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.
1781~
17
96
ISSN: 2088-
8694,
DOI
:
10.11591
/ijpeds.
v10.
i
4.pp1781-1796
1781
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
Pow
er compe
n
sation for vecto
r-based current control of a
modular multilevel conv
erter (MMC) ba
sed
STATCOM
A.
U
.
L
a
wa
n
1
,
S
. Babani
2
,
A
.
A
b
d
ul
k
a
ri
m
3
,
G
.
S
.
S
h
e
h
u
4
,
Y.
J
i
b
ril M,
5
I
. S
.
M
a
du
gu
6
1
Dep
a
rt
m
e
nt
o
f E
l
ectrical & El
e
ct
ro
nic
En
gin
e
e
r
in
g,
Un
i
ve
rsity o
f
N
o
tti
ngh
am, Malay
si
a.
2
Dep
a
rt
m
e
nt of E
l
ect
rical Eng
ineerin
g, Bay
ero U
n
iv
eris
ty Kan
o
,
1
0 Gwarzo
Road
, Nigeri
a
3,
4,
5
D
e
p
a
rtm
e
n
t
o
f
El
ectri
cal E
ngineeri
n
g
,
Ah
m
ad
u
Bell
o
U
n
i
v
eristy
Z
ari
a
, Nigeri
a.
6
D
e
p
a
rtm
e
n
t
o
f El
ectri
cal E
ng
in
eeri
n
g
, Kan
o U
n
i
v
ers
i
ty
of S
c
ienc
e
and
Techn
o
l
o
g
y
, Nig
eria
Art
i
cl
e In
fo
ABSTRACT
A
r
tic
le hist
o
r
y
:
R
e
ce
i
v
e
d
Mar
21,
2
0
1
9
Re
vise
d A
p
r
21,
201
9
Ac
ce
p
t
ed
J
un
2
0
,
2
019
T
h
i
s
p
a
p
e
r
s
t
u
d
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e
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t
h
e
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i
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a
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wh
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lo
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d
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o
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its
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a
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es
po
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h
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cti
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current
a
pp
lic
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t
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ns
o
f
st
ati
c
c
o
m
p
n
es
ato
r
s
(S
TA
T
C
O
M
s).
T
h
e
current
c
o
n
t
r
ol
i
s
mo
s
t
ly
a
c
h
i
e
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us
in
g
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p
o
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l
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integral
(
PI)
co
ntro
ll
ers
becau
se
o
f
th
e
adv
antage
o
f
thei
r
go
od
t
rack
in
g
a
n
d
sm
all
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a
riation
s
.
Ho
wever,
d
u
e
t
o
t
h
e
d
e
pen
d
e
n
c
y
of
P
I
con
t
ro
llers
on
t
he
m
od
ul
ar
m
u
l
t
i
l
e
vel
co
nv
ert
e
r
(MM
C
)
s
y
st
em
d
y
na
mic
s
,
p
e
rform
a
nc
e
va
ria
t
io
ns
a
rise
d
ur
in
g
ste
a
d
y-sta
t
e
S
T
ATCO
M
no
n-id
eal
op
erati
ons
.
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h
i
s
p
aper p
resen
t
s a
n
i
m
p
rov
e
d MMC based
S
TATCO
M
c
ont
ro
l
with
a
d
-q
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om
pe
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s
at
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al
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rit
h
m
ad
ded
t
o
t
he
v
ect
or
b
ased
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ur
re
nt
c
on
tro
l
.
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e
alg
o
rit
h
m
i
s
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ed
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o
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a
ck
le
t
he
e
ff
ects
of
t
he
d
yn
amics
o
f
t
h
e
M
M
C
and
th
e
S
T
ATCOM
ideal
v
ari
a
ti
on
s
wit
hou
t
the
use
of
a
ny
ad
di
tio
nal
con
t
ro
ller.
Th
e
co
ntro
l
is
achi
e
ved
by
p
rovi
d
i
ng
a
p
ow
er
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om
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sation
in
t
he
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hi
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nj
ec
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s
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u
rrent
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at
t
h
e
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n
put
o
f
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on
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ove
t
h
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t
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at
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perf
o
r
m
a
n
ce
of
t
he
S
TAT
C
OM
c
o
n
trol.
K
eyw
ord
s
:
d-q p
o
w
e
r
algo
rithm
DSTA
TC
OM
P
o
w
e
r qua
lit
y (
P
Q
)
Un
iv
ersal
po
wer
q
u
a
li
t
y
conditioner (UPQC)
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
,
D
e
pa
rtem
ent o
f
E
lectr
i
c
a
l E
n
gi
nee
r
in
g,
U
n
i
v
ersi
ty o
f
N
o
tti
n
gham
, Ma
l
ays
i
a Cam
pus,
435
0
0
S
eme
n
y
i
h.
Mala
ysi
a
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
P
o
w
e
r
quali
ty
can
b
e
a
f
fe
cte
d
b
y
ma
ny
t
ype
s
of
d
i
s
tur
b
an
c
e
s
w
h
i
ch
c
a
n
a
ffec
t
o
r
p
re
ci
sel
y
r
e
duc
e
gri
d
pow
er
p
e
r
form
anc
e
[
1].
D
e
pe
nd
i
n
g
o
n
t
he
i
ss
ues
ar
i
s
in
g
suc
h
as
d
ist
o
rt
io
ns,
vo
l
t
a
g
e
t
r
an
sie
n
ts,
sags,
fl
ic
ker
s
,
sw
e
l
ls
, un
b
ala
n
c
e
s
a
m
ong
vo
lta
ge
s,
a
n
d
in
t
e
rrup
t
i
ons.
C
ustom
powe
r
dev
ices
(CPDs) t
e
ch
no
l
o
g
y
has
bee
n
u
s
e
d
t
o
m
it
i
g
at
e
p
o
w
e
r
p
r
o
b
le
ms
a
n
d
i
mprove
pow
er
qua
l
i
t
y
(
P
Q
)
[
1-
3].
S
o
me
o
f
the
s
e
de
vices
s
uch
as
the
u
n
i
v
e
r¬sa
l
p
o
w
e
r
q
u
al
i
t
y
c
ond
i
t
i
one
r
(U
P
Q
C),
stat
ic
c
om
pe
ns
at
o
r
(
S
T
ATCOM
)
,
an
d
dyn
ami
c
v
olt
a
ge
restore
r
(D
V
R
)
c
a
n be i
m
p
le
m
e
nted in v
o
l
t
a
ge
s
o
u
rc
e mul
t
i
l
e
v
e
l
con
v
e
rte
r
s
(V
S
M
Cs)
t
o
pol
ogi
e
s
[
4
-
6].
S
h
u
n
t
dev
i
ce
s
suc
h
a
s
S
T
A
T
CO
M
w
h
i
c
h
is
a
pp
lie
d
in
d
i
s
tri
b
ut
ion
gri
d
s
c
a
l
l
ed
d
i
s
tri
b
u
t
i
o
n
sta
tic
c
om
p
e
nsa
t
o
r
(DS
T
ATC
O
M)
e
xists,
i
t’s
m
a
inly
f
or
r
ea
ct
i
v
e
p
o
wer
c
o
m
p
ensa
t
i
o
n
,
f
lic
k
e
r
m
i
t
i
ga
t
i
o
n
,
v
o
lta
ge
r
eg
ula
t
io
n
,
harm
on
ic filte
r
i
n
g
,
a
n
d l
o
a
d
v
ol
ta
ge
ba
l
a
n
c
i
n
g
.
The
r
e are
se
rie
s c
o
nn
e
c
t
io
n dev
i
ce
s
[
7],
ty
pi
c
a
l
one
is dy
nam
i
c
vo
lta
ge
r
e
s
tore
r
(D
V
R
)
commonl
y
used
f
o
r
s
t
a
b
i
l
i
za
t
i
o
n
o
f
vo
l
t
a
g
e
f
o
r
c
r
i
t
i
c
a
l
e
n
d
u
s
e
r
s
.
D
V
R
n
o
r
m
a
l
l
y
com
e
s
w
ith
a
n
ene
r
gy
fac
i
l
i
t
y
for
st
orage
a
nd
s
h
ort
v
o
lta
ge
-d
ip
c
on
t
r
o
l
o
r
shor
t
l
o
ss
o
f
a
g
e
n
e
r
at
io
n
pla
n
t
f
r
o
m
t
h
e
p
o
w
er
n
et
wo
rk
s
o
u
r
ce
[
7
]
.
Al
so
,
t
h
ere
a
r
e
co
mbi
n
ed
s
hu
nt
a
n
d
se
ries
c
o
nnec
t
io
n
de
v
i
c
e
s
s
u
c
h
as
u
n
i
v
e
r
¬
s
a
l
p
o
w
e
r
q
u
a
l
i
t
y
c
o
n
d
i
t
i
o
n
e
r
(
U
P
Q
C
)
[
8
,
9
]
t
h
a
t
c
a
n
b
e
e
m
p
lo
yed.
E
lec
t
r
i
c
d
r
ives
s
uc
h
as
d
ri
v
i
n
g
pump
s
,
hi
g
h
-p
ow
e
r
c
ons
umi
ng
b
l
ow
e
r
s,et
c
.
n
eed
a
h
i
g
h
le
ve
l
of
e
l
e
c
t
ric
pow
er
c
on
ve
rsio
n
for
th
eir
opera
tio
n.
A
s
the
p
o
w
e
r
le
ve
l
r
eac
hes
ma
ny
m
e
ga
w
a
tts,
t
h
e
i
r
o
p
e
r¬a
t
ions
a
t
ele
c
t
rica
l
m
e
di
um
v
ol
t
a
ge
(
MV
)
up
to
8
k
ilo
vo
l
t
(
kV
)
has
bec
o
m
e
a
dva
n
t
a
g
e
ous
o
p
t
i
o
n.
H
ow
e
v
er,
ot
he
r
i
ssue
s
t
ha
t
ar
e
o
f
l
ess
im
por
tanc
e
for
low
-
p
o
w
e
r
drive
s
b
ec
om
e
is
sues
i
n
m
e
diu
m
-vol
t
a
ge
ope
r
a
tions
[
1
0
,
11].
F
o
r
i
n
s
t
a
n
c
e
,
a
t
h
i
g
h
po
wer
a
n
d
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN: 2088-
8694
I
nt
J
P
ow
Elec
& Dr
i
S
y
st V
ol.
10,
N
o.
4
, Dec
201
9 :
1
7
8
1
– 1
796
1
782
h
i
g
h
-
vo
l
t
a
g
e
l
e
v
el
s,
t
e
c
h
n
i
cal
i
ssu
es
l
i
k
e,
b
i
d
i¬
re
ct
io
n
a
l
p
o
we
r
ex
ch
a
nge
c
apa
b
i
lit
y
or
e
l
e
c
t
rom
a
gne
t
i
c
brak
in
g
i
n
e
lec
t
ric
dr
ive
sy
ste
m
i
s
nee
d
ed
w
hen
a
la
rge
qu
ant
i
t
y
of
k
i
n
et
i
c
pow
er
i
s
pla
c
e
d
i
n
t
h
e
rota
t
i
o
n
ar
y
b
odi
es.
Th
e
s
e
te
c
h
ni
cal
i
ssu
e
s
ca
n
b
e
m
i
t
ig
ate
d
b
y
t
h
e
u
s
e
of
m
ult
i
l
e
v
el
c
o
nver
t
¬ers
due
t
o
t
h
eir
ca
pa
bi
li
ty
t
o
pro
duce
less
d
i
s
t
or
t
e
d
c
u
rr
ents
a
n
d
s
m
a
l
l
s
tep
p
e
d
v
o
l
ta
ge
w
ave
f
orm
s
w
it
h
a
fa
ster
s
w
i
t
c
hin
g
.
M
u
l
tile
ve
l
in
verter
s
ha
ve
a
pp
l
i
ca
ti
o
n
s
w
h
ic
h
i
n
c
l
u
d
e
i
n
f
lex
i
b
l
e
A
.
C
trans
m
i
ss
ion
syst
ems
(FACTs
)
.
T
hese
a
pp
lic
ati
ons
are
p
o
ss
ib
le
d
u
e
t
o
m
u
lt
ile
v
e
l
co
nv
erte
rs’
po
w
e
r
sw
it
c
h
e
s
b
a
l
a
n
c
i
n
g
a
n
d
d
is
trib
u
t
e
d
v
ol
ta
ge
s
tress
tha
t
a
re
share
d
a
m
o
n
g
t
he
p
ow
e
r
s
w
i
t
c
he
s.
T
h
e
m
u
l
t
i
l
e
ve
l
na
t
u
r
e
s
i
g
n
i
fie
s
pr
od
uc
i
n
g
ne
ar
ly
s
te
ppe
d
s
i
n
u
s
o
ida
l
o
u
t
pu
t
AC
v
olta
ge
w
a
v
efor
ms
w
ith
r
educe
d
h
arm
o
n
i
cs.
The
i
r
p
r
i
n
c
i
p
l
es
a
re
b
ased
o
n
gene
r
a
ti
n
g
s
ma
ll
casc
a
d
e
vo
lta
ge
s
te
ps
f
or
b
e
t
te
r
p
o
w
er
qua
lit
y,
t
hus,
o
p
e
r
at
in
g
at
a
r
e
duce
d
s
w
itch
i
ng
fre
que
nc
y
w
i
t
h
r
e
duc
e
d
sw
it
c
h
i
n
g
losse
s,
sm
a
ll vo
l
t
a
g
e r
i
se
cha
nge
a
nd be
tter
ele
c
tr
om
agne
t
i
c
compa
t
ibi
l
ity
[
12].
Th
ere
a
r
e
many
t
i
m
e
d
o
mai
n
b
as
e
d
c
on
t
r
ol
s
c
h
emes
s
u
c
h
a
s
d
i
r
ec
t
a
n
d
i
n
-d
ire
c
t
c
o
nt
rol,
pow
e
r
syn
c
hro
n
i
za
ti
o
n
,
and
d-q
vect
or
c
on
tro
l
m
e
t
h
o
ds.
A
m
ong
the
s
e,
m
ost
co
m
m
only
em
p
l
o
y
e
d
a
re
d
irec
t
p-q
a
n
d
d-q
vec
t
o
r
c
on
tro
l
m
eth
ods
i
n
st
ud
y
i
ng
t
h
e
pe
rform
a
n
c
e
o
f
S
TA
TC
OMs
an
d
VS
Is.
S
T
ATCOM
appl
icat
i
o
n
s
for
pow
er
f
ac
tor
sup
por
t
a
n
d
sta
b
ili
ty
s
t
u
die
s
h
a
v
e
bee
n
e
mplo
y
e
d
i
n
[
13-
17]
b
a
s
e
d
o
n
d
i
ffe
ren
t
c
on
t
r
ol
schem
e
s.
G
rid
in
te
gra
t
i
o
n
a
n
d
V
A
R
s
up
p
o
rt
o
f
MM
C
i
n
v
e
rter
h
as
b
e
e
n
a
cha
l
le
nge
[
1
8
]
.
L
a
w
an
e
t
a
l
.
[1
9]
and
M
u
sta
p
h
a
e
t
al.[2
0
]
ha
v
e
e
m
p
l
o
ye
d
t
h
e
dire
c
t
(
p-q)
a
ppr
oa
ch
t
o
i
m
prove
t
he
g
rid
P
C
C
p
o
w
e
r
fact
or
s
us
i
n
g
M
M
C
b
ase
d
S
TA
TCO
M
a
p
p
l
ica
t
i
ons
.
The
o
ret
i
cal
l
y
,
no
p
o
w
e
r
l
os
s
i
n
a
n
idea
l
S
T
ATC
O
M
system
.
R
e
su
l
t
i
n
g
i
n
n
o a
c
t
iv
e
po
we
r tra
n
sf
er
t
h
r
o
ugh i
t
.
In
th
i
s s
t
at
e
t
h
e
STATCO
M
i
s lo
ssl
e
ss,
i
t
s
out
put
AC vo
lt
a
g
e
w
ill be i
n p
h
as
e w
ith t
h
e
g
r
i
d A
C
v
o
l
t
a
ge.
B
u
t t
h
e d-
ax
is
c
ur
re
nt c
om
pon
e
n
ts co
n
t
ro
l is im
por
tan
t
c
ons
ider
i
n
g
the
fac
t
s
tha
t
dur
ing
tra
n
s
i
e
n
t
opera
t
i
o
n
s
of
t
he
S
TATCO
M
,
d
u
e
to
c
har
g
i
n
g
an
d
d
i
schar
g
in
g
of
t
he
ca
paci
t
o
rs,
the
r
e
is
a
l
w
a
y
s
a
varia
t
i
o
n
i
n
t
he
a
ct
iv
e
p
o
wer
ou
t
pu
t
of
t
he
S
TA
TCO
M
s.
I
n
othe
r
w
o
rds
t
h
ere
i
s
ac
t
i
ve
pow
e
r
t
ra
nsfer
be
t
w
een
t
he
g
ri
d
a
n
d
the
S
T
A
T
CO
M.
I
n
pr
ac
tica
l
,
a
sma
l
l
am
oun
t
o
f
p
ow
e
r
l
oss
is
assoc
i
ate
d
w
i
t
h
t
he
p
roce
ss
[
2
1
]
.
S
pe
cifi
ca
l
l
y
in
t
he
MMC
b
ase
d
I
nve
r
t
ers
an
d
S
T
A
T
CO
Ms
s
t
r
uc
tural
t
o
p
o
l
og
i
e
s.
E
ach
s
ub
m
od
ul
e
s
(
S
M
)
h
a
s
c
e
l
l
c
a
p
aci
to
r
with
a
n
a
c
t
i
ve
e
n
e
rgy
s
t
ore
d
i
n
it
o
r
dc-l
i
n
k
v
o
l
t
a
g
e
source
s
t
r
ans
f
e
r
red
to
t
hem
.
W
i
t
ho
u
t
t
he
p
rope
r
ca
pac
i
t
o
rs
v
o
l
t
a
g
e
ba
l
a
n
c
in
g
a
nd
e
n
erg
y
t
ra
n
s
fer,
t
he
M
M
C
base
d
i
nver
t
e
r
s
o
r
S
TA
TCO
M
s
w
ill
a
l
w
a
y
s
h
ave
s
o
m
e
v
aria
tio
ns
i
n
t
he
ir
o
u
t
pu
t
s
pow
er
,
low
l
i
fe
t
im
e
of
t
he
ce
l
l
s
c
a
pa
c
i
t
o
r
s
d
ue
t
o
vo
lta
ge
u
n
b
ala
n
c
i
ng
an
d
str
e
ss
e
t
c.
[22-
2
4
]
.
O
t
h
e
r
v
a
r
iat
i
ons
a
nd
/or
osc
i
ll
a
t
ions
c
an
occ
u
r
d
u
e
t
o
s
w
i
t
c
hin
g
o
r
from
no
n
–ide
a
l
n
at
ure
c
a
pac
i
t
o
r
s
a
n
d
b
uffe
r
induc
t
o
r
s
[
2
5
].
A
good
t
u
ned
P
I
con
t
ro
l
l
er
c
ou
l
d
yie
l
d
a
re
s
ponse
w
i
t
h
a
z
e
r
o or m
i
n
im
um
s
te
ady
s
tate
er
r
or w
i
t
hi
n a
ver
y
sm
a
ll
r
ise
t
i
m
e
.
B
u
t
w
i
t
h
h
ig
h
set
t
l
i
ng
tim
e
a
nd
ove
rs
ho
o
t
s
(i.e
.
high
e
r
o
s
c
illa
t
i
o
n
p
e
r
io
d
a
nd
a
lo
n
g
er
r
espo
nse
)
[
26-
36]
a
nd
a
b
it
p
o
o
r
d
y
n
a
m
ics
and
t
r
a
n
sie
n
t
r
e
sp
onse
t
y
pica
l
e
x
a
m
ple
i
s
i
n
[3
7].
A
n
o
t
her
op
t
i
o
n
i
s
t
h
e
use
of
P
-
c
o
nt
roll
e
r
s,
e
ven
t
hou
gh
i
s
simp
l
e
r
in
i
mp
l
e
men
t
at
ion
,
i
t
ha
s
a
p
rob
l
em
o
f
a
n
i
ncr
ease
d
over
s
h
oo
ts
c
om
par
e
d
to
P
I
con
t
ro
ll
e
r
s.
T
hi
s
i
n
cr
ease
d
o
versh
o
o
t
cou
l
d
be
s
e
e
n
i
n
[3
7]
w
here
it
w
a
s
e
m
pl
o
y
e
d
t
o
s
y
n
c
h
r
o
n
i
ze
t
he
g
r
id
volt
a
g
e
a
nd
S
TATCO
M
o
utp
u
t
vo
lt
a
g
e
u
s
i
n
g
d
eco
upl
ed
c
u
rren
t
c
o
n
tr
ol. T
h
e vec
t
or con
tro
l
a
l
s
o e
m
pl
o
y
s
the
c
a
sca
d
e
c
o
n
t
ro
l
pri
n
ci
p
l
e.
I
n
MMC
vol
ta
ge
s
ourc
e
i
n
v
er
t
e
rs,
t
h
e
s
w
i
t
c
hi
ng
fre
qu
en
cy
i
s
c
r
u
c
i
a
l
an
d
restric
t
s
how
q
uic
k
t
he
V
S
I
can
r
esp
o
nd
t
o
c
han
g
es
f
rom
one
o
pe
ra
t
i
ng
con
d
iti
on
t
o
a
n
o
t
her,
t
here
fo
re
t
h
e
op
tim
al
a
n
d
a
c
h
ie
va
bl
e
in
ne
r
c
u
rre
nt
c
ont
rol
ba
nd
w
h
ic
h
is
a
l
s
o
r
estr
icte
d
b
y
t
he
e
m
p
lo
ye
d
sw
itc
hin
g
fre
que
nc
y.
T
h
a
t
is,
t
h
e
c
o
n
t
rol
l
er
b
a
n
dw
i
d
th
s
h
o
w
s
h
ow
acc
urate
a
nd
fast
a
c
o
n
tro
l
s
i
gna
l
c
o
u
l
d
cha
nge
i
n
t
i
me
,
h
e
n
c
e
ho
w
ac
cu
ra
t
e
a
n
d
f
a
s
t
th
e
MMC
s
y
s
t
e
m
re
sponse
c
oul
d
b
e
[3
8].
The
M
M
C
base
d
in
verte
r
b
ase
d
o
n
t
h
e
c
a
s
c
a
d
e
c
o
n
t
r
o
l
h
a
s
t
h
e
c
u
r
r
e
n
t
v
e
c
t
o
r
c
o
n
t
r
o
l
a
s
t
h
e
i
n
ner
c
u
rre
nt
c
ontro
l
(I
CC),
a
nd
t
h
e
o
u
ter
c
o
n
t
r
o
l
tha
t
c
om
prises
t
he
a
c
t
ive
an
d
re
ac
t
i
ve
pow
e
r
c
ontro
l.
O
n
t
h
e
o
t
her
h
a
nd,
t
he
a
ct
i
v
e
p
o
w
er
c
ontro
l
is
u
su
a
l
l
y
t
h
e
a
c
t
i
v
e
p
o
w
e
r
r
e
f
eren
ce
o
r
vo
lt
a
g
e
ref
e
ren
c
e
b
a
sed
co
nt
rol
.
The
re
ac
t
i
ve
p
ow
er
c
o
n
t
ro
l
is
e
m
p
loy
e
d
in
term
s
of
a
r
efe
r
enc
e
r
ea
ct
ive
pow
e
r
o
r
re
fere
nce
A
C
v
ol
ta
ge
t
o
pro
v
i
de
t
he
d
es
ire
d
i
nne
r
c
u
rr
ent.
T
he
adva
n
t
a
g
e
o
f
t
he
c
ascade
d
s
c
h
em
es
i
s
the
i
n
c
r
ea
se
d
de
gre
e
of
f
r
ee
dom
f
or
t
he
c
o
n
tro
l
o
f
the
s
y
ste
m
b
an
d
w
i
d
t
h
due
t
o
t
h
e
co
ntro
lle
d
tim
e
sepa
rati
o
n
of
t
he
l
oo
ps.
The
c
o
n
tr
o
l
ler
s
o
f
the
i
n
n
e
r
cur
r
ent
co
n
t
ro
l
ar
e
usu
a
l
l
y
used i
n
V
S
I
are propo
r
t
i
ona
l-i
n
tegra
l
–der
i
va
t
i
ve (
P
I
D
)
,
hyste
resi
s and
ram
p
t
im
e
curre
nt co
n
tr
ol
lers.
The
P
I
D
and
hys
ter
e
s
i
s
are
t
h
e
m
o
st
c
om
m
on
[3
8].
The
hyster
e
s
i
s
c
urre
nt
c
on
tro
l
i
s
the
fa
ste
s
t
con
t
ro
l
i
n
t
e
r
m
s
o
f
d
ynam
i
c
respo
n
se
h
o
w
ever
g
i
v
es
m
or
e
di
s
t
ort
i
ons
d
ue
t
o
th
e
l
o
w
or
de
r
h
a
rm
oni
c
s
,
varia
b
l
e
s
w
itc
h
i
n
g
f
r
e
que
nc
y
an
d
e
f
fec
t
o
f
dea
d
tim
e
[
3
7
-
38].
I
n
the
ot
h
e
r
w
o
r
d
,
P
I
-c
urrent
c
o
n
tr
oller
s
h
a
v
e
ma
int
a
ine
d
s
w
itc
h
i
n
g
fre
que
nc
y
b
u
t
t
h
a
t
a
t
the
e
x
pe
nse
of
a
s
lo
wer
response.
The
P
I-controller
h
as
its
dy
nam
i
c
s
de
p
e
n
d
i
ng
o
n t
h
e dyna
mic
o
f
t
he
s
ys
tem
he
nc
e ha
s m
o
re
co
ntro
lle
d c
o
up
lin
g e
ffe
c
t
tha
t
give
s
be
tte
r
di
st
ort
i
o
n
.
B
u
t
t
h
e
h
y
s
t
e
r
esis
c
on
tro
l
d
oes
n
o
t
depe
n
d
o
n
the
d
y
n
a
m
i
c
s
o
f
t
h
e
s
y
s
t
e
m
p
l
a
n
t
s
,
h
e
n
c
e
h
a
s
h
i
g
h
di
st
ort
i
o
n
s
[3
7
-
38]
c
om
pare
d
to
t
he
ir
P
I-
contr
o
l
c
o
u
n
t
er
pa
rts.
Thi
s
p
r
o
pose
d
d
,q-
c
om
pen
s
at
i
on
a
l
go
rit
h
m
in
t
r
od
uc
es
a
t
r
a
ns
ien
t
s
up
por
t
s
b
a
s
ed
o
n
t
h
e
c
once
p
ts
o
f
a
tra
n
s
i
en
t
im
pa
cts
a
nd
a
p
pl
i
e
d
t
h
e
c
onc
e
p
ts
o
f
t
h
e
S
T
A
T
CO
M
pa
ra
me
ter
var
i
ations
t
o
pro
duce
a
n
a
lg
ori
t
hm
s
t
h
a
t
e
n
h
a
nce
s
t
he
c
o
n
t
r
ol
o
f
t
h
e
S
T
A
T
CO
M.
T
he
in
t
r
od
uc
ed
d
,
q
-
t
ra
n
s
ien
t
c
om
pe
nsa
t
or se
r
vi
ng
onl
y
d
u
ri
ng t
h
e
t
r
a
n
si
en
t
op
era
t
i
o
n
s
.
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
Po
we
r c
o
mp
ensat
i
on
f
o
r
v
ecto
r
-ba
s
e
d
c
u
r
r
e
n
t
c
ont
ro
l
of
a
mo
du
la
r
mul
t
i
l
ev
e
l
…
(A.
U.
La
wan
)
1
783
2.
VECT
OR
C
O
N
TROL
F
OR
MODULAR
MULT
I
LE
VEL CONVERTER (
MM
C) I
NVERTE
R
I
n
t
he
v
ec
tor
c
o
n
t
ro
l
sc
hem
e
d
ep
ic
ted
i
n
F
igur
e
1,
t
he
V
S
C
b
a
s
e
d
MMC
s
ys
tem
in
t
he
t
h
r
ee
pha
se
vo
lta
ge
s
a
n
d
c
u
rrents
are
tra
n
sf
orm
e
d
in
to
a
d
ire
c
t
-qua
dr
at
u
r
e
(d-q)
r
o
t
a
ti
n
g
f
ra
me
.
F
r
om
F
igure
1
a
nd
F
i
g
u
re
2
w
h
i
c
h
pr
ese
n
t
s
t
he
v
ec
t
o
r
c
o
n
t
r
o
l
sc
he
me
,
appl
y
i
n
g
K
irchh
o
f
f
’
s
v
o
lta
ge
p
r
i
nc
i
p
l
e
s,
a
nd
ba
se
d
on
t
h
e
Lap
l
ac
e
trans
f
or
m,
t
he
r
e
l
a
t
i
ons
h
i
p
be
tw
ee
n
the
grid
A
C
vo
lta
ge
fdq
V
a
n
d
t
h
e
M
M
C
s
y
s
t
e
m
v
o
l
t
a
g
e
cd
q
V
o
f
F
i
gure
1 c
a
n
b
e
w
r
i
tten
as fo
llow
s
[
39-4
1
]:
T
d
q
c
dq
f
d
q
T
dq
T
d
q
f
ff
sL
I
V
V
R
I
j
L
I
(1)
W
h
ere:
T
R
is the
e
ffec
t
i
ve
r
esistance
be
t
w
e
e
n the
M
M
C sys
t
em
a
nd t
h
e gr
id
T
L
i
s t
h
e
e
f
fe
cti
v
e i
n
du
c
t
an
ce
b
e
t
we
en
t
h
e
M
MC
sy
s
t
e
m a
n
d
th
e
g
r
i
d.
f
dq
V
i
s
t
h
e
P
C
C ve
ctor v
ol
ta
ge.
cd
q
V
is t
he
MMC o
u
t
p
u
t
v
ec
tor
for
m
o
f the
vo
lta
ge
dq
f
I
is the
o
u
tp
ut v
e
c
tor form
of
the
c
u
rre
n
t
a
t t
h
e
PCC
and
is t
he
fu
n
d
am
enta
l
line
fr
eq
ue
nc
y.
+
+
‐
DC
DC
‐
+
‐ +
‐
+
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
cwp
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
GRI
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
3
SM
N
3
SM
N
3
SM
N
cwn
V
cu
n
V
cv
n
V
un
i
vn
i
wn
i
in
V
SM
2
Vdc
2
Vdc
SM
ip
V
PC
C
F
i
gure
1.
T
he thre
e
p
ha
se i
nt
e
r
na
l circ
u
itry
o
f
a
MM
C
base
d
S
TA
TCO
M
c
on
nec
t
e
d
t
o t
h
e thre
e
phase
g
rid
sys
t
em
.
I
f
a
c
ons
t
a
n
t
p
ow
er
g
r
i
d
sys
t
e
m
i
s
a
s
s
u
m
e
d,
t
he
n
th
e
resu
lta
n
t
syst
e
m
c
oul
d
be
l
i
n
ea
r
a
n
d
a
u
ton
o
m
ou
s.
H
en
c
e
th
e
sy
st
em
i
s
ass
u
me
d
t
o
b
e
an
i
nd
ep
end
e
n
t
v
a
ria
b
le
t
ha
t
i
s
i
n
d
e
p
en
de
nt
o
f
tim
e.
Thi
s
assum
p
ti
o
n
i
s
va
li
d
s
i
n
c
e
ver
y
s
m
a
l
l
d
ev
ia
ti
ons
o
f
fre
q
u
e
n
c
y
ar
e
ex
pe
c
t
e
d
i
n
a
hea
lth
y
and
n
o
rm
al
o
p
e
ra
tion
of
a
g
ri
d
p
o
w
e
r
system
.
The
s
e
sm
all
va
riat
ions
c
a
n
o
n
l
y
hap
p
e
n
e
d
as
g
ri
d
is
e
xp
e
c
t
e
d
t
o
h
av
e
a
cons
t
a
n
t
vo
lta
ge.
F
r
om
F
igure
2
(a
),
a
t
ransfe
r
fu
nc
ti
o
n
f
or
t
he
M
M
C
b
as
e
d
V
I
S
q
-
a
x
i
s
v
o
l
t
a
g
e
c
a
n
b
e
w
r
i
t
t
e
n
a
s
fo
l
l
ow
s
:
2
2
2
2
1
(2
)
()
1
()
fq
T
T
TT
T
cq
T
TT
T
IR
s
L
Rs
L
L
V
R
s
Rs
L
L
Whe
r
e
fq
I
is
t
he
o
ut
p
u
t
A
C
c
u
r
rent
a
t
t
h
e
poi
nt
o
f
c
o
upl
in
g
(P
CC)
i
n
the
q-
a
x
i
s
a
n
d
cq
V
i
s
the
MMC
system
ou
tpu
t
vol
ta
ge
in t
h
e
q-ax
is.
F
r
om
(
2),
the
po
l
e
s
o
f
t
he
g
rid
c
o
n
n
e
c
t
ed
M
MC
s
yste
m
w
i
l
l
b
e
ro
o
t
s
o
f
t
he
s
yst
e
m
c
h
ar
acte
r
ist
i
cs
equa
t
i
o
n
(
de
no
m
i
nat
o
r
of
(
2):
2
2
/
TT
sR
L
,
f
o
r
2,
f
w
h
e
r
e
f
i
s
t
h
e
A
C
n
e
t
w
o
r
k
f
r
e
q
u
e
n
c
y
assum
e
d
50H
z
,
t
hese
p
o
l
es
o
f
the
n
o
n
–
fee
dba
ck
s
ys
t
e
m
w
i
l
l
b
e
c
omple
x
i
n
na
t
u
re,
henc
e
t
h
e
sy
stem
w
ill
be
osc
illa
t
o
r
y
.
In
o
t
h
er
w
or
ds,
the
grid
–
c
o
nn
ecte
d
M
M
C
s
ys
tem
w
i
l
l
h
a
v
e
a
b
ad
d
yn
ami
c
p
e
r
fo
rma
n
ce
a
nd
po
orl
y
dam
pe
d
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN: 2088-
8694
I
nt
J
P
ow
Elec
& Dr
i
S
y
st V
ol.
10,
N
o.
4
, Dec
201
9 :
1
7
8
1
– 1
796
1
784
To
s
o
l
ve
t
hi
s
c
o
n
t
ro
l
pro
b
l
e
m
of
t
he
n
o
n
-fe
e
dbac
k
M
M
C
s
yste
m,
t
he
g
rid
-
con
n
e
c
te
d
M
M
C
c
urrent
a
t
t
h
e
P
C
C
(
i
.
e
.
fd
I
a
n
d
fq
I
)
has
to
b
e
fe
d
bac
k
t
o
t
h
e
d
q
-
c
o
n
tr
ol
l
e
rs
s
o
a
s
t
o
clo
s
e
the
c
u
r
re
n
t
l
oop
s.
T
hi
s
con
t
ro
l
i
m
pro
v
e
me
nt
i
s
show
n
i
n
F
ig
ure
2
(b)
.
T
he
f
e
e
d
b
a
c
k
c
o
n
t
ro
l
l
oops
w
i
l
l
i
m
p
r
ov
e
t
h
e
syst
e
m
r
e
s
p
o
n
s
e
by
s
h
if
tin
g
th
e
po
les
of
t
he
s
ys
t
e
m
ope
n
l
o
op
tra
n
sfer
f
unc
tio
n
o
f
F
i
gure
2
(a
)
tow
a
r
d
s
t
h
e
re
al
a
xi
s,
t
h
e
re
by
pus
hi
ng
t
he
s
ystem
t
o
h
a
v
e
a
crit
ical
l
y
d
am
pe
d
resp
o
n
se[3
2
]
.
A
s
sum
i
ng
t
h
e
d
q-c
o
nt
r
o
ller
to
b
e
a
simp
le
g
a
i
n
con
t
ro
l
l
er
(
P-
c
o
n
t
ro
l
l
e
r
w
i
t
h
()
P
Cs
K
),
t
h
e
n
b
l
o
c
k
d
ia
gram
o
f
F
i
g
u
r
e
2
(b)
can
b
e
in
a
t
r
a
ns
fer
func
tio
n
form
w
i
t
h o
u
te
r
fe
edba
ck l
o
o
p
s g
i
ve
n b
y
:
1
1
(3
)
1
1/
1
P
P
TT
PT
T
T
P
P
TT
K
K
Rs
L
KR
s
L
s
L
K
K
Rs
L
A
ssum
i
ng
the
in
n
e
r
c
u
rre
nt
c
on
t
r
o
lle
r
gai
n
i
s
m
u
ch
m
or
e
h
i
g
h
er
tha
n
t
he
v
a
l
ue
o
f
t
h
e
effect
i
v
e
resista
n
ce
b
et
we
en
t
he
M
M
C
s
yste
m
a
n
d
the
gri
d
(
i
.
e.
P
T
KR
),
t
he
n
the
e
qui
v
a
le
n
t
r
epre
se
nt
ati
on
o
f
t
h
e
gri
d
c
on
nec
t
e
d
MMC
s
ystem
i
n
bl
o
c
k
d
ia
gra
m
w
i
ll
be
a
s
sh
ow
n
in
figure
3
t
herefore
i
t
is
poss
i
ble
to
d
er
ive
a
q-ax
is
c
ur
rent
t
rans
fe
r
func
t
i
on
give
n
b
y
(
4)
a
nd
(5)
.
W
he
re
*
fq
I
i
s
the
grid
–
co
n
n
ec
t
e
d
MM
C
sys
t
em
q
-a
xis
reference
curr
ent
for
the
inne
r
cur
r
ent
con
t
rol
sc
he
me
.The
s
am
e
p
rinci
p
l
e
s
a
p
p
lies
t
o
t
heir
c
orre
spon
di
n
g
d
-
axis current.
*
22
2
2
1
11
(4
)
1
11
1
TT
fq
PP
fq
TT
T
PP
P
T
P
LL
ss
I
KK
I
LL
L
s
KK
K
L
s
K
+
‐
+
‐
‐
+
‐
+
‐
+
‐
‐
+
‐
+
‐
+
‐
+
‐
1
TT
s
LR
1
TT
s
LR
T
L
T
L
T
L
T
L
1
TT
s
LR
1
TT
s
LR
fq
V
fq
V
fq
V
*
cq
V
*
cq
V
*
cd
V
*
cd
V
fd
V
fd
V
fd
V
fd
I
fd
I
fq
I
fq
I
*
fd
I
fd
I
fq
I
*
fq
I
()
Cs
()
Cs
()
a
()
b
F
i
gure
2. Bl
o
c
k
dia
gram
re
p
re
se
n
t
a
t
ion
o
f
t
he
M
M
C
b
a
s
e
d
V
S
I
i
n
a
sta
t
e
spac
e e
qua
ti
o
n
(a
)
w
ith
ou
t fe
edbac
k
con
t
ro
l
l
er
(
b) with
f
ee
d
b
ac
k
con
t
ro
l
l
e
r
*
1
1
(5
)
1
11
T
fq
P
T
TT
TT
fq
P
PP
PP
L
s
I
K
L
LL
LL
I
s
ss
K
KK
KK
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
Po
we
r c
o
mp
ensat
i
on
f
o
r
v
ecto
r
-ba
s
e
d
c
u
r
r
e
n
t
c
ont
ro
l
of
a
mo
du
la
r
mul
t
i
l
ev
e
l
…
(A.
U.
La
wan
)
1
785
+
+
‐
+
+
1
1
T
P
L
s
K
1
1
T
P
L
s
K
T
P
L
K
T
P
L
K
+
f
d
P
V
K
f
q
P
V
K
f
d
I
f
q
I
*
f
d
I
*
f
q
I
F
i
gure
3.
Bloc
k
dia
gra
m
r
epresen
t
at
i
on o
f
t
he
M
MC
b
ase
d
V
SI
in
a
state
spa
c
e
equa
t
i
o
n
w
i
t
h de
co
u
p
l
e
d
in
ner
curr
ent v
e
c
t
or c
ontro
l
l
er
.
The
s
y
s
t
em
o
bta
i
ned
from
t
he
(
5)
i
s
a
f
i
rst
orde
r
s
y
s
t
e
m
,
hence
e
m
p
l
o
y
i
n
g
a
P
-
c
o
n
t
r
o
l
l
e
r
w
i
l
l
impro
v
e
the
s
y
ste
m
r
espo
ns
e
an
d
he
lps
i
n
t
he
s
t
e
ad
y
s
t
ate
by
e
l
i
m
i
n
atin
g
t
h
e
cros
s
c
o
uple
d
i
nte
r
ac
tio
n
betw
ee
n
the
d-a
x
is
a
nd
q
-a
xis
o
f
t
he
s
y
s
tem
.
T
he
c
o
n
t
ro
ller
w
i
l
l
al
so
r
e
duce
t
h
e
e
f
f
ect
o
f
th
e
v
a
ri
at
i
o
n
s
i
n
the
AC
out
p
u
t
vo
lt
a
g
e
at
t
h
e
P
CC[13
]
.
Du
ri
ng
t
h
e
s
t
ead
y
sta
t
e
co
ndi
ti
o
n
(
i.e
.
0
s
)
,
f
o
r
t
h
e
b
l
o
c
k
d
i
a
g
r
a
m
show
n in fi
gur
e
3,
it ca
n gi
ve
(
6) a
s:
*
*
22
2
1
11
1,
,
&
(
6
)
11
1
T
fq
fq
fq
T
PP
fq
fq
P
f
d
P
TT
T
PP
P
L
II
I
L
KK
IV
K
I
K
LL
L
KK
K
Whe
r
e
fq
V
i
s
t
h
e
A
C
v
ol
ta
ge
a
t th
e
P
CC in t
he
q
-axis a
n
d
*
fd
I
is th
e
A
C
c
ur
rent a
t
the P
C
C
in
t
he
d-a
xis.
Al
so
f
ro
m
(6
),
t
h
e
s
ame
p
r
i
n
c
i
pl
e
app
l
i
e
s
to
i
t
s
c
o
rre
spond
ing
d
-
axi
s
q
ua
n
t
i
t
i
e
s
,
wh
ere
t
h
ei
r
g
a
i
n
s
can
be
f
ound
by
j
us
t
s
u
btitling
t
h
e
q-axis
q
uantities
w
i
th
t
he
d-a
x
is
o
nes.
A
na
ly
z
i
n
g
(
6),
if
t
he
t
erm
/1
,
TP
LK
the
MM
C
i
n
ve
rter
o
u
t
p
u
t
d,
q
-
cur
r
ents
w
il
l
fo
llow
t
h
e
refe
re
nce
cur
r
en
ts
(
i.e.
*
fq
fq
II
a
nd
*
f
df
d
II
)
e
xactl
y
,
hen
c
e
th
e
effec
t
s
o
f
t
he
P
CC
A
C
vo
l
t
age
va
ria
t
ions
w
i
l
l
b
e
da
m
p
ed
b
y
t
h
e
c
o
n
t
ro
l
l
e
r
(
i.e
.
/
fq
fq
P
IV
K
).
A
lso
t
h
e
l
a
st
t
e
r
m
of
t
he
(
6)
i
m
p
l
i
es
t
ha
t
t
h
e
li
ttle
v
ari
a
t
i
ons
i
n
t
h
e
d-a
x
i
s
c
u
r
ren
t
r
efe
r
enc
e
ca
nno
t
affec
t
t
he
q-a
x
i
s
curre
nt i
n t
h
e
gri
d
con
ne
c
t
ed
M
M
C
.
Fo
r
a
g
o
o
d
dyn
ami
c
p
erf
o
rma
n
ce
,
i
f
c
arri
er
b
a
s
ed
P
WM
i
s
e
m
p
l
o
y
e
d
(i.
e
.
*
/
fq
T
P
f
d
IL
K
I
),
h
e
n
ce
the
c
r
o
ss
co
u
p
lin
g
effect
s
o
f
t
he
d
-a
x
i
s
a
nd
q-a
x
i
s
a
re
c
a
n
cell
e
d
.
,
t
he
c
lo
sed
–
l
oo
p
ban
d
w
idt
h
,
cl
s
ho
ul
d
no
t
be
g
re
ater
t
ha
n
5
time
s
t
he
s
w
i
t
c
h
i
ng
freque
ncy
,
s
w
A
lso
th
is
s
w
itch
i
ng
f
r
e
que
nc
y
(i.e
.
,
s
w
)
s
u
pp
osed
no
t be
l
ow
e
r
t
h
a
n hal
f
t
he sa
m
pl
i
ng fr
eq
ue
nc
y
,
s
i.e.
(
7)[32]:
(7)
55
(
2
)
sw
s
cl
F
o
r
insta
n
c
e
,
i
f
a
s
w
i
tch
i
ng
f
re
que
nc
y
of
2
.5
k
H
z
o
f
CS
P
W
M
w
h
ic
h
is
1
5.7
0
b
k
ra
d
/
s
(50H
z
=
31
4
rad/s)
i
s
a
p
p
lie
d,
t
he
h
ig
he
st
b
an
d
w
i
d
t
h
of
t
he
s
ystem
,
cl
i
s
e
st
ima
t
ed
a
s
3.1
4kra
d
/s
(
5
00H
z)
.
Then
t
h
e
r
i
s
e
ti
m
e
,
,
r
t
w
h
i
c
h
i
s
d
e
f
i
n
e
d
i
n
a
f
i
r
s
t
o
r
d
e
r
s
y
s
t
e
m
a
s
a
t
i
m
e
t
h
e
s
y
s
t
e
m
o
u
t
p
u
t
g
oes
from
10
to
9
0%
c
a
n
b
e
defi
ne
d
as
(
8)[32]
:
(9)
/
(
8
)
rc
l
tl
n
The
S
T
ATC
O
M
d
e
v
i
ce
i
s
a
s
hu
n
t
d
ev
ice
h
e
nc
e
c
o
n
n
ec
te
d
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ara
l
l
e
l
a
w
i
t
h
t
he
g
ri
d
i
n
t
he
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er
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y
s
t
e
m
v
i
a
a
s
e
r
i
e
s
i
m
p
e
d
a
n
c
e
Z
f
wh
i
c
h
co
nt
ai
ns
a
c
ou
pli
n
g
in
du
ctor
a
n
d
a
n
as
sume
d
n
e
g
l
i
g
i
b
le
s
e
ries
resista
n
ce
a
t
t
h
e
P
C
C[6].
T
h
e
s
h
un
t
gri
d
c
onne
c
t
i
on
is
e
m
p
l
o
ye
d
u
si
ng
t
h
e
M
M
C
t
o
pol
o
g
y
i
s
s
ho
wn
i
n
F
i
gure
1
t
o
d
e
r
ive
t
h
e
pro
p
o
s
ed
c
o
n
t
ro
l
sc
he
me
s.
T
he
b
a
s
ic
f
u
n
c
t
i
on
o
f
t
h
e
s
h
u
n
t
V
A
R
c
om
pe
nsa
tio
n
i
s
t
o
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN: 2088-
8694
I
nt
J
P
ow
Elec
& Dr
i
S
y
st V
ol.
10,
N
o.
4
, Dec
201
9 :
1
7
8
1
– 1
796
1
786
gi
ve
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p
ow
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f
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c
t
o
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c
e
m
ent
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y
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ci
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h
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r
op
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u
nc
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la
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p
l
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pe
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n
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n
c
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r
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m
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o
f
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he
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eq
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er
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ha
t
w
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l
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p
ro
d
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c
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e
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ired
S
TA
T
C
O
M
c
urrent
c
i
t
h
a
t
fl
ow
s
thr
o
u
g
h
t
he
c
ou
p
lin
g
im
peda
nce
f
Z
i
s
de
pen
d
e
n
t
on
t
h
e
d
iffe
renc
e
in
v
o
l
t
a
ge
b
etw
e
e
n
t
he
S
TA
TCO
M
A
C
ou
t
put
v
olt
a
ge
c
V
and
t
h
e
v
o
l
tage
a
t
the
PC
C
f
V
. If
t
h
e
S
T
ATCO
M
is
l
ess i
n
vo
l
tage
t
ha
n gri
d
P
CC
f
V
.
The
S
T
A
T
CO
M
ab
sorb
s
rea
c
ti
ve
pow
er
f
rom
t
h
e
A
C
p
ow
er
s
yste
m.
O
t
he
rw
ise
it
in
j
e
c
t
s
re
act
ive
pow
er
.
F
r
om
F
ig
ure 1, the
cq
i
r
epresents
the
reactiv
e
c
u
rrent of
the
S
T
ATCOM
while
cd
i
i
s t
h
e
act
i
v
e
cu
rren
t
use
d
in
char
gi
n
g
a
nd d
i
sc
har
g
ing
the
tw
o D
C
-li
nk c
a
pac
itor
s
.
(
9
)
fc
ff
c
f
VV
iV
V
Z
Base
d
o
n
t
he
e
qua
t
i
o
n
(
9)
u
si
ng
t
h
e
d
i
a
g
ram
repr
esentat
i
o
n
s
of
F
i
gur
e
1
t
h
e
tra
n
s
f
e
r
f
unc
t
i
o
n
equa
t
i
o
n
s o
f
t
he
S
TA
TCO
M
i
n d,q-
re
p
rese
nt
a
t
i
on
is
g
i
v
e
n
b
y
:
22
2
(1
0)
c
d
fd
fq
fq
f
q
rr
r
ff
f
c
q
f
q
fd
fd
f
d
VV
I
I
I
d
LR
L
LR
L
dt
VV
I
I
I
Whe
r
e
f
L
an
d
f
R
a
r
e
t
h
e
A
.
C
c
o
u
p
l
i
n
g
i
n
d
u
c
t
a
n
c
e
a
n
d
r
e
s
i
s
t
a
n
c
e
r
e
s
p
e
c
t
i
v
e
l
y
w
h
il
e
r
L
a
n
d
r
R
a
r
e
M
M
C
in
verter
b
uffe
r
in
duc
ta
nc
e
a
nd
resis
t
ance
.
The
cum
u
la
tive
M
M
C
i
nv
e
r
t
e
r
t
o
g
ri
d
c
o
nn
ec
t
i
on
i
n
d
u
c
t
a
nc
e
a
n
d
resista
n
ce
is t
h
en
d
efi
n
ed
[2
2
]
by
(1
1) w
he
n the
so
u
r
ce
impe
danc
e is n
eg
l
ect
e
d
.
(1
1
)
22
rr
TT
T
f
f
LR
ZL
R
L
R
Thi
s
i
mp
li
e
s
t
h
a
t
th
ere
is
t
o
t
al
i
mp
e
d
a
n
ce
(
TT
T
Z
LR
)
be
t
w
e
e
n
the
M
M
C
o
u
t
put
a
n
d
t
h
e
gri
d
.Inte
g
ra
ti
n
g
(
1
0
)
be
tw
e
e
n
t
h
e
pre
s
e
n
t
s
a
m
p
le
(
k)
a
n
d
t
he
n
e
xt
s
ample
(k
+1
)
th
e
avera
g
e
es
ti
ma
t
e
d
d
,
q
-
varia
b
l
e
s ca
n b
e
a
s fol
l
ow
s
:
,1
,1
,1
,
1
(
12)
df
d
c
d
T
f
q
T
f
d
kk
kk
kk
k
k
k
QVV
L
I
R
I
,1
,1
,
1
,1
(1
3
)
qf
q
c
q
T
f
d
T
f
q
k
k
kk
kk
k
k
k
QV
V
L
I
R
I
W
h
ere
th
e
re
la
t
i
on
sh
ip
b
et
we
en
r
at
e
s
o
f
c
h
ang
e
o
f
t
h
e
MM
C
Inv
e
r
ter
ou
t
p
u
t
d
q
c
u
r
r
ents,
the
eff
ecti
v
e
in
duc
ta
nce
i
n
w
hic
h
t
h
e
c
urre
n
t
c
ha
n
g
e
s
a
n
d
d
-q va
r
ia
ble
s
re
s
pe
cti
v
e
l
y is
de
f
i
n
e
d
in
(14
)
a
nd (15)
.
,1
(1
4
)
d
fd
T
Qk
k
dI
dt
L
,1
(1
5
)
q
fq
T
Qk
k
dI
dt
L
The
d,q-va
ria
b
les
,1
d
kk
Q
a
nd
,1
q
kk
Q
from
t
he
p
eri
o
d
of
(
k)
t
o
(
k
+1)
c
a
n
b
e
g
o
t
us
i
ng
P
I
con
t
ro
l
l
ers[2
6
]
as:
,
_
(,
)
(
,
)
(
,
)
_
(,
)
(
,
)
(
,
)
,1
1
1
(1
6
)
dq
P
i
dq
f
d
q
f
dq
I
i
dq
f
d
q
f
d
q
k
k
kk
kk
QK
I
I
K
I
I
d
t
The
s
e
d,q-
var
i
able
s
c
a
n
b
e
u
s
ed
t
o
o
b
ta
i
n
t
h
e
ir
c
or
resp
on
din
g
d
,
q
-v
ol
t
a
ge
c
om
p
one
n
t
s
at
t
he
M
M
C
in
verter
dur
in
g
the
sam
e
sa
m
p
l
e
peri
od
((k)
t
o
(
k+
1))
as fo
l
l
o
w
s:
,
1
,1
,1
,1
(1
7
)
cd
f
d
T
f
q
T
fd
d
kk
kk
kk
k
k
k
VV
L
I
R
I
Q
,
1
,1
,1
,1
(1
8
)
cq
f
q
T
f
q
T
f
q
q
kk
k
k
kk
k
k
k
VV
L
I
R
I
Q
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
Po
we
r c
o
mp
ensat
i
on
f
o
r
v
ecto
r
-ba
s
e
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U.
La
wan
)
1
787
Whe
r
e
t
h
e
d,
q
p
r
o
por
t
i
o
n
al
a
nd
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n
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e
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l
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ontro
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ain
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)
P
id
q
K
a
n
d
_(
,
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I
id
q
K
w
h
i
c
h
d
e
p
e
n
d
o
n
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h
e
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C b
a
se
d
S
T
ATCOM
dyna
mi
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u
r
in
g
th
e
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c
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ra
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tive
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2
Tf
Pi
d
q
id
q
LR
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T
_(
,
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f
Ii
d
q
P
i
d
q
i
d
q
T
R
KK
T
L
The
s
e
P
I
c
on
trol
l
e
rs
f
a
s
t
re
sp
onse
ma
kes
t
h
e
S
T
A
T
CO
M
ou
t
p
u
t
d
,
q
c
urr
e
nt
r
efere
n
ces
a
nd
t
h
e
d,q-
PI
v
ariab
l
es
(,
)
,1
dq
kk
Q
o
f
t
h
e
ne
x
t
s
a
m
ple
pe
ri
od
(
i
.e.
(,
)
1
fd
q
k
I
)
m
eet
up
w
i
t
h
t
h
e
c
ur
rent
s
am
p
l
e
d,q
curr
ent r
e
fe
ren
c
es
*
,)
(
q
fd
k
I
and
d,q-P
I
v
ariab
l
es
(,
)
dq
k
Q
r
e
s
pe
cti
v
e
l
y as
f
ol
low
s
:
*
1
(2
1
)
fd
f
d
kk
II
*
1
(2
2
)
fq
f
q
kk
II
(,
)
(
,
)
,1
(
2
3
)
dq
dq
kk
k
QQ
F
o
r
a
li
ne
ar
t
r
a
ns
it
ion
betw
e
e
n
t
he
t
w
o
s
a
m
pl
i
n
g
peri
o
d
s
o
f
(i.
e.
1
kt
o
k
.
(21)
a
nd
(
2
2)
c
an
f
urt
h
er
b
e
repr
esente
d
in
a
n
aver
age
for
m
ove
r the
ti
m
e
k
a
s
:
*
,1
11
(2
4
)
22
fd
fd
f
d
kk
k
k
II
I
,1
11
(2
5
)
22
fq
lq
f
q
kk
k
k
II
I
S
ubst
i
t
ut
i
ng
(
2
1)
a
n
d
(
2
2
)
i
n
t
o
(
1
6
)
an
d
(1
7
)
r
espec
t
i
v
e
l
y
a
n
d
us
ing
(
23)
t
o
be
r
epre
se
n
t
e
d
i
n
a
se
para
te
d
,q
for
m
a
s
:
**
_(
)
_
(
)
(2
6
)
d
P
i
d
fd
fd
I
i
d
f
d
f
d
kk
k
k
k
QKI
I
K
I
I
d
t
**
_(
)
_
(
)
(2
7
)
q
P
i
q
fq
fq
I
i
q
f
q
f
q
kk
k
k
k
QKI
I
K
I
I
d
t
D
u
e
t
o
t
he
a
c
t
i
o
n
of
t
he
P
I
var
i
a
b
l
e
s
repre
s
ente
d
i
n
t
he
(
2
4
)
a
nd
(25).
The
grid
d
,q
v
o
l
ta
ges
an
d
t
h
e
M
M
C ba
sed S
T
A
T
CO
M o
u
t
p
u
t
d,q v
o
l
t
a
g
e
s
a
re a
ssum
ed
t
o
be
t
he
:
(,
)
(
,
)
,1
(2
8
)
fd
q
f
d
q
kk
k
VV
(,
)
(
,
)
*
,1
(2
9
)
cd
q
c
d
q
kk
k
VV
P
u
tt
ing
(
2
1)-(27)
i
n
t
o
(1
7)
a
nd
(
1
8
)
g
i
ves
the
d,
q
vol
tag
e
c
om
p
o
n
e
n
ts
a
t
the
MM
C
STA
T
CO
M
in
verter
,
whic
h
se
rve
a
s
t
he
d
,q
v
o
l
ta
ge
r
efe
r
enc
e
s
em
pl
o
y
e
d
u
s
i
n
g
the
P
I
c
o
n
t
r
o
l
l
e
r
s
.
T
h
e
s
e
v
o
l
t
a
g
e
r
e
f
e
r
e
n
c
e
s
a
r
e
defi
ne
d
b
y
:
**
*
_(
)
_
(
)
*
(
30)
2
T
c
d
P
i
d
f
d
f
d
I
i
d
f
d
f
d
fd
fq
fq
T
f
d
kk
k
k
k
k
k
k
k
L
VK
I
I
K
I
I
d
t
V
I
I
R
I
**
*
_(
)
_
(
)
*
(3
1
)
2
T
cq
P
i
q
f
q
f
q
I
i
q
f
q
fq
fq
fd
f
d
T
f
q
kk
k
k
k
k
k
k
k
L
VK
I
I
K
I
I
d
t
V
I
I
R
I
Eq
uat
i
ons
(
3
0
)
and
(3
1)
w
hi
ch
r
epre
sen
t
t
h
e
d
,q
v
ol
t
a
ge
r
efere
n
ces
t
o
th
e
PWM
co
n
t
ro
l
m
o
d
u
l
at
or
can
b
e
furt
her
re
prese
n
t
e
d
in
t
er
ms
o
f
the
es
t
i
ma
te
d
d,
q
var
i
ab
les
(
i
.e.
d
k
Q
q
k
Q
)
as
p
a
r
t
o
f
t
he
f
ee
d
for
w
ar
d
term
s[
26]
as:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN: 2088-
8694
I
nt
J
P
ow
Elec
& Dr
i
S
y
st V
ol.
10,
N
o.
4
, Dec
201
9 :
1
7
8
1
– 1
796
1
788
*
*
(3
2)
2
T
cd
d
f
d
f
q
T
f
d
fq
kk
k
k
k
k
L
VQ
V
I
I
R
I
*
()
*
(3
3
)
2
T
cq
q
f
q
f
d
f
d
T
f
q
kk
k
k
k
L
VQ
V
I
I
R
I
Wher
e
d
k
Q
a
n
d
q
k
Q
)
de
fi
ned
pre
v
i
o
us
ly
i
n
(
26)
a
n
d
(
27)
r
espe
c
t
ive
l
y
repr
ese
n
t
th
e
s
e
tt
led
d
q
varia
b
l
e
s
(
i
.e.
d
Q
a
n
d
q
Q
)
gi
ve
n
in
(
34)
a
n
d
(
35)
,
ob
ta
ine
d
a
s
o
u
t
p
u
t
s
o
f
t
he
d
,
q
-P
I
con
tro
l
lers
o
f
a
v
o
l
t
a
g
e
con
t
ro
l
d
ynam
i
c
mode
l
base
d
o
n
t
he
d
eco
up
le
d
inne
r
c
u
rrent
c
on
t
ro
l
o
f
t
he
S
TA
TC
O
M
g
ri
d
c
onn
e
c
te
d
syste
m
[2
6
]
,
show
n
i
n
F
igure
4.
**
_(
)
_
(
)
(3
4)
d
P
i
d
fd
fd
I
i
d
f
d
f
d
QK
I
I
K
I
I
d
t
**
_(
)
_
(
)
(3
5
)
qP
i
q
f
q
f
q
I
i
q
f
q
f
q
QK
I
I
K
I
I
d
t
The
refere
nce
M
M
C
base
d
S
T
A
T
CO
M
m
o
du
l
a
tio
n
i
n
dex
m
a
nd
pha
se
s
hi
ft
a
ngle
t
ha
t
p
r
oduc
e
the
i
nver
t
e
r
t
hr
e
e
pha
se
A
C
c
u
rre
n
t
s
(
i
.e.
sin
k
o
iI
t
)
a
nd
v
o
lta
ge
s
(
sin
Ck
o
VV
t
)
has
the
subs
cript
(
i
.
e
.
k
)
re
prese
n
t
i
ng
p
h
a
s
e
n
u
mbe
r
,
and
i
s
the
pha
se
s
hi
ft
b
etw
e
e
n
t
he
o
u
t
p
u
t
c
urre
nt
a
nd
v
o
l
t
a
g
e
in A
C for
m
g
i
v
e
n
by
(3
6) a
nd (37)
re
s
pe
cti
v
e
l
y a
s
f
o
l
lo
w
s
:
**
22
(
36)
cd
cq
mV
V
*
*
1
tan
(
37
)
cq
cd
V
V
F
o
r
the
MMC
i
n
t
er
n
a
l
c
o
n
t
ro
l,
it
em
p
l
oye
d
the
c
o
mm
on
c
ontr
o
l
c
o
nce
p
ts
u
se
d
i
n
[2
7][28]
[2
4]
t
ha
t
u
t
i
l
i
zed
t
h
e
c
on
c
e
pt
s
o
f
C
arri
e
r
s
h
i
ft
e
d
p
ul
se
w
i
d
t
h
m
od
ul
ati
o
n
s
chem
es
(
C
SPWM
)
[6][29][30]
[
13][12
]
.
A
p
p
l
y
i
ng
the
K
i
rc
hh
off’s
t
h
e
o
ry,
t
h
e
A
C
vo
lta
ge
i
n t
h
e
arm
a
s
a
n
outp
u
t
c
a
n
be de
r
i
v
ed
a
s fol
l
o
w
s
:
11
(3
8
)
,
22
k
kO
kn
kp
r
r
k
cc
c
di
VV
V
L
R
i
dt
2(
3
9
)
ci
r
k
r
r
c
i
r
k
d
c
kn
kp
cc
di
LR
i
V
V
V
dt
Where
ci
r
k
i
i
s
t
h
e
circ
u
l
at
ing
c
u
rrent
a
n
d
dc
V
i
s
the
cum
u
l
a
t
i
ve
d
c
–lin
k
v
o
l
t
a
g
e
ac
ross
t
h
e
arm
s
.
r
L
is
t
he
b
u
ffe
r
a
r
m
ind
u
c
t
ance
o
f
eac
h
arm
.
r
R
i
s
t
h
e
s
eri
e
s
resi
st
an
ce
a
s
s
o
c
i
a
t
e
d
wit
h
e
ach
b
u
ffe
r
i
ndu
c
t
ance
tha
t
i
s
usua
lly
a
ssum
e
d
ne
g
l
i
g
i
b
le
[27]
.
f
L
a
nd
f
R
a
re
t
h
e
A
C
c
o
upl
ing
in
duc
t
a
n
c
e
and
it
s
c
o
rre
s
p
ondi
ng
series resis
tanc
e
re
spect
ive
l
y
. If the M
M
C
is con
n
e
c
t
ed
t
o g
r
id
, t
he
n there
w
i
l
l
be a s
our
ce impe
da
nce
s
R
a
nd
s
L
(
i
.
e
.
t
h
e
A
C
s
o
u
r
c
e
ind
u
c
t
an
c
e
a
nd
i
t
s
c
orresp
o
ndi
ng
A
C
s
o
u
r
c
e
s
e
r
i
e
s
re
si
st
an
c
e
r
esp
ect
iv
el
y
)
.
Th
e
ma
them
at
i
c
al
r
elat
i
o
n
s
h
i
p
bet
w
ee
n
the
M
M
C
o
utp
u
t
curr
e
n
t
o
f
e
a
c
h
ph
a
s
e
,
t
h
e
a
rm
u
ppe
r
a
n
d
lo
we
r
cu
rre
n
t
s
,
the c
i
rc
ula
t
in
g cur
r
ent a
n
d
outp
u
t
vo
l
tag
e
c
o
u
l
d
be
as:
(
40)
kk
p
k
n
ii
i
1
(4
1
)
2
ci
r
k
kp
kn
ii
i
1
(4
2
)
2
kk
n
k
p
cc
c
VV
V
Base
d
on
(
3
8)
-
(
39)
,
the
re
gul
a
t
i
o
n
of
t
he
c
ircu
lat
i
ng
c
u
rr
ent
ci
r
k
i
a
n
d
co
nt
ro
l
o
f
t
h
e
out
pu
t
vol
t
a
ge
k
c
V
c
ou
l
d
b
e
ac
h
i
e
v
ed
b
y
c
o
n
t
ro
lli
ng
t
he
u
p
p
er
a
nd
l
ow
er
c
urrents
a
n
d
u
p
p
e
r
a
nd
l
o
w
e
r
v
o
lta
ge
s.
I
n
ot
he
r
w
o
rd,
t
o
a
c
h
ie
ve
t
he
d
es
ired
o
u
t
pu
t
v
o
l
t
a
ge
a
nd
m
ai
nta
i
n
the
ba
la
nce
b
e
t
w
ee
n
the
e
n
t
i
re
D
C
vo
lta
ge
a
nd
e
a
c
h
pha
se
l
eg o
u
t
p
u
t v
o
l
t
a
ge,
the
ou
t
p
u
t
v
ol
t
a
ges
c
a
n
b
e
de
f
i
n
e
d
a
s
:
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
Po
we
r c
o
mp
ensat
i
on
f
o
r
v
ecto
r
-ba
s
e
d
c
u
r
r
e
n
t
c
ont
ro
l
of
a
mo
du
la
r
mul
t
i
l
ev
e
l
…
(A.
U.
La
wan
)
1
789
1
(4
3
)
2
kp
d
c
k
cc
VV
V
1
(4
4
)
2
kn
d
c
k
cc
VV
V
Whe
r
e
k
c
V
i
s
t
h
e
M
M
C
s
y
ste
m
out
pu
t v
o
l
t
a
g
e
of t
he
cor
respo
n
d
i
n
g
p
hase
d
e
fi
ne
d
as:
*
1
sin
(
4
5
)
2
dc
ck
o
k
VV
M
t
3.
T
H
E PROPOSED COMPENS
AT
ING D
Q
-
POWE
R
AL
GORITHMS
(
dc
o
m
Q
AN
D
qc
o
m
Q
) FOR
MMC BASED
S
T
A
T
C
OM
D
E
C
O
U
PLED C
URR
E
N
T CONTROL
.
Ma
ny
a
t
t
e
mp
t
s
t
o
e
nha
nce
gr
id
v
o
l
t
a
ge
o
r
pow
e
r
f
a
c
t
or
w
e
r
e
c
ons
i
d
e
r
ed
u
s
i
ng
e
x
te
r
n
al
r
e
act
i
v
e
curr
ents
s
uch
as
i
n
[
6
],[31].
The
s
e
e
x
t
e
r
n
al
c
om
pe
nsa
t
i
ng
re
ac
t
i
v
e
curre
n
t
s
w
e
r
e
ma
d
e
fro
m
th
e
q
-
axi
s
com
p
o
n
e
n
t
s
o
nl
y
a
nd
in
tro
d
u
ce
d
as
a
m
ea
ns
o
f
com
p
e
n
s
a
tin
g
the
l
i
n
e
l
o
ad
r
ea
ct
i
v
e
c
u
rrent
r
efere
n
c
e
s.
B
u
t
the
d-ax
is
c
o
m
pone
n
t
c
omp
e
nsa
tio
n
ge
ne
r
a
ti
o
n
w
as
n
ot
c
on
si
dere
d
.
G
i
ven
b
y
(
3
4
)
and
(
35)
a
nd
f
i
gur
e
1,
base
d
o
n
t
he
d
,
q
-
fram
e
n
a
t
ur
e
of
d
,q-
in
ner
c
u
rre
nt
v
e
c
t
or
c
on
tro
l
t
he
re
w
il
l
a
l
w
a
ys
b
e
t
h
e
d-axi
s
a
n
d
q
-a
x
i
s
in
its
i
m
p
le
me
nt
a
tio
n.
A
lso
c
ons
ide
r
i
ng
the
fac
t
s
tha
t
t
he
P
I
c
ontr
o
lle
rs
a
r
e
i
mple
me
n
t
e
d
i
n
d,
q
i
n
depe
nde
nt
form
t
o
c
o
ntr
o
l
t
h
e
ac
tiv
e
an
d
re
acti
v
e
pow
er
i
nde
pe
nde
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tin
g
a
d
e
c
o
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p
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d
dy
n
a
mi
c
re
sponse
[
2
2
]
,
th
i
s
w
ork
i
n
tro
duc
e
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a
n
o
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r
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om
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.
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x
i
s
a
c
tive
vec
t
o
r
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o
mp
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t
i
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pe
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at
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d,
q-var
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i.e.
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Q
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q
Q
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plu
s
t
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x
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s
re
acti
v
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or
c
ompe
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a
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p
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t
s
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ompe
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t
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o
r th
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q-a
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g
me
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ia
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e
v
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ri
abl
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s
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d
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d
q
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w
ere
ob
ta
ine
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l
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for
t
h
e
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g
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I
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on
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ler
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of
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he
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e
r
c
urrent
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his
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t
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ompe
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t
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at
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ll
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have
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her
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l
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n
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ce
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nee
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n
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T
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T
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en
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k
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o
ns
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d
e
r
i
n
g
t
h
e w
i
de
ned ca
use
s
and f
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s
o
f S
T
A
T
CO
M va
riat
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n
t
h
e
othe
r
han
d
,
gi
ve
n
by
(
4
0
)
a
n
d
(1
4)
a
nd
F
i
gur
e
8.
I
t
c
a
n
b
e
se
en
t
ha
t
the
ex
trac
t
i
on
o
f
the
q-a
x
is
c
ur
ren
t
c
omp
one
n
t
s
(i.e
.
load
r
e
act
i
v
e
c
u
r
r
ent
lq
k
I
)
i
s
r
e
q
ui
re
d
in
m
ain
t
a
i
n
i
ng
t
h
e
g
ri
d
vol
t
a
g
e
o
r
c
o
rrec
t
t
h
e
p
o
w
e
r
fa
ct
or
o
f
th
e
power
s
ys
tem
at
t
he
P
CC.
T
h
i
s
i
s
u
se
d
as
t
he
r
ea
cti
v
e
cur
r
ent
r
ef
e
r
en
c
e
*
fq
k
I
in
o
rder
t
o
e
f
fe
c
tive
l
y
c
o
ntr
o
l
the
varia
t
i
o
ns.
The
pro
p
o
s
ed
pow
e
r
a
lg
ori
t
hm
(
dc
o
m
Q
a
nd
qc
o
m
Q
)
w
a
s
derive
d
t
o
b
asica
l
l
y
t
o
pr
o
duc
e
a
n
op
p
o
site
d
,q-
pow
er
c
om
ma
nds
t
o
t
h
e
d,
q-
P
I
c
ontro
lle
r’s
o
u
t
pu
t
v
a
r
i
a
b
l
e
s
(
d
Q
a
n
d
q
Q
)
i
n
r
espo
n
s
e
t
o
t
h
e
STA
T
CO
M
ac
t
i
ve
a
n
d
r
ea
c
tive
c
u
rr
ents
v
a
r
iat
i
o
n
d
ur
ing
the
s
t
ead
y
sta
t
e
o
p
er
at
ion.
A
ssumi
ng
t
ha
t
in
t
h
e
S
T
A
T
CO
M
n
on-
ide
a
l
ope
rat
i
on,
t
he
o
ut
p
u
t
c
u
rre
n
t
h
a
s
not
o
n
l
y th
e
rea
c
t
i
v
e
c
u
rre
nt
c
om
po
ne
nt
s
f
q
I
b
ut
a
l
s
o
has a
ve
ry sm
a
ll a
c
t
i
ve
c
urren
t
c
ompo
ne
n
t
s[5]
in i
t
its o
per
a
ti
on
s
f
d
I
t
ha
t
ha
s som
e
v
ariat
i
on
s.
Th
is
s
m
a
l
l
a
c
t
i
v
e
c
u
rre
nt
v
a
r
iat
i
o
n
w
as
c
ons
i
d
er
ed
d
ue
t
o
a
ver
y
sm
all
re
al
pow
er
e
xc
ha
n
g
e
assum
i
ng
a
st
ea
dy
s
t
a
t
e
o
p
e
r
a
tion
of
t
he
S
TA
TCO
M
.
Ther
efor
e,
d
u
r
i
n
g
t
he
t
ra
n
s
i
e
n
t
s
(i.
e
.
F
i
gure
5)
,
the
ac
t
i
ve
c
urrent
f
d
I
flow
s
in
o
r
d
e
r
t
o
m
a
in
ta
i
n
t
he
capa
c
i
t
o
rs
v
o
l
t
a
ge
s
duri
n
g
c
h
ar
g
ing
a
n
d
d
i
s
c
har
g
i
n
g.
H
en
ce,
a
smal
l
int
e
rn
al
a
cti
v
e
po
we
r
l
o
ss
i
s
ass
o
c
i
a
t
e
d
a
c
r
o
ss
t
h
e
re
s
is
t
a
nce
T
R
.
A
l
so
a
ssum
i
n
g
t
h
e
pow
e
r
l
os
s
due
t
o
the
v
o
lta
ge
d
r
o
p
L
v
a
cro
ss
th
e
c
u
mu
l
a
ti
ve
i
n
duct
a
n
c
e
T
L
c
o
n
si
d
e
rin
g
t
he
S
TA
TCO
M
o
pe
rat
i
on
i
n
q
uadra
n
t
I
and
qu
adra
n
t
I
I
r
e
gion
s
o
f
F
i
gure
5.
I
n
t
h
es
e
tw
o
re
gi
o
n
s,
t
h
e
STA
T
CO
M
i
n
jec
t
s
po
si
tive
re
ac
tive
p
o
w
er
t
o
the
gr
id
a
n
d
a
bsor
b
a
ver
y
n
egl
i
g
ib
le
a
m
o
u
n
t
o
f
a
c
t
i
v
e
po
w
e
r
fo
r
t
h
e
d
c
v
olt
a
g
e
m
ai
n
t
en
a
n
c
e
r
esp
e
cti
v
ely
.
Th
en
t
h
e
S
TATC
OM
c
a
n
a
tt
ai
n
a
st
e
a
d
y
st
a
t
e,
w
h
e
n
comp
en
sat
e
d
an
d
it’s
t
h
e
d-
ax
i
s
a
n
d
q
-a
xis
c
u
rrent
com
p
o
n
e
n
t
s
f
d
I
a
n
d
f
q
I
i
n
(34)
a
n
d
(
35)
r
espect
i
v
e
l
y
w
i
l
l
b
e
dri
v
en
t
o
t
h
e
desir
e
d
d
-
ax
is
a
nd
q-a
x
i
s
c
urren
t
com
p
o
n
e
n
t
s
*
fd
I
and
*
fq
I
.
A
f
ter
the
c
o
mpe
n
sat
i
on,
t
ha
t
i
s
i
n
t
h
e
stead
y
s
t
ate,
t
he
r
ate
o
f
ch
ang
e
o
f
th
e
STATCOM current
s
f
d
I
and
f
q
I
w
i
ll
be
ze
r
o a
s
:
(,
)
0
(
46)
fd
q
dI
dt
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN: 2088-
8694
I
nt
J
P
ow
Elec
& Dr
i
S
y
st V
ol.
10,
N
o.
4
, Dec
201
9 :
1
7
8
1
– 1
796
1
790
Assu
mi
ng
al
s
o t
h
e cu
mu
la
tive
ef
f
e
c
tiv
e
re
si
s
t
a
n
c
e
T
R
to be
n
egl
i
g
i
b
le
d
urin
g the
s
t
e
a
d
y
sta
te
, a
nd gi
ve
n b
y
:
0(
4
7
)
2
r
fT
R
RR
The
n
c
urr
e
nt
c
om
pone
nts
*
fd
I
and
*
fq
I
w
ould
ha
ve
t
h
e
ir
r
e
s
ulta
nt
l
i
n
e
cur
r
en
t
va
lu
e
f
I
in
qua
dr
at
ur
e
w
ith
t
he
a
va
ila
ble
A
C
vol
ta
ge
a
t
t
h
e
grid
P
CC
as
s
how
n
i
n
f
i
gur
e
1
.
F
ro
m
t
h
e
Fi
gu
re
1
i
t
c
a
n
b
e
s
een
t
hat
t
h
e
at
t
he
s
te
a
d
y
sta
t
e
c
o
n
d
it
i
o
n
,
t
he
g
rid
A
C
v
olta
ge
d
V
w
i
l
l
a
lso
be
i
n
pha
se
w
ith
S
TA
T
C
O
M
o
u
t
pu
t
A
C
vo
lta
ge
c
V
a
n
d
th
e
co
nt
roll
er
a
t
t
h
i
s
s
t
a
t
e
,
will
g
e
n
erat
e
th
e
d
e
si
red
co
mp
en
sat
i
ng
volt
a
ge
c
o
m
man
d
s
wi
t
h
t
h
ei
r ma
g
n
itu
de
s
*
fd
V
a
nd
*
fq
V
that c
a
n
be a
s:
**
(4
8
)
cd
f
d
T
f
q
VV
L
I
**
(49
)
cq
T
f
d
VL
I
C
ons
ide
r
i
ng
th
a
t
i
n
de
c
o
u
p
l
e
d
c
ur
rent
c
on
trol,
t
h
e
v
o
lta
ge
v
ect
or
i
s
a
l
w
a
ys
a
l
i
gne
d
w
ith
t
he
d
-ax
i
s,
t
h
en
g
rid
q
-
axi
s
v
olt
a
g
e
v
ector
f
q
V
c
an
b
e
ig
nor
e
d
.
H
e
nce
0
fq
V
.O
n
t
h
e
o
t
her
ha
nd,
t
he
g
ri
d
d-a
x
i
s
v
o
l
t
a
ge
vec
t
or
f
q
V
i
s
d
e
fin
e
d
in
i
n
t
e
r
ns
o
f
re
su
l
t
a
nt
S
TATCOM
d
-ax
i
s
vol
t
a
g
e
c
omm
and
*
cd
V
i
n
(4
8)
b
ase
d
o
n
t
h
e
o
ppo
si
ng
v
ol
t
a
g
e
d
r
o
p
sign
a
l
(
i
.
e
.
*
T
fq
L
I
),
*
cq
V
i
s
the
resul
t
an
t
S
T
A
T
CO
M
q-
ax
is
t
ha
t
e
q
uals
t
he
o
p
p
o
s
i
n
g
vo
lta
ge dr
o
p
signa
l
*
T
fd
L
I
.
If
f
rom
(
48 and
49) the
cor
respo
n
d
i
ng
des
i
red
d-ax
is a
nd q-
a
x
is
STATC
OM
cu
rre
n
t
s ca
n b
e
d
ef
in
ed
as:
*
*
(50
)
cq
fd
T
V
I
L
*
*
(5
1
)
cd
f
d
fq
T
VV
I
L
Wher
e
the
cum
u
la
t
i
ve
i
n
t
erfac
i
n
g
i
nd
uc
ta
nc
e
T
L
is gi
ven
by
:
(5
2
)
2
r
Tf
L
LL
The
i
r
com
p
ens
a
ti
n
g
s
te
a
d
y-st
ate
ac
t
i
v
e
a
nd
r
ea
ctive
pow
e
r
v
aria
ble
s
(
dcom
Q
and
qcom
Q
)
a
t
t
h
e
f
i
l
t
e
r
PCC bus is a
s:
*
**
33
(5
3
)
22
cq
f
q
d
c
o
m
fd
fd
fd
T
VV
QV
I
V
I
L
2
*
**
33
3
(5
4
)
22
2
cd
f
d
fd
q
c
o
m
fd
fq
f
d
TT
VV
V
QV
I
V
I
LL
F
r
om
(
9)
s
up
por
ted
by
fi
g
u
re
4
,
ca
n
be
s
ee
n
t
h
a
t
t
he
c
um
ul
a
tive
i
nd
uc
ta
nce
T
L
i
s
no
t
o
n
ly
em
plo
y
e
d
f
or
f
il
t
e
ri
n
g
pur
p
o
se
a
nd
c
o
u
p
lin
g
bu
t
is
a
ls
o
ine
v
ita
b
l
e
fo
r
c
r
ea
t
i
ng
a
v
e
r
y
smal
l
ph
a
s
e
shif
t
δ
th
a
t
w
ill
e
n
a
b
le
t
he
act
i
v
e pow
er
e
xc
ha
n
g
e d
u
rin
g
t
he
c
a
p
aci
t
o
rs
t
ra
n
s
i
e
n
t
s.
O
n the
o
t
he
r
ha
n
d
,
from
t
he
(
9),
it
c
a
n
be
s
een
t
hat
the
v
o
lta
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