Five-Dimensional Switching-Table-Based Direct Torque Control of Six-Phase Drives

This article proposes a novel 5-D switching-table-based direct torque control (5D-DTC) strategy for asymmetrical six-phase induction machines (6PIMs). As is well-known, classical DTC of 6PIM is penalized by significant stator current harmonics, which are mapped into the nonenergy subspace (<inline-formula><tex-math notation="LaTeX">$x-y$</tex-math></inline-formula> subspace). The concept of virtual voltage vectors (VVs) has been frequently perceived as able to tackle the problem of <inline-formula><tex-math notation="LaTeX">$x-y$</tex-math></inline-formula> currents. Such a concept maintains zero average volt-second in the <inline-formula><tex-math notation="LaTeX">$x-y$</tex-math></inline-formula> subspace, which in turn suppresses current harmonics due to discrete pulsewidth modulation implementation to a great extent compared to the classical DTC. However, it still cannot effectively compensate for <inline-formula><tex-math notation="LaTeX">$x-y$</tex-math></inline-formula> currents, mainly arising from dead band effect and machine/converter asymmetry, due to lack of dedicated regulators for <inline-formula><tex-math notation="LaTeX">$x-y$</tex-math></inline-formula> currents. Indeed, the <inline-formula><tex-math notation="LaTeX">$x-y$</tex-math></inline-formula> currents can be fully suppressed only in the presence of active control over them. This article incorporates additional hysteresis regulators of <inline-formula><tex-math notation="LaTeX">$x-y$</tex-math></inline-formula> currents into the direct torque control strategy. In the proposed 5D-DTC scheme, there are in total five indexes for optimal selection of VVs as torque, stator flux, <inline-formula><tex-math notation="LaTeX">$x-y$</tex-math></inline-formula> currents, and stator flux position indexes. In this way, a clustering method is developed for synthesizing VVs to cover all possible cases of the switching table. A beneficial feature in comparison with 3-D switching-table-based DTC is effective suppression of <inline-formula><tex-math notation="LaTeX">$x-y$</tex-math></inline-formula> currents with a rather simple structure, without increasing the switching frequency, and without decreasing dc-link utilization. Experimental results confirm the effectiveness of the proposed technique.


I. INTRODUCTION
T HE concept of more electric systems on the one hand, and the necessity of reliable operation in a range of safetycritical applications on the other hand, have stimulated intensive research interests in the development of multiphase drives [1]- [4]. For high-performance control of multiphase drives, different versions of field-oriented control (FOC) [5]- [8], model predictive control (MPC) [9]- [13], and switching-table-based direct torque control (ST-DTC) [14]- [22] schemes have been proposed. Compared to FOC and MPC strategies, ST-DTC offers extremely high dynamic response besides a straightforward and robust structure [23], [24]. However, a simple extension of classical ST-DTC to a multiphase drive can result in unacceptable performance because of uncontrolled secondary subspaces (several x − y orthogonal subspaces in relation with the number of machine phases) [15]. Indeed, more freedom degrees within multiphase drives cannot be ignored and dedicated techniques are needed for control of these additional freedom degrees in particular applications [1].
To tackle the problems of uncontrolled secondary subspaces, virtual voltage vectors (VVs) have been synthesized in various ways for nonmodulation-based control techniques, i.e., MPC [10]- [13] and ST-DTC [15]- [22] techniques, where voltage vectors are selected from the switching table and cost function minimization, respectively. In the case of six-phase drives, a synthesis of different actual voltage vectors to provide VVs with different amplitudes is well-addressed in [17] and [18]. These schemes can reduce x − y currents to a large extent compared with the classical ST-DTC. Furthermore, they can explore the possibilities of reducing torque ripples by applying a five-level torque hysteresis regulator. However, they may not satisfactorily compensate for x − y currents due to some phenomena such as dead time effect, winding/converter asymmetry, and backelectromotive force (BEMF) distortion. To this end, a remedial solution has been recently proposed in [19] for reducing BEMF harmonics in ST-DTC of permanent-magnet synchronous machine (PMSM) drives, not dead time and asymmetry harmonics. Actually, x − y currents can be effectively compensated only when x − y current controllers are incorporated into the ST-DTC technique. It is worth mentioning here that, as it has been reported in [8], the machine/converter asymmetries in six-phase drives introduce x − y currents at fundamental frequency rotating in the synchronous, antisynchronous, or both directions. The rotation direction of these harmonics depends on the type of asymmetry, where unequal amplitudes for each three-phase set is a very likely asymmetry in six-phase drives because of three-phase modularity. On the other hand, the dead time effect produces x − y currents with dominant harmonic orders of fifth and seventh in the case of six-phase drive with two isolated neutral points.
While the aforementioned problem can be effectively resolved in modulation-based control techniques using appropriate x − y current controllers [6]- [8], they remain problematic in the case of ST-DTC due to lack of x − y current controllers. It is worth mentioning here that the idea of null voltage vectors in the x − y subspace cannot guarantee that there will be no x − y This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ currents in real implementation, because there is no control over x − y currents yet. Therefore, devising an ST-DTC method to suppress x − y currents is still a challenging task and requires more investigation, which has been studied to a limited extent until now [20], [25]. The authors in [20] proposed a modified ST-DTC scheme with closed-loop current regulation based on a proportional-integral-resonant (PIR) controller for dual three-phase PMSMs, where this scheme poses the risk of increasing the average switching frequency and complexity of the ST-DTC method. In [25], a dynamic duty-ratio-based DTC technique has been proposed, which maintains the average switching frequency at the same level of the conventional ST-DTC method [17]. However, it may be still overshadowed by the complexity due to extra PI regulators. Extending the switching table to control the additional degrees of freedom of a multiphase machine may be considered to be a possible solution that can retain most of the useful features of the conventional ST-DTC method, which has not been elaborated until now. The idea of extending the switching table by synthesizing VVs to improve the steady-state performance of three-phase PMSM has been reported in [26], while developing some elegant solutions to fully control the freedom degrees of multiphase drives within ST-DTC still remains challenging.
In this article, a novel 5-D switching table-based direct torque control (5D-DTC) strategy is proposed to fully control the freedom degrees of an asymmetrical six-phase induction machine (6PIM) with two isolated neutral points. The main contribution of this article is incorporating two additional x − y currents hysteresis regulators into the ST-DTC scheme, which provides a 5D-DTC scheme. In the proposed scheme, there are in total five indexes to make a more accurate decision on the selected VVs during every sampling period. These five indexes include electromagnetic torque, stator flux amplitude, angular position of stator flux, and x − y currents indexes. The core of the proposed scheme is based on clustering of VVs, where each VV is replaced by a quadruple cluster of VVs to increase/decrease x − y voltage components during every sampling period. Taking into account that an asymmetrical 6PIM with two isolated neutral points contains four degrees of freedom, i.e., torque, stator flux, x-axis current, and y-axis current, the proposed method can control all degrees of freedom simultaneously using an extended switching table. It is worth mentioning here that in most previous schemes with conventional VVs, only the torque and stator flux are controlled, while the x − y voltages are always kept at zero within each sampling period, which cannot guarantee zero x − y currents due to dead band effect and machine/converter asymmetry. Extensive experimental results are provided to validate the feasibility of the proposed scheme. The proposed scheme is compared to the 3-D switching table-based DTC (3D-DTC), which the latter case is also beneficiary of VVs, without additional control over x − y currents.

II. PROPOSED 5D-DTC
A. Preliminary Remarks 1) 6PIM Model: In this article, the well-known vector space decomposition (VSD) technique [27] is adopted for analytical modeling and control of asymmetrical 6PIM with two isolated neutral points. In this regard, the quantities can be decoupled into three orthogonal subspaces α − β (torque-component subspace), x − y (harmonic-component subspace), and o 1 − o 2 (zero-sequence subspace) using VSD transformation as presented in [27]. The o 1 − o 2 current components cannot flow in the two star-connected stator windings. The stator voltage equations in the α − β and x − y subspaces can be written as where, v, i, Ψ, R, and L illustrate voltage vector, current vector, flux vector, resistance, and inductance, respectively; Subscripts s, α, β, x, y, and l indicate stator, α-axis, β-axis, x-axis, y-axis quantities, and leakage inductance, respectively; p denotes the derivative operator and j is the imaginary unit.
2) Actual Voltage Vectors: In a two-level dual three-phase voltage source inverter (VSI)-fed 6PIM with two isolated neutral points, there are 12 actual large voltage vectors, which can be represented by binary sequences as [S u1 S u2 S w1 S w2 S v1 S v2 ], in turn, decimal numbers referring to the state of upper VSI switches. These actual large voltage vectors with corresponding decimal numbers are shown in Fig. 1. The amplitudes of actual large voltage vectors in the α − β and x − y subspaces are 2V dc cos(π/12)/3 and 2V dc cos(5π/12)/3, respectively, where V dc is the dc-link voltage. It can be seen from these amplitudes that applying the actual large voltage vectors during every sampling period in the classical ST-DTC method stimulates the x − y subspace significantly, and subsequently, high harmonic contents of the stator currents can flow due to low impedance of the x − y subspaces, which causes a significant efficiency decrease.
3) Virtual Voltage Vectors: As reported in [25] and [28], synthesizing the VVs effectively alleviates the problem of large stator current harmonics within classical ST-DTC, where the maximum amplitude of synthesized VVs in the α − β subspace is 0.5977V dc with zero resultant vector in the x − y subspace. Therefore, a feasible maximum rate of dc-link utilization can be calculated as 92.82% [28], which is 7.18% less than full utilization rate of the dc supply, when actual large voltage vector vectors are used. The VVs with maximum voltage level can be synthesized in two ways, by large and medium large voltage vectors (hereinafter labeled 2D-VVs) or by three consecutive actual large voltage vectors (hereinafter labeled 3D-VVs) [25]. In this article, 3D-VVs are employed due to their flexibility to compensate for x − y components, which will be discussed further in the next section. Details of 3D-VVs construction can be found in [25] and [28].
4) Flexibility of 3D-VVs: As shown in [20], zero x − y voltage vectors within conventional VVs may not suppress x − y current harmonics due to machine/converter asymmetry and dead band effect. A principal prerequisite for compensating for such harmonics is that the synthesized VVs have sufficient flexibility to increase/decrease x − y components at every sampling period. In other words, the synthesized VVs should be flexible enough to simultaneously regulate the torque, stator flux, x-axis, and y-axis components. This can be realized only when the projection of actual voltage vectors, which are constituent of VVs, cover all four quadrants of the x − y subspace for all angular positions of the stator flux. Taking a detailed look at the 2D-VVs [25], it can be easily seen that their projection in the xand y-axes is always two-quadrant, and this makes them unsuitable for the proposed 5D-DTC scheme. However, 3D-VVs (based on three consecutive actual large voltage vectors) are flexible and four quadrant, which offer an exciting possibility to regulate all freedom degrees of 6PIM drives, i.e., torque, stator flux, and x − y currents. Therefore, 3D-VVs are employed in this article, where every conventional 3D-VV (throughout this article, conventional 3D-VVs refer to the vectors that have zero average volt-second in the x − y subspace) is replaced by a cluster of four 3D-VVs in its vicinity in the α − β subspace, resulting in the same effect on the torque and stator flux, while it is four quadrant in the x − y subspace with the ability of increase/decrease in the x − y components during every sampling period. The clusters of 3D-VVs are schematically shown in Fig. 1, where the 3D-VVs for increasing or decreasing the x − y components are marked with four different colors. The 3D-VVs within each cluster are exactly bisectors of each quadrant in the x − y subspace with similar predefined amplitudes, which will be explained further in the following sections. In Fig. 1 as well as throughout this article, the superscripts 0-3 for the 3D-VVs of each cluster are defined as follows: B. VVs Clustering 1) Mathematical Derivations: In this article, clustering method refers to the process of extending every conventional 3D-VV for four 3D-VVs in its vicinity in the α − β subspace, which enables simultaneous control of all freedom degrees of 6PIM. Some clusters of the 3D-VVs for different values of the injected x − y voltages are illustrated in Fig. 2. Regarding every 3D-VV made by three consecutive actual voltage vectors, the normalized volt-second equations for every 3D-VV (in total 12 vectors) in the x − y subspace can be written as follows: where v hxy , h = {1, 2, 3} are the normalized x − y components of 3D-VVs, t h is the applied time of actual voltage vectors within 3D-VVs, and v xyi are the injected x − y voltages. Taking t 1 + t 2 + t 3 = 1 into consideration, the applied times t 1 , t 2 , and t 3 can be represented by two action times T 1 and T 2 for experimental realization as follows: Solving (2)-(4) for calculating the action times yields It should be noted that (5) and (6) are valid for all 12 synthesized 3D-VVs. Assuming v xi = v yi = 0 yields which keeps the x − y voltages at zero during each sampling period. This is the principal concept of conventional VVs-based ST-DTC techniques in the most of literature [17], [18]. The related 3D-VVs are shown in Fig. 2(f). However, injecting x − y voltages in four directions can produce clusters of 3D-VVs, as shown in Fig. 2(b)-(e) so that the dispersion degree of 3D-VVs within each cluster in the α − β subspace depends on the amplitude of the injected x − y voltage. In this regard, there is a maximum/minimum allowable limit for the injected x − y voltages, which is specified by TMVCL in this article. For example, in order to synthesize the clusters of 3D-VVs for 5%TMVCL, (5) and (6) should be calculated four times for the injected x − y voltages as 2) Total Maximum Voltage Compensation Limit (TMVCL): It is defined as the maximum absolute value of the injected x − y voltages without erroneous calculation of the action times. For TMVCL calculation, the maximum absolute value of the injected x − y voltages for each 3D-VVs should be first calculated, then the minimum of these values is TMVCL. TMVCL depends on the amplitude of actual large voltage vectors in the x − y subspace. It can be proved that normalized TMVCL for 3D-VVs is as follows: 3) Example of Switching Sequences for Cluster of 3D-V V L7 : Fig. 3 shows an example of switching sequences for synthesizing the cluster of 3D-V V L7 , when the injected x − y voltages are ±50%TMVCL. Fig. 3(a) shows the switching sequence of the actual large voltage vector V L7 , which is specified by decimal number 15 or binary sequence "001111." Fig. 3(b) shows the conventional 3D-V V L7 , where the action times take place ex- Fig. 3(c)-(f), which can increase/decrease x − y voltages by partial displacements of the action times T 1 and T 2 . The actual and virtual voltage vectors corresponding to every switching sequence are also shown in the top side of Fig. 3.

4) Properties of 3D-VVs Clusters:
As can be seen from Fig. 3, a maximum of one jump may take place at every sampling period for each VSI switch, when conventional 3D-VVs are implemented. This increases the average switching frequency of VVs-based ST-DTC in exchange for significant decrease of the current harmonics compared with the classical ST-DTC, which is a very sensible compromise, because the classical ST-DTC is seriously penalized by a substantial amount of stator current harmonics. On the other hand, the cluster of 3D-VVs does not increase the average switching frequency compared to conventional 3D-VVs, because only the action times of inverter jumps change within each cluster, not the number of jumps. These two points are also valid for all remaining clusters of 3D-VVs.
The dc-link utilization rate (η) can be calculated as the ratio of the amplitude of VVs to the amplitude of actual large voltage vector [25], [28]. As already mentioned, this value is constant and equals 92.82% for the conventional 3D-VVs. However, η is not constant anymore for clusters of 3D-VVs when x − y voltages are injected. Changes of η for different injected x − y voltages and different clusters of 3D-VVs are shown in Fig. 4(a). The surfaces of TMVCL, maximum/minimum η, and η of conventional 3D-VVs are also included in this figure. Moreover, the averages of η for all 3D-VVs belonging to the same scenario of x − y voltage injection are shown in Fig. 4(b). From this figure, it can be seen that the average η for the clusters of 3D-VVs is always above η of conventional 3D-VVs. To clarify the issue more, the results of Fig. 4 are discretized in Fig. 5 for some predetermined x − y voltage levels. Fig. 5 shows dc-link utilization rate for all synthesized 3D-VVs in three different injected x − y voltage levels. It can be seen from this figure, the smaller injection of x − y voltages, the closer η to the conventional 3D-DTC method. A reasonable level for the injected x − y voltages can ensure effective suppression of x − y currents and keeping η very close to the conventional 3D-DTC (in this article, 20%TMVCL is chosen).

C. Basics and Principles of 5D-DTC 1) Basics of 5D-DTC:
The block diagram of the proposed 5D-DTC scheme is shown in Fig. 6. The concept of the proposed technique is as straightforward as the classical ST-DTC method.
In the classical ST-DTC scheme, the decoupled control of the electromagnetic torque and stator flux is attained by acting on the tangential and radial components of the stator flux space vector in the α − β subspace, respectively. This will be achieved through optimal selection of the voltage space vectors using a 3-D switching table, where the selection indexes are based on torque, stator flux amplitude, and stator flux angular position.  Such a concept is extended for the 6PIM drive system in this article, to act not only on tangential and radial components of the stator flux space vector in the α − β subspace but also on its radial components in the x − y subspace. The x − y components of the stator flux can be interpreted as the x − y currents regarding (1). In the proposed scheme, the optimal selection of the voltage space vectors is achieved by a 5-D switching table, where the x − y current indexes are considered in addition to the selection indexes of the 3-D switching table.
Thanks to the 6PIM model in the x − y subspace presented in (1), it can be seen that x − y currents are directly impressed by x − y voltages. Hence, to increase and decrease x − y currents in a short period of time, x − y voltages should be increased and decreased, respectively. This means that the stator current space vector in the x − y subspace changes in the same direction of the applied voltage space vector in the x − y subspace. The introduced clusters of 3D-VVs, including four VVs within each cluster, can change x − y voltages in four possible scenarios, i.e., increase v x and increase v y , increase v x and decrease v y , decrease v x and increase v y , and decrease v x and decrease v y . Accordingly, x − y currents can be regulated during every sampling period regarding their required demands arising from the hysteresis regulators.
2) Principles of 5D-DTC: In this article, a three-level torque hysteresis regulator is adopted, while hysteresis regulators for the stator flux, x-axis current, and y-axis current have two levels. Hence, there are in total four hysteresis regulators. As mentioned many times in previous literature [23], the control law for the torque hysteresis regulator is and the control law for the flux hysteresis regulator is and B ψ are the torque and flux hysteresis bands, respectively, d T e and d ψ s are torque and flux indexes, respectively. Both 3D-DTC and proposed 5D-DTC techniques have the same control laws of (11) and (12). Supplementary control laws for x − y currents are defined in the proposed scheme as follows: where Δi x = i * x − i x , Δi y = i * y − i y , B x and B y are the x − y currents hysteresis bands, and d i x and d i y are the x − y currents indexes. Obviously, the x − y current references are set to zero during normal operation of the 6PIM drive. The clusters of 3D-VVs are flexible enough to act on the voltage space vectors, in a way that a complete control over all freedom degrees of the 6PIM drive system is achieved. Table   1) Rules of Operation: The proposed 5-D switching table is shown in Table I. There are five indexes for selecting voltage space vectors within every sampling period as d ψ s , d T e , d i x , d i y , and θ s (angular position of the stator flux). The proposed 5-D switching table functions exactly like the classical switching table for selecting every cluster of 3D-VVs, where all four 3D-VVs belonging to the selected cluster give the same result in the torque and flux. Indeed, the rules for selecting the clusters are the same as conventional ST-DTC schemes [24]. Therefore, the torque and flux indexes are used to select each cluster, then the x − y currents indexes are employed to select the final 3D-VV belonging to the selected cluster. Considering three-level torque and two-level flux hysteresis regulators, there are 3 × 2 = 6 different voltage clusters for every sector (sector number is denoted by m in Table I derived from θ s ), which are distinguished in the four-member groups in Table I, e.g., rows 1-4, rows 5-8, and so on. It should be noted that in the conventional ST-DTC schemes, there are six different actual or virtual voltage vectors, not voltage clusters. Considering two-level x − y currents hysteresis regulators, there are in total 6 × 4 = 24 3D-VVs for every sector to be selected according to d ψ s , d T e , d i x , and d i y indexes. The proposed 5-D switching table can be easily implemented by defining an overall index as follows:

D. Five-Dimensional Switching
For example, if d ψ s = 1 (increase flux), d T e = −1 (decrease torque), d i x = 1 (increase i x ), and d i y = 0 (decrease i y ), the overall index is calculated as d o = 10, which means that V V 2 L(m−2) should be selected. Then, the final 3D-VV is selected according to the sector number m. For example if m = 5, the final selected 3D-VV is V V 2 L3 . As mentioned earlier, the superscripts 0-3 are used to specify four 3D-VVs within each cluster. These four 3D-VVs are also highlighted by the assigned color with bookmark symbol in Table I. Table Calculations for Cluster of 3D-V V L7 : Every element of the proposed 5-D switching table should include information about the duty ratio of the selected 3D-VV and the state of the switching sequence at the beginning of every sampling period. The main calculations of the proposed switching table for a specified TMVCL are performed offline. Hence, the computational burden of the proposed 5D-DTC is not much greater than the conventional 3D-DTC. In this section, the calculations of the switching table elements for the cluster of 3D-V V L7 are described. The switching sequences for this cluster were already presented in Fig. 3. Nevertheless, the same calculations can be extended over all other clusters of 3D-VVs. The calculations are based on the assumption that the injected x − y voltages are ±50%TMVCL. As described in [25], 3D-V V L7 is synthesized by three consecutive actual large vectors V L6 , V L7 , and V L8 , whose corresponding decimal numbers are 14, 15, and 7, respectively, referring to Fig. 1. Accordingly, the binary sequences are "001110," "001111," and "000111," where the phase sequence of

2) Example of Switching
The switching state matrix for 3D-V V L7 can be written as follows: (16) where superscript "T " stands for transposition. Thanks to (2) and (3), the applied time matrix of actual voltage vectors within 3D-V V L7 for ±50%TMVCL is calculated as follows: (17) Using (16), (17), and the duty ratios of 3D-V V L7 for all scenarios of x − y voltage injection can be obtained as follows: (19) It should be noted that (19) was already visualized in Fig. 3. The state of the switching sequence at the beginning of every sampling period is needed for digital implementation in addition to (19). For this purpose, the flag matrix is defined based on the first row of (16) as (20) For easy implementation, the duty flag matrix may be taken into consideration as follows: which results in (22) Finally, df V V L7 matrix can be implemented by the following rules: 1) if df V V L7 = 0, the pulsewidth modulation (PWM) output is low for the whole sampling period; 2) if df V V L7 = 2, the PWM output is high for the whole sampling period; 3) if 0 < df V V L7 < 1, the PWM output is low high, where the transitioning from low to high takes place at the PWM output is high-low, where the transitioning from high to low takes place at (df V V L7 − 1)T s . In the case of conventional 3D-VVs, the applied time matrix of actual voltage vectors only includes one column as follows: The duty ratios of 3D-V V L7 are then obtained as follows: (24) which has been approved by many previous literature, for example, see [29, Table III].

III. EXPERIMENTAL RESULTS
Extensive experiments were performed to validate the effectiveness of the proposed scheme in different steady-state (SS) and dynamic test conditions. To consider the comprehensiveness of the study, experimental results are presented for different speed directions and setpoints under different load torques. Moreover, to highlight the superiority of the proposed scheme, all experiments are compared with the conventional 3D-DTC scheme. It should be noted that both conventional 3D-DTC and proposed 5D-DTC schemes, under study in this article, are beneficiaries of VVs. The experimental results of the classical DTC method based on actual voltage vectors are not presented in this article, because it has been reported in many previous literature [17], [18]. A 6PIM test setup was employed, the details of which can be found in [25] and [28]. The hysteresis bands for torque, flux, and x − y currents are chosen as 0.5 Nm, 0.012 Wb, and 0.1 A, respectively. The x − y currents bands are adjusted based on the trial-and-error method. If these bands are selected too small, the regulators would control the switching noises, which is not obviously their purpose. If the x − y currents bands are selected too big, the x − y currents may not be effectively suppressed. Hence, a compromise is made to appropriately select the x − y currents hysteresis bands. The clusters of 3D-VVs are calculated for 20%TMVCL, which means that the maximum absolute value of the injected x − y voltages is 0.0065 p.u. during each sampling period. The stator flux command is set to 0.6 Wb for all experiments.
The SS performance of the conventional 3D-DTC and the proposed 5D-DTC schemes for a speed command of +250 r/min under no load and rated load torque conditions is shown in Fig. 7(a) and (b), respectively. Each oscillogram includes the traces of the rotor speed, electromagnetic torque, x − y voltages, u 1 − u 2 phase currents, and x − y currents. As reported in [8], if the x − y current controllers of the field-oriented control (FOC) strategy are inactivated by setting the x − y voltages to zero, the machine/converter asymmetry may cause x − y currents to rotate at the fundamental frequency. On the other hand, dead band effect causes additional harmonics with the order of 6k ± 1, k = 1, 3, 5, . . ., with dominant orders of fifth and seventh in the case of 6PIM with two isolated neutral points. The dead band harmonics are mainly mapped in the x − y subspace. In the conventional 3D-DTC, there is no control over x − y currents, which can be understood from the traces of x − y voltages within Fig. 7, where the conventional 3D-VVs offers a zero average volt-second in the x − y subspace. In this  condition, both asymmetry and dead band harmonics appear in the x − y current. However, the proposed 5D-DTC scheme effectively compensate these harmonics under both no load and loading test conditions. It can be seen from Fig. 7 that the x − y voltages are not zero anymore in the case of the proposed method, while the injected x − y voltages are decided according to the proposed 5-D switching table to maintain the x − y currents inside the hysteresis bands. The effect of machine/converter asymmetry on the phase current traces of the conventional 3D-DTC scheme can be clearly observed in Fig. 7, where the amplitude of the phase-u 2 is slightly larger than the amplitude of phase-u 1 . This causes x − y currents at fundamental frequency, which the proposed 5D-DTC alleviates by implementing the hysteresis regulators and clusters of 3D-VVs.
A harmonic analysis is performed to highlight the superiority of the proposed 5D-DTC over the conventional 3D-DTC. The harmonic spectra of x − y currents at speed command of 250 r/min under no load and rated load torques are shown in Fig. 8(a), and (b), respectively. Moreover, the harmonic spectra of phase currents (phase-u 1 and phase-u 2 ) at the same operating conditions are shown in Fig. 9. These harmonic spectra correspond with the experimental results of Fig. 7. The machine/converter asymmetry causes unequal phase currents in the studied 6PIM drive, thereby x − y currents at fundamental frequency. It can be seen from Fig. 8 that the proposed method effectively suppresses x − y currents at fundamental frequency. Accordingly, as shown in Fig. 9, the difference in amplitude of the phase currents in the proposed method is less than the conventional method. On the other hand, the proposed method alleviates dominant fifth and seventh current harmonics in the 6PIM drive, which are mainly caused by dead band effect. Fig. 10 shows the experimental results for SS performance of the conventional 3D-DTC and proposed 5D-DTC schemes at −1000 r/min with/without load torque. From top to bottom, the traces of the phase-u 1 current, x-axis current, torque, and rotor speed are shown within every oscillogram. A considerable reduction in x − y current components can be observed, especially for loading tests, when the proposed 5D-DTC scheme is used. Taking a detailed look at torque traces in Figs. 7 and 10, it can be seen that the proposed 5D-DTC method does not cause any tangible effect on the SS performance of the electromagnetic torque, because the injected x − y voltages are small to the extent that they have no significant effect on the controlled variables in the α − β subspace.
The calculated average switching frequency f sw at speed command of +250 r/min under no load torque is around 2.3 kHz for both 3D-DTC and 5D-DTC schemes, while f sw is about 2.9 kHz under rated load torque for both schemes. The average switching frequency at speed command of −1000 r/min under no load torque and rated load torque is around 3.1 and 3.5 kHz, respectively, for both conventional 3D-DTC and proposed 5D-DTC schemes. These results were predictable, because as previously justified, the proposed method does not increase the average switching frequency compared to the conventional method. The reason behind this reality can be concluded from Fig. 3, where it can be seen that only the action times are partially changed using the proposed technique, while the number of transitioning does not change, and there is a maximum of one transitioning for each phase during every sampling period for all 3D-VVs. It is worth mentioning here that, f sw can be affected by the torque and flux hysteresis bands, not by x − y currents hysteresis bands. The reason is that only torque and flux hysteresis regulators decide about the selected clusters. When a cluster is selected, the x − y currents hysteresis regulators decide about the final 3D-VV among four available subvectors inside the selected cluster. However, there is no difference between these four 3D-VVs from switching frequency viewpoints.
The SS experimental results for the current traces as well as the duty ratios of the 3D-VVs with the conventional and proposed DTC techniques at low speeds under no load torque are shown in  Fig. 11, where the speed command is set to −100 r/min. A zoom of x − y currents in this figure may better demonstrate the ability of the proposed 5D-DTC method to reduce the asymmetry harmonics, which appear as x − y currents at the fundamental frequency when the conventional 3D-DTC method is operated. It can be seen from Fig. 11(b) that the duty ratios of the conventional 3D-VVs take the values of 1 (high PWM output), 0 (low PWM output), 2 − √ 3 = 0.2679, and √ 3 − 1 = 0.7321 (high-low or low-high PWM output depending on the selected 3D-VVs). Such duty ratios always obtain zero x − y voltages, which cannot compensate for dead band and asymmetry harmonics. However, the duty ratios of the 3D-VVs within the proposed 5D-DTC scheme are not exactly 0.2679 or 0.7321 values. They may be a bit larger or smaller than these values according to the x − y currents indexes as the output of x − y currents hysteresis regulators, in turn, the selected 3D-VVs. Such a selection always increases or decreases x − y currents inside their upper and lower hysteresis bands.
The dynamic tests besides the SS ones were carried out to evaluate the validity of the proposed 5D-DTC scheme. To this end, different test scenarios have been defined. Fig. 12 shows the dynamic performance of the conventional and proposed DTC methods for different scenarios of speed changes. The experimental results for speed reversal tests with speed commands of ±500 r/min under no load and rated load torque are shown in Fig. 12(a) and (b), respectively. The experimental results for a step increase in the speed command from 10 r/min (very low speed) to 500 r/min under no load torque are presented in Fig. 12(c). Moreover, the dynamic performances for load torque changes at a speed command of 500 r/min are shown in Fig. 13(a) and (b), for sudden loading and unloading scenarios, respectively, where the rated load torque is applied. In the dynamic test package, the traces of rotor speed, load torque, phase-u 2 current, and y-axis current are shown within each captured oscillogram. From these results, it can be first seen that the proposed 5D-DTC method offers a comparable dynamic performance with the conventional 3D-DTC scheme, because as already mentioned, the controlled variables in the α − β subspace are not significantly impressed by the proposed  method. Moreover, the ability of both conventional and proposed schemes to maintain the stability of the controller under dynamic tests is visible from Figs. 12 and 13. The superiority of the proposed 5D-DTC scheme over the 3D-DTC scheme in reduction of the x − y currents can also be concluded from these results (the x-axis current is not shown because a four-channel oscilloscope was employed). The amount of x − y currents is intensified when a load torque is applied to the 6PIM, while the proposed 5D-DTC scheme effectively compensates for the current harmonics under loading tests as well.
A comparison between the proposed 5D-DTC, conventional 3D-DTC, and [25] helps to clarify the advantages and disadvantages of each technique. Such a comparison is tabulated in Table II, where these methods are evaluated using different items based on experimental results of this article and [25]. As shown in Table II, all three studied methods have relatively similar performance in terms of dc-link utilization rate, average switching frequency, parameter dependence, torque and flux ripples, and transient performance. The reasons for that have been mentioned earlier by analytical descriptions besides the experimental results. The reference to judge every item can be found in the last column of Table II. The average switching frequency of conventional 3D-DTC, method of [25], and proposed 5D-DTC over operating speed range is shown in Fig. 14. It can be seen from this figure that the maximum deviation of Δf sw between these three schemes is around 20 Hz, which is negligible regarding f sw . The proposed 5D-DTC method has a higher ability to suppress x − y currents compared with 3D-DTC, but a lower ability in comparison with [25]. The reason is that the ST-DTC method of [25] employs a dual PI regulator, which can accurately provide the injected x − y voltages. However, this is achieved in exchange for higher complexity. In this regard, the monitored rates of the central processing unit (CPU) utilization (at 150 MHz operating speed of the adopted digital processor) for 3D-DTC, [25], and the proposed 5D-DTC are 20.15%, 39.27%, and 26.32%, respectively. In actual, the extra computational burden, which the algorithm of [25] imposes to 3D-DTC, is more considerable compared with the proposed 5D-DTC. The major reason behind this fact is the online calculation of appropriate duty ratios at every sampling period in [25], while there are limited precalculated duty ratios in the proposed 5D-DTC. Roughly speaking, the proposed method in this article discretizes the method of [25] in a sense. In [25], there are indefinite number of x − y voltage commands at every sampling period, which arises from outputs of PI regulators. To apply these voltage commands, indefinite numbers of duty ratios are possible at every sampling period. However, the proposed method of this article uses only four x-y voltage options at every sampling period. Therefore, the proposed 5D-DTC technique offers an in-between solution with a medium ability to reduce x − y currents and medium complexity.

IV. CONCLUSION
This article proposed a novel 5D-DTC strategy for asymmetrical 6PIM drives to alleviate the contents of x − y currents due to machine/converter asymmetry and dead band effect. To this end, the clusters of 3D-VVs were synthesized for appropriate injecting x − y voltages during every sampling period. It was experimentally proved that the 3D-VVs belonging to every cluster have the same effect on α − β components, while they can increase or decrease x − y components. The proposed 5D-DTC scheme includes a torque hysteresis regulator and a stator flux hysteresis regulator exactly like the conventional 3D-DTC with the same rules for selecting the optimal voltage space vectors, as well as, two x − y currents hysteresis regulators in addition to the conventional scheme to provide a full control over all freedom degrees of the 6PIM drive with two isolated neutral points. Extensive experiments were carried out under different SS and dynamic test scenarios, where the results showed the effectiveness of the proposed 5D-DTC scheme in reduction of x − y currents, due to asymmetry and dead band effect, without increasing the average switching frequency compared with the conventional 3D-DTC scheme.