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Analytical and Experimental Evaluation of a WECS

Based on a Doubly Fed Induction Generator Fed by a

Matrix ConverterR. Cárdenas (1) , R. Peña (1) , G. Tobar (1), R. Blasco-Giménez(2), P. Wheeler (3),

G. Asher (3)

, J. Clare(3)

(1) University of Magallanes, email [email protected](2) Technical University of Valencia, email [email protected].

(3) University of Nottingham, email [email protected]

Abstract- In this paper the control of a grid-connected Wind

Energy Conversion System (WECS), based on a sensorless vector

controlled Doubly-Fed Induction Generator (DFIG) fed by a

matrix converter, is presented. The matrix converter is controlled

using a space vector modulation algorithm. Stability issues relatedto the operation of the WECS connected to the grid using a matrix

converter are also discussed in this work. The influence of a

synchronous rotating filter in the dynamic of the proposed

WECS is analysed. A Model Reference Adaptive System (MRAS)

observer for sensorless control of the proposed WECS is used in

this work. Using the speed estimated by the MRAS observer the

electrical torque of the induction generator is regulated in order

to drive the WECS to the operating point where the aerodynamic

efficiency is maximized. Experimental results, obtained with a

3.5kW prototype are presented, and fully discussed in this work.

I. I NTRODUCTION

ATRIX converters have many advantages, which arewell documented in the literature [1-2]. The MatrixConverter (MC) provides bi-directional power flow,

sinusoidal input/output currents and controllable input powerfactor [3]. The MC allows a reliable and compact design due to

the lack of dc-link capacitors for energy storage. Because of itsrelatively small size, in some applications the matrix converter

can be embedded in the machine itself [4]. Considering its potential robustness, reduced size and reliability, MCs are goodcandidates for wind energy applications.

The advantages of DFIGs, for wind energy applications, arealso well known [5-7]. The DFIG is widely used for variablespeed generation, and is one of the most important generators

for wind energy applications [6]. For a typical DFIG, the power converters are connected to the rotor and, for restrictedspeed range, are rated at a fraction of the machine rated power

[7], typically ±30% of nominal. Moreover, in a vectorcontrolled DFIG, the torque and currents are controlled with afast dynamic response [5]. In a grid-connected DFIG, back to back converters are used to connect the DFIG rotor to the grid.

In this paper a new topology is proposed. The back to backconverters are replaced with a matrix converter. A Space

Vector Modulation (SVM) algorithm [3] is used to control the

F I L T E R

MATRIX

CONVERTER

Modulation

Algorithm

∼

MC

Input

Voltage

DSP Based

Control System

DFIG

Grid

Variable Speed

Wind Turbine

Gear

Box

Fig. 1. WECS proposed in this work.

MC, regulating the torque and the magnetising current in thegenerator. The proposed WECS is shown in Fig. 1. The DFIGis vector controlled using a conventional control systemorientated along the stator flux vector. To avoid the use of a position encoder a rotor current MRAS observer isimplemented [8].

As shown in Fig. 1, in the matrix converter a second order

input filter is required in order to reduce the input voltagedistortion and to improve the input current waveform.However, the filter at the input can produce instability in someoperating point, specially when the output power is increased[9]. In this paper the stability of the proposed WECS isanalysed using a small signal model of the WECS and MC. A

synchronous rotating filter can be applied to improve thestability of the proposed WECS.

II. CONTROL OF THE PROPOSED VARIABLE SPEED WECS

The control system proposed in this work is shown inFig.2. The DFIG is driven by a variable speed wind turbine.

The MC is controlled using the space vector modulationalgorithm discussed in [3]. Zero displacement factor at the

M

2438978-1-4244-1666-0/08/$25.00 '2008 IEEE



matrix converter input is used in this work. However variabledisplacement factor control is also feasible [10]. A rotorcurrent MRAS observer is used to estimate the rotor positionangle and rotational speed. As shown in the top right of Fig. 2,the input voltage vi is filtered using a synchronous rotatingfilter before being used in the SVM algorithm.

A. Vector Control of a Grid-Connected DFIG.

Vector control of grid-connected DFIGs has already beenstudied before and only a brief description is given here. Themachine equations in a d-q synchronous rotating frameoriented along the stator flux vector are [7]:

=

qr

dr

qs

ds

r m

r m

m s

m s

qr

dr

qs

ds

i

i

i

i

L L

L L

L L

L L

00

00

00

00

λ

λ

λ

λ

(1)

−

+

+

=

qs

ds

e

e

qs

ds

qs

ds

s

s

qs

ds

dt

d

i

i

R

R

v

v

λ

λ

ω

ω

λ

λ

0

0

0

0 (2)

−+

+

=

qr

dr

sl

sl

qr

dr

qr

dr

r

r

qr

dr

dt d

ii

R R

vv

λ λ

ω ω

λ λ

00

00 (3)

)(2

3qr dsdr qsme

iiii L p

T −= (4)

where λ s= Lmims is the stator flux and λ r is the rotor flux; L s , Lm and Lr are the stator, magnetising and rotor inductancesrespectively; v s and i s are the stator voltages and currents; vr and ir are the rotor voltages and currents; Rr and R s are the rotor

and stator resistances; ω s and ω r are the synchronous and

rotating angular frequencies respectively, ω sl =ω s-ω r is the slipfrequency and ims is the magnetising current; T e is the electrical

torque and p is the number of poles.

The induction machine is vector controlled using a directvector control orientated along the stator flux vector [7]. Fieldorientation for transforming the machine variables uses the slipangle derived from:

r e slip θ θ θ ˆ−= (5)

wherer θ is the rotor position angle estimated from the MRAS

observer (see Fig. 2).

The position of the stator flux vector (θ e) is obtained from

the stator flux α - β components as:

= −

s

s

e

α

β

λ

λ θ

1tan (6)

the α - β components of the stator flux are obtained from thestator voltages and currents as :

dt i Rv

dt i Rv

s s s s

s s s s

)(

)(

β β β

α α α

λ

λ

−=

−=

∫∫ (7)

In this work a rotor current MRAS observer is used for

tracking the rotor position angle and rotational speed.

B. Rotor Current MRAS Observer.

The Rotor Current MRAS Observer (RCMO) has already

been discussed in [8] and only a brief description is presentedhere. The block diagram of the RCMO is shown in Fig. 3.

For the RCMO, the reference model is the rotor currentmeasured by the transducers. An estimation of ir is obtainedusing i s and v s. In the stationary frame the stator flux isobtained as:

t jr m s s

r ei Li L ω +=s λ (6)

From (6) the rotor current is obtained as:t j

m

s r e L

i Li

ω −−= ss

r

λ (7)

Replacing r ω ˆ in (7), an estimation of the rotor current is

obtained as:

t j

m

s s r e L

i Li

ω λ ˆˆ −−= s

r (8)

The MRAS error is defined as the cross product between ir

and r i :

)sin(ˆˆˆ error qr dr qr dr iiiiii θ ξ r r =−= (9)

Using (6-9), the RCMO is shown in Fig. 3. The error of (9)is the input to a PI controller used to correct the estimated

rotational speed of (8). The error of (9) is driven to zerowhen the reference rotor current and that estimated from (8)are in phase. More information about the RCMO is in [8].

C. Modelling of the Matrix Converter.

The matrix converter topology used in this work is shownin Fig. 4. Nine bidirectional switches, implemented using

2/3-

+

-

+

3/2

-+

Grid

DFIG-

+

ωslip

3/2

MRAS

Observer

e jθ

e-jθ

slip

slip

∧

∧

Matrix

Converter

Space

Vector

Modulation

∼

Input

Filter

e-jθv

1

sτf +1e jθv

vif

θvi ∫

vi

ωe*

idr *

iqr *

vdr *

vqr *

idr

iqr

vα r *

v β r *

iα r i β r iar ibr

ias ,ibsiα s ,i β s

vα r ,v β r

vas ,vbs

∧

ωe

∧θ r

ωr

∧

Fig. 2. Control system proposed.

R s

+

-

i s

∫

i r

+

L s

-

1/L m

| i r| -2

v s ω r

^

θ r

^λ s i r^

e -j θ r

^

Fig. 3. MRAS observer proposed in this work.

2439



IGBTs, are required. A second order input filter is used toreduce the harmonic content in the current supplied to thegrid. A resistor connected in parallel with the input inductance

may be required to improve the input filter dampingcoefficient. However, if the matrix converter is connected toa weak grid, an external inductance may not be necessary andthe filtering is realised by the relatively large inductance of the

grid [3]. In this case the parallel resistors of the input filtercannot be connected.

When the MC is used to control a DFIG, the turn ratio between rotor and stator should be designed in order to operatethe matrix converter with a voltage transfer ratio q close to thenominal value, when the machine is operating with a slip of

≈30%. In this case the KVA rating of the converter switches isminimised.

Assuming that the switching frequency is much higher than

the input/output fundamental frequencies, then theinput/output voltages can be represented by their average

values over a cycle period. The MC input/output relationship isobtained as [3,11 ]:

[ ]d ci

ciio mvmvv +=

2

3 (10)

[ ]d mimii coioi +=

2

3 (11)

The demanded values of d m and im are calculated as:

cif

coi

cif od vvmvvm 33 **** == (12)

where the vectors oo iv , are the output voltage and output

current respectively; the vectors ii iv , are the input voltage and

input current respectively. In (12) “c” stands for complexconjugated and vo

* is the reference output voltage used by the

modulation algorithm. In Fig. 2, vo

*is the output of the current

controllers (i.e. vo*=vdr

*+jvqr * ).

The small signal models of (10-11) are obtained as:

[ ]d c

id ci

cii

cii mV M vmV M vv ∆+∆+∆+∆=∆ 00000

2

3 (13)

[ ]d cc

d iii m I i M m I i M i ∆+∆+∆+∆=∆ 0000002

3 (14)

Using (12) small signal functions for *d m and *

im are obtained

as:*

0*

0 33 co

cif iiif vv M mV ∆=∆+∆ (15)

A

B

C

Bidirectional

SwitchR

R

R

Input Filter Grid input

a b c

To DFIG Rotor

s11 s12 s13

s31 s32s33

s21s22 s23

∼

∼

∼

Fig. 4. Matrix converter topology used in this work.

*0

*0 33 o

cif d d if vv M mV ∆=∆+∆ (16)

The stability of the matrix converter is improved when theinput voltage used in the modulation algorithm is filtered usinga first order filter [11]. In order to avoid a phase shift between

the MC input voltage and the voltage used by the SVMalgorithm, the filtering is carried-out using a synchronous d-qrotating axis. The d and q components of the input voltage are

filtered using a first order low pass filter with a cut-offfrequency of 1/τf . In d-q coordinates the filtered voltage isobtained as (see Fig. 2):

11 +=

+=

f

qiqif

f

didif

s

vv

s

vv

τ τ (17)

A small signal model of the synchronous rotating filter isobtained as:

1+∆

=∆ f

iif

s

vv

τ (18)

A simplified model in d-q coordinates can be obtainedassuming that the DFIG rotor is a high inductance load and

∆io≈0. If the output current does not change, then the output ofthe current controller ∆vo* ≈0.

In the quiescent point it is assumed that V i0=V if0=V i0c= V if0

c and V o=qV i0, where q is the voltage transfer ratio. Using theseassumptions the small signals models of (14) can be modified

to:

[ ]d c

ii m I m I i ∆+∆=∆ 002

3 (19)

the small signal models of (15-16) can be simplified to:cif id

cif ii vV qmvV qm ∆−=∆∆−=∆ )3()3( 0

*0

* (20)

using (20) in (19); assuming ii mm ∆=∆ * ,d d mm ∆=∆ * ; and

considering V i0=Vo/q yields:

[ ] cif

coo

oi v I I

V

qi ∆+−=∆

2

2

(21)

Defining Y o( ω o )= I o /V o as the equivalent admittance connectedat the MC output, in d-q components (21) can be written as:

[ ]

[ ]1

)(Re

1)(Re

2

2

+

∆=∆

+

∆−=∆

f

iooiq

f

iooid

s

vY qi

s

vY qi

τ ω

τ ω

(22)

In (22) some digital effects have been neglected. For instancethe dynamic effects of the zero order hold and the time delay

produced because the duty cycles calculated in a givensampling time are used in the next processing cycle. Theseeffects can be important when the switching frequency is low.

The digital effects can be easily considered if the values of*d m and *

im are calculated as:

−∆=∆ −

sT

eemm

sT s

ii d

1* τ

−∆=∆ −

sT

eemm

sT s

d d d

1* τ (23)

where τ d is the time delay, T is the sampling time and(1-e-sT)/(sT) is the representation of the zero order hold [11].

2440



∼

R f

Lf

∆i g ∆ii

Grid

∆v g

Cf ∆vi

Req

Fig. 5. Single phase models of the proposed WECS considering a strong grid.

When the digital effects are important, (23) can be used to

obtain the small signal model of (19). If the MC is operated atunity power factor, then the matrix converter can berepresented by an equivalent resistance Req. Therefore theinput filter and the MC can be represented by the blockdiagram of Fig. 5. In this graphic it is assumed that the MC is

connected to a strong grid (i.e. the grid impedance≈ 0). It can be shown that the damping coefficient of the system shown in

Fig. 5 can be obtained as:

filter eq

f f

R

C Lζ ζ +=

2 (24)

Where ζfilter is the no-load damping coefficient (i.e. when

Req→∞). If 1/ Req is negative, the damping coefficient may become negative producing unstable operation.

When a small signal model is derived, then the dynamicresistance can be obtained as:

i

i

i

ieq

i

v

i

v R

∆

∆≈

∂

∂= (25)

Therefore for a disturbance ∆vi a current ∆i is produced. If τ f is zero and assuming that the d-q components of the voltages

and currents in the input filter are decoupled, then the dynamicequivalent resistance in the d-q axis can be calculated as:

))(Re(

1

))(Re(

1

02 _

02 _

ω

ω

o

qeq

o

d eq

Y q R

Y q R

=

−=

(26)

Therefore for the matrix converter operating at unity powerfactor there is always a negative resistance in either the d or qaxis (see (22)). Replacing this negative resistance in (25) thedamping coefficient can reach negative values when the

matrix converter is feeding a relatively large load at theoutput. Therefore the system can become unstable when a highload is fed by the matrix converter.

To increase the stability, the synchronous rotating filter is

used. For instance, if τ f → ∞, then the resistance calculated

from (22,25) is a large value (i.e. Req→∞ ) and the WECS isstable in the whole operating range. However, to use a very

high value of τ f is not appropriate because in this case theSVM cannot compensate disturbances/harmonics in the MCinput voltage and the output current can have a relatively high

Host PC

Wind TurbineModel

3ΦInput

DC

Machine

Doubly Fed

induction

Generator

Modulation

and Control

DSP-based

Control and

Emulation system

∼

∼

∼Input Filter Variac

FPGA Based

Switching

ωr

Fig. 6. The experimental system.

harmonic distortion. Therefore a trade-off between system

stability and output current waveform is required in order toregulate the cut-off frequency of the synchronous rotatingfilter.

III. EXPERIMENTAL R ESULTS

The control system of Figs 1 and 2 have been experimentallyimplemented in the experimental rig shown in Fig. 6. Thevariable speed wind turbine is emulated using a speedcontrolled dc machine. To implement the emulation, a wind

speed profile is sent from the host PC to a second order modelof the WECS implemented in the DSP [12-14]. The power

coefficient curve, C p(TSR, β ), of [13] has been discretised andstored in a look-up table. Linear interpolation is used to obtain

the power coefficient from the look-up table.From the wind turbine model the rotational speed of the

WECS generator, ω r *, is calculated in each sampling time. The

driving dc motor forces the DFIG rotational speed to this value[12]. With this emulation technique the DFIG rotates at the

same speed as that of a generator driven by a real wind turbine.A complete discussion of the emulation technique can be found

in [13].The matrix converter is controlled using the SVM algorithm

presented in [3]. A switching frequency of 12.5kHz is used tocontrol the bidirectional switches. The matrix converter iscontrolled from a DSP and FPGA-based external hardware.The commutation is controlled using the four-step method

implemented in the FPGA [1]. A speed encoder of 10,000 pulses per revolution is used for comparison purposes and forthe control of the dc machine. The MC input filter isimplemented using a inductor of 0.625mH, a delta connected

capacitor of 2µF and a parallel resistor of 100Ω.Fig. 7, shows the performance of the proposed WECS when

a wind turbine of 4kW is emulated using the dc machine. Fig.

7a, shows the wind profile used in the emulation. In Fig. 7b thereal and estimated rotational speed (using the MRAS observer)

are shown. The speed tracking is good with a small error.

2441



0 5 10 15 20 25 306

7

8

9

10

11

12

W i n d s p e e d m s - 1

(a)

0 5 10 15 20 25 301200

1250

1300

1350

1400

1450

1500

S p e e d ( r p m )

ωr and ωr

∧

(b)

0 5 10 15 20 25 30-5

-3

-1

1

3

5

S p e e d E r r o r ( r p m ) (c)

0 5 10 15 20 25 30-3

-2

-1

0

1

2

3

Time (s)

(d)

R o t o r p o s i t i o n

e r r o r ( d e g r e e s )

Fig. 7. Variable speed operation of the proposed WECS.a) Wind profile b)Estimated and real rotational speed c) Speed estimation error d) Rotor position

estimation error.

As shown in Fig. 7c the speed error is below ±4rpm even forrapid speed transients. The position error is shown in Fig. 7d.

This error is below ±3 degrees for the whole wind profile. In

order to drive the wind turbine to the point of maximumaerodynamic efficiency, for all the experimental test shown inthis work, the torque current is regulated as:

2*ˆ r opt qr k i ω = (27)

where k opt is dependant on the blade aerodynamic, gear boxratio, DFIG parameters etc.

Fig. 8a shows the power component of the MC inputcurrent (Idi) and the DFIG torque current (Iqr ), for a wind

profile similar to that shown in Fig.7a. For sub-synchronousoperation the MC converter is supplying energy from the grid

to the DFIG rotor. For super- synchronous operation, the DFIGis supplying energy to the grid from the rotor and stator. Therotational speed corresponding to the test of Fig. 8a is shown inFig. 8b. In Fig. 8c the total power generated by the DFIG ( P T )and the power generated/absorbed by the rotor are shown.

For the experimental tests shown in Figs. 7-8, the cut-offfrequency of the synchronous rotating filter has been adjusted

to ≈50Hz to achieve a good performance. In order to test thesystem stability, the DFIG is operated at 600rpm, with a steady

state torque current of 8A.The cut-off frequency of thesynchronous rotating filter is varied from 20Hz to 2000Hz in30s. Fig. 9 shows the experimental results obtained in this test.

In Fig. 9a the variation of the term f c= 1/(2πτf ) is shown.Fig.9b shows the variation of the voltage transfer ratio. Whenthe cut-off frequency is low, the control system is operatingwith reduced oscillations in the voltage transfer ratio. Whenthe term f c is increased, the oscillations in the voltage transfer

ratio increase until the system becomes unstable at a cut-offfrequency of about 900Hz.

Fig. 11 shows the MC input current for a high cut-offfrequency of 700Hz (with this cut-off frequency the system isclosed to instability). In this case the damping coefficient of(24) is low, and harmonics close to the resonant frequency ofthe input filter are produced in the input current and voltage.

0 5 10 15 20 25 30-4

-2

0

2

4

6

8

10

12

Iqr

Idi

Sub-synchronous

`speed

Super

Synchronous

speed

C u r r e n t ( A )

(a)

0 5 10 15 20 25 30700

800

900

1000

1100

1200

1300

R o t a t i o n a l S p e e d ( r p m )

Time (s)

Sub-synchronous

speed

Super synchronous

speed

(b)

0 5 10 15 20 25 30-4500

-3500

-2500

-1500

-500

500

P r

P T

P o w e r ( W )

Time(s)

(c)

Fig. 8. Sub-synchronous and super-synchronous operation. a) Torque current

and MC input power current b) Rotational speed c) DFIG generated power

from the rotor and stator .

2442



0 5 10 15 20 25 300

400

800

1200

1600

2000

C u t o f f f r e q u e n c y ( H z )

f c

(a)

0 5 10 15 20 25 300.5

0.6

0.7

0.8

0.9

V o l t a g e t r a n s f e r r a t i o q

Time (s)

(b)

Unstable operation

Fig. 10. Effect of the variation of the cut off frequency. a) variation of the 1/τ f coefficient. b) Voltage transfer ratio q.

0 5 10 15 20 25 30 35 40-10

-6

-2

2

6

10

Time (ms)

C u r r e n t ( A )

Fig. 11. MC input current for a filter cut- off frequency of ≈700Hz.

IV. CONCLUSIONS

In this paper a control system suitable for the operation of aWECS based on a DFIG fed by a matrix converter has been presented. The control system proposed in this work used arotor current MRAS observer for sensorless vector controlledoperation of the DFIG.

The stability of the proposed WECS is improved by using

a synchronous rotating filter in the input voltage vi. Filteringthe voltage used in the SVM algorithm enhances the dampingcoefficient of the input filter.

Experimental results have been presented in this work. The performance of the WECS have been tested using wind profiles, sub-synchronous and super-synchronous operation

with wind profiles of relatively high variability. For all the teststhe performance of the proposed WECS has been excellent.

V. ACKNOWLEDGEMENT.

This work has been funded by Fondecyt Chile, contract Nr.

1060498. The support of the University of Magallanes is alsoacknowledged.

VI. APPENDIX

Parameters of the experimental rig

DFIM : Stator 220V delta, rotor 250V star, 3.5kW, six poles,

Rr =0.525Ω, R s=0.398, L s=0.0835H, Lm=0.0796, Lr =0.0825.

Turn ratio Nr /Ns ≈ 1.4.

Control loops: iqr and idr control loops, designed with a natural

frequency of ω n≈ 70Hz, damping coefficient ≈ 0.8. MRAS

observer designed for a bandwidth of ≈ 10Hz.Wind turbine emulation: A small wind turbine of ≈4kW is

emulated, nominal speed ≈1200rpm, blade inertia J≈0.8kgm2,

B≈0.01Nms.

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