Internati
o
nal
Journal of P
o
wer Elect
roni
cs an
d
Drive
S
y
ste
m
(I
JPE
D
S)
V
o
l.
6, N
o
. 3
,
Sep
t
em
b
e
r
2015
, pp
. 48
6
~
49
7
I
S
SN
: 208
8-8
6
9
4
4
86
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
Numeri
cal M
e
thod for P
o
wer L
o
ss
es Minimizati
on
of
Vect
or-
Controlled Induction Motor
Alex B
o
risevic
h
Samsung SDI R
&
D Center, Korea
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Apr 12, 2015
Rev
i
sed
Ju
l 12
,
20
15
Accepte
d
J
u
l 28, 2015
The p
a
per d
e
vot
ed to
energ
y
eff
i
cien
c
y
m
a
xim
i
zi
ng problem
of
th
e indu
ction
motor under p
a
rt-lo
a
d cond
itions. The
problem is form
ulated as th
e
minimization of
ohmic losses power
as a fun
c
tion from flux
-producing
current
in field-
oriented motor opera
tion. Contr
o
l input pr
efiltering which
t
r
a
n
sforms t
h
e
dy
na
mic
ti
me
-va
r
y
i
ng op
timizatio
n problem to
stationar
y
one
is introduced. U
pdate ru
le for
control
var
i
able is proposed which speeds-up
the method
conv
ergence in
comparison with
linear variation of
in
put. Finally
a new cont
inu
ous
-tim
e s
earch
algor
ithm for
solving the
problem of
minimizing power consumption was gi
ven.
The statements
on method
behavior
were f
o
rmulated
and
converg
ence
to local minimum was proved.
The method
ver
i
fied
in simulatio
n and
in h
a
rdwar
e
exp
e
rimental s
e
tup.
Keyword:
Fi
el
d-
ori
e
nt
e
d
cont
rol
I
ndu
ctio
n m
o
to
r
Losses
power
On-lin
e op
ti
m
i
zatio
n
Searc
h
c
ontroll
er
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
:
Alex B
o
rise
vic
h
,
Sam
s
ung SD
I R
&
D
C
e
nt
er,
4
6
7
Beon
yeong
-r
o, Seob
uk-
gu
, C
h
eo
n
a
n
-
si,
3
31-
300
,
Ko
r
e
a.
Em
a
il: alex
.b
orysev
ych@g
m
ail.co
m
,
a.b
o
risev
i
ch@sam
su
n
g
.co
m
1.
INTRODUCTION
Cu
rren
tly
domin
atin
g
ap
pro
ach
to
th
e
co
n
t
ro
l of asyn
chro
nou
s m
o
to
rs is th
e v
ecto
r
con
t
ro
l,
i
n
cl
udi
ng
fi
el
d
ori
e
nt
ed c
o
nt
rol
(F
OC
) a
n
d
di
rect
t
o
rq
u
e
co
nt
r
o
l
(
D
T
C
). Im
po
rt
ant
feat
u
r
e
of t
h
e FOC
in
du
ctio
n
m
o
to
r co
n
t
ro
l [1
,
2
]
is th
e po
ssi
b
ility o
f
an
indep
e
nd
en
t m
a
n
i
p
u
l
ation
of quad
r
at
u
r
e stator cu
rren
t
qs
i
wh
ich
lin
early affects th
e m
o
to
r to
rq
ue a
nd
cont
rol
o
f
r
o
t
o
r fl
u
x
r
. In
dep
e
nde
nt
co
nt
r
o
l
of c
u
r
r
ent
qs
i
and fl
ux
r
tran
sfo
r
m
s
th
e asyn
ch
ron
o
u
s
m
ach
in
e to a
DC m
o
to
r with ind
e
p
e
n
d
e
n
t
ex
citatio
n
.
In
th
e literature [3
,
4
,
5
]
th
ere are larg
e
n
u
m
b
er o
f
d
i
fferen
t strateg
i
es to
im
p
r
o
v
e
th
e
efficien
cy
of
i
n
d
u
ct
i
on m
o
t
o
rs,
whi
c
h can
be di
vi
de
d i
n
t
o
t
w
o
gr
ou
ps:
c
ont
rol
b
a
sed
o
n
m
o
t
o
r ene
r
g
y
m
odel
s
(l
oss
m
odel
cont
rol
,
LM
C
)
and m
i
nim
i
zing t
h
e m
easured p
o
we
r co
ns
um
pt
i
on o
n
t
h
e basi
s of
n
u
m
eri
cal
opt
im
izat
i
o
n
al
go
ri
t
h
m
s
(search c
o
nt
r
o
l
,
S
C
). M
o
st
of
t
h
e kn
o
w
n
st
rat
e
gi
es o
f
e
n
er
gy
opt
i
m
i
zati
on
m
a
ni
pul
at
e o
f
rot
o
r
m
a
gnet
i
c
fl
ux
r
set
p
oi
nt
i
n
F
O
C
or
DTC
al
go
ri
t
h
m
.
LM
C
m
odel
-
base
d
cont
rol
ca
n
q
u
i
ckl
y
cal
cul
a
t
e
t
h
e
opt
i
m
al
val
u
e of
fl
u
x
-
p
r
o
du
c
i
ng c
u
r
r
e
n
t
sd
i
based
on m
o
tor
m
echanical load estim
a
tion, shaft s
p
ee
d a
n
d
m
o
to
r p
a
ram
e
ters. Majo
r
drawb
a
ck
of LMC
con
t
ro
l is
a sen
s
itiv
ity to
m
o
t
o
r m
o
d
e
l p
a
rameters v
a
riation
.
Th
e search
contro
l (SC
)
is ano
t
h
e
r techn
i
que th
at do
es no
t
relies on
m
o
to
r m
o
d
e
l an
d
p
a
ram
e
ters. It
consists in algorithm
i
c search
fo
r m
i
nim
u
m
of t
h
e
m
easure
d
i
n
p
u
t
p
o
w
er co
ns
um
pt
ion
in
P
. Th
is is ea
sy to
im
pl
em
ent
and
effect
i
v
e m
e
t
hod
.
A m
a
jor
sh
ort
c
om
i
ng of
S
C
con
t
ro
l is the n
e
ed
for artificial p
e
rturb
a
ti
o
n
of
sd
i
fo
r t
h
e
obt
ai
ni
ng
of t
h
e i
n
put
po
wer
de
ri
vat
i
ve
sd
in
i
P
/
, as well as relativ
ely slo
w
conv
erg
e
n
c
e to
th
e
opt
i
m
u
m
and t
o
r
q
ue ri
ppl
e
d
u
e
t
o
sd
i
s
t
e
p
ch
ang
e
s
.
M
a
ny
sci
e
nt
i
f
i
c
pape
rs we
re dev
o
t
e
d t
o
dev
e
l
opm
ent
and i
m
provem
e
nt
of searc
h
co
nt
ro
l
t
echni
qu
e
s
for m
o
to
r
po
wer lo
sses m
i
n
i
mizatio
n
.
W
e
will co
un
t m
o
st cited
of th
em
an
d relev
a
n
t
t
o
curren
t
work.
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
. 3, Sep
t
em
b
e
r
2
015
:
48
6 – 497
48
7
In pa
pe
rs [
6
,
7
]
t
h
ree
m
e
t
hods we
re st
udi
ed:
m
e
t
hod b
a
sed o
n
t
h
e p
o
we
r-
fl
u
x
g
r
a
d
i
e
nt
, l
i
n
ear
st
epwi
se
(ram
p
) c
h
a
nge
o
f
f
l
ux set
poi
nt
an
d c
o
m
b
i
n
at
i
o
n
of
l
o
ss m
odel
an
d sear
ch c
o
nt
r
o
l
t
o
s
p
ee
d
up
t
h
e
co
nv
erg
e
n
ce,
wh
ere t
h
e lo
ss
m
o
d
e
l p
r
o
v
i
d
e
s in
itial p
o
i
n
t
fo
r
search
con
t
ro
ller.
In n
e
x
t
sectio
n
we
will refer to
one
o
f
m
e
t
hod
descri
bed
t
h
e
r
e
.
The
pa
pe
r [
8
]
pr
o
poses
al
g
o
r
i
t
h
m
based
o
n
t
h
e g
o
l
d
en
sect
i
on t
e
c
h
ni
q
u
e i
n
t
h
e
pr
ocess
o
f
sea
r
chi
n
g
t
h
e o
p
t
i
m
a
l
val
u
e of
r
o
t
o
r
fl
ux c
u
r
r
e
n
t
ref
e
rence
,
f
o
r
w
h
i
c
h t
h
e el
ect
r
i
cal
i
nput
p
o
w
er o
f
t
h
e sy
st
em
i
s
m
i
nim
a
l
.
The al
go
ri
t
h
m
i
s
fast
and e
ffect
i
v
e, h
o
we
ve
r i
t
pr
o
duces
di
scr
e
t
e
seque
nce
o
f
m
a
gnet
i
z
i
ng
cur
r
ent
values
, which
requires l
o
w-pass filteri
ng of algorithm
output. Ve
ry clos
el
y related is m
e
thod
from
[9] where
lo
sses
f
u
n
c
tion lo
cally app
r
ox
im
a
t
ed
b
y
quad
r
atic
p
o
l
ynomial o
n
ev
er
y
iter
a
tio
n
an
d t
h
e
n
e
x
t
sear
ch po
in
t
selected bas
e
d
on analytically cal
cul
a
t
e
d m
i
nim
u
m
of a
p
p
r
o
x
i
m
at
i
on.
In
pa
per
[
10]
a fl
u
x
sea
r
ch
cont
rol
l
e
r i
s
p
r
o
p
o
se
d to i
n
c
r
ease the
efficiency of a
direct torque
-
cont
rol
l
e
d i
n
d
u
ct
i
on m
o
t
o
r
.
The am
pl
i
t
ude of st
at
o
r
c
u
rren
t is u
s
ed
as t
h
e obj
ectiv
e fun
c
tio
n. Also
ad
ap
tiv
e
strateg
y
i
m
p
l
emen
ted
to
d
e
term
in
e th
e p
r
oper flux
step
. Related
resu
lts are pu
b
lish
e
d
in [11
]
wh
ere adap
tiv
e
gra
d
i
e
nt
desce
n
t
m
e
t
hod use
d
fo
r p
o
we
r o
p
t
i
m
i
zat
i
on i
n
di
rect
t
o
rq
ue
cont
r
o
l
of a si
x-
pha
se i
n
d
u
c
t
i
o
n
machine.
An
ot
he
r searc
h
m
e
t
h
o
d
pr
o
pos
ed i
n
[1
2]
based
on
par
t
i
c
l
e
swarm
opt
im
i
zat
i
on fo
r l
o
ss m
odel
esti
m
a
t
i
o
n
wh
i
c
h
pro
d
u
ces i
n
itial p
o
i
n
t
fo
r search
con
t
ro
ller.
Unfortun
ately o
n
l
y sim
u
lati
o
n
resu
lts are
g
i
v
e
n
without ha
rdware tests. Classical s
earch method of ext
r
e
m
u
m
seeking
is used in
[1
3
]
for
po
wer l
o
sses
m
i
nim
i
zat
i
on,
ho
we
ver
t
h
e c
o
n
v
e
r
ge
nce t
o
o
p
t
i
m
u
m
i
s
rat
h
er sl
ow
.
A
not
her
rel
a
t
e
d
t
echni
q
u
e i
s
ri
p
p
l
e
-
cor
r
el
at
i
on c
o
nt
r
o
l
,
w
h
i
c
h
u
s
es i
nhe
rent
ri
ppl
e i
n
po
wer co
nv
erters to ach
iev
e
th
e
op
ti
m
u
m
o
f
ob
jectiv
e
fu
nct
i
o
n [1
4]
.
Ho
we
ver
,
i
n
i
n
duct
i
o
n m
achi
n
e t
h
ere i
s
n
o
i
nhe
rent
ri
p
p
l
e
wi
t
h
desi
re
d f
r
eq
ue
ncy
ran
g
e
, and
t
h
i
s
m
e
t
hod r
e
duce
s
t
o
a
vari
at
i
on
of
ext
r
em
um
seeki
n
g
co
nt
r
o
l
[
15]
.
The p
u
r
p
o
se
of
prese
n
t
e
d pape
r i
s
t
o
m
a
ke fu
rt
he
r
devel
opm
ent
s
of sea
r
c
h
cont
rol
di
rect
opt
i
m
i
zati
on m
e
t
hods
. Pr
op
ose
d
m
e
t
hod
pr
o
duce sm
oo
t
h
t
r
aject
o
r
y
o
f
sd
i
and fa
ster than ram
p
-ba
s
ed
t
echni
q
u
es
. T
h
e m
e
t
hod
ha
s f
o
l
l
o
wi
ng
i
n
gre
d
i
e
nt
s:
-
Th
e
calcu
lated
v
a
lu
e of p
o
wer
lo
sses
loss
P
i
s
use
d
i
n
st
ead of
m
e
asure
d
i
n
p
u
t
po
wer
in
P
,
-
Inpu
t preco
m
p
en
sation
is
p
r
ov
id
ed
wh
ich
tran
sform
s
d
y
n
a
mic o
p
timizati
o
n pro
b
l
em
to
static o
n
e
,
- Depe
ndence
of
sd
i
fr
om
los
s
P
is ad
ded
th
at m
a
k
e
s p
r
op
o
s
ed
algorith
m
co
n
s
id
erab
ly faster th
an ram
p
-
base
d m
e
t
hod,
-
Method operat
es in c
ont
i
n
u
o
u
s t
i
m
e
and pr
od
uces sm
oot
h
t
r
aject
o
r
y
of
sd
i
, t
hus t
h
e t
o
rq
u
e
ri
ppl
es a
r
e
co
m
p
letely el
i
m
in
ated
withou
t an
y
sm
o
o
t
h
i
n
g
filters.
2.
BA
C
KGR
OUN
D
2.
1
Mo
t
o
r Mo
del
Th
e m
o
to
r m
o
d
e
l u
s
ed
in
th
is p
a
p
e
r is th
e
-in
v
e
rse m
o
d
e
l [1
6
]
illu
strated
in
Fig
u
re 1
where
s
u
is
t
h
e st
at
or
v
o
l
t
a
ge
phas
o
r,
s
i
and
r
i
are the
stator and
rot
o
r
current pha
sors, respectively
s
R
and
r
R
are the
stator and rot
o
r re
sistances, respectively.
Also
L
denotes
the stray inductance a
nd
M
L
is th
e
m
a
in
inductance
.
Fi
gu
re 1.
-i
nve
rse e
qui
val
e
nt
ci
rcui
t
o
f
a
n
i
n
duct
i
o
n m
achi
n
e
W
i
t
h
th
e orien
t
atio
n
o
f
th
e ro
tor flux
vecto
r
r
al
ong
t
h
e d-a
x
i
s
o
f
sy
nchr
o
n
o
u
sl
y
rot
a
t
i
n
g
ort
h
ogonal
dq-coordinate syst
e
m
, the st
ate-s
p
ace m
o
tor m
odel can be real
ized
by t
h
e
fourt
h
order syste
m
of
d
i
fferen
tial eq
uatio
n
s
[17
]
:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
N
u
meri
c
a
l
Met
ho
d f
o
r P
o
w
e
r
Losses
Mi
ni
mi
zat
i
o
n
of
Vect
o
r
-C
o
n
t
r
ol
l
e
d
I
n
duct
i
o
n
Mot
o
r
(Alex Borisevich)
48
8
J
T
i
p
p
dt
d
L
u
i
i
L
R
i
L
R
L
L
R
i
dt
d
L
u
i
i
L
R
i
L
R
L
i
dt
d
R
i
L
R
dt
d
m
sq
r
sd
s
sq
sd
s
sd
R
r
M
R
sd
sq
s
sd
sq
R
sq
s
r
sq
R
sd
r
M
R
r
=
=
=
=
(1
)
whe
r
e
r
sq
R
s
i
R
=
is a sy
n
c
hro
nou
s sp
eed
,
is electrical shaft
rotation s
p
eed,
sq
r
e
i
p
T
=
is
an el
ect
r
o
m
a
gnet
i
c
t
o
rq
ue
p
r
o
duce
d
by
t
h
e
m
o
t
o
r.
Not
e
, t
h
at
t
h
e
cur
r
ent
s
a
n
d v
o
l
t
a
ges i
n
m
odel
(
1
) a
r
e m
e
asure
d
usi
n
g
p
o
we
r-i
nva
ri
ant
scal
i
ng
of
Park
-
C
lark
e tr
an
sfo
r
m
s
.
During
all
m
a
t
e
rial o
f
th
e p
a
p
e
r we
will n
e
g
l
ect th
e d
y
n
a
mics o
f
sd
i
and
sq
i
stator curre
nts with
assu
m
p
tio
n
th
at in
FOC con
t
ro
l th
e p
e
rfo
r
m
a
n
ce of PI cu
rrent controllers
are m
u
ch fa
ster than
flux and speed
dynam
i
cs. In t
h
is case, we
ca
n
write the
re
duced m
o
tor m
odel:
J
T
i
p
p
dt
d
R
i
L
R
dt
d
m
sq
r
R
sd
r
M
R
r
=
=
(
2
)
wh
ich
is sub
j
ect o
f
stud
y in
presen
t
wo
rk
.
2.
2.
Power Losses
and Optimal
Regime
For
gi
ve
n c
o
n
s
t
a
nt
m
echani
cal
t
o
r
que
m
T
it is
possible t
o
ca
lculate steady-state power losses a
s
fu
nct
i
o
n of st
eady
-
st
at
e
m
a
gn
et
i
z
i
ng
c
u
r
r
ent
sd
i
:
s
sd
R
s
sd
M
m
sd
ss
loss
R
i
R
R
i
pL
T
i
P
2
2
)
(
=
)
(
(
3
)
whe
r
e
)
/(
sd
M
m
i
pL
T
is a stead
y
-state v
a
l
u
e
o
f
qu
ad
rature cu
rren
t
sq
i
fo
r f
i
x
e
d
sd
i
and
m
T
.
It is kno
wn
[1
-3
], th
at
op
ti
m
a
l
m
a
g
n
e
tizin
g
cu
rren
t t
h
at m
i
n
i
m
i
zes (3
) can
b
e
calcu
lated
as fo
llo
ws
4
=
)
(
s
s
R
M
m
m
opt
sd
R
R
R
p
L
T
T
i
(4
)
2.
3.
Simple Ramp
Method
The ram
p
-ba
s
e
d
m
e
t
hod [
6
]
i
s
a sim
p
l
e
st
ty
pe of sea
r
c
h
cont
rol
l
e
r.
Su
p
pos
e t
h
e di
rect
i
on
of t
h
e
o
p
tim
u
m
search
relativ
e to the pr
esent
val
u
e
o
f
m
a
gnet
i
z
i
ng c
u
r
r
ent
sd
i
i
s
kn
ow
n.
S
u
ch
i
n
fo
rm
ati
on c
oul
d
b
e
provide
d
from
the analysis of
)
(
t
i
sq
tran
sien
t: if n
e
w st
eady-stat
e value of
sq
i
is
h
i
gh
er th
an
prev
iou
s
on
e,
then t
h
e loa
d
t
o
rque
m
T
i
s
i
n
cre
a
sed a
n
d ne
w
opt
i
m
u
m
of
opt
sd
i
is highe
r
tha
n
previous
one as
well (becaus
e
m
opt
sd
T
i
), a
n
d
vi
ce-ve
r
s
a.
In
ori
g
i
n
al
des
c
ri
pt
i
o
n [
6
]
ra
m
p
m
e
t
hod c
o
nsi
s
t
s
o
f
se
que
nt
i
a
l
chan
ges
of
sd
i
b
y
sm
all st
ep
s
u
n
til
measu
r
ed
input p
o
w
er
in
P
starting to inc
r
ease. In co
ntinuous
time the step ch
anges coul
d be re
placed
to
in
teg
r
ation
of a con
s
tan
t
:
c
i
d
t
d
sd
=
(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
. 3, Sep
t
em
b
e
r
2
015
:
48
6 – 497
48
9
u
n
til th
e in
pu
t p
o
wer
in
P
st
ops chan
gi
n
g
nea
r
t
h
e poi
nt
of m
i
nim
u
m
:
|<
|
in
P
, whe
r
e
const
c
=
and
the sign of
c
de
pe
nd
s
on
t
h
e di
rect
i
o
n
o
f
s
earch
,
0
>
is an
arb
itrary sm
all v
a
lu
e.
The sel
ect
i
o
n
of c
o
nst
a
nt
val
u
e
c
coul
d
be
d
one
f
r
om
t
h
e t
w
o
f
o
l
l
o
wi
n
g
si
m
p
le id
eas: if a v
a
lu
e of
c
i
s
t
oo sm
all
,
then t
h
e m
e
t
hod co
nve
rge
s
sl
owl
y
;
i
f
t
h
e
c
i
s
to
o
b
i
g, th
en th
e in
stan
t v
a
lu
e o
f
)
(
t
P
in
is
con
s
idera
b
le d
i
ffere
nt fr
om
s
t
eady-state value for
give
n
sd
i
due t
o
i
nhe
ri
t
i
ng
dy
nam
i
cs
of
po
wer l
o
ss
es
,
wh
ich
leads to
o
p
tim
izat
io
n
error.
3.
OPTIMIZ
A
T
I
ON PROBLEM
FORMULATION
3.
1.
Objec
t
ive Fun
c
tion
Here a
n
d aft
e
r
i
n
st
ead
of
usi
ng t
h
e i
n
put
p
o
we
r
sd
sd
sq
sq
in
u
i
u
i
P
=
as opt
i
m
i
zat
i
on cri
t
e
ri
on
we
will u
s
e t
h
e
o
h
mic p
o
w
er l
o
sses
loss
P
, calculated
from
m
easured value
s
of
sq
i
and
sd
i
:
s
sd
R
s
sq
loss
R
t
i
R
R
t
i
t
P
)
(
)
)(
(
=
)
(
2
2
(
6
)
It's in
tro
d
u
ces so
m
e
m
o
d
e
li
n
g
un
certain
ty
, bu
t u
s
u
a
lly th
e m
easu
r
em
e
n
t no
ise of
loss
P
is
m
u
ch
lo
wer th
an
i
n
in
P
.
3.
2.
Contr
o
l Input Prefiltering
For
searc
h
c
o
n
t
rol
m
e
t
hods i
t
i
s
essent
i
a
l
t
o
kn
o
w
st
eady
-
st
at
e po
wer
)
(
sd
ss
loss
i
P
fo
r
gi
ve
n
sd
i
val
u
e
.
B
u
t
cha
ngi
ng
sd
i
according to some search tra
j
e
c
tory
)
(
t
i
sd
we will
g
e
t on
ly in
stant v
a
lu
e
)
(
t
P
loss
, wh
ich is
ob
vi
o
u
sl
y
di
f
f
e
rent
f
r
om
)
(
sd
ss
loss
i
P
. The sol
u
t
i
on i
s
t
o
cha
nge a
n
d
fi
x
sd
i
an
d
th
en wait so
m
e
t
i
m
e u
n
til
steady-state and only then m
e
asure
loss
P
, whic
h lim
i
ts the speed and accur
acy of m
e
thods.
Here we propose
pre
-
com
p
ensat
i
on
schem
e
wi
t
h
w
h
i
c
h
i
s
po
ssible to estimate the steady
-
state value
)
(
sd
ss
loss
i
P
on
-
t
h
e
-f
ly
,
with
ou
t waiting
for
th
e
steady-state.
Suppose the speed controller is fast
enough to accomm
odate the change
of torque load a
n
d the spee
d
dr
o
p
i
s
cl
ose
t
o
zer
o f
o
r t
o
rq
ue
m
T
and
fl
u
x
r
v
a
riatio
n.
Th
en
it is p
o
s
sib
l
e to
n
e
g
l
ect th
e tran
sient
processes i
n
s
p
eed PI-c
ontroller and as
sume that th
e regulator alwa
ys
m
a
intains appropriate va
lue of
qu
d
r
at
ure
cu
rre
nt
t
o
e
n
s
u
re
co
nst
a
nt
o
u
t
p
ut
t
o
r
q
ue:
)
(
=
)
(
t
p
T
t
i
r
m
sq
(7
)
Th
us t
h
e
m
a
jor
so
urce
o
f
i
nhe
rent
dy
nam
i
cs of
loss
P
is th
e fl
u
x
dyn
amics o
f
m
o
to
r.
Let
'
s i
n
t
r
od
uce
a ne
w m
a
ni
pu
l
a
bl
e vari
a
b
l
e
dt
d
/
, whic
h
determ
ines the t
r
aject
ory of
)
(
t
i
sd
as
fo
llows:
)
(
)
(
=
)
(
)
(
=
)
(
0
t
t
dt
t
t
t
i
r
t
r
sd
(
8
)
The sam
e
can
be
rewritte
n in
Laplace s-domain:
)
(
1
=
)
(
s
s
s
s
I
R
sd
(9
)
whe
r
e
)
(
s
I
sd
and
)
(
s
are
Laplace t
r
ans
f
orm
s
of the
)
(
t
i
sd
and
)
(
t
.
The
n
we ca
n c
onst
r
uct
ne
w e
s
t
i
m
a
ti
on f
o
r
p
o
we
r l
o
sses i
n
fol
l
o
wi
n
g
fo
rm
:
s
R
s
sq
ss
loss
R
t
R
R
t
i
P
2
2
)
(
)
)(
(
=
)
(
(
1
0
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
Nu
merica
l Met
h
od
f
o
r Po
wer
Lo
sses Min
i
miza
tion
o
f
Vector-Con
tro
lled Ind
u
c
tion
Mo
t
o
r
(Alex Borisevich)
49
0
We will sh
ow
th
at (10
)
h
a
s a v
e
ry n
i
ce prop
erty th
at it d
o
esn
'
t d
e
p
e
nd
ant fro
m
flu
x
d
y
n
a
m
i
cs. It i
s
allo
ws conv
ert
th
e o
p
tim
iza
t
i
o
n
p
r
ob
lem with
ti
m
e
-v
arying
fun
c
tion
(6
)
to
si
m
p
le
min
i
mizatio
n
p
r
ob
l
e
m
fo
r
one
-t
o
-
o
n
e st
at
i
c
fun
c
t
i
on
(1
0
)
wi
t
h
negl
ect
i
ng
fl
u
x
dy
nam
i
cs. Ou
r m
a
i
n
resul
t
can be
f
o
rm
ul
at
ed as a sim
p
le
th
eorem
.
The
o
rem 1
. For any
t
r
aj
ect
ory
)
(
t
i
sd
d
e
termined
b
y
wh
en
in
itia
l co
nd
ition
s
(0)
=
(0)
sd
i
and
0
=
(0)
a
r
e sa
tisfied
t
h
e fo
llo
wi
n
g
a
r
e tru
e
:
1. )
(
=
)
(
t
L
t
M
r
2.
op
t
sd
op
t
i
=
whe
r
e
op
t
i
s
a m
i
nim
u
m
poi
nt
of
(1
0)
, i
.
e.
)
(
)
(
ss
los
s
opt
ss
los
s
P
P
.
Proof.
Su
bst
i
t
u
t
i
ng
(
9
) t
o
fl
u
x
e
quat
i
o
n
fr
o
m
m
o
t
o
r m
ode
l
(2
)
gi
ves
)
(
=
)
(
1
1
=
)
(
s
L
s
s
s
s
s
L
s
M
R
R
M
r
(
1
1
)
Fo
r conv
erting to
tim
e d
o
m
a
i
n
no
te, t
h
at
(0
)
=
(0)
sd
i
and the
n
(0)
=
(0
)
M
r
L
f
r
om
th
e f
l
ux
eq
u
a
tion
of th
e m
o
to
r m
o
d
e
l (2
). Th
us, t
h
e fi
rst statem
en
t o
f
th
eo
rem
co
mes obv
iou
s
ly.
Fo
r t
h
e proo
f
of second
statemen
t, let
'
s su
bstitu
te th
e sp
eed
con
t
ro
ller d
y
n
a
m
i
cs (7
) to
(1
0) and
tak
e
to
accoun
t th
e
first statem
en
t o
f
th
eo
rem
)
(
=
)
(
t
L
t
M
r
:
s
R
s
M
m
s
R
s
r
m
ss
loss
R
t
R
R
t
pL
T
R
t
R
R
t
p
T
P
2
2
2
2
)
(
)
(
)
(
=
)
(
)
(
)
(
=
)
(
(1
2)
Fro
m
th
e last eq
u
a
tion
on
e can
no
te th
at the
)
(
ss
los
s
P
is in
fact equ
a
tio
n
fo
r stead
y
-state lo
sses (3)
whe
r
e
sd
i
is formally replaced to
. Th
us
, b
o
t
h
f
unct
i
ons a
r
e i
d
ent
i
cal
an
d
have t
h
e sam
e
ran
g
e a
nd t
h
e
sam
e
m
i
nim
u
m
poi
nt
op
t
sd
op
t
i
=
.
Q.
E.D
.
To
dem
onst
r
a
t
e t
h
e di
f
f
e
r
en
ce bet
w
ee
n m
e
t
h
o
d
s
(1
0)
,
(3
) an
d
(
6
)
fo
r
p
o
we
r l
o
sses es
t
i
m
a
t
i
on w
e
si
m
u
lated
th
e
m
o
to
r m
o
d
e
l un
d
e
r lin
ear i
n
creasin
g of
sd
i
(Fi
g
ure
2
)
.
Figure
2. Diffe
r
ence
bet
w
een
)
(
ss
los
s
P
by
(
1
0),
)
(
sd
ss
los
s
i
P
by
(
3
)
and
)
(
t
P
loss
b
y
(6
)
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
. 3, Sep
t
em
b
e
r
2
015
:
48
6 – 497
49
1
4.
O
P
T
I
M
I
ZA
TI
O
N
A
L
GO
RI
T
H
M
4.
1.
Terminati
on Criteria
and Accur
a
cy
In th
is section we
will ch
ang
e
th
e no
tation
b
y
con
s
i
d
eri
n
g th
e ab
stract
m
i
n
i
m
i
za
tio
n
prob
lem
o
f
conve
x
scalar function
)
(
=
x
f
y
. T
h
e c
o
nnection t
o
problem
above is
:
=
x
,
)
(
=
)
(
=
ss
loss
P
x
f
y
.
Let'
s
*
x
i
s
a m
i
ni
m
u
m
poi
nt
o
f
f
, and
0
x
is in
itial g
u
e
ss.
The al
go
ri
t
h
m
descri
bed
i
n
se
ct
i
on
2.
3
is
form
u
l
a
t
ed
in
t
h
e
n
e
w no
tatio
n as fo
llo
ws:
c
x
dt
x
df
y
=
do
|>
)/
(
|=|
|
while
(1
3)
whe
r
e
)
(
sign
=
sign
0
*
x
x
c
,
0
=
(0)
x
x
.
Let
us st
udy
h
o
w t
h
e
val
u
e
o
f
affects the a
ccuracy
of
sea
r
ch al
gorithm
.
If
sufficien
tly sm
a
ll,
th
en
it is
po
ssi
b
l
e to
exp
a
nd th
e
)
(
x
f
in
to
po
w
e
r
ser
i
es n
ear
t
h
e n
e
igh
bor
hoo
d o
f
*
x
:
3
*
2
*
*
*
)
(
)
)(
(
2
1
)
(
=
)
(
x
x
O
x
x
x
f
x
f
x
f
(1
4)
Fo
r
furth
e
r an
alysis, withou
t lo
ss of g
e
n
e
rality, we
can
assu
m
e
0
=
*
x
and
use q
u
a
d
rat
i
c
app
r
oxi
m
a
t
i
on near
t
h
e
*
x
:
2
*
)
(
2
1
=
)
(
x
x
f
x
f
(1
5)
Hen
c
e th
e tim
e
d
e
riv
a
tiv
e is
x
x
f
c
x
x
x
f
y
)
(
=
)
(
=
*
*
(1
6)
Since
ct
x
t
x
0
=
)
(
, the
n
)
(
)
(
=
0
*
ct
x
c
x
f
y
.
Let
'
s denot
e
y
ˆ
as a num
e
rical
estim
a
tion
of
y
obt
ai
ne
d
by
di
gi
t
a
l
di
ffe
r
e
n
t
i
a
t
i
on. T
h
e ea
si
est
w
a
y
to
ob
tain
y
ˆ
b
y
usin
g first
-
ord
e
r filter in
t
h
e
operato
r fo
rm
;
)
(
1
1
=
)
(
1
=
)
(
ˆ
s
Y
s
s
Y
s
s
s
Y
(1
7)
whe
r
e
)
(
ˆ
s
Y
is Lapl
ace trans
f
orm
of
)
(
ˆ
t
y
,
)
(
s
Y
is a Lapl
ace trans
f
orm
of
)
(
t
y
, and
is a filter
t
i
m
e
const
a
nt
.
Fr
o
m
[
1
8
]
k
now
n, th
at t
h
e
r
e
sp
on
se of
f
i
r
s
t
or
d
e
r
system
to
th
e r
a
m
p
)
(
)
(
=
=
)
(
0
*
ct
x
c
x
f
y
t
u
is
)
(
)
(
)
(
)
(
=
)
(
ˆ
0
*
0
/
*
x
c
ct
x
f
c
x
c
e
x
f
c
t
y
t
(1
8)
Whe
n
is su
fficien
tly s
m
a
ll
th
e exp
o
n
e
n
tial ter
m
in
(1
8
)
rap
i
d
l
y conv
erg
e
s t
o
zero
.
Thu
s
it is
pos
sible to wri
t
e an e
x
pressi
on
for the
steady-state
0
t
:
)
)
(
(
)
(
=
)
(
)
(
)
(
ˆ
*
0
*
c
t
x
x
f
c
c
ct
x
x
f
c
t
y
(1
9)
The al
gorithm
terminates whe
n
|=
ˆ
|
y
or
=
)
)
(
(
)
(
*
c
t
x
x
f
c
(2
0)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
Numeric
a
l Met
h
od f
o
r P
o
wer
Losses
Mini
m
iza
tion
of Vector-Controlled Induction
Mot
o
r
(Alex Borisevich)
49
2
Wh
en
u
s
ing
the h
i
gh
-p
ass
filter to
esti
m
a
te
th
e d
e
ri
v
a
tiv
e
y
ˆ
, there is a
delay between t
h
e actual
val
u
e
y
an
d est
i
m
a
ti
on
y
ˆ
. As
a res
u
lt the se
arch is
stoppe
d wit
h
a
delay
c
. He
nce, i
n
the abs
o
lut
e
accuracy
x
of search
proce
d
ure the error as
s
o
ciated with fi
nite precision
an
d
b
a
nd-li
m
i
t
e
d
filter sho
u
l
d
be t
a
ke
n
wi
t
h
a
di
f
f
ere
n
t
si
g
n
s
:
c
c
x
f
x
)
(
=
*
(
2
1
)
4.
2.
Conver
gence
Speed-Up
To accele
r
ate the sea
r
ch it is
possibl
e to use
the tim
e
deriva
tive of
y
.
Th
us
, we have
u
p
d
ated rule:
y
k
x
ˆ
=
(
2
2
)
whe
r
e
0
>
k
is positive constant.
Since the
acc
u
r
acy
m
i
nim
u
m setp
oint (
2
1)
depe
n
d
s
fr
om
the a
r
g
u
m
e
nt
x
r
a
te of
cha
n
ge,
to en
su
re
that the specifi
ed acc
uracy is
necessary to li
mit the value
x
, i.e.:
}
,
ˆ
{
min
=
c
y
k
x
(2
3)
whe
r
e
1
>
is ratio of the
m
a
xim
u
m
rate of ch
ange
of
x
to ini
tial
c
. Note that this form
ula i
s
written for
t
h
e case
0
>
c
, othe
rwis
e, o
b
v
io
usly
, t
h
e
min
operation
needs t
o
be re
pl
aced to
max
.
To e
n
sure
the
unc
onditional i
m
provem
e
nt of c
o
nverge
nce
rate, it is
necessary to excl
ude
cases
when
|
|<|
ˆ
|
c
y
k
. He
nce, we
obtain the
final e
x
pre
ssio
n
fo
r th
e ar
gu
m
e
n
t
dyn
amics:
}}
,
ˆ
{
min
,
{
max
=
c
y
k
c
x
(2
4)
4.
3.
Final Algorithm
In t
h
is subsection we
are
going
to form
ulate the fi
nal form
of
searc
h
algorith
m
procedure
accordi
n
g
to all disc
ussio
n
s a
b
ove
in
ori
g
inal
nota
tion
for power l
o
sses optimization problem
.
Let's denote:
0
>
1,
>
0,
>
0,
>
k
c
are t
h
e al
gorith
m
param
e
te
rs as it desc
ri
bed before
. L
e
t'
s
introduce additional param
e
te
r
0
t
f
o
r
dur
atio
n
o
f
un
co
nd
itional ch
an
g
e
o
f
x
for initial esti
mation of
y
ˆ
.
And
tem
porary
va
riable
1,1}
{
d
is use
d
fo
r in
dication
o
f
search
di
rectio
n.
He
re is
pse
u
doc
o
d
e
of
alg
o
r
ithm
:
1.
if
0
*
>
x
x
then
1
:=
d
, else
1
:=
d
.
2.
while
0
<
t
t
do
c
d
x
=
3.
while
|>
ˆ
|
y
do
3.
1
if
c
y
k
>
ˆ
then
3.
1.
1
if
c
y
k
<
ˆ
then
y
k
d
x
ˆ
=
else
c
d
x
=
3.
2
else
c
d
x
=
Value of
y
ˆ
is bei
n
g estim
a
t
ed in parallel to algor
ithm
executi
on by
high
pass filter (17).
Algorithm
called in m
echanical
steady-st
ate condition
whe
n
tra
n
si
ent after torque change is
finished. Th
e condition
0
*
>
x
x
is e
qui
valent t
o
(0)
>
*
sq
sq
i
i
whe
r
e
(0)
sq
i
is initial values of m
a
gnetizi
ng
and
quadrature
currents re
spe
c
tiv
ely
(bef
ore
the tor
que c
h
a
nge
),
*
sq
i
is a stea
dy-state quadrature current for
new load torque.
4.
4.
Co
nver
gence
An
al
ys
ys
The
be
havi
or
o
f
the
alg
o
rithm
can
be c
h
a
r
acterized
by the
followi
ng the
o
re
m
.
Theorem 2
If t
h
e followi
ng conditions
are t
r
ue:
|
<|
|
|
0
*
0
x
x
t
c
(2
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
. 3, Sep
t
em
b
e
r
2
015
:
48
6 – 497
49
3
0
t
(2
6)
th
e a
l
g
o
rith
m fro
m
section
4.
3
find
s
a lo
ca
l m
i
n
i
m
u
m
o
f
the fu
n
c
tion
)
(
x
f
with
accuracy
(2
1)
.
Proof.
Th
e first
con
d
ition
|
<|
|
|
0
*
0
x
x
t
c
me
a
n
s
th
a
t
t
h
e
mi
n
i
mu
m p
o
i
n
t
*
x
is no
t w
i
t
h
in
t
h
e
in
terv
al ]
,
[
0
0
0
t
c
x
x
,
whe
r
e the algorithm cannot te
rm
in
ate.
D
u
e t
o
th
e seco
nd
co
nd
ition
0
t
at tim
e
0
t
tran
sien
t pro
cess in
th
e d
e
ri
vativ
e esti
m
a
to
r is
f
i
n
i
s
h
ed
and
)
)
(
(
)
(
=
)
(
ˆ
*
c
t
x
x
f
c
t
y
near the
*
x
wi
t
h
r
a
m
p
dy
nam
i
cs o
f
ar
g
u
m
e
nt
ct
x
t
x
0
=
)
( .
Because the
)
(
x
f
is conve
x
, the
n
t
h
e val
u
e of de
rivative
)
(
x
f
dec
r
ea
ses at interval
]
,
[
*
0
x
x
, and
th
e v
a
lu
e
o
f
time d
e
riv
a
tiv
e
x
x
f
y
)
(
=
decreases as
well in case of noni
ncreasi
n
g
x
. He
nce, aft
e
r the
transition t
o
the accelerated dynam
i
cs
y
k
x
ˆ
=
th
ere is ex
ist
m
o
men
t
o
f
ti
m
e
w
h
ere
y
k
c
ˆ
and t
h
e
search proce
d
ure al
ways switches to
the
ra
m
p
cha
nge
of argum
e
nt
c
x
=
n
ear m
i
nim
u
m
poi
nt
. T
h
u
s
, t
h
e
state
m
ent of the the
o
rem
(21) in
form
of ram
p
search
a
ccuracy is always true
in
the fin
a
l ph
ase
o
f
th
e
alg
o
rith
m
.
Q.E.D.
Fro
m
a practical v
i
ewp
o
i
n
t
t
h
e co
nd
ition
(2
5) m
ean
s th
at
th
e i
n
itial ap
pro
a
ch
o
f
0
x
is f
a
r
en
ough
away from
the desire
d
*
x
. Satisfacto
r
y in
terp
retatio
n
of th
e con
d
itio
n
(26
)
fro
m
an
en
g
i
n
e
ering
v
i
ew
po
in
t
is
a choice
3
0
t
.
5.
E
X
PERI
MEN
T
AL D
A
TA
5.
1.
Simulati
on
For t
h
e veri
fi
cat
i
on o
f
pr
o
p
o
s
ed al
g
o
ri
t
h
m
t
h
e sim
u
l
a
t
i
on was co
nd
uct
e
d. St
r
u
ct
u
r
e o
f
Sim
u
l
i
n
k
m
odel
i
s
sho
w
n at
Fi
gu
re
3.
Fig
u
r
e
3
.
Model o
f
op
timizati
o
n algo
r
ith
m
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
Nu
m
e
rica
l Met
h
od
f
o
r Po
wer
Lo
sses Min
i
m
i
za
tion
o
f
Vector-Con
tro
lled Ind
u
c
tion
Mo
t
o
r
(Alex Borisevich)
49
4
Fi
gu
re
4.
Si
m
u
l
a
t
i
on res
u
l
t
s
f
o
r
p
r
o
p
o
se
d m
e
t
h
o
d
al
o
n
g
wi
t
h
ram
p
an
d
g
o
l
den
searc
h
m
e
t
h
o
d
s
The m
odel
of
m
o
t
o
r (2
) i
s
i
m
pl
em
ent
e
d i
n
si
de bl
oc
k S
u
b
s
y
s
t
e
m
[
m
ot
or
]
.
Param
e
t
e
rs of m
odel
a
r
e
esti
m
a
ted
fro
m
m
o
to
r D
R
S71S4
b
y
SEW
-
Eu
rod
r
iv
e
w
ith
0
.
3
7
kW
rated p
o
w
e
r.
A
l
go
ri
th
m o
f
o
p
tim
i
zatio
n
im
pl
em
ent
e
d as di
scret
e
sy
st
em
wi
t
h
t
h
e bl
o
c
ks M
A
TL
A
B
Fun
c
tio
n and
U
n
it
D
e
lay. Param
e
ters o
f
algo
rith
m
a
r
e
ch
o
s
en
as
fo
llo
ws
:
0.15
=
c
,
0.
0
2
=
k
,
0.5
=
, 0.2
=
0
t
.
During sim
u
la
tion two cases
was conside
r
ed: when
nom
m
m
T
T
=
and
t
h
en l
o
a
d
t
o
r
que
dr
o
ppe
d t
o
nom
m
m
T
T
0.2
5
=
, whe
r
e
nom
m
T
–
rated
lo
ad
t
o
rq
u
e
(2
.6
N
m
). In
itially th
e cu
rren
t
sd
i
was selected as optim
al
fo
r lo
we
r loa
d
nom
m
m
T
T
0.2
5
=
.
For
t
h
e
com
p
ari
s
o
n
of
a
pr
o
pos
ed
ap
pr
oac
h
wi
t
h
ot
he
r s
i
m
i
l
a
r al
go
ri
t
h
m
s
t
h
e pure
ra
m
p
m
e
t
h
o
d
[6,7
] an
d go
ld
en
section
tech
niq
u
e
[8
] w
e
re i
m
p
l
e
m
en
ted
in sim
u
lat
i
o
n
.
Th
e sim
u
latio
n
resu
lts are
p
r
esen
ted at Fig
u
re
3.
For t
h
e
powe
r los
s
e
s analytically calculated
m
i
n
i
mu
m
mi
n
los
s
P
i
s
sho
w
n as wel
l
as a bl
ack l
i
n
e.
The g
r
ee
n l
i
n
e i
s
a t
r
aject
ory
f
o
r
ram
p
m
e
t
hod, a
nd
red l
i
n
e
fo
r g
o
lde
n
sear
ch. T
h
e cu
rre
n
t
step for
ram
p
is chose
n
0.05
=
I
A.
The duration
of curre
n
t steps
for ram
p
m
e
t
hod
was ad
ju
st
ed s
o
t
h
at
t
h
e t
r
an
si
ent
s
h
a
ve t
i
m
e
t
o
be com
p
l
e
t
e
d bef
o
re t
h
e
ne
xt
ch
ange
. F
o
r i
n
c
r
e
a
se o
f
mag
n
e
tizin
g curren
t
0.5
=
T
s was
us
ed a
n
d for dec
r
ease of m
a
gnet
i
zing c
u
rrent
0.2
=
T
s.
5.
2.
Hardw
a
re Im
plementati
on
As a
platf
o
rm
fo
r im
plem
enting
control algorithm
s
was
used t
h
e
co
n
t
ro
l
l
er
dSPA
CE with
D
S
52
02
m
o
t
o
r cont
r
o
l
boa
r
d
(Fi
g
u
r
e
5)
. The
dSP
A
C
E
pl
at
fo
rm
i
s
a sy
st
em
based o
n
D
SP a
nd
FPG
A w
h
i
c
h i
s
use
d
as
har
d
ware t
a
r
g
e
t
for a
u
t
o
m
a
t
i
c
code
gen
e
rat
i
on a
n
d im
pl
em
ent
a
t
i
on o
f
M
A
TLAB
Si
m
u
li
nk m
odel
s
. Fo
r t
h
e
m
o
tor power
s
t
age use
d
a m
odified
SE
W-E
u
rodrive M
o
vi
Axi
s
i
nve
rt
er.
The P
W
M
control signals for three-
pha
se
bri
d
ge c
o
m
e
s di
rect
l
y
from
t
h
e d
S
P
A
C
E
co
nt
r
o
l
l
e
r.
Exp
e
rim
e
n
t
al s
e
tu
p
co
n
s
ist
o
f
D
R
S112
M4
m
o
to
r fro
m
SEW
-
Eurodriv
e w
ith
26
.6
N
m
rated
torqu
e
(4 kW
) coup
led
w
ith
a l
o
ad
mach
in
e fo
r testin
g
v
a
ri
o
u
s l
o
ad cond
itio
n
s
.
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
. 3, Sep
t
em
b
e
r
2
015
:
48
6 – 497
49
5
Fi
gu
re
5.
Ex
pe
ri
m
e
nt
al
set
up
Fi
el
d-
ori
e
nt
e
d
vect
o
r
c
ont
rol
o
f
st
at
o
r
c
u
r
r
e
nt
s i
n
t
h
e
r
o
t
a
t
i
ng
dq
-c
oo
r
d
i
n
at
es was
i
m
pl
em
ent
e
d.
Fo
llow
i
ng
algo
rith
m
p
a
ram
e
ters w
a
s
u
s
ed
:
0.
5
=
c
,
0.01
5
=
k
,
2
=
, 0.5
=
0
t
. For th
e filterin
g
of inp
u
t
po
we
r ri
ppl
e t
h
e co
nt
i
n
u
o
u
s t
i
m
e
3-r
d
o
r
de
r
B
u
t
t
e
rw
ort
h
fi
l
t
er wi
t
h
cut
o
f
f
fre
que
ncy
2
=
c
f
Hz was use
d
.
Th
e m
o
to
r w
a
s p
u
t
to
con
tinu
o
u
s
v
ect
o
r
-con
tro
lled
ro
tating
m
o
d
e
w
ith
th
e sp
eed
100
=
rad/
sec
and two
values
of m
echanical load we
re tested
1
3
.6
=
m
T
Nm
(appr
o
x
5
0
% o
f
rat
e
d t
o
r
q
ue) a
nd
6.8
=
m
T
Nm
(app
ro
x 2
5
% of rat
e
d
t
o
r
que
).
The t
r
an
si
ent
s
of p
o
w
er o
p
t
im
i
zat
i
on obt
ai
ned f
r
om
gr
adi
e
nt
-
b
ase
d
a
l
go
ri
t
h
m
are
prese
n
t
e
d at
Fig
u
r
e
s 5-
6.
Th
e p
o
w
e
r
lo
sses
loss
P
w
e
r
e
c
a
l
cu
late
d
fr
o
m
me
a
s
u
r
ed
cu
rr
en
ts
by (
6
)
.
Fi
gu
re
6.
P
o
we
r l
o
ss
es
loss
P
an
d m
a
gnet
i
z
i
n
g c
u
r
r
e
nt
sd
i
dy
nam
i
cs for
6.8
=
m
T
Nm
Evaluation Warning : The document was created with Spire.PDF for Python.