Field oriented control based on a 24-sector vector space decomposition for dual three-phase PMSM applied on electric ship propulsion

Received Aug 6, 2019 Revised Apr 13, 2020 Accepted Apr 23, 2020 A Field Oriented Control (FOC) strategy based on a 24-Sector Vector Space Decomposition (24SVSD) technique used to control a Dual Three-Phase Permanent Magnet Synchronous Motor (DTP-PMSM) applied on electric ship propulsion prototype is presented in this paper. This machine is supplied by Dual Three-Phase Voltage Source Inverter (DTP-VSI). This study carried out on these multiphase machines has revealed that the large zero sequence harmonic current components on (z1, z2) subspace, constitutes major drawbacks; despite their advantages. The machine’s dynamic model is accomplished in three two-dimensional and orthogonal subspaces. In order to reduce more extra stator harmonic currents that produces losses, the current work is also concering of to consider the three different approaches for the sake of boosting the machine’s efficiency. Thus, the principle selection of the reference voltage vector and the time calculation method are presented in details. The proposed strategy effectiveness is validated by simulation results.


INTRODUCTION
Nowadays, the multiphase machines provide special characteristics compared with the three-phase counterparts. These machines offer the power segmentation and ameliorates the reliability and availability of the drive system. In electrical machines, the winding factors can be augmented, and the torque ripple reduced by using multiphases. Therefore, one of the perspective structures of multiphase systems is a Dual Three Phase (DTP) motor fed by two three-phase voltage source inverters (DTP-VSI).
The synchronous machines are widely used in industry, particularly the permanent magnet synchronous machines (PMSM). Thus, the DTP Machines notably PMSM drives has become very attractive in high performance control [1]. And DTP motor drives are widely used for electric propulsion ships, locomotive traction, electric/hybrid electric vehicles and aircrafts [2][3][4][5][6]. Several Space Vector Pulse Width Modulation (SVPWM) techniques for six-phase synchronous and induction drive system have been studied and developed in [7][8][9], such as conventional SVPWM and vector space decomposition (VSD) theory.
In order to control the voltage vectors components (z1, z2) subspace, a Field Oriented Control (FOC) strategy applying 24-Sector Vector Space Decomposition (24SVSD) is presented and established, which can be divided into 24-sector Modulation strategy. For six-phase induction machine, the conventional VSD technique was reported in [1,10]. Then, the 12-sector Modulation is developed in [2]. According to the VSD control [1], the original machine's characteristics model, is transformed into three two-dimensional orthogonal subspaces in the stationary reference plane, identified as the torque-component (α, β) subspace, the harmonic-component (z 1 , z 2 ) subspace and the zero-sequence (o 1 , o 2 ) subspace. [11][12][13][14][15]. It should be noted, the fundamental components contribute to the conversion of the electromechanical energy. Whereas, the (z 1 , z 2 ) and (o 1 , o 2 ) current components do not contribute to the last conversion. Therefore, they are restricted by the stator resistance and leakage inductance [16][17][18]; also, the extra current harmonics only produce stator losses. To reduce these extra current harmonics, a Field Oriented Control strategy using a 24-Sector VSD (24SVSD) technique is proposed and simulated in this paper. Thus, three approaches have been also considered and developed. Finally, the simulation results show that the adopted approach reduces significally the harmonic components in (z 1 , z 2 ) subspace.

MACHINE MODEL
The structure of the DTP-PMSM windings fed by the VSI-DC sources, presenting spatially dual windings three-phase stator are shifted by 30 electrical degrees with two neutral independent points, as shown in three

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According to the VSD theory [1], the transformation matrix can be expressed as follows: (1) Applying the transformation matrix T, the Dual Three-Phase motor can be represented in three twodimensional decoupled mutually orthogonal planes (α-β), (z 1 , z 2 ) and (o 1 , o 2 ).
The (α-β) current components contribute to the electromechanical conversion. However, the currents in (z 1 , z 2 ) and (o 1 , o 2 ) do not contribute efficiently in the electromechanical conversion which produces stator losses.
The transformation matrix must be used to represent the stationary (α, β) subspace in the rotating frame (d,q) which is shown in the following [20]: The electric and mechanic equations for this model in (d-q) plane are expressed by:  ISSN: 2088-8694 where: p is the number of pair poles.

RESEARCH METHOD "FOC BASED ON VSD TECHNIQUE" 3.1. DTP-SVPWM Strategy
The Pulse Width Modulation (PWM) techniques have been the focus of many researches. The main goal of the PWM is to control the inverter output voltage and to reduce the harmonic content in the output voltage. To increase the output voltage, the space vector pulse width modulation (SVPWM) technique is applied. This technique is used for adjustable speed drives. By using sectors in SVPWM, the SVPWM can identify the position of reference vector and the switches can be operated by the mentioned sectors. Thus, the field orientation approach consists on need of the (α-β) subplane as being reference stator voltage vector produced by the control system. Then, the harmonics in (z 1 -z 2 ) and (o 1 , o 2 ) subspaces produces only losses. So as to reduce these harmonics. The average voltage vectors produced in the two subspaces must be fixed at zero [21].
In the Figure 4, two Proportional Integral regulators are used in the current control structure which generates the direct and indirect voltage reference, especially V d * and V q * . Furthermore, the closed-loop speed regulator yields the indirect-axis current. The speed control can be achieved by controlling the q-axis current iq provided that the d-axis current component i d is kept at zero [22][23][24]. Therefore, the control strategy of id=0 is a relatively simple method in ordinary 3-phase PMSM control strategy. When the d-axis stator current winding id=0, the mechanic (11) becomes:

24-Sector Vector Spae Decomposition (24SVSD)
The FOC technique for the machine drive is shown in Figure 4. The relationship between phase voltages and device switch states is as follows: (14) Figure 3 shows the wingdings and DTP-VSI fed DTP-PMSM. A combinatorial analysis shows 64 switching modes. Therefore, 64 different voltage vectors can be applied to the machine. Each vector is represented by a decimal number, corresponding to binary numbers, considered in the order [S a1 S b1 S c1 S a2 S b2 S c2 ]. Thus, the both figures 5 and 6 shows the schematic diagram of the space vectors based on VSD respectively in (α-β) and (z 1 -z 2 ) subspaces. Regarding the Figures 5 and 6, the (α-β) voltage vectors can be decomposed into four dodecagons with different magnitudes (from innermost to outermost: D 1 , D 2 , D 3 , D 4 ) [25]. Their magnitudes are identified by the following relationship: The maximum amplitude (α-β) voltage vectors become the minimum amplitude (z 1 -z 2 ) voltage vectors, whereas the others keep the same magnitude. In Figure 7, the 12 voltage vectors and 24 ones having respectively maximum and half magnitude divide the (α-β) subspace into 24 sectors. The voltage vectors selected allows having the smallest amplitude vectors in the (z 1 -z 2 ) plane, which guarantees the minimum current harmonics content in the (z 1 -z 2 ) plane. Thus, leading to the decrease losses [3].
The combination of the maximum magnitude (α-β) vectors and the ones with half magnitude allows synthetizing the reference voltage vector (Figure 7). According to the location of the reference voltage vector in the (α-β) subspace ( Figure 5), the proposed strategy uses three vectors having the maximum magnitude and the fourth vector having half magnitude. In the sector S 1 , the voltage vectors (37-36-52) combined with the fourth voltage vector (60) and zero voltage vectors (7-56) are employed. The time corresponding to each voltage vector is determinated by the following relationship: where: • : is the sample time; • t 1 , t 2 , t 3 and t 4 : are the times corresponding to the four voltage vectors; • : is the projection of k th voltage vector on the x-axis (x= α, β, z 1 , z 2 ).

SIMULATION RESULTS AND DISCUSSION
The simulation tests were accomplished by using a 3-KW machine applied on electric ship prototype, whose parameters are indicated in Table 3 [19].
In all the simulation tests, the load torque is kept at 15 N.m, the machine speed is fixed at 300 rpm and the inverter switching frequency is maintained at the same frequency 5 kHz. The figures 9 -14 show the simulation results of the proposed strategy.
As shown in Figure 9, in a period of 24 sectors, it is observed that 6 sectors remain with no switching for approaches (b) and (c). This allows a 25% reduction of the switching compared with approach (a) where there is a continuous switching mode. Thus, a significant reduction of the switching losses is accomplished.  T  I  M  E  S   t 1  -T 1  T 4  -T 6  T 8  -T 10  T 5  -T 11  T 9  -T 12  T 2  -T 3  T 7  t 2  -T 2  -T 1  -T 7  -T 6  T 4  -T 10  T 8  -T 11  T 5  -T 12  T 9  -T 3  t 3  -T 3  -T 5  T 1  -T 9  T 6  -T 2  T 10  -T 7  T 11  T 4  T   In Figure 10, the stator phase current resultant from the three presented approaches is extremely sinusoidal. The Figure 11 shows the harmonic current analysis significantly decreased to 1.05%. Thus, the harmonic analysis shows that the THD is slightly lower in approach (1) (2) and (3). It should be noted that these stator current components will only produce losses since they have no contribution in the elaboration of air gap flux. To be noted that for the three approaches envisaged; THD current and switching number, the following table: Table 4. Swiching sequences evaluation Figure 12 shows that the stator currents in the subspace (z 1 -z 2 ) are too weak (amplitude 0.2 A) compared with the stator currents fundamental (amplitude 8 A). In Figure 13, the harmonic currents do not  Figure 14 shows that the machine speed achieves its reference with satisfying static and dynamic performance.

CONCLUSION
In this paper, a FOC strategy based on the 24-Sector Vector Space Decomposition (24SVSD) appropriate to the control machine drive system applied on electric ship prototype has been presented in order to reduce the current harmonics. The chosen four voltage vectors divide the (α-β) subplane into 24 sectors which is related to electromechanical energy conversion and each sector is π/12 rad. This modulation approach is required to efficiently eliminates the stator current harmonics in (z 1 -z 2 ) and (o 1 , o 2 ) subspaces. The simulation results show that the proposed technique allows the notable reduction current harmonics. Thus, it is confirmed that the suggested strategy offers lower harmonic current analysis. This explains by an effective reduction to 1.05%. Also, it is worth noting that a significant reduction in the stator current harmonics is associated with a 25% decrease in the switching losses corresponding to approaches (2) and (3), is observed at the level of the converter-machine combination. The results attest the effectiveness of the suitable modulation. Performing experimentation by an implementation on Digital Signal Processor (DSP) board of the proposed strategy will be the subject of the further works.