Due to attractive features, including high efficiency, low device stress, and ability to boost voltage, a Vienna rectifier is commonly employed as a battery charger in an electric vehicle (EV). However, the 6k ± 1 harmonics in the acside current of the Vienna rectifier deteriorate the THD of the ac current, thus lowering the power factor. Therefore, the current closedloop for suppressing 6k ± 1 harmonics is essential to meet the desired total harmonic distortion (THD). Fast repetitive control (FRC) is generally adopted; however, the deviation of power grid frequency causes delay link in the six frequency fast repetitive control to become noninteger and the tracking performance to deteriorate. This paper presents the detailed parameter design and calculation of fractional order fast repetitive controller (FOFRC) for the noninteger delay link. The finite polynomial approximates the noninteger delay link through the Lagrange interpolation method. By comparing the frequency characteristics of traditional repetitive control, the effectiveness of the FOFRC strategy is verified. Finally, simulation and experiment validate the steadystate performance and harmonics suppression ability of FOFRC.
Industrialization requires a lot of energy consumption [
The key to restraining current harmonics, reducing total harmonic distortion (THD) and improving power quality lies in the control of AC current [
Repetitive control (RC) can effectively realize the accurate tracking of harmonic signal by accumulating and tracking the error cycle by cycle [
In order to solve the problem that the delay link of fast repetitive control becomes noninteger when the power grid frequency fluctuates, a fractional order fast repetitive control (FOFRC) strategy is proposed. The Vienna rectifier system is modeled, the Lagrange interpolation method is used, and the Lagrange interpolation method is used to replace the noninteger delay link with polynomial, which shortens the periodic delay of the traditional RC and makes the controller to accurately track the expected output. From the perspective of frequency characteristics, it is analyzed and compared with traditional RC and FRC. At the same time, a zero phase shift filter is designed to improve the stability and antiinterference of the system. The simulation and experimental results show that the FOFRC strategy has better steadystate performance, suppresses the THD of AC current at the grid side and has less AC harmonic component.
The rest chapters are arranged as follows: The second part introduces the topology and control strategy of Vienna rectifier; The third part analyzes the problems of rapid repetitive control; In the fourth part, a fractional order fast repetitive control strategy based on Lagrange interpolation polynomial is proposed, and its stability is proved. In the fifth part, the proposed control strategy is verified by simulation and experiment. The sixth part summarizes the full paper.
This article takes the Vienna rectifier as the research object. The topological of Vienna rectifier is shown in
The double closedloop control strategy for Vienna rectifier is shown in
The current inner loop control [
The transfer function of traditional repetitive control is shown in
If the grid frequency is defined as
Aiming at the problem of traditional RC, the sampling frequency is increased to reduce the number of periodic delay points and improve the dynamic performance. The dynamic performance of RC with different sampling frequencies is shown in
In practice, according to the requirements of the controlled object, only the harmonics at a specific frequency need to be suppressed. The harmonic of the threephase Vienna rectifier AC side current in this paper is mainly 6k ± 1 harmonic [
The transfer function of the FRC strategy in
Although FRC strategy improves the dynamic performance of the system and reduces the use of memory in digital implementation, when the designed sampling frequency is unreasonable, N/6 is not an integer and can only be approximately rounded in discrete, resulting in large deviation between discrete form
Even if the sampling frequency has been reasonably designed, because the actual grid frequency
In this paper, a fractional order fast repetitive control strategy is proposed to solve the noninteger delay caused by power grid frequency offset. Firstly, Lagrange interpolation polynomial is used to replace the fractional part of noninteger delay link. Secondly, zero phase shift filter is used to design internal model link to improve the stability and gain of the system. Finally, good steadystate performance is achieved, the THD of AC current at the grid side is reduced, the harmonic suppression ability is improved, and the stable operation of the system is realized.
In order to approximate the noninteger part, fractional order fast repetitive control is introduced, and the delay points are divided into two parts [
According to
1− 
( 
–( 

d( 

The control block diagram of FOFRC strategy is shown in
In order to realize the good control of fundamental and harmonic, the compound control strategy of PI control and fractional order fast repetitive control in parallel is adopted. The transfer function of the controlled object
According to
When the characteristic roots of the system are in the unit circle, the discrete system is stable. According to
1) The roots of
2)
Condition 1) shows that the proportional control is stable when the root of
FOFRC is improved from the basis of the traditional RC, so the FOFRC controller can be designed using the method of the traditional controller.
The internal mode coefficient
Convert it to frequency domain, the expression is as
In order to ensure the stability of fast repetitive controller,
From
Since the zero phase shift filter has high amplitude gain at low frequency and attenuation at high frequency, therefore, when
As shown in
The sampling frequency of the controller is 20 kHz, so the cutoff frequency of the compensator is selected as
After adding the compensator, the phase of the controlled object lags, as shown in
The order of the lead link is determined according to the effect of the lead link. The phase frequency characteristics of
In order to make the system have better stability and dynamic performance, according to the stability condition of the current inner loop, the root of
Assuming that the sampling frequency of the system is 20 kHz, the number of periodic delay points of fast repetitive control is
Since N/6 is not an integer, when the FRC is realized by discrete digitization, the resonant frequency shifts to 298 Hz at 300 Hz and 896 Hz at 900 Hz. This effect is more obvious at higher frequencies, and the deviation of resonant frequency reaches 10 Hz, as shown in
If the sampling frequency of the optimized system is 20.7 kHz, N/6 = 69. If the grid frequency does not shift, the control effect of FRC can be guaranteed. Assuming that the grid frequency offset is +0.6 Hz, the number of periodic delay points of FRC
If the sampling frequency of the optimized system is 20.7 kHz, N/6 = 69. If the grid frequency does not shift, the control effect of FRC can be guaranteed. Assuming that the grid frequency offset is +0.6 Hz, the number of periodic delay points of FRC.
When noninteger d is close to M/2, the interpolation effect is the best [
According to the calculation formula when M = 2 in
Bode diagrams of traditional RC, FRC and FOFRC are shown in
In order to verify the effectiveness of the method proposed in this paper, the simulation model of the Vienna rectifier control system is built in MATLAB, and the simulation parameters are shown in
Parameters  Value 

Phase voltage( 
220 V 
AC voltage rated frequency( 
50 Hz 
AC side inductance( 
3 mH 
Equivalent impedance( 
0.36 Ω 
Switching frequency( 
20 kHz 
DC side voltage( 
800 V 
DC side capacitance(C_{1,}C_{2})  1000 µF 
DC side load( 
50 Ω 
The simulation waveform diagrams in
In order to further compare and analyze the steadystate performance of rectifier, THD of steadystate input current and tracking error, the results are shown in
Harmonic 
Control strategy  

Traditional RC  FRC  FOFRC  
Current steady state THD  8.19%  5.78%  1.84%  
Frequency tracking error  300 Hz  3.6 Hz  10.4 Hz  0.4 Hz 
900 Hz  10.8 Hz  41 Hz  11 Hz  
2.1 kHz  24 Hz  200 Hz  30 Hz 
According to the theoretical analysis and simulation above, an experimental prototype of Vienna rectifier based on TMS320F28335 is built as shown in
The grid frequency offset is 0.6 Hz, the THD diagrams of the traditional RC, FRC and FOFRC strategies captured by the Fluke 43B Power Quality Analyzer are shown in
The grid frequency offset is 0.6 Hz, the steadystate AC current waveforms of traditional RC, FRC and FOFRC strategies are shown in
After the current THD data is derived by the power quality analyzer, the results are shown in
Control strategy  Current steady state THD 

Traditional RC  8.5% 
FRC  6.8% 
FOFRC  2.6% 
Aiming at the widely concerned interaction technology between electric vehicle and power grid, this paper analyzes the typical topology Vienna rectifier suitable for electric vehicle DC charging pile. On this basis, considering the problem that the traditional repetitive control has poor steadystate accuracy and cannot accurately track the expected output in the case of power grid frequency offset, a fractional order fast repetitive control is proposed, the combination of fractional order fast repetitive control and proportional control is applied to the current inner loop.
Firstly, the reason why the repetitive control delay link in power grid frequency offset is noninteger is analyzed. Secondly, the noninteger delay link that cannot be realized by discrete system is approximated by Lagrange interpolation polynomial, and the fractional delay element is used to overcome the influence of noninteger periodic delay, which effectively improves the fitting accuracy. The simulation and experimental results show that the FOFRC controller effectively improves the frequency characteristics of repetitive control, ensures the steadystate accuracy of repetitive control in the case of grid frequency offset, and reduces the current THD of repetitive control in the case of grid frequency offset. Compared with traditional RC and FRC, the steadystate sinusoidal AC current under FOFRC is smoother, and the THD is only 2.6%. Therefore, Vienna rectifier with FOFRC has better steadystate performance and harmonic suppression ability.
However, the parallel combined control has coupling to the control effect. In the next step, the coupling analysis of the combined control to the control effect will be carried out. In addition, the engineering application and popularization of control strategy is also a problem worthy of discussion.
The authors would like to thank the editor and the reviewers for their constructive comments which improve the quality of the paper.