Topology optimization of asymmetric Interior permanent magnet motors using Gaussian basis function

Abstract

Novel rotor structures of an asymmetric interior permanent magnet (IPM) motor are realized by using the topology optimization (TO) in this study. The TO method to be presented in this article takes into account both the configuration of the magnet and the geometry of the magnetic core. In the optimization, the shape of the magnetic core is determined by a linear combination of the Gaussian basis functions, whereas the magnet configuration is determined by a Boolean geometry projection for its manufacturability. It is shown that the optimized asymmetric IPM motor exhibits better torque performance compared to the conventional TO which considers only the magnetic core.

Keywords

asymmetric IPM motor Boolean operation magnet configuration topology optimization

Introduction

Interior permanent-magnet (IPM) motors have been widely used for transportation electrification owing to their high torque density and efficiency. Because the torque characteristics of IPM motors are determined by the configurations of the magnets and the form of the magnetic core within the rotor, it is critical to optimize these geometries to improve motor performance. Although various rotor topologies have been proposed in previous studies, asymmetric IPM motors with asymmetric flux barriers and magnet configurations have recently attracted attention.^1–4 By effectively concentrating the magnetic flux in the direction of rotation, this motor is capable of substantially improving its torque performance without the need to increase the magnet usage. However, it is difficult to obtain a novel machine structure using parameter optimization (PO),^1,2 which relies on the insights of experienced engineers.

In contrast to PO, topology optimization (TO) in which the material distribution is freely determined without presetting the shape parameters, can lead to novel rotor structures. In,^3,4 iron and air in the rotor were optimized using TO, and asymmetric IPM motors with excellent performance were achieved. In prior research, however, solely the magnetic core's shape was taken into account, despite the fact that magnet configurations are critical for further performance enhancement.

In this paper, we simultaneously optimize the magnetic core and magnet configuration in an asymmetric IPM motor to obtain its novel rotor topology. The goal of this study is to clarify the unique magnetic properties of the optimized asymmetric IPM motors. In the optimization, the shape of the magnetic core is determined by TO using the Gaussian basis function,⁵ whereas the magnet configuration is represented by the Boolean geometry projection (BGP) method, which combines PO with Boolean operations to improve for manufacturability.⁶ In the study, the torque performance of symmetric and asymmetric IPM motors are optimized using conventional TO which considers only the magnetic core, and the BGP + TO method to compare the differences in these characteristics.

Optimization method

Boolean geometry projection (BGP) method

The reference and optimization models of the IPM motor are shown in Figure 1. The symmetric and asymmetric IPM motors have both been optimized to compare their performances. The specifications are listed in Table 1. The rotor surfaces with 1.0- and 0.5-mm thicknesses facing the shaft and the air gap were excluded from the design region to account for manufacturability. Assuming that the rotor is mirror symmetric, the red colored quarter- and half-design regions for each optimization model were optimized using the conventional TO and BGP + TO methods.

Figure 1.

Reference and optimization models. (a) Reference model⁷ (b) Optimization model for a symmetric IPM motor (c) Optimization model for an asymmetric IPM motor.

Table 1.

Specifications of the IPM motor.⁷

Core grade	JIS 35A300
Magnetization	1.1 T
Coil current	4243 A $\cdot$ turn
Driving frequency	200 Hz
Rotation speed	3000 r/min

In the BGP method,⁶ the magnet configuration is represented by an elementary bar, as shown in Figure 2(a), to create a shape that satisfies the manufacturing constraints. The magnet is parameterized with five design variables, where $r, θ, l, w,$ and $θ_{m}$ denote the radius and angle of the magnet center, magnet length and width, and magnetization direction, respectively. During the optimization, $p_{BGP} = [r, θ l, w, θ_{m}]^{t} \in R^{5}$ is determined to minimize the objective function. By eliminating upper and lower limits on the design variables and enabling the magnet to move freely within the design region, the BGP method enables the representation of novel magnetic configurations. However, this generation process may result in an unfeasible permanent magnet leaving the design region. To solve this problem, the BGP method applies a Boolean operation to the deviating magnet to obtain a feasible shape, as shown in Figure 2(b).

Figure 2.

Boolean geometry projection method.⁶ (a) parameter setting of the elementary bar (b) Boolean operation for an infeasible solution.

Topology optimization (TO) method

After determining the magnet configuration, the rotor shape excluding the magnet region was determined by TO using a normalized Gaussian function (NGnet).⁵ In the TO, the material distribution in the design region is determined based on the output of the shape function, which is defined as follows:

\begin{matrix} y (x, w) = \sum_{i = 1}^{N_{G}} w_{i} b_{i} (x), (1) \end{matrix}

where

w_{i}

b_{i} (x)

, and

N_{G}

denote the weighting coefficient, the normalized Gaussian function, and the number of Gaussian bases, respectively. In this study, 34 and 68 Gaussian bases for each optimization model were uniformly deployed, as shown in Figure 1(b) and (c). The material type

M_{e}

of the finite element e in the design region is determined as follows:

\begin{matrix} M_{e} = {\begin{matrix} iron, & y (x, w) \geq 0 \\ air, & y (x, w) < 0 \end{matrix} . (2) \end{matrix}

In TO, the weighting coefficients $w = p_{NGnet} = [w_{1}, w_{2}, \dots, w_{N_{G}}]^{t} \in R^{N_{G}}$ are determined to minimize the objective function. That is, TO was reduced to PO for $w$ .

In the BGP + TO method, the design variable $p = (p_{BGP}, p_{NGnet})$ is determined using the covariance matrix adaptation evolution strategy (CMA-ES).⁸ Although the similar hybrid optimization methods have been proposed,^9,10 we have to carefully set the upper and lower limits of the geometric parameters in the conventional approach to avoid popping out of a magnet from the rotor. Since the BGP + TO method freely optimizes the permanent magnet without such geometrical constraints, we can easily find the novel rotor structures. A flowchart of the optimization process is shown in Figure 3.

Figure 3.

Flowchart of the BGP + TO method.

Optimization of the IPM motor

The purpose of the optimization is to maximize the average torque while minimizing the torque ripple. Therefore, the optimization problem is solved as follows:

\begin{aligned} min_{p} F (p), F (p) = - \frac{T_{avg} (p)}{T_{avg}^{Ref}} + 0.2 \frac{T_{rip} (p)}{T_{rip}^{Ref}}, \\ sub . to S_{mag} \leq S_{mag}^{Ref}, \end{aligned}

(3)

where

T_{avg} (p)

T_{rip} (p)

, and

S_{mag}

denote the average torque, torque ripple and magnet area, respectively. The quantities for the reference IPM motor are marked with the superscript “Ref.” In this study, the symmetric and asymmetric IPM motors were optimized at three initial electrical angles (0°, 30°, and 60°) to compare the differences in their optimal shapes. The torque characteristics and magnet area of the reference motors are listed in Table 2. Moreover, only the magnetic core shape is optimized, as in,^3,4 using the conventional TO method,⁵ to validate the effectiveness of the BGP + TO method. During the optimization,

4 \times {4 + 3 \ln (N_{p})}

populations were generated at first iteration according to the recommended setting and the evolution process was continued for over 1000 iterations using CMA-ES,⁸ where

N_{p}

denotes the number of design variables. It took approximately two days to obtain the optimization results using an Intel Xeon CPU (3.5 GHz, 40 threads).

Table 2.

Torque characteristics and magnet area of the reference model.

Initial electrical angle	$T_{avg}^{Ref} [Nm]$	$T_{rip}^{Ref} [%]$	$S_{mag}^{Ref} [{mm}^{2}]$
0 deg	68.8	32.0	360
30 deg	83.6	17.8
60 deg	69.2	25.6

Optimization results

The resulting rotor shapes obtained using the conventional TO and BGP + TO methods are shown in Figures 4 and 5, respectively. The average torque and torque ripple of each model are summarized in the caption. It should be noted that there is only a slight difference in the magnet areas of the optimized IPM motors. From the conventional TO results shown in Figure 4, it can be seen that the optimized symmetric and asymmetric IPM motors have almost the same torque performance for each electrical angle, although their magnetic core shapes are different. In contrast, the optimized asymmetric IPM motors in Figure 5(d)–(f) obtained using the BGP + TO method have substantially higher average torques than the optimized symmetric motors in Figure 5(a)–(c), regardless of the electrical angle. It is noted that the thin magnet region is generated in Figure 5(d)–(f) because the demagnetization was not considered in the optimization.

Figure 4.

Optimized rotor shapes obtained using the conventional TO ( $T_{avg} [Nm], T_{rip} [%]$ ) (a) Sym., 0 deg, (82.8, 3.83) (b) Sym., 30 deg, (89.6, 3.23) (c) Sym., 60 deg, (77.1, 3.09) (d) Asym., 0 deg, (83.5, 2.31) (e) Asym., 30 deg, (90.9, 0.921) (f) Asym., 60 deg, (80.6, 3.62).

Figure 5.

Optimized rotor shapes obtained with the BGP and TO ( $T_{avg} [Nm], T_{rip} [%]$ ) (a) Sym., 0 deg, (83.1, 2.22) (b) Sym., 30 deg, (88.3, 3.25) (c) Sym., 60 deg, (84.5, 4.93) (d) Asym., 0 deg, (95.4, 4.86) (e) Asym., 30 deg, (96.8, 2.35) (f) Asym., 60 deg, (89.7, 2.59).

To interpret the performance differences between these rotor shapes, we compare the decomposed torque characteristics shown in Figure 6, where the magnetic and reluctance torques were computed using the frozen permeability method.¹¹ It can be seen that the magnetic torque in the optimized asymmetric IPM motors obtained using the BGP + TO method is effectively improved at electrical angles of 30° and 60° compared with the other optimized models. This is due to the fact that the magnet configuration shown in Figures 5(d)–(f) is shifted in the rotation direction depending on the electrical angle to maximize the circumferential magnetic flux. However, it can be seen that the improvement in magnetic torque is limited to other optimized IPM motors at higher electrical angles because it is difficult to effectively increase the circumferential magnetic flux under fixed or symmetric magnet configurations. Moreover, the reluctance torque of the asymmetric IPM motor obtained using the BGP + TO method is significantly improved at an electrical angle of 0° because the asymmetric magnetic pole circled in red in Figure 5(d), was generated on the q-axis so that the magnetic flux increased in the rotation direction. This unique result can only be attained by asymmetric optimization. Although similar design principle (magnetic-field-shifting: MFS) to enhance the torque of an asymmetric IPM motor has been reported in the PO,² the variation of an optimal shape for different initial electrical angles was not discussed because of the geometrical complexity of asymmetric shapes. We clarified the above important point by using the BGP + TO method in this study. Finally, the torque ripple is significantly suppressed in the optimization results because the fluctuations in the magnetic and reluctance torques are canceled out, as shown in Figure 6(d). From the above results, it can be concluded that the BGP + TO method, which considers the magnet configuration and the shape of the magnetic core of the asymmetric IPM motor, is effective in improving the torque performance compared to the conventional TO.

Figure 6.

Decomposed torque characteristics of reference and optimized IPM motors (a) electrical angle 0 deg (b) electrical angle 30 deg (c) electrical angle 60 deg (d) Torque wave of optimized model in Figure 4(e).

Conclusion

The novel rotor structures considering the magnetic core and magnet configuration in an asymmetric IPM motor were presented in this study. It has been shown that the superior torque performance of the asymmetric IPM motors optimized by the BGP and TO methods has been demonstrated in comparison to the other optimized configurations. Moreover, we clarified the unique magnetic property of the asymmetric ones to the initial electrical angles. In the future, we intend to extend the BGP + TO method to realize asymmetric design for all rotor poles.

Footnotes

ORCID iD

Yoshitsugu Otomo

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported in part by the JSPS KAKENHI under Grant JP23K13312.

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.

References

Xiao Y, Zhu ZQ, Jewell GW, et al. A novel asymmetric Interior permanent magnet synchronous machine. IEEE Trans Ind Appl 2022; 58: 3370–3382.

Zhu

Xiao

. Novel magnetic-field-shifting techniques in asymmetric rotor pole Interior PM machines with enhanced torque density. IEEE Trans Magn 2022; 58: 8100610.

Cherrière T, Hlioui S, Laurent L, et al. Topology optimization of asymmetric PMSM rotor. In: Proc. ICEM 2022, 2022, pp.469–475.

Niu

. Optimal design of asymmetric rotor pole for Interior permanent magnet synchronous motor using topology optimization. Energies 2022; 15: 8254.

Sato

Watanabe

Igarashi

. Multimaterial topology optimization of electric machines based on normalized Gaussian network. IEEE Trans Magn 2015; 51: 7202604.

Otomo Y, Abe T, Igarashi H, et al. Optimization of permanent magnet configuration using Boolean geometry projection method for IPM motors. In: Proc. CEFC 2022, Oct. 2022. DOI: https://doi.org/10.1109/CEFC55061.2022.9940901.

Virtual Motor Model: 8-poles 12-slots IPMSM, IEEJ Tech. Committee Report, Jun. 2022.

Hansen

. The CMA Evolution Strategy: A Tutorial. arXiv:1604.00772. 2016.

Hiruma S, Ohtani M, Soma S, et al. Novel hybridization of parameter and topology optimizations: application to permanent magnet motor. IEEE Trans Magn 2021; 57: 8204604.

10.

Hayashi

Igarashi

. Parameter-topology hybrid optimization of electric motor with multiple permanent magnets. IJAEM 2023; 71(S1): S245–S255.

11.

Chu

Zhu

. Average torque separation in permanent magnet synchronous machines using frozen permeability. IEEE Trans Magn 2013; 49: 1202–1210.