m
a
n
d
f
o
r
t
h
is
m
od
u
l
a
t
i
o
n
c
o
n
tro
l
i
s
(,
)
kB
i
j
v
.
W
h
e
n
*
(,
)
VV
Ck
i
j
Ck
,
t
h
e
arm
curr
ent
is
c
ha
rgi
ng
the
SM
c
apa
c
i
t
or
s
an
d
the
e
n
erg
y
i
s
be
in
g
t
ra
n
s
f
e
r
r
e
d
f
ro
m
th
e
DC
lin
k
to
t
he
S
M
ca
paci
t
o
rs.
The
p
r
op
ortio
na
l
c
o
n
t
ro
l
l
e
r
5
K
can
b
e e
m
ploye
d
as
i
n
[3
3]
f
or the
i
nd
ivi
d
ual c
o
n
t
r
o
l
.
The
co
n
t
ro
l
bl
oc
k
for
t
h
e
ar
m
ba
l
a
nc
i
ng
c
o
n
t
ro
l
is
s
h
o
w
n
i
n
F
i
g
ure.
4
.
Thi
s
c
o
n
t
r
ol
l
esse
n
s
t
he
di
sc
repa
nc
ies in
v
olta
ges
be
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e
e
n
t
h
e
v
o
l
t
a
g
e
s
o
f
u
p
p
er
a
nd l
o
w
er
a
rm
.
The
P
-
controller
with
a
g
ain
6
K
ca
n
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e
m
p
l
o
ye
d i
n
the
c
o
n
t
ro
l as
i
n
[3
3].
The
vol
t
a
ge
c
omm
a
nd
for
the a
r
m
co
n
t
ro
l
i
s
.
kA
v
3.
T
H
E PROPO
S
ED VOLT
A
GE COMPEN
S
ATION ALGO
RITH
M
(
dc
o
m
v
AND
qc
om
v
)
FOR
MMC
B
A
SED S
T
A
T
COM
To
c
on
tro
l
t
he
r
eac
t
i
ve
a
n
d
a
c
tive
p
o
w
e
rs
f
or
t
he
S
TA
T
C
O
M
a
p
p
l
i
ca
ti
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the
d-ax
is
a
nd
q
-a
x
i
s
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ent
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n
t
ro
l
ha
ve
t
o
be
a
s
fast
a
s
p
o
ss
i
b
l
e
.
The
conv
e
n
ti
ona
l
PI
c
ont
r
o
ller
s
u
sua
l
l
y
e
m
p
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ye
d,
m
ig
ht
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ave
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ho
o
t
s
a
n
d
l
o
n
g
e
r-se
t
t
l
i
n
g
t
i
m
e
[
1
2
].
T
he
S
TA
TCO
M
v
a
r
ia
ti
on
s/
o
s
ci
ll
a
tio
ns
c
a
n
o
ccur
due
t
o
the
PI
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN: 2088-
8694
Int J
P
o
w
El
e
c
&
D
ri S
yst
,
V
ol.
10,
N
o.
3
, S
e
p
2
0
1
9
:
148
3
– 1
499
1
490
con
t
ro
l
l
ers
de
la
yed
re
sponse
,
c
onve
rter
s
w
i
tc
hi
n
g
n
o
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se
,
w
a
vefo
rm
d
is
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rt
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wh
en
u
si
ng
f
und
ame
n
t
a
l
fre
que
nc
y
tec
h
ni
q
u
es,
ripp
l
e
s fr
om
unba
la
nc
e
d
ca
p
ac
i
t
ors,
no
n
-i
dea
l
i
t
y or
m
ism
a
tch
of t
he
c
om
po
ne
n
t
s
[12].
Th
e
p
r
op
o
s
ed
c
on
t
r
ol
w
i
l
l
b
e
i
nt
rodu
ce
d
to
p
ro
vid
e
t
h
e
M
M
C
w
ith
a
dd
i
tio
na
l
c
o
mpe
n
sat
i
on
varia
b
l
e
s us
i
n
g
the us
ua
l CS
P
W
M
.
The
se
c
o
m
pe
nsa
tio
n va
riab
les w
ill be a
dde
d a
s
o
pp
os
ing sig
n
a
l
s to t
he P
I
c
o
nt
rol
out
put
v
a
r
i
a
b
l
es
(
i
.
e.
d
Q
an
d
q
Q
)
[
31]
w
ithi
n
t
he
M
MC
c
o
n
tr
ol
b
andw
id
t
h
f
or
a
f
a
s
t
c
u
rrent
respo
n
se
w
it
h
few
e
r
overs
ho
ot
s.
U
nl
ike
[
2
9]
,
w
h
e
r
e
t
h
e
u
s
e
of
a
ny
o
f
P
I
-
c
ont
r
oller
s
w
as
s
acrifice
d
and
the
st
e
a
dy-s
t
ate
c
o
n
d
i
t
i
o
n
w
a
s
a
ssum
e
d
i
n
t
he
c
urrent
c
o
n
t
rol.
I
t
c
an
b
e
seen
i
n
[
2
9
]
t
h
a
t
th
ere
were
h
i
g
h
curren
t
di
st
ort
i
o
n
s d
u
ri
ng t
h
e
t
r
ans
i
tio
n
of t
he
l
oa
d fr
om
capa
c
i
tive
t
o in
d
u
ct
i
v
e m
ode.
Th
is
p
ro
pose
d
c
on
tro
l
w
ill
u
t
i
l
i
z
e
the
low
-
d
i
st
orti
o
n
f
e
a
t
u
r
e
s
o
f
t
h
e
P
I
c
ontr
o
l
l
er
s
w
h
e
n
c
ompa
red
w
ith
t
he
h
yste
resis
c
u
rre
nt
c
on
tro
l
lers
a
nd
add
s
t
r
a
ns
i
e
nt
s
up
p
ort
i
n
t
he
f
orm
of
c
omp
e
nsa
t
i
o
n
varia
b
le
s
(
dc
om
v
and
qc
o
m
v
)
to
a
c
h
ie
ve
m
i
n
im
al
d
is
tort
io
n
w
i
t
h
a
n
enha
nce
d
r
espo
nse.
S
inc
e
dur
ing
t
h
e
t
r
ans
i
en
ts
i
n
Figure
.
5
,
a
sm
all
act
iv
e
p
o
w
e
r
l
o
ss
i
s
a
ssoc
i
a
te
d
w
i
th
t
he
e
ff
ec
tiv
e
r
e
sista
n
ce
T
R
,
there
will
a
l
s
o
be
add
i
tio
na
l
p
o
w
e
r
loss
w
i
t
h
in
t
he
numbe
r
o
f
S
M
ca
paci
t
o
rs
[
33]
.
A
l
s
o
,
t
h
e
r
e
i
s
a
s
m
a
l
l
p
o
w
e
r
l
o
s
s
f
r
o
m
t
h
e
vo
lta
ge
d
iffer
e
nce
acr
oss
t
h
e
cum
u
lat
i
ve
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n
duc
t
o
r
T
L
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onsi
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er
i
n
g
t
h
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trans
i
e
n
t
o
p
era
t
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ons
i
n
S
T
A
T
CO
M
I
regi
on
a
n
d
S
T
A
T
CO
M
II
regio
n
[
3
1
].
T
he
S
TA
TCO
M
c
a
n
a
tta
i
n
s
tea
d
y-s
t
a
t
e
fas
t
er
w
he
n
t
h
e
ac
t
i
ve
a
n
d
t
he
re
ac
t
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v
e
v
a
r
i
a
tio
n
i
m
p
a
ct
s
a
r
e
co
mp
ens
a
t
e
d
,
a
n
d
t
h
e
STATC
O
M
cu
r
rents
fd
I
a
n
d
fq
I
w
il
l
be
d
ri
ven
t
o
t
he
des
i
red
cur
r
en
ts
*
f
d
I
an
d
*
f
q
I
fas
t
er
.
This
w
ill
ha
ppe
n
w
h
e
n
t
he
i
n
duc
ta
nce
va
r
i
a
t
i
on
t
h
a
t
c
an
b
e
obt
a
i
ned
betw
ee
n
t
h
e
i
n
duc
t
a
nce
a
t
t
h
e
s
t
e
a
dy-s
t
ate
and
the
i
n
duc
t
a
nce
a
t
t
r
a
n
si
en
ts
i
s
co
mp
en
sa
t
e
d
[2
9
]
.
Th
en
,
t
h
e
curr
ent c
o
mp
o
n
en
ts
*
f
d
I
and
*
f
q
I
w
oul
d ha
ve t
he
i
r
re
s
u
l
ta
nt
l
i
n
e curr
ent
f
I
i
n
qua
drat
u
r
e
w
ith t
he a
va
ila
b
l
e A
C
vo
lta
ge
a
t
the
P
C
C.
A
nd
t
h
e
A
C
v
ol
ta
ge
f
V
of
t
he
g
r
i
d
w
i
l
l
a
ls
o
be
i
n
p
h
a
s
e
w
i
t
h
S
TA
TCO
M
o
ut
put
A
C
vo
lta
ge
c
V
.
Thi
s
m
e
a
ns
t
h
e
S
TA
TCO
M
d
-ax
i
s
vo
l
t
age
c
o
m
m
a
nd
*
cd
V
w
il
l
ove
rcom
e
the
o
p
p
os
i
n
g
v
o
l
t
a
g
e
dro
p
(
i
.
e.
*
T
f
q
L
I
)
in
F
i
g
ure
.
5
,
and
*
cq
V
w
hic
h
i
s
the
r
e
sulta
n
t
v
o
lta
g
e
c
om
m
a
nd
of
t
he
S
TA
TCO
M
q
-a
xi
s
wo
ul
d
opp
ose
th
e
v
o
l
t
a
g
e
d
r
o
p
*
T
f
d
L
I
.
F
r
om
F
i
gur
e.
5
,
t
h
e
c
h
a
n
ge
i
n
ind
u
c
t
a
n
c
e
v
al
ue
b
e
t
w
een
t
he
i
nd
u
c
t
anc
e
a
t
stea
d
y
-sta
te
a
n
d
in
duc
ta
nce
at
t
ra
nsie
nt
s
due
t
o
h
i
g
h
s
w
itch
i
ng
m
o
du
l
a
tio
n
w
ill
re
sul
t
t
o
a
sma
l
l
c
u
rre
nt
d
egra
da
t
i
o
n
dur
ing
the tra
n
si
e
n
t
s
(
d
an
d
q
)
give
n i
n
(
31) a
n
d
(32)
(
33)
(
34)
To
c
ou
nt
e
r
ac
t
t
h
e
v
ol
t
a
g
e
v
ari
a
tio
n
s,
w
h
i
c
h
i
s
in
di
sp
e
n
sabl
e
i
n order t
o
s
p
e
e
d
u
p t
h
e
r
e
sponse
of
t
h
e
P
I
c
ont
r
o
l
l
er
,
a
nd
re
duc
e
t
h
e
curre
nt
e
rror
i
n
(
31)
a
n
d
(
3
2
),
a
t
ra
nsie
n
t
v
ol
ta
ge
c
a
n
b
e
der
i
ve
d
from
(
2
)
w
i
t
h
a
com
p
en
sat
i
ng i
n
d
u
c
t
anc
e
co
m
L
a
s
(
35)
The
r
ef
ore,
t
he
t
o
t
a
l
s
tea
dy-
sta
t
e
i
n
terfac
i
ng
i
nd
uc
ta
n
c
e
val
u
e
T
L
w
ill
b
e
mod
i
fi
ed
t
o
Tc
om
L
onl
y
d
u
r
i
n
g
th
e
t
r
a
n
si
en
t
s
a
nd
a
re r
el
at
ed
as
(
36)
A
ssume
the
t
r
a
ns
ien
t
cha
nge i
n
(37),
happe
n
s
duri
ng t
h
e t
i
m
e (
i
.e
.
t
),
putti
ng (3
5)
in (37)
w
il
l
gi
ve
:
(
37)
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
Enha
nc
e
d
de
co
up
le
d c
u
rre
n
t c
o
n
t
r
o
l
wi
t
h
v
o
lta
ge
Com
p
e
n
sat
i
on f
o
r … (
A
bd
urr
a
hm
a
n
U
m
a
r L
a
wa
n)
1
491
F
r
o
m
F
i
g
u
r
e
.
5
,
i
t
c
a
n
b
e
s
e
e
n
t
h
a
t
t
o
c
r
o
s
s
i
n
t
o
t
h
e
r
e
g
i
o
n
o
f
t
he
s
tea
dy-s
t
a
t
e,
t
he
c
om
p
e
nsa
t
i
o
n
c
a
n
be
a
pp
l
i
ed
dur
i
n
g
t
h
e
t
r
a
n
sie
n
t
s
(
r
e
pre
s
e
n
te
d
in
r
e
d
a
nd
b
l
ue
l
i
n
e
i
n
F
i
g
ure.
5
)
by
o
v
e
r
c
o
m
i
n
g
t
he
v
ol
ta
ge
di
ffe
re
nce
due
t
he
e
ffec
tive
ind
u
c
t
a
n
ce
,
t
h
is
w
ill
p
us
h
t
h
e
r
e
sul
t
a
n
t
ou
t
p
ut
v
ol
tage
*
c
V
t
o
r
eac
h
as
c
l
o
se
a
s
pos
si
b
l
e,
i
ts
s
te
a
dy-s
t
a
t
e
va
l
u
e
at
uni
t
y
pow
er
f
a
c
t
or
l
i
n
e,
a
s
s
u
min
g
n
o
var
i
a
tio
n
due
t
o
S
T
A
T
CO
M
n
on-
i
d
eal
opera
tio
n,
and
no ab
sorbe
d
re
a
c
t
i
v
e
p
o
w
e
r
in the
i
n
duc
t
o
rs
(
f
L
and
r
L
).
S
i
nce the tra
n
s
i
e
n
t
var
i
a
t
i
o
n h
a
ppe
ns i
n ever
y sw
i
t
c
h
in
g pe
rio
d
,
the sw
itc
hi
n
g
fre
que
nc
y
(
i
.
e
.
s
f
)
can
be
e
st
i
m
ate
d
a
s
t
h
e
in
verse
of
t
he
tra
nsie
n
t
t
i
m
e for the
in
t
r
o
d
u
ce
d
com
p
e
n
sa
ti
on
[2
9].
The
r
ef
ore,
the
c
om
pen
s
at
i
n
g
vo
lta
ge i
n t
h
e
q-a
x
i
s
i
s
as
(
38)
The
c
o
rrespo
n
d
i
n
g d-a
x
i
s
act
ive
v
o
lta
ge
c
om
pensa
t
i
o
n w
oul
d
b
e
d
eriv
ed
as:
(
39)
S
i
milarl
y
(39) is sim
p
l
i
fie
d
a
s
:
(
40)
D
u
r
i
n
g
t
he
t
ra
ns
i
e
n
t
s,
t
he
v
olta
ge
d
ro
p
v
a
riab
le
s
(
i
.
e
.
co
m
f
d
L
I
and
co
m
f
q
L
I
i
n
(
40)
a
n
d
(
4
2
)
re
sp
ec
ti
v
e
ly
can
b
e
co
mp
e
n
sat
e
d
by
t
h
e
P
I
c
o
nt
rol
l
e
rs
;
t
h
u
s
a
re
i
gn
or
ed.
F
u
r
t
he
rm
ore,
t
he
s
m
a
ll
d
i
s
t
or
t
i
ons
i
n
the
o
u
t
put
A
C
vo
lta
ge
l
e
v
e
l
s
of
t
he
M
M
C
d
ue
t
o
the
v
o
lta
ge
d
is
t
urbanc
e
betw
ee
n
the
u
pper
a
n
d
l
o
w
e
r
ar
ms
in
e
ac
h
leg
/
pha
se
c
an
b
e
c
ons
i
d
ere
d
[
33].
Ta
k
i
n
g
p
h
a
se-
k
f
or
i
n
s
t
a
n
c
e
,
a
s
s
u
m
e
t
h
e
e
n
t
i
r
e
S
M
v
o
l
t
a
g
e
s
w
i
t
h
i
n
the
sa
me
pha
s
e
a
rm
a
re
e
qua
l
(w
hich
i
s
poss
i
b
l
e
i
f
t
he
c
a
p
ac
it
o
r
vol
tag
e
b
ala
n
c
e
c
o
n
t
r
ol
i
n
F
i
g
u
re
.
2-4
a
r
e
em
plo
y
e
d
),
a
s show
n
as
(
41)
wh
ere
Ck
p
V
a
n
d
Ck
n
V
a
r
e
t
h
e
s
u
m
o
f
S
M
c
a
p
a
c
i
t
o
r
s
v
o
l
t
a
g
e
s
i
n
t
h
e
u
p
p
e
r
a
r
m
a
n
d
l
ow
er
a
rm
o
f
t
h
e
pha
se-
k
l
eg,
dc
V
i
s
the
to
ta
l
SM
c
apa
c
i
t
o
rs
v
o
lta
ge
m
ea
sur
e
d
w
h
ic
h
i
s
e
q
u
a
l
t
o
t
h
e
stea
d
y
-sta
te
v
al
ue
o
f
Ck
p
V
and
Ck
n
V
.
N
i
s
t
h
e
n
u
mbe
r
o
f
the
SM
i
n
t
h
e
u
pper
an
d
lower
ar
m,
Ck
p
V
a
nd
Ck
p
V
a
r
e
t
h
e
vol
ta
ge
c
han
g
e
due
t
o
d
i
s
t
ur
ba
nce
s
i
n
t
h
e
up
p
e
r
and
low
e
r
ar
ms
r
espe
ct
ive
l
y.
T
he
c
irc
u
l
a
ti
ng
c
urr
e
n
t
c
a
n
b
e
in
f
orm
of
a
D
C
com
p
o
n
e
n
t
dc
I
a
nd
h
ar
mon
i
c
co
m
p
one
n
t
s
n
i
is gi
ven
as
(
42)
F
o
r
ty
pica
l
S
T
A
T
CO
M
a
p
pli
c
a
tio
ns,
t
h
e
D
C
c
ompo
ne
n
t
dc
I
i
s
zer
o
as
n
o
D
C
v
o
l
ta
ge
s
ource
i
s
con
n
ec
ted.
T
h
e
h
a
r
monic
c
o
mpone
n
t
s
n
i
are
the
n
t
h
order
ha
rmonic
o
f
t
he
c
i
r
cu
lat
i
n
g
c
u
rrent.
The
out
p
ut
A
C
vo
lta
ge
Ck
V
a
nd th
e
o
u
t
p
ut A
C c
u
rre
nt
k
i
i
n t
h
e
pha
se–
k
leg
a
re
gi
v
en
a
s
(
43)
(
44)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN: 2088-
8694
Int J
P
o
w
El
e
c
&
D
ri S
yst
,
V
ol.
10,
N
o.
3
, S
e
p
2
0
1
9
:
148
3
– 1
499
1
492
M
is
t
h
e
v
o
l
t
a
ge
m
odu
la
tio
n
in
dex,
0
I
i
s
t
h
e
pe
a
k
a
m
p
lit
ude
o
utp
u
t
A
C
c
ur
rent
s,
i
s
t
h
e
angle
di
ffe
re
nce
be
tw
e
e
n
t
h
e
M
M
C
o
utp
u
t A
C
c
urre
nt w
it
h t
h
e P
C
C
vol
ta
ge.
Base
d
on (
43)
-(44),
the
insta
n
ta
ne
o
u
s ac
tive
pow
er
of
the
u
pper
a
nd
l
o
w
e
r
a
r
m
a
r
e
:
(
45)
S
u
mm
ation
of
pow
ers
i
n
(
45)
f
or
t
he
t
hre
e
-p
ha
se
s
is
s
up
po
se
d
t
o
b
e
e
q
u
a
l
t
o
t
h
e
p
o
w
e
r
(
PQ
)
at
the gr
i
d
i
n
t
e
r
fa
c
e
.
These
ar
e
relate
d
as
(
46)
The
corr
espo
n
d
i
n
g
acti
v
e
ene
r
gy
stor
ed i
n
t
h
e
upper
an
d
l
o
w
e
r
a
rm
s of
p
h
a
se-
k
base
d
o
n (
45) a
re
a
s foll
ow
s:
(
47)
(
48)
Ba
se
d on
(4
5),
(47),
a
nd (48)
the follow
i
ng can be as
(49)
(
50)
U
s
in
g (
4
5), (49
) and (
5
0), t
he
vo
l
tage
d
i
s
t
u
rb
anc
e
s in
the
a
r
m
s
wh
i
c
h
c
ont
ri
bu
t
e
t
o
t
h
e
MMC
volt
a
g
e
varia
t
i
o
ns c
an be
a
s
(
51)
Th
ere
f
o
r
e
,
c
o
n
sid
e
ri
ng
t
h
e
v
ari
a
ti
on
s
in
(
51
),
t
h
e
r
e
a
l
out
pu
t
vol
t
a
g
e
s
of
t
h
e
upp
e
r
a
nd
l
o
w
e
r
a
rms
ca
n be
a
dj
us
t
e
d to
:
Evaluation Warning : The document was created with Spire.PDF for Python.