Abstract: In order to improve the performance of space vector PWM (pulse width modulation), the optimization is widely used. The switching period optimization space vector PWM can reduce both total harmonic distortion and spectrum peaks, so it has important application prospects and has been widely discussed among several types of optimization strategies. Aiming at the characteristic of uncontrollable switching frequency, the study proposes an improved optimization method that can be used in both steady and non-steady states. The method uses a time window with adaptive width to control the average switching period and divides the value range of reference voltage vectors into two kinds of regions: R and Q based on period coefficients. Switching periods are dependent in the same time window, while independent in different windows. The total extra time between specified periods and optimal periods in region R can be made full use of in region Q. A simulation platform was built in software MATLAB/Simulink. The simulation result verifies excellent performance on harmonic distortion and amplitude peak reduction. The improved strategy has widespread applications in electric vehicles, industry automation and drives, household appliances and so on.
INTRODUCTION
Space Vector Pulse Width Modulation (SVPWM) has gained wide applications because it can make full use of DC voltage, improve harmonic performance and be implemented in many digital chips, such as TMS320F2812 and Infineon TriCore 1797/1767 (Hava, 1998). SVPWM can be divided into deterministic SVPWM and random SVPWM. Due to the principle of PWM, the output voltage harmonics focus on or around the switching frequency and its integer multiple ones in many deterministic SVPWM strategies with fixed switching frequency, fixed zero vector distribution and fixed pulse position method (Holmes, 1995; Holmes and Lipo, 2003). The optimization method has been used to eliminate selective harmonics or improve modulation performance (Bruckner and Holmes, 2005; Dehbonei et al., 2002; Hao et al., 2010; Holtz and Beyer, 1994; Laczynski et al., 2008; Lee et al., 2009; Wu et al., 2005; Yuan et al., 2009). The randomization method has been used to reduce harmonic spectrum amplitude peaks through uniformly spreading harmonic spectrum on frequency axes, so many random SVPWM strategies have been proposed and discussed (Carlosena et al., 2007; Chen et al., 2012a, b, c; Chen et al., 2013). Three fundamental RSVPWM strategies include random zero vector distribution PWM (RZDPWM), Random Pulse Position PWM (RPPPWM) and Random Switching Frequency PWM (RSFPWM) (Kirlin et al., 2002, 1994; Ma et al., 2007a, b, 2008; Na et al., 2002; Oh et al., 2009; Wang and Wang, 2007). Aiming at switching frequency that is a key parameter in SVPWM, randomization and optimization methods have been discussed. RSFPWM has the most excellent performance in the three fundamental random strategies. Randomization may increase harmonic distortion. Holtz and Beyer (1994) presented an optimal SVPWM method that considered switching subcycles as optimization variables and given a prediction method. Chen et al. (2013) discussed the numerical solution of switching cycle duration/period optimization RSVPWM based on infinite cycles in a fundamental. The optimal SVPWM method can get excellent performance and the average switching frequency is controllable if the motor control system works in a steady state, while uncontrollable in a non-steady state. The uncontrollable frequency may damage power switches, so the study proposes an improved optimization method that can be used in both steady and non-steady states.SWITCHING PERIOD OPTIMIZATION SVPWM
The switching state of the classic two-level three-phase inverter forms eight
basic space voltage vectors, as shown in Fig. 1 (Holmes
and Lipo, 2003). The eight vectors include six active voltage vectors
Aiming at harmonic current Root-Mean-Square (RMS) value minimization. Holtz and Beyer (1994) drew the conclusion that the total harmonic distortion reaches minimum when all harmonic current RMS values of all cycles/subcysles/periods are made equal. Based on the above conclusion, Chen et al. (2013) discussed computing formulas and numerical computing algorithms for formulas of optimal period Random Space Vector Pulse Width Modulation (RSVPWM) based on infinite cycles in a fundamental. The optimal cycle durations/periods can be computed using the following equation (Holmes and Lipo, 2003).
| (1) |
where, εk is the kth switching cycle (subcysle)/period coefficient, T0 is the fixed cycle duration, Tk is the kth optimal cycle duration, Ih,min is the same harmonic current RMS value of all subcycles, Ih,k0 is the harmonic current RMS value of the kth cycle and N is the number of cycles.
So:
| (2) |
| Fig. 1: | Voltage vectors, vector summation method and PWM waveforms |
Let:
| (3) |
For N→+∞, Ih,k0 becomes a function of the modulation index M and phase angle θ, that is to say, Ih,k0 becomes Ih(M, θ). Then, Eq. 2 can be expressed as:
| (4) |
where, Ih(M, θ) is RMS value of current ripple to specified modulation index M and phase angle θ, f (M, θ) is the corresponding micro Harmonic Distortion Factor (HDF) and C is the constant factor difference between mean-square value and RMS value.
So, εk becomes a function of M and θ and can be expressed as ε (M, θ). Optimal Tk/T0 for the traditional SVPWM (TSVPWM) and RZDPWM can be computed and shown in Fig. 2.
| Fig. 2(a-b): | Optimal switching period coefficient Tk/T0 between fixed cycle duration T0 and optimal cycle duration Tk in the first sector for (a) TSVPWM and (b) RZDPWM |
IMPROVED OPTIMIZATION METHOD
The hexagon region shown in Fig. 1 can be divided into two kinds of regions: R and Q as shown in Fig. 3. Region R includes six sub regions: R1, R2, R3, R4, R5 and R6, while region Q includes Q1, Q2, Q3, Q4, Q5 and Q6. The region division method is as follows:
| (5) |
The improved switching period optimization method is shown in Fig. 4 and can be described as follows:
| S1: | The optimization method begins to work. Whether the reference
voltage vector |
| S2: | If the return-to-zero condition is satisfied, the count variable k and
the timing variable T1 are set to 0: 0→k, 0→T1,
then go to S3; else, go to S3 directly. The return-to-zero condition is
(1) |
| S3: | k+1→k. Switching period coefficient ε is computed based on the
amplitude and phase angle of |
| S4: | The region where |
| S5: | If kT0≤T1+Ts, go to S6; else, let Ts←T0 and go to S6 |
| S6: | T1+Ts→T1 and go to S7 |
| S7: | The duration time of basic voltages is computed firstly, rising edge time and trailing edge time are computed based on dead time and modulation strategy secondly and corresponding resisters are set finally. The 6 switching signals are automatically generated by the controller |
In order to be realized easily in the digital control system, switching period coefficient ε is computed, discretized as a two-dimension lookup table off-line and stored in read-only memory (ROM). The modulating index M and phase angle θ are discretized with step δM and δθ, respectively and only εs on the grids are stored.
| Fig. 3: | Region division method of R and Q based on the switching period coefficient ε (M, θ) value and ε (M, θ)>1 in R, ε(M, θ)<1 in Q, ε (M, θ) = 1on boundaries between R and Q |
If pairs of (M, θ) is on the grid point, the corresponding coefficient ε can be got from the ROM table directly. If not, the interpolation can be utilized.
The classical bilinear interpolation has more excellent performance with less computation and the principle can explained in Fig. 5. A(M1, θ1), B(M2, θ1), C(M1, θ2) and D(M2, θ2) are the four nearest neighbors of F(M, θ), then ε(M, θ) can be got as:
ε(M, θ) = (θ-θ1)[ε(F)-ε(E)]+ε(E) |
(6) |
where, ε(E) = (M-M1)[ε(B)-ε(A)]+ε(A) and ε(F) = (M-M1) [ε(D)-ε(C)]+ε(C).
The nearest neighbor interpolation is a very simple method that can be used. As shown in Fig. 5, C is the nearest neighbor of F, so ε(C) can also be used as the approximate value of ε(F).
SIMULATION RESULTS AND DISCUSSION
A simulation platform was built in MATLAB/Simulink. The simulation results are shown in Fig. 6-9, with parameters of fundamental frequency 30 Hz, switching frequency 10000 Hz and modulation index 0.9. The Fast Fourier Transform (FFT) algorithm is employed to analyze phase voltage spectra. From the figures, some phenomena can be found as follows:
| • | The improved optimization method, as well as the traditional optimization method, can significantly reduce harmonic distortion and harmonic amplitude peaks, which can be seen from Fig. 6 and 7 |
| • | In the same time window, switching periods are dependent. That is to say,
the kth period is determined by corresponding period coefficient and past
k-1 switching periods. From Fig. 4 and 8,
the switching period can be computed using the corresponding coefficient
ε if |
| Fig. 4: | Flowchart of the improved optimization method |
| Fig. 5: | Bilinear interpolation algorithm to compute F(M, θ) value using four nearest neighbors |
| Fig. 6: | Phase voltage spectrum of TSVPWM |
| Fig. 7: | Phase voltage spectrum of proposed optimization method for TSVPWM |
| Fig. 8: | Switching period of traditional optimization method for TSVPWM |
| Fig. 9: | Switching period of proposed optimization method for TSVPWM |
CONCLUSION
In this study an improved switching period optimization SVPWM strategy is proposed.
The strategy uses a time window with adaptive width to control the average switching
period and divides the value range of
The improved strategy has widespread applications in electric vehicles, industry automation and drives, household appliances and so on.
It is variable and optimal period that reduces harmonic distortion and spectrum peaks. The combination of switching period optimization and randomization can get more excellent performance. But randomization makes it complicated to realize in digital control units. In the future, the study will focus on how to realize the optimization method and control algorithm in one chip, for example Field-Programmable Gate Array (FPGA).
ACKNOWLEDGMENTS
This study is supported by Henan Polytechnic University Doctoral Science Foundation with grant No. 648495 and Key Science and Technology Project of Education Department of Henan Province of China with grant No. 13B460027. The authors would also like to thank the anonymous reviewers.