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
)
Vo
l
.
1
0
, No
.
2
, Ju
n
e
20
1
9
, p
p
.
1
0
3
4
~
1
048
ISSN: 2088-
8694,
DOI
:
10.11591
/ijpeds.
v10.
i
2.pp1034-1048
1034
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
Frequency adaptive Sli
di
n
g
Fourier Transform
for
synchronizing VSI to the
grid
Osama M.
A
rafa,
M. E.
A
b
d
a
llah
,
G
h
a
da
A
. Ab
d
el Azi
z
D
e
partm
e
n
t
o
f
Pow
e
r
El
ectron
i
cs
a
nd
E
n
e
rgy
Conversi
o
n
,
E
lectron
i
c
s Resea
r
ch Inst
i
tute,
Egyp
t
Art
i
cl
e In
fo
ABSTRACT
A
r
tic
le hist
o
r
y
:
R
e
c
e
i
v
e
d
Dec
1
9
,
2
018
Re
vise
d F
e
b 20,
201
9
Ac
ce
p
t
ed
M
ar 8
, 2
0
19
Sl
i
d
i
ng
D
i
screte
F
ourier
T
ransf
o
r
m
(
SD
FT)
is
v
ery
efficient
reg
ardi
ng
com
p
utatio
nal
load
a
nd
i
t
p
o
sses
ses
a
v
e
ry
f
ast
ph
ase
an
gl
e
d
e
t
ecti
o
n
w
ith
excel
lent
h
arm
o
n
i
c
reject
io
n
at
n
om
in
a
l
f
requ
ency
.
How
e
ver,
a
t
o
f
f
-
nomi
n
al
f
r
equ
e
ncy,
S
DFT
gen
e
ra
t
e
s
errors
i
n
b
o
th
m
agnitu
de
a
nd
p
has
e
a
n
gl
e
d
u
e
to
sp
ectral
l
eakag
e
.
T
h
i
s
p
a
per
in
trod
uces
a
w
ork
a
roun
d
f
o
r
F
o
uri
e
r
T
r
ans
f
o
r
m
to
h
andl
e
t
h
is
d
is
abilit
y
under
o
f
f-n
o
m
i
n
al
f
requ
ency
w
h
ile
a
vo
id
i
ng
vari
abl
e
-rat
e
sam
p
li
ng.
S
li
di
ng
F
o
u
r
ier
T
r
a
n
sf
o
r
m
(SF
T
)
is
u
sed
a
s
a
phas
e
d
e
te
c
t
or
f
o
r
a
p
h
a
se
-loc
k
e
d
lo
o
p
w
h
o
se
o
utpu
t
fre
q
ue
n
c
y
is
u
se
d
t
o
driv
e
the
SFT
.
The
paper
revis
i
ts
t
he
m
athem
a
t
i
cs
o
f
Fouri
e
r
Transf
or
m
(F
T)
i
n
a
th
ree-ph
a
s
e
s
e
tt
in
g
v
i
a
a
t
i
m
e
-do
m
ain
app
r
oach
t
o
s
h
o
w
a
n
ewly
p
r
op
osed
filteri
ng
t
echnique
f
o
r
t
h
e
doub
l
e
-f
requency
o
sci
l
l
a
ti
on
j
u
st
b
y
su
m
m
in
g
th
e
F
T
s
i
n
e/co
si
ne
f
ilt
er
o
utp
u
t
s
o
f
th
e
th
ree
in
di
vi
dual
p
h
as
es.
A
l
s
o
,
t
h
e
a
n
a
l
y
s
i
s
aim
s
t
o
determine
and
corre
ct
t
he
p
has
e
a
n
d
m
ag
nit
u
d
e
e
rro
r
s
u
n
de
r
o
f
f-
no
m
i
n
a
l
f
r
eq
uency
o
p
erati
o
n
.
T
h
e
p
rop
o
s
e
d
t
echn
i
qu
e
(S
FT-PL
L
)
i
s
tes
t
ed
i
n
real
t
i
m
e
o
n
d
S
P
A
CE
D
S
1
2
0
2
D
S
P
u
s
in
g
v
o
lt
age
vect
ors
t
h
at
a
re
p
re-
generated
to
s
imulate
the
mos
t
a
dverse
g
ri
d
cond
itions.
T
h
e
tes
t
i
ng
scen
arios
com
p
are
th
e
perf
orm
a
n
ce
of
t
he
S
F
T
-P
LL
wit
h
t
ha
t
o
f
t
he
D
ecou
p
l
ed
S
t
atio
nary
R
e
f
erence
F
r
am
e
P
L
L
(d
αβ
P
L
L
)
.
T
h
e
resu
lts
p
r
ov
e
th
at
S
F
T
-PLL
i
s
s
up
e
r
io
r
to
d
αβ
PLL.
K
eyw
ord
s
:
Discret
e
f
our
i
e
r
transf
o
rm
Fourie
r tra
n
sfo
r
m
Re
new
a
b
l
e
ene
r
gy so
urc
e
s
S
lid
ing
d
i
sc
re
t
e
f
ourie
r
trans
f
or
m
S
ync
hr
on
i
z
a
t
i
o
n
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:
Osam
a
M
. A
r
a
f
a
,
D
e
pa
rtme
nt
o
f
P
o
w
e
r Elec
tro
n
i
c
s and
Ener
g
y
C
on
ver
s
io
n,
Elec
tron
ics
R
e
se
arc
h
Insti
t
u
te
, D
o
k
k
i,
Egy
p
t
.
Em
ail:
oara
fa2
0
0
4
@
ya
h
oo.
co
m
1.
I
N
TR
OD
U
C
TI
O
N
F
u
el
c
el
ls
a
nd
o
the
r
R
e
n
e
w
able
E
nerg
y
S
ourc
e
s
(RES
),
s
uch
a
s
P
V
s
ys
t
e
ms
a
nd
W
i
n
d
E
nerg
y
C
o
n
v
e
r
t
e
r
s
(
W
E
C
S
)
,
u
s
e
i
n
v
e
r
t
e
r
s
t
o
i
n
j
e
c
t
t
h
e
c
a
p
t
u
r
e
d
e
n
e
r
g
y
i
n
to
t
he
u
ti
lit
y
g
r
id
[
1
]
.
Th
e
s
u
cc
essf
ul
in
t
e
grat
io
n
of
t
he
se
s
ourc
e
s
t
o
t
he
e
lec
t
ric
a
l
g
ri
d
i
s
s
t
i
ll
ha
v
i
n
g
a
l
o
t
o
f
i
s
s
u
e
s
a
n
d
t
e
c
h
n
i
c
a
l
c
h
a
l
l
e
n
g
e
s
[
1
]
.
O
n
e
o
f
the
m
o
s
t
cr
it
ica
l
issue
s
o
f the
s
e inte
gr
ati
ons
i
s
the
sy
n
c
h
r
oni
z
a
t
io
n
of
t
h
e
g
ri
d
-
si
de
c
on
v
e
rt
e
r
(
G
S
C)
t
o
t
h
e
el
e
c
t
r
i
cal
g
ri
d
[2
].
T
h
e
s
yn
ch
ro
ni
z
a
t
i
on
o
f
t
h
e
GSC
t
o
t
h
e
e
l
ec
trica
l
g
r
i
d
c
a
n
b
e
def
i
ne
d
as
t
he
p
r
o
cess
o
f
minim
i
z
i
n
g
t
he
d
iffe
renc
e
in
pha
se
a
ng
l
e
,
fr
e
que
nc
y
a
n
d
v
o
l
t
a
g
e
b
e
t
w
e
e
n
t
he
G
S
C
o
ut
p
u
t
a
nd
t
h
e
ut
il
i
t
y
gr
id
wave
form
[
3].
Th
is
s
yn
c
h
ro
n
i
z
a
ti
o
n
p
r
o
cess
has
to
b
e
re
al
iz
ed
b
ef
ore
t
h
e
c
o
n
n
ec
t
i
o
n
b
et
wee
n
t
he
R
ES
o
r
t
h
e
G
S
C a
nd the
u
til
i
t
y
gr
id ta
k
e
s
p
lace
,
t
h
e
n
i
t
is
t
o be
m
a
i
n
t
a
i
n
ed
af
t
e
r
t
he co
n
n
ecti
o
n
is es
t
ablishe
d
.
In
[
4
]
,
t
h
e
id
eal
s
yn
ch
ron
i
z
a
t
i
on
t
echn
i
qu
e
mu
st
i
n
s
t
a
n
t
ly
r
e
s
p
o
nd
t
o
a
n
y
g
r
i
d
c
h
a
n
g
e
s
,
e
ffi
c
ien
t
ly
trac
k
t
h
e
phas
e
a
ng
le
o
f
t
h
e
elec
t
r
i
c
al
g
r
i
d,
c
ompe
t
e
n
t
l
y
d
e
t
ec
t
an
y
va
ria
t
i
on
i
n
t
h
e
g
rid
fr
equ
e
n
c
y
a
n
d
com
p
ete
n
t
l
y
e
lim
i
n
a
t
e
ha
rm
o
n
i
c
c
omp
o
n
en
t
s
a
n
d
d
i
s
t
u
rba
n
ce
from
t
h
e fu
ndam
e
n
t
a
l
w
av
e
f
orm
.
T
here
a
re
l
o
t
s
of
m
et
ho
ds
u
s
e
d
f
o
r
trac
k
i
n
g
t
he
p
ha
se
a
ng
le
o
f
the
gri
d
v
ol
t
a
g
e
s
u
ch
a
s
z
e
ro-
c
rossi
ng
de
tect
i
on
me
t
h
o
d
(ZCD),
S
pace
V
ector
(S
V)
m
et
hod,
K
a
l
m
a
n
fi
lter,
p
ha
se
-loc
ked
l
o
o
p
m
etho
ds
(
P
LL)
a
nd
D
i
sc
re
te
F
o
u
r
i
e
r
Tran
sfo
r
m
(D
FT)
.
T
h
e
p
ro
bl
e
m
i
s
non
e
of
t
h
e
se
m
e
t
ho
ds
c
o
m
p
l
e
t
e
l
y
s
a
t
i
sfies
t
h
e
fu
l
l
r
equ
i
rem
e
nt
s
of
a
n
idea
l sy
nc
hron
iz
ati
o
n tec
h
n
i
qu
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
Freq
uenc
y
a
d
a
p
t
i
ve
sl
idi
n
g fo
ur
ier tr
ansf
o
r
m
for
sy
nch
r
o
n
i
z
i
n
g
VSI
to t
h
e
gri
d
(O
sam
a
M.
Arafa)
1
035
Ze
ro-Cross
ing
D
e
tecti
o
n
(ZCD
)
is
t
he
s
i
m
ples
t
a
n
gl
e
s
ync
hro
n
i
za
tio
n
met
h
od
.
Th
e
z
e
ro
-cro
ssi
ng
po
int
i
s
de
t
ec
t
e
d
tw
ice
e
v
er
y
s
i
n
g
l
e
c
y
c
l
e
a
n
d
t
h
e inte
r
v
al
b
e
t
w
een
e
a
c
h
tw
o
s
u
cce
ssi
ve
c
r
o
ss
in
gs
i
s
m
e
a
s
ured
to
a
n
y
d
es
i
r
ed
r
esol
uti
on.
T
his
me
t
h
o
d
i
s
ve
ry
s
im
p
l
e
b
u
t
i
t
ca
n
be
u
se
d
on
l
y
i
f
the
in
put
s
i
gna
l
is
a
s
tab
l
e
sinus
o
i
da
l
si
g
n
a
l
a
s
i
t
i
s
v
e
ry
s
e
n
si
t
i
v
e
t
o
harm
on
ics
a
n
d
n
o
is
e
w
h
ic
h
ca
n
ca
use
mult
i
p
le
c
on
t
i
guo
us
zero crossi
ngs
[5].
P
h
ase
loc
k
e
d
l
oo
p
i
s
a
w
e
ll-e
s
t
a
blis
hed
tec
hni
que
a
nd
th
er
e
are
a
l
ot
o
f
im
ple
m
e
n
t
a
ti
ons
o
f
P
L
L
syn
c
hro
n
i
za
ti
o
n
.
Ea
ch
i
mp
le
m
e
nta
t
i
o
n
d
i
f
f
er
s
from
t
h
e
othe
r
fro
m
the
p
o
i
n
t
o
f
v
ie
w
of
s
impl
ic
ity,
imm
unit
y
to
d
is
t
u
r
b
a
n
ce
,
robu
stne
ss
i
n
c
a
se
o
f
gri
d
f
a
u
lts,
fa
s
t
c
on
v
e
rg
ence
t
o
the
gr
id
a
n
g
l
e,
a
n
d
h
ow
it
deal
s
wi
t
h
di
st
urba
nces.
S
ync
hr
on
o
u
s
F
r
ame
P
LL
(S
F
-
P
LL)
t
e
c
hn
i
que
i
s
w
i
de
l
y
use
d
i
n
t
h
re
e-pha
se
g
ri
d-c
o
nne
cte
d
pow
er
c
on
ver
t
e
r
s
for
i
t
s
s
i
m
p
le
i
m
p
lem
e
n
t
atio
n,
f
a
s
t
a
n
d
a
ccur
a
te
e
s
t
i
m
atio
n
o
f
t
he
pha
se
a
ng
le
o
f
t
h
e
gri
d
[6].
U
nder
i
d
ea
l
gr
id
c
ond
i
t
i
ons,
w
i
t
h
ou
t
a
ny
d
i
s
t
ur
b
a
nc
e
s
o
r
h
arm
o
n
i
cs,
t
h
is
t
ec
h
n
i
q
ue
e
xh
i
b
i
t
s
fa
st-tra
cki
n
g.
H
o
w
e
ve
r,
i
t
is
n
o
t
a
g
o
od s
o
l
u
ti
o
n
in t
h
e
pres
e
n
ce
of
u
n
b
a
l
a
n
ce
d
gr
id v
o
lta
ges
[7].
Rea
l
a
n
d
i
ma
g
i
na
r
y
p
ow
er
p
ha
se
-l
ocke
d
lo
op
(P
Q
-
P
LL)
i
s
a
robu
st
d
i
g
i
t
al
P
LL
a
l
g
o
r
i
t
h
m
based
o
n
the
i
n
st
a
n
t
a
ne
ous
r
e
a
l
a
n
d
i
m
a
ginar
y
p
ow
e
r
.
Thi
s
a
lgor
ithm
is
i
nt
roduc
ed
t
o
mai
n
t
a
i
n
s
yn
ch
ro
ni
zati
o
n
in
prese
n
ce
of har
m
oni
c
s
an
d
n
e
g
at
i
v
e se
qu
enc
e
u
nba
la
n
c
es.
Ano
t
her
al
g
o
r
ith
m wh
i
c
h
can
d
ea
l
wit
h
un
b
al
an
c
e
d
gri
d
v
o
l
t
a
ges
i
s
t
he
D
o
u
b
l
e
S
ynchr
on
o
u
s
F
r
a
m
e
P
LL
(D
S
F
-
P
LL).
It
i
s
a
very
s
u
ita
ble
so
l
u
ti
on
f
or
g
rid-
con
n
ec
ted c
o
n
v
erter
s
t
ha
t o
p
e
r
a
te
i
n
u
n
st
a
b
l
e
fr
e
quenc
y
gri
d
a
n
d
unb
a
l
an
c
e
d
co
ndi
tion
s
[
8
].
S
i
n
u
s
oid
a
l
si
gn
al
i
nt
e
g
ra
t
o
r
PLL
a
l
go
rith
m
(S
SI-PLL)
h
a
s
t
h
e
a
d
va
n
t
ag
e
of
o
pera
ti
ng
w
ell
u
n
d
e
r
un
ba
lanc
e
d
a
n
d
d
is
t
o
rte
d
g
r
i
d
[
8
].
A
no
t
h
e
r
a
dva
n
t
a
g
e
o
f
t
h
i
s
a
l
gor
i
t
hm
i
s
th
a
t
i
t
can
b
e
use
d
f
or
t
he
s
i
n
g
l
e-
pha
se
s
y
s
tem
w
i
t
h
a
s
m
a
l
l
m
odi
fica
ti
on.
E
n
h
ance
d
pha
se-loc
ke
d
l
o
o
p
(EP
LL)
i
s
a
fr
eque
nc
y-ad
apt
i
v
e
no
n
l
i
n
ea
r
te
ch
ni
q
u
e.
E
P
LL
s
h
ow
s
a
hi
gh
de
gree
o
f
ro
bus
tne
ss
to
th
e
h
a
rmo
n
i
c,
n
o
i
se
,
an
d
unb
a
l
an
ced
g
rid
[9].
T
h
i
s
al
g
o
ri
t
h
m
can be use
d
f
or
s
i
n
g
l
e ph
ase
ap
pl
ica
t
i
o
n
s
.
Q
u
a
d
rature
P
LL
(QP
LL)
i
s
ba
sed on
es
ti
ma
ti
n
g
the
i
n
-p
hase
a
nd
t
h
e
qua
dra
t
ur
e
p
h
ase
val
u
e
of
t
he
f
u
nda
m
e
ntal
c
o
mpon
e
n
t
o
f
t
h
e
i
npu
t
vol
t
a
g
e
s
ig
na
l
.
T
hi
s
me
tho
d
i
s
a
g
o
o
d
s
o
l
uti
o
n
for
dis
t
ri
bute
d
g
e
n
era
t
i
o
n
[1
0
]
.
Pr
ed
ict
i
ve
P
ha
se-Loc
ke
d
Loo
p
(
P
P
LL)
is
a
com
p
le
x
sy
nc
hro
n
iza
t
io
n
m
e
th
o
d
.
H
o
w
e
v
e
r,
P
P
L
L
ha
s
a
w
i
de
-ra
n
g
e
sy
nc
hro
n
i
za
ti
on
c
a
p
a
b
il
i
t
y
and
t
h
e
syn
c
hro
n
i
za
ti
o
n
i
n
f
orm
a
ti
on
c
a
n
be
t
a
k
en
o
u
t
w
ith
i
n
t
w
o
v
o
lta
ge
c
yc
le
s
un
de
r
t
h
e
w
o
r
s
t
c
a
se.
S
t
ati
o
nar
y
Fr
a
m
e
P
LL
(
αβ
P
LL)
i
n
[11],
D
e
c
oup
l
e
d
D
o
u
b
l
e
S
ynchr
o
n
o
u
s
Re
fe
renc
e
F
r
am
e
P
LL
(
dd
srfP
L
L
)
and
De
co
up
l
e
d
St
a
t
io
n
a
ry
R
ef
e
r
en
c
e
F
ra
me
(
d
αβ
P
LL)
t
e
c
hn
iq
ues
have
t
he
w
ide
s
t
u
s
e
in
i
nd
ust
r
y.
T
he
l
im
it
a
t
ion
of the
αβ
P
LL
is i
t
s
d
isa
b
il
i
t
y to t
rac
k
t
he gr
i
d
a
n
g
l
e
i
n
the
c
a
s
e of
u
n
ba
l
a
nce
d
fa
u
l
t
s. The
dd
srfPL
L
te
c
hn
i
que
overc
ome
s
t
h
i
s
lim
i
t
a
t
i
o
n
b
y
d
ec
ou
pl
i
n
g
the
ne
ga
ti
ve
a
n
d
t
he
p
o
si
ti
ve
s
e
que
nce
o
f
t
he
g
ri
d
vol
tage
b
ut
t
he
sever
e
d
raw
b
a
c
k
o
f
t
h
i
s
tec
h
ni
q
u
e
i
s
t
he
h
i
gh
o
v
e
r
sh
oot
i
n
t
h
e
e
s
t
i
m
a
te
d
gri
d
a
ng
le
w
hen
t
h
e
fa
ul
t
o
c
c
u
rs.
The
d
αβ
P
LL
ove
r
c
om
es
t
h
i
s
d
r
aw
bac
k
b
u
t
it
s
p
en
ds
m
ore
t
h
an
t
w
o
c
yc
les
t
o
t
ra
ck
t
he
g
ri
d
fre
q
ue
ncy
in
c
ase
of
f
req
u
e
n
cy c
han
g
e.
D
i
scr
e
te
F
our
ie
r
t
r
ansform
me
tho
d
(
D
F
T)
i
s
o
n
e
o
f
t
he
e
arlies
t
m
e
th
ods
u
se
d
f
o
r
pha
se
a
n
g
l
e
est
i
ma
t
i
o
n
. Th
i
s
m
eth
od pro
v
i
d
es a h
igh de
gre
e
o
f im
mun
i
t
y
a
g
a
i
nst harm
onic
s
[
12
]
.
S
li
ding D
i
sc
rete
F
o
u
rier
Tra
n
sf
orm
(S
D
F
T
)
i
s
an
i
mpro
ve
d
ve
rsio
n
of
t
he
D
F
T
i
n
w
h
ich
o
p
t
im
ized
u
se
o
f
co
m
put
a
tio
n
p
o
w
e
r
and
re
d
u
ced
l
a
t
en
cy
a
re
p
ro
vi
d
e
d
b
y
u
s
ing
a
recursive
form
ulation
o
f
t
h
e
D
F
T
algori
t
hm
.
H
o
w
e
ve
r,
i
f
th
e
t
i
m
e
w
i
n
dow
o
f
t
h
e
S
D
F
T
i
s
not
i
den
t
ica
l
w
it
h
t
h
e
gri
d
p
e
r
i
o
d,
a
pha
s
e
s
h
i
f
t
o
c
c
u
r
s
b
e
t
w
e
e
n
t
h
e
e
s
t
i
m
a
t
e
d
a
n
g
l
e
and
t
h
e
ac
tua
l
g
ri
d
an
gle.
I
n
[12]
s
om
e
st
r
a
t
e
gie
s
h
a
v
e
be
e
n
r
e
por
t
e
d
to
c
om
pen
s
ate
for
t
h
is
pha
se
e
r
r
or
s
uch
as
t
he
u
se
o
f
t
i
m
e
-va
r
yi
ng
F
our
ie
r
c
o
eff
i
c
i
ent
s
a
n
d
pol
yn
om
i
a
l
re
gressi
on
a
n
a
l
ysis.
In
f
ac
t
,
t
hese
s
tr
ateg
ie
s
are
render
e
d
w
ith
n
o
t
i
c
eab
le
c
om
ple
x
i
t
y.
O
the
r
f
i
x
i
n
g
m
e
t
h
o
d
s
f
or
t
he
S
D
F
T
draw
ba
ck
a
re
p
r
e
sente
d
i
n
[1
3
]
.
H
o
w
e
ve
r,
t
he
p
rese
n
t
e
d
m
et
ho
ds
i
n
v
o
l
v
e
t
h
e
use
o
f
a
P
I
con
t
ro
l
l
er
a
nd
a
N
u
m
e
r
i
call
y
C
on
tro
lle
d
O
s
c
i
l
l
a
t
or
(NCO)
.
T
h
e
m
et
hod
a
do
pt
s
a
v
a
ri
ab
l
e
s
a
m
pli
n
g
rat
e
o
f
th
e
g
r
id
v
o
lta
ge
c
o
n
tr
ol
l
e
d
b
y
t
he
N
CO
.
V
a
ria
b
le
r
ate
sam
p
lin
g
a
d
ds
d
i
f
ficu
l
t
y
t
o
e
mbe
d
d
i
ng
t
he
S
DF
T
alg
o
ri
th
m
in
t
he
m
a
i
n
a
ppl
i
cati
on
DS
P
c
o
n
t
roll
e
r
w
h
i
ch
norm
a
l
l
y
ru
ns
a
t
fi
xed
e
x
e
c
u
t
i
o
n
r
ate.
A
l
t
ho
ug
h
the
tw
o
prop
ose
d
met
h
od
s
in
[
13
]
are
si
mpl
e
r
t
h
an
t
ho
se
o
f
[1
0],
bot
h
me
t
h
o
d
s
com
p
e
n
s
a
te
t
he
p
ha
se
e
rror
on
a
cyc
l
e
-
by-cyc
le
b
a
s
i
s
,
w
h
ic
h
is
r
ath
e
r
a
sl
ow
p
roc
e
ss.
T
he
un
des
i
rab
l
e
fe
a
t
ures
o
f
t
h
e
S
D
F
T
i
n
the
ca
se
o
f
off-
nom
i
n
a
l
fre
que
n
c
y
are
no
t
o
n
ly
l
i
m
i
t
e
d
t
o
t
h
e
p
h
ase
er
ror
w
h
ic
h is a fu
n
c
tio
n
of t
he
f
re
q
u
enc
y
d
e
v
ia
t
i
o
n
b
u
t
a
ls
o inc
l
u
d
e
s a
high-frequency oscilla
tion observable in the
ma
gni
tude
a
n
d
p
hase
a
n
g
l
e
e
sti
m
ates.
T
h
is
h
igh-
fre
q
u
e
ncy
osci
ll
a
t
i
o
n
i
s
r
e
f
er
red
to
i
n
the
litera
t
u
r
e
a
s
do
u
b
l
e
-fr
e
que
n
c
y
r
i
pp
le
a
n
d
s
om
e
p
o
s
t
-pr
o
ce
ssin
g
f
or
f
r
e
que
nc
y
an
d
pha
se
a
n
g
l
e
e
s
t
i
m
a
tes
in
t
h
e
o
f
f
-
nomi
n
a
l
o
pera
tio
n
m
ode
a
r
e
p
r
o
po
se
d.
T
he
pos
t-proc
es
sin
g
c
a
n
b
e
d
one
e
ithe
r
by
s
i
m
p
le
d
ig
i
t
al
a
ve
ra
gin
g
fi
l
t
e
r
or
by r
e
sa
m
p
lin
g
fi
l
t
e
r
w
hi
c
h
is re
l
a
tive
l
y
mor
e
c
ompl
ex
i
n imp
l
e
m
e
n
ta
tio
n.
F
r
om
t
he
p
rev
i
o
u
s
sur
v
e
y
,
i
t
a
p
p
ear
s
that
t
her
e
i
s
stil
l
some
r
oom
f
or
i
mprove
me
n
t
i
n
the
p
h
as
e
ang
l
e
tra
c
k
i
ng
issue
es
pec
i
a
l
ly
i
n
t
h
e
case
o
f
v
ar
i
a
b
l
e
a
n
d
of
f
-n
om
in
a
l
g
rid
fre
q
ue
ncy
a
c
c
o
mpan
ie
d
b
y
h
ig
h
harm
on
ic
c
o
n
t
ent
an
d
/
or
i
n
un
ba
lanc
e
d
t
hree
-phase
s
ys
t
e
ms.
This
p
ap
er
r
e
v
i
s
i
t
s
t
h
e
F
T
c
o
n
cep
t
thro
ug
h
a
t
i
me
-do
m
ai
n
an
a
l
y
s
i
s
t
o
f
u
rth
e
r
an
a
l
y
z
e
t
h
e
und
e
s
i
r
ab
l
e
f
ea
tu
re
s
asso
ciate
d
w
it
h
t
h
e
off-
nomi
n
a
l
g
r
i
d
fre
que
nc
y.
A
n
effi
c
i
e
n
t
a
nd
nat
u
ral
fi
l
t
e
r
in
g
me
t
h
od
for
the
do
u
b
l
e-
fre
q
u
e
nc
y
o
s
ci
l
l
a
t
i
ons
i
s
de
v
i
se
d.
T
he
law
s
r
ela
t
i
ng
t
h
e
p
h
ase
a
nd
m
a
gni
t
ude
e
r
r
o
r
s
to
t
he
freq
u
e
nc
y
d
i
f
fer
e
nc
e
a
r
e
de
rive
d.
T
he
f
orm
u
l
a
t
i
o
n
o
f
t
h
e
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N: 2
0
8
8
-
86
94
I
n
t
J Po
w
Elec
&
Dr
i
Sy
st,
Vo
l. 1
0
,
No
. 2
,
Ju
n
e
2
019
:
1
0
34
–
1
048
1
036
F
T
i
n
a
sli
d
in
g
m
o
de
i
s
exp
l
ai
ne
d.
E
xp
e
r
ime
n
ta
l
t
e
st
in
g
i
s
u
se
d
t
o
c
o
nf
ir
m
the
va
l
i
d
i
t
y
of
t
he
pr
opose
d
t
ec
hn
iq
ues.
Th
is
p
aper
i
s
o
r
ga
nize
d
as
f
o
l
lows
:
t
h
e
m
a
them
at
ical
a
nal
y
sis
f
or
d
eter
min
i
n
g
t
he
p
hase
a
nd
ma
g
n
itu
d
e
e
r
r
o
r
s
und
e
r
o
f
f-n
o
m
i
n
al
fre
qu
en
cy
c
on
dit
i
o
n
i
s
p
re
se
nte
d
i
n
sec
t
ion
2.
T
he
r
e
c
ur
si
v
e
im
pl
e
m
e
n
ta
t
i
o
n
i
s
e
x
p
l
a
i
ne
d
in
s
ec
t
i
o
n
3
.
The
str
u
c
t
ur
e
of
t
he
pr
o
p
o
se
d
S
F
T-
P
LL
is
e
xpla
i
ned
i
n
s
e
c
t
ion
4.
Th
e
meth
od
u
sed
fo
r
exp
e
rimen
t
al
t
es
tin
g
a
n
d
t
he
t
es
t
r
e
sult
s
i
n
r
eal-
t
im
e
on
a
D
S
P
pr
o
t
o
t
y
p
i
ng
p
l
at
f
o
r
m
(
D
S1
2
0
2
by
dSPAC
E
TM
)
ar
e
pr
esen
ted
in
s
ecti
o
ns
4
a
n
d
5
,
r
e
specti
v
e
l
y.
S
e
c
tion
6
pr
esent
s
t
he
c
onc
l
u
s
i
o
n
s
of
this p
ap
er.
2.
MAT
H
EMA
T
I
C
AL
ANAL
YSI
S
The
gr
i
d
v
o
lta
ge
s
ar
e
r
e
pr
e
s
e
n
te
d
b
y
a
b
a
l
a
n
ce
d
se
t
o
f
t
hr
e
e
-
p
h
a
s
e
s
i
nus
oi
da
l
vol
ta
ge
with
a
f
r
e
que
n
c
y
(
H
z
)
and
w
i
th
a
p
e
a
k
va
l
u
e
E
(
Volt),
th
u
s
is
g
iv
e
n
b
y
:
s
i
n
2
,
s
i
n
2
(
1
)
s
i
n
2
2
/
3
wh
ere
is
t
he
i
n
itia
l
pha
se
s
h
i
ft
(ra
d)
o
f
t
h
e
v
o
lta
ge
,
thi
s
p
h
a
se
s
hi
f
t
d
e
p
en
ds
o
n
t
h
e
r
a
nd
om
i
ns
ta
nt
a
t
whic
h
the m
e
a
s
urem
ent
i
s
s
ta
rted (
t
=
0
)
.
F
or the
s
a
k
e
of
t
h
i
s
a
n
aly
s
i
s
w
i
ll be
a
ssum
ed
c
o
n
s
t
an
t.
The
F
our
ier
tr
ansf
or
m
is
b
as
ed
o
n
a
p
p
l
yi
n
g
tw
o
or
th
o
g
o
n
a
l
f
i
lt
e
rs,
na
me
l
y
a
s
i
n
e
f
ilter
a
nd
a
c
o
s
i
n
e
fil
t
er
on
the
i
n
pu
t
s
i
g
n
a
l
.
Th
e
po
in
t
her
e
i
s
tha
t
w
e
se
t
t
h
e
p
ha
se
s
e
q
ue
nce
an
d
t
h
e
p
h
a
se
s
hif
t
b
e
t
w
e
e
n
t
he
si
ne/c
os
ine
fi
lte
rs
i
n
a
way t
h
a
t
s
er
ve
a
d
d
iti
o
n
al
f
ilte
ri
ng pu
r
pose
as
w
i
l
l b
e
s
e
e
n l
a
ter
o
n
.
Ther
e
f
or
e
,
t
he
t
hr
e
e
gr
id
v
olta
ge
s
w
i
l
l
b
e
pr
oce
sse
d
us
ing
tw
o
sets
o
f
ba
la
nc
e
d
t
hr
e
e-
pha
se
s
i
gna
ls
a
n
d
hav
i
ng
t
he
s
am
e
pha
se
s
e
q
ue
nc
e
a
s
t
he
g
r
i
d
vol
ta
ge
,
r
unn
i
ng
a
t
a
p
r
e
-
k
n
o
w
n
nom
i
na
l
gr
i
d
f
r
e
q
u
enc
y
(
H
z
)
and
hav
i
ng
a
un
i
t
y
pea
k
v
a
l
ue
.
F
or
t
he
s
ak
e
of
t
h
i
s
ana
l
ys
is
w
il
l
be
a
ssu
m
e
d
cons
tan
t
[
14
]
.
The
fir
s
t
se
t
of
s
i
gna
ls
i
s
gi
ve
n
by
s
i
ne
f
unc
tio
ns
a
n
d
i
s
c
a
l
l
e
d
t
he
s
i
n
e
or
d
i
r
ect
s
e
t:
s
i
n
2
,
s
i
n
2
2
3
s
i
n
2
2
/
3
(2
)
The
sec
o
nd
se
t
of
v
o
l
t
a
ges
i
s gi
ve
n
by
co
si
n
e
f
unc
t
i
o
n
s
an
d
i
s
calle
d
t
h
e
c
o
sine
o
r
qua
dr
atic
s
e
t
:
c
o
s
2
,
c
o
s
2
2
3
c
o
s
2
2
/
3
(3
)
The
F
our
i
e
r
tr
ansf
or
m
r
e
quir
e
s
per
f
o
r
m
in
g
the
fo
l
l
ow
ing
i
n
t
e
gr
a
tion
s to
get
t
h
e
an
g
le
:
1
.
(4
)
1
.
(5
)
(4
)
an
d
(5
)
a
r
e
co
mp
act
fo
rms
f
o
r
th
e
th
r
ee-
p
h
a
se
w
h
e
re
t
h
e
s
u
b
script
a
,
b
,
c
and
1
/
.
F
o
r
a
ny
p
o
s
iti
ve
a
nd
n
o
n-
zer
o
val
u
e
of
, two c
ases
a
re c
onsi
d
er
ed
f
or
calcul
a
ti
ng t
h
ese
in
te
gra
l
s,
the
f
irst ca
s
e is
(
of
f-
nom
i
n
a
l
f
r
e
que
nc
y
ca
se
)
w
h
ich
is
t
he
g
e
n
er
ic
c
ase
a
nd
t
h
e
s
e
c
on
d
ca
se
(
nomin
a
l
freque
nc
y c
a
se) whic
h
i
s
a
spe
cial c
ase
o
f
t
h
e
first c
ase.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t
J
P
o
w
Elec
&
D
r
i
S
y
st
I
S
S
N
:
2088-
86
94
Frequ
e
n
c
y adap
ti
ve
s
lid
in
g
f
o
u
r
i
e
r
t
r
a
n
s
fo
rm f
o
r sy
n
c
hro
n
i
z
i
n
g
VS
I
t
o
th
e g
r
id
(
O
sa
ma
M.
Ara
f
a
)
1
037
2.
1.
T
h
e
f
irst
c
ase:
(o
ff-
n
o
mina
l f
req
uency
ca
se)
Fo
r
,
a
nd
u
s
i
n
g
su
i
t
a
b
le
t
r
i
go
n
o
m
e
tr
ic
i
de
nt
it
ies
i
t
c
an
b
e
pr
ove
n
t
h
at
i
n
(4
)
(let
u
s
c
a
ll
it
t
h
e
d
i
r
e
ct
c
omp
o
n
e
n
t
)
a
nd
i
n
(
5
)
(
l
e
t
u
s
ca
l
l
i
t
t
h
e
qua
dr
at
i
c
c
om
po
ne
nt)
a
r
e
fu
nct
i
o
n
s
of
t
i
m
e
and
freq
u
e
n
c
ies
,
t
hat
ar
e
gi
ven
a
s
f
ol
l
o
w
s
:
F
o
r
pha
se
a
:
co
s
∆
co
s
∆
(6
)
sin
∆
sin
∆
(7
)
F
o
r
pha
se
b
:
co
s
∆
co
s
∆
2
/
3
(
8
)
sin
∆
si
n
∆
2
/
3
(
9
)
And
for
p
h
ase
c:
cos
∆
cos
∆
2
/
3
(
10)
sin
∆
si
n
∆
2
/
3
(
11)
wh
ere
2
(
12)
2
(
13)
∆
2
(
14)
E
s
i
n
∆
2∆
(
15)
∆
2
(
16)
E
s
i
n
∆
2∆
(
17)
F
i
gure
1. Wav
eform
s
of
,
,
,
:
1
V
,
6
0
H
z
,
5
0
H
z
a
nd
π
2
⁄
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.
2
, June
20
1
9
:
103
4 –
1
0
48
1
038
I
t
c
a
n
b
e
not
ic
e
d
t
h
a
t
b
o
t
h
o
f
t
he
d
irec
t
a
nd
qua
dr
at
ic
c
om
po
n
e
nt
a
n
d
f
or
t
he
t
hre
e
-p
ha
se
s
(
a,
b,
c
a
s
show
n
in
(
6
)
t
o
(11)
a
re
c
ompose
d
o
f
tw
o
sinus
o
i
da
l
term
s.
O
n
e
of
t
he
m
is
l
a
r
ger
in
m
a
gnit
ude
a
n
d
its
f
r
e
q
u
enc
y
i
s
t
h
e
d
i
ffe
r
en
c
e
b
etw
e
en
t
h
e
g
r
i
d
a
c
t
ua
l
a
nd
n
o
m
i
na
l
(o
r
fil
t
er)
fr
eque
nc
y
(
i
f
the
tw
o
fre
que
nc
ies
be
com
e
e
qua
l,
t
h
e
n
it
is
a
D
C
v
a
lu
e
a
s
w
i
l
l
be
s
how
n
i
n
(
6),
(
7
)
of
t
he
s
ec
o
nd
c
a
se)
.
T
he
o
the
r
has
a
muc
h
s
m
a
ller
ma
g
n
it
ud
e
and its
f
re
q
u
e
ncy
i
s
t
h
e
s
u
m
of
t
he
g
r
i
d
a
c
t
ua
l
a
nd nom
i
n
al fre
q
u
enc
y
(
if
the
tw
o
freq
u
e
n
c
i
e
s
b
ec
ome
equa
l
,
t
he
n
i
t
v
an
is
hes
a
s
t
h
e
m
agni
t
u
de
b
ec
ome
s
zer
o
as
w
i
ll
be
c
o
n
fi
rm
ed
i
n
t
h
e
se
co
nd
c
a
s
e).
R
e
fe
rri
ng
t
o
t
h
e
se
c
o
n
d
t
erm
i
n
t
h
e
r
i
g
ht
h
a
n
d
si
de
o
f
e
a
c
h
of
(
6),
(8)
and
(1
0),
i
t
i
s
e
v
i
d
e
n
t
tha
t
t
he
t
hree
t
e
r
m
s
c
o
n
st
i
t
ute
a
ba
l
a
nce
d
t
hree
-phase
s
e
t
i
n
c
osine
no
ta
ti
o
n
w
hose
sum
is
z
e
r
o.
T
han
k
s
to
t
he
p
r
op
e
r
s
el
ecti
o
n
o
f
t
h
e
s
i
n
e
a
n
d
cos
i
n
e
f
ilt
e
r
ph
a
s
e
a
n
g
l
es
.
R
e
f
e
rri
n
g
to
t
he
s
e
c
ond
t
e
r
m
i
n
t
h
e
r
i
ght
h
a
n
d
si
de
of
e
a
c
h
o
f
(7)
,
(
9)
a
nd
(11),
the
t
h
re
e
term
s
als
o
c
o
n
sti
t
u
t
e
a
b
ala
n
ce
d
t
h
ree
-
phase
s
e
t
i
n
sine
n
o
t
a
t
i
o
n
w
hos
e
s
u
m
is ze
r
o.
Her
e it bec
om
es
clea
r
the
us
e
f
u
lne
s
s
o
f
the
s
u
m
m
i
n
g p
r
oces
s
com
b
ine
d
w
i
t
h
t
h
e pro
p
er
s
elec
t
i
on
of
t
he
F
T
fil
t
er
pha
se
a
n
g
l
e
s
.
The
s
u
m
m
in
g
proce
s
s
t
hus
r
esu
l
t
s
i
n
fi
lt
e
r
i
n
g
th
e
do
ubl
e-f
r
e
q
u
e
n
c
y
o
s
c
illa
t
i
o
n
s
w
itho
u
t
a
ny
add
i
ti
ona
l
or
post-fil
t
er
i
n
g.
S
ome
re
presentati
ve
v
al
ues
(
1,
6
0
H
z
,
5
0
H
z
a
n
d
π
/
2
a
r
e
used to
plo
t
F
i
gure
1
w
h
e
r
e
the
w
a
vefor
m
s
of
a
nd
a
re
p
l
o
t
t
e
d
in
t
h
e
l
e
f
t
subp
l
o
t
a
n
d
th
e
ri
g
h
t
su
bp
lo
t
respe
c
t
i
ve
l
y
a
g
a
in
st
tim
e
t
o
illu
strate
t
he
d
o
u
b
l
e-
fre
q
u
e
ncy
ri
pp
le,
t
h
e
su
ms
a
nd
a
r
e
a
lso
p
l
o
tte
d
i
n
t
h
e
tw
o
s
u
b
p
l
o
t
s
t
o
s
h
o
w
t
h
e
smoo
t
h
s
i
n
uso
i
da
l
va
ria
t
io
n
s
a
s
the
t
h
re
e
h
i
g
h
e
r
f
r
e
quenc
y
os
cilla
t
i
o
n
s
ru
nn
i
ng
a
t
(
canc
e
l
eac
h
o
t
her.
T
here
fore,
the
sum
of t
h
e
d
i
r
ect
a
nd
t
h
e
qua
drat
ic
c
o
m
pone
n
t
s
f
or
t
h
e
t
hre
e
in
div
i
dua
l p
h
a
s
e
s
, are
given b
y
(18)
a
nd (
19)
, respec
tive
l
y:
x
x
3
k
co
s
ω
t
θ
∆
θ
(
1
8
)
y
y
3
k
si
n
ω
t
θ
∆
θ
(
1
9
)
D
i
v
i
d
i
ng (
1
9) by
(1
8) yie
lds
:
y
x
t
a
n
ω
t
θ
∆
θ
(20)
Th
e
r
efo
r
e:
2π
f
f
t
θ
∆
θ
t
a
n
y
x
(2
1
)
The
pha
se
a
ng
le
of
t
he
g
ri
d
vo
l
t
a
g
e
tha
t
i
s
equa
l
t
o
2
f
r
o
m
(
1
)
i
s
t
h
e
n
g
i
v
e
n
b
y
rearrangi
ng
(
21)
a
s follows
:
θ
2
π
f
t
θ
t
a
n
y
x
2
π
f
t
∆
θ
(2
2
)
S
ubst
i
t
ut
i
ng fr
o
m
(
14) int
o (
2
2
)
yie
lds,
θ
tan
2
π
f
t
(2
3
)
R
e
cal
li
n
g
t
h
at
i
s
c
o
n
s
t
a
n
t
,
t
h
e
n
∆
∆
0
a
n
d
∆
i
s
c
onst
a
n
t
.
i
s
a
l
s
o
c
o
n
s
t
a
n
t
.
T
h
e
r
e
f
o
r
e
,
t
h
e
t
i
me
d
e
r
iv
a
tiv
es
o
f
b
o
t
h
∆
a
nd
a
r
e
t
yp
ica
l
l
y
z
eros.
Conse
q
uen
t
l
y
,
the
va
l
u
e
of
c
an
b
e
c
a
l
c
ul
at
ed
b
y
tak
i
ng t
i
me
der
i
v
a
tive
o
f
bot
h
si
des
of (
22) a
s fol
l
o
w
s
:
f
1
2π
.
d
dt
tan
y
x
f
(24)
On
ce
i
s
e
stima
t
e
d
,
∆
is
d
e
t
e
r
m
i
ne
d
a
n
d
co
nse
q
ue
n
t
l
y
can
b
e
est
i
ma
t
e
d
from
(22)
.
To
e
stim
ate
the m
a
gn
i
t
ude
E
,
it c
a
n be
fou
nd
from
(
15),
(18)
a
n
d
(
1
9
)
th
at:
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
Freq
uenc
y
a
d
a
p
t
i
ve
sl
idi
n
g fo
ur
ier tr
ansf
o
r
m
for
sy
nch
r
o
n
i
z
i
n
g
VSI
to t
h
e
gri
d
(O
sam
a
M.
Arafa)
1
039
E
2
3
.
∆θ
sin
∆θ
.
x
y
(25)
The
ter
m
∆
∆
c
an
b
e
calle
d
t
h
e
m
a
gni
t
ude
c
orre
cti
on
fa
c
t
or
a
nd
the
ter
m
can
b
e
ca
l
l
e
d
t
he u
nc
o
m
pe
n
s
a
t
e
d
m
agn
i
tu
d
e
.
2.2.
S
econ
d
c
ase:
(
n
o
m
i
na
l
f
r
eq
ue
n
c
y
ca
s
e
)
B
y
u
s
i
n
g
s
u
i
t
a
bl
e
tri
g
o
n
o
me
tr
ic
i
de
n
t
i
t
i
e
s,
i
t
c
a
n
b
e
p
rove
n
t
h
at
a
nd
y
a
r
e
i
nd
e
p
e
n
d
e
nt
o
f
t
i
m
e
and
are
gi
ven a
s
fol
l
o
w
s
:
c
o
s
2
(2
6
)
s
i
n
2
(2
7
)
There
f
ore
,
t
he
sum
min
g
for
x
an
d
y
c
om
po
ne
nt
s
of
t
he
three
-phase
ar
e
give
n
by (2
8)
and
(29),
re
sp
e
c
ti
ve
ly
:
3 E c
o
s
2
(2
8
)
3E s
in
2
(
2
9
)
N
o
w
div
i
di
ng
bo
th
s
ide
s
in (2
9)
by t
h
eir
re
spe
c
ti
ve si
d
es
i
n
(2
8
)
y
ie
ld
s
:
sin
co
s
t
a
n
(3
0
)
(3
1
)
Re
ca
l
l
i
ng t
h
e
equa
l
ity
o
f
an
d
, the
pha
se
ang
le
of
grid v
ol
ta
ge
is then given
b
y (32)
:
2
2
(3
2
)
A
c
cordi
n
g
t
o
(
23)
,
t
h
e
la
st
t
er
m
i
n
t
he
r
igh
t
-
h
an
d
si
de
o
f
(23)
v
a
n
i
s
h
e
s
w
h
e
n
t
h
u
s
l
e
a
di
ng
t
o
(32)
a
nd
c
o
n
f
i
r
ms
t
ha
t
th
e
n
o
m
i
na
l
freq
u
e
n
c
y
c
a
s
e
is
a
s
pec
i
a
l
ca
se
o
f
the
of
f-nom
i
n
a
l
f
r
e
que
nc
y
ca
se
.
There
f
ore
,
i
t
is
o
bv
i
o
u
s
t
ha
t
(
2
3)
c
a
n
b
e
ge
n
e
ra
l
i
z
e
d
f
or
b
ot
h
nomi
n
a
l
a
nd
of
f-n
o
m
i
n
a
l
fr
eque
nc
y
c
a
ses.
T
h
u
s,
p
h
a
s
e
a
n
g
l
e
t
r
a
c
k
i
n
g
i
s
a
s
e
a
m
l
e
s
s
p
r
o
c
e
s
s
w
h
e
n
t
h
e
a
c
t
u
a
l
f
r
e
q
ue
ncy
sh
ifts
f
r
o
m
no
m
i
nal
to
o
ff-n
omin
a
l
mode
a
nd v
i
ce
versa
duri
n
g a
c
tua
l
o
pe
rat
i
o
n
.
The
magni
t
ude
E
c
an sim
pl
y
ob
t
a
i
n
e
d
fr
o
m
(28)
a
nd (
29) a
s
fol
l
o
w
s
:
2
3
.
(3
3
)
K
now
i
n
g t
h
a
t
:
li
m
∆
→
∆
∆
1
, the
(25) bec
ome
s
the sam
e
as
(33) a
nd it tu
rns a
s
state
d ea
r
l
ie
r th
a
t
the
nom
ina
l
f
re
que
nc
y
case
is
a
s
pec
i
a
l
c
a
s
e
of
t
he
o
ff-n
o
m
inal
g
e
n
er
ic
case
.
T
he
refore
(
25)
c
an
b
e
a
va
l
i
d
expre
ssi
on
for
bo
t
h
case
s
.
T
hus,
t
h
e
ma
gn
itu
de
t
rac
k
ing
i
s
a
ls
o
a
s
e
a
m
l
ess
pr
oces
s
s
h
oul
d
the
ac
tua
l
g
r
i
d
fre
que
nc
y
shif
ts
f
rom
off-nom
ina
l
t
o
nom
ina
l
m
ode
of
opera
t
i
o
n
a
nd v
i
ce
v
er
sa
.
N
o
w
it
i
s
c
l
ear
t
ha
t by
pr
oper
sele
c
tio
n of the
pha
se
s
e
que
n
c
e
a
nd
p
h
ase
s
h
i
f
t
o
f
the
w
a
v
eform
s
u
se
d
for
s
i
ne
a
n
d
c
o
s
ine
fi
l
t
er
s
of
t
he
F
T
a
nd
a
p
p
l
yi
n
g
t
he
s
um
min
g
p
r
oce
dur
e
we
g
et
a
v
ery
effec
t
i
v
e
fi
l
t
eri
ng
for
the
do
uble
-
fre
que
nc
y
osc
i
l
l
a
t
i
o
n
cha
r
ac
t
e
riz
i
n
g
the
F
T
i
n
o
f
f-n
om
ina
l
f
r
e
que
nc
y
c
a
se.
F
i
g
u
re
2
i
l
l
us
trates
t
he
schem
a
t
i
c
d
i
a
g
r
a
m
of
t
he
p
ro
pos
ed
F
T
imp
l
em
enta
t
i
o
n
b
as
e
d
on
(1
)
t
h
ro
ug
h
(3
3).
In
t
h
e
l
ef
t
m
os
t
b
l
oc
k,
t
he
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.
2
, June
20
1
9
:
103
4 –
1
0
48
1
040
sine
a
nd
cos
i
n
e
f
ilte
rs
a
re
a
pplie
d
to
t
he
t
hre
e
-pha
se
v
olta
ge
,
a
nd
th
e
su
mm
ing
p
r
oce
d
ur
e
i
s
a
pp
l
i
ed
t
o
ge
t
and
.
The
u
n
c
o
mpe
n
sa
t
e
d
pha
se
a
ng
le
g
i
v
en
by
(21)
i
s
es
ti
m
a
ted
in
t
he
l
e
f
tmos
t
b
l
ock
as
w
e
l
l.
T
he
is
est
i
ma
t
e
d a
cco
rdi
ng t
o
(24) in the i
n
ter
m
e
d
ia
te bl
o
ck.
U
s
ing
th
e
e
s
t
i
m
at
e
d
,
the phase ang
l
e
c
orre
ction term
∆
c
a
n
b
e
est
i
ma
t
e
d
i
n
t
he
r
i
ght
m
o
st
b
loc
k
.
S
ubtrac
tin
g
t
h
e
c
o
rre
ct
i
o
n
t
e
rm
fro
m
t
h
e
un
comp
en
s
a
t
e
d
ph
ase
ang
l
e
a
s
p
e
r
(
23)
r
esults
i
n
the
true
phase
a
n
g
le
o
f
t
h
e
gri
d
v
olt
a
g
e
.
The
magn
i
t
u
d
e
corre
cti
o
n
fac
t
or
c
a
n
be
easi
l
y
c
a
l
c
u
la
te
d
a
l
so
f
rom
(25)
i
n
t
h
e
b
o
tt
om
b
l
o
ck
a
nd
fi
n
al
l
y
t
he
c
om
pe
n
s
ate
d
m
agn
i
t
u
de
c
an
b
e
e
s
t
i
m
a
t
ed
.
F
i
gure
2.
S
che
m
a
t
i
c
d
iagra
m
of
the
pro
p
o
s
e
d
F
T
i
m
plem
enta
t
i
o
n
f
or
three
-phase
sys
t
e
m
3.
SLIDING
FT IMPLE
M
E
N
T
ATION
(SF
T
)
The
sl
id
i
ng
i
n
t
e
gra
t
i
o
ns
f
rom
t
t
o
as
m
entione
d
in
(
4)
a
nd
(5)
c
a
n
be
c
a
rri
e
d
o
u
t
num
eri
c
a
lly
i
n
a
d
i
s
c
r
e
t
e
s
a
m
p
l
e
d
s
y
s
t
e
m
.
T
h
e
m
e
t
h
o
d
a
p
p
l
i
e
d
f
o
r
o
b
t
a
i
n
i
n
g
an
d
u
pda
t
i
n
g
t
he
num
er
ic
al
i
nt
e
g
rat
i
on
over
one
c
ycl
e
u
po
n
p
i
ck
i
ng
eac
h
fr
esh
sa
mple
o
f
xi
(
o
r
yi
)
is
t
o
ins
e
rt
a
d
elaying
buff
er
o
f
length
a
f
t
e
r
t
h
e
di
sc
rete
t
i
m
e
i
n
te
gr
at
or
[
15]
.
Thus,
int
e
grati
o
n
over
i
s
up
da
te
d
eve
r
y
ne
w
sam
p
le
o
f
xi
a
nd
i
t
i
s
implem
e
n
te
d
a
s
s
how
n
sc
he
m
a
tica
l
l
y
i
n
F
i
gure
.
In
t
h
i
s
im
plem
en
tat
i
on,
t
he
b
u
f
fer
is
i
nit
i
al
ize
d
w
it
h
ze
roes
a
n
d
i
t
t
ak
es
/
sam
ples t
o b
e
f
ille
d inc
r
em
ent
a
ll
y w
ith va
l
i
d
r
ea
dings as
is e
qua
l t
o
t
he sa
m
pl
i
n
g peri
o
d
.
Ba
se
d
on
this,
the
de
tecte
d
g
r
i
d
angle
a
n
d
fre
q
u
e
nc
y
nee
d
s
o
n
e
com
p
le
te
c
yc
l
e
(
)
t
o
c
onverge
to
t
heir
act
u
a
l
v
alues
for all
. It ca
n be
show
n t
h
a
t
f
o
r
t
he de
t
ec
t
i
o
n
is alm
ost
i
n
sta
n
ta
ne
ou
s.
F
i
gure
3. U
pda
tin
g
N
u
me
rical
i
nte
g
r
a
t
i
on
o
v
e
r
one
c
y
cle
up
on
eac
h ne
w
sa
m
p
le of
4.
T
H
E PROPO
S
ED S
FT
-
P
LL
The
im
p
l
e
m
e
n
ta
tio
n
o
f
S
F
T
s
how
n
in
F
ig
ure
2
w
o
rks
very
f
i
n
e
an
d
g
i
v
es
f
as
t
an
d
a
c
c
u
r
a
t
e
resul
t
s
in
the
nom
ina
l
a
n
d
o
ff-n
om
ina
l
freque
nc
y
case
s
a
s
lo
ng
as
t
h
e
i
n
p
u
t
vo
l
t
age
s
a
r
e
f
re
e
fr
om
h
a
r
monic
di
st
ort
i
o
n
On
c
e
t
h
e
i
n
put
f
requ
e
n
cy
s
hi
ft
s
fro
m
no
mi
na
l
v
a
lu
e
w
h
il
e
h
a
rmo
n
ics
ar
e
exi
s
t
i
ng
t
h
e
e
s
t
i
m
a
ted
p
h
ase
a
n
g
l
e
,
the
es
t
i
ma
t
e
d
ma
gnit
ude
a
n
d
t
he
e
s
tim
ated
freque
nc
y
al
l
s
u
ffer
from
o
s
c
illa
t
i
o
n
s
d
u
e
to
s
pec
t
r
a
l
lea
k
a
g
e
and
the
S
F
T
lose
s
it
s
e
x
ce
ll
e
n
t
harm
on
ic
r
eje
c
t
i
o
n
a
n
d
bec
o
me
s
un
pr
ac
t
i
c
a
l.
T
h
i
s
i
s
s
how
n
usin
g
n
u
m
e
r
ical
si
m
u
lat
i
on
i
n
F
ig
ure
4.
F
i
g
ur
e
4
(
a
)
illus
t
rate
s
tha
t
t
he
i
n
p
u
t
v
o
l
tage
i
s
d
i
st
orte
d
w
i
t
h
h
ar
monic
s
s
t
a
r
t
i
ng
fr
o
m
t
=0
.
1
s
e
c
t
o
t
=
0
.45
sec
a
n
d
the
fre
que
ncy
is
s
te
ppe
d
u
p
the
n
d
ow
n
by
1
0
H
z
at
t
=0.
15
a
nd
t
=0.
35
se
c
respe
c
t
i
ve
l
y
.
Fig
u
re
4
(
b)
a
nd
(
c)
s
how
t
hat
the
S
F
T
r
e
jec
t
s
th
e
ha
rmon
ic
s
a
f
ter
e
x
ac
ty
one
pow
e
r
c
y
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le
a
n
d
gi
ves
ac
c
u
ra
t
e
f
r
e
que
nc
y
an
d
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se
a
n
g
l
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es
t
i
m
a
t
i
o
n
w
h
en
it
w
o
r
k
s
a
t
nomi
n
al
f
re
qu
en
cy
(
5
0
H
z
)
i
.e
.
f
r
o
m
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t
J
P
o
w
Elec
&
D
r
i
S
y
st
I
S
S
N
:
2088-
86
94
Frequ
e
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t
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15
se
c
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n
d
fr
om
t
=
0
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5
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ec
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0
.
5
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e
c
.
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o
w
e
ver
,
d
ur
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ng
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h
e
of
f-
n
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m
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na
l
per
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od
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om
t
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ec
to
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3
5
se
c,
t
he
e
xi
st
e
n
ce
o
f
ha
r
m
onics
c
ha
r
a
c
t
e
r
ize
s
t
he
e
s
t
im
at
io
n
o
f
b
o
t
h
fr
e
que
nc
y
a
n
d
pha
se
an
gl
e by
a su
s
ta
i
n
ed
e
rro
r.
(a)
(b
)
(c)
F
i
gur
e
4.
(
a)
V
ol
ta
ge
o
f
p
h
a
s
e
a
(
o
the
r
pha
se
s
om
i
t
ted
f
o
r
clar
ity
)
,
(b
)
estimated
freq
u
ency
b
y
SF
T
(c)
erro
r
in
p
h
a
se an
g
l
e estimation
b
y
SFT
To
o
ver
c
ome
thi
s
d
ef
ic
it,
a
n
d
c
on
si
d
e
r
i
ng
t
he
a
b
i
lit
y
of
t
he
F
T
t
o
de
te
ct
t
he
phase
d
if
fe
r
e
nc
e
θ
0
be
t
w
ee
n
t
h
e
an
gle
of
t
he
s
i
n
e
/
cos
i
ne
f
i
lter
s
a
nd
the
a
n
g
l
e
o
f
t
he
i
n
p
u
t
v
o
l
t
a
ge
a
s
sh
ow
n
by
(
2
3)
,
the
S
F
T
c
a
n
be
u
se
d
a
s
a
p
hase
d
e
t
ec
t
o
r
in
a
s
ta
n
d
ar
d
p
h
a
s
e
loc
k
e
d
l
o
op.
T
o
do
th
is,
t
h
e
S
F
T
tim
e
w
i
n
dow
(
or
n
u
m
be
r
of
pr
o
c
esse
d
sam
p
l
e
s)
s
h
o
u
l
d
b
e
c
o
nti
n
uo
us
l
y
a
nd
p
r
oper
l
y
ad
j
u
s
t
e
d
t
o
enf
o
rc
e
no
min
a
l-frequ
en
c
y
o
p
e
ration
a
n
an
d
h
e
n
ce
ma
in
t
a
in
i
t
s
e
x
c
el
len
t
h
a
r
mon
i
c
rej
e
c
tion
.
T
hi
s
is
s
c
he
ma
tic
a
l
ly
s
how
n
in
F
i
gur
e
5
a
n
d
F
i
g
u
r
e
6
.
Note
t
ha
t
the
s
h
ade
d
r
e
c
t
an
g
l
es
i
n p
h
ase b
a
n
d
c
in
F
i
gure 5
ar
e
s
i
mi
lar
i
n
c
o
n
str
u
c
t
io
n to tha
t
o
f
pha
se
a
.
The
pha
se
-
l
oc
ke
d
l
o
o
p
P
I
D
c
ontr
o
l
l
er
i
n
F
i
g
u
r
e
6
s
ets
th
e
lo
op
f
r
e
q
ue
nc
y
s
o
a
s
to
e
l
i
mi
na
t
e
t
he
p
h
a
se
d
i
f
fe
r
e
nc
e
∆
.
T
he
f
r
e
quenc
y
a
nd
t
h
e
pha
se
a
ngle
e
stim
ated
b
y
the
P
L
L
ar
e
fe
d
bac
k
t
o
the
var
i
a
b
l
e
w
in
dow
w
i
d
t
h
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.
2
, June
20
1
9
:
103
4 –
1
0
48
1
042
SF
T
to
c
a
l
c
u
la
te
t
he
s
ine
an
d
cosine
f
il
ter
s
w
hose
ou
t
puts
are
use
d
t
o
expl
ore
the
phas
e
d
iffere
nc
e
∆
.
T
h
e
P
LL
there
f
ore
a
d
jus
t
s
t
h
e
fr
eque
nc
y
of
t
he
S
F
T
s
uch
tha
t
w
hen
i
t
loc
k
s,
t
he
S
FT
w
orks
a
t
t
h
e
ac
tua
l
g
ri
d
fre
que
nc
y.
I
n
t
h
i
s
c
a
s
e
,
t
he
f
i
l
ter
fr
equenc
y
s
h
i
f
t
s
t
o
a
nd
t
he
ope
rat
i
on
m
ode
s
h
i
fts
from
o
ff-n
om
ina
l
t
o
nomi
n
a
l
a
n
d
∆
v
a
n
i
s
hes.
T
h
i
s
ar
range
me
nt
p
ro
vide
s
fa
st
p
h
a
se
d
i
f
fere
nc
e
dete
c
tio
n
w
h
ile
m
a
i
n
t
ai
n
i
n
g
the
S
F
T
w
e
l
l
-kn
o
w
n
exc
e
lle
nt
h
arm
onic
r
e
ject
i
on.
F
i
gure
5
a
n
d
F
i
g
u
r
e
6
to
g
e
t
h
er
i
llu
st
rate
t
h
e
s
t
r
u
c
tu
re
o
f
t
h
e
pro
pose
d
S
F
T
-P
LL
t
e
chn
i
que
.
F
i
gure
5.
S
c
h
em
atic r
epre
sent
atio
n
of t
he
pro
pos
ed
v
aria
b
l
e
w
i
n
dow width SF
T
phase detector
F
i
gur
e 6.
P
hase-loc
ke
d l
o
op s
c
he
ma
tic
re
p
rese
nta
t
i
o
n
5.
EXP
E
RIMENTAL
TES
TING METHODOLOGY &
RESULTS
The
ex
perim
e
n
t
al
t
es
t
i
n
g
r
ns
u
r
es
t
he
a
p
p
lica
b
i
l
ity
o
f
t
h
e
prop
o
se
d
tec
h
ni
q
u
e
i
n
r
ea
l
-
tim
e
sett
ing
a
n
d
un
der
prac
tic
a
l
oper
a
t
i
ng
c
o
n
d
i
tio
ns
t
ha
t
ma
y
v
i
o
l
a
t
e
som
e
o
f
t
he
c
o
n
s
i
d
e
re
d
ass
u
m
p
t
i
ons
m
ade
dur
i
ng
the
ma
them
at
i
c
al
a
nal
y
s
i
s.
T
he
t
e
s
t
i
n
g
m
eth
o
d
d
e
p
en
ds
on
ge
ne
rati
ng
t
he
v
o
l
t
a
ge
v
e
c
t
ors
tha
t
c
onta
i
ns
t
he
t
a
r
get
tes
t
cha
rac
t
er
is
t
i
c
s
(D
C
of
f
se
t
, harm
onic dis
t
ort
i
o
n
,
m
a
gn
it
ude
, fre
q
u
e
n
cy
or pha
se j
um
p
a
n
d
un
ba
la
nc
e
of the
three
-
p
h
ases)
u
s
i
ng
o
ffl
ine
MATLA
B
simu
l
a
ti
o
n
.
These
v
o
l
t
a
g
e
ve
c
t
or
s
a
r
e
sam
p
led
at
64
sa
mp
le
p
er
pow
er
cyc
l
e
(0.
0
2 sec
)
a
nd r
e
c
o
rde
d
alo
n
g
w
it
h a
n
a
rra
y of c
orr
e
sp
on
d
in
g
tim
esta
m
p
s.
A
lso
t
h
e
t
r
ue p
hase a
n
g
l
e
,
the
true
f
r
e
q
u
e
n
c
y
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t
h
e
true
m
a
gni
tu
de
w
h
i
c
h
a
r
e
use
d
f
or
g
e
n
era
t
in
g
th
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m
a
re
s
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nd
re
cor
d
ed
f
o
r
s
a
k
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o
m
par
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l
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n.
T
he
n
t
h
ese
v
o
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ge
v
e
c
t
ors
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r
e
pl
ay
e
d
b
ack
i
n
re
al
-ti
m
e
at
t
h
e
s
a
m
e
sa
mp
li
n
g
rate
u
si
n
g
d
S
P
A
C
E
ds12
0
2
D
SP
D
/
A
c
ha
nne
ls,
rea
d
b
a
c
k
via
A
/
D
ch
an
n
e
l
s
a
nd
a
re
u
s
e
d
as
i
n
put
s
t
o
t
he
S
F
T-P
L
L.
T
he
o
ut
p
u
t
o
f
t
he
S
F
T
-
P
LL
a
r
e
th
e
n
r
e
c
orde
d
i
n
M
A
T
f
i
l
es,
the
n
c
om
p
a
r
e
d
w
i
t
h
t
he
t
rue
v
a
lue
s
tha
t
a
re
p
r
e
-r
ec
or
de
d
to
c
omp
u
te
t
he
e
r
r
ors.
T
he
s
am
e
proce
dure
i
s
r
e
p
e
a
t
e
d
w
i
t
h
d
αβ
P
LL
tech
n
i
q
u
e
an
d
the
err
o
r
in
bot
h
ca
ses
are
p
l
ot
t
e
d
on
sam
e
a
xis
t
o
c
om
par
e
t
he
p
er
for
m
a
n
ce
of
b
ot
h
te
chn
i
que
s.
T
he
t
es
tin
g
vo
lta
ge
s a
r
e:
a.
50
H
z
b
ala
n
ce
d
t
h
re
e-pha
se
(
pea
k
E
=
2
20
V
)
w
ith
t
he
f
o
l
low
i
ng
tes
t
i
ng
sce
n
ari
o
s
:
H
a
r
m
oni
c
d
i
s
t
or
ti
on
(com
pose
d
of thir
d,
f
ifth,
seve
nth
a
n
d
n
i
nt
h h
a
rm
onic
c
o
mp
one
n
t
s
wit
h
7
0%,
6
0
%
,
30
%
a
n
d
20
%
o
f
t
he
f
und
a
m
e
n
t
a
l
ma
g
n
it
ud
e
re
sp
ect
iv
el
y
)
s
wit
c
he
d
-
on
a
t
t
=
1
s
ec
an
d
s
w
it
ch
ed-
o
f
f
a
t
t
=
1
.
5
s
ec
.
P
h
ase
jum
p
of
π
/
2
a
t
t
=2
s
ec
,
fr
eque
nc
y
ste
p
-up
o
f
5
H
z
at
t
=2.5
s
e
c
.
M
ag
n
itu
de
s
te
p-
u
p
of
0
.
2
E
a
t
t
=3.5
s
ec.
Ad
di
ng
D
C
o
f
f
s
e
t
s
to
b
al
an
ced
t
h
r
e
e
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h
a
se
a
t
t
=4
s
ec
,
0.
1
E
to
pha
se
A
,
-.
05
E
to
pha
s
e
B
,
-0.05
E
t
o
p
h
a
s
e
C
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t
J
P
o
w
Elec
&
D
r
i
S
y
st
I
S
S
N
:
2088-
86
94
Frequ
e
n
c
y adap
ti
ve
s
lid
in
g
f
o
u
r
i
e
r
t
r
a
n
s
fo
rm f
o
r sy
n
c
hro
n
i
z
i
n
g
VS
I
t
o
th
e g
r
id
(
O
sa
ma
M.
Ara
f
a
)
1
043
b.
U
nba
l
a
nce
d
t
h
r
ee
-
phase
i
npu
t
w
ith
know
in
g
sym
m
e
tr
ica
l
c
om
po
ne
nt
s
stru
c
t
u
r
e
a
s
f
oll
o
ws:
ma
gn
itude
o
f
po
s
i
tiv
e
sequ
e
n
ce
i
s
220
V
a
nd
i
t
s
a
ngl
e
i
s
z
e
r
o
, mag
ni
tud
e
of
n
e
g
at
i
v
e
se
q
u
e
n
ce
i
s
6
0
V
and
i
t
s
a
n
gle
is
-
150
deg.
,
ma
gni
tu
de
o
f
z
e
r
o
s
eq
uenc
e
is
20
V
a
nd
its
a
n
g
l
e
is
1
0
de
g.
I
n
a
l
l
o
f
t
h
e
s
e
t
e
s
t
s
,
t
h
e
e
r
r
o
r
i
s
c
o
m
p
u
t
e
d
a
s
t
h
e
d
i
f
f
e
r
e
n
c
e
be
tw
ee
n
the
tr
ue
q
ua
n
tit
y
a
nd
t
h
e
e
s
tim
ated
o
ne
.
In
t
he
case
o
f
u
n
b
ala
n
ce
d
t
h
ree
-
phase
,
the
true
va
l
u
e
f
o
r
pha
se
a
n
g
le
e
s
tima
t
i
o
n
e
r
r
o
r
is
t
he
pha
se
a
ngle
of
pha
se
A
o
f
the
p
o
s
i
t
i
ve
s
e
que
nce.
Harm
on
i
c
d
ist
o
rtion
te
st:
F
ig
ure
7
(a
)
sh
o
w
s
the
intr
od
u
c
ti
o
n
o
f
ha
rm
oni
c
s
a
t
t
=1
s
ec.
F
igure
7
(b)
a
nd
(
c
)
show
t
hat
d
αβ
P
L
L
gi
ve
s
cont
inu
o
u
s
o
s
ci
lla
t
i
on
in
e
st
im
at
i
o
n
of
b
ot
h
fr
e
que
nc
y
(
±
20
H
z
)
a
nd
angl
e
(
±
4
de
g.
)
w
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