Unknown Input Observer for a Doubly Fed Induction Generator Subject to Disturbances

International Journal of Power Electronics and Drive System (IJPEDS)
Vol.6, No.4, December 2015, pp. 781~787
ISSN: 2088-8694  781
Journal homepage: http://iaesjournal.com/online/index.php/IJPEDS
Unknown Input Observer for a Doubly Fed Induction
Generator Subject to Disturbances
Samir Abdelmalek*,
**, Linda Barazane*, Abdelkader Larabi*
*Industrial and Electrical Systems Laboratory, Faculty of Electronics and Computer, University of Sciences and
Technology HouariBoumediene, B. P. 32 El - Alia, 16111, Bab - Ezzouar. Algiers, Algeria
**Unité de Développement des Equipements Solaires (UDES) EPST CDER, 42415, Tipaza, Algérie
Article Info ABSTRACT
Article history:
Received Jan 22, 2015
Revised Oct 11, 2015
Accepted Oct 26, 2015
This paper deals with the problem design of an unknown input observer
(UIO) for a Doubly Fed Induction Generator (DFIG) subject to disturbances.
These disturbances can be considered as unknown inputs (UI). The state
space model of the DFIG is obtained from the voltage equations of the stator
and rotor. Then, this latter is used for the design of an unknown input
observer (UIO) in order to estimate both the state and the unknown inputs of
the DFIG. Furthermore, the UIO gains are computed by solving a set of
linear matrix inequalities (LMIs). Simulations results are given to show the
performance and the effectiveness of the proposed method.
Keyword:
Doubly fed induction generator
Linear matrix inequalities
State space model
Unknown input observer
Voltage equations Copyright ©2015 Institute of Advanced Engineering and Science.
All rights reserved.
Corresponding Author:
Samir Abdelmalek,
Industrial and Electrical Systems Laboratory, Faculty of Electronics and Computer, University of Sciences
and Technology HouariBoumediene, B. P. 32 El - Alia, 16111, Bab - Ezzouar. Algiers, Algeria
Unité de Développement des Equipements Solaires (UDES) EPST CDER, 42415, Tipaza, Algérie
Email: samir_aut@yahoo.fr
1. INTRODUCTION
Induction generators are one of the most popular electric machines used in wind turbines. Wind
turbines based on the doubly fed induction generator (DFIG) has received increasing attention in the recent
years, due to its remarkable advantages over other wind turbine systems [1], the increase of power capture
[2], their capacity to operate at different range of the wind speed, the ability to control active and reactive
powers [3].However, DFIG-based wind turbines can be subject to disturbances which can have as source:
measurement noises, sensors and actuators faults, especially to voltage dips [4]. These disturbances, can be
considered as unknown inputs, have adverse effects on the normal behavior of the real system and their
estimates can be used to conceive systems of diagnostic and control [5]. Robust observers are proposed to
estimate simultaneously states and actuator faults for various class of linear and nonlinear systems [6]-[10].
Recently, diagnosis and estimation faults are becoming very important to ensure a good supervision
of the systems and guarantee the safety of human operators and equipment’s, even if systems are becoming
more and more complex. In this respect, a large number of diagnostic methods for sensors of induction
machines have proposed in [11]-[13].
Fault diagnosis and State estimation of induction machines has attracted considerable interest, as
they are often used in practical control systems [14]. FDI in sensor faults of induction machines is necessary
since control systems rely on the information provided by measured signals. In [15], the authors have studied
the FDI problem of induction machines. Since DFIG can be subject to different kinds of faults as studied in
[16]. Authors in [17]-[19], focus on current sensor fault detection and isolation (FDI) and control

 ISSN: 2088-8694
IJPEDS Vol. 6, No. 4, December 2015: 781 – 787
782
reconfiguration current for DFIG. They have used two Luenberger observers to generate residuals for the
current sensors. A proposed algorithm for fault identification is designed to isolate current sensor faults
instator or in rotor. In [20]-[21], have studied the effect of current sensor fault on a doubly fed induction
machine (DFIM). In [22], a new FDI algorithm of stator current sensors and speed sensor faults detection
problem has proposed for Permanent Magnet synchronous machine drives where simulation and
experimental results are reported. In [23] presents a signal-based approach to detect and isolate the fault in
stator current and voltage sensors of the DFIG.
The contribution of this paper focuses on the design of the UIO to estimate both of state and
unknown inputs. These unknown inputs are considered in this work, as faults affecte the stator voltages of the
DFIG and their estimates can be used to conceive systems of diagnostic and control.
This work is organized as follows. In Section 2, system description and modeling are presented. In
Section 3, formulation problem is discussed. Then, in Section 4, simulation results are conducted to evaluate
the performance of the proposed observer. Finally, the conclusions and future works are given in Section 5.
2. SYSTEM DESCRIPTION AND MODELING
In a DFIG-based wind turbine, as shown in Fig. 1, the generator is coupled to the wind turbine rotor
through a gearbox. The stator of the DFIG is directly connected to the grid and the rotor side is connected to
a back-to-back converter via slip-rings [24].
Figure 1.Model of DFIG-based wind turbine.
2.1. DFIG model
For the DFIG, the dynamic voltages of the stator ( sV and sV ) and those of the rotor ( rV and
rV ) in the general (   ) reference frame are respectively expressed as [25]:
d
d
d
d
d
d
d
d
s s s s s s
s s s s ss
r r r r r r
r r r r rr
V R I Φ Φ
t
V R I Φ Φ
t
V R I Φ Φ
t
V R I Φ Φ
t
   
  
   
  

   

    


    


    

1
The stator and rotor (   ) fluxes, s , s , r and r are given by:
s s s m r
s s s m r
r r r m s
r r r m s
Φ L i L i
Φ L i L i
Φ L i L i
Φ L i L i
  
  
  
  
 

 

 
  
2

Where, sV , sV , rV and rV stator and rotor in (   ) voltages; sI  , sI , rI  and rI stator
and rotor in (   ) currents; , s , s , r and r stator and rotor in (   ) fluxes; sR , rR stator
and rotor per phase resistance; sL , rL cyclic stator and rotor inductances.
2.2. DFIG State space model
In this paper, themathematical model developed of the DFIG is derived from the voltage equations
of the stator and rotor (for more details see ([17-19]). Based on (1), (2) and (3), the DFIG model is expressed
in the reference (   ) frame, as the following:
Wind Turbine
Gear Box
DC Bus
Control
System
Control
System
DC BusACAC
AC‐DC DC‐AC
DFIG

IJPEDS ISSN: 2088-8694 
Unknown Input Observer for a Doubly Fed Induction Generator Subject to Disturbances (Samir Abdelmalek)
783
2
2
1
( )
1
( )
s s m m r m m m m
s s s r r s r
s s r s r s s s r
s sm m m m r m m
s s s r r s r
s r s s s r s s r
r m s m m r
s s
s r r r
d I R L p R L L p L
I I I I u u
dt L L L L L L L L L
d I RL p L p R L L
I I I I u u
dt L L L L L L L L L
d I L R L p R
I I I
dt L L L L

     

     

  
 
        
     
 
        
     

   
  
1
( )
1
( )
m m
r s r s r
s r r
r m sm m m r m
s s s r r s r
r s r r s r r
p L
I u u
L L L
d I L RL p p R L
I I I I u u
dt L L L L L L L
  

     







        

  
       
     
3
The state space model of the DFIG is given by:
( ) ( ) ( ) ( ) ( )
( ) ( )
m inx t A x t B u t R u t
y t C x t
   



4
Where, m is the mechanical speed of the rotor, p is the number of pole pairs, and the matrices
( )mA  , B and C are expressed as follows :
2
2 2
2 2
( )
1
( )
s m m r m m m
s
s s r s r s
m
s m m m m
r
s r s r
R L p R L L p
I J I J
L L L L L L
A
R L L p p
I J I J
L L L L
                          
     
                
,
2
2
1
m
s r
r
L
I
L L
B
I
L
 
 
 
 
 
 
, 2 2
2
0T
C
I
 
  
 
0 1
1 0
J
 
  
 
The state vector
T
s s r rx I I I I       , consists of the stator currents and rotor current
components. The control inputs
T
s su u u     are the rotor voltage components. The measured
disturbances (Unknown inputs)
T
in r ru u u     are the stator voltage components. It is clear from the
representation as in (4), that the system matrix A is varying time and depends on the mechanical rotor speed
m . In this paper, letus consider that the DFIG operates at a fixed-speed ( m mec   ).
3. PROBLEM FORMULATION
UIO’s goal is to estimate the system states where some inputs are unknown. Authors in [26],
demonstrated that the conventional Luenberger observer is not suitable to overcome unknown inputs. Using
the estimated states and the known inputs, the unknown inputs are reconstructed [3]. The block diagram of a
UIO with reconstruction of the unknown inputs is given in Figure 3. For the system as in (1), the UIO is as
follows[27]:
( ) ( ) ( ) ( )
ˆ( ) ( ) ( )
z t N z t G u t L y t
x t z t E y t
  

 

5
Where is a new state of the observer, y the output vector, u the known input vector, N is a stable
matrix. Matrices N, G, L and E are the observer gains. The matrices N, K, G and E have to be designed in
such a way that ˆ( )x t converges asymptotically to ( )x t . As a consequence, the observer error will converge
to zero.
Figure 2. Unknown Input Observer for a DFIG subject to disturbances.

 ISSN: 2088-8694
784
Let us define the state estimation error as:
ˆ( ) ( ) ( )e t x t x t  6
By using (5) and (6), the state estimation error e(t) becomes:
( ) ( ) ( ) ( )nxe t z t I EC x t   7
By setting ( )nxP I EC  , the dynamics of the estimation error is given by the following equation:
( ) ( ) ( ) ( ) ( ) ( ) ine t N e t G P B u t P N LC N P x t PR u       8
Theorem.1 . The necessary and sufficient conditions for the existence of UIO (5) of system (4) are
[27-28]:
a) N is stable (eig (N) 0);
b) rank(CR) = rank(R) = dim(y);
c) The pair ((Inx + EC)A,C) is observable.
If the following relations are satisfied:
LC P A N C  9a
G PB 9b
( ) 0nxI E C R  9c
Based on (9c) yield to:
( )E R C R 
  10
where (CR)+ is the generalized inverse matrix of (CR) and can be given as :
 ( ) ( ) ( ) ( )
TT T
C R C R C R C R
 11
Finally, all matrices N, K, G and E are defined in the following equations :
 
1
( ) ( ) ( )T T
E R C R C R C R

  12a
 
1
( ) ( ) ( )T T
nxP I R C R C R C R

  12b
G P B 12c
N P A K C  12d
L K N E  12e
The state estimation error is then refined as:
( ) ( )e t N e t
(13)
3.1. Stability and convergence conditions
Based on the above UI observer, the following theorem will give the fault estimation algorithm and
the conditions that guarantee the stability of error system (13).
Theorem. 2. The UIO (5) for a DFIG system with inputs unknown (4) exists and their estimation
error (13) converges asymptotically to zero, if and only if, the pair (A,C) is detectable . This observer is
asymptotically stable if exists a positive definite symmetric matrix P and matrices i iW P K such that the
following LMI holds:
 0, 1,...,T T T
i i i iA P P A C W W C i r      (14)
The solution of the inequality (14) can then be obtained usingLMI conditions. Observer gains can
becalculated from 1
i iK P W
 . Then, consequently ˆ( )x t will asymptotically converge to ( )x t and ( )inu t
to ˆ ( )inu t .
3.2. Unkown input estimation
In order to obtain the unknown inputs ˆ ( )inu t , we combining (4) and (15), the unknown inputs are
expressed as the following:

785
ˆ( )
( ) ( ) ( ) ( )
d x t
N z t L y t G u t E y t
dt
     15
ˆ ˆ( ) ( ) ( )inu t R N R G B u R Ly R E y R A x t    
      16

4. SIMULATION AND DISCUSSION RESULTS
In order to validate the proposed approache, the model (4) with the parameters in [9-10] is used as a
controlled system in the simulation studies. The studies were conducted in Matlab using 4th-order Runge-
Kutta method with the fixed step size of 0.01 s.Figure. 3 represents the measured stator and rotor currents of
the DFIG and their estimated based on the UIO, and in Figure. 4 is represented the dynamic errors of the
states.
Figure 3. Simulation results of original states and their estimated.
Figure 4. The errors between states and their estimated.
It can be clearly observed from the simulation results that the states estimation generated from the
UIO converge rapidly to those simulate by the DFIG system. In addition, we can see in Figures 4, that the
estimation errors are very weak.
4.1. Unknown Input Estimation
Figures. 5 and 6 represent the unknown inputs and their estimates, with their dynamic errors . The
simulation results show the good estimation of these unknown inputs.
0 5 10
-100
-50
0
50
100
Times [s]
I
s
[A]
DFIG
UIO
0 5 10
-40
-20
0
20
40
Times [s]
I
s
[A]
DFIG
UIO
0 5 10
-100
-50
0
50
100
Times [s]
I
r
[A]
DFIG
UIO
0 5 10
-40
-20
0
20
40
Times [s]
I
r
[A]
DFIG
UIO
0 5 10
-2
-1
0
1
2
x 10
-13
Times [s]
r
s
0 5 10
-4
-2
0
2
4
x 10
-13
Times [s]
r
s
0 5 10
-1
-0.5
0
0.5
1
x 10
-12
Times [s]
r
r
0 5 10
-1
-0.5
0
0.5
1
x 10
-12
Times [s]
r
r

 ISSN: 2088-8694
786
Figure 5.Unknown input sV and its estimated ˆin su  .
Figure 6.Unknown input sV and its estimated ˆin su  .
The simulation results show a good estimation of both state and unknown inputs by using the UIO.
5. CONCLUSION
In this paper, the problem of designing the unknown input observer (UIO) for a DFIG which subject
to disturbance is treated. These unknown inputs affects the states of the DFIG. The Unknown Input Observer
(UIO) design problem is formulated as a set of linear constraints which can be easily solved using linear
matrix inequalities (LMIs) technique. Solving a set of LMIs, the UIO can be designed. An application based
on a DFIG is presented to evaluate the performance and the effectiveness of the proposed observer. The
observer is applied to estimate both stator and rotor currents with unknown inputs which described by the
stator voltages. The simulation results show a good estimation of both state and unknown inputs.
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