Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
V
o
l.
6, N
o
. 2
,
A
p
r
il
201
6, p
p
.
53
5
~
54
8
I
S
SN
: 208
8-8
7
0
8
,
D
O
I
:
10.115
91
/ij
ece.v6
i
2.9
387
5
35
Jo
urn
a
l
h
o
me
pa
ge
: h
ttp
://iaesjo
u
r
na
l.com/
o
n
lin
e/ind
e
x.ph
p
/
IJECE
An Extender Kalman Filter-Base
d Induction Machines Faults
Detection
Abdel
g
h
a
ni
C
h
ahmi
*,
M
o
k
h
ta
r B
e
nd
jeb
b
ar
*,
Ber
t
ran
d
Raison**,
Mohamed Be
nbouz
id***
*Ele
ctri
cal
Driv
es
Labor
ator
y
L
D
EE, Univ
ers
i
t
y
of S
c
ience and Techno
log
y
of
Oran,
USTO/MB.Oran,
Algeria
** Grenoble Electrical Eng
i
neer
ing Labor
ator
y
,
G2Elab
***University
of
Brest, LBMS
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Nov 9, 2015
Rev
i
sed
D
ec 16
, 20
15
Accepte
d Ja
n
5, 2016
This paper d
e
a
l
s with the d
e
t
ect
i
on and loc
a
l
i
za
ti
on of ele
c
tr
ica
l
d
r
ives faul
ts,
es
peci
all
y
thos
e
cont
aining
ind
u
ction
m
achin
es
. F
i
rs
t,
th
e
con
t
ext
of th
e
stud
y
is
presen
ted
and an Extend
ed
Ka
lman
Filter is descr
i
bed
fo
r induction
m
achines fault
detec
tion. Th
en th
e modelin
g procedure u
nder facu
lty
conditions
is shown, and
the machine
diagnosis
methods are d
e
v
e
loped
.
Th
e
proposed diagno
sis approach r
e
q
u
ires onl
y
little
experimental d
a
ta, and
more
im
portantl
y
it p
r
ovides effic
i
ent
sim
u
lation tool
s that allow ch
arac
teri
zing
faulty
b
e
havior
.
Fault detection
uses
signal processing techniqu
es in known
operating phas
e
s (fixed
speed)
,
consid
ering and locating
malfun
ctions.
Keyword:
Ex
tend
ed
Kalman
filter
I
ndu
ctio
n m
o
to
r
M
odel
i
n
g
Rotor de
fect
Stator defect
Copyright ©
201
6 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
Ab
del
g
ha
ni
C
h
ahm
i
,
Electrical Drives Laboratory
LDEE
,
Un
i
v
ersity of
Scien
ce an
d Te
c
h
n
o
l
o
gy
of
O
r
a
n
UST
O
/
M
B
El Mn
aou
e
r
,
B
P
150
5, Bir
El
D
j
i
r
,
O
r
an
, A
l
ger
i
a 310
00
Em
a
il: ch
ah
mi.ab
d
e
l
g
h
a
n
i
@gmail.co
m
1.
INTRODUCTION
Larg
e ind
u
s
t
r
ial syste
m
s are wid
e
ly con
fro
nted
to
th
e ch
all
e
n
g
e
s of reliabilit
y an
d
av
ailab
ility o
f
th
e
pr
o
duct
i
o
n
de
vi
ces. El
ect
ri
c
dri
v
e
s
co
nt
ai
ni
n
g
an i
n
d
u
c
tion m
achine are largely
us
ed in the i
n
dustrial
ap
p
lication
s
than
ks to th
eir low co
sts,
h
i
gh
perfo
r
m
a
n
ces an
d rob
u
s
tn
ess
[1
].
As
k
nown,
a syste
m
can realize
t
h
e assi
gne
d t
a
sk, o
n
l
y
un
de
r
con
d
i
t
i
ons en
suri
ng t
h
e sec
u
ri
t
y
. An ea
rl
y
det
ect
i
on o
f
anom
al
i
e
s i
n
elect
ri
cal
m
o
t
o
rs m
a
y
hel
p
a
voi
di
n
g
do
wnt
i
m
es [2]
.
T
hus
, i
n
or
der
t
o
det
ect
a
n
i
n
ci
p
i
ent
fa
ul
t
,
we s
h
o
u
l
d
gi
ve a
s
p
eci
al
atten
tio
n
to th
e sp
ectral an
alysis o
f
stato
r
curren
t
s [3
],
[4]
.
M
o
st o
f
re
cent
researc
h
es
on defects detection of
in
du
ctio
n m
a
c
h
in
es
u
s
e t
h
e
stato
r
cu
rren
ts an
alysis. Several con
d
ition
-
m
o
n
ito
rin
g
m
e
th
od
s [5
] h
a
v
e
b
een
devel
ope
d a
n
d
were
use
d
t
o
hel
p
t
h
e det
e
ct
i
on
of el
ect
r
i
c defect
s i
n
i
n
d
u
ct
i
o
n m
a
chi
n
e.
Am
on
g
t
h
ese
m
e
t
hods
, t
h
e aut
h
ors i
n
[
6
]
consi
d
er t
h
e
vi
brat
i
o
ns m
oni
t
o
ri
ng
, t
o
r
que c
ont
rol
,
t
h
e t
e
m
p
erat
ure m
oni
t
o
ri
ng
,
etc.
In th
is
p
a
p
e
r,
th
e ro
tor
resist
an
ce is esti
m
a
t
e
d
u
s
ing
a m
e
th
od
o
f
Ex
tend
ed Kalm
an
Filter (EKF).
Thi
s
st
at
e obse
r
ve
r i
s
pr
ove
d t
o
be a use
f
ul
t
ool
t
o
det
e
rm
in
e th
e p
a
ram
e
te
r v
a
riation
s
[7
], [8
]. In
th
is p
a
p
e
r, it
is shown that
EKF ca
n be e
m
ployed
to identify the distur
ba
nce in rotor re
sistan
ce.
Recently, the Kalm
an
filter was app
lied
su
ccessfu
lly for d
e
fect
forecastin
g
in
[9
].
Fo
r ex
am
p
l
e it
is u
s
ed
to
i
n
terpo
l
ate th
e trend
s
of
a si
gnal
-
l
e
a
r
ne
d g
fr
om
t
h
e recor
d
e
d
dat
a
an
d t
h
e
n
est
i
m
a
t
e
t
h
e ev
ol
ut
i
o
n of
defect
s i
n
t
h
e pr
op
ose
d
di
a
g
n
o
si
s
m
e
t
hods i
n
[
1
0]
. In t
h
i
s
st
u
d
y
,
t
h
e pr
o
p
o
s
ed ap
pr
oac
h
consi
d
er
s t
h
e v
a
l
u
e of r
o
t
o
r r
e
si
st
ance as fi
xed f
o
r
co
nd
itio
n
m
o
n
ito
ring
. Th
is v
a
lu
e in
th
e d
i
agn
o
s
tic too
l
s wh
ich
on
e u
s
es is n
o
t
fix
e
d
contrary to
th
e classical
app
r
oaches
of
cont
rol
o
f
m
a
chi
n
e.
Hence
,
w
e
pr
op
ose
a method that considers a va
riation
of r
o
to
r resi
stance
in order t
o
give
a good estim
a
t
e as
well as a
good
ro
b
u
st
ne
ss
f
o
r t
h
e c
ont
rol
o
f
t
h
e
i
n
duct
i
o
n m
achi
n
e.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
6, No
. 2, A
p
ri
l
20
16
:
53
5 – 5
4
8
53
6
Thi
s
pa
per i
s
o
r
ga
ni
zed as f
o
l
l
o
ws:
pa
rt
II d
e
scri
bes t
h
e i
n
duct
i
o
n m
achine m
odel
unde
r sim
p
l
i
f
y
i
ng
assu
m
p
tio
n
s
.
In
du
ction
m
ach
in
e m
o
d
e
ls h
a
ve b
een
d
e
v
e
loped
und
er
h
ealth
y and
fau
lty co
nd
itio
ns in
the (ab
c
)
referen
ce
fram
e
. Part III, presen
ts a state o
f
th
e art re
v
i
ew
o
f
Ex
tend
ed
Kal
m
an
Filter. Fin
a
lly, p
a
rt IV
g
i
v
e
s
di
gi
t
a
l
si
m
u
l
a
t
i
ons
of t
h
e
pr
o
p
o
se
d fa
ul
t
y
m
odel
.
T
h
e
o
b
tained
resu
lts
prove th
e effectiv
en
ess
of th
e prop
o
s
ed
m
o
n
ito
rin
g
m
e
th
od
b
a
sed
o
n
an
Ex
tend
ed
Kal
m
an
Filter.
2.
INDUCTION MOTOR DRIVES
2.
1. Over
vi
ew
of
the
Dri
v
e
Th
e d
r
i
v
e un
der
stud
y
is
com
p
o
s
ed
o
f
a mach
in
e w
ith
in
du
ctio
n,
i
n
ver
t
er
o
f
w
i
d
t
h
of
im
p
u
l
se,
or
deri
ng
of
rot
o
r
fl
o
w
- di
rec
t
ed, a l
o
o
p
of
m
easurem
ent
of t
h
e cu
rre
nt
a
nd a l
o
o
p
o
f
sp
eed co
nt
r
o
l
.
Fi
gu
re 1
p
r
esen
ts t
h
e to
t
a
l d
i
agram
o
f
t
h
e
u
n
it.
Fi
gu
re
1.
Sc
he
m
a
t
i
c
di
agram
of
t
h
e sy
st
em
unde
r st
udy
2.
2. M
o
del
of
t
h
e
Ind
u
cti
o
n Mo
t
o
r
Th
e obj
ectiv
e
o
f
th
is section
is to
p
r
esen
t the d
i
fferen
t
m
o
d
e
ls u
s
ed
fo
r
ou
r st
u
d
y
. Th
is
is first o
f
all
to
m
o
d
e
l th
e in
du
ction
m
ach
in
e fin
e
ly en
oug
h
t
o
in
trod
uce a relev
a
n
t
mo
d
e
ling
o
f
d
e
fects. Th
en
we
p
r
esen
t
ho
w t
h
e
defa
ul
t
t
e
m
p
l
a
t
e
s are
not
use
d
i
n
t
h
ei
r si
m
p
l
i
f
i
e
d an
d
bal
a
nce
d
ver
s
i
o
n
di
a
g
n
o
st
i
c
p
u
r
p
oses.
2.2.1.
Models
Simulati
on P
r
oposed i
n
the
Re
ference F
r
ame
(abc)
Th
e i
n
du
ction
mach
in
e is a kn
own
system
-well [1
1
]
;
m
o
dels ex
ist an
d al
lo
wing
, in th
e
maj
o
rity of
the cases a
nd
unde
r certain
sim
p
lifying as
sum
p
tions, a
rep
r
esen
tation
o
f
t
h
e m
ach
in
e with
ou
t fau
l
t
.
Th
ese
assu
m
p
tio
n
s
are th
e
f
o
llow
i
ng on
es
[
12-
13
]:
-
The air-gap is of consta
nt width
without
e
ffect of
notch (not of
ecce
ntricity,
of unbal
a
nce, re
ducti
on
am
ong
st
ha
rm
oni
cs
i
n
t
h
e
de
vel
o
pm
ent
of i
n
d
u
ct
ance
s e
x
p
r
essi
o
n
)
.
-
Assim
i
lation of t
h
e cage t
o
a s
h
ort-circ
uit havi
ng
the
sam
e
nu
m
b
er of phases
a
s
static windi
ng
(sim
p
lificat
io
n
o
f
th
e equ
a
tio
ns of th
e m
ach
in
e b
y
ta
ki
ng int
o
acc
ount the a
v
era
g
e e
ffect
of the
cage
)
.
-
Sinus
oi
dal distribution, along the
air-ga
p
,
of the m
a
gnetic fields of
each windi
ng (simplification
of t
h
e
equat
i
o
ns
o
f
t
h
e m
achi
n
e by
t
a
ki
n
g
i
n
t
o
acc
ou
nt
of
t
h
e e
x
p
r
essi
o
n
s i
n
c
o
m
p
l
e
x of t
h
e c
u
r
r
ent
s
, al
l
o
ws
t
o
use a
vect
ori
a
l
m
odel
)
.
-
Abse
nce
of sat
u
rat
i
o
n i
n
t
h
e
m
a
gnet
i
c
ci
rcu
i
t
(i
dent
i
cal
behavi
or
of eac
h
pha
se, al
l
o
w
s
use o
f
a vect
or
i
a
l
m
odel
)
.
-
The infl
uences
of the effect of
s
k
i
n
an
d t
h
e heat
i
ng o
f
t
h
e d
r
i
v
er
s are negl
ect
e
d
(n
ot
vari
at
i
on
of t
h
e
p
a
ram
e
ters in
terv
en
ing
i
n
th
e
mach
in
e und
er no
rm
al fu
n
c
ti
o
n
i
n
g
).
-
St
ar c
o
u
p
l
i
n
g
o
f
wi
n
d
i
n
gs
(t
he
cu
rre
nt
s f
o
rm
a sy
st
em
bal
a
n
ced
un
de
r
no
r
m
al
funct
i
o
ni
n
g
)
.
Und
e
r th
e cond
itio
n
s
po
in
ted o
u
t
prev
i
o
u
s
ly
, th
e eq
u
a
tion
s
o
f
t
h
e electrical circu
its u
tilize clean
an
d
m
u
t
u
al
i
nduct
a
nces m
a
ki
ng
i
t
pos
si
bl
e t
o
de
fi
ne
fl
o
w
s acc
or
di
n
g
t
o
t
h
e
cu
rr
ent
s
[
1
2]
.
On th
e lev
e
l
o
f
th
e stator, t
h
e
ten
s
ion
s
ch
eck
:
sc
sb
sa
dt
d
isc
isb
isa
R
Vsc
Vsb
Vsa
s
.
(1
)
Wh
ere t
h
e resistan
ces m
a
trix
is written
accord
i
n
g to
t
h
e sup
p
o
s
ition
o
f
symme
try:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
An
Exten
d
er K
a
lma
n
Filter-Ba
sed
Ind
u
c
tion Ma
ch
in
es Fa
ults Detectio
n
(
A
bd
elgh
an
i Cha
h
m
i
)
53
7
s
s
s
s
r
r
r
R
0
0
0
0
0
0
(2
)
On
th
e level of th
e ro
t
o
r, und
er - assu
m
p
ti
o
n
-th
a
t
- ro
tor-is co
m
p
arab
le to
a ro
tor with th
ree-p
h
a
se
ro
lling
s
u
p
, th
e vo
ltag
e
are
written
:
rc
rb
ra
dt
d
irc
irb
ira
R
Vrc
Vrb
Vra
R
.
(3
)
Matrix
R
R
has the sam
e
form
that R
S
, flux vector are expre
ssed accord
ing to the inducta
nces m
a
trix
and
stator
an
d
rot
o
r c
u
rre
nts.
irc
irb
ira
M
isc
isb
isa
L
sc
sb
sa
SR
s
.
.
(4
)
irc
irb
ira
L
isc
isb
isa
M
rc
rb
ra
R
RS
.
.
(5
)
In
or
de
r t
o
be
abl
e
t
o
gi
ve a
n
acco
u
n
t
o
f
t
h
e i
n
t
e
ract
i
o
ns
bet
w
ee
n t
h
e
pha
ses,
we ex
press
e
d ea
c
h
resistance a
n
d
inductance
m
a
trix acc
or
di
n
g
t
o
t
h
e
n
u
m
b
er
o
f
w
h
orl
s
o
f
rol
l
i
ngs
u
p
of
t
h
e
m
achi
n
e.
rc
rb
ra
r
rc
rb
ra
R
sc
sb
sa
s
sc
sb
sa
S
n
n
n
R
R
R
R
R
n
n
n
R
R
R
R
R
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
(6
)
n
s
a
co
rr
esp
onds
to
t
h
e n
u
m
b
e
r
o
f
wh
or
ls o
f
th
e p
h
a
se
a h
a
s stato
r
.
r
l
rc
n
r
m
rc
n
rb
n
r
m
rc
n
ra
n
r
m
rc
n
rb
n
r
l
rb
n
r
m
rb
n
ra
n
r
m
rc
n
ra
n
r
m
rb
n
ra
n
r
l
ra
n
rc
L
rbc
M
rca
M
rbc
M
rb
L
rab
M
rca
M
rab
M
ra
L
R
L
s
l
sc
n
s
m
sc
n
sb
n
s
m
sc
n
sa
n
s
m
sc
n
sb
n
s
l
sb
n
s
m
sb
n
sa
n
s
m
sc
n
sa
n
s
m
sb
n
sa
n
s
l
sa
n
sc
L
sbc
M
sca
M
sbc
M
sb
L
sab
M
sca
M
sab
M
sa
L
S
L
2
2
2
2
2
2
(7
)
)
cos(
)
3
/
2
cos(
3
/
2
cos(
)
3
/
2
cos(
)
cos(
)
3
/
2
cos(
)
3
/
2
cos(
)
3
/
2
cos(
)
cos(
)
(
sr
rc
sc
sr
rb
sc
sr
ra
sc
sr
rc
sb
sr
rb
sb
sr
ra
sb
sr
rc
sa
sr
rb
sa
sr
ra
sa
SR
m
n
n
m
n
n
m
n
n
m
n
n
m
n
n
m
n
n
m
n
n
m
n
n
m
n
n
M
(8
)
T
RS
SR
M
M
)
(
)
(
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
6, No
. 2, A
p
ri
l
20
16
:
53
5 – 5
4
8
53
8
2.
2.
2.
C
age
E
l
ectri
c F
a
i
l
u
res in th
e (
a
bc
)
Mo
del
Th
is m
o
d
e
l th
at we h
a
v
e
ju
st descri
b
e
d
in th
e
prev
iou
s
p
a
rt all
o
ws in
p
a
rticu
l
ar creating
asym
m
e
t
r
i
cal
faul
t
s
i
n
b
o
t
h
st
at
or a
n
d r
o
t
o
r
by
t
h
e
va
ri
at
i
on am
ong
st
w
h
orl
of
t
h
e
rol
l
i
ng
u
p
of
t
h
e
p
h
ase at
fault.
T
h
e de
fe
ct
can be
l
o
calized on
one or m
o
re
phase
s.
It sh
ou
ld
b
e
n
o
ted
th
at th
is
mo
d
e
ling
can
b
e
criticized
:
th
e
d
e
fects of typ
e
sh
ort-ci
rcu
it are represen
ted
in
th
e
m
o
d
e
l lik
e re
m
o
v
a
ls o
f
whorls i.e. o
p
e
n
i
n
g
s p
a
rtial o
f
phase; th
at d
o
e
s
n
o
t
correspon
d co
m
p
letel
y
to
reality
since it would
also
be
necessary to ta
ke acc
ount
of
t
h
e sha
d
ing
rings
[13]. This m
odel constitutes a
progress
co
m
p
ared
to
t
h
e b
a
lan
c
ed
m
o
d
e
ls or /and
con
s
id
ering
on
l
y
one
va
ri
at
i
on
o
f
resi
st
an
ce us
ual
l
y
used t
o
v
a
l
i
d
at
e
th
e algo
rith
m
s
[14
]
.
2.
3. Model “T
he
G
o
od
Per
f
orm
a
nce
”
for Contr
o
l and of Monitorin
g
M
odel
s
a
r
e
ne
cessary
t
o
desi
gn
an
d i
m
pro
v
e
t
h
e co
nt
r
o
l
l
e
rs a
nd t
h
e
obse
r
ve
rs.
They
a
r
e
com
m
onl
y
balance
d
.
O
n
e
passes
fr
om
the re
fere
nce
f
r
am
e (a
bc),
fixes,
with the
f
i
xed
refe
rence
fram
e
(
α
,
β
) by
the
t
r
ans
f
o
r
m
a
ti
on
of
C
o
nco
r
di
a [
15]
.
X
C
Y
U
B
X
A
pX
c
c
c
(9
)
W
i
t
h
X=
[
i
s
α
i
s
β
߮
r
ߙ
߮
r
ߚ
]
T
1
0
1
0
0
0
m
m
c
L
L
K
A
T
s
s
c
L
L
B
0
0
1
0
0
0
0
1
0
0
1
0
0
0
0
1
c
C
and
1
s
s
L
R
r
r
R
L
r
s
m
L
L
L
r
s
m
L
L
L
2
1
2.
4. I
nduc
tion
Machine
Disc
rete
Model
We de
vel
o
p i
n
t
h
i
s
para
gra
p
h
a di
scret
e
m
odel
of a
n
i
n
d
u
c
t
i
on m
achi
n
e i
n
or
der t
o
be
abl
e
t
o
use
th
ese eq
u
a
tion
s
in
th
e
d
e
v
e
lopmen
t o
f
Kalm
a
n
Filter.
We con
s
id
er ind
e
ed
t
h
at th
e equ
a
tion
s
o
f
t
h
e
o
b
serv
er at
en
ds
o
f
d
i
ag
nosis will b
e
in
ev
itab
l
y u
s
ed
i
n
a system
co
n
t
ain
i
n
g
m
i
cro
p
r
o
cesso
r or
o
t
her an
d thu
s
en
t
e
red in
th
e fo
rm
o
f
a
discrete m
o
d
e
l.
T
h
e
di
scret
e
ve
rsi
o
n i
s
o
b
t
a
i
n
ed
by
usi
n
g
t
h
e f
o
rm
ul
a o
f
Eul
e
r:
)
(
)
(
)
(
)
(
)
(
)
1
(
k
x
C
k
y
k
U
B
k
x
A
T
k
x
k
x
c
c
c
s
(1
0)
Of o
r
)
(
)
(
)
(
)
(
)
(
)
1
(
k
x
C
k
y
k
U
B
T
k
x
I
A
T
k
x
c
c
s
c
s
Or
)
(
)
(
)
(
)
(
)
1
(
k
x
C
k
y
k
U
B
k
x
A
k
x
d
d
d
c
d
s
c
d
s
c
d
C
C
T
B
B
T
A
I
A
,
,
2.
5. I
nduc
tion
Machine
Obs
erver-B
a
sed
EKF
This pa
rt prese
n
ts the
propose
d
m
e
thod to s
u
perv
ise the a
pplication.
W
e
will develop the
m
e
thod
by
usi
n
g t
h
e
obse
r
ver
m
odel
pre
v
i
ousl
y
descri
be
d.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
An
Exten
d
er K
a
lma
n
Filter-Ba
sed
Ind
u
c
tion Ma
ch
in
es Fa
ults Detectio
n
(
A
bd
elgh
an
i Cha
h
m
i
)
53
9
2.
5.
1. T
h
e E
x
t
e
nded
Kal
m
a
n
Fi
l
t
er
Th
e b
a
sic eq
u
a
tio
n
s
of th
e ex
t
e
n
d
e
d
Kalm
an
filter are d
e
d
u
c
ed
fro
m
th
at o
f
th
e d
i
screte mo
d
e
l of the
m
achi
n
e i
n
go
od f
o
nct
i
o
nne
m
e
nt
.l
a di
scret
e
m
a
t
r
i
x
of st
at
e
s
c
d
T
A
I
A
. One introduce
s
noises of state
)
(
k
w
and noi
ses of m
easurem
ent
)
(
k
v
[16
]
- [1
7
]
. The fun
c
tio
n
F is d
e
fin
e
d
so
th
at o
n
e
to
ex
tend filter of
Kalm
an
h
a
s al
l th
e p
a
ram
e
te
rs o
f
th
e m
achine. The estimate of rot
o
r res
i
stance is cons
idere
d
like
m
e
ans of
m
oni
t
o
ri
ng
.
)
(
)
(
)
(
)
(
)
(
)
(
)
1
(
k
v
k
x
C
k
y
k
w
k
U
B
k
x
A
k
x
d
d
d
(1
1)
Whe
r
e
)
(
)
(
)
(
),
(
k
U
B
k
x
A
k
U
k
x
f
d
d
(1
2)
The
process
noise w
(k) is c
h
a
r
acterized by
Q
k
w
k
w
E
k
w
E
T
)
(
)
(
0
)
(
(1
3)
Wh
ere Q
is d
e
fin
ite
p
o
sitiv
e.
The
m
easurement
noise v (k) respects:
R
k
v
k
v
E
k
v
E
T
)
(
)
(
0
)
(
(1
4)
R is d
e
fin
ite positiv
e to
o.
If
we call P th
e m
a
trix
of cov
a
rian
ce
o
f
th
e error in estim
a
tio
n
,
t
h
is m
a
trix
is ob
tain
ed
to leav
e:
T
k
x
k
x
k
x
k
x
E
k
k
p
)
(
ˆ
)
(
.
)
(
ˆ
)
(
)
/
(
The
not
at
i
o
n
k
/
k co
rres
p
on
ds
t
o
a
pre
d
i
c
t
i
on at
t
i
m
e
K ba
sed
on
dat
a
up
t
o
an
d i
n
cl
u
d
i
ng t
i
m
e K (
a
p
r
i
o
ri cov
a
riance) and
k
+
1
/
k
co
rresp
ond
s to
a p
r
ed
ictio
n
at ti
m
e
k
+
1
startin
g
fro
m
th
e d
a
ta u
n
til th
e ti
me K (a
post
e
ri
o
r
i
co
va
ri
ance)
.
In
t
h
e
fol
l
o
wi
n
g
e
q
ua
t
i
ons,
t
re
pr
ese
n
t
s
t
h
e
Jac
obi
a
n
m
a
t
r
i
x
o
f
t
h
e
n
onl
i
n
ear
f
unc
t
i
on
f
is
u
s
ed
[7
].
)
(
),
(
ˆ
)
(
k
U
k
x
k
x
f
J
(1
5)
Ap
pl
i
e
d t
o
t
h
e
sy
st
em
unde
r s
t
udy
,
J ca
n
be
exp
r
esse
d as
1
0
0
0
0
0
1
0
0
1
0
ˆ
1
1
0
ˆ
1
0
1
s
s
s
m
s
s
s
m
sq
s
s
s
s
s
sd
s
s
s
s
s
T
T
T
L
T
T
T
L
i
T
L
T
T
T
i
T
L
T
K
T
T
J
At each inte
gra
tion tim
e, the Kalm
an
Filter equations to
be
sol
v
ed are:
1/
C
o
m
put
at
i
o
n
of
t
h
e
o
b
ser
v
er st
at
es
)
(
)
/
(
ˆ
)
/
1
(
ˆ
k
U
B
k
k
x
J
k
k
x
d
(1
6)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
6, No
. 2, A
p
ri
l
20
16
:
53
5 – 5
4
8
54
0
2/
C
o
m
put
at
i
o
n
of
t
h
e a
p
o
st
e
r
i
o
ri
c
o
vari
a
n
c
e
m
a
t
r
i
x
P(
k+
1
/
k)
Q
J
k
k
JP
k
k
P
T
)
/
(
)
/
1
(
(1
7)
3
/
Co
m
p
u
t
ation
o
f
th
e
ou
tpu
t
s of th
e filter
)
/
1
(
ˆ
)
1
(
).
1
(
)
/
1
(
ˆ
)
1
/
1
(
ˆ
k
k
x
C
k
y
k
G
k
k
x
k
k
x
d
d
(1
8)
4
/
Co
m
p
u
t
ation
o
f
th
e feedb
a
ck
g
a
in
m
a
trix
o
f
th
e Kalm
an
Filter
1
)
(
)
/
1
(
)
(
)
1
(
)
/
1
(
1
R
k
C
k
k
P
k
C
k
C
k
k
P
k
G
T
d
d
T
d
d
(1
9)
5/
C
o
m
put
at
i
o
n
of
t
h
e a
p
r
i
o
r
i
cova
ri
ance
m
a
t
r
i
x
P
(
k+
1/
k+
1)
)
/
1
(
)
1
(
)
1
(
)
/
1
(
)
1
/
1
(
k
k
P
k
C
k
G
k
k
P
k
k
P
d
d
(2
0)
We con
s
i
d
er that Q and
R m
a
trix
are co
nstan
t
bu
t th
ey
can b
e
ti
m
e
v
a
riant. Th
e
b
e
h
a
v
i
or and
th
e stab
ility of
th
e Kalm
an
Filter are
fu
n
c
tions of
Q an
d R.
3.
R
E
SU
LTS AN
D ANA
LY
SIS
The test c
o
ndit
ions
are t
h
e
fol
l
owing:
- Fro
m
t=0
to
t=0
.
3s,
flux
app
licatio
n
- F
r
om
0.3 t
o
1.
8s, l
i
ner i
n
cr
ease o
f
t
h
e
s
p
e
e
d
refe
rence
- F
r
om
1.8s
co
nst
a
nt
s
p
ee
d
-
At t=3
s
,
fau
l
t in
th
e m
ach
ine
The observers are
e
n
able
d from
beginning.
3.
1. Go
od
Per
f
orm
a
nce
Our ap
pro
a
ch
in
th
is p
a
rt of
si
m
u
latio
n
is to
s
ee th
e b
e
h
a
v
i
or of th
e i
n
du
ctio
n m
ach
in
e in
h
ealthy
m
ode and with electric defect (rot
o
r,
stato
r
,
in
v
e
rter) in
th
e
m
o
d
e
tran
sitory an
d
p
e
rm
an
en
t, after fro
m
th
ere
will in
trod
u
c
e
an
ob
serv
er
o
n
th
e ou
tlet sid
e
o
f
t
h
e m
ach
in
e (ex
t
en
d
e
d
Kal
m
an
filter) is to
see its reactio
n
.
The
res
u
lts res
u
lt fr
om
roto
r
r
e
si
stance, the s
t
ator c
u
rrent
FFT.
The
f
o
llowi
ng
fig
u
r
e
take
s t
h
e
fo
rm
of t
h
e
cu
rre
nt in a
phase
of the
in
duction m
achine.
One ca
n
d
i
stin
gu
ish th
ere th
e v
a
riou
s ph
ases related
to th
e ord
e
r
fro
m
th
e syste
m
: p
h
a
se
of
f
l
u
x
i
ng
,
ph
ase of
ap
p
lication
o
f
th
e i
n
stru
ction
o
f
coup
le related
to
t
h
e
sl
ope
o
f
s
p
ee
d, t
h
e
n
st
a
b
i
l
i
zat
i
on t
h
e s
p
e
e
d a
n
d
i
n
st
ruct
i
o
n
of
cou
p
l
e
.
Up
t
o
0.
3 sec
o
n
d
, t
h
e
pu
r
pose
o
f
t
h
e appl
i
e
d v
o
l
t
a
ges i
s
t
o
p
o
si
t
i
on t
h
e
vect
or
f
l
ow i
n
t
h
e m
achi
n
e, a
s
fr
om
t
h
i
s
m
o
m
e
nt
, an i
n
st
r
u
ct
i
on
of
co
u
p
l
e
i
s
ap
pl
i
e
d t
o
t
h
e
or
der
an
d
sp
eed
bel
i
e
ves.
Fi
gu
re
2.
St
at
o
r
c
u
r
r
ent
0
1
2
3
4
5
-500
0
500
ti
m
e
(
s
)
c
u
rre
nt
(A
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
An
Exten
d
er K
a
lma
n
Filter-Ba
sed
Ind
u
c
tion Ma
ch
in
es Fa
ults Detectio
n
(
A
bd
elgh
an
i Cha
h
m
i
)
54
1
Th
e fo
llowing
figu
re tak
e
s the form
o
f
stato
r
flux
fo
llowing
th
e ax
is alp
h
a. On
e find
s
well th
e p
h
a
se
of
fl
u
x
i
n
g a
s
wel
l
as t
h
e
ev
ol
ut
i
o
n
of
fl
o
w
acc
or
di
n
g
t
o
the inc
r
ease
num
b
er of
revol
u
tions of
t
h
e machine
and
t
h
us
of
t
h
e
fre
q
u
ency
o
f
t
h
e s
u
ppl
y
vol
t
a
ges.
Fi
gu
re
3.
St
at
o
r
fl
ux
Th
e
fo
llowing
figu
re
p
r
esen
ts th
e ev
o
l
u
tio
n
o
f
t
h
e
p
r
od
u
c
ed
torqu
e
b
y
the in
du
ction
m
a
ch
in
e. One
finds the
phas
e of a
ccelerat
ion
(with a going
beyond
related to the
spee
d control) and the
phas
e of
stab
ilizatio
n
speed
an
d thu
s
co
up
le.
Figure
4. Machine couples
The
followi
ng figure take
s t
h
e
form
the s
p
eed
o
f
t
h
e syste
m
. On
e fi
n
d
s
well th
ere th
e
v
a
riou
s
pha
ses, w
h
i
c
h one
de
scri
be
s pre
v
i
o
usl
y
.
Fi
gu
re
5.
S
p
ee
d
of
t
h
e m
achi
n
e
Th
e
fo
ll
o
w
i
n
g
figu
re tak
e
s t
h
e fo
rm
o
f
ro
to
r resistan
ce at ex
it of th
e filter of
Kalm
an
(for a
h
ealth
y
syste
m
). One
notes a dri
f
t of
roto
r resistance
according to s
p
eed. This
deri
vative was c
o
rrected: accordi
ng t
o
0
1
2
3
4
5
-1
-0.
5
0
0.5
1
ti
m
e
(
s
)
fl
ux
(W
b)
0
1
2
3
4
5
-50
0
50
100
150
200
ti
m
e
(
s
)
t
o
r
que (
N
.
m
)
0
1
2
3
4
5
0
50
100
150
200
ti
m
e
(
s
)
speed (
R
d/
s)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
6, No
. 2, A
p
ri
l
20
16
:
53
5 – 5
4
8
54
2
sp
eed
,
t
h
e est
i
m
a
ted
p
a
rameter is co
m
p
en
sated fo
r so th
at it is cl
o
s
er to
t
h
e ex
p
ected
v
a
lu
e. Th
is
com
p
ensation
has t
h
e form
of a linear relation
be
tween
th
e d
r
ift of
ro
to
r re
sistance
and speed.
Figu
re 6. Esti
mate o
f
ro
tor
resistan
ce b
y
t
h
e filter of
Kal
m
an
witho
u
t
co
m
p
en
satio
n
The followi
ng
figure takes the form
of rotor resi
stance
wit
h
com
p
ensation of
the
drift according to
spee
d
Fig
u
re
7
.
Estimated
ro
t
o
r
resistan
ce b
y
t
h
e
Kalm
an
filter with
co
m
p
en
satio
n
3.
2. R
o
t
o
r
F
a
ul
ts
Fi
gu
re 8 t
a
kes
t
h
e fo
rm
of t
h
e st
at
or current
for an unbalanced de
fect
(4
0
%
of w
h
orl
s
re
m
oved i
n
t
h
e
pha
se at fault), note the a
p
pe
arance
of
t
h
e lin
es in
th
e sp
ectru
m
o
f
th
e cu
rr
en
t. Th
e
d
e
fect with
th
e
ro
tor is
characte
r
ized by the appeara
n
ce of two
f
r
e
qu
en
cies (1±
2g) o
r
g
r
e
pr
esen
ts th
e slip
(
9
2
,
2H
z et1
07H
z)
ar
oun
d
th
e electric freq
u
e
n
c
y f electric (10
0
Hz). If th
e FFT of th
is
sig
n
a
l is m
a
d
e
, th
is d
e
fect
will b
e
v
e
ry d
i
fficu
lt t
o
detect in t
h
e spectrum
of the
curre
nt
because the am
plitude
of the
line
at
the fundam
e
ntal freque
ncy
is very
i
m
p
o
r
tan
t
co
m
p
ared
to
th
at
o
f
th
e
requ
ired
lin
e an
d the
s
e lines are at a frequ
en
cy fram
e
th
e lin
e
with
fund
am
en
tal with
a d
i
fferen
c
e related
t
o
th
e slip
(in g
e
n
e
ral
v
e
ry
weak
).
Fig
u
r
e
8
.
Stator
cu
rr
en
t
un
d
e
r ro
to
r f
a
u
lt
0.5
1
1.5
2
2.5
3
3.5
0.7
0.8
0.9
1
1.1
ti
m
e
(
s
)
es
t
i
m
a
t
e
d r
o
t
o
r
r
e
s
i
s
t
an
c
e
(
p
u
)
0.5
1
1.
5
2
2.5
3
3.5
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
ti
m
e
(
s
)
e
s
ti
mated
rotor re
s
i
s
t
anc
e
c
o
mpe
n
s
a
ted(
pu)
0
1
2
3
4
5
-600
-400
-200
0
200
400
600
ti
m
e
(
s
)
curre
nt
(A
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
An
Exten
d
er K
a
lma
n
Filter-Ba
sed
Ind
u
c
tion Ma
ch
in
es Fa
ults Detectio
n
(
A
bd
elgh
an
i Cha
h
m
i
)
54
3
The f
o
l
l
o
wi
n
g
fi
gu
re t
a
kes t
h
e f
o
rm
of res
i
st
ance est
i
m
a
ted f
o
r a
defec
t
un
bal
a
nce
d
t
o
t
h
e r
o
t
o
r
(4
0%
of
wh
o
r
l
s
rem
oved at
f
a
ul
t
on a
p
h
ase
)
. N
o
t
i
n
g h
o
w
e
v
er t
h
at the estim
a
te of rotor resistance is ass
i
gne
d
to
th
e m
o
m
e
n
t
t=3
seco
nd
(time o
f
th
e app
l
icatio
n
of
defe
ct) from
whe
r
e
appear
a
n
ce
of the low
fre
quency
o
s
cillatio
n
s
. Th
is ex
isten
ce
o
f
t
h
e frequ
e
n
c
y allo
ws t
h
en
on
t
h
e b
a
sis o
f
resistan
ce ro
tor estimated
ch
aracterizin
g
th
at it is well a
d
e
fect
with
t
h
e ro
to
r.
Fig
u
re
9
.
Estimatio
n
resistance ro
to
r fo
r an
asy
m
m
e
trical fau
lt witch
t
h
e
ro
tor
3.
3.
De
fect w
i
t
h
the
S
t
a
t
o
r
Fi
gu
re 1
0
t
a
ke
s t
h
e fo
rm
of t
h
e st
at
or c
u
r
r
e
n
t
fo
r a de
fect
un
bal
a
nce
d
t
o
t
h
e st
at
or (
4
0
%
of
wh
orl
s
rem
oved at fa
ult on a phase
)
. Let us note
the appeara
n
ce
of the freque
ncy, whic
h c
h
aracterizes the
defect,
wh
ich
is
3
f
electric (300
Hz). Th
e app
e
aran
ce
o
f
th
e lin
es in t
h
e s
p
ec
trum
of t
h
e cu
rren
t is du
e
to
the
ap
p
lication
of t
h
e d
e
fect to
t
h
e
m
o
m
e
n
t
t= 3
s
, it is eas
y
t
o
det
ect
t
h
e fre
que
ncy
,
w
h
ich cha
r
acterizes the
defect
in lo
w
fre
que
n
c
y
.
(See
fig
u
r
e
belo
w a
f
ter the
stator
phase c
u
rrent
FFT
).
Fi
gu
re 1
0
. St
at
or
cu
rre
nt
of
t
h
e
m
ach
in
e at fau
lt to
th
e stator
Figure
11.
FFT
of the
static curre
nt of t
h
e
pha
se
Th
e fo
llowing
figu
re
p
r
esen
ts th
e cu
rren
t o
f
th
e ph
ase at exit of the m
achine and the c
u
rrent for the
sam
e
phase at
exi
t
of t
h
e
obs
erve
r, i
t
sho
u
l
d
be speci
fi
e
d
that the two c
u
rrents (m
easure
d
and estim
a
te
d) a
r
e
si
m
ilar. Th
e
ob
serv
er thu
s
m
o
d
i
fied
th
e ro
tor size
re
sist
ance
s
o
that the two cu
rre
nt
s are “identical”. The
err
o
r
,
w
h
i
c
h e
x
i
s
t
s
bet
w
ee
n t
h
e t
w
o cu
rre
nt
s, ex
pl
ai
ns
by
th
e fact wh
y th
e ob
serv
er is a d
i
screte syste
m
: h
e
t
hus
d
o
es
t
h
e c
a
l
c
ul
at
i
ons
o
f
t
h
e si
zes
o
n
t
h
e
basi
s
o
f
m
easure
t
a
ke
n at
p
r
evi
o
us t
i
m
e. Thi
s
e
xpl
ai
n
s
t
h
e l
i
ght
err
o
r
o
f
haul
i
n
g.
0
1
2
3
4
5
-1
-0
.
5
0
0.
5
1
1.
5
2
ti
me
(s)
e
s
tima
te
d
r
e
sis
t
a
n
c
e
r
o
to
r
( p
u
)
0
1
2
3
4
5
-600
-400
-200
0
200
400
600
ti
me
(
s
)
c
u
rre
n
t
(A
)
0
100
200
30
0
40
0
50
0
0
20
40
F(
H
)
am
pl
i
t
ud
e (
db)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
6, No
. 2, A
p
ri
l
20
16
:
53
5 – 5
4
8
54
4
Figu
re
1
2
.
Pha
s
e cu
rre
nt (
I
sa
)
,
O
b
se
rve
d
c
u
r
r
ent
(Isa
k
al)
Fig
u
re
1
2
illustrates th
e app
licatio
n
o
f
t
h
is esti
m
a
to
r on
th
e ex
it of
th
e m
ach
in
e lead
s to
t
h
e
conve
r
ge
nce of
resistance up to
3
sec
o
nd
(
a
pp
licatio
n
of
th
e d
e
f
ect). I
t
is no
ted
w
h
er
eas t
h
e ob
ser
v
er
does n
o
t
manage to m
a
ke converge the resistance est
i
m
a
ted follo
wi
ng t
h
e ap
pl
i
cat
i
on
of t
h
e de
fe
ct
. Thi
s
i
s
ex
pl
ai
ned
because the
model
use
d
for t
h
e e
quations
of the
observe
r
is a balanced
m
odel. Howe
ver the
de
fect re
sults in
an u
nbal
a
nce
d
beha
vi
o
r
o
f
t
h
e
m
achi
n
e. The
obs
er
ver
ha
s o
f
anot
h
e
r ch
oi
ce
t
o
onl
y
m
a
ke
evol
ve dy
nam
i
cal
l
y
rot
o
r
resistance
so t
h
at the
ot
her states
are
c
o
here
nt c
o
m
p
ared to m
easurements.
Fig
u
re
13
. Estimatio
n
ro
tor resistan
ce fo
r
an
asymm
e
trical fault witch t
h
e s
t
ator
3.
4. Sy
nchr
on
ous Dem
o
d
u
l
a
ti
on
Ap
pl
i
e
d
t
o
Ro
to
r
R
e
si
st
ance
Each
defect
ha
s a fre
quency
whic
h cha
r
acte
r
izes it,
th
e idea o
f
t
h
is m
e
t
h
od
co
nsists with
research
th
e a
m
p
litu
d
e
o
f
th
e lin
e asso
ciated
with
this frequ
en
cy
to detect the evolution of
this one. T
h
e pri
n
ci
ple of
th
is d
e
m
o
du
latio
n
is expo
sed
to
th
e
fo
llo
wi
ng
figu
re.
Fi
gu
re 1
4
. Pri
n
ci
pe
o
f
t
h
e de
m
odul
at
i
o
n
Th
e fo
llowing figu
re h
a
s
th
e a
m
p
litu
d
e
o
f
t
h
e
lin
e a
ssociates with t
h
e
fre
que
ncy
of de
fe
ct (2fe). T
h
e
ch
aracteristic is well
d
e
tected b
y
th
is m
o
n
ito
ring
system
. Th
e
d
e
fect is
app
lied
to 3 secon
d
s
.
0
1
2
3
4
5
-600
-400
-200
0
200
400
600
ti
me
(s
)
c
u
rre
n
t
(A
)
Is
a
I
s
ak
al
3.
2
3.
22
3.
24
3.
26
3.
28
3.
3
-4
00
-3
00
-2
00
-1
00
0
10
0
20
0
30
0
ti
me
(s
)
Isa
I
s
ak
al
0
1
2
3
4
5
-1
-0.
5
0
0.5
1
1.5
2
ti
m
e
(
s
)
e
s
t
i
m
a
te
d re
s
i
s
t
a
n
c
e
s
t
at
or
(p
u)
Evaluation Warning : The document was created with Spire.PDF for Python.