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Electric Power Systems Research 79 (2009) 355–364

Contents lists available at ScienceDirect

Electric Power Systems Research

journa l homepage: www.e lsev ier .com/ locate /epsr

ontrol strategy for a Doubly-Fed Induction Generator feeding an unbalancedrid or stand-alone load

ubén Penaa, Roberto Cardenasb,∗, Enrique Escobarb, Jon Clarec, Pat Wheelerc

University of Concepcion, Electrical Engineering Department, P.O. Box 160-C, Concepcion, ChileUniversity of Magallanes, Electrical Engineering Department, P.O. Box 113-D, Punta Arenas, ChileUniversity of Nottingham, School of Electrical and Electronic Engineering, Nottingham NG7 2RD, UK

r t i c l e i n f o

rticle history:eceived 10 February 2008eceived in revised form 15 July 2008ccepted 18 July 2008vailable online 11 September 2008

a b s t r a c t

In this paper, the control systems for the operation of a Doubly-Fed Induction Generator (DFIG), feedingan unbalanced grid/stand-alone load, are presented. The scheme uses two back-to-back PWM invertersconnected between the stator and the rotor, namely the rotor side and stator side converters respec-tively. The stator current and voltage unbalances are reduced or eliminated by injecting compensation

eywords:nduction generatorsower generation controlind energy

ector control

currents into the grid/load using the stator side converter. The proposed control strategy is based ontwo revolving axes rotating synchronously at ±ωe. From these axes, the d–q components of the negativeand positive-sequence currents, in the stator and grid/load, are obtained. The scheme compensates thenegative-sequence currents in the grid/load by supplying negative-sequence currents via the stator sideconverter. Experimental results obtained from a 2-kW experimental prototype are presented and dis-cussed in this work. The proposed control methodology is experimentally validated for stand-alone and

ition

utrifmluiHdt

cast

weak grid-connected cond

. Introduction

The Doubly-Fed Induction Generator (DFIG) is widely used forariable-speed generation, and it is one of the most important gen-rators for Wind Energy Conversion Systems (WECS) [1]. Both grid-onnected and stand-alone operation is feasible [2,3]. For variable-peed operation, the standard power electronics interface consistsf a rotor and stator side PWM inverters that are connected back-to-ack. These inverters are rated, for restricted speed range operation,o a fraction of the machine rated power [2]. Applying vector controlechniques yields current control with high dynamic response [2,3].

In grid-connected applications, the DFIG may be installed inemote, rural areas [4,5] where weak grids with unbalanced volt-ges are not uncommon. As reported in [6,7], induction machinesre particularly sensitive to unbalanced operation since localizedeating can occur in the stator and the lifetime of the machine cane severely affected. Furthermore, negative-sequence currents inhe machine produce pulsations in the electrical torque, increasing

he acoustic noise and reducing the life span of the gearbox, bladessembly and other components of a typical WECS [4,5]. To protecthe machine, in some applications, DFIGs are disconnected from therid when the phase-to-phase voltage unbalance is above 6% [5].

∗ Corresponding author.E-mail address: rcd@ieee.org (R. Cardenas).

ustcaDla

378-7796/$ – see front matter © 2008 Elsevier B.V. All rights reserved.oi:10.1016/j.epsr.2008.07.005

s and the results show the excellent performance of the strategy used.© 2008 Elsevier B.V. All rights reserved.

Control systems for the operation of induction generators innbalanced grids have been reported in [4,5], where it is proposedo inject compensating current in the DFIG rotor to eliminate oreduce torque pulsations. The main disadvantage of this methods that the stator current unbalance is not eliminated [5]. There-ore, even when the torque pulsations are reduced, the induction

achine power output is derated, because the machine currentimit is reached by only one of the stator phases. Compensation ofnbalanced voltages and currents in power systems are addressed

n [8] where a STATCOM is used to compensate voltage unbalances.owever, the application of the control method to DFIGs is notiscussed. No formal methodology for the design of the control sys-ems is presented and only simulation results are discussed in [8].

In this paper a new control system to compensate the statorurrent unbalance in grid-connected and stand-alone DFIG oper-tion is presented. The strategy uses two revolving axes rotatingynchronously at ±ωe to obtain the d–q components of the nega-ive and positive-sequence currents in the stator and grid/load. Thenbalance is compensated by the stator side converter. The statoride converter positive-sequence current is conventionally con-rolled to regulate the dc link voltage, whereas negative-sequence

urrent is regulated to reduce or eliminate the grid voltage unbal-nce. The control system for unbalanced operation of stand-aloneFIGs has been succinctly discussed in a two-page paper pub-

ished by the authors [9]. However, in that publication issues suchs small-signal models, controller design, compensation of volt-

3 ystems

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56 R. Pena et al. / Electric Power S

ge unbalances in grid-connected DFIGs, etc., were not discussed.n addition, this work also introduces new methods, applicationcenarios and insights. These include:

The modelling of a DFIG for unbalanced operation is presented.This model can be used to study the effects of negative-sequencecomponents in the machine, e.g. torque pulsations in the elec-trical torque, ripple in the machine flux, etc., The modellingpresented in Section 2 is suitable for analysing DFIGs feedingunbalanced loads as well as DFIGs connected to unbalanced, weakgrids.The control systems, based on two synchronously rotating axes,are presented and fully analysed. The control systems of Section4 are linearised using small-signal models suitable for designingthe current controllers of the positive/negative-sequence currentcontrol loops.The control systems, suitable for compensating the effects ofnegative-sequence components in a DFIG connected to an unbal-anced grid are discussed. Grid-connected operation is consideredas the most important application of DFIGs.

The rest of this paper is organised as follows. In Section 2 theodelling of the DFIG, for variable-speed stand-alone and grid-

onnected operation, is addressed. In Section 3 the control systemsor balanced operation of DFIGs are briefly reviewed. In Section

the control systems for unbalanced operation of DFIGs are pre-ented. In Section 5 experimental results obtained from a 2 kWrototype are analysed. Finally an appraisal of the proposed controlethod is discussed in the conclusions.

. Modelling of DFIGs

.1. Modelling of DFIGs considering balanced operation

The modelling of DFIGs, for balanced operation is reviewed inhis section. Further details can be found in [2,3]. The nomenclaturesed is shown in Appendix A, with superscript + used to indi-ate that the d–q-axes are rotating at +ωe (positive-sequence). Theachine equations written in a d–q synchronous frame rotating at

he supply frequency +ωe, are [2]:

�+ds

�+qs

�+dr

�+qr

⎤⎥⎥⎥⎥⎦ =

⎡⎢⎢⎢⎣

Ls 0 Lm 0

0 Ls 0 Lm

Lm 0 Lr 0

0 Lm 0 Lr

⎤⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎣

i+ds

i+qs

i+dr

i+qr

⎤⎥⎥⎥⎥⎦ (1)

v+ds

v+qs

]=

[Rs 0

0 Rs

][i+ds

i+qs

]+ d

dt

[�+

ds

�+qs

]+

[0 −ωe

ωe 0

][�+

ds

�+qs

](2)

v+dr

v+qr

]=

[Rr 0

0 Rr

][i+dr

i+qr

]+ d

dt

[�+

dr

�+qr

]+

[0 −ωsl

ωsl 0

][�+

dr

�+qr

](3)

here �s, �r, vs and is are the stator and rotor flux vectors and thetator voltage and current vectors respectively; Ls, Lm, Lr, Rr and Rs

re the stator, magnetising and rotor inductances and the rotor andtator resistance respectively; and ωsl = ωe − ωr is the slip frequencyith ωr the rotational speed. Aligning the d-axis on the stator flux

ector yields:

+qr = − Ls

Lmi+qs (4)

+ds = Lmims; �+

qs = 0 (5)

wds

Research 79 (2009) 355–364

onsidering (1) and (2) the following expression is obtained for theynamics of the equivalent stator magnetising current:

sdims

dt+ ims = i+

dr + 1 + �s

Rsv+

ds (6)

here �s = Ls/Rs and �s = (Ls − Lm)/Lm are the stator time constantnd leakage factor respectively. Therefore, for stand-alone applica-ions, the stator magnetising current, hence the stator voltage, cane controlled using the positive-sequence direct rotor current i+

dr.or grid-connected operation ims can be supplied from the machinetator and/or rotor.

The torque produced by the DFIG is obtained as [2,3]:

e = 3p

2Lm(i+qsi+

dr − i+dsi+qr) (7)

n steady state and balanced operation of the DFIG, the d–q compo-ents of the stator and rotor currents are dc values and the electricalorque is constant.

.2. Modelling of DFIG considering unbalanced operation

Assuming negligible zero-sequence components in the grid, thenbalanced voltage of a weak grid can be described using negativend positive-sequence components:

ˆs = v1s ej(ωet+�v+) + v2s e−j(ωet+�v−) (8)

here v1s and v2s are the moduli of the positive and negative-equence voltages respectively. Referring (8) to d–q-axes rotatingt +ωe and −ωe yields:

ˆ+s = v1s ej�v+ + v2s e−j(2ωet+�v−) (9)

ˆ−s = v1s ej(2ωet+�v+) + v2s e−j�v− (10)

As shown in (9) the negative-sequence voltage produces double-requency components when referred to the frame rotating a + ωe.n the other hand, the positive-sequence voltage also produces aouble-frequency components in the d–q-axes rotating at −ωe (see10)). In general, any unbalanced vector in the stator frame can beritten as

ˆs = x1s ej(ωet+�x+) + x2s e−j(ωet+�x−) (11)

nd in the rotor frame, an unbalanced vector can be written as:

ˆr = x1r ej((ωe−ωr)t+�x+) + x2r ej((−ωe−ωr)t+�x−) (12)

here x1 and x2 are the moduli of positive and negative-sequenceomponents. Using (9)–(12), (2) and (3) are written in the negative-equence frame as

v−ds

v−qs

]=

[Rs 00 Rs

][i−dsi−qs

]+ d

dt

[�−

ds�−

qs

]+

[0 ωe

−ωe 0

][�−

ds�−

qs

](13)

v−dr

v−qr

]=

[Rr 00 Rr

][i−dr

i−qr

]+ d

dt

[�−

dr�−

qr

]

+[

0 −(ωe + ωr)(ωe + ωr) 0

][�−

dr�−

dr

](14)

or stator unbalanced operation the electrical torque, Te, is obtaineds (see (11)):

e = 3p

2Lm[(i1qs − i2s sin(2ωet + �i−))i∗dr

−(i1ds + i2s cos(2ωet + �i−))i∗qr] (15)

here the terms i2s sin(2ωet + �i−) and i2s cos(2ωet + �i−) areouble-frequency current components produced by the negative-equence stator current. In (15) the rotor current is assumed to be

ystems

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sl

3

3

DiDtuf

�

w[b

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Ta

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i

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4g

R. Pena et al. / Electric Power S

qual to the demand i∗r . Even with balanced rotor currents (i.e.∗dr, i∗qr are continuous signals) the electrical torque of (15) has

double-frequency component, which can produce large torqueulsations.

Torque pulsations may be reduced or eliminated by addingompensation components to the rotor currents i∗

dr and i∗qr [4,5].owever, this compensation technique does not eliminate theegative-sequence components of the stator/rotor currents andoltages. Furthermore, according to [5], the unbalance in the sta-or current may even be increased when compensation signals aredded to the rotor current.

Rotor compensating currents can be used to achieve other con-rol targets such as balancing the stator currents. However, usingotor current compensation techniques does not provide enoughegrees of freedom to achieve several control targets simultane-usly, for instance to eliminate both the stator current unbalancend the torque pulsations. Furthermore, the slip velocity for theegative-sequence frequency is −(ωe + ωr) (see (14)), thereforenegative-sequence rotor flux may produce a large machine

ack e.m.f., i.e. relatively large rotor voltages are necessary toegulate ir.

This paper proposes to use the stator side converter to compen-ate the negative-sequence components of the grid or stand-aloneoad.

. Vector control of DFIGs for balanced applications

.1. Control systems for grid-connected applications.

The control system for balanced operation of grid-connectedFIGs has already been discussed in [2] and only a brief discussion

s presented here. The typical control system for a grid-connectedFIG is shown in Fig. 1. The d–q reference frame is orientated along

he stator flux. The demodulation of the rotor currents and mod-lation of the rotor demand voltages uses the slip angle derivedrom:

slip = �e − �r (16)

here �r is the rotor position (sensorless operation is also feasible10]). For balanced operation, the stator flux vector position �e can

e obtained from the stator flux ˛–ˇ components as

e = tan−1

(�ˇs

�˛s

)(17)

Fig. 1. Vector control system for DFIG operation.

4

ccFt(ssttParnaAf

r

[

Research 79 (2009) 355–364 357

he stator flux ˛–ˇ components are obtained from the stator volt-ges and currents as [2]:

�˛s =∫

(v˛s − Rsi˛s) dt

�ˇs =∫

(vˇs − Rsiˇs) dt(18)

As shown in Fig. 1, PI controllers are used to regulate the rotorurrents. The direct component of the rotor current may be usedo supply the magnetising current to the machine. If i∗

dr = 0 theagnetising current is entirely supplied from the grid. The electri-

al torque is controlled using the q-axes current i∗qr (see (7)). If theFIG is used in a variable-speed WECS, the current i∗qr is regulated

o capture the maximum power of a wind turbine for a given windondition [12].

.2. Control system considering stand-alone applications

Stand-alone balanced operation of DFIGs is discussed in [3] andnly a summary is presented here. With some minor modifications,he vector control system of Fig. 1 can be used to supply electricalnergy to a stand-alone load or isolated grid. In this case, the statorux position �e is derived from a free running integral of the stator

requency demand ω∗e (50 Hz):

e =∫

ω∗e dt (19)

Typically, in stand-alone connections, the magnetising currentms is supplied entirely from the rotor. Therefore, the stator flux isontrolled using the d-axes current i∗

dr (see (5)). The torque currentqr is now controlled according to

∗qr = − Ls

Lmiqs (20)

hich forces the orientation of the reference frame along the statorux vector position (see (2)). More information about control sys-ems for the stand-alone operation of DFIGs is presented in [3,10].

. Control system for DFIGs feeding an unbalancedrid/load.

.1. Vector control system for the stator side converter

In the control system proposed in this paper, the stator sideonverter is controlled to supply positive and negative-sequenceurrents to the grid/load. The vector control system is shown inig. 2. The system is orientated along the positive-sequence sta-or voltage vector. Because of the unbalance, a phase locked loopPLL), shown at the bottom of Fig. 2, is implemented to calculate thetator voltage angle �v [11]. A notch filter eliminates the negative-equence from the d–q voltage components. The PI controller forceshe q-axes positive-sequence to zero, ensuring the orientation ofhe reference frame. The parameters of the notch filter and theI controller used in the PLL are given in Appendix B. From +�v

nd −�v, the currents can be referred to two synchronous d–q-axesotating at +ωe and −ωe respectively. Double-frequency compo-ents are produced when the positive/negative-sequence currentsre referred to the d–q-axes rotating in the opposite direction [4–7].

s shown in Fig. 2, notch filters are used to eliminate these high

requency components.The control systems for the stator side positive-sequence cur-

ents i+df and i+

qf are entirely conventional (see Figs. 1 and 2 and

13]). The current i+df regulates the dc link voltage E and the current

358 R. Pena et al. / Electric Power Systems Research 79 (2009) 355–364

tem fo

i

n

TtP(

ia√waacs

c[

4g

cpfuuI˛–ˇ components obtained from (18). The PI controller drives thestator flux positive-sequence q-axes component to zero, ensuringthe correct orientation of the reference frame. The PLL parametersare the same to the ones used in Section 4.

Fig. 2. Proposed control sys

+qf regulates the reactive power supplied to the load. The stator sideegative-sequence currents are regulated to (see Fig. 2)

i− ∗dqf = i−

dqLidqL = (i−

dqs + i−dqf)

(21)

herefore the negative-sequence current demand is a function ofhe grid/load negative-sequence current. In steady state, when theI controller regulates i−

dqf = i−dqL, the stator current i−

dqs = 0 (see21)), and the torque pulsations are eliminated.

The total current supplied by the stator side converter is lim-ted to avoid overloading. Assuming i+

qf = 0, the maximum currentvailable to compensate the negative-sequence stator current is

(i−df)

2 + (i−qf)

2 ≤ Irated − i+df (22)

here Irated is the converter nominal current. In this work it isssumed that the control of the positive-sequence currents hashigher priority. Therefore, in each sampling time the demand

urrent i+ ∗df is calculated first. After that the maximum negative-

equence current is obtained using (22).In a, b, c coordinates, the total voltage demand for the stator side

onverter is obtained as (see Fig. 2):

v∗a(t)

v∗b(t)

v∗c(t)

]=

[v+

a (t)v+

b(t)

v+c (t)

]+

[v−

a (t)v−

b(t)

v−c (t)

](23)

r the stator side converter.

.2. Control of the DFIG considering connection to an unbalancedrid

Fig. 3 shows a DFIG connected to an unbalanced grid. In thisase the stator current has positive and negative-sequence com-onents. The control of the DFIG is carried out using a referencerame aligned with the positive-sequence stator flux vector. For annbalanced or distorted grid, the flux vector position is obtainedsing a PLL [11], with a structure similar to the one shown in Fig. 2.

n this case, the inputs to the demodulator block are the stator flux

Fig. 3. Grid-connected operation of a DFIG.

R. Pena et al. / Electric Power Systems Research 79 (2009) 355–364 359

g

i

wDcTaIn

ovvan

n

V

isvnc

I

Ts

rra[

4s

sgsdU

o[

wtao

i

a

i

titbaapc

nccs

i

Tcuit for the negative-sequence current components and the fullnegative-sequence current circulates through stator side converterwith i−

dqs ≈ 0The control system described in Fig. 2 compensates the

negative-sequence stator current. The zero-sequence component

Fig. 4. Single phase equivalent circuit corresponding to Fig. 3.

The stator side converter is controlled to supply the current ifiven by

f = i1f ej(ωet+�f+) + i2f e−j(ωet+�f−) (24)

here the component i2f is supplied to the grid to compensate theFIG stator voltage unbalance. Fig. 4 shows the Thevenin equivalentircuit per phase of Fig. 3 for the negative-sequence components.he voltage v−

aT is the equivalent grid negative-sequence voltagend L2T is the equivalent negative-sequence inductance of the grid.n Fig. 4 it is assumed that the coupling between the sequenceetworks is low.

The negative-sequence equivalent for the DFIG is dependentn the machine operating point. However, if the stator and rotoroltages/currents are balanced, then the negative-sequence voltage−as ≈ 0. In this work, it is assumed that balanced reference currentsre applied to the rotor. Therefore to balance the DFIG it is onlyecessary to eliminate the negative-sequence stator voltage.

Using the single phase circuit of Fig. 4, it can be shown that theegative-sequence current is eliminated from the DFIG stator when

−af − ωeLfI

−af = 0 (25)

.e. the stator side converter is a short circuit for the grid negative-equence voltage (see Fig. 4). Therefore the negative-sequenceoltage applied to the machine’s stator is zero. Using (25) theegative-sequence current supplied by the stator side converteran be calculated as

−af = − V−

aTωeL2T

(26)

herefore, when the stator side converter supplies the negative-equence current given by (26), the DFIG is in balanced operation.

The control of the stator side converter positive-sequence cur-ent is used to regulate the dc link voltage E (see Fig. 2) and theeactive power supplied to the grid. This is entirely conventionalnd will not be discussed here. The interested reader is referred to13].

.3. Vector control system for a DFIG feeding an unbalancedtand-alone load

Fig. 5 shows a DFIG sourcing an unbalanced load. The machine

tator and the load are star-connected, with the neutral of theenerator connected to the load. This allows the presence of zero-equence currents. The topology of Fig. 5 is similar to that used byiesel driven synchronous generators feeding stand-alone loads.sing Fig. 5, the instantaneous currents in the phases a, b and c are

Fig. 5. DFIG sourcing a stand-alone unbalanced load.

btained as

ia(t)ib(t)ic(t)

]=[

1/(sLa + Ra) 0 00 1/(sLb + Rb) 00 0 1/(sLc + Rc)

][va(t)vb(t)vc(t)

](27)

here va, vb and vc are the instantaneous DFIG line to neutral sta-or voltages and La, Lb, Lc and Ra, Rb, Rc are the load inductancesnd resistances respectively. The negative-sequence current can bebtained from (27) as [14]:

−aL(t) = ia(t) + ib(t) ej2�/3 + ic(t) e−j2�/3 (28)

nd the zero-sequence current is obtained as

0aL(t) = ia(t) + ib(t) + ic(t) (29)

The zero-sequence current does not produce a resulting sta-or flux; therefore this current does not produce torque pulsationsn the machine. If the negative-sequence stator current is rela-ively small, the line to neutral voltages are approximately balancedecause a stator flux control loop regulates idr. Therefore, therere negligible negative/zero-sequence load voltages. For the stand-lone system of Fig. 5, the sequence components are coupled so thatositive-sequence voltages produce negative and zero-sequenceurrents in the load.

A single phase equivalent system of Fig. 5 is shown in Fig. 6. Theegative-sequence current is represented as a current source. Theurrent of (28) can be supplied from the DFIG or the stator sideonverter. Because the aim is to have balanced stator current, thetator negative-sequence component is eliminated when−af = i−aL (30)

herefore, the stator side converter is again similar to a short cir-

Fig. 6. Single phase equivalent circuit corresponding to Fig. 5.

360 R. Pena et al. / Electric Power Systems Research 79 (2009) 355–364

Fa

iTDtavto

4

taf

c

ntt(

t

cro

�

nba

5

ummic

ma1pTct

5s

tvrc respectively (see Fig. 5). The rotational speed is varied from≈1350 to 1650 rpm to illustrate the performance at variable-speed(from below to above synchronous speed). Before t ≈ 1.25 s, thecompensation system is disabled and the stator current has anegative-sequence component (see Fig. 9). At t ≈ 1.25 s the com-

Table 1Parameters of PI controllers

System PI controller Closed loop naturalfrequency (Hz);damping factor

Samplingfrequency (Hz)

Rotor current 11.14z − 0.893734

z − 170; 0.8 2000

Stator side 26.35z − 0.897692

70; 0.8 2000

ig. 7. Small-signal model. (a) Grid-connected applications and (b) stand-alonepplications.

s not compensated for and the stator current is still unbalanced.herefore localized heating is not completely eliminated, and theFIG power may be derated if only one of the phases reaches the sta-

or current limit. Zero-sequence current compensation cannot bechieved with the proposed control system, unless a four-leg con-erter is used [15]. However, the application of four-leg inverters forhe compensation of zero-sequence stator currents is consideredutside the scope of this paper.

.4. Small-signal models

As discussed in Section 4, the proposed control strategy driveso zero the negative- sequence stator current. For grid-connectedpplications the small-signal model is shown in Fig. 7a. The transferunction relating the stator side converter voltage v−

dqf to the stator

urrents i−dqs is

�i−dqs ≈

�v−dqf

sL2

L2T

L2s + L2T

L2 = Lf + L2TL2s

L2T + L2s

(31)

In (31) cross coupling terms between the d- and q-axes areeglected. This is normal, because they are compensated for andheir effect is eliminated at the controller output [2]. The resis-ances of the machine filter and lines have also been neglected in31).

For stand-alone unbalanced loads (see Fig. 7b), the transfer func-ion between the stator side converter voltage v−

dqf and the stator

urrents i−dqs is obtained assuming that the negative-sequence cur-

ent load of (28) is constant. From Fig. 7, the transfer function isbtained as

i−dqs ≈

�v−dqf

s(Lf + L2s)(32)

In (32) the resistances and cross coupling terms are alsoeglected. The small-signal transfer functions of (31) and (32) cane used to design the controllers using conventional root-locusnalysis.

. Experimental results

The control systems of Figs. 1 and 2 have been implemented

sing a 2 kW DFIG driven by a cage machine. This cage inductionachine may be used to emulate a wind turbine or another primeover using the emulation techniques presented in [16]. The exper-

mental rig is shown in Fig. 8. Two PWM back-to-back inverters areonnected to the machine rotor. Current transducers are used to

d

M

Fig. 8. Experimental system.

easure the rotor, stator and stator side converter currents. Volt-ge transducers measure the stator voltage. A position encoder of0,000 pulses per revolution (ppr) is used to measure the rotorosition. The parameters of the whole system are in Appendix B.he PI controller parameters for the rotor and stator side converterurrents, the dc link voltage and the magnetising current, based onhe procedure shown in [2,3], are depicted in Table 1.

.1. Experimental results for a DFIG feeding an unbalancedtand-alone load.

Figs. 9 and 10 show the performance of the proposed con-rol system for negative-sequence current compensation underariable-speed stand-alone operation. The load consists of threeesistors of 25, 154 and 154 , connected to phases a, b and

converter currentz − 1

c link voltage 0.034256z − 0.975753

z − 11.25; 0.8 200

agnetising current 0.788693z − 0.934173

z − 11.5; 0.8 200

R. Pena et al. / Electric Power Systems Research 79 (2009) 355–364 361

Fig. 9. Control system response for the negative-sequence currents.

Fa

p

Fi

dfitietpsbbrricvmee

a

Fig. 12. Unfiltered rotor voltage referred to the d–q positive-sequence axes.

aaipairtfcnctisr

5

ig. 10. Stator and rotor unfiltered currents referred to the d–q positive-sequencexes.

ensation is enabled and the stator current i−dqs is driven to zero.

or t > 1.5 s, i−dqL ≈ i−

dqf and the negative-sequence currents are elim-nated from the machine stator.

Fig. 10 shows the unfiltered machine currents referred to the–q-axes rotating at +ωe (corresponding to the test of Fig. 9). Notchlters are not applied to the currents shown in Fig. 10. For t < 1.25 she DFIG d–q currents have a double-frequency component whichs eliminated after t ≈ 1.25 s when the proposed control system isnabled. The stator voltage, in the ˛–ˇ coordinates (correspondingo the test of Fig. 9), is shown in Fig. 11. Before enabling the com-ensation the voltage is unbalanced (see Fig. 11a). After the controlystem is enabled, the negative-sequence load current is suppliedy the stator side converter and the stator voltage is approximatelyalanced (see Fig. 11b). Fig. 12 shows the unfiltered rotor voltageeferred to the positive-sequence d–q-axes. For t < 1.25 s the d–qotor voltage has a large double-frequency component, because∗dr and i∗qr are constant d–q values without negative-sequenceomponents. Therefore the PI controller injects negative-sequenceoltages in the rotor in order to compensate the negative-sequence

achine back e.m.f. (see (14)). When the compensation system is

nabled, negative-sequence currents and voltages are completelyliminated from the machine.

Figs. 13 and 14 show the control system’s response to an unbal-nced load-step. For t < 4 s the stand-alone DFIG is feeding a load of

Fig. 11. Stator voltage. (a) Before compensation and (b) after compensation.

saibi

Fig. 13. Control system response for a load-step in one phase.

bout 200 per phase. The line to neutral voltage is about 145 Vnd the rotational speed is 1650 rpm. At t ≈ 4 s the load resistancen one phase is changed from 200 to 33 . In Fig. 13, the d–q com-onents of the rotor and stator currents are shown. These currentsre referred to the axes which are rotating at +ωe. No notch filters applied to these signals. When the load-step is produced, theotor quadrature current increases to compensate the increase inhe machine output power. As shown in Fig. 13, there are no double-requency components in any of the machine currents, because theompensation system is enabled during that test. In Fig. 14 theegative-sequence currents of the machine stator and stator sideonverter are shown. When the unbalanced load-step is connected,he converter negative-sequence current is increased, compensat-ng the load-unbalance and driving the current i−

dqs to zero. Ashown in Fig. 14, the proposed control system has a good dynamicesponse.

.2. Experimental results for a DFIG feeding an unbalanced grid

Unbalanced stator voltages can be created using the schemeshown in Fig. 15. In Fig. 15a the unbalanced grid is created by

dding a variable voltage to one of the phases. Another possibilitys shown in Fig. 15b. In this case the voltage unbalance is producedy connecting an unbalanced load to a balanced weak grid. For themplementation of Fig. 15b, the equivalent circuit of Fig. 4 is not

Fig. 14. Negative-sequence currents corresponding to the test of Fig. 13.

362 R. Pena et al. / Electric Power Systems Research 79 (2009) 355–364

Fig. 15. Experimental setup to obtain unbalanced stator voltages. (a) Unbalancingby adding an extra voltage to one phase and (b) unbalancing by connecting a smallimpedance between two phases.

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ompletely correct because the sequence components are coupled,.e. unbalanced stator voltages are produced even when the weakrid is balanced and has only positive-sequence voltages.

Using the experimental system shown in Figs. 8 and 15a,he proposed compensating system has been tested consideringariable-speed grid-connected operation. The rotational speed isaried from ≈1600 rpm to 1400 rpm in 4 s. The machine is mag-etised from the stator and the rotor current idr = 0. During the testhe rotor quadrature current is regulated to a constant value of 10 A.he negative-sequence compensation system is enabled at t ≈ 2 s.voltage of 82 V is added to phase a.Fig. 16 shows the unfiltered machine currents referred to the

–q-axes rotating at +ωe. Notch filters are not applied to theseurrents. In t ≈ 2 s, the compensation system is enabled and theegative-sequence currents are eliminated from the machine. Theegative-sequence currents are shown in Fig. 17. When negative-

ig. 17. Machine stator and stator side converter negative-sequence currents for theest of Fig. 16.

Assc

ig. 18. Electrical torque corresponding to the test of Fig. 16. (a) Compensated andncompensated toque and (b) amplified view of (a).

equence current is injected from the stator side converter, thetator negative-sequence current is driven to zero.

In Fig. 18 the electrical torque is shown. The torque is calculatedsing (7), with the unfiltered d–q components obtained from thexes rotating at +ωe. As shown in Fig. 18a, before the compensatingystem is enabled, there are large torque pulsations of ±5 N m (35%f the dc value). The torque pulsations are eliminated when theompensation is enabled. In Fig. 18b, an amplified view of Fig. 18as shown. The torque pulsations, with a fundamental frequency of00 Hz, are noticeable in this figure.

The stator voltage corresponding to the test of Fig. 16 is shownn Fig. 19. The ˛–ˇ components of the stator voltage are unbalancedor t < 2 s. When the proposed compensation system is enabled thetator voltage is completely balanced and the negative-sequenceomponents are eliminated.

The hardware implementation of Fig. 15b was also used toest the proposed control system. The weak grid is implementedsing a variable 3� transformer and 20 mH line inductances. Theachine is running at 1400 rpm, with a line-to-line stator volt-

ge of 260 V and a resistance of 12.5 is connected betweenhases a and c in t ≈ 2.5 s. Fig. 20a shows the unfiltered rotornd stator machine currents with the compensation strategynabled. When the unbalanced load-step is applied there is amall disturbance in the currents that is compensated for byhe proposed control scheme. Fig. 20b shows the d–q negative-equence currents in the stator side converter and DFIG stator.

fter the load-step is connected the stator side converter negative-equence current is used to drive to zero the unbalance in thetator current. For this test the equivalent circuit of Fig. 4 is notompletely correct, however even in this case the proposed com-

Fig. 19. Stator voltage in ˛–ˇ components.

R. Pena et al. / Electric Power Systems

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ig. 20. Stator side converter and machine currents. (a) Machine positive-sequenceurrents and (b) negative-sequence currents.

ensation system has a good performance with a fast dynamicesponse.

. Conclusions

In this paper a new methodology to compensate the statoroltage unbalance of DFIG has been proposed. The effects of volt-ge unbalances in DFIG have been discussed, equivalent circuitsnd small-signal models, appropriate to design the current controloops, have been proposed.

The control system proposed in this paper uses negative-equence currents supplied from the stator side converter toliminate the current and voltage unbalances in the machine. Thisontrol system is suitable for stand-alone and grid-connected appli-ations.

Experimental results have been presented to validate the pro-osed control methodology. For stand-alone and grid-connectedpplications the performance of the control system has been testedonsidering variable-speed operation, fixed-speed operation andtep connection of unbalanced loads. The experimental results val-date the excellence of the proposed methodology.

For grid-connected applications the control system has beenested considering two experimental implementations to gener-te voltage unbalance in the machine. Using these experimentalmplementations, the performance of the proposed control systemas again tested considering variable-speed operation, fixed-speedperation and step connection of unbalanced loads. For the entireest carried out in this work, the performance of the proposed con-rol system is very good.

cknowledgement

This research is supported by FONDECYT, Grant 1060500.

ppendix A. List of symbols

eneralstator or rotor fluxstator or rotor current

stator or rotor voltagequantity moduleresistanceinductance

s stator leakage coefficient

Research 79 (2009) 355–364 363

total leakage coefficiente electrical torque

Time constantnumber of poles

r induction machine rotational speede stator electrical frequencysl slip frequency

r rotor position angleslip slip anglee electrical anglev voltage vector angle

ms magnetising current

uperscriptsdemanded valuepositive-sequence valuenegative-sequence valuephasor quantity

ubscripts, b, c phase quantities, ˇ two-phase fixed coordinates, q synchronous rotating coordinates, s, m rotor, stator, magnetising quantities respectively, 2 positive and negative-sequence

stator side converter quantityTheveninload

ated nominal quantity

ppendix B. System parameters

Doubly-fed induction machine: 2.0 kW, 1500 rpm, stator 220 Velta, rotor 105 V star, Rr = 0.45 , Rs = 1.7 , Ls = 0.19872 H,m = 0.1899 H, Lr = 0.01646 H, turn ratio = 3.5. 20 mh added to theotor to improve current filtering.

Stator side converter: C = 2000 �F, Lf = 12 mH. Converter switch-ng frequency = 1 kHz.

Phase locked loop:

otch filter = z2 − 1.902113z + 1z2 + 1.812478 + 0.907973

entered at 100 Hz with a bandwidth of 25 Hz and sampling fre-uency of 2 kHz; PI = 0.972168(z − 0.999756/z − 1).

eferences

[1] S. Muller, M. Deicke, R.W. De Doncker, Doubly fed induction generator systemsfor wind turbines, IEEE Ind. Appl. Mag. 8 (3) (2002) 26–33.

[2] R. Pena, J.C. Clare, G.M. Asher, A doubly fed induction generator using back-to-back PWM converters supplying an isolated load from a variable-speed windturbine, IEE Proc. Electr. Power Appl. 143 (5) (1996) 380–387.

[3] R. Pena, J.C. Clare, G.M. Asher, Doubly fed induction generator using back-to-back PWM converters and its application to variable-speed wind-energygeneration, IEE Proc. Electr. Power Appl. 143 (5) (1996) 231–241.

[4] T. Brekken, N. Mohan, A novel doubly-fed induction wind generator controlscheme for reactive power control and torque pulsation compensation underunbalanced grid voltage conditions, in: Proceedings of the IEEE 34th AnnualPower Electronics Specialists Conference, Acapulco, Mexico, 2003, pp. 15–19.

[5] T. Brekken, N. Mohan, Control of a doubly fed induction wind generator underunbalanced grid voltage conditions, IEEE Trans. Energy Conv. 22 (1) (2007)129–135.

[6] E. Muljadi, T. Batan, D. Yildirim, C.P. Butterfield, Understanding the unbalanced-voltage problem in wind turbine generation, in: Proceedings of the IEEE-IAS

Annual Meeting, Phoenix, USA, 1999, pp. 1359–1365.

[7] A.H. Ghorashi, S.S. Murthy, B.P. Singh, B. Singh, Analysis of wind driven gridconnected induction generators under unbalanced grid conditions, IEEE Trans.Energy Conv. 9 (2) (1994) 217–223.

[8] C. Hochgraf, R.H. Lasseter, Statcom controls for operation with unbalanced volt-ages, IEEE Trans. Power Deliv. 13 (2) (1988) 538–544.

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64 R. Pena et al. / Electric Power S

[9] R. Pena, R. Cardenas, E. Escobar, J. Clare, P. Wheeler, Control system for unbal-anced operation of stand-alone doubly fed induction generators, IEEE Trans.Energy Conv. 22 (2) (2007) 544–545.

10] R. Cardenas, R. Pena, J. Proboste, G. Asher, J. Clare, MRAS observer for sensorlesscontrol of standalone doubly fed induction generators, IEEE Trans. Energy Conv.20 (4) (2005) 710–718.

11] V. Kaura, V. Blasko, Operation of a phase locked loop system under distortedutility conditions, IEEE Trans. Ind. Appl. IA-33 (1) (1997) 58–63.

12] R. Cardenas, R. Pena, Sensorless vector control of induction machines for vari-able speed wind energy applications, IEEE Trans. Energy Conv. 19 (1) (2004)196–205.

13] R. Pena, R. Cardenas, J. Clare, G. Asher, Control strategies for voltage control ofa boost type PWM converter, in: Proceedings of the IEEE 32nd Annual PowerElectronics Specialists Conference, Vancouver, Canada, 2001, pp. 730–735.

14] W.D. Stevenson, Elements of Power System Analysis, first ed., McGraw-Hill,Kogakusha, 1975.

15] J.-H. Kim, S.-K. Sul, A carrier-based PWM method for three-phase four-leg volt-age source converters, IEEE Trans. Power Electron 19 (1) (2004) 66–75.

16] R. Cardenas, R. Pena, G. Asher, J. Clare, Emulation of wind turbines and flywheelfor experimental purposes, in: Proceedings of the Ninth European Conferenceon Power Electronics and Applications, Graz, Austria, 2001 (in CD-rom).

ubén Pena was born in Coronel, Chile. He received the electrical engineeringegree from the University of Concepcion, Chile, in 1984 and the M.Sc. and Ph.D.egrees from the University of Nottingham, U.K., in 1992 and 1996 respectively.rom 1985 to 1991 he was a lecturer in the University of Magallanes, Chile. He isurrently with the Electrical Engineering Department, University of Concepcion,hile. His main interests are in control of power electronics converters, A.C. drives

nd renewable energy systems. Dr. Pena is a member of the Institute of Electricalnd Electronic Engineers.

oberto Cardenas was born in Punta Arenas, Chile. He received the electrical engi-eering degree from the University of Magallanes, Chile, in 1988 and the M.Sc. andh.D. degrees from the University of Nottingham in 1992 and 1996 respectively. From

ErcuI

Research 79 (2009) 355–364

989 to 1991 he was a lecturer in the University of Magallanes. He is currently withhe Electrical Engineering Department, University of Magallanes, Chile. His mainnterests are in control of electrical machines, variable-speed drives and renewablenergy systems. Dr. Cardenas is a member of the Institute of Electrical and Electronicngineers.

nrique Escobar was born in Punta Arenas, Chile. He received the electrical engi-eering degree from the University of Magallanes, Chile, in 2006. He is currentlyith the Chilean Antarctic Institute (INACH) spending one year as a field electrical

ngineer in the Antarctica. His main interests are in control of electrical machinesnd variable-speed drives.

on Clare was born in Bristol, England. He received the B.Sc. and Ph.D. degrees in elec-rical engineering from The University of Bristol, U.K. From 1984 to 1990 he workeds a research assistant and lecturer at The University of Bristol involved in teachingnd research in power electronic systems. Since 1990 he has been with the Powerlectronics, Machines and Control Group at the University of Nottingham, U.K. ands currently professor in power electronics and head of research group. His researchnterests are: power electronic converters and modulation strategies, variable-speedrive systems and electromagnetic compatibility. Prof. Clare is a member of the Insti-ution of Electrical Engineers and is an associate editor for the IEEE Transactions onndustrial Electronics.

atrick Wheeler received his Ph.D. degree in electrical engineering for his work onatrix converters at the University of Bristol, England, in 1993. In 1993 he moved

o the University of Nottingham and worked as a research assistant in the Depart-ent of Electrical and Electronic Engineering. In 1996 he was appointed lecturer

lectronics, Machines and Control Group at the University of Nottingham, U.K. Hisesearch interests are in Variable-Speed AC Motor Drives, particularly different cir-uit topologies; power converters for power systems and semiconductor switchse. Dr. Pat Wheeler is a member of the Institution of Electrical Engineers and the

nstitute of Electrical and Electronic Engineers.