1. Introduction
The matrix converter (MC) achieves direct AC–AC power conversion without large DC-link energy storage elements [
1], as shown in
Figure 1. After decades of continuous academic research [
1,
2,
3,
4,
5], its commercialization has gained attention from industry. The Yaskawa Company has launched at least two series of MC products featuring high power quality, high efficiency, high power density, and energy regeneration.
As the MC is composed of nine bidirectional AC switches connecting every input phase to output phase without intermediate energy storage elements, controlling the MC is relatively complicated compared to the typical voltage source converters and has been an important issue since the birth of the MC. Linear control algorithms based on the PWM (Pulse Width Modulation) technique have been well developed for the MC, which can achieve superior control performance and are widely adopted for the MC [
2,
3,
6,
7]. However, the duty cycle calculation is relatively complicated to implement in practice.
Benefiting from the fast development of semiconductor technology, model predictive control (MPC) has become a promising alternative to linear modulation algorithms [
3,
8]. Considering the discrete characteristic of a power converter, MPC calculates the cost function corresponding to each valid switching state and then applies the switching state that minimizes the cost function to the converter. Without the need of duty cycle calculations, MPC is easy to understand and implement. Featuring fast dynamic response and multi-objective optimization, MPC has attracted attention from researchers in various fields of power converters [
9], including MCs. MPC can be applied to achieve normal operations of the MC, such as input reactive power minimization [
10], sinusoidal input and output currents [
11], direct power control [
12], among others [
13]. As MPC has the capability of multi-objective optimization, features such as motor torque control [
14,
15,
16,
17], speed control [
18], common-mode voltage elimination [
19], and efficiency improvement [
20] can be easily included in MPC. Moreover, harmonic reduction [
21], input active damping [
22,
23], over-modulation [
24], and fault diagnosis [
25,
26] can also be obtained based on MPC.
The application of existing MPC schemes to the MC is therefore a good idea, yet their implementation is not optimal. In practice, MPC is usually implemented in a digital signal controller and thus it relies on the discretized prediction model to describe the MC’s behavior in every sampling period. All existing MPC schemes for the MC discretize the input and output circuits separately and consider the input and output prediction models fixed, irrespective of the MC’s switching states. Due to the absence of large energy storage elements, currents and voltages on the input and output sides interact with each other directly. However, existing MPC schemes consider input voltages and output currents constant in one sampling period, ignoring the variations caused by the interaction. Consequently, they fail to accurately characterize the MC’s behavior during each sampling period. This leads to increased prediction errors, especially under a large sampling period. For MPC, larger prediction errors always introduce more harmonics. This in turn requires larger filter components, which decreases the power density of the MC.
In this paper, an improved MPC scheme is proposed for the MC. The proposed MPC is based on a precise prediction model which considers the input circuit, MC, and output circuit as a whole system. The precise prediction model is obtained by discretizing the integral state-space equation of the whole MC system, with the interaction considered. Each valid switching state corresponds to a set of matrices of the precise prediction model. Compared with the conventional MPC which uses separate prediction models, the proposed approach adopts a precise prediction model which has the same eigenvalues as the continuous model of the whole MC system. Therefore, the proposed prediction model accurately characterizes the MC’s behavior in every sampling period, as the interaction between the input and output circuits are considered. The proposed approach can thus achieve lower prediction errors, especially under a large sampling period, which in turn requires smaller filter components.
The rest of this paper is organized as follows.
Section 2 introduces the conventional MPC with separate input and output prediction models.
Section 3 presents the principle of the improved MPC based on the precise prediction model.
Section 4 shows the experimental verification.
Section 5 draws the conclusions.
4. Experimental Verification
Both the conventional and the improved MPC schemes are evaluated by experimental results. A picture of the experimental prototype is shown in
Figure 5. The normal parameters of the prototype are listed in
Table 3. Filter components are selected so that the converter can achieve satisfactory power quality when the sampling time is 20 μs. Values of the filter components are obtained with a high-accuracy LCR meter. The digital controller consists of a digital signal processor (DSP) TMS320F28379 and an FPGA (Field Programmable Gate Array). The DSP has dual CPU cores working at 200 MHz, which provides strong computational capability and enables the completion of all the calculations within 20 μs. The adopted weighting factor is obtained based on the experimental results under various values of it, so that the input and output power quality are optimal. One can conclude that the two MPC schemes have almost the same optimal weighing factor as the one listed in
Table 3. Therefore, the same optimal weighting factors was selected for the two MPC schemes, which guarantees a fair comparison.
The performances of the MC using the conventional and improved MPC schemes were comparatively evaluated under six cases. Experimental conditions of the six cases are summarized in
Table 4. Case 1 evaluates the performance when all working conditions are normal as listed in
Table 3. In Case 2, the sampling time is increased from 20 μs to 40 μs. In Case 3, the parameters used in the calculation of the prediction model’s coefficients are artificially increased or decreased by 5%. In Case 4, the output filter inductance
Lo is reduced to 2.51 mH. In Case 5, source voltages are abnormal and contain 5% unbalanced component and 5% 5th harmonic. In Case 6, the output current amplitude steps between 10A and 5A to evaluate the dynamic performance. The analysis results of the total harmonic distortions (THDs) of source and output currents at the steady state are listed in
Table 5.
Table 6 shows the execution time of the two MPC schemes. It can be seen that the improved MPC scheme consumes slightly more time to complete all the calculations. This is because all the elements in the matrices
ΦP and
ΓP are nonzero for the improved MPC. For the conventional MPC, some elements in matrices
ΦC and
ΓC are zero, which helps to simplify the control algorithm. Considering the reduction of the prediction errors as presented below, such a slight increase in the execution time is worthwhile.
The experimental results of Case 1 are shown in
Figure 6. The prediction errors, which are the subtractions of the actual variables and the predicted ones, are denoted by the symbol ∆. It can be seen that the improved MPC generates smaller prediction errors than the conventional MPC, especially the errors of input voltage
uiA and output current
ioU. Benefiting from the reduction of prediction errors, the improved MPC has reduced the THDs of the source current
isA and output current
ioU from 4.61% and 2.07% to 3.47% and 1.80%, respectively, indicating that a better power quality is obtained.
The experimental results of Case 2 are shown in
Figure 7. One can conclude that, with the increase in sampling time, the conventional MPC generates significant prediction errors. On the contrary, the improved MPC still achieves much smaller prediction errors. With the improved MPC, THDs of
isA and
ioU are reduced from 11.88% and 5.66% to 10.98% and 4.72%, respectively. Differences of the spectral distribution between the two MPC schemes are more significant. It shows that the improved MPC helps to reduce harmonics at almost all frequency bands. The results prove that the improved MPC scheme is more effective under a larger sampling frequency.
The experimental results of Case 3 are shown in
Figure 8. It can be seen that the prediction errors of the two MPC schemes are increased when the parameters used in the prediction models are inaccurate, which also demonstrates the importance of the prediction model’s accuracy. Even under such parameter variations, the improved MPC still achieves smaller prediction errors and a better waveform quality, with the THDs of source and output currents reduced from 5.32% and 2.08% to 3.95% and 1.83%, respectively. Therefore, the improved MPC scheme has robustness with respect to the parameter variations.
The experimental results of Case 4 are shown in
Figure 9. It can be seen that when smaller output filter inductors are adopted, the prediction errors of both the conventional and improved MPC scheme are increased compared with those shown in Case 1. However, the improved MPC scheme still achieves significantly lower errors in this case. The THDs of the source and output currents are still lower than those of the conventional MPC scheme.
The experimental results of Case 5 are shown in
Figure 10. It is clear that under unbalanced and distorted source voltages, prediction errors can still be reduced by the improved MPC, even though more low-order harmonics are contained in the source and output currents. With the improved MPC, a higher power quality is obtained. In particular, the THD of
isA is reduced from 9.04% to 7.65%.
The experimental results of Case 6 are shown in
Figure 11, where the reference amplitude of the output current steps between 10A and 5A. It shows that both the conventional and improved MPC schemes track the reference very fast. Therefore, the improved MPC scheme does not degrade the dynamic performance. It should be noted that the small differences of the reference current come from the different step moment and are not related to the MPC schemes.
In summary, the improved MPC based on the precise prediction model can always obtain a better power quality than the conventional MPC without degrading the dynamic control performance, no matter what the operation condition.
5. Conclusions
Due to the lack of large energy storage elements, the input and output circuits of the MC interact with each other directly. Existing MPC schemes apply the delay compensation and model predictions to the input and output sides of MC separately, ignoring the interaction between the two sides. Therefore, prediction models adopted by conventional MPC schemes cannot accurately describe the MC’s behavior in one sampling period. The improved MPC proposed in this paper integrates the input circuit, MC, and output circuit into one precise prediction model, which accurately characterizes the MC’s behavior. Experimental results have demonstrated that the improved MPC can always obtain a better control performance under various working conditions. In turn, the filter components can be reduced if the same power quality is considered as the design criterion.
The precise prediction model proposed in this paper is also applicable to other MPC schemes for the MC so as to reduce the prediction errors and improve the power quality, but only if the cost functions are set accordingly.