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Abstract—A predictive control scheme for the indirect matrix converter including a method to mitigate the resonance effect of the input filter is presented.
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 59, NO. 1, JANUARY 2012

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Current Control for an Indirect Matrix Converter With Filter Resonance Mitigation Marco Rivera, Student Member, IEEE, Jose Rodriguez, Fellow, IEEE, Bin Wu, Fellow, IEEE, José R. Espinoza, Member, IEEE, and Christian A. Rojas, Student Member, IEEE

Abstract—A predictive control scheme for the indirect matrix converter including a method to mitigate the resonance effect of the input filter is presented. A discrete-time model of the converter, the input filter, and the load is used to predict the behavior of the instantaneous input reactive power and the output currents for each valid switching state. The control scheme selects the state that minimizes the value of a cost function in order to generate input currents with unity power factor and output currents with a low error with respect to a reference. The active damping method is based on a virtual harmonic resistor which damps the filter resonance. This paper shows experimental results to demonstrate that the proposed control method can generate good tracking of the output-current references, achieve unity input displacement power factor, and reduce the input-current distortion caused by the input filter resonance. Index Terms—AC/AC power conversion, current control, digital control, harmonic distortion, matrix converter, predictive control.

N OMENCLATURE is vs ii vi io vo i∗o vdc idc qs Cf Lf Rf RL LL (α, β)

Source current [isA isB isC ]T . Source voltage [vsA vsB vsC ]T . Input current [iA iB iC ]T . Input voltage [vA vB vC ]T . Load current [ia ib ic ]T . Load voltage [van vbn vcn ]T . Output-current reference [i∗a i∗b i∗c ]T . DC-link voltage. DC-link current. Input reactive power. Filter capacitor. Filter inductor. Filter resistor. Load resistance. Load inductance. αβ coordinates.

Manuscript received December 9, 2010; revised April 30, 2011 and July 11, 2011; accepted August 8, 2011. Date of publication August 18, 2011; date of current version October 4, 2011. This work was supported in part by the Centro Científico–Tecnológico de Valparaíso under Grant FB0821 and in part by Universidad Técnica Federico Santa María. M. Rivera, J. Rodriguez, and C. A. Rojas are with the Department of Electronics Engineering, Universidad Técnica Federico Santa María, Valparaiso 2390123, Chile (e-mail: [email protected]; [email protected]; christian.rojas@ usm.cl). B. Wu is with the Department of Electrical and Computer Engineering, Ryerson University, Toronto, ON M5B 2K3, Canada (e-mail: [email protected]). J. R. Espinoza is with the Department of Electrical Engineering, Universidad de Concepción, Concepción 407-0386, Chile (e-mail: [email protected]). Digital Object Identifier 10.1109/TIE.2011.2165311

I. I NTRODUCTION

T

HE indirect matrix converter (IMC) [1], [2] has been subject to investigation for some time. One of the favorable features of an IMC is the absence of a dc-link capacitor, which allows for the construction of compact converters capable of operating at adverse atmospheric conditions such as extreme temperatures and pressures. These features have been explored, and they are the main reasons why the matrix converter family has been investigated for decades [3]. The IMC presents an easy to implement and more secure commutation technique, i.e., the dc-link zero-current commutation [4]. Moreover, the conventional IMC has bidirectional power flow capabilities and can be designed to have small-sized reactive elements in its input filter. These characteristics make the IMC a suitable technology for high-efficiency converters for specific applications, such as military, aerospace, wind turbine generator system, external elevator for building construction, and skin pass mill, as reported in [5]–[7], where these advantages make up for the additional cost of an IMC compared to conventional converters. However, the matrix converter technology presents three main disadvantages: ride-through capability due to the absence of storage elements, gain voltage limited to 0.866, and the inability to control the input currents independently of the output currents, which are imposed by the closed-loop motor control. For all these reasons, the matrix converter is rather restricted to niche applications, as reported in [8] and [9]. IMC uses complex pulsewidth modulation (PWM) and space vector modulation (SVM) schemes to achieve the goal of unity power factor and sinusoidal output current [3], [9]–[15]. Recent improvements in the capabilities of microprocessors have made it possible to use them to control power converters with more accuracy than in the past. As a result, it is now possible to use microprocessors for a predictive control technique to enhance the efficiency of many power electronics applications [16]–[23]. This paper presents the feasibility of using model predictive control as an effective approach for converter modulation and control, as reported in [4] and [24]–[27], where a predictive method has been proposed to handle the output currents in an IMC. The switching states are chosen in order to minimize a cost function that includes the load current error and input displacement factor as performance objectives [4], [24], [25]. An LC filter is required at the input of the IMC to reduce the high-frequency current harmonics caused by the switching operation and the inductive nature of the ac line, as shown in Fig. 1. Since the source inductance usually varies with the

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Fig. 1. General topology of the 3 × 3 IMC.

operating conditions, the variation in the source impedance may move the LC resonant frequency close to a system harmonic, particularly if the converter is used in aerospace applications with a variable frequency source [28]. The filter resonance can result in a line current with substantial distortion due to harmonic pollution from the supply. This issue has been the focus of several publications, where different methods to mitigate the resonance effect in power converters and the associated study of the stability of matrix converters have been reported in [28]–[35]. For instance, in [30]–[34], active damping methods are proposed to mitigate the resonances in three-phase current source converters controlled with classical techniques. In [35], an iterative design algorithm is presented to determine the filter parameters, considering the most significant grid current harmonics, using a PWM modulation technique. Most of these schemes include complex control methods. The basic idea of the active damping approach can be obtained from [31]–[33], where the method is applied to a current source converter. The method consists of emulating a damping resistor placed in parallel with the filter capacitor such that the harmonic currents caused by the resonances flow through this resistor. In doing so, a new modulation index is obtained as a function of the input currents [31]. Similar idea is considered in the control method proposed in this paper, but it does not require an explicit modulation index control because the predictive control does not use this variable [26]. The main contribution of this paper is the improvement of the predictive current control (PCC) operating at a potential resonance frequency in the input filter. To mitigate the potential resonance of the input filter, the predictive controller is modified to include active damping. As will be explained in the following sections, the active damping method does not involve additional measurements or any modification to the predictive algorithm, and thus, it is easy to implement. Experimental results for a three-phase IMC are presented to validate the proposed approach.

II. IMC M ODEL The conventional IMC is shown in Fig. 1. It can be observed that the rectifier side has six bidirectional switches and the inverter side has six insulated-gate bipolar transistors (IGBTs), both connected by a fictitious dc link (without capacitor or

dc/dc converter). The switching function of a single switch for both bridges is defined as  1, Switch Sxj closed , x = r, i; j = 1,. . . ,6. Sxj = 0, Switch Sxj open (1) For the rectifier side, dc-link voltage vdc is obtained as a function of the rectifier switches and the input voltages vi as follows: vdc = [Sr1 − Sr4 Sr3 − Sr6 Sr5 − Sr2 ]vi

(2)

and input currents ii are defined as a function of the rectifier switches and the dc-link current idc as ⎤ ⎡ Sr1 − Sr4 ii = ⎣ Sr3 − Sr6 ⎦ idc . (3) Sr5 − Sr2 For the inverter side, dc-link current idc is determined as a function of the inverter switches and the output currents io as idc = [Si1 Si3 Si5 ]io

(4)

and finally, output voltages are synthesized as a function of the inverter switches and the dc-link voltage vdc as ⎡ ⎤⎡ ⎤ 2 −1 −1 Si1 − Si4 1⎣ −1 2 −1 ⎦ ⎣ Si3 − Si6 ⎦ vdc . (5) vo = 3 −1 −1 2 Si5 − Si2 Equations (2)–(5) correspond to the mathematical model of the IMC for the nine and eight valid switching states for the rectifier and the inverter stages, respectively, such as those reported in [24], [25], [27], and [36], following the restrictions of no short circuits in the input and no open lines in the output. Another operational condition for the IMC is that the dc-link voltage must always be positive vdc > 0. As indicated in (2), the dc-link voltage is synthesized by the rectifier stage switches and the input voltages vi . At any instant, only three of the nine valid switching states that can be applied to the rectifier stage produce a positive dc link. For this reason, at every sampling time, only three of the nine valid switching states are considered [4]. In addition, the rectifier includes an LC filter on the input side, which is needed to prevent overvoltages and to provide

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A. Models in Discrete Time The predicted values of the input side are given as       vi (k) vs (k) vi (k + 1) =Φ +Γ is (k + 1) is (k) ii (k)

(9)

with Φ∼ = eATs

(10)

Γ∼ = A−1 (Φ − I2×2 )B

(11)

where Fig. 2.



PCC scheme.

A=

filtering of the high-frequency components of the input currents produced by the commutations and the inductive nature of the load. The filter consists of a second-order system described by dis 1 Rf = (vs − vi ) − is dt Lf Lf

(6)

dvi 1 = (is − ii ). dt Cf

(7)

The load model is obtained similarly. Assuming an inductive–resistive load as shown in Fig. 1, the following equation describes the behavior of the load: dio 1 RL = vo − io . dt LL LL

(8)

III. C URRENT C ONTROL S CHEME FOR THE IMC The control scheme for the IMC is shown in Fig. 2. The approach pursues the selection of the switching state of the converter that leads the output currents closest to their respective references at the end of the sampling period. In addition, the instantaneous reactive power on the line side of the rectifier must be minimized, and finally, the dc-link voltage must be always positive [24]. First, the control objectives are obtained, and the necessary variables to obtain the prediction model are measured and calculated, respectively. The model of the system and measurements are used to predict the behavior of the variables to be controlled in the subsequent sampling time for each of the valid switching states, and as a final point, the predicted values are used to evaluate a cost function which deals with the control objectives. After that, the valid switching state that produces the minimum value of the cost function is selected for the next sampling period. It is known that most industrial application requires unity power factor in the grid side. For this reason, through the instantaneous reactive power minimization, the system is forced to work with a unity power factor on the input side. In order to compute the differential equations shown in (2)–(8), the general forward-difference Euler formula is used as the derivative discretization used to estimate the variablepredicted value.

0 −1 Lf

1 Cf

Rf



B=

0

−1 Cf

1 Lf

0

(12)

and I2×2 is the 2 × 2 square identity matrix. Similarly, the load current prediction can be obtained as io (k + 1) = d1 vo (k) + d2 io (k)

(13)

where d1 = Ts /LL and d2 = (1 − RL Ts /LL ) are constants dependent on load parameters and the sampling time Ts [21]. Note that the currents is (k + 1) and io (k + 1) depend upon the switch state through (3) and (5), respectively. B. Cost Function With the system discretized model, the predictive algorithm is very straightforward implemented. A quality function is then defined to measure the error between the reference and the predicted load current response. This quality function is then computed every sample period for each commutation state possible on the converter to select the one with the smallest error (minimal value of the cost function), in order to apply this state at the beginning of the next sample period. The quality function can be as simple as d 2 q 2 + (i∗q (14) io = i∗d o − io o − io ) where ido and iqo denote the load current in dq frame for k + 1 ∗q sample time and i∗d o and io denote their respective references. Furthermore, an extra term can be added to this quality function in order to minimize the instantaneous reactive power consumed by the IMC input along with the filter. So, the cost function used to validate the control scheme in this paper is g = io + λq qs .

(15)

In (15), λq is a weighting factor, and qs denotes the error between the reference and the predicted value of the instantaneous reactive power in k + 1 sampling time, expressed as follows: 2 (16) qs = qs∗ − vsd iqs − vsq ids with vsd , vsq , ids , and iqs as the source voltages and currents in d−q coordinates, respectively. The instantaneous reactive

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power reference is established as qs∗ = 0, in order to have a unity power factor on the input side. Noting that g = 0 (for an arbitrary λq ) gives perfect tracking of the load current and unity power factor at the source side; then, by minimizing g, the optimum value for commutation state is guaranteed. In practice, by the appropriate selection of the weighting factor λq , a given total harmonic distortion (THD) of the input and output currents is obtained. The principal method for the selection of the weighting factors has been presented in [23]. C. Required Delay Compensation Several measured and calculated variables are needed, as well as the knowledge of the nine rectifier-side and the eight inverter-side valid switching states, to compute the control scheme algorithm. With these IMC rectifier- and inverter-side valid states, there are 72 (9 × 8) possible switching combinations which must be calculated to select the one resulting in less error in the quality function. At any time, only three of the nine valid rectifier-side switching states provides positive dc-link voltage. If they are determined before the quality function calculation routine, then only 24 (3 × 8) switching combinations must be computed, resulting in saved computation time, but still, a lot of numerical burden is carried out by the microelectronic controller, causing an unwanted delay and producing strong impact in the output predictions. This time delay is well recognized in linear control schemes and may be compensated. As reported in [4], [16], [18], [19], and [27], a simple solution to compensate for this delay is to take into account the calculation time and apply the selected switching state after the next sampling instant. In this way, the algorithm first obtains the measurements of the needed variables and applies the switching state S(k), calculated in the previous interval. With the applied switching state S(k), variables vi (k + 1), is (k + 1), and io (k + 1) are estimated based on (9) and (13). With this, the variables obtained in k + 1 are used as an initial condition for the predictions of qs (k + 2) and io (k + 2) for all 24 possible switching states. The cost function is then evaluated for each prediction, and finally, the switching state that minimizes the cost function is chosen for application in the next sampling instant. For simplicity, the control disturbance vs (k + 1) is considered equal to vs (k) due to its very small change in one sample time [4]. IV. C URRENT C ONTROL S CHEME FOR THE IMC W ITH ACTIVE DAMPING A PPROACH As mentioned before, an input filter is necessary to assist the commutation of switching devices and to mitigate against linecurrent harmonics. However, the filter configuration shown in Fig. 1 presents a resonance frequency, and it can be excited by the utility due to potential harmonics in the ac source and also by the converter itself. To suppress the resonances, different propositions have been reported. For example, it is feasible to choose a proper filter resonant frequency, which may limit performance since the LC resonant frequency is a function of the power system impedance, which usually varies with the power system operating conditions. Also, it is possible to use

Fig. 3.

PCC with active damping scheme.

a high commutation frequency or connect a physical resistor damper with the filter circuit. The first solution results in output currents featuring low THD, but the converter power losses are increased significantly, wasting energy unnecessarily and decreasing the converter efficiency. This cannot be tolerated in static converters where the energy efficiency is an important issue. The second one is the classical solution, where a damping resistor physically connected in parallel to the inductor is used to mitigate a fixed series resonance. There is another strategy with which it is possible to mitigate different resonances, between the series and parallel resonances, as reported in [37], i.e., the active damping control. Active damping is a control technique which achieves the attenuation of system resonance without affecting the efficiency of the converter. The method considers a virtual harmonic resistive damper Rd , which is immune to system parameter variations, in parallel with the input filter capacitor Cf , as shown in Fig. 3, without affecting the fundamental component [26], [30]–[35]. The converter draws a damping current proportional to the capacitor voltage which is extracted by the converter itself, emulating the damping resistance Rd , as indicated by id =

vi . Rd

(17)

As only the harmonics are mitigated and not the fundamental component, the damping current is calculated using the harmonic capacitor voltage vih . To do this, the input voltage vi is considered in dq axes, passing this voltage through a dc-blocker digital filter, deleting the fundamental element and considering only the harmonic components. The converter is required to draw the current that produces the input filter resonance in the matrix converter. The transformation in dq axes was done by the implementation of a synchronous-referenceframe phase-locked loop [38]. Once the voltage harmonics have been obtained, the current damping harmonics are calculated,

RIVERA et al.: CURRENT CONTROL FOR AN IMC WITH FILTER RESONANCE MITIGATION

dq as indicated in (18), where vih corresponds to all harmonic components present in vi

idq dh =

dq vih . Rd

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TABLE I E XPERIMENTAL S ETUP PARAMETERS

(18)

Then, the active damping implementation in the IMC topology is found by passing the harmonic component effect present on the input side to the output side, adding this effect to the load current reference [26]. This is possible because, in the IMC topology, the input current ii is related to the output current by (3) and (4). Thus, the new load current reference can be expressed as  ∗d   ∗d   d  I i io = o∗q + dh (19) i∗q Io iqdh o ∗d ∗q T ∗dq ∗d ∗q T ∗ T where i∗dq is the o = [io io ] , Io = [Io Io ] = [Io 0] required load current, and the damping reference current is q T d given by (18) as idq dh = [idh idh ] . An essential aspect of the active damping control is that it does not require any extra measurements and, furthermore, does not incorporate any modification of the algorithm, where only the output-current reference has been modified, such as that shown in Fig. 3.

V. E XPERIMENTAL R ESULTS A laboratory IMC prototype designed and built by Universidad Federico Santa María, owing to the support of the Power Electronics Systems Laboratory of ETH in Zurich, was used for experimental evaluation. The converter features IGBTs of type IXRH40N120 for the bidirectional switch and standard IGBTs with antiparallel diodes IRG4PC30UD for the inverter stage. The control scheme was implemented in dSPACE 1103, which is connected to additional boards that include the FPGA for the commutation sequence generation and the signal conditioning for the measurement of voltages and currents. Instantaneous reactive power minimization is achieved by considering the value of the weighting factor λq in (15) as λq = 0.003, which has been empirically adjusted, as explained in [23], where, first, it is established in a value that is equal to zero in order to prioritize the control of the output current and, later, it is increased slowly, aiming to obtain unity power factor in the input currents while maintaining a good behavior on the output side. In the IMC, a ratio transfer equal to 0.866 can be achieved, but in this case, a different operation point is considered because the only objective of this experimental validation is to demonstrate the effect of the active damping implementation. A. Experimental Results Without Active Damping The parameters used in the experimental tests are given in Table I, and the sampling time is defined as Ts = 20 μs. Fig. 4(a) shows the measured source current and voltage of phase A, and Fig. 4(b) shows the reference and measured output current of phase a. As expected, the source current fulfills the condition of unitary power factor because it is in phase with

Fig. 4. Experimental results of current control without active damping approach. (a) Source voltage (50 V/div) and current (5 A/div). (b) Output current and reference (5 A/div).

Fig. 5. Experimental results of current control without active damping approach. (a) Spectrum of source voltage [in per unit (p.u.)]. (b) Spectrum of source current (in p.u.). (c) Spectrum of output current (in p.u.).

respect to its voltage, so as a consequence, the instantaneous reactive power is minimized. However, the source current and voltage show a ripple corresponding to the resonance frequency of the input filter and the harmonic distortion of the ac supply, as can be observed in Fig. 5(a) and (b). Fig. 4(a) shows that the voltage source is altered when the system is in resonance. This phenomenon is due to the utilization of a three-phase variac as

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TABLE II E XPERIMENTAL THD R ESULTS W ITHOUT ACTIVE DAMPING

Fig. 7. Experimental results of current control without active damping approach. (a) Spectrum of source voltage (in p.u.). (b) Spectrum of source current (in p.u.). (c) Spectrum of output current (in p.u.).

Fig. 6. Experimental results of current control with active damping approach. (a) Source voltage (50 V/div) and current (5 A/div). (b) Output current and reference (5 A/div).

the ac source, which behaves like a weak ac supply for the system, due to the inductance associated with the autotransformer connection. On the other hand, a very good tracking of the load current to its reference is observed in Fig. 4(b). The THD of source voltage and current and the output current are indicated in Table II. B. Experimental Results With Active Damping In comparison to Fig. 4(a), the improvement in the quality of the source current and voltage is noticeable due to an important reduction of distortion by filter resonance mitigation, as shown in Fig. 6(a). Also, an almost sinusoidal source current is obtained; a distortion harmonic due to the unclean ac supply is still observed, which cannot be mitigated by the active damping method. Similar to the aforementioned case, the output current follows its reference accurately in spite of the distortion added to the reference by the active damping method. In Fig. 4(a), a distorted input current with a THD of 29.24% is observed, but when the resonance mitigation is applied using active damping, 20.84% of THD is obtained. This is a relatively small value considering the polluted source with 13.05% of THD under normal operation. The output-current THD is 7.79% without active damping action, and it is 6.53% with active damping operation. It is expected that, with a clean ac source, the input- and output-current THDs can be decreased. By including the active damping approach, the input- and output-current distortions are attenuated considerably. In the IMC topology, it is possible to observe the series resonance, which is produced by the ac supply and the parallel resonance, which is generated by the converter itself and the control method. According to the filter parameters and as observed in

Fig. 8. Experimental results of current control without/with active damping approach. (a) and (d) Spectra of source voltage (in p.u.). (b) and (e) Spectra of source current (in p.u.). (c) and (f) Spectra of output current (in p.u.).

Fig. 5, the resonance frequency is located around 650 Hz, which is excited by the converter and the distorted ac supply. From Figs. 7 and 8, it is verified that the filter resonance is mitigated by considering the active damping method. This is reflected in a more sinusoidal ac source. Because the 3rd, 5th, and 7th harmonics are from the ac source itself, the active damping method is not able to mitigate these harmonics because the ac source and network models are not included in the control strategy, but the operation of the converter in an industrial application must be independent of the grid parameters, to which it is connected. For this reason, an active damping technique is incorporated in the control, where the only parameter to be adjusted is the virtual resistor regardless of the grid model. As mentioned before, the resonance and distortion effect is reflected in the load current, as can be observed in Fig. 8(f). As shown in Figs. 9 and 10, a different output frequency is considered, which is established in 100 Hz. As it can be

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TABLE III E XPERIMENTAL THD R ESULTS W ITH ACTIVE DAMPING

Fig. 9. Experimental results of current control without active damping approach. (a) Source voltage (50 V/div) and current (5 A/div). (b) Output current and reference (5 A/div) with output frequency reference equal to 100 Hz.

in the performance. This approach reduces power losses as compared to the use of real resistive damping. The control scheme uses the predicted values of the input and output currents to evaluate the best suited converter state considering the output-current error and the input power factor. Our experimental results indicate that the presented strategy allows good tracking of the output current to its reference and minimizes the instantaneous reactive power on the input side at the same time. Active damping improves the quality of the input currents even in the presence of a weakly damped input filter. The ac supply has an important influence in the behavior of the source current, and better results can be expected by optimizing the input filter and also with a clean ac supply. In order to fully evaluate the potential of the control method presented in this paper, a complete assessment must be done in the future with respect to SVM in terms of switching losses, distortion, algorithm complexity, and others. A PPENDIX The parameters of the experimental setup are indicated in Table I, and the THD information is detailed in Tables II and III. R EFERENCES

Fig. 10. Experimental results of current control with active damping approach. (a) Source voltage (50 V/div) and current (5 A/div). (b) Output current and reference (5 A/div) with output frequency reference equal to 100 Hz.

observed, the active damping approach mitigates the resonance of the input filter. However, similar to the previous case and due to a weak ac supply in our laboratory, the source currents are distorted as well. VI. C ONCLUSION Due to technology advances in power semiconductors and processors, matrix converters with predictive control schemes have recently emerged as feasible options for future power converter applications. The use of predictive control in IMCs and power electronics is very new and introduces an important simplification because it is very easy to understand and implement. Our research takes advantage of these advances and proposes a predictive control scheme for a conventional IMC where an active damping method is added to mitigate against the input filter resonance. The resonance of the input filter is still a major concern that directly affects the selection of the design parameters and the modulation method. The reduction of resonances in the input filter, as observed in this paper, is an important improvement

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Marco Rivera (S’10) received the B.Sc. degree in electronics engineering and the M.Sc. degree in electrical engineering from the Universidad de Concepción, Concepción, Chile, in 2007 and 2008, respectively, and the Ph.D. degree from the Department of Electronics Engineering, Universidad Técnica Federico Santa María, Valparaíso, Chile, in 2011. During January and February of 2010, he was a visiting Ph.D. student in the Electrical and Computer Engineering Department, Ryerson University, Toronto, ON, Canada, where he worked on predictive control applied on four-leg inverters. He was also a visiting Ph.D. student in the Departamento de Ingeniería Eléctrica y Computacional, Instituto Tecnológico y de Estudios Superiores de Monterrey, Monterrey, Mexico, where he worked on experimental aspects of a doubly fed induction generator and indirect matrix converter system. He is currently working in a Postdoctoral position with Universidad Técnica Federico Santa María. His research interests include direct and indirect matrix converters, predictive and digital controls for high-power drives, four-leg converters, and development of high-performance control platforms based on field-programmable gate arrays. Dr. Rivera was a recipient of a scholarship from the Marie Curie Host Fellowships for Early Stage Research Training in Electrical Energy Conversion and Conditioning Technology at University College Cork, Cork, Ireland.

Jose Rodriguez (M’81–SM’94–F’10) received the Engineer degree in electrical engineering from Universidad Técnica Federico Santa María (UTFSM), Valparaíso, Chile, in 1977 and the Dr.-Ing. degree in electrical engineering from the University of Erlangen, Erlangen, Germany, in 1985. Since 1977, he has been with the Department of Electronics Engineering, UTFSM, where he is currently a Full Professor and a Rector. He has coauthored more than 300 journal and conference papers. His main research interests include multilevel inverters, new converter topologies, control of power converters, and adjustable-speed drives. Dr. Rodriguez is a member of the Chilean Academy of Engineering. He has been an Associate Editor of the IEEE T RANSACTIONS ON P OWER E LEC TRONICS and the IEEE T RANSACTIONS ON I NDUSTRIAL E LECTRONICS since 2002. He was a recipient of the Best Paper Award from the IEEE T RANS ACTIONS ON I NDUSTRIAL E LECTRONICS in 2007 and the IEEE Industrial Electronics Magazine in 2008.

RIVERA et al.: CURRENT CONTROL FOR AN IMC WITH FILTER RESONANCE MITIGATION

Bin Wu (S’89–M’92–SM’99–F’08) received the Ph.D. degree in electrical and computer engineering from the University of Toronto, Toronto, ON, Canada, in 1993. After being with Rockwell Automation Canada as a Senior Engineer, he joined Ryerson University, Toronto, where he is currently a Professor and the NSERC/Rockwell Industrial Research Chair in Power Electronics and Electric Drives. He has published more than 200 technical papers, authored two Wiley–IEEE Press books, and is the holder of more than 20 awarded/pending patents in the area of power conversion, advanced controls, adjustable-speed drives, and renewable energy systems. Dr. Wu is a Fellow of the Engineering Institute of Canada and the Canadian Academy of Engineering. He is currently an Associate Editor for the IEEE T RANSACTIONS ON P OWER E LECTRONICS and IEEE C ANADIAN R EVIEW. He was the recipient of the Gold Medal of the Governor General of Canada, the Premier’s Research Excellence Award, Ryerson Distinguished Scholar Award, Ryerson Research Chair Award, and the NSERC Synergy Award for Innovation.

José R. Espinoza (S’92–M’97) received the Engineer degree in electronic engineering and the M.Sc. degree in electrical engineering from the Universidad de Concepción, Concepción, Chile, in 1989 and 1992, respectively, and the Ph.D. degree in electrical engineering from Concordia University, Montreal, QC, Canada, in 1997. Since 2006, he has been a Professor with the Department of Electrical Engineering, Universidad de Concepción, where he is engaged in teaching and research in the areas of automatic control and power electronics. He has authored and coauthored more than 100 refereed journal and conference papers. Dr. Espinoza is currently an Associate Editor of IEEE T RANSACTIONS ON I NDUSTRIAL E LECTRONICS and IEEE T RANSACTIONS ON P OWER E LECTRONICS.

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Christian A. Rojas (S’10) received the Engineer degree in electronic engineering from the Universidad de Concepción, Concepción, Chile, in 2009. He is currently working toward the Ph.D. degree in power electronics at Universidad Técnica Federico Santa María, Valparaiso, Chile, with a scholarship from the Chilean research foundation Comisión Nacional de Investigación Científica y Tecnológica, awarded in 2010. His research interests include matrix converters, digital control, and model predictive control of power converters and drives.