TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol.12, No.1, Jan
uary 20
14
, pp. 415 ~ 4
2
1
DOI: http://dx.doi.org/10.11591/telkomni
ka.v12i1.4145
415
Re
cei
v
ed
Jun
e
27, 2013; Revi
sed Aug
u
st
27, 2013; Accepted Sept
em
ber 20, 20
13
Gross Error Elimination Based on the Polynomial Least
Square Method in Integrated Monitoring System of
Subway
Zi-Yu Ma
1
Da
-W
ei Li
2
Fang-Wu Don
g
*
1
1
F
a
cult
y
of el
ec
trical an
d rail tr
affic, Z
heJiang
T
e
xtile & F
a
shi
on Co
lle
ge
2
Ningb
o Cit
y
R
a
il T
r
ansportati
on Group C
o
m
pan
y L
i
mited
o
perati
ng su
bsid
iaries
Ning
B
o cit
y
, Z
h
eJia
ng, Chi
na,
315
21
1, telp 0
86-5
74-8
6
3
2
9
6
228
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: Maz
y
1
9
6
2
@
163.ccom, 15
0
582
91
155
@13
9
.com, dongf
w
01@
163.com
*
A
b
st
r
a
ct
T
he meas
ure
m
ent data
of par
ameter in th
e e
l
ectrical
eq
uip
m
e
n
t
conta
i
ns
ma
ny no
ises i
n
subw
ay
integr
ated
m
o
nitoring system
. To elim
inate the im
pa
ct of gross error in
the measurement dat
a, a
poly
n
o
m
i
a
l le
a
s
t square curv
e fitting al
gorit
hm is
used
i
n
this pap
er. Based o
n
the R
a
jd
a criterio
n, the
algorithm
gives the variance estima
tion of the noises, an
d then
uses dynam
i
c thres
h
old to detect
and
repl
ace the
measur
e
m
ent d
a
ta w
i
th gross error by
statistical esti
mati
on. F
i
nal
ly, a data process
i
n
g
proce
dure
has
bee
n pr
esent
ed to d
e
a
l
w
i
th the gr
oss er
ror. T
he practi
cal a
ppl
icatio
n
ind
i
cates th
at th
e
prop
osed
alg
o
r
i
thm c
an effect
ively e
l
i
m
i
nate
the gross
error
in many types
of meas
ure
m
e
n
t signa
ls so a
s
to ensure the reliability
of the m
o
nitoring syst
em
.
Ke
y
w
ords
:
Su
bw
ay Integrate
d
Monitor
i
ng, l
east s
quar
e, d
y
na
mic thres
h
o
l
d, error ha
ndl
i
n
g
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
To monito
r th
e wo
rk
state
of sub
w
ay e
q
u
ipm
ent
s,
ma
ny
electri
c
al para
m
eters such as
voltage, cu
rrent, and work freq
uen
cy
are m
easure
d
and tran
sm
itted to controlling comp
uter in
real time.
Ne
verthele
ss, th
ere a
r
e m
any
noises
in th
e mea
s
u
r
em
ent data
s
of
the paramete
r
s.
For
example,
variou
s i
n
terferen
ce
so
urce
s exi
s
t in
onsite
monito
ring.
Dete
ctio
n devi
c
e
s
al
so
have
cha
n
ce
to gen
erate
e
rro
rs [1-2]. Besid
e
s,
som
e
sig
nal di
stu
r
ban
ce
s
cann
ot be
avoide
d in
th
e
d
a
t
a
c
o
lle
c
tio
n
pr
oc
ess
an
d r
e
mo
te tr
a
n
s
m
i
ssi
on
process. T
h
erefo
r
e, g
r
o
s
s m
e
a
s
urem
en
t
sign
als somet
i
mes contain
a certai
n amo
unt of e
rro
rs, althoug
h the prob
ability of
the occurren
ce
of those
gro
s
s e
r
rors i
s
l
o
w. Be
cau
s
e
the a
m
plitude
of the
error is rel
a
tively large, the
untre
ated
gro
ss m
e
a
s
u
r
eme
n
t datas with error cannot be di
rectly inputte
d to the co
mputer for d
a
ta
pro
c
e
ssi
ng,
whi
c
h
wo
uld
lead
to in
a
c
curate
inp
u
t data,
wro
n
g
pro
c
e
s
sing
results,
or
e
v
en
misop
e
ration
of the device
s
[3-5]. Th
erefore, dat
a p
r
ocessin
g
of the
mea
s
u
r
e
m
ents sho
u
ld
be
con
d
u
c
ted to eliminate the
error an
d en
sure co
mp
reh
ensive mo
nitoring
system
be in goo
d work
con
d
ition. Ba
sed
on the
st
atistical th
eory and calc
ula
t
ion method,
this pa
per
uses a
polynom
ial
least squ
a
re
method
to automatically
eliminat
e th
e gro
s
s e
rro
rs in the
me
asu
r
em
ents.
The
method is th
en effectively applied in el
ectri
c
al pa
ra
meters acqui
sition and d
a
ta pro
c
e
ssi
ng in a
sub
w
ay statio
n.
2. Gross Err
o
r Processin
g
2.1. Gross E
rror Proces
s
i
ng Method
Whe
n
the eq
uipment
s in the integrated
monitori
ng sy
stem of su
bway
are in no
rmal wo
rk
con
d
ition, the
voltage, cu
rrent, and oth
e
r
dynami
c
p
h
y
sical
param
eters
are u
s
u
a
lly contin
uo
us
with normal dis
t
ributions
res
p
ec
t to time. Ac
c
o
rdin
g to the norm
a
l distributio
n of erro
r theo
ry, any
cha
nge
s of a
physical pa
rameter a
r
e
continuo
us
wi
th no jump [6]
.
In a certain
perio
d of time, a
set of data
can be
obtain
ed by u
s
ing
polynomial
l
e
ast squa
re
curve fitting o
n
a continuo
us
recording
of a mea
s
u
r
ed
para
m
eter [7-8]. Dynami
c
mean val
ue
of the pa
ram
e
ter
can th
en
be
got by avera
g
ing the abo
ve two sets
of data. Ac
cordin
g to the Rajda criteri
on, i.e. the small
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ISSN: 2302
-4046
TELKOM
NIKA
Vol. 12, No. 1, Janua
ry 2014: 415 –
421
416
probability does not exist
[
9
-10], a proper threshold i
s
set for
the gross
error, whi
c
h repl
aces
the
obviou
s
ly wro
ng mea
s
u
r
em
ent datas
with co
rre
s
p
ondi
ng statisti
cal
values [11
-
12
].
2.2 Mean Val
u
e of Dy
namic Noise
In comp
reh
e
n
sive mo
nitoring syste
m
, the
syste
m
m
easure
m
ent signal can be
assume
d
as a
seq
uen
ce of recorded
data with respect to the ti
me (
(T
i
，
Y
i
)
，
i
=1
，
2
…
n).
When ste
p
len
g
t
h
of data sampl
i
ng is set as
T
, the ith data point is gen
e
r
ated in time i
n
stant
T
i
:
1
(1)
The
m
th least squa
re fitting polynomial is:
∑
(2)
The deviation
on each data
point is:
|
ε
|
|
|
(3)
To minimize the summati
on of the square of
the deviation, pol
ynomial least
squa
re
curve i
s
used
to fit the data,
4
Whe
n
(4
) rea
c
he
s its mini
maal value, multiple functi
on value con
d
itions in
dicat
e
:
2
5
After inner p
r
odu
ction sim
p
lification i
s
intr
odu
ce
d in (5), polynomi
a
l (6) an
d (7
) can be
obtaine
d:
…
…
…
…
6
…
…
…
…
7
The fifted value of data poi
nts (
T
i
,PY
i
)(
i
=1,2......n) is then obtained. The dynami
c
noise
value is
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Gro
s
s Erro
r Elim
ination Based o
n
the Polyn
o
m
i
al Least Squa
re M
e
thod in ...
(Fang-Wu Don
g
)
417
2
1
()
1
n
ii
i
x
YP
Y
n
(8)
2.3 Replac
e
m
ent of the
gross err
o
r
Acco
rdi
ng to
the Rajd
a crit
erion, the
r
e i
s
97%
confid
ence of the
d
a
ta wh
en its
absolute
error i
s
beyo
nd 3
ɛ
.
[13] Beca
use the measurement
in t
he integrated monito
ri
ng syste
m
may
contai
n fault i
n
formatio
n, a
nd thu
s
it n
e
e
d
to be
retai
n
ed for furthe
r
fault identifica
t
ion. The
gro
s
s
error threshol
d is set a
s
5
ε
with
co
rrespo
nding
confid
e
n
ce
b
e
ing 9
9
.99%[14] , i.e.:
|
|
5
̂
(9)
Data
sati
sfying (9
) i
s
the
gross e
r
ror,
whi
c
h
i
s
n
eed
ed
to eliminate
a
nd the
n
to
re
place.
Con
s
id
erin
g t
he
contin
uou
s multi
-
poi
nt
gro
s
s e
rro
r
p
r
ocessin
g
, da
ta in i
+
1, i
+
2
point
ca
n b
e
determi
ned.
The re
pla
c
em
ent can b
e
ob
tained a
c
cord
ing to formula
(10) a
nd (1
1) interpolatio
n:
Whe
n
two
ne
arby p
o
ints
a
r
e n
eed t
o
int
e
rpol
ated, i.e.
Y
i
, Y
i+1
are b
o
th gross
errors, th
e
repla
c
e
m
ent value
are:
3
10
Whe
n
a sin
g
l
e
point is ne
e
d
to repla
c
e, i.e.
Y
i
is a gross e
r
ror, the repla
c
eme
n
t value is:
4
11
3. Gross Err
o
r Processin
g
3.1. Gross E
rror Proces
s
i
ng Flo
w
c
h
a
r
t
The flo
w
chart
of the g
r
o
s
s
error
pro
c
e
ssing is sho
w
n
in Figu
re
1.
Detaile
d alg
o
r
ithm i
s
as
follows
:
Step 1: the initial value in the refere
nce se
ct
ion is
set as the first value of the whol
e
gros
s
error data es
timation
ε
2
.
Step 2: estim
a
te the state
value of the
nex
t point. Compa
r
e the e
s
timation a
n
d
the real
value, if the differen
c
e
|
|
is less than 3
x
, this estimati
on is con
s
id
ered
as rea
s
onabl
e.
Otherwise, the poi
nt is t
houg
ht
as
gross e
rro
r a
n
d
nee
d to
repla
c
e. Sin
c
e the
signal
s in
monitori
ng sy
stem are all contin
uou
s a
nd dynami
c
,
the pre
d
ictio
n
can reply on the value in the
last poi
nt. Beca
use the
error of the
predi
ction
grad
ually be
come
s la
rg
e
r
wh
en
sev
e
ral
con
s
e
c
utive gro
s
s
e
r
rors appe
ar,
th
re
shold sh
ould
be set to be large
r
, su
ch
as 5
x
, to avoid
misjud
gme
n
t.
Step 3: updat
e refe
ren
c
e
section.
Whe
n
the
refe
ren
c
e se
ction m
o
ves ba
ckward by one
point, its first value is
rem
o
ved an
d re
p
l
ace
d
by
an
adja
c
ent valu
e in the
reference sectio
n
(in
this mann
er, the mea
s
urem
ent is judge
d as gross
erro
r, and then re
placed with its pre
d
ictio
n
). If
the data pro
c
essing i
s
com
p
leted, swit
ch
to Step4, otherwi
se, go to
Step 3.
Step 4: the data after th
e repl
aceme
n
t
in the last
roun
d is
re
-estimated
ε
2
t
o
judge
wheth
e
r the
r
e are n
e
w g
r
oss erro
r is remove in thi
s
ro
und of d
a
ta. If not,
the com
putatio
n is
compl
e
ted. O
t
herwi
se, g
o
to Step2. This pro
c
e
ss i
s
a
c
tually a loop
of repla
c
em
ent of the gro
s
s
err
o
r
s
. Th
e l
a
rge
r
er
ro
rs
are
firstly
rep
l
ace
d
a
n
d
th
en follo
we
d
by sm
aller e
r
rors. T
he
wh
ole
repla
c
e
m
ent
can g
ene
rally
be finishe
d
when the loo
p
is run fo
r 3 times.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 2302
-4046
TELKOM
NIKA
Vol. 12, No. 1, Janua
ry 2014: 415 –
421
418
Figure 1. Flowchart of gro
ss e
r
ror p
r
o
c
essing
4. Practical
Applica
t
ion
4.1. Gross E
rror Da
ta Pr
ocessin
g
A data
se
rie
of a m
o
tor wo
rkin
g
cu
rre
nt i
s
sho
w
n
in
T
able
1 a
nd
Fi
gure
2. T
h
e
initial
x
is
0.13 from
cal
c
ulatio
n. Fro
m
Figu
re 2,
at l
east th
ree
mea
s
ureme
n
t datas are
obviou
s
ly wrong
and ne
ed to repla
c
e.
Table 1. Mea
s
ureme
n
t data of the curre
n
t for a worki
ng motor
(Uni
t: A)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Gro
s
s Erro
r Elim
ination Based o
n
the Polyn
o
m
i
al Least Squa
re M
e
thod in ...
(Fang-Wu Don
g
)
419
Figure 2. Current mea
s
u
r
e
m
ent data of a workin
g mo
tor (Unit: A)
After the first loop of gross
error elimi
nati
on,
errors a
r
e
basi
c
ally eli
m
inated a
s
shown in
Figure 3. Ho
wever, the figure
contai
ns some bu
rrs.
When
several loop
s are
compl
e
ted, b
u
rrs
are ba
si
cally eliminated a
s
sho
w
n in Fig
u
re 4.
4.2. Algorith
m
Impro
v
ement
Becau
s
e the
prop
osed alg
o
rithm is a
ki
nd of
forward
algorithm, it can only jud
ge the
data in th
e n
e
xt point. Th
erefo
r
e, if the
first d
a
ta
p
o
int is
with g
r
o
s
s erro
r, this
point
cann
ot
be
repla
c
e
d
. In orde
r to solve this proble
m
, it’s need
ed to judg
e wheth
e
r the
first data poi
nt
contai
ns g
r
o
ss e
r
ror. In other words,
if (9) is
not
satisfied, an
other st
arting
point sho
u
ld
be
sele
ct
ed.
Figure 3. Gro
ss e
r
ror elimi
nation re
sult
after the first loop
Figure 4. Final gro
ss e
r
ror elimination result
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 2302
-4046
TELKOM
NIKA
Vol. 12, No. 1, Janua
ry 2014: 415 –
421
420
Whe
n
this al
gorithm i
s
u
s
ed to elimin
ate gro
s
s erro
rs, data
with large
error
ha
ve been
removed in Figure 3, but there
still exist many small burrs, wh
ich affect the smoothness of
curve. Thi
s
p
henom
eno
n is usually tre
a
ted by
usin
g averag
e m
e
thod. Avera
ge method
can
effectively improve the
sm
oothne
ss of the data
cu
rv
e. Ho
wever, i
t
only use
s
th
e avera
ge val
u
e
of the adj
ecti
ve mea
s
ure
m
ents to
re
p
l
ace th
e b
u
rr, and
can
not
fundam
ental
ly eliminate t
h
e
error. T
herefore, in
practi
cal
appli
c
atio
n, ban
dpa
ss
filtering i
s
u
s
ed to fu
rthe
r
improve
the
data
smooth
n
e
ss
after the gro
s
s error
being
repla
c
e
d
.
4.3. Analy
s
is
of Res
u
lts a
nd Discu
ssi
on
In the gro
ss
error p
r
o
c
e
s
sing, accordin
g to Rajda
criterio
n, |
Y
i
-
PY
i
|>
3
ε
is set as the
judgme
n
t co
n
d
ition. Beca
u
s
e fault info
rmation may
exist in the in
tegrated
mon
i
toring
syste
m
of
sub
w
ay, and
the noises b
e
c
ome la
rg
e in
the fault sites, |
Y
i
-
PY
i
|>
3
ε
is
not proper any more. If t
h
is
crite
r
ion i
s
n
o
t
chan
ged,
some u
s
eful i
n
formation
will
be ign
o
red,
whi
c
h affe
cts the reli
ability o
f
the system. T
herefo
r
e, In p
r
ac
ti
cal a
ppli
c
ation, if the 5
ε
su
ccessive m
easur
e
m
ent
s contai
n gross
er
rors
, |
Y
i
-
PY
i
|>
5
ε
is set as
a new
crite
r
io
n.
Thro
ugh
the
analysi
s
of th
e mea
s
u
r
em
ent in Fi
gure
2, figure 3
a
nd Fig
u
re 4 f
r
om
an
equipm
ent o
peratio
n inte
grated
monit
o
ring
syst
e
m
of sub
w
ay,
the pro
p
o
s
ed gross e
r
ror
pro
c
e
ssi
ng
can co
mplete
ly meet the requi
re
m
ent
s of monito
ring
system,
eliminate the
phen
omen
on
of the slow respon
se a
nd the mi
so
peratio
n of the co
mprehe
nsive monito
ring
system. The
maximum rel
a
tive erro
rs o
f
all the
para
m
eters are controlle
d
with
in 2%. Theref
ore,
the polynomi
a
l least squa
re curve fitting
method
with
Rajd
a crite
r
io
n sho
w
s goo
d perfo
rman
ce in
detectin
g
an
d repla
c
in
g gross e
rro
r. It has goo
d practicality in not only eliminating the gross
error, but also improving t
he reliability of the system.
5. Conclusio
n
This pa
per in
trodu
ce
s the polynomial le
ast
sq
uare method into the data pro
c
e
ssi
ng of
the integ
r
ate
d
monito
rin
g
syste
m
of
subway. Th
e
measurement
data
s
a
r
e
firstly
statistically
analyzed, an
d then sm
oot
hed by re
mo
ving the gro
s
s
erro
rs d
e
te
cted u
s
ing
Rajda criteri
on.
The
remove
d me
asu
r
em
ent d
a
tas a
r
e
repl
ace
d
by valu
es o
b
tained f
r
om
statistica
l theory, whi
c
h
improve
s
the
accuracy of
the measurements
and
the relia
bility of the syste
m
. The pract
i
cal
appli
c
ation
shows that, th
e method
ha
s the ch
ara
c
te
ri
stics of
high
reliabilit
y, strong p
r
a
c
tica
b
ility
in monitori
ng
equipm
ent op
eration p
a
ra
meters for th
e metro integ
r
ated mo
nitori
ng syste
m
.
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2302-4
046
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