iecon06_rothenhagen

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Current Sensor Fault Detection by Bilinear Observer

for a Doubly Fed Induction Generator

Kai Rothenhagen Friedrich W. Fuchs

Christian-Albrechts-University of KielChair of Power Electronics and Electrical Drives

Kaiserstr. 2, 24143 Kiel, Germany

[email protected] [email protected]

Abstract- A Bilinear Observer is presented to detect

current sensor faults in a doubly fed induction generator.

The state space model is derived from the voltage

equations of stator and rotor. The bilinear behaviour of

induction machines and its influence on the state space

model is discussed. An open loop model suffers from

model and parameter uncertainty. A closed loop observer

is shown to greatly reduce these effects. Observer pole

placement is mentioned. Feeding back rotor currentsonly, the observer becomes independent from stator

current measurements. The resulting free residuals can

be used for sensor fault detection. The observer is

validated by laboratory experiment.

I. INTRODUCTION

Doubly Fed Induction Generators (DFIG) have become awidely used generator type in wind energy conversion [1].This is not the only application, however, since DFIG arealso used in pump storage plants [2], in flywheel energystorage [3] as well as in marine applications [4] or proposedas aviation generators [5].

Fault detection is becoming more and more important forvariable speed drives to increase availability and reliabilityand reduce downtime. Typical elements of a fault detectionwith subsequent reconfiguration are shown in figure 1. Stateobservers, in this case also called soft sensors, can be used toobserve a state that is already measured [6], [7], [8], [9].Sensor faults can be detected by analysing the residualbetween measurement and observation. It is therefore ofinterest to research state observers for DFIG.

Fault tolerance of induction machines has been covered by[9], [10], and [11]. In both cases, model based fault methodsare used. While an open loop induction drive is consideredby [9], [10], fault tolerant control of closed loop systems is

examined by [11].The focus of this paper is on model based residualgeneration for a field oriented controlled DFIG.

Measure-

ment

Residualgeneration

ChangeDetection

Fault

Diagnosis

Recon-figuration

Figure 1: Necessary steps towards a fault tolerant control. Scope of thispaper is residual generation using an observer.

System Description

Doubly fed induction machines comprise of a wound statorand a wound three phase rotor, where the rotor windings canbe accessed by brushes. Usually, the stator is connected tothe grid, and the rotor is fed by an inverter. This way, therotor can be fed by a variable voltage and frequency. Thepower P

Rtransferred through the rotor windings is

proportional to the slip s (1).

R SP s P= (1)

In applications, where only a limited range of speed isneeded, the inverter can be downsized in comparison to astator fed machine. In wind power applications, the invertermay typically be designed for a speed range of thirty percentaround synchronous speed.

The examined topology is depicted in figure 2. The rotor isfed by an inverter from a voltage DC link. Field orientedcontrol is used for rotor current control loops. Stator activeand reactive power can be controlled independently fromeach other. The machine is loaded by a speed controlled ACload machine.

M3 ~

Field Oriented Control

Rotor

Position

Stator Voltage

Stator Current

PWMDC-Link

Voltage

Rotor

Current

Reference

Values

Inverter

Crowbar

Fault Detection Observer

Figure 2: Overview of the considered doubly fed induction generator withcurrent sensor fault detection observer

Overview of this Paper

The paper is organised as follows. In section II, the statespace model of a DFIG is derived from the voltage equationsof stator and rotor.The model is then used to implement anobserver in section III.Section IV gives measurement results.A Conclusion is presented in section V. Acknowledgementand References finish the paper in sections VI and VII.


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II. BILINEAR OBSERVER FOR A DOUBLY FEDINDUCTION GENERATOR

Model of the DFIG

In the following, a state-space model for the doubly-fedinduction machine shall be presented. State-space models in

general are thoroughly known [8], yet a short review ishelpful for the understanding of this paper.The model is derived from the voltage equations of the statorand rotor. The following assumptions are made:

a)The stator and rotor winding are symmetricalb) Symmetrical feeding of the stator and rotorc) No saturation of inductances (linear magnetisation)d) Sinusoidal distribution of fluxe) No iron losses, no skin effect, no bearing frictionf) Constant winding resistancesThe mechanical differential equation is not included. This

is reasonable if the mechanical system is much slower thanthe electrical system or the load machine keeps the rotationalspeed constant.

The voltages Usand U

ras well as the currents I

sand I

rare

space vectors. Ls

and Lr

denote the stator and rotorinductance, respectively. R

sand R

rare the stator and rotor

resistance, respectively. Lh

denotes the main inductance. Allparameters are referred to the stator.

The voltage equations of stator and rotor are given byequations (2) and (3) for their respective reference frame.Equation (4) transforms (3) from rotor to stator referenceframe. Both voltage equations may also be written in an

arbitrary reference frame A rotating with A

with respect tothe fixed stator (5).

S

S S SSS S dU = R I

dt+JJG

GJG (2)R

R R R

RR R

dU = R I

dt

+ JJG

GJG

(3)

S

S S S R

R RR R m

dU = R I jp

dt

+ JJG

G JJGJG

(4)

A

A A A S

S SS S A

A

A A A R

R RR R A m

dU = R I j

dt

dU = R I j( p )

dt

+ +

+ +

JJG

G JJGJG

JJG

G JJGJG

(5)

Using (6) and splitting the resulting equations into real andimaginary part, one obtains:

A A A

S S RS h

A A A

R S Rh R

= L I L I

= L I L I

+

+

JJG G G

JJG G G

(6)A A

A A A A sd rd

sd s sd A s sq A h rq s h

A A

sq rqA A A A

sq s sq A s sd A h rd s h

A A

A A A A sd rd

rd r rd A m h sq A m r rq h r

A A A A

rq r rq A m h sd A m r rd

dI dIU = R I L I L I L L

dt dt

dI dIU = R I L I L I L L

dt dt

dI dIU = R I ( p )L I ( p )L I L L

dt dt

U = R I ( p )L I ( p )L I

+ +

+ + + +

+ +

+ +

A A

sq rq

h r

dI dIL L

dt dt+ +

Solving for the differential terms and defining

2

h

S R

L1

L L =

one obtains four differential equations, which can be writtenin the known state space equation:

dxAx Bu; y Cxdt = + = (7)

The input vector u is defined by the real and imaginaryparts of the stator and rotor voltages (8), the state vector xcomprises of the real and imaginary parts of the stator androtor currents (9).

T

sd sq rd rqu U U U U =

(8)T

Sd Sq Rd Rqx I I I I =

(9)

The system matrix A, the input matrix B and the outputmatrix C are given in (10), (11), and (12).

2

S h h R hA m m

S S R S R S

2

Sh h h R

A m m

S R S S S R

h S h R

m A m

R S R R

h Sh R

m A m

R R S R

R L L R Lp pL L L L L L

RL L L R ( p ) p

L L L L L LA =

L R L R 1p p

L L L L

L RL R1p ( p )

L L L L

+

+

h

S R S

h

S R S

h

R S R

h

R S R

L10 0

L L L

L10 0

L L LB =

L 10 0

L L L

L 10 0L L L

(10)

(11)

1 0 0 0

0 1 0 0C =

0 0 1 0

0 0 0 1

(12)

Note that the system matrix A still contains the rotational

speed A

of the arbitrary reference frame and therefore is ageneral description for all reference frames. The systemmatrix for the stator-fixed reference frame is obtained by

setting A=0. Likewise, the system matrix for the rotor-fixed

reference frame is obtained by setting A=

m.

When using a reference frame fixed to the stator, the

indices dq are usually replaced by .As can be seen from (10), A contains the rotor speed

mas

a variable input parameter, which means that the system isnon-linear. Therefore, A is split up into A

0which represents

the elements independent ofm, and N, which contains the

elements dependent ofm[9] (13).

0 mA A N= + (13)

For the stator-fixed system, NS

shall be used. A state spacesystem like this is called bilinear, because the non-linear


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input m

can be isolated. Then A0

is linear, and Nm

is linear

as well for constant m, and linearly dependent on

m.

2

h h

S R S

2

h h

S R S

S

h

R

h

R

L L0 0

L L L

L L0 0

L L LN =

L 10 0

LL 1

0 0L

(14)

S h R

S S R

S h R

S S R

0

h S R

R S R

h S R

R S R

R L R0 0

L L L

R L R0 0

L L LA =

L R R0 0

L L L

L R R0 0

L L L

(15)

Modelling of Stator and Rotor Current Sensor Faults

In case of sensor faults, some states are either not

measurable or shall not be used for feedback because of falsemeasurements. This is described by introducing the effectiveoutput matrix C

effaccording to (16). If all sensors are used for

feedback, F becomes the identity matrix.

1

2

eff

3

4

f 0 0 0

0 f 0 0C C F C

0 0 f 0

0 0 0 f

= =

(16)

( )i1 sensorfedback

f i 1, 2,3,40 sensornotfedback

=

(17)

Placement of the Observer Eigenvalues

According to [6] and [7], Luenberger observers can be usedfor residual generation. Using Luenberger observers, theeigenvalues of the closed loop observer can be placed bychoosing an appropriate feedback matrix L.

eff

eff

eff

dx Ax Bu

dt

d x Ax Bu LC ( x x)

dt

d d de x x

dt dt dt

d e Ax Bu (Ax Bu LC ( x x ))

dt

de (A LC ) e

dt

= +

= + +

=

= + + +

=

(18)

By doing so, the dynamical behaviour of the estimationerror e can be defined. For linear systems, the error dynamicsare described by (A-LC

eff) (18), where x and x are the

estimated and measured state vectors, respectively, and e isthe error vector.

Similar to (13), L0

and M are defined by (19), leading to(20). Figure 3 shows the structure of this bilinear observer.

uy

A0

CB1

s

x

NS

+

+

L0

x.

+

+

-

+

mech

A0

CB

1

s

NS

+

+

+

+

.

.

.

++

yxx.

Doubly Fed Induction Machine

Model

u

Residual Generation and

Feedback

Figure 3: Topology of bilinear observer

0 mL L M= + (19)

0 0 eff S eff m

de (A L C (N MC ) )e

dt= +

(20)

As can be seen, the error dynamics are again defined byA

0-L

0C

effif M is defined by MC

eff=N. This is only possible if

Ceff is invertible. This is the case if all sensors are used forfeedback. In this case (21) and (22) will define the

eigenvalues i of the error dynamics independently of the

rotor speed m.

[ ]T

0 0 4 1 2 3 4L A I= + (21)

M N= (22)

Observer without complete set of sensors

Observers are useful for observing states that shall not orcannot be measured. Therefore, the case of an observerwithout a complete set of sensors is more realistic. In this

case, the effective output matrix Ceff (16) is not invertible.Therefore, MC

eff=N is not possible, and placing the

eigenvalues becomes a non-linear function ofm.

In this case, the Matlab-function place is used to placethe eigenvalues according to the algorithm described in [12].The problem of non-linear observer feedback is notaddressed in this paper.

III. IMPLEMENTATION OF THE OBSERVERSTRUCTURES

Referring the Measured Signals to the Stator

The observer is fed with the stator and rotor voltages

referred to the stator. Residuals are computed with additionalinput of stator and rotor current also referred to the stator.Therefore, the rotor signals that are only available in the rotorsystem need to be processed in order to obtain the referredsignals. This requires a change in frequency from rotor tostator frequency. This is achieved by implementing a vectorrotation with the angle of the rotor position. Usually a changein amplitude is necessary because of the transformation ratioof the DFIG. The transformation ratio is derived from thenumber of turns of stator and rotor winding. Therefore, thetransformation ratio is implemented by a gain factor for the


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rotor current and voltage. The measurements need to beconverted into space vectors by concordia transform. Therotor voltage is not measured. Instead, the commanded rotorvoltage is used, decreased by a factor for dead timecompensation. The matrices F and I

4-F are used to separate

the residuals into feedback residuals and detection residuals.

Realisation of a Fault-Detection Observer

The structures depicted in figures 4 to 7 are realised in

Matlab-Simulink for simulation and experiment via adSpace Board. Integration is approximated by a 500sForward-Euler algorithm.

Figure 4: Overview of observer structure

Figure 5: Signal Processing. Referring the rotor signals to the stator system.

Figure 6: Model: Bilinear state space model of the DFIG.

Figure 7: Residual and feedback: Calculation of the error feedback andresiduals for detection

Figure 8: Stator and rotor currents (experiment). Red: observed currents.Green: Measured currents. Activation of error feedback at t=0s, feedback ofstator and rotor currents

Figure 9: Stator and rotor current residuals (experiment). Activation of error

feedback at t=0s, feedback of stator and rotor currents.

IV. EXPERIMENTAL BEHAVIOUR

No Error Feedback vs. Complete Error Feedback

In case of ideal modelling and ideal knowledge of the plant'sparameters, the state space model itself should follow theplant without feedback. These assumptions can hardly everbe met, yet it is important for the model to be as close aspossible to the plant. Therefore the behaviour of the openloop model and the closed loop observer is presented infigure 8. Machine data is given in table 1 in appendix VI.Note that the parameters used in the observer need to be

chosen larger than the identified machine parameters, asstated in table 2. Figure 8 shows the measured and estimatedstator and rotor currents before and after closing the feedbackloop at t=0s. Without feedback, a large subharmonic can benoticed in the open loop model estimates, while thefundamental component is close to the measuredcurrents.With activated feedback, the closed loop observerestimates and the measured currents fit very good. Figure 9

shows the stator and rotor residuals Sand R . As can beseen, the residuals become small with activated feedback.


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Figure 10: Stationary stator and rotor currents (experiment). Red: observedcurrents. Green: Measured currents. Blue: Residuals. Activation of errorfeedback at t=0s, feedback of rotor currents only

Behaviour with Error Feedback of two States

There is no fault detection capability in case all four statesare fed back, because the observer will be affected by anyfalse current measurement. Feeding back only the measuredrotor currents, the stator currents can be checked for sensorfaults. The behaviour of a two state feedback is documentedin figure 10. It is shown that the subharmonic component iseliminated in both the fed back rotor currents and in thestator currents, that are not fed back. The stator currentresiduals increase compared to a four-current feedback infigure 9, but the stator currents residuals are now decoupledfrom the stator current sensors. This makes stator currentsensor fault detection possible.

Behaviour of the Observer During Transients

It is important that the currents estimated by the closed loopobserver follow the measured currents not only in steadystate, but also during transients. To demonstrate this, figure11 shows the behaviour of the observer during a step of thereference current. The rotor current reference steps up from 4to 28 A at t=0s. Note that the current depicted in figure 11 isthe referred current and therefore the actual rotor currentdivided by the transmission ratio of the DFIG. Thecorresponding residuals are shown during the steps. It can beseen that the residuals are almost not responding during bothsteps. This is an important feature, since any fault detectionmust not be triggered by transients.

Figures 12 and 13 show the behaviour of an observer usingonly the rotor currents as feedback. This time, a step upwardsand downwards is shown. The stator current residuals aremore affected than in the case of feeding back all fourcurrents. They are, however, still small and the peak duringthe transients declines fast.

Figure 11: Step response of stator and rotor currents (experiment). Red:observed currents. Green: Measured currents. Blue: Residuals. Activation oferror feedback at t=0s, feedback of stator and rotor currents.

Figure 12: Step response of stator and rotor currents (experiment) Red:observed currents. Green: Measured currents. Step at t=0s and t=2s,

feedback of rotor currents only.

Figure 13: Residuals during the steps. Step at t=0s and t=2s, feedback of

rotor currents only.


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Using the Free Stator Current Residuals for Fault Detection

If only two currents are fed back, the other two may beused for fault detection of the respective current sensors. Inorder to show this capability, the observer stator currentinputs are switched to zero to simulate a sensor fault. Theinputs to the current controllers are not affected, though, tomaintain a working control.

Due to the character of the concordia transform (23), a faultin stator phase U will affect the and component of thestator current. A fault in phase V will affect the -componentonly. Therefore, a fault in stator phase V can be detected by

an increase of the -residual. A fault in phase U will lead to achange in both the and -residual, where the change in theb-component will be smaller than the change resulting from afault of phase V. A fault of both current sensors will also leadto an increase of both residuals.

U

V W V U

V U

I I

I I 2 I II 1.15 I 0.58 I

3 3

=

+= = = +

(23)

Figure 14: Stator and rotor current residuals. Current sensor failure in phaseU (blue flag), phase V (red flag), and both phase U and V. Feedback of

rotor currents only.

This is difficult to distinguish from a fault in phase U, sincethe magnitude of the residuals is only slightly different. Thisbehaviour is shown in figure 14. Detection of a currentsensor fault in phase U may be realised by cyclic swappingof the three phases, so that phase U will only affect a new

residual *. Detection of rotor current sensor fault may berealised by using an observer in the rotor reference frame.

V. CONCLUSION

A bilinear state observer is used to detect current sensorfaults. The observer may be stabilised by using a feedback ofrotor currents only, therefore leaving the stator currentresiduals for detection. The proposed observer may serve twopurposes. Firstly, it serves as a residual generator for sensorfault diagnosis. It is shown that a significant residual can begenerated for a stator current fault in phase V. Residuals forother current sensors can be generated as well. Secondly, theobserver supplies an approximated signal for the faultymeasurement. This may be used to maintain a controlled

operation of the drive, when fault tolerant control is to berealised. Further research in this direction is necessary.

VI. DATA OF EXPERIMENTAL SETUP

Machine VEMSPER 200 LX4

22 kW Stator: 400 V 41 ARotor: 255 V 53 A

Control dSPACE DS1104 250 Mhz

Inverter (DFIG) IGBT 2-level Voltage Source InverterConverter (Grid) 3-phase Diode Rectifier

Table 1: Data of experimental setup

Parameters Identified Used in ObserverLh 48 mH 48 mH

Ls 49.1 mH 51.1 mH

Lr 49.1 mH 51.6 mH

Rs 100 m 200 mRr 250 m 300 mTransmission Ratio 1.5 1.5

Table 2: Doubly Fed Induction Machine Parameters

VII. ACKNOWLEDGEMENT

This work was funded by the DeutscheForschungsgemeinschaft (German Research Foundation).

VIII. REFERENCES

[1] A. Hansen, F. Iov, P. Srensen, F. Blaabjerg. Overall control strategy ofvariable speed doubly-fed induction generator wind turbine. Nordicwind power conference 2004, CD-ROM, Gteborg, Sweden, 2004.

[2] A. Bocquel, J. Janning:Analysis of a 300 MW Variable Speed Drive for

Pump-Storage Plant Applications, EPE'05, CD-ROM Paper, Dresden,Germany, 2005.

[3] H. Akagi, H. Sato: Control and Performance of a Doubly-Fed Induction

Machine Intended for a Flywheel Energy Storage System, IEEETransactions on Power Electronics, Vol. 17, No.1, pp. 109-116, Jan2002.

[4] R. Burschen: Shaft Generator Plants with Slipring Induction Machines,Treffen Schiffsbautechnische Gesellschaft e. V., pp. 18-21, 11, 1987,

Hamburg.[5] F. Khatounian, E. Monmasson, F. Berthereau, E. Delaleau, J. Louis:

Control of a Doubly Fed Induction Generator for Aircraft Application ,IECON'03, pp. 2711-2716, Roanoke, USA, 2003

[6] R. Patton, P. Frank, R. Clark: Issues of Fault Diagnosis for Dynamic

Systems. Springer-Verlag, Berlin, 2000.[7] T. Steffen: Control reconfiguration of dynamical systems : linear

approaches and structural tests. Springer Verlag, Berlin, 2005[8] R. Lorenz: Observers and state filters in drives and power electronics.

Journal of Electrical Engineering, Vol. 2, 2002.

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

[10] S. Bennett, R. Patton, S. Daley, D. Newton: Model Based IntermittentFault Tolerance in an Induction Motor Drive, Symposium on Control,Optimization and Supervision, Vol.1, pp 678-83, Lille, 1996.

[11] C. Thybo: Fault-tolerant Control of Inverter Fed Induction MotorDrives. PhD Thesis, University of Aalborg, 1999.

[12] Kautsky, J. and N.K. Nichols, Robust Pole Assignment in Linear StateFeedback, Int. J. Control, 41 (1985), pp. 1129-1155

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