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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 27, NO. 3, JULY 2012 1485

Dynamic Model and Control of DFIG Wind EnergySystems Based on Power Transfer Matrix

Esmaeil Rezaei, Ahmadreza Tabesh, Member, IEEE, and Mohammad Ebrahimi

AbstractThis paper presents a power transfer matrix modeland multivariable control method for a doubly-fed inductiongenerator (DFIG) wind energy system. The power transfer matrixmodel uses instantaneous real/reactive power components as thesystem state variables. It is shown that using the power transfermatrix model improves the robustness of controllers as the powerwaveforms are independent of a frame of reference. Thesequential loop closing technique is used to design the controllersbased on the linearized model of the wind energy system. Thedesigned controller includes six compensators for capturing themaximum wind power and supplying the required reactive powerto the DFIG. A power/current limiting scheme is also presentedto protect power converters during a fault. The validity and per-formance of the proposed modeling and control approaches areinvestigated using a study system consisting of a grid-connectedDFIG wind energy conversion system. This investigation usesthe time-domain simulation of the study system to: 1) validatethe presented model and its assumptions, 2) show the trackingand disturbance rejection capabilities of the designed controlsystem, and 3) test the robustness of the designed controller to theuncertainties of the model parameters.

Index TermsDoubly fed induction generator (DFIG), dy-namics modeling, instantaneous power, multivariable control,wind energy systems, wind power control, wind turbine generator.

I. INTRODUCTION

W IND ENERGY conversion systems are currentlyamong economically available and viable renewableenergy systems which have experienced rapid growth in recentyears. Increasing the penetration level of wind farms highlightsthe grid integration concerns including power systems stability,power quality (PQ), protection, and dynamic interactions ofthe wind power units in a wind farm [1][3]. Wind energysystems based on doubly fed induction generators (DFIGs)have been dominantly used in high-power applications sincethey use power-electronic converters with ratings less than therating of the wind turbine generators [4][8]. The scope of thispaper is dynamic modelling and control of DFIG wind turbinegenerators.

Modeling and control of DFIGs have been widely investi-gated based on well-established vector control schemes in a

Manuscript received July 31, 2011; revised April 02, 2012; accepted April13, 2012. Date of publication May 30, 2012; date of current version June 20,2012. Paper no. TPWRD-00653-2011.

The authors are with the Department of Electrical and Computer Engineering,Isfahan University of Technology, Isfahan 84156, Iran (e-mail: [email protected]; [email protected]; [email protected]).

Digital Object Identifier 10.1109/TPWRD.2012.2195685

stator field-oriented frame of reference [7][9]. The vector con-trol is a fast method for independent control of the real/reactivepower of a machine. The method is established based on con-trol of current components in a frame of reference using an

transformation. Since the components are not phys-ically available, the calculation of these components requires aphase-locked loop (PLL) to determine synchronous angle [10],[11]. The dynamics of transformations are often ig-nored in the procedure of control design. Thus, any control de-sign approach must be adequately robust to overcome the un-certainties in estimation of machine parameters as well as un-accounted dynamics of the overall system. The proposed powertransfer matrix model for DFIG in this paper presents an alter-native modeling and control approach which is independent of

transformations.Direct torque control (DTC) and direct power control

schemes (DPC) have been presented as alternative methodswhich directly control machine flux and torque via the selectionof suitable voltage vectors [12][14]. It has been shown thatDPC is a more efficient approach compared to modified DTC[15][17]. However, the DPC method also depends on theestimation of machine parameters and it requires a protectionmechanism to avoid overcurrent during a fault in the system.

This paper presents a modelling and control approach whichuses instantaneous real and reactive power instead of compo-nents of currents in a vector control scheme. The main featuresof the proposed model compared to conventional models in the

frame of reference are as follows.1) Robustness: The waveforms of power components are in-

dependent of a reference frame; therefore, this approachis inherently robust against unaccounted dynamics such asPLL.

2) Simplicity of realization: The power components (statevariables of a feedback control loop) can be directly ob-tained from phase voltage/current quantities, whichsimplifies the implementation of the control system.

Using power components instead of current in the model ofthe system, the control system requires an additional protectionalgorithm to prevent overcurrent during a fault. Such an algo-rithm can be simply added to the control system via measuringthe magnitude of current. The sequential loop closing techniqueis adopted to design a multivariable control system includingsix compensators for a DFIG wind energy system. The designedcontrol system captures maximum wind power via adjusting thespeed of the DFIG and injects the required reactive power to thesystem via a grid-side converter.

0885-8977/$31.00 2012 IEEE

1486 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 27, NO. 3, JULY 2012

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

II. MODEL OF A DFIG WIND ENERGY SYSTEM USINGINSTANTANEOUS POWER COMPONENTS

A. Definitions and AssumptionsThe schematic diagram of a DFIG wind turbine generator is

depicted in Fig. 1. The power converter includes a rotor-sideconverter (RSC) to control the speed of generator and a grid-sideconverter (GSC) to inject reactive power to the system. Usinga passive sign convention, the instantaneous real and reactivepower components of the grid-side converter, and ,in the synchronous reference frame, are [18]

(1)

where and are components of the stator voltagesand GSC currents in the synchronous reference frame, respec-tively. Solving (1) for and , we obtain

(2)

where

(3)

Similarly, the instantaneous real/reactive power components ofDFIG can be obtained in terms of stator currents as

(4)

and the stator current components are given by

(5)

The negative sign in (5) complies the direction of the statorpower flow on Fig. 1. The exact dynamic model of an inductionmachine is conventionally expressed by voltage and torqueequations [18]. Herein, we develop a simplified model for theDIFG-based wind turbine of Fig. 1 by substituting currentsin the exact model in terms of instantaneous real and reactivepower. The key assumption to simplify the model is assumingan approximately constant stator voltage for DFIG. This as-sumption can be only used under a steady-state condition wherethe grid voltage at the point of common coupling (PCC) variesin a narrow interval, typically less than 0.05 p.u. Using this

assumption, is approximately constant and derivatives ofcurrents will be proportional to the derivatives of power basedon (2) and (5).

B. Model of DFIG Using Instantaneous Power ComponentsThe voltage and flux equations of a doubly fed induction ma-

chine in the stator voltage synchronous reference frame can besummarized as [18]

(6)

(7)

(8)where and are the stator and rotor resistances, and isthe synchronous (stator) frequency. Subscripts and signifythe stator and rotor variable, and and are the stator,rotor, and magnetization inductances, respectively. The com-plex quantities and represent the voltage, current,and flux vectors, and is the slip frequency defined as

(9)

where is the rotor speed of the induction machine. To obtaina model of DFIG in terms of and , the rotor flux andcurrent are obtained from (8) as

(10)where . Then, by substituting for

and from (10) in (7) and then by solving (6) and (7)for , we obtain

(11)

Using (5) to replace components of in (11) and byrearranging the equation, we obtain

(12)

(13)

where

(14)

REZAEI et al.: DYNAMIC MODEL AND CONTROL OF DFIG WIND ENERGY SYSTEMS 1487

The state equation of the stator flux can be obtained by substi-tuting for and from (5) in (6). Solving the stator voltageequations for yields

(15)

(16)

The electromechanical dynamic model of the machine is [18]

(17)

where and are the number of pole pairs, inertia of therotor, and mechanical torque of the machine, respectively. Theelectric torque is given by [18]

(18)

In (17), the mechanical torque is input to the model and ,based on (18), can be expressed in terms of instantaneous realand reactive power. Substituting for and from (5) in (18)and then replacing in (17), we deduce

(19)

where

(20)

The simplified model of the induction machine is presented in(12)(16) and (19) which is summarized as

(21)

The model of DFIG in (21) is a nonlinear dynamic model sincethe coefficients of the state variables are functions of the statevariables.

Fig. 2. Equivalent circuit of the grid-side filter.

C. Grid-Side Converter and Filter ModelFig. 2 shows the representation of the grid-side converter and

its filter in the synchronous reference frame. The model ofthe grid-side converter and filter is

(22)

where and are the resistance and inductance of the filter,respectively, and subscript signifies the variables at the grid-side converter [19]. Substituting for from (2) in (22) yields

(23)

where

(24)

(25)

The dc-link model can be deduced from the balance of realpower at the converter dc-link node as given by

(26)

where is the real power that the converter delivers to therotor and represents the total power loss, including con-verter switching losses and copper losses of the filter. The de-livered real power to the rotor is [18]

(27)

Using (10) and (5), can be expressed as

(28)

In the high-power converter, the power loss is often less than 1%of the total transferred power, and the impact of in (26) canbe neglected. Substituting in (26),the model of the dc link is deduced as follows:

(29)

Using (28), the right-hand-side quantities in (29) can be ex-pressed in terms of the state variables .


D. Wind Turbine Model

The captured mechanical power by a wind turbine can be ex-pressed with the algebraic aerodynamic equation as [1]

(30)

where are the wind turbine radius, air mass density,and wind speed, respectively. is the wind turbine power co-efficient which is a function of the tip speed ratioand the pitch angle of the turbine blades, . For a high-powerwind turbine, the maximum mechanical power captured atranges from 6 to 8. Theoretically, it can be shown that 0.6and practically at is about 0.5 for high-powerwind turbines [1].

III. LINEARIZED DYNAMIC MODEL OF A DFIGWIND TURBINE GENERATOR

A. DFIG and Wind Turbine Model

For a high-power machine, the stator resistant is small; there-fore, based on (6), a constant stator voltage under normal oper-ation yields slow-varying flux components. Thus, the com-ponents of the stator flux of a DFIG in a field-oriented frame ofreference with 0 can be obtained from (15) and (16) as

(31)

Substituting for from (31) in (12), (13), and (19), thenby linearizing the equations about an operating point, the small-signal model of DFIG can be expressed as

(32)

(33)

(34)

where denotes small-signal quantities, and

(35)

In the linearized model, superscript 0 denotes the quantities atan operating point. To calculate , the power torque equation

is linearized by assuming a constant wind speedas

(36)

where is obtained via linearizing in (30) as given by

Transferring the linearized dynamic model of DFIG and windturbine in the Laplace domain yields

(37)

where

(38)

Using (37), the dynamic model of DFIG and the wind turbinein Laplace domain can be expressed based on a power transferfunction as

(39)

where can be readily obtained from the solution of (37) forand .

B. Model of the Grid-Side Filter and DC LinkThe model of the grid-side filter in Laplace domain can be

obtained by transferring (23) into the Laplace domain as

(40)

where

(41)

Solving (40) for and , the grid-side filter model in theLaplace domain is

(42)

where

(43)

(44)

Using (29), the linearized model of dc link can be obtained as

(45)

where

(46)


Fig. 3. Schematic diagram of the feedback control system for the machine-sideand grid-side converters.

From (45), the dc bus model in the Laplace domain is

(47)

Equations (39), (42), and (47) represent the linearized multi-variable model of a DFIG wind turbine generator.

IV. MULTIVARIABLE CONTROLLER DESIGN FOR ADFIG WIND TURBINE GENERATOR

A. Controller Design Scheme

Fig. 3 depicts the suggested multivariable feedback controlsystem for the machine- and grid-side control schemes. In thisscheme, the control inputs of the linearized model of the systemare to control real/reactive power of the rotor; and

to adjust the dc-link voltage and injected reactivepower to the system. The outputs (feedbacks) of the system arethe rotor speed, dc-link voltage, and the instantaneous real/reac-tive power of the rotor- and grid-side converters. The feedbackcontrol system includes six compensators which are used in twonested loops. The inner loops consist of , and

where the required reactive power of the machine and gridare directly controlled via and control loops as shownFig. 3. The outer control loops include for regulating therotor speed and for adjusting the dc-link voltage level.

The sequential loop closing (SLC) method [20] is adopted todesign six controllers based on the multivariable model of thesystem developed in Section III. In the SLC method, based onphysical relevance of the inputs and outputs, the input-outputpairs are determined. Then, a controller is designed for the firstpair of the input-output by treating the system as a single-inputsingle-output (SISO) system. The second controller is designedfor the next pair of input-output variables using the first con-troller as an integral part of the system. Based on the theory of

the SLC design method [20], the multivariable system is stableif all of the designed subsystems during the sequential controllerdesign procedure are stable.

B. Design of the Machine-Side Controllers1) Stator Real and Reactive Power Controllers: Considering

as the first pair in (39) and, thus, imposing ,we obtain the first SISO subsystem for controller design as

(48)

The first controller to be designed is

(49)

Substituting from (49) in (48), the closed-loop model of the firstsubsystem in Laplace domain is

(50)

Thus, must be designed so that all poles of (50) remain inthe left-half plane (LHP). The design of can be simply per-formed via SISO system design methods, such as frequency re-sponse or root locus. To design for reactive power control,the first controller is considered as a part of the system, thenby substituting for and

in (39), the closed-loop model of the second subsystem isobtained

(51)

where

Thus, must be designed so that the second subsystem in(51) remains stable.

2) Rotor Speed Controller: Speed control of the turbine-gen-erator rotor is performed via control of the real power of thestator. Therefore, the speed controller uses as the con-trol input. Using the control scheme of Fig. 3, is

(52)

Embedding and controllers in the model of the system,the transfer function of rotor speed can be calculated as

(53)

where

Substituting for from (52) in (53) yields

(54)


Thus, must be designed so that the subsystem in (54) re-mains stable.

C. Grid-Side Controller

1) Grid-Side Real and Reactive Power Controllers: The con-troller design procedure for and is quite similar tothat of the rotor-side converter since both controllers have thesame structure. Therefore, and can be simply ob-tained by repeating the design procedure as explained in (48)(51). The only modification is replacing with

. Also, both subscripts and should be re-placed with subscript . For brevity, the details of the designprocedure have been omitted.

2) DC-Link Voltage Controller: Substituting for ,and into (46), we obtain

(55)

where detailed expressions for and are given in theAppendix. Based on (47) and (55), can regulateat its reference value using the dc-link controller in

. Therefore, the closed-loop system foris deduced as

(56)

where detailed expressions for and are given in theAppendix. Finally, must be designed to stabilize thedc-link closed-loop system in (56).

D. Current Limiting During a Fault

The target of the controller design procedure is to improveperformance of wind energy conversion while maintainingthe stability of the system under normal operating conditions.Therefore, the design procedure mainly deals with stability,tracking performance for capturing maximum wind power,disturbance rejection, and robustness against uncertainties andunaccounted dynamics.

During a fault and/or sever transients, additional protectionalgorithms, such as fault ride through (FRT) and startup al-gorithms, must be added to the control system. Various algo-rithms, including active crowbar [21], series dynamic restorer[22], and dynamic voltage restorer [23] have been suggested forFRT. These algorithms are independent of the control approachduring the normal operation; therefore, they can be used withthe proposed transfer power matrix method herein as well.

In addition to FRT algorithms and to mitigate overcurrentduring a transient, an extra feedback loop can be used tosense the converter currents and reduce the power referencecommands during transients. This extra loop only requires themagnitude of the current and it merely becomes operationalduring a fault condition. An example of such a current loopfor the protection of the converter is elaborated in [19] and[23]. This loop does not impact the performance of controllers

Fig. 4. Schematic diagram of the study system.

TABLE ISTUDY SYSTEMS WIND TURBINE GENERATOR DATA

during the normal operation of the system and, therefore, it willnot be included in the design procedure of the controllers.

V. MODEL VALIDATION AND PERFORMANCE EVALUATION OFTHE MULTIVARIABLE CONTROL SYSTEM

Fig. 4 shows the schematic of a study system for validationof the proposed modelling and control approaches. The studysystem includes a 1.5-MW DFIG wind turbine-generator con-nected to a grid. The electrical and mechanical parameters ofthe turbine generator are adopted from [24] and summarizedin Table I. Using the proposed designed method, the followingper-unitized controllers were designed for the study system

(57)

(58)

(59)(60)

The performance of these controllers was investigated based ontime-domain simulations of the study system using the Matlab/Simulink software tool.

A. Tracking and Disturbance Rejection CapabilitiesFig. 5(a) and (b) shows a trapezoidal pattern for wind speed

and a step change in the reactive reference which are applied to


Fig. 5. Reference commands for wind and the stator reactive power.

Fig. 6. Tracking performance of real and reactive stator powers.

the controllers of the study system. The trapezoidal pattern wasselected to examine the system behavior following variation inthe wind speed with both negative and positive slopes. The se-lected wind speed pattern spans an input mechanical wind powerfrom 0.7 to 1 p.u. (70 to 100% of the turbine-generator ratedpower). The reactive power command is a step change of 0.25p.u. and occurs at 3 s when the real power is about 0.6 p.u.

Fig. 6 compares real/reactive power quantities of the DFIGagainst their command signals. Due to the coupling phenom-enon, the variation of each power quantity can be considered asa disturbance to the other one. For instance, the effect of cou-pling can be seen in Fig. 6(a) at 3 s, where the step com-mand in reactive power causes a small deviation in real power.However, as Fig. 6 shows, both real and reactive power quan-tities accurately track their command signals which means thecontrollers successfully mitigate the impact of coupling effect inthe tracking of commands signals. Fig. 7(a) and (b) depicts thedc-link voltage and the rms values of the machine voltage/cur-rent quantities. These figures show that the stator and rotor cur-rents are changing as the real/reactive power changes whereasthe dc link and stator voltages remained fixed as expected fromthe control strategy. Specifically, the and current curvesshow a step change at 3 s, corresponding to the 0.25-p.u.step command in the reactive power. Fig. 7 shows that as the

Fig. 7. RMS values of the stator voltage and currents.

Fig. 8. Robustness of the controllers to variations in .

Fig. 9. Robustness of the controllers to a 40 error in the PLL angle.

power reference commands are within the rated power of theturbine generator, the voltage/current of the machine and con-verter will remain within their limits.

B. Control System RobustnessFig. 8 shows the tracking and disturbance rejection perfor-

mances of real/reactive power when the leakage inductance of


the machine is changed using the same reference commandsas shown in Fig. 5. Since Fig. 8 shows the responses accuratelytrack the commands for and , therefore, the designedcontroller is robust to a variation of this parameter.

Fig. 9 compares tracking performance of the proposed con-trol system with the conventional vector control method as de-scribed in [25]. The PI controllers of the vector control methodwere first tuned for best performance at 0.1 and 2.Then, the synchronous signal of the phase-locked loop (PLL)was deviated via biasing the PLL angle with 40 . As Fig. 9shows, the proposed method accurately follows the referencecommands for real and reactive power whereas the vector con-trol method fails to track the commands. The reason is that thevector control method is significantly sensitive to the frameof reference whereas the proposed control system is less inde-pendent to the reference frame.

VI. SUMMARY AND CONCLUSION

An alternative modeling and controller design approachbased on the notion of the instantaneous power transfer matrixis described for a DFIG wind energy system. The waveformsof the power components remain intact at different referenceframes and can be easily calculated using the phase voltagesand currents. Therefore, this approach facilitates the imple-mentation of the controllers and improves the robustness of thecontrol system. Furthermore, the proposed model can be po-tentially used to simplify the control issues of the wind energysystem under an unbalanced condition since feedback variablesare independent of -components in positive, negative, andzero sequences.

The proposed approach is verified using the time-domainsimulation of a study system for DFIG wind energy systems.The simulation results show that the suggested model and con-trol scheme can successfully track the rotor speed reference forcapturing the maximum power and maintain the dc-link voltageof the converter regardless of disturbances due to changes inreal and reactive power references.

APPENDIXDetails of , and in (55) and (56) are shown in

the equations at the top of the page.

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Esmaeil Rezaei was born in Isfahan, Iran in 1979. Hereceived the B.Sc. degree in electronics and the M.Sc.degree in electrical engineering from Isfahan Univer-sity of Technology (IUT), Isfahan, Iran, in 2001 and2004, respectively, where he is currently pursuing thePh.D. degree in electrical engineering.

He was a Technical Designer with the Informationand Communication Technology Institute (ICTI),Isfahan University of Technology, from 2004 to2007. His current research interests include electricaldrives and energy conversion systems for renewable

energy resources.

Ahmadreza Tabesh (M12) received the B.Sc. de-gree in electronics and the M.Sc. degree in systemscontrol from Isfahan University of Technology, Is-fahan, Iran, in 1995 and 1998, respectively, and thePh.D. degree in energy systems from the Universityof Toronto, Toronto, ON, Canada, in 2005.

From 2006 to 2009, he was Postdoctorate at theMicroengineering Laboratory for MEMS, Depart-ment of Mechanical Engineering, Universit deSherbrooke, Sherbrooke, QC, Canada. Currently,he is an Assistant Professor with the Department of

Electrical and Computer Engineering, Isfahan University of Technology. Hisareas of research include renewable energy systems and micropower energyharvesters (power MEMS).

Mohammad Ebrahimi received the B.Sc. and M.Sc.degrees in electrical engineering from Tehran Univer-sity, Tehran, Iran, in 1984 and 1986, respectively, andthe Ph.D. degree in power systems from the TarbiyatModarres University, Tehran, Iran, in 1996.

Currently, he is an Associate Professor at theIsfahan University of Technology (IUT), Isfahan,Iran. His research interests include electrical drives,renewable energy, and energy savings.

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