Advances in Dual-Three-Phase Permanent Magnet Synchronous Machines and Control Techniques

Multiphase electrical machines are advantageous for many industrial applications that require a high power rating, smooth torque, power/torque sharing capability, and fault-tolerant capability, compared with conventional single three-phase electrical machines. Consequently, a significant number of studies of multiphase machines has been published in recent years. This paper presents an overview of the recent advances in multiphase permanent magnet synchronous machines (PMSMs) and drive control techniques, with a focus on dual-three-phase PMSMs. It includes an extensive overview of the machine topologies, as well as their modelling methods, pulse-width-modulation techniques, field-oriented control, direct torque control, model predictive control, sensorless control, and fault-tolerant control, together with the newest control strategies for suppressing current harmonics and torque ripples, as well as carrier phase shift techniques, all with worked examples.


Introduction
Compared with conventional single-three-phase electrical machines, multiphase electrical machines are advantageous for many industrial applications that require high power rating, smooth torque, power/torque sharing capability, and fault-tolerant capability. In recent decades, there has been a significant and increasing number of published technical and review papers that discuss the topologies, modelling methods, control strategies, pulse-widthmodulation (PWM) techniques, and applications of multiphase electrical machines [1][2][3][4][5][6][7][8][9].
Among these existing reviews, winding layouts, space harmonic cancellation, and equivalent circuits of multiphase induction machines (IMs) are discussed in [1]. The modelling and field-oriented control (vector control) of multi-three-phase (MTP) IMs are reviewed and compared with those of their single-three-phase counterparts in [2]. The PWM techniques and direct torque control (DTC) of dual-three-phase (DTP) IMs are studied in [3]. Additionally, the control strategies of five-phase and asymmetric six-phase IMs, and fault tolerant control strategies are reviewed in [4,5]. The use of multiphase machines in variable-speed applications and automotive traction applications are introduced in [6] and [9], respectively. The research in [7] presents multilevel inverters and corresponding PWM techniques as well as the exploitation of additional degrees of freedom. The review of model predictive control (MPC) in multiphase machine systems is covered in [8] and [9]. However, the existing reviews mainly focus on the development of multiphase IMs without a comprehensive coverage of multiphase permanent magnet (PM) machines. This paper presents an overview of multiphase PM synchronous machines (PMSMs) and drive control techniques. This paper attempts to provide an up-to-date review of MTP PMSM systems, with a focus on DTP PMSMs, as tabulated in Table 1. Several important topics are grouped and introduced with reference to the existing body of research. It should be noted that while the major topics of MTP PMSM systems in the existing review work are covered, this paper provides a comprehensive overview of machine topologies, modelling methods, PWM techniques, field oriented control, direct torque control, model predictive control, sensorless control, and fault-tolerant control, together with the newest control strategies for suppressing current harmonics and torque ripples, as well as carrier phase shift techniques. Worked examples, which were developed at the University of Sheffield, are presented in each section, whenever possible, to support the discussions.

Various MTP and DTP PMSM Topologies
There are many machine topologies for MTP PMSMs. Various surface-mounted and interior PM rotor topologies can be employed [53] and will not be described here, since for MTP PMSMs, the major difference is in the stator winding configurations, which are associated with slot/pole number combinations. In this section, the focus will be on DTP PMSMs according to: • Slot/pole number combinations; • Angle displacement between two sets of three-phase windings; • Coil pitch of windings.
Since there are two sets of three-phase windings in DTP PMSMs, the slot number should be a multiple of six to install the windings. For DTP PMSMs with slot number N s (N s = 6 k, k is an integer number) and pole number 2p (p is the number of pole pairs), various angle displacements between two sets of three-phase windings and coil pitches can be chosen, according to the coil electromotive force (EMF) phasor diagram.
It should be noted that the angle displacements are analyzed based on the unit machine of each machine (the greatest common divisor (N s , 2p) = 1). For a unit machine, the feasible angle displacements can be 0 • or 360 • /N s (60 • /k), where k is an integer. Besides 0 • and 60 • /k, if k is even, 30 • is another feasible angle displacement. Thus, the feasible angle displacements in DTP PMSMs considering different slot/pole number combinations are shown in Table 2. For example, the feasible angle displacements in a 24-slot/20-pole PMSM are 0 • and 30 • , the same as those in a 12-slot/10-pole PMSM. In Table 2, the cells of the machines that share the same unit machine are marked with the same background color, and the relationship between the background colors and the corresponding unit machines are also shown beneath Table 2.    It is well known that a winding factor (Kw) is the product of the distribution factor (Kd) and pitch factor (Kp). For all DTP PMSMs, the winding factor is computed based on its unit machine. For a unit DTP PMSM, the distribution factor is determined by its slot/pole number combination and angle displacement, and the coil pitch factor can be calculated by its slot/pole number combination and coil pitch number.
With a specific slot/pole number combination and angle displacement, the distribution factor in DTP PMSMs can be obtained. Since the symmetrical characteristics of the coil EMF phasor diagrams are different when k is odd or even, the calculation of distribution factors should take "k is odd" and "k is even" into consideration, respectively. When k is odd, take an 18-slot/14-pole PMSM as an example, the coil EMF phasor diagrams with 0° and 20° angle displacements are shown in Figure 2a It is well known that a winding factor (K w ) is the product of the distribution factor (K d ) and pitch factor (K p ). For all DTP PMSMs, the winding factor is computed based on its unit machine. For a unit DTP PMSM, the distribution factor is determined by its slot/pole number combination and angle displacement, and the coil pitch factor can be calculated by its slot/pole number combination and coil pitch number. With a specific slot/pole number combination and angle displacement, the distribution factor in DTP PMSMs can be obtained. Since the symmetrical characteristics of the coil EMF phasor diagrams are different when k is odd or even, the calculation of distribution factors should take "k is odd" and "k is even" into consideration, respectively. When k is odd, take an 18-slot/14-pole PMSM as an example, the coil EMF phasor diagrams with 0 • and 20 • angle displacements are shown in Figure 2a,b, respectively. ** Unfeasible slot/pole combinations for DTP PMSMs.
It is well known that a winding factor (Kw) is the product of the distribution factor (Kd) and pitch factor (Kp). For all DTP PMSMs, the winding factor is computed based on its unit machine. For a unit DTP PMSM, the distribution factor is determined by its slot/pole number combination and angle displacement, and the coil pitch factor can be calculated by its slot/pole number combination and coil pitch number.
With a specific slot/pole number combination and angle displacement, the distribution factor in DTP PMSMs can be obtained. Since the symmetrical characteristics of the coil EMF phasor diagrams are different when k is odd or even, the calculation of distribution factors should take "k is odd" and "k is even" into consideration, respectively. When k is odd, take an 18-slot/14-pole PMSM as an example, the coil EMF phasor diagrams with 0° and 20° angle displacements are shown in Figure 2a It can be seen in Figure 2 that the relative layouts of the EMF phasors used in one three-phase winding set are identical for these two angle displacements. Thus, the distribution factors of the 18 slot/14 pole PMSM with 0° and 20° angle displacements are both 0.945, according to the coil EMF phasor diagram. Similarly, it can be further deduced that when k is odd, the angle displacement between two sets of three-phase windings does not affect the distribution factor. Furthermore, if k is odd, the distribution factor can be calculated as   It can be seen in Figure 2 that the relative layouts of the EMF phasors used in one threephase winding set are identical for these two angle displacements. Thus, the distribution factors of the 18 slot/14 pole PMSM with 0 • and 20 • angle displacements are both 0.945, according to the coil EMF phasor diagram. Similarly, it can be further deduced that when k is odd, the angle displacement between two sets of three-phase windings does not affect the distribution factor. Furthermore, if k is odd, the distribution factor can be calculated as k sin π 6k = sin π 6 N s 6 sin π N s , k is odd. (1) When k is even, the angle displacement can be 0 • , 60 • /k and 30 • . For example, in a 24 slot/14 pole PMSM, 0 • , 15 • , and 30 • displacements are feasible. The EMF phasor diagrams of the machine with these angle displacements are shown in Figure 3.
When k is even, the angle displacement can be 0°, 60°/k and 30°. For example, in a 24 slot/14 pole PMSM, 0°, 15°, and 30° displacements are feasible. The EMF phasor diagrams of the machine with these angle displacements are shown in Figure 3. Comparing the coil EMF phasor diagrams in Figures 2 and 3, it can be observed that the layouts of the coil EMF phasors utilized in one winding set in a 24 slot/14 pole DTP PMSM are very different with different angle displacements. According to Figure 3, the distribution factors of the machine with 0°, 15°, and 30° angle displacements can be computed as 0.958, 0.966, and 0.991, respectively. Based on the analyses above, it could be   Comparing the coil EMF phasor diagrams in Figures 2 and 3, it can be observed that the layouts of the coil EMF phasors utilized in one winding set in a 24 slot/14 pole DTP PMSM are very different with different angle displacements. According to Figure 3, the distribution factors of the machine with 0 • , 15 • , and 30 • angle displacements can be computed as 0.958, 0.966, and 0.991, respectively. Based on the analyses above, it could be inferred that for the N s -slot (N s = 6 k, k is even) DTP PMSMs with β = 0 • , β = 60 • /k and β = 30 • , the numbers of EMF phasors used in the distribution factor calculation are k, k/2, and k/2, respectively, and the angles between the phasors are 60 • /k, 120 • /k, and 60 • /k, respectively. Overall, the distribution factor calculation in this condition can be summarized as From Equation (2), it can be deduced that for the 24 slot/14 pole DTP PMSM, when k is even, In DTP PMSMs with a 30 • angle displacement, the superiority in distribution factors over the counterparts with 0 • angle displacement contributes to higher fundamental EMF and larger output torque. In accordance with Equations (1) and (2), the distribution factors of DTP PMSMs with different slot/pole number combinations considering different angle displacements are provided in Table 3. In DTP PMSMs, the calculation of the pitch factor is the same as that in single-threephase PMSMs. It is well known that a pitch factor can be calculated by the angle difference between the pole-pitch and coil-pitch, which can be determined by the slot/pole number combination and the coil pitch number, y. The diagram of the pole-pitch and coil-pitch in PMSMs is shown in Figure 4. Accordingly, the pitch factor for the v-th harmonic is where y is the coil pitch in terms of the number of slot pitches.

= cos =
where y is the coil pitch in terms of the number o Since the working harmonic is the p-th harm pitch factor of the working harmonic is

= cos
With Equation (4), the pitch factors of DT number combinations considering coil pitch nu Table 4. In Table 4, the high pitch factors (>0.86 color. It should be mentioned that some values a DTP winding configurations are not considered ble-check whether the slot/pole and coil pitch n utilizing them. Since the working harmonic is the p-th harmonic in N s -slot/2p-pole DTP PMSMs, the pitch factor of the working harmonic is With Equation (4), the pitch factors of DTP PM machines with different slot/pole number combinations considering coil pitch numbers varying from 1 to 4 are shown in Table 4. In Table 4, the high pitch factors (>0.866) are marked with a pink background color. It should be mentioned that some values are 0 in Table 4, because the feasibility of DTP winding configurations are not considered in the calculation. It is necessary to double-check whether the slot/pole and coil pitch number combinations are feasible before utilizing them.  Pitch factor = 0 suggests unfeasible winding configuration.

3.
The feasibility of the specific slot/pole and pitch number combination is determined in Tables 2 and 4.
In addition to the calculated results shown in Table 4, the pitch factors in some slot/pole number combinations can be further improved with other coil pitch numbers. For example, in a 12 slot/2 pole PMSM, the pitch factor is 1 with y = 6. Overall, the pitch factors (K p ) of DTP PM machines with different slot/pole number combinations and optimized coil pitch numbers are summarized in Table 5.  To achieve the pitch factors in Table 5, the corresponding coil pitch numbers are shown in Table 6. In Table 6, the cells using the same coil pitch number are marked with the same background color. However, it should be noted that the coil pitch numbers shown in Table 6 ** Unfeasible slot/pole combinations for DTP PMSMs.
To achieve the pitch factors in Table 5, the corresponding coil pitch numbers are shown in Table 6. To achieve the pitch factors in Table 5, the corresponding coil pitch numbers are shown in Table 6. In Table 6, the cells using the same coil pitch number are marked with the same background color. However, it should be noted that the coil pitch numbers shown in Table 6 were chosen to enhance the pitch factor only. In practice, it is preferable to use a lower coil pitch number, especially 1, because a lower coil pitch number means a shorter end-winding length. When y = 1, the tooth coil concentrated windings can be utilized to significantly ease fabrication. Hence, it is necessary to consider coil pitch number thoughtfully in the design of DTP PMSMs. According to the distribution factors and pitch factors calculated above, the winding factors of DTP PMSM with different slot/pole number combinations and different angle displacements are summarized in Table 7.
In Table 6, the cells using the same coil pitch number are marked with the same background color. However, it should be noted that the coil pitch numbers shown in Table 6 were chosen to enhance the pitch factor only. In practice, it is preferable to use a lower coil pitch number, especially 1, because a lower coil pitch number means a shorter end-winding length. When y = 1, the tooth coil concentrated windings can be utilized to significantly ease fabrication. Hence, it is necessary to consider coil pitch number thoughtfully in the design of DTP PMSMs. According to the distribution factors and pitch factors calculated above, the winding factors of DTP PMSM with different slot/pole number combinations and different angle displacements are summarized in Table 7.  Compared with [46] where the winding factors only take a 0° angle displacement into consideration, the winding factors shown considered various possible angle displacements. Moreover, the pitch factors can be further improved with the optimized coil pitch numbers (as shown in Tables 4 and 5) in some slot/pole number combinations. Thus, the winding factors can also be increased in these slot/pole number combinations. For instance, the winding factor for a 36 slot/10 pole DTP PMSM with a 0° angle displacement is 0.924 (y = 3), while it can be improved to 0.942 with y = 4.
Based on the analyses above, the topologies of the DTP PMSMs proposed and/or analyzed in existing research are summarized in Table 8. The angle displacements and coil pitch numbers in these machines are also provided in Table 8. The blank cells in Table 8 suggest more feasible topologies in this area. Compared with [46] where the winding factors only take a 0 • angle displacement into consideration, the winding factors shown considered various possible angle displacements. Moreover, the pitch factors can be further improved with the optimized coil pitch numbers (as shown in Tables 4 and 5) in some slot/pole number combinations. Thus, the winding factors can also be increased in these slot/pole number combinations. For instance, the winding factor for a 36 slot/10 pole DTP PMSM with a 0 • angle displacement is 0.924 (y = 3), while it can be improved to 0.942 with y = 4.
Based on the analyses above, the topologies of the DTP PMSMs proposed and/or analyzed in existing research are summarized in Table 8. The angle displacements and coil pitch numbers in these machines are also provided in Table 8. The blank cells in Table 8 suggest more feasible topologies in this area. winding factors can also be increased in these slot/pole number combinations. For instance, the winding factor for a 36 slot/10 pole DTP PMSM with a 0° angle displacement is 0.924 (y = 3), while it can be improved to 0.942 with y = 4. Based on the analyses above, the topologies of the DTP PMSMs proposed and/or analyzed in existing research are summarized in Table 8. The angle displacements and coil pitch numbers in these machines are also provided in Table 8. The blank cells in Table 8 suggest more feasible topologies in this area.

Modelling of MTP PMSMs
The modelling of MTP PMSMs can be divided into two categories: multiple individual three-phase models [54][55][56][57][58][59][60][61] and vector space decomposition (VSD) models [62][63][64][65][66][67]. The former considers the machine as multiple single-three-phase submachines, and the model is straightforwardly derived from the synchronous dq-axis model of the three-phase machine. The VSD model uses a mathematical transformation to separate the variables of different orders into multiple orthogonal subspaces. In this context, it is convenient to formulate the machine model by using a set of decoupled subspaces. The advantages and disadvantages of the two modelling approaches are summarized in Table 9. 1. Less capable of dealing with the unbalance issues due to asymmetry among the sets. 2. More complex to achieve active power/torque sharing.

Multiple Individual Three-Phase PMSM Model
Multiphase PMSMs are composed of multiple individual three-phase sets, and thus the model of MTP PMSMs can be considered as the sum of the models of all three-phase machines with extra coupling voltage terms. For each three-phase set, the machine model in a synchronous dq frame is conducted. In particular, the two individual three-phase model of a DTP PMSM is shown in Figure 5.
Multiphase PMSMs are composed of multiple individual three-phase sets, a the model of MTP PMSMs can be considered as the sum of the models of all thre machines with extra coupling voltage terms. For each three-phase set, the machine in a synchronous dq frame is conducted. In particular, the two individual thre model of a DTP PMSM is shown in Figure 5. The corresponding voltage equations are as follows [58]: The corresponding voltage equations are as follows [58]: where R s is the stator resistance, ω e is the electrical angular speed, and ψ f d is the PM flux linkage; s represents the derivative; u d1 , u q1 , i d1 , and i q1 represent the dq-axis voltages and currents of the first winding set, and u d2 , u q2 , i d2 , and i q2 represent the voltages and currents of the second winding set. The values L d and L q are the dq-axis inductances. The values M d and M q are the dq-axis mutual inductances, which denote the magnetized coupling between the two winding sets. The electromagnetic torque in the multiple individual three-phase model can be expressed as and the inductances are: where L σ is the leakage inductance.

VSD Machine Model
The vector space decomposition (VSD) approach [62] is prevailing in the modeling of multiphase machines because the complex high-order electromagnetic system can be clearly simplified as twoorder subsystems in many decomposed subspaces, which makes it easy to achieve independent current regulation in these decoupled subspaces. The VSD transformation for the DTP machine is given by: where F can be the voltage, current, or flux linkage. The voltage and current can be decomposed into the αβ and z 1 z 2 subspaces by using (9). (see Figures 6 and 7). After the VSD transformation, the DTP PMSM can be described by the following models: u αβ = R s i αβ + sψ αβ (10) ψ αβ = L sαβ i αβ + ψ f e jθ e (11) where u, i, and ψ are the voltage, current, and flux linkage. The value ψ f is the PM flux linkage. The value θ e is the electrical rotor angle.

Electromagnetic Performance Analysis under Healthy and Open-Circuit Conditions
According to the analyses presented above, a 24 slot/10 pole PMSM and a 24 slot/22 pole PMSM can be equipped with DTP windings with 0°, 15°, and 30° angle displacements, respectively [46,47]. The coil EMF phasors and the winding arrangements of the two machines with different winding configurations are shown in Figures 8 and 9. It should be mentioned that the coil pitch numbers in the two machines are 2 and 1 in order to obtain high winding factors, respectively. The winding connections of Phase A in the machines with different configurations are summarized in Table 10.

Electromagnetic Performance Analysis under Healthy and Open-Circuit Conditions
According to the analyses presented above, a 24 slot/10 pole PMSM and a 24 slot/22 pole PMSM can be equipped with DTP windings with 0°, 15°, and 30° angle displacements, respectively [46,47]. The coil EMF phasors and the winding arrangements of the two machines with different winding configurations are shown in Figures 8 and 9. It should be mentioned that the coil pitch numbers in the two machines are 2 and 1 in order to obtain high winding factors, respectively. The winding connections of Phase A in the machines with different configurations are summarized in Table 10.       Figure 6 shows the equivalent circuits in two subspaces and the definition of the axes. The value e αβ represents the back-EMF mapped in the αβ subspace. It should be noted that the energy conversion occurs in the αβ subspace, and the electromagnetic torque is only relevant to the variables in this subspace. Unlike those in the single three-phase machine, the current harmonics mapped in z 1 z 2 subspace are limited by the resistance and leakage inductance, which are usually small. This causes serious current harmonics in this subspace, as the 5th and 7th values shown in Figure 7.

Electromagnetic Performance Analysis under Healthy and Open-Circuit Conditions
According to the analyses presented above, a 24 slot/10 pole PMSM and a 24 slot/22 pole PMSM can be equipped with DTP windings with 0 • , 15 • , and 30 • angle displacements, respectively [46,47]. The coil EMF phasors and the winding arrangements of the two machines with different winding configurations are shown in Figures 8 and 9. It should be mentioned that the coil pitch numbers in the two machines are 2 and 1 in order to obtain high winding factors, respectively. The winding connections of Phase A in the machines with different configurations are summarized in Table 10.
With the winding arrangements shown in Figures 8 and 9 and Table 10, the winding factors of the DTP PMSMs with different slot/pole number combinations, different angle displacements, and different coil pitch numbers are given in Table 11. From the winding factors of the fundamental components shown in Table 11, it can be seen that 24 slot/22 pole machines feature higher winding factors than their 24 slot/10 pole counterparts. Since the higher winding factor suggests higher fundamental back-EMF components and average output torque, it is assumed that the 24 slot/22 pole DTP PMSM features the highest output torque performance. Furthermore, among the same machines with 0 • , 15 • , and 30 • angle displacements, the 30 • machine features the highest winding factor and the 15 • machine features a higher winding factor than the 0 • machine.
To demonstrate the analyses of the winding factors in this section and to show the electromagnetic performances of DTP PMSMs, the back-EMF and torque characteristics of 24-slot/22-pole DTP PMSMs with tooth-coil (y = 1) and different angle displacements under different conditions are also presented. For the DTP PMSMs with 0 • , 15 • , and EMF components and average output torque, it is assumed that the 24 slot/22 pole DTP PMSM features the highest output torque performance. Furthermore, among the same machines with 0°, 15°, and 30° angle displacements, the 30° machine features the highest winding factor and the 15° machine features a higher winding factor than the 0° machine.
To demonstrate the analyses of the winding factors in this section and to show the electromagnetic performances of DTP PMSMs, the back-EMF and torque characteristics of 24-slot/22-pole DTP PMSMs with tooth-coil (y = 1) and different angle displacements under different conditions are also presented. For the DTP PMSMs with 0°, 15°, and 30° angle displacements under healthy conditions, the stator MMFs and spectra of each condition are shown in Figure 10.   It can be seen that the amplitudes of the working harmonics (the 11th harmonics) are similar in the three machines, and the amplitudes of the different angle displacements correspond to their winding factors. Although the MMF distributions of the three angle displacements look similar, their harmonic contents are different. The (12 k ± 5)-th (k is an integer) harmonics, i.e., the 5th, 7th, 17th, and 19th harmonics, are eliminated in the 30 • angle displacement, while the (24 k ± 1)-th (k is an integer) harmonics, i.e., the 1st, 23rd, and 25th harmonics for the 15 • angle displacement are cancelled. The phase-and line-back-EMFs of the DTP PMSMs with these angle displacements can be obtained by using the finite element method; the values are presented in Figure 11a-c. The spectra of the phase-and line-back-EMFs are presented in Figure 11d. It can be seen that the amplitudes of the working harmonics (the 11th harmonics) are similar in the three machines, and the amplitudes of the different angle displacements correspond to their winding factors. Although the MMF distributions of the three angle displacements look similar, their harmonic contents are different. The (12 k ±5)-th (k is an integer) harmonics, i.e., the 5th, 7th, 17th, and 19th harmonics, are eliminated in the 30° angle displacement, while the (24 k ± 1)-th (k is an integer) harmonics, i.e., the 1st, 23rd, and 25th harmonics for the 15° angle displacement are cancelled. The phase-and lineback-EMFs of the DTP PMSMs with these angle displacements can be obtained by using the finite element method; the values are presented in Figure 11a-c. The spectra of the phase-and line-back-EMFs are presented in Figure 11d. As shown in Figure 11, the displacements between the phase-and line-back-EMFs of the two winding sets are 0°, 15°, and 30°, respectively, in the three machines. The fundamental amplitude in the 30° angle displacement is about 3.53% which is higher than that in the 0° counterpart, and 2.3% higher than that in the 15° counterpart, which is in agreement with their winding factors.
The electromagnetic torque performances of the machines with different angle displacements under healthy conditions are shown in Figure 12. The input current angle of each angle displacement is adjusted to maximize the output torque. It can be seen that the machine with 30° displacement produces the highest output torque, followed by the 15° machine and the 0° machine. Besides the advantages in average torque, the 30° angle displacement features the lowest torque ripple among the three angle displacements because of the cancellation of the 6th torque harmonics. The torque ripple of the 15° angle displacement is slightly lower than that of its 0° counterpart.

Back-EMF (V)
Harmonics As shown in Figure 11, the displacements between the phase-and line-back-EMFs of the two winding sets are 0 • , 15 • , and 30 • , respectively, in the three machines. The fundamental amplitude in the 30 • angle displacement is about 3.53% which is higher than that in the 0 • counterpart, and 2.3% higher than that in the 15 • counterpart, which is in agreement with their winding factors.
The electromagnetic torque performances of the machines with different angle displacements under healthy conditions are shown in Figure 12. The input current angle of each angle displacement is adjusted to maximize the output torque. It can be seen that the machine with 30 • displacement produces the highest output torque, followed by the 15 • machine and the 0 • machine. Besides the advantages in average torque, the 30 • angle displacement features the lowest torque ripple among the three angle displacements because of the cancellation of the 6th torque harmonics. The torque ripple of the 15 • angle displacement is slightly lower than that of its 0 • counterpart. the two winding sets are 0°, 15°, and 30°, respectively, in the three machine mental amplitude in the 30° angle displacement is about 3.53% which is hig in the 0° counterpart, and 2.3% higher than that in the 15° counterpart, whi ment with their winding factors.
The electromagnetic torque performances of the machines with differ placements under healthy conditions are shown in Figure 12. The input cu each angle displacement is adjusted to maximize the output torque. It can be machine with 30° displacement produces the highest output torque, follow machine and the 0° machine. Besides the advantages in average torque, the placement features the lowest torque ripple among the three angle displacem of the cancellation of the 6th torque harmonics. The torque ripple of the 15° a ment is slightly lower than that of its 0° counterpart.    It should also be mentioned that the average torques produced under ( Figure 13) are clearly lower than those produced under healthy conditio To maintain the torque capability of the machine, the input currents need to The variations of average torque with input current amplitudes in the mach ferent configurations under three-phase OC conditions are shown in Figur concluded that the 15° displacement offers the best over-rating capability phase OC conditions, compared with its 0° and 30° counterparts. The sup over-rating capability in the 15° displacement under three-phase OC con explained by its relatively low sub-harmonic distributions [47]. Furthermore  It should also be mentioned that the average torques produced under OC conditions ( Figure 13) are clearly lower than those produced under healthy conditions ( Figure 12). To maintain the torque capability of the machine, the input currents need to be increased. The variations of average torque with input current amplitudes in the machines with different configurations under three-phase OC conditions are shown in Figure 14. It can be concluded that the 15 • displacement offers the best over-rating capability under threephase OC conditions, compared with its 0 • and 30 • counterparts. The superiority of the over-rating capability in the 15 • displacement under three-phase OC conditions can be explained by its relatively low sub-harmonic distributions [47]. Furthermore, it is also suggested that the 15 • winding configuration features the lowest PM demagnetization risk among the three winding configurations, because the peak values of flux densities produced by the armature reaction are the lowest in the 15 • winding configuration [47].
ferent configurations under three-phase OC conditions are shown in Figur concluded that the 15° displacement offers the best over-rating capability phase OC conditions, compared with its 0° and 30° counterparts. The sup over-rating capability in the 15° displacement under three-phase OC cond explained by its relatively low sub-harmonic distributions [47]. Furthermore gested that the 15° winding configuration features the lowest PM demagn among the three winding configurations, because the peak values of flux duced by the armature reaction are the lowest in the 15° winding configura

Field Oriented Control (FOC)
The most common control method in DTP drives is the well-known control (FOC, or vector control). It is comprised of inner current loops and a loop, as shown in Figure 15. The values ωr * , iq * , id * , iz1 * , and iz2 * are the refere the speed and currents, respectively. It should be noted that iq * is the track torque per ampere (MTPA) to operate. The values ωr, iq, id, iz1, iz2 are the me (feedback values) for the speed and currents, respectively. The values ud,q a of the current proportional integral (PI) controllers, and then u α , β are obtaine mation, as the input of the modulation strategy to generate the proper swi

Field Oriented Control (FOC)
The most common control method in DTP drives is the well-known field-oriented control (FOC, or vector control). It is comprised of inner current loops and an outer speed loop, as shown in Figure 15. The values ω r *, i q *, i d *, i z1 *, and i z2 * are the reference values for the speed and currents, respectively. It should be noted that i q * is the tracking maximum torque per ampere (MTPA) to operate. The values ω r , i q , i d , i z1 , i z2 are the measured values (feedback values) for the speed and currents, respectively. The values u d,q are the outputs of the current proportional integral (PI) controllers, and then u α,β are obtained by transformation, as the input of the modulation strategy to generate the proper switching signals S ABC and S XYZ . The value θ s represents the electrical rotor angle. Unlike threephase machines, the DTP machine suffers from large low-order current harmonics due to its inherent low impedance for the 5th and 7th harmonics, which may be caused by back-EMF distortion, inverter non-linearity, modulation strategies, and asymmetry [8,9]. Hence, the recent development of FOC mainly focuses on the current harmonic reduction for DTP machines. SABC and SXYZ. The value θs represents the electrical rotor angle. Unlike three-phase machines, the DTP machine suffers from large low-order current harmonics due to its inherent low impedance for the 5th and 7th harmonics, which may be caused by back-EMF distortion, inverter non-linearity, modulation strategies, and asymmetry [8,9]. Hence, the recent development of FOC mainly focuses on the current harmonic reduction for DTP machines. One way to reduce current harmonics is the use of double synchronous reference frames current control, which is based on the two individual dq-axis modelling method and controls the two sets of windings separately. It has been introduced into DTP IMs [55,56] and PMSMs [54,57,58,[68][69][70][71][72][73][74][75][76][77][78] and provides benefits such as its excellent steadystate performance, easy implementation in practical applications, and its ability to sup- Figure 15. FOC diagram for DTP PMSMs.
One way to reduce current harmonics is the use of double synchronous reference frames current control, which is based on the two individual dq-axis modelling method and controls the two sets of windings separately. It has been introduced into DTP IMs [55,56] and PMSMs [54,57,58,[68][69][70][71][72][73][74][75][76][77][78] and provides benefits such as its excellent steady-state performance, easy implementation in practical applications, and its ability to suppress the current unbalance resulting from asymmetries between the two sets of three-phase windings [54,57]. However, the two-individual current control may develop instability issues if a strong magnetic coupling between two sets exists, as concluded in [58,68]. Therefore, the VSD current control is proposed in FOC to overcome these drawbacks [68], as well as in 0 • and 60 • DTP topologies [79]. Because the PI regulator has the inherent drawbacks of the limited control capability to the AC components, synchronous frame PI controllers [69][70][71] and resonant controllers [68,[72][73][74] are introduced in DTP drives as alternatives through which to suppress the 5th and 7th harmonic components (AC components). It is noted that the resonant (PIR) [72,73] or quasi-resonant (QPIR) [75] controller parallel with PI regulator (PIR) provides infinite gain at the resonant frequency and, therefore, the AC components (the 5th and 7th harmonic components) in the z 1 -z 2 subspace can be suppressed effectively with a resonant controller. Moreover, other alternatives, such as the disturbance observer (DOB) [76], the extended state observer (ESO) [77] and the virtual impedance [78] are also extended to DTP PMSMs to obtain benefits in terms of performance and robustness. In particular, a multiple synchronous reference frame (MSRF) current harmonic regulation technique, proposed in [80,81] is able to decouple not only the 5th and 7th harmonics, but also the 11th and 13th harmonics in DTP PMSMs. This concept of the MSRF technique also features the potential to be extended to other multi three-phase machine systems.
Taking results in [68,78] as examples, it is clear from Figures 16a and 17a that the phase current suffers from large current harmonics, especially the 5th and 7th harmonics, when no compensation approaches are employed in the harmonic-related subspace. By utilizing resonant [68] and the virtual impedance [78] current regulators, the phase current harmonics can be reduced significantly, as shown in Figures 16b and 17b, where i a and i x are the stator currents in phases a and x, respectively. Another relevant research activity is related to torque enhancement. In the DTP machines, injecting the low-order harmonics into the PM shaping and phase currents are both effective ways to increase the torque capability of the machines [82][83][84]. In [82] and [83], zero-sequence current components (the third current harmonics) injection is employed in DTP-IMs [82] and DTP-PMSMs [83], respectively, as shown in Figure 18. Figure 19 shows the finite element analysis (FEA) and experimental results with or without the third current harmonic injection. It is obvious that the torque improves significantly in both 12 slot/8 pole and 12 slot/10 pole machines, and the experimental results validate the torque enhancement as well. Furthermore, 5th and 7th current harmonics injection is investigated in [84]. According to the results of relevant research, the improvement of the overload torque due to the third-order current harmonic injection can be up to 15%. However, the improvement of the overload torque for the 5th and 7th harmonic injection is only 7%, although it offers the benefits of its simple hardware structure and the lack of the extra requirement of current sensors, compared to the method of third current harmonic injection. The value I m represents the amplitude of the phase current.

Spectra ( a)
(b) Figure 16. Measured currents and spectra for PIR method [68]. Top: phase currents in phase a (ia) and x (ix); bottom: spectra. Another relevant research activity is related to torque enhancement. In the DTP m chines, injecting the low-order harmonics into the PM shaping and phase currents are b effective ways to increase the torque capability of the machines [82−84]. In [82] and [8 zero-sequence current components (the third current harmonics) injection is employed DTP-IMs [82] and DTP-PMSMs [83], respectively, as shown in Figure 18. Figure 19 sho the finite element analysis (FEA) and experimental results with or without the third c rent harmonic injection. It is obvious that the torque improves significantly in both slot/8 pole and 12 slot/10 pole machines, and the experimental results validate the torq enhancement as well. Furthermore, 5th and 7th current harmonics injection is investiga in [84]. According to the results of relevant research, the improvement of the overlo torque due to the third-order current harmonic injection can be up to 15%. However, improvement of the overload torque for the 5th and 7th harmonic injection is only 7 although it offers the benefits of its simple hardware structure and the lack of the ex requirement of current sensors, compared to the method of third current harmonic in tion. The value Im represents the amplitude of the phase current.
The ST-DTC was firstly adapted to DTP machines in [85], which employs, when it only employed flux and torque hysteresis regulators and a look-up table to impose machine flux and torque, as shown in Figure 20. The ω r *, Ψ s *, and T e * are the reference values for the speed, stator flux, and torque, respectively. The values ω r , Ψ s , and T e are the measured values (feedback values) for the speed, stator flux, and torque, respectively. The values I ab and I xy represent the measured currents in phases A, B, X, and Y. The value V dc is the dc-link voltage. The values V s and I s represent the stator voltage and current after transformation. The method offers the benefits of fast response and dynamic performance, but suffers from large current harmonics, as concluded in [86][87][88][89][90][91]. The Redesign of the torque and flux regulators is investigated in [86], where it reduced current harmonics and torque ripples effectively, albeit with at the cost of complicating the control structure. In order to reduce current harmonics and maintain a simple structure, the reduction of current harmonics for ST-DTC has recently focused on the definition of switching table and virtual vectors (VVs) selection strategy, which was first investigated in [87]. By employing two groups of space VVs (i.e., virtual voltage vector) and setting the amplitude of the synthetic voltage vector to zero in harmonic-related subspaces [87,88], the stator current harmonics can be significantly reduced. The switching look-up table in [87] is shown in Figure 21 and Table 12. Further, the back-EMF harmonics are considered and selectively [90] and fully [91] compensated in an ST-DTC employing a virtual vector based on two or three groups of VVs, respectively. Additionally, the authors of [100] introduced an additional current loop to determine the optimal dwell times of applied voltage vectors; the current harmonics caused by back-EMF distortion were reduced as well. Furthermore, the reduction of torque ripple using virtual VVs, and the improvement of the performance by employing a non-linear observer are also analyzed in [92] and [93,94], respectively. For reducing the torque ripples, two virtual voltage vectors are synthesized and a five-level torque regulator is employed corresponding in [88]. In summary, the experimental results for conventional ST-DTCs and ST-DTCs proposed in [87,88] are summarized and compared in Figure 22. It is clear that the current harmonics are effectively compensated and torque ripples are reduced as well when virtual voltage vectors are employed, compared to conventional ST-DTC. reducing the torque ripples, two virtual voltage vectors are synthesized and a five-level torque regulator is employed corresponding in [88]. In summary, the experimental results for conventional ST-DTCs and ST-DTCs proposed in [87,88] are summarized and compared in Figure 22. It is clear that the current harmonics are effectively compensated and torque ripples are reduced as well when virtual voltage vectors are employed, compared to conventional ST-DTC.        I  II  III  IV  V  VI   1 1  However, although virtual voltage vector injections are effective at reducing current harmonics and torque ripples, switching-table-based DTCs (ST-DTCs) still suffer from two inherent drawbacks: • The look-up table in DTC leads to variable switching frequency, which is not suitable  Hence, the duty-ratio-based ST-DTC (which divides one sampling period into two intervals and applies active vectors and zero vectors together) was adapted to DTP machines in [96] and obtained the reduction of torque ripples, but it still suffered from drawbacks, such as its variable switching frequency [97]. Therefore, SVM-DTC, presented in Figure 23, was adapted to DTP drives to deal with the torque and flux ripple as well as the variable switching frequency. It employs torque, flux PI controllers and space vector modulation to replace the hysteresis regulators and look-up table in ST-DTC, as shown in Figure 24. The values u m and u T are the outputs of the torque, flux PI controllers; after transformation, u α and u β are obtained as the inputs of SVPWM to generate the switching signals S ABC and S XYZ . The value θ s represents the electrical rotor angle. In this way, the flux and torque ripples can be reduced significantly, and constant switching frequency can be achieved, but this method features the disadvantage of a deteriorating dynamic response, as analyzed in [96,97].

Model Predictive Control (MPC)
Model predictive control (MPC) is based on a model of the real system, also called the "predictive model", and is used to predict its future evolution. The prediction is car-

Model Predictive Control (MPC)
Model predictive control (MPC) is based on a model of the real system, also called the "predictive model", and is used to predict its future evolution. The prediction is carried out for possible switching states to determine which one minimizes a defined cost

Model Predictive Control (MPC)
Model predictive control (MPC) is based on a model of the real system, also called the "predictive model", and is used to predict its future evolution. The prediction is carried out for possible switching states to determine which one minimizes a defined cost function. The basic MPC structure is shown in Figure 25. The values i t+1 d , i t+1 q are the predictive currents at instant t + 1, which are calculated by the measured currents i t d , i t q at instant t. Next, the proper switching signals S ABC and S XYZ are obtained by the cost function. According to the different cost functions, three different control constraints are included, namely the current predictive control (MPCC), the torque predictive control (MPTC), and the flux predictive control (MPFC). The MPCC was firstly adapted to DTP drives in [101], where only the 12 outer vectors of the 64 VVs (the largest) are employed to reduce the computation burden. The cost function is expressed as: where λ is a weighting factor between phase currents in the αβ and z1z2 subspaces; * , * , * , * are the current references in the αβ and z1z2 subspaces, respectively; and , , , are the predictive currents in the αβ and z1z2 subspaces at instant t + 1 calculated by the measured currents at instant t.
It is notable that the computation burden increases significantly for multiphase machines, while the VVs increase exponentially with the number of phases. Furthermore, MPCC methods with the cost function of minimizing current harmonics and reducing computation cost are investigated in [102] and [103], respectively. The selected voltage vector combined with a zero vector, namely, one-step modulation predictive current control (OSPC), or PWM-MPCC, is introduced in, where the current harmonics were suppressed further and the steady-state performance improved effectively. However, the active vectors in [102−104] are the same as those in [101], where only the largest vectors are employed. Although some improved techniques are utilized in [102−104], the system still suffers from current harmonics caused by uncontrolled harmonics-related components. Hence, a virtual VVs strategy synthesized by the largest and the second-largest actual vectors (synthesized by two groups of VVs) was introduced to MPCC for DTP machines, as described in [105]. Similar to the virtual vector approach in the DTC method, the virtual vectors ensure null z1-z2 voltages on average during the sampling period, thereby reducing the current harmonics. Moreover, two groups of virtual vectors (synthesized by three groups of VVs) aiming at harmonic currents reduction are introduced in [106], and the deadbeat current control method is also introduced to reduce the computing time with the increase of the number of employed VVs [106]. This technique for two groups of syn- The MPCC was firstly adapted to DTP drives in [101], where only the 12 outer vectors of the 64 VVs (the largest) are employed to reduce the computation burden. The cost function is expressed as: where λ is a weighting factor between phase currents in the αβ and z 1 z 2 subspaces; i * α , i * β , i * z1 , i * z2 are the current references in the αβ and z 1 z 2 subspaces, respectively; and i t+1 α , i t+1 β , i t+1 z1 , i t+1 z2 are the predictive currents in the αβ and z 1 z 2 subspaces at instant t + 1 calculated by the measured currents at instant t.
It is notable that the computation burden increases significantly for multiphase machines, while the VVs increase exponentially with the number of phases. Furthermore, MPCC methods with the cost function of minimizing current harmonics and reducing computation cost are investigated in [102] and [103], respectively. The selected voltage vector combined with a zero vector, namely, one-step modulation predictive current control (OSPC), or PWM-MPCC, is introduced in, where the current harmonics were suppressed further and the steady-state performance improved effectively. However, the active vectors in [102][103][104] are the same as those in [101], where only the largest vectors are employed. Although some improved techniques are utilized in [102][103][104], the system still suffers from current harmonics caused by uncontrolled harmonics-related components. Hence, a virtual VVs strategy synthesized by the largest and the second-largest actual vectors (synthesized by two groups of VVs) was introduced to MPCC for DTP machines, as described in [105]. Similar to the virtual vector approach in the DTC method, the virtual vectors ensure null z 1 -z 2 voltages on average during the sampling period, thereby reducing the current harmonics. Moreover, two groups of virtual vectors (synthesized by three groups of VVs) aiming at harmonic currents reduction are introduced in [106], and the deadbeat current control method is also introduced to reduce the computing time with the increase of the number of employed VVs [106]. This technique for two groups of synthesized virtual vectors is further refined in [107,108], where a two-step procedure and a dual virtual vector are involved to compensate the harmonics caused by machine asymmetries, dead time effects, and back-EMF harmonics, respectively. Furthermore, some MPCC methods aiming at other optimal goals are also developed to reduce the current harmonics. For example, torque ripple reduction and DC-link voltage utilization increase are analyzed in [109,110]. Moreover, an autoregressive model with exogenous variables and a disturbance observer are employed to compensate for the effects of parameter mismatch errors and unmodelled dynamics in [111,112], respectively.
MPTC aims at minimizing the torque ripple and flux ripple, and the cost function can be expressed as: where λ is a weighting factor between torque and stator flux; T * e and ψ * s are the torque and stator flux references, respectively; and T t+1 e and ψ t+1 s are the predictive torque and stator flux at instant t+1 calculated by measured torque and stator flux at instant t.
The viability of the MPTC is investigated in [113], where the virtual vector strategy is employed to reduce current harmonics in the MPTC. Similarly, the selected voltage vector combined with a zero vector is also introduced in [114] and [115] to reduce the torque ripple and flux ripple, and fixed switching frequency in MPTC, respectively. Further, a deadbeat-based MPTC is introduced in [116] to reduce both the harmonic currents and computation time simultaneously.
A MPFC for DTP machines was first evaluated in [117], and the cost function for MPFC is expressed as: where λ 1 and λ 2 are the weighting factors between torque and stator fluxes in the αβ and z 1 z 2 subspaces, respectively; ψ * z1 and ψ * z2 are the stator fluxes references in the z 1 and z 2 subspaces, respectively; and ψ t+1 z1 and ψ t+1 z2 are the predictive stator fluxes in the z 1 z 2 subspaces at instant t + 1 calculated by measured torque and stator flux at instant t.
In this way, the current harmonics and the torque ripple reduce simultaneously. In summary, these three types of control strategies are compared and summarized in Table 13 with different characteristics.

Linear Region
As DTP machines are attracting interest for their use in high-power applications, various pulse width modulation (PWM) techniques have been developed. They can be classified into three main categories: carrier-based PWM (CPWM), zero-sequence injection-based PWM (ZSPWM), and space vector PWM (SVPWM). The CPWM method is implemented by comparing reference signals to carrier signals to determine the duty cycle for the inverter switches [118]. Particularly if the reference signals are sinusoidal, the CPWM is called sinusoidal PWM (SPWM), which is widely used in common three-phase machine drives.
One of the drawbacks of CPWM is the lack of control degrees with which to achieve a better performance [118][119][120]. Therefore, a phase-shift SPWM and a sawtooth carrier-based PWM were investigated in DTP machines to reduce common-mode voltage in [119,120], respectively, where phase angle shifts and the shape of carrier waves were modified to reduce the common-mode voltage. Details about the carrier shift technique in DTP machines can be found in Appendix A.
Another drawback of CPWM is its low DC bus voltage utilization. This can be improved by injecting zero-sequence components, namely ZSPWM. is the method is implemented by injecting a zero-sequence component into the reference voltage in each three-phase set. A common zero-sequence signal is added to two sets of three-phase sinusoidal signals, designated as single-zero-sequence injection PWM (SZIPWM) [121,122]. Correspondingly, two different zero-sequence signals for the two sets of sinusoidal signals are employed, designated as double-zero-sequence injection PWM (DZIPWM) [56,123,124]. When double-zero-sequences are injected in DTP drives, DC bus utilization and current harmonics improve significantly compared with single-zero-sequence injection, as concluded in [56,123,124]. On the basis of this technique, the different magnitudes of the zero-sequence injection components are further investigated in symmetrical six-phase machines [125] to compensate for unbalanced loads. In [126], "trapezoidal-like" zero-sequence signals are injected to improve the performance in terms of current distortion and common-mode voltage reduction.
In multiphase inverters, the number of switching states increases in accordance with the phase number. For example, there are 64 switching states for DTP inverters. It would be more complicated to analyze and select proper switching states in DTP machines compared with three-phase machines. According to the VSD method in Section 3: Modelling, the space VVs can be transformed into two different subspaces: the αβ subspace and the z 1 − z 2 subspace, as shown in Figure 26. It is notable that the groups of voltage vectors from largest to smallest magnitude in the αβ subspace are defined as L(D 1 ), ML(D 2 ), M(D 3 ), and S(D 4 ), and this αβ subspace magnitude-based definition has no physical meaning, as listed in Table 14. and [120], respectively, where phase angle shifts and the shape of carrier waves were modified to reduce the common-mode voltage. Details about the carrier shift technique in DTP machines can be found in Appendix A. Another drawback of CPWM is its low DC bus voltage utilization. This can be improved by injecting zero-sequence components, namely ZSPWM. is the method is implemented by injecting a zero-sequence component into the reference voltage in each threephase set. A common zero-sequence signal is added to two sets of three-phase sinusoidal signals, designated as single-zero-sequence injection PWM (SZIPWM) [121,122]. Correspondingly, two different zero-sequence signals for the two sets of sinusoidal signals are employed, designated as double-zero-sequence injection PWM (DZIPWM) [56,123,124]. When double-zero-sequences are injected in DTP drives, DC bus utilization and current harmonics improve significantly compared with single-zero-sequence injection, as concluded in [56,123,124]. On the basis of this technique, the different magnitudes of the zerosequence injection components are further investigated in symmetrical six-phase machines [125] to compensate for unbalanced loads. In [126], "trapezoidal-like" zero-sequence signals are injected to improve the performance in terms of current distortion and common-mode voltage reduction.
In multiphase inverters, the number of switching states increases in accordance with the phase number. For example, there are 64 switching states for DTP inverters. It would be more complicated to analyze and select proper switching states in DTP machines compared with three-phase machines. According to the VSD method in Section 3: Modelling, the space VVs can be transformed into two different subspaces: the αβ subspace and the z1 -z2 subspace, as shown in Figure 26. It is notable that the groups of voltage vectors from largest to smallest magnitude in the αβ subspace are defined as L(D1), ML(D2), M(D3), and S(D4), and this αβ subspace magnitude-based definition has no physical meaning, as listed in Table 14.

VVs Group Magnitude
αβ Subspace z 1 z 2 Subspace Where U dc is the amplitude of the dc-link voltage.
The SVPWM based on the VSD method was first analyzed in DTP-IMs in [127], where the 12 largest vectors in the αβ subspace were utilized. However, this approach is incompetent and generates significant lower-order harmonics due to its lack of control over the components in the z 1 z 2 subspace. Hence, the utilization of 4 active VVs in the 2 largest vectors (4 L) to control the components in both subspaces is proposed in [62]. It is termed the 12 sector SVPWM strategy. Further, the SVPWM selected from 2 L + 2 ML [128], 3 L + 1 ML [129], 2 L + 1 ML + 1 M [130] are introduced, namely, the 24 sector SVPWM strategy. As stated in previous research, the 24 sector SVPWM method is easier to implement using DSP microcontrollers and results in less total harmonic distortion (THD) in the output waveforms of the converter, but it has a lower maximum modulation index due to the introduction of VVs with smaller amplitudes. Further, the concepts of discontinuous SVPWM [131], synchronized SVPWM [132], SVPWM using five active vectors [133] are investigated in DTP machines, and modulation restraints for SVPWM, i.e., the linear modulation range of z 1 z 2 subspace under an assured modulation index of the fundamental voltage, are investigated in [134], due to the existence of modulation coupling between the αβ subspace and the z 1 z 2 subspace.
These three types of PWM techniques are compared and summarized in Table 15. It is notable that the SVPWM strategies can be implemented by finding their CPWM equivalent, which is simpler to implement in the hardware. This requires finding the proper zerosequence injection to the original sinusoidal reference signals, as concluded in [135,136].

Overmodulation Region
In [137,138], an overmodulation (OVM) strategy is proposed on the basis of SVPWM in [62], where the desired voltage vector in the αβ subspace was synthesized using two adjacent active vectors. This resulted in large current harmonics in the z 1 -z 2 subspace. OVM strategies require a group of harmonics in line-neutral voltages to attain an increase in the modulation index, which inherently generates significant low-order harmonics in the z 1 − z 2 subspace. To reduce harmonics in the OVM region, two different spacevector-based OVM (SVOVM) techniques are proposed in [139,140], which use four active VVs (2 L + 2 ML) and five active VVs (2 L + 1 M + 2 S), respectively. With these vectors, by formulating an optimization problem to achieve minimum root-mean-square (RMS) voltage in the z 1 − z 2 subspace, the current harmonics can be reduced effectively in DTP drives. Further, an OVM technique modulated by two separately three-phase inverters (TINV-based OVM) rather than the six-phase inverter described in [139,140], is proposed in [141], where the computational cost is significantly reduced without the involving complex six-dimensional transformation discussed in previous studies.

PWM Techniques for ST-DTC
As mentioned above, for the FOC method, two adjacent vectors, or virtual vectors, are employed to synthesize all the VVs in one sector. However, in the ST-DTC method, only one voltage vector or virtual voltage vector is employed to achieve flux linkage and torque control, according to the output of the stator flux and torque hysteresis regulators and the prefixed switching table. Hence, PWM techniques for ST-DTC are totally different from the methods used in DTP drives described in Section 6.
In [85], the ST-DTC is analyzed for DTP machines. Firstly, the αβ subspace is divided into 12 sectors, and appropriate VVs are obtained through instantaneous control of the stator flux and torque. Subsequently, in order to decrease the computation burden and maximize the utilization of DC bus voltage, only the 12 largest VVs in the αβ subspace are employed (L), as shown in Figure 27. In this way, the method proposed in [85] (ST-DTC-L) is easy to implement, but it suffers from large stator current harmonics, since only the requirements of stator flux in the αβ subspace and torque are considered, and stator flux in the z 1 z 2 subspace is uncontrollable. In [85], the ST-DTC is analyzed for DTP machines. Firstly, the αβ subspace is divided into 12 sectors, and appropriate VVs are obtained through instantaneous control of the stator flux and torque. Subsequently, in order to decrease the computation burden and maximize the utilization of DC bus voltage, only the 12 largest VVs in the αβ subspace are employed (L), as shown in Figure 27. In this way, the method proposed in [85] (ST-DTC-L) is easy to implement, but it suffers from large stator current harmonics, since only the requirements of stator flux in the αβ subspace and torque are considered, and stator flux in the z1 z2 subspace is uncontrollable. In order to control the components in the z1z2 subspace and reduce current harmonics, the authors of [87] employed an ST-DTC method based on VVs in L and ML groups (ST-DTC-L + ML). Since the VVs in the L and ML groups have the same direction in the αβ subspace while the opposite direction in the z1z2 subspace, the amplitude of the components in the z1z2 subspace can be set to zero, as shown in Figure 28a,b; subsequently, the corresponding harmonic currents are decreased. Furthermore, in order to implement standard centralization PWM sequences, three adjacent vectors in the L group are employed to synthesize the virtual vectors, which are equivalent to the virtual vectors synthesized by the vectors in the L and ML groups. For example, the virtual vectors synthesized by V9-V11-V27 in the L groups are employed to replace the virtual vectors synthesized by V11-V25 to obtain standard PWM sequences, as shown in Figure 28c,d. In order to control the components in the z 1 z 2 subspace and reduce current harmonics, the authors of [87] employed an ST-DTC method based on VVs in L and ML groups (ST-DTC-L + ML). Since the VVs in the L and ML groups have the same direction in the αβ subspace while the opposite direction in the z 1 z 2 subspace, the amplitude of the components in the z 1 z 2 subspace can be set to zero, as shown in Figure 28a,b; subsequently, the corresponding harmonic currents are decreased. Furthermore, in order to implement standard centralization PWM sequences, three adjacent vectors in the L group are employed to synthesize the virtual vectors, which are equivalent to the virtual vectors synthesized by the vectors in the L and ML groups. For example, the virtual vectors synthesized by V9-V11-V27 in the L groups are employed to replace the virtual vectors synthesized by V11-V25 to obtain standard PWM sequences, as shown in Figure 28c,d. In [90,91], the 5th and 7th back-EMF harmonics are considered in DTP machines. Because the 5th and 7th back-EMF harmonics that possess phase shifts are unaligned with the VVs in the L and ML groups, the ST-DTC based on the VVs in the L and ML groups can only compensate for the selective order back-EMF harmonics, as concluded in [90]. Hence, in order to obtain composite vectors aligned with back-EMF harmonic components, the VVs in the M group are employed to combine with the VVs in the L and ML groups (STDTC-L + ML + M) in [91], due to their different directions with VVs in the L and ML groups, as shown in Figure 29a,b. In this way, the back-EMF harmonics can be fully compensated; consequently, the current harmonics caused by back-EMF distortion can be reduced. Further, proper PWM techniques with VVs selection from the L, ML, and M groups are proposed in [91] to obtain standard PWM sequences. For example, V59 is selected instead of V3 to combine with V10-V27 in order to generate standard PWM sequences, as shown in Figures 29c,d, where TD2, TD3, and TD4 are the dwelling times for the applied voltage vectors. In [90,91], the 5th and 7th back-EMF harmonics are considered in DTP machines. Because the 5th and 7th back-EMF harmonics that possess phase shifts are unaligned with the VVs in the L and ML groups, the ST-DTC based on the VVs in the L and ML groups can only compensate for the selective order back-EMF harmonics, as concluded in [90]. Hence, in order to obtain composite vectors aligned with back-EMF harmonic components, the VVs in the M group are employed to combine with the VVs in the L and ML groups (STDTC-L + ML + M) in [91], due to their different directions with VVs in the L and ML groups, as shown in Figure 29a,b. In this way, the back-EMF harmonics can be fully compensated; consequently, the current harmonics caused by back-EMF distortion can be reduced. Further, proper PWM techniques with VVs selection from the L, ML, and M groups are proposed in [91] to obtain standard PWM sequences. For example, V59 is selected instead of V3 to combine with V10-V27 in order to generate standard PWM sequences, as shown in Figure 29c Table 16. and ML groups, as shown in Figure 29a,b. In this way, the back-EMF harmonics can be fully compensated; consequently, the current harmonics caused by back-EMF distortion can be reduced. Further, proper PWM techniques with VVs selection from the L, ML, and M groups are proposed in [91] to obtain standard PWM sequences. For example, V59 is selected instead of V3 to combine with V10-V27 in order to generate standard PWM sequences, as shown in Figures 29c,d, where TD2, TD3, and TD4 are the dwelling times for the applied voltage vectors.  Table 16.

Sensorless Control
Sensorless control aims to remove the rotor position sensor from the control system,  Table 16.

Sensorless Control
Sensorless control aims to remove the rotor position sensor from the control system, and thus features advantages in PMSM drives in the form of cost reduction, system downsizing, and reliability enhancement [142]. The sensorless control of conventional threephase PMSMs has been thoroughly studied so far, and the control approaches can be clas-

Sensorless Control
Sensorless control aims to remove the rotor position sensor from the control system, and thus features advantages in PMSM drives in the form of cost reduction, system downsizing, and reliability enhancement [142]. The sensorless control of conventional three-phase PMSMs has been thoroughly studied so far, and the control approaches can be classified into two categories: model-based sensorless control and salient-effect-based sensorless control. The former usually utilizes the mathematical model of the machine to estimate, online, the back-EMF [143][144][145][146][147][148] and flux linkage [149][150][151][152][153] associated with the fundamental excitation, from which the rotor position can be further extracted. This kind of rotor position estimation is more applicable in high-speed regions but suffers from inaccuracy at low-speed and zero-speed conditions due to the model's uncertainty and inverter nonlinearity, as well as its low signal-to-noise ratio. The salient-effect-based control usually makes use of the salient effect by injecting high frequency (HF) signals to calculate the rotor position; hence, it is suitable for the low-speed and zero-speed range. With reference to the injected signals, the control approaches can be grouped as rotating [154][155][156] and pulsating [157] HF signal injection; pulsating HF signal injections can be further divided into pulsating sinusoidal and pulsating square-wave HF signal injections.
The sensorless control approaches mentioned above belong to the category of singlethree-phase PMSM systems, and can be also extended to the MTP family. However, regarding MTP PMSMs, thanks to the additional degrees of freedom, many novel techniques of rotor position estimation have been proposed in recent years, which demonstrates the superiority of MTP PMSMs in the area of sensorless control. The author of [158] used an auxiliary resistor network to measure the third harmonic EMFs of two three-phase sets ( Figure 31). According to the phase relationship between the fundamental and the third harmonic EMFs, the estimator in Figure 32 can be employed to calculate the rotor position ( Figure 33). rgies 2021, 14, x FOR PEER REVIEW sified into two categories: model-based sensorless control and salientless control. The former usually utilizes the mathematical model of mate, online, the back-EMF [143−148] and flux linkage [149−153] asso damental excitation, from which the rotor position can be further ext rotor position estimation is more applicable in high-speed regions bu curacy at low-speed and zero-speed conditions due to the model's verter nonlinearity, as well as its low signal-to-noise ratio. The salientusually makes use of the salient effect by injecting high frequency (HF the rotor position; hence, it is suitable for the low-speed and zero-sp erence to the injected signals, the control approaches can be grouped a and pulsating [157] HF signal injection; pulsating HF signal injection vided into pulsating sinusoidal and pulsating square-wave HF signal The sensorless control approaches mentioned above belong to th three-phase PMSM systems, and can be also extended to the MTP fa garding MTP PMSMs, thanks to the additional degrees of freedom niques of rotor position estimation have been proposed in recent y strates the superiority of MTP PMSMs in the area of sensorless control used an auxiliary resistor network to measure the third harmonic phase sets (Figure 31). According to the phase relationship between th the third harmonic EMFs, the estimator in Figure 32 can be employed t position (Figure 33). mate, online, the back-EMF [143−148] and flux linkage [149−1 damental excitation, from which the rotor position can be fu rotor position estimation is more applicable in high-speed re curacy at low-speed and zero-speed conditions due to the m verter nonlinearity, as well as its low signal-to-noise ratio. The usually makes use of the salient effect by injecting high freque the rotor position; hence, it is suitable for the low-speed and erence to the injected signals, the control approaches can be g and pulsating [157] HF signal injection; pulsating HF signal vided into pulsating sinusoidal and pulsating square-wave H The sensorless control approaches mentioned above belo three-phase PMSM systems, and can be also extended to the garding MTP PMSMs, thanks to the additional degrees of niques of rotor position estimation have been proposed in r strates the superiority of MTP PMSMs in the area of sensorles used an auxiliary resistor network to measure the third ha phase sets ( Figure 31). According to the phase relationship be the third harmonic EMFs, the estimator in Figure 32 can be em position ( Figure 33). Unlike [158], the auxiliary resistor network is n quence carrier voltage between the neutral points o injecting HF carrier signals, and the rotor position ca age. The injected HF carrier signals can be classifi rotating injection [160], as shown in Table 17.  Unlike [158], the auxiliary resistor network is not required in [159,160]. The zerosequence carrier voltage between the neutral points of two winding sets are measured after injecting HF carrier signals, and the rotor position can be derived from the measured voltage. The injected HF carrier signals can be classified into pulsating injection [159] and rotating injection [160], as shown in Table 17.
It is clear that after the pulsating and rotating HF signals are injected, the voltage difference between two isolated neutral points includes the zero-sequence carrier voltage u 0sn1n2 , which is relevant to the rotor position and injected signals. Using the demodulation functions, the estimated errors can be calculated and then utilized to compensate for the estimated rotor position. There are clear biases and ripples in the estimated rotor position, which can be eliminated by optimizing the phase shift angle ϕ between the injection signals of two three-phase sets. Specifically, the optimal phase shift angles are π/2 and 2π/3 for the pulsating injection and rotating injection, respectively. The overall control system is shown in Figure 34. , which is relevant to the rotor position and injected signals. Using the demodulation functions, the estimated errors can be calculated and then utilized to compensate for the estimated rotor position. There are clear biases and ripples in the estimated rotor position, which can be eliminated by optimizing the phase shift angle between the injection signals of two three-phase sets. Specifically, the optimal phase shift angles are π/2 and 2π/3 for the pulsating injection and rotating injection, respectively. The overall control system is shown in Figure 34.  Figure 34. Overall sensorless control system using zero-sequence carrier voltage in DTP PM machine [159,160].
In terms of the rotor position errors caused by the parameter uncertainty, [161] proposed a simple correction method to compensate the rotor position error for the DTP PM machine system, as shown in Figure 35.
Adaptive observer Vector control of set2 R s L s R s L s + - Figure 34. Overall sensorless control system using zero-sequence carrier voltage in DTP PM machine [159,160]. Table 17. Rotor position estimation using zero-sequence carrier voltage between two neutral points.
Pulsating Injection [159] Rotating Injection [160] u 0sn1n2 Demodulation * 4 ( − 3 ) Test results without phase shift It is clear that after the pulsating and rotating HF signals are injected, the voltage difference between two isolated neutral points includes the zero-sequence carrier voltage Demodulation * 4 ( − 3 ) Test results without phase shift It is clear that after the pulsating and rotating HF signals are injected, the voltage difference between two isolated neutral points includes the zero-sequence carrier voltage Demodulation * 4 ( − 3 ) Test results without phase shift It is clear that after the pulsating and rotating HF signals are injected, the voltage difference between two isolated neutral points includes the zero-sequence carrier voltage Demodulation * 4 ( − 3 ) Test results without phase shift It is clear that after the pulsating and rotating HF signals are injected, the voltage difference between two isolated neutral points includes the zero-sequence carrier voltage In terms of the rotor position errors caused by the parameter uncertainty, [161] proposed a simple correction method to compensate the rotor position error for the DTP PM machine system, as shown in Figure 35.  Figure 34. Overall sensorless control system using zero-sequence carrier voltage in DTP PM machine [159,160].
In terms of the rotor position errors caused by the parameter uncertainty, [161] proposed a simple correction method to compensate the rotor position error for the DTP PM machine system, as shown in Figure 35.  Figure 35. Rotor position correction using adaptive observers in the DTP PM machine system [161].
The outputs of two adaptive observers represent the two estimated rotor positions in two three-phase subsystems. However, due to the uncertain resistance and inductance, the two estimated positions are not identical, and the difference between the estimated rotor positions is relevant to the current distribution of the two three-phase sets, which indicates that currents with nonidentical amplitudes and phases in two sets amplify that difference. Accordingly, extra fundamental current is injected in [161] to achieve the rotor position correction. The rotor position difference ∆ is then utilized to tune the machine resistance and inductance parameters in adaptive observers. The PI regulators are employed in the updated controller to guarantee that ∆ can be eliminated, and the rotor position error correction is achieved simultaneously. Figure 36 shows the experimental results of rotor position error correction. The extra q-axis current of set 1 is injected at 1.5 s, and subsequently causes additional torque and transient pulsating in position errors, as shown in Figure 36a. As the updated controller works, the rotor position errors are reduced to zero at a steady-state. The extra current is removed from the system at 3.5 s, after which a short period of pulsating occurs; the error can subsequently remain at 0 degrees. Although the position errors are corrected, the injected current in one set causes additional The outputs of two adaptive observers represent the two estimated rotor positions in two three-phase subsystems. However, due to the uncertain resistance and inductance, the two estimated positions are not identical, and the difference between the estimated rotor positions is relevant to the current distribution of the two three-phase sets, which indicates that currents with nonidentical amplitudes and phases in two sets amplify that difference. Accordingly, extra fundamental current is injected in [161] to achieve the rotor position correction. The rotor position difference ∆θ e r is then utilized to tune the machine resistance and inductance parameters in adaptive observers. The PI regulators are employed in the updated controller to guarantee that ∆θ e r can be eliminated, and the rotor position error correction is achieved simultaneously. Figure 36 shows the experimental results of rotor position error correction. The extra q-axis current of set 1 is injected at 1.5 s, and subsequently causes additional torque and transient pulsating in position errors, as shown in Figure 36a. As the updated controller works, the rotor position errors are reduced to zero at a steady-state. The extra current is removed from the system at 3.5 s, after which a short period of pulsating occurs; the error can subsequently remain at 0 degrees. Although the position errors are corrected, the injected current in one set causes additional torque and pulsating in the position estimation. To achieve a smoother correction, the extra currents are injected into both sets, as displayed in Figure 36b. The injected currents feature the same amplitude but the opposite polarity; thus, they cannot introduce any additional torque as the one-set injection does. Consequently, the rotor position errors can be smoothly regulated to 0 deg. in 0.25 s.

Fault Tolerance Control
Fault tolerance is an important property, and makes MTP machines distinct from conventional three-phase machines. Fault tolerance control aims to utilize the additional degrees of freedom in MTP machine systems and keeps the machine continuously operating after one (or more) phase(s) develop faults. The fault usually occurs in the power switches and the machine windings, and faults in both the power switches and the machine windings cause short-circuit or open-circuit in MTP drives. Fault tolerance for shortcircuit faults is only feasible in MTP machines with modular structures and winding ar-

Fault Tolerance Control
Fault tolerance is an important property, and makes MTP machines distinct from conventional three-phase machines. Fault tolerance control aims to utilize the additional degrees of freedom in MTP machine systems and keeps the machine continuously operating after one (or more) phase(s) develop faults. The fault usually occurs in the power switches and the machine windings, and faults in both the power switches and the machine windings cause short-circuit or open-circuit in MTP drives. Fault tolerance for short-circuit faults is only feasible in MTP machines with modular structures and winding arrangements [5,[162][163][164]. In general, fault tolerance control for short-circuit faults involves the isolation of the faulty sets of three-phase windings, followed by the operation of the machine with the remaining sets of windings. Compared to the fault tolerance used for short-circuit faults, the most widely investigated cases in recent research have  .
The purpose of fault tolerance control is usually to generate a rotating flux field to drag the rotor using the residual healthy phases. The control strategies depend on the degrees of freedom of the machine. MTP machines can be grouped into two types, according to their connection of neutral points, namely isolated neutral points and connected neutral points, as shown in Table 18.

Fault Tolerance Control
Fault tolerance is an important property, and makes MTP machines distinct from conventional three-phase machines. Fault tolerance control aims to utilize the additional degrees of freedom in MTP machine systems and keeps the machine continuously operating after one (or more) phase(s) develop faults. The fault usually occurs in the power switches and the machine windings, and faults in both the power switches and the machine windings cause short-circuit or open-circuit in MTP drives. Fault tolerance for shortcircuit faults is only feasible in MTP machines with modular structures and winding arrangements [5,162−164]. In general, fault tolerance control for short-circuit faults involves the isolation of the faulty sets of three-phase windings, followed by the operation of the machine with the remaining sets of windings.  [165−185].
The purpose of fault tolerance control is usually to generate a rotating flux field to drag the rotor using the residual healthy phases. The control strategies depend on the degrees of freedom of the machine. MTP machines can be grouped into two types, according to their connection of neutral points, namely isolated neutral points and connected neutral points, as shown in Table 18.

Fault Tolerance Control
Fault tolerance is an important property, and makes MTP machines distinct from conventional three-phase machines. Fault tolerance control aims to utilize the additional degrees of freedom in MTP machine systems and keeps the machine continuously operating after one (or more) phase(s) develop faults. The fault usually occurs in the power switches and the machine windings, and faults in both the power switches and the machine windings cause short-circuit or open-circuit in MTP drives. Fault tolerance for shortcircuit faults is only feasible in MTP machines with modular structures and winding arrangements [5,162−164]. In general, fault tolerance control for short-circuit faults involves the isolation of the faulty sets of three-phase windings, followed by the operation of the machine with the remaining sets of windings.  [165−185].
The purpose of fault tolerance control is usually to generate a rotating flux field to drag the rotor using the residual healthy phases. The control strategies depend on the degrees of freedom of the machine. MTP machines can be grouped into two types, according to their connection of neutral points, namely isolated neutral points and connected neutral points, as shown in Table 18. In terms of the isolated type, the most common approach is to cut off the three-phase set, in which the fault occurs, from the control system; the required torque can be equally distributed to the healthy sets. To take the DTP machine as an example, when phase A 2 is faulty due to the isolated neutral point of the second set, the currents of phases B 2 and C 2 are identical, and can only generate a pulsating flux field, which leads to poor torque performance. Therefore, the second set can simply be abandoned, and the currents of the first set should be doubled to make up for the lost torque; in other words, the available torque and power are reduced to half of their previous rating. This case does not require the conversion of the control to the specifically designed postfault algorithm if each set is individually regulated, which is quite simple to implement and practical in industrial applications, such as railway traction [168,169] and ship propulsion [10][11][12][13]. Meanwhile, the disadvantage is clear. The healthy phases in the fault set, phases B 2 and C 2 , together with the corresponding switching devices, are not utilized, and the system utilization is not maximized.
To improve the system utilization of the machine with isolated neutral points, the current should be redistributed to the remaining healthy phases. The pulsating flux generated by the fault set can be decomposed into positive-sequence and negative-sequence rotating flux. The positive-sequence flux provides average torque while the negativesequence flux generates the second-order torque ripple. The core purpose of this kind of fault tolerance control is to inject negative-sequence current into the healthy set and to counterbalance the negative-sequence flux; as a result, the average torque remains while the torque ripple is eliminated. To implement this idea, a genetic algorithm is provided in [166] to optimize the stator currents, thereby maximizing the average torque or minimizing the torque ripple. Furthermore, the copper loss of faulty DT-PMSM with isolated neutral points is further reduced by minimizing the d-axis current of the healthy three-phase winding [178][179][180].
It is also possible to use modelling with open-circuit faults to achieve different control purposes. In particular, the VSD technique can decouple motor variables in several orthogonal subspaces, thereby controlling them independently [164,[181][182][183][184][185]. Because the VSD technique can decompose the negative-sequence current in the z subspace, the control structure in [185] is greatly simplified, as shown in Figure 37. The values i q *, i d *, i z *, v q *, v d *, v z1 *, v z2 *, v z3 *, v α2 *, v β2 *, v α *, v β *, v abc *, v x *, and v y * are the reference values for the voltage and currents in different coordinate systems, respectively. The values i abc , i x , i y , i α,β,z1,z2,z3 are the measured currents. The values V ffq_d and V ffd_q are the feedforward voltages to compensate for the d-q coupling. The values Duty abc * and Duty xyz * are the dwelling times for each switching signal.
To improve the system utilization of the machine with isolated neutral points, the current should be redistributed to the remaining healthy phases. The pulsating flux generated by the fault set can be decomposed into positive-sequence and negative-sequence rotating flux. The positive-sequence flux provides average torque while the negative-sequence flux generates the second-order torque ripple. The core purpose of this kind of fault tolerance control is to inject negative-sequence current into the healthy set and to counterbalance the negative-sequence flux; as a result, the average torque remains while the torque ripple is eliminated. To implement this idea, a genetic algorithm is provided in [166] to optimize the stator currents, thereby maximizing the average torque or minimizing the torque ripple. Furthermore, the copper loss of faulty DT-PMSM with isolated neutral points is further reduced by minimizing the d-axis current of the healthy three-phase winding [178−180].
It is also possible to use modelling with open-circuit faults to achieve different control purposes. In particular, the VSD technique can decouple motor variables in several orthogonal subspaces, thereby controlling them independently [164,181−185]. Because the VSD technique can decompose the negative-sequence current in the z subspace, the control structure in [185] is greatly simplified, as shown in Figure 37. The values iq * , id * , iz * , vq * ,  To eliminate the negative-sequence current, a PIR regulator is employed and the fundamental electrical frequency is utilized as the resonant frequency. Figure 38 provides the experimental results under healthy and faulty conditions. To eliminate the negative-sequence current, a PIR regulator is employed and the fundamental electrical frequency is utilized as the resonant frequency. Figure 38 provides the experimental results under healthy and faulty conditions. The phase currents are balanced under healthy conditions, and there are few current harmonics according to the spectrum in Figure 38a. When it is converted to a faulty condition, it is clear that the two phase currents of the fault set are out of phase, and the currents of the healthy set become unbalanced, which is due to the injected negative-sequence current, as mentioned above. In addition, there are also clear odd current harmonics under faulty condition efficient suppression is lacking.
The type with connected neutral points features more degrees of freedom because all the healthy phases are available. In this case, the currents of phases B2 and C2 can be flexibly regulated to follow different references, which means they can generate the rotating flux field and thus provide average torque. However, the post-fault control algorithm is The phase currents are balanced under healthy conditions, and there are few current harmonics according to the spectrum in Figure 38a. When it is converted to a faulty condition, it is clear that the two phase currents of the fault set are out of phase, and the currents of the healthy set become unbalanced, which is due to the injected negativesequence current, as mentioned above. In addition, there are also clear odd current harmonics under faulty condition efficient suppression is lacking. The type with connected neutral points features more degrees of freedom because all the healthy phases are available. In this case, the currents of phases B 2 and C 2 can be flexibly regulated to follow different references, which means they can generate the rotating flux field and thus provide average torque. However, the post-fault control algorithm is different from the normal algorithm used in the healthy case and should be specifically designed for this purpose; the issue becomes how to design the optimal current reference for each healthy phase. The optimization objectives include maximal average torque, minimal torque ripple, minimal copper loss, and uniform distribution of copper loss [170][171][172][173][174][175][176]. The current harmonics may be injected to boost the average torque and to reduce the torque ripple of DTP PMSM under open-fault conditions, as in [174]. To automatically track the condition of an MTPA operation, the injection-based MTPA control in three-phase machine drive is extended to the fault tolerance control of five-phase PMSM systems under single-and double-phase fault in [175]. It should be noted that these optimal current control approaches emphasize designing the current reference, and various current regulators, as studied in FOC, DTC, and MPC, and can be utilized to track the designed current references.
To avoid fault detection and algorithm switching between normal control and fault tolerance control, the robust control attracts the attention of researchers due to its good disturbance rejection capability. The loss of phases can be considered as a disturbance of machine parameter variation. If this disturbance can be easily rejected by the controller, the system can dynamically respond to the fault, and consequently the machine can still be operated without fault detection or algorithm transition. A robust speed regulator is used in [177] to provide average torque under both normal and post-fault conditions. Fuzzy logic is employed to design the current regulator of a nine-phase PMSM system in [186], as shown in Figure 39. The speed PI regulator determines the references, to which the fuzzy logic controller can achieve fundamental current tracking even before and after a fault occurs. This means that the amplitudes and phase angles of the currents in the healthy phases are automatically adjusted to provide the required torque and to cover the fault. In addition, PI regulators are used to suppress the current harmonics in the harmonic subspaces, such as the 3rd, 5th, and 7th. Furthermore, fault-tolerant controls in DTP machines fed by multi-level inverters are also investigated by considering both open-circuit switch faults and phase faults in [98,[187][188][189], which provide greater reliability and fault capability.

Conclusions
Due to their excellent characteristics, such as their high torque/power capability, high efficiency, and low torque ripple, the low requirements of their DC link capacitors, their efficient power/torque-sharing capability, modular configuration, and high fault tolerance capability, multiphase (particularly MTP) PMSMs have been applied to ship propulsion, electric vehicles, wind power generation, and aerospace application. They offer significant potential for even wider adoption in many high-power and high-fault-tolerance applications.
This paper presented a comprehensive overview of the advances in multiphase PMSMs' machine topologies, modelling methods, PWM techniques, field-oriented control, direct torque control, model predictive control, sensorless control, fault-tolerant con-

Conclusions
Due to their excellent characteristics, such as their high torque/power capability, high efficiency, and low torque ripple, the low requirements of their DC link capacitors, their efficient power/torque-sharing capability, modular configuration, and high fault tolerance capability, multiphase (particularly MTP) PMSMs have been applied to ship propulsion, electric vehicles, wind power generation, and aerospace application. They offer significant potential for even wider adoption in many high-power and high-faulttolerance applications.
This paper presented a comprehensive overview of the advances in multiphase PMSMs' machine topologies, modelling methods, PWM techniques, field-oriented control, direct torque control, model predictive control, sensorless control, fault-tolerant control, and the newest control strategies for suppressing current harmonics and torque ripples, as well as carrier phase shift techniques. Many worked examples, presented with the measured results obtained at the University of Sheffield, were provided to illustrate the effectiveness of the developed machines and techniques, as well as their excellent performance. The emphasis was on DTP PMSMs and their control strategies, since the authors strongly believe they are most likely to be widely applied in many industries.
Firstly, the topologies of different DTP PMSMs were summarized according to their slot/pole number, angle displacement, and pitch number combinations. The feasibility of different winding configurations and the calculation of winding factors with different winding configurations were also provided in Section 2. The electromagnetic performance of DTP PMSMs was further introduced in Section 4, based on a 24 slot/22 pole machine considering different angle displacements. It is suggested that even for the dual threephase PMSMs with the same slot/pole number and pitch number combinations, their performances are affected significantly by angle displacement. Next, aspects of MTP/DTP modelling methods were reviewed in Section 3, including multiple individual three-phase modelling with simple structure and implemental simplicity but with coupling in each set of three phase wingdings, and VSD modelling with decoupling results using decoupling transformation matrices. Subsequently, recent advances in control strategies, including FOC, DTC and MPC, were also covered in Section 5. Particularly, current harmonics reduction in DTP drives was on the main focus as it is a significant topic in the study of both FOC and DTC, since DTP machines suffer from large low-order current harmonics due to their inherent low impedance for the 5th and 7th harmonics. Besides, torque enhancement in FOC and torque ripple reduction in DTC have also received significant attention in recent years.
Moreover, PWM strategies for DTP were reviewed in Section 6, followed by some novel aspects of sensorless control, which are successfully adapted from the conventional three-phase sensorless methods. They include: back-EMF/ flux linkage-based methods, which offer simple implementation and accuracy in high-speed regions but also demonstrate inaccuracy under low-and zero-speed conditions due to the model's uncertainty and the inverter's nonlinearity; and the method of injecting high frequency (HF) signals, which is complicated but suitable for low-speed and zero-speed ranges. Finally, one of the most significant merits of MTP/DTP machines, their high fault tolerance capability, was surveyed through the fault-tolerant control discussions in Section 8. The modelling, optimal current references, and control methods in post-fault operation were studied, especially in open-circuit faulty conditions.
The additional degrees of freedom also include the nonidentical PWM carriers. In the control system of MTP PMSMs, the drive topology with multiple individual threephase converters provides the convenience and flexibility for various PWM carrier designs. Carrier phase-shift techniques, also known as the interleaving in Figure A1, is proposed to reduce the common-mode voltage [119,191], and are recently employed in this multiconverter case to reduce the current harmonics, torque ripples, vibration and acoustic noise, as well as the DC link pulsating currents [192][193][194][195][196][197][198][199][200][201]. The performance of the carrier phase-shift techniques is up to the machine winding structure, the converter topology, and the shifted angle of carriers. For the DTP machine fed by two segment three-phase inverter, as studied in [192,193], the vibration and acoustic noise of switching frequency can be reduced to hal after shifting π/2 phase angle of the PWM carrier of the second set. The switching flu generated by the second set is shifted in the airgap, and can counterbalance the one o the first set, which results in the reduction of switching torque ripples and vibration. Nev ertheless, due to the cancellation of air gap flux, the equivalent impedance to the switching voltage harmonics is reduced simultaneously, and further switching current harmonic are increased considerably. By appropriate machine design to minimize the mutual in ductance among sets [194], the switching current harmonics can be suppressed in the car rier phase shift operation.
A DTP PMSM with each phase fed by an H-bridge is studied in [195,196]. The phas axes of the first set are coincident with the ones of the second set. This winding design combined with anti-phase shifting of carrier in the second set, determines the elimination of switching torque ripples, as shown in Figure A2. However, suffering from the sam problem as [192], the coincident axes of two set means more magnetized couple, and th equivalent impedance regarding switching voltage harmonics is greatly reduced by th For the DTP machine fed by two segment three-phase inverter, as studied in [192,193], the vibration and acoustic noise of switching frequency can be reduced to half after shifting π/2 phase angle of the PWM carrier of the second set. The switching flux generated by the second set is shifted π in the airgap, and can counterbalance the one of the first set, which results in the reduction of switching torque ripples and vibration. Nevertheless, due to the cancellation of air gap flux, the equivalent impedance to the switching voltage harmonics is reduced simultaneously, and further switching current harmonics are increased considerably. By appropriate machine design to minimize the mutual inductance among sets [194], the switching current harmonics can be suppressed in the carrier phase shift operation.
A DTP PMSM with each phase fed by an H-bridge is studied in [195,196]. The phase axes of the first set are coincident with the ones of the second set. This winding design, combined with anti-phase shifting of carrier in the second set, determines the elimination of switching torque ripples, as shown in Figure A2. However, suffering from the same problem as [192], the coincident axes of two set means more magnetized couple, and the equivalent impedance regarding switching voltage harmonics is greatly reduced by the anti-phase shifting of carrier. Consequently, the circulating current is serious, and [195,196] connect additionally coupled inductor between inverter and machine to limit the circulating current, which has the disadvantages of increasing system cost, volume, and complexity. The carrier phase shift is extended to a twelve-phase (quadruple-three-phase) PMSM, of which each phase is fed by an H-bridge single phase inverter [197]. A general and analytical framework as well as the mechanism of carrier phase shift are carried out based on the evaluation of radial electromagnetic forces on stator tooth faces near the airgap. To eliminate the torque ripple and vibration of multiple times the switching frequency ( f s , 2 f s , 3 f s , . . . ), [198] proposed a selective torque ripple elimination using carrier phase shift. The shifted phase can be selected from −π/2, π/2, and π in terms of different spatial phase displacement between two winding sets. [195,196] connect additionally coupled inductor be the circulating current, which has the disadvantag and complexity. The carrier phase shift is extended phase) PMSM, of which each phase is fed by an H general and analytical framework as well as the me ried out based on the evaluation of radial electroma the airgap. To eliminate the torque ripple and vibr frequency ( , 2 , 3 , …), [198] proposed a selectiv rier phase shift. The shifted phase can be selected fr ent spatial phase displacement between two windin