Research Article  Open Access
Hao Li, Shuo Chen, Xiang Wu, Guojun Tan, "Model Predictive Control Method with Constant Switching Frequency to Reduce CommonMode Voltage for PMSM Drives", Journal of Electrical and Computer Engineering, vol. 2018, Article ID 1090452, 12 pages, 2018. https://doi.org/10.1155/2018/1090452
Model Predictive Control Method with Constant Switching Frequency to Reduce CommonMode Voltage for PMSM Drives
Abstract
A model predictive control method to reduce the commonmode voltage (MPCRCMV) with constant switching frequency for PMSM drives is proposed in this paper. Four nonzero VVs are adopted in future control period and the switching sequence is designed to ensure the switching frequency is fixed and equal to the control frequency. By substituting the finitecontrol nonzero voltage vectors in the current predictive model, a current predictive error space vector diagram is obtained to determine the adopted four VVs. The duty ratio calculating method for the selected four VVs is studied. Compared with the conventional MPCRCMV method, the current and torque ripples are greatly reduced and the switching frequency is fixed. The simulation and experiment results validate the effectiveness of the proposed method.
1. Introduction
The commonmode voltage (CMV) of the permanent magnet synchronous motor (PMSM) drive system with twolevel threephase voltage source inverter (VSI), as shown in Figure 1, can be approximately calculated according to the voltage between the midpoint of the dclink capacitor and the neutral point of the threephase load [1]. The highfrequency CMV can cause many problems including increasing the leakage currents [2], causing damage of the motor shaft [3], and increasing electromagnetic interference [4].
To address this problem, different solutions by improving the control strategy of the PMSM have been studied. In one aspect, many CMV reduction pulsewidth modulation (CMVRPWM) strategies have been proposed [5–10]. The research results in [7] suggest that only the active zerostate PWM1 (AZSPWM1) [5] and the nearstate PWM (NSPWM) [6] are practical in most cases among the earlier developed CMVRPWM strategies. In addition, the newly developed optimized CMVRPWM strategies with the consideration of the current ripple losses and switching losses minimizing have been proposed in [8]. All of the above CMVRPWM strategies can restrict the amplitude of the CMV within sixth of the dclink voltage. Some CMVRPWM strategies are proposed to reduce the loworder components of CMV [11, 12]. A virtual space vector PWM (VSVPWM) for the reduction of CMV in both magnitude and thirdorder component is studied in [13].
On the other hand, the model predictive control methods [14, 15] to reduce the CMV (MPCRCMV) have also been proposed in [16–22]. The MPCRCMV method proposed in [16] adopts only six nonzero voltage vectors (VVs) to calculate the cost function and the CMV amplitude is reduced without utilizing zero VVs. In [17], the optimal VV is selected from three adjacent nonzero VVs, and this MPCRCMV method needs less compute efforts. However, it has been proven that many spikes may appear in the current ripple with the method in [17]. The problem has been analyzed in [18], and an improved MPCRCMV method with four candidate VVs is developed to eliminate the current spikes. To eliminate the CMV spikes caused by the dead time, the method proposed in [19] preexcludes, from the candidates for future vectors, those voltage vectors which can increase the commonmode voltage during the dead time. A MPCRCMV strategy without calculating the cost function is proposed in [20], where the nonzero optimal future VV is determined according to the basis of an inverse dynamics model. A twovector based MPCRCMV is proposed in [21], where two nonzero VVs are selected at every control period. The satisfactory load current ripple performance can be yielded; however, it is computationally intensive and the average switching frequency is still lower than 1/T_{s}. However, with these above MPCRCMV strategies, the current and torque ripples may be larger than the conventional fieldoriented control (FOC) [23, 24] with the same control frequency (1/T_{s}), mainly owing to that the average switching frequency is far smaller than 1/T_{s}. In addition, the switching frequency of these methods are not fixed, which make it hard to analyze the harmonic features and design the filter. Many model predictive controllers with fixed switching frequency have been studied. The fixed switching frequency schemes for finitecontrolset MPC are studied in [25, 26]. Modulated MPC strategies [27, 28] have been developed to realize the target of constant switching frequency with the aid of a modulator. However, these fixed switching frequency MPC strategies in [25–28] have not considered the target of CMV reduction, and the amplitude of the CMV reaches to half of the dclink voltages owing to the adoption of the zero VVs.
In this paper, a MPCRCMV method with constant switching frequency is proposed. Unlike the conventional MPCRCMV methods where one or two nonzero VVs are applied in future control period, four nonzero VVs are adopted and the switching sequence is designed to ensure the average switching frequency is fixed and equal to 1/T_{s}. A current predictive error space vector diagram is obtained by substituting the finitecontrol nonzero VVs in the current predictive model, and according to it, the sector where the coordinate origin locates is calculated to determine the adopted four VVs. The duty ratio calculating method for the selected four VVs is studied. With the proposed method, the current reference can be fast and accurately tracked and the current ripples are much smaller than the conventional MPCRCMV methods. Compared with the MPC methods with fixed switching frequency studied in [25–28], the proposed method can limit the amplitude of the CMV within sixth of the dclink voltage. The effectiveness of the proposed method is validated by simulation and experiment.
2. Conventional MPCRCMV Method
The conventional MPCRCMV method for PMSM drives is shown in Figure 2. The current predictive model is given in the following equation:where and are the direct axis and quadrature axis inductances, is the permanent magnet flux linkage, , , , and are the load current and output voltage of the VSI at the th sampling instant, and are the predicted direct axis and quadrature axis currents at the th sampling instant, is the stator resistance, is sample period, and is angular speed.
To compensate the delay caused by the sampling and calculation in digital control, the typical method as proposed in [29, 30] is adopted, i.e., the direct axis and quadrature axis currents at the th sampling instant which are predicted as given in Equation (2) are adopted to calculate the cost function as Equation (3).
For the MPC strategy, the voltage vector (VV) that minimized the cost function is chosen to be applied in the next control period. The difference between the conventional MPC and the MPCRCMV method is the VV selection in the finitecontrol set. For the MPCRCMV method, only nonzero VVs are included in the finitecontrol set to limit the amplitude of the CMV within sixth of the dclink voltage. According to the number of the VVs in the finitecontrol set, the conventional MPCRCMV methods can be divided into three categories as shown in Figure 3, i.e., the 6 VV, 3 VV, and 4 VV strategies. In Figure 3, the arrows define the possible changes of the switching states in the next control period. The finitecontrol set of 6 VV strategy includes all the six nonzero VVs. As shown in Figure 3(a), the optimal VV of the next control sample can be selected as any nonzero VV no matter which VV is applied in the current period. The switching frequency of the 6 VV strategy is relatively high due to too many switches between nonadjacent VVs, e.g., all the power switches turn off or turn on with the switching state changing from 100 to 011. As shown in Figure 3(b), the finitecontrol set of 3 VV strategy is determined by the VV applied in the current period and the number of the selectable VVs is 3. The VV that applied in the current period and its two adjacent VVs can be selected for the next control period. Compared with the 6 VV strategy, the switching frequency of the 3 VV strategy is limited since at most one switch change is permitted, in addition, the computation amount is reduced by 50% owing to the number of cost function calculation changes from 6 to 3. However, it has been proven that many spikes may appear in the current ripple when 3 VV strategy is utilized due to the lack of enough alternative VVs [16]. To solve this problem, an improved 4 VV strategy is developed in [16] where four VVs are chosen as candidate VVs at the same time as shown in Figure 3(c). Compared with the 3 VV strategy, a nonadjacent VV is included in the finitecontrol set to ensure that the motor current can be adjusted towards two directions.
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It should be noted that there are some problems for the conventional MPCRCMV methods. Firstly, the switching frequency of these methods is not fixed, which makes the voltage and current spectrum spread over a wide range of frequencies. Secondly, the average switching frequency is smaller than the control frequency (1/T_{s}) which should be set small enough considering the computation time, e.g., the results in [16] indicate that the average frequency of the 4 VV strategy is no more than 2 kHz with the control period set as 100 μs. Thus, the conventional MPCRCMV methods are not suitable in the fields where the average switching sequence needs to be high enough to realize the control targets such as decrease the torque ripples and the noises.
3. Proposed MPCRCMV Method with Constant Switching Frequency
The control diagram of the proposed MPCRCMV method with constant switching frequency is shown in Figure 4. It includes four parts, i.e., current predictive model, delay compensation, voltage vector selection and duty ratio calculation, and switching sequence producing part. Unlike the conventional MPCRCMV method where only one active VV is applied in a single control period, the proposed method adopts 4 active VVs to keep the switching frequency fixed. The current predictive model part calculates and by Equation (4) where and are the average d and qaxis voltages in the whole control period. The and can be calculated according to the selected VVs and their duty ratios which are obtained in the last control period by the proposed MPCRCMV method.
Then, the direct axis and quadrature axis currents at the sampling instant for different active VVs can be calculated by substituting Equation (4) in Equation (2).
The predictive current errors (PCE) are defined as
By combining Equations (2) and (4), the PCE equation can be written aswhere and satisfy
According to Equation (6), the values of PCEs are related with the d and qaxis voltages, and thus, the position of the PCE in the  coordinate system, as shown in Figure 5, are changed by selecting different VVs.
In Figure 5, e_{1}e_{6} represents the PCE vector by applying the VVs V_{1}V_{6}, and e_{0} represents the PCE vector of the zero VVs. Owing to the corresponding d and qaxis voltages for the zero VVs are both 0, the coordinate of e_{0} is (, ) according to Equation (6). The variables of and are determined by the VV that applied in the current period and have no relation with the selected VV in the next period, and thus, they can be viewed as constants in Equation (6).
The reference current tracking problem can be solved by selecting appropriate VVs to make both the and become zero, i.e, the current can track its reference by making the current error vector reach the coordinate origin as shown in Figure 5. In fact, the optimal VV selecting process of the conventional MPCRCMV method can be viewed as determining the nearest PCE from the coordinate origin by minimizing the cost function. As an example, the vector V_{1} will be selected for the case shown in Figure 5 with the conventional MPCRCMV method. Owing to the fact that only single VV is selected during one control period for the conventional MPCRCMV, the selected optimal VV may not ensure the current error vector reach the coordinate origin. In this part, the MPCRCMV method with constant switching frequency is proposed by adopting four nonzero VVs in one control period to ensure the average current error approach to 0 within one period for the case shown in Figure 5.
A new coordinate system ( as shown in Figure 5) is defined by the transformation given in following equation:
The reference current error vector () is defined as the vector , and its coordinate is (−, −). The reference current tracking problem can be solved by synthetizing e^{∗} with the PCE vectors (e_{1}e_{6} as shown in Figure 5).
The vector selection process can be implemented according to the angle of , e_{1}, e_{2}, e_{3}, e_{4}, e_{5}, and e_{6} in the  coordinate system which can be calculated by the following equation:
The sector (S) is defined as shown in Figure 5 and can be obtained according to Table 1, and then, the selected VVs can be determined. In Table 1, the variable (i = 1, 2, 3, 4, 5, and 6) and represent the angle of and , respectively.

According to Equation (8), the current error vector will reach the position defined in Equation (10) if the voltage vector V_{i} is adopted with the duty ratio d_{i}. Equation (10) indicates that the change of the current error vector is proportional to the duty ratio of the adopted VV. Accordingly, the reference current tracking can be realized by synthetizing with linear combination of the PCE vector of the selected VVs.
According to the nearest threevector principle, can be synthetized by e_{0}, e_{1}, and e_{2} for the case that S is 1 as shown in Figure 5. Owing to that the zero vectors should be avoided to restrict the CMV amplitude, two opposing PCEs (e_{3} and e_{6}) with equal duty cycle can be utilized to create equivalent e_{0}. The duty ratios of the adopted VVs can be calculated by Equation (11) if S is 1.However, the duty ratios calculated by Equation (11) needs to be adjusted as Equation (12) for the case where the reference current error vector () is outside of the hexagon enclosed by e_{1}–e_{6} as shown in Figure 6.
If locates in other sectors, the duty ratio calculating method is the same as the case where S is 1. After obtaining the duty ratios of the adopted VVs, the switching sequence can be designed as shown in Table 2.

The flowchart of the proposed method is shown in Figure 7(b). The values of and are firstly calculated by Equation (4), and then, and for different active VVs can be obtained by Equation (5). The sector and the selected VVs can be determined by Table 1 and the duty ratios of the selected VVs can be calculated by Equations (11) and (12). Finally, the switching sequence can be obtained according to Table 2.
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4. Simulation and Experimental Validation
4.1. Simulation Results and Analysis
To validate the effectiveness of the proposed MPCRCMV with constant switching frequency, the strategy is simulated by Matlab/Simulink and the main parameters are given in Table 3.

To minimize the copper losses of the PMSM, the maximal torque per ampere (MTPA) strategy [31, 32] is adopted and the reference of the d and qaxis current can be calculated as Equation (13) where is the reference of the stator current.
In the simulation, is set as 200 A and 300 A at the interval 0–0.5 s and 0.5–1 s, respectively. The speed of the motor is set as 750 r/min in the simulation.
The linetoline voltage (V_{ab}), CMV, and threephase current waveforms for both the conventional 6 VV strategy, and the proposed method are given in Figure 8. The CMV curves in Figure 8 indicate that the amplitude of the CMV has been restricted within sixth of the dclink voltage with both the conventional and the proposed methods.
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The waveforms of the d and qaxis current and the electromagnetic torque () for both the conventional and the proposed methods are shown in Figure 9. It can be seen from Figure 9 that the fluctuations of , , and of the proposed method are far smaller than the conventional method. The comparison results of the peaktopeak of the fluctuation of , , and are given in Table 4. In the case where of these two methods are both set as 100 µs, the fluctuations of , , and with the proposed method in the case where is set 200 A reduce by 95.9%, 97.8%, and 78.2%, respectively. In addition, the fluctuations of , , and in the case where is set 300 A reduce by 95.9%, 98.2%, and 74.8%, respectively. In Table 4, the peaktopeak of the fluctuation of , , and for the conventional method with set as 50 µs is also analyzed, and the fluctuations of , , and are still much bigger than the proposed method with set as 100 µs, which indicate that the control performance of the proposed method is greatly enhanced compared with the conventional method.
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The fast Fourier transform analysis results of the phase current for both the conventional and the proposed methods are shown in Figure 10. The results in Figure 10 indicate that the main current harmonics components of the proposed method concentrate on the switching frequency (10 kHz) and its multiple, which is helpful for the harmonic filter design. In addition, the total harmonic distortion (THD) of the phase current with the proposed method is smaller than the conventional one.
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Accordingly, it can be concluded that the proposed MPCRCMV with constant switching frequency shows great advantages over the conventional 6 VV strategy, mainly in the current and torque ripple reduction.
In order to illustrate the advantage of the proposed method over the MPC controller without considering the target of CMV reduction, the comparison results between the MPC method with constant switching frequency in [27] and the proposed method are shown in Figure 11. In Figure 11, the MPC method in [27] is adopted before 0.2 s and the proposed method is used after 0.2 s. The linetoline voltage, CMV, and threephase currents are shown in Figure 11. The CMV amplitude of the MPC method in [27] reaches to the half of the dclink voltage, but it has been limited within sixth of the dclink voltage with the proposed method. Accordingly, the main advantage of the proposed method is that the CMV can be reduced.
4.2. Experimental Validation
To validate the effectiveness of the proposed MPCRCMV with constant switching frequency, an experimental prototype has been established as shown in Figure 12. A 540 V dclink voltage is obtained by the PWM rectifier. The motor 1 is connected with the inverter 1, and the proposed MPCRCMV with constant switching frequency is implemented by the controller 1. The motor 2 is a load motor which is controlled by the inverter 2. The parameters of the PMSM are the same as the simulation.
In the experiment, a 150MIPS fixedpoint 32bit TMS320F2812 DSP board is adopted to carry out the computation. The execution time for the proposed MPCRCMV methods with necessary protecting and sampling process is 83 µs. To introduce the realtime implementation details, the flowchart of the proposed method in Figure 7(b) is divided into 4 parts, and the execution time of these four parts are 5 µs, 61 µs, 6 µs, and 2 µs, respectively. The other execution time is for the necessary sampling and protecting process.
The d and qaxis current waveforms for the both conventional and proposed methods, which are measured through a digitaltoanalog (DA) chip (TLV5610) on the control board, are shown in Figure 13, where the reference stator current changes from 100 A to 150 A at the middle of the graph. The results in Figure 13(b) indicate that the d and qaxis current reference can be fast tracked with high precision as the conventional method as shown in Figure 13(a). In addition, it can be seen from Figure 13 that the ripples of both the d and qaxis current for the proposed method have been greatly reduced compared with the conventional one.
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The threephase currents and the linetoline voltage curves for the both methods are shown in Figure 14. The THD values for the conventional method where the reference of the stator current is 100 A and 150 A are 6.3% and 5.1%, respectively. The THD values for the proposed method where the reference of the stator current is 100 A and 150 A are 2.4% and 1.6%, respectively. Thus, the current quality of the proposed method has been greatly improved. The CMV curves shown in Figure 15 indicate that the amplitude of the CMV has been successfully restricted within sixth of the dclink voltage. Accordingly, the effectiveness of the proposed method has been verified by the experimental results.
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5. Conclusion
A MPCRCMV method with constant switching frequency for the PMSM drives is studied in detail. In the proposed method, four nonzero VVs are adopted and the switching sequence is designed according to the established current predictive error space vector diagram. Unlike the conventional methods where the switching frequency changes with the work condition of PMSM, the average switching frequency is fixed and equal to 1/T_{s}. The simulation and experiment results indicate that the current reference can be fast and accurately tracked, and the current ripples are greatly reduced.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This paper is partly funded by the National Natural Science Foundation of China under Award U1610113 and National key Research and Development Project of China under award 2016YFC0600804.
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Copyright © 2018 Hao Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.