International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 5, Issue 2, February 2015)
Control by Back Stepping of the DFIG Used in the Wind
Turbine
A. Elmansouri1, J. El mhamdi2, A. Boualouch3
1,2,3
Laboratoire de Génie Electrique, Ecole Normale Supérieur de l’Enseignement Technique, Université Mohamed 5 de Rabat,
Maroc
Abstract— This paper presents the modeling and control
back stepping of Double-Fed Induction Generator (DFIG)
integrated into a wind energy system. The goal is to apply this
technique to control independently the active and reactive
power generated by DFIG. Finally, the system performance
was tested and compared by simulation in terms of following
instructions and robustness with respect to parametric
variations of the DFIG.
II. BACKSTEPPING CONTROL
A. Historical
The back stepping was developed by Kanellakopoulos
(1991) and inspired by the work of Feurer& Morse (1978)
on one hand and Tsinias (1989) and Kokotovit& Sussmann
(1989) on the other.
The arrival of this method gave a new life to the
adaptive control of nonlinear systems .This, at despite the
great progress made, lacked general approaches. The back
stepping presents a promising alternative to methods based
on certain equivalence. It is based on the second Lyapunov
method, which combines the choice of the function with
the control laws and adaptation. This enables it him, in
addition to the task for which the controller is designed
(continuation and / or regulation) to ensure, at all times, the
overall stability of the compensated system [4].
Keywords— Back stepping, Control, DFIG, Wind turbine
I. INTRODUCTION
The development and use of renewable energy have
experienced strong growth in recent years. Among these
energy sources, the wind one has a fairly important
potential; it is not powerful enough to replace the existing
sources, but still can compensate for the increasing
reduction of demand. There is a new solution using
alternating machine operating in a rather unusual way; it is
the double-fed induction machine (DFIG) [1].
DFIG is very popular because it has certain advantages
over all the other variable speed; its use in
electromechanical conversion chain as a wind turbine or
engine has grown dramatically in recent years [2].Indeed,
the energy converter used to straighten and curl alternating
currents of the rotor has a fractional power rating of the
generator [3], which reduces its cost relative to competing
topologies.
In this context, several robust control methods of DFIG
appeared; among them back stepping control is one of
them. This technique is a relatively new control method for
nonlinear systems. It allows sequentially and
systematically, by the choice of a Lyapunov function, to
determine the system's control law. Its principle is to
establish in a constructive manner the control law of the
nonlinear system by considering some state variables as
virtual Drives and develop intermediate control laws [12].
B. Application of back stepping for machine control
Research on the development of the DFIG control
technology has multiplied in recent decades. We currently
find several techniques present in the literature, such as
vector control, DTC, nonlinear controls such as back
stepping and control sliding mode…
Application of back stepping technique to control the
DFIG is to establish a machine control law via a Lyapunov
function selected, ensuring the overall stability of the
system. It has the advantage of being robust vis-à-vis the
parametric variations of the machine and good continuation
references [5]-[6].
C. Power control by back stepping of DFIG
In this section, we present a new approach to
backstepping control applied to the asynchronous machine
dual power supply. This approach is designed in such a way
to keep the same general structure of a vector control power
[7].
472
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 5, Issue 2, February 2015)

D. Step 1: Computation of the reference rotor currents
We define the errors e1 and e2 representing the error
between the actual power Ps and the reference power Pref
and the error between reactive power
v1  k1e12  k 2 e22
So,
and its
Qs
the error between the current quadrature rotor I qr and the
reference current I qrref and the error between the direct
(1)
rotor current I dr and I drref reference.
The derivative of this equation gives:

M 



e
I qr

1  P ref  V s
e1  P ref  P s (2)  
Ls







e2  Qref  Qs
e  Q  V M I dr

2
ref
s

Ls

e3  I qrref  I qr

e4  I drref  I dr
(3)
The first Lyapunov function is chosen so such that:
1
v1  (e12  e22 )
2



1


e 3  I qrref  Vqr  S1
e 3  I qrref  I qr (11) 








e 4  I drref  1 V  S
e 4  I drref  I dr
2

 dr

(4)


v1  e1 e1  e2 e2





(6)


MVs
v1  e1  P ref 
Ls

 1
 
   (7)

MVs
Vqr  Rr I qr  g s I dr  g
Ls

S1 
1
MVs 
(13)
 Rr I qr  gs I dr  g

 
Ls 
S2 
1

 R I
r
dr
 g s I qr 
(14)
Actual control laws of the machine

MVs
Vdr  Rr I dr  gs I qr  1 
 e2 Q ref 
Ls


Vqr and Vdr shown
in (12), then we can go to the final step.
The final Lyapunov function is given by:
The pursuits of goals are achieved by choosing the
references of the current components representing the
stabilizing functions as follows:
v2 
1 2
(e1  e22  e32  e42 )
2
(15)
The derivative of equation is given by:



VM
MVs 

I qrref  X k1e1  Pref  s Vqr  g s I dr  g
Ls 
Ls  (8)






Vs M

I drref  X k 2 e2  Qref  L  Vdr  g s I qr 
s



With: X 
(12)
With:
(5)
v1  e1 ( P ref  P s )  e2 (Qref  Qs )
(10)
The derivative of this equation gives:
Its derivative is:

I qrref and I drref in (8) are asymptotically stable.
E. Stepe2: computation of the control voltages.
We define the errors e3 and e4 respectively representing
reference Qref .
e1  Pref  ps



e2  Qref  Qs
(9)





v2  e1 e1  e2 e2  e3 e3  e4 e4
(16)
Which can be rewritten as follows:

v2  k1e12  k 2 e22  k3e32  k 4 e42
Ls ,
M2 ,
M2
  Ls 
  Lr 
Vs MRr
Ls
Ls

1
1


 

 e3  k3e3  I qrref  Vqr  S1   e4  k 4 e4  I drref  Vdr  S 2 






7)
Where: k1 , k 2 : positive constants.
The derivative of the Lyapunov function becomes:
Where: k 3 , k 4 Positive constants.
473
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Control voltages
The general structure of control by the backstepping of
DFIG is illustrated in Figure (1) the blocks computation
I qrref and I drref represent the fictitious control, respectively
Vqr and Vdr are selected as:




Vqr    k3e3  I qrref  S1 





V    k e  I drref  S 
4 4
2
 dr



(18)
provide current references obtained from the errors of active
and reactive power.
The computation of commands voltages Vqr and Vdr are

based on the error between the currents references and
actual implanted by equation (18)
Which ensures: v2  0
The stability control is obtained by a good choice of
gains: k1 , k 2 , k3 and k 4 .
Fig.1: Block diagram of the control with power loop.
III. MODELING OF THE WIND TURBINE
The total kinetic power available on a wind turbine is
given by [8]:
Pmax
1
1
 Svv3  R 2 vv3
2
2
(19)
 C2

 C3   C4  e

 i

pm
1

C p     R 2v13
1 21
(24)

1
  0.08
Cmec 
(20)
 C5 
 
 i 

0.035
 3 1
 C6 
Cturbine
G
(25)
A. Model of the shaft:
The fundamental equation of dynamics can be written:
(21)
J
With:
Cturbine 
1
Multiplier model:
1
 C p     . .R 2 .vv3
2
C p  f   ,    C1 
(23)
i
The aerodynamic power is given by:
Pmax
1  R2 
2 3
Pmg  C p Pm  C p 
  R v1
2  KV1 
d mec
 Cturbine  f mec
dt
With: mec  Gturbine
(22)
474
(26)
(27)
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Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 5, Issue 2, February 2015)
0.4
power coefficient Cp
Cem  P
B=0
B=2
B=5
B=10
B=20
M
(ds I qr  qs I dr )
Ls
(32)
Mechanical Equation:
0.3
d 1
 (Cem  Cr  f r )
dt
J
0.2
Cem  P  I s  
0.1
T
0
-0.1
0
2
4
6
Tip speed ratio
8
Fig.2: Power coefficient C p
10
(34)
For medium and high power machines used in wind
turbines, we can neglect the stator resistance [8]. Under this
hypothesis, the equation (30) becomes:
12
 ,  
ds  s  Ls ids  MI dr

qs  0  Ls iqs  MI qr
The figure (2) shows the C p curves for multiple values
of  . This curve is characterized by the optimum point; this
value is called the Betz limit, and which is the point
corresponding to maximum power coefficient.
(35)
Equation (32) allows us to write.
s M

ids  L  L idr

s
s

M
I   i
qs
qr

Ls

B. Modeling of the DFIG
The model of DFIG is equivalent to the model of the
induction machine cage. However, the rotor of the DFIG is
not shorted.
Vds  0, Rs  0

Vqs  Vs  sds
Park transformation:
The classical electrical equations of the DFIG in the Park
frame are written as follows [9]:
d

Vsd  Rs I sd  sd   s sq


dt


d

V  R I  sq   s 
sq
s sq
sd

dt

d
( M sr  I r )
d
(33)

Vs
M2
dr  ( Lr  L idr )  M L 

s
s s

2
  ( L  M )i
qr
r
qr

Ls


(28)
(36)
(37)
(38)
Active and reactive power equations then become:
drd 

V

R
I

  s rq
rd
r
rd


dt


V  R I  drq   s 
rq
s
rq
rd

dt


P  Vdsids  Vqsiqs


Q  Vqsids  Vdsiqs
(29)
M

 Ps  Vs L iqr

s


M
Q  V
idr  Vs s
s
s

Ls
Ls

The stator and rotor flux can be expressed as:
ds  Ls ids  MI dr

qs  Ls iqs  MI qr
(30)
dr  Lr idr  MI ds

qr  Ls iqr  MI qs
(31)
(39)
(40)
Replaces equations (31) into (29) we obtain:

M 2 didr
M2
Vdr  Rr idr  ( Lr  L ) dt  gs ( Ls  L )iqr

s
s

2
2
V  R i  ( L  M ) diqr  g ( L  M )i  g MVs
r qr
r
s
s
dr
 qr
Ls dt
Ls
Ls
The electromagnetic torque is given by the following
expression:
475
(41)
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12000
(42)
10000
8000
Active power ( W )
In steady state, so we can write:

M2
)iqr
Vdr  Rr idr  g s ( Ls 
Ls


2
V  R i  g ( L  M )i  g MVs
qr
r qr
s
s
dr

Ls
Ls

The electromagnetic torque is then written:
Cem  P
M
ds I qr
Ls
6000
4000
(43)
2000
The previous equations used to establish a block diagram
of the electrical system to regulate:
0
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.08
0.09
0.1
Time (seconds)
Fig.4: Active power (W).
50
0
-50
Reactive power ( Var )
-100
-150
-200
-250
Fig.3: Block diagram of the DFIG regulate.
-300
-350
0
IV. SIMULATIONS RESULTS.
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Time (seconds)
The Control of The DFIG proposed was simulated using
(Matlab / Simulink). The parameters of the DFIG are
attached. The parameters k1 , k2 , k3 and k 4 of backstepping
Fig.5: Reactive power (Var)
The negative sign of the reactive power shows that the
generator functions in capacitive mode.
control
are
chosen
as
follows: k1  80600 , k2  900600 , k3  5000 , k4  6000 , to
meet the convergence condition.
In the first case, we present the response of the DFIG
under the backstepping control. The following figures
illustrate the active power, reactive, the rotor currents in
quadrature and direct the rotor voltages in quadrature and
direct.
4
x 10
1
Quadrature component of the rotor voltage ( V )
0
-1
-2
-3
-4
-5
-6
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Time (seconds)
Fig.6: Quadrature component of the rotor voltage (V).
476
International Journal of Emerging Technology and Advanced Engineering
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12000
50
10000
0
-50
6000
-100
Reactive power ( Var )
Direct component of the rotor voltage
8000
4000
2000
0
-150
-200
-250
-2000
-300
-4000
-350
-6000
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0
0.1
0.01
0.02
Fig.7: Direct component of the rotor voltage (V).
Quadrature component
of the rotor current (A)
0.05
0.06
0.07
0.08
0.09
0.1
The two figures (9) and (10) show a good behavior of the
machine despite the change in the rotor resistance, active
and reactive power exhibit a good track their orders
-20
-40
V. CONCLUSION
-60
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.07
0.08
0.09
0.1
The first part of this article is devoted to the design of a
backstepping control law for DFIG. This law is established
step by step while ensuring the stability of the loop
machine closed by a suitable choice of the function of
Lyapunov.
The second part presents the modeling of the turbine and
controlling of the machine based on the physical equations,
followed by simulation of operation.
A simulation result allows us to show the proposed
algorithm capabilities, in terms of regulation, tracking and
disturbance rejection, and demonstrating the robustness of
this technique can advantageously replace conventional PI
control.
From a conceptual point of view, we can notice that the
backstepping control is simpler is easier to implement and
has very interesting global stability properties
Time (seconds)
2
Direct component
of the rotor current (A)
0.04
Fig.10: Reactive power (Var) with change of Rr (-50%)
0
1.5
1
0.5
0
0
0.01
0.02
0.03
0.04
0.05
0.06
Time (seconds)
Fig.8: Rotor current Idr and Iqr (A)
To demonstrate the performance of the order by
Backstepping, the DFIG is subject to robustness tests for the
operating conditions, set point change and the parameters of
the machine.
12000
10000
REFERENCES
[1]
8000
Active power ( W )
0.03
Time (seconds)
Time (seconds)
6000
4000
[2]
2000
[3]
0
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Time (seconds)
Fig.9: Active power (W) with change of Rr (-50%)
477
Abdellah Boualouch, Abdellatif Frigui, Tamou Nasser, Ahmed
Essadki, Ali Boukhriss, "Control of a Doubly-Fed Induction
Generator for Wind Energy Conversion Systems by RST
Controller." , International Journal of Emerging Technology and
Advanced Engineering IJETAE, Volume 4, Issue 8, August 2014,
pp 93-99.
Ahmed G. Abo- Khalil, Dong-Choon Lee and Jeong-Ik Jang
―Control of Back-to-Back PWM Converters for DFIGWind Turbine
Systems under Unbalanced GridVoltage‖
B. Beltran, T. Ahmed-Ali, and M.E.H. Benbouzid, ―Sliding mode
power Control of variable speed wind energy conversion
systems,‖IEEE Trans. Energy Convers., vol.23, no.22, pp.551–558,
Jun.2008.
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 5, Issue 2, February 2015)
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à la commandedécentralisée des systèmes non linéaires", Thèse PhD,
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Nihel KHEMIRI1,4, Adel KHEDHER2,5, Mohamed Faouzi
MIMOUNI3,4 ‗‗An Adaptive Nonlinear Backstepping Control of
DFIG Driven by Wind Turbine‘‘
A.MEDJBER, A.MOUALDIA, A.MELLIT et M.A. GUESSOUM
‗‗Commande vectorielle indirecte d‘un générateur asynchrone
double alimenté appliqué dans un système de conversion éolien‘‘
Mohamed Allam,BoubakeurDehiba, Mohamed Abid, Youcef Djeriri
,etRedouaneAdjoudj ‗‗Etude comparative entre la commande
vectorielle directe et indirecte de la Machine Asynchrone à Double
Alimentation(MADA) dédiée à une application éolienne‘‘
Appendix: Power coefficient constants:
c1  0.5176 , c2  116 , c3  0.4 , c4  5 , c5  21 ,
c6  0.0068
Backstepping parameters:
k1  80600 , k2  900600 , k 3  5000 , k 4  6000
Pref  10Kw
Qref  200VAr
List of symbols:
 : Air density
R :Blade radius.
vv : Wind speed (m/s)
Pmax : Wind maximum power.
APPENDIX:
C p : Power coefficient:
Appendix: Wind turbine data
Blade Radius
Power coefficient
Optimal relative wind speed
Moment of inertia

R=3.5 m
0.48
8
J=0.0017 kg/m²
1 : Wind turbine speed(Shaft speed).
Cmec : Wind turbine torque (Nm).
G : Mechanical speed multiplier.
mec : Mechanical speed (rad/s).
Appendix: DFIG parameters
Rated power
Rated stator voltage
Nominal frequency
Number of pole pairs
Rotor resistance
Stator resistance
Stator inductance
Rotor inductance
Mutual inductance
Sliding
1R
: Relative wind speed
V1
10kw
Vs=220V
Ws=2π50
P=2
Rr=0.62Ω
Rs=0.455Ω
Ls=0.084 H
Lr=0.085 H
M=0.078 H
g=0.03
Pmg : Mechanical power (watts).
 2 : Rotation speed multiplier (rad/s).
f : Damping cofficient.
Cem : Electromagnetic torque(Nm).
J : Moment of inertia(Kg.m²)
Vdr , Vqr : Direct and quadrature rotor voltages
Cturbine : Wind turbine torque (Nm).
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