Inter
national
J
our
nal
of
Electrical
and
Computer
Engineering
(IJECE)
V
ol.
8,
No.
2,
April
2018,
pp.
711
–
722
ISSN:
2088-8708
711
I
ns
t
it
u
t
e
o
f
A
d
v
a
nce
d
Eng
ine
e
r
i
ng
a
nd
S
cie
nce
w
w
w
.
i
a
e
s
j
o
u
r
n
a
l
.
c
o
m
Maximum
P
o
wer
Extraction
Method
f
or
Doubly-fed
Induction
Generator
W
ind
T
urbine
Dinh
Chung
Phan
and
T
rung
Hieu
T
rinh
F
aculty
of
Electrical
Engineering,
The
Uni
v
ersity
of
Danang-Uni
v
ersity
of
Science
and
T
echnology
,
V
ietnam
Article
Inf
o
Article
history:
Recei
v
ed:
Aug
11,
2017
Re
vised:
Feb
23,
2018
Accepted:
Mar
5,
2018
K
eyw
ord:
Doubly-fed
induction
generator
L
yapuno
v-function
based
controller
maximum
a
v
ailable
po
wer
MPPT
Speed
control
W
ind
turbine
ABSTRA
CT
This
research
presents
a
ne
w
scheme
to
e
xtract
the
maximal
a
v
ailable
po
wer
from
a
wind
turbine
emplo
ying
a
doubly
fed
induction
generator
(DFIG).
This
scheme
is
de
v
eloped
from
the
wind
turbine’
s
MPPT
-curv
e.
Furthermore,
we
propose
control
la
ws
for
the
rotor
and
grid
side-con
v
erters.
The
stability
of
the
proposed
maximum
a
v
ailable
po
wer
method
and
the
control
la
ws
are
pro
v
ed
mathematically
upon
L
yapuno
v’
s
stability
criterion.
Their
ef
ficienc
y
is
tested
through
the
simulations
of
a
DFIG
wind
turbine
in
Matlab/Simulink.
Simulation
results
are
analyzed
and
compared
with
that
using
a
con
v
entional
scheme.
Thanks
to
the
suggest
ed
scheme,
the
wind
turbine
can
track
its
maximum
po
wer
point
better
and
the
electric
ener
gy
output
is
higher
comparing
with
that
using
the
con
v
entional
scheme.
Furthermore,
by
the
suggested
controllers,
the
rotor
speed
and
current
of
the
DFIG
con
v
er
ged
to
their
desired
v
alues.
In
other
w
ords,
the
wind
turbine
can
achie
v
e
stable
op-
erations
by
the
suggested
control
la
ws.
Copyright
c
2018
Institute
of
Advanced
Engineering
and
Science
.
All
rights
r
eserved.
Corresponding
A
uthor:
Dinh
Chung
Phan
F
aculty
of
Electrical
Engineering,
The
Uni
v
ersity
of
Danang-Uni
v
ersity
of
Science
and
T
echnology
54-Nguyen
Luong
bang
street,
Lien
Chieu
district,
Danang
city
,
V
ietnam
84-988983127
pdchung@dut.udn.vn
1.
INTR
ODUCTION
The
maximum
po
wer
generation
of
a
wind
turbine
has
been
interested
in
se
v
eral
decades
and
man
y
algo-
rithms
ha
v
e
been
suggested.
According
to
[1],
pre
vious
maximum
po
wer
point
tracking
(MPPT)
methods
can
be
listed
into
three
groups
including
indirect
po
wer
controller
,
direct
po
wer
controller
,
and
others.
The
indirect
po
wer
controller
which
aims
to
maximize
mechanical
po
wer
by
using
tip-speed
ratio
[2],
[3],
optimal
torque
[4],
and
po
wer
signal
feedback
MPPT
algorithm
[5]
is
simplicity
and
only
allo
ws
the
wind
turbine
to
track
its
MPPT
-curv
e
quickly
when
a
wind
speed
measurement
is
precise
and
instantaneous.
In
the
case
of
an
una
v
ailable
wind
measurement,
a
wind
turbine
using
the
indirect
po
wer
controller
f
ails
to
track
its
maximum
po
wer
point
quickly
and
accurately
[6],
[7].
The
direct
po
wer
controller
which
maximizes
electric
po
wer
by
using
the
perturbation
and
observ
ation
(P&O)
algorithm
such
as:
Hil
l
cl
imb
search
[8],
incremental
conductance
[9],
optimal-relation
based
MPPT
algorithm
[10],
[11],
and
h
ybrid
MPPT
algorithm
[12]
does
not
require
an
y
wind
turbine
kno
wledge
and
a
v
ailable
anemometer
.
Un-
fortunately
,
a
wind
turbine
using
the
direct
po
wer
controller
cannot
track
its
maximum
po
wer
because
this
controller
cannot
recognize
instantaneously
the
v
ari
ation
in
wind
speed.
Until
no
w
,
this
controller
has
been
implemented
to
adjust
the
v
oltage
and
current
of
DC
circuit
in
a
permanent
magnetic
synchronous
generator
wind
turbine.
The
last
group
is
de
v
eloped
based
on
soft
computing
techniques
lik
e
Fuzzy
[13]
and
Neural
netw
ork
[14].
A
wind
turbine
us-
ing
these
methods
only
has
a
good
performance
when
the
full
information
of
the
wind
turbine
is
a
v
ailable.
Ho
we
v
er
,
these
methods
are
comple
xity
and
lar
ge
memory
requirement.
Hence,
a
ne
w
MPPT
scheme
for
DFI
G
wind
turbines
should
be
researched.
T
o
control
the
generator
-wind
turbine,
proportional-inte
gral
(PI)
control
is
normally
implemented
because
of
its
simplicity
[7],
[15],
[16].
Ho
we
v
er
,
by
using
the
PI
control,
we
cannot
guarantee
the
wind
turbine
system
will
become
stable
operation
[17],
[18].
Recently
,
control
la
ws
based
on
sliding
mode
were
suggested
for
rotor
speed
adjustment
[19],
[20].
Ho
we
v
er
,
these
control
la
ws
require
an
a
v
ailable
wind
speed
measurement.
Hence,
we
need
J
ournal
Homepage:
http://iaescor
e
.com/journals/inde
x.php/IJECE
I
ns
t
it
u
t
e
o
f
A
d
v
a
nce
d
Eng
ine
e
r
i
ng
a
nd
S
cie
nce
w
w
w
.
i
a
e
s
j
o
u
r
n
a
l
.
c
o
m
,
DOI:
10.11591/ijece.v8i2.pp711-722
Evaluation Warning : The document was created with Spire.PDF for Python.
712
ISSN:
2088-8708
Fig.
1.
DFIG
wind
turbine
configuration
to
propose
a
ne
w
control
la
w
for
rotor
speed
in
the
DFIG-wind
turbine.
In
this
research,
we
suggest
a
ne
w
scheme
to
e
xtract
the
maximum
a
v
ailable
po
wer
from
a
wind
turbine
emplo
ying
DFIG.
This
scheme
is
de
v
eloped
from
the
feedback
po
wer
algori
thm
b
ut
in
this
research,
we
do
not
require
an
anemometer
.
Ne
w
control
la
ws
which
are
de
v
eloped
upon
L
yapuno
v
function
for
rotor
speed,
current,
and
v
oltage
are
proposed.
This
scheme
is
v
alidated
through
numerical
simulations
of
a
wind
turbine
emplo
ying
DFIG.
From
simulation
results,
we
will
analyze
and
compare
with
the
simulation
results
of
a
wind
turbine
using
an
old
MPPT
scheme.
2.
DFIG
WIND
TURBINE
A
DFIG-wind
turbine
is
described
in
pre
vious
publications,
as
sho
wn
in
Fig.
1.
Generally
,
it
consists
of
a
wind
turbine,
shaft-gearbox,
doubly-fed
induction
generator
(DFIG),
and
back-to-back
con
v
erter
.
2.1.
W
ind
turbine
When
the
wind
turbine
is
rotating
at
a
s
peed
of
!
r
and
wind
speed
at
the
wind
turbine
is
V
w
,
the
mechanical
po
wer
of
the
turbine
is
calculated
through
blade
length
R
,
air
density
,
and
po
wer
coef
ficient
C
p
(
;
)
[21]
P
m
(
t
)
,
1
2
R
2
C
p
(
;
)
V
3
w
(
t
)
:
(1)
The
wind
turbine’
s
po
wer
coef
ficient
C
p
(
;
)
depends
on
both
pitch
angle
and
tip
speed
ratio
[16]
(
t
)
=
R
!
r
(
t
)
V
w
(
t
)
:
(2)
At
a
constant
,
when
increases
C
p
(
)
will
be
decreased.
In
contrary
,
at
a
constant
,
C
P
(
)
reaches
to
a
maximum
v
alue
C
p
(
opt
)
at
=
opt
.
2.2.
DFIG
The
DFIG’
s
main
objecti
v
e
is
to
c
o
n
v
ert
the
mechanical
po
wer
P
m
on
the
wind
turbine
shaft
to
el
ectricity
po
wer
P
e
.
Relationship
between
P
e
and
P
m
is
described
through
the
DFIG-wind
turbine’
s
inertia
J
J
!
r
(
t
)
d
d
t
!
r
(
t
)
=
P
m
(
t
)
P
e
(
t
)
:
(3)
Generally
,
the
DFIG
is
an
induction
generator
so
its
rotor
slip
is
defined
as
s
(
t
)
,
1
N
p
n
!
r
(
t
)
!
s
;
(4)
where
p
n
is
the
number
of
pole
pairs;
N
is
the
gearbox
ratio;
!
s
is
the
rotational
speed
of
stator
flux.
In
dq
frame,
the
stator
v
oltage
v
s
,
v
sd
v
sq
>
and
rotor
v
oltage
v
r
,
v
r
d
v
r
q
>
are
computed
from
the
stator
current
i
s
,
i
sd
i
sq
>
,
rotor
current
i
r
,
i
r
d
i
r
q
>
,
stator
flux
s
(
t
)
=
sd
(
t
)
sq
(
t
)
>
,
rotor
flux
r
(
t
)
=
r
d
(
t
)
r
q
(
t
)
>
,
rotor
resistance
r
r
,
rotor
inductance
L
r
,
stator
resistance
r
s
,
stator
inductance
IJECE
V
ol.
8,
No.
2,
April
2018:
711
–
722
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
713
L
s
,
magnetizing
inductance
L
m
as
[22]
8
>
>
>
>
>
<
>
>
>
>
>
:
v
s
(
t
)
=
r
s
i
s
(
t
)
+
!
s
s
(
t
)
+
d
d
t
s
(
t
)
v
r
(
t
)
=
r
r
i
r
(
t
)
+
s
(
t
)
!
s
(
r
(
t
)
+
d
d
t
r
(
t
)
;
s
(
t
)
=
L
s
i
s
(
t
)
+
L
m
i
r
(
t
)
r
(
t
)
=
L
r
i
r
(
t
)
+
L
m
i
s
(
t
)
(5)
where
,
0
1
1
0
.
Lemma
1.
If
we
ne
glect
the
stator
resistance,
r
s
=
0
,
and
choose
the
dq
frame
so
that
s
(
t
)
sd
0
>
then
we
can
write
the
state-space
equation
of
the
DFIG
(5)
as
d
d
t
i
r
(
t
)
=
A
r
(
t
)
i
r
(
t
)
+
1
v
r
(
t
)
+
d
(
t
)
;
(6)
where
,
L
r
L
2
m
L
s
;
A
r
(
t
)
,
1
r
r
I
2
!
s
s
(
t
)
;
d
(
t
)
,
L
m
L
s
s
(
t
)
0
V
s
;
I
2
=
1
0
0
1
:
(7)
Pr
oof.
Ob
viously
,
if
we
ensure
s
(
t
)
sd
0
>
then
by
using
(5),
we
ha
v
e
s
(
t
)
=
L
s
i
s
(
t
)
+
L
m
i
r
(
t
)
=
sd
0
>
;
d
d
t
s
(
t
)
=
0
,
L
s
d
d
t
i
s
(
t
)
=
L
m
d
d
t
i
r
(
t
)
:
(8)
By
using
(8)
and
r
s
=
0
in
(5),
we
ha
v
e
v
s
(
t
)
=
!
s
s
(
t
)
=
0
!
s
sd
>
=
0
V
s
>
;
(9)
where
we
used
V
s
=
k
v
s
(
t
)
k
=
j
!
s
sd
j
.
From
(8)
and
(9),
we
ha
v
e
i
s
(
t
)
=
L
m
L
s
i
r
(
t
)
+
1
L
s
!
s
V
s
0
;
d
d
t
i
s
(
t
)
=
L
m
L
s
d
d
t
i
r
(
t
)
:
(10)
From
(10)
and
(5),
we
ha
v
e
r
(
t
)
=
i
r
(
t
)
+
L
m
L
s
!
s
V
s
0
>
;
d
d
t
r
(
t
)
=
d
d
t
i
r
(
t
)
;
(11)
where
we
used
=
L
r
L
2
m
=L
s
.
T
o
use
(11)
and
(5),
we
ha
v
e
v
r
(
t
)
=
r
r
i
r
(
t
)
+
!
s
s
(
t
)
i
r
(
t
)
+
!
s
s
(
t
)
L
m
L
s
!
s
V
s
0
>
+
d
d
t
i
r
(
t
)
;
(12)
From
(12),
we
can
e
xtract
(6)
easily
.
From
[22]
and
by
using
(9)
and
(10),
we
calculate
the
stator
side
acti
v
e
po
wer
P
s
in
the
DFIG
as
P
s
(
t
)
=
v
sd
i
sd
+
v
sq
i
sq
=
L
m
L
s
V
s
i
r
q
(
t
)
:
(13)
2.3.
Grid
side
con
v
erter
F
or
the
DFIG,
the
electricity
frequenc
y
on
the
rotor
side
al
w
ays
depends
on
the
rotor
speed
!
r
.
Hence,
to
interf
ace
to
the
connected
grid,
a
back
to
back
con
v
erter
which
includes
a
rotor
side
con
v
erter(RSC),
a
DC-link,
and
a
grid
side
con
v
erter
(GSC)
must
be
installed
on
t
he
rotor
side
of
the
DFIG
[23].
Normally
,
to
reduce
harmonic
components
generated
by
the
GSC,
a
filter
which
consists
of
a
resistor
R
f
,
an
inductor
L
f
in
series
and
a
po
wer
f
actor
correction
pf
in
parallel
is
used
as
Fig.
1.
Maximum
P
ower
Extr
action
Method
for
Doubly-fed
Induction
...
(Dinh
Chung
Phan)
Evaluation Warning : The document was created with Spire.PDF for Python.
714
ISSN:
2088-8708
In
dq
frame,
the
relationship
of
v
oltage
v
g
=
v
g
d
v
g
q
>
and
current
i
g
=
i
g
d
i
g
q
>
of
the
GSC
is
written
[24]
d
d
t
i
g
(
t
)
=
A
g
i
g
(
t
)
+
d
g
+
1
L
f
v
g
(
t
)
;
(14)
where
A
g
=
L
1
f
R
f
I
2
!
s
;
d
g
=
L
1
f
V
s
0
>
:
(15)
3.
MAXIMUM
WIND
PO
WER
EXTRA
CTION
SCHEME
The
optimal
po
wer
control
re
gion
of
a
wind
turbine
is
limited
by
[25]
D
,
f
(
!
r
;
V
w
)
j
!
r
min
!
r
!
r
rated
;
V
w
min
V
w
V
w
rated
;
=
0
;
and
C
p
(
;
)
>
0
g
;
where
!
r
min
and
!
r
rated
are
the
minimum
and
rated
rotor
speed,
respecti
v
ely;
V
w
min
and
V
w
rated
stand
for
the
minimum
and
rated
wind
speed;
is
the
blade
system’
s
pitch
angle.
Hence,
when
the
wind
turbine
operates
in
D
,
the
tip-speed
ratio
and
the
rotor
speed
reference
are
limited
by
min
,
R
!
r
min
V
w
rated
(
t
)
max
,
max
f
j
C
p
(
;
)
>
0
g
;
!
r
min
!
r
ref
!
r
rated
:
From
(1),
to
e
xtract
the
maximization
of
the
mechanical
po
wer
,
we
must
adjust
!
r
to
obtain
the
maximiza-
tion
of
C
p
(
(
!
r
;
V
w
))
.
F
or
an
y
wind
turbine,
we
ha
v
e
[24]
C
p
max
,
C
p
(
opt
)
;
opt
,
arg
max
C
p
(
)
:
(16)
As
the
wind
turbine
operates
at
opt
,
the
rotor
speed
becomes
the
optimal
rotor
speed
!
r
opt
(
V
w
)
,
opt
V
w
R
(17)
and
the
mechanical
po
wer
becomes
maximal
max
!
r
P
m
(
!
r
;
V
w
)
=
1
2
R
2
C
p
max
V
3
w
=
k
opt
!
3
r
opt
(
V
w
)
;
(18)
k
opt
,
1
2
R
5
C
p
max
3
opt
:
(19)
In
this
paper
,
the
po
wer
coef
ficient
in
D
is
gi
v
en
as
C
p
(
)
=
165
:
2842
16
:
8693
e
21
+
0
:
009
;
(20)
it
has
an
unique
maximum
point
of
C
p
max
=
0
:
4
at
opt
=
6
:
7562
,
and
its
mechanical
po
wer
at
dif
ferent
wind
speed
as
sho
wn
in
Fig.
2.
Fig.
2.
MPPT
curv
e
of
wind
turbine
Remark
1.
From
(1),
(2),
and
(19),
we
ha
v
e
[24]
P
m
(
t
)
k
opt
!
3
r
(
t
)
=
(
t
)
!
r
(
t
)(
!
r
opt
(
t
)
!
r
(
t
))
;
(21)
(
!
r
;
V
w
)
=
R
4
V
w
(
t
)
2
(
t
)
C
p
max
3
opt
3
(
t
)
C
p
(
)
(
t
)
opt
>
0
:
(22)
IJECE
V
ol.
8,
No.
2,
April
2018:
711
–
722
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
715
3.1.
Con
v
entional
MPPT
-cur
v
e
scheme
The
con
v
entional
MPPT
-curv
e
method
[7]
mak
es
!
r
!
!
r
ref
=
3
q
P
e
=k
opt
:
(23)
The
problem
is
this
con
v
entional
method
cannot
track
quickly
the
maximum
po
wer
point.
Hence,
we
need
to
propose
a
ne
w
method
as
ne
xt
subsection.
3.2.
Pr
oposal
of
maximum
a
v
ailable
po
wer
scheme
The
subsection
aims
to
propose
a
ne
w
scheme
to
maximi
ze
P
m
(
!
r
;
V
w
)
or
minimize
the
error
j
!
r
opt
!
r
(
t
)
j
.
T
o
obtain
this
tar
get,
we
propose
!
r
ef
(
t
)
satisfying
k
opt
!
3
r
r
ef
(
t
)
=
P
e
(
t
)
+
k
opt
(
!
3
r
r
ef
(
t
)
!
3
r
(
t
))
+
k
d
d
t
!
2
r
(
t
)
+
q
2
(
q
y
max
k
d
d
t
!
2
r
(
t
))
(24)
where
k
,
,
y
max
are
positi
v
e
constants;
and
q
=
8
>
>
>
>
<
>
>
>
>
:
0
for
j
k
d
d
t
!
2
r
(
t
)
j
<
y
max
1
for
k
d
d
t
!
2
r
(
t
)
>
y
max
1
for
k
d
d
t
!
2
r
(
t
)
<
y
max
:
(25)
From
(23)
and
(24),
we
can
see
that
the
proposed
scheme
is
de
v
eloped
from
the
con
v
entional
MPPT
method.
4.
CONTR
OLLER
DESIGN
FOR
DFIG
4.1.
Rotor
side
contr
ol
The
purpose
of
RSC
controller
is
to
reduce
the
errors
(
i
r
d
i
r
dr
ef
)
and
(
!
r
!
r
ref
)
in
whi
ch
i
r
dr
ef
and
!
r
ref
are
the
reference
of
i
r
d
and
!
r
,
respecti
v
ely
.
From
(3),
(6),
(13),
to
mak
e
!
r
con
v
er
ge
to
!
r
ref
,
we
can
adjust
i
r
q
to
a
reference
v
alue
i
r
q
r
ef
corresponding
!
r
ref
.
In
[7],
this
task
is
carried
by
traditi
o
na
l
PI
controls.
In
this
research,
to
obtain
the
abo
v
e
tar
get,
we
design
v
r
of
the
DFIG
(5)
as
v
r
(
t
)
=
A
r
(
t
)
i
r
(
t
)
+
d
(
t
)
K
r
(
i
r
ref
(
t
)
i
r
(
t
))
d
d
t
i
r
ref
(
t
)
;
(26)
i
r
ref
(
t
)
,
i
r
d
ref
(
t
)
i
r
q
(
t
)
+
k
pr
e
!
r
ref
(
t
)
+
k
ir
R
e
!
r
ref
(
)
d
>
;
(27)
e
!
r
ref
(
t
)
,
!
3
r
ref
(
t
)
!
3
r
(
t
)
;
(28)
where,
k
pr
>
0
,
k
ir
>
0
and
matrix
K
r
>
0
.
Theor
em
1.
When
the
DFIG-wind
turbine
operates
in
D,
if
!
r
ref
and
v
r
of
the
DFIG
(5)
are
designed
as
(24)
and
(26),
respecti
v
ely
and
if
there
e
xist
positi
v
e
constants
1
,
2
,
and
b
satisfying
min
D
2
(
t
)
j
(
t
)
j
1
(
t
)
2
>
b
1
^
J
;
(29)
min
D
2
k
ir
2
(
t
)
q
0
1
K
r
K
>
r
0
1
>
>
b
1
k
pr
;
(30)
K
r
+
K
>
r
q
K
>
r
0
1
>
0
1
K
r
>
b
1
;
(31)
where
^
J
,
J
2
k
(1
q
2
))
;
(
t
)
,
^
J
d
d
t
!
r
opt
(
t
)
^
J
q
y
max
!
r
(
t
)
;
(
t
)
,
(1
)
k
opt
!
r
(
t
)
;
then
lim
t
!1
(
i
r
ref
(
t
)
i
r
(
t
))
=
0
;
lim
t
!1
!
3
r
ref
(
t
)
!
3
r
(
t
)
=
0
;
j
!
r
opt
!
r
j
s
max
!
r
j
(
t
)
j
^
J
b
:
Maximum
P
ower
Extr
action
Method
for
Doubly-fed
Induction
...
(Dinh
Chung
Phan)
Evaluation Warning : The document was created with Spire.PDF for Python.
716
ISSN:
2088-8708
Pr
oof.
Let
define
e
r
(
t
)
=
e
!
r
ref
(
t
)
e
ir
(
t
)
>
=
!
3
r
ref
(
t
)
!
3
r
(
t
)
i
r
r
ef
(
t
)
i
r
(
t
)
>
e
m
(
t
)
=
e
!
r
opt
(
t
)
e
r
(
t
)
>
=
!
opt
(
t
)
!
r
(
t
)
!
3
r
ref
(
t
)
!
3
r
(
t
)
i
r
r
ef
(
t
)
i
r
(
t
)
>
:
By
substituting
(26)
into
(6),
we
ha
v
e
d
d
t
(
i
r
ref
(
t
)
i
r
(
t
))
=
K
r
(
i
r
ref
(
t
)
i
r
(
t
))
:
(32)
(27)
can
be
re
written
as
k
pr
d
d
t
e
!
r
ref
(
t
)
=
k
ir
e
!
r
ref
(
t
)
+
0
1
d
d
t
e
ir
(
t
)
;
(33)
and
by
substituting
(32)
into
(34),
we
ha
v
e
k
pr
d
d
t
e
!
r
ref
(
t
)
=
k
ir
e
!
r
ref
(
t
)
0
1
K
r
e
ir
(
t
)
;
(34)
Then,
E
r
d
d
t
e
r
(
t
)
=
Q
r
e
r
(
t
)
;
(35)
E
r
=
k
pr
0
0
I
2
>
0
;
Q
r
=
k
ir
0
1
K
r
0
K
r
:
(36)
When
we
define
a
L
yapuno
v
function
as
V
r
,
e
>
r
(
t
)
E
r
e
r
(
t
)
;
its
deri
v
ati
v
e
is
d
d
t
V
r
=
e
>
r
(
t
)
E
r
d
d
t
e
r
(
t
)
+
d
d
t
e
r
(
t
)
>
E
r
e
r
(
t
)
:
(37)
By
substituting
(35)
into
(37),
and
noting
that
Q
r
+
Q
>
r
=
~
Q
r
,
we
ha
v
e
d
d
t
V
r
=
e
>
r
(
t
)
Q
r
+
Q
>
r
e
r
(
t
)
=
e
>
r
(
t
)
~
Q
r
e
r
(
t
)
min
(
~
Q
r
)
e
>
r
(
t
)
e
r
(
t
)
:
(38)
Furthermore,
by
substituting
(24)
into
(3)
J
!
r
(
t
)
d
d
t
!
r
(
t
)
=
P
m
(
t
)
k
opt
!
3
r
ref
(
t
)
+
k
opt
(
!
3
r
ref
(
t
)
!
3
r
(
t
))
+
k
d
d
t
!
2
r
(
t
)
+
q
2
(
q
y
max
k
d
d
t
!
2
r
(
t
))
=
P
m
(
t
)
k
opt
!
3
r
(
t
)
+
k
opt
(
!
3
r
(
t
)
!
3
r
ref
(
t
))
+
k
d
d
t
!
2
r
(
t
)
+
k
opt
(
!
3
r
ref
(
t
)
!
3
r
(
t
))
+
q
2
(
q
y
max
k
d
d
t
!
2
r
)
=
(
t
)
!
r
(
t
)
e
!
r
opt
(
t
)
+
(
t
)
!
r
(
t
)
e
!
r
ref
(
t
)
+
q
3
y
max
+
2
k
(1
q
2
)
!
r
(
t
)
d
d
t
!
r
(
t
)
;
(39)
where
we
use
(21)
and
(
1)
k
opt
(
!
3
r
ref
(
t
)
!
3
r
(
t
))
=
(
t
)
!
r
(
t
)(
!
r
ref
(
t
)
!
r
(
t
))
:
By
using
^
J
,
J
2
k
(1
q
2
)
,
(39)
becomes
^
J
d
d
t
!
r
(
t
)
=
(
t
)
e
!
r
opt
(
t
)
+
(
t
)
e
!
r
ref
(
t
)
+
q
3
y
max
!
r
(
t
)
:
(40)
It
means
^
J
d
d
t
e
!
r
opt
(
t
)
=
(
t
)
e
!
r
opt
(
t
)
(
t
)
e
!
r
ref
(
t
)
+
(
t
)
;
(41)
where
we
used
(
t
)
=
^
J
(
d
d
t
!
r
opt
q
3
y
max
!
r
(
t
)
)
.
Hence,
E
m
d
d
t
e
m
(
t
)
=
Q
m
(
t
)
e
m
(
t
)
+
M
m
(
t
)
(42)
IJECE
V
ol.
8,
No.
2,
April
2018:
711
–
722
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
717
where
E
m
=
^
J
0
1
x
3
0
3
x
1
E
r
;
Q
m
(
t
)
=
2
4
(
t
)
(
t
)
0
1
x
2
0
k
ir
0
1
K
r
0
2
x
1
0
2
x
1
K
r
3
5
;
M
m
(
t
)
=
diag
(
(
t
)
;
0
;
0
;
0)
:
W
e
define
L
yapuno
v
function
as
V
m
=
e
m
(
t
)
>
E
m
e
m
(
t
)
;
its
deri
v
ati
v
e
is
d
d
t
V
m
=
e
m
(
t
)
>
E
m
d
d
t
e
m
(
t
)
+
E
m
d
d
t
e
m
(
t
)
>
e
m
(
t
)
:
(43)
By
using
(42)
in
(43),
we
ha
v
e
d
d
t
V
m
=
e
m
(
t
)
>
Q
m
(
t
)
e
r
(
t
)
+
e
m
(
t
)
>
M
m
(
t
)
e
m
(
t
)
>
Q
m
(
t
)
>
e
m
(
t
)
+
M
m
(
t
)
>
e
m
(
t
)
:
(44)
Noted
that
for
1
>
0
e
m
(
t
)
>
M
m
(
t
)
+
M
m
(
t
)
>
e
m
(
t
)
=
2
e
!
r
opt
(
t
)
(
t
)
e
2
!
r
opt
(
t
)
j
(
t
)
j
=
1
+
1
j
(
t
)
j
e
m
(
t
)
>
M
m
1
(
t
)
e
m
(
t
)
+
1
j
(
t
)
j
(45)
where
we
used
M
m
1
(
t
)
=
di
ag
(
j
(
t
)
j
=
1
;
0
;
0
;
0)
.
Hence,
d
d
t
V
m
e
m
(
t
)
>
~
Q
m
(
t
)
e
m
(
t
)
+
1
j
(
t
)
j
;
(46)
where
~
Q
m
=
Q
m
(
t
)
+
Q
m
(
t
)
>
M
m
1
(
t
)
.
Remark
2.
F
or
(29)-(31),
with
2
>
0
,
we
ha
v
e
~
Q
m
(
t
)
diag
2
(
t
)
j
(
t
)
j
1
(
t
)
2
;
2
k
ir
2
(
t
)
s
0
1
K
r
K
>
r
0
1
;
K
r
+
K
>
r
s
K
>
r
0
1
0
1
K
r
!
>
b
1
E
m
:
and
certainly
,
~
Q
r
(
t
)
>
0
.
Hence,
according
to
the
L
yapuno
v
Stability
Theory
,
(38)
and
(46)
gi
v
e
us
lim
t
!1
e
r
(
t
)
=
0
and
j
!
r
opt
!
r
j
q
V
m
=
^
J
q
max
!
r
j
(
t
)
j
=
(
^
J
b
)
.
4.2.
Grid-Side
Contr
ol
In
this
secti
on
,
we
propose
a
ne
w
control
la
w
such
that
V
dc
and
i
g
q
are
maintained
at
their
references
V
dc
ref
and
i
g
q
ref
,
respecti
v
ely
.
T
o
maintain
V
dc
at
V
dc
ref
,
we
need
to
mak
e
i
g
d
con
v
er
ge
to
i
g
d
ref
corresponding
to
V
dc
ref
.
Theor
em
2.
F
or
an
y
V
dc
ref
and
i
g
q
ref
,
if
v
g
of
the
GSC
(14)
are
designed
as
v
g
(
t
)
=
L
f
d
d
t
i
g
r
(
t
)
+
K
g
e
ig
(
t
)
A
g
i
g
(
t
)
+
V
s
0
>
(47)
i
g
r
(
t
)
=
i
g
d
(
t
)
+
k
pg
e
v
(
t
)
+
k
ig
R
e
v
(
)
d
i
g
q
ref
(
t
)
>
(48)
e
v
(
t
)
=
V
2
dc
ref
(
t
)
V
2
dc
(
t
)
;
e
ig
(
t
)
=
i
g
r
(
t
)
i
g
(
t
)
(49)
and
if
there
e
xist
k
pg
>
0
,
k
ig
>
0
,
and
K
g
>
0
with
~
Q
g
=
"
2
k
ig
1
0
K
g
K
>
g
1
0
>
K
>
g
+
K
g
#
>
0
;
(50)
then
lim
t
!1
(
V
2
dc
ref
(
t
)
V
2
dc
(
t
))
=
0
;
lim
t
!1
(
i
g
q
ref
(
t
)
i
g
q
(
t
))
=
0
:
Maximum
P
ower
Extr
action
Method
for
Doubly-fed
Induction
...
(Dinh
Chung
Phan)
Evaluation Warning : The document was created with Spire.PDF for Python.
718
ISSN:
2088-8708
Pr
oof.
Let
define
e
g
(
t
)
=
e
v
(
t
)
e
ig
(
t
)
>
:
By
using
(47)
and
(14),
we
ha
v
e
d
d
t
e
ig
(
t
)
=
K
g
e
ig
(
t
)
:
(51)
Furthermore,
by
taking
time
deri
v
ati
v
e
of
(48)
and
then
using
(51),
we
ha
v
e
k
pg
d
d
t
e
v
(
t
)
=
k
ig
e
v
(
t
)
+
1
0
d
d
t
e
ig
(
t
)
=
k
ig
e
v
(
t
)
1
0
K
g
e
ig
(
t
)
:
(52)
Hence,
E
g
d
d
t
e
g
(
t
)
=
Q
g
e
g
(
t
)
;
(53)
where
E
g
=
k
pg
0
0
I
2
>
0
;
Q
g
=
k
ig
1
0
K
g
0
K
g
:
(54)
If
we
use
a
L
yapuno
v
function
as
V
g
=
e
g
(
t
)
>
E
g
e
g
(
t
)
,
its
time
deri
v
ati
v
e
will
be
d
d
t
V
g
=
e
g
(
t
)
>
E
g
d
d
t
e
g
(
t
)
+
E
g
d
d
t
e
g
(
t
)
>
e
g
(
t
)
:
(55)
By
substituting
(53)
into
(55),
we
ha
v
e
d
d
t
V
g
=
e
g
(
t
)
>
Q
g
e
g
(
t
)
e
g
(
t
)
>
Q
>
g
e
g
(
t
)
=
e
g
(
t
)
>
~
Q
g
e
g
(
t
)
min
(
~
Q
g
)
e
g
(
t
)
>
e
g
(
t
)
:
(56)
Hence,
if
(50)
holds,
then
d
d
t
V
g
<
0
for
all
nonzero
e
g
.
This
completes
the
proof
of
Theorem
2.
5.
SIMULA
TION
RESUL
T
AND
DISCUSSION
T
o
e
v
aluate
the
performance
of
the
suggested
MPPT
scheme,
we
compare
the
simulation
results
of
the
1.5
MW
DFI
G
wind
turbine
with
that
using
the
con
v
ent
ional
MPPT
-curv
e
scheme
with
traditional
PI
controls
[7].
In
this
research,
the
generator
and
turbine
parameters
[22]
as
sho
wn
in
T
able
1
are
used.
T
able
1.
P
arameters
of
wind
turbine
and
DFIG[22]
Name
Symbol
V
alue
Rated
po
wer
P
1.5
MW
The
length
of
blade
R
35.25
m
Rated/minimum
rotor
speed
!
r
rated
=!
r
min
22/11
rpm
Rated
wind
speed
V
w
rated
12
m/s
Rated
stator
v
oltage
V
s
690
V
Rated
stator
frequenc
y
f
50
Hz
Number
of
pole
pairs
p
n
2
p.u
Rotor
winding
resistance
r
r
2.63
m
Stator
winding
inductance
L
s
5.6438
mH
Rotor
winding
inductance
L
r
5.6068
mH
Magnetizing
inductance
L
m
5.4749
mH
Inertia
of
system
J
445
ton.m
2
W
ith
the
po
wer
coef
ficient
(20),
the
re
gion
D
is
1
:
15
!
r
2
:
3
;
1
:
15
!
r
ref
2
:
3
;
5
V
w
12
;
3
:
4
=
min
10
:
239
:
In
this
re
gion,
(
!
r
;
V
w
)
as
Fig.
3,
which
gi
v
es
the
minimum
v
alue
,
min
(
!
r
;
V
w
)
=
1
:
271
10
5
.
IJECE
V
ol.
8,
No.
2,
April
2018:
711
–
722
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
719
(a)
(b)
(c)
(d)
(e)
Fig.
4.
Simulation
results:
(
a)
wind
profile;
(
b)
error
between
!
r
opt
and
!
r
;
(
c)
po
wer
coef
ficient
C
p
;
(
d)
error
between
P
max
and
P
m
;
(
e)
electrical
ener
gy
output
Fig.
3.
(
!
r
;
V
w
)
Here,
we
use
RSC
controller’
s
parameter
as
k
ir
=
0
:
4
J
,
k
pr
=
0
:
65
J
,
K
r
=
J
diag
(0
:
5
;
1)
,
k
=
0
:
3
J
,
=
0
:
2
,
y
max
=
0
:
1
!
r
rated
.
F
or
a
wind
profile
with
d
dt
V
w
0
:
44
m=s
2
,
the
boundary
of
j
!
r
opt
!
r
j
is
determined
as
T
able
2.
T
able
2.
Limitation
of
j
!
r
opt
!
r
j
as
1
=
1
and
2
=
2
Object
q=1
q=0
^
J
4.45
10
5
3.12
10
5
max
(
t
)
0.8673
10
5
0.8673
10
5
max
j
(
t
)
j
0.263
10
5
0.263
b
=
0
:
5915
0.1895
0.5915
j
!
r
opt
!
r
j
1.2248rad/s
0.3775
rad/s
When
the
wind
profile
as
Fig.
4a
is
used,
simulation
results
are
demonstrated
in
Fig.
4.
Fig.
4b
sho
ws
that
when
the
wind
speed
has
an
i
nsignificant
change,
the
turbine
speed
is
almost
k
ept
up
at
its
optimal
v
alue.
Since
the
wind
turbine’
s
lar
ge
inertia,
the
turbine
speed
f
ails
to
respond
instantaneously
to
the
rapid
change
of
the
wind;
this
mak
es
the
turbine
speed
i
mpossible
to
k
eep
up
at
its
optimal
v
alue.
Therefore,
the
error
j
!
r
opt
!
r
j
increases
when
the
wind
v
elocity
changes
rapidly
.
Ho
we
v
er
,
comparing
with
the
turbine
using
the
old
MPPT
scheme,
by
using
the
suggested
scheme,
the
turbine
speed
can
retain
its
optimal
v
alue
more
promptly
because
of
the
decrease
in
inertia
from
J
to
(
J
)
and
the
error
j
!
r
opt
!
r
j
is
smaller
.
As
a
result,
during
a
rapid
change
in
wind
conditions,
in
the
Maximum
P
ower
Extr
action
Method
for
Doubly-fed
Induction
...
(Dinh
Chung
Phan)
Evaluation Warning : The document was created with Spire.PDF for Python.
720
ISSN:
2088-8708
(a)
(b)
Fig.
5.
Error
between
reference
signal
and
actual
output
in
the
controller:
(
a)
errors
of
RSC
controller
and
(
b)
errors
of
GSC
controller
(a)
(b)
(c)
Fig.
6.
Simulation
results
with
5%
measurement
nois
e
:
(a)
po
wer
coef
ficient;
(
b)
error
between
P
max
and
P
m
;
(
c)
electrical
ener
gy
output
proposed
method,
C
p
restores
C
p
max
more
quickly
,
as
sho
wn
in
Fig.
4c.
Ob
viously
,
by
implem
enting
the
old
MPPT
scheme,
the
C
p
can
be
reduced
to
to
0.363
while
by
implementing
the
proposed
scheme,
this
data
is
0.393.
Figure
4
d
sho
ws
the
ef
ficienc
y
of
the
suggested
method
comparing
with
the
con
v
entional
one
in
ter
ms
of
mechanical
po
wer
.
When
the
wind
v
elocity
v
aries
insignificantly
,
the
error
(
P
max
P
m
)
in
the
wind
turbine
using
the
suggested
scheme
is
lik
e
that
using
the
con
v
entional
one.
Ho
we
v
er
,
this
error
becomes
significant
when
the
wind
changes
suddenly;
by
using
the
ne
w
method
this
error
is
significant
smaller
comparing
with
that
using
the
old
one
thanks
to
the
restoration
of
C
p
.
Fig.
4e
indicates
that
to
increase
the
turbine
v
elocity
in
the
period
of
20s-40s,
the
turbine
using
the
of
fered
scheme
requires
a
higher
mechanical
po
wer
comparing
with
that
using
the
old
one.
Ho
we
v
er
,
the
stored
mechanical
po
wer
is
returned
in
the
interv
al
of
60s-75s
in
which
the
rotor
speed
decreases.
Hence,
in
the
period
of
20s-40s,
comparing
with
the
DFIG
using
the
suggested
scheme,
the
electric
ener
gy
generated
by
the
DFIG
using
the
old
MPPT
method
is
little
higher
b
ut
in
the
60s-75s
interv
al,
it
becomes
opposite.
As
a
result,
accumulating
to
the
end
of
simulation,
the
wind
turbine
using
the
con
v
entional
method
f
ails
to
generate
the
electrical
ener
gy
in
total
as
high
as
that
using
the
proposed
method,
as
Fig.
4e.
This
indicates
the
quality
of
the
suggested
MPPT
scheme.
Fig.
5
sho
ws
the
control
quality
of
the
RSC
and
GSC.
Both
(
!
3
r
ref
!
3
r
)
and
(
i
r
d
ref
i
r
d
)
in
Fig.
5a
are
v
ery
small,
it
means
!
r
and
i
r
d
track
their
reference
v
alues;
in
other
w
ords,
the
control
la
w
proposing
for
the
RSC
has
a
good
performance.
Lik
ely
,
from
Fig.
5b,
the
errors
of
(
V
2
dc
ref
V
2
dc
)
and
(
i
g
q
ref
i
g
q
)
are
about
zero;
in
other
w
ords,
the
controller
suggesting
for
the
GSC
has
a
qualified
performance.
When
a
measurement
noise,
5%
of
rated
v
alues,
is
added
to
the
measurement
signals,
i
,
!
r
,
with
the
wind
profile
as
Fig.
4a,
the
simulation
results
are
sho
wn
in
Fig.
6.
This
figure
indicates
that
with
the
abo
v
e
noise
measurement,
the
turbine
using
t
he
recommended
MPPT
scheme
still
tracks
its
maximum
points
more
e
xactly
comparing
wit
h
the
turbine
using
the
old
MPPT
scheme.
Ho
we
v
er
,
comparing
with
the
case
of
pure
measurement
as
Fig.
4,
the
measurement
noise
causes
a
ne
g
ati
v
e
impact
on
the
turbine
performance
b
ut
this
impact
is
insignificant.
Fig.
7
is
the
simulation
results
for
a
wind
profile
which
v
aries
rapidly
as
Fig.
7a.
It
is
easy
to
see
from
Fig.
7b
and
Fig.
7c
that
the
turbine
using
the
suggested
scheme
has
more
qualified
performance
in
both
the
po
wer
coef
ficient
C
p
and
the
electrical
ener
gy
in
total
comparing
with
the
case
using
the
con
v
entional
scheme.
IJECE
V
ol.
8,
No.
2,
April
2018:
711
–
722
Evaluation Warning : The document was created with Spire.PDF for Python.