Three-stage synchronous overmodulation control of five-phase inverters

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Three-stage synchronous overmodulation control of five-phase inverters

INTRODUCTION

inverter modulation voltage frequency

Multiphase power electronic converters and ac drives are a subject of increasing interest in the last years due to some advantages compared with standard three-phase systems, especially in the field of high power/current applications. In particular, multiphase topologies of electric drives allow reduction of amplitude and increasing the frequency of torque pulsation, and also reduction of the rotor harmonic losses in electrical machines. Multiphase inverters and converters allow dividing of the controlled power on more inverter legs, reducing the rated current of power switches 1.-2.. Between different multiphase solutions the more interesting and addressed ones are the five-phase and six-phase power conversion systems 1..

In particular, five-phase adjustable speed drives on the base of five-phase inverters are now ones of the most suitable topologies of multiphase drives for such perspective fields of application, as electric vehicles, ship propulsion, aerospace, etc. So, space-vector-based control and modulation methods and techniques for five-phase systems have been developed intensively during last period 3.-9.. Five-leg voltage source inverters for five-phase drives are characterised by additional degrees of freedom for their control in comparison with conventional three-phase systems. At the same time it is known, that almost all versions of standard space-vector modulation are based on asynchronous principle, which results in sub-harmonics (of the fundamental frequency) in spectrum of the output voltage of inverters, that are very undesirable in drive systems with increased power rating, operating at low switching frequencies 10.,11..

In order to avoid asynchronism of standard space-vector PWM, novel method of synchronized PWM has been recently proposed and developed for control of three-phase inverters 12., three-level converters 13., open-end winding induction motor drives 14., symmetrical six-phase traction systems 15. (including overmodulation control of split-phase drives 16.), five-phase inverters with operation in linear modulation range 17.,18.. So, this paper present results of the development of this new PWM method for synchronous control of five-phase inverters in the zone of overmodulation.

1.SYNCHRONIZED SPACE-VECTOR PWM FOR A FIVE-PHASE SYSTEM IN A LINEAR MODULATION RANGE

Basic topology of power circuit of a five-phase voltage source inverter with a star-connected load with the neutral point n is presented in Fig. 1 7.. In particular, in the case of a five-phase motor as a load, the five stator phases a, b, c, d and e are distributed with a spacing of 72⁰.

Space-vector pulsewidth modulation is one of the most suitable control techniques for five-phase systems. Similar to three-phase space-vector PWM, it can be developed for a five-phase system, as shown in Fig. 2 for a period of the fundamental frequency 8.. In particular, Fig. 2 shows the basic ten large (1, 2, 3, 4, 5, 6, 7, 8, 9, 10) and ten medium (1', 2', 3', 4', 5', 6', 7', 8', 9', 10') switching vectors for a five-phase inverter in accordance with conventional designation (in brackets) of switching state of every switch of the inverter, where “1” corresponds to the switch-on state of the corresponding switch of the upper group of switches of the phases a - e. This control scheme includes also two zero switching states (00000 and 11111), providing zero voltage at the outputs of the inverter.

Although total number of available voltage space vectors in five-phase inverter is equal to 32, the using only of the mentioned above 22 vectors (ten large and ten medium active vectors, plus two zero vectors) allows minimum number of switchings in inverter and minimum switching losses, combined with good spectrum of the phase-to-neutral voltage 6.,8..

Fig. 1. Topology of a five-phase voltage source inverter 7.

Fig. 2. Basic voltage vectors of five-phase inverter

In order to provide synchronisation of the process of vector PWM, a novel approach has been proposed for synthesis of the output voltage of three-phase inverters 12.. The main principles of this method have been recently developed and modified for synchronous space-vector PWM control of five-phase systems in linear modulation range 17.. In particular, with application to five-phase inverters, one of the basic ideas of the method of synchronized PWM is in continuous synchronization of the positions of all central active switching signals in the centres of the 36⁰-clock-intervals (to fix positions of the -signals in the centres of the 36⁰-cycles), and then - to generate symmetrically around the centres of the 36⁰-clock-intervals all others active -signals.

It is necessary to take into consideration during this process, that in the linear modulation region the total duration of each -signal consists from four parts (, see Figs. 4-5), corresponding to four voltage vectors (two large vectors and two medium vectors) 6.. So, all these sub-signals, together with notches between the -signals, have to be generated symmetrically around the centres of the 36⁰-clock-intervals too.

To provide continuous synchronization of the output voltage of five-phase inverters, by analogy with three-phase converters 12., the scheme of synchronized PWM includes, as an important control parameter, boundary frequencies F_i, transient between control sub-zones, which are situated on the axis of the fundamental frequency F of the system.

Table I presents basic control functions for five-phase voltage source inverter with synchronized PWM during linear modulation zone, which are compared here with the corresponding PWM functions for standard three-phase inverter. It is assumed, that both three-phase and five-phase systems operate under standard V/F control; - width of sub-cycle, i - number of notches inside a half of clock-intervals in Table 1.

Basic Control Functions of Synchronized Schemes of PWM

As an example of operation of five-phase inverter with synchronized PWM in linear modulation range, Fig. 3 presents switching signals (pole voltages V_a - V_e) for the phases a - e, line-to-line voltage V_as and phase-to-neutral voltage V_an on a period of the fundamental frequency F.

Fig. 4 and Fig. 5 show more in details switching state sequence and the pole and phase voltages of five-phase inverter for the 2^nd and the 3^rd control sectors. Switching states correspond here to the designations presented in Fig. 2.

Fig. 3. Basic voltages of five-phase inverter in linear modulation zone

Fig. 4. Switching state sequence and basic voltage waveforms of five-phase inverter with synchronized PWM in the second control sector.

Fig. 5. Switching state sequence and basic voltage waveforms of five-phase inverter with synchronized PWM in the third control sector

As an other example of operation of five-phase inverter with synchronized PWM at the boundary of linear modulation zone, Figs. 6-7 show basic voltage waveforms and spectra of the line and phase voltages of five-phase inverter at the fundamental frequency F=41.3Hz, modulation index m=0.826 in the case of scalar control mode. The switching frequency is 3 kHz.

Fig. 6. Basic voltage waveforms of five-phase inverter at the boundary of linear control range (F=41.3Hz, m=0.826)

Fig. 7. Spectra of line and phase-to-neutral voltages of five-phase system at the boundary of linear control range (F=41.3Hz, m=0.826)

2.THREE-STAGE CONTROL OF FIVE-PHASE SYSTEM WITH SYNCHRONIZED PWM IN THE OVERMODULATION ZONE

Сontrol in the zone of overmodulation has some specific peculiarities 19.-22.. Due to features of five-phase systems, rational PWM control of five-phase inverters in the zone of overmodulation, from the boundary of linear control range until the maximum fundamental voltage (at ten-step control mode) can be based on three-stage control scheme.

Maximum Modulation Index

Modulation index is an important parameter of control scheme of PWM inverters. For power conversion systems with standard scalar V/F=const control, the modulation index m is characterizing both the normalized fundamental voltage and the normalized fundamental frequency. In particular, for five-phase inverters with V/F control the modulation index m is defined as

m= F/F_ten-step=V₁/V_1ten-step (1)

where V₁ is the fundamental voltage, and V_1ten-step is the fundamental voltage generated at ten-step operation, which is characterized by the maximum F_ten-step fundamental frequency. So, the maximum modulation index m=m_ten-step=1 can be achieved only at the ten-step operation mode.

The First Stage of Synchronous Overmodulation Control

It is known, that the maximum fundamental peak line-to-neutral voltage of five-phase inverter at ten-step operation is 0.6366V_dc 9.. At the same time it is known, that the maximum achievable peak phase-to-neutral voltage of five-phase system (with high-quality control based on optimal combined application of medium and large space vectors) is limited by 0.5257V_dc 6.. So, in order to obtain linearity of the fundamental voltage versus frequency characteristic of five-phase system with standard scalar control during the whole control range, scaled coefficient equal to 0.6366/0.5257=1.21 should be used in PWM scheme for determination of duration of active switching states (in particular, see the corresponding equation for the basic central active switching state duration in Table I).

In accordance with (1), the modulation index, corresponding to the boundary of linear control range and the overmodulation zone, can be determined as

m_ov1 = 0.5257/0.6366 = 0.826 (2)

In particular, if the maximum fundamental frequency F_ten-step=50Hz, the threshold fundamental frequency , which characterizes the beginning of the zone of overmodulation of five-phase inverter with scalar control, is equal to = m_ov1 F_ten-step=41.3Hz.

The first stage of synchronous overmodulation control of five-phase inverters is characterized by smooth transition from the control scheme, based on combined application of all medium and large space vector (in total 22 vectors, including two zero vectors, are used in this scheme), to the control scheme, based on ten large vectors and two zero vectors. Taking into consideration the fact, that the largest fundamental peak phase-to-neutral voltage, which is characterizing the boundary of this overmodulation sub-zone and corresponds to the radius of the larges circle that can be inscribed within the decagon (see Fig. 2), is equal to 0.6155V_dc 6., the second threshold modulation index in accordance with (1) is determined as

m_ov2 = 0.6155/0.6366 = 0.967 (3)

The second threshold fundamental frequency , which characterizes the upper limit of the first control stage in the zone of overmodulation of five-phase inverter with scalar control (F_ten-step=50Hz), is equal to = m_ov2 F_ten-step=48.34Hz.

In order to provide smooth transition to the PWM scheme with ten large space vectors, basic control correlations include during the first overmodulation sub-zone special linear coefficients of overmodulation K_ov1 (4), connecting modulation index m with two threshold indices m_ov1 and m_ov2

K_ov1 = 1- (m - m_ov1)/( m_ov2 - m_ov1) (4)

So, during the first control stage of the overmodulation zone, between the fundamental frequencies and, a smooth linear decrease of widths of the and active signals (17., see also Table I) is observed in accordance with (5):

(5)

Fig. 8 (F=43Hz, m=0.86) and Fig. 9 (F=46Hz, m=0.92) illustrate smooth synchronous voltage control of five-phase inverter during the first control stage of the overmodulation zone.

Fig. 10 presents basic voltage waveforms of the five-phase inverter with synchronized PWM at the upper boundary of the first stage of inverter control in the zone of overmodulation (F=48.35Hz, m=m_ov2=0.967). It is necessary to mention, that during all stages of overmodulation control of five-phase inverters there are inevitable distortion of output voltage spectra 6.,9.. But and in this case all presented in Figs. 8-10 voltage waveforms have quarter-wave symmetry during over-modulation control, and its spectra do not contain even harmonics and sub-harmonics, that is especially important for high power/high current applications.

Fig. 8. Pole voltages V_a - V_c, line and phase voltages V_ac and V_an, of five-phase system at the first part of the zone of overmodulation (F=43Hz, m=0.86)

Fig. 9. Pole voltages V_a - V_c, line and phase voltages V_ac and V_an, of five-phase system at the first stage of the zone of overmodulation (F=46Hz, m=0.92).

Fig. 10. Pole voltages V_a - V_c, line and phase voltages V_ac and V_an, of five-phase system at the boundary of the first stage of the zone of overmodulation (F=48.35Hz, m=0.967)

The Second Stage of Synchronous Overmodulation Control

To provide full utilization of the available DC-bus voltage in five-phase systems, it is necessary to carry out smooth transition from the control scheme, based on ten large space vectors application, to the ten-step mode. In this case, the next (second) stage of the overmodulation control is connected with smooth decrease until close to zero values of durations of all notches. By the analogy with three-phase inverters, where value of the corresponding threshold modulation index corresponds to the medium part of the corresponding zone of overmodulation, for five-phase systems the rational value of the third threshold modulation index (and relative value of the third threshold fundamental frequency F_ov3) can be determined as: m_ov3=(m_ten-step+m_ov2)/2=(1+0.967)/2=0.984, F_ov3= 0.984F_ten-step. In particular, if F_ten-step=50Hz, F_ov3=49.2Hz.

Basic PWM correlations (6) and (7) include special linear coefficient of overmodulation K_ov2 (8) in this control sub-zone, connecting modulation index m with the threshold modulation indices m_ov2 and m_ov3. So, realization of control functions (6)-(8) provides smooth synchronous decrease of durations of all notches (6) together with simultaneous increasing of widths of the total active -signals (7) until the maximum duration, equal to the duration of sub-cycle .

(6)

(7)

K_ov2 = 1- (m - m_ov2)/( m_ov3 - m_ov2) (8)

As an illustration of the PWM control in the second overmodulation sub-zone, Fig. 11 presents basic voltage waveforms of the five-phase inverter with synchronized PWM (F=48.75Hz, m=0.975). The switching frequency is 3kHz.

Fig. 11. Pole voltages V_a - V_c, line and phase voltages V_ac and V_an, of five-phase system at the second stage of the zone of overmodulation (F=48.75Hz, m=0.975)

The Third Stage of Synchronous Overmodulation Control

At the boundary frequency between the second and the last (the third) stage of overmodulation control of five-phase inverter there are no notches between active switching states, and total durations of these -signals include only the -signals like a minor part of total active switching states (see Figs. 4-5). Fig. 12 illustrates this control stage and shows basic voltage waveforms of the modulated five-phase inverter at the third threshold overmodulation frequency (F=F_ov3=49.2Hz, m=m_ov3=0.984).

In order to provide smooth final pulse dropping process of five-phase system with scalar control, with smooth transition from the PWM control mode to the ten-step mode, the third coefficient of overmodulation K_ov3 (9) is applied for equations (10)-(11), providing smooth decrease until zero of the widths of the -signals in this control sub-zone:

K_ov3=1-(m-m_ov3)/(m_ten-step-m_ov3)=1-(m-m_ov3)/(1-m_ov3) (9)

(10)

(11)

Fig. 13 shows the pole, line and phase-to-neutral voltages of five-phase system operating in the third sub-zone of the overmodulation region (F=49.6Hz, m=0.992). The switching frequency is 3kHz. The presented voltages have quarter-wave symmetry, and its spectra do not include even harmonics and subharmonics. Fig. 14 presents basic voltage waveforms of five-phase inverter at the maximum fundamental frequency =50Hz, at the ten-step operation mode (m=1).

The proposed algorithm of synchronized modulation of five-phase inverter with standard scalar control mode provides both full utilization of the DC bus voltage in the system and linearity of the fundamental voltage during the whole control range. Fig. 15 illustrates this fact and presents variation of magnitude of the first harmonic of the phase-to-neutral voltage versus modulation index m.

Fig. 12. Pole voltages V_a - V_c, line and phase voltages V_ac and V_an, of five-phase system at the third threshold overmodulation frequency (F=49.2Hz, m=0.984)

Fig. 13. Pole voltages V_a - V_c, line and phase voltages V_ac and V_an, of five-phase system at the third stage of the zone of overmodulation (F=49.6Hz, m=0.992)

Fig. 14. Pole voltages V_a - V_c, line and phase voltages V_ac and V_an, of five-phase system at the ten-step operation mode (F=50Hz, m=1)

Fig. 15. Magnitude of the fundamental phase-to-neutral voltage versus modulation index of five-phase inverter with synchronized PWM

CONCLUSION

Novel three-stage algorithm of synchronous PWM control of five-phase inverter in the zone of overmodulation has been proposed and investigated. It provides smooth transition from linear modulation range to the ten-step operation mode of five-phase system at the maximum fundamental frequency. The presented control scheme provides both full utilization of the DC-bus voltage of the system at the maximum fundamental frequency and linearity of the fundamental voltage during the whole control range. The spectra of the phase-to-neutral voltage of five-phase power conversion systems with algorithms of synchronized PWM do not contain even harmonics and sub-harmonics in the zone of overmodulation, which is especially important for high power/high current applications.

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