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Abstract

Surface permanent magnet synchronous motor (SPMSM) is widely used in low-speed and high torque permanent magnet synchronous motor. A multi-objective optimization method of hollow shaft direct drive motor (HSDDM) for double direct drive feed system (DDFS) is proposed in this paper. Several dimensions of HSDDM are considered in the proposed design procedure including pole embrace, thickness of magnet, eccentric distance, and air gap length. A multi-objective optimization model is established to minimize cogging torque, maximize efficiency, and minimize the magnet weight. The Optimized Latin hypercube sampling (OLHS) is used to design the experimental samples, and the sampling results are calculated by the finite element method (FEM). The Kring approximate model is established according to the sampling results. Based on the established approximate model, NSGA II multi-objective genetic algorithm is used to optimize the HSDDM. Compared with that before optimization, the efficiency after optimization is increased by 4.08%, the cogging torque is reduced by 86.78%, and the magnet weight is reduced by 20.24%. Eventually, the integration of NSGA-II and FEM provides an effective approach to obtain the optimal design of HSDDM.

Keywords

Hollow shaft direct drive motor multi-objective optimization cogging torque approximate model genetic algorithm

Introduction

Permanent magnet synchronous motor (PMSM) has the advantages of simple structure, high output torque, high power density, and high efficiency.^1–3 With the development of PMSM in the direction of increasing torque and reducing volume, low speed high torque permanent magnet motor is widely used in direct drive in industry.⁴ The installation methods of permanent magnet on the rotor of PMSM mainly include surface permanent magnet synchronous motor (SPMSM), embedded permanent magnet synchronous motor (EPMSM), and interior permanent magnet synchronous motor (IPMSM). SPMSM is easy to design, install, and produce, with low cost. It can realize the rich combination of slot and magnetic pole, stator slot wedge and magnet shape, and has low cogging torque. Therefore, SPMSM is widely used in low-speed and high torque permanent magnet synchronous motor.^5,6

Many researchers have done a lot of research on the optimal design of PMSM. Ahn et al.⁷ used particle swarm optimization algorithm to design and optimize the SPMSM of unmanned aerial vehicle (UAV), but the particle swarm optimization algorithm needs to define the initial area, which is difficult and time-consuming. Yermaz et al.⁸ used single purpose artificial bee colony algorithm and symbiotic biological search to optimize the design of SPMSM. Edhah et al.⁹ designed fractional slot concentrated winding of a PMSM based on experience and analysis method. It can be seen that these researches are usually far from the robustness required by multi-objective algorithms. The design of a PMSM often depends on multi-objective, and the purposes often conflict with each other. Therefore, multi-objective evolutionary algorithm is used in the optimal design of motor, such as genetic algorithm, fuzzy method, or Taguchi method.^10–12 Multi objective genetic algorithm provides better convergence and solution than other multi-objective evolutionary algorithms. Therefore, multi-objective genetic algorithm has become one of the highly preferred algorithms in motor design optimization.^13,14 Mutluer¹⁵ used multi-objective genetic algorithm to optimize the design of direct drive external rotor permanent magnet synchronous motor. Chowdhury et al.¹⁶ uses GA and Taguchi’s robust design techniques to minimize torque fluctuations while maximizing the average torque of SPMSM at a single operating point. Islam et al.¹⁷ uses the global response surface method to optimize the multi-objective and multi-load point design of IPMSM to achieve low torque ripple and high average torque in the whole speed range. Baek¹⁸ proposed a shape optimization design method of Brushless DC (BLDC) motor for electric continuous variable valve timing (E-CVVT) system of internal combustion engine and hybrid electric vehicle. The proposed design aims to maximize the maximum torque and rated efficiency of BLDC motor and minimize cogging torque.¹⁸ Wang et al.¹⁹ proposed a multi-objective optimization algorithm to determine the rotor design parameters which maximize the output torque of the PMSM over three different load conditions (no load, rated load, and maximum speed load). El-Nemr et al.²⁰ proposed an optimal design methodology for SRM using the non-dominated sorting genetic algorithm (NSGA-II) optimization technique.

In this paper, multi-objective genetic algorithm is applied to the optimal design of HSDDM in the DDFS. It is preferred to determine the design variables and value range that affect the shape of permanent magnet of the HSDDM. A multi-objective optimization model is established to minimize cogging torque, maximize efficiency, and minimize the magnet weight. The OLHS is used to design the experimental samples, and the sampling results are calculated by the FEM. The Kring approximate model is established according to the sampling results. Based on the established approximate model, NSGA II multi-objective genetic algorithm is used to optimize the HSDDM. Compared with that before optimization, the efficiency after optimization is increased by 4.08%, the cogging torque is reduced by 86.78%, and the magnet weight is reduced by 20.24%.

The DDFS and the HSDDM description

The DDFS description

The schematic diagram for the structure of the DDFS based on the nut driven ball screw pair is illustrated in Figure 1.

Figure 1.

The schematic diagram of the DDFS structure.

In the DDFS, the screw shaft and screw nut, both driven by PMSMs, rotate in the same direction at high speed and are superimposed though the ball screw pair to obtain low-velocity of the table. This design ensures that the driven table travels at low velocity with the two motors rotating at high speed. The low velocity of the table is obtained, and the crawling zone for the motors at low speed is avoided by superposing two PMSM rotating speeds at high speed by the ball screw pair.^21,22

The HSDDM description

The HSDDM for the DDFS adopts permanent magnet surface mounting type, and the diameter of the rotating shaft hole is 18 mm, as shown in Figure 2.

Figure 2.

The HSDDM structure: (a) the HSDDM structure and (b) the slot type of the HSDDM.

The parameters of the HSDDM are shown in Table 1. To reduce the torque ripple of the HSDDM, the shape of permanent magnet is optimized in this paper.

Table 1.

Description of the HSDDM.

Parameters	Values	Parameters	Values
Rated power (W)	240	R _s (mm)	0.5
Rated line voltage (V)	48	Outer diameter of stator (mm)	86
Rated speed (rpm)	1000	Inner diameter of stator (mm)	61
Rated efficiency	79.32%	Length of stator (mm)	30
Number of poles	14	Air gap length (mm)	0.5
Number of stator slots	51	Outer diameter of rotor (mm)	60
H _s0 (mm)	0.3	Inner diameter of rotor (mm)	45
H _s1 (mm)	0.2	Type of magnet	Nd2Fe14B
H _s2 (mm)	8.6	Residual flux density (T)	1.18
B _s0 (mm)	1	Coercive force (kA/m)	890
B _s1 (mm)	3.1	Slot fill factor (%)	72.4%
B _s2 (mm)	5	Thickness of magnet (mm)	3.5
Tooth width/(mm)	2.1	Pole embrace	0.8

Multi-objective optimization of the HSDDM

In order to accurately obtain the optimal parameters of HSDDM, the multi-objective optimization design process based on genetic algorithm is shown in Figure 3.

Figure 3.

Optimization design process of the HSDDM.

Analysis and determination of motor optimization objectives and parameters

Determination of design variables

There are many design variables in the design optimization of a PMSM. According to the actual situation, the pole embrace (α), thickness of magnet (h), eccentric distance (l), and air gap length (δ) are selected as the design variable, as shown in Figure 4 in this paper. The final determination of the value range of design variables based on the original design is shown in Table 2.

Figure 4.

Design variables.

Table 2.

Value range of design variables.

Design variables	Minimum	Maximum
Pole embrace α	0.6	1
Thickness of magnet h (mm)	3	5
Eccentric distance l (mm)	0	20
Air gap length δ (mm)	0.4	1

Constraints

The constraint condition is to ensure that the optimization result can meet the performance indexes and basic design requirements of motor design while meeting the objective function. General constraints include slot fill factor, stator-teeth flux density, stator-yoke flux density, thermal load, and magnet weight. If the slot fill factor is too high, it will lead to the difficulty of motor offline and increase the process difficulty of motor design; If the stator-teeth flux density is too high, it will lead to the increase of iron consumption and the increase of heat dissipation requirements for the motor. According to the actual situation, this paper takes the stator-teeth flux density, stator-yoke flux density, and thermal load as the constraints of this optimization.

Due to the difference in the order of magnitude and sensitivity of constraint values between constraint functions, the constraint conditions can be expressed as the following equation (1) by normalizing each constraint condition.

\begin{matrix} {\begin{matrix} g_{1} (x) = \frac{B_{t} - B_{t 0}}{B_{t 0}} \leq 0 \\ g_{2} (x) = \frac{B_{j} - B_{j 0}}{B_{j 0}} \leq 0 \\ g_{3} (x) = \frac{AJ - A J_{0}}{A J_{0}} \leq 0 \end{matrix} \end{matrix}

(1)

Where, B_t is the stator-teeth flux density of the optimized motor; B_j is the stator-yoke flux density of the optimized motor; A_j is the thermal parameters of the optimized motor; B_t₀ is 1.64 T, which represents the stator-teeth flux density of the motor before optimization; B_j₀ is 1.64 T, which represents the stator-yoke flux density of the motor before optimization; A_j₀ is 250 A/mm³, which represents the thermal parameters of the motor before optimization.

Objective function

Cogging torque directly affects the operation performance of motor, especially for high-performance direct drive motor. At the same time, considering the economy of permanent magnet motor, it is hoped to reduce the magnet weight and save the cost without affecting the performance of the motor. The cogging torque T_cog (peak to peak value), the motor efficiency η, the magnet weight M are selected as objective functions in this paper.

The influence of design variables on the objective values is analyzed. The influence of design variables on efficiency is shown in Figure 5. As can be seen from Figure 5, the efficiency first increases and then decreases with the increase of pole embrace, eccentric distance and air gap, and first decreases and then increases with the increase of thickness of magnet.

Figure 5.

Influence of design variables on efficiency: (a) pole embrace, (b) thickness of magnet, (c) eccentric distance, and (d) air gap length.

The influence of design variables on cogging torque is shown in Figure 6. As can be seen from Figure 6(a), the cogging torque increases with the increase of the thickness of magnet. Figure 6(b) shows that when the pole embrace exceeds 0.9, the cogging torque increases sharply with the increase of pole arc coefficient. As shown in Figure 6(c), the cogging torque decreases with the increase of the eccentric distance. This is because after the eccentricity of the permanent magnet, the air gap magnetic field is closer to sinusoidal, the harmonic of the air gap magnetic field decreases, and the cogging torque decreases. Figure 6(d) shows that changing the air gap also has an impact on the cogging torque. When other conditions remain unchanged, the cogging torque is the smallest when the air gap length is 0.7 mm.

Figure 6.

Influence of design variables on cogging torque: (a) thickness of magnet, (b) pole embrace, (c) eccentric distance, and (d) air gap length.

The influence of design variables on the magnet weight is shown in Figure 7. According to Figure 7(a), (b) and (d), the magnet weight changes linearly with the pole embrace, the thickness of magnet, and the air gap length. The magnet weight increases with the increase of the pole embrace and the thickness of magnet, and the mass of the magnet weight decreases with the increase of the air gap length. Figure 7(c) shows that the magnet weight decreases with the increase of the eccentric distance, but there is no nonlinear relationship.

Figure 7.

Influence of design variables on the magnet weight: (a) pole embrace, (b) thickness of magnet, (c) eccentric distance, and (d) air gap length.

In Isight, the objective function is taken as

f_{min} (x_{i}) = - \frac{ω_{1} η (x_{i})}{s f_{1}} + \frac{ω_{2} T_{cog} (x_{i})}{s f_{2}} + \frac{ω_{3} M (x_{i})}{s f_{3}}

(2)

Where, x_i is the design variable; ω₁, ω₂, ω₃ are the weighted values of efficiency, cogging torque, and magnet weight respectively. In this paper, the optimization focuses on improving motor efficiency and reducing cogging torque ω₁ = ω₂ = 0.4, ω₃ = 0.2; sf₁, sf₂, and sf₃ are the scale factors of efficiency, cogging torque, and magnet weight respectively, where sf₁ = 10, sf₂ = 4, and sf₃ = 0.02.

Optimized Latin hypercube sampling

According to the value range of design variables, in order to reduce the number of experiments, the OLHS is selected for experimental design. The sample distribution results obtained with OLHS are shown in Table 3.

Table 3.

Sample distribution using OLHS.

Sample	Design variables
Sample	Pole embrace α	Thickness of magnet h (mm)	Eccentric distance l (mm)	Air gap length δ (mm)
1	0.972	4.034	17.93	0.772
2	0.697	5	13.79	0.855
3	0.628	3.966	11.03	0.938
4	0.614	4.448	15.86	0.607
5	0.89	4.586	13.1	0.959
6	0.655	4.655	4.83	0.566
7	0.752	3.69	2.07	0.462
8	1	4.103	6.9	0.731
9	0.917	4.379	4.14	0.421
10	0.779	4.31	12.41	0.4
11	0.959	3.483	8.97	0.483
12	0.848	4.931	8.28	0.648
13	0.834	4.241	0	0.71
14	0.931	4.862	2.76	0.917
15	0.683	3.276	16.55	0.752
16	0.71	4.793	3.45	0.897
17	0.766	3	9.66	0.524
18	0.669	3.552	17.24	0.441
19	0.986	4.517	14.48	0.503
20	0.738	4.172	20	0.876
21	0.821	4.724	19.31	0.628
22	0.862	3.759	5.52	0.979
23	0.945	3.138	11.72	0.793
24	0.641	3.828	0.69	0.814
25	0.807	3.345	15.17	1
26	0.793	3.897	10.34	0.69
27	0.724	3.069	6.21	0.834
28	0.6	3.621	7.59	0.586
29	0.903	3.207	1.38	0.669
30	0.876	3.414	18.62	0.545

Automatic calculation of parameters of sample points by FEM

When establishing the FEM of the HSDDM in Maxwell software, the script recording function in Maxwell software is used to record the motor modeling and solution process into a script file in VBS format. The script file of the above modeling and solution process is parametrically integrated with simcode component in the Isight software to complete the parametric calculation of sample points of experimental design. The parameterized calculation integration process of the experimental design sample points in the sight software is shown in Figure 8.

Figure 8.

Parametric calculation integration of experimental sample points in the Isight.

According to the experimental samples designed above, calculate the cogging torque, efficiency, magnet weight, heat load, stator-teeth flux density B_T, and stator-yoke flux density B_J corresponding to each sample point, as shown in Table 4.

Table 4.

Characteristics of sample obtained from OLHS using FEM analysis.

Sample	Constraint			Optimization objectives
Sample	B_t (T)	B_j (T)	A_J (A/mm³)	η (%)	T_cog (mNm)	M (kg)
1	1.697	1.491	282.781	78.32	2.3267	0.13723
2	1.686	1.349	453.645	69.8736	6.2087	0.12894
3	1.635	1.218	638.532	62.593	14.5881	0.09431
4	1.764	1.278	457.699	69.681	5.692	0.102678
5	1.656	1.509	300.059	77.402	5.6941	0.149434
6	1.797	1.400	248.236	80.275	17.6751	0.116844
7	1.813	1.609	116.748	88.659	30.3039	0.108792
8	1.728	1.730	125.559	88.021	22.5556	0.155521
9	1.833	1.898	134.042	87.168	38.4114	0.154915
10	1.841	1.633	116.393	88.630	6.034	0.127376
11	1.790	1.814	110.324	88.950	10.3178	0.127802
12	1.750	1.655	137.401	87.215	9.458	0.157306
13	1.743	1.670	125.416	88.066	23.7196	0.136511
14	1.645	1.627	204.966	82.903	16.8207	0.170587
15	1.660	1.284	465.38	69.414	3.4681	0.083612
16	1.686	1.399	350.964	74.765	18.0067	0.12866
17	1.753	1.542	154.842	86.111	11.3378	0.08904
18	1.780	1.377	269.862	79.039	4.4178	0.089505
19	1.805	1.719	117.311	88.531	3.5796	0.162992
20	1.653	1.298	559.885	65.485	4.8669	0.10969
21	1.758	1.484	271.223	78.926	2.3347	0.13829
22	1.628	1.534	253.954	80.009	6.0748	0.123435
23	1.649	1.574	208.396	82.725	7.4228	0.110754
24	1.683	1.291	436.517	70.689	23.2836	0.094989
25	1.580	1.353	461.206	69.618	4.5117	0.099149
26	1.733	1.547	182.891	84.286	7.0876	0.117359
27	1.627	1.369	350.202	74.847	14.0911	0.086157
28	1.756	1.262	409.088	71.926	13.9562	0.0844
29	1.709	1.735	119.359	88.480	19.6683	0.113652
30	1.755	1.529	196.068	83.431	4.0797	0.1063

Kriging approximation model and multi-objective optimization

Kriging approximation model

Kriging approximation model is composed of regression model and correlation model, which has high fitting accuracy. The regression models include zero-order, first-order, and second-order polynomial fitting. The relevant models mainly include Gaussian function, linear function, exponential function, and so on.

Let an m-dimensional design variable S = (s₁s₂…s_m) ^T and the response value Y = (y₁, y₂…y_m) ^T then the response value can be expressed as

Y = β F (s) + z (s)

(3)

Where, F(s) is the global regression model, β is the regression coefficient, z(s) is the correlation function, and obeys the local deviation with zero mean but non-zero variance created on the basis of the regression model.

For any two z(s_i) and z(s_j), the covariance is

Cov [z (s_{i}), z (s_{j})] = σ^{2} r (s_{i}, s_{j})

(4)

r (s_{i}, s_{j}) = Π_{d = 1}^{q} \exp (- θ^{d} {| s_{i}^{d} - s_{j}^{d} |}^{p^{d}})

(5)

In this paper, the correlation function is Guassian function.

Kriging approximate models are established between the objective values and the design parameters, and the prediction accuracy of the approximate model is evaluated by R-square error analysis method and mean value. The approximate model error estimates between constraint values, objective values, and parameter variables are shown in Table 5 and Figure 9.

Table 5.

Approximate model error estimation.

Constraint value and objective value	R ²	Mean value
Stator-teeth flux density B_t	0.99354	0.01677
Stator-yoke flux density B_j	0.99844	0.00832
Motor thermal parameters A_J	0.99857	0.00819
Efficiency η	0.99817	0.00945
Cogging torque T_cog	0.82912	0.07706
Magnet weight M	0.98992	0.02239

Figure 9.

Fitting between constraint values, objective values, and parameter variables: (a) stator-teeth flux density, (b) stator-yoke flux density, (c) motor thermal parameters, (d) efficiency, (e) cogging torque, and (f) magnet weight.

According to Table 4, the R-square of stator-teeth flux density, stator-yoke flux density, thermal parameters, efficiency and magnet weight is more than 0.9, the R-square of cogging torque is slightly lower than 0.829, and the mean error is within 0.2. The established approximate model can accurately simulate the relationship between objective value and constraint value and various parameter variables.

Multi-objective optimization

Based on the Kriging approximate model, NSGA-II is used for multi-objective optimization to find out the optimal parameters of design variables in the feasible interval in this paper. The integration of approximate model and multi-objective genetic algorithm of NSGA-II in Isight is shown in Figure 10.

Figure 10.

Integration of approximate model and multi-objective optimization in Isight.

The population number is defined as 100, the genetic algebra is 2000, the mating probability is 0.9, and the mutation probability is 0.09. After 2000 generations of optimization, the optimal solution and the corresponding optimal design variables found by NSGA-II can be obtained. The comparison of design variables before and after optimization is shown in Table 6.

Table 6.

Comparison of design variables before and after optimization.

Design variable	Before optimization	After optimization
Pole embrace α	0.8	0.778
Thickness of magnet h (mm)	3.5	3.04
Eccentric distance l (mm)	0	16.33
Air gap length δ (mm)	0.5	0.491

Simulation and test verification

Simulation results before and after optimization

The optimized design variable values are substituted into the FEM. The comparison of objective values before and after optimization is shown in Table 7. After optimization, the efficiency is increased by 4.08%, the cogging torque is reduced by 86.78%, and the magnet weight is reduced by 20.24%.

Table 7.

Comparison of results before and after optimization.

Objective values	Before optimization	After optimization	Rate of change
Efficiency η (%)	79.32	82.56	Increase 4.08%
Cogging torque T_cog (mNm)	29.13	3.85	Decrease 86.78%
Magnet weight M (kg)	0.110	0.088	Decrease 20.24%

The model of the HSDDM comparison before and after optimization is shown in Figure 11, the comparison of air gap magnetic density distribution is shown in Figure 12, and the tangential magnetic density and spectrum analysis of air gap are shown in Figure 13. After optimization, the air gap magnetic field distribution is more reasonable and the sinusoidal performance of air gap magnetic density is improved.

Figure 11.

Model before and after optimization: (a) model before optimization and (b) model after optimization.

Figure 12.

Air gap magnetic density distribution before and after optimization: (a) before optimization and (b) after optimization.

Figure 13.

Optimized tangential air gap magnetic density: (a) air gap flux density distribution and (b) harmonic analysis of air gap flux densities.

The comparison of no-load back EMF waveform and harmonic analysis of the motor before and after optimization is shown in Figure 14. After optimization, the third harmonic of no-load back EMF is significantly reduced and the sinusoidal performance of no-load back EMF is improved.

Figure 14.

Comparison of no-load back EMF before and after optimization: (a) back EMF waveform and (b) harmonic analysis of back EMF.

The comparison of cogging torque and rated torque of motor before and after optimization is shown in Figure 15. After optimization, the peak to peak value of cogging torque is 3.85 mNm, and the fluctuation rate of rated torque peak to peak value is 4.16%, which meets the design requirements.

Figure 15.

Comparison of cogging torque and rated torque: (a) cogging torque and (b) rated torque.

Experiment of the HSDDM

To verify the proposed optimum design process, a prototype of the optimum HSDDM was fabricated. The optimized stator and rotor of the HSDDM are shown in Figure 16. The experimental setup is illustrated in Figure 17.

Figure 16.

Optimized stator and rotor of the HSDDM: (a) stator and (b) rotor.

Figure 17.

Experimental setup for the HSDDM.

The motor speed torque curve before and after optimization is shown in Figure 18. The optimized motor torque at the same motor speed is greater than that before optimization.

Figure 18.

Motor speed torque curves before and after optimization.

The comparison of experimental and simulation results of objective values is shown in Table 8. After optimization, the error of objective values of the simulation and experimental results of the HSDDM is less than 5%, which indicate that the multi-objective optimization method proposed in this paper has high accuracy.

Table 8.

Comparison of simulation and experimental results after optimization.

Objective values	Simulation	Experiment	Error (%)
Efficiency η (%)	82.56	81.08	1.79
Cogging torque T_cog (mNm)	3.85	4.01	4.16
Magnet weight M (kg)	0.088	0.089	1.14

Conclusion

In this paper, multi-objective genetic algorithm is applied to the optimal design of the HSDDM in DDFS. It is preferred to determine the design variables and value range that affect the shape of permanent magnet of the HSDDM. A multi-objective optimization model is established to minimize cogging torque, maximize efficiency, and minimize magnet weight. The OLHS is used to design the experimental samples, and the sampling results are calculated by the FEM. The Kring approximate model is established according to the sampling results. Based on the established approximate model, NSGA II multi-objective genetic algorithm is used to optimize the HSDDM. Compared with that before optimization, the efficiency after optimization is increased by 4.08%, the cogging torque is reduced by 86.78%, and the magnet weight is reduced by 20.24%.

Footnotes

Handling Editor: Chenhui Liang

Authors contributions

ZW wrote the manuscript. JG performed the data analyses. BL revised the language of the paper.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research has been supported by the Key projects of natural science research in colleges and universities in Anhui Province (No. KJ2021A0468), Anhui intelligent mine technology and equipment Engineering Laboratory (No. AIMTEEL202204), and university the level key projects of Anhui University of Science and Technology (No. xjzd2020-10).

Consent to publish

I would like to declare on behalf of my co-authors that the work described was original research, and it has not been published previously, meanwhile, the paper is not to under consideration for publication elsewhere. All the authors listed have approved the manuscript that is enclosed.

ORCID iD

Zhaoguo Wang

References

Hyoseok

Niguchi

Hirata

. Characteristic analysis of surface permanent-magnet vernier motor according to Pole Ratio and winding pole number. IEEE Trans Magn 2017; 53: 1–4.

Wang

Yuan

Atallah

. Design optimization of a surface-mounted permanent-magnet motor with concentrated windings for electric vehicle applications. IEEE Trans Vehicular Technol 2013; 62: 1053–1064.

Lipo

. Efficient utilization of rare earth permanent-magnet materials and torque ripple reduction in interior permanent-magnet machines. IEEE Trans Ind Appl 2017; 53: 3485–3495.

Zhongshu

Aijun

Lie

, et al. Design of line-start low speed and high torque PMSM for gearless drive system. In: 2011 international conference on electrical machines and systems, Beijing, 2011, pp.1–4.

Simon-Sempere

Burgos-Payan

Cerquides-Bueno

. Cogging torque cancellation by magnet shaping in surface-mounted permanent-magnet motors. IEEE Trans Magn 2017; 53: 1–7.

Wahab

Setiabudy

Rahardjo

. Cogging torque reduction by modifying stator teeth and permanent magnet shape on a surface mounted PMSG. In: International seminar on intelligent technology and its application, Surabaya, Indonesia, 2017, pp.227–232.

Ahn

J-M

Son

J-C

Lim

D-K

. Optimal design of outer-rotor surface mounted permanent magnet synchronous motor for cogging torque reduction using territory particle swarm optimization. J Electr Eng Technol 2021; 16: 429–436.

Yılmaz

Yenipınar

Sönmez

, et al. Optimization of PMSM design parameters using update meta-heuristic algorithms. In: ICAIAME the international conference on artficial inteligence and applied mathematics in engineering, Antalya, Turkey, pp.914–934.

Edhah

Alsawalhi

Al-Durra

. Multi-objective optimization design of fractional slot concentrated winding permanent magnet synchronous machines. IEEE Access 2019; 7: 162874–162882.

10.

Owatchaiphong

Fuengwarodsakul

. Multi-objective based optimization for switched reluctance machines using fuzzy and genetic algorithms. In: International conference on power electronics and drive systems, Taipei, Taiwan, 2009.

11.

Chui

Chiang

Lee

, et al. Multi-objective optimization design of interior permanent-magnet synchronous motors for improving the effectiveness of field weakening control. In: 17th international conference on electrical machines and systems, Hangzhou, China, 2014.

12.

Marimin

Arkeman

. Solving fuzzy multi-objective optimization using non-dominated sorting genetic algorithm II. In: International conference on advanced computer science and information systems, Malang, Indonesia, 2016.

13.

Lim

D-K

Woo

D-K

. A new surrogate-assisted robust multi-objective optimization algorithm for an electrical machine design. J Electr Eng Technol 2019; 14: 1247–1254.

14.

Huang

Tian

, et al. Multi-physical field optimization analysis of high-speed permanent magnet synchronous motor based on NSGA-II algorithm. In: ICEMS 22nd international conference on electrical machines and systems, Harbin, China.

15.

Mutluer

. Analysis and design optimization of permanent magnet motor with external rotor for direct driven mixer. J Electr Eng Technol 2021; 16: 1527–1538.

16.

Chowdhury

Islam

Gebregergis

, et al. Robust design optimization of permanent magnet synchronous machine utilizing genetic and Taguchi’s algorithm. In: Proceedings of the 2013 IEEE energy conversion congress and exposition, September 2013, Denver, CO, USA, pp.5006–5012.

17.

Islam

Chowdhury

Shrestha

, et al. Multiload point optimization of interior permanent magnet synchronous machines for high-performance variable-speed drives. IEEE Trans Ind Appl 2021; 57: 427–436.

18.

Baek

. Optimum shape design of a BLDC motor for electric continuous variable valve timing system considering efficiency and torque characteristics. Microsyst Technol 2018; 24: 4441–4452.

19.

Wang

Nien

Huang

. Multi-objective optimization design and analysis of V-shape permanent magnet synchronous motor. Energies 2022; 15: 3496.

20.

El-Nemr

Afifi

Rezk

, et al. Finite element based overall optimization of switched reluctance motor using multi-objective genetic algorithm (NSGA-II). Mathematics 2021; 9: 576.

21.

Wang

Feng

, et al. A novel method for smooth low-speed operation of linear feed systems. Precis Eng 2019; 60: 215–221.

22.

Wang

Feng

Yang

, et al. Research on low-speed characteristics of differential double-drive feed system. Mech Sci 2021; 12: 791–802.

Multi-objective optimization of the hollow shaft direct drive motor

Abstract

Keywords

Introduction

The DDFS and the HSDDM description

The DDFS description

The HSDDM description

Multi-objective optimization of the HSDDM

Analysis and determination of motor optimization objectives and parameters

Determination of design variables

Constraints

Objective function

Optimized Latin hypercube sampling

Automatic calculation of parameters of sample points by FEM

Kriging approximation model and multi-objective optimization

Kriging approximation model

Multi-objective optimization

Simulation and test verification

Simulation results before and after optimization

Experiment of the HSDDM

Conclusion

Footnotes

Authors contributions

Declaration of conflicting interests

Funding

Consent to publish

ORCID iD

References