TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol. 12, No. 9, September
2014, pp. 64
9
4
~ 650
1
DOI: 10.115
9
1
/telkomni
ka.
v
12i9.461
3
6494
Re
cei
v
ed O
c
t
ober 2, 20
13;
Revi
se
d May 28, 2014; Accepte
d
Ju
ne
12, 2014
Compressed Sensing High-accuracy Detection for
Electric Power Interharmonics
Tiejun Cao,
Jingfan
g Wa
ng*
Schoo
l of Information Sci
enc
e and En
gi
neer
ing, Hu
na
n Internatio
nal Ec
on
omics Univ
ersit
y
Chan
gsh
a
, Ch
ina, postco
de:4
102
05
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: matlab_
b
y
sj
@12
6
.com
A
b
st
r
a
ct
Interhar
mo
nics
freque
ncies
are n
o
t integ
e
r mu
ltip
le of
the fund
a
m
ental fre
que
n
cy, an
d
interh
ar
mon
i
cs
ampl
itudes
ar
e far less
tha
n
fun
d
a
m
e
n
tal
ampl
itude
an
d har
monics
a
m
p
litu
des, w
h
i
c
h
me
an hi
gh se
nsitivity to des
ynchro
ni
z
a
ti
on
proble
m
s, so
it
’
s
dificult to estim
a
te
i
n
ter
har
mo
ni
cs. In this
pap
er, a new
meth
od b
a
se
d on ra
ndo
m sparse sa
mp
ling
an
d co
mpresse
d sens
i
ng (CS) Bre
g
m
a
n
techni
qu
e w
a
s prop
osed to
es
timate th
e inter
har
mo
nics. T
he rand
o
m
sa
m
plin
g h
a
s foll
o
w
ing adv
antag
es;
alias-fre
e
, samplin
g frequ
enc
y need n
o
t ob
ey the Nyquist
limit, an
d hig
her frequ
ency
resol
u
tion. So
the
rand
o
m
sa
mp
l
i
ng c
an
meas
ure the s
i
g
nal
s w
h
ich
their
freque
ncies
c
o
mpo
nent
are
close, a
nd c
a
n
imple
m
ent the
high
er frequ
e
n
cies
me
asur
e
m
e
n
t w
i
th low
e
r sa
mpl
i
ng fr
equ
ency. How
e
ver, the ran
d
o
m
sampli
ng
exist
s
the n
o
ise
in
spectru
m
a
n
a
l
ysis, so it
’
s
di
fficult to esti
mate the
low
a
m
p
litu
de si
gn
a
l
s.
Co
mpress
ed s
ensi
ng ca
n w
o
rk out this pro
b
le
m by
des
ig
nin
g
obs
ervati
on
matrix a
n
d
w
i
th the spar
sit
y
reconstructi
on
of the s
i
gn
al
i
n
the
F
ouri
e
r
do
ma
in; i
n
a
d
d
itio
n, the
ap
p
licatio
n
of
CS can esti
mate
t
h
e
amplit
udes and
phases of
the signals
exact
l
y. The
results ofexper
im
ents
show
that
the proposed method
can esti
mate t
he inter
har
mon
i
cs exactly,eve
n
if the
interh
a
r
mo
nics freq
ue
ncies ar
e clos
e
the funda
ment
al
freque
ncy an
d interh
ar
mon
i
cs
amp
litu
des ar
e far less
than
funda
me
ntal a
m
p
litu
de an
d can meas
ure hi
g
h
order i
n
terhar
mo
nics w
i
th lo
w
e
r sampl
i
n
g
frequ
ency.
Ke
y
w
ords
: co
mpr
e
sse
d sens
ing, inter
har
mo
nics, rand
o
m
sampli
ng, spars
e
sampl
i
ng, sp
ectrum a
n
a
l
ysi
s
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
With the
wide
spread
u
s
e
of
nonlinea
r
loa
d
s,
power sy
stem, a large numbe
r of frequen
cy
of the funda
mental fre
q
u
ency of a
neu
ploidy
bet
we
en the h
a
rm
o
n
ics [1]. Pollu
tion ca
used b
y
the
pre
s
en
ce of
harm
oni
cs a
n
d
interha
r
mo
nics on t
he p
o
we
r syste
m
environ
ment
must be ri
ght
to
effective gov
erna
nce, and
accurate det
ection
of
harmonics i
s
tha
t
the premi
s
e
of govern
a
n
c
e.
H
o
w
e
ve
r
,
in
te
r
-
h
a
r
mo
n
i
c
c
h
ar
ac
te
r
i
s
t
ics
de
te
r
m
ine its
d
e
t
ec
tion
is
d
i
ffic
u
lt fo
r
ha
r
m
o
n
ic
detectio
n
. First, the inter-harm
oni
c fre
quen
cy of
th
e fundam
ent
al frequ
en
cy of aneu
ploid
y
is
often difficult to determine t
he cycle of the wa
veform
contain
s
interh
armo
nics. Ha
rmoni
c and th
e
fundame
n
tal i
n
ter-harmoni
c freq
uen
cy d
o
main i
s
le
ss
than on
e wo
rking frequ
en
cy, which m
e
a
n
s
that the hig
h
e
r h
a
rm
oni
cs in the
dete
c
tion
of
inter-freq
uen
cy re
solutio
n
. Betwee
n ha
rmo
n
ic
amplitude i
s
often far le
ss than the am
plitude
of the
fundame
n
tal
and ha
rmo
n
i
c
compo
nent
s,
whi
c
h m
ean
s that the
ha
rmonic
comp
onent
of spe
c
tral
lea
k
ag
e
with
high
sensitivity, inter-
harm
oni
c an
d fundame
n
tal and ha
rmo
n
ic compo
n
e
n
ts
the frequ
ency is cl
ose
to, this effect is
more p
r
on
ou
nce
d
. Theref
ore, the inter-ha
rmoni
c an
alysis meth
o
d
shoul
d hav
e the followi
ng
cha
r
a
c
teri
stics: by non-sy
n
c
hrono
us
sa
mpling is
sm
all; samplin
g
time sho
u
ld n
o
t be too long
, so
as
to avoid
b
e
fore and after
the coll
ect
i
on
of
data
from the
same
sign
al; with
high freque
n
c
y
resolution.
Accu
ra
cy du
e
to non
-synch
rono
us sampl
i
ng an
d d
a
ta
truncated, u
s
i
ng the
Fa
st F
ourie
r
Tran
sfo
r
m (F
FT) al
gorithm
for ha
rmoni
c analysi
s
to p
r
odu
ce
sp
ect
r
al lea
k
ag
e an
d fence effect,
the impa
ct o
f
harm
oni
c a
nalysi
s
[2-3]. To
red
u
ce
su
ch
errors,
the sch
o
lars ba
sed
on
the
recta
ngul
ar
windo
w [4], Hanning
wind
o
w
[5], the Ha
mming wi
ndo
w [6] Blackm
an, wind
ows
[7],
Blackman
-Ha
rri
s
wind
ow [8], the Kai
s
e
r
windo
w
[9]
su
ch
as si
gn
al wi
ndo
ws a
nd inte
rpol
ated
FFT algo
rith
m can re
du
ce encounte
r
e
d
alone FFT
spe
c
tru
m
lea
k
ag
e and fen
c
e effect, imp
r
ove
the dete
c
tion
accu
ra
cy of
t
he ha
rmoni
c paramete
r
s,
but ca
n n
o
t
be dete
c
ted
near the inte
ger
harm
oni
cs a
s
ked
ha
rmo
n
ic; adopt
funda
mental
and
h
a
rmo
n
ic pa
ra
meter
estim
a
tion ba
se
d o
n
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Com
p
re
ssed
Sensin
g Hig
h
-
accu
ra
cy De
tection for Ele
c
tri
c
Powe
r…
(Tiejun
Cao
)
6495
highe
r o
r
d
e
r co
sin
e
com
b
ination
win
d
o
w
of the
spectrum [5,
7, 10] o
r
m
o
re li
ne
s [11
-
12]
interpol
ation
FFT alg
o
rith
m, the soluti
on of
hig
her equatio
ns [
13-1
5
], com
puting
compl
e
x;
contin
uou
s wavelet tran
sfo
r
m [16-17] to achieve t
he
detectio
n
of inter / harm
o
nic, but there
is
mutual inte
rfe
r
en
ce of diffe
rent scale
s
of
the
wavel
e
t functio
n
in the
freque
ncy d
o
m
ain, wh
en t
h
e
sign
al to be
detecte
d wit
h
simila
r fre
quen
cy
ha
rm
onic to
dete
c
t method
wi
ll fail; on Prony
method [18
-
19] is h
a
rm
onic, b
e
twe
e
n
harmoni
c
analysi
s
an
d
modelin
g e
ffective way
to
accurately estimate the si
nusoidal
com
pone
nt
of frequen
cy, ampl
itude and p
h
a
se a
ngle, b
u
t it
need
s to
solv
e two
odd
eq
uation a
nd
p
o
lynomial tim
e
, the hig
h
computation
a
l
compl
e
xity and
noise-sen
sitive; there a
r
e
othe
r meth
ods [20
-
22]
,
or limited
frequ
en
cy resolution, o
r
l
a
rge
cal
c
ulatio
n, there a
r
e limitations
in the
sp
ecific a
ppli
c
at
ion.
De
sign
ed a
compresse
d
b
e
twee
n the p
e
rception
of harm
oni
c det
ection m
e
tho
d
s, time
domain i
s
lower tha
n
the Nyquist the
o
ry
of rando
m samplin
g, Bregman, freque
ncy-dom
ain
recon
s
tru
c
tio
n
with high
accuracy d
e
tection
sign
al all the h
a
rmo
n
ics an
d interh
arm
o
nics
freque
ncy, a
m
plitude
and
pha
se. In t
h
is
pape
r,
a
theoretical
analysi
s
a
n
d
cal
c
ulatio
n
of
derivation, ra
ndom sa
mpli
ng coul
d
b
e
circu
m
vented
by Fou
r
ier do
main
spe
c
tral
lea
k
ag
e, pi
cket
fence effe
ct, as well as n
on-inte
ge
r times a
wave
phen
omen
on.
The simul
a
tion re
sults
sh
ow
that: the p
r
op
ose
d
al
gorith
m
can
effecti
v
ely elimi
nate
all the
ha
rmo
n
ics inte
rfere
with e
a
ch oth
e
r
to improve th
e accuracy of
signal a
nalysis, harm
oni
c analysi
s
[23-29].
2. Random S
a
mpling and
Analy
s
is
2.1. The Dra
w
b
a
ck
s of
Uniform Sampling
Uniform sam
p
ling of a fu
nction
of time is
a li
nea
r function
of the sta
nda
rd,
su
ch a
s
sampli
ng tim
e
interval
dist
ribution.
Defi
ne the
sam
p
l
ed si
gnal
)
(
t
x
, the s
a
mpling interval
t
, the
sampli
ng tim
e
point
t
n
t
n
, the
sam
p
ling
freque
ncy
t
f
s
1
, and to me
et th
e samplin
g
theore
m
, is
greater th
an 2 ti
mes th
e hig
h
e
st fre
que
ncy
of the value
sign
al. For a l
i
mited length
of
the sam
p
led
signal di
scretization, ie
)
]
:
1
([
]
[
t
N
x
n
x
, N is sampl
i
ng point
s, sampling
duratio
n
t
N
T
.
By Fou
r
ier tra
n
sform
analy
s
is of
sa
mpl
ed
sign
als
)
256
,
256
,
185
(
),
2
sin(
)
(
Hz
f
N
Hz
f
ft
t
x
s
. Signal spe
c
trum a
nalysi
s
re
sults
sho
w
n
in Figure 1.
Figure 1. Signal Spe
c
tru
m
Analysis of
Uniformly Sa
mpling (f
s=25
6Hz)
It can b
e
see
n
from
Figu
re
1, the
sampli
ng fre
que
ncy
is le
ss than
2 times of th
e sig
nal
real fre
que
ncy value, frequen
cy value
Hz
f
185
of the aliasin
g
sign
al 71
Hz. The re
al si
gnal
spe
c
tru
m
i
s
n
o
t distin
gui
sh
ed b
e
cau
s
e t
he ali
a
si
ng
si
gnal
sp
ect
r
u
m
is eq
ual to
the true
sig
n
a
l.
Also
noted
th
at the
ca
se
s i
n
the
freq
uen
cy reso
lution
of 1Hz, th
e
si
gnal f
r
equ
en
cy is
an i
n
tege
r
multiple of the freque
ncy resol
u
tion, so
the ab
ility to
accurately measure the freque
ncy valu
e.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 9, September 20
14: 64
94 – 650
1
6496
In the previ
ous
ca
se
s, other p
a
ra
m
e
ters
co
nsta
nt, the chan
ging samplin
g frequ
en
cy is
Hz
f
s
512
, to meet the
limitations of
the sam
p
ling
theore
m
. Signal spe
c
trum
analysi
s
resul
t
s
s
h
ow
n
in
F
i
gu
r
e
2
.
Figure 2. Signal Spect
r
um
Analysis of Uniformly Sam
p
ling (fs=5
1
2
H
z)
Figure 2
sho
w
s th
e alia
si
ng si
gnal
(0,
2
/
s
f
) ba
nd, but
due to
cha
n
g
e
s in
sa
mpli
ng
freque
ncy, th
e frequ
en
cy resol
u
tion be
comes
2Hz,
si
gnal the true f
r
equ
en
cy of families
185
Hz is
not an integ
e
r
multiple of the freq
uen
cy
resoluti
on, thus le
adin
g
to spe
c
tral lea
k
ag
e and fe
n
c
e
phen
omen
on,
so th
at the
measured fre
quen
cy value
is
Hz
f
188
which is d
e
viated from
th
e
corre
c
t value.
As ca
n be
se
en from th
e
above a
nalysis, uniform sampling i
s
li
mited by the
sampli
ng
freque
ncy li
mit; aliasing
freque
ncy; freque
ncy resolu
tion is
not
high, there is the p
r
oble
m
of
spe
c
tral le
akage an
d fence phen
omen
a
.
2.2. Random
Sampling and its Fourie
r Trans
f
orm
Ran
dom
sa
mpling,
som
e
times
call
ed
non
-unifo
rm
sam
p
ling, a
s
o
ppo
sed
to unifo
rm
sampli
ng
of
a samplin
g
method. T
h
e
sa
mpling
int
e
rval
ran
dom
sa
mpling
is
rand
om, the
time
interval is ge
nerally set to unequal inte
rvals,
not a linear fun
c
tion
of the samp
ling points a
n
d
sampli
ng tim
e
. Rand
om sampl
e
from
the samplin
g theore
m
limit, increa
si
ng the frequ
ency
detectio
n
ra
n
ge can b
e
de
tected in th
e
sho
r
t l
ength
of the data, l
o
w
sampli
ng
freque
ncy to
the
highe
r o
r
de
r frequ
en
cy, allowin
g
real
-time to q
u
ickly meet the
req
u
ire
m
ent
s of a
pa
rticular
occa
sion.
Th
e mo
st imp
o
rtant thing i
s
t
hat
the
ra
ndo
m samplin
g
of non
-unifo
rm samplin
g
can
eliminate
sig
nal alia
sin
g
p
r
oble
m
s
ca
used by u
n
iform sam
p
ling;
also
ha
s the
advantag
e of
high
freque
ncy
re
solutio
n
, re
d
u
cin
g
the
spectrum le
ak to eliminat
e the p
r
obl
e
m
of the fe
nce
phen
omen
a.
In the exampl
e above,
the
other paramet
ers con
s
tant,
)
(
)
(
),
(
)
1
,
0
(
n
n
t
x
n
x
n
g
T
rand
t
,the switch
to rando
m
sa
mpling,
whe
r
e
ra
nd
(0,1)
rand
om n
u
m
ber b
e
twe
e
n
(0,1),
N
n
,
,
2
,
1
,g (n
) is
a nonli
n
ear fun
c
tion
of n. Fouri
e
r
trans
form:
N
n
n
t
j
n
x
X
1
)
exp(
)
(
)
(
(1)
Figure 3. Signal Spect
r
um
Analysis of Random Sam
p
ling (Avera
ge
fs=25
6
Hz)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Com
p
re
ssed
Sensin
g Hig
h
-
accu
ra
cy De
tection for Ele
c
tri
c
Powe
r…
(Tiejun
Cao
)
6497
Ran
dom
sam
p
le (N
= 2
5
6
)
, Fou
r
ier tran
sform
spe
c
tru
m
analy
s
is re
sults sho
w
n i
n
Figu
re
3. The
u
s
e
o
f
ran
dom
sa
mpling tim
e
sampli
ng i
n
te
rval in
crea
se
s, the
freq
ue
ncy
re
solutio
n
to
eliminate the
phen
omen
on
of the fence. As a resu
lt of random
sa
mpling, the al
iasin
g
sign
al will
no long
er
concentrate
d
on so
me sp
ecial p
o
ints
and the
sa
mpling fre
q
u
ency, but ev
enly
distrib
u
ted to
all of th
e
si
gnal frequ
en
cy ba
nd.
In
addition,
sp
e
c
tral l
e
a
k
ag
e
will
ca
use t
h
e
spe
c
tru
m
noi
se. The
spe
c
t
r
um noi
se
ca
n be red
u
ced
with the incre
a
se of sampli
ng point
s.
3.
Compres
s
e
d Sensing
Principle
3.1. Compre
ssed Sensin
g Repr
esen
tation
Comp
re
ssed
Sensin
g (CS) theory main
idea is: Su
pp
ose
a length
Nof the
signa
l
x
on
an o
r
thog
ona
l basi
s
or tig
h
t
frame
coeffi
cient
s
is spa
r
se
(ie
only a
few n
on-ze
ro
co
efficient),
the coeffici
en
ts of proje
c
tio
n
to another
N
M
:
(
N
M
)not rel
a
ted to a transfo
rm
-ba
s
ed
observation
s
, the colle
ctio
n of the
ob
se
rvations
1
y:M
. Signal
x
is accurately recovered
by solving an
optimizatio
n probl
em
in virtue of these o
b
se
rvation
s
.
First, if the si
gnal
N
R
x
on an
orthogon
al ba
si
s o
r
tight fra
m
e
is com
p
re
ssi
ble,
t
h
e
obtaine
d tran
sform
coeffici
ents
x
T
,
is the equivalent o
r
s
parse a
pproximation of
x
;
the second
step, to de
sig
n
a
stable,
n
o
t relate
d to
the tran
sfo
r
m
-
ba
sed
,
N
M
dime
nsio
n
observation
matrix
to obs
e
rve
x
the
up
comin
g
p
r
oje
c
ted
onto th
e M-dimen
s
i
onal
spa
c
e,
observing
a
colle
ction
x
y
o
f
th
e
pr
oc
es
s fo
r
the
c
o
mp
re
ss
io
n o
f
th
e s
a
mp
lin
g pro
c
ess
,
namely the takin
g
of sam
p
les [26
-
29].
Finally,
the use of optimi
z
ation pro
b
le
m solving the
x
’s
exact or ap
proximate app
roximation
x
ˆ
.
Whe
n
the noi
se z o
b
servat
ions,
x
y
+
z
(2)
It can be tran
sform
ed for t
he sa
ke,
2
1
||
||
.
.
||
||
min
x
y
t
s
x
T
x
(
3
)
Or,
1
2
||
||
||
||
2
1
min
arg
ˆ
x
x
y
x
T
x
(
4
)
3.2. Separab
l
e Bregman I
t
era
t
iv
e Algorithm to Re
s
t
ore the Signal
Problem
(4
) to solve the fi
rst conve
r
ted
to the spa
r
se vector
(5
) to solve,
A
,
then:
1
2
2
||
||
||
||
2
min
arg
ˆ
A
y
(
5
)
Bregma
n
alg
o
rithm [17-20
], specif
ic
ste
p
s are as foll
ows:
1) Cal
c
ulate:
1
)
(
N
T
I
A
A
B
,
N
I
is N-dimensional unit
matrix,
y
A
F
T
;
0
0
,
d
b
are for the N-dime
nsi
onal
zero vecto
r
.
2) Given
)
10
(
, iteration termin
ation co
ndition
s
)
001
.
0
(
, the number of
iteration
s
1
n
.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 9, September 20
14: 64
94 – 650
1
6498
3) Cal
c
ulate:
)
(
1
1
n
n
n
b
d
F
B
,
)
0
,
1
|
max(|
)
(
1
1
n
n
n
n
n
b
b
sign
d
,
4)
n
n
n
n
d
b
b
1
.
5) If
||
||
1
n
n
,
1
nn
, Go to Step 3); Othe
rwise, stop the i
t
eration,
n
ˆ
3.3.
Signal Lo
w
-
speed Samp
ling Design
w
i
th the
Obs
e
rv
ation Matrix
The de
sign
of the obse
r
ver is to design e
fficient observation
matrix can capture the
desi
gn of
a spa
r
se sign
a
l
useful
info
rmation
effi
ciency of
the observation
s
(ie, sam
p
lin
g)
proto
c
ol, whi
c
h
the sp
arse sign
al
i
s
com
p
re
ssed
i
n
to
a sm
all a
m
ou
nt of data.
Th
ese
ag
ree
m
e
n
ts
is non
-a
dapti
v
e, only need
a small am
o
unt of the
fixed wavefo
rm
and the o
r
igi
nal sig
nal lin
king
these fixed
waveform an
d
sign
al to prov
ide a compa
c
t repre
s
e
n
tation of the ba
se. In addition,
the observati
on proce
s
s i
s
inde
pen
de
nt of the
sig
nal itself. Usi
ng an optimi
z
ed
re
con
s
tructed
sign
al ca
n ga
ther a sm
all n
u
mbe
r
of observation
s.
Sampling
int
e
rval [0,
T] in thi
s
i
n
terval
we
re
coll
ecte
d
randomly
M
point
s,
M
i
T
rand
t
i
,
,
2
,
1
,
)
1
,
0
(
,
)
1
,
0
(
rand
are ra
ndom p
o
in
ts between
(0,1)
,
)]
(
,
),
(
),
(
[
2
1
T
M
t
x
t
x
t
x
x
M
R
x
y
Interval reconstructio
n
o
f
the comple
x freque
ncy
domain N-di
mensi
onal
N
M
C
N
,
.
Comp
re
ssed
sen
s
in
g harm
onic dete
c
tio
n
is to find a
mappin
g
:
N
M
C
R
F
:
.
De
sign a ran
dom ob
se
rvation matrix
N
n
M
m
n
T
t
N
l
i
N
n
m
N
N
l
s
m
,
,
2
,
1
;
,
,
2
,
1
,
))
(
2
exp(
1
)
,
(
2
/
1
2
/
(
6
)
s
T
is unifo
rm tim
e
-do
m
ain
re
constructio
n
of
the equivale
nt of N-p
o
int
sampli
ng inte
rval,
is Fourier-based,
N
j
e
N
t
N
jt
i
j
,
,
2
,
1
,
)
(
/
2
2
/
1
. This
desi
gn to me
et irrel
e
vant
and
limitations of iso
m
e
t
ric re
si
stan
ce.
A
, This ran
dom sa
mple
of observati
ons
rand
om ch
aracteri
stics. O
b
se
rvat
ion
m
a
trix of rand
om un
rel
a
ted
ch
ara
c
te
risti
c
i
s
a
suffici
en
t
con
d
ition for
the rig
h
t to restore the
si
gnal, t
he h
e
i
ght of the o
b
se
rvation m
a
trix and
sig
nal
irrel
e
vant to ensu
r
e the effe
ct
ive re
storati
on of the sign
al.
3.4.
Implementation Steps of
the Harmoni
c Detec
t
ion in Compre
ss
ed Sensing
1)
In the time domain
give
n interval
were
colle
cted
rand
omly M
points, the
point
seq
uen
ce for
the observati
on vector;
2)
Re
con
s
tru
c
t the N poi
nts in
this interval
s
T
,
whi
c
h have freque
ncy-dom
ain re
solutio
n
Hz
NT
f
s
1
3)
By (6), the d
e
sig
n
of the
N
M
observation matrix
, the
N
N
or
d
e
r
in
ve
rs
e
Fouri
e
r tra
n
sf
orm matrix
;
4)
Bregma
n
alg
o
rithm for reconstructio
n
of complex N-p
o
int freque
ncy domain;
5)
Given the m
agnitud
e
of the thre
sh
old,
when th
e re
con
s
tru
c
tion
of the freque
ncy
domain i
s
larger tha
n
the
threshold, fi
n
d
the app
rop
r
iate frequ
en
cy, amplitude and
pha
se.
4. Experimental Ev
aluation
Signal
contai
ns th
e fund
a
m
ental, ha
rm
onics
a
nd
ha
rmoni
cs, and
their
param
eters in
Table 1, the e
x
pressio
n
:
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Com
p
re
ssed
Sensin
g Hig
h
-
accu
ra
cy De
tection for Ele
c
tri
c
Powe
r…
(Tiejun
Cao
)
6499
Table 1. Truth Comp
one
nt of t
he Signal & their Testin
g Re
sult
Wavefor
m
Actual value
Detection value
Freq./
H
z
Margin/V
Phase/
Ԩ
Freq./
H
z
Margin/V
Phase/
Ԩ
Fundament
al
50.00
35.0000
0.0000
50.00
34.9816
0.0056
Interhar
monic
75.00
5.0000
155.0000
75.00
4.9715
154.9602
Harmonic
150.00
7.0000
35.0000
150.00
6.9858
34.9871
Interhar
monic
175.00
3.0000
50.0000
175.00
2.9770
49.9232
Harmonic
250.00
4.0000
70.0000
250.00
3.9800
69.9572
Harmonic
350.00
1.1250
115.0000
350.00
0.9840
115.0625
6
1
)
2
cos(
)
(
i
i
i
i
t
f
A
t
x
(
7
)
The high
est
signal freq
uen
cy:
Hz
f
350
max
, Were co
llected rand
o
m
ly in one se
con
d
of time M =
2
56 poi
nts, its
rand
om e
quiv
a
lent samplin
g freq
uen
cy
max
256
f
Hz
f
s
, Ran
dom
equivalent
sa
mpling frequ
e
n
cy is
mu
ch l
e
ss t
han
2 times th
e high
est si
gnal f
r
e
quen
cy, doe
s not
meet the Ny
quist sampli
n
g
theorem; If the frequ
e
n
c
y of su
ch u
n
iform sampl
i
ng, the Fou
r
ier
transfo
rm of the existen
c
e
of a
certain spectrum alia
sing and lea
k
a
ge, is not po
ssi
ble to dete
c
t
tothe si
gnal
h
a
rmo
n
ics. Mi
ning M
=
256
usin
g thi
s
me
thod recon
s
truct the
fre
q
u
ency
domai
n
N
=
768
256
3
point, resol
u
tion 1Hz, te
st re
sults a
r
e
in Table 1
the right sid
e
of Figure 4.
Harmoni
c fre
quen
cy, ampl
itude, initial p
hase of
the true value
s
and mea
s
u
r
ed v
a
lue
s
are plot
ted
on the sam
e
plot, the resul
t
s are very a
c
curate.
Figure 4
sh
o
w
s the o
r
igi
n
al sig
nal, the
sam
p
ling
poi
nts a
nd the
reco
nstructio
n
of time-
domain
sign
al, the picture sh
ows th
e origin
al si
gnal and re
con
s
tru
c
ted
sign
al amplit
ude-
freque
ncy
di
agra
m
, the l
o
we
r p
a
rt
of the o
r
igi
nal
sign
al a
nd
reco
nstructe
d
sig
nal f
r
equ
ency
diagram; time domain reco
nstru
c
tion of t
he sig
nal rel
a
tive error
Rel
a
tiveerror = 0
.
0014.
Figure 4. The
Inter-ha
r
mo
n
i
c Com
p
resse
d
Sensin
g De
tection
5. Conclusio
n
Ran
dom sam
p
ling tech
niq
ue as a no
n-uniform
samp
ling method
can effectively improve
the sam
p
ling
rate of the sampling
syst
em. In r
and
o
m
sampli
ng,
the sam
p
ling
time interval
o
f
non-unifo
rm distrib
u
tion can
not
be col
l
ected
e
nou
g
h
sam
p
le val
ues fo
r sig
n
a
l
recon
s
tru
c
tion
.
The high pre
c
isi
on of
the power ha
rmo
n
ic analy
s
is
for
elect
r
ic m
e
tering,
ha
rm
onic po
we
r fl
ow
cal
c
ulatio
n, equipme
n
t, network te
sting, power
sy
ste
m
harm
oni
c compen
satio
n
, and inhibitio
n
of
great
sig
n
ificance. In thi
s
pap
er, th
e
sign
al i
s
spa
r
se
in
the F
ourie
r tran
sfo
r
m, to d
e
si
g
n
a
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 9, September 20
14: 64
94 – 650
1
6500
rand
om ob
se
rvation matri
x
, then sparse samplin
g; the use of Bregma
n
iterative algorith
m
su
ccessfully restored th
e
si
gnal.
Thi
s
me
thod
without
addin
g
a
n
y h
a
rd
wa
re
co
st
s o
n
the
ba
si
s of
the limited ra
ndom
sampli
ng value reconstructi
o
n
freque
ncy do
main si
gnal.
The expe
rim
ents
sho
w
e
d
that frequ
en
cy-d
o
m
ain sp
arse
sign
al we
ll be
low the sam
p
ling rate of the signal
Nyq
u
ist
freque
ncy sampling, co
mpre
ssed sensi
ng sig
n
a
l
reco
nstructi
on algo
rithm
can accu
ra
tely
recon
s
tru
c
t the freq
uen
cy
-dom
ain
sign
al throu
gh t
he metho
d
of this pa
pe
r, high
-preci
sion
detectio
n
of sign
al of each harm
onic
and in
ter-ha
rmonic freque
ncy, amplitud
e and pha
se
. A
theoreti
c
al a
nalysi
s
and
cal
c
ulatio
n of derivati
on o
f
this metho
d
to circumv
ent the Fou
r
ier
domain
spe
c
trum l
e
a
k
age,
picket fen
c
e
effect, and
no
n-integ
e
r times
a
wave p
h
enome
non.
T
he
prop
osed
alg
o
rithm
ca
n e
ffectively eliminate
all th
e ha
rmo
n
ics interfe
r
e
wit
h
ea
ch
othe
r to
improve the a
c
cura
cy of sig
nal analy
s
is,
su
itabl
e for hi
gh accu
ra
cy harm
oni
c ana
lysis.
Referen
ces
[1]
T
e
sta A, Akram MF
, Burch
R. In
terharmo
nic: theor
y a
n
d
mode
lin
g.
IEEE Transactionson Power
Deliv
ery
. 200
7; 22(4): 233
5-2
348.
[2]
GREGORIO A
,
MARIO S
,
A
M
ERIGO
T.
W
i
ndo
w
s and in
t
e
rpolation algorithms
to improve electrical
measur
ement accurac
y
.
IEEE Trans. on Instrum
e
nt
ation and Meas
urem
ent.
1989; 38(
4): 856-8
63.
[3]
X
U
E H, YANG RG.
Pre
c
ise
alg
o
r
it
h
m
s fo
r
ha
rmo
n
i
cs an
a
l
ysis b
a
s
e
d
on
FFT a
l
g
o
r
it
h
m
.
Procee
din
g
s
of the CSEE. 2002; 22(
12): 10
6-11
0.
[4
]
JAIN VK, COLL
INS WL
, DAVIS DC. Hig
h
-
a
c
cu
ra
cy
a
n
a
log
me
a
s
u
r
e
m
e
n
t
s via
in
te
rp
o
l
a
t
ed
FFT.
IEEE
T
r
ans. Instrum.
Meas.
1979; 2
8
(2): 113-
12
2.
[5]
PAN W
,
QIAN Y SH, Z
H
OU
E. Po
w
e
r harm
onics me
asure
m
ent base
d
on
w
i
nd
o
w
s an
d interp
olate
d
FFF(
Ⅱ
)
dual interpolated FFT algorit
hms.
T
r
ansacti
ons of Chin
a Electrot
echn
ical Soc
i
et
y
. 1994; 2(1)
:
50-5
4
.
[6]
X
I
L,
X
U
WS, YU YL.
A fast har
mo
nic
detectio
n
met
hod
base
d
o
n
reeursiv
e
DF
T
. Electronic
Measur
ement
and Instrume
nt
s. ICEM107. 8th In
ternatio
na
l Confer
ence
on
. 2007; 97
2-97
6.
[7]
QIAN H, Z
H
AO RX, C
H
EN
T
.
Interharmonics
ana
l
y
sis
base
d
o
n
in
terpol
ating
w
i
ndo
w
e
d F
F
T
algorithm.
IEEE Trans. Power De1
. 200
7; 2
2
(2): 106
4-1
0
6
9
.
[8]
HARRIS F
J
. On the us
e o
f
w
i
nd
o
w
s for
harmo
nic a
n
a
l
y
sis
w
i
t
h
the
discrete F
o
ur
ier transform
.
Procee
din
g
s of
the IEEE. 1978; 66(1): 51-8
3
.
[9]
Gao YP,
T
eng
Z
SH, W
en
H, et al. Harmonic a
n
a
l
ysis base
d
on
K
a
is
er w
i
n
d
o
w
pha
se
differenc
e
correctio
n and
its appl
icatio
n.
Scientific Instru
m.
20
09; 30(
4): 767-7
7
3
[10]
PANG H, LI
D
X
, Z
U
Y
X
, et
al.
An im
pr
oved algor
ithm
for
har
moni
c
analysis
of power system
usi
ng
FFT te
ch
n
i
qu
e
.
Proceed
in
gs o
f
the CSEE. 2003; 23(6): 5
0
-5
4.
[11]
AGREZ
D. W
e
ighte
d
muh
i
po
i
n
t interpo
l
ate
d
DFT
to improve ampl
itud
e e
s
ti
mation of m
u
ltifreq
uenc
y
sign
al.
IEEE Tr
ans. Instrum
.
Meas
., 2002; 5
1
(2): 287-
29
2.
[12]
AGREZ
D. D
y
namics
of~
e
q
u
ene
y
estim
a
tio
n
i
n
th
e fre
que
nc
y d
o
mai
n
.
I
EEE Trans. Instrum
.
M
eas
.,
200
7; 56(6): 21
11-2
118.
[13]
LIN H
C
. Inter-
harmo
nic
id
ent
ificatio
n us
in
g
grou
p-harm
oni
c
w
e
ighti
n
g
a
p
p
roac
h
base
d
on th
e F
F
T
.
IEEE Trans. P
o
wer Eleetr.
, 2008; 23(
3): 130
9-13
19.
[14]
LOBOS
T
,
LEONOW
ICZ
Z
,
REZ
M
ER J, et al. Hig
h-res
o
lut
i
on s
pectrum e
s
timation m
e
th
ods for si
gna
l
analy
sis in po
w
e
r s
y
stems.
IEEE Trans. Instr
u
m
.
Meas.
, 20
06, 55(1): 2
19-
225.
[15]
LIGUORI C, PAOLILLO A, PIGNO
T
II A
.
Estimation
of sig
n
a
l p
a
rameters
i
n
the fre
que
nc
y d
o
ma
in i
n
the prese
n
ce
of harmon
i
c in
terf
erence: A compar
ative an
al
ysis.
IEEE Trans. Instrum
.
Meas
., 2006;
55(2): 56
2-5
6
9
.
[16]
Xu
e H, Yan
g
RG. Morlet w
a
vel
e
t
bas
e
d
detecti
on o
f
nonint
eger
harmo
nics.
Power System
T
e
chno
logy
. 2
002; 26(
12): 41
-44.
[17]
Pham V
L
, W
ong
KP.
Wav
e
let-transfor
m
-
based algorithm
fo
r
har
m
onic analysis
of
power syste
m
waveform
s
. IEE Proceed
in
g of Generatio
n, T
r
ansmissi
on a
nd Distri
butio
n. 1999; 1
46(3):
249-
254
[18]
Leo
no
w
i
cz Z
,
Lob
os T
,
Rezm er J. Advan
c
ed sp
ec
trum
estimatio
n
met
hods for si
gn
a
l
an
al
ysis i
n
po
w
e
r electro
n
i
cs.
IEEE T
r
ansactions o
n
Ind
u
strial El
ectron
ics
. 2003; 5
0
(3
): 514-51
9.
[19] Ding YF
, Che
n
g
H Z
H
, Lu GY, etal. Spectrum es
tim ation of harmon
i
cs a
nd inter
harmo
n
i
cs base
d
on
pron
y a
l
gor
ith
m
.
T
r
ansaction
s of China El
ec
trotechnic
a
l So
ciety
. 2005; 2
0
(
10): 94-9
7
.
[20]
Cai T
,
Duan
SH X, L
i
u F
Ri. Po
w
e
r
Har
m
oni
c An
al
ys
i
s
Based
on R
eal-V
alu
ed S
p
ectral MUSIC
Algorit
hm.
T
r
ansactio
n
s of Chin
a Electrotec
hnic
a
l Soci
ety.
200
9; 24(1
2
): 149-1
55.
[21]
W
en H, T
eng
Z
SH, Z
eng
B
,
et al.
Hi
gh
a
ccu
rac
y
ph
ase
estimati
on
al
gorithm
bas
ed
on
sp
ectra
l
leak
age c
ance
l
latio
n
for electri
c
al harm
onic.
Scientific Instru
m. 2009; 30(
11
): 2354-2
3
6
0
.
[22]
Su YX,
Liu Z
H
G, Li K, et al. Electric Ra
il
w
a
y
Harmo
nic
Detectio
n Bas
ed o
n
HHT
Method. R
a
il
w
a
y
Societ
y. 200
9; 31(6): 33-
38.
[23]
CHEN
Ha
n, LI
U Hu
i-ji
n,
et
al
. Appl
icatio
n
of No
nun
iforml
y
Sa
mpli
ng
a
nd Least
S
q
u
a
re T
e
chnique
i
n
Interharmo
nic Measur
ement.
Procee
din
g
s
of
the CSEE. 2009; 29(1
0
): 109
-114.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Com
p
re
ssed
Sensin
g Hig
h
-
accu
ra
cy De
tection for Ele
c
tri
c
Powe
r…
(Tiejun
Cao
)
6501
[24]
Shi GM, L
i
u
D
H
, Gao
DH, L
i
u Z
,
Li
n J, W
a
ng
LJ
.Adv
anc
es i
n
the
o
r
y
an
d a
ppl
icati
on
o
f
compress
ed
sensi
ng.
Chi
n
e
s
e jou
m
a
l
of Eleetro
nics.
20
0
9
; 37(5): 10
70-
108
1. (in Chi
n
e
s
e )
[25]
EJ Cand
e`s J Romb
erg, T
T
a
o. Robust u
n
ce
rtaint
y
princ
i
pl
e
s
: Exact sig
nal
reconstructio
n
from high
l
y
incom
p
lete fre
que
nc
y
inform
ation.
IEEE Trans. Inform
. Theory
. 2006; 5
2
(2
): 489–5
09.
[26]
S Osher, Y M
ao, B
Do
ng,
W
Yin.
F
a
st L
i
near
i
z
e
d
Bre
g
m
a
n
Iterati
on
for C
o
mpress
e
d
Se
nsin
g
an
d
Sparse D
e
n
o
isi
ng.
UC.A CAM
Report (08-
37)
. 2008.
[27]
W Yin, S Os
her, D Goldfarb, J Da
rbo
n
, Bregman
iter
ative al
gor
ith
m
s for
ℓ
1-minimization w
i
t
h
app
licati
ons to
compress
ed se
nsin
g,
SIAM J.
Im
aging Sci.,
2008; 2: 14
3–1
6
8
.
[28]
JF
Cai, S Osher, Z
Shen. Lin
eariz
ed
Bregm
an iterati
ons fo
r compresse
d sensi
ng.
Math.
Comp.
200
9;
78: 151
5-1
536.
[29]
JF
Cai, S
Osh
e
r, Z
She
n
. C
o
nverg
enc
e
of t
he
Lin
ear
ized
Bregma
n
Iterat
i
on
for
ℓ
1-n
o
rm
Minim
i
zati
on,
Math. Comp.
2
009; 21
27-
213.
Evaluation Warning : The document was created with Spire.PDF for Python.