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1444 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 24, NO. 6, JUNE 2009

Design and Implementation of a Robust CurrentController for VSI Connected to the Grid

Through anLCLFilterIvan Jorge Gabe, Vincius Foletto Montagner, and Humberto Pinheiro, Member, IEEE

AbstractThis paper describes the design and implementationof a discrete controller for grid-connected voltage-source inverterswith anLCL filter usually found in wind power generation systems.First, a theorem that relates the controllability of the discrete dy-namic equation of the inverter with LCL filter and the samplingfrequency is derived. Then, a condition to obtain a partial statefeedback controller robust to grid impedance uncertainties andbased on linear matrix inequalities is proposed. This controllerguarantees the stability and damping of the LCL filter resonancefor a large set of grid conditions without requiring self-tuning pro-

cedures. Finally, an internal model controller is added to ensureasymptotic reference tracking and disturbance rejection, signif-icantly reducing the impact of grid background voltage distor-tion on the output currents. Experimental results are presentedto support the theoretical analysis and to demonstrate the systemperformance.

Index TermsGrid-connected inverters, LCL filters, linear ma-trix inequalities, partial state feedback, pulsewidth-modulation(PWM) converters.

I. INTRODUCTION

W

ITH the recent increase of energy demands and con-

cerns about global warming and greenhouse gas emis-

sions, there is growing interest in the use of renewable energy

sources. Wind power is one of these resources that has under-

gone very fast expansion around the world. This growth has

been mainly supported by advances in technologies of variable

speed generation, enabled by the power electronics converters.

The main advantages of generation with variable speed turbines

include the lower stress on mechanical components, the reduced

acoustical noise, and a higher power capture by the turbine. The

two main concepts of variable-speed wind turbines are full and

partial power control topologies [1]. In both concepts, power

electronics converters contribute to raise the power levels as

Manuscript received September 9, 2008; revised December 17, 2008.Current version published May 15, 2009. This work was suppored in partby Coordenacao de Aperfeicoamento de Pessoal de Nvel Superior (CAPES)and in part by National Counsel of Technological and Scientific Development(CNPq). Recommended for publication by Associate Editor M. Ponce-Silva.

I. J. Gabe is with the Power Electronics and Control Research Group(GEPOC), Federal University of Santa Maria (UFSM), 97105-900 Santa Maria,Brazil (e-mail: [email protected]).

V. F. Montagner is with the Power Electronics and Control Research Group(GEPOC), Federal University of Pampa (UNIPAMPA), 97546-550, Alegrete,Brazil (e-mail: [email protected]).

H. Pinheiro is with Department of Electrical Energy Conversion (DPEE),Federal University of Santa Maria (UFSM), 97105-900 Santa Maria, Brazil(e-mail: [email protected]).

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

Digital Object Identifier 10.1109/TPEL.2009.2016097

well as to control the active and reactive power injected into the

grid [2], [3].

In some cases, the utilization of wind resources requires

generation to be located in remote regions, where grids with

high power transference capacity are generally not available [4].

Under these conditions, several issues may be of concern, in-

cluding limited power transference capability [5], thermal re-

strictions [6], and instability of the current controller due to

uncertainty of the grid impedance at the point of common con-nection (PCC).

The converter topology used in most grid-connected high-

power wind turbines is the pulsewidth-modulation (PWM)

voltage-source inverter (VSI). The use of PWM requires an

output filter to limit the harmonic content of the grid-injected

currents. LCL filters allow for a reduction of the current har-

monic, due to PWM, to acceptable levels with lower inductance

values than can be achieved with L filters. The main limitation ofL filters is the need for high switching frequencies to avoid highamounts of reactive power circulation. For high-power wind

turbines (WT), in the range of hundreds of kilowatts to a few

megawatt, low switching and sampling frequency are usually

deployed to limit switching losses, and as a result, LCL filters

offer better performance with reduced reactive power consump-

tion when compared withL filters [7], [8].There are two main issues to be considered in the design

of the current controller for grid-connected converters with an

LCL filter. The first is the grid background voltage distortion

and the second is the resonance of the LCL filter. PI resonant

controllers in stationary frames [9] or PI controllers in rotating

frames [10] have been considered to address the problem of grid

background distortion in the grid-injected currents. Both imple-

mentations are equivalent as shown in [9]. In this paper, the

stationary frame was selected due to its simplicity. The second

issue is the damping of resonance of the LCL filter. There aretwo well-established techniques to damp the resonance ofLCL

filters, which are passive and active damping. Passive damp-

ing [7], [11] consists of introducing additional passive elements

on the filter circuit. Generally, resistors in series with the ca-

pacitors are added. Usually, passive damping results in losses

typically around 1% of the converter nominal power [7]. These

losses may be unacceptable in some applications. For instance,

wind turbines operate typically at 30% of their nameplate ca-

pacity, so the relative damping losses become higher. Alter-

natives of passive damping solutions have also been consid-

ered [11]; however, their performances are highly dependent

on the grid-side impedance. Furthermore, additional passive

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GABEet al.: DESIGN AND IMPLEMENTATION OF A ROBUST CURRENT CONTROLLER FOR VSI CONNECTED 1445

Fig. 1. Three-phase inverter withLCLfilter connected to the grid.

elements are not desirable due to the increase in hardware size

and cost.

On the other hand, active damping can be achieved when

one includes a digital compensator in the current control loop.

Many papers in the literature address the design of active damp-ing controllers. In [12], a filter was added on the reference

voltage for the modulator. Genetic algorithms have been used to

tune filter coefficients in order to optimize resonance damping

for a given system configuration. Dissipative-based controllers

for UPS have been developed in terms of phase variables and

transformed intoabc coordinates [13], [14]. The resulting con-

trollers are similar to a proportional plus resonant controllers

with voltage reference feedforward and current feedback. Al-

though the dissipative-based approach has been demonstrated

to be a powerful design tool, to extend the results to high-power

grid-connected converters, two issues should be addressed. First,

the output filter should be replaced by an LCL filter and thegrid background voltage should be considered as a disturbance.

Second, to ensure the performanceand stability underdiscretiza-

tion, an addition effort should be made since the controller is

designed in the continuous time domain [15]. In [3], the impact

of grid impedance uncertainties on the LCLfilter resonance fre-

quency as well as the stability of the current loop on low- and

high-frequency ranges are addressed in detail. It is suggested

to use a leadlag controller tuned by a startup procedure that

identifies the grid impedance at the PCC. A robust multiloop

control algorithm for grid-connected PWM inverters withLCL

filter is proposed in [16]. In that paper, an inner current capac-

itor loop improved the stability margins while a synchronous

frame PI or a P+ resonant controller was used to provide thedesired steady-state performance. However, additional sensors

were needed for its implementation.

So far, a solution that does not need tuning procedures or

adaptive approaches to ensure robustness for grid uncertainties

at the PCC has not yet been described in the literature.

This paper proposes a robust partial state feedback as active

damping for theLCLfilter resonance. First, a design restriction

on the sampling frequency that guarantees the controllability

of the equivalent discrete-time system that represents the three-

phase grid-connected inverter with LCL filter is derived. Then,

partial state feedback gains are obtained by the solution of a

linear matrix inequality (LMI) [17] based on a theorem that

assures sufficient conditions for a pole location of the system

even under a large grid impedance uncertainty at the PCC. The

proposed controller does not need additional sensors for its im-

plementation since a partial state feedback is used. Finally, an

internal model controller [18] is included in the control loop tomeet the steady-state specifications, which are grid background

voltage disturbance rejection and current reference tracking.

The presented methodology is new to this application. In ad-

dition, its good performance is demonstrated theoretically and

experimentally.

II. CONTROLLABILITY OF THEDISCRETE-TIMESYSTEM

Fig. 1 shows the equivalent circuit of a three-phase three-wire

inverter with an outputLCL filter connected to the grid. The grid

impedance is considered a purely inductive reactance. This is

supported by the fact that WT are often connected to distribution

grids, where the system presents less interaction with residentialurban area loads [3].

The three-phase circuit of Fig. 1 can be transformed into

two single-phase decoupled circuits by the well known abc to transformation. Each of the single phase circuits can berepresented by a linear time-invariant state-space model of the

form

x(t) =Ax(t) +Bu(t) +F (t)

y(t) =C x(t) (1)

where x is the normalized state vector selected as[iL 1/Ibase vCf/Vbase iL 2/Ibase ]

T, u is the normalized in-verter output voltage, w is the normalized grid voltage, andy is the normalized boost inductor current in the referenceframe. A, B, C, and Fare matrices with appropriate dimen-sions given in the Appendix. Note that the impact of the dc bus

voltage (Vcc ) on the loop gain can be eliminated by dividing thecontrol actionu by Vcc . Therefore, the dc-bus voltage does notappear explicitly in the dynamic model. Also, for the stability of

the current loop and the LCL resonance damping analysis, the

grid can be represented by background voltage sources behind

inductances [3]. As a result, it is reasonable to assume that the

voltage sources are exogenous input that are not affected by

the converter currents; therefore, initially it is considered that

(t) = 0.

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TABLE I500-kW WINDTURBINEPARAMETERS

Now, by considering that the PWM voltages are synthesized

in a DSP controller or microcontroller, it is convenient to analyze

and design the system in the discrete-time domain. The repre-

sentation in the discrete-time domain with a sampling periodTsbecomes

x((k+ 1)Ts ) =Gx(kTs ) +Hu(kTs )

y(kTs ) =C x(kTs ) (2)

whereG andHare given by

G= eA Ts H=

Ts0

eA(Ts ) B d.

In addition, (2) can be modified to include the delay present

in the discrete controller implementation, which results in

x((k+ 1)) =Gx(k) +Hu(k)

y(k) =C x(k) (3)

wherex = [x ud ]T andG,H, andCare given by

G=

G H0 0

H= [ 0 0 0 1 ] C= [ C 0 ] .

Note that ud is an additional state variable that has been includedin the state vector to represent the time delay of the digital

implementation.

In order to make the active damping possible, the oscillatory

modes of (1) with a discrete controller and the controllability of

(3), will be investigated. For this purpose, a theorem that relates

the controllability of the discrete system (3) with the parameters

of (1) and the sampling frequency is presented.

Theorem 1: Assume that the dynamic equation given in (1)

is controllable. The necessary and sufficient condition for the

discrete-time dynamic equation given in (3) to be controllable is

that Im[i (A) j (A)]= 2/Ts for= 1, 2, . . ., when-ever Re[i (A) j (A)] = 0.

Proof:Let (1) represent a single-inputsingle-output (SISO)

linear time invariant (LTI) system, where the matrix A is inthe Jordan canonical form with distinct eigenvalues. Ifi is an

eigenvalue ofA, then 1 =e1TS is an eigenvalue ofG. As-

sume that 1 = +jand 2 = j is the pair of complexeigenvalues of A. Whenever = /Ts for = 1, 2 . . .,then 1 = 2 . As a result, G has two Jordan blocks associatedwith the same eigenvalue; hence, the pair {G, H} is not con-trollable [18].

To demonstrate a practical example of Theorem 1, consider

the wind turbine described in [3]. Table I gives the parameters

Fig. 2. State feedback diagram.

of a 500-kW wind turbine, where the grid-side inductance is

assumed as an uncertain parameter belonging to the interval

7.9 HLg 79 H. (4)

The complex eigenvalues of the matrixA are given by

1,2 =j

L1 + LoL1 Lo Cf

(5)

whereLo =Lg +L2 .

If sampling period Ts =1/5000, then whenever the total grid-side inductanceLo reaches 64.6 H the condition Im[i (A) j (A)] = 2/Ts is fulfilled, and accordingly with Theorem 1,the controllability of the discrete equation (3) is lost. Indeed,

by computing the controllability matrix of the discrete system

(3) that is given by[ H H G H G2

H G3

], it is found thatfor this value of total grid-side inductance, the controllability

matrix is not full rank, indicating that the discrete system is not

controllable. Therefore, in order to make it possible to damp

oscillatory modes associated to the LCL filter using a discrete

controller, the following inequality must be satisfied:

fs > 1

L

1+ L

oL1 Lo C

(6)

where fs = 1/Ts . Note that the inequality (6) has been obtainedfrom Theorem 1 and (5) by confining the eigenvalues of (1)

within the primary strip in thes plane, that is,= 1.One way to guarantee that the controllability will not be lost is

to choose a minimum value ofL2 to a given sampling frequencyfs . In this way, even with grid impedance uncertainty, Theorem1 is satisfied within the whole range ofLg . For the previouscase, the minimum value ofL2 must be greater than 64.6 H.

Once this condition is fulfilled, it is possible to actively damp

the oscillatory modes of theLCLfilter with a discrete controller.

In the next section, a robust partial state feedback design for thediscrete system (3) is developed.

III. ROBUSTPARTIALSTATEFEEDBACKDESIGN

The robust partial state feedback aims to assign the discrete-

time system poles to mitigate the resonance of the LCL filter

in the current control loop given in Fig. 2. Another desired

characteristic of the closed-loop system is that all the poles

remain inside the unit circle for any inductance value inside the

interval defined for the operation of the system from weak grid

conditions to stiff grid conditions. One limitation imposed on

the control design is that (iL 2 ) will not be available. Therefore,

a partial state feedback control law is used. This results in a

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reduction in the number of sensors, making the controller design

more challenging.

The LMI conditions presented in [19] and applied to grid-

connected inverters withLCL filters in [20] can assure a robust

pole location for the uncertainty systems under investigation.

Rewriting (1) as a function ofLo , one has that

x(t) =A(Lo )x(t) +Bu(t) +F(Lo )w(t). (7)

The uncertainty in Lg results in variations in almost all entriesof the matricesG and Hof the discrete system, which can beexpressed as

x(k+ 1) =G()x(k) +Hu(k) (8)

where each entry ofG can be included in a convex polyhedralset. As a result,G()belongs to the polytope Pdefined as

P=G() R44 :G() =N

i= 1i Gi ,

Ni=1

i = 1, i 0, i= 1, . . . , N (9)

[21], [22].

Note that the representation of (8) is a version of (3) including

the parametric uncertainty (vector).Let the partial state feedback control law be represented by

u(k) =K x(k), K= [ k11 k12 0 k14 ] . (10)

The design aim is to findk11 ,k12 , andk14 inside the set

S= [ k11 k12 k14 ] R3

with

k11 k11 k11 k12 k12 k12 e k14 k14 k14

that is defined by the control designer.

The vector of gains K has to assure that the closed-loopsystem

x(k+ 1) =Gcl ()x(k), Gcl () =G() +HK, G() P(11)

is stable and that the eigenvalues ofGcl () (close-loop poles)remain inside the circleC, with centerd and radiusr chosena

prioriby the designer, (placed inside the unit circle), shown in

Fig. 3.

The next theorem provides a sufficient LMI condition to solve

the problem stated earlier.

Theorem 2:Given that the values ofr and d define a regioninCto the pole location and given that the gains k11 ,k12 , andk14 of the controller belong to S. If there exists a symmetricpositive definite matrixPR44 such that

rP (Gi+ HK)P dP

P(Gi+ HK) dP rP

> 0,

i= 1, . . . , N (12)

thenGcl ()is stable and the system eigenvalues belong to C.

Fig. 3. Circle with radiusr and centerd for the allocation of the closed-looppoles.

Proof:If Theorem 2 holds, based on Schur complement and

on convexity [21], one has

(G() +HK dI)

r P

(G() +HK dI)

r P

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Fig. 4. Eigenvalue location of the system given in Table I. (a) Eigenvaluelocation without state feedback under stiff and weak grid condition.(b) Eigenvalue location with state feedback under stiff and weak gridconditions.

Fig. 5. Complete block diagram of theclosed-loop system with thestabilizingpartial state feedback controller and the internal model steady-state controller.

asymptotic stability for the closed-loop system but are not suit-

able to cope with tracking of sinusoidal reference signals. Then,

the resonant controller is used to ensure tracking, based on the

internal model principle [9], [23]. This is equivalent to the pro-

portional + resonant structure considered in [3], [9]. The con-

troller structure used to eliminate the grid background voltage

is given by (14), whereiis the compensated harmonic order

Gc (z) =N(z)

(z) =Z

k

s

s2 + (i)2

. (14)

The proposed current control loop is shown in Fig. 5.

The gaink determines the performance of the internal modelcontroller, which means how fast the internal model controller

tracks the reference and/or rejects the disturbance. A single

gaink is considered for all resonant controllers. This simplifies

Fig. 6. Root locus of the closed-loop system assumingGcl (z) = 1.

significantly the design; however, if no satisfactory result is

found, then resonant controllers with different gains give an

additional degree of freedom. The reference current iL 1re f canbe obtained, for example, from the capacitor voltage positive

sequence. For more details see [24] and [25].

V. INTERACTIONBETWEEN THECONTROLLERS

The partial feedback gains belonging to set Swere interac-tively tested in the LMI condition of Theorem 2. Thus, it is

likely that more than one vector K allocates the closed-looppoles inside the circle centered in d and with radius r speci-fied a priori. Assume Q to be the set of all vectors of gainsK= [k11 k12 0 k14 ], with[k11 k12 k14 ] S that sat-isfy the LMI condition (12). The problem now is to select one

of these vectors ofQ that even with the inclusion of the internalmodel controllers does not compromise pole allocation.

In order to gain insight into this problem, the impact of the

closed-loop poles associated with Gc (z) in Gcl (z) and viceversa will be considered. It is also important to consider in this

analysis the closed-loop zeros, that is, the zeros ofGcl (z).Let us assume initially thatGcl (z)does not have dynamics,

that is,Gcl (z) = 1. In this case, it is possible to find gains k ofthe resonant controller such that the closed-loop system stays

stable. Note that the departure angle of the root locus of the

controller complex poles are pointing inside the unit circle, as

shown in Fig. 6. However,Gcl (z)= 1. This means that for thedesign of the internal model controller gain k, as well as to

select the vectorKfrom the setQ, the angular contribution ofGcl (z) in Gc (z) should be considered. It is reasonable to assumethat in the frequency around the internal model controller poles,

the phase of Gcl (z) is decreasing with the frequency. As aresult, ensuring that the departure angles of the root locus of the

complex poles point toward the interior of the unit circle for all

grid conditions, it is possible to guarantee the stability of the

closed-loop poles associated with the internal model controllers

for some gaink.The departure angle of the root locus of the complex

conjugate poles of the internal model controllers is given by

p =+ (15)

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TABLE IISETUPPARAMETERS

wherecan be expressed as

= Gc (z)(z ej (i )Ts )Gcl (z). (16)

To guarantee that the departure angle points to inside the unit

circle,p must belong to the interval given by

(ej (i )Ts ) +

2

< p 0.On the other hand, in Fig. 8(b), the same root locus is drawn for

a weak grid condition. It is possible to see that under the weak

grid condition, the angular margin limits the maximum internal

model controller gain tok = 1.54. This gain is normalized withk= 250, in this way. To avoid instability in all grid conditions,the internal model controller gain k must be less than 250 1.54= 385, ork

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Fig. 9. InductorL 1 current with different L o values. (a) Transient responseon the current ofL1 withLo = 750 H, (within design interval). (b) Current

inL1 toLg = 1000 H(value above the design interval).

TMS320F2812 with the parameters given in Table II. The grid

impedance interval is assumed in a range from Lo = 0 understiff grid conditions, that is, Sk = to Lo =880 Hunder aweak grid condition that results in a short-circuit power at the

PCC ofSk = 15. The filter parameters were chosen to fulfill thecontrollability conditions stated by (6).

The feedback gains obtained under the LMI framework are

K= [ 1.8 0.9 0 0.4 ] . (21)

Gaink of the internal model controllers isk = 250.To demonstrate the system behavior on stability limits, the fil-

ter output is short-circuited, andthe total grid-sideinductance Lois changed. Fig. 9(a) shows the L1 current whenLo =750 H,which is within the assured stability interval. A step in cur-

rent reference from 5 to 25 A demonstrates the system dy-

namic response. On the other hand, a weak grid condition, with

Lo >925 H, which corresponds to Sk =13 is considered inFig. 9(b), which shows a low-frequency oscillation associated

with the seventh harmonic resonant controller, as predicted by

the root locus analysis. Values ofLo lower than the minimum

value of the stability interval can lead to system instability as

Fig. 10. Filter currents in full-power operation. (a)Three-phasecurrentsin theL1 inductors. (b) Spectrum of the current in phase a. (c) Three-phase currentsin theL2 inductor. (d) Spectrum of the current in phase a.

well. However, this case will never be reached, since the mini-

mum value ofLo is ensured by the appropriate selection ofL2 .These results show that the system behaves as predicted by the

theoretical analysis.

Fig. 10 shows the main waveforms of the PWM inverter with

the LCL filter operating at full power connected to the grid.

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Fig. 11. Filter states and grid voltage.

In this case, a filter with lower p.u. values of inductances and

capacitance is considered to demonstrate the generality of the

proposed design procedure, as well as the performance at full

power. In this case, the filter has been designed to meet the

current harmonic requirements of the IEEE 1547 standard. The

current controller parameters have been obtained as describedin Sections II, IV, and V. Fig. 10(a) shows the boost inductors

currents. Note that a higher harmonic content appears as a result

of a lower value of inductance. Fig. 10(b) shows the iL 1 a spec-trum, where the first harmonic group appears at the switching

frequencyfc . Fig. 10(c) shows the iL 2 currents and Fig. 10(d)shows the current spectrum. Note that the filter reduces the out-

put harmonics to meet the standard requirements. These results

demonstrate that the proposed control is feasible using different

filter sizes.

In Fig. 11, it is possible to see that the output current and the

grid voltage are in phase resulting in a power factor close to

unity.

VII. CONCLUSION

This paper has proposed the design of a robust current con-

troller for grid-connected converters with an LCL filter. In ad-

dition, it demonstrates how to obtain the feedback gains for a

given interval of grid inductance that assure the active damping

of theLCLfilter oscillatory modes. Consequently, the controller

does not need self-tuning or adaptive approaches in the case

of grid impedance uncertainty. Moreover, low-frequency con-

trollers are included to assure steady state performance. Further-

more, a procedure for controller design, which avoids instabilityunder grid impedance uncertainty, is described. Two sets of ex-

perimental results are reported. The first one demonstrates that

the experimental stability limits are in very good agreement

with the theoretical predictions. Second, results in full-power

operation are presented to demonstrate the proposed control

performance.

APPENDIX

MATRIXES

The matrices of the continuous- and discrete-state space mod-

els of the three-phase grid-connected inverters with LCL filter

incoordinates are given by

A=

0 1

L1

VbaseIbase

0

1

Cf

IbaseVbase

0 1

Cf

IbaseVbase

0 1

Lo

VbaseIbase

0

B=

1

L1

VbaseIbase

0

0

C= [ 1 0 0 ]

F =

0

0

1

Lo

Vbase

Ibase

.

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[21] S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear MatrixInequalities in System and Control Theory. Philadelphia, PA: SIAM,1994.

[22] P. Gahinet, A. Nemirowskii, A. J. Laub, and M. Chilali, LMI ControlToolbox Users Guide. Natick, MA: The MathWorks Inc., 1995.

[23] F. Botteron and H. Pinheiro, Discrete-time internal model controllerfor three-phase pwm inverters with insulator transformer, in Proc. Inst.

Electr. Eng.Electr. Power Appl., Jan. 2006, vol. 153, no. 1, pp. 5767.[24] R. F. de Camargo and H. Pinheiro, Synchronisation method for three-

phase PWM converters under unbalanced and distorted grid, Proc. Inst.

Electr. Eng.Electr. Power Appl., vol. 153, pp. 763772, Sep. 2006.[25] P. Rodriguez, R. Teodorescu, I. Candela, A. Timbus, M. Liserre, and

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IEEE PESC, Jun. 2006, pp. 17.

IvanJorge Gabe wasbornin Ibiruba,Brazil, in1983.He received the B.S. degree in electrical engineeringin 2006 and the Master degree in 2008 from the Fed-eral University of Santa Maria (UFSM), Santa Maria,Brazil, wherehe is currentlyworkingtowardthe Ph.Ddegree.

His current research interests include wind powergeneration and grid-interconnected converters.

Vincius Foletto Montagnerreceived the Ph.D. andthe Postdoctoral degrees in electrical engineeringfrom the University of Campinas, Campinas, Brazil,in 2005 and 2006, respectively.

Currently, he is a Professor at the Federal Univer-sity of Pampa, Santa Maria, Brazil, and a Researcherwith the Power Electronics and Control ResearchGroup. His current research interests include con-trol theory and applications.

Humberto Pinheiro (M90) was born in SantaMaria, Brazil, in 1960. He received the B.S. degreefrom the Federal University of Santa Maria, SantaMaria, Brazil, in 1983, the M.Eng. degree from theFederal University of Santa Catarina, Florianopolis,Brazil, in 1987,and thePh.D. degree from ConcordiaUniversity, Montreal, QC, Canada, in 1999.

From 1987 to 1990, he was a Research Engineerwith a Brazilian UPS company and then joined the

Pontifcia Universidade Catolica do Rio Grande doSul, Brazil, where he lectured on power electronics.

Since 1991, he has been with Federal University of Santa Maria. His currentresearch interests include uninterruptible power supplies, wind power systems,and control applied to power electronics.

Dr. Pinheiro is currently a member in the IEEE Power Electronics Society.

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