https://doi.org/10.6113/JPE.2019.19.6.1380
ISSN(Print): 15982092 / ISSN(Online): 20934718
Research on Grid Side Power Factor of Unity Compensation Method for Matrix Converters
Yihui Xia^{*}, Xiaofeng Zhang^{*}, Zhihao Ye^{†}, and Mingzhong Qiao^{*}
^{†,*}Department of Electric Engineering, Naval University of Engineering, Wuhan, China
Abstract
Input filters are very important to matrix converters (MCs). They are used to improve grid side current waveform quality and to reduce the input voltage distortion supplied to the grid side. Due to the effects of the input filter and the output power, the grid side power factor (PF) is not at unity when the input power factor angle is zero. In this paper, the displacement angle between the grid side phase current and the phase voltage affected by the input filter parameters and output power is analyzed. Based on this, a new grid side PF unity compensation method implemented in the indirect space vector pulse width modulation (ISVPWM) method is presented, which has a larger compensation angle than the traditional compensation method, showing a higher grid side PF at unity in a wide output power range. Simulation and experimental results verify that the analysis of the displacement angle between the grid side phase current and the phase voltage affected by the input filter and output power is right and that the proposed compensation method has a better grid side PF at unity.
Key words: Grid side power factor (PF), Indirect space vector pulse width modulation (ISVPWM), Input filter, Matrix converter (MC), Output power
Manuscript received Dec. 29, 2018; accepted Jun 21, 2019
Recommended for publication by Associate Editor Huiqing Wen.
^{†}Corresponding Author: yxyx928@126.com
^{*}Dept. of College of Electric Engineering, Naval University of Engineering, China
Ⅰ. INTRODUCTION
The matrix converter (MC), as an ACAC power electronic converter device, can fulfill all of the requirements of the traditionally used AC/DC/AC structures of power converters. In addition, it can provide efficient ways to convert electric power for motor drives, and has the following advantages [1][3].
(1)High power density, with no bulky DC link component.
(2)Sinusoidal input and output current.
(3)Adjustable input power factor.
(4)Regeneration capability.
The input filter in an important component of MCs, It is used to improve the grid side current quality with low harmonic components. It is also able to reduce the input voltage distortion supplied to the grid side. Therefore, there are many studies devoted to input filter designs [4][10]. In [4], an automated design procedure to select and optimize both the topology and the parameters of the input filter for an MC based on a genetic algorithm (GA) was proposed. The GA optimize structure and the parameters of the input filter as a function of different factors can be applied to many converter configurations. In [5] and [6], input filter designs for sliding mode controlled MCs consider the maximum allowable displacement angle introduced by the input filter and the controllable grid side PF capability, as well as the ripple of input voltages. The integration of an MC with filters is presented in [7], which provides a lower electromagnetic interference, lower commonmode current and lower shaft voltage. Input filter optimization for a DTC driven matrix converterfed PMSM drive is analyzed and the optimization is performed using the Cuckoo Search method in [8]. Cuckoo Search has better quality indicators of the input filter than the analytical method. Aiming at the resonances in the input filter of an MC with the classic model predictive control, two methods have been proposed to mitigate the resonances in the input filter in [9]. Both of the methods can significantly reduction the harmonic distortion produced by the input filter.
However, if the input filter is not well designed in conjunction with the control system of the MC, it can generate an instability problem of the whole system [11][16]. Several papers have discussed the stability of an MC affected by input filter parameters. In [11], a smallsignal analysis around a steadystate operating point was introduced to analyze the stability of an MC. It paid more attention to the delay time introduced in the digital controllers. In addition, the system may be unstable since the output power exceeds a limit value. The analyses in [12] and [13] show that the power limit can be increased if the duty cycles of the switching configurations are calculated using filtered input voltages. Although the capability of the MC to compensate input voltage disturbances is affected to some extent, the stability limit can be sensibly improved with a digital low pass filter. In [15], the stability of an MC is analyzed by considering the change of the input impendence with the input current control, which can be used to select control parameters. A general constructive approach to matrix converters is analyzed in [16], and the stability of the MC is improved by constructing a correction term to increase the input impendence of the MC.
In addition, the stability of the MC is affected by the input filter. This also results in a displacement angle between the grid side phase current and phase voltage when the input PF angle is set to 0. In the case of low output power/voltage transfer ratio conditions, the grid side PF decreases significantly. To achieve a grid side PF at unity control, several papers have proposed compensation methods [17][22]. In [17], a closedloop input displacement factor correction is used to adjust the grid side current phase shift according to the load level. In [18], a model predictive control is used and a cost function is built based on the output current error, common mode voltage, and minimum reactive power in the grid side. Since 27 switch states are calculated in a switch period, the computational amount is very large. Two grid side PF compensation algorithms were proposed in [19]. One is based on a mathematical model for the grid side of the MC, and the other is based on closedloop control with a PI controller. The method based on a mathematical model for the grid side of the MC has good steady performance of the grid side PF at unity control under a high output power. However, the performance of the grid side PF at unity control with a low output power is not satisfied. In [20], a novel unity power factor fuzzy battery charger using an ultrasparse matrix converter was proposed. Its advantages are a low harmonic distortion and a near unity power factor over a wide operating output power range. A current control with instantaneous reactive power minimization for an indirect MC was proposed in [21]. The bestsuited switch state considering the output voltage error and the input power factor is evaluated by predicting the values of the input and output current in the control scheme. In essence, it is still a model predictive control method that needs a large number of calculations. A hybrid indirect modulation strategy based on extending the reactive input power operating range of an MC was proposed in [23]. It can be generalized to an arbitrary load condition by using the pulse merging and compensation principal, which leads to an increased reactive power control range. However, the hybrid modulation strategy needs to use appropriate schemes such as twovector, threevector and optimum schemes, which adds to the complexity of the modulation strategy. A novel modulation method based on the mathematical construction method for an MC is proposed in [24], which can enhance the control range of the input reactive power with low computational efforts. The maximum input reactive power with the method proposed in [24] is higher than that with the optimumamplitude method and the indirect SVM method. Two modulation methods for an indirect MC were proposed to extend the input reactive power in [25]. First, they are divided into two independent parts: the output voltage formation and input reactive current formation. In terms of the input reactive power range, they are superior to the traditional indirect SVPWM in most situations.
In addition, an MC has been successfully employed in a DFIGbased VSCF WECS in different operation modes, and its input reactive and active power have been researched in [26][31]. In [26], the maximum active and reactive power capability of a matrix converterfed DFIGbased wind energy conversion system is analyzed, and the maximum possible reactive power support by the MC under different operating conditions is accurately predicted by using a new steadystate model of the MC. In [15], a new source current control strategy that can improve both the source current quality and the input factor was proposed for an indirect MC with an asymmetric mode. However, the control strategy needs a large number of calculations and is complex. In [27][29], several control techniques such as direct power control, vector control and direct torque control are used in DFIGbased VSCF WECS. In [30], a singular value decompositionbased modulation strategy for an MC supplying a permanent magnet synchronous wind generator was proposed, which can significantly improve the input reactive power supply capability of the MC.
To obtain a higher performance of the grid side PF with the unity control method, the effect on the grid side PF is analyzed by building a mathematical model of the MC in this paper. Based on this, a new grid side PF at unity control method is presented by analyzing the grid side reactive power. In addition, the compensation angles with the proposed and traditional compensation methods are compared, and the compensation angle with the proposed method is shown to be larger than that with the traditional compensation method, showing a better grid side PF at unity. Simulation and experimental results verify that theoretical analysis is right and that the proposed method is feasible.
This paper is organized as follows. In section II, the basic structure of a direct MC is introduced and the displacement angle between the grid side phase current and phase voltage affected by the input filter and output power is analyzed. In section III, a new grid side PF with a unity compensation algorithm is proposed based on the analysis in section II, and the compensation angles of the proposed and traditional compensation methods are compared. In section IV, simulations are carried out with the traditional ISVPWM method, the compensation method in [19], and the proposed compensation method in this paper. In section V, experiments are carried out to verify the validity of the analysis of the displacement angle affected by the input filter and output power, and the effectiveness of the proposed compensation method in the paper. Finally, some conclusion are presented in Section VI.
Ⅱ. DISPLACEMENT AFFECTED BY THE INPUT FILTER AND OUTPUT POWER
A. Basic Structure of a Direct Threephase to Threephase MC
Fig. 1 shows the basic topology structure of a threephase to threephase direct MC, which is composed of an input filter, a bidirectional flow switch and a clamping circuit. The input filter is used to improve the grid side current quality with low harmonic components, and to reduce the input voltage distortion supplied to the grid side. The clamping circuit is designed for overvoltage protection, and the bidirectional switch is used to synthesize a desired output voltage and input current.
Fig. 1. Basic structure of an MC.
B. Grid Side Mathematical Model of an MC in the Synchronous Reference Frame
From Fig. 1, a threephase input equivalent circuit can be shown as in Fig. 2. In this figure, the output side is seen as a current source.
Fig. 2. Threephase equivalent circuit of an MC.
From Fig. 2, the grid side mathematical model in the abc coordinates can be obtained as:
Where . The grid side rotates with the input angular frequency ω_{i}. In addition, (1) and (2) can be expressed in the synchronous reference frame as (3) and (4), respectively.
The steady state operation points, such as i_{sd} and i_{sq}, of the whole system can be obtained by letting the derivation of i_{sd}, i_{sq}, U_{id} and U_{iq} be set to 0.
The grid side voltage U_{sj}(j=a,b,c) and the input current i_{ij} are obtained as follows:
Where U_{im }is the grid side phase voltage amplitude, I_{im} is the grid side phase voltage amplitude, and φ_{i} is the input phase current amplitude and input power factor angle. From (6) and (7), U_{sd}, U_{sq}, i_{id} and i_{iq} can be expressed as follows:
Substituting (8) into (5), i_{sd} and i_{sq} can be rewritten as follows:
The grid side active power p and reactive power q can be calculated as follows:
When the input P is set to 0, that is to say, cosφ_{i}=1,sinφ_{i}=0, (10) can be simplified as follows:
From (11), it can be known that the reactive power q has something to do with the input filter parameters and the input current amplitude I_{im}. To conveniently analyze how p and q are affected by the input filter parameters and the input current amplitude, the parameters of the input filter and the input current amplitude are shown in Table I.
Parameter 
Value 
Input phase voltage RMS/frequency 
150V/50Hz 
Input filter

R_{f}=50Ω， L_{f}=2mH， C_{f}=11.25µF 
C. Grid Side PF affected by the Damping Resistor R_{f}
Fig. 3 shows the relationship between the grid side power and the input power factor angle φ_{i} under different damping resistors. The solid line is the active power p and the dotted line is the reactive power q. The grid side active power p and reactive power q have not been greatly changed with R_{f} increased in the condition where φ_{i} is a constant. In particular, when φ_{i }is set to 0°, p and q are near 956W and 239W, respectively. In this condition, the displacement angle is 14°and the grid side PF is 0.97. When φ_{i} decreases from 0° to 90°, p is decreased and q is increased, resulting in a decrease in the grid side PF. When φ_{i} increases from 0° to 14°, both p and q are decreased. However, q is decreased faster than p. When φ_{i} increases from 14° to 90°, p is decreased and q is increased, resulting in a decrease of the grid side. It should be specially noted that when φ_{i }is set to 14°, p is 0 and the grid side PF is achieved at unity. The input PF angle φ_{i} is called the compensation angle in the following sections of the paper.
Fig. 3. Variations of the grid side active power and reactive power with input power factor angle φ_{i} under different damping resistors.
D. Grid Side PF affected by the Filter Inductance L_{f}
Fig. 4 shows the relationship between the grid side power and the input power factor angle φ_{i} under different filter inductances. The solid red line is active power p and the dotted blue line is the reactive power q. The grid side active power and reactive power are not greatly changed with an increase of L_{f} since φ_{i} is a constant. When φ_{i} is set to 0°, the active power p and the reactive power q are near 956W and 239W, respectively. The displacement angle is still 14° and the grid side PF is 0.97. The grid side PF varies with φ_{i}. This is_{ }almost the same as what was just described in part C of Section II when R_{f} is a constant.
Fig. 4. Variations of the grid side power with the input power factor angle φi under different filter inductances.
E. Grid Power Affected by the Filtered Capacity C_{f}
Fig. 5 shows the relationship between the grid side power and the input power factor angle φ_{i} under different filter capacities. The solid line is the active power p and the dotted line is the reactive power q. It can be seen that the grid side active power p is largely unchanged. However, the reactive power q is greatly changed with an increase of C_{f} when φ_{i} is a constant. When φ_{i} is set to 0°, the active power p is near 956W, and the reactive powers q are 106.2W (5µF), 239W (11.25µF) and 425.6W (20µF). The corresponding displacement angles between the grid side phase voltage and the phase current are 6.4°, 14° and 24°, and the grid side PFs are 0.99, 0.97 and 0.914, respectively. The grid side PF varies with φ_{i }almost the same (only the compensation angle is changed) as that described in part C of Section II when C_{f} is a constant.
Fig. 5. Variations of the grid side power with the input power factor angle φi under different filter capacities.
F. Grid Power Affected by Input Current Amplitude I_{im}
Fig. 6 shows the relationship between the grid side power and the input power factor angle φ_{i} under different input current amplitudes (since the input power is equivalent to the load power, when I_{im} varies, it can be seen that the load power varies). The solid red line is the active power p and the dotted blue line is the reactive power q. As can be seen, both the grid side active power p and the reactive power q (except for φ_{i}=0°) is largely changed when I_{im} is increased when φ_{i} is a constant. When φ_{i}=0°, the active power p is largely changed (1A, 319.2W; 3A, 956W; 6A, 1594.4W) and the reactive power does not change and is still 239W. The corresponding displacement angles between the grid side phase voltage and the phase current are 36.8°, 14° and 8.6°, and the grid side PFs are 0.8, 0.97 and 0.989, respectively. The grid side PF varies with φ_{i} which is almost the same (only the compensation angle is changed) as that described in part C of Section II when I_{im} is a constant. In addition, q changes faster when I_{im} is increased and the grid side PF is not at unity control.
Fig. 6. Variations of the grid side power with the input power factor angle φ_{i} under different input current amplitudes.
Ⅲ. PROPOSED GRID SIDE PF WITH THE UNITY COMPENSATION METHOD IMPLEMENTED IN THE INDIRECT SVPWM
A. Proposed Grid Side Power with the Unity Compensation Method
According to (10), to achieve the grid side PF at unity control, the grid side reactive power q must be set to 0, which can be expressed as (12).
To obtain the compensation angle φ_{i},_{ }(12) is rewritten as follows:
Since the input active power is equal to the output active power, the following equation exists:
Where U_{om}, I_{om} and φ_{L} are the output phase voltage amplitude, the output phase current amplitude, and the load power factor angle. In (14), a grid side voltage approximately equal to the filter capacity voltage u_{ij} is used. Substituting (14) into (13), φ_{i} is derived as follows:
In (15), R is so small that it can be ignored, and (15) can be rewritten as (16).
Substituting (14) into (16), (16) can be rewritten as (17).
From (12), it can be known that as long as φ_{i} is set as in (17), a grid side PF at unity can be achieved. Therefore, the proposed grid side PF at unity compensation method is obtained, and it is an online method by measuring and calculating the amplitude of the input voltage and input current. From (17), it is known that the proposed grid side PF at unity compensation method is obviously affected by the output power and filter capacitor. However, it is almost independent of the input filter inductance and damped resistance. This is consistent with section II.
B. Comparative Analysis of the Proposed and Traditional Compensation Methods
In [19], the traditional compensation angle φ_{i} is calculated as follows:
The compensation angles with the proposed method and the traditional method varying with the input current amplitude I_{im} are shown as Fig. 7. The input filter and input voltage parameters are set as in Table. I. It can be seen that the compensation angle with the proposed method is larger than that with the traditional method. This shows that the proposed compensation method has a better grid side PF at unity than the traditional compensation method.
Fig. 7. Compensation angles varying with I_{im}_{ }with the proposed and traditional compensation methods.
The duty ratio of the zero switch has to be positive to validate the grid side power factor under unity control in [19]. At the same time, the method in [19] is only validated if the input current vector leads the input voltage vector to one sector, i.e., φ_{imax} ≤ 60°. In addition, the maximum compensation angle φ_{imax} is defined as follows:
However, when V_{TR }≤ , the duty ratio of the zero switch is positive in most cases. In (26), if d_{0}<0 or d_{0}=0, the duty ratios d_{1}, d_{2}, d_{3} and d_{4} are modified as follows:
Taking into account both the input power factor and the systematic stability, the maximum compensation angle φ_{imax} is restricted as 80° from Fig. 7. In addition, its corresponding V_{TR} =0.14. Therefore, under the condition of a low V_{TR}, the maximum compensation angle can be appropriately enlarged to further improve grid side PF on the premise of ensuring system stability. The enlarged φ_{imax} as follows:
C. Proposed Compensation Method Implemented in the Indirect SVPWM of an MC
An MC can be considered as a fictitious ACDCAC structure, which is shown in the right half plane of Fig. 8. The ACDC can be seen as a voltage source rectifier (VSR) in the input side and the DCAC can be seen as a voltage source inverter (VSI) in the output side. Both the VSR and the VSI use SVPWM. Then the relation between the MC structure and the fictitious ACDCAC structure is utilized to obtain the duty ratio of each switch in the MC.
Fig. 8. Fictitious ACDCAC structure of a matrix converter.
The combination of two switches in the VSR and two switches in the VSI results in four switches in the MC. The final duty ratio of the four active and one zero switching configurations with the traditional SVPWM can be obtained as follows:
Where the voltage transfer ratio V_{TR}=U_{om}/U_{im}, and θ_{j} and θ_{i} are the output voltage vector and the input current vector angles, respectively. From section II, it can be known that the grid side phase current usually leads the phase voltage due to the input filter and the output power. That is to say, when the input PF angle δ=0, the grid side PF is not at unity. To achieve the grid side PF at unity control, the input current vector phase angle must be modified to eliminate the effect of the lowpass input filter. As long as the grid side reactive power q=0, the grid side PF must be at unity, and (22) is modified as (23).
Where (23) θ_{i}=θ_{v}φ_{i}, θ_{v} is the input voltage vector angle.
Ⅳ. SIMULATION RESULTS AND DISCUSSIONS
A simulation model of a direct threephase to threephase MC is built with MATLAB/Simulink software. The load is the RL type. The simulation parameters are listed in Table II.
Parameters 
Values 
Input phase voltage RMS/frequency 
150V/50Hz 
Input filter 
R_{f}=50Ω, L_{f}=2mH, C_{f}=11.25µF 
Switch frequency 
5kHz 
RL Load 
R=25Ω, L=8mH 
A. Grid Side PF Affected by the Input Filter and the Output Power
Fig. 9(a) through Fig. 9(d) show simulation results of the traditional SVPWM method at a referenced output voltage of V_{TR}=0.44, and an output frequency of f_{o}=10Hz under different input filter parameters and output powers.
(a) 
(b) 
(c) 
(d) 
From Fig. 9(a) and Fig. 9(b) it can be seen that the grid side phase current i_{sa} leads the grid side phase voltage u_{sa}, the active power is 495W, and the reactive power is 250W regardless of whether R_{f} is 50Ω or 100Ω. The grid side PF is 0.892. Simulation results indicate that the damping resistor has small impact on the grid side PF.
Fig. 9(c) shows simulation results of the MC with the filter inductance changed to 4mH. When comparing Fig. 9(a) to Fig. 11(c), it can be seen that i_{sa} still leads u_{sa}, the active power is 510W, and the reactive power is 250W when L_{f} is 4mH. The grid side PF is a slightly improved to 0.897. However, it is not greatly changed, which indicates that the filter inductance L_{f} has a small impact on the grid side PF.
Fig. 10. Input/output waveforms with the traditional SVPWM at V_{TR}=0.58 f_{o}=20Hz for R_{f}=50Ω, L_{f}=2mH, C_{f}=11.25µF(PF=0.95).
(a) 
(b) 
(c) 
Fig. 9(d) shows simulation results of the MC with the filter capacity C_{f} increased to 20µF. When comparing Fig. 11(a) to Fig. 9(d), i_{sa} leading u_{sa }is increased, the active power is 500W, and the reactive power is 425W. The grid side PF is decreased from 0.892(11.25µF) to 0.762(20µF), which indicates that grid side PF is decreased with an increase of the filter capacity C_{f}.
The waveforms shown in Fig. 10 are simulation results of an MC with the traditional SVPWM method at a referenced output voltage of V_{TR}=0.58, f_{o}=20Hz. The grid side phase current leading phase voltage is reduced when compared to Fig. 9(a). The active power is increased to 885W, the reactive power is 270W, and the grid power PF is improved to 0.95. It can be known that the grid power PF is increased when the output power is increased and that the reactive power is not greatly changed.
B. Proposed Grid Side Power Factor under Unity Control
Fig. 11 shows dynamic performances of V_{TR} changing from V_{TR}=044, f_{o}=15Hz to V_{TR}=0.58, f_{o}=20Hz with the traditional SVPWM method, the traditional compensation method, and the proposed compensation method, respectively. The displayed waveforms show that the grid side phase current is controlled to be in phase with the grid side phase voltage with the two compensation methods in [19] and the proposed method in this paper. When V_{TR}=0.44, f_{o}=15Hz, the grid side PF is 0.99(p=510W, q=70W) with the compensation method in [19]. The grid side PF is 0.997(p=510W, q=35W) with the proposed compensation method. When V_{TR}=0.58, f_{o}=20Hz, the grid side PF is 0.995(p=910W, q=90W) with the traditional compensation method. The grid side PF is 0.998(p=910W, q=55W) with the proposed compensation method in this paper. Both of the compensation methods have improved the grid side PF from 0.892(V_{TR}=0.44, f_{o}=15Hz) and 0.95(V_{TR}=0.58, f_{o}=20Hz) to the unity. In addition, it can be seen that the proposed method has higher grid side PF by comparing the two compensation methods.
Fig. 12(a) through Fig. 12(c) show steadystate performances at V_{TR}=0.22, f_{o}=7.5Hz of the traditional SVPWM, the traditional compensation method, and the proposed compensation method, respectively. As can be easily seen, the grid side phase current is almost in phase with the phase voltage in Fig. 12(c), and the grid side phase current leads the phase voltage in Fig. 12(a) and Fig. 12(b). In particular, it is exceedingly obvious in Fig. 12(a). The grid side phase currents and phase voltage waveforms in Fig. 11(a) represent the grid side PF at 0.447(p=120W, q=240W), and in Fig. 12(b) they represent the grid side PF at 0.8(p=135W, q=100W). The grid side phase currents and phase voltage waveforms in Fig. 9(c) represent the grid side PF at 0.989(p=135W, q=20W). Although both of the compensation methods have the ability to improve the grid side PF, the proposed compensation method shows better performances in terms of improving the grid side PF.
(a) 
(b) 
(c) 
Fig. 13(a) and Fig. 13(b) show the steadystate and dynamic state performances at V_{TR}=0.44, f_{o}=15Hz as well as the filter capacity C_{f }20µF of both the traditional compensation method and the proposed compensation method, respectively. As can be seen, the grid side active power and reactive power in the steadystate vary immensely, and the grid side power oscillation in the dynamicstate process lasts a long time with the traditional compensation method. The grid side power oscillation in the dynamicstate process is very short, and the active power and reactive power in the steadystate vary slightly. In the steadystate, the grid side PF is 0.975(p is near 530W, q is near 120W) with the traditional compensation method, and grid side PF is 0.998 (p is near 540W and q is near 30W). The proposed compensation method still shows better performances in terms of the grid side PF under unity control with the filter capacity increased.
(a) 
(b) 
Fig. 14 shows comparisons of the grid side PF obtained from the traditional SVPWM method, the traditional compensation method, and the proposed compensation method, respectively. With a constant U/f 220V/50Hz for the RL load, a higher output load corresponds to a higher V_{TR}. Compensation angles at the steady state; obtained from the traditional SVPWM method, the traditional compensation method, and the proposed compensation method; and the maximum compensation angle; obtained from the traditional compensation method and the proposed compensation method; are shown in Fig. 14(b). As can be easily seen from Fig. 14(a), with the high output voltage region V_{TR}≥0.44, the grid side PFs obtained from both the traditional and the proposed compensation methods are significantly increased to unity when compared to the lower values obtained from the traditional SVPWM method. When V_{TR}<0.44, the grid side PF obtained from the proposed compensation method is higher than that of the traditional compensation method and the traditional SVPWM method. In particular, when 0.14≤V_{TR}<0.44, the grid side PF achieves almost still at unity by using the proposed compensation method. The reason the proposed compensation method has better performances of grid side PF under the unity control is shown in Fig. 14(b). The compensation angle with the proposed method in this paper is closer to the desired compensation angle than the traditional SVPWM method and traditional compensation method. In Fig. 14(b), it can be seen that when V_{TR}≤0.22, the desired compensation angle exceeds the traditional maximum compensation angle. To further improve the grid side PF under light load conditions, the maximum compensation angle with the proposed method is enlarged, and is shown as d in Fig. 14(b).
(a) 
(b) 
Fig. 15(a), Fig. 15(b) and Fig. 15(c) show input current waveforms before and after the filter of the MC when V_{TR} changes from V_{TR}=0.44, f_{o}=15Hz to V_{TR}=0.58, f_{o}=20Hz, V_{TR}=0.22 f_{o}=7.5Hz, and to V_{TR}=0.44 f_{o}=15Hz, C_{f=}20µF, respectively. From Fig. 15(a) through Fig. 15(c), it can be seen that the displacement angle between the input current before and after the filter decreases with an increase of the output power. In addition, the displacement angle between the input current before and after the filter increases with an increase of the filter capacitor C_{f}. These observations are consistent with the theoretical analysis in section II.
(a) 
(b) 
(c) 
Fig. 16. Prototype of a matrix converter.
Ⅴ. EXPERIMENTAL RESULTS
A. Prototype of a Direct ThreePhase to ThreePhase MC
To demonstrate the validity of the proposed compensation method for grid side PF under unity control, a direct three phase to threephase MC prototype has been built in the laboratory as shown in Fig. 15. The experimental parameters are the same as the simulation parameters shown in Table II.
The prototype of the matrix converter is based on a DSP (TMS320F28335)+FPGA (EP2C8T144C8). The floatpoint DSP is used to fulfill the traditional SVPWM, the traditional and the proposed compensation methods. In addition, the FPGA is used to finish voltagebased on twostep commutation method and to generate the gating signals for the power circuit.
B. Experimental Results
The experimental results shown in Fig. 17 are the grid side and output waveforms of the MC at V_{TR}=0.44, f_{o}=15Hz under different input filter parameters with the traditional SVPWM method. It can be seen that the grid side phase current leads the phase voltage since the effects of the input filter and the displacement angle between them are not different from those in Fig. 17(a) through Fig. 17(c). In Fig. 17(d) the displacement angle is increased when compared to Fig. 9(a). These results are identical to both the simulation results and the theoretical analysis.
(a) 
(b) 
(c) 
(d) 
Fig. 18 shows input/output waveforms of the MC with V_{TR} changing from V_{TR}=0.44, f_{o}=15Hz to V_{TR}=0.58, f_{o}=20Hz with the traditional SVPWM method. With the output power increasing, the displacement angle between the grid side phase current and phase voltage is reduced. Therefore, the grid side PF can be improved by increasing the output power, which is identical to simulation results, verifying that the theoretical analysis is correct.
Fig. 18. Input/output waveforms of the MC as the load changes from V_{T}_{R}=0.44, f_{o}=15Hz to V_{TR}=0.58, f_{o}=20Hz with the traditional SVPWM method.
Fig. 19 shows input/output waveforms of the MC as V_{TR} changing from V_{TR}=0.44, f_{o}=15Hz to V_{TR}=0.58, f_{o}=20Hz with the traditional compensation method and the proposed compensation method. As can be seen, both of the compensation methods nearly achieve the goal of grid side PF under unity control, and the proposed compensation in the paper has a smaller displacement angle between the grid side phase current and the phase voltage than the traditional compensation method. Experimental results verify that both of the methods are feasible for achieving the grid side PF under unity control with a high V_{TR}.
(a) 
(b) 
Fig. 20 shows input/output waveforms of the MC at the referenced output voltage V_{TR}=0.22, f_{o}=7.5Hz of the traditional SVPWM method, the traditional compensation method, and the proposed compensation method under light load conditions. As can be seen, the grid side phase current leads the phase voltage with the traditional SVPWM method. With the traditional compensation method, the displacement angle between the grid side phase current and the phase voltage is decreased. However, they are still not in phase. With the proposed compensation method, the grid side phase current is almost in phase with phase voltage, and the grid side PF under unity control is achieved. However, the output voltage and the input current waveform qualities with the proposed compensation method are worse than those with the traditional compensation method. This is due to the fact that the proposed compensation method has a larger compensation angle than the traditional method. In other words, the angle of the input current vector with the proposed compensation method is modified to be higher than the conventional method. As a result, the input current harmonic with the proposed compensation method is higher than that with the conventional method. Even if the proposed input power factor compensation method has a higher PF in the light load condition, it is at the expense of the input performance of the MC.
(a) 

(b) 

(c) 
Fig. 21 shows input/output waveforms of the MC at the referenced output voltage V_{TR}=0.44 f_{o}=15Hz and C_{f}_{ =}20µF with both the traditional and the proposed compensation methods. As can be seen, with the proposed compensation method the grid side phase current is in phase with the phase voltage, and the displacement angle between them is slightly smaller than that with the traditional compensation method.
These experimental results are identical to the simulation results, which verifies that the proposed method has better performance in terms of the grid side PF under unity control even if the filter capacity is increased.
(a) 
(b) 
In Fig. 17 through Fig. 21, U_{AB} and i_{A} are the output line voltage and the output current, respectively. It can be seen that the output performances of the MC with the proposed method are slightly worse than those with the traditional method. This is due to the fact that the proposed method has a larger compensation angle than the traditional method, which leads to high harmonics in the input current. This in turn, worsens the output performances. A method to optimize the input and output performances under a high input power factor needs further study.
Ⅵ. CONCLUSION
In this paper, the grid side PF, which is affected by the input filter parameters and the output power, was discussed. A new grid side PF under unity control was proposed by letting the reactive power be 0. The principal and maximum compensation angle of the proposed method implemented in indirect SVPWM were presented. Simulation results indicated that both the filter capacity and output power have a greater impact on the grid side PF. However, both the damping resistor and the filter inductance have less impact on the grid side PF. Simulation results also indicated that the proposed compensation method had better performances in terms of the grid side PF under unity control than that with the traditional compensation method. This was especially true under light load conditions. However, this was at the expense of deteriorating the input current and output voltage waveform qualities. Experimental results verified that the theoretical analysis was correct and that the proposed compensation method was feasible.
REFERENCES
[1] M. Siami, D. A. Khaburi, and J. Rodriguez, “Simplified finite control setmodel predictive control for matrix converterfed PMSM drives,” IEEE Trans. Power Electron., Vol. 33, No. 3, pp. 24382446, Mar. 2018.
[2] T. D. Nguyen and H. H. Lee, “A new SVM method for an indirect matrix converter with commonmode voltage reduction,” IEEE Trans. Ind. Informat., Vol. 10, No. 1, pp. 61726, Feb. 2014.
[3] Y. H. Xia, X. F. Zhang, and M. Z. Qiao, “An improved multiorbit vector weighted overmodulation method of matrix converter,” Proc. the CSEE., Vol. 35, No.23, pp. 61436151, Dec. 2015.
[4] A. Trentin, P. Zanchetta, J. Clare, and P. Wheeler, “Automated optimal design of input filter for direct AC/AC matrix converter,” IEEE Trans. Ind. Electron., Vol. 59, No. 7, pp. 28112822, Jul. 2012.
[5] S. F. Pinto and J. F. Silva, “Input filter design for sliding mode controlled matrix converters,” in Proc. 32^{nd} IEEE Annu. Power Electron. Spec. Conf., Vol. 2, pp. 648653, 2011.
[6] S. F. Pinto and J. F. Silva, “Direct control method for matrix converters with input power factor regulation,” in Proc. 35^{th} IEEE Power Electron . Spec. Conf., Vol. 3, pp. 23662372, 2004.
[7] T. Kume, K. Yamada, T. Higuchi, E. Yamamoto, H. Hara, T. Sawa, and M. M. Swamy, “Integrated filters and their combined effects in matrix converter,” IEEE Trans. Ind. Appl., Vol. 43, No. 2, pp. 571581, Mar./Apr. 2007.
[8] P. Siwek, “Input filter optimization for a DTC driven matrix converterfed PMSM drive,” in Proc. 23^{th} MMAR. Conf., pp. 3540, 2018.
[9] M. Rivera, M. Amirbande, and A. Vahedi, “Predictive control strategies operating at fixed switching frequency for input filter resonance mitigation in an indirect matrix converter,” in Proc. IEEE. Sou. Pow. Ele. Conf., pp. 16, Dec. 2017.
[10] A. K. Sahoo, K. Basu, and A. Mahan, “Systematic input filter design of matrix converter by analytical estimation of RMS current ripple,” IEEE Trans. Ind. Informat., Vol. 62, No. 1, pp. 132143, Jan. 2015.
[11] D. Casadei, G. Serra, A. Tani, A. Trentin, and L. Zarri, “Theoretical and experimental investigation on the stability of matrix converters,” IEEE Trans. Ind. Electron., Vol. 52, No. 5, pp. 14091419, Oct. 2005.
[12] D. Casadei, G. Serra, and A. Tani, “Improvement of the stability of electrical drives fed by matrix converters,” in Pro. IEEE Seminar on Matrix Converters, Birmingham, U.K., pp. 3/13/12, 2003.
[13] D. Casadei, G. Serra, and A. Tani, “Effects on input voltage measurment on stability of matrix coverter drive system,” Proc. IEEElectrElectr. Power Appl., Vol. 151, No. 4, pp. 487497, 2004.
[14] D. Casadei, J. Clare, L. Emprinham, G. Serra, A. Trentin, P. Wheeler, and L. Zarri, “Largesignal model for the stability analysis of matrix converters,” IEEE Trans. Ind. Electron., Vol. 54, No. 2, pp. 939950, Apr. 2007.
[15] N. Han, B. Zhou, and J. Yu, X. Qin, J. Lei, and Yang Yang, “A novel source current control strategy and its stability analysis for an indirect matrix converter,” IEEE Trans. Pow. Eletron., Vol. 32, No. 10, pp. 81818192, Oct. 2017.
[16] Y. Sun, M. Su, and X. Li, H Wang, and W. H. Gui, “A general constructive approach to matrix converter stabilization,” IEEE Trans. Pow. Electron., Vol. 28, No. 1, pp. 418441, Jan, 2013.
[17] L. Huber and D. Borojevic, “Space vector modulated three phase to threephase matrix converter with input power factor correction,” IEEE Trans. Ind. Appl., Vol. 31, No. 6, pp. 12341256, Nov. 1995.
[18] M. Rivera, C. Rojas, J. Rodriguez, P. Wheeler, B. Wu, and J. Espinoza, “Predictive current control with input filter resonance mitigation for a direct matrix converter,” IEEE Trans. Power Electron., Vol. 26, No. 10, pp. 27942803, Oct, 2007.
[19] H. M. Nguyen, H. H. Lee, and T. W. Chun, “Input power factor compensation algorithms using a new directSVM method for matrix converter,” IEEE Trans. Ind. Electron., Vol. 58, No. 1, pp. 232243, Jan. 2011.
[20] R. Metidji, B. Metidji, and B. Metidji, “Design and implementation of a unity power factor fuzzy battery charger using ultrasparse matrix converter,” IEEE Trans. Pow. Electron., Vol. 28, No. 5, pp. 22692276, May 2013.
[21] M. Rivera, J. Rodriguez, J. R. Espinoza, and H. AbuRub, “Instantaneous reactive power minimization and current control for an indirect matrix converter under a distorted AC supply,” IEEE Trans. Ind. Electron., Vol. 8, No. 3, pp. 482590, Aug. 2012.
[22] M. Rivera, J. Rodriguez, and J. R. Espinoza, T. Friedli, J.W. Kolar, A. Wilson, and C. A. Rojas, “Imposed sinusoidal source and load currents for an indirect matrix converter,” IEEE Trans. Ind. Electron., Vol. 59, No. 9, pp. 34273435, Sep. 2012.
[23] F. Schafmeister and J. W. Kolar, “Novel hybrid modulation schemes significantly extending the reactive power control range of all matrix converter topologies with low computational effort,” IEEE Trans. Ind. Electron., Vol. 59, No. 1, pp. 194210, Jan. 2012.
[24] X. Li, M. Su, and Y. Sun, H. B. Dan, and W. J. Xiong, “Modulation strategy based on mathematical construction for matrix converter extending the input reactive power range,” IEEE Trans. Power Electron., Vol. 29, No. 2, pp. 654664, Feb. 2014.
[25] X. Li, Y. Sun, and J. X. Zhang, M. Su, and S. D. Huang, “Modulation methods for indirect matrix converter extending the input reactive power range,” IEEE Trans. Power Electron., Vol. 32, No. 6, pp. 48524863, Jun. 2017.
[26] S. Mondal and D. Kastha, “Maximum active and reactive power capability of a matrix converterfed DFIGbased wind energy conversion system,” IEEE J. Emerg. Sel. Topics Power Electron., Vol. 5, No. 3, pp. 13221333, Sep. 2017.
[27] S. Mondal and D. Kastha, “Improved direct torque and reactive power control of a matrixconverterfed grid connected doubly fed induction generator,” IEEE Trans. Ind. Electron., Vol. 62, No. 12, pp. 75907598, Dec. 2015.
[28] A. YousefiTalouki, E. Pouresmaeil, and B. N. Jorgensen, “Active and reactive power ripple minimization in direct power control of matrix converterfed DFIG,” Int. J. Electr. Power Energy Syst., Vol. 63, pp. 600608, Dec. 2014.
[29] R. Cardenas, R. Pena, P. Wheeler, J. Clare, A. Munoz, and A. Sureda, “Control of a wind generation system based on a brushless doublyfed induction generator fed by a matrix converter,” Electr. Power Syst. Res., Vol. 103, pp. 4960, Oct. 2013.
[30] H. Hojabri, H. Mokhtari, and L. Chang, “Reactive power control of permanentmagnet synchronous wind generator with matrix converter,” IEEE Trans. Power Del., Vol. 28, No. 2, pp. 575584, Apr. 2013.
[31] H. She, H. Lin, B. He, X. Wang, L. Yue, and X. An, “Implementation of voltagebased commutation in space vectormodulated matrix converter,” IEEE Trans. Ind. Eletron., Vol. 59, No. 1, pp.154166, Jan. 2012.
Yihui Xia was born in Henan, China, in 1987. He received his B.S., M.S. and Ph.D. degrees in Electrical Engineering from the College of Electrical Engineering, Naval University of Engineering, Wuhan, China, in 2005, 2011 and 2015, respectively. He is presently working as a Lecturer in the College of Electrical Engineering, Navy University of Engineering. His current research interests include power electronics, electric machine drives, active power filters and matrix converters.
Zhihao Ye received his B.S., M.S. and Ph.D. degrees in Electrical Engineering from the College of Electrical Engineering, Naval University of Engineering, Wuhan, China, in 1997, 2000 and 2005, respectively. Since 2007, he has been working as a Professor in the College of Electrical Engineering, Navy University of Engineering. His current research interests include the automation of shipboard power systems.
Xiaofeng Zhang was born in Jiangsu, China, in 1963. He received his B.S. and M.S. degrees in Electrical Engineering from the College of Electrical Engineering, Naval University of Engineering, Wuhan, China, in 1982 and 1985, respectively. He received his Ph.D. degree from Tsinghua University, Beijing, China, in 1996. Since 1997, he has been working as a Professor in the College of Electrical Engineering, Navy University of Engineering. His current research interests include the automation of shipboard power systems.
Mingzhong Qiao was born in Jiangsu, China, in 1971. He received his B.S., M.S. and Ph.D. degrees in Electrical Engineering from the College of Electrical Engineering, Naval University of Engineering, Wuhan, China, in 1996, 1999 and 2003, respectively. Since 2012, he has been working as a Professor in the College of Electrical Engineering, Navy University of Engineering. His current research interests include power electronics and electric machine drives.