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筒井・遠藤研究室
東京科学大学
Institute of Science Tokyo

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電磁モータの軽量化を目的とした固定子のトポロジー最適化
Topology Optimization of Stator for Weight Reduction of Electromagnetic Motors

源元颯人，遠藤 央，中村裕司，田中真平 / Hayato Minamoto, Mitsuru Endo, Hiroshi Nakamura, Shimpei Tanaka

ロボティクス・メカトロニクス講演会 2023 ROBOMECH 2023

2A1-D21

背景 – Background

　電磁モータの軽量化は高トルク化や高出力密度化が望め，総合的に高性能化へつながる．従来，軽量化は構造の最適化[1]や軽量な材料の利用[2]，電磁モータからの脱却[3]などに取り組まれている．本研究では従来の電磁モータの軽量化を進めるべく，最適化に着目している．形状の最適化はトポロジー最適化が最も自由度が高い設計ができる．これを用いた研究例は，最適化にかかる計算のコストが膨大であることから回転子に限定した研究[4]や固定子だけを対象とした研究[5]など，限られた要素を対象としている．本発表では固定子に着目し，先行研究[5]では固定子の鉄心の形状と形態だけを最適化したものを，巻線まで含めて最適化する手法について提案し，実際の最適化結果より有効性を示した．

The lightweight of electromagnetic motors can lead to increased torque and higher power density, thereby contributing to overall improved performance. Historically, efforts to achieve light-weighting have centered on structural optimization [1], the use of lightweight materials [2], and even initiatives to move away from electromagnetic motors entirely [3]. In this study, our attention is drawn to the optimization in the pursuit of further reducing the weight of traditional electromagnetic motors. Topology optimization allows for the highest level of design freedom when it comes to shape optimization. Due to the extensive computational costs associated with such optimization, past research efforts have been confined to specific components, with studies focusing solely on the rotor [4] or just the stator [5]. In this presentation, our focus is on the stator. While previous research [5] had only optimized the shape and form of the stator’s core, we propose a method that extends the optimization to include the windings. The effectiveness of this approach is validated through actual optimization results.

手法 - Method

\[\mathrm{minimize} ~~ \frac{M_i}{M_{ref}} + \left\{ \begin{matrix} 0 & \mathrm{if} ~ \tau_{ave,i} \geq \tau_{des} \\ k\left(1 - \frac{\tau_{ave,i}}{\tau_{des}} \right) & \mathrm{otherwise} \end{matrix} \right.\]

図1に基本的な正規化ガウス関数ネットワークに基づくデコード化の概念を示す．ガウス分布を持つ波の中央の位置を有限要素解析に用いるメッシュの上に定義する．この各ガウス分布の大きさを設計変数として最適化を実行する．得られた設計変数に基づき，各ガウス分布の大きさを計算し，その”波”を重ねわせることで1つの波を得る．このとき，波の値が閾値（通常は０）よりも大きいところには鉄心があり，小さいところは空気（フラックスバリア）であるとして計算する．

本発表ではこれを，巻線を含む最適化に拡張する．通常は１つの正規化ガウス関数ネットワークにより下記に示す判定をする．

Figure 1 illustrates the basic concept of decoding based on a normalized Gaussian function network. The center positions of waves with a Gaussian distribution are defined on the mesh used for finite element analysis. The size of each Gaussian distribution is optimized as design variables. Based on the obtained design variables, the size of each Gaussian distribution is calculated, and by overlaying these “waves”, a single wave is obtained. At this time, where the wave value is greater than a threshold (usually zero), it is considered to have a core (iron core), and where it is smaller, it is considered to be air (flux barrier).

In this presentation, this is extended to include winding in the optimization. Normally, a decision is made by one normalized Gaussian function network as described below.

\[\left\{ \begin{array}{ll} \mathrm{if}~~ \phi_(\boldsymbol{x}_e) \geq t_h & \mathrm{then ~~ Stator~Core} \\ \mathrm{if} ~~ \phi_(\boldsymbol{x}_e) < t_h & \mathrm{then ~~ Air} \end{array} \right.\]

Fig.1 正規化ガウス関数ネットワークに基づくデコード化 / Decoding based on the normalized Gaussian function network.

これを２つの正規化ガウス関数ネットワークを同時に使うことにより，2 bitで4態表せるようになる．そのうちの３つを用いることで，巻線，鉄心，フラックスバリアの3つを表す．

By using two normalized Gaussian function networks simultaneously, it becomes possible to represent 4 states with 2 bits. By using three of those states, we can represent the winding, iron core, and flux barrier.

\[\left\{ \begin{array}{ll} \mathrm{if}~~ \phi_1(\boldsymbol{x}_e) \geq t_h & \mathrm{then ~~ Coil} \\ \mathrm{if} ~~ \phi_1(\boldsymbol{x}_e) < t_h ~\cap~ \phi_2(\boldsymbol{x}_e) \geq 0 & \mathrm{then ~~ Stator~Core} \\ \mathrm{if} ~~ \phi_1(\boldsymbol{x}_e) < t_h ~\cap~ \phi_2(\boldsymbol{x}_e) <0 & \mathrm{then ~~ Air} \end{array} \right.\]

図２に概念を示す．２つの正規化ガウス関数ネットワークの値がそれぞれ閾値より大きいかを判定することで，３つの状態（図では黄，青，白で示している）を分けることができる．

Figure 2 illustrates the concept. By determining whether the values from the two Normalized Gaussian Networks are greater than a certain threshold, we can differentiate between three states (represented in the figure as yellow, blue, and white).

Fig.2 巻線を含む固定子の正規化ガウス関数ネットワークを用いた最適化の概念 / Concept of stator optimization using normalized Gaussian function network including the winding.

しかしながら実際には，巻線と固定子鉄心の面積は最適化に基づいて変化してしまう．巻線の面積が変化することでモータの性能が著しく変化してしまう．そのため巻線の面積を含めた最適化は複雑になり実行できない．そこで本研究では巻線の面積を決める正規化ガウス関数ネットワークのしきい値を変化させることで面積を最適化開始時から一定にすることを提案した．図3に概念を示す．実際には目標面積を与え，その値になるようにしきい値の値を二分法により最適化する．

However, in reality, the area of both the winding and the stator core changes based on the optimization. When the area of the winding changes, it can drastically alter the motor’s performance. Hence, optimizing with the winding area included becomes too complex to execute. This study proposes to keep the area consistent from the start of optimization by adjusting the threshold of the Normalized Gaussian Network that determines the winding area. Figure 3 illustrates this concept. In practice, we set a target area and optimize the threshold value using the binary search method to achieve that area.

Fig.3 しきい値の変化による面積に調整 / Area adjustment through threshold value changes.

結果 / Result

CMA-ESの個体数とペナルティ関数のゲインの大きさをパラメータとして，それぞれ３回の結果を示した．これらのパラメータは最適化の性能には影響 を与えるが，最適化結果の物理的な設計結果には影響を与えない．形状を見ると今までにない形が得られていることがわかる．さらに質量は軽いものでは50%の低減，大きいものでも30%程度は軽量化に成功している．また，性能は狙い通り変化していないこともわかる．

The number of individuals in CMA-ES and the magnitude of the penalty function gain were taken as parameters, and the results of three runs for each are shown. These parameters influence the performance of the optimization, but they don’t affect the physical design outcome of the optimization. By looking at the shape, it’s clear that we obtained a design unlike any before. Furthermore, in terms of weight, we’ve managed to reduce it by 50% for the lighter designs and by about 30% even for the heavier ones. Also, as targeted, we can see that the performance hasn’t changed.

結論 / Conclusion

本研究ではトポロジー最適化を用いたモータの軽量化に取り組む．本発表では固定子に着目し，巻線の面積を変えずに巻線の形状や形態も含めて最適化する手法を提案し，実際の設計結果より有効性を示した．従来の半分程度の重さになっただけでなく性能は据え置きであることがわかる．このような「今までの技術者には想像もつかない形」を生み出すことが計算機に設計を任せることの醍醐味であり，今後はモータ全体の設計を完全自動化することに踏み込んでいく．

In this study, we tackled motor weight reduction using topology optimization. In this presentation, we focused on the stator and proposed a method to optimize not only the shape but also the form of the winding without changing the winding area. The actual design results demonstrate its effectiveness. The weight was reduced to about half of the conventional one, while performance was maintained. Generating such “shapes beyond the imagination of traditional engineers” is the true beauty of entrusting design to computers. Moving forward, we aim to fully automate the entire motor design process.

参考文献 / Reference

[1] Zhang, X., et al., Mass Optimization Method of a Surface-Mounted Permanent Magnet Synchronous Motor Based on a Lightweight Structure, IEEE Access (2020).

[2] Rallabandi, et al., Coreless Multidisc Axial Flux PM Machine with Carbon Nanotube Windings, IEEE Transactions on Magnetics (2017).

[3] Sakai, K., et al., Ultralightweight motor design using electromagnetic resonance coupling, IEEE-ECCE (2016).

[4] Sato, T., et al., Multi-material Topology Optimization of Electric Machines Based on Normalized Gaussian Network, IEEE Transactions on Magnetics (2015).

[5] Choi, J. S., et al., Topology Optimization of the Stator for Minimizing Cogging Torque of IPM Motors, IEEE Transactions on Magnetics (2011).

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