I. INTRODUCTION
Nowadays, multiphase variable-speed drives are widely discussed used in various industrial applications and transportation systems such as electric/hybrid vehicles, traction locomotives, electric propulsion ships and aerospace applications [1], [2]. When compared with their standard three-phase counterparts, multiphase drives have several advantages [3], [4]; they make it easier to realize high-power levels using existing power switch devices, have lower torque ripple, less dc-link current harmonics, better torque production per ampere for the same machine volume, higher efficiency and reliability.
Among the different multiphase motor drive solutions, one of the most widely discussed is the two-level six-phase voltage source inverter (VSI) fed dual stator induction machine (DSIM), having two sets of three-phase symmetrical windings spatially shifted by 30 electrical degrees with isolated neutral points, as shown in Fig. 1. Over the years, various topics pertinent to this new type of motor drive system have been covered in depth in the literature, including machine modeling and optimal designing [5], [6], high-performance control strategies [7], [8], digital current controller designing [9], [10], harmonic current suppression [11], fault-tolerant operation [12] and various pulse width modulation (PWM) techniques [13]-[22].
Fig. 1.Dual stator induction machine windings.
In general, the existing research related to the PWM control of DSIMs has two main approaches. The most straightforward of these approaches is undoubtedly the utilization of carrier-based PWM methods [13]-[16]. Double zero-sequence injection (DZSI) techniques are extensions of the six-phase case of the three-phase zero-sequence injection techniques [23], [24]. They are simpler and easier to implement in digital systems [14], [15]. Space-vector PWM (SVPWM) techniques are probably the most popular methods. In [14], [15], two sets of conventional three-phase SVPWM, shifted by 30 electrical degrees, are applied to the two three-phase windings of a DSIM. Actually, these dual three-phase SVPWM methods are equivalent to the DZSI technique [14]. In addition, 12-sector SVPWM [17]-[21] and 24-sector SVPWM [22] are presented based on vector space decomposition [17], [18]. The 12-sector SVPWM techniques produce asymmetrical waveforms and usually contain more than two transitions (from high to low or from low to high) per phase of the PWM waveforms within a sampling period, which increases the switching frequency of inverters and is difficult to achieve with digital implementation [22]. The 24-sector SVPWM techniques overcome these drawbacks of the 12-sector SVPWM and offer great advantages from the point of view of harmonic performance and the ease of digital implementation [22]. However, the algorithm is still complicated and requires high computational efforts [14].
In this paper, a modified 24-sector SVPWM technique is proposed and the correlation between this modified 24-sector SVPWM and the carrier-based PWM methods is also established. A comparative study of the harmonic flux analysis among the existing 24-sector SVPWM, modified 24-sector SVPWM and the DZSI techniques is developed and discussed. Simulation and experimental results are obtained from a software simulation system and a 3.0 kW prototype machine using a DSP control board.
II. REVIEW OF MODULATION TECHNIQUES FOR DSIM
A DSIM drive system is shown in Fig. 2. The two threephase stator windings are identical and symmetrical with to the two neutral points being isolated. They are fed by two sets of three-phase voltage source inverters (VSIs), which share the same DC link.
Fig. 2.Six-phase VSI fed DSIM.
A. Double Zero-sequence Injection PWM Technique
The double zero-sequence injection (DZSI) approach has been introduced in [13]-[16], and is shown in Fig. 3. It is actually an extension to the six-phase case of the three-phase zero-sequence injection technique. The double zero-sequence voltages for the two three-phase windings are computed separately with a simple logic algorithm as given below.
where Vmax1* Vmin1* represent the maximum and minimum values of the sinusoidal reference voltages Va,b,c*, while Vmax2* and Vmin2* represent the maximum and minimum values of the sinusoidal reference voltages Vu,v,w*. The DZSI has an advantage in terms of being easily implemented without any computational effort, which is the reason it is widely used [13]-[16].
Fig. 3.Modulation scheme for double zero-sequence injection technique.
B. Existing 24-sector SVPWM Technique
There are a total of 64 switching states for a six-phase VSI, which determines 60 active voltage vectors and 4 zero voltage vectors. Each voltage vector is represented by a decimal number corresponding to a binary number (KwKvKuKcKbKa), which gives the states of the upper switches. Based on the vector space decomposition theory [17], [18], the inverter voltage vectors can be transformed into the (α-β), (x-y) and (o1-o2) subspaces. The electromechanical energy conversion variables are mapped in the (α-β) subspace, while the non-electromechanical energy conversion variables can be found in the other subspaces [18]. The zero sequence components in the (o1-o2) subspace can be omitted from consideration, since the two neutral points are isolated [18], [22].
According to the principles of the six-phase SVPWM techniques proposed in [17]-[19], four variables need to be controlled to make sure that the PWM waveforms can generate the maximum (α-β) and minimum (x-y) voltage amplitudes during each sampling period. Therefore, a set of four active voltage vectors and appropriate zero voltage vectors should be chosen to satisfy these two conditions at the same time.
The 24-sector SVPWM technique (SVPWM24), presented in [22], employs twelve large voltage vectors with maximum magnitudes, twenty four small ones with half magnitudes and zero voltage vectors (0,7,56,63) in the (α-β) subspace to synthesize the reference voltage vectors. These active voltage vectors divide the (α-β) subspace into twenty four sectors, as shown in Fig. 4(a). For example, the reference voltage vector located in sector 1 is obtained by three large voltage vectors 41, 9, and 11 and one small voltage vector 15 in each sampling period.
Fig. 4.Presentation of the voltage vectors having maximum and half magnitude in (α-β) and (x-y) subspaces.
The 24-sector SVPWM method offers greater advantages concerning the reduction of the low order harmonic components in the motor phase currents. However, the algorithm is very complicated and requires greater computational efforts [14], [22]. As shown in Fig. 5, the zero-sequence voltages Vzabc* and Vzuvw* of the SVPWM24 are not identical to the Vz1* and Vz2* of the DZSI, which means that the SVPWM24 is difficult to realize in the carrier-based PWM technique. Then a modified 24-sector SVPWM technique is proposed in this paper. It offers good output performance that is similar to that of the existing 24-sector SVPWM and can also be realized by an equivalent carrier-based PWM method.
Fig. 5.Modulation voltages and zero sequence voltages of SVPWM24.
III. PROPOSED PWM STRATEGY FOR DSIMS
A. Modified 24-sector SVPWM Technique
In the modified 24-sector SVPWM technique (SVPWM24_M), three large active voltage vectors and two small active voltage vectors in the (α-β) subspace are used in each sampling period. For example, in the first sector three large vectors 41, 9, and 11; two small vectors 40 and 15; as well as two zero voltage vectors 56 and 7 are selected. In order to be realized in the carrier-based PWM method, an equal division of the duration times is applied to the two zero voltage vectors (T56 = T7, in sector 1) and two small voltage vectors (T40 = T15, in sector 1). The PWM waveforms in the first sector are given in Fig. 6.
Fig. 6.Switching sequences of SVPWM24_M in sector 1.
B. Correlation between the Modified 24-sector SVPWM and the Carrier-Based PWM Techniques
Fig. 6 illustrates the switching sequence and duration times (T56, T40, T41, T9, T11, T15, T7) of the voltage vectors in the first sector of the modified 24-sector SVPWM technique. The timing of the gating pulses of the carrier-based PWM equivalent to the SVPWM24_M is also presented in Fig. 7. Since the initial levels of the PWM waveforms in winding abc and winding uvw are opposite in the first sector, the equivalent triangular carrier waves are opposite as well. The durations Ta, Tb and Tc of gating pulses for phases A, B and C are expressed as:
Fig. 7.Timing of gating pulses of carrier-based PWM equivalent to modified 24-sector SVPWM technique.
In the modified 24-sector SVPWM technique, the duration times of the two small voltage vectors are equally distributed, as well as the two zero voltage vectors. In the first sector, they can be expressed as:
In Fig.7, it is evident that during the applying times (T41 + T9) and T11, the bus voltage Vdc is applied to the lines ab and bc, respectively [23]. Employing the principle of volt-seconds over the sampling period Ts, it is possible to obtain:
where Va*, Vb* and Vc* (Va,b,c* for short) are the phase voltages of the winding abc. Then the time intervals (T41+T9) and T11 can be obtained as:
To correlate the carrier-based method with the modified 24-sector SVPWM method, firstly a set of new input reference voltages Va,b,c** and Vu,v,w** has to be found. The equation describing the triangle waveform of the winding abc in Fig. 7 is expressed as:
where Vt and Vp=0.5Vdc represent the instantaneous and peak values of the triangle waveform, respectively. By introducing equations (3)-(5) into (6), a new reference voltage for phase A can be calculated as:
The difference between the new reference voltage the original reference voltage Va** and the original reference voltage Va*, which is defined as the zero sequence voltage, is obtained as:
The zero sequence voltages for phase B and phase C can be derived in the same way as above, and the results of the winding abc in the first sector are summarized below.
Extending the same procedure to all of the other sectors, the new reference voltages Va,b,c** and the zero sequence voltage Vz1* for the winding abc can be summarized as follows:
For the winding uvw, although the triangle waveform is opposite to that in the winding abc, shown in Fig. 7, similar results can be achieved as below:
Since new reference voltages Va,b,c** and Vu,v,w** have been obtained, one more factor must be considered. The modified 24-sector SVPWM technique divides the four zero voltage vectors into two sets: (56, 7) and (63, 0). In each sector, the switching sequence employs only one set of zero voltage vectors and they alternate every two sectors. As a result, the initial levels of the PWM waveforms change every two sectors as well. For example, the switching sequences in sector 1 and sector 2 use the zero voltage vectors 56 and 7, as shown in Fig. 6, the initial levels of the phases abc are ‘low’, which means the PWM waveforms are switching from low to high during the switching period. Meanwhile, the initial levels of the phases uvw are ‘high’, which means that the PWM waveforms are switching from high to low. The switching sequences in sector 3 and sector 4 employ the zero voltage vectors 63 and 0, the corresponding initial levels of the phases abc and uvw are both ‘low’. The results for all of the sectors are summarized in Table I.
TABLE IINITIAL LEVELS OF PWM WAVEFORMS IN EVERY SECTOR
In order to obtain an equivalent effect from the SVPWM24_M, the triangular carrier waves for the two windings should be adjusted according to the initial levels of each sector, as in the case of sector 1 shown in Fig. 7. Therefore, the modified 24-sector SVPWM technique can be realized by the DZSI technique with the triangular carrier waves adjusted.
C. Realization of the Modified 24-sector SVPWM in the Equivalent Carrier-Based PWM Method
According to the analysis above, a new carrier-based PWM technique, which is equivalent to the modified 24-sector SVPWM, can be easily realized through the control scheme shown in Fig. 8. The modulation scheme contains two parts: the first part is to obtain the modulation voltages by adding double zero sequence voltages to the reference sinusoidal voltages, which is exactly the same as the method of the existing DZSI [13]-[16]; the second part is to adjust the triangular carrier waves according to the initial levels in each sector.
Fig. 8.Modulation scheme for new carrier-based PWM method.
The simulation waveforms of the modulation voltages Va,b,c** and Vu,v,w** and the zero sequence voltages Vz1* and Vz2* are presented in Fig. 9. Compared with the DZSI, the new carrier-based PWM technique needs to adjust the triangular carrier waves according to the sector number. The triangular waves on the top of the figures represent the triangular carrier waves used in each sector. For example, in winding abc, the triangular carrier waves in black are used in sectors 1-4, 9-12 and 17-20. Meanwhile, the triangular carrier waves in yellow are used in sectors 5-8, 13-16 and 21-24. Therefore, the modified 24-sector SVPWM can be realized in an equivalent carrier-based PWM technique. As is well known, the carrier-based PWM technique is much simpler and easier in terms of calculation complexity and digital signal processor (DSP) implementation than its SVPWM counterpart.
Fig. 9.Modulation voltages and zero sequence voltages of SVPWM24_M.
IV. WAVEFORM QUALITY
Since the switching frequency harmonics determine the torque ripple of a motor load, the line current total harmonic distortion (THD) of a line-connected VSI and so on, the output voltage waveform quality of a PWM-VSI drive is determined by the switching frequency harmonic characteristics [25]-[28]. It can be verified that the harmonic current and harmonic flux trajectories are only different in scale and that the harmonic flux requires no load information, which means it only characterizes the switching frequency behavior of a modulator [25]. From this perspective, harmonic flux is a pretty good evaluation index for comparing different modulation techniques.
A. Calculation of Normalized Per-Fundamental Cycle RMS Harmonic Flux Values
Unlike the situations in conventional three-phase machines [25], the harmonic flux in a DSIM is calculated in both the (α-β) and (x-y) subspaces [26]-[28]. The harmonic flux in the Nth carrier cycle, which is normalized to λb, is calculated as follows:
In the above formulas, Vsαβk and Vsxyk are the inverter output voltage vectors of the kth state in the (α-β) and (x-y) subspaces, respectively. The per-fundamental cycle rms value of the harmonic flux is the most important performance characteristic of a modulation technique because it determines the output waveform quality and harmonic losses of the inverter [25]. Because of symmetry, the per-fundamental cycle rms harmonic flux values in the (α-β) and (x-y) subspaces can be calculated in the following:
where:
In the above formulas, the variables λ2hrms-αβ and λ2hrms-xy are the per-carrier cycle rms values of the harmonic fluxes in the (α-β) and (x-y) subspaces, respectively. Nevertheless, the (x-y) current components are limited by the stator leakage inductance [22] and the introduction of the coefficient kσxy=σLs/Llsxy is necessary to evaluate and compare the performance of the PWM techniques. In the above formula, σ=Ls-Lm2/Lr, Ls , Lr and Lm represent the stator leakage inductance, rotor leakage inductance and mutual leakage inductance between stator and rotor in the (α-β) subspace. Meanwhile, Llsxy is the transformed stator leakage inductance in the (x-y) subspace. Then the total per-fundamental cycle rms harmonic flux normalized to λb is calculated as follows:
The corresponding results for the SVPWM24 [22], SVPWM24_M and DZSI [15] techniques are calculated and summarized in the Appendix.
B. Performance Comparison
The curves of the normalized per-fundamental cycle rms harmonic flux for the SVPWM24, SVPWM24_M and DZSI techniques are plotted in Fig. 10. It is obvious that the rms value of the harmonic flux varies with respect to the PWM techniques. Compared with the SVPWM24 technique, the SVPWM24_M method proposed in this paper provides better harmonic characteristics. The curves in Fig. 10(a) show that the three PWM techniques offer similar performance in the low modulation index range, while in the high modulation index range, the DZSI exhibits the best performance in the (α-β) subspace. From the results shown in Fig. 10(b), the SVPWM24_M presents the best performance in the (x-y) subspace. According to [18], the harmonics of the order k=12n±1, (n=1,2,3…) are mapped in the (α-β) subspace, while the harmonics of the order k=6n±1, (n=1,3,5…) are transformed in the (x-y) subspace. That is to say, the low order harmonic components in the (x-y) subspace have a greater influence on the motor output performance. The results of the total rms harmonic flux are shown in Fig. 10(c), (d) and (e) with the coefficients kσxy equal to 0.5, 1 and 2, respectively. When kσxy<1, as shown in Fig. 10(c), the DZSI technique presents the best harmonic characteristic. When kσxy=1, as shown in Fig. 10(d), the DZSI and SVPWM24_M techniques offer the same performances, which are better than those of the SVPWM24 technique. When kσxy>1, as shown in Fig. 10(e), the SVPWM24_M exhibits the best harmonic performance.
Fig. 10.Normalized per-fundamental cycle rms harmonic flux values for SVPWM24, SVPWM24_M and DZSI techniques.
V. SIMULATION AND EXPERIMENTAL RESULTS
To verify the validity and feasibility of the modified 24-sector SVPWM technique, a simulation system and a 3.0 kW prototype DSIM test bench have been established. The DSIM prototype fed by two three-phase inverters sharing a common DC link, as shown in Fig. 2, is rewound from a conventional three-phase machine. The stator windings employ a type of single layer and diameter pitch, and the detailed motor parameters are listed in Table II.
TABLE IIMOTOR SPECIFICATIONS
Simulation and experimental results for the SVPWM24, SVPWM24_M and DZSI techniques under constant V/f control are presented in this paper. The average switching frequency for all of the PWM techniques is set to 2 kHz. From the simulation results, it can be observed that the SVPWM24 and SVPWM24_M present similar good performance in the (α-β) and (x-y) subspaces. The DZSI technique offers an advantage in the (α-β) subspace. However, it has the worst performance in the (x-y) subspace, which agrees with the results shown in Fig. 10.
Fig. 11.Simulation results of SVPWM24, SVPWM24_M and DZSI techniques with motor operating at 40Hz. From top to bottom: (α-β) and (x-y) subspace current trajectories, phase-A current.
The experimental phase current waveforms and FFT analysis for all of the SVPWM techniques are given in Fig. 12 and Fig. 13. The motor operates at 20Hz and 40Hz, respectively. It needs to be noted that the DSIM prototype used in the experimental test bench was rewound from a traditional three-phase induction machine without much optimization. The experimental results can still show that the proposed SVPWM24_M technique has the smallest THD values and offers the best output current quality.
Fig. 12.Experimental results of SVPWM24, SVPWM24_M and DZSI techniques with motor operating at 20Hz. From top to bottom: phase-A current and FFT analysis of phase-A current.
Fig. 13.Experimental results of SVPWM24, SVPWM24_M and DZSI techniques with motor operating at 40Hz. From top to bottom: phase-A current and FFT analysis of phase-A current.
VI. CONCLUSION
In this paper, a modified 24-sector SVPWM (SVPWM24_M) technique based on the vector space decomposition theory has been presented. It is suitable for six-phase voltage source inverter fed dual stator induction machines. Compared with the existing 24-sector SVPWM technique, the proposed SVPWM24_M offers better harmonic characteristics in the whole voltage range. In addition, an equivalent relationship between the SVPWM24_M and the carrier-based PWM is also demonstrated in this paper, which means that the complexity of the digital implementation process can be greatly simplified. The validity of the discussion is verified by simulation and experimental results.
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