Power control of a doubly fed induction machine via output feedback

Control Engineering Practice 12 (2004) 41–57

Power control of a doubly fed induction machinevia output feedback

Sergei Peresadaa, Andrea Tillib,*, Alberto Toniellib

aDepartment of Electrical Engineering, National Technical University of Ukraine Prospect Pobedy 37, Kiev 252056, UkrainebDepartment of Electronics, Engineering Faculty, Computer and System Sciences (DEIS), University of Bologna, Viale Risorgimento 2,

40136 Bologna, Italy

Received 7 June 2002; accepted 14 November 2002

Abstract

A new output feedback control algorithm for a doubly fed induction machine (DFIM) is presented. The asymptotic

regulation of active and reactive power is achieved by means of direct closed-loop control of active and reactive components

of the stator current vector, presented in a line-voltage-oriented reference frame. To get the maximum generality of the

solution, the usual assumption of negligible stator resistance is not made. A full-order DFIM model is used for the control

algorithm development. The proposed control system is robust with respect to bounded machine parameter variations and

errors on rotor position measurement. In the paper, it is also shown how the proposed current control algorithm

can be modified in order to achieve asymptotic active current tracking and zero reactive current stabilization during

steady state. An extension for the speed control objective and output EMF control during the excitation–synchronization

stage are also presented. Simulation and experimental tests demonstrate high dynamic performance and robustness of the

control algorithm for typical operating conditions. The proposed controller is suitable for both energy generation and electrical

drive application with restricted speed variation range.

r 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Electric machines; Inverter drives; Electric energy generation; Speed control; Output feedback; Lyapunov-based control

1. Introduction

A vector-controlled doubly fed induction machine(DFIM) is an attractive solution for high-performancerestricted speed-range electric drives and energy genera-tion applications (Leonhard, 1995). In Fig. 1, the typicalconnection scheme of this machine is reported. Thestator windings are directly connected to the line grid,while the rotor windings are supplied by a bi-directionalpower converter. This solution is suitable for all of theapplications where limited speed variations around thesynchronous speed are present. Since the power handledby the rotor side (slip power) is proportional to the slip,

the energy conversion requires a rotor-side powerconverter which handles only a small fraction of theoverall system power. Moreover, when the DFIM isused as a variable-speed drive, the slip power isregenerated during motor operating conditions by therotor-side converter to the line grid, resulting in highlyefficient energy conversion. Electric energy generationsystems operating at variable speed have severaladvantages when compared with fixed-speed synchro-nous and induction generation. In generation systemsdriven by a diesel engine, the variable-speed operationdepending on the generated power allows for areduction of fuel consumption. In hydroelectric genera-tion systems it increases the energy efficiency up to 10%.In wind energy generation systems the adjustment of theshaft speed as a function of the wind speed permits ahigher energy capture by maximizing the turbineefficiency. Reduction of the torque ripple in thedrive train due to torsional mode resonance can be

ARTICLE IN PRESS

*Corresponding author. Tel.: +39-051-2093024; fax:+39-051-

2903073.

E-mail addresses: [email protected] (S. Peresada),

[email protected] (A. Tilli), [email protected] (A. Tonielli).

0967-0661/03/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved.

doi:10.1016/S0967-0661(02)00285-X

additionally achieved with variable-speed operation(Nakra & Duke, 1988).An important feature of the vector-controlled DFIM

reported in Leonhard (1995) and Vas (1990) is thepossibility to achieve decoupled control of the stator-side active and reactive power in both motor andgenerator applications. Moreover, if a suitably con-trolled AC/AC converter is used to supply the rotorside, the power components of the overall system can becontrolled with low-current harmonic distortion in thestator and rotor sides.The fundamentals of DFIM vector control are

presented in Leonhard (1995). Different strategies wereproposed to solve the DFIM control problem. The mostimportant results are reported in Leonhard (1995),Pena, Clare, and Asher (1996a) and Hopfensperger,Atkinson, and Lakin (2000). All of them are based onthe classical concept of field orientation (stator or air-gap flux) used as a torque–flux decoupling technique forinduction motor control. Since in DFIM both statorand rotor currents are available from measurement, theflux vectors (stator, air-gap or rotor) can be computedusing flux–current correlation equations. Consequently,the DFIM control problem is typically classified as anonlinear state-feedback problem.Under the assumption of rotor current-fed DFIM and

negligible stator resistance, the torque and stator-sidereactive power control problem is transferred to rotorcurrent control if the rotor currents are defined in afield-oriented reference frame. Torque (active power) orspeed control objective, together with the stator-sidereactive power regulation (stabilization) are typicallyconsidered.The structure of a standard DFIM controller includes

two-axis high-gain rotor current control loops with PIcurrent controllers, implemented in a flux-orientedreference frame. Two rotor current references are usedas scaled references for torque and reactive power. Thesolutions based on direct stator flux orientation,reported in Leonhard (1995), Yamamoto andMotoyoshi (1992), Pena et al. (1996a), Hopfenspergeret al. (2000), Walczyna (1991), rely on some simplifyingassumptions. In particular, stator resistance is usually

considered negligible. This hypothesis, which is typicalfor high-power DFIMs, leads to neglect also the statorflux poorly damped dynamics in the controller designsince the stator flux vector is always assumed constant inquadrature with the line-voltage vector. As far as theauthors know, no analytically proven full-order controlalgorithms based on the stator flux field orientation areavailable in literature. State-feedback linearization hasbeen applied in Bogalecka and Kzreminski (1993) tosolve the DFIM control problem. The assumption ofcurrent-fed rotor is used with an additional first-orderfilter in the control loop. Rotor position sensorlesssolutions have been considered in Xu and Cheng (1995),Hopfensperger et al. (2000) and Bogalecka (1993). Theoperation of a vector-controlled DFIM supplying anisolated load is reported in Pena, Clare, and Asher(1996b). The classical approach for DFIM vectorcontrol (Leonhard, 1995) requires measurements ofstator, rotor currents and rotor position. In order toachieve synchronization with the line-voltage vector forsoft connection to the line grid, the information aboutline voltages is also needed. Exact knowledge ofinduction machine inductances (including the saturationeffect) is required to compute fluxes from currents. Thenecessity for high-precision rotor position measurementhas been addressed in Xu and Cheng (1995). In Penaet al. (1996a, b), the authors use the integration of statorvoltage equations in order to estimate stator fluxes. Thissolution requires particular adjustments to avoid open-loop-integration drift due to variations of statorresistance and measurements offset. However, it mustbe underlined that the compensation of such effect forDFIMs is less difficult than for typical induction motordrives. In fact, the stator flux components in a fixed a–breference frame are sinusoidal with a frequency equal tothat of the line grid and the stator resistances are verysmall in large DFIMs, usually adopted in industrialpractice. Different approaches for the implementationof the stator flux-oriented reference frame are discussedin Hopfensperger et al. (2000).In Peresada, Tilli, and Tonielli (1998), an alternative

approach for the design of DFIM active–reactive powercontrol is proposed. The controller development isbased on implementation of a line-voltage vector-oriented reference frame. Since the line-voltage vectorcan easily be measured with negligible errors, thisreference frame is DFIM parameter-independent incontrast to the field-oriented one. Moreover, informa-tion about line voltage is typically needed in order toperform the soft connection of the DFIM to the line gridduring the preliminary excitation–synchronizationstage. This full-order control algorithm ensures globallyasymptotically stable torque tracking and stator-sideunity-power-factor. It is demonstrated that conditionsof stator flux field orientation and line-voltage vectororientation are equivalent if the stator-side power factor

ARTICLE IN PRESS

DC/AC (Inverter)

AC/DC (Rectifier)

DFIM

Mechanical Load or Mover

Control Unit

Line Grid

Fig. 1. Typical connection scheme of a DFIM.

S. Peresada et al. / Control Engineering Practice 12 (2004) 41–5742

is controlled at unity level. In Peresada, Tilli, andTonielli (1999a), the approach of Peresada et al. (1998)is extended to the control problem of speed tracking andstator-side power factor stabilization of rotor current-fed DFIMs. A rotor speed/position sensor is used in thealgorithms Peresada et al. (1998, 1999a), but noparticular accuracy is required. The above controlalgorithms utilize the concept of indirect flux regulation(similarly to indirect field-oriented control of squirrel-cage induction motors).Both direct and indirect stator flux field-oriented

solutions are open loop with respect to the outputvariables, i.e. torque (active power) and reactive power.Robustness of the above control algorithms with respectto parameter variation and rotor position measurementare based on the natural stability properties of theDFIM electromagnetic subsystem. In order to improverobustness with respect to parameter variations anderrors in rotor position measurements, the outer stator-side reactive power and active power loops are added inmany publications. Nevertheless, no stability analysis isgiven for such solutions. In Peresada, Tilli, and Tonielli(1999b), a new full-order nonlinear control algorithm isproposed. It is shown that direct closed-loop control ofactive and reactive power guarantees global asymptoticregulation of output variables and internal stability,under the condition of measurable stator currents andvoltages, as well as rotor position and speed. Theconcepts of cascaded architecture and field-orientationare not used to develop the controller.The aim of this paper is the generalization of the

preliminary result given in Peresada et al. (1999b) for thecase of torque and speed control. Rigorous stabilityanalysis of the electromagnetic dynamics is given.Relying on this result, a new dynamic second-orderspeed controller with load compensation has beendeveloped. This solution guarantees stator-side reactivepower regulation and asymptotic speed regulation underthe condition of constant load torque provided thatDFIM physical limits are satisfied. In addition, a newclosed-loop synchronization–excitation control algo-rithm is proposed in the paper which guaranteestransient free connection of the DFIM to the line grid.The relative-degree equal to one of the DFIM betweenthe stator currents (outputs) and rotor voltages (inputs)is exploited in order to achieve robustness propertieswith respect to parameter perturbations and particularattention is paid to the residual internal dynamics. Noinformation on rotor currents is required to achieve thecontrol objectives hence, in industrial plants, only arough measurement of these variables is necessary forprotection purposes. It is worth observing that theabsence of rotor current feedback does not imply thatthese variables are ‘uncontrolled’, in fact, as enlightenedin the Lyapunov-based controller development, globalasymptotic stabilization of all state variables is guaranteed.

The proposed nonlinear output feedback controllerdemonstrates strong robustness properties with respectto stator and rotor resistances/inductances variation andto rotor position measurement errors. In addition,owing to the closed-loop structure with true-stator-current feedback, the controller compensates for non-idealities of the electric machine magnetic structure,delivering improved stator current waveforms. Thecontroller is suitable both for drive application andelectric energy generation (e.g. in alternative energyplants) including operation as an autonomous generatorduring the excitation–synchronization stage.The paper is organized as follows. In Section 2, the

DFIM model and the control problem statement arepresented. In particular, the selection of the line-voltage-vector-oriented reference frame is deeply discussed,recalling also the active/reactive power control objectiveconsidered in Peresada et al. (1999b). Stator-side activeand reactive current controllers are designed in Section 3.The extension of the active current control algorithm forspeed control objective is given in Section 4. The resultsof simulation and experimental tests are presented inSection 5. As underlined in Section 3, a variant of theproposed stator current controller, which guaranteesstator active current tracking and stabilization to zero ofthe stator reactive current during steady state, isreported in Appendix A; while, in Appendix B, theabove-mentioned excitation–synchronization algorithm,which replicates the current controller structure ofSection 3, is presented.

2. DFIM model and control objectives

Under the assumption of linear magnetic circuits andbalanced operating conditions, the equivalent two-phasemodel of the symmetrical DFIM, represented in anarbitrary rotating (d2q) reference frame is

’e ¼ o;

’o ¼1

J½Tg � T �; Tg ¼ mðcd iq � cqidÞ;

’id ¼ �gid þ o0iq þ abcd þ bocq þ1

sud � bu2d ;

’iq ¼ �o0id � giq � bocd þ abcq þ1

suq � bu2q;

’cd ¼ �acd þ ðo0 � oÞcq þ aLmid þ u2d ;

’cq ¼ �ðo0 � oÞcd � acq þ aLmiq þ u2q;

’e0 ¼ o0; ð1Þ

where id ; iq; cd ; cq are the components of the statorcurrent and rotor flux vectors, u2d ; u2q are thecomponents of the rotor voltage vector, while ud ; uq

ARTICLE IN PRESSS. Peresada et al. / Control Engineering Practice 12 (2004) 41–57 43

represent the line-voltage components (stator windingsare directly connected to the line grid); e and o are rotorangular position and speed; T is the external torqueapplied to the mechanical system of the DFIM; Tg is thetorque produced by the electrical machine; J is the totalrotor inertia; e0 and o0 are angular position and speedof the (d � q) reference frame with respect to the a-axisof the fixed stator reference frame (a � b).Variablesexpressed in the ðd � qÞ reference frame are given by

x1dq ¼ e�Je0x1ab

x2dq ¼ e�Jðe0�eÞx2uv

where e�Jx ¼cos x sin x

�sin x cos x

" #;

J ¼0 �1

1 0

" #; ð2Þ

where xyz stands for two-dimensional vectors in thegeneric ðy � zÞ reference frame; subscript ‘1’ indicatesstator variables while subscript ‘2’ indicates rotorvariables; ðu � vÞ indicates the rotor reference frameand e is the rotor angle (i.e. the angle between the u- andthe a-axis).Positive constants in (1), related to electrical and

mechanical parameters of the DFIM, are defined asfollows:

s ¼ L1 1�L2

m

L1L2

� �; b ¼

Lm

sL2; m ¼

3

2

Lm

L2;

a ¼R2

L2; g ¼

R1

sþ aLmb

� �;

where R1; R2; L1; L2 are stator/rotor resistances andinductances, respectively, Lm is magnetizing inductance.One pole pair is assumed without loss of generality.Depending on the DFIM application, T in (1) has

different meanings.

* When the DFIM is used as an electric generator,T is the torque produced by a controlledprimary mover. Torque Tg produced by the DFIMis a perturbation for speed control system of theprimary mover. Without loss of generality, it isassumed that the mechanical dynamics of the DFIMis properly stabilized by the primary mover speedcontroller.

* When the DFIM is used as a motor, T is the externalload torque. Usually, in this condition, a speedcontrol loop, acting on the torque Tg; generated bythe DFIM, is present. In wind power plants, thespeed control objective can be formulated in order toadjust the turbine speed as a function of the windspeed.

The main control objective considered is the regula-tion of DFIM stator-side active and reactive powers (i.e.the active and reactive power exchanged between the

line grid and the stator port of the DFIM). The statoractive power control objective is significant for energygeneration applications since, given a prime mover ableto produce a given power at a given speed, it is possibleto compute (Leonhard, 1995) the stator power whichmust be imposed to have a total DFIM active powerequal to that available from the prime mover (keeping inmind the overall efficiency) at the considered speed.Alternatively, the control of stator reactive power isrelevant since it can coincide with the control of the totalreactive power delivered by the DFIM system if avector-controlled rectifier is adopted at rotor sideimposing the exchange of active power only with theline grid.Active and reactive power at stator side are given by

Pa ¼ 32ðud id þ uqiqÞ; Pr ¼ 3

2ðuqid � udiqÞ: ð3Þ

The first step in the mathematical formulation of thecontrol problem is the selection of the reference framefor control algorithm development. It is well known thatvector control of an induction machine with the samespecifications can be designed in any reference frame.The criteria adopted in this case for reference frameselection are:

* reliability of realization under the condition of plantparameter variation and measurement errors;

* simplicity and approximation avoidance in theformulation of control objectives.

Many approaches to DFIM control, (Leonhard,1995), use the field-orientation concept based on theavailability of stator and rotor current measurements.Stator, rotor or air-gap flux can be computed directlyfrom current vectors to organize the flux-orientedreference frame. Nevertheless, one of the two currentvectors should be transformed to rotor or statorreference frame using the rotor position angle in thecoordinate transformation matrix (2). This requires anaccurate position measurement (resolution of standardsensors is usually satisfactory, but even small misalign-ment could be very harmful). Moreover, stator-sidepower factor control implies variations of the fluxmodulus which produces variations of the magnetizinginductance (involved in flux computation) due to thesaturation effect.In order to reduce the effect of the above inaccuracies

in the reference frame generation and in vectortransformations, a line (stator) voltage vector referenceframe (d � q) has been adopted (the d-axis is alignedwith the line-voltage vector). This reference frame isindependent of machine parameters and positionmeasurement accuracy. The space location of DFIMvectors in the line-voltage-vector-oriented referenceframe is shown in Fig. 2.

ARTICLE IN PRESSS. Peresada et al. / Control Engineering Practice 12 (2004) 41–5744

Using the line-voltage vector reference frame, a simpleand smooth connection of the stator windings to the linegrid can be performed during the start-up procedure ofthe DFIM-based system.Measured line (stator) voltages in two-phase presen-

tation are equal to

ua ¼ U cos ðo1t þ j1Þ; ub ¼ U sin ðo0t þ j0Þ: ð4Þ

where U and o0 are the line-voltage amplitude and theangular frequency, j0 is the initial angular position ofthe line-voltage vector.The synchronous stator voltage-oriented reference

frame is defined by setting in (1) and (2)

cos ðe0Þ ¼ua

U; sin ðe0Þ ¼

ub

U; o0 ¼ o1: ð5Þ

Under this transformation ud ¼ U and uq ¼ 0 in theDFIM model (1). In addition, currents id and iq; in theline-voltage-oriented reference frame (see Fig. 2) repre-sent the active and reactive components of the statorcurrent vector. The expressions of active and reactivepowers (3) can be presented as

Pa ¼ 32Uid ; Pr ¼ � 3

2Uiq: ð6Þ

For convenience, a positive power flow is defined in(6) when its direction is from the power source to theelectric machine. Conditions iqo0 and iq > 0 indicate alagging and leading power factor, respectively; iq ¼ 0 isthe condition of unity power factor. From (6), it followsthat active–reactive power control objective is equiva-lent to active–reactive stator currents control. Let P�

a

and P�r be the references for the power components at

stator side for the DFIM. Using (6), references for thecomponents of the stator current, in the voltage-orientedreference frame, are given by

i�d ¼2

3

P�a

U; i�q ¼ �

2

3

P�r

U: ð7Þ

The control problem of the DFIM generator isformulated in terms of stator active–reactive currentregulation as follows.

Proposition. Consider the DFIM model (1) undercoordinate transformation (2), (5). Let us assume that:A.1. The stator voltage amplitude and frequency are

constants (stator windings are directly connected to theline grid).A.2. References for active and reactive stator currents

are constant and bounded, or represent ramp signalswith bounded first derivative and bounded amplitude.A.3. Under the assumption of a properly controlled

primary mover, the rotor speed is time varying,measurable and bounded together with its first timederivative.A.4. Stator currents and voltages as well as rotor

position and speed are available from measurements.Under these conditions a dynamic output feedback

controller exists in the form

u2d

u2q

!¼ f i�d ; i

�q; ’i

�d ; ’i

�q; id ; iq;U ;o0;o; z

� �;

’z ¼ j o0; i�d ; i

�q; id ; iq

� �; ð8Þ

which guarantees:O.1. Asymptotic active–reactive stator current regula-

tion independent of speed variations; i.e.

limt-N

ð*idÞ ¼ 0; limt-N

ð*iqÞ ¼ 0; ð9Þ

where *id ¼ id � i�d ; *iq ¼ iq � i�q are current errors.O.2. Boundness of all internal signals and synchroni-

zation of the control actions with the stator voltagevector; and which has the following qualitative property:O.3. Good robustness with respect to stator and rotor

resistance variations and constant error in rotor positionmeasurement due to position sensor misalignment.

The proof of O.1 and O.2 of the proposition is givenby the controller design and the stability analysispresented in the next section. The validity of O.3 isbased on the general stability arguments for the closed-loop systems having proportional-integral controllersand confirmed by simulation and experimental testsreported in Section 5. In particular, the choice of a line–voltage-oriented reference frame (as discussed before)and the adopted control algorithm structure (asdiscussed in the following paragraphs) play a key rolein achieving good robustness properties.In Section 4, it is shown how the stator active–reactive

current control can be fitted to the speed control of theDFIM.

3. Design of the output feedback control algorithm for the

DFIM

Before starting the design procedure let us make thefollowing remark: the objective of active–reactive

ARTICLE IN PRESS

ua

ub u

i

id

iq 0 a

b

d

q

Fig. 2. Space vectors of a DFIM.

S. Peresada et al. / Control Engineering Practice 12 (2004) 41–57 45

current control must be achieved without violations ofthe physical constraints on the DFIM operation. Theseconstraints require that machine fluxes are bounded androtate synchronously with the stator voltage vectorduring steady state. Let us consider stator fluxequations:

c1d ¼ L1id þ LmL�12 ðcd � LmidÞ;

c1q ¼ L1iq þ LmL�12 ðcq � LmiqÞ: ð10Þ

The stator flux dynamics derived from (1) with u1d ¼U ; u1q ¼ 0 is equal to

’c1d ¼ o0c1q � R1id þ U ;

’c1q ¼ �o0c1d � R1iq: ð11Þ

Achieving the condition of ideal stator currenttracking, i.e. id i�d ; iq i�q; makes the dynamics of thestator fluxes unobservable by the controller. Thisdynamics has oscillatory behavior and it is not suitablefor practical application. Even if in a real system theperfect tracking condition is not achievable due to thelimited power of the rotor converter and line impedance,problems would be encountered if a feedback linearizingstrategy were applied. From this consideration, it ispossible to conclude that the stator active–reactivecurrent error dynamics cannot be independent of fluxdynamics, restricting the controller development on thelevel of the regulation problem.The design procedure is performed in two steps: a flux

control is designed first, to achieve flux regulation, thenthe current control algorithm is developed.Let us define the flux regulation errors as

*cd ¼ cd � c�d ;

*cq ¼ cq � c�q; ð12Þ

where flux references, c�d and c�

q; will be defined lateraccording to stator current control objectives.Using definition (12), the last two equations of the

DFIM model (1) can be rewritten in ‘error form’ as

’*cd ¼ � a ð *cd þ c�dÞ þ o2 ð *cq þ c�

qÞ

þ a Lmð*id þ i�dÞ þ u2d � ’c�d ;

’*cq ¼ � a ð *cq þ c�qÞ � o2 ð *cd þ c�

dÞ

þ a Lmð*iq þ i�qÞ þ u2q � ’c�q; ð13Þ

where o2 ¼ o0 � o is the slip angular frequency.Constructing the flux control algorithm as

u2d ¼ ac�d � o2c

�q � aLmi�d þ ’c�

d þ vd ;

u2q ¼ ac�q þ o2c

�d � aLmi�q þ ’c�

q þ vq; ð14Þ

the flux error dynamics becomes

’*cd ¼ �a *cd þ o2*cq þ aLm

*id þ vd ;

’*cq ¼ �a *cq � o2*cd þ aLm

*iq þ vq; ð15Þ

where vd ; vq will be defined later.Applying the control algorithm (14), the current error

dynamics from the first two equations of (1) can berewritten as

’*id ¼ � g*id þ o0*iq þ ab *cd þ bo *cq � bvdþ

� ’i�d � b ’c�d �

R1

si�d þ o0i

�q þ

1

sU þ bo0c

�q;

’*iq ¼ � g*iq � o0*id þ ab *cq � bo *cd � bvqþ

� ’i�q � b ’c�q �

R1

si�q � o0i

�d � bo0c

�d : ð16Þ

From Eq. (16) it follows that a reasonable choice ofthe flux references dynamics is given by the followingdifferential equations:

’c�d ¼

1

bbo0c

�q þ

1

sU �

R1

si�d þ o0i

�q � ’i�d

� �;

’c�q ¼

1

b�bo0c

�d �

R1

si�q � o0i

�d � ’i�q

� �: ð17Þ

Substituting (17) into (16) the resulting current–fluxerror dynamics becomes

’*id ¼ �g*id þ o0*iq þ ab *cd þ bo *cq � bvd ;

’*iq ¼ �g*iq � o0*id þ ab *cq � bo *cd � bvq;

’*cd ¼ �a *cd þ o2*cq þ aLm

*id þ vd ;

’*cq ¼ �a *cq � o2*cd þ aLm

*iq þ vq: ð18Þ

According to (17), the flux references are given by alinear time-invariant differential equation which is notreally implementable, since it is stable but not asympto-tically (i.e. it is a non-autonomous harmonic oscillator).A particular solution of (17), where oscillating terms areavoided by means of a suitable selection of the initialcondition, is given by

c�d

c�q

24

35 ¼ �

1

sb1

o0J

U

0

" #þ sI�

R1

o0J

� � i�d

i�q

" #(

�R1

XNk¼1

1

o0J

� �kþ1dk

dtk

i�d

i�q

" #" #): ð19Þ

From (19), it follows that for arbitrary trajectories ofcurrent reference all of the time derivatives together withtheir initial conditions should be known. The followingdevelopment is based on the assumption that both

ARTICLE IN PRESSS. Peresada et al. / Control Engineering Practice 12 (2004) 41–5746

current reference signals have bounded first time-derivative with all of the higher-order ones equal tozero. Then the flux references are

c�q ¼

1

bo0

R1

si�d � o0i

�q �

1

sU �

R1

so0

’i�q

� �;

c�d ¼

1

bo0�

R1

si�q � o0i

�d �

R1

so0

’i�d

� �ð20Þ

with .i�d .i�q 0; according to assumption A.2 oncurrent references.To simplify stability analysis, it is worth perfor-

ming the following linear time–invariant coordinatetransformation

*id

*iq

zd

zq

2666664

3777775 ¼

1 0 0 0

0 1 0 0

1 0 b 0

0 1 0 b

2666664

3777775

*id

*iq

*cd

*cq

26666664

37777775: ð21Þ

In the new coordinates, system (18) becomes

’*id ¼ �ðgþ aÞ *id þ o2*iq þ azd þ ozq � bvd ;

’*iq ¼ �ðgþ aÞ *iq � o2*id þ azq � ozd � bvq;

’zd ¼ o0 zq �R1

s*id ; ’zq ¼ �o0 zd �

R1

s*iq: ð22Þ

Setting vd ¼ vq ¼ 0 in (22) an open-loop controlalgorithm is obtained (no current measurement isneeded), based on the natural passivity properties ofthe DFIM. To check stability of the equilibrium pointx1 ¼ ð*id ; *iq; zd ; zqÞ

T ¼ 0 consider the following Lyapunovfunction

V1 ¼ 12xT1P1x1 ð23Þ

with P1 ¼ PT1 > 0 equal to

P1 ¼

1 0 �1 0

0 1 0 �1

�1 0 g1 0

0 �1 0 g1

2666664

3777775; ð24Þ

where g1 > 1 in order to guarantee that P1 > 0:Selecting g1 ¼ 1þ s=R1ð2aþ aLmbÞ; which verifies

the previous inequality owing to positiveness of theDFIM parameters, the derivative of V1 along the systemtrajectories is equal to

’V1 ¼ � a ð1þ LmbÞð*id2 þ *iq2Þ þ ðz2d þ z2qÞ

h io � ajjx1jj22; ð25Þ

where jj � jj2indicates the Euclidean norm.From (23) and (25), according to standard Lyapunov

stability arguments, it can be concluded that theequilibrium point x1 ¼ 0 is globally exponentially stable.

By defining the control signals vd and vq in (22) as

vd ¼1

bki*id ; vq ¼

1

bki*iq; ð26Þ

where ki > 0 is the proportional gain of the currentcontrollers, a closed-loop proportional current control isobtained. Previous stability analysis, based on (23) and(25) is still valid with

g2 ¼ 1þs

R1ð2aþ aLmbþ kiÞ > 1;

’V1 ¼ � a 1þ Lmbþki

a

� �ð*id2 þ *iq

2Þ þ ðz2d þ z2qÞ� �

o � ajjx1jj22: ð27Þ

The proposed nonlinear active–reactive current con-troller, given by (14), (20) and (26) has a proportionalaction based on reliably measured current feedbacksignals. Since the linear time-varying dynamics of theDFIM (22) has relative degree equal to one and it isexponentially stable for any ki > 0; the robustnessproperties of current regulation can be improved byincreasing the current controller gain ki:In order to compensate, during steady-state condi-

tions, for constant perturbation generated by DFIMparameter variations and error in rotor positionmeasurement, the following two-dimensional propor-tional-integral current controller is designed

vd ¼1

bðki

*id þ l*iq � yd Þ; ’yd ¼ �kii*id � l

R1

s*iq;

vq ¼1

bðki

*iq � l*id � yqÞ; ’yq ¼ �kii*iq þ l

R1

s*id ; ð28Þ

where kii > 0 is the integral gain of the currentcontrollers and l ¼ kiio�1

0 is the ‘cross gain’.The resulting flux–current dynamics, with the PI

controllers (28), is the following:

’*id ¼ �ðgþ aþ kiÞ*id þ ðo2 � lÞ*iq þ azd þ ozq þ yd ;

’*iq ¼ �ðgþ aþ kiÞ*iq � ðo2 � lÞ*id þ azq � ozd þ yq;

’zd ¼ o0zq �R1

s*id ; ’zq ¼ �o0zd �

R1

s*iq;

’yd ¼ �kii*id � l

R1

s*iq; ’yq ¼ �kii

*iq þ lR1

s*id : ð29Þ

To investigate stability conditions of the equilibriumpoint x ¼ 0; where x ¼ ð*id ; *iq; zd ; zq; yd ; yqÞ

T; let usconsider the non-negative function

V ¼ 12xTPx ð30Þ

ARTICLE IN PRESSS. Peresada et al. / Control Engineering Practice 12 (2004) 41–57 47

with P ¼ PT > 0 equal to

P ¼

P1

0

0 o�10

�o�10 0

00 �o�1

0

o�10 0

g2 0

0 g2

2666666666664

3777777777775; ð31Þ

where

g1 ¼ ð1þ eÞ þ g11; e > 0; g11 > 0; g2 > 0;

g11g2 >1

o20

ð32Þ

to guarantee that P > 0:The time derivative of V along the trajectories of (29)

is negative semidefinite and equal to

’V ¼ ’V1p0 ð33Þ

with

g1 ¼ 1þs

R1ð2aþ aLmbþ kiÞ þ

1

g2o20

� �: ð34Þ

Note that conditions (32) and (34) give freedom in theselection of the current controller parameters ki and kii;in fact for each ki > 0 and kii > 0; it is possible to find g1and g2 in order to satisfy the above conditions.From (30) and (33), it follows that V is bounded and

limt-N V ðtÞ exists and is bounded. Moreover, since V isradially unbounded, x is bounded too and, according toassumption A.3, also ’x will be bounded. That means .V

is bounded and then, according to Barbalat’s Lemma

(see Khalil, 1994), ’V -t-N

0; hence

limt-N

ð*id ; *iq; zd ; zqÞT ¼ 0: ð35Þ

Global current and flux regulation is achieved withbounded internal signals. In order to show that integralterms yd ; yq converge to zero in ideal conditions (i.e.without perturbations in model (1)), let us consider thedynamics *id ; *iq; zd ; zq given by the first four equations in(29). Since ’x is bounded and, by assumptions A.2, A.3,.id� .iq

� 0 and ’o2 ¼ � ’o is also bounded, it follows

that .*id ;.*iq; .zd ; .zq are bounded. Hence, according to

Barbalat’s Lemma

limt-N

ð’*id ; ’*iq; ’zd ; ’zqÞT ¼ 0: ð36Þ

Combining (35) and (36), it follows from (29) that

limt-N

ðyd ; yqÞT ¼ 0: ð37Þ

The control objectives O.1 and O.2 are globallyachieved according to stability analysis. The complete

equations of the nonlinear current controller are givenby


�q � aLmi�d þ ’c�

d

þ1

bðki

*id þ l*iq � ydÞ;



q

þ1

bðki

*iq � l*id � yqÞ;

c�q ¼

1

bo0�

R1

si�d � o0i

�q �

1

sU �

R1

so0

_iiq�

� �

c�d ¼

1

bo0�

R1

si�q � o0i

�d �

R1

so0

_iid�

� �

’yd ¼ � kii*id � l

R1

s*iq; ’yq ¼ �kii

*iq þ lR1

s*id ð38Þ

with the actual rotor-side voltages

u2u

u2v

" #¼ eJðe0�eÞ

u2d

u2q

" #: ð39Þ

The block diagram of the controller (38) is shown inFig. 3. In Appendix A, it is shown as to how thecontroller can be modified in order to achieve activecurrent tracking and stabilization of zero reactivecurrent during steady state. In Appendix B, a particularexcitation–synchronization algorithm is presented toperform smooth connection to the line grid of a DFIMsystem used to generate electric energy. That algorithmreplicates the current controller structure shown in thepresent paragraph in order to simplify the commutationbetween the two regulators after the connection of thegenerator to the line grid.

4. Design of the speed control algorithm for the DFIM

In this section, the speed regulation—load torquecompensation control algorithm is designed on the basisof the inner current control loops developed in Section 3.The property of global asymptotic stability of the DFIMelectrical subsystem is used for the design of the desireddynamics of the mechanical subsystem. First, the active–reactive currents control problem is transferred to thetorque—reactive current control problem, then an outerspeed control loop is designed with a dynamic speedcontroller, generating the torque reference command forthe inner torque control loop.Consider the DFIM torque Eq. (1)

Tg ¼mðcd iq � cqidÞ ¼ mðc�d i�q � c�

qi�dÞ

þ mð *cd*iq � *cq

*idÞ þ mð *cd i�q � *cqi�dÞ

þ mð *c�d iq � *c�

qidÞ ¼ T�g þ *Tg; ð40Þ

ARTICLE IN PRESS

– – – – – – – – – – – – – – – – – – –

–––––

– – – – – – – –

–––––––––––––


where torque reference is defined as

T�g ¼ mðc�

d i�q � c�qi�dÞ: ð41Þ

Substituting flux references given by (20), Eq. (41)becomes

T�g ¼

mbo0

�R1

sði�2d þ i�2q Þ þ

U

si�d

��

R1

s1

o0

� ð’i�d i�q � ’i�qi�dÞi¼D T�

g1 þ T�g2; ð42Þ

where

T�g1 ¼

mbo0

�R1

sði�2d þ i�2q Þ þ

U

si�d

� �;

T�g2 ¼ �

mbo2

0

R1

s½ð’i�d i�q � ’i�qi�dÞ�: ð43Þ

From the expression for Tg1* , the active current

reference is given by

i�d ¼U=s� Q1=2

2R1=s;

Q ¼U

s

� �2

�4R1

sR1

si�2q þ T�

g1

bo0

m

� �" #: ð44Þ

Eq. (42) establishes the steady-state relation betweenactive current reference and torque command. Oninspecting (40)–(42), it can be seen that (42) reflectsthe power balance condition in the stator side of the

DFIM, which establishes the solvability of (42) with Q > 0:Note that if ’i�q ¼ ’i�d 0 then T�

g2 0: The dynamictorque component T�

g2 can be further expressed using(44) as

T�g2 ¼

mbo2

0

R1

s’i�q i�d �

2

Q1=2

R1

si�2q

� ��

�1

o0

1

Q1=2

R1

si�q’T�

g1: ð45Þ

For practical operating conditions of the DFIM withbounded i�q; T�

g1 and small value of R1; from (44), the

result is: Q1=2DU=s: Under this condition with ’i�q

properly bounded, the dynamic torque component T�g2

is small enough to be neglected.Using definition (40) together with condition T�

g2 ¼ 0;the speed dynamics of the DFIM, given by the firstequation in (1) can be written as

’o ¼1

JðT�

g1 þ *Tg � TÞ; ð46Þ

where T is the load torque.The DFIM speed control problem under stator-side

reactive power control is formulated as follows. Assumethat the speed reference o� is constant and it can assumevalues in a restricted slip speed range around motorsynchronous speed o0: Load torque T is assumed to bebounded unknown and constant. Reference for reactivecurrent component i�q is formed as a ramp signal with a

ARTICLE IN PRESS

iq*

id*

*id

*iq

eq. (20)

ψd*

ψq*

*ψd

* ψq}

iq

iq*

–

+

qi

+

kii

k i

+ –

∫ – +

1 βvq

λ σ RS

1λ

λ σ RS

1λ

+

kii+ ∫

+

+

+ ki

di−

−

id

+

–

vd1 β

id*

eq. (14)

sensors

U,ω0

eJε 2

e Jε− 0

+

–

U

ω

ε

ε0

ε 2

u u2

u q2

u d2

u v2

ia

ib

id

iq

∼

∼

.

.

Fig. 3. Block diagram of the current controller.


properly bounded first-order time derivative. Under thiscondition, it is necessary to design the speed controlalgorithm forming the torque reference command T�

g1

which guarantees asymptotic speed regulation when i�q isconstant, i.e.

limt-N

*o ¼ 0; *o ¼ o� o�: ð47Þ

Before starting the speed controller development, it isimportant to note that current control algorithm (38)requires knowledge of the first-order time derivative ofi�d ; which implies from (44) that ’T�

g1 should be known.To satisfy this condition, the following dynamic speedcontroller is defined

T�g1 ¼ x; ’x ¼ �

1

txþ

1

t½Jð�ko *oþ #TÞ�;

’#T ¼ �koi *o; ð48Þ

where ðko; koiÞ > 0 are the proportional and integralgains of the speed PI controller with #T defined asestimation of the constant quantity T=J;while t is thetime constant of the first-order filter. Substituting (48) in(46), the speed error dynamics becomes

’#T ¼ �koi *o; ’*o ¼1

Jðxþ *Tg � TÞ;

’x ¼ �1

txþ

1

t½Jð�ko *oþ #TÞ�:

ð49Þ

To show that the speed control objective is achievedwith *Tg ¼ 0; the change of coordinates

Z ¼1

Jðx� TÞ

is considered. In addition, the load torque estimationerror is defined as

*T ¼T

J� #T

with ’*T ¼ � ’#T; since T=J is constant. Hence system (49)becomes

’*T ¼ koi *o; ’*o ¼ Zþ1

J*Tg;

’Z ¼ �1

tZ�

1

tko *o�

1

t*T: ð50Þ

The dynamics of linear time-invariant system(50) can be specified by the selection of thethree tuning parameters ko; koi; t in order to gua-rantee asymptotic stability and the desired transientperformance.The composite speed current error dynamics is

given by (50), with *Tg defined in (40), and (29)with an additional perturbation arising from thenon-compensating dynamics in (20) and fromthe second derivative of i�d according to (44).This interconnection term is a small gain feedbackbetween the two subsystems due to the scalingfactor proportional to R1Q

�1=2=o0 which is muchsmaller than 1. Hence the mechanical subsystem given

ARTICLE IN PRESS

id*

iq*

k iω

+

∫

kω

−ω

ω

+

– 1 τω*

eq. (44)

– ∫+

–

Tg1*

*Tg1

*iq

*id

Filter

∼

.

.

_

Fig. 4. Block diagram of the speed controller.

0 1 2 3 4 50

20

40

60

80

100

120

140

Excitation command

Connection command

time [s]

spee

d [r

ad/s

]

Fig. 5. Primary mover speed and sequence of DFIM operation.


by (50) and the electrical one given by (29) are almostdecoupled. This structure of the error dynamics ensuresthat the composite system is practically asymptoticallystable, i.e.

limt-N

ð*id ; *iq; zd ; zq; yd ; yq; *TL; *o; ZÞT ¼ 0: ð51Þ

Remark. If stator resistance is negligible, the proposedsolution can be simplified by setting its value to zero inthe control algorithm computation.

A block diagram representing the proposed speedcontroller is given in Fig. 4.

ARTICLE IN PRESS

0 1 2 3 4 5-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

time [s]0 1 2 3 4 5

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

time [s]

~id

~iq

0 1 2 3 4 5-20

-15

-10

-5

0

5

10

15

curr

ents

[A]

0 1 2 3 4 5-70

-60

-50

-40

-30

-20

-10

0

10

i2d

2qi

u2d

u q2

curr

ents

[A]

volta

ges

[A]

curr

ents

[A]

time [s]time [s]

Fig. 7. Transient during active–reactive current regulation with no parameter errors.

0 1 2 3 4 5-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

time [s]

Flu

xes

[Wb]

0 1 2 3 4 5

-10

-8

-6

-4

-2

0

2

4

6

time [s]

curr

ents

[A]

id*

iq*

ψq

*

ψd

*

(a) (b)

Fig. 6. Stator current and rotor flux references.


5. Simulation and experimental tests

Simulation and experimental tests have been per-formed using a small (5 kW) wound–rotor inductionmachine whose rated data are reported in Appendix C.The first set of simulations is reported to demonstrate

the performance during active–reactive current regula-tion of the DFIM generator.

1. The initial time interval 0–0.95 s is used to start theprimary mover, to perform the excitation andsynchronization of the DFIM with the line-voltagevector and to connect the stator windings to the linegrid. The trajectory of the primary mover speed(125 rad/s in no-load condition) and the sequence ofthe DFIM operation during excitation–synchroniza-tion preliminary stage is reported in Fig. 5.

2. At time t ¼ 0:95 s the active current referencetrajectory is applied, starting from zero initial valueand reaching 90% of the rated value (correspondingto 45Nm of produced torque with reactive currentequal to zero). From Fig. 5, it can be noted that theprimary mover speed reduces to 115 rad/s (no integralaction is adopted in the primary mover speedcontroller). The DFIM generator still operatessuper-synchronously and usually, in this condition,the rotor port delivers active power (Leonhard, 1995),nevertheless, in the proposed example, the rotor portabsorbs power owing to the relevant resistive losses.

3. At time t ¼ 1:3 s, the reference trajectory for reactivecomponent of the stator current is applied.

Both current and flux references computed using (20)are shown in Figs. 6a and b, respectively. The controllergains during all of the tests are set at ki ¼ 200; kii ¼10000: The transients, reported in Fig. 7, demonstratethe dynamic performance of the proposed controllerduring active–reactive current regulation. The current

errors are negligibly small during the above test; the softalmost transient-free connection to the line gridis achieved with the excitation algorithm given inAppendix B.The second set of simulation results shows the

dynamic behavior of the DFIM-based electrical drive.The following operating conditions have been considered:

1. At t ¼ 0 s, the unloaded wound rotor inductionmotor is directly connected to the line grid. Anadditional start up resistor equal to 4 R2N is insertedinto the rotor circuit in order to damp the currenttransient.

2. At t ¼ 1 s, the closed-loop control from rotor side isapplied with the following speed controller para-meters: ko ¼ 80; koi ¼ 3200; t ¼ 0:005 s. The speedreference and the load torque profile equal to therated value and applied at t ¼ 2 s, are shown inFig. 8, together with the enable command for thespeed controller.

ARTICLE IN PRESS

Speed-controller enable command

0 1 2 3 4 50

20

40

60

80

100

120

Enable command for the speed controller

Speed reference

Load torque

time [s]

spee

d [r

ad/s

], to

rque

[Nm

]

Fig. 8. Sequence of operation, speed reference and load torque profile

for speed control application of the DFIM.

0 1 2 3 4 5-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

time [s]

Flu

xes

[Wb]

0 1 2 3 4 5-15

-10

-5

0

5

10

15

20

time [s]

id*

iq*

ψq*

ψd*

(a) (b)

curr

ents

[A]

Fig. 9. Stator current and rotor flux references in speed control application of the DFIM.


3. At t ¼ 4 s, a non-zero reference for the reactivecomponent of the stator current is imposed. Thestator current references as well as the flux referencetrajectories, computed according to (20), are shownin Fig. 9.

Note that the speed, current and flux references andthe related errors are not significant in the interval(0–1) s, since the speed control is disabled. In Figs. 10and 11, the behavior of mechanical and electricalvariables is reported. The speed tracking error is lessthan 0.5 rad/s after the enable transient at t ¼ 1 s. Thestator current tracking error is almost negligible. Someshort transients arise only when a relevant change in thereference speed derivative occurs.It is worth observing that no theoretical constraint

prevents the speed control starting from zero speed. Theimpossibility of controlling low speed derives fromvoltage limits on the rotor inverter given by the usualsizing of this device in this application (Leonhard, 1995).In Figs. 12–14 experimental results are reported. The

proposed controller has been discretized using a simplebackward-derivative method and implemented on aDSP-based control board (TMS320C32) with a sam-pling time of 200 ms. The PWM frequency of the inverteradopted to feed the rotor side of the DFIM is 5 kHz. Asingle-phase diode rectifier has been used to provide theDC supply to the inverter, hence the nominal DC–linkvoltage was 310V. This hardware solution prevents a bi-directional exchange of power with the line grid at rotorside; hence it is not suitable for industrial plants, but itcan be used to test the proposed controller anyway. Theresolution of the encoder mounted on the rotor shaft is1024 ppr. An electric drive based on a two-pole-pairinduction motor with V=f control was used as primarymover. The reference speed was set to 125 rad/s. The aimof this test is to verify the performance of the proposedactive–reactive current controller. In Fig. 12, the beha-vior of the electric variables is reported, the referencesfor both stator currents and rotor fluxes are the same asin Figs. 5 and 6, except for the initial interval from 0 to0.5 s. In fact, in the experimental tests the excitation–synchronization stage is not considered. The initialcondition of the results reported in Fig. 12 is character-ized by stator windings connected to the three-phase linegrid and stator currents regulated to zero by theproposed controller. Some transients can be noted inthe stator current errors when the reference is variable,zero errors are guaranteed in steady-state conditions. Ahigher level of stator current transient errors ascompared to the simulation test can be observed fromFigs. 12 and 7. The reason for such behavior are anencoder misalignment of 0.1 rad (due to inaccuracy inthe rotor magnetic axis definition) and strong saturationof the DFIM magnetic system for required flux levels

(Fig. 6b) with a variation of the mutual inductancegreater than 1.5 times. However, a good stator–currenttracking is achieved confirming the robustness of theproposed solution.In Fig. 13, the real stator phase voltage and current in

a fixed stator reference frame are reported when thereference for the reactive current reference is set to zero,while the active current one is equal to 10A. The resultof the proposed solution (Fig. 13b) is compared with thestator current obtained with the open-loop stator flux

ARTICLE IN PRESS

0 1 2 3 4 5-100

-50

0

50

100

150

200

torq

ue [N

m]

0 1 2 3 4 50

20

40

60

80

100

120

time [s]

spee

d [ra

d/s]

ω

ω*

Produced torque

0 1 2 3 4 5-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

time [s]

spee

d [ra

d/s]

~ω

time [s]

Fig. 10. Velocity and torque responses of the speed-controlled DFIM.


field-oriented control algorithm proposed in Peresadaet al. (1998) under the same conditions (Fig. 13a). InFig. 14, the normalized harmonic content of the currentwaveforms of the previous pictures is considered. Arelevant reduction of the stator current distortion can benoted passing from Peresada et al. (1998) to theproposed solution. This is an important feature of theproposed closed-loop control of the stator currents. Infact, owing to the direct feedback of the stator currenterrors on the rotor voltages, the proposed controller actsin order to compensate for non-idealities of theinduction machine electromagnetic circuit.

6. Conclusions

The new direct active–reactive power controller forthe DFIM provides global asymptotic regulation in thepresence of induction machine parameter variations androtor position measurement errors. In addition, itdelivers an improved stator current waveform compen-sating for non-idealities of the induction machineelectromagnetic circuit. The simulation and experimentaltests confirm the high dynamic performance

and robustness of the proposed controller. Twoextensions of the active–reactive power control algo-rithm are presented. The first one allows the mechanicalspeed control; while the second one is suitable to controlthe autonomous DFIM-based generator during theexcitation–synchronization stage in order to achievetransient-free connection to the line grid. The proposedcontroller is suitable for both energy generation andelectrical drive applications, where restricted variationsof the speed around the synchronous velocity are present.

Appendix A. Variation of the proposed controller: the

active current tracking and unity power factor stabiliza-

tion problem

In this section, a variant of the proposed solution ispresented to address a slightly different control problem,given by the following objectives: (a) active-currentcomponent tracking (i.e. limt-N

*id ¼ 0) and (b) stabi-lization at zero of reactive current during steady state(i.e. limt-N

*iq ¼ 0 if i�d is constant). The active currentreference is assumed to be bounded together with its firstand second time derivatives.

ARTICLE IN PRESS

0 1 2 3 4 5-80

-60

-40

-20

0

20

40

60

80

time [s]

volta

ges

[A]

0 1 2 3 4 5-30

-25

-20

-15

-10

-5

0

5

10

15

20

time [s]

0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

time [s]

curr

ents

[A]

0 1 2 3 4 5-40

-30

-20

-10

0

10

20

30

time [s]

curr

ents

[A]

~,~

i i d q

i 2d

i q2

u2d

u2q

iq

id

curr

ents

[A]

Fig. 11. Currents and control voltages of speed-controlled DFIM.


In order to solve this problem the following choice isadopted

c�d ¼ �

1

bi�d ; i�q ¼ �

1

o0

’i�d : ðA:1Þ

Hence, redefining the control algorithm (14) as


�q � aLmi�d þ vd ;

ARTICLE IN PRESS

0 1 2 3 4 5-90

-80

-70

-60

-50

-40

-30

-20

-10

0

time [s]

volta

ges

[A]

0 1 2 3 4 5-25

-20

-15

-10

-5

0

5

10

15

time [s]

curr

ents

[A]

0 1 2 3 4 5-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

time [s]

curr

ent [

A]

0 1 2 3 4 5-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

time [s]

curr

ent [

A]

~id

~iq

i d2

i q2

u d2

u q2

Fig. 12. Experimental results with active–reactive current regulation.

0 0.01 0.02 0.03 0.04 0.05-400

-300

-200

-100

0

100

200

300

400

Phase Voltage

Phase Current x 10

0 0.01 0.02 0.03 0.04 0.05-400

-300

-200

-100

0

100

200

300

400

Phase Voltage

Phase Current x 10 volta

ge [V

], cu

rren

t [A

]

volta

ge [V

], cu

rren

t [A

]

time [s] time [s](b)

Fig. 13. Experimental results: stator phase voltage and current. (a) Open-loop solution presented in Peresada et al. (1998). (b) Proposed closed-loop

solution.




q þ vq;

c�q ¼ �

1

bo0

R1

si�d � o0i

�q �

1

sUm

� �; ðA:2Þ

the resulting flux–current error dynamics is exactly equalto (18) and the control objectives are fulfilled.It is worth noting that:

1. Reactive current dynamics, according to (A.1), is notfree and asymptotically tends to zero if i�d is constant.

2. Active current tracking is achieved together withstator flux field orientation, since from (A.1) itfollows that z�d ¼ bc�

d þ i�d ¼ 0; i.e. line-voltage vectorand stator flux vector are orthogonal.

Appendix B. Excitation—synchronization control

algorithm

In this section, the DFIM with open stator circuitsand operating as an autonomous generator is consid-ered. This kind of working condition is interesting sinceit is necessary to perform transient-free connection ofthe stator windings to the line grid. The dynamics of themachine in the above conditions can be derived from (1)with id ¼ ’id ¼ iq ¼ ’iq ¼ 0 and it results as follows

Ed ¼Lm

L2ðacd þ ocq � u2dÞ;

Eq ¼Lm

L2ðacq � ocd � u2qÞ;

’cd ¼ �acd þ o2cq þ u2d ;

’cq ¼ �acq � o2cd þ u2q; ðB:1Þ

where Ed ; Eq are stator EMF components, generated byrotor excitation.The control objective during the excitation–synchro-

nization stage is to design rotor control voltages in sucha way that the stator EMF vector is equal to the line-voltage vector, i.e.

limt-N

Ed ¼ �U ; limt-N

Eq ¼ 0: ðB:2Þ

Under this condition, the connection of the DFIM toline grid is transient-free. The open-loop excitationcontrol algorithm can easily be designed using (B.1) as isshown in (Peresada et al., 1998). Otherwise, in thissection, in order to provide fast and robust excitation,the result of Section 3 is adopted to design the closed-loop excitation control algorithm. For this purpose, thefollowing filtered EMF signals xd and xq (given by afirst-order two-dimensional filter) are defined:

’xd ¼ �kxd þ o0xq þ Ed ;

’xq ¼ �kxq � o0xd þ Eq; ðB:3Þ

where t0 ¼ k�1 is time constant of the filter.The reference for output filter variables are defined as

a steady-state solution of (B.3) under condition (B.2)

x�d ¼ �

kU

k2 þ o20

; x�q ¼

o0U

k2 þ o20

: ðB:4Þ

Combining (B.1) and (B.3), the resulting dynamicsubsystem has a similar structure to the model of theDFIM. Following the same conceptual line, the follow-ing definitions are derived

* control voltages


�q þ vd ;


�d þ vq; ðB:5Þ

* flux references

c�d ¼

Lm

L2

1

o0ð�kx�

q þ o0x�dÞ;

c�q ¼

Lm

L2

1

o0ðkx�

d � o0x�qÞ ðB:6Þ

proportional-integral EMF controller given by (28)with *id ; *iq replaced by *xd ; *xq; where

*xd ¼ xd � x�d ; *xq ¼ xq � x�

q: ðB:7Þ

Straightforward computation leads to the dynamicsof the output filter variables in the form of (29).Following the same line of stability analysis as presentedin Section 3, it follows that condition (B.2) is globallyachieved.

ARTICLE IN PRESS

Fig. 14. Experimental results: normalized harmonic spectra of the

stator phase current. White bars: Open-loop solution presented in

Peresada et al. (1998). Black bars: Proposed closed-loop solution.

(fundamental component at 50Hz is normalized to 1 for both cases).


Appendix C. Rated data of the DFIM used for simula-

tions and experiments

Nameplate data and parameters of the DFIMadopted in simulations and experiments are reportedin Table 1.Nominal values of id ; cd ; cq; i2d ; i2q; u2d ; u2q for the

considered DFIM working as a motor in conditions ofnominal speed, nominal torque and zero reactive powerimposed at stator side (iq ¼ 0) are reported in Table 2.

References

Bogalecka, E. (1993). Power control of a double fed induction

generator without speed and position sensors. Proceedings of the

EPE Conference 1993, Brighton, UK (pp. 224–228).

Bogalecka, E., & Kzreminski, Z. (1993). Control system of a doubly

fed induction machine supplied by current controlled voltage

source inverter. Proceedings of the IEE Conference on Electrical

Machines and Drives 1996, London, UK (pp. 168–172).

Hopfensperger, B., Atkinson, D. J., & Lakin, R. A. (2000). Stator–flux

oriented control of a doubly fed induction machine with and

without position encoder. Electric Power Applications, IEE

Proceedings, 147(4), 241–250.

Khalil, H. K. (1994). Nonlinear systems. Upper Saddle River, NJ:

Prentice-Hall.

Leonhard, W. (1995). Control of electric drives. Berlin, Germany:

Springer.

Nakra, H. L., & Duke, B. (1988). Slip power recovery induction

generators for large vertical axis wind turbines. IEEE Transactions

on Energy Conversion, 3(4), 733–737.

Pena, R., Clare, J. C., & Asher, G. M. (1996a). Doubly fed induction

generator using back-to-back PWM converters and its applications

to variable-speed wind-energy generation. Electric Power Applica-

tions, IEE Proceedings, 143(3), 231–241.

Pena, R., Clare, J. C., & Asher, G. M. (1996b). A doubly fed induction

generator using back-to-back PWM converters supplying an

isolated load from a variable-speed wind turbine. Electric Power

Applications, IEE Proceedings, 143(5), 380–387.

Peresada, S., Tilli, A., & Tonielli, A. (1998). Robust active–reactive

power control of a doubly fed induction generator. Proceedings of

the IEEE Annual Conference of Industrial Electronics Society

IECON’98, Aachen, Germany (pp. 1621–1625).

Peresada, S., Tilli, A., & Tonielli, A. (1999a). Dynamic output

feedback linearizing control of a doubly fed induction motor.

Proceedings of the IEEE International Symposium on Industrial

Electronics ISIE’99, Bled, Slovenia (pp. 1256–1260).

Peresada, S., Tilli, A., & Tonielli, A. (1999b). Robust output feedback

control of a doubly fed induction machine. Proceedings of the IEEE

Annual Conference of Industrial Electronics Society IECON’99, San

Jose, CA, USA (pp. 1348–1354).

Vas, P. (1990). Vector control of AC machines. Oxford, UK:

Clarendon.

Walczyna, A. (1991). Comparison of the dynamics of the DFM in field

and rotor axes. Proceedings of the EPE Conference 1991, Florence,

Italy (pp. 231–236).

Xu, L., & Cheng, W. (1995). Torque and reactive power control of a

doubly fed induction machine by position sensorless scheme. IEEE

Transactions on Industry Applications, 31(3), 636–642.

Yamamoto, M., & Motoyoshi, O. (1992). Active and reactive power

control for doubly fed wound rotor induction generator. IEEE

Transactions on Power Electronics, 6(4), 624–629.

ARTICLE IN PRESS

Table 1

Nameplate data and parameters of the adopted DFIM

Nominal voltage 380 VRMS (Y-connected)

Nominal power 5 kW

Nominal speed 100 rad/s

Stator resistance (R1) 0.95ORotor resistance (R2) 1.8OStator inductance (L1) 0.094H

Nominal frequency 50Hz

Nominal torque 50Nm

Pole pairs 3

Rotor inductance (L2) 0.088H

Magnetizing inductance (M) 0.082H

Rotor inertia (J) 0.1 kgm2

Table 2

Nominal values of the electric variables of the adopted DFIM, under

the conditions of zero reactive power at stator side, nominal speed and

nominal torque

id 11.7A

cd �0.22Wb

i2d �13.4Au2d �9.7Viq 0A

cq �1.02Wb

i2q �11.6Au2q �24V


Power control of a doubly fed induction machine via output feedback

Documents

Transcript of Power control of a doubly fed induction machine via output feedback