TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol.12, No.7, July 201
4, pp
. 5078 ~ 50
8
5
DOI: 10.115
9
1
/telkomni
ka.
v
12i7.494
2
5078
Re
cei
v
ed O
c
t
ober 2
4
, 201
3; Revi
se
d March 23, 201
4
;
Accepte
d
April 2, 2014
A Harmonic Analysis Method Based on Harris Window
for Digital Substation
Luo Han
w
u
*
1
, Jiang Guo
y
i
2
, Li Fang
3
, Kang
Kai
4
1,2,
4
East Inner Mong
oli
a
Elect
r
ic Po
w
e
r L
i
mited
Com
pan
y
of State Grid, Hohhot 01
00
20, Chin
a
3
Hena
n Po
w
e
r
Comp
an
y of State Grid, Z
hen
gzho
u, 450
05
2
,
China
*Corres
p
o
ndi
n
g
author, e-ma
i
l
:
w
h
u-li
u
y
a
n
g
@
w
hu.
edu.cn
1
Ab
stra
ct
Accurate a
nd
efficient h
a
rmo
nics an
alysis
i
s
an i
m
p
o
rtant
pre
m
ise
a
nd
b
a
sis
for control
ling th
e
har
m
o
nics in
power system
.
The sampli
ng f
r
equency
of the grid
voltage and
curre
nt in
digit
a
l s
ubstation
nor
mal
l
y
must
be a
co
nstant
valu
e as r
e
q
u
ir
ed by
t
he st
an
dard
of IEC6
18
50, thus
async
h
ron
ous s
a
mpl
i
n
g
cause
d
by the
fluctuatin
g gri
d
frequ
ency a
nd this
co
nsta
nt sampli
ng fr
equ
ency w
ill r
e
sult in s
pectr
al
leak
age
a
nd
a
liasi
ng w
h
ich
w
ould
affect the
accuracy
o
f
har
mon
i
c a
n
a
lysis. In th
is
pap
er, a
dou
b
l
e-
spectru
m
-li
ne i
n
terpo
l
atio
n F
a
st F
ourier T
r
ansform
ap
pro
a
c
h
base
d
on
multi-ter
m
Harris
w
i
ndow
for gri
d
freque
ncy
and
har
monic
me
asure
m
ent w
a
s pro
pose
d
, a
nd
th
e si
mpl
e
and
practic
a
l
a
d
just
me
nt for
m
ula
s
for the frequ
e
n
cy, amplit
ude
and p
has
e a
ngl
e w
e
re
der
ived by c
u
rve
fitting. T
he correctness
an
d
effectiveness
o
f
the pro
pos
ed
method
w
e
re
valid
ated
by si
mu
lati
on
and
fi
eld test. T
h
e
p
r
opos
ed
metho
d
has hi
gher acc
u
racy in b
o
th the an
alysis of the har
m
onic a
m
p
litu
de an
d p
hase a
ngl
e as
compar
ed w
i
th th
e
conve
n
tio
nal w
i
nd
ow
ed
meth
od a
nd, is
mor
e
ap
plic
ab
le
to
the h
a
rmon
i
c
ana
lysis i
n
the
situatio
n of d
i
gi
tal
substation.
Ke
y
w
ords
:
ha
rmo
n
ic a
nalys
i
s
, DF
T
,
spectrum l
eak
age, H
a
rris w
i
ndow
Copy
right
©
2014 In
stitu
t
e o
f
Ad
van
ced
En
g
i
n
eerin
g and
Scien
ce. All
rig
h
t
s reser
ve
d
.
1. Introduc
tion
The a
ccu
rate
and efficie
n
t detectio
n
of powe
r
syste
m
harm
oni
c ha
s great si
gnificance to
the ha
rmo
n
ic so
urce
lo
ca
tion, ha
rmoni
c
cont
ro
lling,
safe
an
d e
c
on
omic op
e
r
ation
of p
o
w
er
system. The
pre
s
ent ha
rm
onic d
e
tectio
n method
s in
clud
e ze
ro
-crossing te
chni
que [1], Taylor
detectio
n
me
thod [2], wa
velet tran
sfo
r
m [3],
lea
s
t
sq
uare met
hod [4
-5], Neural
net
work
techni
que [6
-7] and
Di
screte Fou
r
ie
r
Tran
sfo
r
m (DFT) [8
-9]. Ha
rmoni
c p
a
ra
meters
could
be
cal
c
ulate
d
a
c
curately
by harm
oni
c a
nalysi
s
ba
se
d on
DFT i
n
co
ndition
s of syn
c
hro
nou
s
sampli
ng.
Wh
ile in a
c
tual
p
o
we
r
system,
the sy
st
em freque
ncy i
s
u
s
ually flu
c
tua
n
t co
ntinuou
sly,
and thi
s
will
make it
difficult to reali
z
e
sync
hro
nou
s
sampli
ng eve
n
usi
ng the
p
hase lo
cked l
oop
techni
que [10
]. The standa
rd IEC 6185
0 [11] for di
gital sub
s
tatio
n
provide
s
th
at the sampli
ng
rate of di
gital
sign
al mu
st b
e
20, 4
0
, 80
or 2
00 time
s
the po
we
r sy
stem fun
dam
ental fre
que
n
c
y.
Hen
c
e, the f
l
uctuatin
g gri
d
freq
uen
cy and thi
s
co
nstant
sam
p
l
i
ng fre
quen
cy will re
sult
in
asyn
chrono
u
s
sa
mpling,
and this
will
cau
s
e the
fence effe
ct and sp
ect
r
um lea
k
age
[12]
unavoid
ably
whe
n
u
s
ing
the DFT t
o
analy
z
e
h
a
rmo
n
ic. At
pre
s
e
n
t, the
win
d
o
w
ed
and
interpol
ation
FFT al
gorith
m
is u
s
ually
a
dopted
to
re
stra
in th
e fen
c
e effect
and
spectrum l
e
a
k
age
and, the wid
e
ly used
win
dow fun
c
tion
s incl
ude Bla
c
kman
windo
w, Han
n
ing
wind
ow, Nuttall
wind
ow [13
-
1
4
] and so on
. The side
-lo
be ch
arac
te
ri
stics of these
windo
w fun
c
tions, ho
wev
e
r,
are not ide
a
l and the effect
s of re
straini
n
g spe
c
tru
m
le
aka
ge ne
ed to be improve
d
.
In
this pap
er,
the Harri
s
windo
w which has
better si
d
e
-lob
e
sup
p
ressing
chara
c
teri
stics
is a
pplie
d, an
d the
sid
e
-l
o
be p
e
rfo
r
man
c
e i
s
i
m
pr
ove
d
furth
e
r
by i
n
crea
si
ng
the
order of
Ha
rris
wind
ow fun
c
tion to inhibit the spectru
m
leaka
ge more effectiv
ely. The simple and practical
adju
s
tment fo
rmula
s
of the
freque
ncy, a
m
plitude
a
n
d
pha
se a
ngle
are d
e
rive
d
by curve fittin
g
.
The digital si
mulation of complex
p
o
we
r sign
als
an
d
the harm
oni
c
a
nalysi
s
of the
a
c
tual sig
nal
are
presente
d
in thi
s
p
aper.
The
e
x
perime
n
tal
results sho
w
that in
the
co
ndition
s
of
asyn
chrono
u
s
sampli
ng
a
nd n
on-i
n
teg
e
r-peri
od t
r
un
cation, th
e p
r
opo
sed
algo
ri
thm ca
n b
e
u
s
ed
to calcul
ate the fundam
en
tal frequen
cy, harmoni
c a
m
plitude an
d
phase a
ngle
more accu
ra
tely
and effectivel
y.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
A Harm
onic
Analysis M
e
thod Based o
n
Harri
s
Win
d
o
w for
Digital Substation
(L
uo Ha
nwu)
5079
2. Spectrum
Leaka
ge due
to No-s
y
n
chronous Sam
p
ling
Whe
n
the total sampli
ng time is not an
integer
multi
p
le of the act
ual sig
nal cy
cle (i.e.
asyn
chrono
u
s
samplin
g),
burrs
will a
p
p
ear in
t
he p
o
s
t-sampli
ng
signal d
ue to t
he fact that
DF
T
will
repe
at th
ese
samplin
g
data
to fo
rm
a
co
ntinuou
s
cycle
wave
form, a
s
ill
ustrated i
n
Fi
gu
re
1(b
)
. While i
n
con
d
itions
of synch
r
on
o
u
s samplin
g, this pheno
menon
woul
d not appea
r (as
sho
w
n in
Fig
u
re 1
(
a
)). Th
e appa
re
nt discontinu
o
u
s
burrs in
Figu
re 1
(
b)
will sprea
d
into th
e
freque
ncy
sp
ectru
m
, and result in the so calle
d sp
ectrum lea
k
ag
e
phenom
eno
n. Figure 2 i
s
the
freque
ncy
sp
ectru
m
of th
e asyn
ch
ron
ous
sam
p
lin
g sig
nal sho
w
n in Fi
gure
1(b
)
, the a
c
tual
spe
c
tru
m
i
s
pre
s
ente
d
by
the da
sh
ed
line. In
co
ndition
s of a
s
ynchro
nou
s sam
p
ling, t
he
freque
ncy sp
ectru
m
will b
e
located at
both side
s of
the real spe
c
trum lin
e, su
ch a
s
the line
k
1
and
k
2
in Fi
g
u
re 2.
Windo
wed fun
c
tion
can
sup
p
re
ss the lon
g
-ra
nge fre
que
ncy leakag
e (a
s
sho
w
n i
n
Fig
u
re 1
(
c)) by
eliminating th
e sig
n
ifica
n
t discontin
uou
s burrs, a
nd i
n
terpol
ation
can
sup
p
re
ss the sho
r
t ran
ge spectral lea
k
a
ge.
Figure 1. Asynch
ron
o
u
s
an
d Synchrono
us
Sampling of the Same Sig
nal
Figure 2. Fre
quen
cy Spect
r
um of
Asynch
ro
nou
s Samplin
g Signal
3. Multi-term
Harris
w
i
ndo
w
s
3.1. Ideal Charac
teris
t
ics
of Windo
w
Function
An ideal
wind
ow fun
c
tion n
eed
s to satisf
y the followin
g
co
ndition
s: 1) Th
e wi
dth
of the
main
side
-lo
b
e
mu
st b
e
a
s
sm
aller a
s
p
o
ssible
to g
u
a
rante
e
high
freque
ncy
re
solutio
n
; 2
)
T
h
e
pea
k level
of
sid
e
-l
obe
m
u
st b
e
as lo
wer
as po
ssibl
e
to
en
sure
g
ood
ability of
noi
se
dete
c
ti
o
n
and inhi
biting
; 3) The atte
nuation
rate
of side
-lob
e should b
e
larg
e enou
gh. A wind
ow fun
c
ti
on
that meets t
he ab
ove th
ree
crite
r
ia
can elim
in
ate
the sp
ect
r
u
m
lea
k
age
a
nd fen
c
e eff
e
ct
con
s
id
era
b
ly. However, these
crite
r
ia a
r
e usually
co
ntradi
ctory a
nd non
e
of the cu
rrently use
d
wind
ow fun
c
t
i
ons
ca
n sat
i
sfy these th
ree
crite
r
ia
simultan
eou
sl
y. So the key probl
em
in
desi
gning th
e
windo
w fun
c
tion is to bala
n
ce the mai
n
lobe with the
side
-lob
e perf
o
rma
n
ce.
The ch
ara
c
te
ristic a
nd pro
c
e
ssi
ng re
qui
reme
nt
s of the signal a
r
e the key facto
r
s to be
con
s
id
ere
d
fo
r the
choi
ce
o
f
windo
w fun
c
tion. Fo
r
exa
m
ple, if only the det
e
c
tion accuracy of
the
sign
al freq
ue
ncy is of the
most con
c
e
r
ned th
an th
at of the sig
nal amplitu
d
e
, su
ch a
s
the
measurement
of the natural frequ
en
cy of a sign
al, a
recta
ngul
ar
wind
ow i
s
sui
t
able due to t
h
e
fact that the width of the
main lobe i
s
narrow to
b
e
disting
u
ish
ed ea
sily. If
the sign
al to
be
analyzed ha
s narrow b
and
width an
d strong interfe
r
e
n
ce n
o
ise, the wind
ow fun
c
tion with
sm
all
side
-lob
e sh
o
u
ld be cho
s
e
n
.
As for the
a
nalysi
s
of vol
t
age a
nd
cu
rrent
sig
nals t
hat contain
variou
s ha
rmo
n
ics in
power
syste
m
ca
used by
the acce
ssin
g of num
erou
s no
nline
a
r l
o
ads, m
o
re
attention
sho
u
ld
be
paid to the si
de-lo
be cha
r
a
c
teri
stics in choo
sing the
wind
ow fun
c
ti
on.
3.2, Multi-ter
m
Harris Wi
ndo
w
Harri
s
win
d
o
w
is a combi
nation of co
sine
win
dows,
and the gen
eral exp
r
e
ssi
on is a
s
following:
0
10
0
20
0
300
400
50
0
60
0
-1
0
1
N
A
m
pl
i
t
ude
(
a
) S
y
n
c
hr
onou
s
S
a
m
p
l
i
n
g
0
10
0
20
0
300
400
50
0
60
0
-1
0
1
N
Am
p
l
i
t
u
d
e
(
b
) A
s
y
n
c
h
rono
us
S
a
m
p
l
i
n
g
0
10
0
20
0
300
400
50
0
60
0
-5
0
5
N
A
m
pl
i
t
ude
(c
) W
i
nd
ow
e
d
A
s
y
n
c
h
ro
nous
S
a
m
p
l
i
n
g
0
20
40
60
80
10
0
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
F
F
T
of
t
he s
i
n
e
w
a
v
e
F
r
equ
enc
y
(
Hz
)
A
m
p
lit
u
d
e
A
c
t
ual
s
pec
t
r
al
c
o
n
t
e
n
t
k
0
S
pec
t
r
al
Lea
k
ages
M
a
x
i
m
u
m
s
pec
t
r
a
l
l
i
nes
k
1
T
he s
e
c
ond l
a
rge
s
t
l
i
nes
k
2
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 7, July 201
4: 5078 – 50
85
5080
0
2
()
(
1
)
c
o
s
(
)
,
0
,
1
,
,
1
K
m
m
m
wn
b
m
n
n
N
N
K
(1)
22
22
sinh
(
)
,1
m
m
bm
m
(2)
1
0
1
2
m
m
cb
b
(3)
0
0
2
m
m
bb
aa
cc
(4)
Coeffici
ents
b
m
of Harri
s wi
ndo
w functio
n
are sho
w
n i
n
Table 1.
Figure 3
and
Figure 4
sh
o
w
the tim
e
-d
o
m
ain a
nd fre
quen
cy cha
r
a
c
teri
stics of t
he 5
-
, 7-
and 9-te
rm
Harri
s
wind
o
w
functio
n
s
respe
c
tively.
It can be se
en that the better sid
e
-l
o
d
e
perfo
rman
ce
can b
e
obtain
ed by increa
si
ng the term o
f
Harri
s wi
ndo
w functio
n
.
Table 1. Co
efficients of Harris
Windo
w
Terms
b
0
b
1
b
2
b
3
b
4
b
5
b
6
b
7
4
0.3499
0.4850
0.1501
0.0150
/
/
/
/
5
0.3136
0.4661
0.1844
0.0339
0.0020
/
/
/
6
0.2867
0.4468
0.2070
0.0530
0.0063
0.0002
/
/
7
0.2657
0.4285
0.2217
0.0705
0.0125
0.0010
2.40E-05
/
8
0.2487
0.4117
0.2313
0.0857
0.0198
0.0026
1.52E-04
2.40E-06
Figure 3. Harris
Windo
w in
Time domain
Figur
e 4. Fre
quen
cy Ch
aracteri
stic of
Harri
s
Wind
ow
Table 2 give
s the com
p
a
r
ison of si
de-l
obe pe
rform
a
nce b
e
twe
e
n
Harris
wind
ow with
other re
gula
r
windo
ws. It can be le
arn
ed that Harri
s
wind
ow fun
c
tion
s are of
good sid
e
-lo
be
cha
r
a
c
t
e
ri
st
ic
s.
0
10
20
30
40
50
60
-0
.
2
0
0.
2
0.
4
0.
6
0.
8
1
H
a
r
r
i
s
W
i
nd
o
w
i
n
T
i
m
e
do
m
a
i
n
w
(
n)
5t
er
m
H
a
r
r
i
s
7t
er
m
H
a
r
r
i
s
9t
er
m
H
a
r
r
i
s
0
100
200
300
400
500
-
200
-
150
-
100
-5
0
0
F
r
equenc
y
B
i
ns
(
K
)
F
r
equenc
y
Res
p
ons
e
of
Har
r
i
s
W
i
ndow
W
(
K)
5ter
m
7ter
m
9ter
m
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
A Harm
onic
Analysis M
e
thod Based o
n
Harri
s
Win
d
o
w for
Digital Substation
(L
uo Ha
nwu)
5081
Table 2. Co
m
pari
s
on of the
Side-lob
e
Performa
nce of Multi Windo
w Function
s
w
i
ndo
w t
y
pe
side lobe level (dB)
asymptotic decay (dB/oct)
Hanning
-32
18
Black-
har
r
i
s -
9
2
6
Nuttall -82.6
30
3-term R-V
-46.8
30
4-term R-V
-61
42
5-term R-V
-82.6
54
6-term R-V
-88
66
7-term R-V
-101.1
78
8-term R-V
-114
90
9-term R-V
-126.8
114
6-term Harris
-130.9
24
7-term Harris
-156.7
18
8-term Harris
-182.7
12
9-term Harris
-209
6
3.3. Harmoni
c Analy
s
is M
e
thod
Bas
e
d
on Multi-te
r
m
Harris Wi
ndo
w
The specifi
c
step
s of the harm
oni
c ana
lysi
s metho
d
based on mul
t
i-term Harris windo
w
prop
osed i
n
this p
ape
r a
r
e
:
Firstly, the
samp
l
ed
discrete
sign
al is wind
owed to
obtain
a dat
a
seq
uen
ce
with finite length; Secon
d
ly, DFT is
used
to analyze th
e data se
que
nce to get th
e
discrete
spectrum of the
windo
wed
sig
n
a
l; Finally,
the amplitud
es
of the two
sp
ectral
line
s
th
at
are the
close
s
t to the actu
al spe
c
tral lin
e are inte
rpol
ated.
Take
the
spe
c
trum
sh
own
in Figu
re 2
a
s
an
exampl
e
,
the actu
al spectral line
is
k
0
, and
th
e
tw
o
c
l
ose
s
t line
s
ar
e
k
1
and
k
2
,
and th
e a
m
p
litudes of th
ese
two
line
s
a
r
e
y
1
an
d
y
2
respe
c
tively.
Two pa
ram
e
ters a
r
e introd
uce
d
as:
01
21
0.
5
21
yy
kk
yy
，
(5)
After getting
the value
of
β
, para
m
eter
α
c
a
n b
e
res
o
lve
d
b
y
us
in
g
th
e le
as
t
s
q
ua
r
e
curve
fitting
method to
p
e
rform
polyn
omial a
ppr
oximation, an
d t
he fre
que
ncy
,
amplitude
a
n
d
pha
se an
gle
of the comple
x harmoni
c si
gnal can be o
b
tained ultim
a
tely.
The ab
ove si
mple and
pra
c
tical a
d
ju
stment
formul
a
s
for the fre
q
uen
cy, amplitude an
d
pha
se a
ngle
derived
by po
lynomial ap
proximati
on an
d dou
ble pe
a
ks li
ne
corre
c
tion method
are
suitabl
e to general win
d
o
w
functi
o
n
s. For exampl
e, param
eter
α
of the
9-term Harris
win
dow
can b
e
ded
uced as:
35
=
5
.
931
+
0
.
729
+
0
.
3
2
7
(6)
The adju
s
tme
n
t formula
s
of the 6-term
Harri
s wi
ndo
w
are:
01
(0
.
5
)
f
kf
(7)
12
4
(
1
2)
(
1
0.
39
6
.
26
19
.
9
9
)
AN
y
y
(8)
arg
(
)
(
(
1
)
0
.
5
)
(
1
,
2
)
i
i
Xk
f
i
(
9
)
4. Simulation Verificatio
ns
The time-d
o
m
ain expression of the sig
nal used in si
mulation is:
11
1
1
()
s
i
n
(
2
)
ii
i
xt
A
f
i
t
(10
)
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 7, July 201
4: 5078 – 50
85
5082
Whe
r
e
f
1
=5
0.5Hz is the fu
ndame
n
tal freque
ncy of
the power sy
stem, and the sampli
ng
freque
ncy i
s
10kHz a
s
p
r
o
v
ided in IEC6185
0. The t
r
un
cated
dat
a length
N
i
s
2048, i.e. 1
0
.
34
fundame
n
tal
cy
cle
s
,
t
o
g
u
a
rante
e
a
s
yn
chrono
us sa
mpling. Th
e
amplitude
A
i
and p
h
a
s
e
θ
i
of
each harmoni
c are
sho
w
n i
n
Table 3.
Table 3. Para
meters of the Harmoni
c Sig
nals
harm
oni
c
1 2 3
4
5
6 7
8
9
10
11
A
i
(V
)
θ
i
(º)
220
0
.
05
4.4
39
10
60.
5
3
12
3
6
-
5
2.
7
2.1
146
3.2
97
1.9
56
2.3
43.
1
0.8
-19
1.1
4.1
The inp
u
t sig
nal
x
(
t
) i
s
p
r
o
c
e
s
sed respe
c
tively by 3-t
e
rm
Nuttall windo
w, 5- a
n
d 6-te
rm
Rife-Vin
ce
nt wind
ow, 6-,
7-, 8 and 9-term Ha
rri
s wind
ow. The
discrete sp
ectru
m
ca
n be
obtaine
d by FFT. Then th
e freque
ncy, amplit
ude a
n
d
pha
se angl
e of each ha
rmoni
c can b
e
cal
c
ulate
d
according to the
double
spe
c
t
r
um line inte
rpolation form
ula of the win
dow fun
c
tion.
The ado
pted
adju
s
tment fo
rmula fo
r Nut
t
all
windo
w a
nd R-V
wind
ow are prese
n
ted in
Referen
c
e [1
3] and
[14]
re
spe
c
tively. T
he
simula
tio
n
re
sult
s a
r
e
shown in
Ta
ble 4
and
Ta
bl
e 5.
Whe
r
e
D
Ai
an
d
D
φ
i
are the
relative devia
tion betwe
en
the
mea
s
ure
d
amplitud
e a
nd pha
se
an
gle
of each h
a
rm
onic
with th
e
actual
value
re
spe
c
tively;
D
f
0
i
s
the
rel
a
tive deviatio
n
between
th
e
measured val
ue of fundam
ental frequ
en
cy with the actual value.
Table 4. Co
m
pari
s
on
s of Relative Erro
rs
in Calculatin
g Amplitude a
nd Fre
que
ncy
Window
ty
p
e
D
f
0
D
A
1
D
A
2
D
A
3
D
A
4
D
A
5
D
A
6
D
A
7
D
A
8
D
A
9
D
A
10
D
A
12
Nuttall(III-
4)
3.E-8
9.E-
7
5.E-
6
5.E-
6
3.E-
7
2.E-
6
4.E-
6
4.E-
8
6.E-
7
5.E-
6
5.E-7
1.E-6
5-term R-V
3.E-
10
1.E-
7
7.E-
8
4.E-
6
3.E-
8
1.E-
0
2.E-
6
5.E-
8
7.E-
8
8.E-
7
4.E-8
8.E-9
6-term
Harri
s
2.E-7
3.E-
7
7.E-
6
2.E-
7
3.E-
7
4.E-
6
3.E-
8
2.E-
7
2.E-
6
1.E-
7
3.E-8
9.E-
10
7-term
Harri
s
2.E-
11
2.E-
7
2.E-
7
5.E-
6
1.E-
7
2.E-
7
2.E-
6
2.E-
8
1.E-
7
1.E-
6
5.E-8
1.E-8
8-term
Harri
s
5.E-
12
1.E-
7
1.E-
7
3.E-
6
7.E-
8
1.E-
7
2.E-
6
1.E-
8
7.E-
8
7.E-
7
4.E-8
4.E-9
9-term
Harri
s
4.E-
13
8.E-
8
9.E-
8
1.E-
6
5.E-
8
8.E-
8
5.E-
7
9.E-
9
5.E-
8
3.E-
7
3.E-8
2.E-9
Table 5. Co
m
pari
s
on
s of Relative Erro
rs in Calculatin
g Phase An
gl
e
Window
ty
p
e
D
φ
1
D
φ
2
D
φ
3
D
φ
4
D
φ
5
D
φ
6
D
φ
7
D
φ
8
D
φ
9
D
φ
10
D
φ
11
Nuttall(III-
4)
9.E-
7
5.E-
6
5.E-
6
3.E-7
2.E-
6
4.E-6
4.E-8
6
.
E
-
7
5.E-6
5.E-
7
1.E-
6
5-
te
r
m
R
-
V
9.E-
6
8.E-
5
5.E-
7
9.E-7
2.E-
6
-2.E
-6
-6.E
-7
8
.
E
-
6
-4.E
-6
-5.E-
6
3.E-
5
6-
t
e
rm
Ha
r
r
is
1.E-
4
8.E-
6
4.E-
6
-1.E
-6
-2.E-
6
4.E-7
2.E-7
9
.
E
-
8
7.E-7
-3.E-
6
1.E-
5
7-
t
e
rm
Ha
r
r
is
-9.E-
7
-3.E-
7
-6.E-
8
-4.E
-8
1.E-
7
-1.E
-7
-6.E
-8
-7.E-
8
-3.E
-7
2.E-
6
-7.E-
6
8-
t
e
rm
Ha
r
r
is
2.E-
7
5.E-
8
5.E-
9
-5.E
-9
-1.E-
8
1.E-9
6.E-
10
3.E-
10
5.E-9
-3.E-
8
9.E-
8
9-
t
e
rm
Ha
r
r
is
-2.E-
8
-5.E-
7
-3.E-
9
-8
.E-
11
-5.E-
9
-5.E-
10
-1.E-
10
-8.E-
9
-9.E-
10
4.E-
9
-2.E-
7
The cu
rve
s
of the measu
r
e
m
ent errors o
f
amplitude a
nd pha
se a
n
g
l
e are sho
w
n
in
Figure 5 and
Figure 6 re
sp
ectively.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
A Harm
onic
Analysis M
e
thod Based o
n
Harri
s
Win
d
o
w for
Digital Substation
(L
uo Ha
nwu)
5083
Figure 5. Co
mpari
s
o
n
s of
Relative Errors of
Harmoni
c Am
plitude
Figure 6. Co
mpari
s
o
n
s of
Relative Errors of
Harmoni
c Ph
ase Angl
e
As
can
be
seen f
r
om th
e
above
results that
th
e
ca
lculatio
n e
r
ro
r of th
e fun
d
a
mental
freque
ncy u
s
i
ng 9-te
rm Ha
rri
s win
d
o
w
interpol
ation i
s
4×10
-13
, and the cal
c
ulat
ion error of t
h
e
amplitude
an
d ph
ase a
ngl
e of fun
dame
n
tal compo
n
e
n
t are
8
×
10
-8
and 2×10
-8
res
p
ec
tively. The
cal
c
ulatio
n error
of the
am
plitude a
nd p
hase of 11
ha
rmoni
c a
r
e
2
×
10
-9
a
nd 2
×
10
-7
res
p
ec
tiv
e
ly.
The a
c
cura
cy of the harmonic
analy
s
is usi
ng
Harri
s win
d
o
w
is several ord
e
rs
of magni
tude
highe
r tha
n
t
hat of the
Nut
t
all wind
ow a
nd R-V
wi
nd
o
w
inte
rpol
atio
n metho
d
. In
the condition
s of
the same
term an
d d
egre
e
of th
e
co
rre
ction fo
rm
ul
a, the
algo
rithm
propo
se
d in
this p
ape
r
ha
s a
highe
r accu
ra
cy, which ca
n
achieve hi
gh
accura
cy
in the analy
sis of
complex ha
rmonic
sign
al.
5. Field test i
n
digital sub
s
ta
tion
Harmoni
c a
nalysi
s
by usin
g the propo
se
d mul
t
i-term Ha
rri
s win
d
ow
method is
perfo
rmed o
n
the data coll
ected by mon
i
toring devi
c
e
s
in a digital subs
tation, to f
u
rther verify the
accuracy and practi
cability
of the proposed me
thod. The experi
m
ental pr
ocedure is shown
i
n
Figure 7.
S
i
gn
al A
c
q
u
i
s
i
t
i
on
D
a
t
a
P
r
o
c
e
s
s
i
ng
a
nd
P
ack
ag
i
n
g
B
y
M
e
r
g
in
g
Un
i
t
s
P
o
w
e
r Q
u
al
it
y
M
o
ni
to
r
i
ng
D
e
v
i
c
e
D
i
g
i
t
a
l
Subs
t
a
ti
o
n
Dis
p
l
ay /
Con
t
r
o
l
L
aye
r
V
o
l
t
a
g
e
/
C
u
rren
t
S
i
g
n
a
l
M
u
l
t
i
-
t
erm
H
a
rri
s
W
i
ndo
w
D
o
u
b
l
e S
p
ect
ru
m
I
n
t
e
r
p
olat
ion
F
F
T
Ca
l
c
u
l
a
t
i
o
n
Out
put
o
f
F
u
nda
m
e
nt
a
l
a
nd
H
a
r
m
o
n
i
c
F
r
e
que
nc
y
,
A
m
pl
i
t
ude
, P
h
a
s
e
Figure 7. Flow Ch
art of Ha
rmoni
c Analy
s
is
The sa
mplin
g
frequen
cy of the colle
cted
data is 10
kHz a
n
d the total sampli
ng
time is
0.2s. The
an
alyzed fu
nda
mental fre
q
u
ency of a
c
tu
a
l
sign
al is 5
0
.052
Hz. Th
e h
a
rmo
n
ic
anal
ysis
results a
r
e sh
own in Ta
ble
7.
The
relative
error of fun
d
a
mental
amp
litude a
n
d
p
hase a
ngle
i
s
3
×
1
0
-8
an
d 3
×
1
0
-11
respe
c
tively, and the
rel
a
tive error
of 4
9
ha
rmoni
c
a
m
plitude
and
pha
s
e
angle
is 8
×
1
0
-8
an
d
9×1
0
-10
re
sp
ectively. The
s
e
re
sults
show that
the
pro
p
o
s
ed
m
e
thod
ca
n p
e
rform
ha
rm
onic
analysi
s
with
very high pre
c
isi
on to the actual
sign
als in digital sub
s
tation.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 7, July 201
4: 5078 – 50
85
5084
Table 7. Ha
rmonic An
alysis Re
sult
s of the Re
al Sign
al
Harmonic
Amplitude (V)
Relative errors of
Am
plitude
Phase (°)
Relative errors of
Phase
1 10500
3.E-8
-42.2
3.E-11
2 5
9.E-8
13.9
2.E-5
3 6
4.E-8
-87
1.E-8
5 65
1.E-08
82.9
2.E-10
7 86
6.E-8
37
2.E-10
9 10
5.E-9
34.9
7.E-10
10 5
8.E-8
149
1.E-10
11 13
7.E-8
113
6.E-10
13 12
2.E-8
104
3.E-11
14 1
7.E-8
89
7.E-9
15 7
8.E-8
-60
3.E-9
16 1
2.E-6
107
3.E-7
17 5
4.E-8
-78.2
1.E-10
19 12
8.E-8
133
1.E-10
21 5
6.E-8
30
9.E-10
22 1
6.E-8
124
1.E-9
23 5
9.E-8
143
1.E-09
25 5
8.E-8
152
5.E-11
26 1
5.E-8
-162
2.E-09
27 2
9.E-8
62.3
8.E-09
28 1
9.E-7
138
-1.E-07
29 2
9.E-8
-82.6
-3.E-10
30 1
4.E-8
-12.1
1.E-08
31 6
9.E-8
-9.28
1.E-08
32 2
7.E-7
91.9
-1.E-07
33 3
1.E-7
133
-7.E-11
34 1
3.E-8
-54.4
-2.E-09
35 2
8.E-8
-27.3
2.E-8
37 2
1.E-7
2.16
5.E-9
38 1
2.E-8
-122
1.E-9
39 2
8.E-8
0.32
4.E-7
40 1
4.E-7
172
4.E-8
41 2
1.E-7
-84.1
4.E-10
43 1
7.E-8
-0.38
1.E-7
44 1
2.E-7
-139
4.E-8
45 1
1.E-7
53.8
1.E-9
46 1
8.E-9
-169
3.E-10
47 2
6.E-8
38.6
8.E-9
48 1
1.E-7
0.57
2.E-6
49 1
8.E-8
-56.3
-9.E-10
6. Conclusio
n
A method of frequ
en
cy me
asu
r
em
ent an
d harm
oni
c a
nalysi
s
ba
sed
on multi-te
rm Ha
rri
s
wind
ow is p
r
o
posed in this pape
r. This m
e
thod
ca
n pe
rform fund
am
ental frequ
en
cy tracking a
nd
harm
oni
c ana
lysis with
hig
h
pre
c
i
s
ion e
v
en in the
sit
uation
s
of asynchrono
us
sampling, thu
s
is
more
suita
b
le to be ap
p
lied to ha
rm
onic
analysi
s
in digital substatio
n
. Th
e prin
cipl
e a
nd
reali
z
ation of
the double
spe
c
trum li
ne interp
olati
on algo
rithm
based o
n
multi-term
Harri
s
wind
ow i
s
int
r
odu
ce
d in d
e
tail, and the
simple
pr
act
i
cal inte
rpol
ation form
ula i
s
calculated
by
usin
g the
curving fitting. T
he
re
sults of
simulatio
n
a
n
d
p
r
a
c
tical
ap
plicatio
n in
di
gital sub
s
tation
sho
w
that the pre
s
ente
d
method is
of less
com
put
ation, highe
r accuracy an
d
better pra
c
ti
cal
value in engi
neeri
ng.
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