Int
ern
at
i
onal
Journ
al of
P
ower E
le
ctr
on
i
cs a
n
d
Drive
S
ystem
s
(
IJ
PEDS
)
Vo
l.
12
,
No.
1
,
M
a
r 202
1
, p
p.
88
~
98
IS
S
N:
20
88
-
8694
,
DOI: 10
.11
591/
ij
peds
.
v12.i
1
.
pp88
-
98
88
Journ
al h
om
e
page
:
http:
//
ij
pe
ds
.i
aescore.c
om
Investig
ation of f
au
lt
y be
havior o
f the sen
sorless
contr
ol
switched r
eluctance m
oto
r driv
es
Alex
an
der
Kr
as
ovs
ky
Depa
rtment
o
f
E
le
c
tri
c
al E
ngin
eering
and
Pow
er El
e
ct
roni
cs
Bau
ma
n
Mos
cow
St
at
e
Techni
ca
l
U
nive
rsity
,
Mos
co
w,
Russ
ia
n
Feder
a
tion
Art
ic
le
In
f
o
ABSTR
A
CT
Art
ic
le
history:
Re
cei
ved
J
ul
2
,
20
20
Re
vised
Jan
20
, 202
1
Accepte
d
Fe
b
1
, 2
0
2
1
The
pap
er
pre
se
nts
th
e
result
s
of
studi
es
of
abnor
ma
l
co
ndit
ions
i
n
a
sw
itched
rel
uc
ta
nc
e
d
rive
(SRD
)
due
to
fa
i
lure
s
in
con
trol
al
gorit
h
ms.
It
di
scuss
es
one
of
the
most
common
cont
ro
l
al
g
orit
hms
for
the
s
e
driv
es
without
the
us
e
of
sensors
,
which
i
s
simpl
e
and
ea
s
y
to
conf
igu
re.
The
p
racti
c
al
ap
pli
c
at
ion
of
thi
s
al
gori
thm
o
f
con
trol
show
e
d
th
at
var
ious
a
noma
lous
pheno
me
na
cou
ld
occ
ur
in
it,
expr
e
ss
ed,
for
ex
am
pl
e,
in
a
sharp
incr
ea
se
in
the a
mpli
tude
o
f
the
phase
cur
ren
t
of
the
mot
or,
and
v
iol
ation
of
th
e
s
witc
hing
mode.
The
r
ea
sons
for
th
ese
ph
eno
me
na
ar
e
no
t
e
vide
nt
and
har
d
to
an
al
yz
e
du
e
to
the
non
-
li
ne
ari
ty
of
thi
s
drive
.
To
id
ent
if
y
the
se
ca
uses
a
nd
sea
rch
for
m
ea
sures
to
el
iminate
them,
we
used
simul
a
ti
on
in
the
envir
onme
nt
of
MA
TL
AB/S
im
uli
nk
.
The
ade
qu
ac
y
and
eff
ec
t
iv
ene
ss
of
the
ap
pli
c
at
ion
of
the
develope
d
s
im
ulation
mod
e
ls
we
conf
irm
e
d
by
a
com
par
i
son
of
the
simul
ation
resu
lt
s
and
the
f
ull
-
sca
le
exp
erime
nt
on
re
al
equ
ipm
en
t.
The
ore
ti
c
al
stud
ie
s
and
si
mul
a
tion
result
s
ar
e
i
n
good
agr
e
eme
nt
with
th
e
expe
ri
me
nt
al
res
ult
s.
Ke
yw
or
d
s
:
Anomal
ous
phenomena
Power syste
m
modeli
ng
Sensorless
con
trol
Sw
it
che
d reluc
ta
nce moto
r
Var
ia
ble sp
ee
d d
rives
This
is an
open
acc
ess arti
cl
e
un
der
the
CC
BY
-
SA
l
ic
ense
.
Corres
pond
in
g
Aut
h
or
:
Alexa
nd
e
r Kra
so
vsk
y
Dep
a
rtme
nt of
Ele
ct
rical
En
gi
neer
i
ng and
Power
Elec
tro
nic
s
Ba
um
a
n Mosc
ow Stat
e Tec
hnic
al
Unive
rsity, M
os
c
ow, Russi
an
Fe
de
rati
on
Emai
l:
k
ras
ovs
ky@
bm
stu
.ru
1.
INTROD
U
CTION
Among
the
pro
blems
so
l
ve
d
duri
ng
the
desig
n
of
t
he
el
ect
ri
c
dri
ve
,
one
of
the
m
os
t
i
mpo
rtant
is
the
determi
nation
of
t
he
permi
ss
ible
de
viati
on
s
of
the
value
s
of
it
s
pa
rame
te
rs
an
d
c
ontr
ol
sig
nals
f
r
om
their
cal
culat
ed
valu
es,
at
w
hic
h
it
mainta
ins
it
s
w
orkin
g
ca
pacit
y.
B
y
wor
king
capaci
ty,
we
unde
rstan
d
t
he
a
bili
ty
of
an
el
ect
ric
dri
ve
t
o
i
mp
le
m
ent
the
r
eq
uire
d
la
ws
of
c
ha
nge
i
n
it
s
a
djust
able
c
oor
din
at
es
(t
orque
a
nd
sp
ee
d)
with
the
ma
xi
mu
m
pe
rmiss
i
ble
de
viati
on
s
est
ablished
i
n
the
te
ch
nical
do
cume
ntati
on
for
it
s
desig
n.
Al
l
oth
er
modes
of
it
s
operati
on
are
a
bnormal
,
w
hich,
in
ad
diti
on
to
the
vio
la
ti
on
s
erv
ic
e
d
by
the
el
ect
ric
dr
ive
of
th
e
te
chnolo
gical
process
,
in
t
he
mo
st
se
ve
re
cases
ca
n
le
ad
to
emer
ge
ncies.
F
or
tra
diti
on
al
el
ect
ri
c
dr
i
ve
sy
ste
ms
,
a
lot
of
li
te
ratu
re
is
devoted
t
o
thes
e
issues,
f
or
ex
ample
[
1
]
-
[
4]
and
oth
e
rs.
T
he
switc
he
d
reluct
anc
e
dr
i
ve
(
SRD
),
wh
ic
h
has
bee
n
ra
pid
l
y
dev
e
lop
in
g
i
n
rece
nt
yea
rs
[5
]
-
[
8],
has
a
num
be
r
of
s
pecific
fe
at
ur
es,
therefo
re,
the
i
ssu
es
of
el
imi
nating
t
he
ab
nor
mal
modes
of
it
s
ope
rati
on
re
quire
s
pec
ia
l
stud
y.
A
m
ong
the
po
s
sible r
easo
ns f
or
t
he
a
bnor
mal o
per
at
in
g mo
des of t
he S
RD w
it
hi
n
the
fr
ame
w
ork of t
his p
a
pe
r,
we r
est
rict
ourselve
s
on
l
y t
o
the
reas
ons
ass
ociat
ed wit
h fail
ur
e
s in
con
trol alg
or
it
hm
s
.
The
mo
st
sp
ec
ific
el
ement
of
these
dri
ves
is
a
switc
he
d
reluctance
m
oto
r
(S
R
M).
A
n
imp
or
ta
nt
adv
a
ntage
of
t
his
m
otor
is
it
s
high
reli
abili
ty,
w
hic
h
is
due
to
it
s
e
xtre
mely
sim
ple
de
sign.
A
n
SR
M
has
con
ce
ntrate
d
phase
windin
gs
on
sta
tor
te
et
h
and
a
gea
r
p
ass
ive
fe
rroma
gn
e
ti
c
ro
to
r
[
9
]
-
[
11]
.
Th
e
oper
at
ion
of
the
SRM
c
onsist
s
of
al
te
rn
at
e
ly
co
nn
ect
in
g
i
ts
ph
ase
winding
s
by
mea
ns
of
a
n
in
ver
te
r
t
o
the
volt
age
s
ource
Evaluation Warning : The document was created with Spire.PDF for Python.
In
t J
P
ow Elec
& Dri S
ys
t
IS
S
N: 20
88
-
8
694
In
vest
ig
atio
n o
f f
au
lt
y be
havi
or
of the
senso
rle
ss control
s
wi
tc
hed
rel
uctan
ce
…
(
Ale
xa
nd
e
r Kra
so
vs
k
y
)
89
in
stric
tl
y
def
i
ned
an
gula
r
po
sit
ion
s
of
t
he
r
otor
relat
ive
t
o
the
sta
to
r.
For
this
reas
on,
i
n
a
ty
pical
ve
rsion
of
SRD,
we
us
e
a
sp
eci
al
ro
t
or
posit
ion
se
nsor
(
PS)
to
deter
mine
the
po
sit
io
n
of
the
ro
t
or
at
the
switc
hi
ng
points
of
the
phase
w
ind
in
gs
[9].
H
ow
e
ve
r,
we
kn
ow
t
hat
the
presence
of
a
re
al
PS
in
t
he
struct
ur
e
of
the
el
ect
ric
dr
i
ve
re
du
ce
s
it
s
reli
abili
ty,
increase
s
wei
ght,
dimensi
ons,
and
c
os
ts
[12
]
,
[
13]
.
T
o
date,
var
i
ou
s
a
utho
r
s
ha
ve
dev
el
op
e
d
qu
it
e
a
few
met
hods
f
or
i
nd
irect
l
y
deter
mini
ng
the
posit
ion
of
the
ro
t
or
in
orde
r
to
excl
ud
e
PS
from
t
he
SR
D
co
ntr
ol
str
uctu
re
[
14
]
-
[
17]
.
Well
-
known
sens
orl
ess
con
t
ro
l
methods
us
e
paramet
er
measu
reme
nts
of
var
i
ous
sig
na
ls
in
the
excit
ed
or
une
xcite
d
phases
of
S
RM.
I
n
the
firs
t
case,
real
cu
r
ves
of
ph
a
se
c
urren
ts
an
d
volt
ages
or
rec
onstr
ucted
si
gn
al
s
are
us
ua
ll
y
us
e
d
i
n
a
math
emat
ic
al
model
of
a
dri
ve
op
e
rati
ng
in
r
eal
-
ti
me
in
pa
rall
el
with
a
r
eal
sy
ste
m
[
14
]
,
[
15]
.
In
th
e
seco
nd
case
,
the
reacti
on
of
th
e
un
e
xcite
d p
has
e to
var
i
ou
s
ty
pes of t
est
sig
na
ls i
s an
al
yz
ed
in [1
6]
.
The
main
disa
dv
a
ntage
of
s
uc
h
a
c
on
trol
is
the
diff
ic
ulty
of
it
s
im
pleme
ntati
on
at
high
s
peed
wh
e
n
the
cu
rr
e
nt
in
t
he
f
unct
ion
i
ng
ph
a
se
la
sts
al
mo
st
the
e
ntire
pe
rio
d,
a
nd
t
here
is
no
ti
me
for
te
st
pu
lse
s.
I
n
their
pap
e
r,
M.
E
hs
a
ni
an
d
B
.
Fa
hi
mi
[
16]
e
xami
ned
in
detai
l
th
e
ad
van
ta
ges
a
nd
di
sa
dvanta
ge
s
of
the
se
a
nd
oth
e
r
con
t
ro
l
met
hods
,
as
well
as
the
tren
ds
f
or
their
f
ur
t
her
i
mpro
veme
nt.
On
e
of
t
he
m
ost
popu
la
r
an
d
widely
known
ap
proa
ches
of
sens
orl
ess
co
ntr
ol
of
SRM
is
t
he
m
et
hod,
w
hich
i
s
know
n
in
t
he
li
te
ratur
e
as
t
he
fl
ux
li
nk
age
/
c
urre
n
t
method
[9
]
,
[
17
]
-
[
19].
In
it
,
the
phase
s
witc
hin
g
posit
ion
c
om
corres
ponds
to
the
m
ome
nt
equ
al
it
y
cal
c
ulate
d
val
ue
of
f
lux
li
nka
ge
of
the
phase
c
alc
an
d
it
s
required
va
lue
in
the
phas
e
of
switc
hi
ng
po
sit
io
n
c
om
,
as
s
how
n
in
Fig
ure
1.
The
val
ue
of
c
al
c
as
a
functi
on
of
ti
me
t
is
determi
ned
f
rom
the
act
ual
values
of the
phase c
urre
nt
ph
(
)
, phase
volt
age
ph
(
)
, a
nd volt
age
dr
op
s
acr
os
s
acti
ve ph
a
se r
e
sist
ance
ph
c
alc
(
)
=
∫
[
ph
(
)
−
ph
(
)
ph
]
(1)
In
the
ge
ne
ral
case,
the
famil
y
of
c
har
act
eri
sti
cs
c
om
f
or
diff
e
r
ent
values
of
t
he
phase
c
urre
nt
ph
(
)
and
the
s
witc
hi
ng
posit
io
ns
c
om
,
usual
ly
cal
le
d
switc
hi
ng
li
ne
s,
pr
e
-
dete
rm
ined
one
wa
y
or
an
othe
r
an
d
store
d
as
a
n
a
r
ray
of
num
bers
in
the
co
rr
es
pondin
g
st
or
a
ge
de
vice
of
c
ontr
ol
s
ys
te
m
S
RD.
I
n
pr
act
ic
e,
s
uch
con
t
ro
l
is
ge
ne
rall
y
reali
zed
by
softwa
re.
It
is
imp
or
ta
nt
to
emp
hasize
he
r
e
that
in
this
m
et
hod
of
deter
minin
g
the
phase
swit
chin
g
posit
ions
of
the
SR
M
,
as
in
an
y
othe
r
se
nsorless
c
on
t
ro
l
var
ia
nt,
these
are
de
te
r
mined
ind
irect
ly
by
s
pecial
proce
ssing
of
el
ect
rical
sig
nals
that
de
pend
on
th
e
r
ot
or
posit
ion.
T
he
refor
e
,
the
ide
ntit
y
of
the
sen
sor
con
t
ro
l
a
nd
s
e
ns
orl
ess
co
ntr
ol
is
ac
hieva
bl
e
only
i
f
wel
l
-
def
i
ned
a
dd
it
ion
al
c
onditi
ons
a
nd
restrict
i
on
s
a
re
met.
He
nce,
w
hen
thes
e
restri
ct
ion
s
are
exce
eded,
t
he
w
ork
ing
capaci
ty
of
the
el
ect
ric
dr
i
ve
is
vio
la
te
d.
Figure
1. Deter
minati
on of t
he
sw
it
chi
ng posi
ti
on
of the
SRM (
Θ
on
–
s
witc
hing
–
on posit
ion
,
Θ
com
-
switc
hing
posit
ion
s
,
Θ
max
–
fu
l
l al
ign
ed
posit
ion st
at
or an
d r
otor tee
th)
The
a
uthor
of
this
pa
per
has
s
et
himself
the
go
al
of
sho
wing
with
a
sp
e
ci
f
ic
exam
ple
tha
t
even
w
hen
us
in
g
a
ve
rsi
on
of
the
SRD
se
ns
orl
ess
c
on
t
rol
that
is
simple
from
the
point
of
view
of
ph
ys
ic
al
interp
ret
at
ion
,
var
i
ou
s
non
-
obvi
ous
a
nomal
ou
s
phen
om
e
na
,
a
nd
dri
ve
malfu
nctio
ns
can
occ
ur.
To
do
this
,
t
he
auth
or
cond
ucted
a
de
ta
il
ed
analysi
s
of
the
featu
r
es
of
the
c
ontr
ol
al
gorith
m
unde
r
c
onsidera
ti
on
a
nd
i
de
ntifie
d
the
mo
st
c
har
act
er
ist
ic
causes
tha
t
cou
ld
le
a
d
to
it
s
fail
ur
es.
Gi
ven
t
he
s
pecifi
cs
of
SRD
,
a
to
ol
f
or
researc
h
in
the
form of
sim
ula
ti
on
m
od
el
s is
justi
fied,
a
br
ie
f
desc
riptio
n of t
he
giv
e
n
e
xp
e
rimental
equip
ment. After t
ha
t, the
researc
h
res
ults
an
d
their
an
al
ys
is
are
pr
es
ented,
wh
ic
h
al
low
us
to
i
de
ntify
the
ph
ysi
cal
causes
of
the
occurre
nce
of
abno
rmal
ph
e
nome
na.
Base
d on this,
r
el
ev
an
t reco
mme
nd
at
ion
s
and c
oncl
us
io
ns
a
re
dr
a
wn.
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2088
-
8
694
In
t J
P
ow
Ele
c
&
D
ri
S
ys
t,
V
ol
.
12
, N
o.
1
,
Ma
rch
20
21
:
88
–
98
90
2.
FEATU
RES
OF
CONSI
D
ERED
SENS
ORLESS
CO
NTRO
L
S
RD
Figure
2
pr
ese
nts
the
f
un
ct
io
nal
sc
heme
of
the
SRD
bl
ock
ge
ner
at
in
g
a
s
ign
al
of
t
he
po
sit
ion
of
t
he
ro
t
or
with
the
consi
der
e
d
al
gorith
m
of
se
nsorless
co
ntr
ol.
In
it
,
the
c
omp
arator
K
deter
mines
t
he
s
wit
chin
g
po
sit
io
n
w
hen
the
sig
nals
c
alc
(
ph
,
c
om
)
an
d
c
om
(
ph
,
c
om
)
are
eq
ual.
T
he
flu
x
li
nka
ge
c
alc
is
cal
c
ulate
d
base
d
on
the
real
values
of
ph
(
)
and
ph
(
)
,
as
we
ll
as
the
valu
es
of
t
he
scal
ing
c
oeffici
ent
s
of
t
he
al
gorithm:
by
vo
lt
age
, curr
e
nt
, and ph
a
se re
sist
ance
c
alc
(
)
=
∫
[
ph
(
)
−
ph
(
)
ph
]
(2)
Wh
e
n
us
in
g
a
dir
ect
sen
sor
induce
d
in
the
ph
a
se
of
the
E
M
F,
or
with
ful
l
ph
ph
com
pe
ns
at
ion,
th
e
current
valu
e
of
c
al
c
de
pends o
n o
nly
one c
oeffic
ie
nt by
E
M
F
-
:
c
alc
=
∫
Е
Е
,
(3)
Wh
e
re
=
(
ph
−
ph
ph
)
is
the
E
M
F
phase,
t
he
c
or
r
e
c
t
op
e
r
a
t
i
on
o
f
t
he
a
l
go
r
i
t
hm
i
nv
ol
ve
s
t
he
us
e
of
a
c
ur
ve
c
om
(
ph
,
c
om
)
,
i
de
nt
i
c
a
l
t
o
t
he
r
e
a
l
de
pe
nd
e
nc
e
,
a
s
w
e
l
l
as
t
he
r
e
pr
e
s
e
n
t
a
ti
on
of
t
he
c
ur
ve
s
c
alc
(
ph
,
c
om
)
a
nd
c
om
(
ph
,
c
om
)
on
on
e
s
c
a
l
e
.
I
n
t
he
ge
ne
r
a
l
c
a
s
e
,
t
he
a
d
j
us
t
a
bl
e
pa
r
a
m
e
t
e
r
s
of
t
he
c
o
nt
r
ol
s
ys
t
e
m
w
he
n
s
e
t
ti
ng
of
t
he
S
R
D
ph
a
s
e
s
s
w
i
t
c
hi
ng
a
l
go
r
i
t
h
m
a
r
e
t
he
c
oe
f
f
i
c
i
e
nt
s
,
,
(
e
i
th
e
r
t
he
o
nl
y
c
oe
f
f
i
c
i
e
nt
)
,
a
s
we
l
l
a
s
t
he
s
ha
pe
of
t
he
s
w
i
t
c
hi
ng
l
i
ne
c
om
(
ph
,
c
om
)
.
Wi
t
ho
ut
t
ou
c
hi
ng
up
on
t
he
s
pe
c
i
f
i
cs
of
c
ho
os
i
n
g
s
p
e
c
i
f
i
c
va
l
ue
s
of
,
,
c
oe
f
f
i
c
i
e
nt
s
a
nd
t
he
i
r
c
on
ne
c
t
i
on
w
i
t
h
pa
r
a
m
e
t
e
r
s
of
t
he
S
RM
c
on
t
r
ol
pa
r
t
,
w
e
on
l
y
no
t
e
t
ha
t
t
he
y
a
r
e
,
i
n
e
s
s
e
nc
e
,
s
c
a
l
e
f
a
c
t
or
s
t
ha
t
de
t
e
r
m
i
ne
th
e
pr
op
or
t
i
on
of
t
he
c
or
r
e
s
p
on
di
ng
ph
ys
i
c
a
l
qu
a
nt
i
t
i
e
s
i
n
t
he
c
al
cu
l
a
t
i
on
pr
oc
e
d
ur
e
de
t
e
r
m
i
ni
n
g
t
he
va
l
ue
s
of
c
alc
and
c
om
.
B
y
c
ha
ng
i
n
g
t
he
v
a
l
ue
s
of
t
he
s
e
c
oe
f
f
i
c
i
e
nt
s
,
i
t
is
po
s
s
i
bl
e
t
o
c
ha
ng
e
t
he
s
ha
pe
a
nd
s
c
a
l
e
of
r
e
pr
e
s
e
nt
a
t
i
on
of
c
alc
and
c
om
.
Ce
rtai
nly
,
with
the
same
pa
rameters
of
t
he
SR
M
phas
es,
co
rr
ect
ly
determi
ned
va
lues
of
the
coeffic
ie
nts
,
,
(
or
on
l
y
t
he
val
ue
of
the
coeffic
ie
nt
)
a
nd
th
e
pr
e
ci
sel
y
sp
eci
fied
s
ha
pe
of
t
he
Sw
it
chin
g
li
ne
c
om
(
ph
,
c
om
)
,
the
c
onditi
ons
f
or
s
witc
hing
the
SR
M
phas
es
in
the
se
ns
orl
ess
c
on
t
ro
l
va
rian
t
do
no
t
dif
fe
r
from
the
se
ns
or
ver
si
on
for
t
he
sam
e
s
ource
da
ta
.
M
il
le
r
T.
[
9]
prese
nted
re
comme
ndat
io
ns
o
n
tun
in
g
this
se
ns
orl
ess
co
ntr
ol
al
go
rithm.
Violati
on
of
t
he
a
bove
c
on
diti
on
s
le
ad
s
to
dev
ia
ti
on
of
t
he
par
a
mete
rs
of
t
he
s
witc
hi
ng
of
th
e
SR
M
f
rom
the
set
.
T
he
refor
e
,
it
is
ve
r
y
im
porta
nt
to
est
ablis
h
acc
ep
ta
ble
li
mit
s
for
c
ha
nges
i
n
eac
h
of
t
hese
pa
rameter
s
of
t
he
al
gorithm
that
do
not
vio
la
te
the
nor
mal
ope
rati
on
of
the
SRD.
We
s
houl
d
not
e
an
oth
e
r
im
portant
feat
ur
e
of
the
pr
act
ic
al
impleme
ntati
on
of
this
c
ontrol
meth
od.
T
o
a
vo
i
d
f
a
l
s
e
a
l
a
r
m
s
of
t
he
a
l
go
r
i
t
hm
f
or
s
m
a
l
l
va
l
ue
s
of
t
he
p
ha
s
e
c
ur
r
e
nt
s
i
n
t
he
s
w
i
t
c
hi
ng
l
i
ne
,
a
“
de
a
d
zo
ne
”
or
l
i
m
i
t
a
t
i
on
of
com
a
t
a
c
e
r
t
a
i
n
m
i
ni
m
um
l
e
ve
l
min
,
w
a
s
i
nt
r
od
uc
e
d,
a
s
s
ho
w
n
i
n
F
i
g
ur
e
3.
I
n
th
e
reg
i
on
of
small
values
of
ph
ase
c
urre
nt
I
ph
fl
ux
li
nkag
e
is
con
sta
nt,
i.
e.
c
o
m
m
i
n
c
o
n
s
t
=
=
.
The
c
orrect
cho
ic
e
of
the
value
of
min
is
of
fun
dame
ntal
importa
nce
f
or
the
c
onside
r
ed
SRD
c
ontr
ol
al
go
rith
m.
A
s
can
yo
u
see,
i
n
the
con
t
ro
l
al
gorit
hm
un
der
c
on
s
iderati
on
t
her
e
are
a
lot
of v
a
riab
le
p
ara
mete
r
s,
on
t
he
c
orrec
t
ch
oice o
f
w
hi
ch
it
s
work
i
ng
ca
pac
it
y
de
pe
nd
s
,
a
nd
i
n
case
of
er
rors,
the
pro
ba
bili
ty
of
occ
urr
ence
of
fail
ur
e
s
an
d
false
pos
it
ives
is
high
[
18].
H
ow
e
ve
r,
the
s
pe
ci
fic
causes
a
nd
mecha
nism
s
of
de
velo
pme
nt
of
these
processes
rem
ai
n
fu
ll
y
un
e
xplo
red
a
nd
not
co
ver
e
d
in
the
li
te
rature.
To
ide
ntif
y
the
causes
of
su
c
h
phen
ome
na,
it
is
necess
ary
to
cond
uct
a
detai
le
d
analysis
of
the
featu
res
of
the
w
ork,
the
pract
ic
al
set
ti
ng
s
of
this
al
go
rithm
a
nd
to
de
ve
lop
un
i
ver
sal
t
oo
ls
for
it
s st
ud
y wit
h
the
maxi
mum ap
pro
ximati
on to real
c
ondi
ti
on
s.
G
i
ve
n
t
he
n
on
-
l
i
ne
a
r
i
t
y
of
S
R
M,
t
he
pr
e
s
e
nc
e
of
c
o
nt
i
nu
ou
s
a
n
d
di
s
c
r
e
t
e
s
i
gn
a
l
s
,
t
he
m
a
t
he
m
a
t
i
c
al
de
s
c
r
i
pt
i
on
o
f
t
he
op
e
r
a
t
i
on
o
f
t
he
s
e
ns
or
l
e
s
s
S
R
D
c
on
t
r
ol
s
ys
t
e
m
un
de
r
c
on
s
i
de
r
a
t
i
on
i
s
r
a
t
he
r
c
um
be
r
s
om
e
,
un
i
nf
or
m
a
t
i
ve
,
a
nd
i
nc
o
nv
e
n
i
e
nt
f
or
s
uc
h
s
t
ud
i
e
s
[
9
]
,
[
1
0]
.
T
her
e
fore,
w
he
n
de
bugg
ing
an
d
tu
ning
t
his
al
gorithm,
it
is
highly
desira
bl
e
to
hav
e
a
si
mu
la
ti
on
m
ode
l
of
the
dri
ve.
I
n
order
to
re
flect
the
real
feat
ur
es
of
the
co
ns
ide
re
d
par
t
of
t
he
SR
D
c
o
ntr
ol
s
ys
te
m
in
t
he
m
od
el
,
it
is
ad
visable
to
ma
ke
the
st
ru
ct
ur
e
of
the
model
as
cl
os
e
a
s
po
s
sible
to
t
he
str
uctu
re
of
a
real
de
vice
with
imi
ta
ti
on
of
t
he
log
ic
a
nd
ti
me
seq
ue
nce
of
w
ork
of
it
s
el
ements.
T
he
m
os
t
c
onve
nient
to
ol
for
t
his
is
the
MA
TLAB
math
em
at
ic
al
pack
a
ge
with
the
SIM
U
LINK
env
i
ronme
nt
[20
]
,
[
21]
.
Evaluation Warning : The document was created with Spire.PDF for Python.
In
t J
P
ow Elec
& Dri S
ys
t
IS
S
N: 20
88
-
8
694
In
vest
ig
atio
n o
f f
au
lt
y be
havi
or
of the
senso
rle
ss control
s
wi
tc
hed
rel
uctan
ce
…
(
Ale
xa
nd
e
r Kra
so
vs
k
y
)
91
Figure
2.
F
un
ct
ion
al
sc
heme
of the
roto
r po
si
ti
on
sign
al
-
ge
ner
at
i
ng b
l
ock w
it
h s
ens
or
le
ss c
ontr
ol of
SRM
Figure
3. Re
al
sh
a
pe of swit
c
hing li
ne
3.
DESCRIPTI
ON
OF THE
SRD SI
M
UL
ATIO
N MO
D
EL
The
ge
ne
ral
pr
inciple
s
f
or
c
onstr
ucting
t
he
SRD
sim
ulati
on
m
od
el
are
c
on
si
der
e
d
in
a
fairly
la
r
ge
numb
e
r
of
wor
ks
,
i
n
pa
rtic
ula
r,
their
de
sc
rip
ti
on
ca
n
be
found
in
[
22
]
-
[
25]
a
nd
ot
her
s
.
The
mai
n
di
ff
e
ren
ce
betwee
n
the
m
od
el
c
re
at
ed
i
n
the
pr
ocess
of
perf
or
mi
ng
thi
s
w
ork
a
nd
the
know
n
ones
is
in
the
desi
gn
of
t
he
SRM
phase
volt
age
ge
ner
at
in
g
blo
c
k.
A
f
ull
model
of
the
vo
lt
age
-
on
-
pha
se
ge
ner
at
io
n
bl
ock
with
sens
or
le
s
s
SRD
c
on
t
ro
l
is
show
n
i
n
Fig
ur
e
4.
T
he
in
puts
of
th
e
bl
oc
k
recei
ve
si
gnal
s:
about
s
witc
hing
po
sit
io
n
of
t
his
ph
a
se
(
port
In
1)
;
a
bout
s
witc
hing
the
pre
vio
us
phase
(
bloc
k
F
rom
1)
;
a
bout
the
insta
nt
aneous
value
of
t
he
ph
a
se c
urren
t
(
blo
c
k
F
r
om
2)
;
abou
t
volt
age
powe
r
s
upply (
blo
c
k
F
r
om
3).
Figure
4.
Sim
ul
at
ion
m
odel
of
the
ph
a
se
vo
lt
age
-
ge
ner
at
in
g bloc
k
The
ph
a
se
c
urren
t
li
mit
is
se
t
by
t
he
blo
c
k
Gain
1.
Bl
oc
ks
Gain
1,
S
um
1,
a
nd
Re
la
y
prov
i
de
a
n
imi
ta
ti
on
of
th
e
cu
rr
e
nt
-
li
mit
ing
m
od
e
in
t
he
mo
t
or
phase
[
9].
T
he
ph
as
e
tur
ns
on
a
nd
tur
ns
off,
res
pecti
vely,
wh
e
n
t
he
t
rig
ge
rs
Fli
p
-
Fl
op
1
a
nd
Fli
p
-
Fl
op
2
a
re
tri
gger
ed.
The
re
mainin
g
blo
c
ks
in
cl
ud
e
d
i
n
the
model,
perform
au
xili
ary
f
unct
ion
s
.
To
sim
ulate
t
he
se
nsorless
con
t
ro
l
m
od
e
,
this
model
prov
i
des
f
or
a
s
pecial
su
bsyste
m
f
or
cal
culat
ing
t
he
switc
hi
ng
pos
it
ion
CSP
,
wh
i
ch
repr
oduces
the
mec
ha
nism
desc
ribe
d
a
bo
ve
for
ind
irect
determ
inati
on
of
the
po
sit
io
n
of
the
switc
hing
of
th
e
ph
ase
SRM
.
The
up
per
par
t
of
Fi
gure
4
s
hows
a
var
ia
nt
of
bu
il
ding
a
s
ubs
ys
te
m
CSP.
T
he
log
ic
of
it
s
op
erati
on
re
peats
the
op
e
rati
on
of
a
real
de
vice,
the
functi
onal
diag
ram
of
wh
ic
h
s
how
n
in
Fi
gur
e
2.
T
his
bl
ock
receives
ph
as
e
vo
lt
age
,
cu
rr
e
nt,
an
d
pu
lse
si
gn
al
s
to
reset
the
in
te
gr
at
or,
wh
ic
h
is
par
t
of
it
.
The
s
witc
hing
li
ne
c
om
(
ph
,
c
om
)
is
set
in
the
f
orm
of
the
corres
pondin
g t
able of the
L
ook
-
U
p Table
el
ement sett
in
gs
.
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2088
-
8
694
In
t J
P
ow
Ele
c
&
D
ri
S
ys
t,
V
ol
.
12
, N
o.
1
,
Ma
rch
20
21
:
88
–
98
92
The
i
nteg
rato
r
pe
rforms
the
cal
culat
ion
of
the
c
urren
t
value
of
t
he
f
lux
li
nk
a
ge
of
the
phase
cal
c
(
ph
,
co
m
)
acco
rd
i
ng
to
e
xpressi
on
s
(
2)
or
(
3).
T
he
val
ues
of
the
tu
ni
ng
coe
ff
ic
ie
nts
of
the
al
gorith
m
K
U
,
,
K
R
are
set
by
the
tra
ns
mis
sion
coe
ff
ic
ie
nt
s
of
the
el
em
ents
Gain
1,
Gain
2,
an
d
Gain
3.
T
he
phase
switc
hing
sig
na
l
is
ge
ne
rated
by
the
Re
la
ti
onal
Op
e
rato
r
bl
ock
a
nd
is
t
ransmi
tt
ed
to
the
ou
t
pu
t
of
O
ut
1
at
the
tim
e
of
e
qu
al
i
ty
co
m
(
ph
,
co
m
)
an
d
cal
c
(
ph
,
co
m
)
.
T
o
el
imi
nate
the
accum
ulati
on
of
er
rors
i
n
posit
ion
determi
nation
as
well
a
s
i
n
a
real
de
vice,
it
is
neces
sar
y
to
reset
the
integ
rator
at
t
he
en
d
of
eac
h
phase
switc
hing
c
ycl
e.
T
he
trig
ge
r
Fli
p
-
Flo
p
2
gen
e
rates
th
e
reset
sig
nal
at
the
ti
me
of
switc
hing
t
he
blo
c
k
Re
la
ti
on
al
Ope
rator.
To
e
nsure
a
unif
or
m
scal
e
of
cha
nge
of
pa
rameters
a
nd
va
riables
in
sim
ulati
on
models
of
SRD
,
as
well
as
t
o
giv
e
ge
ner
al
it
y
to
the
sim
ulati
on
r
esults,
we
us
e
d
the
t
ran
sit
io
n
to
relat
ive
val
ue
s.
T
he
basic
va
lue
of
the
mag
neti
c
cond
uctivit
y
of
ph
a
se
is
eq
ual
to
it
s
value
in
the
co
ordin
at
ed
posit
ion
of
the
te
et
h
of
s
ta
tor
and
ro
t
or,
i.e.,
bas
e
=
ma
x
.
The
basic
va
lues
of
the
a
ngula
r
inter
vals
of
m
oveme
nt
of
th
e
r
otor,
e
xcep
t
of
int
erv
al
of
s
witc
hin
g
on
phas
e
on
,
e
qu
al
t
o
the
an
gu
la
r
inter
va
l
of
pa
rtia
l
overlap
of
the
i
nteracti
ng
te
et
h
of
the
sta
to
r
a
nd
ro
t
or
,
i.e.
,
bas
e
=
over
lap
.
The
base
val
ue
of
the
inter
val
of
switc
hing
on
t
he
phase
on
base
,
is
equ
al
t
o
the
a
ngular
i
nter
val
betwee
n
the
a
ng
le
of
t
he
be
ginnin
g
of
ov
e
rlap
ping
of
the
te
et
h
u
na
l
an
d
the
ang
le
0
wh
e
re
0
i
s
the
a
ng
le
of
i
ntersecti
on
of
t
he
c
onti
nuat
io
n
of
t
he
gro
wing
sect
io
n
of
t
he
de
pende
nce
(
)
with
the
-
a
xi
s,
i.e.
at
(
)
=0
.
The
base
volt
age
value
is
eq
ual
to
t
he
r
at
ed
volt
age
of
the
mo
to
r
bas
e
=
r
at
ed
. Th
e
base
valu
es o
f
c
urre
nt,
tor
que, a
nd
s
pe
ed
are
ta
ken
as
their v
al
ues
at
the point of tra
ns
it
ion
of
t
he
mag
netic
sy
ste
m
of
t
he
SRM
f
rom
l
inear
mode
t
o
local
sat
urat
ion
of
i
nteracti
ng
te
et
h
bas
e
=
sa
t
,
wh
e
re
sa
t
is
the
c
urren
t
at
wh
ic
h
is
be
gi
ns
loc
al
sat
ur
at
io
n
of
te
et
h.
T
hus,
t
he
base
value
of
t
he
inter
val
of
switc
hing
on
t
he
phase
on
base
,
ens
ur
es
t
he
co
ns
t
ancy
of
t
he
phase
cu
rr
e
nt
with
the
am
plit
ud
e
I
sat
in
t
he
ov
e
rlap
ping
z
one
of
t
he
sta
to
r
an
d
r
oto
r
te
et
h
at
the
rated
vo
lt
age
r
at
ed
and
the
c
orrespo
nd
i
ng
sp
ee
d
valu
e
r
at
ed
. A
ll
relat
ive val
ues of
pa
ram
et
ers
an
d va
riables are
s
ub
se
quently
ma
rk
e
d wit
h
a
n
ast
eris
k (*).
4.
EXPERI
MEN
TAL EQ
UIP
MENT
FO
R VERIF
IC
ATI
ON OF
SIMU
LATION
R
E
S
ULTS
To
ver
if
y
the
simulat
ion
res
ults
an
d
co
ncl
us
io
ns
ma
de
on
their
basis,
we
use
d
e
xper
imenta
l
SRD
samples.
The
s
e
samples
a
re
an
inte
gr
al
par
t
of
a
co
mputer
iz
ed
te
st
com
pl
ex
f
or
ex
pe
rim
ental
stud
ie
s
of
SRD
with
a
capaci
ty
of
up
to
15
kW.
The
co
mpu
te
r
-
ai
de
d
te
st
c
omplex
al
lows
to
m
easu
re
up
to
16
a
nalo
g
s
ign
al
s;
a
uto
mati
cal
ly
instal
l
and
ma
intai
n
a
giv
e
n
load;
aut
om
at
ic
al
ly
reg
ist
er
fast
an
d
slo
w
dynamic
proce
sses,
choose
the
values
neces
sar
y
for
rec
ordin
g,
save
the
res
ults
of
ex
pe
rimen
ts
in
va
rio
us
da
ta
f
ormat
s.
I
nverte
r
with
powe
r
m
odules
BS
M75GB1
201
has
t
he
f
oll
owin
g
pa
r
amet
ers:
powe
r
up
t
o
15
kW,
su
ppl
y
volt
age
-
380
V,
f
re
que
ncy
-
50
Hz
, pow
e
r
-
up
t
o
15 k
W
, rat
ed
c
urren
t
-
50A
. U
ni
ver
s
al
micro
c
ontrolle
r
I
ntel 8Х
C
196MH
.
The
ex
per
i
me
ntal
f
our
-
phas
e
SR
M
with
a
rate
d
powe
r
of
5
kW
ha
s
the
f
ollo
wing
main
co
ns
tr
ucti
ve
par
a
mete
rs:
t
he
num
be
r
of
st
at
or
pole
s
8,
th
e
num
be
r
of
r
ot
or
pole
s
6,
the
oute
r
dia
mete
r
of
t
he
act
i
ve
pa
rt
of
the stat
or is
206
m
m,
t
he ou
te
r diame
te
r of t
he rot
or
is
11
6 mm, a
nd th
e ai
r gap
is 0.4
m
m.
To
obta
in
c
urr
ent
sig
nals
us
e
d
in
the
SRD
con
t
ro
l
al
gorit
hm
a
nd
duri
ng
te
st
reg
ist
rati
on,
the
te
st
com
plex
is
prov
i
ded
with
c
urren
t
se
nsors
include
d
in
ea
ch
phase
of
t
he
conve
rter,
in
the
DC
li
nk
a
nd
t
he
capaci
ti
ve
filt
er
ci
rcu
it
of
the
conve
rter.
C
urren
t
se
nsors
a
r
e
base
d
on
LE
M
ty
pe
LT
100
-
P
Hall
se
ns
or
s.
The
sign
al
s
f
r
om
t
he
sen
sors
e
nter
the
cu
rrent
/
vo
lt
age
c
onve
r
sion
bo
a
r
d,
a
nd
the
n
to
the
analo
g
in
pu
ts
of
th
e
microc
ontrolle
r
a
nd
i
nto
a
com
pu
te
rized
measu
reme
nt
recordi
ng
s
ys
t
em.
The
re,
a
sign
al
is
tr
ans
mit
te
d
pro
portion
al
t
o t
he vo
lt
ag
e in
the D
C
li
nk
, o
btained
em
ployin
g
a
volt
age
sens
or
.
5.
RESU
LT
S
A
ND
DI
SCUS
S
ION
We
co
nducte
d
the
main
stu
di
es
on
SRD
si
mu
la
ti
on
mode
ls,
and
t
hen
w
e
com
par
e
d
s
ome
modeli
ng
resu
lt
s
with
e
xperime
ntal
res
ults.
Give
n
a
s
uffici
ently
la
rge
va
riet
y
of
po
ssible
c
ombina
ti
on
s
of
de
viati
on
s
of
the
pa
ramete
rs
of
the
co
ns
id
ered
al
gorith
m
from
the
c
al
culat
ed
values
in
this
arti
cl
e,
at
the
fi
rst
sta
ge,
we
consi
der
the
i
nf
l
uen
ce
of
only
de
viati
ons
the
t
unin
g
c
oeffici
ents
of
the
sens
orl
es
s
co
ntr
ol
al
gorithm.
M
ore
ov
e
r,
due
to
the
wide
va
riet
y
of
po
s
sibl
e
com
bin
at
io
ns
of
t
he
c
oeffici
ents
,
,
we
will
consi
der
f
or
a
sim
pler
case
-
we
will
c
onsider
on
l
y
de
viati
on
s
of
the
c
oe
ff
ic
ie
nt
,
w
hic
h
c
orres
ponds
to
t
he
us
e
of
t
he
sign
al
directl
y
by E
M
F
unde
r t
he
ex
pressi
on
(3).
Be
sides,
for
de
finite
ness
,
we
co
nd
it
io
nally
assume
d
t
hat
the
ide
ntit
y
of
the
SR
M
c
on
t
r
ol
m
od
el
i
n
the
prese
nce
of
a
posit
ion
se
ns
or
a
nd
a
m
ode
with
ou
t
a
s
ens
or
occ
ur
s
a
t
a
coe
ff
ic
ie
nt
=
1
.
T
he
n,
by
changin
g
the val
ues
of
in
bo
t
h
di
recti
on
s r
el
at
ive
to
this
va
lue,
we
ca
n
e
va
luate
it
s
eff
ect
on
t
he
co
ndit
ion
s
of
s
witc
hing
S
RM.
T
he
er
ror
in
the
cal
culat
ion
c
alc
by
the
sig
na
l
of
the
E
M
F
sens
or
ma
y
be
du
e
t
o
er
rors
i
n
the
co
rr
e
spo
nding
s
ens
ors
a
nd
inte
rme
diate
conve
rters
at
t
he
in
put
of
t
he
co
ntr
ol
micr
opr
ocess
or
syst
em.
As
fo
ll
ows
from
(
3),
the
value
of
the
c
oeffici
en
t
aff
ect
s
the
ra
te
of
inc
rease
i
n
ti
me
of
the
c
alc
value,
w
hich
Evaluation Warning : The document was created with Spire.PDF for Python.
In
t J
P
ow Elec
& Dri S
ys
t
IS
S
N: 20
88
-
8
694
In
vest
ig
atio
n o
f f
au
lt
y be
havi
or
of the
senso
rle
ss control
s
wi
tc
hed
rel
uctan
ce
…
(
Ale
xa
nd
e
r Kra
so
vs
k
y
)
93
accor
dingly
a
f
fects
the
po
sit
i
on
of
the
phas
e
switc
hing
c
om
.
As
a
n
exa
mp
l
e,
in
Fi
gure
5
(
a)
with
t
he
s
olid
li
nes
s
how
n
t
he
cu
r
ves
c
alc
∗
(
∗
)
an
d
c
om
∗
(
∗
)
are
obta
ined
by
modeli
ng
pr
ocesses
in
the
SRM
at
=
1
and
∗
=
2
.
I
n
this ca
se, the
posit
ion
∗
c
om
is
determi
ned w
it
hout e
rro
r.
Fr
om
relat
ion
(
2)
it
f
ollow
s
th
at
with
a
n
i
ncrea
se
in
t
he
v
al
ue
of
(
>
1
),
the
ra
te
of
inc
rease
of
c
alc
∗
(
∗
)
increase
s
.
Th
e
eq
ualit
y
between
c
alc
∗
(
∗
)
and
c
om
∗
(
∗
)
co
mes
earli
er
,
as
a
res
ult
of
w
hich
t
he
work
i
ng
phase
of
the
SR
M
a
nd
t
he
subse
quent
phase
are
s
witc
hed
on
with
ad
va
nce
.
A
n
earli
er
i
nclusi
on
of
the
phase
le
ads
to
an
inc
rease
in
the
cu
rr
e
nt
in
it
and
,
acc
or
dingly,
with
th
e
same
sh
a
pe
of
the
switc
hi
ng
li
ne,
it
increases
t
he
flu
x
li
nka
ge
c
om
∗
.
As
a
res
ult,
a
c
ertai
n
new
sta
bl
e
sta
te
of
the
mode
of
switc
hing
of
the
SR
M
occurs
c
al
c
∗
=
c
om
∗
w
hen
t
he
po
sit
io
n
∗
c
om
is
sh
ifte
d
by
a
c
ertai
n
phase
a
ng
le
∗
adv
in
the
di
recti
on
of
adv
a
nce.
This
is
confirme
d
by
the
flu
x
li
nkage
cu
r
ves
c
alc
∗
(
∗
)
and
c
om
∗
(
∗
)
at
=
1
,
2
,
sho
wn
i
n
dashed
li
nes
in
Fig
ur
e
5
(a).
We
fou
nd
that
a
n
over
est
imat
ion
of
the
valu
e
by
20%
le
d
to
an
a
ppr
ox
imat
el
y
60
%
increase i
n
a
m
plit
ud
e
of the
phase c
urre
nt
ph
∗
(
∗
)
.
On
the
co
ntra
r
y,
it
ca
n
al
so
be
seen
f
rom r
el
at
ion
(
2)
t
hat, w
it
h
a
dec
reas
e
in
t
he
value
of
,
the
rate
of
i
ncr
ease
of
c
al
c
∗
(
∗
)
decr
ease
s.
T
he
eq
ualit
y
betw
een
c
alc
∗
(
∗
)
and
c
om
∗
(
∗
)
com
es
with
a
delay
an
d
the
work
i
ng
phase
is
switc
he
d
off
an
d
t
he
s
ub
se
qu
e
nt
ph
ase
is
tur
ned
on
a
nd
al
so
la
gs
c
once
rn
i
ng
t
he
po
sit
i
on
of
∗
co
m
by
the
ph
a
se
an
gle
∗
l
a
g
.
Fi
gur
e
5(
b)
by
das
hed
li
nes
s
ho
ws
the
cu
r
ves
cal
c
∗
(
∗
)
an
d
co
m
∗
(
∗
)
corres
pondin
g
to
this
switc
hi
ng
mode
wh
e
n
=
0
,
8
.
In
t
he
same
place,
f
or
c
omparis
on,
by
a
na
logy
with
Figure
5
(a)
,
s
olid
li
nes
sho
w
simi
la
r
c
urv
es
at
=
1
.
I
n
t
his
case,
t
he
dec
re
as
e
in
the
a
mpl
it
ud
e
of
t
he
ph
a
se c
urren
t
ph
∗
(
∗
)
is ab
out 2
0%.
(a)
(b)
Figure
5.
Dis
placement
of the
switc
hing
posit
ion
∗
c
om
at
(a
)
in
cre
asi
ng
an
d
(b)
decr
easi
ng
A
se
ries
of
ad
diti
on
al
e
xperi
ments
on
the
S
RD
sim
ulati
on
model
made
it
possible
t
o
de
te
rmin
e
t
he
bounda
r
y
valu
es
of
the
coe
f
fici
ent
and
to
cl
arify
the
be
hav
i
or
of
the
dr
i
ve
w
hen
going
be
yond
th
ese
bounda
ries.
As
a
n
exa
mp
l
e,
Fig
ur
e
6
s
hows
the
famil
y
o
f
me
cha
nic
al
char
act
erist
i
cs
in
the
for
m
of
a
dep
e
ndence
of
the
ave
rag
e
val
ue
of
t
he
to
rqu
e
on
sp
e
ed
av
∗
(
∗
)
f
or
dif
fer
e
nt
val
ue
s
of
the
coe
ffi
ci
ent
for
a
view
with
a
config
ur
at
io
n
of
8/6
with
t
ypic
al
par
amet
er
s.
The
se
cha
ra
ct
erist
ic
s
ref
le
ct
the
b
asi
c
prop
e
rtie
s
of the
SRD
w
he
n
the
changes
.
An
al
ys
is Fig
ur
e 6
all
ows u
s t
o
co
nclu
de
th
at
the S
RD
w
ork
ing
a
rea for
a
ny ch
a
racteri
sti
c
av
∗
(
∗
)
(at
any
value
of
)
in
the
zo
ne
of
relat
ively
hi
gh
s
pee
ds
is
l
imi
te
d
by
a
c
ertai
n
ma
ximum
value
of
spe
e
d
corres
pondin
g
to
this
c
har
act
e
risti
c,
w
hic
h
i
nc
reases
with
in
creasin
g
of
val
ue
.
W
he
n
<
1
a
nd
>
1
,
the
wor
king
a
r
ea
of
the
dri
ve
ha
s
a
li
mit
at
ion
in
the
z
one
of
lo
w
s
pee
ds.
Wh
e
n
we
go
be
yond
the
w
orkin
g
area
of
eac
h
of
the
c
har
act
e
r
ist
ic
s,
abno
r
m
al
SRD
ope
rati
on
m
od
es
occ
ur,
w
hich
are
manifeste
d
ei
ther
i
n
a
fail
ur
e
of
the
s
witc
hing
m
od
e
or
in
a
sh
a
r
p
su
r
ge
of
the
phase
cu
rr
e
nt
of
t
he
SRM
.
N
ext,
we
c
on
si
de
r
t
he
causes
of
t
he
a
bove
li
mit
at
ions
of
t
he
w
orki
ng
a
rea
of
the
SRD
on
it
s
me
chan
ic
al
c
har
a
ct
erist
ic
s
in
the
fiel
d
of
high
an
d
lo
w
sp
ee
ds
a
nd
analyze
the
po
ssible
de
velo
pme
nt
of
an
a
bnormal
sit
uation
i
n
the
dri
ve
wh
e
n
le
aving
t
he
w
orkin
g
area
.
F
igure
7
(a
)
shows
a
n
enla
rged
scal
e
of
th
e
hi
gh
-
s
pee
d
zon
e
of
mec
ha
nical
char
act
e
risti
cs
sh
ow
n
i
n
Fi
gure
6.
Y
ou
can
see
that
for
a
ny
of
the
c
ha
ra
ct
erist
ic
s
pr
ese
nted,
the
mini
mu
m
value
of
t
he
to
rque
do
es
not
r
emai
n
c
on
sta
nt,
but
cha
nges
r
el
at
ively
li
tt
le
.
The
sim
ulati
on
made
it
possi
bl
e
t
o
est
ablish
that
t
he
rea
son
for
t
he
li
mit
at
ion
s
of
t
he
SR
D
work
i
ng
a
rea
i
n
th
e
high
-
sp
ee
d
re
gion
is
the
li
mit
at
ion
of
the
switc
hi
ng
li
ne
c
om
(
ph
,
c
om
)
at
the
l
evel
of
c
om
=
min
=
co
nst
as
sho
wn
i
n
Fig
ur
e
3.
As
s
oon
as
th
e
am
plit
ud
e
of
t
he
ph
ase
curre
nt,
dec
r
easi
ng
with
i
ncr
easi
ng
sp
e
ed,
reache
s
a
value
at
whic
h
it
s
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2088
-
8
694
In
t J
P
ow
Ele
c
&
D
ri
S
ys
t,
V
ol
.
12
, N
o.
1
,
Ma
rch
20
21
:
88
–
98
94
corres
pondin
g
value
∗
c
om
om
be
comes
eq
ual
to
∗
min
as
show
n
in
Fig
ure
8,
t
he
po
sit
io
n
of
∗
c
o
m
is
determi
ned
wi
th
an
e
rro
r
in
the
directi
on
of
the
la
g
of
t
he
switc
hing
po
sit
ion
of
eac
h
ph
a
se
by
a
bout
the
switc
hing
posit
ion
of
the
pr
e
vi
ou
s
phase.
As
a
res
ult,
the
phase
an
gle
bet
w
een
tw
o
a
djace
nt
phases
switc
hing
∗
c
om
increase
s
with
each ne
w
s
witc
hing c
ycle
∗
c
om
,1
<
∗
c
om
,2
,
..
.
.
Figure
6. The
mecha
n
ic
al
cha
racteri
sti
cs
**
av
(
ω)
T
of
the SR
D
at
c
ha
ng
i
ng the
val
ue
of
c
oeffici
ent
E
K
(a)
(b)
Figure
7. Fr
a
gme
nts
of mec
ha
nical
ch
a
racteri
sti
cs
av
∗
(
∗
)
of the
S
RD in
the z
one
of
high (
a
)
a
nd lo
w (
b)
sp
ee
ds
wh
e
n
c
hangin
g
t
he va
lue of
c
oeffici
ent
The
a
ngular
d
i
sp
la
ceme
nt of t
he
s
witc
hing
posit
ion
of the
phase
windin
gs
of
SRM
in
the
directi
on
of
delay
with
e
a
ch
new
switc
hi
ng
c
ycle
al
s
o
increase
s.
As
a
res
ult,
the
po
sit
io
n
of
th
e
phase
s
witc
hing
is
gr
a
dual
ly
sh
i
fted
to
wards
th
e
gen
e
rato
r
m
od
e
of
operati
on
of
t
he
SR
M
.
In
the
firs
t
of
the
ph
ase
s,
the
switc
hing
cycl
e
of
w
hich
(f
a
ll
s
into
the
ge
ner
at
or
m
ode,
we
obser
ve
a
sh
ar
p
s
urge
of
cu
rr
e
nt,
as
s
how
n
in
Figure
9
(a
).
T
he
ex
pe
riment
al
curves
of
th
e
ph
a
se
cu
rr
e
nt
s
of
the
SRM
,
sh
ow
n
in
Fig
ure
9
(b),
c
onfir
m
the
above
-
desc
ribe
d
mec
han
is
m
f
or
th
e
de
velo
pme
nt
of
the
a
nomalo
us
sit
uati
on
i
n
the
dr
ive
in
the
re
gion
of
hig
h
sp
ee
ds
. A
s can
be
seen
,
the
na
ture
of
the
ch
ang
e
in phase cu
r
ren
ts
obta
in
ed
by
m
od
el
i
ng
as
sho
wn
in Fig
ure
9
(a)
a
nd
the
ex
per
ime
ntal
de
pe
nd
e
nce
a
s
s
how
n
i
n
Fi
gure
9
(b)
a
re
quit
e
simi
la
r
in
ap
pear
a
nce.
T
o
e
xclu
de
su
c
h
an
ab
nor
mal
mode
of
operati
on
of
the
SRD,
an
a
ppr
opriat
e
ch
oice
of
t
he
val
ue
∗
min
.
Thro
ughout
t
he
entire
op
e
rati
ng
ra
nge
of
sh
a
ft
loa
ds
of
SRM,
t
he
al
go
rith
m
of
se
nsorles
s
switc
hi
ng
of
the
SR
M
s
houl
d
not
fall
into
t
he
“
de
ad
z
one” of
th
e
switc
hi
ng
li
ne
.
T
he
m
os
t rel
ia
ble
up
per
esti
mate
of
∗
min
,
ca
n
be
su
c
h
a
va
lue
that
ens
ur
e
s
st
able
switc
hi
ng
of
t
he
vie
w
in
real
idle
mod
e
of
t
he
dri
ve
(the
loa
d
is
ca
us
e
d
only
by
the
tota
l
losses i
n
the
dr
ive).
The
lo
w
-
s
pee
d
zon
e o
f
c
ha
rac
te
risti
c
av
∗
(
∗
)
on
an
enlar
ged
scal
e is
sh
ow
n
in
Fi
gure
7
(
b)
. Yo
u
can
see
that
at
values
s
uffici
ently
cl
os
e
to
un
it
y,
t
he
m
od
e
of
sta
ble
switc
hi
ng
of
the
SRM
is
obser
ve
d
al
mo
s
t
to
zer
o
s
pee
d.
H
owe
ver,
as
th
e
de
viati
on
s of
value
s
f
rom unit
y
inc
rease
i
n
both d
irect
io
ns,
w
he
n
the sp
e
ed
reaches
a
ce
rtai
n
minim
um
value,
s
witc
hi
ng
fail
ur
e
occ
ur
s
.
More
over
,
the
mecha
nis
ms
of
de
velo
pme
nt
of
abno
rmal
sit
ua
ti
on
s
in
–
SR
D
an
d
their
ma
ni
festat
ion
at
>
1
and
at
<
1
a
re
dif
fe
ren
t.
At
<
1
,
the
Evaluation Warning : The document was created with Spire.PDF for Python.
In
t J
P
ow Elec
& Dri S
ys
t
IS
S
N: 20
88
-
8
694
In
vest
ig
atio
n o
f f
au
lt
y be
havi
or
of the
senso
rle
ss control
s
wi
tc
hed
rel
uctan
ce
…
(
Ale
xa
nd
e
r Kra
so
vs
k
y
)
95
po
sit
io
n
of
s
wi
tc
hin
g
t
he
ph
as
e
of
t
he
SR
M
ph
a
se
sh
i
fts
to
ward
the
delay
with
res
pect
t
o
the
posit
ion
at
=
1
.
I
f
t
he
phase
does
not
hav
e
ti
me
to
tu
rn
off
befor
e
the
SR
M
e
nters
the
ge
ner
at
or
m
od
e
,
this
ca
us
es
a
s
harp
increase
i
n
the
current
i
n
it
and
a
switc
hi
ng
fail
ur
e
(eme
rgency
m
od
e
SRD)
.
The
mec
ha
nism
of
tra
ns
it
ion
of
the SR
D
int
o
e
mer
gen
c
y mo
de
at
=
0
,
7
s
how
n
in
Figure
10 (
a
)
a
nd Fig
ure
10 (b
).
Figure
8.
Determinati
on
of s
witc
hing
posit
ion
∗
c
om
at
low
valu
es Y
com
.
(a)
(b)
Figure
9. A
no
malous SR
D
m
od
e
in
t
he hig
h
-
sp
ee
d
z
one
(a) si
mu
la
ti
on, (
b) ex
per
ime
nt
The
s
olid
li
nes
in
the
fig
ur
e
s
sh
ow, r
es
pecti
vely,
t
he
phase
current
c
urves
in
Figure 1
0
(
a),
as
well
as
the
depen
de
nc
ie
s
co
m
∗
(
∗
)
an
d
cal
c
∗
(
∗
)
in
Figure
10
(
b),
at
t
he
boun
da
ry
of
the
em
erg
e
nc
y
mode
at
∗
=
0
,
6
.
As
can
you
s
ee,
the
e
qual
it
y
betwee
n
c
om
∗
(
∗
)
an
d
cal
c
∗
(
∗
)
,
on
the
basis
of
wh
ic
h
a
c
omman
d
i
s
forme
d
to
tu
rn
off
t
he
ph
a
se
cu
rr
e
nt
ph
∗
(
∗
)
ta
kes
place
in
the
vi
ci
nity
of
the
agr
ee
d
posit
ion
of
t
he
te
et
h
(close t
o
the
bounda
ry of the
tran
sit
io
n of t
he
SRM f
rom m
ot
or
mode t
o
the
g
e
ner
at
or
o
perat
ing
m
ode
[10
])
.
Howe
ver,
with
a
sli
gh
t
dec
re
ase
in
s
peed
t
o
*
ω
0
,
5
8
=
,
t
he
rate
of
inc
rease
i
n
cu
rr
e
nt,
a
s
well
as
co
m
∗
(
∗
)
and
cal
c
∗
(
∗
)
inc
rease,
bu
t
to
a
diff
e
re
nt
e
xtent.
In
t
hi
s
case,
the
nat
ur
e
of
the
c
ha
nge
in
the
c
urve
s
of
co
m
∗
(
∗
)
and
cal
c
∗
(
∗
)
(d
a
sh
e
d
li
nes)
bec
ome
s
su
c
h
that
th
e
conditi
on
f
or
disco
nn
e
ct
ing
the
phase
is
not
fu
lfil
le
d
(t
her
e
is
no
i
ntersect
ion
point
of
th
ese
dep
e
n
den
c
ie
s).
As
a
res
ul
t,
the
phase
s
hu
t
dow
n
c
omman
d
is
no
t
receive
d.
The
SR
M
go
e
s
into
the
z
one
of
t
he
ge
ne
rato
r
m
od
e
with
a
po
sit
ive
volt
ag
e
on
phase
a
nd
,
as
a
resu
lt
,
the
re
is
a
sh
a
rp
i
ncr
e
ase
in
the
c
urr
ent
ph
∗
(
∗
)
in
it
as
s
how
n
i
n
das
he
d
li
ne
i
n
Fig
u
r
e
10
(a
).
T
he
simulat
ion
s
ho
wed
t
hat
at
>
1
,
a
s
the
sp
ee
d
de
creases
in
t
he
sta
ble
switc
hi
ng
m
od
e
,
the
rati
o
of
the
SR
D
par
a
mete
rs
bec
om
es
s
uch
tha
t
the
cal
c
∗
(
∗
)
and
co
m
∗
(
∗
)
curv
es
f
or
eac
h
ph
ase
are
de
f
orm
ed
t
ow
a
rds
the
ir
appr
oach.
A
si
tuati
on
arises
i
n
wh
ic
h
e
ve
n
a
sli
gh
t
e
xter
na
l
disturba
nce
le
ads
to
a
pr
e
mature
interse
ct
ion
of
the
cal
c
∗
(
∗
)
and
co
m
∗
(
∗
)
c
urve
s
an
d,
as
a
res
ult,
to
false
ph
ase
switc
hi
ng
of
the
SRM
,
i.
e.,
to
eme
rg
e
nc
y
mode
SRD
.
A
s
an
exa
mp
le
,
by
anal
ogy
with
the
pr
e
vi
ou
s
c
ase,
we
c
o
nsi
de
r
the
op
e
rati
on
of
the
SR
D
on
t
he
bor
der
of
t
he
e
mer
gen
c
y
m
od
e
at
1
,
4
E
K
=
as
show
n
in
zo
ne
I
in
F
igure
11.
At
th
e
sp
ee
d
*
ω4
=
,
as
can
be
seen
f
r
om
Fi
gure
15,
t
he
cal
c
∗
(
∗
)
and
co
m
∗
(
∗
)
curves
a
re
cl
ose
st
to
each o
th
er
at
the
be
ginnin
g
of
the pha
se
switc
hing
c
ycle
(in
the
vicini
ty
of
the
po
i
nt
“a”
-
the
br
ea
kpoin
t
of
t
he
co
m
∗
(
∗
)
curve
at
the
le
ve
l
*
m
in
).
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2088
-
8
694
In
t J
P
ow
Ele
c
&
D
ri
S
ys
t,
V
ol
.
12
, N
o.
1
,
Ma
rch
20
21
:
88
–
98
96
Nev
e
rtheless
,
we
ha
ve
the
m
od
e
of
sta
ble
s
witc
hing
of
the
SRM
with
t
he
correspo
nd
i
ng
la
w
of
t
he
cha
ng
e
i
n
ph
a
se
cu
rr
e
nts.
This
m
ode
co
r
respo
nd
s
to
the
tradit
ion
al
al
te
rn
at
io
n
of
phas
e
switc
hing
of
a
four
-
ph
as
e
S
RM
-
1,2,3,
4,1, ...
with the
same
current
sh
a
pe
ph
∗
(
∗
)
, a
s sho
wn in Fi
gure
12 in
zon
e I.
(a)
(b)
Figure
10.
Illus
trat
ion
of the
possibili
ty s
witc
hing the
SR
M
from mot
or m
ode
(a)
t
o gen
e
r
at
or
mode
(b)
i
n
the
low
-
sp
ee
d
z
on
e
Figure
11. E
xp
la
nation o
f
the
dev
el
opment
mecha
nism
of
the anomal
ous
SRD mo
de
i
n
the lo
w
-
s
pee
d zon
e
at
=
1
,
4
Figure
12.
T
he
calc
ulate
d p
ha
se cu
rr
e
nt c
urv
es at
=
1
,
4
A
f
ur
t
her
sli
gh
t
decr
ease
in
s
peed
f
or
a
ny
r
easo
n
(fo
r
exa
mp
le
,
to
∗
=
0
,
38
)
le
a
ds
to
a
dd
it
io
na
l
deformat
ion
of
the
c
urves
c
al
c
∗
(
∗
)
an
d
c
om
∗
(
∗
)
,
an
d
their
in
te
rsecti
on
,
acc
ordin
g
t
o
wh
ic
h
the
s
witc
hing
mo
me
nt
is
det
ermine
d
by
the
ad
op
te
d
al
gorithm,
is
sh
i
fted
by
the
a
ng
ular
interval
adv
∗
to
t
he
break
point
of
the
c
om
∗
(
∗
)
cur
ve
(
po
i
nt
“b”
i
n
z
on
e
II
i
n
Fig
ure
11
.
T
his
w
ould
le
ad
t
o
a
premat
ur
e
switc
hing
of
t
he
currents
ph
∗
(
∗
)
of
the
three
of
t
he
ne
arest
phases
wi
th
a
hi
gh
fr
e
qu
ency.
A
s
a
re
s
ult,
the
ph
a
se
duri
ng
w
hich
the
fir
st
false
switc
hing
of
the
al
go
rithm
oc
curred
(“
phas
e
2”
i
n
Fi
gure
12
is
s
witc
he
d
on
agai
n
befor
e
t
he
current
ph
∗
(
∗
)
in it
ha
s ti
me to dr
op t
o
ze
ro fro
m the
previ
ou
s
s
witc
hing c
ycle.
The
validit
y
of
the
co
ncl
us
io
ns
made
re
gardi
ng
the
ca
us
es
a
nd
mec
han
is
m o
f
t
he
dev
el
opment o
f
t
he
anomal
ous
SR
M
m
ode
un
der
the
co
ns
ide
re
d
co
ndit
ion
s
c
an
be
c
onfir
me
d
by
the
e
xperi
mental
phase
c
urren
t
ph
(
)
curves
sho
wn
in
Fig
ur
e
13.
As
you
ca
n
see
,
the
natu
re
of
the
cha
nge
in
t
he
ph
ase
cu
rr
e
nts
of
the
SR
M
in
this
fig
ur
e
is
cl
os
e
t
o
the
c
urves
of
t
he
c
urren
ts
ph
∗
(
∗
)
in
zo
ne
I
I
in
Fig
ure
12
ob
ta
ine
d
by
m
od
el
in
g.
T
he
com
bin
at
io
n
of
the
re
su
lt
s
of
modeli
ng
the
be
hav
i
or
of
the
dr
i
ve
in
t
he
lo
w
-
s
pee
d
z
on
e
for
>
1
and
<
1
al
lo
ws
us
to
det
ermine
the
lo
wer bou
nd
a
r
y
of
sta
ble
s
witc
hing
of
t
he
SR
M
in
the
f
orm
o
f
the
d
e
pende
nce
of
the mini
mum r
eal
iz
able sp
ee
d
m
in
on t
he val
ue of
,
show
n
i
n
Fi
gure
14.
Evaluation Warning : The document was created with Spire.PDF for Python.
In
t J
P
ow Elec
& Dri S
ys
t
IS
S
N: 20
88
-
8
694
In
vest
ig
atio
n o
f f
au
lt
y be
havi
or
of the
senso
rle
ss control
s
wi
tc
hed
rel
uctan
ce
…
(
Ale
xa
nd
e
r Kra
so
vs
k
y
)
97
Figure
13. T
he
experime
ntal
ph
a
se c
urren
t c
urves
at
=
1
,
4
Figure
14. L
ower
bo
undary
of the
mode
of
sta
ble sw
it
chi
ng
of SRM
6.
CONCL
US
I
O
N
In
the
m
ode
of
se
nsorless
c
on
t
ro
l
of
the
SRD
un
der
c
onside
rati
on,
ab
normal
co
ndit
ion
s
ca
n
be
induced
by
de
viati
on
s
in
t
he
val
ues
of
the
al
gorith
m
pa
r
amet
ers
-
t
un
i
ng
coe
ff
ic
ie
nts
a
nd
th
e
s
ha
pe
of
the
switc
hing
li
ne.
With
certai
n
c
ombinati
ons
of
par
a
mete
rs
of
the
sen
sorle
ss
con
t
ro
l
a
lg
ori
thm
of
t
he
SR
D,
the
cycle
of
it
s
s
w
it
ching
sh
i
fts
towa
r
d
the
zo
ne
of
cre
at
ion
of
the
braki
ng
t
orq
ue
in
the
m
od
e
of
t
he
generator
with a s
harp inc
rease in
t
he
phase c
urren
t.
T
his wo
uld
happ
en
due to t
he
a
ccum
ulati
on
of an
er
ror
in t
he
an
gle
of
s
witc
hin
g
on
t
he
phase
of
the
SR
M
.
T
he
reas
on
f
or
th
e
s
hift
of
t
he
SRM
s
witc
hing
c
ycle
to
the
zon
e
of
br
a
king
to
r
qu
e
in
the
reg
ime
s
cl
os
e
to
i
dle
(no
loa
d)
ma
y
be
the
presen
ce
of
a
“dea
d
zon
e"
i
ntr
oduc
ed
f
or
sta
ble
op
e
rati
on
of
the
al
gori
thm
in
the
fiel
d
of
small
sig
nals.
With
a
n
i
ncr
ease
i
n
the
load
of
the
SRD,
the
main
reas
on
f
or
the
tran
sit
ion
of
the
SR
M
to
t
he
ge
ner
at
or
m
od
e
is
a
n
unfa
vora
ble
c
ombinati
on
of
scal
e
factors i
n ca
lc
ul
at
ing
the
curre
nt v
al
ue of
pha
se
flu
x
li
nka
ge.
REFERE
NCE
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utomat
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at
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Evaluation Warning : The document was created with Spire.PDF for Python.