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Published in IET Renewable Power GenerationReceived on 18th October 2007Revised on 6th December 2007doi: 10.1049/iet-rpg:20070104

ISSN 1752-1416

Specification of rotor side voltage sourceinverter of a doubly-fed induction generatorfor achieving ride-through capabilityS. Chondrogiannis* M. BarnesUniversity of Manchester, School of Electrical and Electronic Engineering, Manchester, PO Box 88, M60 1QD, UK*Present address: 4 Agiou Pavlou, Kavala 65302, GreeceE-mail: mike.barnes@manchester.ac.uk

Abstract: The ride-through capability of a doubly-fed induction generator under three-phase balanced voltagesags is examined, under the condition that the machine should have the capability of real and reactive powercontrol during the external fault. Mathematical formulae for the peak rotor fault current and the requiredrotor voltage output under vector control are derived. Moreover, the DC link dynamics are incorporated intothe analysis and it is shown that they can have a decisive impact on fault behaviour of the machine duringvoltage sags. Combined, a design methodology for the rotor side voltage source inverter aiming to achieve aride-through capability at the lowest cost is described. Simulation results in PSCAD/EMTDC show very goodagreement with the theoretical analysis.

1 IntroductionThe doubly-fed induction generator (DFIG) wind turbine iscurrently the most widely used topology in wind farms. Oneof its main advantages over the other main variable speedtechnology, the synchronous generator coupled to the gridthrough a frequency converter in series, is the reducedrating of the power electronics employed [1, 2]. Themaximum power that the frequency converter of the DFIGhas to handle under steady-state conditions is only afraction of the rated power output. This fraction isapproximately equal to the maximum slip (in absolute values).

The above applies to steady-state conditions. At theoccurrence of a fault in the power supply network, a gridconnected DFIG experiences serious rotor overcurrents [3,4], which could damage the rotor side voltage sourceinverter (VSI). A popular method to protect the powerelectronics during sudden changes of the stator voltage isthe use of a crowbar circuit, which short-circuits the rotorwhen a fault occurs. After fault clearance, the rotor sideVSI is reconnected [5, 6]. However, control of the real andreactive power output of the DFIG is only possible as longas the rotor side VSI remains connected to the rotor

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circuit. Crowbar triggering turns the machine into asquirrel cage induction machine that draws reactive power,suppressing the voltage further during voltage sags.Moreover, triggering of the crowbar protection makes thereal power output of the machine unpredictable [7].

The discussion above shows that the acceptability of a ride-through technique depends fundamentally on the definitionof the ‘ride-through capability’. If the main purpose is forthe wind turbine to remain connected to the grid duringthe voltage sag and support the recovery of the system afterfault clearance, a crowbar scheme can provide acceptableperformance. However, if the wind turbine is to behavesimilarly to a classical directly connected synchronousgenerator, supporting the system with real and reactivepower injection during the voltage sag, disconnection of theexcitation circuit (i.e. the rotor side VSI) of the DFIGduring faults becomes problematic.

This paper contributes to this discussion by providing arelatively simple mathematical formula for the calculation ofthe DFIG peak rotor fault current under a three-phasebalanced voltage sag. The calculations refer to a DFIGwhich employs vector control, since this is the most popular

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scheme [4, 8–12]. In order that the rotor side VSI be able torestrain the rotor current within certain acceptable limits, itsvoltage output should not saturate during the fault [13, 14].Accordingly, the required rotor voltage output is alsomathematically estimated. Together, these two expressionscan provide a prediction of the required rating of the rotorside VSI of a DFIG for achieving a ride-through capabilitywhen control of the real and reactive power outputs of themachine during voltage sags is required.

It should be noted that unbalanced faults could induceenhanced DFIG rotor overcurrents in comparison withbalanced voltage sags [14]. There is a significant differencebetween the fault behaviour of a DFIG and that of adirectly coupled synchronous generator, since in the lattercase a three-phase bolted fault is the worst case fault for itstransient stability [15]. However, the paper concentrates onthree-phase balanced voltage sags, since the longer term‘ride-through capability’ of the GB Grid Code refersmainly to such kinds of fault conditions [16].

2 Description of the DFIG vectorcontrolIn this paper the fault behaviour of a DFIG employing statorvoltage orientated (SVO) (or grid-flux oriented) vectorcontrol is examined [17]. The T-equivalent circuit of theDFIG in the synchronous reference frame is shown inFig. 1. The exact configuration of the inner currentcontroller is shown in Fig. 2 in a block diagram form. Theexact configuration of the inner current controller along

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with the employed feed-forward compensation terms is thesame as presented in [18].

Throughout this work the positive direction of the statorcurrent is out of the stator, and that of the rotor current is intothe rotor. All quantities are expressed in pu. The base values ofthe pu followed system are provided in Appendix 2. InAppendix 3 the notations employed in this paper are provided.

Referring to Fig. 2, the transfer function of the main loopis (x denotes either the d or the q axis)

F (s) ¼ Ixr=Ixr,ref ¼kPs þ (kI=vb)

Lsrs2 þ (Rr þ kP )s þ (kI=vb)

(1)

The inverse of the dynamic stiffness function or ‘noise’function is

N (s) ¼ Ixr=Cxy ¼s

Lsrs2 þ (Rr þ kP )s þ (kI=vb)

(2)

where Cxy denotes any of Cd2, Cd3, (Cq2 � Cq2), Cqc3.

Figure 1 T-equivalent of DFIG in the synchronous referenceframe

Figure 2 Inner current loop of employed vector control (time in pu)

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Expressing the second order system in terms of thedamping ratio z and natural frequency vn, (1) and (2) become

F (s) ¼(2Lsrzvn � Rr)s þ Lsrv

2n

Lsr(s2 þ 2zvns þ v2

n)(3)

N (s) ¼s

Lsr(s2 þ 2zvns þ v2

n)(4)

where

vn ¼

ffiffiffiffiffiffiffiffiffiffiffiffikI

Lsrvb

s¼) kI ¼ Lsrvbv

2n (5)

z ¼kp þ Rr

2Lsrvn

¼) kP ¼ 2Lsrzvn � Rr (6)

The characteristic polynomial is

C(s) ¼ s2 þ 2zvn þ v2n (7)

Note that since in all equations time is expressed in pu, thenatural frequency vn should be perceived in ‘pu’ values withbase value the fundamental angular frequency vb.

3 Fundamental assumptionsThe mathematical analysis in this work is based on theassumption that the stator resistance is negligible. Thissimplifies greatly the calculation of the stator flux responseunder a three-phase voltage sag. The calculation of thestator flux response is the necessary first step for theestimation of the DFIG rotor fault current and the vectorcontrol rotor voltage output required.

It should be made clear that the simplified mathematicalexpressions derived do not provide the true response overtime of the stator flux, rotor current and required rotorvoltage output under a three-phase voltage sag. Rather, theyrepresent the response of these quantities for the case of nodamping in the DFIG system, i.e. in the case that a three-phase voltage sag triggers perfectly undamped stator fluxoscillations. However, it is considered that the amplitude ofthe ‘idealised’ undamped stator flux oscillation can beexpected to match with great accuracy the magnitude of thefirst peak of the actual damped stator flux oscillation. This isbecause the DFIG system shows very low damping [19].Furthermore, the aforementioned postulation is evaluatedthrough simulation results in Section 4.

Throughout the analysis the following assumptions aremade:

1. There is negligible phase-angle jump. Previous research [19]on the ride-through capability of aDFIGcame to the conclusionthat the magnitude of the phase-angle jump has a negligibleeffect on the magnitude of the peak fault rotor current.

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2. The mechanical dynamics are considered much slowerthan the inner current loop dynamics and, thus, the rotorspeed is considered as a fixed parameter.

3. The system is in steady state before the occurrence of a fault.

4 Stator flux response under athree-phase voltage sagUnder the aforementioned assumptions, the stator fluxresponse when a three-phase voltage sag occurs at t ¼0 is

Cs ¼

V (0�)=vs, t � 0

[DV cos(vst)þ V (0þ)]=vs

�jDV sin(vst)=vs, t . 0

8<: (8)

The mathematical proof is given in Appendix 4.1 and DV ;DVqs because of the orientation frame chosen. The vectordiagrams of the DFIG stator voltage and flux for steadystate and for transient conditions are provided in Fig. 3. Inboth diagrams, the (usually) very small voltage drop acrossthe stator resistance is neglected.

Figure 3 Vector diagrams of DFIG stator voltage and fluxunder steady-state and transient fault conditions

Stator voltage orientated

Figure 4 Comparison between theoretical calculations andsimulated results for the stator flux

Three-phase bolted fault

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In Fig. 4 theoretical and simulation results in PSCAD/EMTDC for the stator flux response to a three-phasebolted fault at the 33 kV terminals of the examinedmachine are depicted. The stator flux has been calculatedbased on the flux-current relationship of the machine, Fig. 1

Cs ¼ �LssIs þ LmIr (9)

Discrepancies between the theoretical calculations and thesimulation results are expected. In the theoretical calculations,the damping caused by the stator resistance has not beentaken into account. However, the important point is that thetheoretical calculations predict with very good accuracy themaximum amplitude of the stator flux oscillation, both for thed- and the q-axis.

5 Rotor current response under athree-phase voltage sagFrom the methodology outlined in Appendix 4.2, the rotorcurrent response under a three-phase voltage sag can be found as

Ir(t) ¼ Id r(t)þ jIqr(t)

¼ Id r,1(t)þ Idr,2(t)þ Id r,3(t)

þ j[Iqr,1(t)þ Iqr,2(t)þ Iqr,3(t)] (10)

where

Id r,1(t) ¼ L�1 Lsrs þ Rr

C(s)Idr(0�)þ F (s)Id r,ref (s)

� �

¼ Id r,ref (0�)þ L�1 F (s)Idr,ref (0þ)� Id r,ref (0�)

s

� �(11)

Idr,2(t) ¼ L�1 1

C(s)L

dCd2

dt

� �� �

¼ �SLarDV (1=vs)L�1{N (s)L {sin(vst)}} (12)

Idr,3(t) ¼ L�1 1

C(s)L

dCd3

dt

� �� �

¼ DVarL�1{N (s)L {sin(vst)}} (13)

Iqr,1(t) ¼ L�1 Lsrs þ Rr

C(s)Iqr(0�)þ F (s)Iqr,ref (s)

� �

¼ Iqr,ref (0�)þ L�1 F (s)Iqr,ref (0þ)� Iqr,ref (0�)

s

� �

(14)

Iqr,2(t) ¼ L�1

�1

C(s)L

dCq2

dt

� ��

1

C(s)L

�dCq2

dt

��

¼ �SLarDV (1=vs)L�1{N (s)L{cos(vst)}} (15)

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Iqr,3(t) ¼ L�1 1

C(s)L

dCq3

dt

� �� �

¼ DVarL�1{N (s)L{cos(vst)}} (16)

The mathematical proof is given in Appendix 4.According to (10)–(16), the rotor current response in eachaxis can be analysed physically into three components:

1. Idr,1 and Iqr,1 represent the response to the change in thereference current value after the occurrence of a fault.

2. Idr,2 and Iqr,2 represent the response caused by the changeof the stator flux (effectively the change in the back e.m.f.).

3. Idr,3 and Iqr,3 represent the response caused by thealteration in the rate of change of the stator flux (effectivelythe direct impact of the stator transients).

In (12), (13), (15) and (16), it can be seen that the rotorcurrent response in each axis caused by the change in theback e.m.f. (Idr,2 and Iqr,2) and directly by the statortransients (Idr,3 and Iqr,3) are in each axis either inopposition when the machine operates at sub-synchronousspeeds or in alignment when the machine operates atsuper-synchronous speeds. Furthermore, these tworesponses in each axis can be understood as the frequencyresponse of the ‘filter’ N(s) to a sinusoidal input of thenetwork’s frequency, weighted by a factor.

Further simplification of the rotor current response to athree-phase voltage sag can be conducted if the followingassumptions are made:

1. The transient response because of the step change in thereference current value in each axis is neglected.

2. The transient response of the filter N(s) to a sinusoidalinput of the network’s frequency is neglected.

Based on these assumptions the final equations for therotor current response are derived

Ir(t) ¼ Id r,ref (0þ)þ jIqr,ref (0þ)þ DVarLsr

1�SLvs

� �

� G[sin(vst þ f)þ j cos(vst þ f)] (17)

with

G ¼ LsrjN ( jvs)j (18)

f ¼ angle{N ( jvs)} (19)

According to (17), the rotor current response in each axisinvolves an oscillation centred to the new reference value.The amplitude of the oscillation:

1. Depends on the pre-fault rotor speed. The worst-casefault is for maximum super-synchronous speed. For wind

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turbine applications, this point corresponds to the rated realpower output pre-fault, since the rotor speed is controlledfor optimum wind power extraction [4].

2. Increases linearly to the magnitude of the voltage drop DV.

3. Depends on the machine’s inductances as expressed by thefactor ar/Lsr. With increased ar/Lsr, the amplitude ofoscillation rises, leading to a worsemaximum rotor fault current.

4. Depends on the factorG, or equivalently to the magnitudeof the frequency response of the ‘noise’ function N(s) to asinusoidal input of the network’s frequency.

6 Calculation of the requiredrotor voltage outputFollowing similar steps as the calculation of the rotor currentresponse under a three-phase voltage sag, the rotor voltageoutput required by the vector control is

Vr(t)¼Vd r(0�)þ j[Vqr(0�)�SLarDV =vs]

þL�1 kP þk0Is

� �[1�F (s)]

Idr,ref (0þ)� Id r,ref (0�)

s

� �� �

þ jL�1 kP þk0Is

� �[1�F (s)]

Iqr,ref (0þ)� Iqr,ref (0�)

s

� �� �

� arDV (1�SL=vs)

�L�1 kP þk0Is

� �N (s)L{sin(vst)þ j cos(vst)}

� �

�SLLsrL�1 F (s)

Iqr,ref (0þ)� Iqr,ref (0�)

s

� �

þ jSLLsrL�1 F (s)

Id r,ref (0þ)� Idr,ref (0�)

s

� �

þSLLsrarDV (1�SL=vs)

�L�1{N (s)L{�cos(vst)þ j sin(vst)}} (20)

Equation (20) can be simplified under the followingassumptions:

1. The transient response resulting from the step change inthe value of the rotor reference current is neglected, sincethe resulting overshoot is considerably lower than theamplitude of induced oscillations for severe faults.

2. The transient response of ‘filters’ N(s) and (kPþ kI0/s)N(s)

to a sinusoidal input of the network frequency is neglected.As will be discussed in the following section, the transientpart of the response of these ‘filters’ to a sinusoidal input isnegligible for most values of the damping ratio and naturalfrequency.

3. It is considered that 2zvn � Rr=Lsr. For most values ofthe damping ratio and natural frequency, this assumption is

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true, since the factor Rr/Lsr is very small. For example, forthe examined machine Rr/Lsr ¼ 0.012.

Under these conditions, the rotor voltage output requiredby the inner current control law under a three-phasebalanced fault is

Vr(t) ¼ Vd r,ss(0þ)þ jVqr,ss(0þ)� ar(1� SL=vs)

� DV [G1sin(vst þ f1)þ SLG cos (vst þ f)]

� jar(1� SL=vs)DV [G1 cos(vst þ f1)

� SLG sin(vst þ f)] (21)

with

Vdr,ss(0þ) ¼ RrId r,ref (0þ)� SLLsrIqr,ref (0þ) (22)

Vqr,ss(0þ) ¼ RrIqr,ref (0þ)þ SLLsrIdr,ref (0þ)þ SLarV (0þ)=vs

(23)

G1 ¼ Lsr kP þk0Ijvs

� �N (jvs)

�������� (24)

f1 ¼ angle kP þk0Ijvs

� �N (jvs)

� �(25)

According to the above expressions, the rotor voltageresponse in each axis can be analysed as the summation oftwo oscillations, centred to the new steady-state output (i.e.the voltage output if the rotor current oscillations did notexist). The maximum rotor voltage depends on:

1. The pre-fault rotor speed. With increased rotor speed, therotor voltage output requirements also increase.

2. The magnitude of the voltage drop DV. Under moresevere faults, the control law requires increased rotor voltageoutput capabilities.

3. The machine’s inductances as expressed by the factors Lsr

and ar.

4. The control parameters of the proportional-integral (PI)controllers. The control parameters affect the magnitudesof the frequency responses of the ‘noise’ function N(s) andthe ‘output’ function G1(s) for the network frequency, alongwith the respective phases f and f1.

7 Evaluation of the theoreticalanalysisIn Figs. 5 and 6, the theoretically calculated peak rotor faultcurrent and required rotor voltage are compared withsimulation results in PSCAD/EMTDC.

In general, the agreement between the theoreticalpredictions and simulation results is very good. Relativelylarge discrepancies are recorded only for low natural

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frequencies (less than 0.3 pu) and low damping ratios (lessthan 0.4), especially for the required rotor voltage output.The basic reason behind this is that for such controlvalues, the transient response of the ‘filters’ N(s) and G1(s)to a sinusoidal input are enhanced. This is shown inFigs. 7 and 8.

Overall it can be argued that the approximate theoreticalexpressions for the peak rotor fault current and therequired rotor voltage output can adequately predict therating requirements of the rotor side VSI for achieving aride-through capability. This is especially true consideringthat tuning of the inner current closed loop with very lowvalues of the natural frequency and/or of the damping

Figure 5 Evaluation of theoretical calculations

z ¼ 1.5, vr(02) ¼1.21 puThree-phase bolted faultSolid line: theoretical calculations; crosses: simulation results

Figure 6 Evaluation of theoretical calculations

vn ¼0.5 pu, vr(02) ¼1.21 puThree-phase bolted faultSolid line: theoretical calculations; crosses: simulation results

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ratio is not recommended for a satisfactory tracking of thereference current.

8 Discussion on the ratingspecifications for achieving ride-through capabilityBased on the developed mathematical expressions, thecontours of the peak rotor fault current and required rotorvoltage output can be drawn with respect to the controlparameters. These are shown in Figs. 9 and 10 for the

Figure 7 Response of N(s) and G1(s) to a sinusoidal input offundamental frequency

z ¼ 1.5Upper plots: vn ¼ 0.1 pu; lower plots: vn ¼1 puSolid line: simplified (steady state) response; dotted line: fullresponse

Figure 8 Response of N(s) and G1(s) to a sinusoidal input offundamental frequency

vn ¼ 0.5 puUpper plots: z ¼0.1 pu; lower plots: z ¼1 puSolid line: Simplified (steady state) response; dotted line: fullresponse

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examined DFIG, for a three-phase bolted fault andmaximum steady-state speed pre-fault.

The main trend indicated in Figs. 9 and 10 is that byincreasing the control effort, the peak rotor fault currentreduces. However, in order to do so, the required rotorvoltage output increases. Furthermore, a region with verylarge current and/or voltage output exists in each plot. Thecontrol parameters corresponding to these two regionsshould be avoided, since they lead to unnecessarily largerotor fault current and/or rotor voltage output.

For the examined machine, the steady-state rotor VSIrating is 0.34 pu (Ir,max ¼ 1.05 pu, Vr,max ¼ 0.32 pu). Hereas the rating of the VSI, the product of the peak rotor faultcurrent times the required peak rotor voltage output isconsidered. In an actual case, the required rating will be thepeak rotor fault current (plus a safety margin) times thenecessary DC link voltage (plus a safety margin) in orderto achieve the required rotor voltage output.

Figure 10 Required rotor voltage output as a function ofthe control parameters

Figure 9 Peak rotor fault current as a function of thecontrol parameters

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The theoretical minimum rotor side VSI rating can becalculated by considering the limit case corresponding to aninfinite damping ratio and natural frequency, orequivalently infinite proportional and integral gains of thePI controllers. In such a case, the peak rotor fault currentwill be equal to the new reference value after the occurrenceof a fault (if the relatively small overshoot because of thestep change in the reference value is neglected). The rotorvoltage output can be approximated according to

Vr(t) ¼ Vd r,ss(0þ)þ jVqr,ss(0þ)� ar 1�SLvs

� �DV

� [sin(vst þ j cos(vst)] (26)

Note that the behaviour of the system with an infinite controleffort is the same as that with a ‘perfect compensationscheme’ where the load disturbances Cd1, Cd2, Cd3 andCq1, Cq2, Cq3 are perfectly feed-forward compensated.Equation (26) shows that if the fault rotor current is toremain inside its steady-state maximum limits, the rotorside VSI must:

1. Provide the necessary voltage injection in order to track thenew current reference value – terms Vdr,ss(0þ), and Vqr,ss(0þ).

2. Counteract perfectly the induced e.m.f. associated withthe stator flux and stator flux transients. Counteraction ofthese terms will increase the required rotor voltage outputbeyond its steady-state maximum absolute value.

The theoretical calculations show that for the examinedmachine the ideal minimum rotor side VSI rating is1.33 pu (Ir,peak ¼ 1.05 pu, Vr,peak ¼ 1.27 pu). Thus, thetheoretical minimum rotor side VSI short-term rating forachieving a ride-through capability under three–phasebolted faults is 3.9(¼1.33/0.34 pu) times larger than therequired steady-state rating.

Overall, the analysis shows that if the rotor side VSI isrequired to be fully operational during severe voltage sags,then its short-term current and voltage withstandingcapability must be considerably larger than its continuouscapability. However, for the inverter switches typicallyemployed in DFIGs at present, ‘short-term’ here issufficiently long to define the steady-state semiconductordevice rating. In such a case, the argument that the DFIGtopology permits the use of relatively low-rating powerelectronic converters is true only for the grid side inverter.

9 Impact of DC link dynamicsTo complete the analysis on the ride-through capability ofthe DFIG DC link dynamics have to be included, sincethe rotor voltage output capability during the voltage sagdepends also on the DC link voltage response. This in itsturn depends both on the operation of the grid side VSIand the behaviour of the rotor side VSI during the fault inthe power supply system.

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The power handling capability of the grid side inverterand, thus, its capability to regulate the DC link voltageeffectively, is linear with the magnitude of the retainednetwork voltage during voltage sags [20]. For a three-phasebolted fault, the grid side inverter becomes practicallyincapable of ‘pumping’ out or in any power at all. As aresult, for the worst-case voltage sag, the DC link responseduring the fault depends only on the behaviour of the rotorside VSI.

The power entering the DC link from the rotor is (positivedirection of the rotor current is into the rotor)

Pin ¼ �(Vd rIdr þ VqrIqr) (27)

A typical waveform of the power entering the DC linkvoltage during a fault is shown in Fig. 11. At theoccurrence of a fault, the DC link capacitor is starting tocharge as a result of the incoming power from the rotorcircuit. To protect the DFIG converter from DC linkovervoltages, a braking resistor is usually employed, whichis activated when the DC link voltage reaches apredetermined safety limit [21]. When the power flow isreversed, power starts to flow out of the DC link capacitorand into the rotor circuit. The DC voltage drops and thebraking resistor is de-activated. If the energy dischargingout of the capacitor is larger than the net energy charginginto the capacitor, the DC link voltage will drop below itspre-fault value. Reduction of the DC link voltage will havea detrimental effect on the ride-through capability of theDFIG, since the rotor voltage injection capability dropsproportionally.

10 Example of calculating therating specification for ride-throughcapabilityThe design of the rotor side VSI of the examined DFIG isexamined in this section. It is considered that the rotor side

Figure 11 Conceptual study on the issues regarding the DClink dynamics during voltage sags

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VSI employs IGBTs and diodes that have a nominalvoltage rating of 1.2 kV (or 1.47 pu) and a nominalcurrent rating of 2.4 kA (or 0.65 pu). Each ‘switch’ involvesM IGBTs (and diodes) in series and N in parallel. Thus,the design objective is the minimisation of the productM � N.

The current withstanding capability of the inverter is

Ir,max,acceptable ¼ N � 0:65 pu (28)

Assuming that a space vector pulse width modulationswitching technique is employed, the maximum permissiblerotor voltage output is

Vr,max,acceptable ¼ 0:577� 1:47�M ¼ 0:848�M pu

(29)

In Fig. 12, the design methodology is depicted. Peak rotorfault current contours corresponding to the currentwithstanding capability of the rotor side VSI are drawn forN equal from one to four. The same is done for therequired rotor voltage output for M equal to one and two.

According to the results seven ‘tuning areas’ are defined.Tuning of the closed loop with parameters inside each ofthese areas will result in different specifications of the rotorside VSI to achieve the ride-through capability. The resultsare summarised in Table 1.

According to the results, setting the tuning parametersinside area five provides the best choice, as long as anadequately large DC link capacitor is employed.

In Fig. 13 simulation results in PSCAD/EMTDC areshown when z ¼ 2.5 and vn ¼ 3.5 pu. With these controlparameters, the theoretically expected peak rotor faultcurrent and required rotor voltage outputs are 4.55 and

Figure 12 Rating specification methodology for fault ride-through under the worst case fault (three-phase bolted,maximum rotor speed)

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1.06 kV, respectively. Thus, the reference DC link voltage isset to 1.84 kV (¼1.06 kV/0.577). The braking resistor isactivated when the DC link voltage reaches 2 kV. Thelower left hand plot depicts the ratio Vr,command/(0.577 Vdc). When this becomes larger than one, the rotorside VSI cannot follow the rotor voltage command (i.e. itsaturates).

When the DC link capacitor is 50 mF, the poweroscillations have a considerable effect on the DC linkvoltage. As a result, the rotor side VSI saturates and largerotor overcurrents are recorded. When the DC linkcapacitor doubles in size, the power oscillations have amuch smaller effect on the DC link voltage. The rotor sideVSI saturates just for the first power swing and onlymarginally. The recorded peak rotor current is very close tothe predicted value of 4.55 kA (the error is 0.1 kA or 2%).Overall, in the case of the large DC link capacitor, theDFIG achieves the ride-through capability successfully.

Figure 13 Behaviour of the DFIG for the worst case fault(three-phase bolted fault lasting 140 ms, maximum realpower production)

Dotted line Cdc ¼ 50 mF; solid line Cdc ¼ 100 mF

Table 1 Analysis of the results in Fig. 12

Area, # Ir, pu Vr, pu N M N � M

1 .2.6 ,0.85 .4 1 .4

2 .2.6 0.852 1.7 .4 2 .8

3 1.952 2.6 0.852 1.7 4 2 8

4 1.32 1.95 0.852 1.7 3 2 6

5 ,1.3 0.852 1.7 2 2 4

6 .2.6 .1.7 .4 .2 .8

7 1.952 2.6 .1.7 4 .2 .8

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11 ConclusionsThis paper examined the conditions necessaryfor a DFIG toachieve a fault ride-through capability under three-phasebalanced voltage sags. In the ‘ride-through capability’definition employed in this work, the machine has toprovide assistance in the recovery of the power supplysystem during a voltage sag by injections of real andreactive power. Thus, the employment of a crowbartechnique has been considered unacceptable.

The fundamental point in this work is that the faultbehaviour of the DFIG should not concentrate solely on themagnitude of the peak rotor fault current. The real issue isthe magnitude of the required rotor voltage output in orderto restrain the peak rotor current within certain acceptablelimits. Moreover, the DC link dynamics during the fault canhave a decisive effect on fault behaviour, since the voltageinjection of the rotor side VSI depends fundamentally onthe instantaneous magnitude of the DC link voltage.

This work described a methodology for examining theseissues based on a robust mathematical analysis. Simulationresults in PSCAD/EMTDC confirm the findings of thetheoretical investigation. The main advantage of themathematical approach followed in this work is that itpermits a rapid study of the impact of the tuning of the PIcontrollers on the design specification of the rotor side VSIto achieve the ride-through capability. Thus, it circumventsthe necessity for repeated time-consuming simulations forevery different set of tuning parameters of the DFIG innercurrent closed loop to calculate the peak rotor fault currentand required rotor voltage output.

The main limitation of this work is that it addresses thebehaviour of a DFIG solely for three-phase balanced voltagesags. The mathematical analysis could include also unbalancedfault conditions based on the principle of superimposition[22]. The positive, negative and zero sequence componentsthat exist in the power supply system when an unbalancedfault occurs can be understood as a positive, negative and zerosequence voltage sag, which will induce three correspondingstator flux ‘oscillations’. Calculating the stator flux responseunder the assumption of negligible stator resistance can set thebasis for the calculation of the peak rotor fault current andrequired rotor voltage output for each component as shown inthis work for a three-phase balanced voltage sag (i.e. in thecase that there is only a positive sequence voltage componentduring the fault and, thus, only a positive sequence stator fluxoscillation). The expansion of the mathematical methodologydeveloped in this work for incorporating the analysis ofunbalanced fault conditions is the subject of further work.

12 References

[1] PETERSSON A., THIRINGER T., HARNEFORS L., PETRU T.: ‘Modelingand experimental verification of grid interaction of a DFIG

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wind turbine’, IEEE Trans. Energy Convers., 2005, 20, (4),pp. 878–886

[2] CARRASCO J.M., FRANQUELO L.G., BIALASIEWICZ J.T., GALVAN E.,GUISADO R.C.P., PRATS M.A.M., ET AL.: ‘Power-electronic systemsfor the frid integration of renewable energy sources:A survey’, IEEE Trans. Ind. Electron., 2006, 53, (4),pp. 1002–1016

[3] AKHMATOV V.: ‘Variable-speed wind turbines withdoubly-fed induction generators. II. Power systemstability’, Wind Eng., 2002, 26, (3), pp. 171–188

[4] EKANAYAKE J.B., HOLDSWORTH L., WU X., JENKINS N.: ‘Dynamicmodeling of doubly fed induction generator windturbines’, IEEE Trans. Power Syst., 2003, 18, (2),pp. 803–809

[5] AKHMATOV V.: ‘Variable-speed wind turbines withdoubly-fed induction generators. Part IV: uninterruptedoperation features at grid faults with converter controlcoordination’, Wind Eng., 2003, 27, (6), pp. 519–529

[6] MOREN J., DE HAAN S.W.H.: ‘Ridethrough of wind turbineswith doubly-fed induction generator during a voltage dip’,IEEE Trans. Energy Convers., 2005, 20, (2), pp. 435–441

[7] KAYIKCI M., MILANOVIC J.V.: ‘Contribution of DFIG basedwind plants to voltage and frequency recovery followingsystem disturbances’. Sixth Int. Workshop on Large-ScaleIntegration of Wind Power and Transmission Networks forOffshore Wind Farms, Delft, The Netherlands, 2006

[8] CARTWRIGHT P., HOLDSWORTH L., EKANAYAKE J.B., JENKINS N.: ‘Co-ordinated voltage control strategy for a doubly-fedinduction generator (DFIG)-based wind farm’, IEE Proc.,Gener. Transm. Distrib., 2004, 151, (4), pp. 495–502

[9] EKANAYAKE J., HOLDSWORTH L., JENKINS N.: ‘Control of DFIGwind turbines’, IEE Power Eng., 2003, 17, (1), pp. 28–32

[10] HANSEN A.D., SORENSEN P., LOV F., BLAABJERG F.: ‘Control ofvariable speed wind turbines with doubly-fed inductiongenerators’, Wind Eng., 2004, 28, (4), pp. 411–432

[11] HOPFENSPERGER B., ATKINSON D.J., LAKIN R.A.: ‘Stator-flux-oriented control of a doubly-fed induction machine withand without position encoder’, IEE Proc., Electr. PowerAppl., 2000, 147, (4), pp. 241–250

[12] PENA R., CLARE J.C., ASHER G.M.: ‘Doubly fed inductiongenerator using back-to-back PWM converters and itsapplication to variable-speed wind-energy generation’, IEEProc., Electr. Power Appl., 1996, 143, (3), pp. 231–241

[13] NUNES M.V.A., LOPES J.A.P., ZURN H.H., BEZERRA U.H., ALMEIDA R.G.:‘Influence of the variable-speed wind generators intransient stability margin of the conventional generators

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integrated in electrical grids’, IEEE Trans. Energy Convers.,2004, 19, (4), pp. 692–701

[14] DAWEI X., LI R., TAVNER P.J., YANG S.: ‘Control of a doubly fedinduction generator in a wind turbine during grid fault ride-through’, IEEE Trans. Energy Convers., 2006, 21, (3),pp. 652–662

[15] KUNDUR P.: ‘Power system stability and control’(McGraw-Hill, Inc., 1994)

[16] National Grid Transmission Electricity plc.: ‘The gridcode’. Issue 3. Revision 20, April 2007

[17] PETERSSON A., HARNEFORS L., THIRINGER T.: ‘Comparisonbetween stator-flux and grid-flux-oriented rotorcurrent control of doubly-fed induction generators’. IEEE35th Annual Power Electronics Specialists Conf., vol. 1,2004

[18] ANAYA-LARA O., WU X., CARTWRIGHT P., EKANAYAKE J.B., JENKINS N.:‘Performance of doubly-fed induction generator(DFIG) during network faults’, Wind Eng., 2005, 29,pp. 49–66

[19] PETERSSON A.: ‘Analysis, modeling and control of doubly-fed induction generators for wind turbines’, Ph.D. Thesis,Chalmers University of Technology, Sweden, 2005

[20] SCHAUDER C.: ‘Vector analysis and control of advancedstatic VAr compensators’, IEE Proc., Gener. Transm.Distrib., 1993, 140, (4), pp. 299–306

[21] ABBEY C., WEI L., OWATTA L., JOOS G.: ‘Power electronicconverter control techniques for improved low voltageride through performance in WTGs’. 2006 IEEE PowerElectronics Specialists Conf., 2006

[22] GRAINGER J.J., STEVENSON W.D.: ‘Power system analysis’(McGraw-Hill, 1994)

13 Appendix 1: DFIG parametersStator voltage: 1 kV, rating: 4.5 MVA, speed range ofoptimum power 0.75–1.21, Rs ¼ 0.00488, Rr ¼ 0.00549,Ls ¼ 0.09241, Lr ¼ 0.09955, Lm ¼ 3.95279 pu.

Three-winding transformer: 4.5 MVA.V1 ¼ 1 kV,V2 ¼ 0.4 kV,V3 ¼ 33 kV; L13 ¼ 0.08 pu, L23 ¼ 0.08 pu, L12 ¼ 0.001 pu.

Rotor side VSI: Lcoupling ¼ 0.2 pu and Rcoupling ¼ 0 pu.

Grid side VSI: Lcoupling ¼0.16 pu and Rcoupling ¼0.0027 pu.

Reactive power capability: 0.9 pf leading to 0.95 pf lagging (at33 kV terminals).

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Pu system base values: Vb ¼ 0.82 kV, Ib ¼ 3.67 kA,vb ¼ 314.16 rad/s.

The machine is connected to a power supply network withshort circuit level of 45 MVA and X/R ratio of 14.3.

14 Appendix 2: base values offollowed pu systemBase voltage (Vb): rated peak phase voltage, kV

Base current (Ib): rated peak phase current, kA

Base angular frequency (vb): rated angular frequency ofpower supply system, rad/s

Base flux linkage: Cb ¼ Vb=vb, kWb-turns

Base impedance: Zb ¼ Vb=Ib, V

Base inductance: Lb ¼ Zb=vb, H

Base power: Sb ¼ (3=2)VbIb ¼ Rated power, MVA

Base for time: 1/vb, s

15 Appendix 3: followednotations

Cd1 ¼ SLLsrIqr (30)

Cq1 ¼ �SLLsrIdr (31)

Cd2 ¼ SLarCqs (32)

Cq2 ¼ �SLarCds (33)

Cq2 ¼ �SLarVqs=vs (34)

Cd3 ¼ �ardCd s

dt(35)

Cq3 ¼ �ardCqs

dt(36)

Lsr ¼ Lrr � L2m=Lss (37)

ar ¼ Lm=Lss (38)

Lss ¼ Ls þ Lm (stator self inductance) (39)

Lrr ¼ Lr þ Lm (rotor self inductance) (40)

SL ¼ vs � vr (slip speed) (41)

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where Vqs is q-axis stator voltage, Vdr the d-axis rotor voltage,Vqr the q-axis rotor voltage,Cds the d-axis stator flux linkage,Cqs the q-axis stator flux linkage, Id r the d-axis rotor current,Iqr the q-axis rotor current, Rr the rotor resistance, Lm themagnetising inductance, Ls the stator leakage inductance,Lr the rotor leakage inductance, Lss the stator self-inductance, Lrr the rotor self-inductance, vs thesynchronous frequency and vr the rotor electrical frequency.

16 Appendix 4: mathematicalcalculations16.1 Calculation of the stator fluxresponse under a three-phase voltage sag

In SVO a three-phase voltage sag can be described as

V ¼jV (0�), t � 0

jV (0þ) ¼ j[V (0�)� DV ], t . 0

�(42)

The stator flux response to a three-phase balanced fault canbe found by the solution of the stator voltage equation ofthe DFIG, which refers to Fig. 1

V ¼ �Rs(Id s þ jIqs)þ vs(�Cqs þ jCd s)þd

dtCds þ j

d

dtCqs

ffi vs(�Cqs þ jCd s)þd

dtCds þ j

d

dtCqs, Rs ffi 0

(43)

Solving (43) derives (8).

16.2 Calculation of the stator fluxresponse under a three-phase voltage sag

Inspection of Fig. 2 reveals the differential equationdescribing the rotor current response

Lsr

d2Id rdt2

þ jd2Iqr

dt2

!þ (kP þ Rr)

dId rdt

þ jdIqr

dt

� �

þ k0I (Id r þ jIqr )

¼ kPdIdr,refdt

þ jdIqr,ref

dt

� �þ k0I (Idr,ref þ jIqr,ref )

þdCd2

dtþdCd3

dtþ j

dCq2

dt�dCq2

dtþdCd3

dt

!(44)

where

k0I ¼ kI=v b (45)

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Applying the Laplace transform to (44) and (45) yields

[Lsrs2þ (kP þ Rr)s þ k0I ][Idr(s)þ jIqr(s)]

¼ (Lsrs þ Rr)[Id r(0�)þ jIqr(0�)]þ (kPs þ k0I )

� [Id r,ref (s)þ jIqr,ref (s)]þ LdCd2

dt

� �þ L

dCd3

dt

� �

þ j LdCq2

dt

� �� L

dCq2

dt

( )þ L

dCq3

dt

� �" #(46)

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The reference rotor current response at the occurrence of afault can be approximated as a step command

I r,ref (s) ¼Idr,ref (0þ)þ jIqr,ref (0þ)

s(47)

Combining (46) and (47) with (1)–(8) and (30)–(41)yields the rotor current response under a three-phasevoltage sag.

IET Renew. Power Gener., 2008, Vol. 2, No. 3, pp. 139–150doi: 10.1049/iet-rpg:20070104