Multi-Objective Optimization Design and Multi-Physics Analysis a Double-Stator Permanent-Magnet Doubly Salient Machine

The double-stator permanent-magnet doubly salient (DS-PMDS) machine is an interesting candidate motor for electric vehicle (EV) applications because of its high torque output and flexible working modes. Due to the complexity of the motor structure, optimization of the DS-PMDS for EVs requires more research efforts to meet multiple specifications. Effective multi-objective optimization to increase torque output, reduce torque ripple, and improve PM material utilization and motor efficiency is implemented in this paper. In the design process, a multi-objective comprehensive function is established. By using parametric sensitivity analysis (PSA) and the sequential quadratic programming (NLPQL) method, the influence extent of each size parameter for different performance is effectively evaluated and optimal results are determined. By adopting the finite element method (FEM), the electromagnetic performances of the optimal DS-PMDS motor is investigated. Moreover, a multi-physical field analysis is included to describe stress, deformation of the rotor, and temperature distribution of the proposed motor. The theoretical analysis verified the rationality of the motor investigated and the effectiveness of the proposed optimization method.


Introduction
Permanent magnet (PM) motors are gaining popularity in electric vehicle (EV) propulsion applications due to features such as high efficiency and high power density [1]. Of the various topologies of PM motors, stator-PM motors (and their types) have attracted a lot of attention [2,3]. The stator-PM doubly salient (SPM-DS) motor, which is a type of stator-PM motor, features special topology with the permanent magnet located in the yoke of the stator. This differs from the PMs that are sandwiched in the stator teeth in stator-PM flux switching (SPM-FS) motors. This feature offers the SPM-DS motor the merit of a simple and robust salient rotor, in addition to effectively realizing heat dissipation and resisting the irreversible risk of demagnetization [4]. And yet, due to the limited space in the yoke, a relatively small number of permanent magnets are used; torque density is lower than that of the SPM-FS motor, where a large number of permanent magnets are embedded in the stator teeth [5].
Several hybrid excitation SPM-DS motors have been proposed to improve torque density. By using the additional dc field excitation windings, the enhanced torque can be successfully obtained in a flux-strengthening mode [6]. However, due to the existence of additional dc excitation winding in Figure 1 depicts the topology of the DS-PMDS motor, where three parts of the outer stator, middle rotor and inner stator are involved. First, as shown in the figure, the two stators and the middle rotor make up the salient pole structure. The middle rotor is sandwiched by the outer and inner stator. The winding housed in the outer stator couples with the middle rotor, comprising a 24-slot/16-pole outer motor. Meanwhile, a 12-slot/8-pole inner motor is formed through the middle rotor and the inner stator. Since there are neither permanent magnets nor windings in the middle rotor, it results in a simple and reliable rotor structure, which is similar to that of the stator permanent magnet motor. Furthermore, the yoke of both stators contains tangential magnetized permanent magnets. The armature windings in both stators are non-overlapping concentrated windings, which can lead to the reduction of copper loss and a relatively higher efficiency. Finally, it is evident that the internal space of the machine is efficiently used by adding the inner stator, which offers the possibility to improve power and torque levels.

Motor Structure and Operating Principle
Energies 2018, 11, x FOR PEER REVIEW 3 of 16 is evident that the internal space of the machine is efficiently used by adding the inner stator, which offers the possibility to improve power and torque levels.  Figure 2 shows the operating principle of the DS-PMDS motor, in which the magnetic field distribution follows the principle of minimum magnetic resistance [12]. To reduce the electromagnetic coupling degree of the inner and outer motors, a non-magnetic ring is added to the middle rotor. Consequently, the magnetic circuit of the inner motor and the outer motor is parallel with low magnetic coupling degree, which makes control of the inner and outer motors more flexible and offers flexible switching between multiple driving modes of the DS-PMDS motor. Generally, the driving cycles of vehicles are complex and changeable in actual road conditions. As shown in Figure 3, the new European driving cycle (NEDC) contains several typical driving cycle units [13]. These driving conditions includes frequent start and stop, normal and high-speed cruise, acceleration and deceleration, and climbing with heavy load. Hence, multi-operating modes are required for the EV traction motor to meet the various requirements of driving cycle conditions. Owing to the two sets of armature windings, the proposed DS-PMDS motor has a variety of operating modes and can be flexibly switched to suit different working conditions. Figure 4 illustrates the corresponding powertrain, based on the DS-PMDS motor in EVs [14].  Figure 2 shows the operating principle of the DS-PMDS motor, in which the magnetic field distribution follows the principle of minimum magnetic resistance [12]. To reduce the electromagnetic coupling degree of the inner and outer motors, a non-magnetic ring is added to the middle rotor. Consequently, the magnetic circuit of the inner motor and the outer motor is parallel with low magnetic coupling degree, which makes control of the inner and outer motors more flexible and offers flexible switching between multiple driving modes of the DS-PMDS motor. is evident that the internal space of the machine is efficiently used by adding the inner stator, which offers the possibility to improve power and torque levels.  Figure 2 shows the operating principle of the DS-PMDS motor, in which the magnetic field distribution follows the principle of minimum magnetic resistance [12]. To reduce the electromagnetic coupling degree of the inner and outer motors, a non-magnetic ring is added to the middle rotor. Consequently, the magnetic circuit of the inner motor and the outer motor is parallel with low magnetic coupling degree, which makes control of the inner and outer motors more flexible and offers flexible switching between multiple driving modes of the DS-PMDS motor. Generally, the driving cycles of vehicles are complex and changeable in actual road conditions. As shown in Figure 3, the new European driving cycle (NEDC) contains several typical driving cycle units [13]. These driving conditions includes frequent start and stop, normal and high-speed cruise, acceleration and deceleration, and climbing with heavy load. Hence, multi-operating modes are required for the EV traction motor to meet the various requirements of driving cycle conditions. Owing to the two sets of armature windings, the proposed DS-PMDS motor has a variety of operating modes and can be flexibly switched to suit different working conditions. Figure 4 illustrates the corresponding powertrain, based on the DS-PMDS motor in EVs [14]. Generally, the driving cycles of vehicles are complex and changeable in actual road conditions. As shown in Figure 3, the new European driving cycle (NEDC) contains several typical driving cycle units [13]. These driving conditions includes frequent start and stop, normal and high-speed cruise, acceleration and deceleration, and climbing with heavy load. Hence, multi-operating modes are required for the EV traction motor to meet the various requirements of driving cycle conditions. Owing to the two sets of armature windings, the proposed DS-PMDS motor has a variety of operating modes and can be flexibly switched to suit different working conditions. Figure 4 illustrates the corresponding powertrain, based on the DS-PMDS motor in EVs [14].  Using two sets of windings and converters, power transmission is realized from the battery to the inner and outer motors. Both motors then drive the middle rotor, which is connected to the final driveline. Based on the law of electromechanical energy conversion, the output torque of the DS-PMDS motor can be further expressed as: where Toutput is the total output torque of the DS-PMDS motor. Tload is the vehicle load torque, which varies with driving conditions. J is the total moment of inertia of the DS-PMDS motor. ω (rad/s) and n (rpm) are rotation angular velocity and speed of the motor, respectively. G is the weight of the middle rotor. D is the outer diameter of the intermediate rotor.
Tinner and Touter are torque of inner motor and outer motor. iinner and iouter are phase current of inner and outer motor. The coefficients k1 and k2 are functions of the phase current of the inner and outer motors, respectively. Therefore,  Using two sets of windings and converters, power transmission is realized from the battery to the inner and outer motors. Both motors then drive the middle rotor, which is connected to the final driveline. Based on the law of electromechanical energy conversion, the output torque of the DS-PMDS motor can be further expressed as: where Toutput is the total output torque of the DS-PMDS motor. Tload is the vehicle load torque, which varies with driving conditions. J is the total moment of inertia of the DS-PMDS motor. ω (rad/s) and n (rpm) are rotation angular velocity and speed of the motor, respectively. G is the weight of the middle rotor. D is the outer diameter of the intermediate rotor.
Tinner and Touter are torque of inner motor and outer motor. iinner and iouter are phase current of inner and outer motor. The coefficients k1 and k2 are functions of the phase current of the inner and outer motors, respectively. Therefore, Using two sets of windings and converters, power transmission is realized from the battery to the inner and outer motors. Both motors then drive the middle rotor, which is connected to the final driveline. Based on the law of electromechanical energy conversion, the output torque of the DS-PMDS motor can be further expressed as: where T output is the total output torque of the DS-PMDS motor. T load is the vehicle load torque, which varies with driving conditions. J is the total moment of inertia of the DS-PMDS motor. ω (rad/s) and n (rpm) are rotation angular velocity and speed of the motor, respectively. G is the weight of the middle rotor. D is the outer diameter of the intermediate rotor. T inner and T outer are torque of inner motor and outer motor. i inner and i outer are phase current of inner and outer motor. The coefficients k 1 and k 2 are functions of the phase current of the inner and outer motors, respectively. Therefore, according to different requirements of load torque, the drive mode of the DS-PMDS motor can be flexibly switched by controlling the current of the two sets of windings. For the DS-PMDS motor to satisfy different driving cycle conditions, the control coefficients k 1 and k 2 are adjusted accordingly to switch the operating mode of the motor; this is listed in Table 1 in detail. When the vehicle is in normal cruise or deceleration driving cycle, the power required is relatively low. Consequently, the independent drive mode of the inner motor with 0 ≤ k 1 ≤ 1, k 2 = 0 can be adopted to meet the required driving power. When the EV is in high-speed cruise, the low torque output of the inner motor cannot meet driving requirements. Thus, to obtain the desired speed and torque output, the outer motor drive mode is adopted with 0 ≤ k 2 ≤ 1, k 1 = 0. For climbing, starting and acceleration, the demand for power and torque is further enhanced. At this point, the dual drive mode is required to obtain higher output power and torque.
High torque High PM material utilization

Multi-Objective Design Optimization
For the proposed DS-PMDS motor, due to its complex configuration with a large number of design variables, the conventional optimal design method that often uses discrete parameter scanning for single objective cannot be effectively applied. Besides, based on the above analysis, the proposed operating modes make the multi-objective optimization design of the motor more complicated. To achieve higher PM material utilization ratio and motor efficiency, higher torque output with lower torque ripple, an effective multi-objective optimization strategy is implemented and introduced in detail in this paper [15,16]. A multi-objective comprehensive function in design strategy is established. Then, by adopting parametric sensitivity analysis (PSA) and the sequential quadratic programming (NLPQL) method, the influence extent of each design parameter for different design objectives is effectively evaluated and the optimal results are determined. The flow diagram of the proposed optimal design is shown in Figure 5.

Optimization Model
As the DS-PMDS motor has a double-salient structure, low cogging torque needs to be first considered. In addition, as a traction motor, high torque output is the preferable design requirement to meet the vehicle's multiple operation conditions of acceleration, deceleration and overload climbing conditions. Moreover, the performances of high PM material utilization ratio and motor efficiency are also required to be satisfied. Consequently, the optimization model of DS-PMDS motor can be presented as: According to the potential application area in EVs, the added boundary constraints are given as design examples: where, T m is average output torque, T ri is torque ripple, δ PM is PM material utilization ratio (which is defined as the ratio of output torque to PM volume), η is motor efficiency, x i is the vector of the optimization parameters, which can be written as: Figure 6 shows the selected nine design variables, which are several in number owing to the structure of a double-stator. The main dimensions are: magnetic thickness of the PM in outer stator h oPM and magnetic thickness of PM in inner stator h iPM . δ o and δ i are outer and inner air gaps, respectively. β is , β os , β ro and β ri is the tooth width of the inner stator, outer stator and middle rotor. h ry is the middle rotor yoke height. In order to simplify the multi-objective optimization calculation and consider feasibility of manufacturing, the constraint ranges of these parameters are listed in Table 2.

Optimization Model
As the DS-PMDS motor has a double-salient structure, low cogging torque needs to be first considered. In addition, as a traction motor, high torque output is the preferable design requirement to meet the vehicle's multiple operation conditions of acceleration, deceleration and overload climbing conditions. Moreover, the performances of high PM material utilization ratio and motor efficiency are also required to be satisfied. Consequently, the optimization model of DS-PMDS motor can be presented as: According to the potential application area in EVs, the added boundary constraints are given as design examples: where, Tm is average output torque, Tri is torque ripple, δPM is PM material utilization ratio (which is defined as the ratio of output torque to PM volume), η is motor efficiency, xi is the vector of the optimization parameters, which can be written as: hry is the middle rotor yoke height. In order to simplify the multi-objective optimization calculation and consider feasibility of manufacturing, the constraint ranges of these parameters are listed in Table 2.

Multi-Objective Comprehensive Function
From the above, there are several goals in the optimization model of the DS-PMDS motor. To reduce the conflict of multiple targets, we simplify the complication of trade-off analysis and improve the multi-objective optimization efficiency; the comprehensive objective function of the DS-PMDS motor is built as follows: where, T ri , T m , η and δ PM are the initial values of the torque ripple, output torque, efficiency and PM material utilization ratio, respectively; T ri (x i ), T m (x i ), η(x i ) and δ PM (x i ) are the functions of the design variables of x i ; λ 1 , λ 2 , λ 3 and λ 4 are the four weight coefficients that need to meet the relationship of

Parameters Sensitivity Analysis and Multi-Objective Optimization
After the comprehensive objective function is proposed in Equation (6), the multi-objective optimization with tradeoff analysis is implemented, where high-dimension calculation and a time-consuming optimization process is involved. To simplify the multi-objective optimization of the proposed complex motor, the PSA approach is used, integrating multiple optimization targets to obtain a global optimal solution, intuitively and efficiently. Besides, in this way, the sensitivity of each parameter to various optimal goals can also be identified and the key size parameters can be easily selected to improve the efficiency of further optimization and adjustment. Figure 7 shows the sensitivities of four optimal objects to the nine design variables, taking into consideration the interaction among the different design parameters. Several conflicts exist among the four design objectives. The most critical design parameters affecting T ri , T m , η and δ PM are the outer stator tooth width β os , the outer air gap δ o , the outer tooth width of middle rotor β ro , and the magnetized thickness of PM in outer stator h oPM . The three-dimensional response surface between four targets and nine chosen design parameters can be also obtained; the four typical ones are shown in Figure 8. The above two figures show that the high sensitivity parameters are different for various application requirements and optimization targets, and different key parameters can be conveniently chosen to further optimization and adjustment.

Multi-Objective Comprehensive Function
From the above, there are several goals in the optimization model of the DS-PMDS motor. To reduce the conflict of multiple targets, we simplify the complication of trade-off analysis and improve the multi-objective optimization efficiency; the comprehensive objective function of the DS-PMDS motor is built as follows: where, T'ri, T'm, η' and δPM' are the initial values of the torque ripple, output torque, efficiency and PM material utilization ratio, respectively; Tri (xi), Tm(xi), η(xi) and δPM (xi) are the functions of the design variables of xi; λ1, λ2, λ3 and λ4 are the four weight coefficients that need to meet the relationship of λ1 + λ2 + λ3 + λ4 = 1 and here the values are 0.3, 0.3, 0.2 and 0.2 separately.

Parameters Sensitivity Analysis and Multi-Objective Optimization
After the comprehensive objective function is proposed in Equation (6), the multi-objective optimization with tradeoff analysis is implemented, where high-dimension calculation and a time-consuming optimization process is involved. To simplify the multi-objective optimization of the proposed complex motor, the PSA approach is used, integrating multiple optimization targets to obtain a global optimal solution, intuitively and efficiently. Besides, in this way, the sensitivity of each parameter to various optimal goals can also be identified and the key size parameters can be easily selected to improve the efficiency of further optimization and adjustment.  Figure 7 shows the sensitivities of four optimal objects to the nine design variables, taking into consideration the interaction among the different design parameters. Several conflicts exist among the four design objectives. The most critical design parameters affecting Tri, Tm, η and δPM are the outer stator tooth width βos, the outer air gap δo, the outer tooth width of middle rotor βro, and the magnetized thickness of PM in outer stator hoPM. The three-dimensional response surface between four targets and nine chosen design parameters can be also obtained; the four typical ones are shown in Figure 8. The above two figures show that the high sensitivity parameters are different for various application requirements and optimization targets, and different key parameters can be conveniently chosen to further optimization and adjustment. Based on the defined multi-objective comprehensive function F GOAL in Equation (6), Figure 9 shows the effects of nine design parameters on comprehensive objective function F GOAL . In addition, as shown in Figure 10, based on the NLPQL approach, after 17 generations of iterative optimization, the optimal solution for the comprehensive objective function F GOAL can be efficiently achieved. An optimal tradeoff can be obtained for engineering practice, based on comprehensive objective function and special boundary conditions. The corresponding optimal results of F GOAL , the four optimization targets, and nine design parameters are listed in Table 3. Based on the defined multi-objective comprehensive function FGOAL in Equation (6), Figure 9 shows the effects of nine design parameters on comprehensive objective function FGOAL. In addition, as shown in Figure 10, based on the NLPQL approach, after 17 generations of iterative optimization, the optimal solution for the comprehensive objective function FGOAL can be efficiently achieved. An optimal tradeoff can be obtained for engineering practice, based on comprehensive objective function and special boundary conditions. The corresponding optimal results of FGOAL, the four optimization targets, and nine design parameters are listed in Table 3.   Based on the defined multi-objective comprehensive function FGOAL in Equation (6), Figure 9 shows the effects of nine design parameters on comprehensive objective function FGOAL. In addition, as shown in Figure 10, based on the NLPQL approach, after 17 generations of iterative optimization, the optimal solution for the comprehensive objective function FGOAL can be efficiently achieved. An optimal tradeoff can be obtained for engineering practice, based on comprehensive objective function and special boundary conditions. The corresponding optimal results of FGOAL, the four optimization targets, and nine design parameters are listed in Table 3.      Figure 11 shows the flux distribution of the proposed DS-PMDS motor, with and without a non-magnetic ring. It can be observed from Figure 11a that when all the PMs in the two stators work together, some flux lines are always directly closed, without passing through the middle rotor. That is, in addition to the parallel main magnetic circuit, there is also a series magnetic circuit, and the electromagnetic coupling degree of the inner and outer motors is relatively high. When the non-magnetic ring is added, it can be seen from Figure 11b that the main magnetic circuit of the inner motor and the outer motor is parallel. The electromagnetic coupling degree of the inner motor and the outer motor is then obviously decreased, which is consistent with the previous theoretical analysis.

No-Load Back EMF
The no-load back EMF of the DS-PMDS motor is studied in this paper, considering that the no-load back EMF directly affects the output torque performance of the motor. For two sets of armature windings, at the rated speed of 750 r/min, the no-load phase EMF waveforms and their harmonic spectrum are compared, before and after optimization, as shown in Figures 12 and 13. It was found that the performance of the no-load back EMF was improved. Figure 12a and Figure 13a show that, compared to the back EMF of the inner winding, the amplitude of the back EMF of the outer winding increases more obviously from 142.3 V to 155.8 V, indicating the enhancement of output torque and proving the validity of the motor optimization method. In addition, as shown in Figure 12b and Figure 13b, after optimization, both inner and outer windings have more sinusoidal

No-Load Back EMF
The no-load back EMF of the DS-PMDS motor is studied in this paper, considering that the no-load back EMF directly affects the output torque performance of the motor. For two sets of armature windings, at the rated speed of 750 r/min, the no-load phase EMF waveforms and their harmonic spectrum are compared, before and after optimization, as shown in Figures 12 and 13. It was found that the performance of the no-load back EMF was improved. Figures 12a and 13a show that, compared to the back EMF of the inner winding, the amplitude of the back EMF of the outer winding increases more obviously from 142.3 V to 155.8 V, indicating the enhancement of output torque and proving the validity of the motor optimization method. In addition, as shown in Figures 12b and 13b, after optimization, both inner and outer windings have more sinusoidal back EMF waveforms with decreased low harmonic content. The high sinusoidal back EMF does not only lay a good foundation for the subsequent reduction of torque ripple, but also indirectly verifies the effectiveness of torque ripple optimization of the motor. Figure 11. Flux distributions of the proposed machine (a) without non-magnetic ring, (b) with non-magnetic ring.

No-Load Back EMF
The no-load back EMF of the DS-PMDS motor is studied in this paper, considering that the no-load back EMF directly affects the output torque performance of the motor. For two sets of armature windings, at the rated speed of 750 r/min, the no-load phase EMF waveforms and their harmonic spectrum are compared, before and after optimization, as shown in Figures 12 and 13. It was found that the performance of the no-load back EMF was improved. Figure 12a and Figure 13a show that, compared to the back EMF of the inner winding, the amplitude of the back EMF of the outer winding increases more obviously from 142.3 V to 155.8 V, indicating the enhancement of output torque and proving the validity of the motor optimization method. In addition, as shown in Figure 12b and Figure 13b, after optimization, both inner and outer windings have more sinusoidal back EMF waveforms with decreased low harmonic content. The high sinusoidal back EMF does not only lay a good foundation for the subsequent reduction of torque ripple, but also indirectly verifies the effectiveness of torque ripple optimization of the motor.

Core Loss and Efficiency
The loss of motor directly affects efficiency; this is analyzed in this section. Core loss is the main component of no-load loss of the DS-PMDS motor. It can be estimated by the following model [17]: where Ps is the core less of the motor, ke and kh, respectively, represent the eddy current and hysteresis loss coefficients. f is the frequency of sinusoidal alternating flux density. Bm is the amplitude of flux density. Figure 14 shows the no-load core loss distribution of the proposed machine. It can be seen that the core loss distribution is not uniform in the outer stator, middle rotor and inner stator. In areas with high magnetic flux density, such as the tip where the stator and rotor overlap and the yoke of the middle rotor, core loss is relatively high. As shown in Figure 15, after the optimization and adjustment of some key size parameters, core loss decreased under different rotating speed conditions. When the rated speed was 750 rpm, the average of core loss decreased from 59.94 W to 34.42 W, which is conducive to improvement of motor efficiency.

Core Loss and Efficiency
The loss of motor directly affects efficiency; this is analyzed in this section. Core loss is the main component of no-load loss of the DS-PMDS motor. It can be estimated by the following model [17]: where P s is the core less of the motor, k e and k h , respectively, represent the eddy current and hysteresis loss coefficients. f is the frequency of sinusoidal alternating flux density. B m is the amplitude of flux density. Figure 14 shows the no-load core loss distribution of the proposed machine. It can be seen that the core loss distribution is not uniform in the outer stator, middle rotor and inner stator. In areas with high magnetic flux density, such as the tip where the stator and rotor overlap and the yoke of the middle rotor, core loss is relatively high. As shown in Figure 15, after the optimization and adjustment of some key size parameters, core loss decreased under different rotating speed conditions. When the rated speed was 750 rpm, the average of core loss decreased from 59.94 W to 34.42 W, which is conducive to improvement of motor efficiency.
where Ps is the core less of the motor, ke and kh, respectively, represent the eddy current and hysteresis loss coefficients. f is the frequency of sinusoidal alternating flux density. Bm is the amplitude of flux density. Figure 14 shows the no-load core loss distribution of the proposed machine. It can be seen that the core loss distribution is not uniform in the outer stator, middle rotor and inner stator. In areas with high magnetic flux density, such as the tip where the stator and rotor overlap and the yoke of the middle rotor, core loss is relatively high. As shown in Figure 15, after the optimization and adjustment of some key size parameters, core loss decreased under different rotating speed conditions. When the rated speed was 750 rpm, the average of core loss decreased from 59.94 W to 34.42 W, which is conducive to improvement of motor efficiency.

Torque Capability
Figures 16-18 shows comparisons of torque capability. As illustrated in Figure 16, the cogging torque peak values of the optimized machine are respectively smaller than those of the initial design. Figure 17 shows the output torque waveform at the rated speed when the amplitude of phase current is 8 A. After optimization and adjustment, the average output torque increased by 15%, while torque ripple reduced by 42%. From Figure 18, it is evident that the optimized motor has a higher torque output capability. This study found that the proposed optimization method can significantly improve torque performance.  Figures 16-18 shows comparisons of torque capability. As illustrated in Figure 16, the cogging torque peak values of the optimized machine are respectively smaller than those of the initial design. Figure 17 shows the output torque waveform at the rated speed when the amplitude of phase current is 8 A. After optimization and adjustment, the average output torque increased by 15%, while torque ripple reduced by 42%. From Figure 18, it is evident that the optimized motor has a higher torque output capability. This study found that the proposed optimization method can significantly improve torque performance.

Torque Capability
torque peak values of the optimized machine are respectively smaller than those of the initial design. Figure 17 shows the output torque waveform at the rated speed when the amplitude of phase current is 8 A. After optimization and adjustment, the average output torque increased by 15%, while torque ripple reduced by 42%. From Figure 18, it is evident that the optimized motor has a higher torque output capability. This study found that the proposed optimization method can significantly improve torque performance.     Figure 16, the cogging torque peak values of the optimized machine are respectively smaller than those of the initial design. Figure 17 shows the output torque waveform at the rated speed when the amplitude of phase current is 8 A. After optimization and adjustment, the average output torque increased by 15%, while torque ripple reduced by 42%. From Figure 18, it is evident that the optimized motor has a higher torque output capability. This study found that the proposed optimization method can significantly improve torque performance.

Structural Analysis
As the proposed DS-PMDS motor has a special structure with double air gap and cupped rotor, a structural simulation is implemented to identify the stress caused by centrifugal forces and verify the structural robustness of the designed rotor [18,19]. Figure 19 shows the 2D stress distribution of the middle rotor when the motor speed reaches 10,000 r/min. As can be seen from the figure, the stress distribution is not uniform. Since the mass distribution of the rotor is not uniform and the yoke of the middle rotor is relatively narrow, the maximum stress appears at the yoke part of the middle rotor, which is 126.5 MPa and less than the stress limit of the material. In addition, simulation results for deformation of the DS-PMDS motor is shown in Figure 20. It was observed that under a maximum speed of 10,000 rpm, the maximum deformation of the rotor is no more than 0.05 mm, which is also within safe operating limits.

Structural Analysis
As the proposed DS-PMDS motor has a special structure with double air gap and cupped rotor, a structural simulation is implemented to identify the stress caused by centrifugal forces and verify the structural robustness of the designed rotor [18,19]. Figure 19 shows the 2D stress distribution of the middle rotor when the motor speed reaches 10,000 r/min. As can be seen from the figure, the stress distribution is not uniform. Since the mass distribution of the rotor is not uniform and the yoke of the middle rotor is relatively narrow, the maximum stress appears at the yoke part of the middle rotor, which is 126.5 MPa and less than the stress limit of the material. In addition, simulation results for deformation of the DS-PMDS motor is shown in Figure 20. It was observed that under a maximum speed of 10,000 rpm, the maximum deformation of the rotor is no more than 0.05 mm, which is also within safe operating limits.
As the proposed DS-PMDS motor has a special structure with double air gap and cupped rotor, a structural simulation is implemented to identify the stress caused by centrifugal forces and verify the structural robustness of the designed rotor [18,19]. Figure 19 shows the 2D stress distribution of the middle rotor when the motor speed reaches 10,000 r/min. As can be seen from the figure, the stress distribution is not uniform. Since the mass distribution of the rotor is not uniform and the yoke of the middle rotor is relatively narrow, the maximum stress appears at the yoke part of the middle rotor, which is 126.5 MPa and less than the stress limit of the material. In addition, simulation results for deformation of the DS-PMDS motor is shown in Figure 20. It was observed that under a maximum speed of 10,000 rpm, the maximum deformation of the rotor is no more than 0.05 mm, which is also within safe operating limits.

Thermal Analysis
The heat flow in the optimal DSPM is analyzed based on the electromagnetic-thermal coupling method [20,21]. Initially, internal heat generation in the motor is obtained by the calculation of total losses, which is then imported to steady-state thermal analysis. During this electromagnetic thermal coupling simulation, key parameters such as heat source, residual magnetic flux density and intrinsic coercive force are updated in real time. In the process of the analysis, the boundary condition of stator housing is set to a temperature of 40 °C with water cooling. The general thermal field distribution and the temperature of every part of the motor is calculated successfully and shown in Figure 21. The outer armature winding reaches the maximum temperature of 107.44 °C, which is still within the acceptable range due to proper water cooling measures. The PMs in the inner and outer stators are at about 80 °C and 65 °C, respectively. They are both lower than the maximum allowable working temperature, avoiding demagnetization of the PMs caused by high temperature.

Thermal Analysis
The heat flow in the optimal DSPM is analyzed based on the electromagnetic-thermal coupling method [20,21]. Initially, internal heat generation in the motor is obtained by the calculation of total losses, which is then imported to steady-state thermal analysis. During this electromagnetic thermal coupling simulation, key parameters such as heat source, residual magnetic flux density and intrinsic coercive force are updated in real time. In the process of the analysis, the boundary condition of stator housing is set to a temperature of 40 • C with water cooling. The general thermal field distribution and the temperature of every part of the motor is calculated successfully and shown in Figure 21. The outer armature winding reaches the maximum temperature of 107.44 • C, which is still within the acceptable range due to proper water cooling measures. The PMs in the inner and outer stators are at about 80 • C and 65 • C, respectively. They are both lower than the maximum allowable working temperature, avoiding demagnetization of the PMs caused by high temperature. field distribution and the temperature of every part of the motor is calculated successfully and shown in Figure 21. The outer armature winding reaches the maximum temperature of 107.44 °C, which is still within the acceptable range due to proper water cooling measures. The PMs in the inner and outer stators are at about 80 °C and 65 °C, respectively. They are both lower than the maximum allowable working temperature, avoiding demagnetization of the PMs caused by high temperature. Figure 21. 3D simulation results of the general thermal field distribution of the motor.

Conclusions
In this paper, a multi-objective optimization design procedure with multi-physics field analysis is presented to provide comprehensive optimization of a new type of DS-PMDS motor.

Conclusions
In this paper, a multi-objective optimization design procedure with multi-physics field analysis is presented to provide comprehensive optimization of a new type of DS-PMDS motor. Firstly, a multi-objective function is established. Secondly, a design optimization method using parametric sensitivity analysis (PSA) and sequential quadratic programming (NLPQL) is discussed in detail. The significance of the parameters is also effectively evaluated and the optimal structure size parameters are determined. The performance of the proposed DS-PMDS motor, including electromagnetism, structure and heat, is then calculated by multi-physics field analysis. Finally, the proposed machine is shown to offer good electromagnetic performance characteristics of high output torque, low torque ripple and high efficiency. The simulation results of stress and deformation verify the robust rotor structure. The thermal analysis also shows that the proposed DS-PMDS motor can operate at a reasonable temperature. Both the theoretical analysis and multi-physics field simulation verify the validity of the motor design and the effectiveness of the proposed optimization method.