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.
7, N
o
. 1
,
Mar
c
h
20
16
,
pp
. 20
2
~
21
6
I
S
SN
: 208
8-8
6
9
4
2
02
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
Criti
c
al
Condition of S
e
ns
orl
e
ss
Induction Generator using Flux
Weakening in Wind Turbine Application
Na
nd
a
Avi
a
nt
o Wi
c
a
ks
on
o,
Abd
u
l
H
a
l
i
m
,
Ari
e
s Su
bi
an
t
o
ro
, Feri
Yusi
var
Department o
f
Electrical E
ngin
e
ering, Univ
ersity
of
In
don
esia (U
I)
,
I
ndo
nesia
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Oct 10, 2015
Rev
i
sed
D
ec 21
, 20
15
Accepte
d
Ja
n 11, 2016
This paper was intend
ed to examine
thoroughly a critical condition of th
e
sensorless induction gen
e
rator
using flux weaken
ing in w
i
nd turbine
application. Th
e critic
al condition would hap
p
en when the
rotor speed
reach
ed th
e cr
iti
cal ro
tor speed
r
e
feren
c
e
.
Th
e cr
itic
al ro
tor spee
d refer
e
nc
e
was
the highes
t
of the rotor
s
p
eed ref
e
ren
ce
that s
t
i
ll
caus
e
d
the s
t
ab
l
e
response. It was
obtained b
y
in
creasi
ng
the rotor speed refer
e
n
ce until the
s
y
stem response became uns
table. In
the
low speed r
a
nge of
wind showed
that th
ere w
a
s
no uns
table
cond
ition what
ever
a
rotor s
p
eed r
e
f
e
renc
e was
s
e
t. On the o
t
h
e
r hand,
ther
e
was
a crit
ic
al r
o
tor s
p
eed ref
e
r
e
nce
in the
medium and hig
h
speed range of wind.
The unstable condition was caused b
y
the induction generator th
at r
eceived a
power higher than its cap
acity
, so its
rotor speed
coul
dn't
be m
a
in
tain
ed at
ref
e
ren
ce
value
.
Th
e first
solution was
suggested that th
e stable conditio
n woul
d be made b
y
setting the
rotor speed
referen
c
e at
th
e m
i
nim
u
m
critical r
e
fer
e
nce. The second s
o
lution was
suggested that th
e contro
lling
rot
o
r speed
in
tri
a
n
g
le ar
ea between
the
crit
ical
condition
and th
e operat
i
on that
used the m
i
nim
u
m
critica
l
refer
e
nce for the
rotor s
p
eed r
e
fer
e
nce
.
In
the t
r
ia
ngle ar
ea
, th
e ro
tor s
p
eed was
c
ontrolled
b
y
setting
the
tip
sp
eed r
a
tio
.
Keyword:
G
e
n
e
r
a
ted power
Rotor s
p
eed
Sens
orl
e
ss
i
n
d
u
ct
i
o
n
ge
ne
rat
o
r
Stab
ility
Wi
n
d
s
p
e
e
d
Copyright ©
201
6 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
N
a
nd
a Av
ian
t
o W
i
caksono
,
Depa
rt
m
e
nt
of
El
ect
ri
cal
Engi
neeri
n
g
,
U
n
i
v
er
sity of
In
don
esia,
Kam
pus
U
I
De
po
k,
Ja
kart
a, 1
6
4
2
4
In
d
onesi
a
.
Em
a
il: n
a
n
d
a
av
ian
t
o@g
m
ai
l.co
m
1.
INTRODUCTION
Th
er
e
w
a
s a
po
ten
c
y of
w
i
nd
en
er
g
y
in m
a
ny locations whic
h we
re fa
r from
cities, areas with t
h
e
sm
all and low density of
population, a
n
d
were
ha
ving
s
m
all econom
ic activities. Th
e
locations
fitted to
be
d
e
v
e
l
o
p
e
d
b
y
u
s
ing
sm
all win
d
turb
i
n
es.
Th
e sm
all win
d
tu
rb
in
es
shou
ld
h
a
v
e
low cost, h
i
g
h
reliab
i
lity an
d
lo
w m
a
in
ten
a
nce.
For e
ffi
ci
ent
c
o
st
an
d si
m
p
l
e
pu
rp
ose a
ppl
i
cat
i
on, t
h
e sm
al
l
wi
nd t
u
r
b
i
n
e
was desi
gne
d
by
usi
n
g (a
)
a sens
orless
sq
uirrel ca
ge in
d
u
ction
ge
nerat
o
r
(SCI
G
)
,
(b
)
fi
xe
d pi
t
c
h
an
g
l
e of
bl
ades
, a
n
d (c
) fi
xe
d gea
r
rat
i
o
.
Th
is configu
r
atio
n
of th
e
wind
turb
i
n
e was
ch
osen
b
ecau
s
e o
f
its sim
p
le
co
nstru
c
ti
o
n
,
si
m
p
le
m
a
in
te
n
a
n
c
e,
reliab
l
e op
eratio
n, an
d also low
p
r
ice.
The fixe
d pitc
h angle of
blades cause
d tha
t
the
wind turbine coul
dn’t lim
it a received power from
wind. The rece
ived power tha
t
wa
s highe
r than the rate
d powe
r cause
d the gene
rator becam
e over-voltage
s
and
o
v
e
r-c
ur
re
nt
. T
o
p
r
ot
ect
t
h
e
gene
rat
o
r
d
a
m
a
ge cause
d
of
t
h
e
o
v
er
-v
ol
t
a
ges a
n
d
t
h
e
o
v
er
-cu
rre
nt
,
t
h
e wi
n
d
t
u
r
b
i
n
e o
p
e
r
at
i
on m
u
st
be st
op
pe
d w
h
en a
wi
nd s
p
ee
d b
ecam
e
hi
gher
t
h
an i
t
s
rat
e
d
spee
d. T
h
e st
o
ppi
n
g
cause
d t
h
e
ge
n
e
rat
e
d
p
o
we
r c
oul
dn
’t
be
pr
o
duce
d
.
To m
a
ke t
h
e wi
nd t
u
rbi
n
e abl
e
t
o
gene
rat
e
p
o
we
r i
n
a hi
gh
spee
d of
wi
n
d
,
t
h
e sy
st
em
was equi
pp
e
d
by
a fl
ux
wea
k
eni
n
g
co
nt
r
o
l
al
go
ri
t
h
m
.
Th
e fl
u
x
weake
n
i
n
g
co
nt
r
o
l
w
o
ul
d
kee
p
t
h
e
v
o
l
t
a
ges a
n
d c
u
rre
nt
s
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
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94
I
J
PED
S
Vo
l. 7,
No
.
1,
Mar
c
h
2
016
: 2
0
2
–
21
6
20
3
gene
rat
o
r ra
n
g
e
d wi
t
h
i
n
t
h
e
rat
e
d
val
u
es.
I
n
[
1
]
,
Y
u
si
va
r
et al.
h
a
d
sim
u
la
ted
th
e win
d
turb
in
e ind
u
c
tion
gene
rat
o
r
usi
n
g fl
ux
wea
k
eni
n
g
.
T
h
e si
m
u
l
a
t
i
on s
h
o
w
e
d
the u
n
stab
le cond
itio
n
at th
e over-s
peed operation.
To
d
e
fin
e
t
h
e
v
a
riab
le wh
ich cau
sed
th
e
u
n
stab
le con
d
ition
,
th
e
Yu
siv
a
r
et al.'s research
shou
ld
b
e
con
tin
u
e
d
by
anal
y
z
i
ng a
bo
u
nda
ry
bet
w
een t
h
e st
abl
e
and t
h
e
un
st
ab
le con
d
itions. Th
e
bo
und
ar
y b
e
tw
een
th
e stab
le
and the
unstabl
e
conditions
wa
s called the
critical condition.
Th
e m
a
in
co
ntrib
u
tion
o
f
this p
a
p
e
r
was
to
ex
am
in
e tho
r
ou
gh
ly th
e
critical co
nd
itio
n of th
e
sens
orl
e
ss i
n
d
u
ct
i
on
ge
nerat
o
r
usi
n
g fl
ux
weake
n
i
n
g i
n
wi
n
d
t
u
r
b
i
n
e
appl
i
cat
i
o
n an
d de
fi
ne t
h
e
v
a
ri
abl
e
wh
ich
cau
s
ed
t
h
e
u
n
stab
le con
d
ition
.
2.
R
E
SEARC
H M
ETHOD
This researc
h
consisted of four stages, i.e
.
(1
) m
a
th
e
m
a
tical m
o
d
e
ll
in
g
,
(2
)
im
p
l
e
m
en
tatio
n
,
(3)
si
m
u
latio
n
,
and
(4) an
alysis.
2
.
1
.
The
Ma
thematica
l
Modelling
The m
odel
of
t
h
e sy
st
em
consi
s
t
e
d
of
t
h
re
e pa
rt
s,
i.e. (1) the m
echanical of
w
i
nd
tu
rb
in
e, (2
) the
in
du
ctio
n gen
e
rato
r, an
d (3
) th
e co
n
t
ro
ller (see Figu
re 1).
First, th
e m
ech
an
ical o
f
wind
tu
rb
in
e was u
s
ed
fo
r calcu
latin
g
a lo
ad
torq
ue th
at referred
to
th
e ro
tor
spee
d of
i
n
d
u
ct
i
on ge
nerat
o
r a
n
d
t
h
e wi
n
d
sp
eed.
Seco
nd
, t
h
e m
odel
o
f
i
n
d
u
ct
i
o
n
g
e
ne
rat
o
r
w
a
s use
d
t
o
p
r
o
duce
t
h
e
st
at
or
cu
rre
nt
s t
h
at
r
e
fer
t
o
st
at
o
r
v
o
ltag
e
s an
d a lo
ad torqu
e
. Th
e stator
vo
ltag
e
s
were th
e
ou
tpu
t
of t
h
e con
t
ro
ller,
wh
ile th
e lo
ad
t
o
rq
ue was
receive
d from
the m
echanical
of wi
nd turbine.
Thi
r
d,
t
h
e c
o
nt
rol
l
e
r
co
nsi
s
t
e
d
of
(
1
)
a t
h
re
e p
h
ases
t
o
t
w
o
pha
ses t
r
a
n
s
f
o
r
m
and
vi
ce
versa
,
(2
)
a
pul
se
wi
dt
h m
o
d
u
l
a
t
i
on
(P
W
M
) gene
rat
o
r,
(3
) a rot
o
r fl
ux
orientatio
n co
ntr
o
l (RFOC
)
,
(4
) a flu
x
wea
k
eni
n
g
,
(5
) a s
p
ee
d c
ont
roller,
an
d
also (
6
) an observe
r to estimate the rot
o
r
spee
d. The b
l
ock di
ag
ram
of
t
h
e
cont
rol
l
e
r
was
sho
w
n i
n
Fi
gu
r
e
2.
Fi
gu
re 1.
The
part
s o
f
sy
st
em
Th
is research
u
s
ed
th
e wind tu
rb
i
n
e typ
e
o
f
h
o
rizo
ntal axis. Its bla
d
es
were c
o
nnecte
d
to the low
spee
d shaft
t
h
a
t
wasn’t
di
rect
l
y
connect
e
d
t
o
t
h
e i
nduct
i
o
n gene
rat
o
r. T
h
e
r
e was a gear
b
ox
bet
w
ee
n t
h
e l
o
w
spee
d s
h
aft
a
n
d t
h
e
i
n
duct
i
on
ge
nerat
o
r.
T
h
e
gea
r
b
o
x
wa
s
used t
o
inc
r
ease
the s
p
eed
of the low s
p
eed shaft.
The m
odel
of
t
h
e m
echani
cal
wi
n
d
t
u
r
b
i
n
e
use
d
si
x e
q
ua
t
i
ons
(1
-
6
).
T
h
e eq
uat
i
o
ns
w
a
s use
d
t
o
calculate the capacity factor
o
f
th
e
wind
turb
i
n
e (C
p
), th
e tu
rb
i
n
e sp
eed (
wt
), th
e tip
sp
eed
ratio
(
),
th
e
tu
rb
in
e po
wer (P
wt
), and t
h
e l
o
ad torque
(T
L
)
[2
],
[
3
].
,
0
.
22
116
0
.
4
5
e
x
p
12.5
(1
)
(2
)
(3
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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S
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208
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9
4
Critica
l
Co
nd
itio
n
o
f
S
e
nso
r
less
In
du
ction
Gen
e
ra
to
r u
s
i
n
g Flu
x
Wea
k
en
ing
…
(
N
an
d
a
Av
i
ant
o
Wi
caks
o
no)
20
4
Flu
x
Weak
en
in
g
RF
O
C
Sp
ee
d
Con
t
ro
lle
r
3
2
PWM
Ge
ner
a
t
o
r
3
2
O
b
s
e
r
ver
IG
G
e
a
r
box
B
l
ad
es
Fi
gu
re
2.
B
l
oc
k
di
ag
ram
of t
h
e co
nt
r
o
l
l
e
r
1
1
0.08
0.035
1
(4
)
0
.
5
(5
)
(6
)
The m
odel
of t
h
e i
n
d
u
ct
i
o
n g
e
nerat
o
r u
s
ed s
e
ven
di
fferen
tial eq
u
a
tion
s
, i.e.
th
e d
i
fferen
tial eq
u
a
tion
of the
stator c
u
rrent in
d-a
x
i
s
(i
sd
) (
7
), t
h
e
diffe
re
ntial equation
o
f
the s
t
ator cu
rre
nt in q
-
axis
(i
sq
)
(8
),
th
e
d
i
fferen
tial eq
u
a
tio
n
o
f
th
e ro
tor cu
rren
t in
d
-
ax
is (i
rd
) (
9
)
,
t
h
e di
ffe
rent
i
a
l
equat
i
on
of
t
h
e rot
o
r cu
rre
nt
i
n
q-
axis (i
rq
) (10
)
,
th
e d
i
fferen
tial eq
u
a
tion
o
f
t
h
e angu
lar speed
of th
e stator vo
ltag
e
(
e
, the
differential
eq
u
a
tion
of th
e ro
to
r sp
eed (
r
) (12
)
, a
n
d t
h
e di
f
f
ere
n
t
i
a
l
equat
i
o
n
o
f
t
h
e
angl
e
o
f
r
o
t
o
r
(
1
3
)
[4]
,
[
5
]
.
1
1
1
1
(7
)
1
1
1
1
(8
)
(9
)
(1
0)
(1
1)
1
.
(1
2)
(1
3)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
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-86
94
I
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PED
S
Vo
l. 7,
No
.
1,
Mar
c
h
2
016
: 2
0
2
–
21
6
20
5
Before using the differe
n
tial
equations
, the stator
voltages
received from
the PW
M
generator
were
con
v
e
r
t
e
d by
t
h
e t
h
ree
phase
s (abc
) t
o
t
w
o
pha
ses (d
q
-
axi
s
) t
r
ans
f
o
r
m
a
t
i
on
. A
nd aft
e
r usi
n
g t
h
e di
f
f
e
r
ent
i
a
l
equat
i
o
ns
, t
h
e
rot
o
r c
u
rre
nt
s
pr
o
duce
d
wer
e
co
nve
rt
ed
b
y
t
h
e t
w
o
p
h
a
s
es (
d
q
-
a
x
i
s
) t
o
t
h
ree
pha
ses
(abc
)
t
r
ans
f
o
r
m
a
ti
on.
To
conv
ert th
e stato
r
vo
ltag
e
fro
m
th
e th
ree p
h
a
ses
(a
bc-a
x
i
s) t
o
t
h
e t
w
o
pha
ses (
d
q
-
a
x
i
s
) was
use
d
th
e Clark
e
transform
(1
4
)
and th
en
th
e Pa
rk trans
f
orm
(15). Viceversa, to
conve
r
t the stator curre
nt from the
t
w
o
phase
s (
d
q-a
x
i
s
) t
o
t
h
e
t
h
ree
phase
s (
a
bc-a
xi
s)
was
use
d
t
h
e Pa
rk
i
nve
rse t
r
a
n
sf
o
r
m
(16) a
nd t
h
en t
h
e
C
l
arke i
n
verse
t
r
ans
f
o
r
m
(17)
.
The co
n
v
ert
i
n
g v
o
l
t
a
ges
or
c
u
rrents
from
th
e three phases
(abc
-axis
)
to the t
w
o
pha
ses
(d
q-a
x
i
s
) a
n
d
vi
ce
vers
a we
re al
so
u
s
e
d
i
n
t
h
e c
o
nt
r
o
l
l
e
r [
6
]
,
[
7
]
.
2
3
1
0.5
0.5
00
.
5
√
3
0.5
√
3
(1
4)
co
s
sin
s
i
n
cos
(1
5)
cos
s
i
n
sin
cos
(1
6)
2
3
10
0.5
0.5
√
3
0.5
0.5
√
3
(1
7)
The P
W
M
ge
nerat
o
r
was c
o
m
p
ari
ng t
h
e
abs
o
l
u
t
e
val
u
e
of t
h
e st
at
or
vol
t
a
ge
ref
e
re
nces wi
t
h
t
h
e
trian
g
l
e carrier. If th
e ab
so
lu
t
e
v
a
lu
e was h
i
g
h
e
r th
an
th
e trian
g
l
e carrier, th
e switch
was
o
n
. Vice v
e
rsa, if th
e
ab
so
lu
te
v
a
lu
e
was l
o
wer th
an th
e tr
ian
g
l
e carrier, th
e switch
was
o
f
f.
Wh
ile th
e swi
t
ch
was on
and
th
e stato
r
vo
ltag
e
re
feren
c
e was
po
sitiv
e, th
e stator
voltag
e
of th
e
i
n
d
u
ct
i
on
ge
ne
rat
o
r
was t
h
e
sam
e
as t
h
e PWM
am
pl
i
t
ude of +
V
dc/
2
.
Vi
ce ve
rsa, t
h
e st
at
or v
o
l
t
a
g
e
of t
h
e
in
du
ctio
n g
e
n
e
rato
r was -Vd
c
/2
wh
ile th
e
switch
was
on
and the
stator
voltage
re
fere
nce was
ne
gative (see
Fi
gu
re 3)
[
8
]
.
Fi
gu
re
3.
Ge
ne
rat
i
n
g
P
W
M
si
gnal
The stato
r
volt
a
ge re
fere
nces
(v
sd
an
d v
sq
) t
h
at were
received
by the
PWM gene
rator c
ontaine
d t
h
e
linier re
fere
nc
es (
u
sdlin
and
u
sqlin
) a
n
d
the
n
onlinie
r
dec
o
u
p
lin
g
refe
re
nces
(u
sddec
and
u
sqdec
). Th
e
lin
ier
refe
rences
was
adjuste
d
from
two PI controllers. T
h
e adju
st
in
g
PI con
t
ro
llers referred
t
o
th
e error b
e
tween
th
e
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9
4
Critica
l
Co
nd
itio
n
o
f
S
e
nso
r
less
In
du
ction
Gen
e
ra
to
r u
s
i
n
g Flu
x
Wea
k
en
ing
…
(
N
an
d
a
Av
i
ant
o
Wi
caks
o
no)
20
6
rot
o
r c
u
rrent
refere
nces a
nd
the actual val
u
es.
On th
e ot
her ha
nd
,
t
h
e no
nl
i
n
i
e
r dec
o
upl
i
n
g refe
ren
ces
wer
e
cal
cul
a
t
e
d
by
e
quat
i
o
ns
(
2
2
)
a
n
d
(
2
3)
[
8
]
.
(1
8)
(1
9)
(2
0)
(2
1)
1
(2
2)
1
(2
3)
The equations
(20) and (21) show
ed
th
at t
h
e calcu
latin
g lin
ier refe
re
nces used the s
t
ator curre
n
t
refe
rences (i
sdref
a
nd i
sqref
)
fr
o
m
t
h
e fl
ux
wea
k
eni
n
g
.
T
h
e fl
ux
wea
k
eni
ng
l
i
m
i
t
e
d t
h
e st
at
or c
u
r
r
e
n
t
refe
rence
s
of the inducti
on
gene
rator by usin
g equa
tions (24-28). The flux we
ak
eni
ng received the stator current
refe
rence in
q-
axis (i
sqref
)
fr
o
m
the speed co
ntr
o
ller of
PI c
ont
roller.
Th
e
ad
ju
sting
PI con
t
ro
ller referred
to
the
err
o
r
bet
w
een
the rot
o
r spee
d refe
rence
(
rref
) a
n
d the act
ual rotor s
p
eed (
r
)
(
s
ee equ
a
tion (2
9)
) [9
],
[10
]
.
(2
4)
(2
5)
(2
6)
(2
7)
(2
8)
(2
9)
The eq
uat
i
o
ns (2
2) a
nd (
2
3) s
h
o
w
e
d
t
h
at
t
h
e cal
cul
a
t
i
ng no
nl
i
n
i
e
r dec
o
upl
i
ng re
fere
nces
neede
d
t
h
e
stator s
p
eed
(
e
) an
d
th
e ro
t
o
r m
a
g
n
e
tizin
g cu
rren
t
(i
mr
) form
RFOC. Besides the stat
or s
p
ee
d and t
h
e rotor
mag
n
e
tizin
g
cu
rren
t, R
F
OC
also
calcu
lated th
e
el
ect
ri
c t
o
rq
ue
of t
h
e i
n
d
u
ct
i
on
ge
nerat
o
r
(T
e
)
and
th
e s
t
a
t
or
angl
e (
e
[8
].
1
(3
0)
1
(3
1)
(3
2)
1
1
(3
3)
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.
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Mar
c
h
2
016
: 2
0
2
–
21
6
20
7
The equation
(29) s
h
owe
d
t
h
at the spee
d
cont
roller
nee
d
ed t
h
e actual
rotor s
p
eed
(
) fr
om
the
in
du
ctio
n m
o
to
r. Bu
t t
h
e actual ro
tor sp
eed
co
u
l
d
n
’t b
e
ob
t
a
in
ed
form
th
e sen
s
o
r
less i
n
ductio
n
g
e
n
e
rator. Th
e
feed b
a
ck
si
g
n
als form
th
e the sen
s
orless ind
u
c
tio
n
ge
nera
t
o
r
were
m
a
i
n
ly
t
h
e st
at
or
c
u
r
r
ent
s
.
To
sub
titu
te the actu
a
l
ro
t
o
r sp
eed (
)
,
th
e c
o
n
t
ro
lle
r
w
a
s
equ
i
pp
e
d
w
ith
an ob
s
e
rv
er
.
T
h
e
o
b
s
e
r
v
e
r
was use
d
t
o
es
t
i
m
a
t
e
t
h
e rot
o
r spee
d by
usi
ng e
quat
i
o
n (
3
4)
. The est
i
m
at
ed rot
o
r s
p
ee
d was cal
cul
a
t
e
d by
using t
h
e estimated rot
o
r
fl
ux (
rdest
and
rqest
) an
d the
er
ror
(e
isd
and e
is
q
) bet
w
een the estim
a
ted stator
currents
and t
h
e actual
values
[11].
(34)
(35)
In
t
h
is research
, t
h
e estim
at
i
o
n of th
e ro
tor m
a
g
n
e
tizin
g flux
es and
the stato
r
currents u
s
ed
the
Lue
nbe
rge
r
ob
serve
r
.
T
h
e L
u
enbe
rger
ob
serv
er
was a
b
le to estim
a
t
e the state variable
s (
X
) a
n
d
t
h
e
out
pu
t
vari
a
b
l
e
s (
Y
) i
n
the state s
p
a
ce (36) and
(37)
[11]. T
h
e e
s
tim
a
ted state
varia
b
les (
X
est
) and the estimated
out
put
va
ri
abl
e
s
(
Y
est
)
co
ul
d
b
e
cal
cul
a
t
e
d
by
usi
n
g
eq
uat
i
o
n
s
(
3
7
)
a
n
d (
3
8)
[
12]
,
[
13]
.
(36)
(37)
(38)
(39)
B
y
usi
ng t
h
e equat
i
o
ns (
3
8)
and (
3
9
)
, t
h
e
Luen
ber
g
e
r
o
b
ser
v
e
r
i
n
t
h
e equat
i
o
n (
4
1)
was use
d
t
o
esti
m
a
te
th
e state v
a
riab
les
o
f
th
e i
n
du
ctio
n
g
e
n
e
rator (i
sd
, i
sq
,
rd
, and
rq
) in
th
e eq
u
a
tion
(40
)
. Th
e
estim
a
ted state
va
riables
were
i
sdest
, i
sqest
,
rdest
, and
rqest
[11
]
,
[12
]
,
[13
]
.
1
1
0
0
1
1
1
0
0
1
0
0
0
0
(4
0)
10
01
00
00
00
00
10
01
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I
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PED
S
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6
9
4
Critica
l
Co
nd
itio
n
o
f
S
e
nso
r
less
In
du
ction
Gen
e
ra
to
r u
s
i
n
g Flu
x
Wea
k
en
ing
…
(
N
an
d
a
Av
i
ant
o
Wi
caks
o
no)
20
8
1
1
0
0
1
1
1
0
0
1
0
0
0
0
2
(4
1)
10
01
00
00
00
00
10
01
Th
e equ
a
tio
n (4
1)
u
s
ed
th
e g
a
in
s
g
1
, g
2
, g
3
, a
n
d g
4
th
at
were written as
b
e
low:
1
(4
2)
1
(4
3)
1
1
(4
4)
1
1
(4
5)
2.2.
Implementation
Th
e m
a
th
e
m
at
ics
m
o
d
e
l of
wind
turb
i
n
e
syste
m
was written
in
th
e
C MEX S-Fun
c
tio
ns and
si
m
u
lated
b
y
usin
g MATLAB/SIMULINK. Th
e realizatio
n
of
wi
nd
t
u
rb
i
n
e m
odel
wa
s
sh
o
w
n
i
n
Fi
g
u
re
4
.
Th
e ind
u
c
ti
o
n
g
e
n
e
rator was
a sq
u
i
rrel cag
e in
du
ctio
n
m
ach
in
e
with
th
e rated
po
wer of
1
hp
power, the rated
spee
d of
14
0 r
a
d/
s, an
d t
h
e
m
a
xim
u
m
t
o
rque o
f
5 Nm
. The pa
ram
e
t
e
rs
of t
h
e
wi
n
d
t
u
r
b
i
n
e an
d t
h
e i
n
duct
i
o
n
gene
rat
o
r
were
l
i
s
t
e
d i
n
T
a
bl
e
1.
Fi
gu
re
4.
The
r
eal
i
zat
i
on o
f
w
i
nd t
u
r
b
i
n
e
m
odel
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I
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No
.
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Mar
c
h
2
016
: 2
0
2
–
21
6
20
9
Tabl
e
1.
Wi
n
d
t
u
r
b
i
n
e a
n
d i
n
d
u
ct
i
o
n
ge
ne
rat
o
r
pa
ram
e
t
e
rs
Sy
m
bols
Descr
i
ption
Value
Units
R Blades
Radius
0.
95
m
K
G
Gearbox ratio
6.
65
-
Np
Nu
m
b
e
r
of
Pole Pa
irs
2
-
L
s
Stator
I
nductance 234.
9
m
H
L
r
Rotor
I
nductance
234.
9
m
H
L
m
M
u
tual
I
nductance
227.
9
m
H
R
s
Stator
Resistance
2.75
R
r
Rotor
Resistance
2.9
2.
3.
T
e
st
Sce
n
ari
o
The t
e
st
i
n
g
of
t
h
e sy
st
em
was d
o
n
e
by
gi
vi
n
g
a
co
nst
a
n
t
wi
n
d
s
p
ee
d
and
a c
o
nst
a
nt
r
o
t
o
r
s
p
e
e
d
refe
rence
.
The
r
e we
re two s
e
ries of
win
d
spee
d that
were tested. The
first series, the wind spee
ds
were
bet
w
ee
n
3 m
/
s an
d
20 m
/
s w
i
t
h
i
n
t
e
r
v
al
o
f
1 m
/
s. The sec
o
n
d
se
ri
es, t
h
e
wi
n
d
s
p
ee
ds
were
bet
w
een
25
m
/
s
and 4
5
m
/
s
wi
th
i
n
t
e
r
v
al
of 5 m
/
s.
For
eac
h wi
n
d
s
p
eed
, the increm
ental of
rot
o
r
spee
d refe
rence was applie
d
u
n
til th
e system resp
on
se
b
e
ca
m
e
u
n
s
tab
l
e. Th
e
h
i
gh
est
of th
e ro
tor sp
eed
referen
ce t
h
at cau
sed
th
e stab
le
resp
o
n
se was n
a
m
e
d
the
c
r
itical
rot
o
r spee
d r
e
fere
nce.
3.
R
E
SU
LTS AN
D ANA
LY
SIS
3
.
1
.
The Sy
stem Sta
b
ility
The si
m
u
l
a
ti
on sh
owe
d
t
h
at
t
h
ere were t
w
o o
p
e
r
at
i
ng
con
d
i
t
i
on r
e
sp
ons
es, i
.
e. (a)
t
h
e unst
a
bl
e
co
nd
itio
n respo
n
s
e and
(b) t
h
e stab
le con
d
itio
n
respo
n
s
e. Th
e ex
am
p
l
e o
f
t
h
e
u
n
s
tab
l
e resp
on
se
was
at th
e
rot
o
r s
p
ee
d
ref
e
rence
o
f
24
0
r
a
d/
s a
n
d
wi
nd
spee
d c
h
an
ge
d
fr
om
6 m
/
s t
o
12
m
/
s (see Fi
gu
re
5)
.
On
t
h
e
ot
he
r
han
d
,
t
h
e e
x
a
m
pl
e of st
abl
e
resp
o
n
se
was a
t
t
h
e r
o
t
o
r
spe
e
d re
fere
nce
o
f
2
70
ra
d/
s an
d
wi
n
d
s
p
ee
d c
h
an
ge
d
fro
m
2
2
m/s to
3
0
m
/
s (see Fig
u
re
6
)
. B
o
th
co
nd
itio
n
showed
th
at th
e syst
e
m
stab
ilit
y d
i
d
n
’t on
ly d
e
p
e
n
d
on
either the
rotor spee
d
refe
renc
e or the
wi
nd s
p
eed.
(a)
Win
d
spee
d
cha
nge
d
fr
om
6 m
/
s to 1
2
m
/
s
(b
) R
o
tor
s
p
ee
d: the
refe
re
nc
e (
g
ree
n
)
,
the actual
(re
d), and t
h
e estimated (blue
)
Fi
gu
re
5.
The
unst
a
bl
e re
sp
o
n
se at
t
h
e r
o
t
o
r
spee
d
refe
renc
e o
f
24
0
rad/
s
and
t
h
e wi
nd
s
p
eed
cha
nge
d
fr
om
6 m
/
s t
o
1
2
m
/
s
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I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
Critica
l
Co
nd
itio
n
o
f
S
e
nso
r
less
In
du
ction
Gen
e
ra
to
r u
s
i
n
g Flu
x
Wea
k
en
ing
…
(
N
an
d
a
Av
i
ant
o
Wi
caks
o
no)
21
0
(a)
Win
d
spee
d
cha
nge
d
fr
om
22
m
/
s to 30
m
/
s
(b
) R
o
tor
s
p
ee
d: the
refe
re
nc
e (
g
ree
n
)
,
the actual
(re
d), and t
h
e estimated (blue
)
Figu
re
6.
The
s
t
able res
p
o
n
se
at the r
o
to
r s
p
e
e
d
refe
rence
o
f
2
7
0
ra
d/s a
n
d
t
h
e wi
nd
s
p
eed
cha
nge
d
fr
om
22
m
/
s t
o
30
m
/
s
3.
2. An
al
ysi
s
Th
e test resu
lted
th
at th
ere were two
op
erati
n
g
ran
g
e
s
.
Firs
t, the low
spee
d ra
nge of wi
nd wa
s at the
wi
n
d
s
p
ee
d l
o
wer
t
h
a
n
8 m
/
s. Sec
o
nd
, t
h
e
m
e
di
um
and
h
i
gh
spee
d
ra
ng
es o
f
wi
n
d
we
re at
t
h
e
wi
n
d
spee
d
eq
u
a
l
or
h
i
gh
er th
an 8 m
/
s.
In the low
spee
d ra
nge of wi
nd, the
r
e
wasn'
t
unsta
ble condition
whate
v
er a
rot
o
r s
p
ee
d re
ference
was
set
.
The hi
ghe
st
gene
rat
e
d p
o
we
r was
reac
hed
by
t
h
e hi
g
h
est
capaci
t
y
fact
or o
f
4
3
.
8
%
(see Fi
g
u
re
7)
. Th
e
h
i
gh
est cap
acity facto
r
was n
a
med
th
e
o
p
timu
m
o
p
e
ratio
n
co
nd
itio
n. Th
e
h
i
gh
est cap
acity facto
r
was reach
ed
b
y
th
e
o
p
tim
u
m
tip
sp
eed
ratio
o
f
6.35
(see Figu
re
8
)
. At th
e op
ti
m
u
m
co
nd
itio
n, th
e
g
e
n
e
rated
po
wer was
still lo
wer t
h
an th
e
p
o
wer capacity o
f
th
e indu
ctio
n g
e
n
e
rato
r.
Fig
u
re
7
.
Th
e
o
p
tim
al co
n
d
itio
n in
t
h
e low
sp
eed rang
e
o
f
wind
In
Fi
g
u
re
8, t
h
e opt
i
m
u
m
op
erat
i
on c
o
ndi
t
i
on
was s
h
ow
n
by
p
o
i
n
t
M
.
At
t
h
e c
onst
a
n
t
wi
nd
spee
d,
the increa
sing
rot
o
r s
p
ee
d re
ference
ca
use
d
the increa
sing
tip spee
d ra
t
i
o
, t
h
e
n
i
t
wo
ul
d ca
use de
crea
si
ng
capaci
t
y
fact
o
r
(see a
r
r
o
w a i
n
Fi
gu
re
8)
.
O
n
t
h
e
ot
her
ha
nd
, t
h
e
dec
r
ea
si
ng
r
o
t
o
r
s
p
ee
d re
fe
rence
ca
use
d
a
d
ecreasi
n
g tip
sp
eed
ratio, and
th
en
it
woul
d cause a
decrea
sing ca
pacity fact
or
t
o
o (see arr
o
w b
i
n
Fi
g
u
re 8)
.
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I
S
SN
:
2
088
-86
94
I
J
PED
S
Vo
l. 7,
No
.
1,
Mar
c
h
2
016
: 2
0
2
–
21
6
21
1
Both dec
r
easing capacity factor m
a
de the lower ge
ne
rate
d
po
we
r. The
r
ef
ore
,
the rot
o
r s
p
eed r
e
fe
rence
that
was hi
ghe
r o
r
l
o
we
r t
h
a
n
t
h
e
opt
i
m
u
m
reference w
oul
d de
crease t
h
e ge
ne
rat
e
d p
o
w
er
, so t
h
e ge
nerat
e
d
po
we
r
neve
r e
x
cee
de
d t
h
e
p
o
w
er
ca
paci
t
y
of
t
h
e i
n
duct
i
o
n
ge
nera
t
o
r.
In t
h
e m
e
di
u
m
and
hi
g
h
s
p
e
e
d ra
n
g
e o
f
w
i
nd, t
h
ere
was
a critical rotor spee
d
refe
re
nce.
A rotor
sp
eed
referen
c
e th
at was h
i
g
h
e
r th
an
th
e
critical reference wou
l
d
cau
s
e u
n
s
tab
l
e con
d
ition
.
Th
e critica
l
co
nd
itio
n o
c
curred
wh
ile th
e ro
t
o
r
sp
ee
d
referen
ce was between
23
0
and
4
0
6
rad
/
s, the lo
ad
t
o
rqu
e
o
f
t
h
e
i
n
d
u
ct
i
o
n
g
e
ne
rat
o
r
wa
s bet
w
een -2
.5
an
d -
5
.
2
Nm
.,
t
h
e
g
e
nerat
e
d po
we
r
w
a
s bet
w
ee
n 72
5
a
n
d 88
0 W.,
a
n
d
t
h
e capaci
t
y
fa
ct
or wa
s bet
w
e
e
n 0
.
1
9
a
nd
42
.4
4 % (see
Fi
gu
re
9
)
. Th
ese sh
owed
th
at th
e lo
ad
torqu
e
and
th
e
capaci
t
y
fact
or
of t
h
e cri
t
i
cal
con
d
i
t
i
on
were
depe
ndi
ng
on
t
h
e rot
o
r s
p
eed
refere
nce.
On
t
h
e ot
he
r ha
nd
,
t
h
e
g
e
n
e
r
a
ted
po
wer
of
th
e cr
itical co
nd
itio
n
was a r
e
lativ
e co
n
s
tan
t
aro
und th
e r
a
ted
p
o
wer
of
1
h
p
o
r
7
46
W
(see Figu
re
9
c
). It sh
owed
th
at un
stab
le co
nd
itio
n
was
cau
sed
b
y
th
e in
du
ctio
n
g
e
nerato
r t
h
at receiv
ed
a
powe
r
higher t
h
an its capacit
y
, so its
rotor s
p
eed
couldn'
t
be m
a
intained at
refe
re
nce
value.
In Fi
gu
re
8, t
h
e cri
t
i
cal
con
d
i
t
i
on
was s
h
ow
n
by
po
in
ts in
C rang
e.
Th
e in
creasing ro
tor sp
eed
refe
rence ca
us
ed inc
r
easing t
i
p spee
d rati
o
and t
h
en it
would ca
use inc
r
e
a
sing ca
pacity factor
(see a
r
row d i
n
Fi
gu
re
8)
. T
h
e
i
n
creasi
n
g
ca
paci
t
y
fact
or
c
a
use
d
t
h
e i
n
cr
easi
ng
ge
nerat
e
d
po
wer
t
h
e
n
i
t
m
a
de t
h
e g
e
nerat
e
d
po
we
r w
a
s
hi
g
h
er
t
h
a
n
t
h
e
rat
i
ng
p
o
we
r m
o
t
o
r
,
s
o
t
h
e
sy
st
em
woul
d
be
u
n
s
t
a
bl
e.
Fi
gu
re
8.
R
e
l
a
t
i
on
bet
w
ee
n c
a
p
aci
t
y
fact
o
r
a
n
d
t
i
p
s
p
ee
d rat
i
o
The si
m
u
l
a
ti
on
sho
w
n i
n
Fi
g
u
re 5 c
oul
d be
descri
be
d by
t
w
o
poi
nt
s A and B
.
P
o
i
n
t
A
rep
r
ese
n
t
e
d
the condition
at the rotor s
p
eed
refe
rence
of
240 ra
d/
s an
d th
e
w
i
nd
sp
eed 6 m
/
s. Po
in
t B
r
e
pr
esen
ted
t
h
e
condition at the rotor spee
d refere
nce of 240 ra
d/s a
nd
wi
nd s
p
ee
d 1
2
m
/
s. The st
abl
e
con
d
i
t
i
on at
p
o
i
n
t
A
ch
ang
e
d
in
t
o
t
h
e
u
n
stab
le con
d
ition
at
po
in
t
B (see arrow
AB
in Figu
re 9a).
On t
h
e
ot
he
r h
a
nd
, t
h
e si
m
u
lat
i
on sh
o
w
n i
n
Fi
gu
re 6 co
ul
d be de
scri
be
d
by
t
w
o p
o
i
n
t
s
C
and D.
Po
in
t
C represen
ted th
e con
d
itio
n
at the ro
t
o
r sp
eed refe
re
nce of
27
0 ra
d
/
s
an
d
t
h
e wi
n
d
s
p
ee
d 2
2
m
/
s.
P
o
i
n
t
D
represen
ted
th
e cond
itio
n at th
e ro
tor sp
eed
referen
c
e
of 2
7
0
rad
/
s
an
d wind
sp
eed
3
0
m/s.
Bo
th
con
d
itio
n
s
at
poi
nt
C
a
n
d
D
were
st
abl
e
(see arrow C
D
i
n
Fi
gure
9a).
Figure 9 als
o
showe
d
the m
i
nim
u
m
critical roto
r s
p
eed
refere
nce that
was 230 ra
d/s
at the wind
sp
eed
of 12
m
/
s. A so
lu
tion
was sug
g
e
sted
that th
e stab
le con
d
ition
wou
l
d
b
e
m
a
d
e
b
y
settin
g
th
e ro
tor sp
eed
refe
rence at the
m
i
nim
u
m
critical refe
rence
of
23
0
rad/
s.
If
t
h
e rot
o
r s
p
ee
d re
fere
nce w
a
s set
const
a
nt
l
y
at
23
0
r
a
d
/
s, th
e g
e
n
e
r
a
ted
po
w
e
r
n
e
v
e
r ex
ceed
e
d
t
h
e
p
o
w
e
r
cap
a
city o
f
t
h
e inductio
n
g
e
n
e
r
a
tor (
s
ee Figu
r
e
1
0
)
.
Th
e
si
m
u
latio
n
sh
owed
th
at th
e syste
m
re
spons
e
was stable at the constant
rot
o
r spee
d re
f
e
rence o
f
2
3
0
rad/s
al
t
hou
g
h
wi
n
d
spee
d
c
h
an
ge
d i
n
ra
n
g
e 6-
45
m
/
s
(see
Fi
g
u
r
e
1
1
)
.
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