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The Application of Resonant Controllers to Four-Leg Matrix Converters Feeding Unbalanced or Nonlinear Loads Roberto C´ardenas, Senior Member, IEEE, Carlos Juri, Student Member, IEEE, Rub´en Pe˜na, Member, IEEE, Patrick Wheeler, Member, IEEE, and Jon Clare, Senior Member, IEEE

Abstract—Matrix converters (MC) have some advantages when compared to conventional back-to-back pulsewidth modulation voltage-source converters. The MC may be considered more reliable and is smaller because the bulky dc capacitor is eliminated from the topology. Therefore, when MCs are used in ac–ac power conversion, the size and weight of the whole generation system is reduced. To interface a MC-based generation system to an unbalanced three-phase stand-alone load, a four-leg MC is required to provide an electrical path for the zero-sequence load current. Moreover, to compensate for the voltage drops in the output filter inductances, nonlinearities introduced by the four-step commutation method and voltage drops in the semiconductor devices, closed-loop regulation of the load voltage is required. In this paper, the design and implementation of a resonant control system for four-leg MCs is presented. The application of this control methodology when the four-leg MC is feeding, a linear/nonlinear unbalanced load is also presented in this study. High-order resonant controllers are also analyzed. Experimental results, obtained from a small prototype, are discussed. Index Terms—AC–AC power conversion, power generation. Fig. 1.

Control system for a variable-speed diesel generation system.

I. INTRODUCTION ATRIX converters (MCs) have many advantages, which are well documented in the literature [1]–[6]. The MC provides bidirectional power flow, sinusoidal I/O currents and controllable input displacement factor [1]. When compared to back-to-back converters, the MC also has additional advantages. For instance, due to the absence of electrolytic capacitors, the MC can be more robust and reliable [4], [5]. The space saved by a MC, compared to a conventional back-to-back converter, has been estimated as a factor of three [2], [3]. Therefore, it becomes feasible to integrate the power converter into the electrical gen-

M

Manuscript received October 26, 2010; revised January 15, 2011; accepted February 20, 2011. Date of current version February 7, 2012. This work was supported by the Fondecyt Chile under Contract 1110984. Recommended for publication by Associate Editor S. Williamson. R. C´ardenas and C. Juri are with the Electrical Engineering Department, University of Chile, Santiago 8370451, Chile (e-mail: [email protected]; carlos. [email protected]). R. Pe˜na is with the Electrical Engineering Department, University of Concepci´on, Concepci´on 4074580, Chile (e-mail: [email protected]). P. Wheeler and J. Clare are with the Department of Electrical and Electronic Engineering, University of Nottingham, Nottingham, NG7 2RD, U.K. (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPEL.2011.2128889

erator, for example [2]. A comparison between back-to-back power converters and MCs is presented in [6]. The advantages of variable-speed generation are also known [7]–[10]. For instance, a relatively new topology, suitable for wind–diesel systems or variable-speed diesel generation to a stand-alone load, has been reported [8], [9]. In this case, a standard diesel generation system is operated (see Fig. 1), following an optimal power–speed characteristic. For this topology, the efficiency is improved because diesel engines have high-fuel consumption when operated at light loads and relatively high speed [8], [10]. Moreover, a diesel engine operating at highrotational speeds can increase its power output well beyond that obtained at the synchronous velocity [8]. Therefore, a mobile generation system can be implemented using a relatively small variable-speed engine [8]. This allows for considerable weight and size reduction. Moreover, if a four-leg MC is used to feed the stand-alone unbalanced load, a further reduction in the size of the generation system is accomplished, because the bulky dc-link capacitors are eliminated from the topology [1]. Control systems and modulation algorithms for four-leg MC have been recently reported in the literature [11]–[16]. Space vector modulation (SVM) algorithms are presented by the authors in [12]–[14] and a standard d—q voltage control system has been reported in [12]. However, the control loop proposed in [12] supplies balanced voltages at the MC output, which is

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experimental results are presented. Finally, an appraisal of the proposed control system is presented in Section IV. The SVM algorithm used in this study is similar to that discussed for conventional voltage-source pulsewidth modulation (PWM) inverters [24]–[27]. SVM algorithms proposed for fourleg MCs have been presented in [12]–[14], and [16] and the interested reader is referred to these publications for further information. II. RESONANT CONTROLLERS FOR FOUR-LEG MC

Fig. 2.

MC and I/O filters.

appropriate only for regulating the voltage of slightly unbalanced loads. When a standard d—q control system is applied, the voltage drop in the output inductance has to be relatively small in order to give good performance. In [15], a resonant control system was presented by Cardenas et al.. However, in that publication, high-order, high-complexity resonant control systems were not addressed, control of a nonlinear load was not discussed and most of the results presented in [15] were obtained from simulations. Resonant controllers have already been applied to several areas, for instance to distributed generation [15]–[17]; wind energy and photovoltaic applications [18], [19]; harmonic cancellation [20]; fuel cells [21], etc. In this paper, the design, implementation, and experimental verification of a control system based on resonant controllers, for four-leg MCs, is presented. As discussed in this paper, resonant controllers can be designed to significantly reduce the harmonic distortion of strongly nonlinear loads, to control highly unbalanced loads and to obtain variable frequency at the MC output. Fig. 2 shows the four-leg MC and filters used in the simulation and experimental work discussed in this paper. A second-order L-C filter is used at the MC input to improve the quality of the input currents [1]. The input filter capacitors also provide the essential decoupling to minimize the commutation inductance between phases. Usually, a resistor in parallel with the filter inductance improves the damping of the system [22], [23]. At the MC output a second-order L-C filter is provided to reduce the effects of the switching harmonic in the load voltages. Fig. 2 also shows the star-connected stand-alone load fed from the four-leg MC. The rest of this paper is organized as follows. In Section II, the proposed resonant control system is analyzed. In Section III,

Unbalanced three-phase loads may draw currents with positive-, negative-, and zero-sequence components [28]. To regulate the line to neutral voltage in each phase, conventional d–q control can be used. However, as mentioned earlier, standard d–q controllers synthesize balanced MC output voltages, which are not appropriate to regulate the voltages of heavily unbalanced or strongly nonlinear loads. Vector control systems based on two revolving axis systems, rotating clockwise, and counter clockwise can be used to both regulate the positive-sequence voltage and to regulate (normally to zero) the negative-sequence voltage at the load [28]. However, this control methodology may have difficulty in eliminating the zero-sequence components, unless a zero-sequence voltage controller is added. However, it may be troublesome to apply this rather complex approach when the elimination of harmonic distortion created by non-linear loads is desired, since several d–q-axes systems rotating at different speeds may be required. In this paper, the application of resonant controllers, for the generation topology shown in Figs. 1 and 2 is investigated. A resonant controller can be used to regulate the output voltage of each phase with respect to the neutral connection “n.” The controller has a couple of purely imaginary poles (in the s-plane) with a resonant frequency of ω o , where ω o is the desired output frequency. In the s-plane, a typical resonant controller has the transfer function [15]–[17] Gc (s) = Kc

s2 + 2ζωn s + ωn2 s2 + ωo2

(1)

where Kc is the controller gain. In the numerator of (1), zeros located close to the resonant poles are used to improve the dynamic response. Considering Fig. 2 the transfer function, between the MC output and the line to neutral voltage of phase a, is as follows: RL a VL a (s) = 2 Voa (s) s RL a Cf Lf + sLf + RL a

(2)

where VL a is the load voltage, Voa is the MC output voltage, RL a is the load resistance, and Cf and Lf are the capacitance and inductance of the second-order output filter, respectively. A resistive load has been assumed in (2); however, the control system presented in this study can be used with both leading and lagging power factor loads. A transfer function similar to (2) can be defined for phases “b” and “c.” In the z-plane, the transfer function of the resonant

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Fig. 3.

IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 27, NO. 3, MARCH 2012

Poles and zeros of the resonant controller and output filter.

converter may be obtained as follows [29]: Gc (z) = Kcz

z 2 + a1 z + a2 . z 2 + b1 z + b2

(3)

In this case, the resonant poles are located along the unit circle with an angle with respect to the real axis of ωo · Ts rads, where Ts is the sampling time and ω o is the output frequency. From the dynamic point of view, one of the worst case situations is when no load is connected at the MC output (i.e., RL →∞) and the damping coefficient of the output stage transfer function is zero (assuming lossless filter components for worst case). This is shown in Fig. 3. Notice that the output filter poles are located along the unit circle for the no-load case. A. Control System Design The SVM algorithm used in this study is able to synthesize a set of independent voltages (van , vbn , and vcn ) obtained at the output of the resonant controllers. Therefore, if it is required, three resonant controllers designed for different specifications can be used to regulated the load voltages. The control system proposed in this paper is shown in Fig. 4(a). Using the prism and tetrahedron selection algorithms discussed in [16], three active vectors are selected. The duty cycles corresponding to these three active vectors plus the duty cycles corresponding to three zero vectors are calculated using the procedure discussed in [12], [14], and [16]. Finally, the voltage is synthesized using a double-sided switching pattern modulation [16]. A simplified single-phase block diagram of Fig. 4(a) is shown in Fig. 4(b). For design purposes, the SVM algorithm may be represented by a sample time delay and a zeroorder hold device. In Fig. 4, the label “RC” stands for “resonant controller.” The resonant controller could be designed using conventional root locus in the z-plane (see Fig. 5). As mentioned earlier, the controllers should have an appropriate dynamic response in the worst case situation, i.e., with no load connected to the output. In this case, to improve the dynamic performance of the control system, a second-order lead-lag network is required. For instance, if Ts ≈ 100 μs, ω o = 50 Hz, Cf = 40 μF, and Lf = 4 mH, a suitable controller is as follows: Cont(z) = 4.9

(z 2 − 1.926z + 0.9276) (z 2 − 1.336z + 0.45) . (z 2 − 1.99937z + 1) (z 2 − 0.024z + 0.042) (4)

Fig. 4. Proposed control system. (a) Resonant controllers, SVM algorithm, and plants. (b) Simplified single-phase diagram. The SVM algorithm is replaced by a plant delay and a zero-order hold device.

The first term after the gain is the resonant controller and the last term is the lead-lag network. In Fig. 5, the resonant controller is shown inside the dashed box at the right side. The controller of (4) has been designed considering a damping coefficient of at least 0.3 for the closed-loop poles shown in Fig. 5. Beside of the root locus shown in Fig. 5, tuning of the controllers is improved using a numerical model of the system, where the performance of a given resonant controller is evaluated through simulation before being experimentally tested. B. Design of the Resonant Controller to Eliminate Harmonic Distortion A resonant control system can be also used to eliminate harmonic distortion from the load. In a parallel implementation, the controller has the following transfer function: Gc (s) = Kc1

2 s2 + 2ζωn s + ωn2 s2 + 2ζωna s + ωna + Kc2 2 2 2 2 s + ωo s + ωoa

+ Kc3

2 s2 + 2ζωnb s + ωnb 2 2 s + ωob

+ · · · + Kcm

2 s2 + 2ζωnm s + ωnm 2 2 s + ωom

(5)

where the poles of each transfer function are tuned to track a particular frequency. The block diagram of the controller, for a single phase, is shown in Fig. 6. In this graphic, the demand ∗ , a sinusoidal signal signal for the fundamental frequency is van of the desired frequency, amplitude, and phase. To eliminate the unwanted harmonics, the additional reference signals are set to

´ CARDENAS et al.: APPLICATION OF RESONANT CONTROLLERS TO FOUR-LEG MC FEEDING UNBALANCED OR NONLINEAR LOADS

Fig. 5.

Root locus of the proposed control system.

Fig. 6.

Proposed control system.

zero. In order to obtain a good dynamic performance, lead-lag networks also have to be included in the transfer function of (5). For simplicity, the control system depicted in Fig. 6 corresponds to a parallel representation of the controller. In the practical rig, the control system is implemented using a state-space matrix representation, which simplifies the control code. The root-locus design of a resonant control system, designed to track a 50 Hz reference signal and to eliminate the components of 0, 100, 150, 200, and 250 Hz, is shown in Fig. 7. Considering Ts ≈ 200 μs, ω o = 50 Hz, the RC of Fig. 7 has the following transfer function:   2  z − 0.9872 z − 1.942z + 0.9462 Gc (z) = 2 · · z−1 z 2 − 1.996z + 1  2  2  z − 1.954z − 0.9681 z − 1.86z + 0.8832 · · z 2 − 1.984z + 1 z 2 − 1.965z + 1  2  2  z − 1.938z + 0.9743 z − 1.906z + 0.9741 · · z 2 − 1.937z + 1 z 2 − 1.902z + 1  2  z − 1.863z + 0.9508 · . (6) z 2 − 1.295z + 0.6889

Fig. 7.

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Root locus for a high-order resonant controller.

In Fig. 7, the high-order resonant controller is shown inside the dashed box at the right side. In Fig. 8, the gain plots of the resonant controllers [corresponding to (4) and (6)] are shown. These frequency responses

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Fig. 10.

Fig. 8. Bode plots showing frequency versus gain for two topologies of resonant controller. (a) ode plot of the resonant controller of (4) including a lead-lag network. (b) The bode plot of a resonant controller with the topology of (6). The lead-lag terms are also included.

Fig. 9.

Experimental system used in this study.

are calculated using the same numbers and resolution as the final DSP implementation. For the controller of (4), it is relatively simple to achieve a high gain in the frequency response of the controller [see Fig. 8(a)]. For the controller topology of (6), the numerical resolution of the DSP implementation [29], [30] as well as the interaction between the poles and zeros located at different frequencies, produce gains, which are smaller [see Fig. 8(b)]. However, according to the experimental results shown in the next section, the controller gains obtained from (8) are demonstrated to be adequate for the control of unbalanced and nonlinear loads. III. EXPERIMENTAL RESULTS The control methodology discussed in this study, has been validated using the 2.5 kW experimental system shown in Fig. 9. The SVM algorithm and proposed resonant control systems are implemented using a DSP-based control board and an fieldprogrammable gate array, the latter implementing the four-step commutation method [1] and the switching signals for the insu-

Step response of the control system.

lated gate bipolar transistor gate drivers. The DSP board used in this application is a high-performance TI TMS320C6713, capable of a peak performance of 1350MFLOPS. For data acquisition purposes, an external board with ten ADC channels of 14 bits, 1 μs conversion time each, is interfaced to the DSP. This board also has four digital to analog (D/A) channels available. Hall-effect transducers are used to measure the input currents, input voltages, and output currents. Antialiasing filters are applied to the signals before being sampled by the ADCs. For the experimental tests involving frequency and voltage variations at the MC input, a 5.5 kW, 2000 r/min permanent magnet generator (PMG) is connected to the four-leg MC input. A commercial inverter and a speed-controlled cage machine are used to drive the PMG. A high-resolution position encoder of 10,000 pulses per revolution is used to measure the generator speed. At the MC output, a star-connected three-phase load and a second-order power filter are connected. This output filter is used to reduce the harmonic content in the voltages and currents. A nonlinear load is implemented using a rectifier diode connected in series with the load of a particular phase. Unless otherwise stated, the sampling time used by the SVM and control algorithm is 100 μs. Further information about the parameters of the experimental rig, are presented in the Appendix. In Fig. 10, the response of the control system to a balanced step change in the output voltage demand is shown. At t ≈ 0.4 s, the reference voltages are stepped from 7 to 100 V peak. At t ≈ 1.6 s, the reference voltages are stepped from 100 to 30 V. For both steps, the four-leg MC feeds a resistive balanced load (RL ≈ 7 Ω). For simplicity, the magnitude of the voltage vector is shown in Fig. 10. This value is calculated from |Vout | =



Vα2 + Vβ2 .

(7)

In Fig. 11, the instantaneous voltages corresponding to the step transients in Fig. 10 are shown. Fig. 11(a) shows the step variation from 7 to 100 V (peak). The settling time for the step change is about 30 ms (1.5 cycles), which is considered adequate for this application. Fig. 11(b) shows the step variation from 100 to ≈30 V, again the performance of the control system is relatively fast with a settling time of about 30 ms. Fig. 12 shows the performance of the control system considering load step variations in two of the phases. At t ≈ 0.16 s, the loads in phases “a” and “b” are disconnected using a solid-state relay fired from a D/A output. After the load step, only the second-order power filter remains connected between phases a, b, and the neutral connection. The performance of the

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Fig. 14. Performance of the system for a load step variation in the three phases. (a) MC output currents. (b) Load voltages. Fig. 11. Instantaneous voltages corresponding to the step transient. (a) Step response from 7 to 100 V. (b) Step response from 100 to 30 V. All the phases are shown.

Fig. 12. Output currents for disconnection and reconnection of the load in two phases. (a) Phase a current. (b) Phase b current. (c) Phase c current.

Fig. 15. Performance of the control system for an inductive load step in one of the phases. (a) Phase to neutral voltage. (b) MC output current. (c) Phase angle between the load voltage and current. (d) MC output voltage.

Fig. 13. Neutral current and load voltages corresponding to the test of Fig. 12. (a) Neutral current. (b) Phase to neutral load voltages.

system is good considering that the load damping coefficient is almost zero in two of the phases. Fig. 13 shows the neutral current and the load voltages corresponding to the test shown in Fig. 12. Because of the unbalanced load, the neutral current is relatively high. Fig. 13(b) shows the load voltages. As shown inside the dashed box, the variation in the phase voltages is controlled in less than 20 ms. Fig. 14 shows the performance of the control system when the load is disconnected, and then reconnected in all the phases.

This is the most drastic change from the control viewpoint, because the transfer function between the MC output voltage and the load voltage has a damping coefficient of ζ ≈ 0 for all the phases during the no-load condition. However, the performance of the control system is very good with a relatively small overshoot (disconnection) and dip (reconnection) in the load voltage [see Fig. 14(b)]. Fig. 15 shows the performance of the control system when a capacitive load step is applied to phase a. In this graphic, the magnitudes of the voltages and currents are calculated using the α–β components. Fig. 15(a) shows the phase to neutral voltage, which is little affected by the load step. Fig. 15(b) shows the phase current and Fig. 15(c) shows the phase angle between the load voltage and MC output current. Finally, Fig. 15(d) shows the MC output voltage for phase a. Because of the capacitive nature of the load, the voltage shown in Fig. 15(d) is decreased between t ≈ 1.1 and 2.4 s. As shown in Fig 15, the resonant

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Fig. 16. Performance of the control system for variable-speed operation of the phase modulation (PM)generator. (a) Rotational speed. (b) Input voltage. (c) MC input current.

Fig. 18. phase.

Instantaneous phase to neutral output voltage corresponding to one

Fig. 19.

Load current considering a rectifier diode in one phase.

overshoot produced when the load step is applied. Fig. 17(b) shows the MC output current, which is also well regulated. The voltage transfer ratio is defined using the input and output voltage magnitude [14], [23] as follows: q= Fig. 17. Performance of the control system for variable-speed operation of the PM generator. (a) Load output voltage. (b) Load current. (c) Voltage transfer ratio.

control system proposed in this study can be used to regulate the voltage of unbalanced loads with non-unity power factor. A. Control System Performance for Variable Frequency/Variable Voltage at the MC Input The control system has been tested feeding the MC input with a variable-speed PMG operating between 1060 to ≈1940 r/min, corresponding to a relatively large acceleration of ≈235 r/min/s. For this speed range, the approximate frequency range is 70 Hz to 130 Hz. In t ≈ 2.5 to 5 s, a step variation is applied changing the three-phase resistive load from R = 7.4 to 5.4 Ω per phase. Fig. 16 shows the magnitude of the voltage and current vectors at the MC input side. Fig. 16(a) shows the speed of the PMG. The speed variation is obtained by controlling the commercial inverter shown in Fig. 9. Fig. 16(b) and (c) show the magnitude of the input voltages and currents calculated from the α—β components [see (7)]. When the voltage increases, the current decreases in order to maintain the power demanded by the output load approximately constant. In Fig. 17(a), the load voltage corresponding to the test of Fig. 16 is shown. The regulation is good with a low dip and

|Vout | . |Vin |

(8)

Fig. 17(c) shows the voltage transfer ratio q, which reduces when the input voltages increases. The instantaneous voltage waveform corresponding to phase a (for the experimental test of Figs. 16 and 17) is shown in Fig. 18. As shown in this graphic, the waveform has little distortion. B. Control of a Nonlinear Load To experimentally test the control topology proposed to eliminate harmonic distortion, a nonlinear load is implemented by connecting a rectifier diode in series with the resistive load of one of the phases (see Fig. 2). In Fig. 19, the open-loop phase current of the nonlinear load is shown. Notice that the current is highly distorted because of the half-wave rectification. Two controllers have been implemented and tested. The first one has a topology similar to that shown in Fig. 8(a) with a single-gain peak located at 50 Hz. The second controller was designed for multiple-gain peaks at dc 50, 100, 150, 200, and 250 Hz [see (6)]. This control system has been designed to regulate the 50 Hz component of the load voltage and to eliminate the zero, second, third, fourth, and fifth harmonics. Because of processing time constraints, the sampling period is increased to 200 μs for this case. Fig. 20(a) shows the voltage waveform corresponding to open-loop operation. In this case, the total harmonic distortion [14] (THD) is about 15%. As shown in Fig. 21(a), the spectrum has large components at dc and the second, fourth, and fifth

´ CARDENAS et al.: APPLICATION OF RESONANT CONTROLLERS TO FOUR-LEG MC FEEDING UNBALANCED OR NONLINEAR LOADS

Fig. 20. Load voltage for nonlinear operation. (a) Open-loop voltage. (b) Load voltage for a resonant controller with a single-gain peak. (c) Load voltage for a resonant controller with multiple-gain peaks.

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been discussed. As shown by the experimental results, singlegain resonant controllers are appropriate to control the voltage of highly unbalanced linear loads, eliminating the negative- and zero-sequence voltages. Multi-gain resonant controllers have also been tested in this study. The performance of these high-order controllers is able to eliminate the most important harmonics in the load voltage. However, according to the experimental results, for the implementation of these controllers, the required processing time is likely to be increased. Therefore, a tradeoff between switching frequency and harmonic distortion has to be considered when a high-order controller is required. The control systems proposed in this study have been experimentally tested considering voltage steps, frequency steps, nonlinear loads, balanced, and unbalanced load steps. For most of the experimental tests presented in this study, the settling time is below 30 ms . For the nonlinear load case, a low THD of about 3% has been achieved using a rather complex high-order resonant control system. The performance of the proposed control system has also been tested considering a fast variation of about 235 r/min/s in the speed of the PMG feeding the MC input. The results demonstrated that the proposed control system has a good performance with variable frequency/variable voltage at the MC input. Therefore, in all the cases, the performance obtained has confirmed the suitability of the proposed control methodology, when applied to generation systems based on four-leg MCs. APPENDIX PARAMETERS OF THE EXPERIMENTAL RIG

Fig. 21. Spectra corresponding to the tests of Fig. 20. (a) Voltage spectrum for open-loop operation. (b) Voltage spectrum for the waveform of Fig. 20 (b). (c) Voltage spectrum for the waveform of Fig. 20(c).

harmonics. The dc component is about 9% of the fundamental value. To increase the resolution, the value of the fundamental component has been clipped at 15% in Fig. 21. Fig. 20(b) shows the performance of the single-gain resonant controller [see (4)]. In this case, the waveform is less distorted and the measured THD is about 10%. As shown in Fig. 21(b), the magnitude of the dc component and the second, fourth, and fifth harmonic components have been reduced. Fig. 21(c) shows the performance of the multigain resonant controller of (6). In this case, most of the targeted harmonic components have been eliminated [see Fig. 21(c)], with only some harmonics barely noticeable. The THD for this test has been reduced to about 3%, which is about a 7% reduction when compared to that obtained with a single-gain resonant controller and a 12% reduction when compared to that obtained with openloop operation. IV. CONCLUSION In this paper, resonant control systems for the regulation of unbalanced or nonlinear loads, fed by four-leg MCs, have been presented. Single-gain and multigain resonant controllers have

MC: Input filter Lf = 0.625 mH, Cf = 2 μF, (delta-connected capacitors), Rf = 100 Ω, four-step commutation method implemented with a 0.7 μs for each step. MC controlled with a 12.5 kHz switching frequency. MC output filter: Output filter implemented with 4 mH inductances and 40 μF capacitors. Output load: The filter capacitors are connected in parallel with a resistive load. In the experimental test, the load is implemented using switchable resistor banks of 140 Ω per phase. Permanent magnet machine: A “control techniques” PM machine of 8 poles, 2000 r/min, 5 kW has been used in the experimental tests related to variable-speed operation. Prime mover: The prime mover is a “Marelli” cage induction machine, 2 poles, 2910 r/min, 5.5 kW, and 380 V. A commercial inverter is used for speed control purposes. REFERENCES [1] J. Rodriguez, J. C. Clare, L. Empringham, A. Weinstein, and P. W Wheeler, “Matrix converters: A technology review,” IEEE Trans. Ind. Electron., vol. 49, no. 2, pp. 276–288, Apr. 2002. [2] J. Clare, D. Lampard, S. Pickering, K. Bradley, L. Empringham, and P. Wheeler, “An integrated 30 kW matrix converter-based induction motor drive,” in Proc. IEEE Power Electron. Spec. Conf., Jun. 2005, pp. 2390– 2395. [3] D. C. Katsis, P. W. Wheeler, J. C. Clare, L. Empringham, M. Bland, and T. F. Podlesak, “A 150-kVA vector-controlled matrix converter induction motor drive,” IEEE Trans. Ind. Appl., vol. 41, no. 3, pp. 841–847, May/Jun. 2005.

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[4] Sangshin Kwak, “Fault-tolerant structure and modulation strategies with fault detection method for matrix converters,” IEEE Trans. Power Electron., vol. 25, no. 5, pp. 1201–1210, May 2010. [5] P. W. Wheeler, J. C. Clare, L. De Lillo, K. J. Bradley, M. Aten, C. Whitley, and G. Towers, “A comparison of the reliability of a matrix converter and a controlled rectifier-inverter,” in Proc. Eur. Conf. Power Electron. Appl., 2005, p. 7. [6] T. Friedli and J. W. Kolar, “Comprehensive comparison of three-phase AC–AC matrix converter and voltage DC-link back-to-back converter systems,” in Proc. Int. Power Electron. Conf., Jun. 21–24, 2010, pp. 2789– 2798. [7] H. Geng and G. Yang, “Output power control for variable-speed variablepitch wind generation systems,” IEEE Trans. Energy Convers., vol. 25, no. 2, pp. 494–503, Jun. 2010. [8] R. Cardenas, J. Proboste, J. Clare, G. Asher, and R. Pe˜na, “Wind–diesel generation using doubly fed induction machines,” IEEE Trans. Energy Convers., vol. 23, no. 1, pp. 202–214, Mar. 2008. [9] T. Waris and C. V. Nayar, “Variable speed constant frequency diesel power conversion system using doubly fed induction generator (DFIG),” in Proc. IEEE Power Electron. Spec. Conf., Jun. 15–19, 2008, pp. 2728–2734. [10] R. Pena, R. Cardenas, J. Proboste, J. Clare, and G. Asher, “A hybrid topology for a variable speed wind–diesel generation system using wound rotor induction machines,” in Proc. 31st IEEE Annu. Conf. Ind. Electron. Soc., 2005, pp. 6–10. [11] P. W. Wheeler, N. Mason, L. Empringham, J. Clare, and Yue Fan, “A new control method of single-stage four-leg matrix converter,” in Proc. Eur. Conf. Power Electron. Appl., Sep. 2007, pp. 1–10. [12] P. Zanchetta, J. Clare, L. Empringham, M. Bland, D. Katsis, and P. W. Wheeler, “Utility power supply based on a four-output leg matrix converter,” IEEE Trans. Ind. Appl., vol. 44, no. 1, pp. 174–186, Feb. 2008. [13] J. Clare, N. Mason, and P. Wheeler, “Space vector modulation for a fourleg matrix converter,” in Proc. Power Electron. Spec., 2005, pp. 31–38. [14] R. Cardenas, R. Pena, P. Wheeler, and J. Clare, “Experimental validation of a space vector modulation method for a four-leg matrix converter,” in Proc. 5th IET Int. Conf. Power Electron Mach. Drives, Apr. 19–21, 2010, pp. 1–6. [15] R. Cardenas, R. Pe˜na, P. Wheeler, and J. Clare, “Resonant controllers for four-leg matrix converters,” in Proc. Int. Symp. Ind. Electron., Bari, Italy, 2010, pp. 1027–1032. [16] R. Cardenas-Dobson, R. Pena, P. Wheeler, and J. Clare, “Experimental validation of a space vector modulation algorithm for four-leg matrix converters,” IEEE Trans. Ind. Electron., vol. 58, no. 4, pp. 1282–1293, Apr. 2011. [17] A. Timbus, M. Liserre, R. Teodorescu, P. Rodriguez, and F. Blaabjerg, “Evaluation of current controllers for distributed power generation systems,” in IEEE Trans. Power Electron., no. 3. vol. 24, Mar. 2009, pp. 654–664. [18] Y. He, D. Sun, and P. Zhou, “Improved direct power control of a DFIGbased wind turbine during network unbalance,” IEEE Trans. Power Electron., vol. 24, no. 11, pp. 2465–2474, Nov. 2009. [19] R. Teodorescu, F. Blaabjerg, and M. Liserre, “Stability of photovoltaic and wind turbine grid-connected inverters for a large set of grid impedance values,” IEEE Trans. Power Electron., vol. 21, no. 1, pp. 263–272, Jan. 2006. [20] R. Teodorescu, F. Blaabjerg, and M. Liserre, “Multiple harmonics control for three-phase grid converter systems with the use of PI-RES current controller in a rotating frame,” IEEE Trans. Power Electron., vol. 21, no. 3, pp. 836–841, May 2006. [21] C.-L. Chen, J. S. J. Lai, and S.-Y. Park, “A wide range active and reactive power flow controller for a solid oxide fuel cell power conditioning system,” IEEE Power Electron., vol. 23, no. 6, pp. 2703–2709, Nov. 2008. [22] R. Cardenas, R. Pena, J. Clare, and P. Wheeler, “Analytical and experimental evaluation of a WECS based on a cage induction generator fed by a matrix converter,” IEEE Trans. Energy Convers., vol. 26, no. 1, pp. 204–215, Mar. 2011. [23] D. Casadei, G. Serra, A. Tani, and L. Zarri, “Matrix converter modulation strategies: A new general approach based on space vector representation of the switch state,” IEEE Trans. Ind. Electron., vol. 49, no. 2, pp. 370–381, Apr. 2002. [24] I. Vechiu, O. Curea, and H. Camblong, “Transient operation of a four-leg inverter for autonomous applications with unbalanced load,” IEEE Trans. Power Electron., vol. 25, no. 2, pp. 399–407, Feb. 2010. [25] F. Zhang and Y. Yan, “Selective harmonic elimination PWM control scheme on a three-phase four-leg voltage source inverter,” IEEE Trans. Power Electron., vol. 24, no. 7, pp. 1682–1689, Jul. 2009.

[26] N. Prabhakar and M. K. Mishra, “Dynamic hysteresis current control to minimize switching for three-phase four-leg VSI topology to compensate nonlinear load,” IEEE Trans. Power Electron., vol. 25, no. 8, pp. 1935– 1942, Aug. 2010. [27] X. Li, Z. Deng, Z. Chen, and Q. Fei, “Analysis and simplification of threedimensional space vector PWM for three-phase four-leg inverters,” IEEE Trans. Ind. Electron., vol. 58, no. 2, pp. 450–464, Feb. 2011. [28] R. Pena, R. Cardenas, E. Escobar, J. Clare, and P. Wheeler, “Control system for unbalanced operation of stand-alone doubly fed induction generators,” IEEE Trans. Energy Convers., vol. 22, no. 2, pp. 544–545, Jun. 2007. [29] A. G. Yepes, F. D. Freijedo, J. Doval-Gandoy, O. Lopez, J. Malvar, and P. Fernandez-Comesa˜na, “Effects of discretization methods on the performance of resonant controllers,” IEEE Trans. Power Electron., vol. 25, no. 7, pp. 1692–1712, Jul. 2010. [30] S. Ben-Yaakov and M. M. Peretz, “Digital control of resonant converters: Resolution effects on limit cycles,” IEEE Trans. Power Electron., vol. 25, no. 6, pp. 1652–1661, Jun. 2010.

Roberto C´ardenas (S’95–M’97–SM’07) was born in Punta Arenas, Chile. He received the B.S. degree from the University of Magallanes, Punta Arenas, Chile, in 1988 and the M.Sc. and Ph.D. degrees both from the University of Nottingham, Nottingham, U.K., in 1992 and 1996, respectively. From 1989 to 1991 and from 1996 to 2008, he was a Lecturer in the University of Magallanes. From 1991 to 1996, he was with the Power Electronics Machines and Control Group, University of Nottingham, U.K.. He is currently an Associate Professor in power electronics and drives with the Electrical Engineering Department, University of Chile, Santiago, Chile. His research interests include electrical machines, variable-speed drives, and renewable energy systems. Prof. C´ardenas received the Best Paper Award from the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS in 2004 and the “Ramon Salas Edward” Award for research excellence from the Chilean Institute of Engineers in 2009.

Carlos Juri (S’10) was born in Santiago, Chile, in 1984. He received the B.Sc. degree in electrical and electronic engineering from the University of Chile, Santiago, Chile, in 2010. He is currently a Research Assistant in the Electrical Engineering Department, University of Chile, Santiago, Chile. His research interests include renewable generation systems, distributed generation, and power electronics converters.

˜ (S’95–M’97) was born in Coronel, Rub´en Pena Chile. He received the B.Sc. degree in electrical and electronic engineering from the University of Concepci´on, Concepci´on, Chile, in 1984, and the M.Sc. and Ph.D. degrees from the University of Nottingham, Nottingham, U.K., in 1992 and 1996, respectively. From 1985 to 2008, he was a Lecturer in the University of Magallanes, Punta Arenas, Chile. He is currently with the Electrical Engineering Department, University of Concepci´on. His research interests include power electronics converters, ac drives, and renewable energy systems.

´ CARDENAS et al.: APPLICATION OF RESONANT CONTROLLERS TO FOUR-LEG MC FEEDING UNBALANCED OR NONLINEAR LOADS

Patrick Wheeler (M’00) received the B.Eng. degree in electrical engineering in 1990 and the Ph.D. degree in matrix converters both from the University of Bristol, Bristol, England, U.K., in 1994. In 1993, he joined as a Research Assistant in the Department of Electrical and Electronic Engineering, University of Nottingham, Nottingham, U.K. In 1996, he was appointed Lecturer (subsequently Senior Lecturer in 2002 and Professor in Power Electronic Systems in 2007) with the power electronics, machines, and control group at the University of Nottingham. His research interests include variable-speed ac motor drives, particularly different circuit topologies, power converters for power systems, and semiconductor switch use. Prof. Wheeler is a member of the Institution of Engineering Technology.

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Jon Clare (M’90–SM’04) was born in Bristol, England. He received the B.Sc. and Ph.D. degrees from the University of Bristol, Bristol, England, U.K., in electrical engineering. From 1984 to 1990, he was a Research Assistant and a Lecturer at the University of Bristol, where he was involved in teaching and research in power electronic systems. Since 1990, he has been with the Power Electronics, Machines and Control Group, University of Nottingham, Nottingham, U.K., where he is currently a Professor in power electronics. His research interests include power electronic converters and modulation strategies, variable-speed drive systems, and electromagnetic compatibility. Prof. Clare is a member of the Institution of Engineering Technology.