1. Introduction

2. Motor Topology

3. Mechanical Modeling and Analysis of Motor Rotors

4. Coupled Modeling and Analysis of Electromagnetic Field and Temperature Field in Electric Motors

4.1. Finite Element Modeling of Electromagnetic Fields

Due to the symmetry characteristics of the motor, the 2-D finite element model adopted in this paper can fully meet the accuracy requirements. In order to further reduce the amount of calculation, the 1/2 model is adopted. Figure 9 is the mesh division diagram of the 1/2 Motor 2D finite element model. The mesh division quality is good, and the number of division elements is 8000, which can meet the calculation accuracy and efficiency.

For the two-dimensional electromagnetic finite element model, the potential function is generally used to calculate the electromagnetic field of the motor. The potential function includes magnetic vector potential $A$ and magnetic scalar potential $\varphi$, so the electromagnetic field problem on the plane field Ω can be expressed as:

(3)

where, $\zeta$s Magnetoresistance, $A$ is magnetic vector potential, only z-axis component, $J_Z$ is Source current density, $H_{\tau }$ is Tangential component of magnetic field intensity, ${\Gamma }_1$ is the first type boundary, ${\Gamma }_2$ is the second type boundary.

(4)

The PM adopts the surface current model, as shown in Figure 10. After being magnetized, the PM shall meet the characteristics, as shown in Equation (5). The right end of the equation reflects the excitation characteristics of the PM, as shown in Equation (6), which is the surface current density.

(5)

(6)

In this paper, the PM adopts radial magnetization, as shown in Figure 11, the current layer along the BC and AD sides is:

(7)

The current layer along the AB and CD sides is:

(8)

By solving the electromagnetic field problem, A can be obtained, and then the performance parameters of each cylinder of the motor can be obtained by using A.

The performance parameters of the motor can be obtained by using A. Figure 12 shows the magnetic field cloud diagram of the finite element model of the motor when no load.

4.2. Finite Element Modeling of Temperature Field

The ultra-high-speed motor adopts water cooling. The basic assumptions for fluid flow and heat transfer calculations in various regions of the motor are as follows:

(1) Basic assumptions and boundary conditions for fluid flow and thermal coupling calculations in electric machines

(2) Assuming that the upper and lower stator windings are of the same width.

(3) It is believed that the components in the motor are in close contact and have no contact thermal resistance.

(4) Assuming that the heat of the motor is only removed by convection through the fluid in the cooling water jacket, and that there is no heat transfer between the motor and the air.

(5) Ignoring the influence of heat dissipation at the end of the stator.

(6) Due to the high Reynolds number of the fluid inside the water jacket, the flow belongs to turbulent flow. A turbulence model is adopted for the fluid and solved, the fluid is incompressible

Based on the above assumptions, the boundary conditions for the calculation model in this article are as follows:

(1) Since the fluid in the water jacket is incompressible, the rated water velocity is 20L/min, and the inlet temperature is 40°C.

(2) The ambient temperature of the motor is set to 25°C.

(3) All solid surfaces are set as wall boundary conditions, and coupling is added between surfaces to allow temperature transfer.

(4) The heat sources in the motor include the stator teeth, the stator yoke, and the stator winding.

Figure 13 is a mesh division diagram of the finite element model of heat transfer of the motor. The division quality is good and can meet the following temperature field calculation.

In order to obtain the temperature distribution in the medium, it is necessary to solve the heat flow equation in an appropriate form. However, this solution depends on the physical conditions at the boundary of the medium. If the process changes with time, this solution also depends on the internal conditions of the medium at a certain starting time. When it comes to boundary conditions, there are several common physical effects at the boundary, which can be simply expressed in mathematical formulas. In order to determine the temperature distribution inside the object, for the above heat conduction equation, it is necessary to give appropriate boundary conditions. Common boundary conditions are divided into three categories.

(1) Temperature boundary conditions (first type of boundary conditions):

(9)

where $T_c$ is the given temperature on the boundary surface $S_1$, $f\left(x,y,z,t\right)$ is a known temperature function, and $S_1$ is the object boundary.

(2) The boundary condition for heat flow (second type boundary condition):

(10)

where, the heat conduction at the boundary is a known value $q_0$ (outward), refers to the known normal gradient of the boundary. When $q_0=0$, there is no heat conduction on this surface, which is called an adiabatic boundary condition.

In this equation, $q_0$ represents the boundary heat flow input on the $S_2$ surface,$\mathrm{g}\left(x,y,z,t\right)$ is the heat flow density function, and $\mathsf{\lambda }$ is the coefficient of heat conduction perpendicular to the surface of the object.

(3) Thermal exchange boundary condition (third type boundary condition):

The heat flow rate transmitted from the interior of the object to the boundary on boundary $S_3$ is equal to the heat flow rate dissipated into the surrounding medium through this boundary, This equation is shown below:

(11)

where $T_0$ represents the temperature of the surrounding medium, $\mathsf{\alpha }$ is the heat transfer coefficient of the $S_3$ surface, and both $T_0$ and $\mathsf{\alpha }$ can be constants or functions of time and position.

When using the finite element method to calculate temperature rise in an ultra-high speed motor, determining the physical parameters of the model is crucial. Physical properties are all reflected by material properties, so the material must be assigned to each entity. When conducting temperature field analysis, motor materials include parameters such as density, specific heat capacity, and coefficient of heat conduction. The main components in this analysis involve permanent magnets, non-magnetic alloy steel, stator steel laminations, copper windings, insulation, the casing and so on. The following are some material property values:

Table 4. Data sheet of materials used in the model.

Table 4. Data sheet of materials used in the model.

Name	material	density (kg/m3)	thermal conductivity (W/m°C)	Specific heat capacity (J/kg°C)
core	Silicon steel sheets	7800	42.5	502.4
winding	copper	8900	387.6	504
Air inside the air gap	Air	1.18	0.026	1042
Permanent magnet protective sleeve	carbon fibre	1600	0.59	1000
PM	NdFeB	8400	8	504
axle	steel	7800	50	504
housing	aluminum alloy	2790	168	833
Slot insulation	Polyimide	1430	0.35	1130

The losses in the motor thermal circuit model are calculated using an electromagnetic model. When the motor speed is 100,000 r/min and the DC bus voltage is 540V with a peak phase current of 35A, the electromagnetic model calculates the iron loss distribution shown in Figure 14. It can be seen that the iron loss mainly occurs around the stator slots, while the rotor loss mainly occurs on the surface of the permanent magnets and iron core. The loss distribution values are shown in Table 5. These losses are directly imported into the motor thermal field model for temperature field simulation calculations.

Figure 16 show the temperature distribution of the motor calculated using finite element analysis. The high-temperature area is mainly concentrated in the stator winding, with a maximum temperature of no more than $130°\mathrm{C}$, and a maximum temperature of no more than $120°\mathrm{C}$ in the rotor area

5. Coupling Model Based on Mechanical Field Limitations

6. Optimized Design of High-Speed Motor

6.2. Optimization Process

Based on the ultra-high-speed motor model that couples electromagnetic and thermal fields, an optimization design algorithm based on a surrogate model is adopted to optimize the design of the ultra-high-speed motor. The specific optimization process is shown in Figure 21. Firstly, the electromagnetic-thermal coupling model is used to conduct sensitivity analysis of the motor, providing parameters for motor optimization design. At the same time, constraints on motor design parameters are given based on the analysis of the motor’s mechanical field. Then, an optimization design model based on multi-physical field coupling is established. MaxPro experimental design is used to obtain initial samples. After calculation using the multi-physical field coupling model, a surrogate model is fitted using Kriging. Based on this surrogate model, genetic algorithms are used for optimization design, and the optimization results are verified. If the requirements are not met, new samples are added to rebuild a New Kriging Surrogate model and perform optimization design until the accuracy is met.

6.2.1. Experimental Design

In 2015, Joseph, Gul, and Ba introduced a Maximum Projection (MaxPro) experimental design that incorporates a weight factor to guarantee accurate predictions across all subspaces[32]. The underlying concept is outlined as follows:

$$\underset{D}{\min }\sum _{i=1}^{n-1}\sum _{j=i+1}^n{\displaystyle \frac{1}{d^k\left(x_i,x_j;w\right)}}p\left(w\right)dw$$

(15)

$$d\left(x_i,x_j;w\right)={\left({\sum }_{l=1}^p{\mathrm{w}}_{\mathrm{i}\mathrm{l}}{\left|{\mathrm{x}}_{\mathrm{i}\mathrm{l}}-{\mathrm{x}}_{\mathrm{j}\mathrm{l}}\right|}^{\mathrm{s}}\right)}^{1/\mathrm{s}}$$

(16)

where, $\mathrm{w}=\left({\mathrm{w}}_1,\cdots ,w_p\right)$，${\sum }_{l=1}^p{w}_l=1$ ，$w_1,\cdots w_l>0$,$w_l$ is the weight, $p\left(w\right)$ represents the priority distribution factor of the weight. When $w$ is uniformly distributed and $w=s\times p$, formula (15) can be simplified to formula (17), where $s$ is typically set to 2.

$$d\underset{D}{\min }\sum _{i=1}^{n-1}\sum _{j=i+1}^n{\displaystyle \frac{1}{{\prod }_{l=1}^p{\left|x_{il}-x_{jl}\right|}^s}}$$

(17)

The MaxPro experimental design does not rely on an even spacing distribution; instead, it aims to reach the boundaries as extensively as possible, making it exceptionally well-suited for surrogate models with a high number of dimensions.

Figure 22 depicts the partial factor distribution of the initial 500 samples acquired through the MaxPro experimental design. It illustrates that the samples are spread uniformly across the entire design space, ensuring comprehensive coverage.

6.2.2. Kriging Surrogate Model

The Kriging model, illustrated in equation (18), integrates a global model with local deviations. Due to its exceptional global approximation and error estimation abilities, it is particularly well-suited for addressing nonlinear problems[33].

(18)

where, $x$represents the design parameter, $y\left(x\right)$ denotes the response function, $f\left(x\right)$ employs a global approximation model of polynomial response surface, and $z\left(x\right)$ signifies local error.

Among them, $z\left(x\right)$ represents a static Gaussian random process with an expected value of 0 and a variance of${\sigma }^2$. The covariance is shown in (19).

(19)

where, $\mathit{R}$ represents the correlation matrix, and $R\left(x_i,x_j\right)$ denotes the correlation function between two samples.

Using the Gaussian correlation function, as shown in equation (20)

(20)

where, $m$ represents the number of design parameters, and ${\theta }_k$ denotes the unknown correlation coefficient used for fitting the approximate model.

The estimated value at $x$ is:

(21)

where, $\mathit{f}$ represents a unit column vector of length $n$, $y$ denotes the response column vector of $n$ sample points, and $r^T\left(x\right)$ signifies the correlation vector between the $n$ sample points and $x$. Among them:

(22)

(23)

The estimated variance is:

(24)

The maximum likelihood estimation used for the Kriging model is:

(25)

The Kriging approximation model is obtained by solving the unconstrained optimization problem using numerical optimization algorithms.

Establish a Kriging surrogate model that maps design parameters to motor performance, as shown in equation (26)

$$\left\{\begin{array}{c}x=\left\{\begin{array}{c}bs0,b_{embrace},L_{Airgap},h_{Nd},h_{Sleeve},L_{ef},D_{o1,}\\ h_{tooth},D,I_{Phase},h_{Jacket},Nstrands,hs0,hs2\end{array}\right\}\\ \left[k_p,k_s,\eta ,P_{out},T_{cog},T_{coilMax}\right]=F\left(x\right)\end{array}\right.$$

(26)

Relative Maximum Absolute Error (RMAE) and Relative Average Absolute Error (RAAE) are adopted as evaluation metrics for the optimization model. The calculation formulas for RAAE and RMAE are shown in Equations (27) and (28), respectively.

$$\mathrm{R}\mathrm{A}\mathrm{A}\mathrm{E}={\displaystyle \frac{\frac{1}{N}\sum _{i=1}^N\left|y_i-{\widehat{y}}_i\right|}{\sqrt{\frac{1}{N-1}\sum _{i=1}^N{\left(y_i-{\widehat{y}}_i\right)}^2}}}$$

(27)

$$\mathrm{R}\mathrm{M}\mathrm{A}\mathrm{E}={\displaystyle \frac{\frac{1}{N}\underset{i=1:N}{\mathit{max}}\left(\left|y_i-{\widehat{y}}_i\right|\right)}{\sqrt{\frac{1}{N-1}\sum _{i=1}^N{\left(y_i-{\widehat{y}}_i\right)}^2}}}$$

(28)

where, $N$ represents the number of test points, $y_i$ denotes the true value of the $i^{th}$ test point, and ${\widehat{y}}_i$ signifies the predicted value of the $i^{th}$ test point.

RMAE and RAAE serve as metrics to assess the local and global precision of the model, respectively. Lower values indicate higher model accuracy.

Figure 23 displays partial outcomes from fitting the Kriging surrogate model. While the surrogate model generally captures the global traits of the samples, its local prediction accuracy is moderate, as depicted in Figure 24. Specifically, the RAAE is below 1 and the RMAE is close to 0. Hence, during the optimization design process, it is crucial to conduct adaptive iterations on the samples to enhance accuracy.

6.2.3. NSGA-2

The schematic diagram of NSGA-2 algorithm is shown in Figure 25 [34]. Its main features are: using fast sorting based on Pareto dominance level can quickly calibrate the dominance level of each individual in the population, and using crowding distance instead of fitness sharing mechanism can obtain a uniformly distributed Pareto optimal solution set. At the same time, using elite retention mechanism can make it easier to obtain excellent next generation. Among them, the crowding distance, as shown in equation (29), is the average value of the Euclidean distance.

$$PCD\left(p_k\right)={\displaystyle \frac{1}{m}}{\displaystyle \sum \frac{f_i\left(p_{k+1}\right)-f_i\left(p_{k-1}\right)}{f_i^{\max }-f_i^{\min }}}$$

(29)

where, $f\left(p_{k+1}\right)$, $f\left(p_k\right)$, and $f\left(p_{k-1}\right)$ are three consecutive Pareto front points and $p_{k+1}$, $p_k$, $p_{k-1}$ are the correspond ding Pareto solution; And $f_i^{min}$, $f_i^{max}$are the minimum and maximum values of the $i^{th}$objective function of the Pareto front, respectively.

6.2.4. Optimization Result Analysis

The MaxPro design is utilized to obtain an initial set of 500 samples. The number of correction points amounts to 50, while there are 200 non-dominant solutions considered. The process involves 500 iterations, and the convergence criteria has been set as

$$\left\{\begin{array}{c}RMAE\le 0.01\\ RAAE\le 0.1\end{array}\right.$$

(30)

Figure 26 shows that the RMAE and RAAE of the output parameters are less than 0.01 and 0.1 at the fifth generation, and the samples are only 750.

Figure 27 shows the optimized Pareto front, as indicated by the red line in the figure. In this project, five schemes were selected from the Pareto front according to different requirements, as shown in Table 8. The performance parameters achieved the optimization design goals, with efficiency reaching 97% and power density exceeding 4kW/kg.

Then, finite element analysis was used to verify the accuracy of the five optimization schemes. The verification results are shown in Table 9. The efficiency error of the five optimization schemes is within 1%, and the power density error does not exceed 7%, which meets the requirements. Consider the efficiency and power density of the motor comprehensively, case 2 is chosen as the final design solution.

7. Experiments and Verification

(31)

(32)

8. Conclusion

References

1. ZHANG Fengge, DU Guanghui, WANG Tianyu, et al. Review on Development and Design of High Speed Machines [J]. TRANS ACTIONS OF CHINA ELECTＲOTECHNICAL SOCIETY, 2016, 31(7):18. [CrossRef]
2. Berton J J, Kim H D, Singh R, et al. Turboelectric Distributed Propulsion Benefits on The N3-X Vehicle[J]. Aircraft Engineering and Aerospace Technology: An International Journal, 2014, 86 (6): 558-561.
3. Clarke S, Lin Y, Kloesel K, et al. Enabling Electric Propulsion for Flight-Hybrid Electric Aircraft Research at AFRC[C]. Atlanta: AIAA Aviation Technology, Integration, and Operations Conference, 2014.
4. Welstead J, Felder J, Guynn M, et al. Overview of the NASA STARC-ABL (rev. B) Advanced Concept[C]. Washington:One Boeing NASA Electric Aircraft Workshop. 2017.
5. Williamson M. Air Power the Rise of Electric Air-craft[J]. Engineering & Technology, 2014, 9(10): 77-79.
6. Whurr J, Hart J. A Rolls-Royce Perspective on Concepts and Technologies for Future Green Propulsion Systems[J]. Green Aviation, 2016: 95.
7. ZHAN Jian, ZHANG Zhiming, ZHANG Juannan, et al. Analysis and improvement of rotor dynamics for super high-speed air compressors applied in fuel cell vehicles [J]. Journal of China University of Metrology, 2017(2). [CrossRef]
8. Cho H W，Ko K J，Choi J Y，et al．Rotor natural frequency in high-speed permanent magnet synchronous motor for turbo-compressor application[J]．IEEE Transactions on Magnetics，2011，47(10)：4258-4261.
9. Lee J G , Lim D K .A Stepwise Optimal Design Applied to an Interior Permanent Magnet Synchronous Motor for Electric Vehicle Traction Applications[J].IEEE Access, 2021. [CrossRef]
10. Credo A , Fabri G , Villani M ,et al.Adopting the Topology Optimization in the Design of High-Speed Synchronous Reluctance Motors for Electric Vehicles[J].IEEE Transactions on Industry Applications, 2020, PP(99):1-1. [CrossRef]
11. Tae-Woo Lee; Do-Kwan Hong, Rotor Design, Analysis and Experimental Validation of a High-Speed Permanent Magnet Synchronous Motor for Electric Turbocharger, IEEE Access, vol.10, pp. 21955- 21969, Feb. 2022.
12. WAN G Feng-xiang. Study on design feature and related technology of high speed electrical machines [J]. Journal of Shenyang University of Technology, 2006, 28(3):7. [CrossRef]
13. ZHANG Fengge, DU Guanghui, WANG Tianyu, et al. Rotor Strength Analysis of High-speed Permanent Magnet Under Different Protection Measures [J]. Proceedings of the CSEE, 2013(S1): 8.DOI: CNKI: SUN: ZGDC. 0.2013-S1-031.
14. ZHANG Fengge, DU Guanghui, WANG Tianyu, et al. Ｒotor containment sleeve study of high-speed PM machine based on multi-physics fields, 2014, 18(6):7. [CrossRef]
15. ZHOU Fengzheng. Investigation of Rotor Eddy-Current Loss in High-Speed PM BLDC Motors. [D],Hangzhou：Zhejiang University, 2008.
16. ZHOU Fengzheng, MENG Qinglin, MENG Zhengzheng, et al. Optimal Design of Rotor Structure and Experimental Study on High Speed PM BLDC Motors [J]. MICROMOTORS, 2019, 52(3): 4.DOI: CNKI: SUN: WDJZ. 0. 2019-03-002.
17. Weili L , Hongbo Q , Xiaochen Z ,et al.Influence of Copper Plating on Electromagnetic and Temperature Fields in a High-Speed Permanent-Magnet Generator[J].IEEE Transactions on Magnetics, 2012, 48(8):2247-2253. [CrossRef]
18. Li W , Qiu H , Zhang X ,et al.Analyses on Electromagnetic and Temperature Fields of Superhigh-Speed Permanent-Magnet Generator With Different Sleeve Materials[J].IEEE Transactions on Industrial Electronics, 2013, 61(6):3056-3063. [CrossRef]
19. Li W, Qiu H, Zhang X,et al. Influence of Rotor-Sleeve Electromagnetic Characteristics on High-Speed Permanent-Magnet Generator[J].IEEE Transactions on Industrial Electronics, 2013, 61(6):3030-3037. [CrossRef]
20. GAO Pengfei, FANG Jiancheng, HAN Bangcheng, et al. Analysis of Rotor Eddy-current Loss in High-speed Permanent Magnet Motors [J]. MICROMOTORS, 2013, 46(5):7. [CrossRef]
21. DONG Jianning, HUANG Yunkai, JIN Long, et al. Review on High Speed Permanent Magnet Machines Including Design and Analysis Technologies [J]. Proceedings of the CSEE, 2014, 34(27):14. [CrossRef]
22. GUO Si, GUO Hong, XU Jinquan. Integrated design method of six-phase fault-tolerant permanent magnet in-wheel motor based on multi-physics fields [J]. Journal of Beijing University of Aeronautics and Astronautics, 2019(3):9. [CrossRef]
23. Zhang F , Dai R , Liu G ,et al.Design of HSIPMM based on multi-physics fields[J].IET electric power applications, 2018(8):12. [CrossRef]
24. HAN Bang-Cheng, XUE Qing-Hao, LIU Xu. Multi-physics analysis and rotor loss optimization of high-speed magnetic suspension PM macine[J].Optics and Precision Engineering, 2017, 025(003):680-688.
25. Kun,Wang,Xin, et al. Multiphysics Global Design and Experiment of the Electric Machine With a Flexible Rotor Supported by Active Magnetic Bearing[J].Mechatronics IEEE/ASME Transactions on, 2019. [CrossRef]
26. Jung Y H , Park M R , Kim K O ,et al. Design of High-Speed Multilayer IPMSM Using Ferrite PM for EV Traction Considering Mechanical and Electrical Characteristics[J].IEEE Transactions on Industry Applications, 2020, PP(99):1-1. [CrossRef]
27. Koronides A , Krasopoulos C , Tsiakos D ,et al. Particular Coupled Electromagnetic, Thermal, Mechanical Design of High-Speed Permanent Magnet Motor[J].IEEE Transactions on Magnetics, 2020, PP(99):1-1. [CrossRef]
28. Gu Y , Wang X , Gao P ,et al. Mechanical Parametric Sensitivity Analysis of High-Speed Permanent Magnet Synchronous Machine[J].IEEE Transactions on Applied Superconductivity, 2021, PP(99):1-1. [CrossRef]
29. Kim J H , Kim D M , Jung Y H ,et al. Design of Ultra-High-Speed Motor for FCEV Air Compressor Considering Mechanical Properties of Rotor Materials[J].IEEE Transactions on Energy Conversion, 2021, (Early Access). [CrossRef]
30. Wu S , Sun X , Tong W .Optimization Design of High-speed Interior Permanent Magnet Motor with High Torque Performance Based on Multiple Surrogate Models[J]. CES TEMS, 2022(003):006. [CrossRef]
31. Issah Ibrahim, Rodrigo Silva, David A. Lowther. Application of Surrogate Models to the Multiphysics Sizing of Permanent Magnet Synchronous Motors[J], IEEE Transactions on Magnetics,vol.58, no.9,pp.,sept. 2022.
32. V. R. Joseph, E. Gul, and S. Ba, Maximum projection designs for computer experiments, Biometrika,vol. 102, no. 2, pp. 371–380, Jun. 2015.
33. M. Li, F. Gabriel, M. Alkadri, and D. A. Lowther, Kriging-assisted multi-objective design of permanent magnet motor for position sensorless control, IEEE Trans. Magn.,vol. 52, no. 3, Mar. 2016, Art. no. 7001904.
34. K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, A fast and elitist multiobjective genetic algorithm: NSGA-II, IEEE Trans. Evol. Comput., vol. 6, no. 2, pp. 182–197, Apr. 2002.