I. INTRODUCTION
The conventional rectification, uncontrolled rectifiers and phase controlled rectifiers, may bring about a large number of harmonics, which have become the main ‘pollution’ of the grid. In addition, such ‘pollution’ makes other electric devices work improperly. Matrix Converters (MC) can be traced to the late 1970s [1]. The three phase to two phase AC-DC MC, i.e. matrix rectifier, can be deduced from the conventional three phase to three phase MC. The MR has several advantages: (1) operation in all four quadrants; (2) no need for a large value DC capacitor on the output side; (3) easy safe current commutation; (4) intrinsic buck converter and (5) power factor regulation [2]-[3]. Because of the advantages mentioned above, the MR has become a strong candidate for potential applications in plug-in automotive, aircraft, DC motor drives, etc. [4]. Moreover, the MR is the first stage of a two-stage matrix converter, and it can be used in the field of wind power generation and has wide application prospects and potential economic value.
Commonly there are several modulation algorithms for the MR, and they are the Alesina–Venturini optimum (A-V optimum) method [5], the Roy method [6] and the Space Vector Modulation (SVM) [7]. However, these algorithms are open-loop control. When applying these modulation algorithms on a MR, the performance of the DC output voltage becomes bad, since such modulation algorithms cannot resist disturbances.
An input filter is necessary for a sinusoidal input current because it can filter the harmonic components of the input current. However, the input filter’s characteristics make the input power factor (IPF) remain high in the presence of high output loads, and the IPF degrades significantly under light-load conditions [8]. What is more, due to the presence of a grid side unbalanced input voltage and voltage disturbances, a MR cannot provide a ripple-free DC output voltage, and periodic oscillations appear.
In order to mitigate the fluctuations of the DC output voltage cause by the grid side unbalanced input voltage and to improve the IPF, scholars have carried out a lot of research in these fields. In [9], the static and power characteristics of the MR have been given. This has been a crucial support in terms of further research on control strategies. In [10], the designs of a feedforward control strategy was proposed to guarantee the stability of the DC output voltages. In [11], the designs of an opened-loop control, and in [12]-[13], the designs of closed-loop control strategies were proposed to compensate the power factor. As a result, applying feedforward control strategies to improve the performance of DC output voltage regulation can restrain disturbances of the unbalanced three-phase input to a certain extent. However, the performance is not obvious. In [14], a new-direct SVM based on an API controller was shown, and its correctness and effectiveness had been verified. However, PI control is a linear control, and its dynamic properties is poor. In [15], in order to improve dynamic performance, the conventional SMC was applied to compensate the input power factor and the DC output. A PI controller and a SMC controller were employed separately in the prototype experiment. The experiment results reveal that the dynamic properties of the SMC are superior. However, the output ripple voltage and input current harmonics are much more serious than those regulated by a PI controller.
In this paper, a novel approach to achieve both a tight DC voltage and a grid side power factor is proposed. The DC output voltage is controlled by a RL-SMC and the grid side power factor is compensated by the conventional SMC. Simulation and experimental results show that the proposed algorithm can fulfill the presupposed goals and alleviate chattering of the sliding mode.
II. MATHEMATICAL MODEL OF A PMSM FOR BACK-EMF ESTIMATION
A. Space Vector Modulation Method
Fig. 1 shows the topology of the MR and this topology consists of 6 parts: a three-phase power source, input damping filters, a main circuit with six bidirectional switches, an output filter, and a load.
Fig. 1.Topology of the MR.
Space vector modulation (SVM) is used to provide sinusoidal input currents for the MR. The SVM must satisfy the following two constrains simultaneously:
(1) The existence theorem of the matrix converter theories in [16]-[17].
(2) The maximization of the DC output voltage to achieve full use of the input line–line voltages.
To satisfy the above constrains, only nine feasible switching states of the six bidirectional switches exist, and these nine states determine nine vectors which can be divided into two categories: active vectors ( I1 - I6 ) and zero vectors （I7 - I9）.
In addition, the space vector hexagon is divided into six sectors by the active vectors, as shown in Fig. 2. The reference current vector Iref, synthesized from the two adjacent vectors Iα and Iβ, and a zero vector, as shown in Fig. 3. θ is the angle between Iref and its right adjacent active vector Iα, and θ∈[0,π/3]. The reference current vector Iref can be calculated as the following expression:
where Tα(θ), Tβ(θ) and T0(θ) are the durative times of Iα, Iβ and I0 in one switching period TS. The duty cycle dα(θ), dβ(θ) and d0(θ) can be calculated as follows:
where m is the modulation index, m∈[0,1]. The averaged output voltage in a switching period can be calculated as:
where Vim is the amplitude of the input phase voltage, and φi is the desired input current displacement angle.
Fig. 2.Space vector diagram of input current.
Fig. 3.Reference input-current space vector with the adjacent vectors.
B. Input filter Analysis of the MR
The input filter, as shown in Fig. 4, is designed to filter the high harmonic components of the input current and to reduce the input voltage distortion supplied to the MR. In this topology, to damp the oscillations resulting from the matrix rectifier hard switching, the resistance is connected in parallel with the inductance, and this topology provides the minimization of power losses [18].
Fig. 4.The input filter of the MR.
In the Fig. 4, Us and Is denote the input voltage and current at the grid side, and Ui and Ii denotes the input voltage and current of the MR. The IPF issue is addressed only when the input filter is considered. At the grid side, let Us =Usm∠α and Is = Ism∠β. At the output side of the filter, let Iin = Iim∠(α-φi). To analyze Fig. 4, it should be known that:
It should be noted that is so small that it can be ignored. Eq. (6) can be simplified as:
According to Eq. (7) and Fig. 5, the grid side power factor can be calculated as follows:
where α and β are the phase angles of the input voltage and input current at the grid side, respectively, φs is the grid side power factor angle, and φi is the switch matrix power factor angle. From Eq.(8), the phase shift φs is imposed by the MR’s input filter, and the phase shift relates to the whole circuit including the DC loads.
Fig. 5.Vector diagram of the Eq.(7).
C. Impact of the DC Output Voltage Caused by an Imbalanced Input Voltage
According to the symmetrical components method, an unbalanced three-phase input voltage Vs can be decomposed into positive sequence component Vsp and negative sequence component Vsn as follows:
In Eq. (9), VPM and VNM denote the amplitudes of the positive and negative sequence input phase voltages, respectively, and ω is the input voltage angular frequency. Let the initial phase angles of the positive and negative sequence components become zero.
In [19]:
Hence, the DC output voltage can be shown as:
In Eq. (11), under the imbalanced three-phase voltage input condition, the output DC voltage of the Matrix Rectifier contain a ripple wave with a frequency that is double that of the input ripple wave except for the desired DC output voltage (as shown in Fig. 6).
Fig. 6.Input and Output voltage waveform of the MR.
III. SLIDING MODE CONTROL SYSTEM BASED ON THE REACHING LAW
The sliding mode control is one of the variable structure control strategies (VSC). The SMC has several advantages such as a quick response, strong robustness, and simple implementation. In addition, it is not affected by system parameter and external disturbances and it has good dynamic performance. However, the SMC makes the system switch between two kinds of states, which results in a tremendous chattering around the sliding surface when the system operates in the sliding mode. These problems have been tackled by using the Reaching Law on the SMC.
A. Mathematical Model of the System
After tangent calculus is applied to Eq. (8), it can be shown as:
where φs is the grid side power factor angle. Assuming φs < π/6 and φi < π/6, the output power of the MR is shown as Therefore, Eq. (8) can be shown as:
From Fig. 1, according to Kirchhoff’s law, the following two equations are known:
Thus, the equation for the state of the output voltage and current can be given as:
Eqs. (5), (14) and (15) are the key factors for the proposed mathematical and control models.
B. Design of the Slide Mode Controller
According to the analysis above, the principle diagram of the slide mode controller operational principle is shown in Fig. 7. The working process is as follows: the sensor detects the DC output voltage Vo that will be compared with the desired output voltage Vref. This determines the error value. Then, the error value is sent to the RL-SMC controller based on the reaching law. Then, the modulation index m will be calculated. The sensor detects the phases of the input voltages and currents at the grid side. This determines φs, the power factor at the grid side. The phase difference φs is sent to the PF SMC controller. Then the phase difference φi is determined, and the compensation angle is calculated.
Fig. 7.The principle diagram.
B. DC Output Voltage Regulation by the RL-SMC
The control system is designed to suppress the DC output voltage fluctuations caused by an unbalanced input three-phase voltage. The controller allows the DC output voltage Vo track the reference voltage Vref.
Where ev is the feedback voltage error value. In addition, Thus, Eq.(16) is shown as:
The slide mode function is shown as:
where c1 is a positive constant.
In Eq. (19), when c1 ≈ RLCO, or Thus, Eq. (19) can be simplified as:
In order to achieve a higher dynamic performance of the DC output voltage, the exponent reaching law is applied:
Where ε1 and k are positive constants. From Eq. (20) and Eq. (21), it can be know that:
From Eq. (5) and Eq. (22), the modulation index m can be shown as:
C. Grid Side Power Factor Compensated by the Constant Reaching Law
The control system is also designed to compensate the power factor at the grid side. This control algorithm is able to compensate the phase of the input current at the grid side by applying the SMC. Thus, it can guarantee that the phase φs is zero in theory. The switching function is set as:
Where, is the average rate of change. Meanwhile the control function is as follows:
where, φref is the pre-compensation angle. The phase δ is the disturbance resist factor and |φi|≤π/6 . To achieve the local reachability of theSMC, Eq. (25) should be satisfied as:
where Ṡ2 is the time derivative of S2.
If c2 is small enough, S2 = φs and Eq. (27) can be simplified as follows:
From Eq. （13）, and Because φs and φi are nonlinear functions, and are the change rates of φs and φi in the control cycle. As a result, and are also the change rates of and .
From Eq. (25), in the next control cycle, there is φi = φref-δ . Meanwhile = (-δ - δ)fs=-2×104δ and = (-2δfs - 2δfs)fs = -4×108δ, where fs is the switching frequency and fs=104. Substituting the above-mentioned expressions of and into the Eq. (28), inequality (26) holds, which means that the sliding mode exists [20].
From Eq. (25), in the next control cycle, there is φi = φref + δ. Meanwhile = (δ + δ)fs= 2×104δ and = (2δfs + 2δfs)fs = 4×108δ. Substituting and into Eq. (28), inequality (26) also holds, which means that the sliding mode exists.
The value range discussion of the sliding mode coefficient: if c2 is small enough to make the equality , then , consequently:
From Eq. (29), the value of the sliding mode coefficient is related to the switching frequency and the control function (25).
IV. SIMULATION AND EXPERIMENTAL RESULTS
In this section, a detailed simulation is built on the MATLAB/SIMULINK platform to verify the proposed control strategy.
A. Simulation and its Analysis
The circuit parameters listed in Table I are used in the small-scale simulation model. Three comparative simulations were carried out in this section: (1) balance input with different values of c1; (2) three-phase balanced input voltage using the conventional SMC; (3) three-phase balanced input voltage using the SMC based on the reaching law; and (4) three-phase unbalanced input voltage using the SMC based on the reaching law. The simulation waveforms and the results of the total harmonic distortions are shown in Fig. 9 to Fig. 11.
TABLE IPARAMETERS OF THE SIMULATION
Fig. 8 shows a simulation with different values of c1 under the balance input condition. In simulation (a) c1=6×10-4=RLCO and in simulation (b) c1=2×10-6 Fig. 8.Experimental results when the input is balanced but with different value of c1: (a) c1=6×10-4; (b)c1=2×10-6. A comparative simulation study was conducted using the same circuit parameters but with different control strategies. Both the conventional SMC and the SMC based on the reaching law were used. According to Fig. 9 and Fig. 10, using the two above-mentioned approached can achieve two goals, ripple-free output voltage and a near unity power factor. Moreover the chattering of the waveform is acceptable. At 50 ms, the target output voltage dropped from 80 V to 50 V. The simulation results proved that the dynamic performance was excellent regardless of which control algorithm was used. Fig. 9.Experimental results of using conventional SMC when the input voltage is balanced: (a) simulation waveform; (b) THD analysis. Fig. 10.Experimental results of using SMC based on reaching law when the input voltage is balanced: (a) simulation waveform;
(b) THD analysis. According to Fig. 11, when the three-phase input voltage is unbalanced, applying the RL-SMC algorithms on the MR can provide a ripple-free output voltage. Fig. 11.Simulation waveform using SMC based on reaching law when the input voltage is unbalanced. An experimental prototype was built to further illustrate and validate the proposed control approach. Three comparative experiments were carried out: (1) a three-phase balanced input voltage using the conventional SMC; (2) a three-phase balanced input voltage using the SMC based on the reaching law; and (3) a three-phase unbalanced input voltage using the SVM algorithm without any compensation and using the SVM algorithm compensated by the SMC based on the reaching law, respectively. From Fig. 12(a), using the conventional SMC algorithm can achieve both output voltage regulation and a higher power factor. However, it also has a drawback since it can only be switch to two modes and cannot change the states according to the running condition. In the dynamic process it requires the system mode switch in a wide range, but for the steady state operation this kind of wide range switch is not conducive to the output performance. As shown in Fig. 12(b), the frequency of the current harmonic is mainly located near 1.4kHz. Using the same circuit parameters, but applying the reaching law can solve this problem effectively through changing the switch state to adapt to different working condition. The experimental waveform applying the RL-SMC are as shown in Fig. 13(a) and Fig. 13(b). In addition, Fig. 13 (a) shows the target voltage undergo a mutation from 80V to 60V, with a response time of 4ms. Fig. 12.Experimental results of using conventional SMC compensation algorithm:(a) AC supply at the grid side and the DC output voltage; (b) spectrum analysis of the A-phase input current. According to Fig. 14(a), using the SVM algorithm without any compensation control approach, to control the MR results in a ripple in the output voltage and a decrease in the power factor at the grid side in the case of unbalanced sinusoidal supply voltages. However, under the same conditions, using the SVM algorithm with compensation by the RL-SMC based on the reaching law and NSMC can effectively achieve both output voltage regulation and power factor control, as shown in the Fig. 13(b). The experimental results show good agreement with the theoretical analysis. Fig. 13.Experimental results of using SMC compensation based on the reaching law when the three-phase input supply voltage is balanced:(a) AC supply at the grid side and the DC output voltage; (b) spectrum analysis of the A-phase input current. Fig. 14.Experimental results when the AC voltage supply is unbalanced: (a) using the SVM algorithm; (b) using the RL-SMC algorithm. However, the RL-SMC algorithm proposed in this paper cannot alleviate chattering well, and the simulation results and experiment results are not the same, due to non-ideal bidirectional switches operating in the real experiment instead of the simulation environment. The novel approach of achieving both a tight DC voltage regulation and direct power factor control by the SVM compensated by the SMC based on the reaching law has been theoretically justified and experimentally verified. A simple single-phase model of a three-phase ac-DC matrix rectifier has been proposed. Its effectiveness has been verified through simulation and experimental results. The experimental results also demonstrate that applying a sliding mode control based on the reaching law can improve the robustness and dynamic performance of the matrix rectifier. Furthermore, the proposed control method is also capable of wide leading or lagging power factor operation which may broaden its application rang.
B. Experimental Results and their Analysis
V. CONCLUSION
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