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Oscillatory Stability and Eigenvalue Sensitivity Analysis of A DFIG Wind TurbineSystem

Yang, Lihui; Xu, Zhao; Østergaard, Jacob; Dong, Zhao Yang; Wong, Kit Po; Ma, Xikui

Published in:I E E E Transactions on Energy Conversion

Link to article, DOI:10.1109/TEC.2010.2091130

Publication date:2011

Document VersionPublisher's PDF, also known as Version of record

Link back to DTU Orbit

Citation (APA):Yang, L., Xu, Z., Østergaard, J., Dong, Z. Y., Wong, K. P., & Ma, X. (2011). Oscillatory Stability and EigenvalueSensitivity Analysis of A DFIG Wind Turbine System. I E E E Transactions on Energy Conversion, 26(1), 328-339. https://doi.org/10.1109/TEC.2010.2091130

https://doi.org/10.1109/TEC.2010.2091130https://orbit.dtu.dk/en/publications/727e4785-a67d-4a70-8524-aeec687c5aa3https://doi.org/10.1109/TEC.2010.2091130

328 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 26, NO. 1, MARCH 2011

Oscillatory Stability and Eigenvalue SensitivityAnalysis of A DFIG Wind Turbine System

Lihui Yang, Zhao Xu, Member, IEEE, Jacob Østergaard, Senior Member, IEEE,Zhao Yang Dong, Senior Member, IEEE, Kit Po Wong, Fellow, IEEE, and Xikui Ma

Abstract—This paper focuses on modeling and oscillatory stabil-ity analysis of a wind turbine with doubly fed induction generator(DFIG). A detailed mathematical model of DFIG wind turbinewith vector-control loops is developed, based on which the loci ofthe system Jacobian’s eigenvalues have been analyzed, showingthat, without appropriate controller tuning a Hopf bifurcation canoccur in such a system due to various factors, such as wind speed.Subsequently, eigenvalue sensitivity with respect to machine andcontrol parameters is performed to assess their impacts on systemstability. Moreover, the Hopf bifurcation boundaries of the keyparameters are also given. They can be used to guide the tuningof those DFIG parameters to ensure stable operation in practice.The computer simulations are conducted to validate the developedmodel and to verify the theoretical analysis.

Index Terms—Doubly fed induction generator (DFIG), eigen-value sensitivity, Hopf bifurcation, stability.

I. INTRODUCTION

DOUBLY fed induction generator (DFIG) is a popular windturbine system due to its high energy efficiency, reducedmechanical stress on the wind turbine, and relatively low powerrating of the connected power electronics converter. The DFIG isalso complex involving aerodynamical, electrical, and mechan-ical systems. With increasing penetration level of DFIG-typewind turbines into the grid, the stability issue of DFIG is ofgreat importance to be properly investigated.

A DFIG system, including induction generator, two-massdrive train, power converters, and feedback controllers, is a

Manuscript received November 30, 2009; revised June 21, 2010 and August31, 2010; accepted October 29, 2010. Date of publication January 6, 2011; dateof current version February 18, 2011. Paper no. TEC-00508-2009.

L. Yang and X. Ma are with School of Electrical Engineering, Xi’anJiaotong University, Xi’an 710049, China (e-mail: [email protected];[email protected]).

Z. Xu was with the Center for Electric Technology, Department of Elec-trical Engineering, Technical University of Denmark, DK-2800 Lyngby,Denmark. He is now with the Department of Electrical Engineering, TheHong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong (e-mail:[email protected]).

J. Østergaard is with Center for Electric Technology, Department of Electri-cal Engineering, Technical University of Denmark, DK-2800 Lyngby, Denmark(e-mail: [email protected]).

Z. Y. Dong is with the Department of Electrical Engineering, The HongKong Polytechnic University, Hung Hom, Kowloon, Hong Kong (e-mail:[email protected]).

K. P. Wong is with the Department of Electrical Engineering, TheHong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, andalso with the School of Electrical, Electronic, and Computer Engineering,The University of Western Australia, W.A. 6009, Perth, Australia (e-mail:[email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TEC.2010.2091130

multivariable, nonlinear, and strongly coupled system. Bifurca-tion phenomena in such a nonlinear system may occur undercertain conditions, leading to oscillatory instability. Therefore,practical analysis of DFIG stability will have to involve the bi-furcation phenomena. In recent years, some researchers studiedstability of industrial motor drives with a wealth of nonlinear dy-namics according to the bifurcation and chaos theories [1]–[4].However, earlier studies mainly deal with dc and simple ac mo-tor drives. The stability analysis of DFIG from a bifurcationperspective is absent.

Eigenvalue analysis of the DFIG wind turbine system hasbeen discussed in [5]–[8], where the participation factor, fre-quency, and damping ratio analysis are focused. The compre-hensive analysis of eigenvalue locus and the eigenvalue sen-sitivity, which can provide useful guidance in tuning systemparameters, have not been carried out earlier.

The Yang et al. have investigated the Hopf bifurcation in avector-controlled DFIG with one-mass drive train [9]. The mainpurpose of this paper is to study the oscillatory stability of aDFIG system with respect to varying wind speed, and to ana-lyze the eigenvalue sensitivity as well. A more comprehensivesystem model, incorporating two-mass drive train, pitch con-trol, etc., is developed. Based on this model, the eigenvalueloci are analyzed, revealing that with inappropriate controllerparameters, Hopf bifurcation is likely to happen in the systemunder certain conditions, such as variation of wind speed. Then,eigenvalue sensitivity analysis is carried out to identify possiblesources of instability, as well as the key influential parameterswith respect to system oscillatory stability. Furthermore, in or-der to obtain the overview of system oscillatory stability, Hopfbifurcation boundaries with regard to some key parameters areanalyzed, in order to facilitate optimal design of the DFIG windturbine system. This paper focuses on the small-signal-stabilityanalysis of the DFIG wind turbine system itself. The impact ofthe DFIG on the power system stability will be considered inour future research.

II. MODELING OF DFIG WIND TURBINE SYSTEMFOR OSCILLATORY STABILITY ANALYSIS

As shown in Fig. 1 [10], the DFIG system utilizes a wound ro-tor induction generator in which the stator windings are directlyconnected to the three-phase grid and the rotor windings arefed through three-phase back-to-back bidirectional pulsewidthmodulation (PWM) converters. The back-to-back PWM con-verters consist of two three-phase six-switch converters, i.e., therotor- and the grid-side converter, between which a dc-link ca-pacitor is placed. For the wind turbine control level, two stage

0885-8969/$26.00 © 2011 IEEE

YANG et al.: OSCILLATORY STABILITY AND EIGENVALUE SENSITIVITY ANALYSIS 329

Fig. 1. Schematic diagram of DFIG wind turbine system.

control strategies, based on the electric power versus wind speedcurve, are used for DFIG wind turbines: power optimizationstrategy below rated wind speed and power limitation strategyabove rated wind speed [10]. For the DFIG control level, vectorcontrol is used for both the rotor- and the grid-side convertersto achieve decoupled control of active and reactive power.

It is recognized that the wind power generations involvingDFIG often experience different oscillations resulted from theDFIG and its auxiliary systems [9], [11]. In order to study theoscillatory behavior of the system, small-signal-stability analy-sis, especially the Hopf bifurcation, is needed. The modeling ofDFIG has been studied in [5]–[13]; however, there is currently alack of a systematic comprehensive modeling approach suitablefor small-signal stability. In the following section, we develop acomprehensive model for the DFIG wind turbine system. Thismodel particularly enables small-signal-stability analysis of theoverall system.

A. Generator

According to the voltage- and flux-linkage equations of theinduction generator [13], [14], the differential equations of thestator and rotor circuits of the induction generator with stator androtor current as state variables can be given in a d–q referenceframe rotating at synchronous speed (we define this referenceframe as the generator reference frame in this paper) as follows:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

didsdt

= D[RsLr ids + (ωs − ωr )L2m iqs − ωsLsLr iqs

−RrLm idr − ωrLrLm iqr − Lruds + Lm udr ]diqsdt

= D[−(ωs − ωr )L2m ids + ωsLsLr ids + RsLr iqs

+ωrLrLm idr − RrLm iqr − Lruqs + Lm uqr ]didrdt

= D[−RsLm ids + ωrLsLm iqs + RrLsidr

+ωsL2m iqr − (ωs − ωr )LsLr iqr + Lm uds − Lsudr ]diqrdt

= D[−ωrLsLm ids − RsLm iqs − ωsL2m idr

+(ωs − ωr )LsLr idr + RrLsiqr + Lm uqs − Lsuqr ](1)

where is = ids + jiqs and ir = idr + jiqr are the stator androtor current vectors, respectively; us = uds + juqs and ur =udr + juqr are the stator and rotor voltage vectors, respectively;D = ωb /(Lm 2 −LsLr ). This paper adopts the motor conventionmeaning that stator and rotor currents are positive when flowinginto the generator. The quantities in the system model are in perunit except the time t.

B. Drive Train

When studying the stability of DFIG wind turbine, the two-mass model of the drive train is important, as the wind turbineshaft is relatively softer than the typical steam turbine shaft inconventional power plants [15]. The equations, which representthe two-mass model of the drive train, are expressed as follows:

dωrdt

=1

2Hg(Tsh − Te − Bωr ) (2)

dθtdt

= ωb(ωt − ωr ) (3)

dωtdt

=1

2Ht(Tm − Tsh) (4)

where ωb , ωr , and ωt are the base, generator, and wind turbinespeeds, respectively. Hg and Ht [SI unit(s)] are the generatorand turbine inertias, respectively. θt is the shaft twist angle.The electromagnetic torque Te , the shaft torque Tsh , and themechanical torque Tm , which are the power input of the windturbine, are as follows:

Te = Lm (idsiqr − iqsidr ) (5)Tsh = Kshθt + Dshωb(ωt − ωr ) (6)

Tm =0.5ρπR2Cp(λ, β)V 3w

ωt(7)

where Cp is the power coefficient as follows:

Cp = 0.22(

116λi

− 0.4β − 5)

e−12.5/λi (8)

λi =1

1/(λ + 0.08β) − 0.035/(β3 + 1) (9)

where λ = ωtR/Vw is the blade tip speed ratio. Cp (λ, β) has amaximum Cmaxp for a particular tip speed ratio λopt and pitchangle βopt . The aim for variable wind turbine at wind speedslower than rated value is to adjust the rotor speed at varying windspeeds; therefore, λ and Cp are always maintained at the opti-mal and maximum value, respectively. The speed control of theDFIG is achieved by driving the generator speed along the opti-mum power-speed characteristic curve [10], which correspondsto the maximum energy capture from the wind. In this curve,when generator speed is less than the low limit or higher thanthe rated value, the reference speed is set to the minimal valueor rated value, respectively. When generator speed is betweenthe lower limit and the rated value, the rotor speed reference canbe obtained by substituting λ = ωtR/Vw into (7) as follows:

ωref =

√Tm

Kopt(10)


Fig. 2. Schematic diagrasm of the pitch control.

Fig. 3. Control scheme of the rotor-side converter.

where Kopt =ρπR5 C m a xp

2λ3o p t

is the optimal constant of wind tur-

bine. Equation (10) is an easy and direct way to get ωref fromthe academic perspective while it implies the mechanical torqueobserver is needed. Although mechanical torque observation isnot popularly used in industrial application, due to some en-gineering problems, it is available in practice and can obtainimproved optimum operating point tracking [16], [17].

C. Pitch Control

The pitch angle of the blade is controlled to optimize thepower extraction of wind turbine as well as to prevent overratedpower production in strong wind. The pitch servo is modeled asfollows:

dβ

dt=

1Tβ

(βref − β). (11)

For the sake of simplicity, the reference of the pitch angle βrefis kept zero when wind speed is below rated value. When windspeed is higher than rated value, the power limitation is active byadjusting the pitch angle using the pitch-control scheme shownin Fig. 2 [10], and⎧⎪⎨

⎪⎩βref = KP β (Pg − Pref ) + xβ

ẋβ =KP βTIβ

(Pg − Pref ).(12)

D. Rotor-Side Converter

The generic control scheme of the rotor-side converter is il-lustrated in Fig. 3. In order to decouple the electromagnetictorque and the rotor excitation current, the induction generatoris controlled in the stator-flux-oriented reference frame, whichis a synchronously rotating reference frame, with its d-axis ori-ented along the stator-flux vector position [17]. The typicalproportional–integral (PI) controllers are used for regulation inboth the rotor speed and the terminal voltage (outer) control loop

Fig. 4. Control scheme of the grid-side converter.

and the rotor current (inner) control loop. In Fig. 3, superscriptϕ denotes the variable is in the stator-flux-oriented referenceframe.

Based on the stator-flux orientation, the stator flux can be de-scribed as ψϕds = Ψs and ψ

ϕqs = 0 [17]. Accordingly, the rotor-

voltage equations can be expressed as follows:⎧⎪⎪⎪⎨⎪⎪⎪⎩

uϕdr = Rriϕdr − σLr iϕqr (ωs − ωr ) +

σLrωb

diϕdrdt

uϕqr = Rriϕqr +

(σLr i

ϕdr +

LmLs

Ψs

)(ωs − ωr ) +

σLrωb

diϕqrdt

(13)where σ = 1 − (L2m /LsLr ) is the leakage factor.

Usually, the bandwidth of the inner current-control loop ismuch wider than the outer speed-control loop [17]. Hence, thefast dynamics of the current-control loop does not affect thelow-frequency oscillations. On account of this, we assume thatthe rotor current can well track the reference current, and thus,omit the dynamics of the rotor current-control loop. Under thisassumption and according to the control scheme of the rotor-side converter shown in Fig. 3, the equations with respect to thecontrol of the rotor-side converter become⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

ẋω =KP ωTIω

(ωref − ωr )

ẋus =KP usTIus

(Usref − Us)

iϕdr = iϕdrref = KP us(Usref − Us) + xus

iϕqr = iϕqrref = −

LsLm Ψs

[KP ω (ωref − ωr ) + xω ]

(14)

where KP ω and TIω are the proportional gain and the integraltime constant of the rotor-speed controller, respectively.

The rotor voltage in the generator reference frame can bederived by the following: [18][

udr

uqr

]=

[cos ϕ − sin ϕsin ϕ cos ϕ

] [uϕdr

uϕqr

](15)

where ϕ = arctan(ψqs/ψds) is the angle between the stator-flux vector and the d-axis of the generator reference frame.

E. Grid-Side Converter

Fig. 4 shows the control scheme of the grid-side converter.In order to obtain the independent control of active and reactivepower flowing between the grid and the grid-side converter, the


converter control operates in the grid-voltage-oriented referenceframe, which is a synchronously rotating reference frame, withits d-axis oriented along the grid-voltage vector position [17].Similarly, the typical PI controllers are used for regulation inboth dc-link voltage (outer) control loop and grid-side inductorcurrent (inner) control loop. In Fig. 4, superscript ε denotes thevariable is in the grid-voltage-oriented reference frame.

Under the grid-voltage-oriented reference frame, the equa-tions of the grid-side converter are given by [17] the following:⎧⎪⎪⎨

⎪⎪⎩uεds = Us = RLi

εdL +

L

ωb

diεdLdt

− ωsLiεqL + uεda

uεqs = 0 = RLiεqL +

L

ωb

diεqLdt

+ ωsLiεdL + uεqa

(16)

where iL = idL + jiqL is the grid-side-inductor-current vector,and ua = uda + juqa is the grid-side converter voltage vector.

Similar to the derivation of the rotor-side controller, based onthe same simplification, which omits the fast dynamics in theinner current-control loop, and according to the control schemeof the grid-side converter shown in Fig. 4, the equations withrespect to the control of the grid-side converter are described asfollows:⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

ẋv =KP vTIv

(Udcref − Udc)

iεqL = IqLref

iεdL = iεdLref =

√2√

3m[KP v (Udcref − Udc) + xu ]

(17)

where KP v and TIv are the proportional gain and the integraltime constant of the dc-link voltage controller, respectively.

The relationship between the generator reference frame andthe grid-voltage-oriented reference frame can be given by [18]the following:[

Vda

Vqa

]=

[cos ε − sin εsin ε cos ε

] [V εda

V εqa

](18)

where ε = arctan(uqs/uds) is the angle between the grid-voltage vector and the d-axis of the generator reference frame;V can be the variable of voltage u or current i.

F. DC-link Capacitor

The equation, which describes the energy balance of the dc-link capacitor can be expressed as follows:

CdcUdcωb

dUdcdt

= pa − pr

=32(udaidL + uqa iqL − udr idr − uqr iqr ) (19)

where Udc is the dc-link voltage, and pa and pr are the powerssupplied to the grid-side converter and the rotor circuit, respec-tively.

From (1)–(19), we can obtain a set of state equations to presentthe DFIG wind turbine system. They can be written in a compactform as follows:

ẋ = f(x,u) (20)

where x and u are the vectors with respect to the state and theinput variables, which are defined as x = [ids iqs idr iqr ωr Udcxω xu θt ωt β xus xβ ]T , u = [uds uqs IqL ref Udcref Vw βref ]T .

III. SMALL-SIGNAL-STABILITY ANALYSIS

A DFIG wind turbine system, modeled by (1)–(19) or sim-ply (20), can be linearized to form the linear model around anequilibrium point for small-signal-stability analysis.

A. System Jacobian

The Jacobian matrix is of great importance to stability anal-ysis of dynamical systems. In order to analyze the Jacobianmatrix, the equilibrium point X0 of the system needs to be cal-culated by solving equation f (x, u) = 0. With X0 , the Jacobianmatrix of the system evaluated at the equilibrium point is givenin (21), shown at the bottom of this page, where {Ji,j} (i =

J(X0) = A =∂f

∂x

∣∣∣∣x=X 0

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

J11 J12 J13 J14 J15 0 J17 0 J19 J110 J111 J112 0

J21 J22 J23 J24 J25 0 J27 0 J29 J210 J211 J212 0

J31 J32 J33 J34 J35 0 J37 0 J39 J310 J311 J312 0

J41 J42 J43 J44 J45 0 J47 0 J49 J410 J411 J412 0

J51 J52 J53 J54 J55 0 0 0 J59 J510 0 0 0

J61 J62 J63 J64 J65 J66 J67 J68 J69 J610 J611 J612 0

0 0 0 0 J75 0 0 0 0 J710 J711 0 0

0 0 0 0 0 J86 0 0 0 0 0 0 0

0 0 0 0 J95 0 0 0 0 J910 0 0 0

0 0 0 0 J105 0 0 0 J109 J1010 J1011 0 0

J1101 J1102 0 0 0 J116 0 J118 0 0 J1111 0 J11130 0 0 0 0 0 0 0 0 0 0 0 0

J131 J132 0 0 0 J136 0 J138 0 0 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(21)


1, 2, . . . , 4 and j = 1, 2, . . . , 13) represents the linearized dy-namics of the generator from (1); {Ji,j} (i = 5, 9, 10 and j= 1, 2, . . . , 13) represents the linearized dynamics of the drivetrain from (2)–(4); {Ji,j} (i = 6, 7, 8, 12 and j = 1, 2, . . . , 13)represents the linearized dynamics of the dc-link capacitor andcontrollers of the back-to-back converters from (14), (17), and(19); {Ji,j} (i = 11, 13 and j = 1, 2, . . . , 13) represents thelinearized dynamics of the pitch control from (11) and (12). Theelements of the Jacobian matrix are given in the Appendix.

In this paper, we focus on the oscillatory stability analysisof the DFIG itself, the studied DFIG is directly connected tothe infinite bus and the dynamic behavior of the grid is notconcerned. Therefore, in (14), we have Us = Us ref , and the dif-ferential equation associated with the voltage-control loop canbe omitted. Under this assumption, 12th row and 12th columnof the Jacobian matrix can be removed. When wind speed islower than the rated value, the power limitation is not active;therefore, 13th row and 13th column of matrix J(X0), whichare associated with pitch-control loop, can be deleted.

B. Hopf Bifurcation

This paper concentrates on the analysis of local bifurcations,particularly Hopf bifurcation that can occur in a DFIG system.

Hopf bifurcation corresponds to emergence of a periodic so-lution from an equilibrium point of (20); in this way, the HFBis responsible for system oscillatory behavior. According to theHopf bifurcation theorem [19], a HFB can be supercritical orsubcritical. A supercritical HFB has the initially stable periodicsolution branch and will result in a smooth transition to oscil-lations. On the other hand, a subcritical HFB is associated withan unstable periodic solution branch and will lead to a hardtransition to large amplitude oscillations.

The DFIG wind turbine system works in power-optimizationoperation mode at most of the time. Under this operation mode,the rotor speed of DFIG usually changes along with the vari-ation of wind speed [10]. We will consequently focus on theeffect of the variation of wind speed as well as rotor speed onthe dynamical behavior of DFIG under the power optimizationoperation mode in the following section.

C. Eigenvalue Sensitivity

Eigenvalue sensitivity, defined as the rate and direction ofeigenvalue movement in the s-plane due to the variation insystem parameters is an efficient tool for designing the controlsystem and parameterizing the system, especially for the higherorder systems. Two types of eigenvalue sensitivities are studied:eigenvalue sensitivity with respect to the entry of system statematrix and system parameter.

The participation factor is a special group of eigenvalue sen-sitivity with respect to the system states [20] as follows:

Pki =∂λi∂akk

= ukivki(i, k = 1, 2, ..., n) (22)

where akk is the kth row and kth column of A, ui, vi ∈ Rndenote the normalized right and left eigenvectors correspondingto λi , respectively.

TABLE ISYSTEM PARAMETERS USED IN SIMULATIONS

TABLE IIEIGENVALUES OF DFIG WIND TURBINE SYSTEM (Vw = 12 M/S, KP ω = 1)

The first-order sensitivity of an eigenvalue λi with regard toa system-operating parameter α can be given by the following:

∂λi∂α

=uTi (∂A/∂α) vi

uTi vi. (23)

The magnitude and the sign of the real part of the eigenvaluesensitivity Sλ,σα are defined as the size and direction of move-ment of eigenvalue λi in the horizontal direction in the s-planedue to the small perturbation of a general parameter α, respec-tively, whereas the imaginary part of the eigenvalue sensitivitySλ,ωα are associated with the movement of eigenvalue λi in thevertical direction.

IV. THEORETICAL SYSTEM EIGENVALUE ANALYSIS

Using the Jacobian matrix derived in Section III, eigenvalueanalysis of the DFIG wind turbine system is given in this section.

A. Eigenvalue Loci

The system parameters, set as the standard value from MAT-LAB vR2007b Demo, are detailed in Table I. Using the Jacobianmatrix (21), eigenvalues of the DFIG system can be calculated.All the eigenvalues at rated wind speed (Vw = 12 m/s, ωr =1.1 p.u., KP ω = 1) are listed in Table II. The eigenvalue loci ofcorresponding oscillatory modes are plotted in Figs. 5 and 6.

Fig. 5(a)–(d) shows the eigenvalue loci of λ1,2 , λ3,4 , λ5,6 , andλ7,8 as wind speed increases when KP ω = 60. The arrows in


Fig. 5. Eigenvalue loci as wind speed increases from 8 to 15 m/s (KP ω =60) for (a) λ1 ,2 , (b) λ3 ,4 , (c) λ5 ,6 , and (d) λ7 ,8 .

Fig. 6. Eigenvalue loci as the proportional gain of rotor speed controllerKP ω increases from 1 to 50 (Vw = 12 m/s) for (a) λ1 ,2 , (b) λ3 ,4 , (c) λ5 ,6 , and(d) λ7 ,8 .

the figures indicate the directions of the eigenvalue movementas Vw increases from 8 to 15 m/s. When Vw is higher thanthe rated value, the power and generator speed limitation areactivated, resulting in slower movement of all the eigenvalues.As shown in this figure, λ1,2 move to the imaginary axis and theoscillation frequency is increased as Vw increases. For λ3,4 , theymove away from the imaginary axis. The oscillation frequencyincreases up to a point (Vw = 9 m/s), then decreases again. Forλ5,6 , they move to right and the oscillation frequency decreasesup to a point (Vw = 10.2 m/s), then suddenly moves towardleft. When Vw is higher than the rated value, they move to rightfrom a new position, which is far away from imaginary axis.For λ7,8 , they move to left up to a point (Vw = 11 m/s), thenthey move toward right. The oscillation frequency is decreasedas Vw increases. When Vw is higher than the rated value, theymove to left from a new position close to imaginary axis.

TABLE IIIEIGENVALUES FOR VARIATION VALUE OF Vw AND ωr (KP ω = 70)

Fig. 6(a)–(d) shows the eigenvalue loci of λ1,2 , λ3,4 , λ5,6and λ7,8 as the proportional gain of rotor speed controller KP ωincreases from 1 to 50, respectively. λ1,2 and λ5,6 move towardthe imaginary axis, while λ3,4 and λ7,8 move away from theimaginary axis as KP ω increases. The oscillation frequencies ofλ1,2 , λ3,4 , and λ5,6 are decreased, while the oscillation frequencyof λ7,8 is increased as KP ω increases.

The aforementioned analyses show that the operation stabilityof a DFIG wind turbine system can vary very much due toreasons, such as varying wind speed and control parameters.For the studied wind turbine, it is observed that as wind speedvaries, λ1,2 and λ5,6 tend to move to the right half of the s-plane, if KP ω is inappropriately selected above a critical value.This indicates that they are the key modes for inducing theoscillatory instability, especially when wind speed is lower thanrated value. Moreover, it is obvious from the results in Fig. 5 thatthe system is more stable and oscillation can hardly happen whenthe wind speed is higher than the rated value due to activenessof the power limitation. Therefore, the oscillatory instabilityand eigenvalue sensitivity at higher wind speed will not beenanalyzed in the following section.

B. Hopf Bifurcation

Table III shows the effect of Vw and ωr variations on theeigenvalues of the studied system when KP ω is set improperly(KP ω = 70). It shows that there totally exist four pairs ofcomplex conjugate eigenvalues and two real eigenvalues, asVw and ωr vary. When ωr is around the synchronous speed,all these eigenvalues have negative real parts. As ωr increasesat a critical value (ωr = 1.0564 p.u.), a simple pair of pureimaginary eigenvalues λ5,6 = 0 ± j19.4 of around 3 Hz ap-pears, while other eigenvalues remain in the left half plane, and((d(Re[λ(μ)]))/dμ) |μ∗ < 0.

A supercritical HFB, therefore, occurs and a stable limit cy-cle emerges, leading to a smooth transition to time-periodicoscillations in the studied DFIG [19]. As ωr increases further,the real part of the complex eigenvalues changes to positive,so the system loses stability and oscillates periodically. Whenωr decreases at a critical value (ωr = 0.9593 p.u.), a simplepair of pure imaginary eigenvalues λ5,6 = 0 ± j16.7 of around3 Hz appears, while other eigenvalues remain in the left-half


TABLE IVEIGENVALUES AND PARTICIPATION FACTORS (Vw = 12 m/s, KP ω = 1)

plane, and ((d(Re[λ(μ)])/dμ)) |μ∗ < 0. Therefore, a supercrit-ical HFB occurs. As ωr decreases further, the real part of thecomplex eigenvalues changes to positive and the system losesstability with periodical oscillation.

The analysis reveals that the Hopf bifurcation can happenin a DFIG wind turbine with inappropriate tuning of controlparameters. For this studied system, this is essentially causedby the shift of the real part of λ5,6 from negative to zero.

C. Eigenvalue Sensitivity

The eigenvalues and participation factors of the studied sys-tem when ωr = 1.1 p.u. (Vw = 12 m/s, KP ω = 1) are shownin Table IV. We can see that λ1,2 are associated with the statorflux; λ3,4 are associated with the rotor and turbine mechanical;λ5,6 are associated with the rotor flux; λ7,8 are associated withrotor and turbine mechanical; λ9 and λ10 are associated withdc-link voltage; λ11 is associated with dynamics of pitch angle.The first-order eigenvalue sensitivities with respect to some ma-chine and control parameters at different rotor speeds are listedin Table V. As the required perturbed parameters appear explic-itly in state matrix A, the analytical approach can be applied tocompute the eigenvalue sensitivities [21].

It is obvious from Table V that a DFIG parameter differs muchin their sensitivities to different eigenvalues. Furthermore, thesensitivities also vary at different rotor speeds. This observationimplies that simply adjusting only one DFIG parameter can-not ensure damping enhancements of several critical eigenvaluepairs at different rotor speeds. Correspondingly, the coordinatedtuning of system parameters using advanced optimization tech-nique should be considered to improve system stability in futurework.

For λ1,2 , the most sensitive parameters are Rs , Lls , and Llras the real part of the sensitivities of λ1,2 with respect to Rs ,Lls , and Llr are larger than the others. The increase in Rsand decrease in Lls and Llr make λ1,2 move toward left inthe s-plane. For λ3,4 , the most sensitive parameter is Hg . Asmall positive perturbation in Hg makes λ3,4 shift toward theimaginary axis. However, as λ3,4 are not the key modes for theoscillatory stability of the studied system, the increase in Hg willnot deteriorate the system stability. For λ5,6 , the most criticalparameters are Rs , Rr , Lls , and Llr . The increase in Rr anddecrease in Lls and Llr will lead to λ5,6 moving toward left inthe s-plane. The decrease in Rs at subsynchronous speed, whileincrease at synchronous and supersynchronous speed makesλ5,6 shift toward left in the s-plane. For λ7,8 , only at synchronous

speed, λ7,8 is sensitive to the variation of Rr and Lls . For λ9and λ10 , the most sensitive parameters are Rr . The increase inRr leads to λ9 shifting toward right and λ10 moving toward leftin the s-plane. λ11 is insensitive to all these parameters listed inTable V.

V. COMPUTER SIMULATION STUDY

The preceding section presents theoretical analysis based onthe mathematical model. In this section, we will present a seriesof computer simulations to verify the theoretical analysis. Inparticular, we will focus on the qualitative change of dynamicsas Vw is varied, as analyzed in Section IV. MATLAB/Simulink isused to establish the simulation model of DFIG system describedin the foregoing section. All the components of the simulationmodel are built with standard electrical component blocks fromthe SimPowerSystems block set in MATLAB/Simulink library.

A. Stable Operation

Fig. 7(a)–(f) shows the time-domain waveforms of rotor speedωr , dc-link voltage Udc , active power P , reactive power Q, statorcurrent of phase A isa and rotor current of phase A ira whenwind speed Vw = 11.1 m/s, rotor speed set point ωref = 1.015. Itis shown that ωr , Udc , P , and Q are nearly constant, isa and iraare sinusoidal. The system is stable and there is no oscillatorybehavior.

B. Oscillatory Instability

When Vw varies above a critical value, oscillatory behaviorcan occure in the studied system. Fig. 8 shows the correspondingsteady-state time-domain waveforms, after a step increase of0.9 m/s (8.1%) is applied to Vw (Vw = 12 m/s, ωref ≈ 1.1).It is shown that ωr , Udc , P , and Q are no longer constant,but oscillate around the frequency of 3 Hz. isa and ira are nomore sineusoidal. In Section III, the nature of such oscillation isanalyzed from a Hopf bifurcation perspective. It can be observedthat Hopf bifurcation takes place at approximately the samewind speed condition as it does in our theoretical analysis, andthe simulated periodic oscillations also match the earlier Hopfbifurcation analysis.

C. Hopf Bifurcation Boundary

The analysis in Section IV already shows that the oscillatoryinstability is essentially Hopf bifurcation induced one, throughthe eigenvalues λ5,6 . Sensitivity analysis also indicates the pa-rameters, which have significant effects on the movement ofλ5,6 , are Rs , Rr , Lls , and Llr . In this section, the stabilityboundary curves with respect to those critical parameters withinthe space of KP ω versus ωr , where the conjugate eigenvaluesλ5,6 intersecting with the imaginary axis are mapped, whichcorrospond to the occurence of Hopf bifurcation. The Hopfbifurcation boundaries can be readily obtained by using the an-alytical means described in the earlier section. On the otherhand, the boundaries obtained from simulations performed inMATLAB/Simulink are also given to verify the results from thetheoretical analysis.


TABLE VFIRST-ORDER EIGENVALUES SENSITIVITIES GIVEN DIFFERENT ROTOR SPEED: (A) ωr = 0.9 SUBSYNCHRONOUS SPEED;

(B) ωr = 1 SYNCHRONOUS SPEED; (C) ωr = 1.1 SUBSYNCHRONOUS SPEED

Fig. 9(a)–(d) shows the Hopf bifurcation boundaries in theparameter space of KP ω versus ωr under different values ofRs , Rr , Lls , and Llr , respectively, which clearly illustrates theeffect of those sensitive parameters on the Hopf bifurcationboundaries. Area below the curves corresponds to stable opera-tion and above that to unstable operation. On top of these curves,the system loses stability via Hopf bifurcation.

As shown in Fig. 9, the simulation results agree well with theanalytical results. Also, we can generally observe that aroundsynchronous speed, the critical value of KP ω is the largest, anddoes not change so significantly in different system parameterconditions. When ωr is away from synchronous speed (e.g., in0.7–0.9 or around 1.1), the critical value of KP ω decreases asthe value of ωr increases, and Hopf bifurcation boundaries haveconsiderable changes in different system parameter conditions.The point marked with “∗” is the stable operating point beforethe bifurcation occurs. At subsynchronous speed, the Hopf bi-furcation margin becomes smaller as Rs increases. While atsupersynchronous speed, the Hopf bifurcation margin becomeslarger as Rs increases. For all studied region of rotor speed, theHopf bifurcation margin becomes larger, as Rr increases, whileLls and Llr decrease.

D. Discussion

The simulation results have confirmed the theoretical analy-sis based on derived model and Jacobian matrix. The simulation

results show that the oscillatory behavior with the nature ofHopf bifurcation can happen due to reason like varying windspeed. The observed oscillation is primarily due to the vary-ing electromagnetic torque, since the mechanical one is fixed.Such oscillation of the electromagnetic torque is related to thevariations of the magnitude and direction of the stator and rotorflux-linkage vectors [22], [23], which is not focused herein, butwill be investigated in our future scope.

Besides the wind speed and control parameters, Hopf bifurca-tion is also sensitive to other system parameters, and the impactof different parameters on the Hopf bifurcation margin at dif-ferent rotor speeds is different. For the studied system, Hopfbifurcation boundaries for the eigenvalues λ5,6 show that suchbifurcation can happen particularly when Rs , Lls , and Llr in-crease, while Rr decreases at subsynchronous speed. At super-synchronous speed, the increase of Lls and Llr , while decreaseof Rs and Rr may lead to such Hopf bifurcation. Hence, simplyincreasing Rs cannot enhance the Hopf bifurcation margin atdifferent rotor speeds, and it is recommended to choose largevalue of Rr , while small value of Lls and Llr to enlarge suchHopf bifurcation margin. These are the important characteristicsfor a DFIG wind turbine system, and very useful for the oper-ators of such system to be careful in situations, where criticalparameters may be changed. Hopf bifurcation boundaries forother eigenvalues have also been studied. Though different bi-furcation boundaries will be exhibited for different eigenvalues,


Fig. 7. Simulation results when DFIG works in stable operation. (a) Rotorspeed ωr . (b) DC-link voltage Udc . (c) Active power P . (d) Reactive power Q.(e) Phase A of stator current isa . (f) Phase A of rotor current ir a .

Fig. 8. Simulation results when oscillatory instability occurs. (a) Rotor speedωr . (b) DC-link voltage Udc . (c) Active power P . (d) Reactive power Q.(e) Phase A of stator current isa . (f) Phase A of rotor current ir a .

Fig. 9. Hopf bifurcation boundaries in the parameter space of KP ω versusωr for (a) different values of stator resistance Rs , (b) mutual inductance Rr ,(c) stator leakage inductance Lls , and (d) rotor leakage inductance Llr . Re-sults from analysis are denoted by the solid line. Results from simulations arerepresented by “•”, “+,” and “◦”. The dot “∗” is the stable operating point.

in general, a varying style along with rotor speed can be clearlyobserved.

When the wind speed is higher than the rated value, the powerlimitation is activated. It is obvious from the results in Fig. 5 thatthe system is more stable and Hopf bifurcation can hardly hap-pen. Therefore, the bifurcation boundary at higher wind speedhas not been analyzed in this paper.

VI. CONCLUSION

DFIG has been one of the popular types for high-power ap-plications of wind power generation. However, the detailednonlinear dynamics of this system, so far, has not been thor-oughly investigated. In this paper, a detailed DFIG wind turbinemodel including two-mass drive train, pitch control, inductiongenerator, back-to-back PWM converters, and vector-controlloops was developed. The Jacobian matrix was also derived forsmall-signal-stability analysis purpose. Bifurcation and eigen-value sensitivity analysis based on both theoretical analysis andcomputer simulations showed that DFIG wind turbine can loosestability via a Hopf bifurcation. Further analysis showed that theimpact of different DFIG parameters on different critical eigen-value pairs at different rotor speeds was different. The mostsensitive parameters to the Hopf bifurcation of a DFIG windturbine system can be identified through eigenvalue sensitivityanalysis. Moreover, the Hopf bifurcation boundaries with re-spect to those critical parameters have also been analyzed thatcan facilitate parameterizing the DFIG wind turbine system toensure stable operation. The analyses in this paper provide in-sights into DFIG oscillatory stability that can be important forboth manufacturer and system operators in designing or practi-cally operating such wind turbines concerning their impact onpower system small-signal stability.


APPENDIX

JACOBIAN MATRIX

J11 = D(

RsLr + Lm∂udr∂ids

)J112 = DLm

∂udr∂xus

J12 = D[(ωs − ωr0)L2m − ωsLsLr + Lm

∂udr∂iqs

)]

J13 = D(−RrLm + Lm

∂udr∂idr

)

J14 = D[−ωr0LrLm + Lm

∂udr∂iqr

)]

J15 = D[−L2m Iqs0 − LrLm Iqr0 + Lm

∂udr∂ωr

)]

J17 = DLm∂udr∂xω

J19 = DLm∂udr∂θω

J110 = DLm∂udr∂ωt

J111 = DLm∂udr∂β

J21 = D[−(ωs − ωr0)L2m + ωsLsLr + Lm

∂uqr∂ids

)]

J22 = D(

RsLr + Lm∂uqr∂iqs

)

J23 = D[ωr0LrLm + Lm

∂uqr∂idr

)]

J24 = D(−RrLm + Lm

∂uqr∂iqr

)J27 = DLm

∂uqr∂xω

J29 = DLm∂uqr∂θω

J25 = D[L2m Ids0 + LrLm Idr0 + Lm

∂uqr∂ωr

)]

J210 = DLm∂uqr∂ωt

J211 = DLm∂uqr∂β

J212 = DLm∂uqr∂xus

J31 = D(−RsLm − Ls∂udr∂ids

)

J32 = D[ωr0LsLm − Ls

∂udr∂iqs

)]

J33 = D(

RrLs − Ls∂udr∂idr

)

J34 = D[ωsL

2m − (ωs − ωr0)LsLr − Ls

∂udr∂iqr

)]

J35 = D[LsLm Iqs0 + LsLrIqr0 − Ls

∂udr∂ωr

)]

J37 = −DLs∂udr∂xω

J39 = DLm∂udr∂θω

J310 = −DLs∂udr∂ωt

J311 = −DLs∂udr∂β

J312 = −DLs∂udr∂xus

J41 = D(−ωr0LsLm − Ls

∂uqr∂ids

)

J42 = D(−RsLm − Ls

∂uqr∂iqs

)J44 = D

(RrLs − Ls

∂uqr∂iqr

)

J43 = D[−ωsL2m + (ωs − ωr0)LsLr − Ls

∂uqr∂idr

)]

J45 = D[−LsLm Ids0 − LsLrIdr0 − Ls

∂uqr∂ωr

)]

J47 = −DLs∂uqr∂xω

J49 = −DLs∂uqr∂θω

J410 = −DLs∂uqr∂ωt

J411 = −DLs∂uqr∂β

J412 = −DLs∂uqr∂xus

J51 = −Lm2Hg

Iqr0

J52 =Lm2Hg

Idr0 J53 =Lm2Hg

Iqs0 J54 = −Lm2Hg

Ids0

J55 = −Dshωb + B

2HgJ59 =

Ksh2Hg

J510 =Dshωb2Hg

J61 = −1

G0

(Idr0

∂udr∂ids

+ Iqr0∂uqr∂ids

)

J62 = −1

G0

(Idr0

∂udr∂iqs

+ Iqr0∂uqr∂iqs

)

J63 = −1

G0

(Udr0 + Idr0

∂udr∂idr

+ Iqr0∂uqr∂idr

)

J64 = −1

G0

(Uqr0 + Idr0

∂udr∂iqr

+ Iqr0∂uqr∂iqr

)

J65 = −1

G0

(Idr0

∂udr∂ωr

+ Iqr0∂uqr∂ωr

)

J66 = −W0G20

∂G

∂Udc+

∂W

G0∂Udc

J67 = −1

G0

(Idr0

∂udr∂xω

+ Iqr0∂uqr∂xω

)

J68 = −W0G20

∂G

∂xu+

∂W

G0∂xu

J69 = −1

G0

(Idr0

∂udr∂θω

+ Iqr0∂uqr∂θω

)

J610 = −1

G0

(Idr0

∂udr∂ωt

+ Iqr0∂uqr∂ωt

)

J611 = −1

G0

(Idr0

∂udr∂β

+ Iqr0∂uqr∂β

)

J612 = −1

G0

(Idr0

∂udr∂xus

+ Iqr0∂uqr∂xus

)

J75 = −KP ωTIω

J710 =KP ω2TIω

(TmKopt

)−0.5∂Tm∂ωt


J86 = −KP vTIv

J711 =KP ω2TIω

(Tm

Kopt

)−0.5∂Tm∂β

J95 = −ωb J910 = ωb

J105 =Dshωb2Ht

J109 = −Ksh2Ht

J1010 =1

2Ht

(∂Tm∂ωt

− Dshωb)

J1011 =1

2Ht∂Tm∂β

J1101 =1Tβ

∂βref∂ids

J1102 =1Tβ

∂βref∂iqs

J116 =1Tβ

∂βref∂Udc

J118 =1Tβ

∂βref∂xu

J1111 = −1Tβ

J1113 =1Tβ

∂βref∂xβ

J131 =KP βTIβ

∂Pg∂ids

J132 =KP βTIβ

∂Pg∂iqs

J136 =KP βTIβ

∂Pg∂Udc

J138 =KP βTIβ

∂Pg∂xu

,

where G =2Cdc3ωb

Udc −√

2LKP v√3mωb

iεdL

W = iεdL

[Us − RLiεdL −

√2LKP v√

3mωbTIv(Udcref − Udc)

+ ωsLIqLref

]− IqLref (RLIqLref + ωsLiεdL )

− (udr idr + uqr iqr ).

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[13] L. Rouco and J. L. Zamora, “Dynamic patterns and model order reductionin small-signal models of doubly fed induction generators for wind powerapplications,” in Proc. IEEE PES Gen. Meeting, 2006, pp. 1–8.

[14] P. C. Krause, O. Wasynczuk, and S. D. Sudhoff,, Analysis of ElectricMachinery and Drive Systems. Piscataway, NJ: IEEE Press, 2002.

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[17] R. Pena, J. C. Clare, and G. M. Asher, “Doubly fed induction generatorusing back-to-back PWM converters and its application to variable speedwind-energy generation,” in Proc. Inst. Elect. Eng. Electr. Power Appl.,May 1996, vol. 143, no. 3, pp. 231–241.

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[19] Z. Y. Dong, “Advanced methods for power system small signal stabilityand control,” Ph.D. thesis, Sydney Univ., Sydney, Australia, 1999, pp. S4–S57.

[20] J. Ma and Z. Y. Dong, “Eigenvalue Sensitivity Analysis for ProbabilisticSmall Signal Stability Assessment,” presented at the IET APSCOM’06,Hong Kong, China.

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Lihui Yang received the Ph.D. degree in electricalengineering from Xi’an Jiaotong University, Xi’an,Shaanxi, China, in 2010.

During 2008–2009, she was a Visiting Ph.D. Stu-dent at the Center for Electric Technology, TechnicalUniversity of Denmark. She is currently an AssistantProfessor at Xi’an Jiaotong University. Her researchinterests include stability and control of wind powergeneration.

Zhao Xu (S’00–M’06) received the Ph.D. degreein electrical engineering from The University ofQueensland, Australia, in 2006.

From 2006–2009, he was an Assistant at the Cen-ter for Electric Technology, Technical University ofDenmark, where he became an Associate Professor.Since 2010, he has been with Hong Kong PolytechnicUniversity, Kowloon, Hong Kong. His research in-terests include demand side, grid integration of windpower, electricity market planning and management,and AI applications.


Jacob Østergaard (M’95–SM’09) received theM.Sc. degree in electrical engineering from the Tech-nical University of Denmark, Lyngby, Denmark, in1995.

He is currently a Professor and theHead of the Center for Electric Technology,Department of Electrical Engineering, Technical Uni-versity of Denmark. His research interests includeintegration of renewable energy, control architecturefor future power system, and demand side.

Prof. Østergaard is engaged with several profes-sional organizations, including the EU SmartGrids Advisory Council.

Zhao Yang Dong (M’99–SM’06) received the Ph.D.degree from The University of Sydney, Sydney,Australia, in 1999.

He was engaged in various academic positions atThe University of Queensland, Australia and NationalUniversity of Singapore. He is currently engagedat Hong Kong Polytechnic University, Kowloon,Hong Kong. He is also engaged in various indus-trial positions with Powerlink Queensland, Virginia,and Transend Networks, Tasmania, Australia (bothare transmission network service providers in corre-

sponding states). His research interests include power system planning, powersystem security assessment, power system stability and control, power systemload modeling, electricity market, and computational intelligence and its appli-cation in power engineering.

Kit Po Wong (M’87–SM’90–F’02) received theM.Sc., Ph.D., and D.Eng. degrees from the Instituteof Science and Technology, University of Manch-ester, Manchester, U.K., in 1972, 1974, and 2001,respectively.

From 1974 to 2004, he was at The Universityof Western Australia, Perth, Australia, where he iscurrently an Adjunct Professor. He was the Head ofthe Department of Electrical Engineering, The HongKong Polytechnic University, Kowloon, Hong Kong,where he has been the Chair Professor, since 2002.

His research interests include computation intelligence applications to powersystem analysis, planning and operations, as well as power market analysis.

Prof. Wong received three Sir John Madsen Medals (1981, 1982, and 1988)from the Institution of Engineers, Australia (IEAust), the 1999 OutstandingEngineer Award from IEEE Power Chapter Western Australia, and the 2000IEEE Third Millennium Award. He was a Co-Technical Chairman of the IEEEInternational Conference on Machine Learning and Cybernetics (ICMLC) 2004and the General Chairman of IEEE/CSEE PowerCon2000. He was an Editor-in-Chief of IEE Proceedings in Generation, Transmission and Distribution and theEditor (Electrical) of the Transactions of Hong Kong Institution of Engineers.He is a Fellow of the Institution of Engineering and Technology (IET), the HongKong Institution of Engineers (HKIE), and the IEAust.

Xikui Ma received the B.E. and M.Sc. degrees inelectrical engineering from Xi’an Jiaotong Univer-sity, Xi’an, China, in 1982 and 1985, respectively.

In 1985, he joined the Xi’an Jiaotong Univer-sity as a Lecturer, where he has been a Professor,since 1992. His current research interests includeelectromagnetic field theory and its application, nu-merical methods, modeling of magnetic components,chaotic dynamics and its applications in power elec-tronics, and applications of digital control in powerelectronics.

oscillatory stability and eigenvalue sensitivity analysis of a dfig …1... · digital object...

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