Internati
o
nal
Journal of P
o
wer Elect
roni
cs an
d
Drive
S
y
ste
m
(I
JPE
D
S)
Vol
.
6
,
No
. 2,
J
une
2
0
1
5
,
pp
. 28
2~
29
2
I
S
SN
: 208
8-8
6
9
4
2
82
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
/
IJPEDS
Vector Control of Three-Phas
e Induction Motor with Two
Stator Phases Open-Circuit
Sey
e
d
Hesa
m A
s
ga
ri
*
,
Mo
ha
mma
d
Ja
nnat
i*
,
To
le
Sut
i
kno*
*, N
i
k R
u
mzi
N
i
k
Idris*
* UTM-PROTON Future Driv
e
Laborator
y
,
Faculty
of
Elec
trical Engin
eering
,
U
n
iv
ersiti Teknologi Malay
s
ia
Johor Bahru, Malay
s
ia
** Departmen
t
o
f
Electr
i
cal
Engin
eering
,
Faculty
of Industrial
Tec
hnolog
y
,
Univ
er
sitas Ahmad Dahlan
Yog
y
ak
arta
, Ind
ones
i
a
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Feb 16, 2015
Rev
i
sed
Ap
r
27
, 20
15
Accepted
May 10, 2015
Variabl
e
frequ
en
c
y
dr
ives
ar
e us
ed to provid
e
rel
i
abl
e
d
y
nam
i
c s
y
s
t
em
s
and
significant redu
ction in usage of
ener
g
y
and
costs of the indu
ction motors.
Modeling and
co
ntrol of f
a
ulty
o
r
an
unbalanced three-phase induction motor
is obviously
different from health
y
three-phas
e
induction mo
tor. Using
conventional vector control techn
i
ques su
ch as Field-Orien
t
ed Co
ntrol (FOC)
for faul
t
y
thr
ee-
phase indu
ction
m
o
tor, resul
t
s in a sign
ific
ant
torque
an
d
speed oscillation
.
This re
search
presented a nov
el method for vector contro
l
of three-ph
ase in
duction motor under fa
ult condition
(two-
phase open circu
it
fault)
. The prop
osed method for vector con
t
rol o
f
faulty
machin
e is based on
rotor FOC method. A comparison be
tween
convention
a
l and
modified
controll
er s
hows
that
th
e m
odifi
ed con
t
roll
er h
a
s
been s
i
gn
ific
a
n
tl
y r
e
duced
the
torque
and
speed osc
ill
ations
.
Keyword:
Fau
lt cond
itio
n
Fi
el
d-
ori
e
nt
e
d
cont
rol
Three
-
phase
i
n
duct
i
o
n m
o
t
o
r
To
rqu
e
an
d speed
o
s
cillatio
ns
Two stator
pha
ses ope
n
-circ
u
i
t
Vector c
ontrol
Copyright ©
201
5 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
:
M
oham
m
ad Jannat
i
,
Facu
lty of Electri
cal Engineering,
Un
i
v
ersiti Tekn
o
l
o
g
i
Malaysia,
U
T
M Sku
d
a
i,
8
131
0 Joho
r,
Malaysia.
Em
a
il: j
a
n
n
a
ti
m
9
4
@
yah
o
o
.
co
m
1.
INTRODUCTION
Th
ree-p
h
a
se ind
u
c
tion
m
o
to
rs are
u
tilized
in wid
e
ran
g
e
of app
licatio
n
s
as a m
ean
s o
f
tran
sform
i
n
g
electric power to
m
echanica
l
power. T
h
e
alternating c
u
rren
t is prov
ided
to
th
e stat
o
r
wind
ing
d
i
rectly
wh
ereas sup
p
l
y th
e vo
ltag
e
t
o
th
e
ro
t
o
r
wind
ing
is b
y
i
n
ductio
n
;
con
s
eq
uen
tly it is n
a
med
indu
ctio
n
m
ach
in
e.
Th
e ind
u
c
tion
mach
in
e h
a
s th
e ab
ility to
fu
n
c
tion
as a m
o
to
r an
d
as
a g
e
n
e
rator. Nev
e
rt
h
e
less, it is rarely
em
pl
oy
ed as
a
gene
rat
o
r
pr
o
v
i
di
ng
el
ectrical powe
r to a loa
d
. T
h
e
overall
perform
a
nce fe
atures a
s
a
ge
nerator
are n
o
t
go
o
d
eno
u
gh
fo
r m
o
st
usa
g
e. Th
e i
nduct
i
o
n m
achi
n
e i
s
br
oa
dl
y
appl
i
e
d as
a
m
o
t
o
r i
n
m
a
ny
ap
p
lication
s
. Th
e ind
u
c
tion
m
o
to
r is e
m
p
l
o
y
ed
in
d
i
fferen
t
sizes. Sm
a
ll sin
g
l
e-ph
ase in
du
ctio
n
m
o
to
rs are
ap
p
lied in
m
a
n
y
d
o
m
estic ap
plian
ces, su
ch as lawn m
o
wers
, j
u
i
ce m
i
xers,
bl
en
ders
, washing
m
achines, stereo
tur
n
tables, a
n
d
refri
gerat
o
rs
.
Large t
h
ree
-
p
h
a
se ind
u
ctio
n
m
o
t
o
rs (i
n 1
0
'
s
or
10
0'
s of
ho
rsep
o
w
er
) are
appl
i
e
d
in
fans, co
m
p
resso
rs, pu
m
p
s, tex
tile
mills, p
a
p
e
r m
ills an
d
so
fo
rt
h
.
Th
e lin
ear typ
e
o
f
the in
du
ctio
n
m
ach
in
e
h
a
s
b
e
en
created
m
a
in
ly in
o
r
der to u
s
e in
t
r
an
sportatio
n
syste
m
s [1
].
Ove
r
the
past
decade
s
, m
a
ny control tec
hni
que
s
ha
ve bee
n
propose
d
fo
r induction m
o
tors
drive
system
. One
of the m
o
st well-known control m
e
thod fo
r
cont
rolling the
spee
d and torque
of the induction
m
o
t
o
r i
s
Fi
el
d-O
r
i
e
nt
ed C
o
n
t
rol
(F
OC
)
[2]
.
M
odel
i
n
g a
n
d co
nt
r
o
l
o
f
fa
ul
t
y
i
nduct
i
o
n
m
o
t
o
r, i
s
ob
vi
ousl
y
di
ffe
re
nt
fr
om
t
h
e co
nv
ent
i
o
n
a
l
bal
a
nce
d
t
h
r
ee-p
h
ase i
n
d
u
c
t
i
on m
o
t
o
r.
As
suc
h
,
new m
odel
i
ng
an
d c
o
n
t
rol
approaches
ha
ve to
be a
p
plied at the i
n
stance the fa
u
lty is d
e
tected
. By ap
p
l
ying
th
e co
nv
en
tio
n
a
l
b
a
lan
c
ed
th
ree-ph
ase ind
u
c
tion
m
o
to
r
co
n
t
ro
l strategy, su
ch
FOC t
o
fau
lty in
du
ctio
n
m
o
to
r, si
gn
ifican
t
o
s
cillatio
n
s
in
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-86
94
I
J
PED
S
Vo
l. 6,
No
.
2,
Ju
ne 20
15
:
282
–
2
92
2
83
th
e torqu
e
ou
tp
u
t
will b
e
p
r
esen
ce; th
is is b
ecau
s
e
o
f
the un
equ
a
l indu
ctan
ces in
t
h
e d
and
q
ax
is o
f
the
un
bal
a
nce
d
i
n
d
u
ct
i
o
n
[
3
]
,
[4]
Im
port
a
nt
wo
r
k
s
has
been
de
vel
o
ped
co
nce
r
ni
ng t
h
e i
m
plem
ent
a
t
i
on o
f
vect
o
r
co
nt
r
o
l
m
e
t
hods
f
o
r
el
ect
ri
cal
m
achi
n
es un
de
r o
p
e
n
-
p
hase fa
ul
t
[5]
-
[
1
8]
. M
o
st
of t
h
e pe
r
v
i
o
us
wor
k
s h
a
ve f
o
cuse
d on
devel
opi
ng
vect
o
r
c
ont
r
o
l
m
e
t
hods
o
f
fa
u
l
t
y
m
u
l
t
i
-
phas
e
i
n
d
u
ct
i
o
n m
o
t
o
rs
(
f
i
v
e a
n
d si
x
phase
s)
[
5
]
-
[
9
]
,
fa
ul
t
y
Perm
anent
Mag
n
e
t Syn
c
hrono
u
s
Mo
t
o
rs (PMSMs) [1
0]-[12
]
, and
th
ree-ph
ase ind
u
c
tio
n
m
o
to
r u
nder 2
-
ph
ase cond
itio
n
(o
ne-
p
hase
op
en ci
rc
ui
t
faul
t
)
[
3
]
,
[
4
]
,
[
1
3]
-[
18]
but
no
ne
of t
h
em
prese
n
t
e
d i
n
t
h
e ca
se of
vect
or c
ont
rol
m
e
t
hod
f
o
r t
h
r
ee-p
h
ase i
n
d
u
c
t
i
on m
o
t
o
r
dri
v
e wi
t
h
t
w
o
st
at
or
p
h
ases
o
p
e
n
-ci
r
cui
t
.
Thi
s
researc
h
prese
n
t
e
d
a ne
w
m
e
t
hod
f
o
r
vect
o
r
c
ont
rol
of
t
h
ree-
pha
se i
n
d
u
ct
i
o
n
m
o
t
o
r un
de
r
f
a
ul
t
co
nd
itio
n. Main
obj
ectiv
es of th
is re
search
are as fo
llo
ws: (1): To
d
e
v
e
l
o
p
a m
o
d
e
l of a fau
lty th
ree-ph
ase
i
n
d
u
ct
i
on m
o
t
o
r
whe
n
t
w
o-
p
h
ases o
f
t
h
e st
at
or are
ope
n
ci
rcui
t
,
w
h
i
c
h
can be c
ont
r
o
l
l
ed usi
n
g r
o
t
o
r
FOC
t
echni
q
u
e,
(
2
):
To
m
odi
fy
a co
nve
nt
i
o
nal
rot
o
r
FOC
o
f
in
du
ction
m
o
to
r, so th
at it
can
b
e
app
lied
for
un
bal
a
nce
d
t
h
r
ee-p
h
ase i
n
d
u
c
t
i
on m
o
t
o
r
(w
h
i
l
e
two-phase
of stator
a
r
e op
en
c
i
r
c
u
it)
.
2.
THREE-PHASE INDUCT
ION
MOTOR MODEL
WITH TWO STATOR PHASES OPE
N
-
C
I
RCU
IT
In
t
h
is section th
e
d
-
q
m
o
d
e
l o
f
th
ree-p
h
a
se in
du
ction
mo
tor wh
en
two ph
ases
o
f
stato
r
are
op
en-
circu
it is presen
ted
.
Figure
1
sh
ow
s
the
d-q
axes a
n
d stator a-a
x
is.
Figure
1. d-q a
x
es a
n
d stator
a-axis
I
n
Figur
e 1
,
f
as
can be cu
rre
nt
,
vol
t
a
ge o
r
fl
u
x
an
d
θ
e
is the
angle bet
w
een
q-a
x
is and a-a
x
is varia
b
le
of
st
at
or
. B
a
se
d
on
Fi
g
u
r
e
1,
st
at
or a
-
axi
s
va
ri
abl
e
i
s
fo
rm
ul
at
ed i
n
t
e
rm
s of
d
an
d
q a
x
es
as f
o
l
l
o
ws:
,
(
1
)
C
onsi
d
eri
n
g
t
h
at
t
h
e
d a
n
d
q
axes a
r
e
ort
h
o
g
o
n
al
, c
o
nse
q
u
e
nt
l
y
t
h
ei
r
dot
pr
o
duct
has t
o
be e
qual
t
o
zero. As a res
u
lt,
θ
e
can be e
q
ual to zero or (
π
/2
). In
th
is st
u
d
y
it is assu
med
,
θ
e
i
s
e
qual
t
o
zero
t
h
en d and q
ax
es can
b
e
written
as equ
a
tion
(2):
0
,
(
2
)
Th
erefo
r
e, th
e
stato
r
tran
sformatio
n
m
a
trix
in
th
e fa
u
lt situatio
n
(t
wo
stato
r
ph
ases
o
p
e
n-circu
it) can
be obt
ai
ne
d as equat
i
o
n (
3
):
→
1
(
3
)
Th
e ro
tor transform
a
t
i
o
n
m
a
tr
ix
is th
e sa
m
e
as ro
to
r tran
sform
a
t
i
o
n
m
a
trix
in
th
e b
a
lan
c
ed
con
d
ition.
In
eq
uat
i
o
n (
5
)
,
γ
is th
e ang
l
e
b
e
tween
ro
t
o
r
a-axis va
riable and
d-a
x
is.
120
120
120
120
(
4
)
B
y
appl
y
i
ng st
at
or a
nd
rot
o
r t
r
ans
f
orm
a
t
i
on t
o
v
o
l
t
a
ge eq
ua
t
i
on o
f
t
h
e m
o
t
o
r
,
t
h
e v
o
l
t
a
ge
equat
i
o
n
of
faul
t
y
m
o
t
o
r i
n
st
at
i
onary
ref
e
rence
f
r
am
e can
be s
h
ow
n as
fol
l
o
ws
[
14]
:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
Vect
or C
ont
r
o
l
of
T
h
ree
-
P
has
e I
n
d
u
ct
i
o
n M
o
t
o
r w
i
t
h
Tw
o
St
at
or
Ph
ases
O
p
en
-C
i
r
cui
t
(
M
oh
a
m
m
a
d J
a
n
n
a
t
i
)
28
4
0
0
0
0
0
0
(
5
)
0
0
0
0
0
0
0
0
(
6
)
whe
r
e,
t
h
e
para
m
e
t
e
r of t
h
e m
odel
a
r
e
defi
ne
d as
t
h
e
val
u
es
of
(
7
):
0
,
,
,
0
,
,
0
,
(7
)
Th
e electro
m
a
g
n
e
tic torq
u
e
of fau
lty th
ree-ph
ase i
n
du
ction
m
o
to
r can
b
e
sh
own
as
fo
llows:
(
8
)
In (
5
)-
(8
),
V
ds
s
and
V
qs
s
are st
at
or d a
nd
q a
x
es v
o
l
t
a
ges
i
ds
s
, i
qs
s
,
i
dr
s
and
i
qr
s
represe
n
t
t
h
e st
at
or a
nd
r
o
to
r
d
and
q
ax
e
s
cu
rr
en
ts
λ
ds
s
,
λ
qs
s
,
λ
dr
s
and
λ
qr
s
den
o
t
e
t
h
e
st
at
or a
n
d
r
o
t
o
r d
an
d
q a
x
es
L
ms
, L
mr
, L
ls
and
L
lr
represe
n
t the s
t
ator and rot
o
r,
m
u
tual and leakage i
n
ducta
n
ces.
r
sd
,
r
sq
and
r
r
are the stator a
n
d rotor d
and
q
axes re
sistance
s. As
presente
d, the m
odel
of three
-
pha
se in
du
ction
m
o
to
r with
t
w
o
ph
ases op
en-circu
i
t
h
a
s
the sam
e
structure
of equations com
p
ared
wi
th bala
nced t
h
ree-phase i
n
du
ctio
n
m
o
to
r ex
cep
t
th
e
v
a
lu
e
of th
e
param
e
t
e
rs of
t
h
e m
odel
.
3.
ROTO
R F
O
C OF T
H
RE
E-PH
ASE I
N
DU
CTIO
N
MOTO
R
MO
DEL WITH
TWO STAT
OR
PHASES OPEN
-C
IRCU
IT
To a
p
pl
y
t
h
e
r
o
t
o
r F
O
C
st
rat
e
gy
, t
h
e
e
quat
i
ons
of t
h
e i
n
d
u
c
t
i
on m
o
t
o
r
sh
oul
d
be t
r
a
n
s
f
o
r
m
e
d t
o
t
h
e
rot
o
r
refe
re
nce
fram
e
. F
o
r t
h
is p
u
r
p
o
s
e, t
h
e r
o
tationa
l tran
sform
a
t
i
o
n
matrix
as sh
own in
(9
)
shou
ld
b
e
ap
p
lied to
t
h
e
v
a
riab
les of t
h
e m
o
to
r [1
9
]
.
(
9
)
In
th
is tran
sformatio
n
m
a
trix
,
θ
mr
i
s
t
h
e angl
e bet
w
ee
n rot
a
t
i
onal
refe
renc
e fram
e
and t
h
e st
at
i
onary
refe
rence
fram
e
. Also t
h
e su
persc
r
ipt "
mr
" sh
ows th
at the v
a
riab
les are ex
pressed
in
ro
tation
a
l referen
ce
fram
e
. In
t
h
e op
en ph
ase
fau
l
t
,
th
is tran
sformatio
n
m
a
trix
can
no
t
b
e
app
l
ied
to
th
e m
o
t
o
r
v
a
riab
les, si
n
ce the
m
o
to
r is un
balan
ced
(M
srd
≠
M
srq
and
L
sd
≠
L
sq
).
A
p
pl
y
i
ng
t
h
i
s
m
a
t
r
i
x
gene
rat
e
s
f
o
r
w
ar
d a
n
d
b
ackwa
r
d
com
pone
nt
s i
n
t
h
e
m
o
t
o
r eq
uat
i
o
n
s
[4]
.
T
o
sol
v
e t
h
i
s
pr
obl
em
, i
n
t
h
i
s
researc
h
, i
t
i
s
pr
o
pose
d
u
n
b
a
l
anced
t
r
ans
f
o
r
m
a
ti
on
m
a
t
r
i
ces. The
pu
rp
ose
of
u
s
i
ng t
h
ese t
r
a
n
sf
orm
a
t
i
on m
a
t
r
i
ces i
s
chan
gi
n
g
t
h
e
un
bal
a
nced
faul
t
y
m
o
t
o
r equat
i
o
ns t
o
t
h
e
bal
a
nced e
qua
t
i
ons. S
o
i
t
i
s
pos
si
bl
e t
o
co
nt
r
o
l
t
h
e faul
t
y
i
nduct
i
o
n m
o
tor b
y
u
s
ing
so
m
e
ch
an
g
e
s in
th
e con
v
e
n
tion
a
l co
ntro
ller. Th
e id
ea o
f
u
s
i
n
g
th
ese tran
sfo
r
m
a
tio
n
m
a
trices is a
d
ap
ted
fr
om
equi
val
e
nt
ci
rcui
t
of si
ngl
e
-
p
h
ase i
n
d
u
ct
i
on m
o
t
o
r.
Thi
s
m
o
t
o
r i
s
ty
pi
cal
l
y
unbal
a
nced
wi
t
h
t
w
o st
at
or
windings, m
a
i
n
a
nd a
u
xiliary windings
which a
r
e act
ua
lly
displaced ort
h
ogonal. Figure 2(a
)
shows
t
h
e
eq
u
i
v
a
len
t
circu
it o
f
m
a
in
an
d
aux
iliary win
d
i
n
g
s of stator fo
r sing
le-phase in
du
ction
m
o
to
r [2
]. The v
o
ltag
e
eq
u
a
tion
s
of the m
a
in
and
auxiliary win
d
i
n
g
s
are
d
e
fi
n
e
s
b
y
(10
)
and
(1
1).
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
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94
I
J
PED
S
Vo
l. 6,
No
.
2,
Ju
ne 20
15
:
282
–
2
92
2
85
Fig
u
re
2
.
(a) Eq
u
i
v
a
len
t
ci
rcuit o
f
m
a
in
and
au
x
iliary
wind
in
g
s
of stat
o
r
for sing
le-ph
a
se i
n
du
ction
m
o
to
r,
(b) Sim
p
lified
eq
u
i
v
a
len
t
circu
it of m
a
in
an
d aux
iliary
windin
g
s
of stator
fo
r sing
le-p
h
a
se in
du
ction
m
o
to
r
(
1
0
)
,
(
1
1
)
I
n
(1
0)
and
(11
)
,
Z
lm
,
Z
la
,
Z
f
a
n
d
Z
b
, are
the leaka
g
e
im
pedance
of
main
wind
ing
,
th
e leak
ag
e
i
m
p
e
d
a
n
ce
o
f
au
x
iliary wind
ing
,
th
e im
p
e
d
a
n
ce
of
fo
rward
d
i
rectio
n an
d
t
h
e im
p
e
d
a
n
ce
of b
a
ck
ward
d
i
rection
resp
ectiv
ely.
E
fa
,
E
ba
,
E
fm
and
E
bm
are t
h
e v
o
l
t
a
ge
i
nduce
d
by
i
t
s
ow
n fl
u
x
es
λ
fa
,
λ
ba
,
λ
fm
, and
λ
bm
,
wh
ich
are fo
rward
an
d
b
a
ck
ward
fl
u
x
e
s
for m
a
in
an
d aux
iliary wind
ing
s
resp
ecti
v
ely. Moreov
er, the
vari
a
b
l
e
s o
f
N
a
and
N
m
are the aux
iliary and m
a
in
wind
ings nu
m
b
er of the stato
r
. By u
s
in
g
so
m
e
ch
ang
e
o
f
vari
a
b
l
e
s, t
h
e s
i
m
p
l
i
f
i
e
d equi
val
e
nt
ci
rc
ui
t
of si
ngl
e
-
p
h
as
e i
n
d
u
ct
i
o
n ca
n be
o
b
t
a
i
n
e
d
.
The
defi
n
e
d
v
a
ri
abl
e
s
are
V
f
,
I
f
,
V
b
a
nd
I
b
whi
c
h ar
e t
h
e f
o
r
w
a
r
d
vol
t
a
ge
, f
o
rwa
r
d c
u
r
r
e
n
t
,
bac
k
wa
r
d
v
o
l
t
a
ge
and
bac
k
war
d
cur
r
en
t
respectively.
,
,
,
(1
2)
To get the simplified equi
valent circ
uit of the single-phas
e
induction
m
o
tor, it is necess
a
ry to defi
ne
V
m
,
I
m
,
V
a
an
d
I
a
as fo
llows:
,
,
,
(
1
3
)
b
y
sub
s
titu
ting (1
3) in (10
)
, (1
1), it can
b
e
co
n
c
l
u
d
e
d
t
h
at:
1
2
2
,
1
2
2
,
2
2
,
(
1
4
)
B
a
sed o
n
t
h
e
(1
4)
, t
h
e si
m
p
l
i
f
i
e
d eq
ui
val
e
nt
ci
rcui
t
o
f
m
a
i
n
and a
uxi
l
i
a
ry
wi
ndi
n
g
s
of st
at
o
r
f
o
r
si
ngl
e-
p
h
ase i
n
duct
i
o
n m
o
t
o
r can be sh
o
w
n
as Fi
gu
re 2(
b
)
.
Accor
d
i
n
g t
o
Fi
gu
re 2
(
b
)
, i
f
we ne
gl
ect
Z
d
, then
th
e eq
u
i
v
a
len
t
circu
it will
b
e
d
i
v
i
d
e
d in
to two circu
its,
wh
ich bo
th of t
h
em
in
d
i
cate a b
a
lan
ced m
o
t
o
r wit
h
fo
rwa
r
d
di
rect
i
o
n
an
d
bac
k
wa
rd
di
rect
i
o
n.
T
h
ere
f
o
r
e,
base
d
o
n
V
f
and
I
f
in
(12
)
, it is easy
to
write:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
Vect
or C
ont
r
o
l
of
T
h
ree
-
P
has
e I
n
d
u
ct
i
o
n M
o
t
o
r w
i
t
h
Tw
o
St
at
or
Ph
ases
O
p
en
-C
i
r
cui
t
(
M
oh
a
m
m
a
d J
a
n
n
a
t
i
)
28
6
1
,
1
(
1
5
)
In fact
(
1
5),
d
e
m
onst
r
at
es a t
r
ans
f
o
r
m
a
ti
on
m
a
t
r
i
x
from
un
bal
a
nce
d
si
t
u
at
i
on
(i
.e.
V
m
and
V
a
) to
b
a
lan
ces situ
atio
n (i.e.
V
f
and
jV
f
).
B
a
se
d
on
t
h
ese e
q
uat
i
ons
we a
r
e a
b
l
e
t
o
use s
o
m
e
subst
i
t
u
t
i
ons as
f
o
l
l
o
ws:
↔
,
1
↔
,
↔
,
↔
,
↔
,
↔
,
↔
,
↔
,
↔
,
↔
,
≃
≃
(
1
6
)
Based
o
n
th
ese sub
s
titu
tion
s
,
th
e stator
ro
tatio
n
a
l t
r
ansfo
r
m
a
tio
n
for
t
h
e variab
les fro
m
statio
n
a
ry
t
o
rot
o
r
refe
re
nce
fram
e
is as f
o
l
l
ows:
,
(1
7)
In o
r
der t
o
t
r
a
n
sf
orm
t
h
e
m
o
del
of t
h
e fa
ul
t
y
i
nduct
i
o
n m
o
t
o
r t
o
r
o
t
o
r
ref
e
rence f
r
am
e, fi
rst
t
h
e ne
w
trans
f
orm
a
tion
m
a
trices for the stator
v
a
riab
les (17) are ap
p
lied. As a resu
lt th
e stato
r
v
o
ltag
e
equ
a
tio
n
s
i
n
rot
o
r
refe
re
nce
fram
e
can
be a
s
f
o
llows:
0
0
(
1
8
)
whe
r
e,
i
ds
-mr
and
i
qs
-mr
are the
backwa
rd com
p
one
n
ts
of t
h
e
stat
or
c
u
rrents
that
are obtained from
:
cos
s
i
n
cos
s
i
n
cos
sin
(
1
9
)
To
tran
sform
t
h
e ro
tor
v
o
ltage equ
a
tio
n
t
o
ro
tor re
feren
ce
fram
e
th
e ro
t
o
r tran
sform
a
tio
n
m
a
trix
(9),
m
u
st b
e
ap
p
lied
to
ro
tor vo
ltag
e
equ
a
tion
.
As a resu
lt
th
e ro
tor vo
ltag
e
equ
a
tio
ns in
ro
tor referen
ce frame can
b
e
as fo
llo
ws:
0
0
(2
0)
Th
e electro
m
a
g
n
e
tic torq
u
e
of th
e fau
lty m
o
to
r
i
n
ro
tor
referen
c
e fram
e will b
e
as
fo
ll
o
w
s:
(
2
1
)
The
rot
o
r
flu
x
equatio
n i
n
rot
o
r
re
fere
nce
fr
am
e can b
e
s
h
o
w
n
as
follo
ws:
0
0
0
0
(
2
2
)
In
o
r
de
r t
o
a
p
ply
r
o
to
r fiel
d
-
o
r
iented
c
ontr
o
l stra
teg
y
, the d-ax
is of t
h
e ro
tatio
n
a
l
frame
m
u
st b
e
situ
ated
o
n
ro
t
o
r flux
, its m
e
a
n
s:
|
|
0
(
2
3
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-86
94
I
J
PED
S
Vo
l. 6,
No
.
2,
Ju
ne 20
15
:
282
–
2
92
2
87
therefore:
|
|
,
(
2
4
)
By su
b
s
titu
ting
(23
)
and
(2
4) in (1
8), th
e
stato
r
v
o
ltage
eq
u
a
tion
s
fo
r ro
tor field-o
r
ien
t
ed
con
t
ro
l
strategy can be
obtained as
fol
l
ows:
|
|
(2
5)
|
|
(2
6)
whe
r
e,
,
(
2
7
)
The
rot
o
r
v
o
l
t
a
ge e
quat
i
o
ns
f
o
r r
o
t
o
r
fi
el
d-
ori
e
nt
ed c
o
nt
r
o
l
s
t
rat
e
gy
can
be
obt
ai
ne
d a
s
f
o
l
l
ows:
|
|
|
|
0
,
|
|
0
(
2
8
)
an
d th
e electromag
n
e
tic to
rque equ
a
tio
n will
b
e
as fo
llows:
|
|
(
2
9
)
The st
at
o
r
v
o
l
t
a
ge eq
uat
i
o
ns
can be
di
vi
ded
i
n
t
o
dec
o
upl
i
n
g
,
re
fere
nce a
nd
bac
k
wa
rd c
o
m
pone
nt
s
as:
,
(
3
0
)
|
|
,
,
(3
1)
|
|
,
,
(
3
2)
Defi
ni
n
g
t
h
e
s
e
va
ri
abl
e
s
hel
p
us
t
o
desi
gn
t
h
e c
o
nt
rol
bl
oc
ks.
F
o
r
t
h
i
s
p
u
r
p
o
se,
V
ds
d
an
d
V
qs
d
can
be
gene
rat
e
d
by
t
h
e
dec
o
u
p
l
i
n
g
ci
rcui
t
an
d
V
ds
b
and
V
qs
b
ca
n
be
gen
e
rat
e
d
b
y
back
wa
rd
bl
ock
.
M
o
re
ov
er
,
V
ds
ref
and
V
qs
ref
can
b
e
ge
nerat
e
d by
t
w
o PI co
nt
r
o
l
bl
oc
ks
as
f
o
l
l
o
ws:
Δ
,
Δ
(
3
3
)
C
onse
q
uent
l
y
, t
h
e rot
o
r fi
el
d
-
ori
e
nt
ed co
nt
r
o
l
bl
ock
di
agra
m
of faul
t
y
t
h
r
ee-p
h
ase i
n
d
u
c
t
i
on m
o
t
o
r i
s
rep
r
ese
n
t
e
d i
n
Fi
gu
re
3.
Acc
o
r
d
i
n
g t
o
t
h
i
s
fi
g
u
re t
h
e re
d
bl
oc
ks s
h
o
w
t
h
e p
a
rt
s i
n
t
h
e
con
v
e
n
t
i
o
nal
vect
o
r
cont
rol
t
h
at
m
u
st
be c
h
a
nge
d
f
o
r
i
n
or
de
r t
o
b
e
use
d
fo
r t
h
e
un
bal
a
nce
d
o
r
faul
t
y
t
h
re
e-
ph
ase i
n
duct
i
o
n
m
o
t
o
r.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
Vect
or C
ont
r
o
l
of
T
h
ree
-
P
has
e I
n
d
u
ct
i
o
n M
o
t
o
r w
i
t
h
Tw
o
St
at
or
Ph
ases
O
p
en
-C
i
r
cui
t
(
M
oh
a
m
m
a
d J
a
n
n
a
t
i
)
28
8
Fi
gu
re
3.
B
l
oc
k
di
ag
ram
of p
r
o
p
o
sed
r
o
t
o
r
FOC
of
fa
ul
t
y
t
h
ree
-
p
h
ase
i
n
d
u
ct
i
o
n
m
o
t
o
r
4.
SIMULATION RESULTS
In
th
is section
,
si
m
u
latio
n
resu
lts are d
i
scu
s
sed
in
o
r
der t
o
veri
fy
t
h
e ef
fe
ct
i
v
eness
of t
h
e pr
op
ose
d
cont
rol
m
e
t
hod f
o
r a
faul
t
y
t
h
ree-
p
h
ase i
n
duct
i
o
n m
o
t
o
r.
To sh
o
w
t
h
e
dy
nam
i
c behavi
o
r
o
f
fa
ul
t
y
m
o
t
o
r,
si
m
u
latio
n
is co
ndu
cted
b
y
usin
g
M
A
TLAB (M-File) so
ftware.
Th
e fo
urth
o
r
d
e
r Run
g
e-Ku
tta algo
rith
m
is
use
d
f
o
r s
o
l
v
i
ng t
h
e bal
a
n
c
ed an
d fa
ul
t
y
i
n
d
u
ct
i
on m
o
t
o
r e
quat
i
o
ns
. The pa
ram
e
t
e
rs and
rat
i
n
g
s
of t
h
e
sim
u
l
a
t
e
d m
o
tor
are a
s
Ta
bl
e
1.
Tabl
e
1. T
h
e
P
a
ram
e
t
e
rs and
R
a
t
i
ngs
of
t
h
e
Sim
u
l
a
t
e
d M
o
t
o
r
Voltage=125 v
f=50 Hz
Pole=4
r
20.6
Ω
r
19.15
Ω
L
L
0
.0814
H
L
0
.851
H
J
0.
0038
kg
.
m
Fi
gu
re 4
sh
ow
s t
h
e com
p
ari
s
on
bet
w
ee
n c
o
n
v
e
n
t
i
onal
a
n
d m
odi
fi
ed ve
ct
or c
ont
r
o
l
l
e
r
fo
r fa
ul
t
y
th
ree-ph
ase i
n
d
u
c
tion
m
o
to
r un
d
e
r
n
o
-lo
a
d
co
nd
itio
n. In
th
is case the sp
eed
refe
ren
ce is set at
5
00rp
m
.
Accord
ing
to
t
h
is Figure, t
h
e referen
ce sp
eed
is set at
500
rp
m
.
Fro
m
t=0
s
to t=1
s
, m
o
to
r is
wo
rk
ing un
d
e
r
heal
t
h
y
con
d
i
t
i
on a
nd f
r
o
m
t=1s t
o
t
=
3s m
o
t
o
r i
s
wor
k
i
n
g u
nde
r fa
ul
t
y
con
d
i
t
i
on. T
h
i
s
Fi
gu
re sh
ow
s usi
n
g
m
o
d
i
fied
con
t
ro
ller in
co
m
p
arison
with
co
nv
en
tion
a
l co
n
t
ro
ller t
h
e torqu
e
an
d sp
eed
o
s
cillatio
n
h
a
s b
e
en
red
u
ce
d.
(a)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
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-86
94
I
J
PED
S
Vo
l. 6,
No
.
2,
Ju
ne 20
15
:
282
–
2
92
2
89
(b
)
(c)
(d
)
(e)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
Vect
or C
ont
r
o
l
of
T
h
ree
-
P
has
e I
n
d
u
ct
i
o
n M
o
t
o
r w
i
t
h
Tw
o
St
at
or
Ph
ases
O
p
en
-C
i
r
cui
t
(
M
oh
a
m
m
a
d J
a
n
n
a
t
i
)
29
0
(
f)
Fig
u
re
4
.
Sim
u
latio
n
resu
lts
of th
e th
ree-p
h
a
se ind
u
c
tion
m
o
tor
v
ecto
r
contro
l
u
n
d
e
r fau
lty co
nd
itio
n (no lo
ad
co
nd
itio
n); (a):
Torqu
e
(co
n
v
e
n
tio
n
a
l
)
, (b):
To
rqu
e
(m
o
d
i
fied
), (c): Sp
eed
(co
n
v
e
n
tio
n
a
l
)
,
(d): Sp
eed
(m
odified)
,
(e
)
:
Stator
A-
A
x
is
Cur
r
e
n
t (c
on
v
e
n
t
i
onal
)
, (f
):
S
t
at
or A-
Axi
s
C
u
r
r
ent
(m
odi
fi
e
d
)
Fi
gu
re 5 sh
o
w
s vect
or c
ont
r
o
l
of t
h
ree-
ph
ase i
ndu
ct
i
on
m
o
t
o
r based o
n
m
odi
fi
ed co
nt
r
o
l
l
e
r an
d
u
n
d
e
r lo
ad
cond
itio
n
.
Acco
rd
i
n
g
t
o
th
is Figure fro
m
t=0
s
t
o
t=1
s
m
o
to
r is work
ing
un
der h
ealth
y cond
ition
and
fr
om
t
=
1s t
o
t
=
3s m
o
t
o
r i
s
wo
rki
ng
un
d
e
r faul
t
y
co
ndi
t
i
on. T
h
e spee
d refe
re
nce i
s
set
at
500r
pm
. Whi
l
e
th
e m
o
to
r is
wo
rk
ing
un
d
e
r
fau
lty co
nd
ition at t=2
s
a lo
ad equ
a
l to
0
.
2N.m
is ap
p
lied
.
Fig
u
re
5
sho
w
s th
e
go
o
d
per
f
o
r
m
a
nce
of t
h
e p
r
o
pos
ed c
o
nt
r
o
l
l
e
r f
o
r
vect
or c
ont
rol
of
fa
ul
t
y
i
n
d
u
ct
i
on m
o
t
o
r eve
n
u
n
d
e
r l
o
a
d
co
nd
itio
n.
(a)
(b
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-86
94
I
J
PED
S
Vo
l. 6,
No
.
2,
Ju
ne 20
15
:
282
–
2
92
2
91
(c)
(d
)
Fig
u
re
5
.
Sim
u
latio
n
resu
lts
of th
e th
ree-p
h
a
se ind
u
c
tion
m
o
tor
v
ecto
r
contro
l
u
n
d
e
r fau
lty co
nd
itio
n
(lo
a
d
co
nd
itio
n); (a): Sp
eed, (b
): Zoo
m
o
f
Speed
, (c): Torque,
(d
): Stato
r
A-Ax
is
Curren
t
5.
CO
NCL
USI
O
N
Thi
s
pa
pe
r ha
s prese
n
t
e
d a vect
o
r
co
nt
r
o
l
m
e
t
hod f
o
r f
a
ul
t
y
t
h
ree-
ph
ase i
n
d
u
ct
i
o
n
m
o
t
o
r (t
hree
-
pha
se i
n
duct
i
o
n
m
o
t
o
r
whe
n
t
w
o
p
h
ases
of
t
h
e st
at
or
are
ope
n ci
rc
ui
t
)
.
I
t
i
s
sho
w
n t
h
e
d-
q m
odel
o
f
faul
t
y
th
ree-ph
ase i
n
d
u
c
tion
m
o
to
r
h
a
s th
e sam
e
struct
ure
of equa
tions as
the
bal
a
nced three-phase induction
m
o
tor.
In t
h
i
s
st
u
d
y
,
by
usi
n
g
som
e
m
odi
fi
cat
i
ons
t
o
t
h
e c
o
nv
ent
i
onal
c
ont
rol
l
e
r, a
n
ovel
t
e
c
h
ni
q
u
e f
o
r t
h
ree
-
p
h
ase
in
du
ctio
n
m
o
t
o
r wh
ile
two
ph
ases o
f
t
h
e st
ator are ope
n
circuit has
bee
n
pr
esen
ted. A co
m
p
arison between
con
v
e
n
t
i
onal
a
nd m
odi
fi
ed c
ont
rol
l
e
r i
n
di
c
a
t
e
d t
h
at
, t
h
e
m
odi
fi
ed cont
r
o
l
l
e
r has si
gni
fi
cant
l
y
red
u
ce
d t
h
e
to
rq
u
e
and
sp
eed
o
s
cillatio
n
s
. Besid
e
th
e imp
l
em
en
tatio
n
of th
is con
t
ro
l strateg
y
fo
r th
ree-ph
ase i
n
du
ctio
n
m
o
to
r, th
is m
e
th
od
can
b
e
also
u
s
ed
for
v
ecto
r
con
t
ro
l
o
f
sin
g
l
e-ph
ase i
ndu
ctio
n m
o
to
r
wh
en
wo
rk
s
with
ju
st
main
wind
ing
an
d v
ect
o
r
contro
l
o
f
asymme
trical
two
-
ph
ase
in
du
ction
m
o
to
r un
d
e
r o
p
e
n-ph
ase fau
lt.
REFERE
NC
ES
[1]
P.C. SEN, “Principles
of
Electric Machin
es and
Power El
ectronics
”, John
Wiley
& Sons,
1997.
[2]
K. Saty
anar
ay
an
a, P. Surekh
a,
and
P. Vijay
a
Pr
asuna, “A New FOC Approach of Induction Mo
tor Drive Using
DTC Strateg
y
f
o
r the Min
i
m
i
z
a
tion of
CMV”,
Internationa
l Journal of Power
Electronics a
nd Drive System
(
I
JPEDS)
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,
no
. 2
,
pp
. 241–
250, 2013
.
[3]
Z. Yifan and T.A. Lipo, “An approach
to modeling and field-or
iented con
t
rol of a three phas
e
in
duction machin
e
with s
t
ruc
t
ural
i
m
b
alance
”,
In
P
r
oc. A
PEC
, San
Jos
e
, T
X
, 1996
,
pp. 380–386
.
[4]
M. Jannati, N.R
.
N. Idris,
and
M.J.A. Aziz, “A New
Method for RFOC of I
nduction Motor
Under Open-Phase
Fault”
,
in
Industrial Electroni
cs Society,
IECON 2013
, 2013
, pp
.
2530–2535.
[5]
Y. Zhao and T.A. Lipo, “Modeling and control
of a
m
u
lti-phas
e
induction m
a
chine with struct
ural unbal
a
nce”,
IEEE Transactio
ns on Energ
y
Co
nversion
, vo
l. 11
, no
. 3
,
pp
. 570–
577, 1996
.
[6]
Huangsheng Xu, H.A. Toli
y
a
t
,
and L.J.
Petersen
, “Resilient Current Contro
l of Five-Phase Induct
i
on Motor under
Asy
mmetrical Fa
ult Conditions”,
App
lied
Pow
e
r
Ele
c
tronics Con
f
erence
and
Expo
sition (
APEC)
, 2
002, pp
. 64–71
.
Evaluation Warning : The document was created with Spire.PDF for Python.