Control strategy for a Doubly-Fed Induction Generator feeding an unbalanced grid or stand-alone load

Electric Power Systems Research 79 (2009) 355–364

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Control strategy for a Doubly-Fed Induction Generator feeding an unbalanced grid or stand-alone load Rubén Pena a , Roberto Cardenas b,∗ , Enrique Escobar b , Jon Clare c , Pat Wheeler c a

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

a r t i c l e

i n f o

Article history: Received 10 February 2008 Received in revised form 15 July 2008 Accepted 18 July 2008 Available online 11 September 2008 Keywords: Induction generators Power generation control Wind energy Vector control

a b s t r a c t In this paper, the control systems for the operation of a Doubly-Fed Induction Generator (DFIG), feeding an unbalanced grid/stand-alone load, are presented. The scheme uses two back-to-back PWM inverters connected between the stator and the rotor, namely the rotor side and stator side converters respectively. The stator current and voltage unbalances are reduced or eliminated by injecting compensation currents into the grid/load using the stator side converter. The proposed control strategy is based on two revolving axes rotating synchronously at ±ωe . From these axes, the d–q components of the negative and positive-sequence currents, in the stator and grid/load, are obtained. The scheme compensates the negative-sequence currents in the grid/load by supplying negative-sequence currents via the stator side converter. Experimental results obtained from a 2-kW experimental prototype are presented and discussed in this work. The proposed control methodology is experimentally validated for stand-alone and weak grid-connected conditions and the results show the excellent performance of the strategy used. © 2008 Elsevier B.V. All rights reserved.

1. Introduction The Doubly-Fed Induction Generator (DFIG) is widely used for variable-speed generation, and it is one of the most important generators for Wind Energy Conversion Systems (WECS) [1]. Both gridconnected and stand-alone operation is feasible [2,3]. For variablespeed operation, the standard power electronics interface consists of a rotor and stator side PWM inverters that are connected back-toback. These inverters are rated, for restricted speed range operation, to a fraction of the machine rated power [2]. Applying vector control techniques yields current control with high dynamic response [2,3]. In grid-connected applications, the DFIG may be installed in remote, rural areas [4,5] where weak grids with unbalanced voltages are not uncommon. As reported in [6,7], induction machines are particularly sensitive to unbalanced operation since localized heating can occur in the stator and the lifetime of the machine can be severely affected. Furthermore, negative-sequence currents in the machine produce pulsations in the electrical torque, increasing the acoustic noise and reducing the life span of the gearbox, blade assembly and other components of a typical WECS [4,5]. To protect the machine, in some applications, DFIGs are disconnected from the grid when the phase-to-phase voltage unbalance is above 6% [5].

∗ Corresponding author. E-mail address: [email protected] (R. Cardenas). 0378-7796/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2008.07.005

Control systems for the operation of induction generators in unbalanced grids have been reported in [4,5], where it is proposed to inject compensating current in the DFIG rotor to eliminate or reduce torque pulsations. The main disadvantage of this method is that the stator current unbalance is not eliminated [5]. Therefore, even when the torque pulsations are reduced, the induction machine power output is derated, because the machine current limit is reached by only one of the stator phases. Compensation of unbalanced voltages and currents in power systems are addressed in [8] where a STATCOM is used to compensate voltage unbalances. However, the application of the control method to DFIGs is not discussed. No formal methodology for the design of the control systems is presented and only simulation results are discussed in [8]. In this paper a new control system to compensate the stator current unbalance in grid-connected and stand-alone DFIG operation is presented. The strategy uses two revolving axes rotating synchronously at ±ωe to obtain the d–q components of the negative and positive-sequence currents in the stator and grid/load. The unbalance is compensated by the stator side converter. The stator side converter positive-sequence current is conventionally controlled to regulate the dc link voltage, whereas negative-sequence current is regulated to reduce or eliminate the grid voltage unbalance. The control system for unbalanced operation of stand-alone DFIGs has been succinctly discussed in a two-page paper published by the authors [9]. However, in that publication issues such as small-signal models, controller design, compensation of volt-

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age unbalances in grid-connected DFIGs, etc., were not discussed. In addition, this work also introduces new methods, application scenarios and insights. These include:

Considering (1) and (2) the following expression is obtained for the dynamics of the equivalent stator magnetising current: s

• The modelling of a DFIG for unbalanced operation is presented. This model can be used to study the effects of negative-sequence components in the machine, e.g. torque pulsations in the electrical torque, ripple in the machine flux, etc., The modelling presented in Section 2 is suitable for analysing DFIGs feeding unbalanced loads as well as DFIGs connected to unbalanced, weak grids. • The control systems, based on two synchronously rotating axes, are presented and fully analysed. The control systems of Section 4 are linearised using small-signal models suitable for designing the current controllers of the positive/negative-sequence current control loops. • The control systems, suitable for compensating the effects of negative-sequence components in a DFIG connected to an unbalanced grid are discussed. Grid-connected operation is considered as the most important application of DFIGs. The rest of this paper is organised as follows. In Section 2 the modelling of the DFIG, for variable-speed stand-alone and gridconnected operation, is addressed. In Section 3 the control systems for balanced operation of DFIGs are briefly reviewed. In Section 4 the control systems for unbalanced operation of DFIGs are presented. In Section 5 experimental results obtained from a 2 kW prototype are analysed. Finally an appraisal of the proposed control method is discussed in the conclusions.

2.1. Modelling of DFIGs considering balanced operation The modelling of DFIGs, for balanced operation is reviewed in this section. Further details can be found in [2,3]. The nomenclature used is shown in Appendix A, with superscript + used to indicate that the d–q-axes are rotating at +ωe (positive-sequence). The machine equations written in a d–q synchronous frame rotating at the supply frequency +ωe , are [2]: + ds

⎤

⎡L

0

s

⎢ +⎥ ⎢ ⎢ qs ⎥ ⎢ 0 Ls ⎢ ⎥ ⎢ + ⎥ = ⎢ ⎣ dr ⎦ ⎣ Lm 0  

+ qr

v+ ds v+ qs v+ dr v+ qr

0



=



=

Lm 0 Lr 0

Lm

Rs

0

0

Rs

Rr

0

0

Rr





+ ids + iqs + idr + iqr



⎤ ⎡ i+ ⎤ ds ⎢ +⎥ ⎢ i Lm ⎥ ⎥ ⎢ qs ⎥ ⎥⎢ + ⎥ ⎥ 0 ⎦ ⎣ idr ⎦ 0

Lr +

 d dt



 +

d dt

+ iqr

+ ds + qs + dr + qr

Ls + =− i Lm qs

= Lm ims ; + ds

(1)



+



+

0

−ωe

ωe

0

0

−ωsl

ωsl

0



(7)

In steady state and balanced operation of the DFIG, the d–q components of the stator and rotor currents are dc values and the electrical torque is constant. 2.2. Modelling of DFIG considering unbalanced operation Assuming negligible zero-sequence components in the grid, the unbalanced voltage of a weak grid can be described using negative and positive-sequence components: vˆ s = v1s ej(ωe t+v+ ) + v2s e−j(ωe t+v− )

(8)

where v1s and v2s are the moduli of the positive and negativesequence voltages respectively. Referring (8) to d–q-axes rotating at +ωe and −ωe yields: +

vˆ s = v1s ejv+ + v2s e−j(2ωe t+v− ) = v1s e

j(2ωe t+v+ )

+ v2s

(9)

e−jv−

(10)

As shown in (9) the negative-sequence voltage produces doublefrequency components when referred to the frame rotating a + ωe . On the other hand, the positive-sequence voltage also produces a double-frequency components in the d–q-axes rotating at −ωe (see (10)). In general, any unbalanced vector in the stator frame can be written as xˆ s = x1s ej(ωe t+x+ ) + x2s e−j(ωe t+x− )

(11)

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

+ ds + qs



+ dr + qr

(2)

(3)

(5)

(12)

where x1 and x2 are the moduli of positive and negative-sequence components. Using (9)–(12), (2) and (3) are written in the negativesequence frame as





(4) + qs = 0

p + + + + Te = 3 Lm (iqs idr − ids iqr ) 2

xˆ r = x1r ej((ωe −ωr )t+x+ ) + x2r ej((−ωe −ωr )t+x− )

where s , r , vs and is are the stator and rotor flux vectors and the stator voltage and current vectors respectively; Ls , Lm , Lr , Rr and Rs are the stator, magnetising and rotor inductances and the rotor and stator resistance respectively; and ωsl = ωe − ωr is the slip frequency with ωr the rotational speed. Aligning the d-axis on the stator flux vector yields: + iqr

(6)

where  s = Ls /Rs and  s = (Ls − Lm )/Lm are the stator time constant and leakage factor respectively. Therefore, for stand-alone applications, the stator magnetising current, hence the stator voltage, can + be controlled using the positive-sequence direct rotor current idr . For grid-connected operation ims can be supplied from the machine stator and/or rotor. The torque produced by the DFIG is obtained as [2,3]:

− vˆ s

2. Modelling of DFIGs

⎡

dims 1 + s + + + vds + ims = idr Rs dt



v− ds v− qs v− dr v− qr





=



Rs 0

0 Rs

Rr 0

0 Rr

=

+





− ids − iqs − idr − iqr

0 (ωe + ωr )





d + dt d + dt





− ds − qs − dr − qr

−(ωe + ωr ) 0



+

0 −ωe

ωe 0



− ds − qs



(13)





− dr − dr

 (14)

For stator unbalanced operation the electrical torque, Te , is obtained as (see (11)): p ∗ Te = 3 Lm [(i1qs − i2s sin(2ωe t + i− ))idr 2 ∗ −(i1ds + i2s cos(2ωe t + i− ))iqr ]

(15)

where the terms i2s sin(2ωe t +  i− ) and i2s cos(2ωe t +  i− ) are double-frequency current components produced by the negativesequence stator current. In (15) the rotor current is assumed to be

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equal to the demand i∗r . Even with balanced rotor currents (i.e. ∗ , i∗ are continuous signals) the electrical torque of (15) has idr qr a double-frequency component, which can produce large torque pulsations. Torque pulsations may be reduced or eliminated by adding ∗ and i∗ [4,5]. compensation components to the rotor currents idr qr However, this compensation technique does not eliminate the negative-sequence components of the stator/rotor currents and voltages. Furthermore, according to [5], the unbalance in the stator current may even be increased when compensation signals are added to the rotor current. Rotor compensating currents can be used to achieve other control targets such as balancing the stator currents. However, using rotor current compensation techniques does not provide enough degrees of freedom to achieve several control targets simultaneously, for instance to eliminate both the stator current unbalance and the torque pulsations. Furthermore, the slip velocity for the negative-sequence frequency is −(ωe + ωr ) (see (14)), therefore a negative-sequence rotor flux may produce a large machine back e.m.f., i.e. relatively large rotor voltages are necessary to regulate ir . This paper proposes to use the stator side converter to compensate the negative-sequence components of the grid or stand-alone load.

The control system for balanced operation of grid-connected DFIGs has already been discussed in [2] and only a brief discussion is presented here. The typical control system for a grid-connected DFIG is shown in Fig. 1. The d–q reference frame is orientated along the stator flux. The demodulation of the rotor currents and modulation of the rotor demand voltages uses the slip angle derived from: (16)

where  r is the rotor position (sensorless operation is also feasible [10]). For balanced operation, the stator flux vector position  e can be obtained from the stator flux ˛–ˇ components as

e = tan

ˇs

˛s

Fig. 1. Vector control system for DFIG operation.



˛s =



ˇs =

(v˛s − Rs i˛s ) dt (18) (vˇs − Rs iˇs ) dt

As shown in Fig. 1, PI controllers are used to regulate the rotor currents. The direct component of the rotor current may be used ∗ = 0 the to supply the magnetising current to the machine. If idr magnetising current is entirely supplied from the grid. The electri∗ (see (7)). If the cal torque is controlled using the q-axes current iqr ∗ is regulated DFIG is used in a variable-speed WECS, the current iqr to capture the maximum power of a wind turbine for a given wind condition [12]. 3.2. Control system considering stand-alone applications Stand-alone balanced operation of DFIGs is discussed in [3] and only a summary is presented here. With some minor modifications, the vector control system of Fig. 1 can be used to supply electrical energy to a stand-alone load or isolated grid. In this case, the stator flux position  e is derived from a free running integral of the stator frequency demand ωe∗ (50 Hz): e =

3.1. Control systems for grid-connected applications.

−1

The stator flux ˛–ˇ components are obtained from the stator voltages and currents as [2]:



3. Vector control of DFIGs for balanced applications

slip = e − r

357

ωe∗ dt

(19)

Typically, in stand-alone connections, the magnetising current ims is supplied entirely from the rotor. Therefore, the stator flux is ∗ (see (5)). The torque current controlled using the d-axes current idr iqr is now controlled according to ∗ iqr =−

Ls iqs Lm

(20)

which forces the orientation of the reference frame along the stator flux vector position (see (2)). More information about control systems for the stand-alone operation of DFIGs is presented in [3,10]. 4. Control system for DFIGs feeding an unbalanced grid/load. 4.1. Vector control system for the stator side converter

(17) In the control system proposed in this paper, the stator side converter is controlled to supply positive and negative-sequence currents to the grid/load. The vector control system is shown in Fig. 2. The system is orientated along the positive-sequence stator voltage vector. Because of the unbalance, a phase locked loop (PLL), shown at the bottom of Fig. 2, is implemented to calculate the stator voltage angle v [11]. A notch filter eliminates the negativesequence from the d–q voltage components. The PI controller forces the q-axes positive-sequence to zero, ensuring the orientation of the reference frame. The parameters of the notch filter and the PI controller used in the PLL are given in Appendix B. From +v and −v , the currents can be referred to two synchronous d–q-axes rotating at +ωe and −ωe respectively. Double-frequency components are produced when the positive/negative-sequence currents are referred to the d–q-axes rotating in the opposite direction [4–7]. As shown in Fig. 2, notch filters are used to eliminate these high frequency components. The control systems for the stator side positive-sequence cur+ + rents idf and iqf are entirely conventional (see Figs. 1 and 2 and + [13]). The current idf regulates the dc link voltage E and the current

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Fig. 2. Proposed control system for the stator side converter. + iqf regulates the reactive power supplied to the load. The stator side negative-sequence currents are regulated to (see Fig. 2) −∗ − = idqL idqf

(21)

− − idqL = (idqs + idqf )

Therefore the negative-sequence current demand is a function of the grid/load negative-sequence current. In steady state, when the − − − = idqL , the stator current idqs = 0 (see PI controller regulates idqf (21)), and the torque pulsations are eliminated. The total current supplied by the stator side converter is lim+ ited to avoid overloading. Assuming iqf = 0, the maximum current available to compensate the negative-sequence stator current is



2

2

− − + (idf ) + (iqf ) ≤ Irated − idf

4.2. Control of the DFIG considering connection to an unbalanced grid Fig. 3 shows a DFIG connected to an unbalanced grid. In this case the stator current has positive and negative-sequence components. The control of the DFIG is carried out using a reference frame aligned with the positive-sequence stator flux vector. For an unbalanced or distorted grid, the flux vector position is obtained using a PLL [11], with a structure similar to the one shown in Fig. 2. In this case, the inputs to the demodulator block are the stator flux ˛–ˇ components obtained from (18). The PI controller drives the stator flux positive-sequence q-axes component to zero, ensuring the correct orientation of the reference frame. The PLL parameters are the same to the ones used in Section 4.

(22)

where Irated is the converter nominal current. In this work it is assumed that the control of the positive-sequence currents has a higher priority. Therefore, in each sampling time the demand +∗ current idf is calculated first. After that the maximum negativesequence current is obtained using (22). In a, b, c coordinates, the total voltage demand for the stator side converter is obtained as (see Fig. 2):



v∗a (t) v∗b (t) v∗c (t)





=



v+ v− a (t) a (t) + vb (t) + v− (t) b v+ v− c (t) c (t)



(23) Fig. 3. Grid-connected operation of a DFIG.

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

359

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

obtained as

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

The stator side converter is controlled to supply the current ˆif given by ˆif = i1f ej(ωe t+f+ ) + i2f e−j(ωe t+f− )

(24)

where the component i2f is supplied to the grid to compensate the DFIG stator voltage unbalance. Fig. 4 shows the Thevenin equivalent circuit per phase of Fig. 3 for the negative-sequence components. The voltage v− is the equivalent grid negative-sequence voltage aT and L2T is the equivalent negative-sequence inductance of the grid. In Fig. 4 it is assumed that the coupling between the sequence networks is low. The negative-sequence equivalent for the DFIG is dependent on the machine operating point. However, if the stator and rotor voltages/currents are balanced, then the negative-sequence voltage v− as ≈ 0. In this work, it is assumed that balanced reference currents are applied to the rotor. Therefore to balance the DFIG it is only necessary to eliminate the negative-sequence stator voltage. Using the single phase circuit of Fig. 4, it can be shown that the negative-sequence current is eliminated from the DFIG stator when − − Vaf − ωe Lf Iaf =0

(25)

i.e. the stator side converter is a short circuit for the grid negativesequence voltage (see Fig. 4). Therefore the negative-sequence voltage applied to the machine’s stator is zero. Using (25) the negative-sequence current supplied by the stator side converter can be calculated as − Iaf =−

− VaT

ωe L2T

(26)

Therefore, when the stator side converter supplies the negativesequence current given by (26), the DFIG is in balanced operation. The control of the stator side converter positive-sequence current is used to regulate the dc link voltage E (see Fig. 2) and the reactive power supplied to the grid. This is entirely conventional and will not be discussed here. The interested reader is referred to [13].



ia (t) 1/(sLa + Ra ) ib (t) = 0 ic (t) 0

0 1/(sLb + Rb ) 0

0 0 1/(sLc + Rc )



va (t) vb (t) vc (t)

(27)

where va , vb and vc are the instantaneous DFIG line to neutral stator voltages and La , Lb , Lc and Ra , Rb , Rc are the load inductances and resistances respectively. The negative-sequence current can be obtained from (27) as [14]: − (t) = ia (t) + ib (t) ej2/3 + ic (t) e−j2/3 iaL

(28)

and the zero-sequence current is obtained as 0 (t) = ia (t) + ib (t) + ic (t) iaL

(29)

The zero-sequence current does not produce a resulting stator flux; therefore this current does not produce torque pulsations in the machine. If the negative-sequence stator current is relatively small, the line to neutral voltages are approximately balanced because a stator flux control loop regulates idr . Therefore, there are negligible negative/zero-sequence load voltages. For the standalone system of Fig. 5, the sequence components are coupled so that positive-sequence voltages produce negative and zero-sequence currents in the load. A single phase equivalent system of Fig. 5 is shown in Fig. 6. The negative-sequence current is represented as a current source. The current of (28) can be supplied from the DFIG or the stator side converter. Because the aim is to have balanced stator current, the stator negative-sequence component is eliminated when − − = iaL iaf

(30)

Therefore, the stator side converter is again similar to a short circuit for the negative-sequence current components and the full negative-sequence current circulates through stator side converter − ≈0 with idqs The control system described in Fig. 2 compensates the negative-sequence stator current. The zero-sequence component

4.3. Vector control system for a DFIG feeding an unbalanced stand-alone load Fig. 5 shows a DFIG sourcing an unbalanced load. The machine stator and the load are star-connected, with the neutral of the generator connected to the load. This allows the presence of zerosequence currents. The topology of Fig. 5 is similar to that used by diesel driven synchronous generators feeding stand-alone loads. Using Fig. 5, the instantaneous currents in the phases a, b and c are



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

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

Fig. 7. Small-signal model. (a) Grid-connected applications and (b) stand-alone applications.

is not compensated for and the stator current is still unbalanced. Therefore localized heating is not completely eliminated, and the DFIG power may be derated if only one of the phases reaches the stator current limit. Zero-sequence current compensation cannot be achieved with the proposed control system, unless a four-leg converter is used [15]. However, the application of four-leg inverters for the compensation of zero-sequence stator currents is considered outside the scope of this paper. 4.4. Small-signal models As discussed in Section 4, the proposed control strategy drives to zero the negative- sequence stator current. For grid-connected applications the small-signal model is shown in Fig. 7a. The transfer function relating the stator side converter voltage v− to the stator dqf currents

− idqs

is

v− dqf

L2T sL2 L2s + L2T L2T L2s L2 = Lf + L2T + L2s − idqs ≈

(31)

In (31) cross coupling terms between the d- and q-axes are neglected. This is normal, because they are compensated for and their effect is eliminated at the controller output [2]. The resistances of the machine filter and lines have also been neglected in (31). For stand-alone unbalanced loads (see Fig. 7b), the transfer function between the stator side converter voltage v− and the stator dqf

− currents idqs is obtained assuming that the negative-sequence current load of (28) is constant. From Fig. 7, the transfer function is obtained as − idqs ≈

v− dqf s(Lf + L2s )

(32)

In (32) the resistances and cross coupling terms are also neglected. The small-signal transfer functions of (31) and (32) can be used to design the controllers using conventional root-locus analysis. 5. Experimental results The control systems of Figs. 1 and 2 have been implemented using a 2 kW DFIG driven by a cage machine. This cage induction machine may be used to emulate a wind turbine or another prime mover using the emulation techniques presented in [16]. The experimental rig is shown in Fig. 8. Two PWM back-to-back inverters are connected to the machine rotor. Current transducers are used to

Fig. 8. Experimental system.

measure the rotor, stator and stator side converter currents. Voltage transducers measure the stator voltage. A position encoder of 10,000 pulses per revolution (ppr) is used to measure the rotor position. The parameters of the whole system are in Appendix B. The PI controller parameters for the rotor and stator side converter currents, the dc link voltage and the magnetising current, based on the procedure shown in [2,3], are depicted in Table 1. 5.1. Experimental results for a DFIG feeding an unbalanced stand-alone load. Figs. 9 and 10 show the performance of the proposed control system for negative-sequence current compensation under variable-speed stand-alone operation. The load consists of three resistors of 25, 154 and 154 , connected to phases a, b and c 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, the compensation system is disabled and the stator current has a negative-sequence component (see Fig. 9). At t ≈ 1.25 s the comTable 1 Parameters of PI controllers System

PI controller

Closed loop natural Sampling frequency (Hz); frequency (Hz) damping factor

Rotor current

11.14

z − 0.893734 z−1

70; 0.8

2000

Stator side converter current

z − 0.897692 26.35 z−1

70; 0.8

2000

dc link voltage

0.034256

z − 0.975753 z−1

1.25; 0.8

200

Magnetising current 0.788693

z − 0.934173 z−1

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

Fig. 10. Stator and rotor unfiltered currents referred to the d–q positive-sequence axes. Fig. 13. Control system response for a load-step in one phase.

pensation is enabled and the stator current

− idqs

is driven to zero.

− − For t > 1.5 s, idqL ≈ idqf and the negative-sequence currents are eliminated from the machine stator. Fig. 10 shows the unfiltered machine currents referred to the d–q-axes rotating at +ωe (corresponding to the test of Fig. 9). Notch filters are not applied to the currents shown in Fig. 10. For t < 1.25 s the DFIG d–q currents have a double-frequency component which is eliminated after t ≈ 1.25 s when the proposed control system is enabled. The stator voltage, in the ˛–ˇ coordinates (corresponding to the test of Fig. 9), is shown in Fig. 11. Before enabling the compensation the voltage is unbalanced (see Fig. 11a). After the control system is enabled, the negative-sequence load current is supplied by the stator side converter and the stator voltage is approximately balanced (see Fig. 11b). Fig. 12 shows the unfiltered rotor voltage referred to the positive-sequence d–q-axes. For t < 1.25 s the d–q rotor voltage has a large double-frequency component, because ∗ and i∗ are constant d–q values without negative-sequence idr qr components. Therefore the PI controller injects negative-sequence voltages in the rotor in order to compensate the negative-sequence machine back e.m.f. (see (14)). When the compensation system is enabled, negative-sequence currents and voltages are completely eliminated from the machine. Figs. 13 and 14 show the control system’s response to an unbalanced load-step. For t < 4 s the stand-alone DFIG is feeding a load of

about 200 per phase. The line to neutral voltage is about 145 V and the rotational speed is 1650 rpm. At t ≈ 4 s the load resistance in one phase is changed from 200 to 33 . In Fig. 13, the d–q components of the rotor and stator currents are shown. These currents are referred to the axes which are rotating at +ωe . No notch filter is applied to these signals. When the load-step is produced, the rotor quadrature current increases to compensate the increase in the machine output power. As shown in Fig. 13, there are no doublefrequency components in any of the machine currents, because the compensation system is enabled during that test. In Fig. 14 the negative-sequence currents of the machine stator and stator side converter are shown. When the unbalanced load-step is connected, the converter negative-sequence current is increased, compensat− ing the load-unbalance and driving the current idqs to zero. As shown in Fig. 14, the proposed control system has a good dynamic response.

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

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

5.2. Experimental results for a DFIG feeding an unbalanced grid Unbalanced stator voltages can be created using the schemes shown in Fig. 15. In Fig. 15a the unbalanced grid is created by adding a variable voltage to one of the phases. Another possibility is shown in Fig. 15b. In this case the voltage unbalance is produced by connecting an unbalanced load to a balanced weak grid. For the implementation of Fig. 15b, the equivalent circuit of Fig. 4 is not

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

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

Fig. 16. Rotor and stator currents.

completely correct because the sequence components are coupled, i.e. unbalanced stator voltages are produced even when the weak grid is balanced and has only positive-sequence voltages. Using the experimental system shown in Figs. 8 and 15a, the proposed compensating system has been tested considering variable-speed grid-connected operation. The rotational speed is varied from ≈1600 rpm to 1400 rpm in 4 s. The machine is magnetised from the stator and the rotor current idr = 0. During the test the rotor quadrature current is regulated to a constant value of 10 A. The negative-sequence compensation system is enabled at t ≈ 2 s. A voltage of 82 V is added to phase a. Fig. 16 shows the unfiltered machine currents referred to the d–q-axes rotating at +ωe . Notch filters are not applied to these currents. In t ≈ 2 s, the compensation system is enabled and the negative-sequence currents are eliminated from the machine. The negative-sequence currents are shown in Fig. 17. When negative-

Fig. 17. Machine stator and stator side converter negative-sequence currents for the test of Fig. 16.

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

sequence current is injected from the stator side converter, the stator negative-sequence current is driven to zero. In Fig. 18 the electrical torque is shown. The torque is calculated using (7), with the unfiltered d–q components obtained from the axes rotating at +ωe . As shown in Fig. 18a, before the compensating system is enabled, there are large torque pulsations of ±5 N m (35% of the dc value). The torque pulsations are eliminated when the compensation is enabled. In Fig. 18b, an amplified view of Fig. 18a is shown. The torque pulsations, with a fundamental frequency of 100 Hz, are noticeable in this figure. The stator voltage corresponding to the test of Fig. 16 is shown in Fig. 19. The ˛–ˇ components of the stator voltage are unbalanced for t < 2 s. When the proposed compensation system is enabled the stator voltage is completely balanced and the negative-sequence components are eliminated. The hardware implementation of Fig. 15b was also used to test the proposed control system. The weak grid is implemented using a variable 3␾ transformer and 20 mH line inductances. The machine is running at 1400 rpm, with a line-to-line stator voltage of 260 V and a resistance of 12.5 is connected between phases a and c in t ≈ 2.5 s. Fig. 20a shows the unfiltered rotor and stator machine currents with the compensation strategy enabled. When the unbalanced load-step is applied there is a small disturbance in the currents that is compensated for by the proposed control scheme. Fig. 20b shows the d–q negativesequence currents in the stator side converter and DFIG stator. After the load-step is connected the stator side converter negativesequence current is used to drive to zero the unbalance in the stator current. For this test the equivalent circuit of Fig. 4 is not completely correct, however even in this case the proposed com-

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

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

 Te  p ωr ωe ωsl r  slip e v ims

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

pensation system has a good performance with a fast dynamic response. 6. Conclusions In this paper a new methodology to compensate the stator voltage unbalance of DFIG has been proposed. The effects of voltage unbalances in DFIG have been discussed, equivalent circuits and small-signal models, appropriate to design the current control loops, have been proposed. The control system proposed in this paper uses negativesequence currents supplied from the stator side converter to eliminate the current and voltage unbalances in the machine. This control system is suitable for stand-alone and grid-connected applications. Experimental results have been presented to validate the proposed control methodology. For stand-alone and grid-connected applications the performance of the control system has been tested considering variable-speed operation, fixed-speed operation and step connection of unbalanced loads. The experimental results validate the excellence of the proposed methodology. For grid-connected applications the control system has been tested considering two experimental implementations to generate voltage unbalance in the machine. Using these experimental implementations, the performance of the proposed control system was again tested considering variable-speed operation, fixed-speed operation and step connection of unbalanced loads. For the entire test carried out in this work, the performance of the proposed control system is very good. Acknowledgement This research is supported by FONDECYT, Grant 1060500. Appendix A. List of symbols

General  stator or rotor flux i stator or rotor current v stator or rotor voltage x quantity module R resistance L inductance stator leakage coefficient s

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total leakage coefficient electrical torque Time constant number of poles induction machine rotational speed stator electrical frequency slip frequency rotor position angle slip angle electrical angle voltage vector angle magnetising current

Superscripts * demanded value + positive-sequence value − negative-sequence value ˆ phasor quantity Subscripts a, b, c phase quantities ˛, ˇ two-phase fixed coordinates d, q synchronous rotating coordinates r, s, m rotor, stator, magnetising quantities respectively 1, 2 positive and negative-sequence f stator side converter quantity T Thevenin L load rated nominal quantity Appendix B. System parameters Doubly-fed induction machine: 2.0 kW, 1500 rpm, stator 220 V delta, rotor 105 V star, Rr = 0.45 , Rs = 1.7 , Ls = 0.19872 H, Lm = 0.1899 H, Lr = 0.01646 H, turn ratio = 3.5. 20 mh added to the rotor to improve current filtering. Stator side converter: C = 2000 ␮F, Lf = 12 mH. Converter switching frequency = 1 kHz. Phase locked loop: Notch filter =

z 2 − 1.902113z + 1 z 2 + 1.812478 + 0.907973

centered at 100 Hz with a bandwidth of 25 Hz and sampling frequency of 2 kHz; PI = 0.972168(z − 0.999756/z − 1). References [1] S. Muller, M. Deicke, R.W. De Doncker, Doubly fed induction generator systems for 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-toback PWM converters supplying an isolated load from a variable-speed wind turbine, IEE Proc. Electr. Power Appl. 143 (5) (1996) 380–387. [3] R. Pena, J.C. Clare, G.M. Asher, Doubly fed induction generator using backto-back PWM converters and its application to variable-speed wind-energy generation, IEE Proc. Electr. Power Appl. 143 (5) (1996) 231–241. [4] T. Brekken, N. Mohan, A novel doubly-fed induction wind generator control scheme for reactive power control and torque pulsation compensation under unbalanced grid voltage conditions, in: Proceedings of the IEEE 34th Annual Power Electronics Specialists Conference, Acapulco, Mexico, 2003, pp. 15–19. [5] T. Brekken, N. Mohan, Control of a doubly fed induction wind generator under unbalanced grid voltage conditions, IEEE Trans. Energy Conv. 22 (1) (2007) 129–135. [6] E. Muljadi, T. Batan, D. Yildirim, C.P. Butterfield, Understanding the unbalancedvoltage 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 grid connected 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 voltages, IEEE Trans. Power Deliv. 13 (2) (1988) 538–544.

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[9] R. Pena, R. Cardenas, E. Escobar, J. Clare, P. Wheeler, Control system for unbalanced 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 sensorless control 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 distorted utility conditions, IEEE Trans. Ind. Appl. IA-33 (1) (1997) 58–63. [12] R. Cardenas, R. Pena, Sensorless vector control of induction machines for variable 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 of a boost type PWM converter, in: Proceedings of the IEEE 32nd Annual Power Electronics 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 voltage 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 flywheel for experimental purposes, in: Proceedings of the Ninth European Conference on Power Electronics and Applications, Graz, Austria, 2001 (in CD-rom). ˜ was born in Coronel, Chile. He received the electrical engineering Rubén Pena degree from the University of Concepcion, Chile, in 1984 and the M.Sc. and Ph.D. degrees from the University of Nottingham, U.K., in 1992 and 1996 respectively. From 1985 to 1991 he was a lecturer in the University of Magallanes, Chile. He is currently with the Electrical Engineering Department, University of Concepcion, Chile. His main interests are in control of power electronics converters, A.C. drives ˜ is a member of the Institute of Electrical and renewable energy systems. Dr. Pena and Electronic Engineers. Roberto Cardenas was born in Punta Arenas, Chile. He received the electrical engineering degree from the University of Magallanes, Chile, in 1988 and the M.Sc. and Ph.D. degrees from the University of Nottingham in 1992 and 1996 respectively. From

1989 to 1991 he was a lecturer in the University of Magallanes. He is currently with the Electrical Engineering Department, University of Magallanes, Chile. His main interests are in control of electrical machines, variable-speed drives and renewable energy systems. Dr. Cardenas is a member of the Institute of Electrical and Electronic Engineers. Enrique Escobar was born in Punta Arenas, Chile. He received the electrical engineering degree from the University of Magallanes, Chile, in 2006. He is currently with the Chilean Antarctic Institute (INACH) spending one year as a field electrical engineer in the Antarctica. His main interests are in control of electrical machines and variable-speed drives. Jon Clare was born in Bristol, England. He received the B.Sc. and Ph.D. degrees in electrical engineering from The University of Bristol, U.K. From 1984 to 1990 he worked as a research assistant and lecturer at The University of Bristol involved in teaching and research in power electronic systems. Since 1990 he has been with the Power Electronics, Machines and Control Group at the University of Nottingham, U.K. and is currently professor in power electronics and head of research group. His research interests are: power electronic converters and modulation strategies, variable-speed drive systems and electromagnetic compatibility. Prof. Clare is a member of the Institution of Electrical Engineers and is an associate editor for the IEEE Transactions on Industrial Electronics. Patrick Wheeler received his Ph.D. degree in electrical engineering for his work on matrix converters at the University of Bristol, England, in 1993. In 1993 he moved to the University of Nottingham and worked as a research assistant in the Department of Electrical and Electronic Engineering. In 1996 he was appointed lecturer (subsequently senior lecturer in 2002) in power electronic systems with the Power Electronics, Machines and Control Group at the University of Nottingham, U.K. His research interests are in Variable-Speed AC Motor Drives, particularly different circuit topologies; power converters for power systems and semiconductor switch use. Dr. Pat Wheeler is a member of the Institution of Electrical Engineers and the Institute of Electrical and Electronic Engineers.