Direct Torque Control of Dual Three-Phase Permanent Magnet Synchronous Motors Based on Master–Slave Virtual Vectors

: In order to further reduce the torque, flux-linkage fluctuation, and current harmonic content of dual three-phase permanent magnet synchronous motors, this paper proposes a direct torque control strategy combined with a master–slave virtual vector duty cycle assignment. Two types of virtual voltage vectors with different amplitudes are used to form a harmonic suppression switching table. The virtual vectors are classified into master and slave virtual vectors according to the degree of influence on the torque and the flux-linkage. Then, the duty cycle of the master and slave virtual vectors is recalculated and allocated through the evaluation function to achieve accurate control of the torque and the flux-linkage. Finally, the switching sequences of the master and slave virtual vectors that act together in one control cycle are rearranged into a symmetrical waveform. It is experimentally verified that the phase current THD of the proposed strategy is reduced by 69.4%, the 5th and 7th current harmonics content is significantly reduced, and the torque fluctuation and flux-linkage fluctuation can also be effectively suppressed, which provides better dynamic performance and steady-state performance.


Introduction
With the increasing demand for the power and safety of motor systems in engineering applications such as aerospace and electric vehicles, the research on multiphase motors with high power and high reliability has attained significant proportions in recent years.Among various multiphase drives, the dual three-phase permanent magnet synchronous motor (DTP-PMSM) drives have the advantages of both multiphase motor drives and permanent magnet motor drives.Compared to three-phase drives, the DTP-PMSM drives offer superior features of reduced torque ripple and remarkable fault tolerance capability [1][2][3].
The direct torque control (DTC) scheme has a simpler structure and faster dynamic response in comparison with field-oriented control (FOC).Meanwhile, for DTP-PMSMs, the six-phase inverter that drives the motor operation can generate more voltage vectors, which provides a rich vector control set for the DTC scheme and enhances the flexibility of the control strategy, but also increases the difficulty of the algorithm design [4,5].In addition, it is noted that only closed-loop control of the fundamental subspace will cause a larger harmonic current due to the relatively small impedance of harmonic subspace in the mathematical model of vector space decomposition (VSD) [6,7].
The voltage vectors controlling the DTP-PMSM drives contain 60 effective vectors and 4 zero vectors, which can be divided into four groups according to the magnitude of the vectors.So as to decrease the complexity of the switching table and increase the voltage utilization, the traditional DTC switching table of DTP-PMSMs only selects the 12 large vectors with the largest magnitude on the fundamental subspace.For the sake of solving the problem of current harmonics in a large vector-based DTC scheme, the modified intermediate vectors are applied to suppress the 5th and 7th harmonics in the harmonic subspace.The modified vector was centralized for simplifying the hardware implementation [8].In some of the literature [9][10][11][12], the non-harmonic voltage vector switching table is applied to reduce the current harmonics.Moreover, the high-order torque hysteresis controller is used to divide the hysteresis width into several adjustment intervals, which can suppress the torque fluctuation and steady-state error, but the voltage utilization of this strategy will be reduced to a certain extent.To suppress the torque fluctuation of the DTC of DTP-PMSMs, some scholars have proposed vector space modulation technology.Scholars propose a direct torque control strategy of DTP-PMSMs based on vector space modulation, which not only reduces harmonic current and torque fluctuation but also ensures constant switching frequency [13].In [14], a hybrid DTC strategy suitable for DTP-PMSMs is proposed.The strategy adopts different torque control modes in dynamic and steady-state modes, respectively, and an observer is designed to realize the tracking control of the flux-linkage.Some scholars also combine DTC with model predictive control (MPC) to reduce the predicted voltage vectors [15,16].In [17], based on 36 effective vectors, the two-step table lookup method is proposed to reduce the calculated vectors.Firstly, 13 vectors are screened out in the fundamental subspace by the positive and negative of the torque error, and then the vectors are further reduced according to the principle of reducing the flux of the harmonic subspace, which simplifies the computational process.In [18], an enhanced DTC strategy based on discrete virtual vectors is proposed to improve the steady-state performance of PMSMs.The proposed strategy maintains the simple control structure and ideal dynamic performance of the DTC strategy and improves the steady-state performance by reducing torque ripple and current distortion.The experiment verifies the effectiveness of the proposed DTC strategy.In [19], the article considers the potential voltage imbalance in the DC link between two DC voltage source inverters of DTP-PMSMs.In order to solve these problems, a DTC strategy considering unbalanced DC side voltage is proposed in this study to improve the performance of DTP-PMSMs using virtual vectors.
Compared with vector control strategies, traditional DTC control saves the complicated coordinate transformation, and can realize the fast response adjustment of the motor by controlling the torque.This method is simple in structure and is not sensitive to the parameters of the motor and external influences.However, in order to ensure the fast dynamic performance of the motor torque, the large vectors are selected for voltage synthesis.The large vector of the α-β subspace will generate a large harmonic current in the harmonic subspace, so the traditional DTC control strategy will have some problems such as a large torque ripple.
Aiming to address the limitations of traditional DTC control for DTP-PMSM drives, a master-slave virtual vector direct torque control strategy based on duty cycle allocation is put forward in this paper to enhance the accuracy of torque/flux-linkage control accuracy and suppress current harmonics.Firstly, different methods are employed to synthesize the virtual vector to suppress current harmonics under various motor operating conditions.Then, considering the impact of torque and flux-linkage, the synthesized virtual vector is divided into master and slave virtual vectors.The duty cycle of each virtual vector is calculated and redistributed based on specific deviations in torque and flux-linkage, enabling precise control over both parameters.Finally, the switch sequence generated by the master and slave virtual vectors is modified to achieve a symmetrical waveform output for motor control.The simulation and experimental results show that compared with the traditional DTC control strategy, the proposed strategy can not only effectively suppress the flux-linkage and torque ripple under different operating conditions but also reduce the difficulty of hardware implementation and ensure the effectiveness of harmonic suppression through the reallocation of the master-slave virtual vector duty cycle.

Mathematical Model of Dual Three-Phase PMSMs
The stator of a DTP-PMSM consists of two sets of Y-connected three-phase symmetrical winding, which are spatially separated by an electrical angle of 30 • and are driven by a six-phase voltage source inverter, as shown in Figure 1.
World Electr.Veh.J. 2024, 15, x FOR PEER REVIEW 3 of 15 reduce the difficulty of hardware implementation and ensure the effectiveness of harmonic suppression through the reallocation of the master-slave virtual vector duty cycle.

Mathematical Model of Dual Three-Phase PMSMs
The stator of a DTP-PMSM consists of two sets of Y-connected three-phase symmetrical winding, which are spatially separated by an electrical angle of 30° and are driven by a six-phase voltage source inverter, as shown in Figure 1.The static coordinate transformation matrix of DTP-PMSMs is shown in (1).
According to the VSD theory, the mathematical model in the static coordinate system of DTP-PMSMs can be decoupled into three two-dimensional orthogonal subspaces, respectively, α-β, x-y, and o1-o2.The components on the α-β subspace comprise the fundamental and harmonic components with orders of 12 m ± 1 (m = 1, 3, 5, •••), which are related to torque generation.The components on the x-y subspace comprise the harmonic components with orders of 6 m ± 1 (m = 1, 3, 5, •••), which do not participate in torque generation.The components on the o1-o2 subspace are the zero-sequence components, which comprise the harmonic components with orders of 6 m ± 3 (m = 1, 3, 5, •••).
The transformation matrix of the rotating coordinate system of DTP-PMSMs is shown in (2).
When the neutral points of the two winding sets are isolated from each other, the zero-sequence subspace can be ignored, then the voltage and torque equations in the rotating coordinate system are as follows, respectively: The static coordinate transformation matrix of DTP-PMSMs is shown in (1).
According to the VSD theory, the mathematical model in the static coordinate system of DTP-PMSMs can be decoupled into three two-dimensional orthogonal subspaces, respectively, α-β, x-y, and o 1 -o 2 .The components on the α-β subspace comprise the fundamental and harmonic components with orders of 12 m ± 1 (m = 1, 3, 5, •••), which are related to torque generation.The components on the x-y subspace comprise the harmonic components with orders of 6 m ± 1 (m = 1, 3, 5, •••), which do not participate in torque generation.The components on the o 1 -o 2 subspace are the zero-sequence components, which comprise the harmonic components with orders of 6 m ± 3 (m = 1, 3, 5, •••).
The transformation matrix of the rotating coordinate system of DTP-PMSMs is shown in (2).
When the neutral points of the two winding sets are isolated from each other, the zerosequence subspace can be ignored, then the voltage and torque equations in the rotating coordinate system are as follows, respectively: where u d , u q and i d , i q represent the dq-axes stator voltages and currents, respectively; u x , u y and i x , i y represent the xy-axes stator voltages and currents, respectively; L d , L q , and L z denote the dq-axes inductances and the leakage inductance, respectively; ψ s is the stator flux-linkage; θ is the rotor position; T e is the electromagnetic torque; n p is the number of pole pairs; and ω is the electric angular velocity.

Traditional Direct Torque Control Scheme
The traditional DTC block diagram of DTP-PMSMs is shown in Figure 2 [20,21].The electromagnetic torque T e , flux-linkage amplitude |ψ s |, and flux-linkage angle θ s are obtained through the torque and flux observer on the α-β subspace.The error between the observed value and the given value is transmitted to the hysteresis comparator to obtain the increase and decrease signal of torque flux-linkage.
s q q q q p q d s s x x p x s y y p y s 0 0 where ud, uq and id, iq represent the dq-axes stator voltages and currents, respectively; ux, uy and ix, iy represent the xy-axes stator voltages and currents, respectively; Ld, Lq, and Lz denote the dq-axes inductances and the leakage inductance, respectively; ψs is the stator flux-linkage; θ is the rotor position; Te is the electromagnetic torque; np is the number of pole pairs; and ω is the electric angular velocity.

Traditional Direct Torque Control Scheme
The   In order to reduce the complexity of the DTC switching table, the traditional strategy selects the twelve outermost vectors VL of α-β subspace.Taking the stator flux-linkage in sector I as an example, from the projection of each vector on the flux-linkage axis in Figure 3a, the vectors for torque increase are V64, V66, V26, and V22, where the vectors that have the greatest influence on torque increase are V66 and V26.Similarly, the vectors for torque reduction are V55, V51, V11, and V13, where the ones that have the greatest influence on torque In order to reduce the complexity of the DTC switching table, the traditional strategy selects the twelve outermost vectors V L of α-β subspace.Taking the stator flux-linkage in sector I as an example, from the projection of each vector on the flux-linkage axis in Figure 3a, the vectors for torque increase are V 64 , V 66 , V 26 , and V 22 , where the vectors that have the greatest influence on torque increase are V 66 and V 26 .Similarly, the vectors for torque reduction are V 55 , V 51 , V 11 , and V 13 , where the ones that have the greatest influence on torque reduction are V 66 and V 26 .Therefore, according to the fact that the two conditions of the increase and decrease of the torque and the flux-linkage are satisfied at the same time, there will always be an effective vector that is selected in the switching table.Taking sector I as an example, the direct torque switching table for DTP-PMSMs is shown in Table 1.The traditional DTC strategy only considers the voltage vector effect in the α-β subspace and does not consider the harmonics caused by the vector in the x-y subspace, so there is a large harmonic current.To solve the above problems, two kinds of virtual vector synthesis methods are proposed, which can control the voltage vector amplitude to zero in the harmonic subspace, so as to suppress current harmonics.The two kinds of virtual vectors can make full use of the rich voltage vectors set of DTP-PMSM and can also increase the control accuracy of motor torque under different states because of the different amplitude of the two kinds of virtual vectors.
As shown in Figure 3, V L , V ML , and V S have the same direction in the α-β subspace but opposite directions in the x-y subspace.Therefore, the amplitude of the voltage vectors on the harmonic subspace can be controlled to zero by adjusting the action time.Let the action time of V L , V ML , and V S be t 1 , t 2 , and t 3 , respectively.The first kind of virtual vector is synthesized via large vectors V L and medium-large vectors V ML in the same direction and satisfies the following relation: The operation time t 1 and t 2 of large and medium-large vectors can be obtained from Equation (6) as follows: The second kind of virtual vector is synthesized via medium-large vectors V ML and small vectors V S in the same direction: Similarly, the action time t 2 and t 3 of medium-large and small vectors can be solved as follows: The first kind of virtual vector voltage amplitude is 0.597U dc , which is very close to the V L amplitude.Under its action, the motor has larger torque and flux-linkage change, and the dynamic performance is better.The second kind of virtual vector voltage amplitude is 0.345U dc , about half of the V L amplitude, which can reduce torque fluctuations and has better steady-state performance, but it is not suitable for larger transient adjustment processes.
In the traditional strategy, regardless of the torque error value, the switching table always outputs a fixed amplitude vector, but this paper chooses the corresponding synthesis method according to the different operating states of the motor.The first kind of virtual vector VV 1-12 is used in the transient adjustment with a large error, and the second kind of virtual vector VV 13-24 is used in the steady adjustment with a small error.The distribution of two kinds of virtual vectors in the α-β subspace is shown in Figure 4.
as follows: The first kind of virtual vector voltage amplitude is 0.597Udc, which is very the VL amplitude.Under its action, the motor has larger torque and flux-linkage and the dynamic performance is better.The second kind of virtual vector voltage tude is 0.345Udc, about half of the VL amplitude, which can reduce torque fluctuati has better steady-state performance, but it is not suitable for larger transient adju processes.
In the traditional strategy, regardless of the torque error value, the switchin always outputs a fixed amplitude vector, but this paper chooses the correspondi thesis method according to the different operating states of the motor.The first virtual vector VV1-12 is used in the transient adjustment with a large error, and the kind of virtual vector VV13-24 is used in the steady adjustment with a small error.T tribution of two kinds of virtual vectors in the α-β subspace is shown in Figure 4.

Master-Slave Virtual Vector Determination and Duty Cycle Allocation
The two types of virtual vectors of DTP-PMSMs are projected on the stator flux-linkage rotating coordinate system x ψ -y ψ , and the torque evaluation function λ T , flux evaluation function λ ψ , and back electromotive force evaluation function λ e are defined as follows [22]: (11) where V xψ and V yψ are the components of the virtual voltage vector in the stator fluxlinkage coordinate system x ψ -axis and y ψ -axis, respectively.The flux-linkage position angle θ s determines the magnitude of the torque and flux-linkage evaluation function, while the back electromotive force evaluation function is related to the electric angular velocity ω.
In order to enhance the torque control effect of DTP-PMSMs, the selection of voltage vectors is no longer limited to only 12 fixed directions.Depending on the sector in which the stator flux is located, the master virtual vector is chosen based on different evaluation functions for each voltage vector, considering its greater influence on torque.The slave virtual vector is selected based on its greater influence on flux-linkage.Both virtual vectors comply with the switching table requirements for torque and flux-linkage increases or decreases; however, their effects differ.Additionally, under different operating states, distinct types of virtual vectors should be selected.
For example, when the stator flux-linkage is in sector I, ∆T e and ∆ψ s are greater than zero.In the transient adjustment of the motor, the evaluation functions corresponding to all virtual vectors are calculated.The first kind of virtual vector VV 1-3 can increase the flux-linkage as well as the torque.The virtual vector VV 3 with the largest increase in torque is selected as the master virtual vector, and the virtual vector VV 1 with the largest increase in flux-linkage is selected as the slave virtual vector.Similarly, the second kind of virtual vector VV 15 is selected as the master virtual vector and vector VV 13 as the slave virtual vector during the steady state adjustment of the motor.According to the above theory, the master-slave virtual vectors satisfying the corresponding conditions can be selected in each sector.
The selected master-slave virtual vectors need to act together in a control period according to a certain duty cycle, and the torque and flux-linkage evaluation function can more accurately represent the degree of influence of the selected master-slave virtual vectors on the torque and flux-linkage of DTP-PMSM.If the duty cycles of the master-slave virtual vectors are d m and d s , respectively, the torque change equation in the entire control period is shown as follows: where T s is the control period; ∆T e is the torque error; L T is the torque coefficient; and ∆T em , ∆T es , and ∆T e0 are the master-slave virtual vector torque changes and the zero vector torque changes, respectively.Similarly, the flux-linkage equation can be expressed as: where ∆ψ s is the flux-linkage error; L ψ is the flux-linkage coefficient; and λ ψm and λ ψs are the flux evaluation functions of master-slave virtual vectors, respectively.The master-slave virtual vector duty cycles dm and ds can be calculated by Equations ( 12) and ( 13).When the stator flux-linkage is located in sector I, the adjustment range of the master-slave virtual vectors when they act together is shown in Figure 5.
World Electr.Veh.J. 2024, 15, x FOR PEER REVIEW 8 of 15 where Δψs is the flux-linkage error; Lψ is the flux-linkage coefficient; and λψm and λψs are the flux evaluation functions of master-slave virtual vectors, respectively.The masterslave virtual vector duty cycles dm and ds can be calculated by Equations ( 12) and ( 13).
When the stator flux-linkage is located in sector I, the adjustment range of the masterslave virtual vectors when they act together is shown in Figure 5. Due to the large fluctuation of the torque and flux-linkage of DTP-PMSMs, and the imprecision of the torque coefficient and flux-linkage coefficient, the calculated value of the duty cycle of the two virtual vectors may not be between 0 and 1, so it is necessary to reassign the duty cycle of the master-slave virtual vectors.According to the principle of master virtual vector priority, it can be divided into the following distribution methods: (1) When dm and ds are both less than zero, then the zero vector acts on the whole period; (2) When dm is greater than zero or ds is less than zero, then the main virtual vector acts on the whole period; (3) When the sum of dm and ds is less than 1, the master-slave virtual vectors and zero Due to the large fluctuation of the torque and flux-linkage of DTP-PMSMs, and the imprecision of the torque coefficient and flux-linkage coefficient, the calculated value of the duty cycle of the two virtual vectors may not be between 0 and 1, so it is necessary to the symmetric waveform via the equivalent substitution of the intermediate vector within the above three value ranges of d m .Similarly, the switching sequence of the master-slave virtual vector in synthesis mode 2 can also be modified symmetrically according to this law.The modified master-slave virtual vector switch sequence is symmetrical in the center, which is convenient for hardware implementation, and also ensures that the power device operates once in one period, reducing the switching loss.
vectors V04, V64, and V67.By analogy, the correction methods of switch sequences in different parity sectors can be obtained.The action time of the high level of each phase of the modified switching sequence does not change, and the modified synthetic vector does not change, which ensures that the effect of the master-slave virtual vector in the two subspaces remains unchanged.Only when the duty cycle of the master-slave virtual vector changes is the individual action vector replaced with the intermediate vector, and each parity sector can be modified to the symmetric waveform via the equivalent substitution of the intermediate vector within the above three value ranges of dm.Similarly, the switching sequence of the masterslave virtual vector in synthesis mode 2 can also be modified symmetrically according to this law.The modified master-slave virtual vector switch sequence is symmetrical in the center, which is convenient for hardware implementation, and also ensures that the power device operates once in one period, reducing the switching loss.
The overall block diagram of the master-slave virtual vector DTC of DTP-PMSMs based on duty cycle allocation is shown in Figure 7.

Experiment System
In order to verify the feasibility of the strategy proposed in this paper, a DTP-PMSM system experiment platform is built, as shown in Figure 8.The rated speed of the dual three-phase permanent magnet synchronous motor , which is manufactured by Hebei Electric Machinery Factory in China, used in the experiment is 1000 r/min, the rated torque is 10 Nm, the permanent magnet flux is 0.22 Wb, the pole-pairs number is 5, the d-axis

Experiment System
In order to verify the feasibility of the strategy proposed in this paper, a DTP-PMSM system experiment platform is built, as shown in Figure 8.The rated speed of the dual three-phase permanent magnet synchronous motor, which is manufactured by Hebei Electric Machinery Factory in China, used in the experiment is 1000 r/min, the rated torque is 10 Nm, the permanent magnet flux is 0.22 Wb, the pole-pairs number is 5, the d-axis inductance is 29 mH, and the q-axis inductance is 42 mH, respectively.The control algorithm is implemented with the TMS320F28377D DSP produced by TI, and Cyclone V FPGA produced by Intel.DSP is applied to execute the algorithm, and FPGA is applied to implement the high-precision analog-to-digital conversion (ADC) sampling, digital-toanalog conversion (DAC) conversion, and PWM pulse generation.The switching frequency of the IPM is 10 kHz.

Experiment System
In order to verify the feasibility of the strategy proposed in t system experiment platform is built, as shown in Figure 8.The three-phase permanent magnet synchronous motor , which is Electric Machinery Factory in China, used in the experiment is 100 is 10 Nm, the permanent magnet flux is 0.22 Wb, the pole-pairs inductance is 29 mH, and the q-axis inductance is 42 mH, respec rithm is implemented with the TMS320F28377D DSP produce FPGA produced by Intel.DSP is applied to execute the algorithm implement the high-precision analog-to-digital conversion (ADC alog conversion (DAC) conversion, and PWM pulse generation. of the IPM is 10 kHz.The experimental platform consists of a power loop in the s a control loop in the weak electricity part.After the experimenta and debugged, Code Composer Studio (ccs10.0) is used to buil program.The experiment simulates the variable load of the mot ductance of the resistance box.In the strong electricity part of th the DC bus voltage of the system is powered with the KEYSIGH the pulse transmission process of the system, the duty cycle sig main control panel to the CPLD panel.In the CPLD, the signal is The experimental platform consists of a power loop in the strong electricity part and a control loop in the weak electricity part.After the experimental platform is powered on and debugged, Code Composer Studio (ccs10.0) is used to build the DSP experimental program.The experiment simulates the variable load of the motor through the series inductance of the resistance box.In the strong electricity part of the experimental platform, the DC bus voltage of the system is powered with the KEYSIGHT DC power supply.In the pulse transmission process of the system, the duty cycle signal is first sent from the main control panel to the CPLD panel.In the CPLD, the signal is converted into an optical signal through the optocoupler module, and then the 12-channel PWM wave is sent to the IPM through the optical fiber.The reference voltage is synthesized by controlling the opening and closing of the IGBT switch tube in the IPM module.The control loop of the weak electricity part of the experimental platform is composed of the DSP chip and the FPGA chip.The DSP chip mainly performs the operation of the basic algorithm of the motor.The DSP converts the algorithm into a duty cycle and controls the FPGA chip to generate a pulse signal.The pulse signal is transmitted to the CPLD board to generate a 12-channel PWM wave and is converted into the optical signal to control the opening and closing of the IPM.

Analysis of Steady-State Experimental Results
To verify the steady-state performance of the proposed strategy, the motor runs at 300 r/min with a load of 4 Nm.After steady-state operation, the experimental waveforms of the traditional strategy, the virtual vector strategy, and the proposed strategy are shown in Figure 9.By referring to the literature on DTC strategy in recent years, most of the improved DTC strategies are mainly analyzed from the three perspectives of the harmonic content analysis of motor current, torque ripple, and flux-linkage ripple.The traditional DTC strategy does not consider the voltage vector synthesis of the harmonic subspace, but the By referring to the literature on DTC strategy in recent years, most of the improved DTC strategies are mainly analyzed from the three perspectives of the harmonic content analysis of motor current, torque ripple, and flux-linkage ripple.The traditional DTC strategy does not consider the voltage vector synthesis of the harmonic subspace, but the proposed control strategy can suppress the voltage synthesis of the harmonic subspace by redistributing the vector operation time.By analyzing the experimental results in Figure 9, under steady-state conditions, the phase current THD of the traditional DTC strategy is 58.8%, the torque ripple is 3.6 Nm, and the flux-linkage ripple is 0.13 Wb.However, the phase current THD of the proposed strategy is 18.0%, the torque ripple is 2.2 Nm, and the flux-linkage ripple is 0.05 Wb.Compared with the traditional DTC control strategy, the current THD of the proposed strategy is reduced by 69.4%, the torque ripple is reduced by 39%, and the flux-linkage ripple is reduced by 62%, which verifies the effectiveness of the proposed control strategy.Moreover, the experimental analysis results show that the proposed control strategy can effectively suppress the harmonic subspace current i x , i y .By analyzing the basic mathematical Equations ( 3)-( 6) of the motor, it can be concluded that the harmonic subspace current will affect the flux-linkage of the harmonic subspace, which will adversely affect the flux-linkage φ s of the motor.
In addition, by analyzing other works in the literature [23], in order to further analyze the improvement effect of the proposed control strategy on motor flux-linkage and torque, two new variables are introduced: the torque standard deviation σ Te and the flux-linkage standard deviation σ φs .By introducing these two new evaluation variables, the effect of the proposed control strategy is further verified.After the analysis, it can be obtained that σ Te is reduced by 43% and σ φs is reduced by 27% compared with the traditional control strategy under 300 r/min with a load of 4 Nm condition.
In order to further verify the steady-state performance of the proposed strategy under different conditions, the phase current THD, torque ripple values, and flux-linkage ripple values of the three control strategies under different load values are shown in Table 3 at the reference speed of 300 r/min.From Table 3, it is obvious that compared to the traditional DTC strategy and virtual vector strategy, the strategy proposed in this paper reduces flux-linkage fluctuation and torque fluctuation under various conditions.After using two sets of virtual vectors, the phase current waveform is greatly improved.

Analysis of Dynamic Experimental Results
As shown in Figures 10 and 11, so as to verify the dynamic tracking performance of the proposed strategy, an experimental analysis was carried out under three different working conditions: the reference speed in the first working condition jumped from 300 r/min to 500 r/min and the second condition is that the motor is suddenly loaded by 4 Nm during no-load time until stable operation.
From Figure 10, it is obvious that the two control strategies can quickly track the reference value of the speed after the motor is started, there is no large overshoot phenomenon, and the speed of the proposed control strategy reaches the given reference value after a short adjustment when the speed changes at different speeds.It can be seen that the proposed strategy has good speed dynamic control performance.
As can be seen from Figure 11, both control strategies quickly reach the same torque value after the abrupt addition of a 4 Nm load.Therefore, the proposed strategy retains the advantages of the traditional strategy of fast response, and the proposed strategy can effectively suppress the torque fluctuation and flux-linkage fluctuation of DTP-PMSMs.From Table 3, it is obvious that compared to the traditional DTC strategy and virtual vector strategy, the strategy proposed in this paper reduces flux-linkage fluctuation and torque fluctuation under various conditions.After using two sets of virtual vectors, the phase current waveform is greatly improved.

Analysis of Dynamic Experimental Results
As shown in Figure 10 and Figure 11, so as to verify the dynamic tracking performance of the proposed strategy, an experimental analysis was carried out under three different working conditions: the reference speed in the first working condition jumped from 300 r/min to 500 r/min and the second condition is that the motor is suddenly loaded by 4 Nm during no-load time until stable operation.From Figure 10, it is obvious that the two control strategies can quickly track the reference value of the speed after the motor is started, there is no large overshoot phenomenon, and the speed of the proposed control strategy reaches the given reference value after a short adjustment when the speed changes at different speeds.It can be seen that the proposed strategy has good speed dynamic control performance.
As can be seen from Figure 11, both control strategies quickly reach the same torque
traditional DTC block diagram of DTP-PMSMs is shown in Figure 2 [20,21].The electromagnetic torque Te, flux-linkage amplitude |ψs|, and flux-linkage angle θs are obtained through the torque and flux observer on the α-β subspace.The error between the observed value and the given value is transmitted to the hysteresis comparator to obtain the increase and decrease signal of torque flux-linkage.

Figure 2 .
Figure 2. Block diagram of direct torque control for dual three-phase PMSMs.A DTP-PMSM is driven by a six-phase inverter, where each switching state combination corresponds to a voltage vector in the α-β and x-y subspaces, with 2 6 = 64 different spatial voltage vectors, including 4 zero vectors and 60 effective vectors.The voltage vector distribution of the α-β and x-y subspaces is shown in Figure 3. Numbers of the basic voltage vectors in the figure are, respectively, converted to octal numbers according to the order of bridge arms of ABC and UVW.Effective vectors can be divided into four groups: large vectors, medium-large vectors, medium vectors, and small vectors according to their different amplitudes on the α-β subspace.The amplitudes of voltage vectors of each group are |VL| = 0.644Udc, |VML| = 0.471Udc, |VM| = 0.333Udc, and |VS| = 0.173Udc.

Figure 2 . 15 Figure 3 .
Figure 2. Block diagram of direct torque control for dual three-phase PMSMs.A DTP-PMSM is driven by a six-phase inverter, where each switching state combination corresponds to a voltage vector in the α-β and x-y subspaces, with 2 6 = 64 different spatial voltage vectors, including 4 zero vectors and 60 effective vectors.The voltage vector distribution of the α-β and x-y subspaces is shown in Figure 3. Numbers of the basic voltage vectors in the figure are, respectively, converted to octal numbers according to the order of bridge arms of ABC and UVW.Effective vectors can be divided into four groups: large vectors, medium-large vectors, medium vectors, and small vectors according to their different amplitudes on the α-β subspace.The amplitudes of voltage vectors of each group are |V L | = 0.644U dc , |V ML | = 0.471U dc , |V M | = 0.333U dc , and |V S | = 0.173U dc .World Electr.Veh.J. 2024, 15, x FOR PEER REVIEW 5 of 15

Figure 5 .
Figure 5. Adjustment range of master-slave virtual vectors.

Figure 5 .
Figure 5. Adjustment range of master-slave virtual vectors.

Figure 6 .
Figure 6.Schematic diagram of switch sequence before and after correction.

Figure 6 .
Figure 6.Schematic diagram of switch sequence before and after correction.The overall block diagram of the master-slave virtual vector DTC of DTP-PMSMs based on duty cycle allocation is shown in Figure7.

Figure 7 .
Figure 7. Direct torque control block diagram based on master-slave vector duty cycle allocation.

Figure 7 .
Figure 7. Direct torque control block diagram based on master-slave vector duty cycle allocation.

Figure 8 .
Figure 8. Experimental platform for the DTP-PMSM control system.

Figure 8 .
Figure 8. Experimental platform for the DTP-PMSM control system.

Figure 10 .
Figure 10.Experimental waveform of step speed response: (a) virtual vector strategy; (b) proposed strategy.

Table 1 .
Sector I direct torque control switch table.

Slave Virtual Vector Direct Torque Control Strategy Based on Duty Cycle Allocation
4.1.Virtual Vector Synthesis Method

Table 2 .
Duty cycle of each phase for sectors I and II.

Table 3 .
Steady-state performance comparison under different load conditions.