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. 4, Decem
ber
2015, pp. 819~
830
I
S
SN
: 208
8-8
6
9
4
8
19
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
Low Speed Estimation in Sensor
less Di
rect
Torque Cont
roll
ed
Induction Motor Drive Using Extended Kalman Filter
Mini R
1
, S
a
r
a
nya
C
2
, B. Hariram Sathees
h
3
,
Dine
sh M.
N
4
1
VTU Res
ear
ch
s
c
holar,
F
acu
lt
y
of Am
rita Vis
h
w
a
Vid
y
ap
ee
tham
Univers
i
t
y
,
Indi
a
2
Amrita Vishwa
Vidy
apee
tham
U
n
iversit
y
,
Indi
a
3
ABB GISL, Bangalore, Ind
i
a
4
R V College of
Engineering, Bangalore, Ind
i
a
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Apr 27, 2015
Rev
i
sed
Sep
29
, 20
15
Accepted Oct 10, 2015
Sensorless Direct
Torque Con
t
r
o
l (DTC)
is a p
o
werful con
t
rol
scheme for
high performan
ce
control of
in
duction
motor (
I
M)
drives, which provid
e
s
ver
y
quick d
y
namic response with simp
le stru
cture and
a d
ecou
p
led
contro
l
of torque and flu
x
.
The p
e
rform
ance of the DTC dr
ive greatly
dep
e
nds on the
accur
a
c
y
of th
e estim
ated flu
x
com
ponents, torque and speed, using
monitored stato
r
voltag
e
s and
current
s
.
Low s
p
eed es
t
i
m
a
tion
is
a gre
a
t
chal
lenge be
cau
s
e
of the pres
enc
e
of tr
ansient off
s
et, drift and do
mination of
ohmic voltag
e
d
r
op.Extended K
a
lman fi
lter
(EK
F
) is a non linear ad
aptiv
e
filte
r which
perf
orm
s
the proces
s of finding
the
best estim
a
t
e fro
m
the noi
s
y
data
based on
state sp
ace tech
nique
and re
cur
s
ive algo
rithm
.
This
pap
e
r
m
a
inl
y
focus
e
s
on the accur
a
te
es
tim
ation of s
p
eed ranging fro
m
very
low
s
p
eed to ra
ted s
p
eed us
ing th
e e
quation of m
o
tio
n. A new s
t
at
e s
p
ace m
ode
l
of
the
IM
is dev
e
loped for estimation in
EKF, with load torque
as an input
variab
le
and not
as
an
es
tim
at
ed
quanti
t
y
which
is the
case in m
o
st previous
studies.Th
e
dev
e
loped
algor
ith
m is
validated
using MATLAB-Simulink
platform for speeds ranging
fro
m low speed to r
a
ted sp
eed
at r
a
ted torqu
e
an
d
at var
i
ous torque conditions. An
exhaustiv
e an
alysis is carried
out to validate
the p
e
rform
ance
of DTC Indu
cti
on m
o
tor drive
e
s
pecia
l
l
y
at
the
l
o
w s
p
eeds
.
The res
u
l
t
s
are p
r
om
is
ing for acc
urate
es
tim
ation
of s
p
eed ranging
from
low
s
p
eed to
ra
ted
s
p
eed us
ing
EKF
.
Keyword:
Di
rect
t
o
rq
ue
c
ont
rol
Ex
tend
ed
Kalman
filter
Indu
ctio
n m
o
to
r
d
r
iv
e
Low s
p
eed estimation
Sens
orl
e
ss
co
n
t
rol
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
:
Min
i
R,
Depa
rtem
ent of Electrical and
Electronics
E
ngi
neeri
n
g,
Am
rita Vishwa
Vidy
a
p
eetham
,
Ban
g
alo
r
e Ca
m
pus,
Carm
alram
,
Of
f Sa
rja
p
ur
Roa
d
, Ba
n
g
alo
r
e-56
003
5, K
a
rn
at
aka, India.
Em
a
il: min
i
_
s
uj
ith
@b
lr.am
r
it
a.edu
,
m
i
n
i
r_
su
j
ithr@yah
o
o
.
co
m
1.
INTRODUCTION
Devel
opm
ents in powe
r electronics
in the
last decades
resulted i
n
an unprece
de
nte
d
growth
of
ad
ju
stab
le sp
eed
dr
iv
es.
I
nductio
n
Mach
in
es (
I
M
)
ar
e
w
i
dely u
s
ed
in
m
o
st o
f
th
ese dr
ives, b
ecau
s
e th
ey ar
e
relatively chea
p a
n
d rugge
d
mach
in
es an
d th
eir con
s
tru
c
tio
n is rea
lized
with
ou
t co
mmu
t
ato
r
s. B
u
t ind
u
c
tion
m
o
to
r con
t
ro
l is co
m
p
lex
du
e
to
ro
tatin
g stat
o
r
field and
also t
h
e
rot
o
r current ca
nnot
be m
easured
di
rec
t
l
y
.
I
n
vect
o
r
c
ont
r
o
l
schem
e
s, t
h
e r
o
t
o
r fl
ux
an
d t
h
e t
o
r
que
p
r
o
d
u
ci
n
g
st
at
o
r
c
u
rre
nt
are
co
nt
r
o
l
l
e
d i
n
de
pe
nd
ent
l
y
,
hence
fast
e
r
t
o
rq
ue a
n
d s
p
eed
co
nt
r
o
l
can
be
achi
e
ve
d
[
1]
,
[
2
]
.
Sens
orl
e
ss
Di
rect
To
rq
ue
C
ont
r
o
l
(
D
TC
) i
s
a
po
wer
f
u
l
vect
or c
o
n
t
rol
sc
hem
e
whi
c
h
gi
ves
i
n
st
ant
a
ne
o
u
s t
o
r
q
ue an
d
fl
u
x
cont
rol
usi
n
g
opt
i
m
u
m
i
nver
t
er o
u
t
p
ut
v
o
l
t
a
ge ve
ct
or
s t
o
obt
ai
n
sha
r
p t
o
rq
ue
resp
o
n
se wi
t
h
great
er e
ffi
ci
en
cy
[3]
-
[
5
]
.
Sel
ect
i
on o
f
i
nve
r
t
er vol
t
a
ge
vec
t
or usi
ng S
p
ac
e Vect
or M
o
d
u
l
a
t
i
o
n
(SVM
) ha
s t
h
e adva
nt
age
o
f
re
duci
ng t
h
e
fl
uct
u
at
i
o
ns i
n
t
o
r
q
ue,
fl
u
x
and s
p
ee
d [
6
]
.
St
at
or v
o
l
t
a
ges an
d
currents at t
h
e
m
o
tor term
inals are
u
s
ed
t
o
ex
tract th
e
ro
t
o
r sp
eed
b
y
esti
m
a
tin
g
th
e mag
n
itud
e
and
sp
atial
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
.
4
,
D
ecem
b
er
2
015
:
81
9 – 830
82
0
p
o
s
ition
o
f
t
h
e revo
lv
i
n
g
m
a
g
n
e
tic fl
u
x
in
th
e stator
or in
th
e ro
tor
[7
]. Sen
s
o
r
less DTC
n
eed
s
a g
r
eat
kn
o
w
l
e
d
g
e o
f
t
h
e dy
nam
i
c prope
rt
i
e
s o
f
t
h
e
i
n
d
u
ct
i
on m
o
tor for t
h
e estimation of s
p
eed, flux and torque [8],
[9]
,
[1
0]
. Vari
o
u
s est
i
m
a
t
i
on techni
que
s l
i
k
e ope
n l
o
o
p
est
i
m
a
t
o
rs, cl
ose
d
l
o
o
p
est
i
m
ators
or o
b
se
rve
r
s a
n
d
adaptive m
ode
ls are use
d
in
high
perform
a
nce dri
v
es [11]
,[12]. T
h
e m
a
in
diffe
re
nce of
closed l
o
op est
i
m
a
tor
fro
m
o
p
e
n
loop
estim
a
t
o
r
is th
e in
clu
s
i
o
n
of esti
m
a
tio
n
error correction
term
to
ad
ju
st
th
e resp
on
se
of th
e
est
i
m
a
t
o
r. O
p
e
n
l
o
op
est
i
m
ators a
r
e
not
usi
ng t
h
i
s
co
rr
ect
i
on t
e
rm
but
cl
ose
d
l
o
op e
s
t
i
m
at
ors are
usi
ng t
h
i
s
correction term and hence t
h
ey are ca
l
l
e
d as obser
ve
rs.
The est
i
m
a
t
o
rs or o
b
ser
v
e
r
s use
d
vary
i
n
t
e
rm
s of
accuracy, robustness a
n
d se
ns
itivity against m
odel param
e
ter va
riations
.
The spee
d se
nsorless control
give
s
go
o
d
per
f
o
rm
ance i
n
t
h
e hi
g
h
er ra
n
g
es of
spee
d but
pe
rf
orm
a
nce det
e
ri
orat
es at
low s
p
eed incl
udi
ng zero
sp
eed
.
Th
e
p
o
o
r
p
e
rfo
rm
an
ce o
f
esti
m
a
to
rs at lo
w sp
eed
s
is
m
a
in
ly d
u
e
to
th
e
v
a
riation
in
m
easu
r
ed
m
o
to
r
vol
t
a
ge
a
n
d
c
u
r
r
ent
d
u
e t
o
t
h
e
dom
i
n
at
i
on
of
dc
of
fset
o
f
el
ect
r
oni
c
com
p
o
n
ent
s
i
n
v
o
l
v
e
d
,
vari
a
t
i
on i
n
mach
in
e p
a
rameters d
u
e
to th
e ch
ang
e
in
wind
ing
te
m
p
erature, drift problem
s a
ssociated with direc
t
in
teg
r
ation
and no
ise in th
e l
o
w sp
eed
rang
e
[7
],[13
]
.
In t
h
i
s
pa
per a
cl
osed l
o
o
p
(
o
bser
ve
r) t
y
pe e
s
t
i
m
a
ti
on t
ech
ni
q
u
e i
s
so
u
ght
t
o
i
nve
st
i
g
at
e and c
o
m
e
u
p
wi
t
h
a rel
i
a
bl
e sol
u
t
i
on
fo
r t
h
e l
o
w s
p
eed
est
i
m
a
ti
on i
ssues i
n
hi
g
h
p
e
rf
orm
a
nce i
nduct
i
o
n m
o
t
o
r
dri
v
e
.
Kalm
an
filter tak
e
s a sto
c
h
a
st
ic ap
pro
ach to
th
is prob
lem
,
wh
ile
o
t
h
e
r
o
b
serv
ers are
d
e
t
e
rm
in
ist
i
c [14
]
,[1
5
]
.
It
is called
sto
c
h
a
stic b
ecau
s
e o
f
t
h
e
fact th
at it tak
e
s in
to
accou
n
t
t
h
e
n
o
i
se i
n
th
e system
, falsity in
measu
r
em
en
ts as well as th
e
u
n
c
ertain
ty in
esti
m
a
t
i
o
n
dur
in
g
its filterin
g
p
r
o
cess,
to
g
i
ve
an
o
p
tim
al
es
ti
m
a
te
of the state. Kal
m
an filter uses the state space
m
odel of
t
h
e syste
m
which give
s an
insight into
internal/non
measurable
va
riables which are to
be
estim
a
ted. State s
p
ace
re
prese
n
tation
o
f
IM
with
c
u
rre
nt, flu
x
a
n
d spee
d
as state v
a
riables in
vo
lv
e non
lin
ear d
i
fferen
tial equ
a
tio
n
s
; h
e
n
c
e an
ex
ten
d
e
d
v
e
rsion
o
f
th
e
Kalm
an
filter
k
nown as
Ex
ten
d
e
d
Kalm
an
Filter (EKF)
ap
p
licab
le to
non
lin
ear systems is
u
s
ed as esti
m
a
to
r in
IM
driv
es.
After exh
a
u
s
tiv
e literatu
re surv
ey,
EKF is ch
o
s
en
t
o
in
v
e
st
ig
ate fo
r m
itig
atin
g
th
e low sp
eed estim
a
tio
n
issu
es
i
n
D
T
C
I
n
duct
i
on
M
o
t
o
r
D
r
i
v
es [
16]
-
[
21]
.
Th
is
p
a
p
e
r m
a
i
n
ly aim
s
at acc
u
r
ate l
o
w sp
eed
estim
at
io
n
un
d
e
r
v
a
riou
s l
o
ad
con
d
ition
s
ran
g
i
n
g
fro
m
n
o
lo
ad
to
fu
ll lo
ad. Literature su
rv
ey shows th
at t
h
e
s
p
eed is estim
ate
d
in
EKF by c
onsi
d
eri
n
g the
rate of
change
of s
p
ee
d as
ne
gligible
[1].
Th
is
corresp
ond
s t
o
infinite in
ertia o
f
the m
ach
in
e wh
i
c
h
is
n
o
t
practically
realizable.Sinc
e
the spee
d ca
nnot be
norm
ally treated as a consta
nt [2],
an alternative
and m
o
re efficient
ap
pro
ach
for sp
eed
estim
atio
n
is
m
e
n
tio
n
e
d
in
literatu
res u
s
ing
th
e equatio
n
of m
o
tio
n
wh
ich
relates th
e
spee
d t
o
l
o
a
d
t
o
r
que a
n
d el
e
c
t
r
om
agnet
i
c
t
o
r
q
ue.T
he st
u
d
y
i
n
[
2
2]
al
so u
s
es t
h
e e
q
u
a
t
i
on
of m
o
t
i
on,
but
co
nsid
ers lo
ad to
rqu
e
requ
ired
for sp
eed
esti
m
a
tio
n
as
a
co
nstan
t
.
Th
is
ap
pro
ach of
f
e
rs goo
d p
e
r
f
o
r
man
c
e
du
ri
n
g
t
h
e
hi
g
h
spee
d
ope
ra
t
i
on o
f
t
h
e
dri
v
e,
but
t
h
e
re
sponse is c
o
mparativ
ely sluggish at low s
p
eeds
esp
ecially wh
en
th
ey are sub
j
ected
to
h
i
gh
lo
ads. Th
e m
a
i
n
con
t
ribu
tion
in
th
is p
a
p
e
r is th
e v
a
lid
atio
n o
f
a
new a
p
proach
for the estim
a
tion of sp
eed
wh
ere th
e lo
ad
p
r
o
f
ile is g
i
v
e
n
as an
in
pu
t to
th
e EKF esti
mato
r.
EKF
uses t
h
i
s
l
o
ad t
o
r
q
ue i
n
p
u
t
f
o
r est
i
m
at
i
ng spee
d u
s
i
ng t
h
e e
quat
i
on
of m
o
t
i
on.
Thi
s
m
e
t
hod h
a
s t
h
e
b
e
n
e
fit o
f
p
r
ovid
i
n
g
fast respo
n
s
e and
h
i
g
h
esti
m
a
t
i
o
n
accu
r
acy und
er all lo
ad
cond
itio
ns o
v
e
r a wi
d
e
ran
g
e
of s
p
eed
from
rated to low s
p
eed incl
uding zero sp
eed. Sim
u
la
tio
n
is carried
o
u
t in M
A
TL
AB-S
IM
ULI
N
K
platform
to va
lidate the effe
ctiven
ess
o
f
m
odi
fi
ed EK
F
fo
r l
o
w
spee
d est
i
m
ati
on a
t
vari
o
u
s
l
o
a
d
t
o
r
que
co
nd
itio
ns.
2.
SENSO
R
LES
S DT
C
CO
NT
ROL
Fig
u
r
e
1
show
s t
h
e
D
T
C
str
u
ct
u
r
e usin
g EK
F.
DT
C
schem
e
of
fers
di
rect
co
nt
r
o
l
o
f
t
h
e
el
ect
rom
a
gnet
i
c
t
o
r
que
an
d st
at
or
fl
u
x
l
i
n
ka
ge
of t
h
e m
o
t
o
r
t
h
r
o
u
g
h
opt
i
m
um
i
nvert
er
v
o
l
t
a
ge vect
o
r
sel
ect
i
on
usi
n
g S
V
M
.
Fi
gu
re
1.
St
r
u
c
t
ure
of
D
T
C
u
s
i
ng
EK
F
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I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
Low
Sp
eed
Est
i
mat
i
o
n i
n
Se
n
s
orl
e
ss
Di
rect
Tor
que
Con
t
rolled
In
du
ction
Mo
to
r Drive .... (Min
i R.)
82
1
The c
h
oice of
the voltage
ve
ctor is
decide
d by the
co
mm
a
n
d sign
als from
th
e to
rqu
e
an
d f
l
ux
PI con
t
r
o
ller
s
.
The
refe
re
nce
and
act
ual
val
u
es
of
fl
u
x
a
n
d
t
o
r
q
ue
req
u
i
r
e
d
fo
r t
h
e P
I
c
o
nt
r
o
l
l
e
rs a
r
e
ge
nerat
e
d
by
t
h
e
spee
d
co
n
t
ro
ller an
d EKF estim
a
t
o
r
resp
ectively. EKF
u
s
es th
e
m
oni
t
o
re
d st
at
or v
o
l
t
a
ges
and st
at
o
r
c
u
r
r
ent
t
o
estim
a
te
the ac
tual flux and torque. The s
p
e
e
d contro
ller needs actual speed of the m
o
tor and the re
fe
rence
spee
d fo
r ge
n
e
rating fl
ux a
nd to
rq
ue re
fe
rence v
a
lues.
The actual rot
o
r s
p
eed f
o
r t
h
e spee
d co
ntr
o
ller is
obt
ai
ne
d fr
om
EKF est
i
m
a
t
o
r usi
n
g
t
h
e
e
quat
i
on o
f
m
o
t
i
on.
3.
STATE SPACE MODEL OF
IN
D
UCTI
O
N
MOTO
R
The two a
x
is state space
m
ode
l of three phas
e inductio
n m
o
tor in stationary reference fra
m
e
consists
of st
at
o
r
cu
rre
nt
s an
d r
o
t
o
r f
l
ux l
i
n
kages a
s
t
h
e st
at
e var
i
abl
e
s. The
rot
o
r s
p
ee
d si
g
n
a
l
i
s
requi
re
d f
o
r t
h
e
est
i
m
a
ti
on of s
t
at
or cur
r
e
n
t
s
and r
o
t
o
r fl
u
x
l
i
nka
ges a
nd al
s
o
fo
r t
h
e ge
ner
a
t
i
on o
f
refe
re
nce t
o
r
q
ue an
d
fl
ux
by
t
h
e spee
d cont
rol
l
e
r.
I
n
EK
F, t
h
e r
o
t
o
r s
p
e
e
d i
s
augm
en
ted as the fifth st
ate variable and is estim
a
ted using
th
e equ
a
tion
of m
o
tio
n
.
Lo
ad
torqu
e
requ
i
r
ed
i
n
th
e eq
uatio
n
of m
o
tio
n
is fed
in
t
h
e form
o
f
lo
ad
p
r
o
f
ile.
B
a
sed o
n
t
h
ese
deri
vat
i
o
ns, t
h
e
m
a
t
h
em
ati
cal represe
n
t
a
t
i
o
n
of t
h
e i
n
d
u
ct
i
on m
achi
n
e i
nvol
vi
n
g
t
h
e fi
v
e
st
at
e
vari
a
b
l
e
s i
s
e
x
press
e
d
bel
o
w
.
(1)
(
2
)
The m
odel
of the induction machine
given by (1) a
n
d (2) are in the
state s
p
ace
form
as given bel
o
w
(3
)
(4)
whe
r
e
i
s
t
h
e st
at
e vect
or, u i
s
t
h
e i
nput
vec
t
or, y
i
s
t
h
e ou
t
put
vect
o
r
, A
i
s
t
h
e sy
st
em
m
a
t
r
i
x
,B
i
s
t
h
e i
nput
matrix
and
C is th
e
o
u
t
p
u
t
m
a
trix
.
The el
ect
rom
a
gnet
i
c
t
o
r
q
ue devel
ope
d by
t
h
e m
o
t
o
r i
s
est
i
m
a
t
e
d i
n
EK
F usi
n
g st
at
or
cur
r
ent
s
a
n
d
rot
o
r
fl
u
x
l
i
n
ka
ges as
gi
ven
i
n
(
5
).
(
5
)
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
.
4
,
D
ecem
b
er
2
015
:
81
9 – 830
82
2
The t
e
rm
s
i
n
equat
i
o
ns (
1
)
,
(
2
) an
d (
5
) are
defi
ned as,
and
: direct and quadrature compone
n
ts
of stator c
u
r
r
e
n
ts in stationar
y
referen
ce fra
m
e
.
and
: direct and quadrature
co
m
p
on
en
ts o
f
ro
to
r
fl
ux
l
i
nkage
s i
n
st
at
i
ona
ry
ref
e
re
nc
e fram
e
.
: ro
tor
sp
eed
.
and
: d
i
rect an
d qu
ad
r
a
t
u
r
e
co
m
p
on
en
t
s
of
stator
voltage
s in s
t
ationary
re
fe
rence
fram
e
.
lo
ad
t
o
rqu
e
.
and
:stator resistance and inductance
respectively.
and
rot
o
r re
sistance and induct
ance refe
rre
d t
o
st
at
or si
de,
r
e
s
pect
i
v
el
y
.
: mag
n
e
tizi
ng
inductance
.
L
s
’
=
L
s
–
(
L
m
2
/L
r
’
):
tran
sien
t ind
u
c
tan
ce.
T
r
=
L
r
’
/ R
r
’:
ro
t
o
r time co
n
s
tan
t
.
:
num
ber o
f
pol
es.
m
o
men
t
o
f
in
ertia.
Accord
ing
to t
h
is ap
proach, t
h
e l
o
ad torqu
e
p
r
o
f
ile
d
a
ta
n
e
ed
s t
o
b
e
g
i
v
e
n
to th
e EKF esti
m
a
to
r for
t
h
e est
i
m
a
ti
on
of r
o
t
o
r spe
e
d
.
To i
n
c
o
r
p
o
r
at
e
t
h
i
s
l
o
ad p
r
o
f
i
l
e
i
nput
i
n
t
h
e
m
a
t
h
em
at
i
c
al
m
odel
,
l
o
ad t
o
r
que i
s
co
nsid
ered
as th
e th
ird
elem
en
t in
t
h
e inpu
t m
a
trix
in
ad
d
i
tio
n
to stato
r
vo
ltag
e
s. Electro
m
a
g
n
e
tic to
rqu
e
o
f
th
e m
o
to
r is esti
m
a
ted
b
y
EKF u
s
ing
th
e
v
a
l
u
es of stator
cu
rren
ts an
d
ro
to
r
flux
lin
k
a
g
e
s. Un
lik
e th
e st
u
d
y
in
[18
]
, th
is
p
a
p
e
r do
es no
t co
nsid
er l
o
ad torqu
e
as an
esti
mated
qu
an
tity i
n
EKF t
h
ereb
y
redu
ci
n
g
th
e
ord
e
r
of
th
e filter t
o
fi
ve and
h
e
n
c
e red
u
c
i
n
g th
e
bu
rd
en of co
m
p
u
t
atio
n
b
u
t
it limits th
e app
licatio
n
s
.
4.
DEVELOPMENT OF EKF ALGORIT
H
M
Th
e EKF is an o
p
tim
u
m
esti
mato
r b
ecau
s
e
o
f
its sto
c
hasti
c
and rec
u
rsi
v
e nature and it
can
be us
ed
fo
r j
o
i
n
t
st
at
e
and
pa
ram
e
t
e
r
est
i
m
a
ti
on o
f
a no
n-l
i
n
ear
dynamic syste
m
. The al
go
rithm proces
ses the state
vari
a
b
l
e
s by
t
a
ki
n
g
i
n
t
o
acc
ou
nt
t
h
e
noi
s
y
envi
r
o
nm
ent of t
h
e system
. The statistics of the
noise are
in
corpo
r
ated
in
th
e algo
rithm u
s
in
g
th
e m
a
trices P, Q,
R
u
and Re which are the c
o
va
riance m
a
trice
s
of the
st
at
e vari
abl
e
s,
sy
st
em
noi
se, vol
t
a
ge m
easur
em
ent
noi
se and curre
nt m
eas
urem
ent noise
respectively. T
h
es
e
matrices take into acc
ount the
noises a
nd errors
in
m
easurem
ent and
inaccuracies due to c
o
m
putational
m
odel
i
ng er
r
o
r
s
.The
t
w
o m
a
in st
a
g
es
of
t
h
e
al
g
o
ri
t
h
m
are the
prediction stage
and
t
h
e estim
a
tion
sta
g
e. In
pre
d
iction sta
g
e, the value
s
of state va
riables are pr
e
d
icted usi
ng the discretized state space m
odel of IM
cont
ai
ni
ng
t
h
ei
r est
i
m
at
ed va
l
u
es at
t
h
e
pre
v
i
o
us i
n
st
ant
.
Est
i
m
a
t
i
on i
s
t
h
en
d
o
n
e by
a
ddi
ng
t
h
e
wei
ght
e
d
di
ffe
re
nce
bet
w
een
t
h
e m
easure
d
a
n
d
pre
d
i
c
t
e
d o
u
t
p
ut
si
g
n
al
s t
o
t
h
e
pr
ed
i
c
t
e
d val
u
es.
For
use of the
algorithm
with
a di
gital proce
ssor, the state space m
odel of the m
achine
give
n in (1)
an
d (2
) h
a
s to
b
e
d
i
scretized
with
a
sam
p
lin
g
tim
e
T
whi
c
h
can
be e
x
pres
s
e
d as
sh
o
w
n
be
l
o
w.
(6)
(7)
The
di
scret
i
zed
m
odel
gi
ve
n i
n
(6
) a
n
d
(
7
)
c
a
n
be e
x
p
r
esse
d i
n
a
fo
rm
m
e
nt
i
one
d
bel
o
w.
(
k
+1)
=
A
d
x(k
)
+ B
d
u(
k)
(8
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
Low
Sp
eed
Est
i
mat
i
o
n i
n
Se
n
s
orl
e
ss
Di
rect
Tor
que
Con
t
rolled
In
du
ction
Mo
to
r Drive .... (Min
i R.)
82
3
y(k
)
=
C
d
x(
k)
(9)
Th
e step
s i
n
the EKF algo
rithm
to
ob
tain
t
h
e state esti
m
a
tes
are
d
e
scrib
e
d
b
e
low.
Step1. Pre
d
iction
of the st
ate
vector using
previou
s e
s
timated value
s
and
meas
ured stat
or v
o
ltages
The pre
d
icted value of
states at
(k+
1
)
th
instan
t
x*(k+1)
i
s
obt
ai
ne
d
usi
n
g
t
h
e f
o
l
l
o
wi
n
g
e
quat
i
o
n
x*(
k
+
1
)
=
A
d
(k
)
+
B
d
u(
k)
(1
0)
whe
r
e
(k
) is t
h
e estim
ated
v
a
lu
e of states at p
r
ev
iou
s
samp
lin
g ti
m
e
;u
(k) is th
e inp
u
t
vecto
r
;
The di
sc
ret
i
zed m
odel
of t
h
e
m
achi
n
e gi
ve
n i
n
(
6
) a
nd (
7
) i
s
used
fo
r p
r
edi
c
t
i
n
g t
h
e v
a
l
u
es of t
h
e
state varia
b
les.
Step
2.
Estima
t
io
n o
f
co
va
riance
ma
trix
o
f
p
r
ed
ictio
n
Th
e cov
a
rian
ce m
a
trix
of
p
r
edictio
n
is
p
r
ed
icted
as
P*(k+1)
= Fe(k+1)
(k )
Fe(k+
1
)
T
+ Fu(
k
+
1
)
Fu(k+1)
T
+
Q
(1
1)
whe
r
e
Q
re
pre
s
ents the covariance of the sy
ste
m
level noise;
represe
n
ts the covaria
n
ce of stator voltage
measurem
ent noise;
P
repre
s
e
n
ts the covaria
n
ce of estim
a
ted state vector
.
Th
e grad
ien
t
matrix
of the s
t
ates
and
i
n
put
s
are
gi
ve
n
bel
o
w.
Fe(
k
+1)
=
(A
d
+ B
d
u )
│
x =
(k
)
(12)
Fu(k+1)
=
(A
d
+ B
d
u)
│
u
=
k)
(1
3)
Step
3
.
K
a
lman filter ga
in comp
ut
a
tio
n
a
n
d upda
tio
n
of co
varia
n
ce ma
trix
Kalm
an
g
a
in
K(k
+
1
)
i
s
com
p
ut
ed usi
n
g
t
h
e equat
i
o
n gi
ven
bel
o
w
K(
k+1)
= P*(
k
+1)
He(
k
+1)
T
[He(k+1)
P*(k
+1)
He(k+1)
T
+Re
]
-1
(14)
whe
r
e
represe
n
ts the c
ova
riance of st
ator c
u
rrent m
easurement noise
;He(
k
+
1)
represen
ts
th
e v
a
riatio
ns i
n
pre
d
i
c
t
e
d st
at
o
r
c
u
r
r
ent
s
d
u
e t
o
unce
r
tain
ty in
p
r
ev
iou
s
estimated
v
a
lu
es.
He(
k
+1)
=
(C
d
)
│
x =x*(
k+
1)
(1
5)
State vector c
o
varia
n
ce m
a
trix
P
i
s
est
i
m
at
ed
usi
n
g t
h
e
Kal
m
an gai
n
as
gi
ven
bel
o
w
(
k
+
1
) =
P
*
(
k
+
1
) –
K
(
k
+
1
)H
e
(
k
+
1
)
P
*
(
k
+
1
)
(
1
6)
Step
4. Estimation of st
ate ve
ctor
The estim
ated value
s
of st
ate vector at
k+1
th
i
n
stan
t i
s
ob
tain
ed
b
y
su
mmin
g
u
p
th
e
p
r
ed
icted
values
of the s
t
ates with a correc
tion term
to minim
i
ze
the cova
riance
of the state va
riables.T
h
e correction
t
e
rm
i
s
t
h
e wei
ght
e
d
di
ffe
re
nc
e bet
w
ee
n t
h
e
m
easured
o
u
t
p
ut
vect
or
y(k)
and
p
r
e
d
i
c
t
e
d
o
u
t
p
ut
vect
or
(k
)
.
(
k
+1)
= x
*
(
k
+1)
+
(k+
1
)
H
e
(k+
1
)
T
Re
-
1
[y(
k
)
-
(
k
)
]
(17)
whe
r
e
y(k)
represents the m
easure
d
stator c
u
rrents;
(k
)
re
p
r
esen
ts th
e esti
m
a
ted
stat
or
currents at pre
v
ious
in
stan
t
wh
ich
i
s
calcu
lated
u
s
i
n
g as
(k
)
=
C
d
│
x
=x*(k+1)
Fi
gu
re
2 s
h
ow
s t
h
e
bl
oc
k
wi
se re
prese
n
t
a
t
i
o
n
o
f
t
h
e est
i
m
at
i
on p
r
oce
d
ure
i
n
E
K
F
.
T
h
e est
i
m
at
ed
values
of the
s
t
ates at pre
v
ious instant a
n
d
the stator
volta
ges a
r
e
use
d
for
predicting t
h
e
values
of the state
varia
b
les at the present
inst
ant. T
h
e
predi
c
ted stator c
u
rrents a
n
d t
h
e
measured stator
curre
nts a
r
e the
n
com
p
ared to c
a
lculate the va
riation
of
predicted valu
es fro
m
th
e actu
a
l v
a
lu
es. Th
is error is tun
e
d
usin
g
a
correction
factor to obtain
an accurate esti
mate of the states, whic
h is
the characte
r
istic of
Kalm
an filter
.
The
corrected
error is the
n
s
u
mme
d
up with
th
e
pred
icted v
a
l
u
es fo
r esti
m
a
tin
g
the val
u
es
of the state varia
b
le
s.
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
.
4
,
D
ecem
b
er
2
015
:
81
9 – 830
82
4
Fi
gu
re
2.
B
l
oc
k
di
ag
ram
repr
esent
a
t
i
o
n
o
f
e
s
t
i
m
a
ti
on
pr
oc
edu
r
e i
n
E
K
F
5.
R
E
SU
LTS AN
D ANA
LY
SIS
Si
m
u
latio
n
o
f
th
e DTC d
r
i
v
e u
s
in
g
2
0
h
p
m
o
to
r is carried
ou
t in
Matla
b
Sim
u
lin
k
to
ev
alu
a
te th
e
ef
f
ectiv
en
ess of
th
e
pr
opo
sed
EK
F algo
r
ith
m.
Th
e p
a
ram
e
te
rs
of t
h
e m
o
tor are listed i
n
t
h
e Table
1.
Tabl
e 1.
M
a
c
h
i
n
e param
e
t
e
rs
Power(kw)
15
(o
h
m
)
0.
2147
Fr
equency
(
Hz) 50
(o
h
m
)
0.
2205
0.
102
(H)
0.
0009
91
B(
Nm
/r
ad/s) 0.
0095
41
(H)
0.
0009
91
P 2
(H)
0.
0641
9
Voltage(
v
) 400
(rp
m
)
1460
Cu
rren
t
(A)
3
6
(N
m
)
98
P,
Q, R
u
and
Re m
a
trices are in
itialized
as d
i
agon
al m
a
tr
ices to
o
b
t
ain
o
p
tim
u
m
p
e
rform
a
n
ce at
steady state and tra
n
sient
state. T
h
e
v
a
lu
es
of th
ese m
a
trice
s
u
s
ed
i
n
th
e si
m
u
la
tio
n
are g
i
v
e
n b
e
l
o
w.
P
=
di
ag {
3
0
30
3
0
3
0
30}
Q
=
di
ag {
3
.
4
e
-
1
2
2e-
1
2
3.
2e-
1
2
5.
3e-
1
2
7.
6e-
1
4}
R
u
=
di
ag
{1e
-
2
1e-
2
}
Re =
diag {
4
.
6
e
-
7 4.
6e-
7
}
5.
1.
Speed es
tim
a
tion using E
K
F consideri
ng
load
torq
ue
as
a c
o
ns
tant.
An
alysis is carried
ou
t to
inves
tigate the efficiency of
an e
x
i
s
t
i
ng s
p
ee
d e
s
t
i
m
a
ti
on m
e
t
h
od i
n
w
h
i
c
h
th
e lo
ad
t
o
rqu
e
req
u
i
red fo
r sp
eed co
m
p
u
t
atio
n is estim
a
t
ed
i
n
EKF b
y
t
r
eatin
g
it as a co
n
s
tan
t
.
Sim
u
l
a
tio
n
s
are pe
rf
orm
e
d
at
di
ffe
rent
s
p
e
e
ds
wi
t
h
va
ry
i
ng l
o
ad
co
n
d
i
t
i
ons
an
d t
h
e t
i
m
e
t
a
ken t
o
re
ach t
h
e st
ea
dy
st
at
e is
m
o
n
ito
red
for all th
e scen
arios. Th
e sp
eed
esti
m
a
ted
b
y
EKF fo
llows t
h
e referen
ce v
a
l
u
e
in
few m
illisec
o
n
d
s
.
Bu
t th
e tim
e ta
k
e
n b
y
t
h
e m
o
to
r t
o
ram
p
u
p
t
o
th
e set sp
eed is relativ
ely very larg
e wh
ich
b
r
ing
s
t
h
e
d
e
lay in
attain
in
g
t
h
e st
ead
y state. Th
e ob
serv
ation
s
are listed
in Table 2
.
Tabl
e
2. Ti
m
e
t
a
ken
f
o
r t
h
e
dr
i
v
e t
o
reac
h t
h
e
st
eady state at
vari
ous
s
p
eeds
and loa
d
c
o
ndi
tions
Speed(
r
p
m
)
At 5% load(
5
N
m
)
At 25% load(
24.
5 Nm
)
At 50% load(
49 Nm
)
At full load(
98 Nm
)
1460
0.
9 s
1.
3 s
1.
65 s
2.
6 s
1000
1.
02 s
1.
45 s
2.
08 s
3.
2 s
750
1.
04 s
1.
63 s
2.
27 s
4.
4 s
500
1.
19 s
2.
1 s
2.
7 s
5.
5 s
100
3.
58 s
9 s
11 s
15 s
Th
e
d
a
ta sho
w
s v
a
riatio
n in settlin
g
ti
m
e
with
ch
ang
e
i
n
op
erating
sp
eeds and
l
o
ad co
nd
itio
n
s
. For
th
e sam
e
o
p
e
ratin
g
sp
eed, t
h
e tim
e
tak
e
n to
reach
th
e
steady state is found t
o
inc
r
ease with i
n
crease in
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
Low
Sp
eed
Est
i
mat
i
o
n i
n
Se
n
s
orl
e
ss
Di
rect
Tor
que
Con
t
rolled
In
du
ction
Mo
to
r Drive .... (Min
i R.)
82
5
lo
ad
.Th
e
settlin
g tim
e is also
ob
serv
ed
t
o
increase
with
t
h
e red
u
c
tion in
sp
eed ev
en if t
h
e lo
ad
ap
p
lied
to
th
e
m
o
to
r is
m
a
in
t
a
in
ed
eq
ual in
all scen
ario
s. Fo
r sp
eed
s
bel
o
w 1
0
0
r
p
m
,
t
h
e resp
o
n
se i
s
v
e
ry
sl
ug
gi
s
h
a
nd t
h
e
per
f
o
r
m
a
nce furt
her det
e
ri
or
at
es whe
n
t
h
e dri
v
e i
s
exp
o
se
d t
o
fre
q
u
ent
l
o
ad
vari
at
i
o
ns.
Thi
s
m
e
t
hod
ens
u
re
s
fairly better c
o
nve
r
ge
nce at
high s
p
ee
ds, but the
low s
p
e
e
d
perform
ance is
acce
ptable
only if t
h
e
drive is
sub
j
ect
e
d
t
o
v
e
ry
l
i
ght
l
o
a
d
s
l
e
ss t
h
an
5%
of
rat
e
d t
o
r
que
. Si
nce m
o
t
i
v
at
i
on be
hi
nd t
h
e wo
r
k
i
s
l
o
w
spee
d
esti
m
a
t
i
o
n
under
all lo
ad
co
nd
ito
ns, a
n
e
w
ap
pro
ach
is ado
p
t
ed
wh
er
e the lo
ad
p
r
o
f
ile
o
f
t
h
e app
licatio
n
is
use
d
for s
p
eed
estim
a
tion in E
K
F.
5.
2.
New
speed
est
i
mati
o
n
me
th
od
usi
n
g
E
K
F
w
i
th l
o
ad
tor
q
ue pr
ofi
l
e
gi
ve
n as
a
n
i
n
p
u
t
.
Accord
ing
to
th
is app
r
o
a
ch
, th
e lo
ad
profile o
f
th
e
ap
p
licat
io
n
is prov
id
ed as an
in
pu
t to
EKF
wh
ich
is used for t
h
e estim
a
tion of spee
d.
This
m
e
thod
provi
d
es
quic
k
re
s
p
onse
, accurat
e
spee
d a
nd t
o
rque
esti
m
a
t
i
o
n
o
v
er th
e en
tire
sp
eed
rang
e i
n
clud
ing
v
e
ry
lo
w sp
eed
s
u
n
d
e
r all lo
ad
co
nd
itio
ns t
h
ereb
y
ove
rc
om
i
ng t
h
e sh
ort
c
om
i
ngs o
f
t
h
e
pre
v
i
ous
m
e
t
hod.
T
h
e c
o
n
v
e
r
ge
nc
e t
i
m
e
t
a
ken
b
y
t
h
e
DTC
dri
v
e
fo
r
di
ffe
re
nt
scen
a
r
i
o
s
usi
n
g
t
h
i
s
app
r
o
ach is tabu
lated
in Tab
l
e 3
.
Tabl
e
3. Ti
m
e
t
a
ken
f
o
r t
h
e
dr
i
v
e t
o
reac
h t
h
e
st
eady state at
vari
ous
s
p
eeds
and loa
d
c
o
ndi
tions
Speed(
r
p
m
)
At 5% load(
5
N
m
)
At 25% load(
24.
5 Nm
)
At 50% load(
49 Nm
)
At full load(
98 Nm
)
1460
0.
53 s
0.
53 s
0.
53 s
0.
54 s
1000
0.
38 s
0.
38 s
0.
38 s
0.
38 s
500
0.
22 s
0.
21 s
0.
212 s
0.
213 s
100
0.
133 s
0.
12 s
0.
113 s
0.
11 s
10
0.
11 s
0.
12 s
0.
125 s
0.
1 s
7
0.
145 s
0.
122 s
0.
115 s
0.
09 s
5
0.
123 s
0.
11 s
0.
12 s
0.
096 s
3
0.
122 s
0.
126 s
0.
127 s
0.
11 s
2
0.
12 s
0.
129 s
0.
126 s
0.
092 s
1
0.
13 s
0.
1304 s
0.
12 s
0.
139 s
The dat
a
sh
o
w
n i
n
Tabl
e 3 i
n
di
cat
e si
gni
fi
ca
nt
red
u
ction in respons
e tim
e
com
p
ared to the previ
ous
m
e
thod for all
the operating c
o
nditions. T
h
e
settling tim
e
rem
a
ins alm
o
st
constant fo
r a
particula
r
operating
sp
eed
reg
a
r
d
less of
t
h
e ch
ang
e
i
n
lo
ad
. The lo
w sp
eed
per
f
o
r
m
a
n
ce is
also
ex
tr
em
ely
ad
equ
a
te under
all
v
a
lu
es of applied
lo
ad
wh
i
c
h
is an im
p
o
rtan
t ach
iev
e
men
t
u
s
ing
t
h
is m
e
th
o
d
.
In ord
e
r t
o
p
r
ov
id
e
an
illustrative com
p
arison, the
traces
of spee
d
obtai
ned usin
g bot
h
the
a
p
proac
h
es a
r
e shown in Fi
gure 3 a
nd
Fi
gu
re 4 fo
r dri
v
e ope
rat
i
o
n
at
1
0
0
r
p
m
.
(a
)
(b
)
Fi
gu
re
3.
Trace
s o
f
s
p
ee
ds
usi
n
g
(a
) e
x
i
s
t
i
n
g
and
(
b
)
p
r
op
os
ed m
e
t
hods
at
10
0
r
p
m
unde
r
a l
o
ad
o
f
5
Nm
(a)
(
b)
Fi
gu
re
4.
Trace
s o
f
s
p
ee
ds
usi
n
g
(a
) e
x
i
s
t
i
n
g
and
(
b
)
p
r
op
os
ed m
e
t
hods
at
10
0
r
p
m
unde
r
ful
l
l
o
a
d
of
9
8
Nm
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
.
4
,
D
ecem
b
er
2
015
:
81
9 – 830
82
6
Figure
3(a
)
a
n
d Figure 4(a) s
h
ows t
h
e trace
s of
sp
eed
ob
tain
ed
u
s
ing
t
h
e ex
istin
g sp
eed
estim
a
tio
n
m
e
t
hod
wi
t
h
t
h
e dri
v
e set
t
o
r
un at
1
0
0
r
p
m
wi
t
h
a l
o
a
d
o
f
5 Nm
and 9
8
Nm
respect
i
v
el
y
.
The t
i
m
e
t
a
ken f
o
r
th
e m
o
to
r to attain
th
e referen
ce sp
eed
is
3
.
5
8
s i
n
th
e
firs
t case and is
m
o
re than
10
s in t
h
e sec
ond case
,
whic
h conveys an inc
r
ease in
settling ti
m
e
with increase i
n
load.
Figure
3(
b) a
n
d Figure
4(b) shows the
traces
o
f
sp
eed
fo
r t
h
e sam
e
scen
ari
o
u
s
i
n
g
t
h
e prop
o
s
ed
m
e
th
o
d
.
Th
e settlin
g
time is co
m
p
arativ
ely v
e
ry less
o
f
the
or
der
0.
12 s i
rres
p
ect
i
v
e o
f
t
h
e chan
ge i
n
ap
pl
i
e
d l
o
a
d
. The pl
ot
s re
veal
t
h
at
t
h
e pr
o
pose
d
m
e
tho
d
o
f
pr
o
v
i
d
i
n
g l
o
a
d
pr
ofi
l
e
i
n
p
u
t
f
o
r est
i
m
at
i
on h
a
s a key
adva
n
t
age of ra
pi
d s
t
eady
st
at
e conver
g
e
n
ce wi
t
h
very
h
i
gh
p
r
ecision
in
estim
a
tio
n
.
Fi
gu
re 5 s
h
ow
s
t
h
e t
r
aject
ory
of st
at
o
r
fl
ux
o
b
t
a
i
n
ed
by pl
otting the
direct axes stator flux against the
qua
d
r
at
ure a
x
e
s
st
at
or fl
u
x
u
s
i
ng
bot
h t
h
e sp
eed est
i
m
a
t
i
on
m
e
t
hods. Fi
g
u
r
e 5
(
b
)
sh
o
w
s
t
h
e im
pro
v
em
ent
i
n
the stator fl
ux traject
ory
of IM
drive
usi
n
g m
odi
fi
ed E
K
F
f
o
r s
p
eed
est
i
m
ati
on.
(a
)
(b
)
Fi
gu
re
5.
St
at
o
r
fl
ux
t
r
a
j
ect
o
r
y
usi
n
g
(a)e
xi
s
t
i
ng a
n
d
(
b
)
p
ro
pos
ed
m
e
t
hods
at
1
4
6
0
r
p
m
wi
t
h
f
u
l
l
l
o
ad
In
ord
e
r to v
a
l
i
d
a
te th
e effectiv
en
ess
of th
e p
r
op
osed algo
r
ith
m
d
u
r
i
ng
steady state and tra
n
sient
ope
rat
i
o
ns, t
h
e
con
d
i
t
i
on
of s
p
eed a
n
d t
o
r
q
u
e
reve
rsal
ar
e tested
fo
r all valu
es of sp
eed with
fu
ll lo
ad. Th
e
accuracy
of est
i
m
a
t
i
on is found to
be
very
hi
gh under all conditions. The
t
r
aces
of speed
and torque
de
picting
t
h
i
s
si
t
u
at
i
on a
r
e sh
o
w
n i
n
Fi
gu
re
6 f
o
r
dri
v
e ope
rat
i
o
n at
5 r
p
m
.The p
r
ofi
l
e
o
f
re
fere
n
ce spee
d an
d a
ppl
i
e
d
lo
ad
at
wh
ich driv
e
i
s
o
p
erat
e
d
i
s
gi
ve
n i
n
T
a
bl
e 4
.
Tabl
e
4.
Pr
ofi
l
e o
f
re
fere
nce
s
p
eed
an
d a
p
pl
i
e
d l
o
a
d
T
i
m
e
(
s
) 0
1
T
i
m
e
(
s
)
0
0.
5
1.
5
Speed(
r
p
m
)
5
-
5
T
o
r
que(
N
m
)
0
98
-
98
(a
)
(b
)
Fi
gu
re
6.
Trace
s o
f
(
a
) s
p
ee
ds
and
(
b
)
t
o
rq
ues
i
n
DTC
-
SVM
at
5
rpm
un
der
ful
l
l
o
a
d
Fi
gu
re 6 s
h
o
w
s t
h
at
t
h
e val
u
es of spee
d an
d t
o
r
q
u
e
est
i
m
a
t
e
d by
EKF i
s
very
cl
ose t
o
t
h
e act
ual
v
a
lu
es and
th
e
d
r
i
v
e is fou
n
d
to
ru
n at referen
ce sp
eed and
to
rq
u
e
. Th
is sho
w
s th
e ab
ility
o
f
EKF to
prov
ide
accurate e
s
timation at
very l
o
w s
p
ee
ds
whe
n
expo
sed to
fre
que
nt
reve
rsals
of s
p
eed and t
o
rque.
To
test th
e resp
on
se
o
f
t
h
e driv
e un
d
e
r v
a
ry
in
g
lo
ad
torque co
nd
itio
ns, th
e driv
e is set
to
run
at fu
ll
lo
ad
, 3
/
4
th
lo
a
d
,
1
/
2
th
l
o
ad a
n
d 1/
4
th
lo
ad
and
th
e sp
eed
and
torqu
e
estimatio
n
is fo
und
to
b
e
satisfact
o
r
y for
th
e en
tire speed
ran
g
e
. Th
e
wav
e
fo
rm
s il
lu
stratin
g
th
is
con
d
ition
are sh
own
i
n
Figu
re
7 fo
r t
h
e op
erati
o
n
o
f
dri
v
e at
1 r
p
m
.
DTC
dri
v
e i
s
s
e
t
t
o
w
o
rk
at
t
h
e refe
re
nce s
p
e
e
d a
n
d
t
h
e l
o
a
d
t
o
r
q
ue as
gi
ve
n i
n
Ta
bl
e 5
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
Low
Sp
eed
Est
i
mat
i
o
n i
n
Se
n
s
orl
e
ss
Di
rect
Tor
que
Con
t
rolled
In
du
ction
Mo
to
r Drive .... (Min
i R.)
82
7
Tabl
e
5.
Pr
ofi
l
e o
f
re
fere
nce
s
p
eed
an
d a
p
pl
i
e
d l
o
a
d
T
i
m
e
(
s
) 0
T
i
m
e
(
s
)
0
0.
5
1
1.
5
Speed(
r
p
m
)
1
T
o
r
que(
N
m
)
98
73.
5
49
24.
5
(a)
(b)
Fig
u
re
7
.
Traces of (a) sp
eed
s
an
d (b
) torqu
e
s in
DTC-SVM at
1
rp
m
u
n
d
e
r v
a
rying
l
o
ad
co
nd
itio
ns
Fi
gu
re 7 c
o
nfi
r
m
s
t
h
at
t
h
e speed a
n
d t
o
r
q
ue est
i
m
a
tion is extrem
ely e
ffective at ve
ry low spee
ds
with
v
a
rying
l
o
ad
co
nd
ition
s
.
To
furthe
r s
ubstantiate the si
gni
ficance
of t
h
e
new
s
p
eed
estim
a
tion a
pproac
h
e
s
pecially at ve
ry low
spee
ds, va
rious perform
a
nce param
e
ters are evaluated
during the steady state operation of
DTC drive. The
facto
r
s th
at
are assessed
con
s
titu
te sp
eed est
i
m
a
t
i
o
n
erro
r,
actu
a
l sp
eed error an
d
p
e
rcentag
e
ri
p
p
l
e in
sp
eed
as well as torque. Spee
d esti
mation error is calculate
d as
the pe
rcenta
ge
differe
n
ce
between t
h
e actual and
estim
a
ted speed which is a measure of proxim
i
ty between
real and est
i
m
a
ted values. Actual speed
error is
considere
d
as t
h
e pe
rce
n
tage
diffe
re
nce bet
w
een t
h
e re
ference and act
ua
l speed a
n
d
it specifies how much t
h
e
actual drive c
h
aracteristics
are aligned with re
quire
d
settin
g
s
. Th
e
resu
lts are
fo
un
d to
rem
a
in
with
i
n
satisfactory limit for a wi
de range
of s
p
eeds. T
h
e
para
m
e
ters are re
prese
n
ted
gr
aph
i
cally to
sh
ow th
ei
r
v
a
riation
s
o
v
er th
e en
tire sp
eed
rang
e
un
d
e
r
fu
ll lo
ad cond
itio
n.
Fi
gu
re
8.
Trace
o
f
vari
at
i
o
n
of
spee
d est
i
m
at
ion error a
n
d ac
tual spee
d e
r
ror wit
h
s
p
eed
Fi
gu
re
8
sh
o
w
s t
h
e
va
ri
at
i
o
n
s
o
f
s
p
ee
d est
i
m
a
t
i
on er
ro
r a
n
d
act
ual
spee
d e
r
r
o
r
f
o
r
var
i
ous
spe
e
ds
.
Th
e sp
eed
esti
matio
n
error is less th
an
1
0
%
for all th
e sp
e
e
ds
ran
g
i
n
g
fr
om
5 rpm
t
o
t
h
e rat
e
d s
p
ee
d,
b
u
t
i
s
sl
i
ght
l
y
hi
ghe
r fo
r spee
ds bel
o
w 5 r
p
m
.
Sim
i
larl
y
,
t
h
e
actu
a
l sp
eed
error is with
in
10
% limit
fo
r sp
eed
s
fro
m
3
r
p
m
to
14
60
r
p
m
.
Belo
w 3 rp
m
,
th
e actual sp
eed
er
ro
r
is foun
d to b
e
in
creasing
.
Hen
ce t
h
is m
e
th
o
d
of
est
i
m
a
ti
on i
s
s
u
i
t
a
bl
e f
o
r
a
wi
de
ran
g
e
of
s
p
e
e
d es
pecially for low s
p
eeds
above
5
rpm
.
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
.
4
,
D
ecem
b
er
2
015
:
81
9 – 830
82
8
Fi
gu
re
9.
Trace
o
f
vari
at
i
o
n
of
pe
rcent
a
ge
ri
ppl
e i
n
act
ual a
n
d estim
a
t
ed s
p
eed for va
rious spee
ds
Fi
gu
re
9
sh
o
w
s t
h
e
pe
rcent
a
ge
ri
p
p
l
e
i
n
ac
t
u
al
an
d e
s
t
i
m
a
t
e
d s
p
eed
st
ar
t
i
ng
fr
om
very
l
o
w
s
p
eed
upt
o t
h
e
rat
e
d
val
u
e.
Pe
rcent
a
ge
of
ri
p
p
l
e
i
n
act
ual
sp
ee
d a
n
d estim
ated speed is
w
ith
in
10
% for
all
th
e sp
eeds
starting from
5
rpm
.
This appreciates
the perform
a
nce of E
K
F estim
a
t
or at
very
l
o
w s
p
ee
ds.
At
hi
g
h
er s
p
ee
d
s
al
so, t
h
e
spee
d ri
ppl
e i
s
ve
ry
m
i
nim
a
l
.
Thi
s
sh
o
w
s t
h
a
t
t
h
e ne
w spe
e
d est
i
m
at
i
on
schem
e
usi
n
g
EK
F
main
tain
s th
e sp
eed ri
p
p
l
e
with
in
n
a
rro
w
to
leran
c
e b
a
nd
.
Fi
gu
re
1
0
. T
r
a
ce o
f
vari
at
i
o
n
of
pe
rcent
a
ge
r
i
ppl
e i
n
act
ual
and
est
i
m
at
ed t
o
r
q
ue f
o
r
vari
o
u
s s
p
ee
ds
Fig
u
re 10
shows th
e p
e
rcen
t
a
g
e
ripp
le in
electro
m
a
g
n
e
tic to
rqu
e
and
esti
m
a
ted
to
rqu
e
fo
r a wide
rang
e
o
f
speeds at fu
ll lo
ad co
nd
itio
n. Th
e
rip
p
l
es i
n
actu
a
l
and
estim
ated
to
rq
u
e
s are v
e
ry less and
are
with
in
6%
fo
r al
l
t
h
e
l
o
w
spee
ds
u
p
t
o
10
0
r
p
m
.
For
spe
e
ds
a
b
o
v
e
1
0
0
r
p
m
,
t
h
e m
a
xim
u
m
percent
a
ge
of
ri
ppl
e
i
s
10
%.
The gra
phical
represe
n
tations
sugg
est th
at the EKF estim
at
i
o
n
ho
ld
s
g
ood
for all th
e sp
eed
s
rang
ing
fr
om
very
l
o
w spee
d o
f
5
rpm
t
o
t
h
e rat
e
d spee
d
with speed esti
mation error and actual speed error
main
tain
ed
with
in
10
% and
sp
eed and
t
o
rque ripp
le limited
with
in 10
%.
6.
CO
NCL
USI
O
N
In t
h
i
s
pa
per
,
a new m
a
t
h
em
at
i
cal
m
odel
i
s
devel
ope
d
fo
r
spee
d est
i
m
ati
on
usi
n
g E
K
F
i
n
o
r
de
r t
o
im
prove the perform
a
nce of
DTC IM dri
v
e
at low speed
s
.
Speed is estimated usi
ng t
h
e equat
i
o
n o
f
m
o
ti
on
an
d th
e lo
ad
t
o
rqu
e
req
u
i
red fo
r sp
eed estimatio
n
is fed
t
o
EKF in
t
h
e
form
o
f
lo
ad
p
r
o
f
ile d
a
ta. Si
n
ce
lo
ad
to
rq
u
e
is
n
o
t
an
estim
ated
q
u
an
tity in
EKF,
th
e
o
r
d
e
r
of
t
h
e
filter
is red
u
ced
th
ereb
y lo
weri
n
g
th
e bu
rd
en
on
com
put
at
i
on.
Al
so, t
h
i
s
m
e
tho
d
has fast
con
v
e
r
ge
nce
whe
n
c
o
m
p
ared to th
e sp
eed
estim
a
tio
n
in
EKF
consideri
n
g loa
d
torque
as a c
onsta
nt.
Sim
u
l
a
t
i
on i
s
carri
ed
out
f
o
r
wi
de ra
n
g
e of
spee
d at
vary
i
ng l
o
a
d
co
n
d
i
t
i
ons
wi
t
h
re
ver
s
al
s i
n
speed
and t
o
r
q
ue.T
h
e
resul
t
s
p
r
ov
es t
h
e ef
fect
i
v
ene
ss o
f
t
h
e
new a
p
pr
oac
h
usi
ng
EK
F
i
n
spee
d a
nd
t
o
r
que
Evaluation Warning : The document was created with Spire.PDF for Python.