Steady-state stability analysis of doubly-fed inductiongenerators under decoupled P–Q control

H. Banakar, C. Luo and B.T. Ooi

Abstract: Based on the standard mathematical model of the doubly-fed induction generator(DFIG) and steady-state stability analysis, DFIGs operating under decoupled P–Q control andmaintaining fixed real power outputs are expected to be unstable. Yet actual field experiencessuggest to the contrary. The research presented here resolves this contradiction and confirms thatDFIGs have extensive steady-state stability regions. The research reveals, for the first time, that aDFIG’s stable region is bounded by unstable regions on both the low- and high-speed ends. Theeigenvalue predictions of the unstable regions have been confirmed by simulations.

1 Introduction

The doubly-fed induction generator (DFIG) [1–8] is acompetitive choice in the burgeoning wind power industryfor allowing direct connection of the stator to the AC gridunder variable rotor speed as well as for providingdecoupled P–Q control capability and offering costadvantages. When operating as part of a wind farm,DFIGs may be required to deliver constant power to thepower grid or extract maximum power from the wind. Boththese operating modes are to be supported by DFIGs in thepresence of continually fluctuating wind velocity. Further-more, the transmission line connecting the wind farm to thepower grid may be long and therefore weak. In that case,DFIGs are required to regulate their reactive power outputto support the wind farm bus voltage. To fulfil theserequirements, DFIGs need to have a high level ofcontrollability and sufficient margins of stability.

As shown in Fig. 1, in addition to the stator-side powerthe DFIG transmits slip power to and from the AC grid viaback-to-back voltage-source converters (VSCs) that areconnected to its rotor terminals. Since the rotor power isproportional to the slip and in most designs the slip islimited to 70.3, the required VSC kVA rating is reducedroughly to one-third, which translates to cost advantagesover other VSC configurations.

2 Motivation

For a host of practical reasons, utility control centreswould like wind farms operating into their systemsto behave simply as conventional power plants. This impliesthat the DFIGs forming the wind farms must be capableof following their contractual fixed real and reactive powerreferences, which raises the question whether DFIGscan remain stable while maintaining fixed output powerlevels?

One has to be concerned about DFIG stability inthis case since a DFIG is in essence a woundrotor induction machine and, as such, the sign of thetorque gradient relative to the rotor mechanical speed,om, determines its stability. When operating at constantoutput power �Pg, provided that the losses are negligible,the DFIG electromechanical torque, Te, can be approxi-mated by �Pg/om. Since @Te=@om’ Pg=o2

m is positive, oneexpects all DFIG operating points to be unstable.

Because large wind turbines with large DFIGs arealready in operation, the implication is that the integratedwind-turbine-DFIG constitutes a stable system. Butwhere does the stability come from? Does it comefrom the DFIG itself, from the wind-turbine aero-dynamic damping, or from the way the DFIG controlsare realised?

Clearly, practice has outpaced theory. As is typical in afiercely competitive market environment, very little of theengineering knowledge is disclosed. Therefore, this paperis a report on university research that seeks to fill theinformation gap in this area and lay the DFIG stabilityanalysis on a sound footing.

3 Objective, modelling and methodology

The objective of this study is to provide answers to thequestions raised in the previous Section. For these answersto be concrete, they are to be based on the basic DFIGcharacteristics and controls. The induction machine non-linear dynamic equations, formulated in the synchronously

wind farmlocal bus

DFIG

converters

transformer

=

=

Fig. 1 Doubly-fed induction generator with VSC controlE-mail: hadi_banakar@yahoo.com

The authors are with the ECE Department, McGill University, 3480 RueUniversity, Montreal, QC, Canada, H3A 2A7

r IEE, 2006

IEE Proceedings online no. 20050388

doi:10.1049/ip-epa:20050388

Paper first received 20th September and in final revised form 13th December 2005

300 IEE Proc.-Electr. Power Appl., Vol. 153, No. 2, March 2006

rotating g–d frame, are

vSg

vSd

vRg

vRd

0BBBB@

1CCCCA ¼

RS þddt

LS �ocLSddt

Lm

ocLS RS þddt

LS ocLm

ddt

Lm ðom � ocÞLm RR þddt

LR

ðoc � omÞLmddt

Lm0 ðoc � omÞLR

0BBBBBBBBBBB@

�ocLm

ddt

Lm

ðom � ocÞLR

RR þddt

LR

1CCCCCCCCCCCA

iSg

iSd

iRg

iRd

0BBBB@

1CCCCA ð1Þ

Here the stator current and voltage components are,respectively, (iSg, iSd) and (vSg, vSd) and those of the rotorare (iRg, iRd) and (vRg, vRd). Furthermore, the per-phasestator winding resistance and inductance are, respectively,RS and LS while RR and LR are their rotor counterparts.The machine’s magnetising inductance is Lm and oc

represents the synchronous speed. The mechanical equationof motion, in the absence of aerodynamic damping, is

Jmdom

dt¼ Te � Tw ð2Þ

In (2), Jm is the machine’s moment of inertia and Tw is thetorque due to the wind. For a machine with Npp pole pairs,the electromechanical torque, Te, is

Te ¼ NppLmðiSg iRd � iRg iSdÞ ð3ÞTraditionally, to perform stability analysis, the above modelis joined with DFIG VSC details, with representation of theDFIG proportional and integral feedback control loop,and with the wind farm local controls and power gridcharacteristics. But, in this case, taking such an approach iscounterproductive since it would obscure sources of DFIGstability by allowing interactions between variables internaland external to the DFIG.

As the objective here is to study the intrinsic stability ofthe DFIG, we proceed by detaching the DFIG from itsexternal world and connecting the following models to thedetachment points:

(a) Power grid model: On the stator side, the DFIG isassumed to be connected to a three-phase 60Hz, infiniteAC bus. This allows treating stator voltages,vT

S ¼ ðvSg; vSdÞ, as time invariant.

(b) Converter model: On the rotor side, irrespective of thecomplexity of the control strategy employed to drive theVSCs, the DFIG is affected by the values of rotorvoltages, vT

R ¼ ðvRg; vRdÞ. Thus, for our purpose, itsuffices to have appropriate rotor voltages impressed atthe DFIG rotor terminals.

(c) Wind-turbine model: On the DFIG mechanical side,the wind turbine torque, Tw, is assumed to bebalanced by an equal and opposite electromechanicaltorque, Te, making Te�Tw¼ 0 at every operating rotorspeed, o0

m.

The analysis is carried out by first establishing the steady-state operation of the DFIG for given rotor speed andoutput power levels. In this step, specific rotor voltages that

are needed to enforce power targets are calculated. Then,using the small perturbation method, we examine how thesign and magnitude of computed damping factors relate tothe DFIG key parameters and the way its controls areimplemented.

The values used for DFIG parameters in this paper arebased on the 1.0MW DFIG reported in [4]. Forcompleteness, they are listed in Table 1.

4 Review of previous research

Interest in induction machine controls and stability analysiswas fairly high from the mid 1960s to the mid 1970s, wheneigenvalue analysis was first applied to the g–d formulationof the induction machine dynamic equations [9–13]. Aninduction machine is stable when all the eigenvalues of itslinearised 5� 5 characteristic matrix lie on the left-hand sideof the complex s-plane. It has been shown that two pairsof complex conjugate eigenvalues constitute the machine’selectrical modes, while its fifth eigenvalue is real, defines theelectromechanical mode, and can be closely linked to itstorque–speed curve gradient [11, 12].

Steady-state stability analysis of induction machinesbegins with (1)–(3). Writing the state vector as xT ¼½iSg; iSd; iRg; iRd;om� and uT ¼ ½vSg; vSd; vRg; vRd; Tw� as theinput vector, and by performing small perturbationlinearisation on the three equations about an equilibriumstate x0, i.e. by setting x¼ x0+Dx and ignoring second andhigher order terms, one obtains the linearised equation:

ddtDx ¼ ½A5�5ðx0Þ�Dxþ ½B5�5ðo0

mÞ�Du ð4Þ

The [A5� 5(x0)] matrix has the basic structure

½A5�5ðx0Þ� ¼½A4�4ðo0

mÞ� aðx0Þ

bT ðx0Þ 0

!ð5Þ

The 4� 4 sub-matrix ½A4�4ðo0mÞ� in (5) is defined by

½A4�4ðo0mÞ�

¼ D0

RSXR r0�~o0m �RR �~o0

mXR

~o0m�r0 RSXR ~o0

mXR �RR

�RS ~o0mXS RRXS r0þ~o0

mXSXR

�~o0m �RS �r0�~o0

mXSXR RRXS

0BBBBBB@

1CCCCCCAð6Þ

where ~o0m ¼ o0

m=oc, Xk¼ocLk for k¼ (m,S,R), and thequantities that are normalised by Xm are identified by thesuperscript ‘4’. The constants r0 and D0 in (6) are definedby the expressions

r0 ¼ 1� XSXR

D0 ¼ oc=r0

Table 1: DFIG information used in computations

Nameplate data Stator and rotor electricalparameters (p.u.)

Type MVA kV Hz RS RR Lm LS LR

Wound 1.0 3.3 60 0.00662 0.01 3.1 3.185 3.21

IEE Proc.-Electr. Power Appl., Vol. 153, No. 2, March 2006 301

The 4� 1 arrays b(x0) and a(x0) in (5) are expressed in

terms of the 2� 1 arrays F 0, E 0R and E 0

S ; that is,

bðx0Þ ¼ CbE0

R

�E0S

" #; aðx0Þ ¼ Ca

F0

�XSF0

" #ð7Þ

In (7), the constants Cb and Ca are defined as Cb¼Npp/

Jmoc and Ca¼D0Npp/oc, while F 0, E 0R and E 0

S are given by

E0S ¼ Xm

�i0Sd

i0Sg

" #; E0

R ¼ Xm

�i0Rd

i0Rg

" #ð8aÞ

F 0 ¼ E 0S þ XRE 0

R ð8bÞ

Prior to viewing [A5� 5(x0)] eigenvalues, it is helpful to

examine eigenvalues of ½A4�4ðo0mÞ�. Note that while a(x0)

and b(x0) are functions of the stator and rotor currents,½A4�4ðo0

m� is current-independent and varies only with o0m.

4.1 ½A4�4ðo0mÞ� Eigenvalues

Figure 2 shows on the complex s-plane the loci of thetwo pairs of complex conjugate eigenvalues of ½A4�4ðo0

m�for rotor operating speed o0

m in the range o0m¼ 0–2p.u.

These are the ‘electrical eigenvalues’, distinct from the‘mechanical eigenvalue’, which will appear as part of[A5� 5(x

0)] eigenvalues.

4.2 Stability analysisFeigenvalues of½A5�5ðx0Þ�In the stability analysis of the conventional inductionmotor, the input vector is set to uT ¼ ½VS ; 0; 0; 0; Tw�, whereTw is chosen to balance the electromechanical torque Te in(3). For a rotor operating at speed o0

m, the steady-state

currents i0Sg; i0Sd; i

0Rg and i0Rd are solved numerically from (1)

and substituted in (5). The eigenvalue loci are shown inFig. 2b.

From analytical continuity with the loci from Fig. 2a, thetwo complex conjugate eigenvalue pairs are considered aselectrical modes since they clearly originate from l1; l

�1 and

l2; l�2 of ½A4�4ðo0

m�. As the electrical eigenvalues consis-tently lie on the left-hand side of the s-plane, in this case andin all case studies carried out on the DFIG later on, theycannot be the source of instability. This enables that thestability study focuses on the fifth eigenvalue.

4.3 Electromechanical modeThe fifth eigenvalue of [A5� 5(x

0)] is real (l5¼ s5) anddefines the DFIG electromechanical mode. Assuming that@Tw=@om ¼ 0, the perturbed form of (2), the mechanicalequation of motion, provides

JmddtDom ¼

@Te

@om

� �x0Dom ð9Þ

Taking Laplace’s transform of (9), one obtains

Jms DOmðsÞ ¼@Te

@om

� �x0DOmðsÞ ð10Þ

where DOm(s) is the Laplace transform of Dom(t). From(10) it follows that

s ¼ s5 ¼ ð1=JmÞ@Te

@om

� �x0

ð11Þ

Numerical evaluation of s5 shows that it generally agreeswith prediction of s5 based on eigenvalue analysis. For a

−400

−200

0

200

400

−35 −30 −25 −20 −15 −10 −5 0

real part

imag

inar

y pa

rt

�m = 0

�m = 0.7

�m = 0.7�m = 1

�m = 0 �m = 0

�m = 0

�m = 1.5

�m = 1.5

�m = 0.5 to 2

�m = 0.5 to 2

�m = 2

�m =2

�1

�5

�2

�1∗ �2

∗

a

−400

−200

0

200

400

−35 −30 −25 −20 −15 −10 −5 0 5 10 15

real part

imag

inar

y pa

rt

�m = 0

�m = 0

�m = 0

�m = 0�1�2

�1∗

�2∗

�m = 0.7

�m = 0.7

�m = 1.5

�m = 1.5

�m = 0.5 to 2

�m = 0.5 to 2

�m = 0

�m = 1

�m = 1

�m = 1

�5

�m = 2

�m = 2

b

Fig. 2 Loci of l1, l�1, l2, l�2 and s5

a ½A4�4ðo0mÞ�

b ½A5�5ðx0Þ�

−7.5

−5.0

−2.5

0.0

2.5

5.0

7.5

torq

ue, k

N m

stable

unstable

unstable

a

−20

−15

−10

−5

0

5

10

15

0.45 0.65 0.85 1.05 1.25 1.45

dam

ping

fact

or, 1

/s unstable

stable

unstable

b

rotor speed, p.u.

Fig. 3a Torque variations with speedb s5 variations with speed

302 IEE Proc.-Electr. Power Appl., Vol. 153, No. 2, March 2006

case study, using parameter values of Table 1, Figs. 3a andb show graphs of Te–vs–om as well as its related s5–vs–om,obtained from eigenvalue analysis.

It is well known that stable operation corresponds to theregion in Fig. 3a where ð@Te=@omÞ is negative. This is thesame region in Fig. 3b where l5¼ s5 is negative andtherefore stable.

5 DFIG operation under decoupled P�Q control

Advances in decoupled P–Q control techniques applied toDFIG [5–8, 14] allow real and reactive power references,Pref (om) and Qref (om), to be independently defined anddelivered for any rotor speed om. The following subsectionsoutline fundamentals of the DFIG decoupled P–Q control.

5.1 Basic relationsIn the g–d frame, the stator real and reactive powers aregiven by

PS ¼ vSgiSg þ vSdiSd ð12aÞ

QS ¼ �vSgiSd þ vSdiSg ð12bÞwhere vSg and vSd satisfy

V 2S ¼ v2Sg þ v2Sd ð12cÞ

As shown in Fig. 1, the slip rotor power is transferred to theAC grid through the back-to-back VSCs. Thus, it isnecessary to distinguish between the real power on thestator side, PS, on the rotor side, PR, and at the DFIGoutput, Pg. Then, letting PL represent real power losseswithin DFIG, one has

Pg ¼ PS þ PR þ PL ð13ÞFor small PL, one can readily show that PR’ð~om � 1ÞPS[13]. Substituting this relation for PR in (13) and neglectingPL, one arrives at the approximate relation

Pg’ ~omPS ð14ÞBy introducing a fixed power factor, fS, a simple linearrelation is established between the stator reactive power, QS,and PS; that is

QS ¼ xSPS ð15Þ

where xS ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffif 2

S � 1q

. The appropriate sign for xS is set

by the fS lead/lag designation. It is assumed here that thegrid-side VSC can be tasked to provide sufficient reactivepower for the DFIG to meet its Qref (om) requirement.

5.2 Resulting DFIG currentsDecoupled P–Q control requires an absolute positionencoder [1] or a ‘sensorless’ method [5–8, 14] to align therotor flux axis. This alignment sets vSg¼VS and vSd¼ 0,allowing PS and QS to be independently controlled by thestator current reference, i�S . Forcing vSg¼VS and vSd¼ 0and using (15), one can solve (12) for the stator currents:

i�S ¼i�Sg

i�Sd

" #¼ ð�P �

S=VSÞ

�1

xS

" #ð16Þ

where P �S will be defined in (18).Under steady-state condition, from the first and the

second rows of (1), the rotor current reference, i�R can besolved in terms of i�S and expressed in the form

i�R ¼i�Rg

i�Rd

" #¼ ðVS=XmÞ

0

1

" #�

RS �XS

XS RS

" #i�S ð17Þ

5.3 Constant power requirementFor a given speed, ~o0

m, with the requirement thatPg¼Pref (om), (14) can be solved for P �S to obtain

P �S ¼ Pref ð~o0mÞ=~o0

m ð18Þ

To keep the derivations simple, PL is assumed to benegligible here. Note that, by letting PL to represent DFIG’scopper losses, (18) becomes a quadratic equation in P �S .

5.4 Rotor voltage calculationsOnce P �S is determined from (18), it is deployed in (16) toobtain i�S , which in turn allows the calculation of i�R from(17). From (1), under steady-state conditions, the rotorvoltages are given by

v�R ¼v�Rgv�Rd

� �¼ ð1� ~omÞXm

0 �11 0

� �i�S

þ Xm

�RR �ð1� ~omÞXR

ð1� ~omÞXR RR

�i�R ð19Þ

The computed values of i�S and i�R are used in (19) to obtainv�R. Figure 4 shows the rotor voltages, v�RgðomÞ, v�RdðomÞ,computed as continuous functions of the rotor speed om.Here the DFIG is required to maintain a lagging powerfactor of 0.8 (i.e. fS ¼ 0.8 and xS¼�0.75) while generatingPg¼�0.9MW.

6 Stability analysisFcontinous time

6.1 Modified characteristic matrix ½A^

5�5ðx0Þ�As mentioned earlier, in induction motor analysis [9–12], Duhas been assumed traditionally to be 0 in the small signalperturbation equations:

ddtDx ¼ ½A5�5ðx0Þ�Dxþ ½B5�5ðo0

mÞ�Du ð20Þ

However, as shown in Fig. 4, associated with the rotorvoltage v�RgðomÞ, v�RdðomÞ, Du can have non-zero entries for

ð@u=@omÞDom. Substituting the ð@u=@omÞDom term in

(20), one has a modified matrix, ½A^

5�5ðx0Þ�, such that:

ddtDx ¼ ½A

^

5�5ðx0Þ�Dx ð21Þ

Entries of ½A^

5�5ðx0Þ� are computed as those of the matrix[A5� 5(x

0)], except that the expression for F0 in (8b) is now

−0.4

−0.2

0

0.2

0.4

0.65 0.75 0.85 0.95 1.05 1.15 1.25 1.35

roto

r vo

ltage

com

pone

nts,

p.u

.

V ∗R

V ∗R

rotor speed, p.u.

Fig. 4 Variations of v�Rg and v�Rd with rotor speed, om

IEE Proc.-Electr. Power Appl., Vol. 153, No. 2, March 2006 303

replaced by

F0 ¼ E0S þ XRE0

R þ@vR

@omð22Þ

6.2 Stability prediction based on (@T^

e=@om)As previously argued and shown in Fig. 5a, for a constantgenerator power strategy, the electromechanical torque is

T^

e’ � Pg=om. From the resulting positive torque gradient,

ð@Te

^=@omÞ’ Pg=o2

m, also shown in Fig. 5b, one expects all

operating points to be unstable.

6.3 Stability prediction based on s5 of

[A^

5�5ðx0Þ]Figure 5b also shows the s5–vs–om curve, with s5 calculated

from ½A^

5�5ðx0Þ�. The prediction of instability based on

ð@T^

e=@omÞ agrees with s5 of the eigenvalue analysis.

6.4 Contradiction with DFIG operationAs the constant power strategy is already used in decoupledP–Q control of DFIGs, the instability, predicted by the

gradient ð@T^

e=@omÞ and confirmed by the eigenvalues of

the modified ½A^

5�5ðx0Þ� matrix, is perplexing. Is theintegrated wind-turbine–DFIG system stable because thepositive s5 is overwhelmed by the wind-turbine aerody-namic damping? Or has there been a special design of thecontrol to provide positive damping? A close examinationof the control schematic is therefore in order.

7 Decoupled control implemenation

Figure 6 shows the block diagram for the decoupled P–Qcontrol implementation. Pertinent to this paper is the blockthat generates the real and reactive power references

Pref (om) and Qref (om). The input to this block is the rotorspeed om measured by a ‘sensorless means’ [14]. Themeasurement om addresses lookup tables from whichprestored values of Pref (om) and Qref (om) are output tothe next block, which computes the g–d frame rotorvoltages, v�RgðomÞ and v�RdðomÞ. These reference voltages

are then transformed to v�Ra, v�Rb and v�Rc, which the VSCapplies to the DFIG rotor slip ring terminals.

7.1 Objections to modified [A^

5�5ðx0Þ]The reference values, Pref (om) and Qref (om), are stored inthe lookup tables of Fig. 6 in discretised form. The tablesare addressed by the rotor speedom. The machine full speedrange is divided into M incremental steps of Dn and theindex m refers to the rotor speed range over which(mDn)romr (m+1)Dn. The corresponding referencefrom the lookup table, Pref (om) and Qref (om), are thereforeconstant values in each incremental step. Thus, as shown inFig. 7a, the computed v�RgðmÞ and v�RdðmÞ remain constant

over each incremental step. Since there is no analyticalcontinuity in the rotor voltages, v�RgðmÞ and v�RdðmÞ, thecontrol vector, Du, does not have the information related to@vRg=@om; @vRd=@om in (20). Therefore for stabilityanalysis, one has to return to the eigenvalues of the originalmatrix, ½A5�5ðx0Þ�, given by (6).

7.2 Stability of Pref (om)¼�Pg based on[A5�5ðx0Þ]Figure 7b shows s5–vs–m based on [A5� 5(x

0)]. In theextensive range where s5 is negative, the DFIG is stable.

8 Stability based on torque gradient

Throughout this paper, a distinction has been made

between the electromechanical torque Te and T^

e.

8.1 Te–vs–om curveTe is the torque when the stator voltages vSg, vSd and therotor voltage vRg, vRd are held constant as the rotorspeed om changes. Fig. 3a, an example of a Te–vs–om

curve, is obtained for the excitation vector uT¼ [vSg¼VS,vSd¼ 0, vRg¼ 0, vRd¼ 0,TL]. In this example, the stabilitypredicted by ð@Te=@omÞ=Jm agrees with the eigenvalues of[A5� 5(x

0)].

−4

−3

−2

−1

torq

ue, k

Nm

a

T e = − Pg / � m

0

1

2

3

4

5

6

0.65 0.75 0.85 0.95 1.05 1.15 1.25 1.35

dam

ping

fact

or, 1

/s

σ5

b

rotor speed, p.u.

∂T e / ∂ωm

Fig. 5

a T^

e variation with speed

b ð@T^

e=@omÞ and s5–vs–om, based on ½A^

5�5ðx0Þ�

VSa VSb VSc

VRa,VRb,VRc

VR�,VR�

iR�∗

iR�∗

wm

�mrotorvoltagecontrol

rotorspeed

measurement

rotorposition

measurement

VSCG VSCR

grid

+−

Pref∗

Qref∗

DFIG

lookuptables

Pref∗, Qref

∗

, ,

,

Fig. 6 Block diagram of decoupled P–Q implementation

304 IEE Proc.-Electr. Power Appl., Vol. 153, No. 2, March 2006

8.2 T^

e–vs–om curve

The electromechanical torque T^

e is defined for excitationvector of the form uT¼ [vSg¼VS,vSd¼ 0,vRg¼ vRg(om),vRd¼ vRd(om),TL]. One sees that the rotor voltages varycontinuously with the rotor speed. In the example of theDFIG controlled to deliver a constant power Pref (om)¼�Pg

at 0.9 power factor, the computed rotor voltages, v�RgðomÞand v�RdðomÞ, take the form of Fig. 4 and the T

^

e–vs–om

curve is illustrated in Fig. 5a. As already discussed, theanalytical continuity of v�RgðomÞ, v�RdðomÞ with respect to om,

requires the ð@u=@omÞ=Dom terms to be accounted for and

this alters the small-signal linearisation matrix to ½A^

5�5ðx0Þ�.The stability predicted by ð@T

^

e=@omÞ=Jm agrees with the

eigenvalues of ½A^

5�5ðx0Þ�. One, therefore, concludes that theestimation of stability based on ð@Te=@omÞ=Jm agrees with

the eigenvalues of [A5� 5(x0)], while ð@T

^

e=@omÞ=Jm agrees

with the eigenvalues of ½A^

5�5ðx0Þ�.

9 Control using lookup tables

Since v�RgðmÞ and v�RdðmÞ, computed from lookup tables,

consist of voltage steps, as illustrated in Fig. 7b, each stephas its own Te–vs–om curve. Thus one can look upon the

T^

e–vs–om curve of Fig. 5a as being synthesised from thefamily of Te–vs–om curves of the voltages from Fig. 7a. Thissynthesis is illustrated in Fig. 8, where it can be seen that the

T^

e–vs–om curve of the constant power strategy (shownbroken) is realised by the family of solid Te–vs–om curves.

At a point where the broken curve intersects a solidcurve, there are two gradients: the gradient ð@Te=@omÞ=Jm,which agrees with the s5 of [A5� 5(x

0)], and the gradient

ð@T^

e=@omÞ=Jm, which agrees with s5 of ½A^

5�5ðx0Þ�. Inparticular, one notices that most of the machine speed rangeð@Te=@omÞ is negative, showing that the DFIG is stable.For speed omZ1.5p.u., ð@Te=@omÞ is positive and theeigenvalue s5 of [A5� 5(x

0)] is positive, both agreeing thatthe operation is unstable.

10 DFIG stability boundaries

10.1 Mapping stability boundariesAsmodern control utilises lookup tables, stability analysis isbased on the eigenvalues of the [A5� 5(x

0)] matrix. Figure 9shows the boundaries within which the DFIG, operatingunder decoupled P–Q control, is stable. The y-axisrepresents the DFIG real power output, Pg. Positive andnegative signs denote motoring and generating, respectively.The x-axis represents om normalised to the synchronousspeed. At each point in Fig. 9, the stator real and reactivepowers are related by the selected power factor. Thestability boundaries, which enclose an extensive stableoperating region, are drawn for leading and lagging powerfactor of 0.8.

Each point in the stability map has been computed bytaking the following steps:

1 Set the value of stator voltages vSg¼VS, vSd¼ 0.

2 Set the values of Pref, fS, o0m and calculate PS.

−0.50

−0.25

0

0.25

0.50

roto

r vo

ltage

com

pone

nts,

p.u

. m = 1

2

109

8

76

54

3

14

11

1312

v∗R� (m )

a

−28

−24

−20

−16

−12

−8

−4

0

4

0.65 0.75 0.85 0.95 1.05 1.15 1.25 1.35

dam

ping

fact

or, 1

/s

m = 1

variations within discrete rangeaverage value

8

765

43

211

12 m = 1413

10

9

b

rotor speed, p.u.

v∗R�(m )

Fig. 7a Discrete rotor voltagesb Variations in electromechanical damping factor for discrete rotorvoltages in a

−6

−5

−4

−3

−2

−1

0

1

0.3 0.5 0.7 0.9 1.1 1.3 1.5rotor speed, p.u.

torq

ue,

kN m

�m = 1.0

�m = 0.5

0.6

0.7

0.8

1.1

0.9

1.51.41.3

1.2

Fig. 8 Family of constant rotor voltage torque–speed curvessuperimposed on constant power output torque–speed curve (broken)

−1.2

−0.8

−0.4

0

0.4

0.8

1.2

0 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

outp

ut p

ower

, p.u

.

lag

lag

lag

lead

lead

lead

stable

stable

unstable

unstable

unstable

rotor speed, p.u.

Fig. 9 Map of DFIG stability boundaries based on eigenvalues of[A5� 5(x0)], for fS¼ 0.8 (lead and lag)

IEE Proc.-Electr. Power Appl., Vol. 153, No. 2, March 2006 305

3 Compute operating currents i�S and i�R.

4 Form the matrix [A5� 5(x0)].

5 Perform the eigenvalue analysis.

This map applies to any decoupled P–Q control strategy,including: (i) constant power control; (ii) optimal windpower acquisition control.

10.2 Verification by time simulationUsing the software that has been developed for sensorlessdecoupled P–Q control of DFIGs [14], a digital simulationexperiment was performed by which the rotor of the DFIGwas accelerated from 0.65p.u. to 1.5p.u. speed, as shown inFig. 10b. The parameters of the DFIG were the same as inthe eigenvalue analysis.

For low speed, 0.65p.u.oomo0.75p.u., the windturbine was unstable. At any speed om, the wind torqueand generator counter torque are TW¼PW/om andTe¼Pe/om, respectively. (Fig. 8 shows Te of the generator.)An operating speed om0 is stable when (Te�TW)o0 forsmall speed perturbation om0+Dom and (TW)40 forsmall speed perturbation om0�Dom. A family of wind-torque TW– vs–om for different wind velocities VW is easilyconstructed from the Cp–vs–l curve.

The acceleration from 0.65 p.u. was caused by the windturbine power PW being greater the 0.6MW set by thereference power Pref of the decoupled P–Q control. Becauseof instability of the wind turbine, the generated outputpower Pg was unable to track Pref ¼ 0.6MW until a speedof around 0.75 p.u. was reached. When Pg4PW, as shownin Fig. 10a, om in Fig. 10b flattened and decreased slightly.When the stable region of wind turbine was reached, Pg

converged abruptly to Pref. The abrupt drop in Pg changedthe accelerating torque (Te�TW). The change caused thechange in the gradient of om in Fig. 10b.

Stability was lost when the speed approached 1.35 p.u.Comparing the stability limits with those in the map inFig. 9, very good agreement exists at the high-speed end.The instability of the wind turbine has overshadowed theinstability of the DFIG at the low-speed end of Fig. 9.

The simulation results confirm that the DFIG, underdecoupled P–Q control, has an upper speed stabilityboundary.

11 Conclusions

This paper has shown that the DFIG, operating underdecoupled P–Q control, has an extensive stable operatingregion. This contradicts the conclusion that would havebeen reached had the stability analysis been pursued withmathematical rigour. This paper draws attention to the factthat, because lookup tables are used in implementing thedecoupled P–Q control, the discretisation does not conveythe ð@u=@omÞDom terms of rigorous mathematical analysis.It is the fortuitous exclusion of the ð@u=@omÞDom termsthat leads to the extensive stable operating region.

The paper reaffirms that the real eigenvalue, l5¼ s5,which is recognisable as the mechanical mode, can beestimated from ð@Te=@omÞ, the gradient of the torque–speed curve. The remaining eigenvalues of the DFIGconsist of two pairs of complex conjugate electrical modes.Since the electrical modes are always on the left-hand side ofthe complex plane, they cannot be the causes of instability.As s5’ð@Te=@omÞ=Jm can be positive or negative, stabilitycan be estimated from the sign of ð@Te=@omÞ.

12 Acknowledgment

The authors acknowledge financial support from NationalScience and Engineering Council of Canada (NSERC).

13 References

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0

0.5

1.0

1.5

2.0

a

0 2 4 6 8 10 120.6

0.8

1.0

1.2

1.4

1.6

b

pow

er,

MW

roto

r sp

eed,

p.u

.

time, s

Pw

Pg

Pref

Fig. 10a Pg response to changes in wind power, Pw, with Pref ¼ 0.6MWb Corresponding rotor speed changes in time

306 IEE Proc.-Electr. Power Appl., Vol. 153, No. 2, March 2006