IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 27, NO. 3, JULY 2012 1485
Dynamic Model and Control of DFIG Wind Energy Systems Based on Power Transfer Matrix
Esmaeil Rezaei, Ahmadreza Tabesh, Member, IEEE, and Mohammad Ebrahimi
Abstract—This paper presents a power transfer matrix model and multivariable control method for a doubly-fed induction generator (DFIG) wind energy system. The power transfer matrix model uses instantaneous real/reactive power components as the system state variables. It is shown that using the power transfer matrix model improves the robustness of controllers as the power waveforms are independent of a frame of reference. The sequential loop closing technique is used to design the controllers based on the linearized model of the wind energy system. The designed controller includes six compensators for capturing the maximum wind power and supplying the required reactive power to the DFIG. A power/current limiting scheme is also presented to protect power converters during a fault. The validity and per-formance of the proposed modeling and control approaches are investigated using a study system consisting of a grid-connected DFIG wind energy conversion system. This investigation uses the time-domain simulation of the study system to: 1) validate the presented model and its assumptions, 2) show the tracking and disturbance rejection capabilities of the designed control system, and 3) test the robustness of the designed controller to the uncertainties of the model parameters.
Index Terms—Doubly fed induction generator (DFIG), dy-namics modeling, instantaneous power, multivariable control, wind energy systems, wind power control, wind turbine generator.
IND ENERGY conversion systems are currently Wamong economically available and viable renewable energy systems which have experienced rapid growth in recent years. Increasing the penetration level of wind farms highlights the grid integration concerns including power systems stability, power quality (PQ), protection, and dynamic interactions of the wind power units in a wind farm –. Wind energy systems based on doubly fed induction generators (DFIGs) have been dominantly used in high-power applications since they use power-electronic converters with ratings less than the rating of the wind turbine generators –. The scope of this paper is dynamic modelling and control of DFIG wind turbine generators.
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 April 13, 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: r[email protected] iut.ac.ir; [email protected]; [email protected]).
Digital Object Identifier 10.1109/TPWRD.2012.2195685
stator field-oriented frame of reference –. The vector con-trol is a fast method for independent control of the real/reactive power 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 a phase-locked loop (PLL) to determine synchronous angle , . 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 power transfer 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 methods which directly control machine flux and torque via the selection of suitable voltage vectors –. It has been shown that DPC is a more efficient approach compared to modified DTC –. However, the DPC method also depends on the estimation of machine parameters and it requires a protection mechanism to avoid overcurrent during a fault in the system.
This paper presents a modelling and control approach which uses instantaneous real and reactive power instead of compo-nents of currents in a vector control scheme. The main features of 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 approach is inherently robust against unaccounted dynamics such as PLL.
2) Simplicity of realization: The power components (state variables of a feedback control loop) can be directly ob-
tained from phase voltage/current quantities, which simplifies the implementation of the control system.
Using power components instead of current in the model of the system, the control system requires an additional protection algorithm to prevent overcurrent during a fault. Such an algo-rithm can be simply added to the control system via measuring the magnitude of current. The sequential loop closing technique is adopted to design a multivariable control system including six compensators for a DFIG wind energy system. The designed control system captures maximum wind power via adjusting the speed of the DFIG and injects the required reactive power to the system via a grid-side converter.
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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 USING
INSTANTANEOUS POWER COMPONENTS
A. Definitions and Assumptions
The schematic diagram of a DFIG wind turbine generator is depicted in Fig. 1. The power converter includes a rotor-side converter (RSC) to control the speed of generator and a grid-side converter (GSC) to inject reactive power to the system. Using a passive sign convention, the instantaneous real and reactive power components of the grid-side converter, and , in the synchronous reference frame, are 
assumption, is approximately constant and derivatives of currents will be proportional to the derivatives of power based on (2) and (5).
B. Model of DFIG Using Instantaneous Power Components
The voltage and flux equations of a doubly fed induction ma-chine in the stator voltage synchronous reference frame can be summarized as 
where and are the stator and rotor resistances, and is the synchronous (stator) frequency. Subscripts and signify the 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
where and are components of the stator voltages and GSC currents in the synchronous reference frame, respec-
tively. Solving (1) for and , we obtain
Similarly, the instantaneous real/reactive power components of DFIG can be obtained in terms of stator currents as
and the stator current components are given by
where is the rotor speed of the induction machine. To obtain a model of DFIG in terms of and , the rotor flux and current are obtained from (8) as
where . Then, by substituting for
and from (10) in (7) and then by solving (6) and (7)
for , we obtain
Using (5) to replace components of in (11) and by
(5) rearranging the equation, we obtain
The negative sign in (5) complies the direction of the stator
power flow on Fig. 1. The exact dynamic model of an induction
machine is conventionally expressed by voltage and torque
equations . Herein, we develop a simplified model for the
DIFG-based wind turbine of Fig. 1 by substituting currents where
in the exact model in terms of instantaneous real and reactive
power. The key assumption to simplify the model is assuming
an approximately constant stator voltage for DFIG. This as-
sumption can be only used under a steady-state condition where
the grid voltage at the point of common coupling (PCC) varies
in a narrow interval, typically less than 0.05 p.u. Using this
REZAEI et al.: DYNAMIC MODEL AND CONTROL OF DFIG WIND ENERGY SYSTEMS 1487
Fig. 2. Equivalent circuit of the grid-side filter.
The state equation of the stator flux can be obtained by substi-
tuting for and from (5) in (6). Solving the stator voltage
equations for yields C. Grid-Side Converter and Filter Model
(15) Fig. 2 shows the representation of the grid-side converter and
its filter in the synchronous reference frame. The model of
the grid-side converter and filter is
The electromechanical dynamic model of the machine is 
where and are the resistance and inductance of the filter,
(17) respectively, and subscript signifies the variables at the grid-
side converter . Substituting for from (2) in (22) yields
where and are the number of pole pairs, inertia of the rotor, and mechanical torque of the machine, respectively. The electric torque is given by 
In (17), the mechanical torque is input to the model and , based on (18), can be expressed in terms of instantaneous real and reactive power. Substituting for and from (5) in (18) and then replacing in (17), we deduce
The simplified model of the induction machine is presented in (12)–(16) and (19) which is summarized as
The dc-link model can be deduced from the balance of real power at the converter dc-link node as given by
where is the real power that the converter delivers to the rotor 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 
Using (10) and (5), can be expressed as
In the high-power converter, the power loss is often less than 1% of the total transferred power, and the impact of in (26) can
be neglected. Substituting in (26),
the model of the dc link is deduced as follows:
The model of DFIG in (21) is a nonlinear dynamic model since the coefficients of the state variables are functions of the state variables.
Using (28), the right-hand-side quantities in (29) can be ex-pressed in terms of the state variables .
1488 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 27, NO. 3, JULY 2012
D. Wind Turbine Model
The captured mechanical power by a wind turbine can be ex-pressed with the algebraic aerodynamic equation as 
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 ratio and the pitch angle of the turbine blades, . For a high-power wind turbine, the maximum mechanical power captured at ranges from 6 to 8. Theoretically, it can be shown that 0.6 and practically at is about 0.5 for high-power wind turbines .
III. LINEARIZED DYNAMIC MODEL OF A DFIG
WIND 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 of
reference with 0 can be obtained from (15) and (16) as
Substituting for from (31) in (12), (13), and (19), then by linearizing the equations about an operating point, the small-signal model of DFIG can be expressed as
Transferring the linearized dynamic model of DFIG and wind turbine in the Laplace domain yields
Using (37), the dynamic model of DFIG and the wind turbine in Laplace domain can be expressed based on a power transfer function as
where can be readily obtained from the solution of (37) for and .
B. Model of the Grid-Side Filter and DC Link
The model of the grid-side filter in Laplace domain can be obtained by transferring (23) into the Laplace domain as
(32) Solving (40) for and , the grid-side filter model in the
Laplace domain is
where denotes small-signal quantities, and
In the linearized model, superscript 0 denotes the quantities at an operating point. To calculate , the power torque equation is linearized by assuming a constant wind speed
as Using (29), the linearized model of dc link can be obtained as
where is obtained via linearizingin (30) as given by
REZAEI et al.: DYNAMIC MODEL AND CONTROL OF DFIG WIND ENERGY SYSTEMS 1489
Fig. 3. Schematic diagram of the feedback control system for the machine-side and grid-side converters.
the SLC design method , the multivariable system is stable if all of the designed subsystems during the sequential controller design procedure are stable.
B. Design of the Machine-Side Controllers
1) 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
The first controller to be designed is
Substituting from (49) in (48), the closed-loop model of the first
subsystem in Laplace domain is
Thus, must be designed so that all poles of (50) remain in
the left-half plane (LHP). The design ofcan be simply per-
formed via SISO system design methods, such as frequency re-
From (45), the dc bus model in the Laplace domain is sponse or root locus. To design for reactive power control,
the first controlleris considered as a part of the system, then
(47) by substituting for and
in (39), the closed-loop model of the second subsystem is
Equations (39), (42), and (47) represent the linearized multi- obtained
variable model of a DFIG wind turbine generator. (51)
IV. MULTIVARIABLE CONTROLLER DESIGN FOR A where
DFIG WIND TURBINE GENERATOR
A. Controller Design Scheme
Fig. 3 depicts the suggested multivariable feedback control system for the machine- and grid-side control schemes. In this scheme, the control inputs of the linearized model of the system are to control real/reactive power of the rotor; and to adjust the dc-link voltage and injected reactive power to the system. The outputs (feedbacks) of the system are the rotor speed, dc-link voltage, and the instantaneous real/reac-tive power of the rotor- and grid-side converters. The feedback control system includes six compensators which are used in two nested loops. The inner loops consist of , and where the required reactive power of the machine and grid are directly controlled via and control loops as shown Fig. 3. The outer control loops include for regulating the rotor speed and for adjusting the dc-link voltage level.
The sequential loop closing (SLC) method  is adopted to design six controllers based on the multivariable model of the system developed in Section III. In the SLC method, based on physical relevance of the inputs and outputs, the input-output pairs are determined. Then, a controller is designed for the first pair of the input-output by treating the system as a single-input single-output (SISO) system. The second controller is designed for the next pair of input-output variables using the first con-troller as an integral part of the system. Based on the theory of
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 the stator. Therefore, the speed controller uses as the con-trol input. Using the control scheme of Fig. 3, is
Embedding and controllers in the model of the system, the transfer function of rotor speed can be calculated as
Substituting for from (52) in (53) yields
1490 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 27, NO. 3, JULY 2012
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 to that of the rotor-side converter since both controllers have the same structure. Therefore, and can be simply ob-
tained by repeating the design procedure as explained in (48)
Fig. 4. Schematic diagram of the study system.
–(51). The only modification is replacing with
. Also, both subscripts and should be re- TABLE I
placed with subscript . For brevity, the details of the design
STUDY SYSTEM’S WIND TURBINE GENERATOR DATA
procedure have been omitted.
2) DC-Link Voltage Controller: Substituting for ,
and into (46), we obtain
where detailed expressions for
and are given in the
Appendix. Based on (47) and (55), can regulate
at its reference value using the dc-link controller in
. Therefore, the closed-loop system for
is deduced as
during the normal operation of the system and, therefore, it will
not be included in the design procedure of the controllers.
where detailed expressions for and are given in the Appendix. Finally, must be designed to stabilize the dc-link closed-loop system in (56).
D. Current Limiting During a Fault
The target of the controller design procedure is to improve performance of wind energy conversion while maintaining the 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 and unaccounted dynamics.
During a fault and/or sever transients, additional protection algorithms, such as fault ride through (FRT) and startup al-gorithms, must be added to the control system. Various algo-rithms, including active crowbar , series dynamic restorer , and dynamic voltage restorer  have been suggested for FRT. These algorithms are independent of the control approach during the normal operation; therefore, they can be used with the proposed transfer power matrix method herein as well.
In addition to FRT algorithms and to mitigate overcurrent during a transient, an extra feedback loop can be used to sense the converter currents and reduce the power reference commands during transients. This extra loop only requires the magnitude of the current and it merely becomes operational during a fault condition. An example of such a current loop for the protection of the converter is elaborated in  and . This loop does not impact the performance of controllers
V. MODEL VALIDATION AND PERFORMANCE EVALUATION OF
THE MULTIVARIABLE CONTROL SYSTEM
Fig. 4 shows the schematic of a study system for validation of the proposed modelling and control approaches. The study system includes a 1.5-MW DFIG wind turbine-generator con-nected to a grid. The electrical and mechanical parameters of the turbine generator are adopted from  and summarized in Table I. Using the proposed designed method, the following per-unitized controllers were designed for the study system
The performance of these controllers was investigated based on time-domain simulations of the study system using the Matlab/ Simulink software tool.
A. Tracking and Disturbance Rejection Capabilities
Fig. 5(a) and (b) shows a trapezoidal pattern for wind speed and a step change in the reactive reference which are applied to
REZAEI et al.: DYNAMIC MODEL AND CONTROL OF DFIG WIND ENERGY SYSTEMS 1491
Fig. 5. Reference commands for wind and the stator reactive power.
Fig. 7. RMS values of the stator voltage and currents.
Fig. 6. Tracking performance of real and reactive stator powers.
Fig. 8. Robustness of the controllers to variations in .
the controllers of the study system. The trapezoidal pattern was selected to examine the system behavior following variation in the wind speed with both negative and positive slopes. The se-lected wind speed pattern spans an input mechanical wind power from 0.7 to 1 p.u. (70 to 100% of the turbine-generator rated power). The reactive power command is a step change of 0.25 p.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 DFIG against their command signals. Due to the coupling phenom-enon, the variation of each power quantity can be considered as a 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 the controllers successfully mitigate the impact of coupling effect in the tracking of commands signals. Fig. 7(a) and (b) depicts the dc-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 whereas the dc link and stator voltages remained fixed as expected from the control strategy. Specifically, the and current curves show 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. 9. Robustness of the controllers to a 40 error in the PLL angle.
power reference commands are within the rated power of the turbine generator, the voltage/current of the machine and con-verter will remain within their limits.
B. Control System Robustness
Fig. 8 shows the tracking and disturbance rejection perfor-mances of real/reactive power when the leakage inductance of
1492 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 27, NO. 3, JULY 2012
the machine is changed using the same reference commands as shown in Fig. 5. Since Fig. 8 shows the responses accurately track the commands for and , therefore, the designed controller 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 . The PI controllers of the vector control method were 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. 9 shows, the proposed method accurately follows the reference commands for real and reactive power whereas the vector con-trol method fails to track the commands. The reason is that the vector control method is significantly sensitive to the frame of reference whereas the proposed control system is less inde-pendent to the reference frame.
VI. SUMMARY AND CONCLUSION
An alternative modeling and controller design approach based on the notion of the instantaneous power transfer matrix is described for a DFIG wind energy system. The waveforms of the power components remain intact at different reference frames and can be easily calculated using the phase voltages and currents. Therefore, this approach facilitates the imple-mentation of the controllers and improves the robustness of the control system. Furthermore, the proposed model can be po-tentially used to simplify the control issues of the wind energy system under an unbalanced condition since feedback variables are independent of -components in positive, negative, and zero sequences.
The proposed approach is verified using the time-domain simulation 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 for capturing the maximum power and maintain the dc-link voltage of the converter regardless of disturbances due to changes in real and reactive power references.
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Esmaeil Rezaei was born in Isfahan, Iran in 1979. He received 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 and 2004, respectively, where he is currently pursuing the Ph.D. degree in electrical engineering.
He was a Technical Designer with the Information and Communication Technology Institute (ICTI), Isfahan University of Technology, from 2004 to 2007. His current research interests include electrical drives and energy conversion systems for renewable
Ahmadreza Tabesh (M’12) received the B.Sc. de-gree in electronics and the M.Sc. degree in systems control from Isfahan University of Technology, Is-fahan, Iran, in 1995 and 1998, respectively, and the Ph.D. degree in energy systems from the University of Toronto, Toronto, ON, Canada, in 2005.
From 2006 to 2009, he was Postdoctorate at the Microengineering Laboratory for MEMS, Depart-ment of Mechanical Engineering, Université de Sherbrooke, Sherbrooke, QC, Canada. Currently, he is an Assistant Professor with the Department of
Electrical and Computer Engineering, Isfahan University of Technology. His areas of research include renewable energy systems and micropower energy harvesters (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, and the Ph.D. degree in power systems from the Tarbiyat Modarres University, Tehran, Iran, in 1996.
Currently, he is an Associate Professor at the Isfahan University of Technology (IUT), Isfahan, Iran. His research interests include electrical drives, renewable energy, and energy savings.
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