analysis of stator voltage observers for a doubly …...analysis of stator voltage observers for a...

Analysis of Stator Voltage Observers for a Doubly Fed Induction Generator

Sönke Thomsen, Kai Rothenhagen, Friedrich W. Fuchs Christian-Albrechts-University of Kiel

Institute for Power Electronics and Electrical Drives Kaiserstr. 2, 24143 Kiel, Germany

Tel.: +49 / 431 – 880 6107 Fax: +49 / 431 – 880 6103

E-Mail: [email protected], [email protected], [email protected] URL: http://www.tf.uni-kiel.de/etech/LEA

Acknowledgements This work was partly funded by Deutsche Forschungsgemeinschaft (German Research Foundation).

Keywords «Doubly fed induction motor», «AC machine», «Estimation technique», «Modelling».

Abstract Observers for the stator voltage estimation of a Doubly Fed Induction Generator (DFIG) are described and investigated in this analysis. These are the Disturbance Observer (DO) and the Unknown Input Observer (UIO) with input reconstruction. They are analysed by simulations and laboratory experiments in two different reference frames. Both observers are extended to observe a bilinear system. A pseudo synchronised reference frame is proposed to improve performance. The results of the observers are compared to the results of an inverse model of the DFIG. It can be stated, that the stator voltage is well observed.

1 Introduction Condition monitoring and fault detection has become very important for electrical drives. Detection of sudden or developing faults which occur in actuators, sensors, or other components may be economically reasonable and may contribute to a safe operation or provide fault ride through capabilities. Doubly fed induction machines are generally used for applications with high power and limited speed range, for example as wind turbine generators with variable speed. For a standard voltage oriented control of a doubly fed induction machine, the stator voltage is measured for stator power calculation and stator voltage angle generation. The stator and rotor voltages are usually considered to be an input value to the system. Instead of measuring the stator voltage the voltage can be detected by an observer. Observation of the stator voltage is important for two reasons: Firstly, it enables a fault detection of the stator voltage sensors by comparing observed and measured voltages [1]. Secondly, operation without stator voltage measurement is an interesting goal. Two observers are described, analysed and verified in this analysis: The disturbance observer and the unknown input observer with input reconstruction. Both observers are analysed in a stationary and in a rotating reference frame. Since voltage measurement shall be avoided, the use of a pseudo synchronized reference frame is proposed. Both observers are extended to observe a bilinear system with variable speed. The measurement results of the observers are compared to each other and to the results of an inverse model of the DFIG. While unknown input observers are usually used for state reconstruction with unknown inputs, they may also be used for input reconstruction [2], what is done here. Unknown input observers are described by [2], [3], [4], [5] and [6]. Most publications treat unknown input observers theoretically without an implementation on an electrical system. In this analysis the UIO is implemented to observe

Analysis of Stator Voltage Observers for a Doubly Fed Induction Generator THOMSEN Sönke

EPE 2007 - Aalborg ISBN : 9789075815108 P.1

the stator voltage of a DFIG. The principle of the state observer can be applied to observe the state of the considered system. Disturbances can be observed by extending the state observer with a disturbance model. Therefore the class of disturbances has to be known. This extended state observer (disturbance observer) treats the disturbances as an additional state, which is then observed [7], [8]. The considered DO for observation of the stator voltage treats the stator voltage as unknown input, which can be seen as a disturbance. An introduction was given in this section. In the second section the system of the DFIG is described and a mathematical model with known and unknown inputs is introduced. The third section deals with the observation of the stator voltage of the DFIG. A disturbance observer and a UIO with input reconstruction are designed for the considered system. The UIO is extended to observe a bilinear system. Input calculation with an inverse model of the DFIG is also presented for comparison. A pseudo synchronized reference frame is introduced in section four to eliminate the need for a voltage oriented reference frame, while section five shows measurements results from the laboratory. The paper is finished by a conclusion, an appendix and a list of references.

2 System Description The general control scheme of a doubly fed induction generator is shown in figure 1. The stator of the DFIG is directly connected to the grid and the rotor is fed by an inverter from a DC voltage link. Thus the rotor can be fed by a variable voltage and frequency. Field oriented control is used for rotor current control loops. Stator active and reactive power can be controlled independently from each other. The machine is loaded by a speed controlled AC load machine for laboratory tests.

M3 ~

Rotor Position

Stator Voltage

Stator Current

PWM PWM

Grid Voltage

DC-LinkVoltage

Rotor Current

ReferenceValues

Inverter Current

Inverter

Crowbar

Field Oriented Control

Stator Voltage Observer

Figure 1: Overview of the DFIG

2.1 System Model The system of the DFIG is described using a state space model [9], [10]. The stator and rotor voltages

SU and RU as well as the currents SI and RI are space vectors with real and imaginary parts. The

state space model of the electrical system is shown in equation (1) where stator and rotor currents form the state vector x. Rotor voltage is the known input vector u and stator voltage is the unknown input vector v, as shown in equation (2). The resulting system matrix A is shown in equation (3). Matrix B represents the input matrix of the known inputs u, matrix D represents the input matrix of the unknown inputs v, and C is the output matrix of the system model, as can be seen in equation (4). Only stator voltage observation is considered in this analysis. Observation of the rotor voltage can be achieved by swapping B and D . In this case the known input vector u consists of the stator voltage and the unknown input vector consists of the rotor voltage.



The rotational speed ωm is the mechanical speed of the rotor, p is the number of pole pairs and ωA is the rotational speed of the reference frame. Explicitly, a stator-fixed reference frame is described by ωA = 0 and a stator-voltage-fixed reference frame is described by ωA = 2πfS. CxyDvBuAxx =++= ; (1)

[ ] [ ] [ ] TSqSd

TRqRd

TRqRdSqSd UUvUUuIIIIx === ;;

(2)

−+−

−−−

−−

+−

+−

=

R

RmA

RS

SHm

R

H

mAR

Rm

R

H

RS

SH

RS

RHm

S

H

S

Sm

RS

HA

mS

H

RS

RHm

RS

HA

S

S

LRp

LLRL

pL

L

pL

RpL

LLLRL

LLRLp

LL

LRp

LLL

pL

LLLRLp

LLL

LR

A

σω

σω

σω

σ

ωσ

ωσ

ωσσ

σω

σσω

σω

ωσσ

ωσ

ωσ

1

1

2

2

(3)

=

−

−=

−

−

=

1000010000100001

;

0

0

10

01

;

10

01

0

0

C

LLL

LLL

L

L

D

L

L

LLL

LLL

B

RS

H

RS

H

S

S

R

R

RS

H

RS

H

σ

σ

σ

σ

σ

σ

σ

σ

(4)

As can be seen in equation (3) the system matrix A depends on the mechanical rotor speed, which can be regarded as a variable input parameter of the system. Due to the dependence on this variable parameter the considered system is non-linear. The matrix A can be split into one matrix A1 depending on the rotational speed and one matrix A0 independent of ωm [10]. The influence of the chosen reference frame is modelled by a separate matrix AA, as seen in equations (5) and (6). The matrix A0 is linear as is the term A1pωm for constant rotational speed ωm. Systems like this are called bilinear systems [11]. AAm ApAAA ωω ++= 10 (5)

−

−=

−−

−−=

−

−

−

−

=

0100100000010010

;

010

100

00

00

;

00

00

00

00

2

2

10 A

R

H

mR

H

S

H

RS

H

S

H

RS

H

R

R

RS

SH

R

R

RS

SH

RS

RH

S

S

RS

RH

S

S

A

LL

pL

LL

LLL

LL

LLL

L

A

LR

LLRL

LR

LLRL

LLRL

LR

LLRL

LR

A

σσ

σω

σ

σσ

σσ

σσ

σσ

σσ

σσ

(6)

3. Observation of Unknown Inputs 3.1 Disturbance Observer Using a disturbance observer, the stator voltage is considered as a disturbance. For the design of the disturbance observer, the system matrix A has to be extended with the disturbance process, in this case the influence of the stator voltage [7]. Assuming that these disturbances are nearly constant in a synchronous reference frame, dv/dt ≈ 0 holds, yielding to a simple extended state space model (7). Therefore, ωA = 2πfS has to be used.



3.1.1 Extended State Space Model

The state space model of equation (1) is a fourth order system. Extended by the disturbance process, a sixth order system is derived (7). Under the assumption that the pair ),( ** CA is observable, which is true for DFIG, a stable observer can be designed using the conventional methods.

[ ]

=

+

=

vx

CyuB

vxDA

vx

CBxAx

*

****

0;000

(7)

3.1.2 State Space Model of the Disturbance Observer

The state space model of the disturbance observer can be written as in equation (8) [15].

( )******* ˆˆˆ xxLCuBxAx −++= (8)

( ) ( ) ( ) ( ) eLCAxxLCAxxe ******** ˆˆ −=−−=−= (9) As can be seen from equation (9) the observer dynamics are given by )( ** LCA − . The feedback matrix L has to be chosen so that the observer error e declines. Therefore the eigenvalues of

)( ** LCA − have to be placed in the left half plane, i.e. with negative real parts. The feedback matrix L can be calculated using Ackermann’s formula [14] or by the MATLAB-function place. The unknown inputs v are now a part of the observer state vector [ ] Tvxx ˆˆˆ* = . The block diagram of the disturbance observer is shown in figure 2. Results for this observer strategy are shown in section five.

Figure 2: Block Diagram of the Disturbance Observer

Figure 3: Block Diagram of the Unknown Input Observer with Input Reconstruction

3.2 Unknown Input Observer Unknown input observers are designed to decouple the observer error e from disturbances, which are considered unknown inputs. That means UIO generally observe the states of systems where some inputs are unknown. Moreover to reconstruct the unknown inputs, as it is extended here, the measured outputs of the system and the known inputs are needed. Using the estimated states and the known inputs, the unknown inputs are reconstructed [2]. The block diagram of a UIO with reconstruction of the unknown inputs is given in figure 3. The following observer is considered: GuKyNzz ++= (10) Eyzx −=ˆ (11)



Where z is a new state of the observer, y the output vector and u the known input vector. The observer error e is defined as the difference between the state vector x and the estimated state vector x . The matrices N, K, G and E have to be designed in such a way that x converges asymptotically to x . As a consequence, the observer error will converge to zero. The dynamics of the observer error are given in (12), where I4 is the identity matrix. To make the error independent of the vectors x , u and v , conditions (13) to (16) have to be met [12].

( ) ( )

( ) ( ) ( ) DvIECuBECBGxAECAKCNECNNee

DvBuAxECDvECBuECAxGuKCxNECxNxNxxNeDvBuAxDvBuAxECGuKyNze

xyEzexxe

4

ˆ

ˆ

+−−−+−−+++=

−−−−−−++++−=++−++−++=

−−=−=

(12)

AECAKCNECN −−++=0 (13) ( )BECBG −−=0 (14) ( ) DIEC 40 += (15) N has to be a stable matrix (eigenvalues with negative real part) (16) Equation (14) and (15) yield to (17) and (18), where ( )+CD is the generalized inverse matrix of ( )CD and can be calculated by (19).

( ) BECIG += 4 (17)

( )+−= CDDE (18)

( ) ( ) ( )( ) ( ) TT CDCDCDCD 1−+ = (19) Next matrices N and K have to be designed. Solving equation (13) for N yields to equation (20). The matrix M is introduced for comprehensibility (21). NECKCECAAN −−+= (20)

NEKM += (21)

( ) MCAECIN −+= 4 (22) The eigenvalues of N have to be chosen in the left half plane. Assuming that the pair ( )CAECI ,)( 4 + is observable, the eigenvalues of ( )MCAECI −+ )( 4 can be placed freely using M, again using Ackermann’s formula [14] or MATLAB’s place. Matrix K is then derived from equation (21). Finally all matrices N, K, G and E are defined. In order to obtain the unknown inputs v, equation (10) and (11) are combined to form equation (23). Combining the state space model (1) and (23), the unknown inputs are calculated by equation (24).

yEGuKyNzx −++=ˆ (23)

( ) xADyEDuBGDKyDNzDv ˆˆ +++++ −−−++= (24)

3.3 Extending the Observer for the Bilinear System The system of the DFIG is bilinear. The dynamics and consequently the eigenvalues of the system depend on the rotor speed ωm. The discussed observers are yet designed only for one fixed rotor speed. For a variable rotor speed the observers have to be adapted to a variable ωm.



3.3.1 Bilinear Unknown Input Observer

To design a bilinear UIO, the equation (5) is used. To simplify calculations, the rotational speed of the reference frame Aω is set to zero, leading to (25). The matrices G and E are defined as before. Again, matrix N needs to be stabilised. By splitting up M as shown in (27), (26) becomes (28).

mpAAA ω10 += (25)

( ) ( ) MCpAECIAECIN m −+++= ω1404 (26)

mpMMM ω10 += (27)

( )( ) ( )( ) mpCMAECICMAECIN ω114004 −++−+= (28) The second term of equation (28) shall be zero to eliminate the rotor speed mω . Therefore 1M has to be chosen according to (29), yielding to (30). With the choice of matrix 0M the eigenvalues of N can be placed in the left half plane. Substituting (27) into (21), and solving for K yields to (31). Finally the bilinear UIO for a variable rotor speed mω is designed.

( ) 1141

−+= CAECIM (29)

( ) CMAECIN 004 −+= (30)

NEpMMNEMK m −+=−= ω10 (31)

3.3.2 Bilinear Disturbance Observer

Similar to the UIO, the DO can be made bilinear to compensate for the nonlinear influence of the rotor mechanical speed. Similar to (5), (8) is made bilinear by adding a matrix *

1A and a bilinear part of the feedback, 1L , which is calculated using the generalized inverse according to (19) as seen in (32) and (33). The observation error will then decay to zero and be independent of the rotational speed (34).

( ) ( )0 1 0 1= + ω + + − + ω −* * * * * * * * * * * *m mˆ ˆ ˆ ˆ ˆx A x A p x B u L C x x L p C x x (32)

1 1* *L A C += ⋅

(33)

( ) ( )( ) ( ) ( ) ( )0 0 1 1 0 0* * * * * * * * * * * *

mˆ ˆ ˆe x x A L C x x A L C p x x A L C e= − = − − + − ω − = − (34)

3.4 Inverse Model The unknown input can be calculated with an inverse model of the doubly fed induction machine. The state space model (1) of the system can be transformed into:

[ ] [ ] ( )AxxBDuv T −= −1 (35) The state vector x including the stator and rotor currents is measurable but noisy. The derivation of noisy signals tends to worse signals. The results of the observers are compared with the results of the inverse model in section four.

4 Pseudo Synchronous Reference Frame The disturbance observer requires a synchronous reference frame. Also, the UIO achieves better results when transferred into a stator voltage oriented reference frame [1]. For the orientation to the stator voltage, the stator voltage angle is needed. Since the observer is meant to reconstruct the stator voltage, the measurement may not be used. It is known, however, that the stator voltage has a fixed frequency, in most cases of 50 or 60 Hz. In order to achieve v 0≈ , a transformation using this information is suggested as shown as a block diagram in figure 4. In reality, any approximation will suffer from a drift, as shown in equation (36), where ωS is the grid angular frequency, for example 2π50 Hz.



The inputs UR, IR, and IS of the considered observers are transformed using the pseudo angle PseudoS ,γ . The stator current in stator fixed reference frame can be written as in (37), where ϕ is the phase angle between stator voltage and current. Using a pseudo synchronous angle as of equation (36) yields to pseudo stationary values changing with the drift frequency equation (41).

Figure 4: Pseudo Synchronous Reference Frame

( ) tt DriftSPseudoS ⋅+= ωωγ )(, (36)

ϕω

βααβ+=+= tj

SSSSSeIjIII ,,, (37)

)(,)(,)(, PseudoqSPseudodSPseudodqS jIII +=

(38)

ϕωγϕωγαβ

+−−+− =⋅== tjS

jtjS

jPseudoSPseudodqS

DriftPseudoSSPseudoS eIeeIeII ,,)(,)(,

(39)

( )ϕω += tII DriftSPseudodS cos)(,

(40)

( )ϕω += tII DriftSPseudoqS sin)(,

(41)

The closer the real stator frequency is approximated ( Drift 0ω ≈ ), the better v 0≈ is be achieved, without being exactly synchronous. The presented results have been obtained using this technique. Exact synchronism is not needed, since the same angle is used for back transformation, thereby cancelling the drift. The presented results have been obtained using this technique, eliminating the need for voltage sensors for angle generation. The use of a pseudo synchronous reference frame makes the use of a disturbance observer possible in the first place, since v 0≈ is required.

5 Measurements at a Test Drive Measurements were taken from a 22 kVA test drive (for data see appendix) via dSPACE-Controldesk® and then plotted using MATLAB-Simulink®. The DFIG was operated at various rotational speeds, and 5 kW stator power. The results of the disturbance observer, the unknown input observer with input reconstruction and the inverse model can be seen in figures 5 to 10. The voltages are calculated in synchronous coordinates, and then transformed back to stator fixed reference frame. The experimental results for both observers are quite good. The observed voltages are shown as continuous line and are close to the real voltages, shown as dashed line. Phase and magnitude of the stator voltage are observed very well, at various speeds showing the bilinear compensation works well. The phase angle of the disturbance observer is a little worse than the angle estimated by the UIO. The phase difference between the measured and observed stator voltage amounts to 1º for the UIO and to 3.6º for the DO. Compared to that, the inverse model delivers a noisy approximation and a phase shift of 9º, as can be seen in figure 7. The noisy behaviour is due to the derivative of the currents, which is directly added to the output. In case of the UIO’s reconstruction, the derivative of the currents is also used, but it contributes to a smaller extend since weighted by the matrix E, thus limiting the influence of noisy current measurement on the estimated voltage. The DO does not use any derivative. The



maximum relative error of the peak values for the UIO amounts to 3% at a rotational speed of 1000min-1 and to 4% at 1300min-1. The DO yields to a maximum relative error of 2% at rotational speed of 1000min-1 and 4% at 1300min-1, at a rotational speed of 1600min-1 the error amounts to 4.6%. The results of the inverse model show a relative error of 16%. Therefore, the observed voltages of the observers show much better results than the inverse model.

Figure 5: UIO in pseudo synchronous reference frame N=1000 min-1 , experimental results

Figure 6: UIO in pseudo synchronous reference frame N=1300 min-1 , experimental results

Figure 7: Inverse Model in pseudo synchronous reference frame at N=1000 min-1 , experimental results

Figure 8: Bilinear Disturbance Observer in pseudo synchronous reference frame N=1000 min-1 , experimental results

Figure 9: Bilinear Disturbance Observer in pseudo synchronous reference frame at N=1300 min-1, experimental results

Figure 10: Bilinear Disturbance Observer in pseudo synchronous reference frame at N=1600 min-1, experimental results

6 Conclusion DFIG are important for applications with high power. They are especially used for on- and off-shore wind farms. The stator voltage as an input to the DFIG is usually measured for power and angle calculation, which are needed for the control of the generator. It has been shown that it is possible to use both observers, a Disturbance Observer or an Unknown Input Observer with input reconstruction to acquire the stator voltage without measurement. This may be useful for control without these sensors or for model based fault detection.



Design methods for unknown input observer and disturbance observer are presented and analysed in this work. While the design of the disturbance observer is simpler, it only provides accurate estimates in a synchronous reference frame where the input voltages are almost constant. The design of the UIO is more complex, yet it does also work in a stationary reference frame. Using the UIO in a rotating reference frame gives better results than in a stationary reference frame. A pseudo synchronised reference frame is introduced to avoid the measurement of the stator voltage for stator angle generation, thereby making the use of a disturbance observer possible at all and increasing the quality of the UIO. Measurements are provided to support the theory and show a good conformance with the simulations. In general, a comprehensive way is presented to acquire good estimates of the stator voltage of DFIG.

Appendix

Table I: Machine Parameters Table II: Data of Experimental Setup Parameters: Identified: LH 48 mH LS 49.1 mH LR 49.1 mH RS 100 mΩ RR 250 mΩ Transmission Ratio 1.5

Machine VEM 22 kW Stator: 400 V 41 A SPER 200 LX4 Rotor: 255 V 53 A Control dSPACE DS1006 2800 Mhz Inverter (DFIG) IGBT 2-level Voltage Source Inverter

References [1] Rothenhagen K., Thomsen S., Fuchs F. W.: Voltage Sensor Fault Detection and Reconfiguration for a

Doubly Fed Induction Generator, SDEMPED’07, Krakow, Poland, in press. [2] Sfaihi B., Boubaker O.: Full Order Observer Design for Linear Systems with Unknown Inputs, IEEE

International Conference on Industrial Technology, pp. 1233-38, 2004 [3] Hou M., Müller P. C.: Design of Observers for Linear Systems with Unknown Inputs, IEEE Transactions on

Automatic Control, pp. 871-875, 1992. [4] Krzemiński S., Kaczorek T.: Perfect reduced-order unknown-input observer for standard systems, Bulletin

of the Polish Academy of Sciences, Vol. 52, 2004 [5] Hui S., Żak S. H.: Observer Design for Systems with Unknown Inputs, International Journal of Applied

Mathematics and Computational Science, Vol. 15, pp. 431-446, 2005 [6] Chen J., Patton R. J., Zhang H.: Design of Unknown Input Observers and Robust Fault Detection Filters,

International Journal of Control, Vol. 68, pp. 85-105, 1996 [7] Lunze J.: Regelungstechnik 2, Spinger, 2005 [8] Kailath T.: Linear Systems, Prentice Hall, 1980

[9] Mohan N.: Advanced Electric Drives; Analysis, Control and Modeling using Simulink®, Mnpere, 2001

[10] Rothenhagen K., Fuchs F. W.: Current Sensor Fault Detection by Bilinear Observer for a Doubly Fed Induction Generator, IECON 2006, CD-ROM

[11] Bennett S.: Model Based Methods for Sensor Fault-Tolerant Control of Rail Vehicle Traction, PhD-Thesis, University of Hull, 1998

[12] Boubaker O.: Robust Observers for Linear Systems with Unknown Inputs: a Review, International Journal on Automatic Control and System Engineering, Vol. 5, pp. 45-51, 2005

[13] Rothenhagen K., Fuchs F. W.: Implementation of State and Input Observers for Doubly-Fed Induction Generators, EUROCON 2007, in press, Warsaw, Poland

[14] Åström K. J., Wittenmark B.: Computer-Controlled Systems, Theory and Design, Prentice Hall, 1997

[15] Xiong Y., Robust Fault Diagnosis in Linear and Nonlinear Systems based on Unknown Input, and Sliding Mode Functional Observer Methodologies, PHD-Thesis, Simon Fraser University, Canada, 2001



analysis of stator voltage observers for a doubly …...analysis of stator voltage observers for a...

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