modeling and stability analysis of a dfig

1504 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 24, NO. 3, JULY 2009

Modeling and Stability Analysis of a DFIG-BasedWind-Power Generator Interfaced With a

Series-Compensated LineAmir Ostadi, Student Member, IEEE, Amirnaser Yazdani, Member, IEEE, and Rajiv K. Varma, Member, IEEE

AbstractThis paper deals with modeling and stability analysisof a doubly-fed induction generator (DFIG)-based wind-powerunit that is interfaced with the grid via a series-compensatedtransmission line. A detailed mathematical model is developed inthis paper that takes into account dynamics of the flux observer,phase-locked loop (PLL), controllers of the power-electronicconverter, and wind turbine. Using the model and based oneigenvalue/participation-factor analysis, the system and controllerparameters that substantially influence the system stability havebeen identified. The developed model is validated through a com-prehensive set of simulation studies in the Matlab/Simulink andPSCAD/EMTDC software environments.

Index TermsControl, doubly-fed induction generator (DFIG),eigenvalue analysis, participation factor, power electronics, seriescompensation, small-signal analysis, wind power.

I. NOMENCLATURE

Stator flux.Stator current.Rotor current.

Stator voltage.Rotor voltage.Ac-side terminal voltage of the grid-side converter.Ac-side current of the grid-side converter.Grid current.Grid voltage.Series capacitor voltage.Stator flux phase-angle.Stator flux angular frequency.PLL reference frame angle.PLL reference frame angular frequency.Stator leakage inductance.Rotor leakage inductance.

Manuscript received May 06, 2008. Current version published June 24, 2009.Paper no. TPWRD-00324-2008.

The authors are with the University of Western Ontario, London, ON, Canada(e-mail: [email protected]; [email protected]; [email protected]).

Digital Object Identifier 10.1109/TPWRD.2009.2013667

Magnetizing inductance.Stator resistance.Rotor resistance.Effective ac-side resistance of the grid-sideconverter.

Effective ac-side inductance of the grid-sideconverter.

Transmission line inductance.Transmission line resistance.Series compensation capacitance.Wind-power unit shunt (filter) capacitance.Dc-link capacitance.Turbine mechanical power.Drive-train friction coefficient.Y-to- side turns ratio of the transformer

-to-Y side turns ratio of the transformerGearbox ratio.Effective moment of inertia.Stator real power.Stator reactive power.Rotor real power.Rotor reactive power.Grid-side converter terminal real power.Grid-side converter terminal reactive power.Real power delivered by the grid-side converter.Reactive power delivered by the grid-side converter.Peak value of the stator line-to-neutral voltage.Steady-state angular frequency of the grid.Subscript denoting the direct-axis component of avariable.Subscript denoting the quadrature-axis componentof a variable.

Differentiation operator .

0885-8977/$25.00 2009 IEEE

OSTADI et al.: MODELING AND STABILITY ANALYSIS OF A DFIG-BASED WIND-POWER GENERATOR 1505

Complex frequency.Superscript denoting the small-signal perturbationof a variable.

: Transfer function of a generic PIcontroller .Wind speed.Turbine pitch angle.

II. INTRODUCTION

I N recent years, penetration of wind energy into the powergrid has been increasing worldwide, at a significant rate[1]. Presently, the majority of wind farms have been based onthe constant-speed technology, for its simplicity, low cost, andruggedness. However, many recent and planned installations areadopting the variable-speed technology due to its better energycapture, smoother operation, lower flicker, and superior control-lability.

One important class of variable-speed wind-power systems isthat based on the doubly-fed induction generator (DFIG). Thisclass has gained significant attentions due to its technical andeconomical advantages. However, control of the DFIG rotor cir-cuit through an electronic power converter, direct connectionbetween the DFIG stator and the grid, and existence of a so-phisticated, multi-loop, control structure makes a DFIG-basedwind-power system prone to undesirable dynamic interactionwith the grid and/or similar units in a farm [2]. This is espe-cially of concern since wind farms are typically located in re-mote areasdue to favorable wind conditionsand are there-fore connected to the power system via weak transmission lines.In addition, to evacuate large amounts of electrical power fromthe wind farms, it is quite likely that the transmission lines willbe series-compensated [3]. Thus, to avoid financial and tech-nical complications, issues need to be identified, fully under-stood, and systematically addressed during the planning and de-sign phases of a wind-power system. Although some issues arealready observed through simulation studies [3], [4], systematicanalysis and design frameworks are required to ensure stability,desired performance, and robustness; the analysis is required toidentify and quantify the impacts of controllers, grid stiffness,system characteristics, and internal interactions on the stabilityof the wind-based generating units [5].

Thus far, a few published works have dealt with the mod-eling and behavior of the DFIG-based wind-power systems. In[6] a state-space model has been developed for a DFIG-basedwind-power system, and a participation-factor analysis has beenconducted on the model to reduce the model order. However, astiff grid has been assumed. In [7] a state-space model has beendeveloped, and the (open-loop) electrical, electromechanical,and mechanical modes of the DFIG-based wind-power systemare identified. Further, impacts of the machine parameters, op-erating point, drive-train parameters, and grid strength on thesystem eigenvalues are investigated. The analysis however doesnot consider the controllers and their impacts on the systemmodes. Moreover, a constant turbine power has been assumed.

This paper investigates the stability of a DFIG-basedwind-power system that is connected to a series-compensatedtransmission line. The analysis is based on a linearized modelobtained from a nonlinear mathematical model of the system.The model takes into account the flux observer, phase-lockedloop (PLL), controllers of the back-to-back voltage-sourcedconverter (VSC), and wind-turbine dynamics. Eigenvalue andparticipation-factor analyses are carried out to identify the con-troller and network parameters that exhibit significant impactson the system (closed-loop) stability.

The rest of this paper is organized as follows. Section III in-troduces the structure of the DFIG-based wind-power system. InSection IV, a mathematical model is developed for the DFIG-based wind-power system. In Section V, the small-signal anal-ysis on the developed model is presented. The linearized modelis validated in Section VI. Section VII concludes the paper.

III. STRUCTURE OF DFIG-BASED WIND-POWER SYSTEMFig. 1 illustrates a schematic diagram of a DFIG-based

wind-power unit, interfaced with the grid through the interfacetransformer and a transmission line. The high-voltageside of , i.e., where the transmission line and meet,is referred to as the Point of Common Coupling (PCC). TheDFIG stator circuit is directly connected to the low-voltage sideof . However, the DFIG rotor circuit is interfaced with thesame side through an acdcac, electronic, power converter.For the electronic converter, the transformer providesvoltage matching. The shunt capacitor primarily preventsthe switching harmonics generated by the power converter fromentering the transmission system. However, it also providesa modest reactive-power support. The transmission line isrepresented by a series - branch, and is series-compensatedby the capacitor . The grid is represented by the ideal voltagesource .

Fig. 1 also shows that the ac-dc-ac converter consists of theback-to-back connection of the rotor-side converter VSC1and the grid-side converter VSC2. The rotor-side and grid-side converters are paralleled from their dc sides with the dc-linkcapacitor . The function of VSC1 is to control the DFIGtorque through the control of the rotor current ; the controlis exercised in the DFIG stator-flux coordinate system. How-ever, VSC2 regulates the dc-link voltage , by controllingthe real power component . VSC2 can also control the re-active power and thus be employed for power-factor regu-lation. VSC2 is controlled in a -frame synchronized to thevoltage .

IV. CONTROL OF DFIG-BASED WIND-POWER SYSTEMTo avoid time-varying inductances in the DFIG model, to deal

with dc signals in the feedback/control loops, to decouple theDFIG torque and flux, and to independently control the real andreactive power of the grid-side converter, the rotor-side and thegrid-side converters are controlled in two respective coordi-nate systems, as detailed in the following subsections. It shouldbe pointed out that the 0-axis components and the correspondingequations are disregarded due to the three-wire configuration ofthe system.


Fig. 1. Schematic diagram of the study DFIG-based wind-power system.

A. Rotor-Side Converter ControlIn an arbitrary -frame making an angle of with the -axis

and rotating with an angular frequency of , the - and -axiscomponents of the generator stator voltage are expressed interms of , and as [8]

(1)where

(2)and . The DFIG torque is

(3)

To maximize the turbine power, the DFIG torque must be con-trolled through the control of and . However, based on (3),this is not a straightforward task since and are also func-tions of and . To simplify the control, the reference-frameangle is selected in such a way that . Hence,

and , and (1) can be rewritten as

(4)

On the other hand, based on (2), and can be expressed as

(5)

Eliminating and between (4) and (5), one obtains

(6)

Fig. 2. Block diagram of the stator flux observer.

where is the stator time-constant. Equation (6)constitutes the base for the stator flux observer of Fig. 2 thatcalculates and delivers for the PWM scheme of the rotor-sideconverter [9].

In a steady state, , and the magnitude of the statorvoltage, i.e., , assumes the constant valueof . On the other hand, if is small, based on (4) oneconcludes that and thus . It also followsfrom (4) that . Hence, (3) can be simplifiedto

(7)

Equation (7) shows that, approximately, the generator torqueis linearly proportional to if the control is exercised in thestator-flux reference frame. Equation (7) is used to calculatethe reference for the rotor -axis current, based on the desiredtorque. To maximize the turbine power, must be changed inproportion to the square of the rotor speed [10], as

(8)


Fig. 3. Block diagram of the rotor current-control scheme.

Fig. 4. Block diagram of the phase-locked loop (PLL).

where is a constant whose expression is given in [10]. There-fore, based on (7) and (8), is calculated as

(9)

The stator reactive power is expressed as

(10)

It then follows from substitution of in (10) (from (5)) andthat

(11)

Equation (11) suggests that the command can be calcu-lated based on the desired stator reactive power, as

(12)

In this paper, is set to zero, and therefore.

Once and are calculated, respectively, from (9)and (12), they are delivered to the decoupled current-controlscheme of Fig. 3. Thus, the outputs of the current-controlscheme, i.e., and , are transformed to their -frameequivalents and fed to the PWM scheme of the rotor-sideconverter, as shown in Fig. 1. It should be noted that inFig. 3 is defined as

(13)

where .

B. Grid-Side Converter Control

The grid-side converter is controlled in a -frame synchro-nized to the stator voltage . The angle for the frame iscalculated by the PLL of Fig. 4. It is shown that, is firsttransformed from the frame to the frame, using the trans-formation angle . Then, the compensator processes the-axis component of , i.e., , and adjusts to force

to zero [11]. If the stator voltage is considered to have theangle and the amplitude , its - and -axis compo-nents assume the forms

(14)

Based on (14), if then, in a steady state,and . Thus, and become, respectively,proportional to and .

It should be noted that the - and -axis voltage componentsin the PLL coordinate system are different than those in thestator-flux coordinate system. However, the components are re-lated to each other through

(15)

Fig. 5 illustrates a block diagram of the current-controlscheme of the grid-side converter which enables independentcontrols of and through their respective reference com-mands [12]. Fig. 5 shows that and are included inthe control process as feed-forward signals. The outputs ofthe current-control scheme, i.e., and , are convertedto their -frame equivalents, using the angle(see Fig. 4), and fed to the PWM scheme of the grid-sideconverter, as shown in Fig. 1; the reason for adding tois to compensate for the phase difference between the primaryand secondary voltages of .

In the wind-power system of Fig. 1, the dc-link voltage iscontrolled by , through the control of . This is based onthe principle of power balance, as

(16)

where and are the control input and the disturbance input,respectively. If the power exchanged with the inductor andthat dissipated in the resistor are ignored, is approxi-mately equal to . Thus,

(17)

Therefore, as shown in Fig. 6(a), can be controlled by, based on (17). Fig. 6(a) also shows that a filtered mea-

sure of is included in the control as a feed-forward signal tomitigate the impact of on .


Fig. 5. Block diagrams of the grid-side current-control scheme.

Fig. 6. Block diagrams of: (a) dc-link voltage regulator and (b) reactive-powerreference generator.

Fig. 6(b) shows that is calculated based on the referencecommand . In this paper, is set to a constant value,corresponding to zero net reactive-power exchange between thewind-power unit and the transmission line. However, maybe actively determined by a feedback control mechanism to reg-ulate the power-factor, reactive-power flow, or PCC voltage.

V. STABILITY ANALYSIS

A. Model Linearization

The nonlinear equations describing the wind-power systemof Fig. 1, although accurate, offer little insight about the systemstability and sensitivity to parameters variations. Moreover, itcan not be easily understood from the nonlinear equations howeach mode is influenced by the controllers and/or network pa-rameters. Therefore, a linear system analysis is conducted on alinearized model of the overall system.

The equations of the uncontrolled system can be written as

(18)

where is vector of output(s), and

(19)

where the vectors , and include the system statevariables

(20)

where

(21)

In (18), and are, respectively, the vector of disturbances/reference-commands and the vector of controllers outputs, as

(22)

The control signals are the outputs of the following dynamicsystem:

(23)

where is the vector of the controllers state variables, andwhere we have

(24)

In (24), depends on the orders of the linear transfer functionsthat process signals in different control schemes of the wind-power unit.

Substituting for from (23) in (18), one deduces the equa-tions of the closed-loop system as

(25)

where

(26)


TABLE IPARTICIPATIONS OF STATE VARIABLES IN THE DOMINANT MODE WHEN , AND

and is the number of entries of , while is the number ofoutputs. The total number of the state-variables is .

The equilibrium point is the solution of

(27)

in which is constant. Linearizing (25) around and , oneobtains the following set of linear equations [13]:

(28)

where and are, respectively, the Jacobian matricesof and with respect to . Similarly, and ,respectively, are the Jacobian matrices of and with re-spect to . The Jacobian matrices are evaluated at and .

It should be pointed out that (18) [and therefore (25)] embedthe following equation for the drive-train dynamics:

(29)

where the nonlinear function describes the power-speedcharacteristic of the wind turbine. The linearization process in-volves calculation of at the operating point, which iscumbersome due to the highly nonlinear dependence of on

. However, in our case, the task is straightforward since thepower maximization strategy of (8) implies that .

For the DFIG-based wind-power system of Fig. 1,and . Therefore, the linearized system has 29 eigen-modes.

B. Participation-Factor Analysis

The linearized model of Section IV provides a basis forstability analysis and parameters selection for the DFIG-basedwind-power system. To that end, the behavior of the dominanteigenvalue, i.e., the one closest to the axis, is studies. More-over, the contribution of each state variable in the dominantmode is evaluated through a participation-factor analysis [8],[14], [15]. The analysis permits identification of system andcontrollers parameters that have major impacts on the dominant

mode. Table I presents participations of the state variablesin the dominant mode, under different series-compensationlevels. The series-compensation level (percentage) is defined as

where and are respectively the capacitiveand inductive reactances of the transmission line, at 60 Hz. Forthe study, , and ; the othercontrollers are introduced in Appendix. The wind speed andthe dc-link voltage are 13 m/s and 1200 V, respectively.

The first row of Table I presents different levels of series com-pensation; the second through the ninth rows show the participa-tion of each state variable in the dominant mode, correspondingto a series-compensation level. As shown in Table I, the statevariables , and exhibit remarkableparticipations in the dominant mode, whereas the other statevariables weakly participate in the dominant mode. Table I in-dicates that the dominant mode is predominantly affected byand . As understood from Fig. 2, these two state variables areclosely related, through the flux estimation process, to and

which themselves actively participate in the dominant mode.It is, therefore, expected that the parameters of the rotor-sidecurrent controller, i.e., and , exhibit significant impactson the location of the dominant eigenvalue. In addition, and

, i.e., the and -axis voltage components of the series ca-pacitor, indicate the largest participations in the dominant mode.Hence, one expects that the series-compensation level has a con-siderable impact on the system stability. Furthermore, Table Iindicates a considerable participation of and in the dom-inant mode, suggesting that the grid stiffness affects the domi-nant mode to a considerable extent.

To verify the findings, the real part of the dominant eigen-value is plotted in Figs. 7 and 8, where the line series com-pensation percentage and rotor-side current-controller gains areused as the parameters. The grid stiffness of the transmissionline is characterized by the ratio of the transmission line.Figs. 7(a) to (c) provide the real parts of the dominant eigenvaluefor ratios of 3, 5 and 9, respectively. Each curve corre-sponds to a value of , i.e., the proportional gain of the PIcontroller of the rotor-side current controller. However, for allcurves the integral gain is kept unchanged at 333.33. As il-lustrated in Fig. 7, the dominant eigenvalue of the system movestowards the right-half plane as the series-compensation level in-creases. It is further observed that, for a given series-compensa-tion level, the dominant eigenvalue moves to right as theratio becomes larger. It is also observed that, for a given level of


Fig. 7. Real part of the dominant eigenvalue as a function of the compensationpercentage, for and different values of ; (a) ,(b) , (c) .

Fig. 8. Real part of the dominant eigenvalue as a function of the compensationpercentage, for and different values of ; (a) , (b) , (c) .

series compensation, an increase in renders the dominantmode unstable.

Figs. 8(a) to (c) demonstrate the effect of the integral constanton the real part of the dominant eigenvalue, for three

ratios of 3, 5, and 9, respectively. The curves suggest that anincrease in the compensation percentage moves the dominanteigenvalue towards the right-half plane. It is observed that upto a certain series-compensation percentage, an increase inrenders the dominant mode more stable; and beyond that levelthe trend is reversed. This level itself is a function of

Fig. 9. Real part of the dominant eigenvalue as a function of: (a) and(b) , for different values of compensation percentage.

ratio and becomes lower as increases. For example, for, the level in question falls beyond 100%; it however

is about 70% and 50% for ratios of 5 and 9, respectively,as Figs. 8(b) and (c) indicate.

As already discussed, among the controllers of theDFIG-based wind-power unit, the rotor-side controller showsthe largest impact on the dominant mode and thus the systemstability. By contrast, as Table I indicates, the state variablesof the PLL, the grid-side current controllers, and the dc-linkvoltage regulator do not exhibit major participations in thedominate mode. Consequently, the parameters of the afore-mentioned building blocks are expected to have insignificantimpact of the system stability. Figs. 9(a) and (b) plot the realpart of the dominant eigenvalue as a function of, respectively,

and , for different series-compensation levels. It isobserved that, at any given series-compensation percentage, thereal part of the dominant eigenvalue is almost independent of

or .

Fig. 10 illustrates the impact of the line inductance, as a mea-sure of the grid strength, on the real-part of the dominant eigen-value. For this case study, a wind speed of 13 m/s and aratio of 5 are assumed, with p.u. corresponding toa line length of 10 km. Fig. 10 shows that the dominant modebecomes less stable as the series-compensation level increases.It is further observed that at low series-compensation levels, thedominant mode becomes more stable as increases. How-ever, beyond a certain level of series compensation, an increasein renders the dominant mode less stable and more sensitiveto .

Fig. 11 illustrates the patterns of variations of the real part ofthe dominant eigenvalue as a function of the series-compensa-tion percentage, for different wind speeds. Figs. 11(a), (b), and(c) correspond to , and ,respectively. The following can be deduced from Fig. 11:

in general, the dominant mode tends to become more un-stable as the wind speed decreases;

at higher wind speeds, the real part of the dominant eigen-value becomes less sensitive to the variations of the windspeed;


Fig. 10. Real part of the dominant mode as a function of compensation per-centage for different values of .

Fig. 11. Real part of the dominant mode versus compensation percentage fordifferent wind speeds; (a) , (b) , (c) .

the dominant mode becomes more unstable as the series-compensation percentage increases.

VI. MODEL VERIFICATION

In this section, the linearized model developed for theDFIG-based wind-power system of Fig. 1 is verified by meansof simulation studies. The linearized model is implementedin the MATLAB/Simulink environment and its results arecompared with those obtained from a detailed, nonlinear,averaged model of the overall system in the PSCAD/EMTDCenvironment. The accuracy of the linearized model is evaluatedfor various disturbances and operational scenarios. For thesimulations, a wind speed of 13 m/s is assumed. The systemand controllers parameters are given in the Appendix.

A. Step Change in Grid VoltageIn this case study, at s, a 10% step change

is introduced in the grid voltage amplitude, while thesystem is in a steady-state condition. For this study,

, and the lineis not compensated. Figs. 1215 illustrate the responses of

Fig. 12. Responses of the rotor speed and dc-link voltage for a step changeimposed on the grid voltage amplitude; (a) linearized model, and (b) detailedmodel.

Fig. 13. Responses of the flux and torque for a step change imposed on the gridvoltage amplitude; (a) linearized model, and (b) detailed model.

and . In each figure, the columns(a) and (b) show the results of the linearized and the detailedmodels, respectively. It can be observed that the results obtainedfrom the linearized model closely agree with those from thedetailed model. Fig. 12 shows the responses of the per-unit rotorspeed as well as the dc-link voltage. As shown in Fig. 12,reduces subsequent to the disturbance. In addition, the dc-linkvoltage undergoes excursions, but retrieves its pre-disturbancevalue in about 150 ms.

Fig. 13 shows the responses of the stator flux and the elec-trical torque, and that a 10-percent increase in the grid voltageamplitude causes the stator flux to increase by approximately10%. The reason is that the steady-state value of the stator flux isproportional to the stator voltage magnitude. Fig. 13 also showsthat, subsequent to the disturbance, the electrical torque settlesat a higher steady-state value.

Fig. 14 illustrates the responses of the real power delivered bythe rotor-side and grid-side converters and that the magnitudeof the rotor real power decreases subsequent to the disturbance.This is because the shift of the operating point to a lower-speedvalue results in a suboptimum power capture. The real power ofthe grid-side converter also exhibits a similar response since, inthe steady state, it is approximately equal to the real power ofthe rotor. Fig. 15 illustrates the responses of and .


Fig. 14. Responses of the real power components of the rotor-side and grid-side converters for a step change imposed on the grid voltage amplitude; (a)linearized model, and (b) detailed model.

Fig. 15. Responses of the reactive power components of the rotor-side and grid-side converters for a step change imposed on the grid voltage amplitude; (a)linearized model and (b) detailed model.

B. Step Change in the DC-Link Voltage ReferenceIn this case study, the dc-link voltage reference is subjected

to a step change from 1200 to 1350 V, at s. The con-troller parameters are the same as those in the previous casestudy, and the line is not compensated. Figs. 16 and 17 pro-vide the responses of , and . The columns (a) and(b) show the results of the linearized and the detailed models,respectively. As with the previous case, a close agreement be-tween the models is noticed.

As Fig. 16 illustrates, subsequent to the reference change, thedc-link voltage rises and settle at 1350 V, in about 150 ms. Thisrequires that real power be drawn by the grid-side converter.Therefore, undergoes an undershoot as illustrated in Fig. 16.The power surge disturbs the DFIG stator voltage (not shown)which, in turn, impacts the DFIG torque. Consequently, un-dergoes transient excursions as shown in Fig. 17. The statorvoltage disturbance also impacts the estimated flux through theflux observer of Fig. 2. The disturbance in the estimated flux, inturn, impacts the rotor voltage generated by VSC1 which resultsin a transient excursion in , as illustrated in Fig. 17. Once thedc-link voltage settles at its final value, and retrieve theircorresponding pre-disturbance values.

Fig. 16. Responses of the dc-link voltage and real power of the grid-side con-verter, for a step change imposed on the dc-link voltage reference; (a) linearizedmodel and (b) detailed model.

Fig. 17. Responses of the rotor speed and real power of the rotor-side converter,for a step change imposed on the dc-link voltage reference; (a) linearized modeland (b) detailed model.

TABLE IIREAL AND IMAGINARY PARTS OF THE DOMINANT EIGENVALUE FOR DIFFERENT

SERIES-COMPENSATION LEVELS

C. Introduction of Series CompensationIn this case study, the impact of series compensation is sim-

ulated for three compensation percentages of 50%, 63%, and70%, while . The PI controller gains areand . Table II provides the real and imaginaryparts of the dominant eigenvalue, for three aforementioned se-ries-compensation levels. Table II indicates that the system isstable for 50% of compensation, is oscillatory for 63% of com-pensation, and is unstable for 70% of compensation. In the nextsections, the validity of the developed model is demonstrated bycomparing the simulation results with those of the eigenvalueanalysis.

1) 50% Compensation: For 50% of series compensation, aseries capacitor of F (0.034 p.u.) is switched in


Fig. 18. Responses of: (a) the stator flux and (b) the q-axis component of thestator voltage, for 50% of compensation.

Fig. 19. Responses of: (a) the stator flux and (b) the q-axis component of thestator voltage, for 63% of compensation.

the transmission line, at s. Fig. 18 shows that andstart to oscillate subsequent to the disturbance. However,

the oscillations decay in about 1.1 s. This duration should cor-respond to about four times the time-constant of the dominantmode, i.e., the inverse of (the absolute value of) the real part ofthe eigenvalue. In other words, the simulation result suggeststhat , which is close to the resultreported in Table II. In Fig. 18, the frequency of the oscillationsis measured as about 35.09 Hz, corresponding to an imaginarypart of rad/s of the dominant eigenvalue.This also agrees with the result of Table II.

2) 63% Compensation: This level of series compensation isrealized by inserting F (0.043 p.u.) in the line, at

s. Fig. 19 shows that and exhibit sustained oscil-lations with a period of about 0.0313 s, corresponding to an an-gular frequency of rad/s of the dominanteigenvalue. This confirms the close agreement with the resultsof Table II.

3) 70% Compensation: For 70% of series compensation, aseries capacitor of F (0.048 p.u.) is switched on,at s. Fig. 20 shows that and become oscillatory andunstable. In this case the period of oscillations is 0.0331 s, whichcorresponds to an angular frequency of

Fig. 20. Responses of: (a) the stator flux, and (b) the q-axis component of thestator voltage, for 70% of compensation.

TABLE IIITURBINE AND GENERATOR PARAMETERS

rad/s. This closely agrees with the results of Table II for 70% ofcompensation.

VII. CONCLUSION

In this paper, stability of a DFIG-based wind-power systemconnected to a series-compensated transmission line is studied.First, a nonlinear mathematical model is developed taking intoconsideration dynamics of the DFIG flux observer, PLL, con-trollers of the back-to-back VSC system, and wind turbine. Aneigenvalue and a participation-factor analysis are conducted ona linearized model of the overall system to identify the impactof the network parameters, controllers, and wind speed on thesystem stability. It is demonstrated that the parameters of therotor-side current-control scheme exhibit the most significantimpacts on the system stability. It is further shown that at agiven series compensation level, the system stability margin isreduced as the line ratio increases. It should be pointed outthat high-voltage (HV) and extra high-voltage (EHV) transmis-sion lines typically have large ratios. Thus, the study re-ported in this paper is relevant in view of the fact that large-scale


TABLE IVPOWER-ELECTRONIC INTERFACE AND LINE PARAMETERS

wind farms are more likely to be connected to EHV lines whichcould also be series-compensated. In addition, it is shown thateven at realistic levels of series compensation, stability of suchDFIG-based wind plants can be of a concern, and appropriatechoice of controllers parameters is necessary.

APPENDIXSYSTEM AND CONTROLLERS PARAMETERS

The controllers of the wind-power system of Fig. 1 are

The parameters are presented in Tables III and IV.

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[12] C. Schauder and H. Mehta, Vector analysis and control of advancedstatic VAR compensators, IEE Proc.Gener., Transmiss., Distrib.,vol. 140, no. 4, pp. 299306, July 1993.

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Amir Ostadi (S05) received the B.Sc. degree inelectrical engineering, from Sharif University ofTechnology, Tehran, Iran, in 2005 and the M.E.Sc.degree in electrical engineering from the Universityof Western Ontario (UWO), London, ON, Canada,in 2008.

His research interests include wind power genera-tion, power system restructuring, and power systemstability.

Amirnaser Yazdani (S02M05) received thePh.D. degree in electrical engineering from theUniversity of Toronto, Toronto, ON, Canada, in2005.

He was with Digital Predictive Systems (DPS)Inc., Mississauga, ON. Presently, he is an AssistantProfessor with the University of Western Ontario(UWO), London, ON. His research interests includedynamic modelling and control of switching powerconverters, distributed generation, and microgrids.

Rajiv K. Varma (M96) received the B.Tech. andPh.D. degrees in electrical engineering from IndianInstitute of Technology (IIT), Kanpur, India, in 1980and 1988, respectively.

He is currently an Associate Professor at theUniversity of Western Ontario (UWO), London,ON, Canada. He was a faculty member in the Elec-trical Engineering Department at IIT Kanpur from1989 to 2001. While in India, he was awarded theGovernment of India BOYSCAST Young ScientistFellowship in 19921993 to conduct research on

flexible ac transmission systems (FACTS) at the UWO. He also receivedthe Fulbright Grant of the U.S. Educational Foundation in India, to conductresearch in FACTS at Bonneville Power Administration (B.P.A.), Portland, OR,during 1998. His research interests include FACTS, power systems stability,and grid integration of wind and photovoltaic solar power systems.

Dr. Varma is the Chair of the IEEE Working Group on FACTS and HVDCBibliography and is active of several IEEE working groups. He has receivedseveral Teaching Excellence Awards, both at the Faculty of Engineering and atthe university level at UWO.

modeling and stability analysis of a dfig

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