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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
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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)
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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 .
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1488 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 27, NO. 3, JULY
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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)
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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-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)
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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
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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. 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
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1492 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 27, NO. 3, JULY
2012
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.